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data-dom-id="Pill-react-component-f006e070-8373-47a9-bf16-9616873eda65"></div> <div id="Pill-react-component-f006e070-8373-47a9-bf16-9616873eda65"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="24636766" href="https://www.academia.edu/Documents/in/Brain_and_Cognitive_Development"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Brain and Cognitive Development"]}" data-trace="false" data-dom-id="Pill-react-component-422bedba-6274-421d-9d45-1350ece1a01f"></div> <div id="Pill-react-component-422bedba-6274-421d-9d45-1350ece1a01f"></div> </a></div></div><div class="external-links-container"><ul class="profile-links new-profile js-UserInfo-social"><li><a class="ds2-5-text-link ds2-5-text-link--small" href="https://oliverknill.academia.edu/"><span class="ds2-5-text-link__content"><i class="fa fa-laptop"></i></span></a></li><li class="profile-profiles js-social-profiles-container"><i class="fa fa-spin fa-spinner"></i></li></ul></div></div></div><div class="right-panel-container"><div class="user-content-wrapper"><div class="uploads-container" id="social-redesign-work-container"><div class="upload-header"><h2 class="ds2-5-heading-sans-serif-xs">Uploads</h2></div><div class="documents-container backbone-social-profile-documents" style="width: 100%;"><div class="u-taCenter"></div><div class="profile--tab_content_container js-tab-pane tab-pane active" id="all"><div class="profile--tab_heading_container js-section-heading" data-section="Papers" id="Papers"><h3 class="profile--tab_heading_container">Papers by Oliver Knill</h3></div><div class="js-work-strip profile--work_container" data-work-id="125874999"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874999/On_the_arithmetic_of_graphs"><img alt="Research paper thumbnail of On the arithmetic of graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/119841159/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874999/On_the_arithmetic_of_graphs">On the arithmetic of graphs</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, 2017</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The Zykov ring of signed finite simple graphs with topological join as addition and compatible mu...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by taking graph complements, it becomes isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the integers by demonstrating that the strong ring is homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth formula holds on the strong ring so that the Poincare polynomial is compatible with the ring structure. The Zykov ring has the clique number as a ring homomorphism. Furthermore, the Cartesian ring has the property that the functor which attaches to a graph the spectrum of its connection Laplacian is multiplicative. The reason is that the connection Laplacians do tensor under multiplication, similarly to what the adjacency matrix does for the weak ring. The strong ring product of two graphs contains both the weak and direct product graphs as subgraphs. The Zykov, Sabidussi or Stanley-Reisner rings are so manifestations of a network arithmetic which has remarkable cohomological properties, dimension and spectral compatibility but where arithmetic questions like the complexity of detecting primes or factoring are not yet studied well. We illustrate the Zykov arithmetic with examples, especially from the subring generated by point graphs which contains spheres, stars or complete bipartite graphs. While things are formulated in the language of graph theory, all constructions generalize to the larger category of finite abstract simplicial complexes.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fa4caf439dee62d313a74f059a5336aa" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841159,"asset_id":125874999,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841159/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874999"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874999"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874999; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874998"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874998/On_a_theorem_of_Grove_and_Searle"><img alt="Research paper thumbnail of On a theorem of Grove and Searle" class="work-thumbnail" src="https://attachments.academia-assets.com/119841157/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874998/On_a_theorem_of_Grove_and_Searle">On a theorem of Grove and Searle</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jun 21, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with ci...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with circular symmetry group of dimension 2d ≤ 8 have positive Euler characteristic χ(M): the fixed point set N consists of even dimensional positive curvature manifolds and has the Euler characteristic χ(N) = χ(M). It is not empty by Berger. If N has a co-dimension 2 component, Grove-Searle forces M to be in {RP 2d , S 2d , CP d }. By Frankel, there can be not two codimension 2 cases. In the remaining cases, Gauss-Bonnet-Chern forces all to have positive Euler characteristic. This simple proof does not quite reach the record 2d ≤ 10 which uses methods of Wilking but it motivates to analyze the structure of fixed point components N and in particular to look at positive curvature manifolds which admit a U (1) or SU (2) symmetry with connected or almost connected fixed point set N. They have amazing geodesic properties: the fixed point manifold N agrees with the caustic of each of its points and the geodesic flow is integrable. In full generality, the Lefschetz fixed point property χ(N) = χ(M) and Frankel's dimension theorem dim(M) < dim(N k) + dim(N l) for two different connectivity components of N produce already heavy constraints in building up M from smaller components. It is possible that S 2d , RP 2d , CP d , HP d , OP 2 , W 6 , E 6 , W 12 , W 24 are actually a complete list of even-dimensional positive curvature manifolds admitting a continuum symmetry. Aside from the projective spaces, the Euler characteristic of the known cases is always 1, 2 or 6, where the jump from 2 to 6 happened with the Wallach or Eschenburg manifolds W 6 , E 6 which have four fixed point components N = S 2 + S 2 + S 0 , the only known case which are not of the Grove-Searle form N = N 1 or N = N 1 + {p} with connected N 1 .</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="94c18585b6be469b5493baacc8f11bdc" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841157,"asset_id":125874998,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841157/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874998"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874998"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874998; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874995"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874995/Some_Fundamental_Theorems_in_Mathematics"><img alt="Research paper thumbnail of Some Fundamental Theorems in Mathematics" class="work-thumbnail" src="https://attachments.academia-assets.com/119841155/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874995/Some_Fundamental_Theorems_in_Mathematics">Some Fundamental Theorems in Mathematics</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jul 22, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 250 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were written down. Since [570] stated "a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The number of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 "tweetable" theorems with included proofs. More comments on the choice of the theorems is included in</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2dab254f36da74efef97aa759ad39097" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841155,"asset_id":125874995,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841155/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874995"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874995"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874995; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874994"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874994/A_Dehn_type_quantity_for_Riemannian_manifolds"><img alt="Research paper thumbnail of A Dehn type quantity for Riemannian manifolds" class="work-thumbnail" src="https://attachments.academia-assets.com/119841153/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874994/A_Dehn_type_quantity_for_Riemannian_manifolds">A Dehn type quantity for Riemannian manifolds</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, May 25, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) =...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) = (2d)!(d!) −1 (4π) −d T d k=1 K t 2k ,t 2k+1 (x)dt involves products of d sectional curvatures K ij (x) averaged over the space T ∼ O(2d) of all orthonormal frames t = (t 1 ,. .. , t 2d). A discrete version γ d (M) with K d (x) = (d!) −1 (4π) −d σ d k=1 K σ(2k−1),σ(2k) sums over all permutations σ of {1,. .. , 2d}. Unlike Euler characteristic which by Gauss-Bonnet-Chern is M K GBC dV = χ(M), the quantities γ or γ d are in general metric dependent. We are interested in δ(M) = γ(M) − χ(M) because if M has curvature sign e, then γ(M)e d and γ d (M) are positive while χ(M)e d > 0 is only conjectured. We compute γ d in a few concrete examples like 2d-spheres, the 4-manifold CP 2 , the 6 manifold SO(4) or the 8-manifold SU (3).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="48c34090f67db369998d9a1c5456a3f1" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841153,"asset_id":125874994,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841153/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874994"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874994"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874994; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874993"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874993/A_Dehn_type_invariant_for_Riemannian_manifolds"><img alt="Research paper thumbnail of A Dehn type invariant for Riemannian manifolds" class="work-thumbnail" src="https://attachments.academia-assets.com/119841150/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874993/A_Dehn_type_invariant_for_Riemannian_manifolds">A Dehn type invariant for Riemannian manifolds</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, May 25, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) =...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) = (2d)!(d!) −1 (4π) −d T d k=1 K t 2k ,t 2k+1 (x)dt involves products of d sectional curvatures K ij (x) averaged over the space T ∼ O(2d) of all orthonormal frames t = (t 1 ,. .. , t 2d). A discrete version γ d (M) with K d (x) = (d!) −1 (4π) −d σ d k=1 K σ(2k−1),σ(2k) sums over all permutations σ of {1,. .. , 2d}. Unlike Euler characteristic which by Gauss-Bonnet-Chern is M K GBC dV = χ(M), the quantities γ or γ d are in general metric dependent. We are interested in δ(M) = γ(M) − χ(M) because if M has curvature sign e, then γ(M)e d and γ d (M) are positive while χ(M)e d > 0 is only conjectured. We compute γ d in a few concrete examples like 2d-spheres, the 4-manifold CP 2 , the 6 manifold SO(4) or the 8-manifold SU (3).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7bb682826c57ba277adff488a80d7bdf" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841150,"asset_id":125874993,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841150/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874993"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874993"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874993; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874992"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874992/Characteristic_Topological_Invariants"><img alt="Research paper thumbnail of Characteristic Topological Invariants" class="work-thumbnail" src="https://attachments.academia-assets.com/119841152/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874992/Characteristic_Topological_Invariants">Characteristic Topological Invariants</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Feb 5, 2023</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The higher characteristics w m (G) for a finite abstract simplicial complex G are topological inv...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The higher characteristics w m (G) for a finite abstract simplicial complex G are topological invariants that satisfy k-point Green function identities and can be computed in terms of Euler characteristic in the case of closed manifolds, where we give a new proof of w m (G) = w 1 (G). Also the sphere formula generalizes: for any simplicial complex, the total higher characteristics of unit spheres at even dimensional simplices is equal to the total higher characteristic of unit spheres at odd dimensional simplices.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="abe4411ef737d785ea8872ee803f0f56" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841152,"asset_id":125874992,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841152/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874992"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874992"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874992; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874989"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874989/One_can_hear_the_Euler_characteristic_of_a_simplicial_complex"><img alt="Research paper thumbnail of One can hear the Euler characteristic of a simplicial complex" class="work-thumbnail" src="https://attachments.academia-assets.com/119841146/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874989/One_can_hear_the_Euler_characteristic_of_a_simplicial_complex">One can hear the Euler characteristic of a simplicial complex</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Nov 26, 2017</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite abstract simplicial complex G matches the number b of even dimensional simplices in G and that the number n of negative eigenvalues matches the number f of odd-dimensional simplices in G. The Euler characteristic χ(G) of G therefore can be spectrally described as χ(G) = p − n. This is in contrast to the more classical Hodge Laplacian H which acts on the same Hilbert space, where χ(G) is not yet known to be accessible from the spectrum of H. Given an ordering of G coming from a build-up as a CW complex, every simplex x ∈ G is now associated to a unique eigenvector of L and the correspondence is computable. The Euler characteristic is now not only the potential energy x∈G y∈G g(x, y) with g = L −1 but also agrees with a logarithmic energy tr(log(iL))2/(iπ) of the spectrum of L. We also give here examples of isospectral but non-isomorphic abstract finite simplicial complexes. One example shows that we can not hear the cohomology of the complex.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="00ef16d39c69400cff5c4080d6469e50" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841146,"asset_id":125874989,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841146/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874989"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874989"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874989; 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We give examples of finite simple graphs which do not allow for any constant µ-curvature and prove that for one-dimensional connected graphs, there is a convex set of constant curvature configurations with dimension of the first Betti number of the graph. In particular, there is always a unique constant curvature solution for trees.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7ee47e2e98f8ab671eb800efc6a0aca4" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841142,"asset_id":125874987,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841142/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874987"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874987"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874987; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874985"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874985/A_parametrized_Poincare_Hopf_Theorem_and_Clique_Cardinalities_of_graphs"><img alt="Research paper thumbnail of A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/119841139/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874985/A_parametrized_Poincare_Hopf_Theorem_and_Clique_Cardinalities_of_graphs">A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jun 15, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E)...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E), we prove the Poincaré-Hopf formula f G (t) = 1 + t x∈V f Sg(x) (t), where S g (x) = {y ∈ S(x), g(y) < g(x)} and f G (t) = 1 + f 0 t + • • • + f d t d+1 is the f-function encoding the f-vector of a graph G, where f k counts the number of k-dimensional cliques, complete sub-graphs, in G. The corresponding computation of f reduces the problem recursively to n tasks of graphs of half the size. For t = −1, the parametric Poincaré-Hopf formula reduces to the classical Poincaré-Hopf result [5] χ(G) = x i g (x), with integer indices i g (x) = 1−χ(S g (x)) and Euler characteristic χ. In the new Poincaré-Hopf formula, the indices are integer polynomials and the curvatures K x (t) expressed as index expectations K x (t) = E[i x (t)] are polynomials over Q. Integrating the Poincaré-Hopf formula over probability spaces of functions g gives Gauss-Bonnet formulas like f G (t) = 1+ x F S(x) (t), where F G (t) is the anti-derivative of f [4, 14]. A similar computation holds for the generating function f G,H (t, s) = k,l f k,l (G, H)s k t l of the f-intersection matrix f k,l (G, H) counting the number of intersections of k-simplices in G with l-simplices in H. Also here, the computation is reduced to 4n 2 computations for graphs of half the size: f G,H (t, s) = v,w f Bg(v),Bg(w) (t, s)− f Bg(v),Sg(w) (t, s) − f Sg(v),Bg(w) (t, s) + f Sg(v),Sg(w) (t, s), where B g (v) = S g (v) + {v} is the unit ball of v.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5f233808661a6ab97f6a50f52e645742" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841139,"asset_id":125874985,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841139/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874985"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874985"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874985; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125874985]").text(description); $(".js-view-count[data-work-id=125874985]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125874985; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125874985']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "5f233808661a6ab97f6a50f52e645742" } } $('.js-work-strip[data-work-id=125874985]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125874985,"title":"A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs","translated_title":"","metadata":{"publisher":"Cornell University","ai_abstract":"This paper presents a parametrized Poincare-Hopf theorem and its application to the computation of clique cardinalities in graphs. It introduces a formula for sub-exponential computation of the f-vector of a graph, utilizing the expected distribution of vertices in a random function. The study connects this approach to classical curvature polynomials and applies it to derive specific cases, establishing a link between graph theoretic properties and topological characteristics.","ai_title_tag":"Parametrized Poincaré-Hopf Theorem for Graphs","grobid_abstract":"Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E), we prove the Poincaré-Hopf formula f G (t) = 1 + t x∈V f Sg(x) (t), where S g (x) = {y ∈ S(x), g(y) \u003c g(x)} and f G (t) = 1 + f 0 t + • • • + f d t d+1 is the f-function encoding the f-vector of a graph G, where f k counts the number of k-dimensional cliques, complete sub-graphs, in G. The corresponding computation of f reduces the problem recursively to n tasks of graphs of half the size. For t = −1, the parametric Poincaré-Hopf formula reduces to the classical Poincaré-Hopf result [5] χ(G) = x i g (x), with integer indices i g (x) = 1−χ(S g (x)) and Euler characteristic χ. In the new Poincaré-Hopf formula, the indices are integer polynomials and the curvatures K x (t) expressed as index expectations K x (t) = E[i x (t)] are polynomials over Q. Integrating the Poincaré-Hopf formula over probability spaces of functions g gives Gauss-Bonnet formulas like f G (t) = 1+ x F S(x) (t), where F G (t) is the anti-derivative of f [4, 14]. A similar computation holds for the generating function f G,H (t, s) = k,l f k,l (G, H)s k t l of the f-intersection matrix f k,l (G, H) counting the number of intersections of k-simplices in G with l-simplices in H. Also here, the computation is reduced to 4n 2 computations for graphs of half the size: f G,H (t, s) = v,w f Bg(v),Bg(w) (t, s)− f Bg(v),Sg(w) (t, s) − f Sg(v),Bg(w) (t, s) + f Sg(v),Sg(w) (t, s), where B g (v) = S g (v) + {v} is the unit ball of v.","publication_date":{"day":15,"month":6,"year":2019,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":119841139},"translated_abstract":null,"internal_url":"https://www.academia.edu/125874985/A_parametrized_Poincare_Hopf_Theorem_and_Clique_Cardinalities_of_graphs","translated_internal_url":"","created_at":"2024-11-26T23:03:25.998-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":24636766,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":119841139,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841139/thumbnails/1.jpg","file_name":"1906.pdf","download_url":"https://www.academia.edu/attachments/119841139/download_file","bulk_download_file_name":"A_parametrized_Poincare_Hopf_Theorem_and.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841139/1906-libre.pdf?1732691530=\u0026response-content-disposition=attachment%3B+filename%3DA_parametrized_Poincare_Hopf_Theorem_and.pdf\u0026Expires=1738661624\u0026Signature=LO~ZQ7m4KHtgSpdqA1sviFbngp31gNCelRLxNHs3g4yD6mUuKl3mpMYghZLkdhRLExG0-dbxyPYAYo4H4dsEFtxBgLW3N6dEU1A5Q205v74rm-6ehnMoXHnWuvNNUvM8Ym3h-89NWrTg4ECrSQxWVUzSjXN4d4H1N6ifiUF7-BKKybBpbj8PYLJ4UoMSYTmQgXmpVn1Umvkp8FXS18JINCeSZtWYSSAeoP4~bLvwne4aGVR3z7v5t~-YPmq552N4I4T7AutoJd1eEdAdgisR3YZdT1MxBVDWzHQQcKZXyPVII3V7UByqwDSkUxegUCftIsEq4UbfS2GCt52qIVgm0A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_parametrized_Poincare_Hopf_Theorem_and_Clique_Cardinalities_of_graphs","translated_slug":"","page_count":12,"language":"en","content_type":"Work","summary":"Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E), we prove the Poincaré-Hopf formula f G (t) = 1 + t x∈V f Sg(x) (t), where S g (x) = {y ∈ S(x), g(y) \u003c g(x)} and f G (t) = 1 + f 0 t + • • • + f d t d+1 is the f-function encoding the f-vector of a graph G, where f k counts the number of k-dimensional cliques, complete sub-graphs, in G. The corresponding computation of f reduces the problem recursively to n tasks of graphs of half the size. For t = −1, the parametric Poincaré-Hopf formula reduces to the classical Poincaré-Hopf result [5] χ(G) = x i g (x), with integer indices i g (x) = 1−χ(S g (x)) and Euler characteristic χ. In the new Poincaré-Hopf formula, the indices are integer polynomials and the curvatures K x (t) expressed as index expectations K x (t) = E[i x (t)] are polynomials over Q. Integrating the Poincaré-Hopf formula over probability spaces of functions g gives Gauss-Bonnet formulas like f G (t) = 1+ x F S(x) (t), where F G (t) is the anti-derivative of f [4, 14]. A similar computation holds for the generating function f G,H (t, s) = k,l f k,l (G, H)s k t l of the f-intersection matrix f k,l (G, H) counting the number of intersections of k-simplices in G with l-simplices in H. Also here, the computation is reduced to 4n 2 computations for graphs of half the size: f G,H (t, s) = v,w f Bg(v),Bg(w) (t, s)− f Bg(v),Sg(w) (t, s) − f Sg(v),Bg(w) (t, s) + f Sg(v),Sg(w) (t, s), where B g (v) = S g (v) + {v} is the unit ball of v.","owner":{"id":24636766,"first_name":"Oliver","middle_initials":null,"last_name":"Knill","page_name":"OliverKnill","domain_name":"harvard","created_at":"2015-01-10T20:13:49.803-08:00","display_name":"Oliver Knill","url":"https://harvard.academia.edu/OliverKnill","email":"OHZleDRSNERtZE1razZCZjJUbW9NS0dLQTRPVzIzS3RpS2FnUHdpYW01bWJzNkdvNTRCV2tUMFRVSjJONk5xVy0tR0RKUzZWdTF6OEZrOGdKWkIySVlYQT09--5f1a89e051b22046870626144adf26be0d9aee0d"},"attachments":[{"id":119841139,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841139/thumbnails/1.jpg","file_name":"1906.pdf","download_url":"https://www.academia.edu/attachments/119841139/download_file","bulk_download_file_name":"A_parametrized_Poincare_Hopf_Theorem_and.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841139/1906-libre.pdf?1732691530=\u0026response-content-disposition=attachment%3B+filename%3DA_parametrized_Poincare_Hopf_Theorem_and.pdf\u0026Expires=1738661624\u0026Signature=LO~ZQ7m4KHtgSpdqA1sviFbngp31gNCelRLxNHs3g4yD6mUuKl3mpMYghZLkdhRLExG0-dbxyPYAYo4H4dsEFtxBgLW3N6dEU1A5Q205v74rm-6ehnMoXHnWuvNNUvM8Ym3h-89NWrTg4ECrSQxWVUzSjXN4d4H1N6ifiUF7-BKKybBpbj8PYLJ4UoMSYTmQgXmpVn1Umvkp8FXS18JINCeSZtWYSSAeoP4~bLvwne4aGVR3z7v5t~-YPmq552N4I4T7AutoJd1eEdAdgisR3YZdT1MxBVDWzHQQcKZXyPVII3V7UByqwDSkUxegUCftIsEq4UbfS2GCt52qIVgm0A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":119841140,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841140/thumbnails/1.jpg","file_name":"1906.pdf","download_url":"https://www.academia.edu/attachments/119841140/download_file","bulk_download_file_name":"A_parametrized_Poincare_Hopf_Theorem_and.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841140/1906-libre.pdf?1732691531=\u0026response-content-disposition=attachment%3B+filename%3DA_parametrized_Poincare_Hopf_Theorem_and.pdf\u0026Expires=1738661624\u0026Signature=R1U-OxyAIhrkfZU7ZEiCpFacWkr8WUTErn60fiJH7r3vC1L24jYE-MRDfo1S~UEvvJ9cCF9k6Q5Gv6MMnJQgyGOGAFuDbGAIaIyUgTzbqodNYBlsAM3-WNX4rGITNDpBikq-4iWtQ4SOmfz-PUNJYIKb8CjOFgzRTX59XW3BjiusFcciNBSSQTYE9Fxsa16qr-eoRXCZIOysukVcWqA0AEzDdVrlJj42hnTlkiPtAupNJtAY9EdN5K7JUQsFGEMLWMswZghNtN6~wqHVoy0mB11CuvS~YfZ~OFZdskjZkvW5kORlRSg2jzaRCRJ~js5ulLeSaiZR4RQJMnG7Ha96YQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":3193313,"name":"arXiv","url":"https://www.academia.edu/Documents/in/arXiv"}],"urls":[{"id":45805008,"url":"http://arxiv.org/pdf/1906.06611"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874983"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874983/The_hydrogen_identity_for_Laplacians"><img alt="Research paper thumbnail of The hydrogen identity for Laplacians" class="work-thumbnail" src="https://attachments.academia-assets.com/119841135/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874983/The_hydrogen_identity_for_Laplacians">The hydrogen identity for Laplacians</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Mar 4, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identit...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identity |H| = L−L −1 holds, where |H| = (|d| + |d| *) 2 is the sign-less Hodge Laplacian defined by the sign-less incidence matrix |d| and where L is the connection Laplacian. Having linked the Laplacian spectral radius ρ of G with the spectral radius of the adjacency matrix its connection graph G allows for every k to estimate ρ ≤ r k − 1/r k , where r k = 1 + (P (k)) 1/k and P (k) = max x P (k, x), where P (k, x) is the number of paths of length k starting at a vertex x in G. The limit r k − 1/r k for k → ∞ is the spectral radius ρ of |H| which by Wielandt is an upper bound for the spectral radius ρ of H = (d + d *) 2 , with equality if G is bipartite. We can relate so the growth rate of the random walks in the line graph G L of G with the one in the connection graph G of G. The hydrogen identity implies that the random walk ψ(n) = L n ψ on the connection graph G with integer n solves the 1-dimensional Jacobi equation ∆ψ = |H| 2 ψ with ∆u(n) = u(n + 2) − 2u(n) + u(n − 2) and assures that every solution is represented by such a reversible path integral. The hydrogen identity also holds over any finite field F. There, the dynamics L n ψ with n ∈ Z is a reversible cellular automaton with alphabet F G. By taking products of simplicial complexes, such processes can be defined over any lattice Z r. Since L 2 and L −2 are isospectral, by a theorem of Kirby, L 2 is always similar to a symplectic matrix if the graph has an even number of simplices. By the implicit function theorem, the hydrogen relation is robust in the following sense: any matrix K with the same support than |H| can still be written as K = L − L −1 with a connection Laplacian satisfying L(x, y) = L −1 (x, y) = 0 if x ∩ y = ∅.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a5d175e5b1035d7e8510ab6c74fe2d6f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841135,"asset_id":125874983,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841135/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874983"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874983"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874983; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125874983]").text(description); $(".js-view-count[data-work-id=125874983]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125874983; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125874983']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a5d175e5b1035d7e8510ab6c74fe2d6f" } } $('.js-work-strip[data-work-id=125874983]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125874983,"title":"The hydrogen identity for Laplacians","translated_title":"","metadata":{"publisher":"Cornell University","ai_abstract":"This paper investigates the intricate relationship between the largest eigenvalue of the Kirchhoff graph Laplacian and the vertex degrees of a graph, leveraging connections between Hodge Laplacians and adjacency matrices. It provides new insights into spectral estimates through random walk analysis, illustrating the utility of the Schur inequality and defining the connection Laplacian within finite abstract simplicial complexes. The research enhances understanding of the interplay between graph structures and spectral properties, particularly in relation to the Euler characteristic and unimodularity of associated matrices.","grobid_abstract":"For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identity |H| = L−L −1 holds, where |H| = (|d| + |d| *) 2 is the sign-less Hodge Laplacian defined by the sign-less incidence matrix |d| and where L is the connection Laplacian. Having linked the Laplacian spectral radius ρ of G with the spectral radius of the adjacency matrix its connection graph G allows for every k to estimate ρ ≤ r k − 1/r k , where r k = 1 + (P (k)) 1/k and P (k) = max x P (k, x), where P (k, x) is the number of paths of length k starting at a vertex x in G. The limit r k − 1/r k for k → ∞ is the spectral radius ρ of |H| which by Wielandt is an upper bound for the spectral radius ρ of H = (d + d *) 2 , with equality if G is bipartite. We can relate so the growth rate of the random walks in the line graph G L of G with the one in the connection graph G of G. The hydrogen identity implies that the random walk ψ(n) = L n ψ on the connection graph G with integer n solves the 1-dimensional Jacobi equation ∆ψ = |H| 2 ψ with ∆u(n) = u(n + 2) − 2u(n) + u(n − 2) and assures that every solution is represented by such a reversible path integral. The hydrogen identity also holds over any finite field F. There, the dynamics L n ψ with n ∈ Z is a reversible cellular automaton with alphabet F G. By taking products of simplicial complexes, such processes can be defined over any lattice Z r. Since L 2 and L −2 are isospectral, by a theorem of Kirby, L 2 is always similar to a symplectic matrix if the graph has an even number of simplices. By the implicit function theorem, the hydrogen relation is robust in the following sense: any matrix K with the same support than |H| can still be written as K = L − L −1 with a connection Laplacian satisfying L(x, y) = L −1 (x, y) = 0 if x ∩ y = ∅.","publication_date":{"day":4,"month":3,"year":2018,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":119841135},"translated_abstract":null,"internal_url":"https://www.academia.edu/125874983/The_hydrogen_identity_for_Laplacians","translated_internal_url":"","created_at":"2024-11-26T23:03:24.294-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":24636766,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":119841135,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841135/thumbnails/1.jpg","file_name":"1803.pdf","download_url":"https://www.academia.edu/attachments/119841135/download_file","bulk_download_file_name":"The_hydrogen_identity_for_Laplacians.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841135/1803-libre.pdf?1732691541=\u0026response-content-disposition=attachment%3B+filename%3DThe_hydrogen_identity_for_Laplacians.pdf\u0026Expires=1738613721\u0026Signature=BLrU6Cr32gOUTVcfGm2wv3tffMwyy2lvzU0sVYg8UjVQ42ddcXhYgdxg7A5Aojs2r93okOyb23IWYuyyUhylXKq0jTqP7thKB4lMjsQfbpVuKkwt7vclzx3kLp0tRx8y7nw9ozP2MIjElTKOJicxwYXg~JSlqAN~sp9pBuILZEKsKPgXYbo7dwE2yutNAWcrwBRGJzMu6cW6PyR9l6-ht9yVvBHCzNt-0xEYsCFjUKWrZwY-DtjWl6rlleZVvJgwrjsZAibTd80VNTdLaLO~9pc7YxlGT1TSiXfDrbitto4gnIY1dKK7v8Ws5TvW~dYiAOzkVSE2HZOwtMZ9f-hfBg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_hydrogen_identity_for_Laplacians","translated_slug":"","page_count":29,"language":"en","content_type":"Work","summary":"For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identity |H| = L−L −1 holds, where |H| = (|d| + |d| *) 2 is the sign-less Hodge Laplacian defined by the sign-less incidence matrix |d| and where L is the connection Laplacian. Having linked the Laplacian spectral radius ρ of G with the spectral radius of the adjacency matrix its connection graph G allows for every k to estimate ρ ≤ r k − 1/r k , where r k = 1 + (P (k)) 1/k and P (k) = max x P (k, x), where P (k, x) is the number of paths of length k starting at a vertex x in G. The limit r k − 1/r k for k → ∞ is the spectral radius ρ of |H| which by Wielandt is an upper bound for the spectral radius ρ of H = (d + d *) 2 , with equality if G is bipartite. We can relate so the growth rate of the random walks in the line graph G L of G with the one in the connection graph G of G. The hydrogen identity implies that the random walk ψ(n) = L n ψ on the connection graph G with integer n solves the 1-dimensional Jacobi equation ∆ψ = |H| 2 ψ with ∆u(n) = u(n + 2) − 2u(n) + u(n − 2) and assures that every solution is represented by such a reversible path integral. The hydrogen identity also holds over any finite field F. There, the dynamics L n ψ with n ∈ Z is a reversible cellular automaton with alphabet F G. By taking products of simplicial complexes, such processes can be defined over any lattice Z r. Since L 2 and L −2 are isospectral, by a theorem of Kirby, L 2 is always similar to a symplectic matrix if the graph has an even number of simplices. By the implicit function theorem, the hydrogen relation is robust in the following sense: any matrix K with the same support than |H| can still be written as K = L − L −1 with a connection Laplacian satisfying L(x, y) = L −1 (x, y) = 0 if x ∩ y = ∅.","owner":{"id":24636766,"first_name":"Oliver","middle_initials":null,"last_name":"Knill","page_name":"OliverKnill","domain_name":"harvard","created_at":"2015-01-10T20:13:49.803-08:00","display_name":"Oliver Knill","url":"https://harvard.academia.edu/OliverKnill"},"attachments":[{"id":119841135,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841135/thumbnails/1.jpg","file_name":"1803.pdf","download_url":"https://www.academia.edu/attachments/119841135/download_file","bulk_download_file_name":"The_hydrogen_identity_for_Laplacians.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841135/1803-libre.pdf?1732691541=\u0026response-content-disposition=attachment%3B+filename%3DThe_hydrogen_identity_for_Laplacians.pdf\u0026Expires=1738613721\u0026Signature=BLrU6Cr32gOUTVcfGm2wv3tffMwyy2lvzU0sVYg8UjVQ42ddcXhYgdxg7A5Aojs2r93okOyb23IWYuyyUhylXKq0jTqP7thKB4lMjsQfbpVuKkwt7vclzx3kLp0tRx8y7nw9ozP2MIjElTKOJicxwYXg~JSlqAN~sp9pBuILZEKsKPgXYbo7dwE2yutNAWcrwBRGJzMu6cW6PyR9l6-ht9yVvBHCzNt-0xEYsCFjUKWrZwY-DtjWl6rlleZVvJgwrjsZAibTd80VNTdLaLO~9pc7YxlGT1TSiXfDrbitto4gnIY1dKK7v8Ws5TvW~dYiAOzkVSE2HZOwtMZ9f-hfBg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":119841134,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841134/thumbnails/1.jpg","file_name":"1803.pdf","download_url":"https://www.academia.edu/attachments/119841134/download_file","bulk_download_file_name":"The_hydrogen_identity_for_Laplacians.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841134/1803-libre.pdf?1732691543=\u0026response-content-disposition=attachment%3B+filename%3DThe_hydrogen_identity_for_Laplacians.pdf\u0026Expires=1738613721\u0026Signature=TBMGe65XFdywn3oW5VNs4XxMjWwlc4z5pu867Ulv2Sc2y2DS7z97WFSYhIzMx1y~xvN5EdEwLaQfyklV0HYik8P6toh8OvkwkB0f1g6jQl0JrpH-k-HaR6zbBZqMdS6zTp8emRnhfaoNO7XI2th1IDUFGVhSMCAXNCwhilOONHoem6Jty4Zr7jk6IMyb2xllkqK2bfX69hRZq~Bj6Jpzr7qVazzQO9l4B5vUzmiNdmsej60boYEkalCeM4OwMMkGuac2xzirVt1vub0~0Lh1kqfcfd9sdjnhViTvY1BcElQsvHg9elXp8z~mVZrFXrm61~KYloDJzYA5BQ6DwKNmFA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":1806143,"name":"Laplace operator","url":"https://www.academia.edu/Documents/in/Laplace_operator"},{"id":1978690,"name":"Laplacian matrix","url":"https://www.academia.edu/Documents/in/Laplacian_matrix"},{"id":3193313,"name":"arXiv","url":"https://www.academia.edu/Documents/in/arXiv"}],"urls":[{"id":45805005,"url":"http://arxiv.org/pdf/1803.01464"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874981"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874981/On_Primes_Graphs_and_Cohomology"><img alt="Research paper thumbnail of On Primes, Graphs and Cohomology" class="work-thumbnail" src="https://attachments.academia-assets.com/119841136/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874981/On_Primes_Graphs_and_Cohomology">On Primes, Graphs and Cohomology</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 21, 2016</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomo...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the Poincaré-Hopf indices at critical points correspond to the values of the Moebius function. The Morse inequalities link number theoretical quantities like the prime counting functions relevant for the distribution of primes with cohomological properties of the graphs. The just given picture is a special case of a discrete Morse cohomology equivalent to simplicial cohomology. The special example considered here is a case where the graph is the Barycentric refinement of a finite simple graph.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2ecd8a0b4f1825b52e54f8f8349d1bc3" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841136,"asset_id":125874981,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841136/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874981"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874981"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874981; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874979"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874979/On_Atiyah_Singer_and_Atiyah_Bott_for_finite_abstract_simplicial_complexes"><img alt="Research paper thumbnail of On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes" class="work-thumbnail" src="https://attachments.academia-assets.com/119841131/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874979/On_Atiyah_Singer_and_Atiyah_Bott_for_finite_abstract_simplicial_complexes">On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 20, 2017</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A linear or multi-linear valuation on a finite abstract simplicial complex can be expressed as an...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A linear or multi-linear valuation on a finite abstract simplicial complex can be expressed as an analytic index dim(ker(D)) −dim(ker(D *)) of a differential complex D : E → F. In the discrete, a complex D can be called elliptic if a McKean-Singer spectral symmetry applies as this implies str(e −tD 2) is t-independent. In that case, the analytic index of D is χ(G, D) = k (−1) k b k (D), where b k is the k'th Betti number, which by Hodge is the nullity of the (k + 1)'th block of the Hodge operator L = D 2. It can also be written as a topological index v∈V K(v), where V is the set of zero-dimensional simplices in G and where K is an Euler type curvature defined by G and D. This can be interpreted as a Atiyah-Singer type correspondence between analytic and topological index. Examples are the de Rham differential complex for the Euler characteristic χ(G) or the connection differential complex for Wu characteristic ω k (G). Given an endomorphism T of an elliptic complex, the Lefschetz number χ(T, G, D) is defined as the super trace of T acting on cohomology defined by D and G. It is equal to the sum v∈V i(v), where V is the set of zerodimensional simplices which are contained in fixed simplices of T , and i is a Brouwer type index. This Atiyah-Bott result generalizes the Brouwer-Lefschetz fixed point theorem for an endomorphism of the simplicial complex G. In both the static and dynamic setting, the proof is done by heat deforming the Koopman operator U (T) to get the cohomological picture str(e −tD 2 U (T)) in the limit t → ∞ and then use Hodge, and then by applying a discrete gradient flow to the simplex data defining the valuation to push str(U (T)) to the zero dimensional set V , getting curvature K(v) or the Brouwer type index i(v).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="faefc4dc12c269cdcbaab05bfe51abce" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841131,"asset_id":125874979,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841131/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874979"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874979"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874979; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874976"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874976/The_graph_spectrum_of_barycentric_refinements"><img alt="Research paper thumbnail of The graph spectrum of barycentric refinements" class="work-thumbnail" src="https://attachments.academia-assets.com/119841127/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874976/The_graph_spectrum_of_barycentric_refinements">The graph spectrum of barycentric refinements</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 9, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If λ0 = 0 ≤ λ1 ≤ λ2 ≤ • • • ≤ λn are the eigenvalues of the Laplacian of G, define the spectral function F (x) = λ [nx] on the interval [0, 1], where [r] is the floor function giving the largest integer smaller or equal than r. The graph G1 is known to be homotopic to G with Euler characteristic χ(G1) = χ(G) and dim(G1) ≥ dim(G). Let Gm be the sequence of barycentric refinements of G = G0. We prove that for any finite simple graph G, the spectral functions FG m of successive refinements converge for m → ∞ uniformly on compact subsets of (0, 1) and exponentially fast to a universal limiting eigenvalue distribution function F d which only depends on the clique number respectively the dimension d of the largest complete subgraph of G and not on the starting graph G. In the case d = 1, where we deal with graphs without triangles, the limiting distribution is the smooth function F (x) = 4 sin 2 (πx/2). This is related to the Julia set of the quadratic map T (z) = 4z − z 2 which has the one dimensional Julia set [0, 4] and F satisfies T (F (k/n)) = F (2k/n) as the Laplacians satisfy such a renormalization recursion. The spectral density in the d = 1 case is then the arc-sin distribution which is the equilibrium measure on the Julia set. In higher dimensions, where the limiting function F still remains unidentified, F appears to have a discrete or singular component. We don't know whether there is an analogue renormalization in d ≥ 2. The limiting distribution has relations with the limiting vertex degree distribution and so in 2 dimensions with the graph curvature distribution of the refinements Gm.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="05d33912315697ab0c84caa75aaf6376" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841127,"asset_id":125874976,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841127/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874976"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874976"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874976; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125874976]").text(description); $(".js-view-count[data-work-id=125874976]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125874976; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125874976']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "05d33912315697ab0c84caa75aaf6376" } } $('.js-work-strip[data-work-id=125874976]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125874976,"title":"The graph spectrum of barycentric refinements","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If λ0 = 0 ≤ λ1 ≤ λ2 ≤ • • • ≤ λn are the eigenvalues of the Laplacian of G, define the spectral function F (x) = λ [nx] on the interval [0, 1], where [r] is the floor function giving the largest integer smaller or equal than r. The graph G1 is known to be homotopic to G with Euler characteristic χ(G1) = χ(G) and dim(G1) ≥ dim(G). Let Gm be the sequence of barycentric refinements of G = G0. We prove that for any finite simple graph G, the spectral functions FG m of successive refinements converge for m → ∞ uniformly on compact subsets of (0, 1) and exponentially fast to a universal limiting eigenvalue distribution function F d which only depends on the clique number respectively the dimension d of the largest complete subgraph of G and not on the starting graph G. In the case d = 1, where we deal with graphs without triangles, the limiting distribution is the smooth function F (x) = 4 sin 2 (πx/2). This is related to the Julia set of the quadratic map T (z) = 4z − z 2 which has the one dimensional Julia set [0, 4] and F satisfies T (F (k/n)) = F (2k/n) as the Laplacians satisfy such a renormalization recursion. The spectral density in the d = 1 case is then the arc-sin distribution which is the equilibrium measure on the Julia set. In higher dimensions, where the limiting function F still remains unidentified, F appears to have a discrete or singular component. We don't know whether there is an analogue renormalization in d ≥ 2. The limiting distribution has relations with the limiting vertex degree distribution and so in 2 dimensions with the graph curvature distribution of the refinements Gm.","publication_date":{"day":9,"month":8,"year":2015,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":119841127},"translated_abstract":null,"internal_url":"https://www.academia.edu/125874976/The_graph_spectrum_of_barycentric_refinements","translated_internal_url":"","created_at":"2024-11-26T23:03:20.347-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":24636766,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":119841127,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841127/thumbnails/1.jpg","file_name":"1508.pdf","download_url":"https://www.academia.edu/attachments/119841127/download_file","bulk_download_file_name":"The_graph_spectrum_of_barycentric_refine.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841127/1508-libre.pdf?1732691550=\u0026response-content-disposition=attachment%3B+filename%3DThe_graph_spectrum_of_barycentric_refine.pdf\u0026Expires=1738661625\u0026Signature=e8c4Jyk3s82j-ES8mDCu7-8K0hcHHG0U4UORjyimUTBYp0Mmg6Lw3PywnWBTQ9AGIM8bNZhNGgs-sSuQdHdLwVXoP2fEay55R5H7FmrIc3HYBOet9vsdkNPzC5aBZEdZbsSJNJzfbuAekwReTnTTNU61S9505ZHd1btzd6avaQUrUoTzG6UrQFsZiy9Ffcw5vy2xPc0srPeBjc1BoA8-ifutuH7srCNPvipQ02yhnT20KHv-bs8znegkf9zft7WR8W6-l9xK2uroZ~oBqR0hsumlukBkel~5EPfujMb3P7Kadv3Uebh-FDuEDWGJnhF02JNECkeDvJGP5jZpGnPI6A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_graph_spectrum_of_barycentric_refinements","translated_slug":"","page_count":20,"language":"en","content_type":"Work","summary":"Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If λ0 = 0 ≤ λ1 ≤ λ2 ≤ • • • ≤ λn are the eigenvalues of the Laplacian of G, define the spectral function F (x) = λ [nx] on the interval [0, 1], where [r] is the floor function giving the largest integer smaller or equal than r. The graph G1 is known to be homotopic to G with Euler characteristic χ(G1) = χ(G) and dim(G1) ≥ dim(G). Let Gm be the sequence of barycentric refinements of G = G0. We prove that for any finite simple graph G, the spectral functions FG m of successive refinements converge for m → ∞ uniformly on compact subsets of (0, 1) and exponentially fast to a universal limiting eigenvalue distribution function F d which only depends on the clique number respectively the dimension d of the largest complete subgraph of G and not on the starting graph G. In the case d = 1, where we deal with graphs without triangles, the limiting distribution is the smooth function F (x) = 4 sin 2 (πx/2). This is related to the Julia set of the quadratic map T (z) = 4z − z 2 which has the one dimensional Julia set [0, 4] and F satisfies T (F (k/n)) = F (2k/n) as the Laplacians satisfy such a renormalization recursion. The spectral density in the d = 1 case is then the arc-sin distribution which is the equilibrium measure on the Julia set. In higher dimensions, where the limiting function F still remains unidentified, F appears to have a discrete or singular component. We don't know whether there is an analogue renormalization in d ≥ 2. The limiting distribution has relations with the limiting vertex degree distribution and so in 2 dimensions with the graph curvature distribution of the refinements Gm.","owner":{"id":24636766,"first_name":"Oliver","middle_initials":null,"last_name":"Knill","page_name":"OliverKnill","domain_name":"harvard","created_at":"2015-01-10T20:13:49.803-08:00","display_name":"Oliver Knill","url":"https://harvard.academia.edu/OliverKnill","email":"U1hzcHFOZXVySkxQdjJyaHU4YVR3R2U3dzd5V3FlWDFFa1NzRkV3aWoraXF0UGtXMDhMaGErbmpRR2w5aWpjSC0tcVFRR3Z5RjhYcEFxOStRMElvNWlZZz09--11977d43c8b27d2886fcd478836516192becfe25"},"attachments":[{"id":119841127,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841127/thumbnails/1.jpg","file_name":"1508.pdf","download_url":"https://www.academia.edu/attachments/119841127/download_file","bulk_download_file_name":"The_graph_spectrum_of_barycentric_refine.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841127/1508-libre.pdf?1732691550=\u0026response-content-disposition=attachment%3B+filename%3DThe_graph_spectrum_of_barycentric_refine.pdf\u0026Expires=1738661625\u0026Signature=e8c4Jyk3s82j-ES8mDCu7-8K0hcHHG0U4UORjyimUTBYp0Mmg6Lw3PywnWBTQ9AGIM8bNZhNGgs-sSuQdHdLwVXoP2fEay55R5H7FmrIc3HYBOet9vsdkNPzC5aBZEdZbsSJNJzfbuAekwReTnTTNU61S9505ZHd1btzd6avaQUrUoTzG6UrQFsZiy9Ffcw5vy2xPc0srPeBjc1BoA8-ifutuH7srCNPvipQ02yhnT20KHv-bs8znegkf9zft7WR8W6-l9xK2uroZ~oBqR0hsumlukBkel~5EPfujMb3P7Kadv3Uebh-FDuEDWGJnhF02JNECkeDvJGP5jZpGnPI6A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":119841129,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841129/thumbnails/1.jpg","file_name":"1508.pdf","download_url":"https://www.academia.edu/attachments/119841129/download_file","bulk_download_file_name":"The_graph_spectrum_of_barycentric_refine.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841129/1508-libre.pdf?1732691547=\u0026response-content-disposition=attachment%3B+filename%3DThe_graph_spectrum_of_barycentric_refine.pdf\u0026Expires=1738661625\u0026Signature=HIXMSlaiw-zVYczF3Kru1AjJGJUg1svGVSiaRdk33XsMmJ5WnZg9Qhjfjb9BBQ5cxKmG~SO0FTnJY3Wde5HFtHmOVvqiK4mWKR4zVNHmfH8o78d5Tp~82fAD1Diz9gyCbDmv7Fl9IYW4liLMvR7Er1TR3tKTv7MS7f6CyyKM5lZSI80BoX2L7PXGYaNPQq3OPmLc0WUdGnukwNO6TUCbVWmp6qBt4QoXYbzzD-c2EYFNEpW8tHvHXWVobi91454gC-5FBowpS5hNgLOR8mIT5ldmavqnWDiU81wl1lSXBf3upcoJqjhvSOZ8LiSofA7ltLktArcC4RsTftvdF1ILmA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":266831,"name":"Graph","url":"https://www.academia.edu/Documents/in/Graph"},{"id":891011,"name":"Eigenvalues and Eigenvectors","url":"https://www.academia.edu/Documents/in/Eigenvalues_and_Eigenvectors"},{"id":3193313,"name":"arXiv","url":"https://www.academia.edu/Documents/in/arXiv"}],"urls":[{"id":45805001,"url":"http://arxiv.org/pdf/1508.02027"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874974"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874974/Complexes_Graphs_Homotopy_Products_and_Shannon_Capacity"><img alt="Research paper thumbnail of Complexes, Graphs, Homotopy, Products and Shannon Capacity" class="work-thumbnail" src="https://attachments.academia-assets.com/119841126/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874974/Complexes_Graphs_Homotopy_Products_and_Shannon_Capacity">Complexes, Graphs, Homotopy, Products and Shannon Capacity</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 13, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A finite abstract simplicial complex G defines the Barycentric refinement graph φ(G) = (G, {(a, b...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A finite abstract simplicial complex G defines the Barycentric refinement graph φ(G) = (G, {(a, b), a ⊂ b or b ⊂ a}) and the connection graph ψ(G) = (G, {(a, b), a ∩ b = ∅}). We note here that both functors φ and ψ from complexes to graphs are invertible on the image (Theorem 1) and that G, φ(G), ψ(G) all have the same automorphism group and that the Cartesian product of G corresponding to the Stanley-Reisner product of φ(G) and the strong Shannon product of ψ(G), have the product automorphism groups. Second, we see that if G is a Barycentric refinement, then φ(G) and ψ(G) are graph homotopic (Theorem 2). Third, if γ is the geometric realization functor, assigning to a complex or to a graph the geometric realization of its clique complex, then γ(G) and γ(φ(G)) and γ(ψ(G)) are all classically homotopic for a Barycentric refined simplicial complex G (Theorem 3). The Barycentric assumption is necessary in Theorem 2 and 3. There is compatibility with Cartesian products of complexes which manifests in the strong graph product of connection graphs: if two graphs A, A are homotopic and B, B are homotopic, then A • B is homotopic to A • B (Theorem 4) leading to a commutative ring of homotopy classes of graphs. Finally, we note (Theorem 5) that for all simplicial complexes G as well as product G = G 1 × G 2 • • • × G k , the Shannon capacity Θ(ψ(G)) of ψ(G) is equal to the number f 0 of zero-dimensional sets in G. An explicit Lowasz umbrella in R f 0 leads to the Lowasz number θ(G) ≤ f 0 and so Θ(ψ(G)) = θ(ψ(G)) = f 0 making Θ compatible with disjoint union addition and strong multiplication.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="21949b0496f2c4baf39909d9756bb203" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841126,"asset_id":125874974,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841126/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874974"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874974"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874974; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125874974]").text(description); $(".js-view-count[data-work-id=125874974]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125874974; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125874974']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "21949b0496f2c4baf39909d9756bb203" } } $('.js-work-strip[data-work-id=125874974]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125874974,"title":"Complexes, Graphs, Homotopy, Products and Shannon Capacity","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A finite abstract simplicial complex G defines the Barycentric refinement graph φ(G) = (G, {(a, b), a ⊂ b or b ⊂ a}) and the connection graph ψ(G) = (G, {(a, b), a ∩ b = ∅}). We note here that both functors φ and ψ from complexes to graphs are invertible on the image (Theorem 1) and that G, φ(G), ψ(G) all have the same automorphism group and that the Cartesian product of G corresponding to the Stanley-Reisner product of φ(G) and the strong Shannon product of ψ(G), have the product automorphism groups. Second, we see that if G is a Barycentric refinement, then φ(G) and ψ(G) are graph homotopic (Theorem 2). Third, if γ is the geometric realization functor, assigning to a complex or to a graph the geometric realization of its clique complex, then γ(G) and γ(φ(G)) and γ(ψ(G)) are all classically homotopic for a Barycentric refined simplicial complex G (Theorem 3). The Barycentric assumption is necessary in Theorem 2 and 3. There is compatibility with Cartesian products of complexes which manifests in the strong graph product of connection graphs: if two graphs A, A are homotopic and B, B are homotopic, then A • B is homotopic to A • B (Theorem 4) leading to a commutative ring of homotopy classes of graphs. Finally, we note (Theorem 5) that for all simplicial complexes G as well as product G = G 1 × G 2 • • • × G k , the Shannon capacity Θ(ψ(G)) of ψ(G) is equal to the number f 0 of zero-dimensional sets in G. An explicit Lowasz umbrella in R f 0 leads to the Lowasz number θ(G) ≤ f 0 and so Θ(ψ(G)) = θ(ψ(G)) = f 0 making Θ compatible with disjoint union addition and strong multiplication.","publication_date":{"day":13,"month":12,"year":2020,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":119841126},"translated_abstract":null,"internal_url":"https://www.academia.edu/125874974/Complexes_Graphs_Homotopy_Products_and_Shannon_Capacity","translated_internal_url":"","created_at":"2024-11-26T23:03:19.503-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":24636766,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":119841126,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841126/thumbnails/1.jpg","file_name":"2012.pdf","download_url":"https://www.academia.edu/attachments/119841126/download_file","bulk_download_file_name":"Complexes_Graphs_Homotopy_Products_and_S.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841126/2012-libre.pdf?1732691545=\u0026response-content-disposition=attachment%3B+filename%3DComplexes_Graphs_Homotopy_Products_and_S.pdf\u0026Expires=1738759467\u0026Signature=SRZZWotKtBokcFht0d7zpFBsYRCaGmo768dSLL0NxsdE8D5-Q7bCgyRtAWlcQwCDKh63C1lPchFMgm2b11J7TpLzPuJkHb0XTfeU9AW8HLgHK9reGjWPzlobFxx1dhwfjIsulW8UbAOADtIYLsHaSoP8zXyYThvg0~c4CwLixk4ARRZK-rN6dv7h27bj21Em52xl9ZnMi4WnfOk3uRAovTjGYXp933bnc4ZR0sbccFNHmrJzr4vaGq4ucaktaJv48uW3rrSVhW4EXWTijF8hjyNTFyCaArczVrO7zrEiPTnc-DxflbPi-jtXncTXYjobhs9pk-9PE2AmNHnuigCVpQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Complexes_Graphs_Homotopy_Products_and_Shannon_Capacity","translated_slug":"","page_count":26,"language":"en","content_type":"Work","summary":"A finite abstract simplicial complex G defines the Barycentric refinement graph φ(G) = (G, {(a, b), a ⊂ b or b ⊂ a}) and the connection graph ψ(G) = (G, {(a, b), a ∩ b = ∅}). We note here that both functors φ and ψ from complexes to graphs are invertible on the image (Theorem 1) and that G, φ(G), ψ(G) all have the same automorphism group and that the Cartesian product of G corresponding to the Stanley-Reisner product of φ(G) and the strong Shannon product of ψ(G), have the product automorphism groups. Second, we see that if G is a Barycentric refinement, then φ(G) and ψ(G) are graph homotopic (Theorem 2). Third, if γ is the geometric realization functor, assigning to a complex or to a graph the geometric realization of its clique complex, then γ(G) and γ(φ(G)) and γ(ψ(G)) are all classically homotopic for a Barycentric refined simplicial complex G (Theorem 3). The Barycentric assumption is necessary in Theorem 2 and 3. There is compatibility with Cartesian products of complexes which manifests in the strong graph product of connection graphs: if two graphs A, A are homotopic and B, B are homotopic, then A • B is homotopic to A • B (Theorem 4) leading to a commutative ring of homotopy classes of graphs. Finally, we note (Theorem 5) that for all simplicial complexes G as well as product G = G 1 × G 2 • • • × G k , the Shannon capacity Θ(ψ(G)) of ψ(G) is equal to the number f 0 of zero-dimensional sets in G. An explicit Lowasz umbrella in R f 0 leads to the Lowasz number θ(G) ≤ f 0 and so Θ(ψ(G)) = θ(ψ(G)) = f 0 making Θ compatible with disjoint union addition and strong multiplication.","owner":{"id":24636766,"first_name":"Oliver","middle_initials":null,"last_name":"Knill","page_name":"OliverKnill","domain_name":"harvard","created_at":"2015-01-10T20:13:49.803-08:00","display_name":"Oliver Knill","url":"https://harvard.academia.edu/OliverKnill"},"attachments":[{"id":119841126,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841126/thumbnails/1.jpg","file_name":"2012.pdf","download_url":"https://www.academia.edu/attachments/119841126/download_file","bulk_download_file_name":"Complexes_Graphs_Homotopy_Products_and_S.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841126/2012-libre.pdf?1732691545=\u0026response-content-disposition=attachment%3B+filename%3DComplexes_Graphs_Homotopy_Products_and_S.pdf\u0026Expires=1738759468\u0026Signature=B~5z~urUnLkiUW5n0SkccBxtc~YEZIj0RznHMpJTUX-H~U4AD1m98flqcyvjT~tPxWBGmsUQTh~CJy-C~yrCSF-8HTMK5LfvLPqzkS7oVyEz7QV~YPOYpYnFqaTS3Uqe2PWlkihFAlwNd1-1cqiF5I51rIOuLDGPuxDM6MZmNIWzoes5cvSqmoLc8DH9wsBfELSoE0MOzZNhVP73KPntPmejymbvshS~lGyYE-wft9bDHTt7rkj9CHYt~I-b1t5GOtDwEGuxbkUOvShSBODNpdTptxsYQzYh3dpATYpDyQny1HuRpaxC2xxya2uNMpBtdY7-zcA6-T34otPLh~NaAw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":119841130,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841130/thumbnails/1.jpg","file_name":"2012.pdf","download_url":"https://www.academia.edu/attachments/119841130/download_file","bulk_download_file_name":"Complexes_Graphs_Homotopy_Products_and_S.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841130/2012-libre.pdf?1732691548=\u0026response-content-disposition=attachment%3B+filename%3DComplexes_Graphs_Homotopy_Products_and_S.pdf\u0026Expires=1738759468\u0026Signature=Dbu52YjRTv6z0G31UOs69iRbQE2vKH8WZ0dJNZj7t5dycbiUP3UbIHEK0Ac0cWRCwAkgBVivQZgNEm4eLijIpcSyf-luLQWIEV0uDuHy4j8PpatPzKOW6dQKpWrhCWPvjEyWG9BwUiJbym43JG7DLPSTz-GSHlP7Ool2YpHAGxl7Kgmp92pHwpVXhrMquD~dXJcO9BZnbze4omtpp6RH2SS240Ge~Kk~2nGuuVEVN0r4a8WnkccXoy65dWnqEgGxoABj5alQzjF3iz3Y~5Uf3sHTPTNX711xtsjNkE1oy~DBt9p2IvQFtUM2OthGWsnI1aR0lUQRaNQvoNq~ZJS6lA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":171869,"name":"HOMOTOPY","url":"https://www.academia.edu/Documents/in/HOMOTOPY"},{"id":2902615,"name":"Cartesian product","url":"https://www.academia.edu/Documents/in/Cartesian_product"},{"id":3193313,"name":"arXiv","url":"https://www.academia.edu/Documents/in/arXiv"},{"id":4120544,"name":"Simplicial complex","url":"https://www.academia.edu/Documents/in/Simplicial_complex"}],"urls":[{"id":45804999,"url":"http://arxiv.org/pdf/2012.07247"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874972"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874972/Green_functions_of_Energized_complexes"><img alt="Research paper thumbnail of Green functions of Energized complexes" class="work-thumbnail" src="https://attachments.academia-assets.com/119841124/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874972/Green_functions_of_Energized_complexes">Green functions of Energized complexes</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Oct 18, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">G to a ring K with conjugation x * defines χ(A) = x∈A h(x) and ω(G) = x,y∈G,x∩y =∅ h(x) * h(y). D...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">G to a ring K with conjugation x * defines χ(A) = x∈A h(x) and ω(G) = x,y∈G,x∩y =∅ h(x) * h(y). Define L(x, y) = χ(W − (x) ∩ W − (y)) and g(x, y) = ω(x)ω(y)χ(W + (x) ∩ W + (y)), where W − (x) = {z | z ⊂ x}, W + (x) = {z | x ⊂ z} and ω(x) = (−1) dim(x) with dim(x) = |x| − 1. 1.2. The following relation [8] only requires the addition in K Theorem 1. χ(G) = x,y∈G g(x, y) 1.3. The next new quadratic energy relation links simplex interaction with multiplication in K. Define |h| 2 = h * h = N(h) in K. Theorem 2. ω(G) = x,y∈G ω(x)ω(y)|g(x, y)| 2. 1.4. The next determinant identity holds if h maps G to a division algebra K and det is the Dieudonné determinant [1]. The geometry G can here be a finite set of sets and does not need the simplical complex axiom stating that G is closed under the operation of taking non-empty finite subsets. Theorem 3. det(L) = det(g) = x∈G h(x). 1.5. If h : G → K takes values in the units U(K) of K, like i.e. Z 2 , U(1), SU(2), S 7 of the division algebras R, C, H, O, the unitary group U(H)∩K of an operator C *-algebra K ⊂ B(H) for some Hilbert space H or the units in a ring K = O K of integers of a number field K, and if G is a simplicial complex, then: Theorem 4. If h(x) * h(x) = 1 for all x ∈ G, then g * = L −1 .</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="e90a9c498c224681608711cb79988683" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841124,"asset_id":125874972,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841124/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874972"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874972"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874972; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874971"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874971/More_on_Poincare_Hopf_and_Gauss_Bonnet"><img alt="Research paper thumbnail of More on Poincare-Hopf and Gauss-Bonnet" class="work-thumbnail" src="https://attachments.academia-assets.com/119841128/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874971/More_on_Poincare_Hopf_and_Gauss_Bonnet">More on Poincare-Hopf and Gauss-Bonnet</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 1, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We illustrate connections between differential geometry on finite simple graphs G = (V, E) and Ri...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We illustrate connections between differential geometry on finite simple graphs G = (V, E) and Riemannian manifolds (M, g). The link is that curvature can be defined integral geometrically as an expectation in a probability space of Poincaré-Hopf indices of coloring or Morse functions. Regge calculus with an isometric Nash embedding links then the Gauss-Bonnet-Chern integrand of a Riemannian manifold with the graph curvature. There is also a direct nonstandard approach [18]: if V is a finite set containing all standard points of M and E contains pairs which are closer than some positive number. One gets so finite simple graphs (V, E) which leads to the standard curvature. The probabilistic approach is an umbrella framework which covers discrete spaces, piecewise linear spaces, manifolds or varieties.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="70c54ca846fdc0cab679d14e424fd919" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841128,"asset_id":125874971,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841128/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874971"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874971"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874971; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874970"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874970/Remarks_about_the_Arithmetic_of_Graphs"><img alt="Research paper thumbnail of Remarks about the Arithmetic of Graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/119841121/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874970/Remarks_about_the_Arithmetic_of_Graphs">Remarks about the Arithmetic of Graphs</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of fin...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We illustrate the compatibility with topology and spectral theory. Multiplicative linear functionals like Euler characteristic, the Poincare polynomial or the zeta functions can be extended naturally. These functionals can also help with number theoretical questions. The story of primes is a bit different as the integers are not a unique factorization domain, because there are many additive primes. Most graphs are multiplicative primes.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fee80caf04e915b93b27cf6b1ba11cec" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841121,"asset_id":125874970,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841121/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874970"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874970"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874970; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874969"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874969/The_counting_matrix_of_a_simplicial_complex"><img alt="Research paper thumbnail of The counting matrix of a simplicial complex" class="work-thumbnail" src="https://attachments.academia-assets.com/119841118/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874969/The_counting_matrix_of_a_simplicial_complex">The counting matrix of a simplicial complex</a></div><div class="wp-workCard_item"><span>arXiv: Combinatorics</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is t...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is the number of subsimplices in the intersection of x and y. We call it the counting matrix of G. Similarly as the connection matrix L which is L(x,y)=1 if x and y intersect and 0 else, the counting matrix K is unimodular. Actually, K is always in SL(n,Z). The inverse of K has the Green function entries K^(-1)(x,y)=w(x) w(y) |W^+(x) intersected W^+y|, where W^+(x) is the star of x, the sets in G which contain x and w(x)=(-1)^dim(x). The matrix K is always positive definite. The spectra of K and K^(-1) always agree so that the matrix Q=K-K^(-1) has the spectral symmetry spec(Q)=-spec(Q) and the zeta function z(s) summing l(k)^(-s) with eigenvalues l(k) of K satisfies the functional equation z(a+ib)=z(-a+ib). The energy theorem in this case tells that the sum of the matrix elements of K^(-1)(x,y) is equal to the number sets in G. In comparison, we had in the connection matrix case the identit...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="22e08a88ae0f2bc7ce620faa0ba572b7" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841118,"asset_id":125874969,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841118/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874969"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874969"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874969; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874967"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874967/A_Reeb_sphere_theorem_in_graph_theory"><img alt="Research paper thumbnail of A Reeb sphere theorem in graph theory" class="work-thumbnail" src="https://attachments.academia-assets.com/119841114/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874967/A_Reeb_sphere_theorem_in_graph_theory">A Reeb sphere theorem in graph theory</a></div><div class="wp-workCard_item"><span>ArXiv</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definit...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definitions of spheres in graph theory. We also reformulate Morse conditions in terms of the center manifolds, the level surface graphs {f=f(x)} in the unit sphere S(x). In the Morse case these graphs are either spheres, the empty graph or the product of two spheres.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7cc3b647afb6a2c938d60b3a0d090b9a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841114,"asset_id":125874967,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841114/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874967"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874967"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874967; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874966"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874966/Listening_to_the_cohomology_of_graphs"><img alt="Research paper thumbnail of Listening to the cohomology of graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/119841111/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874966/Listening_to_the_cohomology_of_graphs">Listening to the cohomology of graphs</a></div><div class="wp-workCard_item"><span>ArXiv</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined g...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we prove that L-L^(-1) is similar to the Hodge Laplacian H of G which is in one dimensions the direct sum of the Kirchhoff Laplacian H0 and its 1-form analog H1. The spectrum of a single choice of H0,H1 or H alone determines the Betti numbers b0,b1 of G as well as the spectrum of the other matrices. It follows that b0 is the number of eigenvalues 1 of L and that b1 is the number of eigenvalues -1 of L. For a general abstract finite simplicial complex G, we express the matrix entries g(x,y) = w(x) w(y) X( St(x) cap St(y) ) of the inverse of L using stars St(x)= { z in G | x subset of z } of x and w(x)=(-1)^dim(x) and Euler characteristic X. One can see W+(x)=St(x) and W-(x)={ z in G | z subset x } as stable and unstable manifolds of a simplex x in G and g(x,y) =w(x) w(y) X(W+(x) cap W+(y)) as heteroclinic inter...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0162c50e77bb20b5d259e0c1cb8f7006" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841111,"asset_id":125874966,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841111/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874966"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874966"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874966; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div><div class="profile--tab_content_container js-tab-pane tab-pane" data-section-id="2395126" id="papers"><div class="js-work-strip profile--work_container" data-work-id="125874999"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874999/On_the_arithmetic_of_graphs"><img alt="Research paper thumbnail of On the arithmetic of graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/119841159/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874999/On_the_arithmetic_of_graphs">On the arithmetic of graphs</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, 2017</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The Zykov ring of signed finite simple graphs with topological join as addition and compatible mu...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by taking graph complements, it becomes isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the integers by demonstrating that the strong ring is homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth formula holds on the strong ring so that the Poincare polynomial is compatible with the ring structure. The Zykov ring has the clique number as a ring homomorphism. Furthermore, the Cartesian ring has the property that the functor which attaches to a graph the spectrum of its connection Laplacian is multiplicative. The reason is that the connection Laplacians do tensor under multiplication, similarly to what the adjacency matrix does for the weak ring. The strong ring product of two graphs contains both the weak and direct product graphs as subgraphs. The Zykov, Sabidussi or Stanley-Reisner rings are so manifestations of a network arithmetic which has remarkable cohomological properties, dimension and spectral compatibility but where arithmetic questions like the complexity of detecting primes or factoring are not yet studied well. We illustrate the Zykov arithmetic with examples, especially from the subring generated by point graphs which contains spheres, stars or complete bipartite graphs. While things are formulated in the language of graph theory, all constructions generalize to the larger category of finite abstract simplicial complexes.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fa4caf439dee62d313a74f059a5336aa" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841159,"asset_id":125874999,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841159/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874999"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874999"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874999; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874998"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874998/On_a_theorem_of_Grove_and_Searle"><img alt="Research paper thumbnail of On a theorem of Grove and Searle" class="work-thumbnail" src="https://attachments.academia-assets.com/119841157/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874998/On_a_theorem_of_Grove_and_Searle">On a theorem of Grove and Searle</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jun 21, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with ci...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with circular symmetry group of dimension 2d ≤ 8 have positive Euler characteristic χ(M): the fixed point set N consists of even dimensional positive curvature manifolds and has the Euler characteristic χ(N) = χ(M). It is not empty by Berger. If N has a co-dimension 2 component, Grove-Searle forces M to be in {RP 2d , S 2d , CP d }. By Frankel, there can be not two codimension 2 cases. In the remaining cases, Gauss-Bonnet-Chern forces all to have positive Euler characteristic. This simple proof does not quite reach the record 2d ≤ 10 which uses methods of Wilking but it motivates to analyze the structure of fixed point components N and in particular to look at positive curvature manifolds which admit a U (1) or SU (2) symmetry with connected or almost connected fixed point set N. They have amazing geodesic properties: the fixed point manifold N agrees with the caustic of each of its points and the geodesic flow is integrable. In full generality, the Lefschetz fixed point property χ(N) = χ(M) and Frankel's dimension theorem dim(M) < dim(N k) + dim(N l) for two different connectivity components of N produce already heavy constraints in building up M from smaller components. It is possible that S 2d , RP 2d , CP d , HP d , OP 2 , W 6 , E 6 , W 12 , W 24 are actually a complete list of even-dimensional positive curvature manifolds admitting a continuum symmetry. Aside from the projective spaces, the Euler characteristic of the known cases is always 1, 2 or 6, where the jump from 2 to 6 happened with the Wallach or Eschenburg manifolds W 6 , E 6 which have four fixed point components N = S 2 + S 2 + S 0 , the only known case which are not of the Grove-Searle form N = N 1 or N = N 1 + {p} with connected N 1 .</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="94c18585b6be469b5493baacc8f11bdc" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841157,"asset_id":125874998,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841157/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874998"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874998"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874998; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874995"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874995/Some_Fundamental_Theorems_in_Mathematics"><img alt="Research paper thumbnail of Some Fundamental Theorems in Mathematics" class="work-thumbnail" src="https://attachments.academia-assets.com/119841155/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874995/Some_Fundamental_Theorems_in_Mathematics">Some Fundamental Theorems in Mathematics</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jul 22, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 250 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were written down. Since [570] stated "a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The number of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 "tweetable" theorems with included proofs. More comments on the choice of the theorems is included in</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2dab254f36da74efef97aa759ad39097" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841155,"asset_id":125874995,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841155/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874995"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874995"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874995; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874994"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874994/A_Dehn_type_quantity_for_Riemannian_manifolds"><img alt="Research paper thumbnail of A Dehn type quantity for Riemannian manifolds" class="work-thumbnail" src="https://attachments.academia-assets.com/119841153/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874994/A_Dehn_type_quantity_for_Riemannian_manifolds">A Dehn type quantity for Riemannian manifolds</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, May 25, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) =...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) = (2d)!(d!) −1 (4π) −d T d k=1 K t 2k ,t 2k+1 (x)dt involves products of d sectional curvatures K ij (x) averaged over the space T ∼ O(2d) of all orthonormal frames t = (t 1 ,. .. , t 2d). A discrete version γ d (M) with K d (x) = (d!) −1 (4π) −d σ d k=1 K σ(2k−1),σ(2k) sums over all permutations σ of {1,. .. , 2d}. Unlike Euler characteristic which by Gauss-Bonnet-Chern is M K GBC dV = χ(M), the quantities γ or γ d are in general metric dependent. We are interested in δ(M) = γ(M) − χ(M) because if M has curvature sign e, then γ(M)e d and γ d (M) are positive while χ(M)e d > 0 is only conjectured. We compute γ d in a few concrete examples like 2d-spheres, the 4-manifold CP 2 , the 6 manifold SO(4) or the 8-manifold SU (3).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="48c34090f67db369998d9a1c5456a3f1" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841153,"asset_id":125874994,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841153/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874994"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874994"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874994; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874993"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874993/A_Dehn_type_invariant_for_Riemannian_manifolds"><img alt="Research paper thumbnail of A Dehn type invariant for Riemannian manifolds" class="work-thumbnail" src="https://attachments.academia-assets.com/119841150/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874993/A_Dehn_type_invariant_for_Riemannian_manifolds">A Dehn type invariant for Riemannian manifolds</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, May 25, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) =...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) = (2d)!(d!) −1 (4π) −d T d k=1 K t 2k ,t 2k+1 (x)dt involves products of d sectional curvatures K ij (x) averaged over the space T ∼ O(2d) of all orthonormal frames t = (t 1 ,. .. , t 2d). A discrete version γ d (M) with K d (x) = (d!) −1 (4π) −d σ d k=1 K σ(2k−1),σ(2k) sums over all permutations σ of {1,. .. , 2d}. Unlike Euler characteristic which by Gauss-Bonnet-Chern is M K GBC dV = χ(M), the quantities γ or γ d are in general metric dependent. We are interested in δ(M) = γ(M) − χ(M) because if M has curvature sign e, then γ(M)e d and γ d (M) are positive while χ(M)e d > 0 is only conjectured. We compute γ d in a few concrete examples like 2d-spheres, the 4-manifold CP 2 , the 6 manifold SO(4) or the 8-manifold SU (3).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7bb682826c57ba277adff488a80d7bdf" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841150,"asset_id":125874993,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841150/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874993"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874993"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874993; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874992"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874992/Characteristic_Topological_Invariants"><img alt="Research paper thumbnail of Characteristic Topological Invariants" class="work-thumbnail" src="https://attachments.academia-assets.com/119841152/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874992/Characteristic_Topological_Invariants">Characteristic Topological Invariants</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Feb 5, 2023</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The higher characteristics w m (G) for a finite abstract simplicial complex G are topological inv...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The higher characteristics w m (G) for a finite abstract simplicial complex G are topological invariants that satisfy k-point Green function identities and can be computed in terms of Euler characteristic in the case of closed manifolds, where we give a new proof of w m (G) = w 1 (G). Also the sphere formula generalizes: for any simplicial complex, the total higher characteristics of unit spheres at even dimensional simplices is equal to the total higher characteristic of unit spheres at odd dimensional simplices.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="abe4411ef737d785ea8872ee803f0f56" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841152,"asset_id":125874992,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841152/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874992"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874992"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874992; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874989"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874989/One_can_hear_the_Euler_characteristic_of_a_simplicial_complex"><img alt="Research paper thumbnail of One can hear the Euler characteristic of a simplicial complex" class="work-thumbnail" src="https://attachments.academia-assets.com/119841146/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874989/One_can_hear_the_Euler_characteristic_of_a_simplicial_complex">One can hear the Euler characteristic of a simplicial complex</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Nov 26, 2017</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite abstract simplicial complex G matches the number b of even dimensional simplices in G and that the number n of negative eigenvalues matches the number f of odd-dimensional simplices in G. The Euler characteristic χ(G) of G therefore can be spectrally described as χ(G) = p − n. This is in contrast to the more classical Hodge Laplacian H which acts on the same Hilbert space, where χ(G) is not yet known to be accessible from the spectrum of H. Given an ordering of G coming from a build-up as a CW complex, every simplex x ∈ G is now associated to a unique eigenvector of L and the correspondence is computable. The Euler characteristic is now not only the potential energy x∈G y∈G g(x, y) with g = L −1 but also agrees with a logarithmic energy tr(log(iL))2/(iπ) of the spectrum of L. We also give here examples of isospectral but non-isomorphic abstract finite simplicial complexes. One example shows that we can not hear the cohomology of the complex.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="00ef16d39c69400cff5c4080d6469e50" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841146,"asset_id":125874989,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841146/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874989"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874989"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874989; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874987"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874987/Constant_index_expectation_curvature_for_graphs_or_Riemannian_manifolds"><img alt="Research paper thumbnail of Constant index expectation curvature for graphs or Riemannian manifolds" class="work-thumbnail" src="https://attachments.academia-assets.com/119841142/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874987/Constant_index_expectation_curvature_for_graphs_or_Riemannian_manifolds">Constant index expectation curvature for graphs or Riemannian manifolds</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 24, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">An integral geometric curvature K µ is defined as the index expectation K(x) = E µ [i(x)] if a pr...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">An integral geometric curvature K µ is defined as the index expectation K(x) = E µ [i(x)] if a probability measure µ is given on vector fields on a Riemannian manifold or on a finite simple graph. We give examples of finite simple graphs which do not allow for any constant µ-curvature and prove that for one-dimensional connected graphs, there is a convex set of constant curvature configurations with dimension of the first Betti number of the graph. In particular, there is always a unique constant curvature solution for trees.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7ee47e2e98f8ab671eb800efc6a0aca4" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841142,"asset_id":125874987,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841142/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874987"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874987"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874987; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874985"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874985/A_parametrized_Poincare_Hopf_Theorem_and_Clique_Cardinalities_of_graphs"><img alt="Research paper thumbnail of A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/119841139/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874985/A_parametrized_Poincare_Hopf_Theorem_and_Clique_Cardinalities_of_graphs">A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jun 15, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E)...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E), we prove the Poincaré-Hopf formula f G (t) = 1 + t x∈V f Sg(x) (t), where S g (x) = {y ∈ S(x), g(y) < g(x)} and f G (t) = 1 + f 0 t + • • • + f d t d+1 is the f-function encoding the f-vector of a graph G, where f k counts the number of k-dimensional cliques, complete sub-graphs, in G. The corresponding computation of f reduces the problem recursively to n tasks of graphs of half the size. For t = −1, the parametric Poincaré-Hopf formula reduces to the classical Poincaré-Hopf result [5] χ(G) = x i g (x), with integer indices i g (x) = 1−χ(S g (x)) and Euler characteristic χ. In the new Poincaré-Hopf formula, the indices are integer polynomials and the curvatures K x (t) expressed as index expectations K x (t) = E[i x (t)] are polynomials over Q. Integrating the Poincaré-Hopf formula over probability spaces of functions g gives Gauss-Bonnet formulas like f G (t) = 1+ x F S(x) (t), where F G (t) is the anti-derivative of f [4, 14]. A similar computation holds for the generating function f G,H (t, s) = k,l f k,l (G, H)s k t l of the f-intersection matrix f k,l (G, H) counting the number of intersections of k-simplices in G with l-simplices in H. Also here, the computation is reduced to 4n 2 computations for graphs of half the size: f G,H (t, s) = v,w f Bg(v),Bg(w) (t, s)− f Bg(v),Sg(w) (t, s) − f Sg(v),Bg(w) (t, s) + f Sg(v),Sg(w) (t, s), where B g (v) = S g (v) + {v} is the unit ball of v.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5f233808661a6ab97f6a50f52e645742" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841139,"asset_id":125874985,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841139/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874985"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874985"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874985; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125874985]").text(description); $(".js-view-count[data-work-id=125874985]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125874985; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125874985']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "5f233808661a6ab97f6a50f52e645742" } } $('.js-work-strip[data-work-id=125874985]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125874985,"title":"A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs","translated_title":"","metadata":{"publisher":"Cornell University","ai_abstract":"This paper presents a parametrized Poincare-Hopf theorem and its application to the computation of clique cardinalities in graphs. It introduces a formula for sub-exponential computation of the f-vector of a graph, utilizing the expected distribution of vertices in a random function. The study connects this approach to classical curvature polynomials and applies it to derive specific cases, establishing a link between graph theoretic properties and topological characteristics.","ai_title_tag":"Parametrized Poincaré-Hopf Theorem for Graphs","grobid_abstract":"Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E), we prove the Poincaré-Hopf formula f G (t) = 1 + t x∈V f Sg(x) (t), where S g (x) = {y ∈ S(x), g(y) \u003c g(x)} and f G (t) = 1 + f 0 t + • • • + f d t d+1 is the f-function encoding the f-vector of a graph G, where f k counts the number of k-dimensional cliques, complete sub-graphs, in G. The corresponding computation of f reduces the problem recursively to n tasks of graphs of half the size. For t = −1, the parametric Poincaré-Hopf formula reduces to the classical Poincaré-Hopf result [5] χ(G) = x i g (x), with integer indices i g (x) = 1−χ(S g (x)) and Euler characteristic χ. In the new Poincaré-Hopf formula, the indices are integer polynomials and the curvatures K x (t) expressed as index expectations K x (t) = E[i x (t)] are polynomials over Q. Integrating the Poincaré-Hopf formula over probability spaces of functions g gives Gauss-Bonnet formulas like f G (t) = 1+ x F S(x) (t), where F G (t) is the anti-derivative of f [4, 14]. A similar computation holds for the generating function f G,H (t, s) = k,l f k,l (G, H)s k t l of the f-intersection matrix f k,l (G, H) counting the number of intersections of k-simplices in G with l-simplices in H. Also here, the computation is reduced to 4n 2 computations for graphs of half the size: f G,H (t, s) = v,w f Bg(v),Bg(w) (t, s)− f Bg(v),Sg(w) (t, s) − f Sg(v),Bg(w) (t, s) + f Sg(v),Sg(w) (t, s), where B g (v) = S g (v) + {v} is the unit ball of v.","publication_date":{"day":15,"month":6,"year":2019,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":119841139},"translated_abstract":null,"internal_url":"https://www.academia.edu/125874985/A_parametrized_Poincare_Hopf_Theorem_and_Clique_Cardinalities_of_graphs","translated_internal_url":"","created_at":"2024-11-26T23:03:25.998-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":24636766,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":119841139,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841139/thumbnails/1.jpg","file_name":"1906.pdf","download_url":"https://www.academia.edu/attachments/119841139/download_file","bulk_download_file_name":"A_parametrized_Poincare_Hopf_Theorem_and.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841139/1906-libre.pdf?1732691530=\u0026response-content-disposition=attachment%3B+filename%3DA_parametrized_Poincare_Hopf_Theorem_and.pdf\u0026Expires=1738661624\u0026Signature=LO~ZQ7m4KHtgSpdqA1sviFbngp31gNCelRLxNHs3g4yD6mUuKl3mpMYghZLkdhRLExG0-dbxyPYAYo4H4dsEFtxBgLW3N6dEU1A5Q205v74rm-6ehnMoXHnWuvNNUvM8Ym3h-89NWrTg4ECrSQxWVUzSjXN4d4H1N6ifiUF7-BKKybBpbj8PYLJ4UoMSYTmQgXmpVn1Umvkp8FXS18JINCeSZtWYSSAeoP4~bLvwne4aGVR3z7v5t~-YPmq552N4I4T7AutoJd1eEdAdgisR3YZdT1MxBVDWzHQQcKZXyPVII3V7UByqwDSkUxegUCftIsEq4UbfS2GCt52qIVgm0A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_parametrized_Poincare_Hopf_Theorem_and_Clique_Cardinalities_of_graphs","translated_slug":"","page_count":12,"language":"en","content_type":"Work","summary":"Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E), we prove the Poincaré-Hopf formula f G (t) = 1 + t x∈V f Sg(x) (t), where S g (x) = {y ∈ S(x), g(y) \u003c g(x)} and f G (t) = 1 + f 0 t + • • • + f d t d+1 is the f-function encoding the f-vector of a graph G, where f k counts the number of k-dimensional cliques, complete sub-graphs, in G. The corresponding computation of f reduces the problem recursively to n tasks of graphs of half the size. For t = −1, the parametric Poincaré-Hopf formula reduces to the classical Poincaré-Hopf result [5] χ(G) = x i g (x), with integer indices i g (x) = 1−χ(S g (x)) and Euler characteristic χ. In the new Poincaré-Hopf formula, the indices are integer polynomials and the curvatures K x (t) expressed as index expectations K x (t) = E[i x (t)] are polynomials over Q. Integrating the Poincaré-Hopf formula over probability spaces of functions g gives Gauss-Bonnet formulas like f G (t) = 1+ x F S(x) (t), where F G (t) is the anti-derivative of f [4, 14]. A similar computation holds for the generating function f G,H (t, s) = k,l f k,l (G, H)s k t l of the f-intersection matrix f k,l (G, H) counting the number of intersections of k-simplices in G with l-simplices in H. Also here, the computation is reduced to 4n 2 computations for graphs of half the size: f G,H (t, s) = v,w f Bg(v),Bg(w) (t, s)− f Bg(v),Sg(w) (t, s) − f Sg(v),Bg(w) (t, s) + f Sg(v),Sg(w) (t, s), where B g (v) = S g (v) + {v} is the unit ball of v.","owner":{"id":24636766,"first_name":"Oliver","middle_initials":null,"last_name":"Knill","page_name":"OliverKnill","domain_name":"harvard","created_at":"2015-01-10T20:13:49.803-08:00","display_name":"Oliver Knill","url":"https://harvard.academia.edu/OliverKnill","email":"OHZleDRSNERtZE1razZCZjJUbW9NS0dLQTRPVzIzS3RpS2FnUHdpYW01bWJzNkdvNTRCV2tUMFRVSjJONk5xVy0tR0RKUzZWdTF6OEZrOGdKWkIySVlYQT09--5f1a89e051b22046870626144adf26be0d9aee0d"},"attachments":[{"id":119841139,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841139/thumbnails/1.jpg","file_name":"1906.pdf","download_url":"https://www.academia.edu/attachments/119841139/download_file","bulk_download_file_name":"A_parametrized_Poincare_Hopf_Theorem_and.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841139/1906-libre.pdf?1732691530=\u0026response-content-disposition=attachment%3B+filename%3DA_parametrized_Poincare_Hopf_Theorem_and.pdf\u0026Expires=1738661624\u0026Signature=LO~ZQ7m4KHtgSpdqA1sviFbngp31gNCelRLxNHs3g4yD6mUuKl3mpMYghZLkdhRLExG0-dbxyPYAYo4H4dsEFtxBgLW3N6dEU1A5Q205v74rm-6ehnMoXHnWuvNNUvM8Ym3h-89NWrTg4ECrSQxWVUzSjXN4d4H1N6ifiUF7-BKKybBpbj8PYLJ4UoMSYTmQgXmpVn1Umvkp8FXS18JINCeSZtWYSSAeoP4~bLvwne4aGVR3z7v5t~-YPmq552N4I4T7AutoJd1eEdAdgisR3YZdT1MxBVDWzHQQcKZXyPVII3V7UByqwDSkUxegUCftIsEq4UbfS2GCt52qIVgm0A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":119841140,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841140/thumbnails/1.jpg","file_name":"1906.pdf","download_url":"https://www.academia.edu/attachments/119841140/download_file","bulk_download_file_name":"A_parametrized_Poincare_Hopf_Theorem_and.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841140/1906-libre.pdf?1732691531=\u0026response-content-disposition=attachment%3B+filename%3DA_parametrized_Poincare_Hopf_Theorem_and.pdf\u0026Expires=1738661624\u0026Signature=R1U-OxyAIhrkfZU7ZEiCpFacWkr8WUTErn60fiJH7r3vC1L24jYE-MRDfo1S~UEvvJ9cCF9k6Q5Gv6MMnJQgyGOGAFuDbGAIaIyUgTzbqodNYBlsAM3-WNX4rGITNDpBikq-4iWtQ4SOmfz-PUNJYIKb8CjOFgzRTX59XW3BjiusFcciNBSSQTYE9Fxsa16qr-eoRXCZIOysukVcWqA0AEzDdVrlJj42hnTlkiPtAupNJtAY9EdN5K7JUQsFGEMLWMswZghNtN6~wqHVoy0mB11CuvS~YfZ~OFZdskjZkvW5kORlRSg2jzaRCRJ~js5ulLeSaiZR4RQJMnG7Ha96YQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":3193313,"name":"arXiv","url":"https://www.academia.edu/Documents/in/arXiv"}],"urls":[{"id":45805008,"url":"http://arxiv.org/pdf/1906.06611"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874983"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874983/The_hydrogen_identity_for_Laplacians"><img alt="Research paper thumbnail of The hydrogen identity for Laplacians" class="work-thumbnail" src="https://attachments.academia-assets.com/119841135/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874983/The_hydrogen_identity_for_Laplacians">The hydrogen identity for Laplacians</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Mar 4, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identit...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identity |H| = L−L −1 holds, where |H| = (|d| + |d| *) 2 is the sign-less Hodge Laplacian defined by the sign-less incidence matrix |d| and where L is the connection Laplacian. Having linked the Laplacian spectral radius ρ of G with the spectral radius of the adjacency matrix its connection graph G allows for every k to estimate ρ ≤ r k − 1/r k , where r k = 1 + (P (k)) 1/k and P (k) = max x P (k, x), where P (k, x) is the number of paths of length k starting at a vertex x in G. The limit r k − 1/r k for k → ∞ is the spectral radius ρ of |H| which by Wielandt is an upper bound for the spectral radius ρ of H = (d + d *) 2 , with equality if G is bipartite. We can relate so the growth rate of the random walks in the line graph G L of G with the one in the connection graph G of G. The hydrogen identity implies that the random walk ψ(n) = L n ψ on the connection graph G with integer n solves the 1-dimensional Jacobi equation ∆ψ = |H| 2 ψ with ∆u(n) = u(n + 2) − 2u(n) + u(n − 2) and assures that every solution is represented by such a reversible path integral. The hydrogen identity also holds over any finite field F. There, the dynamics L n ψ with n ∈ Z is a reversible cellular automaton with alphabet F G. By taking products of simplicial complexes, such processes can be defined over any lattice Z r. Since L 2 and L −2 are isospectral, by a theorem of Kirby, L 2 is always similar to a symplectic matrix if the graph has an even number of simplices. By the implicit function theorem, the hydrogen relation is robust in the following sense: any matrix K with the same support than |H| can still be written as K = L − L −1 with a connection Laplacian satisfying L(x, y) = L −1 (x, y) = 0 if x ∩ y = ∅.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a5d175e5b1035d7e8510ab6c74fe2d6f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841135,"asset_id":125874983,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841135/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874983"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874983"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874983; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125874983]").text(description); $(".js-view-count[data-work-id=125874983]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125874983; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125874983']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a5d175e5b1035d7e8510ab6c74fe2d6f" } } $('.js-work-strip[data-work-id=125874983]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125874983,"title":"The hydrogen identity for Laplacians","translated_title":"","metadata":{"publisher":"Cornell University","ai_abstract":"This paper investigates the intricate relationship between the largest eigenvalue of the Kirchhoff graph Laplacian and the vertex degrees of a graph, leveraging connections between Hodge Laplacians and adjacency matrices. It provides new insights into spectral estimates through random walk analysis, illustrating the utility of the Schur inequality and defining the connection Laplacian within finite abstract simplicial complexes. The research enhances understanding of the interplay between graph structures and spectral properties, particularly in relation to the Euler characteristic and unimodularity of associated matrices.","grobid_abstract":"For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identity |H| = L−L −1 holds, where |H| = (|d| + |d| *) 2 is the sign-less Hodge Laplacian defined by the sign-less incidence matrix |d| and where L is the connection Laplacian. Having linked the Laplacian spectral radius ρ of G with the spectral radius of the adjacency matrix its connection graph G allows for every k to estimate ρ ≤ r k − 1/r k , where r k = 1 + (P (k)) 1/k and P (k) = max x P (k, x), where P (k, x) is the number of paths of length k starting at a vertex x in G. The limit r k − 1/r k for k → ∞ is the spectral radius ρ of |H| which by Wielandt is an upper bound for the spectral radius ρ of H = (d + d *) 2 , with equality if G is bipartite. We can relate so the growth rate of the random walks in the line graph G L of G with the one in the connection graph G of G. The hydrogen identity implies that the random walk ψ(n) = L n ψ on the connection graph G with integer n solves the 1-dimensional Jacobi equation ∆ψ = |H| 2 ψ with ∆u(n) = u(n + 2) − 2u(n) + u(n − 2) and assures that every solution is represented by such a reversible path integral. The hydrogen identity also holds over any finite field F. There, the dynamics L n ψ with n ∈ Z is a reversible cellular automaton with alphabet F G. By taking products of simplicial complexes, such processes can be defined over any lattice Z r. Since L 2 and L −2 are isospectral, by a theorem of Kirby, L 2 is always similar to a symplectic matrix if the graph has an even number of simplices. By the implicit function theorem, the hydrogen relation is robust in the following sense: any matrix K with the same support than |H| can still be written as K = L − L −1 with a connection Laplacian satisfying L(x, y) = L −1 (x, y) = 0 if x ∩ y = ∅.","publication_date":{"day":4,"month":3,"year":2018,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":119841135},"translated_abstract":null,"internal_url":"https://www.academia.edu/125874983/The_hydrogen_identity_for_Laplacians","translated_internal_url":"","created_at":"2024-11-26T23:03:24.294-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":24636766,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":119841135,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841135/thumbnails/1.jpg","file_name":"1803.pdf","download_url":"https://www.academia.edu/attachments/119841135/download_file","bulk_download_file_name":"The_hydrogen_identity_for_Laplacians.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841135/1803-libre.pdf?1732691541=\u0026response-content-disposition=attachment%3B+filename%3DThe_hydrogen_identity_for_Laplacians.pdf\u0026Expires=1738613721\u0026Signature=BLrU6Cr32gOUTVcfGm2wv3tffMwyy2lvzU0sVYg8UjVQ42ddcXhYgdxg7A5Aojs2r93okOyb23IWYuyyUhylXKq0jTqP7thKB4lMjsQfbpVuKkwt7vclzx3kLp0tRx8y7nw9ozP2MIjElTKOJicxwYXg~JSlqAN~sp9pBuILZEKsKPgXYbo7dwE2yutNAWcrwBRGJzMu6cW6PyR9l6-ht9yVvBHCzNt-0xEYsCFjUKWrZwY-DtjWl6rlleZVvJgwrjsZAibTd80VNTdLaLO~9pc7YxlGT1TSiXfDrbitto4gnIY1dKK7v8Ws5TvW~dYiAOzkVSE2HZOwtMZ9f-hfBg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_hydrogen_identity_for_Laplacians","translated_slug":"","page_count":29,"language":"en","content_type":"Work","summary":"For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identity |H| = L−L −1 holds, where |H| = (|d| + |d| *) 2 is the sign-less Hodge Laplacian defined by the sign-less incidence matrix |d| and where L is the connection Laplacian. Having linked the Laplacian spectral radius ρ of G with the spectral radius of the adjacency matrix its connection graph G allows for every k to estimate ρ ≤ r k − 1/r k , where r k = 1 + (P (k)) 1/k and P (k) = max x P (k, x), where P (k, x) is the number of paths of length k starting at a vertex x in G. The limit r k − 1/r k for k → ∞ is the spectral radius ρ of |H| which by Wielandt is an upper bound for the spectral radius ρ of H = (d + d *) 2 , with equality if G is bipartite. We can relate so the growth rate of the random walks in the line graph G L of G with the one in the connection graph G of G. The hydrogen identity implies that the random walk ψ(n) = L n ψ on the connection graph G with integer n solves the 1-dimensional Jacobi equation ∆ψ = |H| 2 ψ with ∆u(n) = u(n + 2) − 2u(n) + u(n − 2) and assures that every solution is represented by such a reversible path integral. The hydrogen identity also holds over any finite field F. There, the dynamics L n ψ with n ∈ Z is a reversible cellular automaton with alphabet F G. By taking products of simplicial complexes, such processes can be defined over any lattice Z r. Since L 2 and L −2 are isospectral, by a theorem of Kirby, L 2 is always similar to a symplectic matrix if the graph has an even number of simplices. By the implicit function theorem, the hydrogen relation is robust in the following sense: any matrix K with the same support than |H| can still be written as K = L − L −1 with a connection Laplacian satisfying L(x, y) = L −1 (x, y) = 0 if x ∩ y = ∅.","owner":{"id":24636766,"first_name":"Oliver","middle_initials":null,"last_name":"Knill","page_name":"OliverKnill","domain_name":"harvard","created_at":"2015-01-10T20:13:49.803-08:00","display_name":"Oliver Knill","url":"https://harvard.academia.edu/OliverKnill"},"attachments":[{"id":119841135,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841135/thumbnails/1.jpg","file_name":"1803.pdf","download_url":"https://www.academia.edu/attachments/119841135/download_file","bulk_download_file_name":"The_hydrogen_identity_for_Laplacians.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841135/1803-libre.pdf?1732691541=\u0026response-content-disposition=attachment%3B+filename%3DThe_hydrogen_identity_for_Laplacians.pdf\u0026Expires=1738613721\u0026Signature=BLrU6Cr32gOUTVcfGm2wv3tffMwyy2lvzU0sVYg8UjVQ42ddcXhYgdxg7A5Aojs2r93okOyb23IWYuyyUhylXKq0jTqP7thKB4lMjsQfbpVuKkwt7vclzx3kLp0tRx8y7nw9ozP2MIjElTKOJicxwYXg~JSlqAN~sp9pBuILZEKsKPgXYbo7dwE2yutNAWcrwBRGJzMu6cW6PyR9l6-ht9yVvBHCzNt-0xEYsCFjUKWrZwY-DtjWl6rlleZVvJgwrjsZAibTd80VNTdLaLO~9pc7YxlGT1TSiXfDrbitto4gnIY1dKK7v8Ws5TvW~dYiAOzkVSE2HZOwtMZ9f-hfBg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":119841134,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841134/thumbnails/1.jpg","file_name":"1803.pdf","download_url":"https://www.academia.edu/attachments/119841134/download_file","bulk_download_file_name":"The_hydrogen_identity_for_Laplacians.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841134/1803-libre.pdf?1732691543=\u0026response-content-disposition=attachment%3B+filename%3DThe_hydrogen_identity_for_Laplacians.pdf\u0026Expires=1738613721\u0026Signature=TBMGe65XFdywn3oW5VNs4XxMjWwlc4z5pu867Ulv2Sc2y2DS7z97WFSYhIzMx1y~xvN5EdEwLaQfyklV0HYik8P6toh8OvkwkB0f1g6jQl0JrpH-k-HaR6zbBZqMdS6zTp8emRnhfaoNO7XI2th1IDUFGVhSMCAXNCwhilOONHoem6Jty4Zr7jk6IMyb2xllkqK2bfX69hRZq~Bj6Jpzr7qVazzQO9l4B5vUzmiNdmsej60boYEkalCeM4OwMMkGuac2xzirVt1vub0~0Lh1kqfcfd9sdjnhViTvY1BcElQsvHg9elXp8z~mVZrFXrm61~KYloDJzYA5BQ6DwKNmFA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":1806143,"name":"Laplace operator","url":"https://www.academia.edu/Documents/in/Laplace_operator"},{"id":1978690,"name":"Laplacian matrix","url":"https://www.academia.edu/Documents/in/Laplacian_matrix"},{"id":3193313,"name":"arXiv","url":"https://www.academia.edu/Documents/in/arXiv"}],"urls":[{"id":45805005,"url":"http://arxiv.org/pdf/1803.01464"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874981"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874981/On_Primes_Graphs_and_Cohomology"><img alt="Research paper thumbnail of On Primes, Graphs and Cohomology" class="work-thumbnail" src="https://attachments.academia-assets.com/119841136/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874981/On_Primes_Graphs_and_Cohomology">On Primes, Graphs and Cohomology</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 21, 2016</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomo...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the Poincaré-Hopf indices at critical points correspond to the values of the Moebius function. The Morse inequalities link number theoretical quantities like the prime counting functions relevant for the distribution of primes with cohomological properties of the graphs. The just given picture is a special case of a discrete Morse cohomology equivalent to simplicial cohomology. The special example considered here is a case where the graph is the Barycentric refinement of a finite simple graph.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2ecd8a0b4f1825b52e54f8f8349d1bc3" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841136,"asset_id":125874981,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841136/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874981"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874981"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874981; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874979"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874979/On_Atiyah_Singer_and_Atiyah_Bott_for_finite_abstract_simplicial_complexes"><img alt="Research paper thumbnail of On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes" class="work-thumbnail" src="https://attachments.academia-assets.com/119841131/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874979/On_Atiyah_Singer_and_Atiyah_Bott_for_finite_abstract_simplicial_complexes">On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 20, 2017</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A linear or multi-linear valuation on a finite abstract simplicial complex can be expressed as an...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A linear or multi-linear valuation on a finite abstract simplicial complex can be expressed as an analytic index dim(ker(D)) −dim(ker(D *)) of a differential complex D : E → F. In the discrete, a complex D can be called elliptic if a McKean-Singer spectral symmetry applies as this implies str(e −tD 2) is t-independent. In that case, the analytic index of D is χ(G, D) = k (−1) k b k (D), where b k is the k'th Betti number, which by Hodge is the nullity of the (k + 1)'th block of the Hodge operator L = D 2. It can also be written as a topological index v∈V K(v), where V is the set of zero-dimensional simplices in G and where K is an Euler type curvature defined by G and D. This can be interpreted as a Atiyah-Singer type correspondence between analytic and topological index. Examples are the de Rham differential complex for the Euler characteristic χ(G) or the connection differential complex for Wu characteristic ω k (G). Given an endomorphism T of an elliptic complex, the Lefschetz number χ(T, G, D) is defined as the super trace of T acting on cohomology defined by D and G. It is equal to the sum v∈V i(v), where V is the set of zerodimensional simplices which are contained in fixed simplices of T , and i is a Brouwer type index. This Atiyah-Bott result generalizes the Brouwer-Lefschetz fixed point theorem for an endomorphism of the simplicial complex G. In both the static and dynamic setting, the proof is done by heat deforming the Koopman operator U (T) to get the cohomological picture str(e −tD 2 U (T)) in the limit t → ∞ and then use Hodge, and then by applying a discrete gradient flow to the simplex data defining the valuation to push str(U (T)) to the zero dimensional set V , getting curvature K(v) or the Brouwer type index i(v).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="faefc4dc12c269cdcbaab05bfe51abce" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841131,"asset_id":125874979,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841131/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874979"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874979"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874979; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874976"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874976/The_graph_spectrum_of_barycentric_refinements"><img alt="Research paper thumbnail of The graph spectrum of barycentric refinements" class="work-thumbnail" src="https://attachments.academia-assets.com/119841127/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874976/The_graph_spectrum_of_barycentric_refinements">The graph spectrum of barycentric refinements</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 9, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If λ0 = 0 ≤ λ1 ≤ λ2 ≤ • • • ≤ λn are the eigenvalues of the Laplacian of G, define the spectral function F (x) = λ [nx] on the interval [0, 1], where [r] is the floor function giving the largest integer smaller or equal than r. The graph G1 is known to be homotopic to G with Euler characteristic χ(G1) = χ(G) and dim(G1) ≥ dim(G). Let Gm be the sequence of barycentric refinements of G = G0. We prove that for any finite simple graph G, the spectral functions FG m of successive refinements converge for m → ∞ uniformly on compact subsets of (0, 1) and exponentially fast to a universal limiting eigenvalue distribution function F d which only depends on the clique number respectively the dimension d of the largest complete subgraph of G and not on the starting graph G. In the case d = 1, where we deal with graphs without triangles, the limiting distribution is the smooth function F (x) = 4 sin 2 (πx/2). This is related to the Julia set of the quadratic map T (z) = 4z − z 2 which has the one dimensional Julia set [0, 4] and F satisfies T (F (k/n)) = F (2k/n) as the Laplacians satisfy such a renormalization recursion. The spectral density in the d = 1 case is then the arc-sin distribution which is the equilibrium measure on the Julia set. In higher dimensions, where the limiting function F still remains unidentified, F appears to have a discrete or singular component. We don't know whether there is an analogue renormalization in d ≥ 2. The limiting distribution has relations with the limiting vertex degree distribution and so in 2 dimensions with the graph curvature distribution of the refinements Gm.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="05d33912315697ab0c84caa75aaf6376" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841127,"asset_id":125874976,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841127/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874976"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874976"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874976; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125874976]").text(description); $(".js-view-count[data-work-id=125874976]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125874976; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125874976']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "05d33912315697ab0c84caa75aaf6376" } } $('.js-work-strip[data-work-id=125874976]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125874976,"title":"The graph spectrum of barycentric refinements","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If λ0 = 0 ≤ λ1 ≤ λ2 ≤ • • • ≤ λn are the eigenvalues of the Laplacian of G, define the spectral function F (x) = λ [nx] on the interval [0, 1], where [r] is the floor function giving the largest integer smaller or equal than r. The graph G1 is known to be homotopic to G with Euler characteristic χ(G1) = χ(G) and dim(G1) ≥ dim(G). Let Gm be the sequence of barycentric refinements of G = G0. We prove that for any finite simple graph G, the spectral functions FG m of successive refinements converge for m → ∞ uniformly on compact subsets of (0, 1) and exponentially fast to a universal limiting eigenvalue distribution function F d which only depends on the clique number respectively the dimension d of the largest complete subgraph of G and not on the starting graph G. In the case d = 1, where we deal with graphs without triangles, the limiting distribution is the smooth function F (x) = 4 sin 2 (πx/2). This is related to the Julia set of the quadratic map T (z) = 4z − z 2 which has the one dimensional Julia set [0, 4] and F satisfies T (F (k/n)) = F (2k/n) as the Laplacians satisfy such a renormalization recursion. The spectral density in the d = 1 case is then the arc-sin distribution which is the equilibrium measure on the Julia set. In higher dimensions, where the limiting function F still remains unidentified, F appears to have a discrete or singular component. We don't know whether there is an analogue renormalization in d ≥ 2. The limiting distribution has relations with the limiting vertex degree distribution and so in 2 dimensions with the graph curvature distribution of the refinements Gm.","publication_date":{"day":9,"month":8,"year":2015,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":119841127},"translated_abstract":null,"internal_url":"https://www.academia.edu/125874976/The_graph_spectrum_of_barycentric_refinements","translated_internal_url":"","created_at":"2024-11-26T23:03:20.347-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":24636766,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":119841127,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841127/thumbnails/1.jpg","file_name":"1508.pdf","download_url":"https://www.academia.edu/attachments/119841127/download_file","bulk_download_file_name":"The_graph_spectrum_of_barycentric_refine.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841127/1508-libre.pdf?1732691550=\u0026response-content-disposition=attachment%3B+filename%3DThe_graph_spectrum_of_barycentric_refine.pdf\u0026Expires=1738661625\u0026Signature=e8c4Jyk3s82j-ES8mDCu7-8K0hcHHG0U4UORjyimUTBYp0Mmg6Lw3PywnWBTQ9AGIM8bNZhNGgs-sSuQdHdLwVXoP2fEay55R5H7FmrIc3HYBOet9vsdkNPzC5aBZEdZbsSJNJzfbuAekwReTnTTNU61S9505ZHd1btzd6avaQUrUoTzG6UrQFsZiy9Ffcw5vy2xPc0srPeBjc1BoA8-ifutuH7srCNPvipQ02yhnT20KHv-bs8znegkf9zft7WR8W6-l9xK2uroZ~oBqR0hsumlukBkel~5EPfujMb3P7Kadv3Uebh-FDuEDWGJnhF02JNECkeDvJGP5jZpGnPI6A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"The_graph_spectrum_of_barycentric_refinements","translated_slug":"","page_count":20,"language":"en","content_type":"Work","summary":"Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If λ0 = 0 ≤ λ1 ≤ λ2 ≤ • • • ≤ λn are the eigenvalues of the Laplacian of G, define the spectral function F (x) = λ [nx] on the interval [0, 1], where [r] is the floor function giving the largest integer smaller or equal than r. The graph G1 is known to be homotopic to G with Euler characteristic χ(G1) = χ(G) and dim(G1) ≥ dim(G). Let Gm be the sequence of barycentric refinements of G = G0. We prove that for any finite simple graph G, the spectral functions FG m of successive refinements converge for m → ∞ uniformly on compact subsets of (0, 1) and exponentially fast to a universal limiting eigenvalue distribution function F d which only depends on the clique number respectively the dimension d of the largest complete subgraph of G and not on the starting graph G. In the case d = 1, where we deal with graphs without triangles, the limiting distribution is the smooth function F (x) = 4 sin 2 (πx/2). This is related to the Julia set of the quadratic map T (z) = 4z − z 2 which has the one dimensional Julia set [0, 4] and F satisfies T (F (k/n)) = F (2k/n) as the Laplacians satisfy such a renormalization recursion. The spectral density in the d = 1 case is then the arc-sin distribution which is the equilibrium measure on the Julia set. In higher dimensions, where the limiting function F still remains unidentified, F appears to have a discrete or singular component. We don't know whether there is an analogue renormalization in d ≥ 2. The limiting distribution has relations with the limiting vertex degree distribution and so in 2 dimensions with the graph curvature distribution of the refinements Gm.","owner":{"id":24636766,"first_name":"Oliver","middle_initials":null,"last_name":"Knill","page_name":"OliverKnill","domain_name":"harvard","created_at":"2015-01-10T20:13:49.803-08:00","display_name":"Oliver Knill","url":"https://harvard.academia.edu/OliverKnill","email":"U1hzcHFOZXVySkxQdjJyaHU4YVR3R2U3dzd5V3FlWDFFa1NzRkV3aWoraXF0UGtXMDhMaGErbmpRR2w5aWpjSC0tcVFRR3Z5RjhYcEFxOStRMElvNWlZZz09--11977d43c8b27d2886fcd478836516192becfe25"},"attachments":[{"id":119841127,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841127/thumbnails/1.jpg","file_name":"1508.pdf","download_url":"https://www.academia.edu/attachments/119841127/download_file","bulk_download_file_name":"The_graph_spectrum_of_barycentric_refine.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841127/1508-libre.pdf?1732691550=\u0026response-content-disposition=attachment%3B+filename%3DThe_graph_spectrum_of_barycentric_refine.pdf\u0026Expires=1738661625\u0026Signature=e8c4Jyk3s82j-ES8mDCu7-8K0hcHHG0U4UORjyimUTBYp0Mmg6Lw3PywnWBTQ9AGIM8bNZhNGgs-sSuQdHdLwVXoP2fEay55R5H7FmrIc3HYBOet9vsdkNPzC5aBZEdZbsSJNJzfbuAekwReTnTTNU61S9505ZHd1btzd6avaQUrUoTzG6UrQFsZiy9Ffcw5vy2xPc0srPeBjc1BoA8-ifutuH7srCNPvipQ02yhnT20KHv-bs8znegkf9zft7WR8W6-l9xK2uroZ~oBqR0hsumlukBkel~5EPfujMb3P7Kadv3Uebh-FDuEDWGJnhF02JNECkeDvJGP5jZpGnPI6A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":119841129,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841129/thumbnails/1.jpg","file_name":"1508.pdf","download_url":"https://www.academia.edu/attachments/119841129/download_file","bulk_download_file_name":"The_graph_spectrum_of_barycentric_refine.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841129/1508-libre.pdf?1732691547=\u0026response-content-disposition=attachment%3B+filename%3DThe_graph_spectrum_of_barycentric_refine.pdf\u0026Expires=1738661625\u0026Signature=HIXMSlaiw-zVYczF3Kru1AjJGJUg1svGVSiaRdk33XsMmJ5WnZg9Qhjfjb9BBQ5cxKmG~SO0FTnJY3Wde5HFtHmOVvqiK4mWKR4zVNHmfH8o78d5Tp~82fAD1Diz9gyCbDmv7Fl9IYW4liLMvR7Er1TR3tKTv7MS7f6CyyKM5lZSI80BoX2L7PXGYaNPQq3OPmLc0WUdGnukwNO6TUCbVWmp6qBt4QoXYbzzD-c2EYFNEpW8tHvHXWVobi91454gC-5FBowpS5hNgLOR8mIT5ldmavqnWDiU81wl1lSXBf3upcoJqjhvSOZ8LiSofA7ltLktArcC4RsTftvdF1ILmA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":266831,"name":"Graph","url":"https://www.academia.edu/Documents/in/Graph"},{"id":891011,"name":"Eigenvalues and Eigenvectors","url":"https://www.academia.edu/Documents/in/Eigenvalues_and_Eigenvectors"},{"id":3193313,"name":"arXiv","url":"https://www.academia.edu/Documents/in/arXiv"}],"urls":[{"id":45805001,"url":"http://arxiv.org/pdf/1508.02027"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874974"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874974/Complexes_Graphs_Homotopy_Products_and_Shannon_Capacity"><img alt="Research paper thumbnail of Complexes, Graphs, Homotopy, Products and Shannon Capacity" class="work-thumbnail" src="https://attachments.academia-assets.com/119841126/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874974/Complexes_Graphs_Homotopy_Products_and_Shannon_Capacity">Complexes, Graphs, Homotopy, Products and Shannon Capacity</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 13, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A finite abstract simplicial complex G defines the Barycentric refinement graph φ(G) = (G, {(a, b...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A finite abstract simplicial complex G defines the Barycentric refinement graph φ(G) = (G, {(a, b), a ⊂ b or b ⊂ a}) and the connection graph ψ(G) = (G, {(a, b), a ∩ b = ∅}). We note here that both functors φ and ψ from complexes to graphs are invertible on the image (Theorem 1) and that G, φ(G), ψ(G) all have the same automorphism group and that the Cartesian product of G corresponding to the Stanley-Reisner product of φ(G) and the strong Shannon product of ψ(G), have the product automorphism groups. Second, we see that if G is a Barycentric refinement, then φ(G) and ψ(G) are graph homotopic (Theorem 2). Third, if γ is the geometric realization functor, assigning to a complex or to a graph the geometric realization of its clique complex, then γ(G) and γ(φ(G)) and γ(ψ(G)) are all classically homotopic for a Barycentric refined simplicial complex G (Theorem 3). The Barycentric assumption is necessary in Theorem 2 and 3. There is compatibility with Cartesian products of complexes which manifests in the strong graph product of connection graphs: if two graphs A, A are homotopic and B, B are homotopic, then A • B is homotopic to A • B (Theorem 4) leading to a commutative ring of homotopy classes of graphs. Finally, we note (Theorem 5) that for all simplicial complexes G as well as product G = G 1 × G 2 • • • × G k , the Shannon capacity Θ(ψ(G)) of ψ(G) is equal to the number f 0 of zero-dimensional sets in G. An explicit Lowasz umbrella in R f 0 leads to the Lowasz number θ(G) ≤ f 0 and so Θ(ψ(G)) = θ(ψ(G)) = f 0 making Θ compatible with disjoint union addition and strong multiplication.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="21949b0496f2c4baf39909d9756bb203" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841126,"asset_id":125874974,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841126/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874974"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874974"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874974; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125874974]").text(description); $(".js-view-count[data-work-id=125874974]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125874974; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125874974']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "21949b0496f2c4baf39909d9756bb203" } } $('.js-work-strip[data-work-id=125874974]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125874974,"title":"Complexes, Graphs, Homotopy, Products and Shannon Capacity","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A finite abstract simplicial complex G defines the Barycentric refinement graph φ(G) = (G, {(a, b), a ⊂ b or b ⊂ a}) and the connection graph ψ(G) = (G, {(a, b), a ∩ b = ∅}). We note here that both functors φ and ψ from complexes to graphs are invertible on the image (Theorem 1) and that G, φ(G), ψ(G) all have the same automorphism group and that the Cartesian product of G corresponding to the Stanley-Reisner product of φ(G) and the strong Shannon product of ψ(G), have the product automorphism groups. Second, we see that if G is a Barycentric refinement, then φ(G) and ψ(G) are graph homotopic (Theorem 2). Third, if γ is the geometric realization functor, assigning to a complex or to a graph the geometric realization of its clique complex, then γ(G) and γ(φ(G)) and γ(ψ(G)) are all classically homotopic for a Barycentric refined simplicial complex G (Theorem 3). The Barycentric assumption is necessary in Theorem 2 and 3. There is compatibility with Cartesian products of complexes which manifests in the strong graph product of connection graphs: if two graphs A, A are homotopic and B, B are homotopic, then A • B is homotopic to A • B (Theorem 4) leading to a commutative ring of homotopy classes of graphs. Finally, we note (Theorem 5) that for all simplicial complexes G as well as product G = G 1 × G 2 • • • × G k , the Shannon capacity Θ(ψ(G)) of ψ(G) is equal to the number f 0 of zero-dimensional sets in G. An explicit Lowasz umbrella in R f 0 leads to the Lowasz number θ(G) ≤ f 0 and so Θ(ψ(G)) = θ(ψ(G)) = f 0 making Θ compatible with disjoint union addition and strong multiplication.","publication_date":{"day":13,"month":12,"year":2020,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":119841126},"translated_abstract":null,"internal_url":"https://www.academia.edu/125874974/Complexes_Graphs_Homotopy_Products_and_Shannon_Capacity","translated_internal_url":"","created_at":"2024-11-26T23:03:19.503-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":24636766,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":119841126,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841126/thumbnails/1.jpg","file_name":"2012.pdf","download_url":"https://www.academia.edu/attachments/119841126/download_file","bulk_download_file_name":"Complexes_Graphs_Homotopy_Products_and_S.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841126/2012-libre.pdf?1732691545=\u0026response-content-disposition=attachment%3B+filename%3DComplexes_Graphs_Homotopy_Products_and_S.pdf\u0026Expires=1738759467\u0026Signature=SRZZWotKtBokcFht0d7zpFBsYRCaGmo768dSLL0NxsdE8D5-Q7bCgyRtAWlcQwCDKh63C1lPchFMgm2b11J7TpLzPuJkHb0XTfeU9AW8HLgHK9reGjWPzlobFxx1dhwfjIsulW8UbAOADtIYLsHaSoP8zXyYThvg0~c4CwLixk4ARRZK-rN6dv7h27bj21Em52xl9ZnMi4WnfOk3uRAovTjGYXp933bnc4ZR0sbccFNHmrJzr4vaGq4ucaktaJv48uW3rrSVhW4EXWTijF8hjyNTFyCaArczVrO7zrEiPTnc-DxflbPi-jtXncTXYjobhs9pk-9PE2AmNHnuigCVpQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Complexes_Graphs_Homotopy_Products_and_Shannon_Capacity","translated_slug":"","page_count":26,"language":"en","content_type":"Work","summary":"A finite abstract simplicial complex G defines the Barycentric refinement graph φ(G) = (G, {(a, b), a ⊂ b or b ⊂ a}) and the connection graph ψ(G) = (G, {(a, b), a ∩ b = ∅}). We note here that both functors φ and ψ from complexes to graphs are invertible on the image (Theorem 1) and that G, φ(G), ψ(G) all have the same automorphism group and that the Cartesian product of G corresponding to the Stanley-Reisner product of φ(G) and the strong Shannon product of ψ(G), have the product automorphism groups. Second, we see that if G is a Barycentric refinement, then φ(G) and ψ(G) are graph homotopic (Theorem 2). Third, if γ is the geometric realization functor, assigning to a complex or to a graph the geometric realization of its clique complex, then γ(G) and γ(φ(G)) and γ(ψ(G)) are all classically homotopic for a Barycentric refined simplicial complex G (Theorem 3). The Barycentric assumption is necessary in Theorem 2 and 3. There is compatibility with Cartesian products of complexes which manifests in the strong graph product of connection graphs: if two graphs A, A are homotopic and B, B are homotopic, then A • B is homotopic to A • B (Theorem 4) leading to a commutative ring of homotopy classes of graphs. Finally, we note (Theorem 5) that for all simplicial complexes G as well as product G = G 1 × G 2 • • • × G k , the Shannon capacity Θ(ψ(G)) of ψ(G) is equal to the number f 0 of zero-dimensional sets in G. An explicit Lowasz umbrella in R f 0 leads to the Lowasz number θ(G) ≤ f 0 and so Θ(ψ(G)) = θ(ψ(G)) = f 0 making Θ compatible with disjoint union addition and strong multiplication.","owner":{"id":24636766,"first_name":"Oliver","middle_initials":null,"last_name":"Knill","page_name":"OliverKnill","domain_name":"harvard","created_at":"2015-01-10T20:13:49.803-08:00","display_name":"Oliver Knill","url":"https://harvard.academia.edu/OliverKnill"},"attachments":[{"id":119841126,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841126/thumbnails/1.jpg","file_name":"2012.pdf","download_url":"https://www.academia.edu/attachments/119841126/download_file","bulk_download_file_name":"Complexes_Graphs_Homotopy_Products_and_S.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841126/2012-libre.pdf?1732691545=\u0026response-content-disposition=attachment%3B+filename%3DComplexes_Graphs_Homotopy_Products_and_S.pdf\u0026Expires=1738759468\u0026Signature=B~5z~urUnLkiUW5n0SkccBxtc~YEZIj0RznHMpJTUX-H~U4AD1m98flqcyvjT~tPxWBGmsUQTh~CJy-C~yrCSF-8HTMK5LfvLPqzkS7oVyEz7QV~YPOYpYnFqaTS3Uqe2PWlkihFAlwNd1-1cqiF5I51rIOuLDGPuxDM6MZmNIWzoes5cvSqmoLc8DH9wsBfELSoE0MOzZNhVP73KPntPmejymbvshS~lGyYE-wft9bDHTt7rkj9CHYt~I-b1t5GOtDwEGuxbkUOvShSBODNpdTptxsYQzYh3dpATYpDyQny1HuRpaxC2xxya2uNMpBtdY7-zcA6-T34otPLh~NaAw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":119841130,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119841130/thumbnails/1.jpg","file_name":"2012.pdf","download_url":"https://www.academia.edu/attachments/119841130/download_file","bulk_download_file_name":"Complexes_Graphs_Homotopy_Products_and_S.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119841130/2012-libre.pdf?1732691548=\u0026response-content-disposition=attachment%3B+filename%3DComplexes_Graphs_Homotopy_Products_and_S.pdf\u0026Expires=1738759468\u0026Signature=Dbu52YjRTv6z0G31UOs69iRbQE2vKH8WZ0dJNZj7t5dycbiUP3UbIHEK0Ac0cWRCwAkgBVivQZgNEm4eLijIpcSyf-luLQWIEV0uDuHy4j8PpatPzKOW6dQKpWrhCWPvjEyWG9BwUiJbym43JG7DLPSTz-GSHlP7Ool2YpHAGxl7Kgmp92pHwpVXhrMquD~dXJcO9BZnbze4omtpp6RH2SS240Ge~Kk~2nGuuVEVN0r4a8WnkccXoy65dWnqEgGxoABj5alQzjF3iz3Y~5Uf3sHTPTNX711xtsjNkE1oy~DBt9p2IvQFtUM2OthGWsnI1aR0lUQRaNQvoNq~ZJS6lA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":171869,"name":"HOMOTOPY","url":"https://www.academia.edu/Documents/in/HOMOTOPY"},{"id":2902615,"name":"Cartesian product","url":"https://www.academia.edu/Documents/in/Cartesian_product"},{"id":3193313,"name":"arXiv","url":"https://www.academia.edu/Documents/in/arXiv"},{"id":4120544,"name":"Simplicial complex","url":"https://www.academia.edu/Documents/in/Simplicial_complex"}],"urls":[{"id":45804999,"url":"http://arxiv.org/pdf/2012.07247"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874972"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874972/Green_functions_of_Energized_complexes"><img alt="Research paper thumbnail of Green functions of Energized complexes" class="work-thumbnail" src="https://attachments.academia-assets.com/119841124/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874972/Green_functions_of_Energized_complexes">Green functions of Energized complexes</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Oct 18, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">G to a ring K with conjugation x * defines χ(A) = x∈A h(x) and ω(G) = x,y∈G,x∩y =∅ h(x) * h(y). D...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">G to a ring K with conjugation x * defines χ(A) = x∈A h(x) and ω(G) = x,y∈G,x∩y =∅ h(x) * h(y). Define L(x, y) = χ(W − (x) ∩ W − (y)) and g(x, y) = ω(x)ω(y)χ(W + (x) ∩ W + (y)), where W − (x) = {z | z ⊂ x}, W + (x) = {z | x ⊂ z} and ω(x) = (−1) dim(x) with dim(x) = |x| − 1. 1.2. The following relation [8] only requires the addition in K Theorem 1. χ(G) = x,y∈G g(x, y) 1.3. The next new quadratic energy relation links simplex interaction with multiplication in K. Define |h| 2 = h * h = N(h) in K. Theorem 2. ω(G) = x,y∈G ω(x)ω(y)|g(x, y)| 2. 1.4. The next determinant identity holds if h maps G to a division algebra K and det is the Dieudonné determinant [1]. The geometry G can here be a finite set of sets and does not need the simplical complex axiom stating that G is closed under the operation of taking non-empty finite subsets. Theorem 3. det(L) = det(g) = x∈G h(x). 1.5. If h : G → K takes values in the units U(K) of K, like i.e. Z 2 , U(1), SU(2), S 7 of the division algebras R, C, H, O, the unitary group U(H)∩K of an operator C *-algebra K ⊂ B(H) for some Hilbert space H or the units in a ring K = O K of integers of a number field K, and if G is a simplicial complex, then: Theorem 4. If h(x) * h(x) = 1 for all x ∈ G, then g * = L −1 .</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="e90a9c498c224681608711cb79988683" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841124,"asset_id":125874972,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841124/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874972"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874972"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874972; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874971"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874971/More_on_Poincare_Hopf_and_Gauss_Bonnet"><img alt="Research paper thumbnail of More on Poincare-Hopf and Gauss-Bonnet" class="work-thumbnail" src="https://attachments.academia-assets.com/119841128/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874971/More_on_Poincare_Hopf_and_Gauss_Bonnet">More on Poincare-Hopf and Gauss-Bonnet</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 1, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We illustrate connections between differential geometry on finite simple graphs G = (V, E) and Ri...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We illustrate connections between differential geometry on finite simple graphs G = (V, E) and Riemannian manifolds (M, g). The link is that curvature can be defined integral geometrically as an expectation in a probability space of Poincaré-Hopf indices of coloring or Morse functions. Regge calculus with an isometric Nash embedding links then the Gauss-Bonnet-Chern integrand of a Riemannian manifold with the graph curvature. There is also a direct nonstandard approach [18]: if V is a finite set containing all standard points of M and E contains pairs which are closer than some positive number. One gets so finite simple graphs (V, E) which leads to the standard curvature. The probabilistic approach is an umbrella framework which covers discrete spaces, piecewise linear spaces, manifolds or varieties.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="70c54ca846fdc0cab679d14e424fd919" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841128,"asset_id":125874971,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841128/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874971"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874971"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874971; 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A single network completes to the Wiener algebra. We illustrate the compatibility with topology and spectral theory. Multiplicative linear functionals like Euler characteristic, the Poincare polynomial or the zeta functions can be extended naturally. These functionals can also help with number theoretical questions. The story of primes is a bit different as the integers are not a unique factorization domain, because there are many additive primes. Most graphs are multiplicative primes.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="fee80caf04e915b93b27cf6b1ba11cec" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841121,"asset_id":125874970,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841121/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874970"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874970"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874970; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874969"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874969/The_counting_matrix_of_a_simplicial_complex"><img alt="Research paper thumbnail of The counting matrix of a simplicial complex" class="work-thumbnail" src="https://attachments.academia-assets.com/119841118/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874969/The_counting_matrix_of_a_simplicial_complex">The counting matrix of a simplicial complex</a></div><div class="wp-workCard_item"><span>arXiv: Combinatorics</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is t...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is the number of subsimplices in the intersection of x and y. We call it the counting matrix of G. Similarly as the connection matrix L which is L(x,y)=1 if x and y intersect and 0 else, the counting matrix K is unimodular. Actually, K is always in SL(n,Z). The inverse of K has the Green function entries K^(-1)(x,y)=w(x) w(y) |W^+(x) intersected W^+y|, where W^+(x) is the star of x, the sets in G which contain x and w(x)=(-1)^dim(x). The matrix K is always positive definite. The spectra of K and K^(-1) always agree so that the matrix Q=K-K^(-1) has the spectral symmetry spec(Q)=-spec(Q) and the zeta function z(s) summing l(k)^(-s) with eigenvalues l(k) of K satisfies the functional equation z(a+ib)=z(-a+ib). The energy theorem in this case tells that the sum of the matrix elements of K^(-1)(x,y) is equal to the number sets in G. In comparison, we had in the connection matrix case the identit...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="22e08a88ae0f2bc7ce620faa0ba572b7" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841118,"asset_id":125874969,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841118/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874969"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874969"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874969; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874967"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874967/A_Reeb_sphere_theorem_in_graph_theory"><img alt="Research paper thumbnail of A Reeb sphere theorem in graph theory" class="work-thumbnail" src="https://attachments.academia-assets.com/119841114/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874967/A_Reeb_sphere_theorem_in_graph_theory">A Reeb sphere theorem in graph theory</a></div><div class="wp-workCard_item"><span>ArXiv</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definit...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definitions of spheres in graph theory. We also reformulate Morse conditions in terms of the center manifolds, the level surface graphs {f=f(x)} in the unit sphere S(x). In the Morse case these graphs are either spheres, the empty graph or the product of two spheres.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7cc3b647afb6a2c938d60b3a0d090b9a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841114,"asset_id":125874967,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841114/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874967"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874967"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874967; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125874966"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125874966/Listening_to_the_cohomology_of_graphs"><img alt="Research paper thumbnail of Listening to the cohomology of graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/119841111/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125874966/Listening_to_the_cohomology_of_graphs">Listening to the cohomology of graphs</a></div><div class="wp-workCard_item"><span>ArXiv</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined g...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we prove that L-L^(-1) is similar to the Hodge Laplacian H of G which is in one dimensions the direct sum of the Kirchhoff Laplacian H0 and its 1-form analog H1. The spectrum of a single choice of H0,H1 or H alone determines the Betti numbers b0,b1 of G as well as the spectrum of the other matrices. It follows that b0 is the number of eigenvalues 1 of L and that b1 is the number of eigenvalues -1 of L. For a general abstract finite simplicial complex G, we express the matrix entries g(x,y) = w(x) w(y) X( St(x) cap St(y) ) of the inverse of L using stars St(x)= { z in G | x subset of z } of x and w(x)=(-1)^dim(x) and Euler characteristic X. One can see W+(x)=St(x) and W-(x)={ z in G | z subset x } as stable and unstable manifolds of a simplex x in G and g(x,y) =w(x) w(y) X(W+(x) cap W+(y)) as heteroclinic inter...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0162c50e77bb20b5d259e0c1cb8f7006" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119841111,"asset_id":125874966,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119841111/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125874966"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125874966"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125874966; 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