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is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Huang%2C+A&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Huang%2C+A&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Huang%2C+A&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.08608">arXiv:2406.08608</a> <span> [<a href="https://arxiv.org/pdf/2406.08608">pdf</a>, <a href="https://arxiv.org/format/2406.08608">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Integral representation and approximation of L-functions associated to Hecke cusp eigenforms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Spinelli%2C+K">Kamryn Spinelli</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.08608v1-abstract-short" style="display: inline;"> We derive a family of approximations for L-functions of Hecke cusp eigenforms, according to a recipe first described by Matiyasevich for the Riemann xi function. We show that these approximations converge to the true L-function, and along the way we demonstrate error formulas which may be used to investigate analytic properties of the L-function and its derivatives. Together with the Euler product… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.08608v1-abstract-full').style.display = 'inline'; document.getElementById('2406.08608v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.08608v1-abstract-full" style="display: none;"> We derive a family of approximations for L-functions of Hecke cusp eigenforms, according to a recipe first described by Matiyasevich for the Riemann xi function. We show that these approximations converge to the true L-function, and along the way we demonstrate error formulas which may be used to investigate analytic properties of the L-function and its derivatives. Together with the Euler product expansion of the L-function, the family of approximations also encodes some of the key features of the L-function such as its functional equation. Finally, we derive via Mellin transforms a convolution-type formula which leads to precise error bounds in terms of the incomplete gamma function. This formula can be interpreted as an alternative definition for the approximation and sheds light on Matiyasevich's procedure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.08608v1-abstract-full').style.display = 'none'; document.getElementById('2406.08608v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11F11 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.08394">arXiv:2405.08394</a> <span> [<a href="https://arxiv.org/pdf/2405.08394">pdf</a>, <a href="https://arxiv.org/ps/2405.08394">ps</a>, <a href="https://arxiv.org/format/2405.08394">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Weak solutions to the steady compressible Euler equations with source terms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Anxiang Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.08394v1-abstract-short" style="display: inline;"> In this paper, we showed that for some given suitable density and pressure, there exist infinitely many compactly supported solutions with prescribed energy profile. The proof is mainly based on the convex integration scheme. We construct suitable subsolutions and localized plane-wave solutions to the reformulated system, and weak solutions are obtained by iterating these subsolutions. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.08394v1-abstract-full" style="display: none;"> In this paper, we showed that for some given suitable density and pressure, there exist infinitely many compactly supported solutions with prescribed energy profile. The proof is mainly based on the convex integration scheme. We construct suitable subsolutions and localized plane-wave solutions to the reformulated system, and weak solutions are obtained by iterating these subsolutions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.08394v1-abstract-full').style.display = 'none'; document.getElementById('2405.08394v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35D30; 35Q31; 76N10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.08390">arXiv:2405.08390</a> <span> [<a href="https://arxiv.org/pdf/2405.08390">pdf</a>, <a href="https://arxiv.org/ps/2405.08390">ps</a>, <a href="https://arxiv.org/format/2405.08390">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Weak solutions to the steady incompressible Euler equations with source terms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Anxiang Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.08390v1-abstract-short" style="display: inline;"> In this paper, we prove the non-uniqueness of stationary solutions to steady incompressible Euler equations with source terms. Based on the convex integration scheme developed by De Lellis and Sz茅kelyhidi, the Euler system is reformulated as a differential inclusion. The key point is to construct the corresponding plane-wave solutions via high frequency perturbations. Then we use iteration and Bai… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.08390v1-abstract-full').style.display = 'inline'; document.getElementById('2405.08390v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.08390v1-abstract-full" style="display: none;"> In this paper, we prove the non-uniqueness of stationary solutions to steady incompressible Euler equations with source terms. Based on the convex integration scheme developed by De Lellis and Sz茅kelyhidi, the Euler system is reformulated as a differential inclusion. The key point is to construct the corresponding plane-wave solutions via high frequency perturbations. Then we use iteration and Baire category argument to conclude that there exist a large amount of weak solutions with given energy profile. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.08390v1-abstract-full').style.display = 'none'; document.getElementById('2405.08390v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 76B03 35D05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.02804">arXiv:2311.02804</a> <span> [<a href="https://arxiv.org/pdf/2311.02804">pdf</a>, <a href="https://arxiv.org/ps/2311.02804">ps</a>, <a href="https://arxiv.org/format/2311.02804">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computational Complexity">cs.CC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Last fall degree of semi-local polynomial systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+M+A">Ming-Deh A. Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.02804v1-abstract-short" style="display: inline;"> We study the last fall degrees of {\em semi-local} polynomial systems, and the computational complexity of solving such systems for closed-point and rational-point solutions, where the systems are defined over a finite field. A semi-local polynomial system specifies an algebraic set which is the image of a global linear transformation of a direct product of local affine algebraic sets. As a specia… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.02804v1-abstract-full').style.display = 'inline'; document.getElementById('2311.02804v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.02804v1-abstract-full" style="display: none;"> We study the last fall degrees of {\em semi-local} polynomial systems, and the computational complexity of solving such systems for closed-point and rational-point solutions, where the systems are defined over a finite field. A semi-local polynomial system specifies an algebraic set which is the image of a global linear transformation of a direct product of local affine algebraic sets. As a special but interesting case, polynomial systems that arise from Weil restriction of algebraic sets in an affine space of low dimension are semi-local. Such systems have received considerable attention due to their application in cryptography. Our main results bound the last fall degree of a semi-local polynomial system in terms of the number of closed point solutions, and yield an efficient algorithm for finding all rational-point solutions when the prime characteristic of the finite field and the number of rational solutions are small. Our results on solving semi-local systems imply an improvement on a previously known polynomial-time attack on the HFE (Hidden Field Equations) cryptosystems. The attacks implied in our results extend to public key encryption functions which are based on semi-local systems where either the number of closed point solutions is small, or the characteristic of the field is small. It remains plausible to construct public key cryptosystems based on semi-local systems over a finite field of large prime characteristic with exponential number of closed point solutions. Such a method is presented in the paper, followed by further cryptanalysis involving the isomorphism of polynomials (IP) problem, as well as a concrete public key encryption scheme which is secure against all the attacks discussed in this paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.02804v1-abstract-full').style.display = 'none'; document.getElementById('2311.02804v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.01610">arXiv:2311.01610</a> <span> [<a href="https://arxiv.org/pdf/2311.01610">pdf</a>, <a href="https://arxiv.org/format/2311.01610">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> Stabilization of codimension of persistence barcodes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Allman%2C+J">Justin Allman</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">Anran Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.01610v1-abstract-short" style="display: inline;"> Persistent homology has become a ubiquitous tool in topological data analysis. Throughout this paper, we consider persistent homology of a fixed dataset which is assumed to be a point cloud in a Euclidean space. Further, we use the Rips construction for computation of persistent homology which, in our setup, takes as a parameter a step size $h$. For a fixed $h$, one important data visualization fo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.01610v1-abstract-full').style.display = 'inline'; document.getElementById('2311.01610v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.01610v1-abstract-full" style="display: none;"> Persistent homology has become a ubiquitous tool in topological data analysis. Throughout this paper, we consider persistent homology of a fixed dataset which is assumed to be a point cloud in a Euclidean space. Further, we use the Rips construction for computation of persistent homology which, in our setup, takes as a parameter a step size $h$. For a fixed $h$, one important data visualization for the persistent homology is the persistence barcode. From another point of view, each persistence barcode corresponds to a specific isomorphism class of quiver representations of an equioriented A-type Dynkin quiver. Using algebro-geometric facts regarding type-A quiver representations, we define the codimension of a persistence barcode as the codimension of the corresponding isomorphism class in the space of all quiver representations. Further, we prove a new formula for the codimension and, as an application we prove that for a fixed dataset, the value of the codimension of the persistence barcode stabilizes as the step size parameter tends to zero. As a consequence, we define a new topological statistic for a dataset which we call the quiver codimension of the dataset, and which is equal to the stabilized value of the barcode codimensions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.01610v1-abstract-full').style.display = 'none'; document.getElementById('2311.01610v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.11818">arXiv:2308.11818</a> <span> [<a href="https://arxiv.org/pdf/2308.11818">pdf</a>, <a href="https://arxiv.org/format/2308.11818">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Incorporating Nonlocal Traffic Flow Model in Physics-informed Neural Networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A+J">Archie J. Huang</a>, <a href="/search/math?searchtype=author&query=Biswas%2C+A">Animesh Biswas</a>, <a href="/search/math?searchtype=author&query=Agarwal%2C+S">Shaurya Agarwal</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2308.11818v1-abstract-short" style="display: inline;"> This research contributes to the advancement of traffic state estimation methods by leveraging the benefits of the nonlocal LWR model within a physics-informed deep learning framework. The classical LWR model, while useful, falls short of accurately representing real-world traffic flows. The nonlocal LWR model addresses this limitation by considering the speed as a weighted mean of the downstream… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.11818v1-abstract-full').style.display = 'inline'; document.getElementById('2308.11818v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.11818v1-abstract-full" style="display: none;"> This research contributes to the advancement of traffic state estimation methods by leveraging the benefits of the nonlocal LWR model within a physics-informed deep learning framework. The classical LWR model, while useful, falls short of accurately representing real-world traffic flows. The nonlocal LWR model addresses this limitation by considering the speed as a weighted mean of the downstream traffic density. In this paper, we propose a novel PIDL framework that incorporates the nonlocal LWR model. We introduce both fixed-length and variable-length kernels and develop the required mathematics. The proposed PIDL framework undergoes a comprehensive evaluation, including various convolutional kernels and look-ahead windows, using data from the NGSIM and CitySim datasets. The results demonstrate improvements over the baseline PIDL approach using the local LWR model. The findings highlight the potential of the proposed approach to enhance the accuracy and reliability of traffic state estimation, enabling more effective traffic management strategies. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.11818v1-abstract-full').style.display = 'none'; document.getElementById('2308.11818v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.12337">arXiv:2302.12337</a> <span> [<a href="https://arxiv.org/pdf/2302.12337">pdf</a>, <a href="https://arxiv.org/format/2302.12337">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A+J">Archie J. Huang</a>, <a href="/search/math?searchtype=author&query=Agarwal%2C+S">Shaurya Agarwal</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.12337v1-abstract-short" style="display: inline;"> Since its introduction in 2017, physics-informed deep learning (PIDL) has garnered growing popularity in understanding the evolution of systems governed by physical laws in terms of partial differential equations (PDEs). However, empirical evidence points to the limitations of PIDL for learning certain types of PDEs. In this paper, we (a) present the challenges in training PIDL architecture, (b) c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.12337v1-abstract-full').style.display = 'inline'; document.getElementById('2302.12337v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.12337v1-abstract-full" style="display: none;"> Since its introduction in 2017, physics-informed deep learning (PIDL) has garnered growing popularity in understanding the evolution of systems governed by physical laws in terms of partial differential equations (PDEs). However, empirical evidence points to the limitations of PIDL for learning certain types of PDEs. In this paper, we (a) present the challenges in training PIDL architecture, (b) contrast the performance of PIDL architecture in learning a first order scalar hyperbolic conservation law and its parabolic counterpart, (c) investigate the effect of training data sampling, which corresponds to various sensing scenarios in traffic networks, (d) comment on the implications of PIDL limitations for traffic flow estimation and prediction in practice. Detailed in the case study, we present the contradistinction in PIDL results between learning the traffic flow model (LWR PDE) and its variation with diffusion. The outcome indicates that PIDL experiences significant challenges in learning the hyperbolic LWR equation due to the non-smoothness of its solution. On the other hand, the architecture with parabolic PDE, augmented with the diffusion term, leads to the successful reassembly of the density data even with the shockwaves present. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.12337v1-abstract-full').style.display = 'none'; document.getElementById('2302.12337v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.12336">arXiv:2302.12336</a> <span> [<a href="https://arxiv.org/pdf/2302.12336">pdf</a>, <a href="https://arxiv.org/format/2302.12336">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Physics Informed Deep Learning: Applications in Transportation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A+J">Archie J. Huang</a>, <a href="/search/math?searchtype=author&query=Agarwal%2C+S">Shaurya Agarwal</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.12336v1-abstract-short" style="display: inline;"> A recent development in machine learning - physics-informed deep learning (PIDL) - presents unique advantages in transportation applications such as traffic state estimation. Consolidating the benefits of deep learning (DL) and the governing physical equations, it shows the potential to complement traditional sensing methods in obtaining traffic states. In this paper, we first explain the conserva… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.12336v1-abstract-full').style.display = 'inline'; document.getElementById('2302.12336v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.12336v1-abstract-full" style="display: none;"> A recent development in machine learning - physics-informed deep learning (PIDL) - presents unique advantages in transportation applications such as traffic state estimation. Consolidating the benefits of deep learning (DL) and the governing physical equations, it shows the potential to complement traditional sensing methods in obtaining traffic states. In this paper, we first explain the conservation law from the traffic flow theory as ``physics'', then present the architecture of a PIDL neural network and demonstrate its effectiveness in learning traffic conditions of unobserved areas. In addition, we also exhibit the data collection scenario using fog computing infrastructure. A case study on estimating the vehicle velocity is presented and the result shows that PIDL surpasses the performance of a regular DL neural network with the same learning architecture, in terms of convergence time and reconstruction accuracy. The encouraging results showcase the broad potential of PIDL for real-time applications in transportation with a low amount of training data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.12336v1-abstract-full').style.display = 'none'; document.getElementById('2302.12336v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.04169">arXiv:2208.04169</a> <span> [<a href="https://arxiv.org/pdf/2208.04169">pdf</a>, <a href="https://arxiv.org/format/2208.04169">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> A Stable Mimetic Finite-Difference Method for Convection-Dominated Diffusion Equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Adler%2C+J+H">James H. Adler</a>, <a href="/search/math?searchtype=author&query=Cavanaugh%2C+C">Casey Cavanaugh</a>, <a href="/search/math?searchtype=author&query=Hu%2C+X">Xiaozhe Hu</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">Andy Huang</a>, <a href="/search/math?searchtype=author&query=Trask%2C+N">Nathaniel Trask</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.04169v2-abstract-short" style="display: inline;"> Convection-diffusion equations arise in a variety of applications such as particle transport, electromagnetics, and magnetohydrodynamics. Simulation of the convection-dominated regime for these problems, even with high-fidelity techniques, is particularly challenging due to the presence of sharp boundary layers and shocks causing jumps and discontinuities in the solution, and numerical issues such… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.04169v2-abstract-full').style.display = 'inline'; document.getElementById('2208.04169v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.04169v2-abstract-full" style="display: none;"> Convection-diffusion equations arise in a variety of applications such as particle transport, electromagnetics, and magnetohydrodynamics. Simulation of the convection-dominated regime for these problems, even with high-fidelity techniques, is particularly challenging due to the presence of sharp boundary layers and shocks causing jumps and discontinuities in the solution, and numerical issues such as loss of the maximum principle in the discretization. These complications cause instabilities, admitting large oscillations in the numerical solution when using traditional methods. Drawing connections to the simplex-averaged finite-element method (S. Wu and J. Xu, 2020), this paper develops a mimetic finite-difference (MFD) discretization using exponentially-averaged coefficients to overcome instability of the numerical solution as the diffusion coefficient approaches zero. The finite-element framework allows for transparent analysis of the MFD, such as proving well-posedness and deriving error estimates. Numerical tests are presented confirming the stability of the method and verifying the error estimates. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.04169v2-abstract-full').style.display = 'none'; document.getElementById('2208.04169v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35M12; 65N06; 65N30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.01217">arXiv:2202.01217</a> <span> [<a href="https://arxiv.org/pdf/2202.01217">pdf</a>, <a href="https://arxiv.org/ps/2202.01217">ps</a>, <a href="https://arxiv.org/format/2202.01217">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Quadratic reciprocity from a family of adelic conformal field theories </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Stoica%2C+B">Bogdan Stoica</a>, <a href="/search/math?searchtype=author&query=Zhong%2C+X">Xiao Zhong</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.01217v1-abstract-short" style="display: inline;"> We consider a deformation of the two-dimensional free scalar field theory by raising the Laplacian to a positive real power. It turns out that the resulting non-local generalized free action is invariant under two commuting actions of the global conformal symmetry algebra, although it is no longer invariant under the full Witt algebra. Furthermore, there is an adelic version of this family of conf… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.01217v1-abstract-full').style.display = 'inline'; document.getElementById('2202.01217v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.01217v1-abstract-full" style="display: none;"> We consider a deformation of the two-dimensional free scalar field theory by raising the Laplacian to a positive real power. It turns out that the resulting non-local generalized free action is invariant under two commuting actions of the global conformal symmetry algebra, although it is no longer invariant under the full Witt algebra. Furthermore, there is an adelic version of this family of conformal field theories, parameterized by the choice of a number field, together with a Hecke character. Tate's thesis gives the Green's functions of these theories, and ensures that these Green's functions satisfy an adelic product formula. In particular, the local $L$-factors contribute to the prefactors of these Green's functions. Quadratic reciprocity turns out to be a consequence of an adelic version of a holomorphic factorization property of this family of theories on a quadratic extension of $\mathbb{Q}$. We explain that at the Archimedean place, the desired holomorphic factorization follows from the global conformal symmetry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.01217v1-abstract-full').style.display = 'none'; document.getElementById('2202.01217v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> nuhep-th/21-10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.00653">arXiv:2201.00653</a> <span> [<a href="https://arxiv.org/pdf/2201.00653">pdf</a>, <a href="https://arxiv.org/ps/2201.00653">ps</a>, <a href="https://arxiv.org/format/2201.00653">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> </div> <p class="title is-5 mathjax"> On product decomposition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+M+A">Ming-Deh A. Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.00653v1-abstract-short" style="display: inline;"> Given a finite set $W$ in $\bar{k}^n$ where $\bar{k}$ is the algebraic closure of a field $k$ one would like to determine if $W$ can be decomposed as $\prod_{i=1}^n V_i$ where $V_i \subset \bar{k}$ under a linear transformation, that is, $W\stackrel位{\to} \prod_{i=1}^n V_i$ where $位\in Gl_n (\bar{k})$. We assume that $W$ is presented as $W=Z(\mathcal{F})$, the zero set of a polynomial system… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.00653v1-abstract-full').style.display = 'inline'; document.getElementById('2201.00653v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.00653v1-abstract-full" style="display: none;"> Given a finite set $W$ in $\bar{k}^n$ where $\bar{k}$ is the algebraic closure of a field $k$ one would like to determine if $W$ can be decomposed as $\prod_{i=1}^n V_i$ where $V_i \subset \bar{k}$ under a linear transformation, that is, $W\stackrel位{\to} \prod_{i=1}^n V_i$ where $位\in Gl_n (\bar{k})$. We assume that $W$ is presented as $W=Z(\mathcal{F})$, the zero set of a polynomial system $\mathcal{F}$ in $n$ variables over $k$. We study algebraic characterization of such product decomposition. For decomposition into component sets of the same cardinality we obtain a stronger characterization and show that the decomposition in this case is essentially unique (up to permutation and scalar multiplication of coordinates). We investigate computational problems that arise from the decomposition problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.00653v1-abstract-full').style.display = 'none'; document.getElementById('2201.00653v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2110.14055">arXiv:2110.14055</a> <span> [<a href="https://arxiv.org/pdf/2110.14055">pdf</a>, <a href="https://arxiv.org/format/2110.14055">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Polynomial-Spline Neural Networks with Exact Integrals </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Actor%2C+J+A">Jonas A. Actor</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">Andy Huang</a>, <a href="/search/math?searchtype=author&query=Trask%2C+N">Nathaniel Trask</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2110.14055v1-abstract-short" style="display: inline;"> Using neural networks to solve variational problems, and other scientific machine learning tasks, has been limited by a lack of consistency and an inability to exactly integrate expressions involving neural network architectures. We address these limitations by formulating a novel neural network architecture that combines a polynomial mixture-of-experts model with free knot B1-spline basis functio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.14055v1-abstract-full').style.display = 'inline'; document.getElementById('2110.14055v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2110.14055v1-abstract-full" style="display: none;"> Using neural networks to solve variational problems, and other scientific machine learning tasks, has been limited by a lack of consistency and an inability to exactly integrate expressions involving neural network architectures. We address these limitations by formulating a novel neural network architecture that combines a polynomial mixture-of-experts model with free knot B1-spline basis functions. Effectively, our architecture performs piecewise polynomial approximation on each cell of a trainable partition of unity. Our architecture exhibits both $h$- and $p$- refinement for regression problems at the convergence rates expected from approximation theory, allowing for consistency in solving variational problems. Moreover, this architecture, its moments, and its partial derivatives can all be integrated exactly, obviating a reliance on sampling or quadrature and enabling error-free computation of variational forms. We demonstrate the success of our network on a range of regression and variational problems that illustrate the consistency and exact integrability of our network architecture. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.14055v1-abstract-full').style.display = 'none'; document.getElementById('2110.14055v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages + 4 pages Technical Appendix. Contact authors regarding supplementary multimedia and code appendices</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2109.14669">arXiv:2109.14669</a> <span> [<a href="https://arxiv.org/pdf/2109.14669">pdf</a>, <a href="https://arxiv.org/ps/2109.14669">ps</a>, <a href="https://arxiv.org/format/2109.14669">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> </div> </div> <p class="title is-5 mathjax"> Pursuit-evasion games on latin square graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ahirwar%2C+S">Shreya Ahirwar</a>, <a href="/search/math?searchtype=author&query=Bonato%2C+A">Anthony Bonato</a>, <a href="/search/math?searchtype=author&query=Gittins%2C+L">Leanna Gittins</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">Alice Huang</a>, <a href="/search/math?searchtype=author&query=Marbach%2C+T+G">Trent G. Marbach</a>, <a href="/search/math?searchtype=author&query=Zaidman%2C+T">Tomer Zaidman</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2109.14669v1-abstract-short" style="display: inline;"> We investigate various pursuit-evasion parameters on latin square graphs, including the cop number, metric dimension, and localization number. The cop number of latin square graphs is studied, and for $k$-MOLS$(n),$ bounds for the cop number are given. If $n>(k+1)^2,$ then the cop number is shown to be $k+2.$ Lower and upper bounds are provided for the metric dimension and localization number of l… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.14669v1-abstract-full').style.display = 'inline'; document.getElementById('2109.14669v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.14669v1-abstract-full" style="display: none;"> We investigate various pursuit-evasion parameters on latin square graphs, including the cop number, metric dimension, and localization number. The cop number of latin square graphs is studied, and for $k$-MOLS$(n),$ bounds for the cop number are given. If $n>(k+1)^2,$ then the cop number is shown to be $k+2.$ Lower and upper bounds are provided for the metric dimension and localization number of latin square graphs. The metric dimension of back-circulant latin squares shows that the lower bound is close to tight. Recent results on covers and partial transversals of latin squares provide the upper bound of $n+O\left(\frac{\log{n}}{\log{\log{n}}}\right)$ on the localization number of a latin square graph of order $n.$ <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.14669v1-abstract-full').style.display = 'none'; document.getElementById('2109.14669v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.11587">arXiv:2106.11587</a> <span> [<a href="https://arxiv.org/pdf/2106.11587">pdf</a>, <a href="https://arxiv.org/ps/2106.11587">ps</a>, <a href="https://arxiv.org/format/2106.11587">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Product Expansions of q-Character Polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Balachandran%2C+A">Adithya Balachandran</a>, <a href="/search/math?searchtype=author&query=Gadish%2C+N">Nir Gadish</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">Andrew Huang</a>, <a href="/search/math?searchtype=author&query=Sun%2C+S">Siwen Sun</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.11587v1-abstract-short" style="display: inline;"> The ring of q-character polynomials is a q-analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain class functions defined simultaneously on the groups $Gl_n(F_q)$ for all n, which we also interpret as statistics on matrices. Here we evaluate these statistics on all matrices and work towards computing the structure constants of the product in t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.11587v1-abstract-full').style.display = 'inline'; document.getElementById('2106.11587v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.11587v1-abstract-full" style="display: none;"> The ring of q-character polynomials is a q-analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain class functions defined simultaneously on the groups $Gl_n(F_q)$ for all n, which we also interpret as statistics on matrices. Here we evaluate these statistics on all matrices and work towards computing the structure constants of the product in this ring. We show that the statistics are periodically polynomial in q, and governed by universal polynomials $P_{位,渭}(q)$ which we compute explicitly, indexed by pairs of integer partitions. The product structure is similarly polynomial in q in many cases, governed by polynomials $R_{位,渭}^谓(q)$ indexed by triples of partitions, which we compute in some cases. Our calculations seem to exhibit several unexpected patterns. Mainly, we conjecture that certain indecomposable statistics generate the whole ring, and indeed prove this for statistics associated with matrices consisting of up to 2 Jordan blocks. Furthermore, the coefficients we compute exhibit surprising stability phenomena, which in turn reflect stabilizations of joint moments as well as multiplicities in the irreducible decomposition of tensor products of representations of $Gl_n(F_q)$ for $n\gg 1$. We use this stabilization to compute the correlation of the number of unipotent Jordan blocks of two sizes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.11587v1-abstract-full').style.display = 'none'; document.getElementById('2106.11587v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, 1 table</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G40; 05E05; 05A10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.12956">arXiv:2104.12956</a> <span> [<a href="https://arxiv.org/pdf/2104.12956">pdf</a>, <a href="https://arxiv.org/ps/2104.12956">ps</a>, <a href="https://arxiv.org/format/2104.12956">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> $\ell$-adic Tautological Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B">Bong Lian</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+D">Dingxin Zhang</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+X">Xinwen Zhu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2104.12956v1-abstract-short" style="display: inline;"> Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $\ell$-adic tautological systems for the case where the ground field is of characteristic $p$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.12956v1-abstract-full" style="display: none;"> Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $\ell$-adic tautological systems for the case where the ground field is of characteristic $p$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.12956v1-abstract-full').style.display = 'none'; document.getElementById('2104.12956v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14F20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.11799">arXiv:2012.11799</a> <span> [<a href="https://arxiv.org/pdf/2012.11799">pdf</a>, <a href="https://arxiv.org/format/2012.11799">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Enforcing exact physics in scientific machine learning: a data-driven exterior calculus on graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Trask%2C+N">Nathaniel Trask</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">Andy Huang</a>, <a href="/search/math?searchtype=author&query=Hu%2C+X">Xiaozhe Hu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2012.11799v1-abstract-short" style="display: inline;"> As traditional machine learning tools are increasingly applied to science and engineering applications, physics-informed methods have emerged as effective tools for endowing inferences with properties essential for physical realizability. While promising, these methods generally enforce physics weakly via penalization. To enforce physics strongly, we turn to the exterior calculus framework underpi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.11799v1-abstract-full').style.display = 'inline'; document.getElementById('2012.11799v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.11799v1-abstract-full" style="display: none;"> As traditional machine learning tools are increasingly applied to science and engineering applications, physics-informed methods have emerged as effective tools for endowing inferences with properties essential for physical realizability. While promising, these methods generally enforce physics weakly via penalization. To enforce physics strongly, we turn to the exterior calculus framework underpinning combinatorial Hodge theory and physics-compatible discretization of partial differential equations (PDEs). Historically, these two fields have remained largely distinct, as graphs are strictly topological objects lacking the metric information fundamental to PDE discretization. We present an approach where this missing metric information may be learned from data, using graphs as coarse-grained mesh surrogates that inherit desirable conservation and exact sequence structure from the combinatorial Hodge theory. The resulting data-driven exterior calculus (DDEC) may be used to extract structure-preserving surrogate models with mathematical guarantees of well-posedness. The approach admits a PDE-constrained optimization training strategy which guarantees machine-learned models enforce physics to machine precision, even for poorly trained models or small data regimes. We provide analysis of the method for a class of models designed to reproduce nonlinear perturbations of elliptic problems and provide examples of learning $H(div)/H(curl)$ systems representative of subsurface flows and electromagnetics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.11799v1-abstract-full').style.display = 'none'; document.getElementById('2012.11799v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.07503">arXiv:2011.07503</a> <span> [<a href="https://arxiv.org/pdf/2011.07503">pdf</a>, <a href="https://arxiv.org/format/2011.07503">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> On arbitrarily underdispersed Conway-Maxwell-Poisson distributions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Alan Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2011.07503v1-abstract-short" style="display: inline;"> We show that the Conway--Maxwell--Poisson distribution can be arbitrarily underdispersed when parametrized via its mean. More precisely, if the mean $渭$ is an integer then the limiting distribution is a unit probability mass at $渭$. If the mean $渭$ is not an integer then the limiting distribution is a shifted Bernoulli on the two values $\floor渭$ and $\ceil渭$ with probabilities equal to the fracti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.07503v1-abstract-full').style.display = 'inline'; document.getElementById('2011.07503v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.07503v1-abstract-full" style="display: none;"> We show that the Conway--Maxwell--Poisson distribution can be arbitrarily underdispersed when parametrized via its mean. More precisely, if the mean $渭$ is an integer then the limiting distribution is a unit probability mass at $渭$. If the mean $渭$ is not an integer then the limiting distribution is a shifted Bernoulli on the two values $\floor渭$ and $\ceil渭$ with probabilities equal to the fractional parts of $渭$. In either case, the limiting distribution is the most underdispersed discrete distribution possible for any given mean. This is currently the only known generalization of the Poisson distribution exhibiting this property. Four practical implications are discussed, each adding to the claim that the (mean-parametrized) Conway--Maxwell--Poisson distribution should be considered the default model for underdispersed counts. We suggest that all future generalizations of the Poisson distribution be tested against this property. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.07503v1-abstract-full').style.display = 'none'; document.getElementById('2011.07503v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, 1 figure, 1 table</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.06063">arXiv:2011.06063</a> <span> [<a href="https://arxiv.org/pdf/2011.06063">pdf</a>, <a href="https://arxiv.org/format/2011.06063">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.37236/10011">10.37236/10011 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> H-chromatic symmetric functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Eagles%2C+N+M">Nancy Mae Eagles</a>, <a href="/search/math?searchtype=author&query=Foley%2C+A+M">Ang猫le M. Foley</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">Alice Huang</a>, <a href="/search/math?searchtype=author&query=Karangozishvili%2C+E">Elene Karangozishvili</a>, <a href="/search/math?searchtype=author&query=Yu%2C+A">Annan Yu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2011.06063v2-abstract-short" style="display: inline;"> We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.06063v2-abstract-full').style.display = 'inline'; document.getElementById('2011.06063v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.06063v2-abstract-full" style="display: none;"> We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementary symmetric functions, can be realized as $H$-chromatic symmetric functions. We end with some conjectures and open problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.06063v2-abstract-full').style.display = 'none'; document.getElementById('2011.06063v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">38 pages; corrected typos and clarified some details</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E05; 05C15 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Electron. J. Comb. 29 (2022) 1 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.06716">arXiv:2006.06716</a> <span> [<a href="https://arxiv.org/pdf/2006.06716">pdf</a>, <a href="https://arxiv.org/format/2006.06716">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Bounds on the Ricci curvature and solutions to the Einstein equations for weighted graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Stoica%2C+B">Bogdan Stoica</a>, <a href="/search/math?searchtype=author&query=Xia%2C+X">Xuyang Xia</a>, <a href="/search/math?searchtype=author&query=Zhong%2C+X">Xiao Zhong</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.06716v1-abstract-short" style="display: inline;"> This is a preliminary study of the equation of motion of Euclidean classical gravity on a graph, based on the Lin-Lu-Yau Ricci curvature on graphs. We observe that the constant edge weights configuration gives the unique solution on an infinite tree w.r.t. the asymptotically constant boundary condition. We study the minimum and maximum of the action w.r.t. certain boundary conditions, on several t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.06716v1-abstract-full').style.display = 'inline'; document.getElementById('2006.06716v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.06716v1-abstract-full" style="display: none;"> This is a preliminary study of the equation of motion of Euclidean classical gravity on a graph, based on the Lin-Lu-Yau Ricci curvature on graphs. We observe that the constant edge weights configuration gives the unique solution on an infinite tree w.r.t. the asymptotically constant boundary condition. We study the minimum and maximum of the action w.r.t. certain boundary conditions, on several types of graphs of interest. We also exhibit a new class of solutions to the equations of motion on the infinite regular tree. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.06716v1-abstract-full').style.display = 'none'; document.getElementById('2006.06716v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> nuhep-th/20-04 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.07923">arXiv:2002.07923</a> <span> [<a href="https://arxiv.org/pdf/2002.07923">pdf</a>, <a href="https://arxiv.org/ps/2002.07923">ps</a>, <a href="https://arxiv.org/format/2002.07923">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Cryptography and Security">cs.CR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Algebraic blinding and cryptographic trilinear maps </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+M+A">Ming-Deh A. Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2002.07923v2-abstract-short" style="display: inline;"> It has been shown recently that cryptographic trilinear maps are sufficient for achieving indistinguishability obfuscation. In this paper we develop algebraic blinding techniques for constructing such maps. An earlier approach involving Weil restriction can be regarded as a special case of blinding in our framework. However, the techniques developed in this paper are more general, more robust, and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.07923v2-abstract-full').style.display = 'inline'; document.getElementById('2002.07923v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.07923v2-abstract-full" style="display: none;"> It has been shown recently that cryptographic trilinear maps are sufficient for achieving indistinguishability obfuscation. In this paper we develop algebraic blinding techniques for constructing such maps. An earlier approach involving Weil restriction can be regarded as a special case of blinding in our framework. However, the techniques developed in this paper are more general, more robust, and easier to analyze. The trilinear maps constructed in this paper are efficiently computable. The relationship between the published entities and the hidden entities under the blinding scheme is described by algebraic conditions. Finding points on an algebraic set defined by such conditions for the purpose of unblinding is difficult as these algebraic sets have dimension at least linear in $n$ and involves $惟(n^2)$ variables, where $n$ is the security parameter. Finding points on such algebraic sets in general takes time exponential in $n^2\log n$ with the best known methods. Additionally these algebraic sets are characterized as being {\em triply confusing} and most likely {\em uniformly confusing} as well. These properties provide additional evidence that efficient algorithms to find points on such algebraic sets seems unlikely to exist. In addition to algebraic blinding, the security of the trilinear maps also depends on the computational complexity of a trapdoor discrete logarithm problem which is defined in terms of an associative non-commutative polynomial algebra acting on torsion points of a blinded product of elliptic curves. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.07923v2-abstract-full').style.display = 'none'; document.getElementById('2002.07923v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2001.01776">arXiv:2001.01776</a> <span> [<a href="https://arxiv.org/pdf/2001.01776">pdf</a>, <a href="https://arxiv.org/ps/2001.01776">ps</a>, <a href="https://arxiv.org/format/2001.01776">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> On the Sum of Ricci-Curvatures for Weighted Graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bai%2C+S">Shuliang Bai</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lu%2C+L">Linyuan Lu</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2001.01776v2-abstract-short" style="display: inline;"> In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph. A weighted graph $G=(V,E,d)$ is an undirected graph $G=(V,E)$ associated with a distance function $d\colon E\to [0,\infty)$. By redefining the weights if possible, without loss of generality, we assu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.01776v2-abstract-full').style.display = 'inline'; document.getElementById('2001.01776v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2001.01776v2-abstract-full" style="display: none;"> In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph. A weighted graph $G=(V,E,d)$ is an undirected graph $G=(V,E)$ associated with a distance function $d\colon E\to [0,\infty)$. By redefining the weights if possible, without loss of generality, we assume that the shortest weighted distance between $u$ and $v$ is exactly $d(u,v)$ for any edge $uv$. Now consider a random walk whose transitive probability from an vertex $u$ to its neighbor $v$ (a jump move along the edge $uv$) is proportional to $w_{uv}:=F(d(u,v))/d(u,v)$ for some given function $F(\bullet)$. We first generalize Lin-Lu-Yau's Ricci curvature definition to this weighted graph and give a simple limit-free representation of $魏(x, y)$ using a so called $\ast$-coupling functions. The total curvature $K(G)$ is defined to be the sum of Ricci curvatures over all edges of $G$. We proved the following theorems: if $F(\bullet)$ is a decreasing function, then $K(G)\geq 2|V| -2|E|$; if $F(\bullet)$ is an increasing function, then $K(G)\leq 2|V| -2|E|$. Both equalities hold if and only if $d$ is a constant function plus the girth is at least $6$. In particular, these imply a Gauss-Bonnet theorem for (unweighted) graphs with girth at least $6$, where the graph Ricci curvature is defined geometrically in terms of optimal transport. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.01776v2-abstract-full').style.display = 'none'; document.getElementById('2001.01776v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2001.01725">arXiv:2001.01725</a> <span> [<a href="https://arxiv.org/pdf/2001.01725">pdf</a>, <a href="https://arxiv.org/format/2001.01725">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> From $p$-adic to Archimedean Physics: Renormalization Group Flow and Berkovich Spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Mao%2C+D">Dan Mao</a>, <a href="/search/math?searchtype=author&query=Stoica%2C+B">Bogdan Stoica</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2001.01725v1-abstract-short" style="display: inline;"> We introduce the $p$-adic particle-in-a-box as a free particle with periodic boundary conditions in the $p$-adic spatial domain. We compute its energy spectrum, and show that the spectrum of the Archimedean particle-in-a-box can be recovered from the $p$-adic spectrum via an Euler product formula. This product formula arises from a flow equation in Berkovich space, which we interpret as a space of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.01725v1-abstract-full').style.display = 'inline'; document.getElementById('2001.01725v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2001.01725v1-abstract-full" style="display: none;"> We introduce the $p$-adic particle-in-a-box as a free particle with periodic boundary conditions in the $p$-adic spatial domain. We compute its energy spectrum, and show that the spectrum of the Archimedean particle-in-a-box can be recovered from the $p$-adic spectrum via an Euler product formula. This product formula arises from a flow equation in Berkovich space, which we interpret as a space of theories connected by a kind of renormalization group flow. We propose that Berkovich spaces can be used to relate $p$-adic and Archimedean quantities generally. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.01725v1-abstract-full').style.display = 'none'; document.getElementById('2001.01725v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages, 3 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2001.01721">arXiv:2001.01721</a> <span> [<a href="https://arxiv.org/pdf/2001.01721">pdf</a>, <a href="https://arxiv.org/format/2001.01721">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Green's Functions for Vladimirov Derivatives and Tate's Thesis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Stoica%2C+B">Bogdan Stoica</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a>, <a href="/search/math?searchtype=author&query=Zhong%2C+X">Xiao Zhong</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2001.01721v2-abstract-short" style="display: inline;"> Given a number field $K$ with a Hecke character $蠂$, for each place $谓$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $蠂$. These theories appear in the study of $p$-adic string theory and $p$-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Four… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.01721v2-abstract-full').style.display = 'inline'; document.getElementById('2001.01721v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2001.01721v2-abstract-full" style="display: none;"> Given a number field $K$ with a Hecke character $蠂$, for each place $谓$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $蠂$. These theories appear in the study of $p$-adic string theory and $p$-adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of $蠂$. We find that the Green's function is given by the local functional equation for Zeta integrals. Furthermore, considering all places $谓$, the field theory two-point functions corresponding to the Green's functions satisfy an adelic product formula, which is equivalent to the global functional equation for Zeta integrals. In particular, this points out a role of Tate's thesis in adelic physics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.01721v2-abstract-full').style.display = 'none'; document.getElementById('2001.01721v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">43 pages, matches published version</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.13290">arXiv:1909.13290</a> <span> [<a href="https://arxiv.org/pdf/1909.13290">pdf</a>, <a href="https://arxiv.org/ps/1909.13290">ps</a>, <a href="https://arxiv.org/format/1909.13290">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1155/2018/2954695">10.1155/2018/2954695 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+C">Caishi Wang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+S">Shuai Lin</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">Ailing Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1909.13290v1-abstract-short" style="display: inline;"> The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let $Z$ be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of gener… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.13290v1-abstract-full').style.display = 'inline'; document.getElementById('1909.13290v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.13290v1-abstract-full" style="display: none;"> The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let $Z$ be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of $Z$. Then, we use the Fock transform to define some fundamental operators on generalized functionals of $Z$, and apply the above mentioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of $Z$, and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of $Z$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.13290v1-abstract-full').style.display = 'none'; document.getElementById('1909.13290v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 60H40; Secondary: 47B38 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Function Spaces, Volume 2018, Article ID 2954695, 9 pages </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1908.06891">arXiv:1908.06891</a> <span> [<a href="https://arxiv.org/pdf/1908.06891">pdf</a>, <a href="https://arxiv.org/ps/1908.06891">ps</a>, <a href="https://arxiv.org/format/1908.06891">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Cryptography and Security">cs.CR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Weil descent and cryptographic trilinear maps </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+M+A">Ming-Deh A. Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1908.06891v3-abstract-short" style="display: inline;"> It has recently been shown that cryptographic trilinear maps are sufficient for achieving indistinguishability obfuscation. In this paper we develop a method for constructing such maps on the Weil descent (restriction) of abelian varieties over finite fields, including the Jacobian varieties of hyperelliptic curves and elliptic curves. The security of these candidate cryptographic trilinear maps r… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.06891v3-abstract-full').style.display = 'inline'; document.getElementById('1908.06891v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1908.06891v3-abstract-full" style="display: none;"> It has recently been shown that cryptographic trilinear maps are sufficient for achieving indistinguishability obfuscation. In this paper we develop a method for constructing such maps on the Weil descent (restriction) of abelian varieties over finite fields, including the Jacobian varieties of hyperelliptic curves and elliptic curves. The security of these candidate cryptographic trilinear maps raises several interesting questions, including the computational complexity of a trapdoor discrete logarithm problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.06891v3-abstract-full').style.display = 'none'; document.getElementById('1908.06891v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 August, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.07891">arXiv:1903.07891</a> <span> [<a href="https://arxiv.org/pdf/1903.07891">pdf</a>, <a href="https://arxiv.org/ps/1903.07891">ps</a>, <a href="https://arxiv.org/format/1903.07891">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00220-020-03708-1">10.1007/s00220-020-03708-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Existence of Solutions to Mean Field Equations on Graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+Y">Yong Lin</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1903.07891v1-abstract-short" style="display: inline;"> In this paper, we prove two existence results of solutions to mean field equations $$螖u+e^u=蟻未_0$$ and $$螖u=位e^u(e^u-1)+4 蟺\sum_{j=1}^{M}{未_{p_j}}$$ on an arbitrary connected finite graph, where $蟻>0$ and $位>0$ are constants, $M$ is a positive integer, and $p_1,...,p_M$ are arbitrarily chosen vertices on the graph. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.07891v1-abstract-full" style="display: none;"> In this paper, we prove two existence results of solutions to mean field equations $$螖u+e^u=蟻未_0$$ and $$螖u=位e^u(e^u-1)+4 蟺\sum_{j=1}^{M}{未_{p_j}}$$ on an arbitrary connected finite graph, where $蟻>0$ and $位>0$ are constants, $M$ is a positive integer, and $p_1,...,p_M$ are arbitrarily chosen vertices on the graph. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.07891v1-abstract-full').style.display = 'none'; document.getElementById('1903.07891v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.00162">arXiv:1903.00162</a> <span> [<a href="https://arxiv.org/pdf/1903.00162">pdf</a>, <a href="https://arxiv.org/format/1903.00162">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Are profile likelihoods likelihoods? No, but sometimes they can be </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Alan Huang</a>, <a href="/search/math?searchtype=author&query=Kim%2C+A+S">Andy Sangil Kim</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1903.00162v2-abstract-short" style="display: inline;"> We offer our two cents to the ongoing discussion on whether profile likelihoods are "true" likelihood functions, by showing that the profile likelihood function can in fact be identical to a marginal likelihood in the special case of normal models. Thus, profile likelihoods can be "true" likelihoods insofar as marginal likelihoods are "true" likelihoods. The prior distribution that achieves this e… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.00162v2-abstract-full').style.display = 'inline'; document.getElementById('1903.00162v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.00162v2-abstract-full" style="display: none;"> We offer our two cents to the ongoing discussion on whether profile likelihoods are "true" likelihood functions, by showing that the profile likelihood function can in fact be identical to a marginal likelihood in the special case of normal models. Thus, profile likelihoods can be "true" likelihoods insofar as marginal likelihoods are "true" likelihoods. The prior distribution that achieves this equivalence turns out to be the Jeffreys prior. We suspect, however, that normal models are the only class of models for which such an equivalence between maximization and marginalization is exact. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.00162v2-abstract-full').style.display = 'none'; document.getElementById('1903.00162v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">short note; 4 pages with references;</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1901.02013">arXiv:1901.02013</a> <span> [<a href="https://arxiv.org/pdf/1901.02013">pdf</a>, <a href="https://arxiv.org/format/1901.02013">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> General relativity from $p$-adic strings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Stoica%2C+B">Bogdan Stoica</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1901.02013v2-abstract-short" style="display: inline;"> For an arbitrary prime number $p$, we propose an action for bosonic $p$-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum $p$-adic strings, similar to the ordinary bosonic strings case. It turns out that certain $p$-adic automorphic forms are the plane wave modes of the bosonic fields o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1901.02013v2-abstract-full').style.display = 'inline'; document.getElementById('1901.02013v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1901.02013v2-abstract-full" style="display: none;"> For an arbitrary prime number $p$, we propose an action for bosonic $p$-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum $p$-adic strings, similar to the ordinary bosonic strings case. It turns out that certain $p$-adic automorphic forms are the plane wave modes of the bosonic fields on $p$-adic strings, and that the regularized normalization of these modes on the $p$-adic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum field theory, while also exhibiting some new features that make loop diagrams much simpler. Assuming a certain product relation, we also observe that the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann Zeta function. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1901.02013v2-abstract-full').style.display = 'none'; document.getElementById('1901.02013v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 January, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> BRX-TH-6643, Brown HET-1778 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.03646">arXiv:1810.03646</a> <span> [<a href="https://arxiv.org/pdf/1810.03646">pdf</a>, <a href="https://arxiv.org/ps/1810.03646">ps</a>, <a href="https://arxiv.org/format/1810.03646">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Cryptography and Security">cs.CR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Trilinear maps for cryptography II </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+M+A">Ming-Deh A. Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.03646v6-abstract-short" style="display: inline;"> We continue to study the construction of cryptographic trilinear maps involving abelian varieties over finite fields. We introduce Weil descent as a tool to strengthen the security of a trilinear map. We form the trilinear map on the descent variety of an abelian variety of small dimension defined over a finite field of a large extension degree over a ground field. The descent bases, with respect… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.03646v6-abstract-full').style.display = 'inline'; document.getElementById('1810.03646v6-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.03646v6-abstract-full" style="display: none;"> We continue to study the construction of cryptographic trilinear maps involving abelian varieties over finite fields. We introduce Weil descent as a tool to strengthen the security of a trilinear map. We form the trilinear map on the descent variety of an abelian variety of small dimension defined over a finite field of a large extension degree over a ground field. The descent bases, with respect to which the descents are performed, are trapdoor secrets for efficient construction of the trilinear map which pairs three trapdoor DDH-groups. The trilinear map also provides efficient public identity testing for the third group. We present a concrete construction involving the jacobian varieties of hyperelliptic curves. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.03646v6-abstract-full').style.display = 'none'; document.getElementById('1810.03646v6-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 February, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.10325">arXiv:1803.10325</a> <span> [<a href="https://arxiv.org/pdf/1803.10325">pdf</a>, <a href="https://arxiv.org/ps/1803.10325">ps</a>, <a href="https://arxiv.org/format/1803.10325">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Cryptography and Security">cs.CR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Trilinear maps for cryptography </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+M+A">Ming-Deh A. Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1803.10325v2-abstract-short" style="display: inline;"> We construct cryptographic trilinear maps that involve simple, non-ordinary abelian varieties over finite fields. In addition to the discrete logarithm problems on the abelian varieties, the cryptographic strength of the trilinear maps is based on a discrete logarithm problem on the quotient of certain modules defined through the N茅ron-Severi groups. The discrete logarithm problem is reducible to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.10325v2-abstract-full').style.display = 'inline'; document.getElementById('1803.10325v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.10325v2-abstract-full" style="display: none;"> We construct cryptographic trilinear maps that involve simple, non-ordinary abelian varieties over finite fields. In addition to the discrete logarithm problems on the abelian varieties, the cryptographic strength of the trilinear maps is based on a discrete logarithm problem on the quotient of certain modules defined through the N茅ron-Severi groups. The discrete logarithm problem is reducible to constructing an explicit description of the algebra generated by two non-commuting endomorphisms, where the explicit description consists of a linear basis with the two endomorphisms expressed in the basis, and the multiplication table on the basis. It is also reducible to constructing an effective $\mathbb{Z}$-basis for the endomorphism ring of a simple non-ordinary abelian variety. Both problems appear to be challenging in general and require further investigation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.10325v2-abstract-full').style.display = 'none'; document.getElementById('1803.10325v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1801.08261">arXiv:1801.08261</a> <span> [<a href="https://arxiv.org/pdf/1801.08261">pdf</a>, <a href="https://arxiv.org/ps/1801.08261">ps</a>, <a href="https://arxiv.org/format/1801.08261">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Jacobian rings for homogenous vector bundles and applications </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B">Bong Lian</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a>, <a href="/search/math?searchtype=author&query=Yu%2C+C">Chenglong Yu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1801.08261v1-abstract-short" style="display: inline;"> In this note, we examine the Jacobian ring description of the Hodge structure of zero loci of vector bundle sections on a class of ambient varieties. We consider a set of cohomological vanishing conditions that imply such a description, and we verify these conditions for some new cases. We also observe that the method can be directly extended to log homogeneous varieties. We apply the Jacobian rin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.08261v1-abstract-full').style.display = 'inline'; document.getElementById('1801.08261v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.08261v1-abstract-full" style="display: none;"> In this note, we examine the Jacobian ring description of the Hodge structure of zero loci of vector bundle sections on a class of ambient varieties. We consider a set of cohomological vanishing conditions that imply such a description, and we verify these conditions for some new cases. We also observe that the method can be directly extended to log homogeneous varieties. We apply the Jacobian ring to study the null varieties of period integrals and their derivatives, generalizing a result in [9] for projective spaces. As an additional application, we prove the Hodge conjecture for very generic hypersurfaces in certain generalized flag varieties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.08261v1-abstract-full').style.display = 'none'; document.getElementById('1801.08261v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 32G20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1801.01194">arXiv:1801.01194</a> <span> [<a href="https://arxiv.org/pdf/1801.01194">pdf</a>, <a href="https://arxiv.org/ps/1801.01194">ps</a>, <a href="https://arxiv.org/format/1801.01194">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Period integrals of local complete intersections and tautological systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B">Bong Lian</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a>, <a href="/search/math?searchtype=author&query=Yu%2C+C">Chenglong Yu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1801.01194v1-abstract-short" style="display: inline;"> Tautological systems developed in [6],[7] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to local complete intersections, which are zero loci of global sections of vector bundles over Fano varieties. In particular, we obtain similar criterion as [6, 7] about holonomicity and regularity of the system. We also p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.01194v1-abstract-full').style.display = 'inline'; document.getElementById('1801.01194v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.01194v1-abstract-full" style="display: none;"> Tautological systems developed in [6],[7] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to local complete intersections, which are zero loci of global sections of vector bundles over Fano varieties. In particular, we obtain similar criterion as [6, 7] about holonomicity and regularity of the system. We also prove solution rank formulas and geometric realizations of solutions following the work of hypersurfaces [5, 4]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.01194v1-abstract-full').style.display = 'none'; document.getElementById('1801.01194v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 32G20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1801.01189">arXiv:1801.01189</a> <span> [<a href="https://arxiv.org/pdf/1801.01189">pdf</a>, <a href="https://arxiv.org/ps/1801.01189">ps</a>, <a href="https://arxiv.org/format/1801.01189">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Hasse-Witt matrices, unit roots and period integrals </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B">Bong Lian</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a>, <a href="/search/math?searchtype=author&query=Yu%2C+C">Chenglong Yu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1801.01189v1-abstract-short" style="display: inline;"> Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [14, 15]. The higher Hasse-Witt matric… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.01189v1-abstract-full').style.display = 'inline'; document.getElementById('1801.01189v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.01189v1-abstract-full" style="display: none;"> Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [14, 15]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz's congruence relations in terms of expansion coefficients [15] to Dwork's congruence relations [8] about periods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.01189v1-abstract-full').style.display = 'none'; document.getElementById('1801.01189v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14G10; 32G20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.00713">arXiv:1709.00713</a> <span> [<a href="https://arxiv.org/pdf/1709.00713">pdf</a>, <a href="https://arxiv.org/ps/1709.00713">ps</a>, <a href="https://arxiv.org/format/1709.00713">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Differential zeros of period integrals and generalized hypergeometric functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+J">Jingyue Chen</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B+H">Bong H. Lian</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.00713v4-abstract-short" style="display: inline;"> In this paper, we study the zero loci of local systems of the form $未螤$, where $螤$ is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space $X$, and $未$ is a given differential operator on the space of sections $V^\vee=螕(X,K_X^{-1})$. Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.00713v4-abstract-full').style.display = 'inline'; document.getElementById('1709.00713v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.00713v4-abstract-full" style="display: none;"> In this paper, we study the zero loci of local systems of the form $未螤$, where $螤$ is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space $X$, and $未$ is a given differential operator on the space of sections $V^\vee=螕(X,K_X^{-1})$. Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of $未螤$. As applications, we prove that the locus is algebraic and in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.00713v4-abstract-full').style.display = 'none'; document.getElementById('1709.00713v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages, minor corrections made</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1609.01351">arXiv:1609.01351</a> <span> [<a href="https://arxiv.org/pdf/1609.01351">pdf</a>, <a href="https://arxiv.org/ps/1609.01351">ps</a>, <a href="https://arxiv.org/format/1609.01351">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> The finite dimensions and determining modes of the global attractor for 2d Boussinesq equations with fractional Laplacian </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Aimin Huang</a>, <a href="/search/math?searchtype=author&query=Huo%2C+W">Wenru Huo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1609.01351v1-abstract-short" style="display: inline;"> In this article, we prove the finite dimensionality of the global attractor and estimate the numbers of the determining modes for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1609.01351v1-abstract-full" style="display: none;"> In this article, we prove the finite dimensionality of the global attractor and estimate the numbers of the determining modes for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.01351v1-abstract-full').style.display = 'none'; document.getElementById('1609.01351v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.07715">arXiv:1605.07715</a> <span> [<a href="https://arxiv.org/pdf/1605.07715">pdf</a>, <a href="https://arxiv.org/format/1605.07715">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Harmonic maps of punctured surfaces to the hyperbolic plane </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A+C">Andy C. Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1605.07715v1-abstract-short" style="display: inline;"> In this paper, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps from once-punctured spheres to the hyperbolic plane have been parameterized by ho… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.07715v1-abstract-full').style.display = 'inline'; document.getElementById('1605.07715v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.07715v1-abstract-full" style="display: none;"> In this paper, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps from once-punctured spheres to the hyperbolic plane have been parameterized by holomorphic quadratic differentials on the complex plane. Our harmonic maps, mapping a genus g>1 domain to a k-sided polygon, correspond to meromorphic quadratic differentials having one pole of order (k+2) and (4g+k-2) zeros (counting multiplicity). In this way, we can associate to these maps a holomorphic quadratic differential on the punctured Riemann surface domain. As an example, we specialize our theorems to obtain a harmonic map from a punctured square torus to an ideal square, and deduce the five possibilities for the divisor of its Hopf differential. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.07715v1-abstract-full').style.display = 'none'; document.getElementById('1605.07715v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 May, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">45 pages, 9 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1512.05111">arXiv:1512.05111</a> <span> [<a href="https://arxiv.org/pdf/1512.05111">pdf</a>, <a href="https://arxiv.org/ps/1512.05111">ps</a>, <a href="https://arxiv.org/format/1512.05111">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.nuclphysb.2015.07.009">10.1016/j.nuclphysb.2015.07.009 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Holonomic Systems for Period Mappings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+J">Jingyue Chen</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B+H">Bong H. Lian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1512.05111v2-abstract-short" style="display: inline;"> Period mappings were introduced in the sixties [G] to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently [LSY,LY] to understand period integrals of algebraic manifolds. In this paper, we give an explicit construction of a tautological system for each component of a period mapping. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1512.05111v2-abstract-full" style="display: none;"> Period mappings were introduced in the sixties [G] to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently [LSY,LY] to understand period integrals of algebraic manifolds. In this paper, we give an explicit construction of a tautological system for each component of a period mapping. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1512.05111v2-abstract-full').style.display = 'none'; document.getElementById('1512.05111v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 December, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Nuc. Phys. B. Vol 898, 693-706, 2015 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1508.00406">arXiv:1508.00406</a> <span> [<a href="https://arxiv.org/pdf/1508.00406">pdf</a>, <a href="https://arxiv.org/ps/1508.00406">ps</a>, <a href="https://arxiv.org/format/1508.00406">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Chain Integral Solutions to Tautological Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B+H">Bong H. Lian</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+X">Xinwen Zhu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1508.00406v2-abstract-short" style="display: inline;"> We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system $蟿$, arising from the universal family of Calabi-Yau hypersurfaces $Y_a$ in a $G$-variety $X$ of dimension $n$. First, we construct a natural topological correspondence between relative cycles in $H_n(X-Y_a,\cup D-Y_a)$ bounded by the union of $G$-invariant divisor… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1508.00406v2-abstract-full').style.display = 'inline'; document.getElementById('1508.00406v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1508.00406v2-abstract-full" style="display: none;"> We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system $蟿$, arising from the universal family of Calabi-Yau hypersurfaces $Y_a$ in a $G$-variety $X$ of dimension $n$. First, we construct a natural topological correspondence between relative cycles in $H_n(X-Y_a,\cup D-Y_a)$ bounded by the union of $G$-invariant divisors $\cup D$ in $X$ to the solution sheaf of $蟿$, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in $Y_a$, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of $H_n(X-Y_a,\cup D-Y_a)$. In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann-Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1508.00406v2-abstract-full').style.display = 'none'; document.getElementById('1508.00406v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 August, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 August, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Revision made and references added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14D07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.02532">arXiv:1505.02532</a> <span> [<a href="https://arxiv.org/pdf/1505.02532">pdf</a>, <a href="https://arxiv.org/ps/1505.02532">ps</a>, <a href="https://arxiv.org/format/1505.02532">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Symbolic Computation">cs.SC</span> </div> </div> <p class="title is-5 mathjax"> On the last fall degree of zero-dimensional Weil descent systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+M+A">Ming-Deh A. Huang</a>, <a href="/search/math?searchtype=author&query=Kosters%2C+M">Michiel Kosters</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yun Yang</a>, <a href="/search/math?searchtype=author&query=Yeo%2C+S+L">Sze Ling Yeo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1505.02532v2-abstract-short" style="display: inline;"> In this article we will discuss a new, mostly theoretical, method for solving (zero-dimensional) polynomial systems, which lies in between Gr枚bner basis computations and the heuristic first fall degree assumption and is not based on any heuristic. This method relies on the new concept of last fall degree. Let $k$ be a finite field of cardinality $q^n$ and let $k'$ be its subfield of cardinality… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.02532v2-abstract-full').style.display = 'inline'; document.getElementById('1505.02532v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.02532v2-abstract-full" style="display: none;"> In this article we will discuss a new, mostly theoretical, method for solving (zero-dimensional) polynomial systems, which lies in between Gr枚bner basis computations and the heuristic first fall degree assumption and is not based on any heuristic. This method relies on the new concept of last fall degree. Let $k$ be a finite field of cardinality $q^n$ and let $k'$ be its subfield of cardinality $q$. Let $\mathcal{F} \subset k[X_0,\ldots,X_{m-1}]$ be a finite subset generating a zero-dimensional ideal. We give an upper bound of the last fall degree of the Weil descent system of $\mathcal{F}$, which depends on $q$, $m$, the last fall degree of $\mathcal{F}$, the degree of $\mathcal{F}$ and the number of solutions of $\mathcal{F}$, but not on $n$. This shows that such Weil descent systems can be solved efficiently if $n$ grows. In particular, we apply these results for multi-HFE and essentially show that multi-HFE is insecure. Finally, we discuss that the degree of regularity (or last fall degree) of Weil descent systems coming from summation polynomials to solve the elliptic curve discrete logarithm problem might depend on $n$, since such systems without field equations are not zero-dimensional. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.02532v2-abstract-full').style.display = 'none'; document.getElementById('1505.02532v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, changed definition of tau and revised Section 5</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 13P10; 13P15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1504.00716">arXiv:1504.00716</a> <span> [<a href="https://arxiv.org/pdf/1504.00716">pdf</a>, <a href="https://arxiv.org/ps/1504.00716">ps</a>, <a href="https://arxiv.org/format/1504.00716">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> The global attractor of the 2D Boussinesq equations with fractional Laplacian in Subcritical case </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Aimin Huang</a>, <a href="/search/math?searchtype=author&query=Huo%2C+W">Wenru Huo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1504.00716v1-abstract-short" style="display: inline;"> We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1504.00716v1-abstract-full" style="display: none;"> We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.00716v1-abstract-full').style.display = 'none'; document.getElementById('1504.00716v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 April, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1503.00283">arXiv:1503.00283</a> <span> [<a href="https://arxiv.org/pdf/1503.00283">pdf</a>, <a href="https://arxiv.org/ps/1503.00283">ps</a>, <a href="https://arxiv.org/format/1503.00283">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s10884-015-9507-1">10.1007/s10884-015-9507-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The nonlinear 2D supercritical inviscid shallow water equations in a rectangle </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Aimin Huang</a>, <a href="/search/math?searchtype=author&query=Petcu%2C+M">Madalina Petcu</a>, <a href="/search/math?searchtype=author&query=Temam%2C+R">Roger Temam</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1503.00283v1-abstract-short" style="display: inline;"> In this article we consider the inviscid two-dimensional shallow water equations in a rectangle. The flow occurs near a stationary solution in the so called supercritical regime and we establish short term existence of smooth solutions for the corresponding initial and boundary value problem. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1503.00283v1-abstract-full" style="display: none;"> In this article we consider the inviscid two-dimensional shallow water equations in a rectangle. The flow occurs near a stationary solution in the so called supercritical regime and we establish short term existence of smooth solutions for the corresponding initial and boundary value problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.00283v1-abstract-full').style.display = 'none'; document.getElementById('1503.00283v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1409.6194">arXiv:1409.6194</a> <span> [<a href="https://arxiv.org/pdf/1409.6194">pdf</a>, <a href="https://arxiv.org/ps/1409.6194">ps</a>, <a href="https://arxiv.org/format/1409.6194">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> On cohomology theory of (di)graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1409.6194v1-abstract-short" style="display: inline;"> To a digraph with a choice of certain integral basis, we construct a CW complex, whose integral singular cohomology is canonically isomorphic to the path cohomology of the digraph as introduced in \cite{GLMY}. The homotopy type of the CW complex turns out to be independent of the choice of basis. After a very brief discussion of functoriality, this construction immediately implies some of the expe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.6194v1-abstract-full').style.display = 'inline'; document.getElementById('1409.6194v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1409.6194v1-abstract-full" style="display: none;"> To a digraph with a choice of certain integral basis, we construct a CW complex, whose integral singular cohomology is canonically isomorphic to the path cohomology of the digraph as introduced in \cite{GLMY}. The homotopy type of the CW complex turns out to be independent of the choice of basis. After a very brief discussion of functoriality, this construction immediately implies some of the expected but perhaps combinatorially subtle properties of the digraph cohomology and homotopy proved very recently \cite{GLMY2}. Furthermore, one gets a very simple expected formula for the cup product of forms on the digraph. On the other hand, we present an approach of using sheaf theory to reformulate (di)graph cohomologies. The investigation of the path cohomology from this framework, leads to a subtle version of Poincare lemma for digraphs, which follows from the construction of the CW complex. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.6194v1-abstract-full').style.display = 'none'; document.getElementById('1409.6194v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C38; 55N99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1409.5853">arXiv:1409.5853</a> <span> [<a href="https://arxiv.org/pdf/1409.5853">pdf</a>, <a href="https://arxiv.org/format/1409.5853">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Graph invariants from ideas in physics and number theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a>, <a href="/search/math?searchtype=author&query=Yueh%2C+M">Mei-Heng Yueh</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1409.5853v3-abstract-short" style="display: inline;"> We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green's function on a graph studied in \cite{CY}. We show that this gives rise to a graph invariant, which is closely related to the 2-dim Weisfeiler-Lehman algorithm for graph isomorphism testing. We complement this invariant by another type of graph invariants, coming from viewing graphs as quadratic… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.5853v3-abstract-full').style.display = 'inline'; document.getElementById('1409.5853v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1409.5853v3-abstract-full" style="display: none;"> We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green's function on a graph studied in \cite{CY}. We show that this gives rise to a graph invariant, which is closely related to the 2-dim Weisfeiler-Lehman algorithm for graph isomorphism testing. We complement this invariant by another type of graph invariants, coming from viewing graphs as quadratic forms over the integers. We explain that the combination of these two ideas give rise to an interesting approach to the graph isomorphism problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.5853v3-abstract-full').style.display = 'none'; document.getElementById('1409.5853v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages, new results added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C60; 05C10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1406.3165">arXiv:1406.3165</a> <span> [<a href="https://arxiv.org/pdf/1406.3165">pdf</a>, <a href="https://arxiv.org/ps/1406.3165">ps</a>, <a href="https://arxiv.org/format/1406.3165">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/0951-7715/28/3/625">10.1088/0951-7715/28/3/625 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The primitive equations of the atmosphere in presence of vapor saturation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zelati%2C+M+C">Michele Coti Zelati</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">Aimin Huang</a>, <a href="/search/math?searchtype=author&query=Kukavica%2C+I">Igor Kukavica</a>, <a href="/search/math?searchtype=author&query=Temam%2C+R">Roger Temam</a>, <a href="/search/math?searchtype=author&query=Ziane%2C+M">Mohammed Ziane</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1406.3165v1-abstract-short" style="display: inline;"> A modification of the classical primitive equations of the atmosphere is considered in order to take into account important phase transition phenomena due to air saturation and condensation. We provide a mathematical formulation of the problem that appears to be new in this setting, by making use of differential inclusions and variational inequalities, and which allows to develop a rather complete… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1406.3165v1-abstract-full').style.display = 'inline'; document.getElementById('1406.3165v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1406.3165v1-abstract-full" style="display: none;"> A modification of the classical primitive equations of the atmosphere is considered in order to take into account important phase transition phenomena due to air saturation and condensation. We provide a mathematical formulation of the problem that appears to be new in this setting, by making use of differential inclusions and variational inequalities, and which allows to develop a rather complete theory for the solutions to what turns out to be a nonlinearly coupled system of non-smooth partial differential equations. Specifically we prove global existence of quasi-strong and strong solutions, along with uniqueness results and maximum principles of physical interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1406.3165v1-abstract-full').style.display = 'none'; document.getElementById('1406.3165v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 June, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">48 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1405.2631">arXiv:1405.2631</a> <span> [<a href="https://arxiv.org/pdf/1405.2631">pdf</a>, <a href="https://arxiv.org/ps/1405.2631">ps</a>, <a href="https://arxiv.org/format/1405.2631">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> The 2D Euler-Boussinesq equations in planar polygonal domains with Yudovich's type data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Aimin Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1405.2631v1-abstract-short" style="display: inline;"> We address the well-posedness of the 2D (Euler)-Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich's type data, which gives a positive answer to part of the questions raised in 2011 [Lai-Pan-Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 739-760]. Our anal… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.2631v1-abstract-full').style.display = 'inline'; document.getElementById('1405.2631v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1405.2631v1-abstract-full" style="display: none;"> We address the well-posedness of the 2D (Euler)-Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich's type data, which gives a positive answer to part of the questions raised in 2011 [Lai-Pan-Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 739-760]. Our analysis on the the polygonal-like domains essentially relies on the recent elliptic regularity results for such domains proved in 2013 [Bardos-Plinio-Temam, The Euler equations in planar nonsmooth convex domains, J. Math. Anal. Appl. 407 (2013), no. 1, 69-89.] and [Plinio-Temam, Grisvard's shift theorem near $l^\infty$ and yudovich theory on polygonal domains, arXiv:1310.5444] <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.2631v1-abstract-full').style.display = 'none'; document.getElementById('1405.2631v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 May, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1403.1351">arXiv:1403.1351</a> <span> [<a href="https://arxiv.org/pdf/1403.1351">pdf</a>, <a href="https://arxiv.org/ps/1403.1351">ps</a>, <a href="https://arxiv.org/format/1403.1351">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> The global well-posedness and global attractor for the solutions to the 2D Boussinesq system with variable viscosity and thermal diffusivity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Aimin Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1403.1351v1-abstract-short" style="display: inline;"> Global well-posedness of strong solutions and existence of the global attractor to the initial and boundary value problem of 2D Boussinesq system in a periodic channel with non-homogeneous boundary conditions for the temperature and viscosity and thermal diffusivity depending on the temperature are proved. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1403.1351v1-abstract-full" style="display: none;"> Global well-posedness of strong solutions and existence of the global attractor to the initial and boundary value problem of 2D Boussinesq system in a periodic channel with non-homogeneous boundary conditions for the temperature and viscosity and thermal diffusivity depending on the temperature are proved. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1403.1351v1-abstract-full').style.display = 'none'; document.getElementById('1403.1351v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">40 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q30; 34A12; 34D45 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1310.5757">arXiv:1310.5757</a> <span> [<a href="https://arxiv.org/pdf/1310.5757">pdf</a>, <a href="https://arxiv.org/ps/1310.5757">ps</a>, <a href="https://arxiv.org/format/1310.5757">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> The linear hyperbolic initial and boundary value problems in a domain with corners </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Aimin Huang</a>, <a href="/search/math?searchtype=author&query=Temam%2C+R">Roger Temam</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1310.5757v1-abstract-short" style="display: inline;"> In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the sys… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1310.5757v1-abstract-full').style.display = 'inline'; document.getElementById('1310.5757v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1310.5757v1-abstract-full" style="display: none;"> In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1310.5757v1-abstract-full').style.display = 'none'; document.getElementById('1310.5757v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 October, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">41 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1303.2560">arXiv:1303.2560</a> <span> [<a href="https://arxiv.org/pdf/1303.2560">pdf</a>, <a href="https://arxiv.org/ps/1303.2560">ps</a>, <a href="https://arxiv.org/format/1303.2560">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Period Integrals and the Riemann-Hilbert Correspondence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B+H">Bong H. Lian</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+X">Xinwen Zhu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1303.2560v2-abstract-short" style="display: inline;"> A tautological system, introduced in [16][17], arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [4], and was verified for the case of projective homogeneous s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1303.2560v2-abstract-full').style.display = 'inline'; document.getElementById('1303.2560v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1303.2560v2-abstract-full" style="display: none;"> A tautological system, introduced in [16][17], arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [4], and was verified for the case of projective homogeneous space under an assumption. In this paper, we prove this conjecture in full generality. By means of the Riemann-Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1303.2560v2-abstract-full').style.display = 'none'; document.getElementById('1303.2560v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 October, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 March, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">44 pages; revisions made, new results added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14D07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1302.4481">arXiv:1302.4481</a> <span> [<a href="https://arxiv.org/pdf/1302.4481">pdf</a>, <a href="https://arxiv.org/format/1302.4481">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> On the Holonomic Rank Problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bloch%2C+S">Spencer Bloch</a>, <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Lian%2C+B+H">Bong H. Lian</a>, <a href="/search/math?searchtype=author&query=Srinivas%2C+V">Vasudevan Srinivas</a>, <a href="/search/math?searchtype=author&query=Yau%2C+S">Shing-Tung Yau</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1302.4481v1-abstract-short" style="display: inline;"> A tautological system, introduced in \cite{LSY}\cite{LY}, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. In this article, we introduce two formulas -- one purely algebraic, the other geometric -- to compute the rank of the solution she… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1302.4481v1-abstract-full').style.display = 'inline'; document.getElementById('1302.4481v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1302.4481v1-abstract-full" style="display: none;"> A tautological system, introduced in \cite{LSY}\cite{LY}, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold $X$, equipped with a suitable Lie group action. In this article, we introduce two formulas -- one purely algebraic, the other geometric -- to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic version gives the local solution space as a Lie algebra homology group, while the geometric one as the middle de Rham cohomology of the complement of a hyperplane section in $X$. We use both formulas to find certain degenerate points for which the rank of the solution sheaf becomes 1. These rank 1 points appear to be good candidates for the so-called large complex structure limits in mirror symmetry. The formulas are also used to prove a conjecture of Hosono, Lian and Yau on the completeness of the extended GKZ system when $X$ is $露^n$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1302.4481v1-abstract-full').style.display = 'none'; document.getElementById('1302.4481v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 February, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">36 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14D07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1209.3194">arXiv:1209.3194</a> <span> [<a href="https://arxiv.org/pdf/1209.3194">pdf</a>, <a href="https://arxiv.org/ps/1209.3194">ps</a>, <a href="https://arxiv.org/format/1209.3194">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00205-013-0702-0">10.1007/s00205-013-0702-0 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Linearized 2D Inviscid Shallow Water Equations in a Rectangle: Boundary Conditions and Well-Posedness </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">Aimin Huang</a>, <a href="/search/math?searchtype=author&query=Temam%2C+R">Roger Temam</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1209.3194v2-abstract-short" style="display: inline;"> We consider the linearized 2D inviscid shallow water equations in a rectangle. A set of boundary conditions is proposed which make these equations well-posed. Several different cases occur depending on the relative values of the reference velocities $(u_0,v_0)$ and reference height $蠁_0$ (sub- or super-critical flow at each part of the boundary). </span> <span class="abstract-full has-text-grey-dark mathjax" id="1209.3194v2-abstract-full" style="display: none;"> We consider the linearized 2D inviscid shallow water equations in a rectangle. A set of boundary conditions is proposed which make these equations well-posed. Several different cases occur depending on the relative values of the reference velocities $(u_0,v_0)$ and reference height $蠁_0$ (sub- or super-critical flow at each part of the boundary). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1209.3194v2-abstract-full').style.display = 'none'; document.getElementById('1209.3194v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 September, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 September, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages</span> </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" 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