Symbolic Computation

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rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> <dl id='articles'> <h3>New submissions (showing 2 of 2 entries)</h3> <dt> <a name='item1'>[1]</a> <a href ="/abs/2503.13640" title="Abstract" id="2503.13640"> arXiv:2503.13640 </a> [<a href="/pdf/2503.13640" title="Download PDF" id="pdf-2503.13640" aria-labelledby="pdf-2503.13640">pdf</a>, <a href="https://arxiv.org/html/2503.13640v1" title="View HTML" id="html-2503.13640" aria-labelledby="html-2503.13640" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13640" title="Other formats" id="oth-2503.13640" aria-labelledby="oth-2503.13640">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> LSU factorization </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Malaschonok,+G">Gennadi Malaschonok</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 7 pages, 1 figures </div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> 2023 International Conference on Computational Science and Computational Intelligence (CSCI), Las Vegas, NV, USA, 2023, pp. 472-478 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Symbolic Computation (cs.SC)</span>; Commutative Algebra (math.AC) </div> <p class='mathjax'> The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm decomposes any matrix A into a product of three matrices A=LSU, where each element of the triangular matrices L and U is a minor of matrix A. The number of non-zero elements in matrix S is equal to rank(A), and each of them is the inverse of the product of a specific pair of matrix A minors. The algorithm's complexity is equivalent to that of matrix multiplication. </p> </div> </dd> <dt> <a name='item2'>[2]</a> <a href ="/abs/2503.14264" title="Abstract" id="2503.14264"> arXiv:2503.14264 </a> [<a href="/pdf/2503.14264" title="Download PDF" id="pdf-2503.14264" aria-labelledby="pdf-2503.14264">pdf</a>, <a href="https://arxiv.org/html/2503.14264v1" title="View HTML" id="html-2503.14264" aria-labelledby="html-2503.14264" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.14264" title="Other formats" id="oth-2503.14264" aria-labelledby="oth-2503.14264">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Positivity Proofs for Linear Recurrences with Several Dominant Eigenvalues </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Ibrahim,+A">Alaa Ibrahim</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Symbolic Computation (cs.SC)</span> </div> <p class='mathjax'> Deciding the positivity of a sequence defined by a linear recurrence and initial conditions is, in general, a hard problem. When the coefficients of the recurrences are constants, decidability has only been proven up to order 5. The difficulty arises when the characteristic polynomial of the recurrence has several roots of maximal modulus, called dominant roots of the recurrence. We study the positivity problem for recurrences with polynomial coefficients, focusing on sequences of Poincar茅 type, which are perturbations of constant-coefficient recurrences. The dominant eigenvalues of a recurrence in this class are the dominant roots of the associated constant-coefficient recurrence. Previously, we have proved the decidability of positivity for recurrences having a unique, simple, dominant eigenvalue, under a genericity assumption. The associated algorithm proves positivity by constructing a positive cone contracted by the recurrence operator. We extend this cone-based approach to a larger class of recurrences, where a contracted cone may no longer exist. The main idea is to construct a sequence of cones. Each cone in this sequence is mapped by the recurrence operator to the next. This construction can be applied to prove positivity by induction. For recurrences with several simple dominant eigenvalues, we provide a condition that ensures that these successive inclusions hold. Additionally, we demonstrate the applicability of our method through examples, including recurrences with a double dominant eigenvalue. </p> </div> </dd> </dl> <dl id='articles'> <h3>Cross submissions (showing 1 of 1 entries)</h3> <dt> <a name='item3'>[3]</a> <a href ="/abs/2503.13512" title="Abstract" id="2503.13512"> arXiv:2503.13512 </a> (cross-list from stat.ML) [<a href="/pdf/2503.13512" title="Download PDF" id="pdf-2503.13512" aria-labelledby="pdf-2503.13512">pdf</a>, <a href="https://arxiv.org/html/2503.13512v1" title="View HTML" id="html-2503.13512" aria-labelledby="html-2503.13512" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13512" title="Other formats" id="oth-2503.13512" aria-labelledby="oth-2503.13512">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Positivity sets of hinge functions </div> <div class='list-authors'><a href="https://arxiv.org/search/stat?searchtype=author&query=Schicho,+J">Josef Schicho</a>, <a href="https://arxiv.org/search/stat?searchtype=author&query=Tewari,+A+K">Ayush Kumar Tewari</a>, <a href="https://arxiv.org/search/stat?searchtype=author&query=Warren,+A">Audie Warren</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Machine Learning (stat.ML)</span>; Discrete Mathematics (cs.DM); Machine Learning (cs.LG); Symbolic Computation (cs.SC); Combinatorics (math.CO); Functional Analysis (math.FA) </div> <p class='mathjax'> In this paper we investigate which subsets of the real plane are realisable as the set of points on which a one-layer ReLU neural network takes a positive value. In the case of cones we give a full characterisation of such sets. Furthermore, we give a necessary condition for any subset of $\mathbb R^d$. We give various examples of such one-layer neural networks. </p> </div> </dd> </dl> <div class='paging'>Total of 3 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/cs.SC/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> </div> </div> </div> </main> <footer style="clear: both;"> <div class="columns is-desktop" role="navigation" aria-label="Secondary" style="margin: -0.75em -0.75em 0.75em -0.75em">  <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; 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Symbolic Computation