Mathematics of Computation最新文献

{"title":"Doubly isogenous genus-2 curves with 𝐷₄-action","authors":"V. Arul, J. Booher, Steven R. Groen, Everett W. Howe, Wanlin Li, Vlad Matei, R. Pries, Caleb Springer","doi":"10.1090/mcom/3891","DOIUrl":"https://doi.org/10.1090/mcom/3891","url":null,"abstract":"<p>We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\u0000 <mml:semantics>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">C</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C prime\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mo>′</mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C’</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are curves over a finite field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-rational base points <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P\">\u0000 <mml:semantics>\u0000 <mml:mi>P</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">P</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P prime\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>P</mml:mi>\u0000 <mml:mo>′</mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">P’</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\u0000 <mml:semantics>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D prime\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo>′</mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">D’</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> map","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"1 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44511508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability and convergence analysis of a fully discrete semi-implicit scheme for stochastic Allen-Cahn equations with multiplicative noise","authors":"Can Huang, Jie Shen","doi":"10.1090/mcom/3846","DOIUrl":"https://doi.org/10.1090/mcom/3846","url":null,"abstract":"We consider a fully discrete scheme for stochastic Allen-Cahn equation in a multi-dimensional setting. Our method uses a polynomial based spectral method in space, so it does not require the elliptic operator \u0000\u0000 \u0000 A\u0000 A\u0000 \u0000\u0000 and the covariance operator \u0000\u0000 \u0000 Q\u0000 Q\u0000 \u0000\u0000 of noise in the equation commute, and thus successfully alleviates a restriction of Fourier spectral method for stochastic partial differential equations pointed out by Jentzen, Kloeden and Winkel [Ann. Appl. Probab. 21 (2011), pp. 908–950]. The discretization in time is a tamed semi-implicit scheme which treats the nonlinear term explicitly while being unconditionally stable. Under regular assumptions which are usually made for SPDEs, we establish strong convergence rates in the one spatial dimension for our fully discrete scheme. We also present numerical experiments which are consistent with our theoretical results.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48263626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing the spectral gap of a family of matrices","authors":"N. Guglielmi, V. Protasov","doi":"10.1090/mcom/3856","DOIUrl":"https://doi.org/10.1090/mcom/3856","url":null,"abstract":"For a single matrix (operator) it is well-known that the spectral gap is an important quantity, as well as its estimate and computation. Here we consider, for the first time in the literature, the computation of its extension to a finite family of matrices, in other words the difference between the joint spectral radius (in short JSR, which we call here the first Lyapunov exponent) and the second Lyapunov exponent (denoted as SLE). The knowledge of joint spectral characteristics and of the spectral gap of a family of matrices is important in several applications, as in the analysis of the regularity of wavelets, multiplicative matrix semigroups and the convergence speed in consensus algorithms. As far as we know the methods we propose are the first able to compute this quantity to any given accuracy.\u0000\u0000For computation of the spectral gap one needs first to compute the JSR. A popular tool that is used to this purpose is the invariant polytope algorithm, which relies on the finiteness property of the family of matrices, when this holds true.\u0000\u0000In this paper we show that the SLE may not possess the finiteness property, although it can be efficiently approximated with an arbitrary precision. The corresponding algorithm and two effective estimates are presented. Moreover, we prove that the SLE possesses a weak finiteness property, whenever the leading eigenvalue of the dominant product is real. This allows us to find in certain situations the precise value of the SLE. Numerical results are demonstrated along with applications in the theory of multiplicative matrix semigroups and in the wavelets theory.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48996200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convergence of a regularized finite element discretization of the two-dimensional Monge–Ampère equation","authors":"D. Gallistl, Ngoc Tien Tran","doi":"10.1090/mcom/3794","DOIUrl":"https://doi.org/10.1090/mcom/3794","url":null,"abstract":"<p>This paper proposes a regularization of the Monge–Ampère equation in planar convex domains through uniformly elliptic Hamilton–Jacobi–Bellman equations. The regularized problem possesses a unique strong solution <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u Subscript epsilon\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">u_varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and is accessible to the discretization with finite elements. This work establishes uniform convergence of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u Subscript epsilon\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">u_varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to the convex Alexandrov solution <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u\">\u0000 <mml:semantics>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">u</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to the Monge–Ampère equation as the regularization parameter <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\">\u0000 <mml:semantics>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> approaches <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\u0000 <mml:semantics>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. A mixed finite element method for the approximation of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u Subscript epsilon\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">u_varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is proposed, and the regularized finite element scheme is shown to be uniformly convergent. The class of admissible right-hand sides are the functions that can be approximated from below by positive continuous functions in the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inlin","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48415841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Enumeration of left braces with additive group 𝐶₄×𝐶₄×𝐶₄","authors":"A. Ballester-Bolinches, R. Esteban-Romero, V. P'erez-Calabuig","doi":"10.1090/mcom/3871","DOIUrl":"https://doi.org/10.1090/mcom/3871","url":null,"abstract":"<p>We show that the number of isomorphism classes of left braces of order <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"64\">\u0000 <mml:semantics>\u0000 <mml:mn>64</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">64</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with additive group isomorphic to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C 4 times upper C 4 times upper C 4\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">C_4times C_4times C_4</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 515 429\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mspace width=\"thinmathspace\" />\u0000 <mml:mn>515</mml:mn>\u0000 <mml:mspace width=\"thinmathspace\" />\u0000 <mml:mn>429</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">1,515,429</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2022-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44654405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Functional graphs of families of quadratic polynomials","authors":"B. Mans, M. Sha, I. Shparlinski, Daniel Sutantyo","doi":"10.1090/mcom/3838","DOIUrl":"https://doi.org/10.1090/mcom/3838","url":null,"abstract":"We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobenius traces of elliptic curves. We also present extensive numerical results which we hope may shed some light on the distribution of leaves for larger families of polynomials.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42604905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of weakly symmetric mixed finite elements for elasticity","authors":"P. Lederer, R. Stenberg","doi":"10.1090/mcom/3865","DOIUrl":"https://doi.org/10.1090/mcom/3865","url":null,"abstract":"We consider mixed finite element methods for linear elasticity where the symmetry of the stress tensor is weakly enforced. Both an a priori and a posteriori error analysis are given for several known families of methods that are uniformly valid in the incompressible limit. A posteriori estimates are derived for both the compressible and incompressible cases. The results are verified by numerical examples.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42338068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the enumeration of finite 𝐿-algebras","authors":"C. Dietzel, P. Mench'on, L. Vendramin","doi":"10.1090/mcom/3814","DOIUrl":"https://doi.org/10.1090/mcom/3814","url":null,"abstract":"We use Constraint Satisfaction Methods to construct and enumerate finite \u0000\u0000 \u0000 L\u0000 L\u0000 \u0000\u0000-algebras up to isomorphism. These objects were recently introduced by Rump and appear in Garside theory, algebraic logic, and the study of the combinatorial Yang–Baxter equation. There are 377,322,225 isomorphism classes of \u0000\u0000 \u0000 L\u0000 L\u0000 \u0000\u0000-algebras of size eight. The database constructed suggests the existence of bijections between certain classes of \u0000\u0000 \u0000 L\u0000 L\u0000 \u0000\u0000-algebras and well-known combinatorial objects. We prove that Bell numbers enumerate isomorphism classes of finite linear \u0000\u0000 \u0000 L\u0000 L\u0000 \u0000\u0000-algebras. We also prove that finite regular \u0000\u0000 \u0000 L\u0000 L\u0000 \u0000\u0000-algebras are in bijective correspondence with infinite-dimensional Young diagrams.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2022-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47501775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling","authors":"Katy Craig, Karthik Elamvazhuthi, M. Haberland, O. Turanova","doi":"10.1090/mcom/3841","DOIUrl":"https://doi.org/10.1090/mcom/3841","url":null,"abstract":"As a counterpoint to classical stochastic particle methods for linear diffusion equations, such as Langevin dynamics for the Fokker-Planck equation, we develop a deterministic particle method for the weighted porous medium equation and prove its convergence on bounded time intervals. This generalizes related work on blob methods for unweighted porous medium equations. From a numerical analysis perspective, our method has several advantages: it is meshfree, preserves the gradient flow structure of the underlying PDE, converges in arbitrary dimension, and captures the correct asymptotic behavior in simulations.\u0000\u0000The fact that our method succeeds in capturing the long time behavior of the weighted porous medium equation is significant from the perspective of related problems in quantization. Just as the Fokker-Planck equation provides a way to quantize a probability measure \u0000\u0000 \u0000 \u0000 \u0000 ρ\u0000 ¯\u0000 \u0000 \u0000 bar {rho }\u0000 \u0000\u0000 by evolving an empirical measure \u0000\u0000 \u0000 \u0000 \u0000 ρ\u0000 N\u0000 \u0000 (\u0000 t\u0000 )\u0000 =\u0000 \u0000 1\u0000 N\u0000 \u0000 \u0000 ∑\u0000 \u0000 i\u0000 =\u0000 1\u0000 \u0000 N\u0000 \u0000 \u0000 δ\u0000 \u0000 \u0000 X\u0000 i\u0000 \u0000 (\u0000 t\u0000 )\u0000 \u0000 \u0000 \u0000 rho ^N(t) = frac {1}{N} sum _{i=1}^N delta _{X^i(t)}\u0000 \u0000\u0000 according to stochastic Langevin dynamics so that \u0000\u0000 \u0000 \u0000 \u0000 ρ\u0000 N\u0000 \u0000 (\u0000 t\u0000 )\u0000 \u0000 rho ^N(t)\u0000 \u0000\u0000 flows toward \u0000\u0000 \u0000 \u0000 \u0000 ρ\u0000 ¯\u0000 \u0000 \u0000 bar {rho }\u0000 \u0000\u0000, our particle method provides a way to quantize \u0000\u0000 \u0000 \u0000 \u0000 ρ\u0000 ¯\u0000 \u0000 \u0000 bar {rho }\u0000 \u0000\u0000 according to deterministic particle dynamics approximating the weighted porous medium equation. In this way, our method has natural applications to multi-agent coverage algorithms and sampling probability measures.\u0000\u0000A specific case of our method corresponds to confined mean-field dynamics of training a two-layer neural network for a radial basis activation function. From this perspective, our convergence result shows that, in the overparametrized regime and as the variance of the radial basis functions goes to zero, the continuum limit is given by the weighted porous medium equation. This generalizes previous results, which considered the case of a uniform data distribution, to the more general inhomogeneous setting. As a consequence of our convergence result, we identify conditions on the target function and data distribution for which convexity of the energy landscape emerges in the continuum limit.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48924036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the computation of general vector-valued modular forms","authors":"Tobias Magnusson, Martin Raum","doi":"10.1090/mcom/3847","DOIUrl":"https://doi.org/10.1090/mcom/3847","url":null,"abstract":"We present and discuss an algorithm and its implementation that is capable of directly determining Fourier expansions of any vector-valued modular form of weight at least \u0000\u0000 \u0000 2\u0000 2\u0000 \u0000\u0000 associated with representations whose kernel is a congruence subgroup. It complements two available algorithms that are limited to inductions of Dirichlet characters and to Weil representations, thus covering further applications like Moonshine or Jacobi forms for congruence subgroups. We examine the calculation of invariants in specific representations via techniques from permutation groups, which greatly aids runtime performance. We explain how a generalization of cusp expansions of classical modular forms enters our implementation. After a heuristic consideration of time complexity, we relate the formulation of our algorithm to the two available ones, to highlight the compromises between level of generality and performance that each them makes.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"1 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42068582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}