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Singularities of caustics of surfaces in non-flat Riemannian 4-space form $$^\dag $$ Res. Math. Sci. (IF 1.2) Pub Date : 2024-03-16 Liang Chen, Ying Jiang, Haibo Yu
We propose a way to study the caustics of surfaces in non-flat Riemannian 4-space form from the viewpoint of singularity theory in this paper. As an application of the theory of Lagrangian singularity, we study the contact of surfaces with the families of hyperspheres, which is measured by the singularities of functions defined on the surfaces.
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Synchrony patterns in Laplacian networks Res. Math. Sci. (IF 1.2) Pub Date : 2024-03-14
Abstract A network of coupled dynamical systems is represented by a graph whose vertices represent individual cells and whose edges represent couplings between cells. Motivated by the impact of synchronization results of the Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to
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Reeb spaces of smooth functions on manifolds II Res. Math. Sci. (IF 1.2) Pub Date : 2024-03-14 Osamu Saeki
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On real algebraic links in the 3-sphere associated with mixed polynomials Res. Math. Sci. (IF 1.2) Pub Date : 2024-03-08 Raimundo N. Araújo dos Santos, Eder L. Sanchez Quiceno
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Bi-Hölder equivalence of real analytic functions Res. Math. Sci. (IF 1.2) Pub Date : 2024-03-07
Abstract In this work, we show that Hölder equivalence of analytic functions germs \(({\mathbb {R}}^2,0)\rightarrow ({\mathbb {R}},0)\) admits continuous moduli.
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Global planar dynamics with a star node and contracting nonlinearity Res. Math. Sci. (IF 1.2) Pub Date : 2024-03-07 Begoña Alarcón, Sofia B. S. D. Castro, Isabel S. Labouriau
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Milnor fibration theorem for differentiable maps Res. Math. Sci. (IF 1.2) Pub Date : 2024-03-05 José Luis Cisneros-Molina, Aurélio Menegon
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De Rham-Witt KZ equations Res. Math. Sci. (IF 1.2) Pub Date : 2024-03-04
Abstract In this paper we propose a de Rham-Witt version of the derived KZ equations and their hypergeometric realizations.
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Duality and geometry of horocyclic evolutes in hyperbolic plane Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-29 Liang Chen, Shyuichi Izumiya, Masatomo Takahashi
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Zariski invariant for quasi-ordinary hypersurfaces Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-28 R. A. Barbosa, M. E. Hernandes
We introduced an \(\tilde{\mathcal {A}}\)-invariant for quasi-ordinary parameterizations, and we consider it to describe quasi-ordinary surfaces with one generalized characteristic exponent admitting a countable moduli.
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Image of iterated polynomial maps of the real plane Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-28 Tat Thang Nguyen
Let \(F: {\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\) be a polynomial mapping. We consider the image of the compositions \(F^k\) of F. We prove that under some condition then the image of the iterated map \(F^k\) is stable when k is large.
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On the topology of complex projective hypersurfaces Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-27 Laurenţiu G. Maxim
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$$\omega $$ -Symplectic algebra and Hamiltonian vector fields Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-26 Patrícia H. Baptistelli, Maria Elenice R. Hernandes, Eralcilene Moreira Terezio
The purpose of this paper is to present an algebraic theoretical basis for the study of \(\omega \)-Hamiltonian vector fields defined on a symplectic vector space \((V,\omega )\) with respect to coordinates that are not necessarily symplectic. We introduce the concepts of \(\omega \)-symplectic and \(\omega \)-semisymplectic groups, and describe some of their properties that may not coincide with the
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Reconstruction of a hypersurface singularity from its moduli algebra Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-26 João Hélder Olmedo Rodrigues
In this paper we present a constructive method to characterize ideals of the local ring \({\mathscr {O}}_{{\mathbb {C}}^n,0}\) of germs of holomorphic functions at \(0\in {\mathbb {C}}^n\) which arise as the moduli ideal \(\langle f,{\mathfrak {m}}\, j(f)\rangle \), for some \(f\in {\mathfrak {m}}\subset {\mathscr {O}}_{{\mathbb {C}}^n,0}\). A consequence of our characterization is an effective solution
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The variance and correlations of the divisor function in $${\mathbb {F}}_q [T]$$ , and Hankel matrices Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-17 Michael Yiasemides
We prove an exact formula for the variance of the divisor function over short intervals in \({\mathcal {A}}:= {\mathbb {F}}_q [T]\), where q is a prime power; and for correlations of the form \(d(A) d(A+B)\), where we average both A and B over certain intervals in \({\mathcal {A}}\). We also obtain an exact formula for correlations of the form \(d(KQ+N) d (N)\), where Q is prime and K and N are averaged
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On a new absolute version of Siegel’s lemma Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-15
Abstract We establish a new version of Siegel’s lemma over a number field k, providing a bound on the maximum of heights of basis vectors of a subspace of \(k^N\) , \(N \ge 2\) . In addition to the small-height property, the basis vectors we obtain satisfy certain sparsity condition. Further, we produce a nontrivial bound on the heights of all the possible subspaces generated by subcollections of these
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On Ribet’s lemma for GL $$_2$$ modulo prime powers Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-14
Abstract Let \(\rho :G\rightarrow {{\,\textrm{GL}\,}}_2(K)\) be a continuous representation of a compact group G over a complete discretely valued field K with ring of integers \(\mathcal {O}\) and uniformiser \(\pi \) . We prove that \({{\,\textrm{tr}\,}}\rho \) is reducible modulo \(\pi ^n\) if and only if \(\rho \) is reducible modulo \(\pi ^n\) . More precisely, there exist characters \(\chi _1
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Extensions of MacMahon’s sums of divisors Res. Math. Sci. (IF 1.2) Pub Date : 2024-02-05 Tewodros Amdeberhan, George E. Andrews, Roberto Tauraso
In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so, we reveal a wealth of divisibility theorems and unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves rational function approximation to MacMahon-type generating
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Singularity properties of Lorentzian Darboux surfaces in Lorentz–Minkowski spacetime Res. Math. Sci. (IF 1.2) Pub Date : 2024-01-30 Yanlin Li, Xuelian Jiang, Zhigang Wang
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On Kleinian mock modular forms Res. Math. Sci. (IF 1.2) Pub Date : 2024-01-22 Claudia Alfes, Michael H. Mertens
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Identification of unbounded electric potentials through asymptotic boundary spectral data Res. Math. Sci. (IF 1.2) Pub Date : 2023-12-28 Mourad Bellassoued, Yavar Kian, Yosra Mannoubi, Éric Soccorsi
We prove that the real-valued electric potential \(q \in L^{\max (2,3 n /5)}(\Omega )\) of the Dirichlet Laplacian \(-\Delta +q\) acting in a bounded domain \(\Omega \subset \mathbb {R}^n\), \(n \ge 3\), is uniquely determined by the asymptotics of the eigenpairs formed by the eigenvalues and the boundary observation of the normal derivative of the eigenfunctions.
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Multi-component separation, inpainting and denoising with recovery guarantees Res. Math. Sci. (IF 1.2) Pub Date : 2023-12-29 Van Tiep Do
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Analyticity of Steklov eigenvalues of nearly hyperspherical domains in $${\mathbb {R}}^{d + 1}$$ Res. Math. Sci. (IF 1.2) Pub Date : 2023-12-22 Chee Han Tan, Robert Viator
We consider the Dirichlet-to-Neumann operator (DNO) on nearly hyperspherical domains in dimension \(> 3\). Treating such domains as perturbations of the ball, we prove the analytic dependence of the DNO on the shape perturbation parameter for fixed perturbation functions. Consequently, we conclude that the Steklov eigenvalues are analytic in the shape perturbation parameter as well. To obtain these
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Towards the Erdős-Hajnal conjecture for $$P_5$$ -free graphs Res. Math. Sci. (IF 1.2) Pub Date : 2023-12-19 Pablo Blanco, Matija Bucić
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Operator shifting for noisy elliptic systems Res. Math. Sci. (IF 1.2) Pub Date : 2023-11-12 Philip A. Etter, Lexing Ying
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Estimate for the largest zeros of the D’Arcais polynomials Res. Math. Sci. (IF 1.2) Pub Date : 2023-11-15 Bernhard Heim, Markus Neuhauser
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Certain invertible operator-block matrices induced by $$C^{*}$$ -algebras and scaled hypercomplex numbers Res. Math. Sci. (IF 1.2) Pub Date : 2023-10-21 Daniel Alpay, Ilwoo Cho
The main purposes of this paper are (i) to enlarge scaled hypercomplex structures to operator-valued cases, where the operators are taken from a \(C^{*}\)-subalgebra of an operator algebra on a separable Hilbert space, (ii) to characterize the invertibility conditions on the operator-valued scaled-hypercomplex structures of (i), (iii) to study relations between the invertibility of scaled hypercomplex
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The tension determination problem for an inextensible interface in 2D Stokes flow Res. Math. Sci. (IF 1.2) Pub Date : 2023-10-20 Po-Chun Kuo, Ming-Chih Lai, Yoichiro Mori, Analise Rodenberg
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Inverse problems for some fractional equations with general nonlinearity Res. Math. Sci. (IF 1.2) Pub Date : 2023-10-17 Pu-Zhao Kow, Jenn-Nan Wang
Inspired by some interesting equations modeling anomalous diffusion and nonlinear phenomena, we will study the inverse problems of uniquely identifying coefficients in nonlinear terms from over-determined data. Precisely, we consider a semilinear fractional Schrödinger operator \((-\Delta )^su+Q(x,u)=0\) in \(\Omega \) with \(0
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The finite and solvable genus of finitely generated free and surface groups Res. Math. Sci. (IF 1.2) Pub Date : 2023-10-05 Andrei Jaikin-Zapirain
Let \({\mathcal {C}}\) be the pseudovariety \({\mathcal {F}}\) of all finite groups or the pseudovariety \({\mathcal {S}}\) of all finite solvable groups and let \(\Gamma \) be either a finitely generated free group or a surface group. The \({\mathcal {C}}\)-genus of \(\Gamma \), denoted by \({\mathcal {G}}_{{\mathcal {C}}}(\Gamma )\), consists of the isomorphism classes of finitely generated residually-\(\mathcal
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A kernel formula for regularized Wasserstein proximal operators Res. Math. Sci. (IF 1.2) Pub Date : 2023-10-04 Wuchen Li, Siting Liu, Stanley Osher
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Stability and certain $$\mathbb {P}^n$$ -functors Res. Math. Sci. (IF 1.2) Pub Date : 2023-10-03 Fabian Reede
Let X be a K3 surface. We prove that Addington’s \(\mathbb {P}^n\)-functor between the derived categories of X and the Hilbert scheme of points \(X^{[k]}\) maps stable vector bundles on X to stable vector bundles on \(X^{[k]}\), given some numerical conditions are satisfied.
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Hook length biases and general linear partition inequalities Res. Math. Sci. (IF 1.2) Pub Date : 2023-09-29 Cristina Ballantine, Hannah E. Burson, William Craig, Amanda Folsom, Boya Wen
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Exact evaluations and reciprocity theorems for finite trigonometric sums Res. Math. Sci. (IF 1.2) Pub Date : 2023-09-25 Bruce C. Berndt, Sun Kim, Alexandru Zaharescu
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first method uses contour integration and extends a previous method used by two of the authors. In the second, we work in two cyclotomic fields to evaluate new sums involving roots of unity, which lead to the evaluations of several sums involving trigonometric functions. Reciprocity theorems for
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Covering sumsets of a prime field and class numbers Res. Math. Sci. (IF 1.2) Pub Date : 2023-09-23 Emre Alkan
We study covering sumsets of a prime field based on its multiplicative structure. By developing various sufficient analytic and algebraic criteria for their existence, it is shown that covering sumsets arise in two main families, namely in the form of complementary sumsets and in the form of double sumsets. In each case, the abundance of covering sumsets is supported by providing asymptotically growing
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Explicit transformations for generalized Lambert series associated with the divisor function $$\sigma _{a}^{(N)}(n)$$ and their applications Res. Math. Sci. (IF 1.2) Pub Date : 2023-09-22 Soumyarup Banerjee, Atul Dixit, Shivajee Gupta
Let \(\sigma _a^{(N)}(n)=\sum _{d^{N}|n}d^a\). An explicit transformation is obtained for the generalized Lambert series \(\sum _{n=1}^{\infty }\sigma _{a}^{(N)}(n)e^{-ny}\) for \(\text {Re}(a)>-1\) using the recently established Voronoï summation formula for \(\sigma _a^{(N)}(n)\) and is extended to a wider region by analytic continuation. For \(N=1\), this Lambert series plays an important role in
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Twisted Neumann–Zagier matrices Res. Math. Sci. (IF 1.2) Pub Date : 2023-09-05 Stavros Garoufalidis, Seokbeom Yoon
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A second-order nonlocal approximation for Poisson model with Dirichlet boundary Res. Math. Sci. (IF 1.2) Pub Date : 2023-08-11 Yajie Zhang, Zuoqiang Shi
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Combinatorial multiple Eisenstein series Res. Math. Sci. (IF 1.2) Pub Date : 2023-07-25 Henrik Bachmann, Annika Burmester
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Operator theory on generalized Hartogs triangles Res. Math. Sci. (IF 1.2) Pub Date : 2023-07-17 Sameer Chavan, Shubham Jain, Paramita Pramanick
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$$\lambda $$ -Invariant stability in families of modular Galois representations Res. Math. Sci. (IF 1.2) Pub Date : 2023-07-12 Jeffrey Hatley, Debanjana Kundu
Consider a family of modular forms of weight 2, all of whose residual \(\pmod {p}\) Galois representations are isomorphic. It is well known that their corresponding Iwasawa \(\lambda \)-invariants may vary. In this paper, we study this variation from a quantitative perspective, providing lower bounds on the frequency with which these \(\lambda \)-invariants grow or remain stable.
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On character varieties of singular manifolds Res. Math. Sci. (IF 1.2) Pub Date : 2023-06-27 Ángel González-Prieto, Marina Logares
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Corrigendum to “Harmonic analysis and statistics of the first Galois cohomology group” Res. Math. Sci. (IF 1.2) Pub Date : 2023-06-22 Brandon Alberts, Evan O’Dorney
In our earlier paper, there are special cases in which the main theorem could not hold for reasons related to the Grunwald–Wang theorem. We correct the statement and its proof, and we include a short discussion of the added hypothesis of “viability” needed to make our theorem true.
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High-dimensional density estimation with tensorizing flow Res. Math. Sci. (IF 1.2) Pub Date : 2023-06-21 Yinuo Ren, Hongli Zhao, Yuehaw Khoo, Lexing Ying
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Peacock patterns and resurgence in complex Chern–Simons theory Res. Math. Sci. (IF 1.2) Pub Date : 2023-06-19 Stavros Garoufalidis, Jie Gu, Marcos Mariño
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The topology of equivariant Hilbert schemes Res. Math. Sci. (IF 1.2) Pub Date : 2023-06-12 Dori Bejleri, Gjergji Zaimi
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Iwasawa invariants for symmetric square representations Res. Math. Sci. (IF 1.2) Pub Date : 2023-06-12 Anwesh Ray, R. Sujatha, Vinayak Vatsal
Let \(p\ge 5\) be a prime, and \({\mathfrak {p}}\) a prime of \({\overline{{\mathbb {Q}}}}\) above p. Let \(g_1\) and \(g_2\) be \({\mathfrak {p}}\)-ordinary, \({\mathfrak {p}}\)-distinguished and p-stabilized cuspidal newforms of nebentype characters \(\epsilon _1, \epsilon _2\), respectively, and even weight \(k\ge 2\), whose associated newforms have level prime to p. Assume that the residual representations
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Computing $$\mathbb {A}^1$$ -Euler numbers with Macaulay2 Res. Math. Sci. (IF 1.2) Pub Date : 2023-06-10 Sabrina Pauli
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The distribution of G-Weyl CM fields and the Colmez conjecture Res. Math. Sci. (IF 1.2) Pub Date : 2023-05-24 Adrian Barquero-Sanchez, Riad Masri, Frank Thorne
Let G be a transitive subgroup of \(S_d\), and let E be a CM field of degree 2d with a maximal totally real G-field. If the Galois group of the Galois closure of E is isomorphic to the wreath product \(C_2 \wr G\), then we say that E is a G-Weyl CM field. Let \(N_{2d}^{\text {Weyl}}(X,G)\) count the G-Weyl CM fields E of degree 2d with discriminant \(|d_E| \le X\), and define $$\begin{aligned} N_{2d}^{\text
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A Kronecker limit formula for indefinite zeta functions Res. Math. Sci. (IF 1.2) Pub Date : 2023-05-22 Gene S. Kopp
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The Mori–Zwanzig formulation of deep learning Res. Math. Sci. (IF 1.2) Pub Date : 2023-05-21 Daniele Venturi, Xiantao Li
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On the quantitative variation of congruence ideals and integral periods of modular forms Res. Math. Sci. (IF 1.2) Pub Date : 2023-05-14 Chan-Ho Kim, Kazuto Ota
We prove the conjecture of Pollack and Weston on the quantitative analysis of the level lowering congruence à la Ribet for modular forms of higher weight. It was formulated and studied in the context of the integral Jacquet–Langlands correspondence and anticyclotomic Iwasawa theory for modular forms of weight two and square-free level for the first time. We use a completely different method based on
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Bitangents to plane quartics via tropical geometry: rationality, $$\mathbb {A}^1$$ -enumeration, and real signed count Res. Math. Sci. (IF 1.2) Pub Date : 2023-05-06 Hannah Markwig, Sam Payne, Kris Shaw
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Hochschild cohomology of symmetric groups and generating functions, II Res. Math. Sci. (IF 1.2) Pub Date : 2023-05-05 David Benson, Radha Kessar, Markus Linckelmann
We relate the generating functions of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups with those of blocks of the symmetric groups. We show that the first Hochschild cohomology of a positive defect block of a symmetric group is nonzero, answering in the affirmative a question of the third author. To do this, we prove a formula expressing the dimension of degree
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Generative modeling via tree tensor network states Res. Math. Sci. (IF 1.2) Pub Date : 2023-04-28 Xun Tang, YoonHaeng Hur, Yuehaw Khoo, Lexing Ying
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Electromagnetic Steklov eigenvalues: existence and distribution in the self-adjoint case Res. Math. Sci. (IF 1.2) Pub Date : 2023-04-26 Martin Halla
In previous works, it was suggested to use Steklov eigenvalues for Maxwell equations as target signature for nondestructive testing, and it was recognized that this eigenvalue problem cannot be reformulated as a standard eigenvalue problem for a compact operator. Consequently, a modified eigenvalue problem with the desired properties was proposed. We report that apart for a countable set of particular
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Global classical solutions near vacuum to the initial-boundary value problem of isentropic flows through divergent ducts Res. Math. Sci. (IF 1.2) Pub Date : 2023-04-25 Jay Chu, John M. Hong, Hsin-Yi Lee, Ying-Chieh Lin
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Total Stiefel Whitney classes for real representations of $${{\,\textrm{GL}\,}}_n$$ over $$\mathbb F_q, \mathbb R$$ and $$\mathbb C$$ Res. Math. Sci. (IF 1.2) Pub Date : 2023-03-31 Jyotirmoy Ganguly, Rohit Joshi
We compute the total Stiefel Whitney class for a real representation \(\pi \) of \({{\,\textrm{GL}\,}}_n(\mathbb F_q)\), where q is odd in terms of character values of \(\pi \) on order 2 diagonal elements. We also compute the total Stiefel Whitney classes of real representations of \({{\,\textrm{GL}\,}}_n(\mathbb R)\) and \({{\,\textrm{GL}\,}}_n(\mathbb C)\).
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On string functions and double-sum formulas Res. Math. Sci. (IF 1.2) Pub Date : 2023-03-07 Eric T. Mortenson, Olga Postnova, Dmitry Solovyev
String functions are important building blocks of characters of integrable highest modules over affine Kac–Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type \(A_{1}^{(1)}\) in terms of Dedekind eta functions. We obtain new symmetries for string functions by exploiting their natural setting of Hecke-type double-sums, where special double-sums are expressed in
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Jacobi forms, Saito-Kurokawa lifts, their Pullbacks and sup-norms on average Res. Math. Sci. (IF 1.2) Pub Date : 2023-02-24 Pramath Anamby, Soumya Das
We formulate a precise conjecture about the size of the \(L^\infty \)-mass of the space of Jacobi forms on \(\mathbb {H}_n \times \mathbb C^{g \times n}\) of matrix index S of size g. This \(L^\infty \)-mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such n, g, S and prove the upper bound in the k aspect when \(n=1\), \(g \ge 1\). When \(n=1\)