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Leonid G. Fel

Publications and source records attributed to Leonid G. Fel.

At least 19 recordsLinked to original sources
arXiv2025

Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials

Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series $F(X)$. Using the weight 1 form $θ(q)^2$ and $F(X)=\exp(X/2)$, we obtain a sequence $\{Y_n(q)\}$ of weight $n$ quasimodular forms on $Γ_0(4)$ whose symmetric function avatars $\widetilde{Y}_n(\pmb{x}^k)$ are the symmetric polynomials $T_n(\pmb{x}^k)$ that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the $T_n(\pmb{x}^k).$ Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch $\widehat{A}$-genus for spin manifolds, where one identifies power sum symmetric functions $p_i$ with Pontryagin classes.

math.NT↗
arXiv2025

Wilf's question in numerical semigroups $S_3$ revisited and inequalities for rescaled genera

We consider numerical semigroups $S_3 = \langle d_1,d_2,d_3\rangle$, minimally generated by three positive integers. We revisit the Wilf question in $S_3$ and, making use of identities for degrees of syzygies of such semigroups, give a short proof of existence of an affirmative answer. We find also the lower bound for Frobenius numbers of $S_3$ and upper and lower bounds for rescaled genera.

math.AC↗
arXiv2020

Genera of numerical semigroups and polynomial identities for degrees of syzygies

We derive polynomial identities of arbitrary degree $n$ for syzygies degrees of numerical semigroups S_m= and show that for n>=m they contain higher genera G_r=\sum_{s\in Z_>\setminus S_m}s^r of S_m. We find a number g_m=B_m-m+1 of algebraically independent genera G_r and equations, related any of g_m+1 genera, where B_m=\sum_{k=1}^{m-1}β_k and β_k denote the total and partial Betti numbers of non-symmetric semigroups. The number g_m is strongly dependent on symmetry of S_m and decreases for symmetric semigroups and complete intersections.

math.AC↗
arXiv2020

Symmetric polynomials associated with numerical semigroups

We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums E_k=\sum_{j=1}^m x_j^k. We observe a visual similarity between normalized polynomials P_n(x_1,...,x_m)/χ_m, where χ_m=\prod_{j=1}^m x_j, and a polynomial part of a partition function W(s,{d_1,...,d_m}), which gives a number of partitions of s\ge 0 into m positive integers d_j, and put forward a conjecture about their relationship.

math.CO↗
arXiv2018

Symmetric (not Complete Intersection) Numerical Semigroups Generated by Six Elements

We consider symmetric (not complete intersection) numerical semigroups S_6, generated by a set of six positive integers {d_1,...,d_6}, gcd(d_1,...,d_6)=1, and derive inequalities for degrees of syzygies of such semigroups and find the lower bound for their Frobenius numbers. We show that this bound may be strengthened if S_6 satisfies the Watanabe lemma.

math.AC↗
arXiv2015

Proof of the theorem that a surface area of a ball is smaller than of any other body of the same volume, by Hermann Schwarz

We present English translation of the classical article of Hermann Amadeus Schwarz (1843--1921) "Proof of the theorem that a surface area of a ball is smaller than of any other body of the same volume" which was published in 1884, in Proceedings of the K"onigliche Gesellschaft der Wissenschaften and the Georg-Augusts-Universit"at, G"ottingen. We preserved the author notations throughout the text and tried to follow his grammar construction of the sentences.

math.HO↗
arXiv2014

Converting Neutron Stars into Strange Stars: Instanton Model

We estimate the quasiclassical probability of the homogeneous nuclear matter transition to a strange matter when a detonation wave propagates radially inside a sphere of nuclear matter. For this purpose we make use of instanton method which is known in the quantum field theory.

astro-ph.HE↗
arXiv2014

Stability of Axisymmetric Pendular Rings

Based on the Weierstrass representation of second variation we develop a non-spectral theory of stability for isoperimetric problem with minimized and constrained two-dimensional functionals of general type and free endpoints allowed to move along two given planar curves. We apply this theory to the axisymmetric pendular ring between two solid bodies without gravity to determine the stability of menisci with free contact lines. For catenoid and cylinder menisci and different solid shapes we determine the stability domain. The other menisci (unduloid, nodoid and sphere) are considered in a simple setup between two plates. We find the existence conditions of stable unduloid menisci with and without inflection points.

physics.flu-dyn↗
arXiv2012

Theory of Pendular Rings Revisited

We present the theory of liquid bridges between two axisymmetric solids, sphere and plane, with prescribed contact angles in a general setup, when the solids are non-touching, touching or intersecting, We give a detailed derivation of expressions for curvature, volume and surface area of pendular ring as functions of the filling angle ψfor all available types of menisci: catenoid Cat, sphere Sph, cylinder Cyl, nodoid Nod and unduloid Und (the meridional profile of the latter may have inflection points). The Young-Laplace equation with boundary conditions can be viewed as a nonlinear eigenvalue problem. Its unduloid solutions, menisci shapes z_n^s(r) and their curvatures H_n^s(ψ), exhibit a discrete spectrum and are enumerated by two indices: the number n of inflection points on the meniscus meridional profile M and the convexity index s=\pm 1 determined by the shape of a segment of M contacting the solid sphere: the shape is either convex, s=1, or concave, s=-1. For the fixed contact angles the set of the functions H_n^s(ψ) behaves in such a way that in the plane (ψ,H) there exists a bounded domain where H_n^s(ψ) do not exist for any distance between solids. The curves H_n^s(ψ) may be tangent to the boundary of domain which is a smooth closed curve. This topological representation allows to classify possible curves and introduce a saddle point notion. We observe several types of saddle points, and give their classification.

physics.flu-dyn↗
arXiv2011

Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

We consider a wide class of summatory functions F{f;N,p^m}=\sum_{k\leq N}f(p^m k), m\in \mathbb Z_+\cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k\in \mathbb Z_+, where p is a prime number. Assuming an asymptotic behavior of summatory function, F{f;N,1}\stackrel{N\to \infty}{=}G_1(N) [1+ {\cal O}(G_2(N))], where G_1(N)=N^{a_1}(log N)^{b_1}, G_2(N)=N^{-a_2}(log N)^{-b_2} and a_1, a_2\geq 0, -\infty < b_1, b_2< \infty, we calculate a renormalization function defined as a ratio, R(f;N,p^m)=F{f;N,p^m}/F{f;N,1}, and find its asymptotics R_{\infty}(f;p^m) when N\to \infty. We prove that the renormalization function is multiplicative, i.e., R_{\infty}(f;\prod_{i=1}^n p_i^{m_i})= \prod_{i=1}^n R_{\infty}(f;p_i^{m_i}) with n distinct primes p_i. We extend these results on the others summatory functions \sum_{k\leq N}f(p^m k^l), m,l,k\in \mathbb Z}_+ and \sum_{k\leq N}\prod_{i=1}^n f_i(k p^{m_i}), f_i\neq f_j, m_i\neq m_j. We apply the derived formulas to a large number of basic summatory functions including the Euler ϕ(k) and Dedekind ψ(k) totient functions, divisor σ_n(k) and prime divisor β(k) functions, the Ramanujan sum C_q(n) and Ramanujan τ(k) Dirichlet series, and others.

math.NT↗
arXiv2011

Rarefaction Shock Waves in Collisionless Plasma with Electronic Beam

We show that an electronic beam passing through the collisionless plasma of the "cold" ions and the "hot" Boltzmann electrons can give rise to the propagation of the supersonic ion-acoustic rarefaction shock waves. These waves are analogous to those predicted by Zeldovich [5] in gasodynamics and complementary to the ion-acoustic compression shock waves in collisionless plasma described by Sagdeev [3].

physics.plasm-ph↗
arXiv2009

New Identities for Degrees of Syzygies in Numerical Semigroups

We derive a set of polynomial and quasipolynomial identities for degrees of syzygies in the Hilbert series H(d^m;z) of nonsymmetric numerical semigroups S(d^m) of arbitrary generating set of positive integers d^m={d_1,...,d_m}, m\geq 3. These identities were obtained by studying together the rational representation of the Hilbert series H(d^m;z) and the quasipolynomial representation of the Sylvester waves in the restricted partition function W(s,d^m). In the cases of symmetric semigroups and complete intersections these identities become more compact.

math.AC↗
arXiv2009

Effects of Material Symmetry on the Coefficients of Transport in Anisotropic Porous Media

The objective of this article is to highlight certain features of a number of coefficients that appear in models of phenomena of transport in anisotropic porous media, especially the coefficient of dispersion, the 2nd rank tensor D_{ij}, and the dispersivity coefficient, the 4th rank tensor a_{ijkl}, that appear in models of solute transport. Although we shall focus on the transport of mass of a dissolved chemical species in a fluid phase that occupies the void space, or part of it, the same discussion is also applicable to transport coefficients that appear in models that describe the advective mass flux of a fluid and the diffusive transport of other extensive quantities, like heat. The case of coupled processes, e.g. the simultaneous transport of heat and mass of a chemical species, are also considered. The entire discussion will be at the macroscopic level, at which a porous medium domain is visualized as homogenized continuum.

physics.flu-dyn↗
arXiv2009

Dispersion and Dispersivity Tensors in Saturated Porous Media with Uniaxial Symmetry

The coefficients of dispersion, D_{ij}, and the dispersivity, a_{ijkl}, appear in the expression for the flux of a solute in saturated flow through porous media. We present a detailed analysis of these tensors in an axially symmetric porous medium and show that in such a medium, the dispersivity is governed by 6 independent moduli. We present also the constraints that have to be satisfied by these moduli. We also show that at least two independent experiments are required in order to obtain the values of these coefficients for any three-dimensional porous medium domain.

physics.flu-dyn↗