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Leonid G. Fel

Publications and source records attributed to Leonid G. Fel.

At least 37 records · Page 2Linked to original sources
arXiv2009

On Numerical Experiments with Symmetric Semigroups Generated by Three Elements and Their Generalization

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87-4k) and S(9,3+9k,85-9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r_1^2,r_1r_2+r_1^2k,r_3-r_1^2k), k\in{\mathbb Z}, r_1,r_2,r_3\in{\mathbb Z}^+, r_1\geq 2 and \gcd(r_1,r_2)=\gcd(r_1,r_3)=1, and calculate their universal Frobenius number Phi(r_1,r_2,r_3) for the wide range of k providing semigroups be symmetric. We show that this kind of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r_1^2,r_3-r_1^2k} for sporadic values of k and find these values by solving the quadratic Diophantine equation.

math.NT↗
arXiv2008

Duality Relation for the Hilbert Series of Almost Symmetric Numerical Semigroups

We derive the duality relation for the Hilbert series H(d^m;z) of almost symmetric numerical semigroup S(d^m) combining it with its dual H(d^m;z^{-1}). On this basis we establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F_j, generators d_i and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately. We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give the necessary and efficient conditions for minimal set d^m to generate such semigroups.

math.AC↗
arXiv2008

Symmetric Numerical Semigroups Generated by Fibonacci and Lucas Triples

The symmetric numerical semigroups S(F_a,F_b,F_c) and S(L_k,L_m,L_n) generated by three Fibonacci (F_a,F_b,F_c) and Lucas (L_k,L_m,L_n) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish necessary and sufficient conditions for both semigroups to be symmetric and calculate their Hilbert generating series, Frobenius numbers and genera.

math.NT↗
arXiv2008

On Summatory Totient Functions

The lower and upper bounds are found for the leading term of summatory totient function $\sum_{k\leq N}k^uϕ^v(k)$ in various ranges of $u\in{\mathbb R}$ and $v\in{\mathbb Z}$.

math.NT↗
arXiv2007

Gaps in Nonsymmetric Numerical Semigroups

There exist two different sorts of gaps in the nonsymmetric numerical additive semigroups finitely generated by a minimal set of positive integers {d_1,...,d_m}. The h-gaps are specific only for the nonsymmetric semigroups while the g-gaps are possessed by both, symmetric and nonsymmetric semigroups. We derive the generating functions for the corresponding sets of gaps, Delta_H({\bf d}^m) and Delta_G({\bf d}^m), and prove several statements on the minimal and maximal values of the h-gaps. Detailed description of both sorts of gaps is given for three special kinds of nonsymmetric semigroups: semigroups with maximal embedding dimension, semigroups of maximal and almost maximal length, and pseudo--symmetric semigroups.

math.AC↗
arXiv2005

Arnold's Conjectures on Weak Asymptotics and Statistics of Numerical Semigroups S(d_1,d_2,d_3)

Three conjectures #1999--8, #1999--9 and #1999--10 which were posed by V. Arnold [2] and devoted to the statistics of the numerical semigroups are refuted for the case of semigroups generated by three positive integers d_1,d_2,d_3 with gcd(d_1,d_2,d_3)=1. Weak asymptotics of conductor C(d_1,d_2,d_3) of numerical semigroup and fraction p(d_1,d_2,d_3) of a segment [0;C(d_1,d_2,d_3)-1] occupied by semigroup are found.

math.NT↗
arXiv2005

On the Pólya Enumeration Theorem

Simple formulas for the number of different cyclic and dihedral necklaces containing $n_j$ beads of the $j$-th color, $j\leq m$ and $\sum_{j=1}^mn_j=N$, are derived.

math.CO↗
arXiv2005

Analytic Representations in the 3-dim Frobenius Problem

We consider the Diophantine problem of Frobenius for semigroup ${\sf S}({\bf d}^3)$ where ${\bf d}^3$ denotes the tuple $(d_1,d_2,d_3)$, $\gcd(d_1,d_2,d_3)=1$. Based on the Hadamard product of analytic functions we have found the analytic representation for the diagonal elements $a_{kk}({\bf d}^3)$ of the Johnson's matrix of minimal relations in terms of $d_1,d_2,d_3$. Bearing in mind the results of the recent paper this gives the analytic representation for the Frobenius number $F({\bf d}^3)$, genus $G({\bf d}^3)$ and the Hilbert series $H({\bf d}^3;z)$ for the semigroups ${\sf S}({\bf d}^3)$. This representation does complement the Curtis' theorem on the non-algebraic representation of the Frobenius number $F({\bf d}^3)$. We also give a procedure to calculate the diagonal and off-diagonal elements of the Johnson's matrix.

math.NT↗
arXiv2004

Plateau Instability of Liquid Crystalline Cylinder in Magnetic Field

The capillary instability of a LC cylinder in magnetic field is considered using an energy approach. The boundary problem is solved in the linear approximation of the anisotropy $χ_a$ of the magnetic susceptibility $χ$. The effect of anisotropy, in the region $1\gg|χ|>|χ_a|\gg χ^2$, can be strong enough to counteract and even reverse the tendency of the field to enhance stabilization by enlarging the cut--off wave number $k_s$ beyond the conventional one set by Rayleigh.

physics.flu-dyn↗
arXiv2004

Frobenius Problem for Semigroups ${\sl S}(d_1,d_2,d_3)$

The matrix representation of the set $Δ({\bf d}^3)$, ${\bf d}^3=(d_1,d_2, d_3)$, of the integers which are unrepresentable by $d_1,d_2,d_3$ is found. The diagrammatic procedure of calculation of the generating function $Φ({\bf d}^3;z)$ for the set $Δ({\bf d}^3)$ is developed. The Frobenius number $F({\bf d}^3)$, genus $G({\bf d}^3)$ and Hilbert series $H({\bf d}^3;z)$ of a graded subring for non--symmetric and symmetric semigroups ${\sf S}({\bf d}^3)$ are found. The upper bound for the number of non--zero coefficients in the polynomial numerators of Hilbert series $H({\bf d}^m;z)$ of graded subrings for non--symmetric semigroups ${\sf S} ({\bf d}^m)$ of dimension, $m\geq 4$, is established.

math.NT↗
arXiv2003

Magnetocapillary Instability of Non--Conducting Liquid Jets Revisited : Plateau versus Chandrasekhar

The magnetostatic and magnetocapillary instability problems of isothermal incompressible and inviscid non--conducting liquid jets in a uniform magnetic field, is considered. The equivalence between static and dynamic approaches at the onset of instability and cut--off wavelength is shown. It is established that in the absence of electric currents the stability of permeable jets can be changed by the magnetic field. A new dispersion relation for magnetocapillary instability in such jets is derived. This relation differs from that given by Chandrasekhar. The existence of critical magnetic field which stabilizes jets with finite susceptibility is established. It is shown that the jet is stabilized by the field irrespective of its being para-- or diamagnetic, but the extent of stabilization is different.

cond-mat.soft↗
arXiv2003

Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order

Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The formulas are represented in a form of a finite sum over Bernoulli and Euler polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of the finite groups is discussed.

math.NT↗
arXiv2003

On Effective Conductivity on ${\mathbb Z}^d$ Lattice

We study the effective conductivity $σ_e$ for a random wire problem on the $d$-dimensional cubic lattice ${\mathbb Z}^d, d \geq 2$ in the case when random conductivities on bonds are independent identically distributed random variables. We give exact expressions for the expansion of the effective conductivity in terms of the moments of the disorder parameter up to the 5th order. In the 2D case using the duality symmetry we also derive the 6th order expansion. We compare our results with the Bruggeman approximation and show that in the 2D case it coincides with the exact solution up to the terms of 4th order but deviates from it for the higher order terms.

math-ph↗
arXiv2002

Rayleigh Instability in Liquid Crystalline Jet

The capillary instability of liquid crystalline (LC) jets is considered in the framework of linear hydrodynamics of uniaxial nematic LC. The free boundary conditions of the problem are formulated in terms of mean surface curvature ${\cal H}$ and Gaussian surface curvature ${\cal G}$. The static version of capillary instability is shown to depend on the elasticity modulus $K$, surface tension $σ_0$, and radius $r_0$ of the LC jet, as expressed by the characteristic parameter $\varkappa=K/σ_0 r_0$. The problem of capillary instability in LC jets is solved exactly and a dispersion relation, which reflects the effect of elasticity, is derived. It is shown that increase of the elasticity modulus results in a decrease of both the cut off wavenumber $k$ and the disturbance growth rate $s$. This implies enhanced stability of LC jets, compared to ordinary liquids. In the specific case, where the hydrodynamic and orientational LC modes can be decoupled, the dispersion equation is given in closed form.

cond-mat.soft↗
arXiv2002

Relation between Effective Conductivity and Susceptibility of Two -- Component Rhombic Checkerboard

The heterogeneity of composite leads to the extra charge concentration at the boundaries of different phases that results essentially nonzero effective electric susceptability. The relation between tensors of effective electric susceptability $\hatχ_{ef}$ and effective conductivity $\hatσ_{ef}$ of the infinite two--dimensional two--component regular composite with rhombic cells structure has been established. The degrees of electric field singularity at corner points of cells are found by constructing the integral equation for the effective conductivity problem. The limits of weak and strong contrast of partial conductivities $σ_1,σ_2$ are considered. The results are valid for thin films and cylindrical samples.

cond-mat.mtrl-sci↗
arXiv2002

Crystallographic Restrictions for Colour Lattices with Modular Sublattices

The {\em d} -- dimensional $n$ -- colour lattice ${\bf \Bbb{L}}^d$ with modular sublattices are studied, when the only one crystallographic type of sublattices does exist and the only one of the colours occupies a sublattice, which is still invariant under $k$--fold rotation $C_k$. Such kind of colouring always preserves an equal fractions of the colours composed ${\bf \Bbb{L}}^d$. The $n$ -- colour lattice with modular sublattices allow to exist the crystallographic rotations $C_k$ for $k=p^r, r\geq 1$ and $n\leq p$, where $p$ is a prime number.

math-ph↗
arXiv2002

Self-Dual Symmetric Polynomials and Conformal Partitions

A conformal partition function ${\cal P}_n^m(s)$, which arose in the theory of Diophantine equations supplemented with additional restrictions, is concerned with {\it self-dual symmetric polynomials} -- reciprocal ${\sf R}^{\{m\}}_ {S_n}$ and skew-reciprocal ${\sf S}^{\{m\}}_{S_n}$ algebraic polynomials based on the polynomial invariants of the symmetric group $S_n$. These polynomials form an infinite commutative semigroup. Real solutions $λ_n(x_i)$ of corresponding algebraic Eqns have many important properties: homogeneity of 1-st order, duality upon the action of the conformal group ${\sf W}$, inverting both function $λ_n$ and the variables $x_i$, compatibility with trivial solution, {\it etc}. Making use of the relationship between Gaussian generating function for conformal partitions and Molien generating function for usual restricted partitions we derived the analytic expressions for ${\cal P}_n^m(s)$. The unimodality indices for the reciprocal and skew-reciprocal equations were found. The existence of algebraic functions $λ_n(x_i)$ invariant upon the action of both the finite group $G\subset S_n$ and conformal group ${\sf W}$ is discussed.

math.NT↗