arXiv subjects
Publications and source records attributed to Leonid G. Fel.
The Keller theorem in the problem of effective conductivity in anisotropic two-dimensional (2D) many-component composites makes it possible to establish a simple inequality $σ_{\sf is}^e(σ^{-1}_i)\cdot σ_{\sf is}^e(σ_k)> 1$ for the isotropic part $σ_{\sf is}^e(σ_k)$ of the 2-nd rank symmetric tensor ${\widehat σ}_{i,j}^e$ of effective conductivity.
The duality approach in 2-{\it dim} two-component regular checkerboards was extended onto piezoelectricity and piezomagnetism problems. There are found a relation for effective piezoelectric and piezomagnetic modules for the checkerboard with $p6^{\prime} mm^{\prime}$-plane symmetry group ({\em dichromatic triangle}). \pacs{Pacs: 73.50.Bk,Jt, 75.70.Ak, 77.65.-j, 77.84.Lf}
The effective dc-conductivity problem of isotropic, two-dimensional (2D), three-component, symmetric, regular composites is considered. A simple cubic equation with one free parameter for $σ_{e}(σ_1,σ_2,σ_3)$ is suggested whose solutions automatically have all the exactly known properties of that function. Numerical calculations on four different symmetric, isotropic, 2D, three-component, regular structures show a non-universal behavior of $σ_{e}(σ_1,σ_2,σ_3)$ with an essential dependence on micro-structural details, in contrast with the analogous two-component problem. The applicability of the cubic equation to these structures is discussed. An extension of that equation to the description of other types of 2D three-component structures is suggested, including the case of random structures. Pacs: 72.15.Eb, 72.80.Tm, 61.50.Ah
A new recursive procedure of the calculation of partition numbers function $W(s,{\bf d}^m)$ is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of $W(s,{\bf d}^m)$ and their periods $τ(G)$ for the irreducible Coxeter groups and a list for the first ten symmetric group ${\cal S}_m$ are presented. A {\it least common multiple} ${\cal L}(m)$ of the series of the natural numbers 1,2,..,$m$ plays a role of the period $τ({\cal S}_m)$ of $W(s,{\bf d}^m)$ in ${\cal S}_m$. An asymptotic behaviour of ${\cal L}(m)$ with $m \to \infty$ is found.