Given a proper edge coloring φ of a graph G, we define the palette S_G(ν,φ) of a vertex ν∈ V(G) as the set of all colors appearing on edges incident with v. The palette index š(G) of G is the minimum number of distinct palettes occurring in a proper edge coloring of G. In this paper we give an upper bound on the palette index of a graph G in terms of cyclomatic number cyc(G) of G and maximum degree Δ(G) of G. We also give a sharp upper bound for the palette index of unicycle and bicycle graphs.

MSC2010: 05C15.

Keywords: palette, edge coloring, unicycle graph, bicycle graph.

We have proved that any automorphism of the free Burnside group B(3) of period 3 and an arbitrary rank is induced by an automorphism of the free group of the same rank.

MSC2010: 20F50, 20F28, 20D45.

Keywords: automorphism, finitary lifting property, free Burnside group, free groups.

We study a singular nonlinear integral equation on the real line that appear in p-adic string theory. A uniqueness theorem for this equation in certain class of odd functions is proved. At the end of the paper we give examples, satisfying
the conditions of the formulated theorem.

MSC2010: Primary 45G05; Secondary 65R20.

Keywords: nonlinearity, singularity, bounded solution, p-adic string theory.

This work was supported by SCS of MES RA, in the frame of the research project No. 18T-1A004.

Let Fⁿ_q be an n-dimensional vector space over a finite field F_q. Let C( Fⁿ_q) denote the set of all cosets of linear subspaces in  Fⁿ_q. Cosets H₁,H₂,...,H_s are called exclusive if H_i  ⊆ H_j , 1≤i<j≤s. A permutation f of C( Fⁿ_q) is called a C-permutation, if for any exclusive cosets H,H₁,H₂,...,H_s  such that H⊆H₁∪H₂∪H_s we have:

i) cosets f (H), f (H₁), f (H₂),...,f (H_s) are exclusive;

ii) cosets  f^-1 (H), f^-1 (H₁), f^-1 (H₂),...,f^-1 (H_s) are exclusive;

iii) f (H) ⊆ f (H₁)∪ f (H₂)∪...∪f (H_s);

vi)  f^-1 (H)⊆ f^-1 (H₁) ∪f^-1 (H₂)∪...∪f^-1 (H_s) .

In this paper we show that the set of all C-permutations of C( Fⁿ_q) is the General Semiaffine Group of degree n over  F_q.

MSC2010: 97H60, 14N20.

Keywords: finite field, coset, covering, bijection, linearized disjunctive normal form, general affine group, general semiaffine group.

David  S. Sargsyan           davit.sargsyan.1993@gmail.com 

In this paper the canonical notion of δ-reduction is considered. Typed λ-terms use variables of any order and constants of order ≤ 1, where the constants of order 1 are strongly computable, monotonic functions with indeterminate
values of arguments. The canonical notion of δ-reduction is the notion of δ-reduction that is used in the implementation of functional programming languages. It is shown that for canonical notion of δ-reduction SI-property is the necessary and sufficient condition for the uniqueness of βδ-normal form of typed λ-terms.

MSC2010: 68N18.

Keywords: canonical notion of d-reduction, bd-reduction, bd-normal form.

Lusine E. Budaghyan            budaghyan.le@gmail.com   

David  A. Grigoryan             david.grigoryan.a@gmail.com 

Lusine H. Torosyan              lusin.torosyan95@gmail.com 

In this paper we consider a substitution and inheritance property, which is the necessary and sufficient condition for the uniqueness of βδ-normal form of typed λ-terms, for canonical notion of δ-reduction. Typed λ-terms use variables of any order and constants of order ≤1, where the constants of order 1 are strongly computable, monotonic functions with indeterminate values of arguments. The canonical notion of δ-reduction is the notion of δ-reduction that is used in the implementation of functional programming languages.

MSC2010: 68N18.

Keywords: canonical notion of δ-reduction, SI-property, βδ-normal form.

David A. Grigoryan                        david.grigoryan.a@gmail.com 

The general formalism describing polyelectrolyte behavior in presence of sequence disorder is presented. The Edwards and Poisson–Boltzmann equations are obtained. The possible effect of the boundary conditions is discussed. Comparison between replica and constrained annealing approaches is made.

MSC2010: 82B44.

Keywords: polyelectrolyte, DNA, RNA, replicas, constrained annealing.

 Aleksandr Ye.   Mamasakhlisov               amamasakhlisov@gmail.com 

The main goal of this analysis is to study momentum (or kinetic energy) distribution of the backward going protons using data from CLAS EG2 experiment at Jefferson Lab. In this experiment scattering of a 5.014 GeV electron beam off various nucleus targets, ranging from deuterium to lead, have been recorded. The analysis includes selection of events in the reaction A(e, e′, P_back)X, where P_back is a proton scattered above 90° either in the lab coordinate frame or with respect to the direction of the interacting virtual photon, then performing required corrections and studying the protons momentum distribution as a function of energy transfer. In this paper identification of electron-proton events is presented.

MSC2010: 70G10.

Keywords: backward going protons, CLAS EG2, coincidence electron-proton events.

Microwave stripline resonators of rectangular and double rectangular shapes for non-invasive sensing have been designed and investigated experimentally and numerically. The resonator of single rectangular shape has had a resonance at about 4.32 GHz for S₂₁ and the resonator of double rectangular shape has had resonances at about 3.5 GHz for S₂₁ and 5.85 GHz for S₁₁. The simulation results for the sensitivity of the single rectangular and double rectangular resonators were -0.023 dB/(mg/dL) (glucose concentration) and 0.0143 dB/(mg/dL) (NaCl concentration), respectively. The electromagnetic field distribution of the resonators was visualized and compared using computer simulation software HFSS. The obtained data have shown the accuracy of resonators as a non-invasive sensor for biophysical applications.

MSC2010: Primary 00A79; Secondary 74M05.

Keywords: microwaves stripline resonator, glucose sensing.

This work was supported by SCS of MES RA, in the frame of the research project No. 18T-1C114.