Vol. 53 No. 1 (248) (2019)

Given a proper edge coloring $ \phi $ of a graph $ G $, we define the palette $ S_G (\nu, \phi) $ of a vertex $ \nu \mathclose{\in} V(G) $ as the set of all colors appearing on edges incident with $ \nu $. The palette index $ \check{s} (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 $ \Delta (G) $ of $ G $. We also give a sharp upper bound for the palette index of unicycle and bicycle graphs.

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.

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.

Let $ F^{n}_{q} $ be an $ n $-dimensional vector space over a finite field $ F_q $ . Let $ C(F^{n}_{q} ) $ denote the set of all cosets of linear subspaces in $ F^{n}_{q} $. Cosets $ H_1, H_2, \ldots H_s $ are called exclusive if $ H_i \nsubseteq H_j $, $ 1 \mathclose{\leq} i \mathclose{<} j \mathclose{\leq} s $. A permutation $ f $ of $ C(F^{n}_{q} ) $ is called a $ C $-permutation, if for any exclusive cosets $ H, H_1, H_2, \ldots H_s $ such that $ H \subseteq H_1 \cup H_2 \cup \cdots \cup H_s $ we have:
i) cosets $ f(H), f(H_1), f(H_2), \ldots, f(H_s) $ are exclusive;
ii) cosets $ f^{-1}(H), f^{-1}(H_1), f^{-1}(H_2), \ldots, f^{-1}(H_s) $ are exclusive;
iii) $ f(H) \subseteq f(H_1) \cup f(H_2) \cup \cdots \cup f(H_s) $;
vi) $ f^{-1}(H) \subseteq f^{-1}(H_1) \cup f^{-1}(H_2) \cup \cdots \cup f^{-1}(H_s) $.
In this paper we show that the set of all $ C $-permutations of $ C(F^{n}_{q} ) $ is the General Semiaffine Group of degree $ n $ over $ F_q $.

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

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

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.

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^{\prime}, 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.

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 $ S_{21} $ and the resonator of double rectangular shape has had resonances at about 3.5 GHz for $ S_{21} $ and 5.85 GHz for $ S_{11} $. 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.