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An edge-coloring of a graph $G$ with colors $1,...,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if $G$ has an interval $t$-coloring for some positive integer $t$. Let $\mathfrak{N}$ be the set of all interval colorable graphs. For a graph $G\in \mathfrak{N}$, the least and the greatest values of $t$ for which $G$ has an interval $t$-coloring are denoted by $w(G)$ and $W(G)$, respectively. In this paper we first show that if $G$ is an $r$-regular graph and $G\in \mathfrak{N}$, then $W(G\square P_{m})\geq W(G)+W(P_{m})+(m-1)r$ ($m\in \mathbb{N}$) and $W(G\square C_{2n})\geq W(G)+W(C_{2n})+nr$ ($n\geq 2$). Next, we investigate interval edge-colorings of grids, cylinders and tori. In particular, we prove that if $G\square H$ is planar and both factors have at least 3 vertices, then $G\square H\in \mathfrak{N}$ and $w(G\square H)\leq 6$. Finally, we confirm the first author's conjecture on the $n$-dimensional cube $Q_{n}$ and show that $Q_{n}$ has an interval $t$-coloring if and only if $n\leq t\leq \frac{n(n+1)}{2}$., Comment: 18 pages

An interval t-coloring of a graph G is a proper edge-coloring of G with colors 1,2,...,t such that at least one edge of G is colored by i, i=1,2,...,t, and the edges incident to each vertex v\in V(G) are colored by d_{G}(v) consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In this paper interval edge-colorings of various graph products are investigated., Comment: 4 pages

We consider codes over the alphabet Q={0,1,..,q-1}intended for the control of unidirectional errors of level l. That is, the transmission channel is such that the received word cannot contain both a component larger than the transmitted one and a component smaller than the transmitted one. Moreover, the absolute value of the difference between a transmitted component and its received version is at most l. We introduce and study q-ary codes capable of correcting all unidirectional errors of level l. Lower and upper bounds for the maximal size of those codes are presented. We also study codes for this aim that are defined by a single equation on the codeword coordinates(similar to the Varshamov-Tenengolts codes for correcting binary asymmetric errors). We finally consider the problem of detecting all unidirectional errors of level l., Comment: 22 pages,no figures. Accepted for publication of Journal of Armenian Academy of Sciences, special issue dedicated to Rom Varshamov

A method is proposed to find the wave function of an electron moving infinitely in the field of an arbitrary 1D layer structure with two different homogeneous semi-infinite boundaries. It is shown that in general the problem reduces to solution of a set of two linear difference equations. The proposed approach is discussed on a base of two cases: a structure of periodically placed identical rectangular potentials and a non-ordered structure with certain distortion of periodicity and potential identity., Comment: Submitted in Supperlattices and Microstructures

A method for determination of bound state energies for an asymmetric quantum well with an arbitrary shape of the bottom is suggested. It is shown that how the equation determining the energy levels can be easily derived if one knows the electron transmission and reflection amplitudes corresponding to the part of potential inside the well. The results are applied to three difference test problems., Comment: a file with three figures

The problem of determination of the maximum of second harmonic generation in the potential well containing a rectangular barrier is considered. It is shown that, in general, the problem of finding the ensemble of structures with equidistant first three levels has two types of solutions. For the first type the second and third energy levels are located above a rectangular barrier, and for the second type the third level is located above the barrier only. It is also shown, that generation corresponding to the second type of solution always is less than generation for the first one. Taking into account the effective mass changes the problem of finding the generation maximum for a finite depth well is exactly solved., Comment: Submitted in Optical and Quantum Electronics

In the present paper we investigate the transmission and reflection band behavior for a plane electromagnetic wave falling obliquely on an ideal layered structure. The dependence of this behavior on the problem parameters and wave incident angle is considered. It is shown, that in general case the band width is a non-monotonous function of the problem parameters. A condition is found, which defines the possibility of the contact of the transmission bands. This condition has the same form for s and p waves. It is also shown that irrespective of the wave polarization, the transmission coefficient equals to the unit at the contact points., Comment: a file with five figures (submitted in Optics Communication)

Heavy ion, focused X-ray, pulsed laser single event transient experiments are performed on a silicon epitaxial diode. Collected charge, transient rise times, and transient fall times are calculated and compared between the different sources. The transient shape characteristics depend on the source (ion, X-ray, or laser), even when similar amounts of charge are generated. The observed differences are examined and explained in terms of basic charge collection mechanisms.

The linear system of differential equations for determination of transmission and reflection amplytudes of scattered electron in the field of one dimensional arbitrary potential is obtained. It is shown that in general the scattering problem can be reduced to the Caushy problem for the stationary Shrodinger equation. An explicit functional for a dependance of transmission amplitude from scattered field is found., Comment: 13 pages, no figures, to be submitted to Phys. Rev. B

Recurrent representations for an electron transmission and reflection amplitudes for a one-dimensional chain are obtained. The linear differential equations for scattering amplitudes of an arbitrary potential are found., Comment: 6 pages, no figs. To be submitted to Phys. Lett.A