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We consider the Euler--Darboux equation with parameters modulo 1/2 and generalization to the space 3D analogue. Due to the fact that the Cauchy problem in its classical formulation is incorrect for such parameter values, the authors propose formulations and solutions of modified Cauchy-type problems with parameter values: a) $\alpha=\beta=\displaystyle\frac{1}{2}$, b) $\alpha=-\,\displaystyle\frac{1}{2}$, $\beta=+\,\displaystyle\frac{1}{2}$, c) $\alpha=\beta=-\,\displaystyle\frac{1}{2}$. The obtained result is used to formulate an analogue of the $\Delta_1$ problem in the first quadrant with the setting of boundary conditions with displacement on the coordinate axes and non-standard conjugation conditions on the singularity line of the coefficients of the equation $y=x$. The first of these conditions glues the normal derivatives of the desired solution, the~second contains the limit values of the combination of the solution and its normal derivatives. The problem was reduced to a uniquely solvable system of integral equations., Comment: 3 pages, Preamble to the reports at The STEMM2019 Conference, Uzbek-Israel Scientific Conference Contemporary Problems in Mathematics and Physics (2017) and XI All-Russian Scientific Conference with International Participation Mathematical Modeling and Boundary Value Problems (2019)

Degenerate hyperbolic equations are dealing with many important issues for applied nature. While a variety of degenerate equations and boundary conditions, successfully matched to these differential equation, most in the characteristic coordinates reduced to Euler-Darboux one. Some boundary value problems, in particular Cauchy problem, for the specified equation demanded the introduction of special classes in which formulae are simple and can be used to meet the new challenges, including Delta-problems in squares that contain singularity line for equation coefficients with data on adjacent or parallel sides of the square. In this short communication the generalized Euler-Darboux equation with negative parameters in the rectangular region is considered., Comment: 5 pages, 1 figure, Uzbek-Israel Scientific Conference Contemporary Problems in Mathematics and Physics (October 6-10, 2017), International Conference Actual problems of differential equations and their applications. TASHKENT. UZBEKISTAN. December 15-17, 2017 and J. Samara State Tech. Univ., Ser. Phys. Math. Sci., Forthcoming paper (Mi vsgtu1557)

The present issue of the series <> represents the Proceedings of the Students Training Contest Olympiad in Mathematical and Theoretical Physics and includes the statements and the solutions of the problems offered to the participants. The contest Olympiad was held on May 21st-24th, 2010 by Scientific Research Laboratory of Mathematical Physics of Samara State University, Steklov Mathematical Institute of Russia's Academy of Sciences, and Moscow Institute of Physics and Technology (State University) in cooperation. The present Proceedings is intended to be used by the students of physical and mechanical-mathematical departments of the universities, who are interested in acquiring a deeper knowledge of the methods of mathematical and theoretical physics, and could be also useful for the persons involved in teaching mathematical and theoretical physics., Comment: 68 pages, Proceedings of the statements and solutions of the problems of the Students Training Contest Olympiad in Mathematical and Theoretical Physics. The subjects covered by the problems include classical mechanics, integrable nonlinear systems, probability, integral equations, PDE, quantum and particle physics, cosmology, and other areas of mathematical and theoretical physics; Ser. Modern Problems of Mathematical Physics Spec. Iss. 3. - Samara : Samara University Press, 2010

In this paper we consider a generalization of analysis on $p$-adic numbers field to the $m$ case of $m$-adic numbers ring. The basic statements, theorems and formulas of $p$-adic analysis can be used for the case of $m$-adic analysis without changing. We discuss basic properties of $m$-adic numbers and consider some properties of $m$-adic integration and $m$-adic Fourier analysis. The class of infinitely divisible $m$-adic distributions and the class of $m$-adic stochastic Levi processes were introduced. The special class of $m$-adic CTRW process and fractional-time $m$-adic random walk as the diffusive limit of it is considered. We found the asymptotic behavior of the probability measure of initial distribution support for fractional-time $m$-adic random walk., Comment: 18 pages

The Higgs boson effective self-couplings $\lambda_{hhh}$ and $\lambda_{HHH}$ are calculated in the framework of the Minimal Supersymmetric Standard Model (MSSM) for the complete set of one-loop diagrams with one of the Higgs bosons off-shell. The comparison with previous results, where only the leading correction terms in the limiting case of large masses of virtual particles were calculated, is carried out. We analyse the dependence of the self-couplings on the energy and ${\tt tan} \beta$; it is demonstrated that the tree-level self-couplings could acquire substantial one-loop corrections, which could be phenomenologically important., Comment: 6 pages, 4 figures, The International Workshop QFTHEP'2003