Kazan Federal University Science Tatarstan / Каза́нский федера́льный университе́т Science Tatarstan (E-Journal)
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Developing Efficient Implementations of Connected Component Algorithms for NEC SX-Aurora TSUBASA
No full textModern vector architectures are tend to be equipped with high-bandwidth memory, what makes them an interesting candidate for solving large-scale graph processing problems. However, highly irregular structure of real-world graphs makes it extremely challenging to map fundamental graph-processing problems on vector systems. This paper describes the world- first attempt, aimed to create efficient vector- friendly implementations of various connected components algorithms for modern NEC SX-Aurora TSUBASA architecture, which provides high performance computational power together with a world-highest bandwidth memory. In order to develop fast implementations, supercomputer co-design principles are used, including: the selection of vector-friendly graph algorithms, adapting these algorithms for target archi- tecture, selecting vectorized graph storage format and applying various optimisations aimed to improve the efficiency of using memory hierarchy of target platform. In addition, current paper analyses if similar implementation approaches can be used for modern NVIDIA GPU architectures, which have many common properties and features with SX-Aurora TSUBASA. Finally, a comprehensive comparative performance analysis is presented for all algorithms, architectures and optimisations, discussed in the paper
Basis Property of Root Functions System for a Problem with Spectral Parameter in the Boundary Condition
No full textIn this paper we study a spectral problem with spectral parameter in the boundary condition. This problem arises when solving boundary value problems for mixed type equations using the spectral method. We analyze the system of root functions of this problem and proof two theorems about basis properties of this system. The biortogonal system is constructed and uniform convergence of spectral expansions is studied
Generalized discrimination between symmetric coherent states for eavesdropping in quantum cryptography
No full textSymmetric coherent states are of interest in quantum cryptography, since for such states there is an upper bound for unambiguous state discrimination (USD) probability, which is used to resist USD attack. But it is not completely clear what an eavesdropper can do for shorter channel length, when USD attack in not available. We consider the task of generalized discrimination between symmetric coherent states and construct an operation which enlarges the information content of the states with fixed failure probability. We apply this transformation to develop a zero-error eavesdropping strategy for quantum cryptography on symmetric coherent states
Symmetries and Differential Invariants for Inviscid Flows on a Curve
No full textSymmetries and the corresponding fields of differential invariants of the inviscid flows on a curve are given. Their dependence on thermodynamic statesof media is studied, and a classification of thermodynamic states is given
Bernstein-Type Inequalities Involving Polar Derivative of a Polynomial
No full textIn this paper, we extend some polynomial inequalities to the polar derivative of a polynomial having all its zeros inside or outside a circle of radius k, k > 0 and thereby present some compact generalizations and improvements of certain well-known inequalities concerning the maximum modulus of the polar derivative of a complex polynomial
Numerical simulation of the shock wave propagation in a two-fraction gas-suspension with a nonuniform particles concentration distribution of the one of the fractions
No full textThe effect of the nonuniform concentration of the one of the fractions of a two-fraction gas-suspension on the parameters of a shock wave moving from pure gas to a gas-suspension is numerically studied. The motion of a direct shock wave in a two-fraction gas suspension was simulated. The finely dispersed fraction of the gas-suspension had a uniform initial mass content, while larger particles had a nonuniform initial mass content along the y coordinate. It was revealed that in the case of a uniform concentration of all fractions, the shock wave in the channel propagates at a lower speed. It was also found that an nonuniform concentration of the larger particles fraction along the y coordinate affects the spatial distribution of the gas velocity
Invariants of symbols of the linear differential operators
No full textIn this paper we classify the symbols of the linear differentialoperators of order , which act from the module tothe module , where isvector bundle over the smooth manifold , bundle is either with fiber or with fiber and, are the modules of their smoothsections. To find invariants of the symbols we associate with everynon-degenerated symbol the tuple of linear operators acting on space and reduce our problem to the classification of such tuples withrespect to some orthogonal transformations. Using the results of C.Procesi, we find generators for the field of rational invariants ofthe symbols and in terms of these invariants provide a criterion ofequivalence of non-degenerated symbols
Non-commutative Graphs in the Fock Space over One-particle Hilbert Space
No full textIn the present paper we continue our studyof non-commutative operator graphs in infinite-dimensional spaces.We consider examples of the non-commutative operator graphsgenerated by resolutions of identity corresponding to theHeisenberg--Weyl group of operators acting on the Fock space overone-particle state space. The problem of quantum error correctionfor such graphs is discussed
On a Boundary value Problem for Boussinesq type Nonlinear Integro-Differential Equation with Reflecting Argument
No full textIn the three-dimensional domain a Boussinesq type nonlinear partial integro-differential equation of the fourth order with a degenerate kernel, integral form conditions, spectral parameters and reflecting argument is considered. The solution of this partial integro-differential equation is studied in the class of generality functions. The method of separation of variables and the method of a degenerate kernels are used. Using these methods, the nonlocal boundary value problem is integrated as a countable system of ordinary differential equations. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving countable system of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed and we obtained the countable system of nonlinear integral equations for each of five cases. To establish the unique solvability of this countable system of nonlinear integral equations we use the method of successive approximations and the method of compressing mappings. Using the Cauchy-Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution of the boundary value problem with respect to given functions in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed as Fourier series. For other cases, the absence of nontrivial solutions of the problem is proved. The corresponding theorems are formulated
On unique solvability of a nonlocal boundary value problem for a loaded multidimensional Chaplygin's equation in the Sobolev space
No full textBoundary value problems for loaded equations in the plane in the case of loaded parts consist of traces of an unknown solution or its first normal derivatives, are well studied. The multidimensional loaded differential equations are relatively less investigated. Moveover, when the loaded part consists of not only the traces of the solution or its first normal derivatives, but also second derivatives of the solutions, the classical methods are not effective. Therefore, inthis paper, we propose a method which overcomes these difficulties.Under some conditions on coefficients of the loaded multidimensional Chaplygin's equation, we prove existence and uniqueness of a solution of a nonlocal boundary value problem in the Sobolev space