Journal of Applied and Industrial Mathematics最新文献

The paper considers the following problem. Given a set of Euclidean vectors, find severalclusters with a restriction on the maximum scatter of each cluster so that the size of the minimumcluster would be maximum. Here the scatter is the sum of squared distances from the clusterelements to its centroid. The NP-hardness of this problem is proved in the case where thedimension of the space is part of the input.

The aim of this paper is to acquaint applied mathematicians interested in the possibilitiesof applying methods for solving inverse problems in mathematical modeling in natural sciences andengineering to economic problems with our papers in this field. These papers refer to the problemof verifying the market demand theory, developed by the first author based on the revision of theunrealistic axiomatic neoclassical theory of individual demand within the framework of generalscientific methodology. At the same time, the artificial object of individual theory—arational and independent individual who maximizes his/her utility function—was replacedby a “statistical ensemble of consumers” of the market under study, and the postulates ofindividual theory became scientific hypotheses of the new theory. The verification of this theoryconsists in elucidating the question of rationalizing the statistical market demand by the collectiveutility function. This problem refers to the inverse problems of mathematical theories of realphenomena, which are usually ill posed and have many solutions. The solution of such problemsconsists in their regularization with involvement of additional information about the desiredsolution. Our method for verifying the market demand theory is a development of thenonparametric Afriat–Varian demand analysis with using “economic indices” of market demand asadditional information, which allows obtaining solutions with various substantive properties.

The problem of identifying Volterra kernels is an important stage in the construction ofintegral models of nonlinear dynamical systems based on the tool of Volterra series. The paperconsiders a new class of two-dimensional integral equations that arise when recoveringnonsymmetric kernels in a Volterra polynomial of the second degree, where( x(t) ) is the input vector function of time. The strategy for choosing test signalsused to solve this problem is based on applying piecewise linear functions (with a rising edge). Anexplicit inversion formula is constructed for the selected type of Volterra equations of the first kindwith variable integration limits. The questions of existence and uniqueness of solutions of thecorresponding equations in the class( C_{[0,T]} ) are studied.

The paper considers a mathematical model of the filtration of two immiscibleincompressible fluids in deformable porous media. This model is a generalization of theMusket–Leverett model, in which porosity is a function of the space coordinates. The model understudy is based on the equations of conservation of mass of liquids and porous skeleton, Darcy’s lawfor liquids, accounting for the motion of the porous skeleton, Laplace’s formula for capillarypressure, and a Maxwell-type rheological equation for porosity and the equilibrium condition ofthe “system as a whole.” In the thin layer approximation, the original problem is reduced to thesuccessive determination of the porosity of the solid skeleton and its speed, and then theelliptic-parabolic system for the “reduced” pressure and saturation of the fluid phase is derived. Inview of the degeneracy of equations on the solution, the solution is understood in a weak sense.The proofs of the results are carried out in four stages: regularization of the problem, proof of themaximum principle, construction of Galerkin approximations, and passage to the limit in terms ofthe regularization parameters based on the compensated compactness principle.

The paper examines the problem of the distribution of public space. We use the methodsof cooperative game theory to solve this problem. Players are districts, while the value of thecharacteristic function is the total number of people interested in a particular type of public spacein the areas under consideration. The axioms that are characteristic of the problem of division arecompiled. A special value of the cooperative game is derived that depends on the weights of theplayers. It is shown how to choose the weights by optimization methods.

A general method for the enumeration of various classes of chord diagrams with even andodd numbers of chord intersections is considered. The method is based on the calculation of thePfaffian and Hafnian of the constraint matrix characterizing a class of diagrams.

The aim of this paper is to study the solvability of inverse problems of determining,together with the solution of the heat equation, its right-hand side or an unknown externalinfluence. The specific feature of the problems studied is that the unknown external influence isdetermined by two functions of which one depends only on the spatial variable and the other, onlyon the time variable.

This paper is devoted to the construction of a time-optimal method for measuring themaximum permissible (critical) value of the supply voltage (as well as other parameters) of thedevice on which the cryptographic algorithm is performed. Knowledge of these values is necessaryfor successful conduct of error injection, as part of a fault attack. The method is based ondynamic programming.

Rusanov’s scheme for solving hydrodynamic equations is one of the most robust in theclass of Riemann solvers. It was previously shown that Rusanov’s scheme based on piecewiseparabolic reconstruction of primitive variables gives a low-dissipative scheme relevant to Roe andHarten–Lax–Van Leer solvers when using a similar reconstruction. Moreover, unlike these solvers,the numerical solution is free from artifacts. In the case of equations of special relativisticmagnetohydrodynamics, the spectral decomposition for solving the Riemann problem is quitecomplex and does not have an analytical solution. The present paper proposes the development ofRusanov’s scheme using a piecewise parabolic reconstruction of primitive variables to use in theequations of special relativistic magnetohydrodynamics. The developed scheme was verified usingeight classical problems on the decay of an arbitrary discontinuity that describe the main featuresof relativistic magnetized flows.

A convex continuation of an arbitrary Boolean function to the set( [0,1]^n ) is constructed. Moreover, it is proved that for any Boolean function( f(x_1,x_2,dots ,x_n) ) that has no neighboring points on the set( mathrm{supp} f ), the constructed function( f_C(x_1,x_2, dots ,x_n) ) is the only totally maximally convex continuation to( [0,1]^n ). Based on this, in particular, it is constructively stated that the problem ofsolving an arbitrary system of Boolean equations can be reduced to the problem of minimizing afunction any local minimum of which in the desired region is a global minimum, and thus for thisproblem the problem of local minima is completely resolved.