A Stratification of Integer Solutions to the Triple Equations 𝑥 + 𝑦 = 𝑧𝟐, 𝟐𝑥 + 𝑦 = 𝟐𝑧𝟐 + 𝑤𝟐, 𝑥 + 𝟐𝑦 = 𝟏𝟎𝑝𝟑

The binary quadratic Diophantine equation represented by is considered and analyzed for its non-zero distinct integer solutions for the choices of t given by (i) \(t=1\) (ii) \(t=3\) (iii) \(t=5\) (iv) \(t=2k\) and (v) \(t=2k+5\). A few interesting relations among the solutions are presented. Further, recurrence relations on the solutions are obtained. Keywords : Pell equation, integer solutions of Pell equation, binary quadratic Diophantine equation.

The Non-homogenous biquadratic equation with four unknowns given by is analyzed for obtaining different sets of non-zero distinct integer solutions through employing the linear transformations.

The ternary quadratic Diophantine equation 5(x2 +y2) - 6xy = 4z2 is analyzed for its non-zero distinct integer solutions. Five different patterns of integral solutions are obtained. Few interesting relations among the solutions and some special polygonal numbers are presented.

Nonzero positive weak solutions of systems of [formula omitted]-Laplace equations

The subject of this work is the development of a nonlinear model of the interaction of an unsteady fluid flow with a structure and finding analytical solutions for the system of equations that correspond to the specified model. The convection effect of the fluid velocity field was already considered in the previous works of the authors of this paper. These studies are devoted to the water hammer without considering the "flow-structure" interaction. This work expands the possibilities of modeling and considers four equations instead of two equations of the theory of the water hammer (equations of conservation of mass and momentum), two of which relate to the motion of particles of a solid body (pipes or structures). The novelty of this work lies in the consideration of the model that describes the interaction of the flow with the structure, the convection in the velocity field, and the effect, together with the friction of the fluid against the solid wall, on the dynamics of the shock pulse propagation process both in the fluid and in the solid body. It should be noted that the solution of the nonlinear system of differential equations as a whole is carried out by an analytical method, which makes it possible to obtain an exact (rather than numerical) solution of the problem. Since the effects of various factors should be evaluated by comparison with the main components, dimensionless equations containing six parameters (dimensionless combinations) were obtained in this study. Two of these parameters were named after scientists – Darcy and Weisbach (steady friction) and Bruno (unsteady friction). Particular cases of the general (full) model were considered, and the effects of various factors on the dynamics of the interaction of the flow with the structure during the propagation of the shock pulse were determined. Research methods are purely theoretical. The concepts of a self-similar equation and a system of equations, balances of forces acting on particles of a fluid and a solid body, and a standard method of reducing a system of equations to a single equivalent equation are used. Conclusions. An extended model of the interaction between the unsteady fluid flow and the structure is proposed. The transition to a self-similar variable makes it possible to solve a nonlinear system of differential equations and obtain an analytical (exact) solution. The functions of longitudinal stress in a solid body, pressure disturbance, and velocity of motion of particles in a solid body (pipe) are linearly expressed by the velocity of shock pulse propagation in the fluid. It should also be noted that the results for the particular case of the linear model completely agree with the already known ones. The advantage of using a self-similar solution is that it is easy to obtain. The results of previous studies on the water hammer problem were qualitatively consistent. As the fluid viscosity increases, the shock pulse domain becomes more concentrative.

Multiple periodic solutions for lattice dynamical systems with superquadratic potentials

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

101 - Hodograph Transformation

Quasi-linear equations of order higher than 1 and systems of quasi-linear equations of any order can be subjected to a classification akin to that of the single second-order equation.

A Schwartz homogeneous problem for vector functions being analytic as per Douglis in the final domain on plane has been considered. The function data are the solutions to a homogeneous elliptic partial-derivatives system of the first order, which depends on the matrix with complex coefficients. It is assumed that the determinant of the complex part of this matrix is different from zero. In the beginning of the article the solution to the problem under consideration is given for a bivariate case in the form of a vector polynomial of the second degree. Further transformations are made to find the general method of building such solutions. It is demonstrated that the real part of the function being analytic as per Douglis will be a solution for a certain homogeneous partial-derivatives system of the second order. Meanwhile, if we know the solution to the Dirichlet problem for this system, we can build the solution to the Schwartz problem, corresponding to the initial matrix. We are searching for the required solution to the Dirichlet problem in the form of a vector polynomial of the second degree with linearly-dependent components. After this function's substitution into the obtained system of partial-derivatives equations, we get a real homogeneous algebraic system. Such system will have nonzero solutions only in case its determinant is equal to zero. By setting the relevant determinant to zero, we get a two-variable algebraic equation. Further on, we prove a fundamental theorem that the existence of a real arbitrary nonzero solution to this algebraic equation is a necessary and sufficient condition of the existence the solution to the Schwartz homogeneous problem in the form of a vector polynomial of the second degree, corresponding to the initial matrix. In the end of the article we give an example of using the fundamental theorem to build two solutions to the studied problem, which correspond to the set matrix

An approximation theory of matrix rank minimization and its application to quadratic equations

Let $ \\gamma = \\frac{1}{2}\\left( {1 + \\sqrt {5} } \\right) $ denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant c > 0 with the property that, for arbitrarily large values of X, the inequalities$ \\left| {{x_0}} \\right| \\leqslant X,\\,\\,\\,\\left| {{x_0}\\xi - {x_1}} \\right| \\leqslant c{X^{{{{ - 1}} \\left/ {\\gamma } \\right.}}}\\,\\,\\,{\\text{and}}\\,\\,\\,\\left| {{x_0}{\\xi^2} - {x_2}} \\right| \\leqslant c{X^{{{{ - 1}} \\left/ {\\gamma } \\right.}}} $admit no non-zero solution $ \\left( {{x_0},{x_1},{x_2}} \\right) \\in {\\mathbb{Z}^3} $. Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each X ≥ 1. Such extremal numbers are transcendental and their set is stable under the action of $ {\\text{G}}{{\\text{L}}_2}\\left( \\mathbb{Z} \\right) $ on $ \\mathbb{R}\\backslash \\mathbb{Q} $ by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf<sub>q→∞</sub>q||qξ|| is $ \\frac{1}{3} $, the largest possible value for a non-quadratic number, and that there is a natural bijection between the $ {\\text{G}}{{\\text{L}}_2}\\left( \\mathbb{Z} \\right) $-equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.

The Riemann problem for the shallow water equations with horizontal temperature gradients

Malware attacks cause great harms in the contemporary information systems and that requires analysis of computer networks reaction in case of malware impact. The focus of the present study is on the analysis of the computer network’s states and reactions in case of malware attacks defined by the susceptibility, exposition, infection and recoverability of computer nodes. Two scenarios are considered – equilibrium without secure software and not equilibrium with secure software in the computer network. The behavior of the computer network under a malware attack is described by a system of nonhomogeneous differential equations. The system of the nonhomogeneous differential equations is solved, and analytical expressions are derived to analyze network characteristics in case of susceptibility, exposition, infection and recoverability of computer nodes during malware attack. The analytical expressions derived are illustrated with results of numerical experiments. The conception developed in this work can be applied to control, prevent and protect computer networks from malware intrusions.

may be reduced to a single linear integral equation whose kernel is defined on a?x ? a+n(b-a), a_s <a+n(b-a). Greggi? considered a system of the form (1) and by use of the transformation introduced by Fredholm showed the form of the resolvent matrix for the system; for the symmetric system where KKj(x; s) = Ksi(s; x) (i, j = 1, 2, , n) he also stated theorems analogous to those proved by Schmidtll for a single integral equation with symmetric kernel. System (1) also comes under the class of systems treated by Platrier.? Weatherburn** has treated the system (1) without using the transformation introduced by Fredholm, but by vector method throughout. He states all results for the case n = 3, but his method of procedure is equally applicable to the general case. In the present paper a special system of the form (1), to which is applied the term self-adjoint, is considered and the existence of a countable infinity of real characteristic numbers is established, together with expansion theorems in terms of the characteristic solutions of the system of integral equations. A definitely self-adjoint system of integral equations includes as a special case the symmetric system with closed matrix kernel. It also includes the system of integral equations to which a boundary value problem for a system of ordinary linear differential equations of the first order which