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The paper is a continuation of the paper where the so-called coordinate and spectral equations were derived for finding the edge Green's functions on a branched surface having branch points of order two. The coefficients of those equations contain unknown constants. To find these constants, it is necessary to state restrictions for the solutions of the equations. After that, finding the unknown constants becomes possible, for example, by a numerical procedure of determining zeros of discrepancies. The paper is devoted to the statement of the problem of finding the unknown constants. As an example, the problem of scattering by two perpendicular half-lines is considered. As the result of using a rather subtle property of the spectral equation (symmetry associated with the reciprocity theorem), one can give a set of restrictions, in which the number of unknowns is equal to the number of restrictions. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]

We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

In this paper, we determine the general solution of the functional equation f (kx+ y)+ f (kx-y) = g(x+y)+g(x-y)+h(x)+h(y) for fixed integers k with k ≠= 0,±1 without assuming any regularity condition on the unknown functions f ,g,h,h. The method used for solving these functional equations is elementary but exploits an important result due to Hosszú. The solution of this functional equation can also be determined in certain type of groups using two important results due to Székelyhidi. The results improve and extend some recent results. [ABSTRACT FROM AUTHOR]

The PDE Δ u = u| u|, with $${m\in\mathbb{N}}$$ and $${p\in(1,\infty)}$$, serves as a paradigm for a large class of higher-order elliptic equations with power-like nonlinearities. In studying radially symmetric solutions of this PDE and their asymptotic behavior, it is of critical importance to understand the dynamics of the associated ODE, u = u| u|, which is the subject of the present paper. Most solutions of the ODE blow up in finite time, diverging to either ∞ or −∞. In the phase space $${\mathbb{R}^{2m}}$$, the orbits of these two types of solutions are separated by a (2 m−1)-dimensional manifold $${\fancyscript{M}}$$, which is unordered and homeomorphic to each of the coordinate hyperplanes under orthogonal projection. Solutions with orbits on $${\fancyscript{M}}$$ do not blow up or, else, are unbounded from above and from below (oscillatory blow-up). In the second-order case, $${\fancyscript{M}}$$ coincides with the stable manifold of the trivial equilibrium. In the fourth-order case, which is our main focus, $${\fancyscript{M}}$$ contains a two-dimensional manifold $${\fancyscript{M}_0}$$ of periodic orbits. There exists a unique periodic solution $${\bar u}$$ with $${\bar u(0)=1}$$ and $${\bar u'(0)=0}$$, which shares most of the symmetry properties of the common cosine; all nontrivial periodic solutions are obtained from $${\bar u}$$ via scaling and phase-shifting. Every solution with orbit on $${\fancyscript{M}\setminus\fancyscript{M}_0}$$ converges to either the trivial equilibrium or one of the nontrivial periodic orbits; oscillatory blow-up does not occur. In higher-order cases, the structure of the manifold $${\fancyscript{M}}$$ remains largely open. [ABSTRACT FROM AUTHOR]

Under the proper structure conditions on the nonlinear term f( u) and weight function b( x), the paper shows the uniqueness and asymptotic behavior near the boundary of boundary blow-up solutions to the porous media equations of logistic type −Δ u = a( x) u1/ m − b( x) f( u) with m > 1. [ABSTRACT FROM AUTHOR]

In this paper, we give an analysis and a general procedure for 4D variational data assimilation (4D-Var). In functional partial differential equation setting, the adjoint equation method, sensitivity analysis, and multicomponent operator splitting are discussed. Nonlinear optimization methods and convergence analysis are also investigated for 4D-Var. [ABSTRACT FROM AUTHOR]

Master equations of different types describe the evolution (reduced dynamics) of a subsystem of a larger system generated by the dynamic of the latter system. Since, in some cases, the (exact) master equations are relatively complicated, there exist numerous approximations for such equations, which are also called master equations. In the paper, we develop an exact master equation describing the reduced dynamics of the Wigner function for quantum systems obtained by a quantization of a Hamiltonian system with a quadratic Hamilton function. First, we consider an exact master equation for first integrals of ordinary differential equations in infinite-dimensional locally convex spaces. After this, we apply the results obtained to develop an exact master equation corresponding to a Liouville-type equation (which is the equation for first integrals of the (system of) Hamilton equation(s)); the latter master equation is called the master Liouville equation; it is a linear first-order differential equation with respect to a function of real variables taking values in a space of functions on the phase space. If the Hamilton equation generating the Liouville equation is linear, then the vector fields that define the first-order linear differential operators in the master Liouville equations are also linear, which in turn implies that for a Gaussian reference state the Fourier transform of a solution of the master Liouville equation also satisfies a linear differential equation. [ABSTRACT FROM AUTHOR]

The paper considers the problem of finding a bounded solution of a one-parametric family of systems of ordinary differential equations. Using the parametrization method, the author proves necessary and sufficient conditions for the existence of a unique solution of the problem considered that is bounded on the whole axis in terms of a two-sided, infinite block-band matrix composed with respect to integrals over intervals of length h > 0 of the matrix of the system of differential equations. Also, the author constructs a family of two-point boundary-value problems on a finite interval that approximates the problem of finding the bounded solution and finds an interconnection between the correct solvability of the initial and approximating problems. [ABSTRACT FROM AUTHOR]

Classical chaos refers to the property of trajectories to diverge exponentially as time t → ∞. It is characterized by a positive Lyapunov exponent. There are many different descriptions of quantum chaos. One description related to the notion of generalized (quantum) Lyapunov exponent is based either on qualitative physical considerations or on the so-called symplectic tomography map. The purpose of this paper is to show how the definition of quantum Lyapunov exponent naturally arises in the framework of the Moyal phase-space formulation of quantum mechanics and is based on the notions of quantum trajectories and the family of quantizers. The role of the Heisenberg uncertainty principle in the statement of the criteria for quantum chaos is made explicit. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem on localization of the spectrum of non-self-adjoint differential operators on unbounded domains with power coefficients. To find the location of spectrum points in the complex plane, we use isospectral deformations of differential operators and the properties of families of closed operators analytic in the Kato sense. [ABSTRACT FROM AUTHOR]

In the paper, we obtain sufficient solvability conditions of a certain boundary-value problem for a nonlinear mixed-type equation and find upper and lower estimates for Kolmogorov widths of the solution set. [ABSTRACT FROM AUTHOR]

In this paper, we obtain representations of solutions of the stochastic Schrödinger equations using Feynman path integrals in the configurational and state spaces. [ABSTRACT FROM AUTHOR]

In this paper, the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation. This procedure of resolution is based on a canonical form of the metric. According to this procedure, the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations, which are much convenient for the resolution. [ABSTRACT FROM AUTHOR]

The following differential equation is considered in this paper: . For positive solutions of this equation defined on the closed interval [ a, b], the author obtains an upper estimate using a power function of ( x − a) with coefficient depending only on n, k, and p. [ABSTRACT FROM AUTHOR]

Development of the modern theory of fully nonlinear, second-order partial differential equations has amazingly enriched the classical collection of ideas and methods. In this paper, we construct first-order interior a priori estimates of new type for solutions of Hessian equations. These estmates are applied to present the most transparent version of Krylov’s method and its tight connection with fully nonlinear equations. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

In this paper we study a problem concerning the compulsory behavior of the solutions of systems of discrete equations u( k + 1) = F( k, u( k)), k ∈ N( a) = { a, a + 1, a + 2, . . . }, a ∈ ℕ, ℕ = {0, 1, . . . } and F : N( a) × ℝ n → ℝ n . A general principle for the existence of at least one solution with graph staying for every k ∈ N( a) in a previously prescribed domain is formulated. Such solutions are defined by means of the corresponding initial data and their existence is proved by means of retract type approach. For the development of this approach a notion of egress type points lying on the defined boundary of a given domain and with respect to the system considered is utilized. Unlike previous investigations, the boundary can contain points which are not points of egress type, too. Examples are inserted to illustrate the obtained result. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity of solutions for two–dimensional Cahn–Hilliard equation with non–constant mobility. Basing on the L p type estimates and Schauder type estimates, we prove the global existence of classical solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Darboux transformation of the Darboux-Treibich-Verdier equation. On the basis of this transformation, we construct a generalization of the Darboux transformation to the case of the Heun equation and to other linear ordinary differential equations of second order. Examples are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the initial value problems of the nonlinear Kaup–Kupershmidt equations $$ \frac{{\partial u}} {{\partial t}} + a_{1} \frac{{u\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0,$$ ( x, t) ∈ R2, and $$ \frac{{\partial u}} {{\partial t}} + a_{2} \frac{{\partial u}} {{\partial x}}\frac{{\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0, $$ ( x, t) ∈ R2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup–Kupershmidt equations. The results show that a local solution exists if the initial function u0( x) ∈ Hs ( R), and s ≥ 5/4 for the first equation and s ≥ 301/108 for the second equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the growth and fixed points of solutions and their 1st, 2nd derivatives, differential polynomial of second order linear differential equations with meromorphic coefficients, and obtain that the exponents of convergence of these fixed points are all equal to the order of growth. [ABSTRACT FROM AUTHOR]

In this paper, a class of global optimization problems is considered. Corresponding to each local minimizer obtained, we introduced a new modified function and construct a corresponding optimization subproblem with one constraint. Then, by applying a local search method to the one-constraint optimization subproblem and using the local minimizer as the starting point, we obtain a better local optimal solution. This process is continued iteratively. A termination rule is obtained which can serve as stopping criterion for the iterating process. To demonstrate the efficiency of the proposed approach, numerical examples are solved. [ABSTRACT FROM AUTHOR]

In this paper, we study and discuss the existence of multiple solutions of a class of non-linear elliptic equations with Neumann boundary condition, and obtain at least seven non-trivial solutions in which two are positive, two are negative and three are sign-changing. The study of problem (1.1):is based on the variational methods and critical point theory. We form our conclusion by using the sub-sup solution method, Mountain Pass Theorem in order intervals, Leray-Schauder degree theory and the invariance of decreasing flow. [ABSTRACT FROM AUTHOR]

A large class of problems in economics, ecology, medicine, and other fields can be described by models of developing systems introduced by Glushkov and described in detail in [1]. Systems of Volterra nonlinear equations are the main tool for describing such models. In the present paper, we suggest a numerical method for solving systems of nonlinear Volterra integral equations of special form describing two- and n-commodity economy models. [ABSTRACT FROM AUTHOR]