On extensions of the Bessel operator on a finite interval and a half-line

The minimum and maximum operators generated by a Bessel differential expression are studied. The boundary triplets for the maximum operator A _??,max are constructed, and the corresponding Weyl functions are found. All non-negative self-adjoint extensions of the minimum operator A _??,min are described.     

On a Gradient Flow with Exponential Rate of Convergence

We present an evolution equation governed by a maximal monotone operator with exponential rate of convergence to a zero of the maximal monotone operator. When the maximal monotone operator is the subdifferential of a convex, proper, and lower semicontinuous function, we show that the trajectory of solutions of the evolution equation converges exponentially to the minimum value of the convex function.     

Fourier method in a mixed problem for the wave equation on a graph

A mixed problem for the wave equation on the simplest geometric graph consisting of two ring edges that touch at a point is considered. The approach used is based on the contour integration of the operator’s resolvent. With the help of a special transformation of a formal series, a classical solution of the problem is obtained under minimum conditions imposed on the initial data. This approach makes it possible to do without an expensive analysis of improved asymptotics for the eigenvalues and eigenfunctions of the operator and to avoid the difficulties associated with the possible multiplicity of the operator’s spectrum.     

On generalized resolvents and spectral functions of differential operators of even order in a space of vector-valued functions

We consider a self-adjoint differential expression of even order in a space of vector-valued functions. We prove that an arbitrary generalized resolvent of the minimal operator L₀ induced by this differential expression is an integral operator and we derive a formula for all of the spectral functions (orthogonal and nonorthogonal) of the operator L₀.     

Reduction of a Minimization Problem of a Separable Convex Function Under Linear Constraints to a Fixed Point Problem

The paper is devoted to studying a constrained nonlinear optimization problem of a special kind. The objective functional of the problem is a separable convex function whose minimum is sought for on a set of linear constraints in the form of equalities. It is proved that, for this type of optimization problems, the explicit form can be obtained of a projection operator based on a generalized projection matrix. The projection operator allows us to represent the initial problem as a fixed point problem. The explicit form of the fixed point problem makes it possible to run a process of simple iteration. We prove the linear convergence of the obtained iterative method and, under rather natural additional conditions, its quadratic convergence. It is shown that an important application of the developed method is the flow assignment in a network of an arbitrary topology with one pair of source and sink.     

The stability radius of a semi-Fredholm operator

We prove an asymptotic formula for the stability radius of a semi-Fredholm operator on a Banach space in terms of the reduced minimum modulus. In particular, this gives a new proof of the Fredholm case studied by Förster and Kaashoek [4].     

The reduced minimum modulus of Drazin inverses of linear operators on Hilbert spaces

In this article, we study the reduced minimum modulus of the Drazin inverse of an operator on a Hilbert space and give lower and upper bounds of the reduced minimum modulus of an operator and its Drazin inverse, respectively. Using these results, we obtain a characterization of the continuity of Drazin inverses of operators on a Hilbert space.

     

Subcritical perturbation of a locally periodic elliptic operator

We consider a singularly perturbed Dirichlet spectral problem for an elliptic operator of second order. The coefficients of the operator are assumed to be locally periodic and oscillating in the scale ? . We describe the leading terms of the asymptotics of the eigenvalues and the eigenfunctions to the problem, as the parameter ? tends to zero, under structural assumptions on the potential. More precisely, we assume that the local average of the potential has a unique global minimum point in the interior of the domain and its Hessian is non‐degenerate at this point. Copyright © 2016 John Wiley & Sons, Ltd.     

Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity

A new continuity theorem of minimum selection is presented for a continuous set-valued operator from a topological space into a Banach space with some uniform convexity. As applications, some problems concerning minimum right inverses for linear operators and minimum fixed points for condensing set-valued nonlinear operators are discussed. Also, the existence of minimum solutions for an integral inclusion is proved.     

Two-Stage Method of Construction of Regularizing Algorithms for Nonlinear Ill-Posed Problems

For an equation with a nonlinear differentiable operator acting in a Hilbert space, we study a two-stage method of construction of a regularizing algorithm. First, we use the Lavrentiev regularization scheme. Then we apply to the regularized equation either Newton’s method or nonlinear analogs of α-processes: the minimum error method, the minimum residual method, and the steepest descent method. For these processes, we establish the linear convergence rate and the Fejér property of iterations. Two cases are considered: when the operator of the problem is monotone and when the operator is finite-dimensional and its derivative has nonnegative spectrum. For the two-stage method with a monotone operator, we give an error bound, which has optimal order on the class of sourcewise representable solutions. In the second case, the error of the method is estimated by means of the residual. The proposed methods and their modified analogs are implemented numerically for three-dimensional inverse problems of gravimetry and magnetometry. The results of the numerical experiment are discussed.     

On the conditions of solvability and the resolvent of a second-order deferential boundary operator in a space of vector functions

A boundary differential operator generated by the Sturm-Liouville differential expression with bounded operator potential and nonlocal boundary conditions is considered. The conditions for a considered operator to be a Fredholm and solvable operator are established and its resolvent is constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 517–524, April, 1995.     

Simulation of soliton solutions for the propagation of a femtosecond pulse in a medium with a cubic nonlinearity

Soliton solutions are constructed numerically for the problem of propagation of a femtosecond pulse in a medium with a cubic nonlinearity. The problem is posed as an eigenvalue problem with an operator nonlinear in the eigenfunctions. For given values of the propagation parameter we find the real eigenvalue λ and the corresponding eigenvector. This eigenvector is a soliton, i.e., a solution that does not vary in the coordinate of propagation of the light pulse. An algorithm is proposed to find the minimum eigenvalue and the corresponding eigenfunctions that satisfy given conditions. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 63–68, 1999.     

Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution

We consider the spectral problem for a model second-order differential operator with an involution. The operator is given by the differential expression Lu = ?u??(?x) and boundary conditions of general form. We obtain a criterion for the basis property of the systems of eigenfunctions of this operator in terms of the coefficients in the boundary conditions.     

最小数学期望与倒向随机微分方程

本文讨论了如何用倒向随机微分方程(BSDE)来计算一类最小数学期望；证明了对Brown运动，最小数学期望算子仍然保留了数学期望算子的某些性质，指出了最小数学期望算子与数学期望算子的某些不同之处。     

Hierarchical-matrix method for a class of diffusion-dominated partial integro-differential equations

A hierarchical matrix approach for solving diffusion-dominated partial integro-differential problems is presented. The corresponding diffusion-dominated differential operator is discretized by a second-order accurate finite-volume scheme, while the Fredholm integral term is approximated by the trapezoidal rule. The hierarchical matrix approach is used to approximate the resulting algebraic problem and includes the implementation of an efficient preconditioned generalized minimum residue (GMRes) solver. This approach extends previous work on integral forms of boundary element methods by taking into account inherent characteristics of the diffusion-dominated differential operator in the resultant algebraic problem. Numerical analysis estimates of the accuracy and stability of the finite-volume and the trapezoidal rule approximation are presented and combined with estimates of the hierarchical-matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical accuracy and convergence estimates, and demonstrate the almost optimal computational complexity of the proposed solution procedure.     

Transformation of Quasiconvex Functions to Eliminate Local Minima

Quasiconvex functions present some difficulties in global optimization, because their graph contains “flat parts”; thus, a local minimum is not necessarily the global minimum. In this paper, we show that any lower semicontinuous quasiconvex function may be written as a composition of two functions, one of which is nondecreasing, and the other is quasiconvex with the property that every local minimum is global minimum. Thus, finding the global minimum of any lower semicontinuous quasiconvex function is equivalent to finding the minimum of a quasiconvex function, which has no local minima other than its global minimum. The construction of the decomposition is based on the notion of “adjusted sublevel set.” In particular, we study the structure of the class of sublevel sets, and the continuity properties of the sublevel set operator and its corresponding normal operator.     

Existence results to a nonlinear p(k)-Laplacian difference equation

In the present paper, by using variational method, the existence of non-trivial solutions to an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The main technical tools applied here are the two local minimum theorems for differentiable functionals given by Bonanno.     

Reduced energy functionals for a three-dimensional fast rotating Bose Einstein condensates

We prove that in the fast rotating regime, the three-dimensional Gross–Pitaevskii energy describing the state of a Bose Einstein condensate can be reduced to a two-dimensional problem and that the vortex lines are almost straight. Additionally, we prove that the minimum of this two-dimensional problem can be sought in a reduced space corresponding to the first eigenspace of an elliptic operator. This space is called the Lowest Landau level and is of infinite dimension     

Sufficient conditions that the minimum and maximum of partial differential operators should coincide and that their spectra should be discrete

An expression of the form $$l(u) = ( - 1)^m \sum\nolimits_{j = 1}^m {D_j^{2m} } u + [q(x)] + ir(x)]u$$ is considered. Sufficient conditions are found such that the minimum operator, formally conjugate tol(u), generated by the expression and the maximum operator generated by the expressionl(u) in ?₂ (E_n) should coincide. It is proved that if q(x)→∞ or q(x)+ r(x)→∞, ¦x¦→∞, then the operator generated byl(u) in ?₂ (E_n) has a discrete spectrum.     

一类具有可修故障和不可修故障的两部件并联可修系统的系统主算子的性质

研究了一类具有可修故障和不可修故障的两部件并联可修系统.运用C_0半群理论,通过修复率均值的观念,对系统主算子的谱上界进行了估值,并得到该谱上界即为各修复率均值的最小者的相反数.然后运用了共尾的概念及相关的理论,得到了系统主算子的谱上界与系统主算子产生的半群的增长界相等,从而得到其增长界也是各修复率均值的最小者的相反数.