Solvability and Construction of a Solution to the Fredholm Integral Equation of the First Kind

The issues of solvability and construction of a solution of the Fredholm integral equation of the first kind are considered. It is done by immersing the original problem into solving an extremal problem in Hilbert space. Necessary and sufficient conditions for the existence of a solution are obtained. A method of constructing a solution of the Fredholm integral equation of the first kind is developed. A constructive theory of solvability and construction of a solution to a boundary value problem of a linear integrodifferential equation with a distributed delay in control, generated by the Fredholm integral equation of the first kind, has been created.

On Fixed-Parameter Solvability of the Minimax Path Location Problem

The minimax path location problem is to find a path P in a graph G such that the maximum distance d_(G)(v,P)from every vertex v∈V(G)to the path P is minimized.It is a well-known NP-hard problem in network optimization.This paper studies the fixed-parameter solvability,that is,for a given graph G and an integer k,to decide whether there exists a path P in G such that max v∈V(G)d_(G)(v,P)≤k.If the answer is affirmative,then graph G is called k-path-eccentric.We show that this decision problem is NP-complete even for k=1.On the other hand,we characterize the family of 1-path-eccentric graphs,including the traceable,interval,split,permutation graphs and others.Furthermore,some polynomially solvable special graphs are discussed.

SOlvaBILITY OF HIGHER INDEX TIME-VARYING LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS

Linear differential-algebraic equations (DAEs) with time-varying coefficients A(t)x’(t) + B(t)x(t) = q(t), which are tractable with a higher index, are discussed. Their essential properties are investigated. Some equivalent systems are given. Using them the paper shows how to state properly initial and boundary conditions for these DAEs. The eaistence and uniqueness theory of the solution of the initial and boundary value problems for higher index DAEs are proposed.

On the Approximation Solvability of the Abstract Differential Inclusion

differential inclusion d u d t∈f(t,u) is presented. The convergence of the approximation solution and the existence of the solution for abstract evolution multivalued problem are discussed.

Painleve Integrability, Consistent Riccati Expansion Solvability and Interaction Solution for the Coupled mKdV-BLMP System

The integrability of the coupled, modified KdV equation and the potential Boiti-Leon-Manna-Pempinelli(mKdVBLMP) system is investigated using the Painleve analysis approach. It is shown that this coupled system possesses the Painleve property in both the principal and secondary branches. Then, the consistent Riccati expansion(CRE)method is applied to the coupled mKdV-BLMP system. As a result, it is CRE solvable for the principal branch while non-CRE solvable for the secondary branch. Finally; starting from the last consistent differential equation in the CRE solvable case, soliton, multiple resonant soliton solutions and soliton-cnoidal wave interaction solutions are constructed explicitly.

THE SUFFICIENT CONDITIONS FOR SOLVABILITY OF ALGEBRAIC INVERSE EIGENVALUE PROBLEMS

Applying constructed homotopy and its properties,we gel some sufficient conditions for the solvability of algebraic inverse eigenvalue problems,which are better than that of the paper [4] in some cases. Inverse eigenvalue problems,solvability,sufficient conditions.

Topological Conditions for Unique Solvability of RLC OP Amp Networks

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THE SOLVABILITY CONDITIONS FOR AN INVERSE EIGEN-PAIR PROBLEM

This paper discusses problem IEP:Given n×m matrix X and m×m diagonal matrix A, find an n×n matrix A such that AX=XA.The new solvablily conditions for the problem IEP are obtained. The eigenvalue dislribulaion of the solutions for the problem IEP are described in detail.

Solvability of the Diophantine equations x^p+2^(2m)y^4=p^kz^2 and x^p+y^2=p^kz^4

We study the solvability of two classes of Diophantine equations by using some new methods and new results in this paper. Let p be an odd prime and B n denote nth Bernoulli number. We prove that if p≡1(mod 4) and pB (p-1)/2 , then the equation x p+2 2m n 4=p ky 2, m,n,k∈[FK(W+3mm\.3mm][TPP107A,+3mm?3mm,BP], k>1, gcd (x,py)=1, and the equation x p+y 2=p kz 4, k∈[FK(W+3mm\.3mm][TPP107A,+3mm?3mm,BP], gcd (x,y)=1, k>1, 2|y have no integral solutions respectively.

Solvability of Nonlinear Sequential Fractional Dynamical Systems with Damping

In this paper, we are concerned with the solvability for a class of nonlinear sequential fractional dynamical systems with damping infinite dimensional spaces, which involves fractional Riemann-Liouville derivatives. The solutions of the dynamical systems are obtained by utilizing the method of Laplace transform technique and are based on the formula of the Laplace transform of the Mittag-Leffler function in two parameters. Next, we present the existence and uniqueness of solutions for nonlinear sequential fractional dynamical systems with damping by using fixed point theorems under some appropriate conditions.

Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations

In this paper, we investigate the effect of weight function in the nonlinear part on global solvability of the Cauchy problem for a class of semi-linear hyperbolic equations with damping.

Solvability of the Economic Input-Output Equation by Time Irreversibility

This paper reinterprets the economic input-output equation as a description of a realized situation without considering decision making. This paper uses the equation that the self-sufficiency rate is added to the Leontief type, and discusses its solvability. The equation has a unique solution if and only if each part of the relevant society satisfies the space-time openness condition. This condition means that commodities which a part of the relevant society possesses are not all inputted to its inside. Moreover, if the process of input and output is time irreversible, each part of the relevant society satisfies the space-time openness condition. Therefore, the solvability of the equation is guaranteed by time irreversibility. This proposition seems to be relevant to the grandfather paradox which is a type of time paradox.

Solvability of a class of PT-symmetric non-Hermitian Hamiltonians：Bethe ansatz method

We use the Bethe ansatz method to investigate the Schrdinger equation for a class of PT-symmetric non-Hermitian Hamiltonians. Elementary exact solutions for the eigenvalues and the corresponding wave functions are obtained in terms of the roots of a set of algebraic equations. Also, it is shown that the problems possess sl(2) hidden symmetry and then the exact solutions of the problems are obtained by employing the representation theory of sl(2) Lie algebra. It is found that the results of the two methods are the same.

Linear Quadratic Leader-Follower Stochastic Differential Games:Closed-Loop Solvability

In this paper,a leader-follower stochastic differential game is studied for a linear stochastic differential equation with quadratic cost functionals.The coefficients in the state equation and the weighting matrices in the cost functionals are all deterministic.Closed-loop strategies are introduced,which require to be independent of initial states;and such a nature makes it very useful and convenient in applications.The follower first solves a stochastic linear quadratic optimal control problem,and his optimal closed-loop strategy is characterized by a Riccati equation,together with an adapted solution to a linear backward stochastic differential equation.Then the leader turns to solve a stochastic linear quadratic optimal control problem of a forward-backward stochastic differential equation,necessary conditions for the existence of the optimal closed-loop strategy for the leader is given by a Riccati equation.Some examples are also given.

Solvability identification and feasibility restoring of divergent optimal power flow problems

Optimal power flow (OPF) has been considered as an important problem in power systems. Although several excellent algorithms, such as Newton method and interior point method, have been developed to solve the OPF problem, divergences still often occur. Till now, few works have focused on the solv- ability identification and feasibility restoring of divergent OPF problems. In this paper, we propose a systematic approach to identify the solvability of divergent OPF problems, and restore a feasible solu- tion for unsolvable OPF cases. The proposed approach consists of two phases: solvability identifica- tion phase (SIP) and feasibility restoring phase (FRP). In SIP, a novel methodology based on problem transformation and active set is adopted to identify the solvability of divergent OPF problem. If a fea- sible solution can be obtained in SIP, then this divergent OPF problem is solvable, otherwise, FRP is used to restore a feasible or optimal solution by relaxing soft constraints and load shedding. In FRP, a feasibility restoring model is presented, and a priority-listing strategy of restoring actions is proposed to restore the unsolvable OPF problems. Numerical studies indicate that the proposed SIP and FRP are reliable to diagnose the solvability of the divergent OPF problems, give an index to measure the un- solvability, and restore an unsolvable OPF case.

ON THE SOLVABILITY OF GENERAL LINEAR METHODS FOR DISSIPATIVE DYNAMICAL SYSTEMS

The main purpose of the present paper is to examine the existence and local uniqueness of solutions of the implicit equations arising in the application of a weakly algebraically stable general linear methods to dissipative dynamical systems, and to extend the existing relevant results of Runge-Kutta methods by Humphries and Stuart(1994).

THE SOLVABILITY CONDITIONS FOR THE INVERSE PROBLEM OF MATRICES POSITIVE SEMIDEFINITE ON A SUBSPACE

This paper studies the following two problems:Problem Ⅰ. Given X, B ∈ Rnxm, find A ∈ Ps,n, such that AX = B, where Ps, n = {A ∈ SRnxn|XTA>0,STX = 0, for given S ∈ R}.Problem Ⅱ. Given A* ∈ Rnxn, find A ∈ SE, such that ||A*-A|| = infA∈SE||A* -A|| where SE denotes the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of Problem Ⅰ, the expression of the general solution of Problem Ⅰ and the solution of Problem Ⅱ are given for two cases. For the general case, the equivalent form of conditions for the solvability of Problem Ⅰ is given.

ON THE p-SOLVABILITY OF THE FINITE GROUPS WITH A T.I. SYLOW p-SUBGROUP

W. Feit has ever beautifully proved the following theorem by the abstract group theoretic method.Theorem 1. Let G be an arbitrary finite group with a cyclic Sylow p-subgroup. If there exists a normal subgroup N of G such that P|(|N|, |G/N|), then G is psolvable.

SOLVABILITY AND WELL-POSEDNESS OF HIGHERORDER ABSTRACT CAUCHY PROBLEMS

We consider the higher-order Cauchy problem （ACPn） xn（t）=sum from i=0 to n-1 Bixi（t）1xi（0）=xi for 0≤i≤n-1,where Bi（0≤i≤n-1） are closed linear operators on a Banach space X such that D=∩ i=0 n-1 D（Bi）is dense in X. It is well known that the solvability and the well-posedness of （ACPn）were studied only in some special cases, such as D(Bn-1)?D（Bi） for 0≤i≤n-2 by F. Neu-brander and a factoring case by J. T. Sandefur. In this paper, by using some new results ofvector valued Laplace transforms given by W. Arenddt, we obtain some characterizations ofthe solvability and some sufficiency conditions of the well-posedness for general （ACPn）,which generalize F. Neubrander’s results and the famous results for （ACP1）

SOLVABILITY OF QUADRATIC OPTIMAL CONTROL VIA OUTPUT FEEDBACK

The solvability of quadratic optimal control via output feedback is studied. It has been concluded that every optimal output feedback is a derivative solution to the corresponding optimal-state feedback, and the linear matrix equation that determines the existence of the optimal output feedback is generally unsolvable. The output matrix C with some parameters is discussed, and the necessary condition for the existence of the optimal output feedback is given. Moreover, for the single-input system, the condition is proved almost sufficient.