Solvability Of Equations Research Articles

An accretive operator approach to ergodic zero-sum stochastic games

We study some ergodicity property of zero-sum stochastic games with a finite state space and possibly unbounded payoffs. We formulate this property in operator-theoretical terms, involving the solvability of an optimality equation for the Shapley operators (i.e., the dynamic programming operators) of a family of perturbed games. The solvability of this equation entails the existence of the uniform value, and its solutions yield uniform optimal stationary strategies. We first provide an analytical characterization of this ergodicity property, and address the generic uniqueness, up to an additive constant, of the solutions of the optimality equation. Our analysis relies on the theory of accretive mappings, which we apply to maps of the form $Id - T$ where $T$ is nonexpansive. Then, we use the results of a companion work to characterize the ergodicity of stochastic games by a geometrical condition imposed on the transition probabilities. This condition generalizes classical notion of ergodicity for finite Markov chains and Markov decision processes.

О первой смешанной задаче в банаховых пространствах для вырождающихся параболических уравнений с меняющимся направлением времени

The article is devoted to studying one of the sections of nonclassical differential equations, namely, matters concerned with solvability of parabolic equations with changing second-order time direction. As is known, in ordinary boundary-value problems for strictly parabolic equations, the smoothness of the initial and boundary conditions completely ensures that the solutions belong to the Holder spaces, but in the case of equations with changing time direction, the smoothness of the initial and boundary conditions does not ensure that the solutions belong to these spaces. S.A. Tersenov (for a model parabolic equation with changing time direction) and S.G. Pyatkov (for a more general second-order equation) obtained the necessary and sufficient conditions for solvability of the corresponding mixed problems in Holder spaces. In so doing, they always assumed the initial and boundary conditions being equal to zero. Cases in which the initial and boundary conditions belong to Banach spaces are considered. The functional spaces in which the solutions must be sought are introduced. Relevant a priori estimates, which make it possible to obtain the solvability conditions for these problems, are obtained. The properties of the obtained solutions have been studied. In particular, the equivalence of the Riesz and Littlewood-Paley conditions similar to the conditions for solutions of strictly elliptic and strictly parabolic second order equations is established. A unique solvability of the first mixed problem with boundary and initial functions from the Banach space has been proved.

Разрешимость неоднородного автономного дифференциального уравнения с последействием на отрицательной полуоси

Разрешимость неоднородного автономного дифференциального уравнения с последействием на отрицательной полуоси

Solvability and optimal controls of non-instantaneous impulsive stochastic neutral integro-differential equation driven by fractional Brownian motion

In this manuscript, a new class of non-instantaneous impulsive stochastic neutral integrodi fferential equation driven by fractional Brownian motion (fBm, in short) with state-dependent delay and their stochastic optimal control problem is studied. We utilize the theory of the resolvent operator and a fixed point technique to present the solvability of the stochastic system. Then, the existence of optimal controls is discussed for the proposed stochastic system. Finally, an example is offered to demonstrate the obtained theoretical results.

On the High-Order Quasi Exactly Solvable Differential Equations

In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.

Solvability of Difference Equations with Nonuniformly Contracting Operators in the Space of Two-Sided Sequences

We establish existence and uniqueness conditions for the solutions of difference equations with contracting compact operators in the metric space of two-sided sequences.

Existence and Stability of Equilibrium of DC Microgrid With Constant Power Loads

The problem of existence and stability of equilibria of DC microgirds with constant power loads methods is addressed in this paper. Constant power loads (CPLs) often cause instability due to its negative impedance characteristics. What is the worse, CPLs may cause that the system admits no equilibria for its nonlinearity. The purpose of paper can be summarized as: A) designing a controller to overcome CPLs instability; B) obtaining the sufficient conditions to guarantee the existence of the equilibria; C) Under the conditions of equilibria existing, obtaining the sufficient conditions of local stability. For the first objective, the method based on virtual resistance and virtual inductance is proposed. For the second objective, we transform the problem of solvability of quadratic equations into the existence of the fixed point of a mapping. Thus, the analytical sufficient conditions is obtained based on Tarski fixed point theorem. Moreover, a numerical method is also provided to reduce conservatism. For the third objective, the small-signal model is established to predict the system qualitative behavior around equilibria. The stability conditions are obtained by using quadratic eigenvalue problem (GEP) theorem. These explicit conditions are obtained as a function of system parameters and it lead to build reliable microgrid.

Non-Markovianity of the post-Markovian master equation

An easily solvable quantum master equation has long been sought that takes into account memory effects induced on the system by the bath, i.e., non-Markovian effects. We briefly review the Post-Markovian master equation (PMME), which is relatively easy to solve, and analyze a simple example where solutions obtained exhibit non-Markovianity. We apply the distinguishability measure introduced by Breuer et al., and we also explicitly analyze the divisibility of the associated quantum dynamical maps. We give a mathematical condition on the memory kernel used in the PMME that guarantees non-CP-divisible dynamics.

O-Groups of finite rank and divisibility in their groups of o-automorphisms

We provide a full description of totally ordered groups of finite archimedean rank and study solvability of equations of the form $$x^n=g$$ in the group of ordered automorphisms of an a-closed totally ordered group G of finite archimedean rank. We also give a full description of these groups of o-automorphisms and a characterization of the elements of a particular group that have an n-th root.

Solvability and Stability for Neutral Stochastic Integro-differential Equations Driven by Fractional Brownian Motion with Impulses

The paper is devoted to the existence, uniqueness and asymptotic behaviors of mild solution to neutral impulsive stochastic integro-differential equations driven by fractional Brownian motion with $$H\in (\frac{1}{2},1)$$ by the theory of resolvent operator and contraction mapping principle. An example is provided to demonstrate the results of the proposed results.

Method of Riesz potentials applied to solution to nonhomogeneous singular wave equations

Riesz potentials are convolution operators with fractional powers of some distance (Euclidean, Lorentz or other) to a point. From application point of view, such potentials are tools for solving differential equations of mathematical physics and inverse problems. For example, Marsel Riesz used these operators for writing the solution to the Cauchy problem for the wave equation and theory of the Radon transform is based on Riesz potentials. In this article, we use the Riesz potentials constructed with the help of generalized convolution for solution to the wave equations with Bessel operators. First, we describe general method of Riesz potentials, give basic definitions, introduce solvable equations and write suitable potentials (Riesz hyperbolic $B$-potentials). Then, we show that these potentials are absolutely convergent integrals for some functions and for some values of the parameter representing fractional powers of the Lorentz distance. Next we show the connection of the Riesz hyperbolic B-potentials with d’Alembert operators in which the Bessel operators are used in place of the second derivatives. Next we continue analytically considered potentials to the required parameter values that includes zero and show that when value of the parameter is zero these operators are identity operators. Finally, we solve singular initial value hyperbolic problems and give examples.

Solvability of Equations by Quadratures and Newton’s Theorem

Picard–Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order n by quadratures in terms of its Galois group. It is based on the differential Galois theory and is rather involved. Liouville in 1839 found an elementary criterium for such solvability for $$n=2$$ . Ritt simplified Liouville’s theorem (1948). In 1973 Rosenlicht proved a similar criterium for arbitrary n. Rosenlicht work relies on the valuation theory and is not elementary. In these notes we show that the elementary Liouville–Ritt method based on developing solutions in Puiseux series as functions of a parameter works smoothly for arbitrary n and proves the same criterium.

The equation solvability problem over supernilpotent algebras with Mal’cev term

In 2011, Horváth gave a new proof that the equation solvability problem over finite nilpotent groups and rings is in P. In the same paper, he asked whether his proof can be lifted to nilpotent algebras in general. We show that this is in fact possible for supernilpotent algebras with a Mal’cev term. However, we also describe a class of nilpotent, but not supernilpotent algebras with Mal’cev term that have co-NP-complete identity checking problems and NP-complete equation solvability problems. This proves that the answer to Horváth’s question is negative in general (assuming P[Formula: see text]NP).

Solvability of Abstract Semilinear Equations by a Global Diffeomorphism Theorem

In this work we develop a method towards unique solvability of abstract semilinear equations. We use a global diffeomorphism theorem for which we provide a simplified proof. Applications to second order partial differential equations are given. Some additional technical tools about the properties of the Niemytskij operator are also given.

On Strongly $$J_n$$ J n -Clean Rings

A ring R is said to be strongly $$J_n$$ -clean if n is the least positive integer such that every element a is strongly $$J_n$$ -clean, that is, there exists an idempotent e such that $$ea=ae$$ , $$a-e$$ is a unit and $$ea^n$$ is in the Jacobson radical J(R). It is proved that strongly $$J_n$$ -clean ring is a strongly clean ring with stable range one. If R is an abelian ring (a ring in which all idempotents are central), then R is strongly $$J_n$$ -clean if and only if R / J(R) is strongly $$\pi $$ -regular and idempotents lift modulo J(R). Some examples and basic properties of these rings are studied. Some criterions in terms of solvability of the characteristic equation are obtained for such a $$2\times 2$$ matrix to be strongly $$J_2$$ -clean.

Space–time spectral collocation method for one-dimensional PDE constrained optimisation

This paper solves an optimal control problem governed by a parabolic PDE. Using Lagrangian multipliers, necessary conditions are derived and then space–time spectral collocation method is applied to discretise spatial derivatives and time derivatives. This method solves partial differential equations numerically with errors bounded by an exponentially decaying function which is dependent on the number of modes of analytic solution. Spectral methods, which converge spectrally in both space and time, have gained a significant attention recently. The problem is then reduced to a system consisting of easily solvable algebraic equations. Numerical examples are presented to show that this formulation has exponential rates of convergence in both space and time.

Taylor approximation to treat nonlocality in the scattering process

Study of scattering process in the nonlocal interaction framework leads to an integro-differential equation. The purpose of the present work is to develop an efficient approach to solve this integro-differential equation with high degree of precision. The method developed here employs Taylor approximation for the radial wave function which converts the integro-differential equation in to a readily solvable second-order homogeneous differential equation. This scheme is found to be computationally efficient by a factor of 10 when compared to the iterative scheme developed in J.~Phys.~G~Nucl.~Part.~Phys.~{\bf 45},~015106~(2018). The calculated observables for neutron scattering off $^{24}$Mg, $^{40}$Ca, $^{100}$Mo and $^{208}$Pb with energies up to 10 MeV are found to be within at most 8$\%$ of those obtained with the iterative scheme. Further, we propose an improvement over the Taylor scheme that brings the observables so close to the results obtained by iterative scheme that they are visually indistinguishable. This is achieved without any appreciable change in the run time.

On a solvable nonlinear difference equation of higher order

In this paper we consider the following higher-order nonlinear difference equation $$ x_{n}=\alpha x_{n-k}+\frac{\delta x_{n-k}x_{n-\left( k+l\right) }}{\beta x_{n-\left( k+l\right) }+\gamma x_{n-l}},\ n\in \mathbb{N} _{0}, $$ where $k$ and $l$ are fixed natural numbers, and the parameters $\alpha $, $ \beta $, $\gamma $, $\delta $ and the initial values $x_{-i}$, $i=\overline{ 1,k+l}$, are real numbers such that $\beta ^{2}+\gamma ^{2}\neq 0$. We solve the above-mentioned equation in closed form and considerably extend some results in the literature. We also determine the asymptotic behavior of solutions and the forbidden set of the initial values using the obtained formulae for the case $l=1$.

Zeros of rational functions and solvable nonlinear evolution equations

Recently a convenient technique to relate the time evolution of the N zeros of a time-dependent polynomial pNz;t of degree N in the (complex) variable z to the time evolution of its N coefficients has been exploited to identify large classes of dynamical systems solvable by algebraic operations. These models also include N-body problems that evolve in the complex plane (or, equivalently, in the real Cartesian plane) according to systems of nonlinearly-coupled equations of motion of Newtonian type (“accelerations equal forces”). Many of these models feature remarkable properties: for instance, they are Hamiltonian and integrable and/or multiply periodic or even isochronous (featuring completely periodic solutions with a fixed period largely independent of the initial data), or asymptotically isochronous (featuring isochrony only up to corrections vanishing in the remote future). In this paper, an analogous technique is introduced that focuses instead on the time evolution of the N zeros of an appropriate class of time-dependent rational functions RNz;t, thereby opening large vistas of new dynamical systems solvable by algebraic operations and featuring remarkable properties. A few examples are reported.

Integrability of exponential process and its application to backward stochastic differential equations

Abstract We consider the integrability problem of an exponential process with unbounded coefficients. The integrability is established under weaker conditions of Kazamaki type, which complements the results of Yong obtained under a Novikov type condition. As applications, we consider the solvability of linear backward stochastic differential equations (BSDEs) and market completeness, the solvability of a Riccati BSDE and optimal investment, all in the setting of unbounded coefficients.