1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
Preface. Part I. EVOLUTION EQUATIONS IN DISCRETE TIME. Preliminaries. Main Results on Stability. Operator Splitting Problems. Equations with Memory. Part II. RUNGE-KUTTA METHODS. Discretization by Runge-Kutta methods. Analysis of Stability. Convergence Estimates. Variable Stepsize Approximations. Part III. OTHER DISCRETIZATION METHODS. The/theta-method. Methods with Splitting Operator. Linear Multistep Methods. Part IV. INTEGRO-DIFFERENTIAL EQUATIONS UNDER DISCRETIZATION. Integro-Differential Equations. APPENDIX. A Functions of Linear Operators. B Cauchy Problems in Banach Space.
The aim of the present paper is to establish some new finite difference inequalities involving functions of two independent variables which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain partial finite difference equations.
High Performance Computing (HPC) applications are essential for scientists and engineers to create and understand models and their properties. These professionals depend on the execution of large sets of computational jobs that explore combinations of parameter values. Avoiding the execution of unnecessary jobs brings not only speed to these experiments, but also reductions in infrastructure usage—particularly important due to the shift of these applications to HPC cloud platforms. Our hypothesis is that data generated by these experiments can help users in identifying such jobs. To address this hypothesis we need to understand the similarity levels among multiple experiments necessary for job elimination decisions and the steps required to automate this process. In this paper we present a study and a machine learning-based tool called JobPruner to support parameter exploration in HPC experiments. The tool was evaluated with three real-world use cases from different domains including seismic analysis and agronomy. We observed the tool reduced 93% of jobs in a single experiment, while improving quality in most scenarios. In addition, reduction in job executions was possible even considering past experiments with low correlations.
. We prove a version of Pontryagin's maximum principle for time and norm optimal control of linear diffusion processes. This result includes both necessary and sufficient conditions and implies a ``concentration principle'' for the optimal measure-valued controls.
Preface Chapter 0 Preliminaries Chapter I Different Families of Sets in Bitopological Spaces Chapter II Different Relations between Two Topologies on a Set and Bitopological Insertions Chapter III Dimension of Bitopological Spaces Chapter IV Baire-Like Properties of Bitopological Spaces Chapter V Dynamics of Bitopological Relations, Baire-Like Properties and Dimensions Chapter VI Generalized Boolean Algebras and Related Problems. Representation Theorems Chapter VII Applications of Bitopologies Bibliography List of Special Symbols and Notations Index
PREFACE CHAPTER 1: INTRODUCTIONP> 1.1 Finite dimensional systems: the maximum principle. 1.2. Finite dimensional systems: existence and uniqueness. 1.3. Infinite dimensional systems. CHAPTER 2: SYSTEMS WITH STRONGLY MEASURABLE CONTROLS, I 2.1. The reachable space and the bang-bang property 2.2. Reversible systems 2.3. The reachable space and its dual, I 2.4. The reachable space and its dual, II 2.5. The maximum principle 2.6. Vanishing of the costate and nonuniqueness for norm optimal controls 2.7. Vanishing of the costate for time optimal controls 2.8. Singular norm optimal controls 2.9. Singular norm optimal controls and singular functionals CHAPTER 3: SYSTEMS WITH STRONGLY MEASURABLE CONTROLS, II 3.1. Existence and uniqueness of optimal controls 3.2. The weak maximum principle and the time optimal problem 3.3. Modeling of parabolic equations 3.4. Weakly singular extremals 3.5. More on the weak maximum principle 3.6. Convergence of minimizing sequences and stability of optimal controls CHAPTER 4: OPTIMAL CONTROL OF HEAT PROPAGATION 4.1. Modeling of parabolic equations 4.2. Adjoints 4.3. Adjoint semigroups 4.4. The reachable space 4.5. The reachable space and its dual, I 4.6. The reachable space and its dual, II 4.7. The maximum principle 4.8. Existence, uniqueness and stability of optimal controls 4.9. Examples and applications CHAPTER 5: OPTIMAL CONTROL OF DIFFUSIONS 5.1. Modeling of parabolic equations 5.2. Dual spaces 5.3. The reachable space and its dual 5.4. The maximum principle 5.5. Existence of optimal controls uniqueness and stability of supports 5.6. Examples and applications. CHAPTER 6: APPENDIX 6.1 Self adjoint operators, I 6.2 Self adjoint operators, II 6.3 Related research REFERENCES NOTATION AND SUBJECT INDEX.
A complete proof of the Dvoretzky theorem, accessible to graduate students, is given.
The repeated compact–normal differentiability of the Euler–Lagrange functional is proved. The sufficient extreme conditions for such a functional in the case of one and two variables are obtained.
In this paper we consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary conditions. We associate this problem with a self-adjoint operator in a Hilbert or in a Pontryagin space. Using this operator-theoretic formulation and analytic methods, we investigate locations, multiplicities and the asymptotic behaviour of the eigenvalues.
Direct and inverse problem of scattering theory (IPST) are investigated for the differential equation−y”+q(x)y=λ2y on half line containing a spectral parameter in the boundary condition y’(0)+(α0+iα1λ+α2λ2)y(0)=0
The notion of median, or Fermat point, of a finite set, has been recently generalized in two ways (see P.L. Papini and J. Puerto, preprint 2002; E. Alvoni, preprint 2002). Here we study conditions on the underlying space related to the existence of solutions concerning the second generalization. Let X be a real Banach space; consider a finite subset A of X containing n elements and let k be an integer between 2 and n. For x in X, consider the distances among x and k points of A nearest to x; set μk(A,x)=average of these numbers. We want to minimize μk(A,x) (for x∈X): a solution of this problem, if it exists, will be called a k-medium of A. The function μk(A,x) is neither convex (or quasi-convex), nor concave; therefore the existence of solutions does not follow from standard results on convex functions. Here we shall discuss existence of k-media; we will show that if the space X is such that every finite set A has a median (also called a Fermat point), then the same is true for k-media; in particular, in reflexive spaces, as well as in several classical spaces, k-media always exist.
We investigate the new definition of analytic functional calculus in the terms of representation theory of SL2(R). We avoid any usage of its algebraic homomorphism property and replace it by the demand to be an intertwining operator. The related notion of spectrum and spectral mapping theorem are given. The construction is illustrated by a simple example of calculus and spectrum of non-normal n x n matrix. Keywords: Functional calculus, spectrum, intertwining operator, spectral mapping theorem, jet spaces.
The aim of this paper is to show that operators closed to normal ones with spectrum on a curve possess certain maximal invariant subspaces, which are related to the interior and the exterior of the curve. We generalize the corresponding results (for the real axis and the unit circle) established by Naboko and Makarov.
Publisher Summary This chapter describes some events and persons connected with Lviv mathematics (1892–1945). One of the first Lviv mathematicians having a substantial influence on the development of mathematics in Lviv was Jozef Puzyna. Stefan Banach was a student of engineering Lviv Polytechnica. The main achievements of the Lviv School are the creation of the fundamentals of functional analysis introducing topological methods and interpretation of probability as a measure and application of probability methods to Fourier series. A characteristic feature of Lviv mathematicians was masterful manipulation with non-constructive methods: the axiom of choice, Baire categories, and Lebesgue integral.
We present some new first-order differential equations invariant under continuous subgroups of the generalized Poincaré group P(1,4) and defined in the spaces M(1,3)×R(u) and M(1,4)×R(u).
By means of the symmetric norming functions and the abstract concept of approximation numbers of an element in a normed algebra, we present a general view of approximation ideals. Some properties and applications are also presented.
This paper is an invitation to look what occurs when tools of Robinson's and Nelson’s Nonstandard Analysis act simultaneoysly. For instance, in this case we have several kinds of infinitesimals in our disposal.
