Chapter 17 Euclidean structure in finite dimensional normed spaces

We study the asymptotic behavior of fixed point free Hilbert metric nonexpansive maps on bounded convex domains. For such maps, we prove that the omega limit sets are contained in a convex subset of the boundary when the domain is either polyhedral or two dimensional. Similar results are obtained for several classes of positive operators defined on closed cones, including linear maps, affine linear maps, max-min operators, and reproduction-decimation operators. We discuss the relationship between these results and other Denjoy-Wolff type theorems. In particular, we investigate the interaction of nonexpansive maps with the horofunction boundary in the Hilbert geometry and in finite dimensional normed spaces.

The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.

Since the pioneering work of Landau and Nakanishi on general Feynman integrals, it has been known that the singularities of the S-matrix, causal Green's functions and related functions are described by the so called Landau equations. These Landau singularities were physically interpreted as the macroscopic causality by the theoretical physicists working in 5-matrix theory, and the notion of essential support was obtained (Chandler-Stapp [3], lagolnitzer-Stapp [5]). In the branch of mathematics, on the other side, the theory of microfunction has evolved and has been applied powerfully to the general theory of partial differential equations (Sato-Kawai-Kashiwara [11]). It contained the essential support theory as the singularity spectrum of a function (i.e. its support viewed as a microfunction), and was far-reaching because of its close connection with the theory of differential equations. Namely the method of microlocal analysis, based on the theory of holonomic systems, has provided a systematic way of handling functions with natural background and found most effective applications to various problems of mathematics, such as the theory of ^-functions and Fourier transformations (Sato [10], Kashiwara [6]). It was then recognized that the Landau equations give holonomic varieties, which led one to the holonomicity postulate of ^-matrix and related quantities (Sato [10]). In the present paper we shall study the holonomy structure of Landau singularities and Feynman integrals from this standpoint. First we review the notion of Landau varieties. In contrast with

At present there can be no doubt to say that the quantum group theory has been one of the most important, modern and rapidly developing directions of mathematics and mathematical physics in the end of the twentieth century. Although initially (in the early 1980’s) the quantum group theory was formulated for solving problems in the theory of integrable systems and statistical physics, later the surprising connection of this theory with many branches of mathematics, and the theoretical and mathematical physics was discovered. Today the quantum group theory is connected with such mathematical fields as special functions (especially, with (q-orthogonal polynomials and basic hypergeometric series), the theory of difference and differential equations, combinatorial analysis and representation theory, matrix and operator algebras, noncommutative geometry, knot theory, topology, category theory and so on. From point of view of the mathematical physics there exists interconnection of the quantum group theory with the quantum inverse scattering method, conformai and quantum fields theory and so on. It is expected that the quantum groups will provide deeper understanding of concept of symmetry in physics.KeywordsQuantum GroupSimple RootBasic Hypergeometric SeriesReduction ChainGeneral Projection OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Partial differential equations of the second- and higher-order are encountered in a number of areas of importance to engineering and mathematical physics. As the processes that are accounted for in the development of the governing differential equations increase in complexity, the differential equations themselves tend to acquire a higher-order. The theory of second-order partial differential equations has found extensive applications in the study of problems in fluid mechanics, flow in porous media, heat conduction in solids, diffusive transport of chemicals in porous media, wave propagation in strings and membranes, and in mechanics of solids. Higher-order partial differential equations have been developed particularly in relation to studies in mechanics of solids. These include the formulation of the biharmonic equation for the solution of problems in plane elasticity and in the flow of viscous fluids, the fourth-order partial differential equations associated with the flexural mechanics of prismatic plates, the sixth-order partial differential equations used to describe the mechanics of shell structures and other fourth-order partial differential equations which describe the mechanics of plastic solids. In this chapter we shall restrict attention to the examination of second-order partial differential equations.

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.

Based on the superposition theory of partial differential equation (PDE), a new method is introduced here to solve complicated problems of flow field. In this method, a problem of flow field with complex boundary conditions and source items is divided into several problems of flow fields with simple boundary conditions and source items, and then the solution of a complicated problem of flow field can be determined by the solutions’ superposition of several simple problems of flow fields. For an instance, by using a software in computational fluid dynamics (CFD), a linear seeping problem in porous media is solved by numerical simulation as well as solutions’ superposition of two simpler problems deriving from the proto-problem, and thus the application feasibility of superposition theory in numerical simulation of flow field is verified. More importantly, the application of the new method in CFD can decrease the number of necessary numerical examination greatly.

Although the theory of partial differential equations (PDEs) is not a mere generalization of the theory of ordinary differential equations (ODEs), there are many points of contact between both theories.

A review of the works of S.L. Sobolev that have played a funda mental role in the development of the theory of partial differential equations and mathematical analysis in the second part of the 20th century. Supplied with a short biographical note.KeywordsNauk SSSRMathematical PhysicWave EquationWeak SolutionCauchy ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

There has been considerable progress in the theory and computer implementation of symbolic computation algorithms to automatically determine and exploit exact symmetries of exact differential equations. Such programs usually apply a finite number of exact differentiations and eliminations to the overdetermined linearized equations for the unknown symmetries (the symmetry defining equations), to complete them to certain involutive or standard forms. The symmetry properties can be determined from these involutive forms. In many applications, however, the differential equations describing a model are only known approximately. For example they may contain parameters that are only known approximately. Symbolic methods are unstable if applied to the symmetry defining equations directly, and indirect techniques (e.g. replacing approximate parameters by symbolic ones) might not be practical in cases with many parameters. Discussion and examples are given of such difficulties. A new generation of symbolic-numeric methods is described and applied to the problem of determining symmetries of differential equations. We introduce a class of differential-elimination methods which uses Numerical Linear Algebra, and in particular the Singular Value Decomposition, to perform the elimination process on the symmetry defining equations. Our approach uses symbolic differentiations but not symbolic eliminations. Substitution of an appropriate random point in the independent variables, followed by numerical projection is used to test for the conditions of completion to a projective involutive form. We prove that this form is equivalent to the involutive form of the Cartan-Kuranishi theory of partial differential equations. Our method is applied to determining symmetry properties of 50 ode from the collection in Kamke’s book.

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

From a fresh data science perspective, this thesis discusses the prediction of coronary artery disease based on Single-Nucleotide Polymorphisms (SNPs) from the Ontario Heart Genomics Study (OHGS). First, the thesis explains the k-Nearest Neighbour (k-NN) and Random Forest learning algorithms, and includes a complete proof that k-NN is universally consistent in finite dimensional normed vector spaces. Second, the thesis introduces two dimensionality reduction techniques: Random Projections and a new method termed Mass Transportation Distance (MTD) Feature Selection. Then, this thesis compares the performance of Random Projections with k-NN against MTD Feature Selection and Random Forest for predicting artery disease. Results demonstrate that MTD Feature Selection with Random Forest is superior to Random Projections and k-NN. Random Forest is able to obtain an accuracy of 0.6660 and an area under the ROC curve of 0.8562 on the OHGS dataset, when 3335 SNPs are selected by MTD Feature Selection for classification. This area is considerably better than the previous high score of 0.608 obtained by Davies et al. in 2010 on the same dataset.

Various definitions of surface area and volume are possible in finite dimensional normed linear spaces (= Minkowski spaces). Using a Bonnesen-style inequality, we investigate the ratio of the Holmes-Thompson surface area of the unit ball to its volume. In particular, in the planar case a stronger lower bound for this ratio is established when the area is defined in the sense of Holmes-Thompson, or is given by the definition of mass. From this we obtain some (characteristic) properties of Radon curves. Mathematics subject classification (2000): 52A10, 52A21, 52A40, 46B20.

The material presented below amounts roughly to the so-called fundamental theorems of Lie and their implications concerning Lie algebras, Lie subgroups and subalgebras. By straightforward and fairly elementary steps, we shall extend the concept of Lie group to include groups admitting coordinate systems in which the functions defining ab, the group product of elements a, b, possess continuous first order derivatives in the coordinates of a and satisfy a Lipschitz condition in those of b. That such groups are equivalent to classical (i.e. analytical) Lie groups was announced in 1936 by van Kampen, although apparently van Kampen published nothing by way of proof beyond a certain decisive uniqueness theorem concerning systems of ordinary differential equations. (This is contained in our theorem (12.1); the proof given below is essentially that of van Kampen [3]). It is conceivable that the condition relative to coordinates of b could be weakened or dispensed with entirely. For, the only part the condition in question plays in our development is to make it certain that the inverse of a certain mapping-the mapping into canonical coordinates-is single-valued. In this connection mention should be made of the paper [1J of Garrett Birkhoff in which it is shown that the existence of continuous first order derivatives of ab with respect to the coordinates of both the a's and the b's is sufficient for defining a Lie group. We shall consider only groups with a finite number of real parameters (whereas the groups considered by Birkhoff are not necessarily finite dimensional). This makes it possible to use the standard existence theorems for systems of ordinary real differential equations. We make no use whatever of the theory of partial differential equations. We maintain throughout a purely local point of view in the sense that we consider only what happens in the neighborhood of the identity. Historical notes. The proof of the uniqueness of square roots (8.5) is due to Claude Chevalley (unpublished). The proof given below of Lie's theorem that a linear system of vector fields which is closed under commutation defines a Lie group is, we believe, due to van der Waerden [4]. We have supplied a number of preliminary lemmas which make the proof rigorously applicable to the case in which the vector fields are only assumed to possess continuous first order derivatives. The theorem in ?17 that every local subgroup of a Lie group is a Lie group is due essentially to E. Cartan [2]. 481

SynopsisTechniques from the theory of partial differential equations are employed to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations of the second order.