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Short Time Geometry Of Random Heat Kernels
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Book Synopsis Short-Time Geometry of Random Heat Kernels by : Richard Bucher Sowers
Download or read book Short-Time Geometry of Random Heat Kernels written by Richard Bucher Sowers and published by American Mathematical Soc.. This book was released on 1998 with total page 145 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume studies the behaviour of a random heat kernel associated with a stochastic partial differential equation, and gives short-time expansion of this heat kernel. The author finds that the dominant exponential term is classical and depends only on the Riemannian distance function. The second exponential term is a work term and also has classical meaning. There is also a third non-negligible exponential term which blows up. The author finds an expression for this third exponential term which involves a random translation of the index form and the equations of Jacobi fields. In the process, he develops a method to approximate the heat kernel to any arbitrary degree of precision.
Book Synopsis Short-Time Geometry of Random Heat Kernels by : Richard Bucher Sowers
Download or read book Short-Time Geometry of Random Heat Kernels written by Richard Bucher Sowers and published by Oxford University Press, USA. This book was released on 2014-09-11 with total page 145 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume studies the behaviour of a random heat kernel associated with a stochastic partial differential equation, and gives short-time expansion of this heat kernel. The author finds that the dominant exponential term is classical and depends only on the Riemannian distance function. The second exponential term is a work term and also has classical meaning. There is also a third non-negligible exponential term which blows up. The author finds an expression for this third exponential term which involves a random translation of the index form and the equations of Jacobi fields. In the process, he develops a method to approximate the heat kernel to any arbitrary degree of precision.
Book Synopsis Controllability, Stabilization, and the Regulator Problem for Random Differential Systems by : Russell Johnson
Download or read book Controllability, Stabilization, and the Regulator Problem for Random Differential Systems written by Russell Johnson and published by American Mathematical Soc.. This book was released on 1998 with total page 63 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume develops a systematic study of time-dependent control processes. The basic problem of null controllability of linear systems is first considered. Using methods of ergodic theory and topological dynamics, general local null controllability criteria are given. Then the subtle question of global null controllability is studied. Next, the random linear feedback and stabilization problem is posed and solved. Using concepts of exponential dichotomy and rotation number for linear Hamiltonian systems, a solution of the Riccati equation is obtained which has extremely good robustness properties and which also preserves all the smoothness and recurrence properties of the coefficients. Finally, a general version of the local nonlinear feedback stabilization problem is solved.
Book Synopsis Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies by : Wolfgang Bulla
Download or read book Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies written by Wolfgang Bulla and published by American Mathematical Soc.. This book was released on 1998 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, the authors provide a self-contained discussion of all real-valued quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies of completely integrable evolution equations. The approach utilizes algebro-geometric methods, factorization techniques for finite difference expressions, as well as Miura-type transformations. Detailed spectral theoretic properties of Lax pairs and theta function representations of the solutions are derived. Features: Simple and unified treatment of the topic. Self-contained development. Novel results for the Kac-van Moerbeke hierarchy and its algebro-geometric solutions.
Book Synopsis Semiclassical Analysis for Diffusions and Stochastic Processes by : Vassili N. Kolokoltsov
Download or read book Semiclassical Analysis for Diffusions and Stochastic Processes written by Vassili N. Kolokoltsov and published by Springer. This book was released on 2007-12-03 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: The monograph is devoted mainly to the analytical study of the differential, pseudo-differential and stochastic evolution equations describing the transition probabilities of various Markov processes. These include (i) diffusions (in particular,degenerate diffusions), (ii) more general jump-diffusions, especially stable jump-diffusions driven by stable Lévy processes, (iii) complex stochastic Schrödinger equations which correspond to models of quantum open systems. The main results of the book concern the existence, two-sided estimates, path integral representation, and small time and semiclassical asymptotics for the Green functions (or fundamental solutions) of these equations, which represent the transition probability densities of the corresponding random process. The boundary value problem for Hamiltonian systems and some spectral asymptotics ar also discussed. Readers should have an elementary knowledge of probability, complex and functional analysis, and calculus.
Book Synopsis Featured Reviews in Mathematical Reviews 1997-1999 by : Donald G. Babbitt
Download or read book Featured Reviews in Mathematical Reviews 1997-1999 written by Donald G. Babbitt and published by American Mathematical Soc.. This book was released on 2000-05-05 with total page 762 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second volume of Featured Reviews makes available special detailed reviews of some of the most important mathematical articles and books published from 1997 through 1999. Also included are excellent reviews of several classic books and articles published prior to 1970. Among those reviews, for example, are the following: Homological Algebra by Henri Cartan and Samuel Eilenberg, reviewed by G. Hochschild; Faisceaux algebriques coherents by Jean-Pierre Serre, reviewed by C. Chevalley; and On the Theory of General Partial Differential Operators by Lars Hormander, reviewed by J. L. Lions. In particular, those seeking information on current developments outside their own area of expertise will find the volume very useful. By identifying some of the best publications, papers, and books that have had or are expected to have a significant impact in applied and pure mathematics, this volume will serve as a comprehensive guide to important new research across all fields covered by MR.
Book Synopsis Time-Dependent Subdifferential Evolution Inclusions and Optimal Control by : Shouchuan Hu
Download or read book Time-Dependent Subdifferential Evolution Inclusions and Optimal Control written by Shouchuan Hu and published by American Mathematical Soc.. This book was released on 1998 with total page 81 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume studies multivalued evolution equations driven by time-dependent subdifferential operators and optimal control problems for such systems. The formulation is general enough to incorporate problems with time varying constraints. For evolution inclusions, existence relaxation and structural results for the solution set are proved. For optimal control problems, a general existence theory is developed, different forms of the relaxed problem are introduced and studied, well-posedness properties are investigated and the precise relation between the properties of relaxability and well-posedness is established. Various examples of systems which fit in the abstract framework are analyzed.
Book Synopsis Squared Hopf Algebras by : Volodymyr V. Lyubashenko
Download or read book Squared Hopf Algebras written by Volodymyr V. Lyubashenko and published by American Mathematical Soc.. This book was released on 1999 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended for graduate students and research mathematicians interested in associative rings and algebras.
Book Synopsis Rank 3 Amalgams by : Bernd Stellmacher
Download or read book Rank 3 Amalgams written by Bernd Stellmacher and published by American Mathematical Soc.. This book was released on 1998 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended for graduate students and research mathematicians working in classical linear algebraic
Book Synopsis Algebraic and Strong Splittings of Extensions of Banach Algebras by : William G. Bade
Download or read book Algebraic and Strong Splittings of Extensions of Banach Algebras written by William G. Bade and published by American Mathematical Soc.. This book was released on 1999 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume, the authors address the following: Let $A$ be a Banach algebra, and let $\sum\:\ 0\rightarrow I\rightarrow\frak A\overset\pi\to\longrightarrow A\rightarrow 0$ be an extension of $A$, where $\frak A$ is a Banach algebra and $I$ is a closed ideal in $\frak A$. The extension splits algebraically (respectively, splits strongly) if there is a homomorphism (respectively, continuous homomorphism) $\theta\: A\rightarrow\frak A$ such that $\pi\circ\theta$ is the identity on $A$. Consider first for which Banach algebras $A$ it is true that every extension of $A$ in a particular class of extensions splits, either algebraically or strongly, and second for which Banach algebras it is true that every extension of $A$ in a particular class which splits algebraically also splits strongly. These questions are closely related to the question when the algebra $\frak A$ has a (strong) Wedderburn decomposition. The main technique for resolving these questions involves the Banach cohomology group $\cal H2(A,E)$ for a Banach $A$-bimodule $E$, and related cohomology groups. Later chapters are particularly concerned with the case where the ideal $I$ is finite-dimensional. Results are obtained for many of the standard Banach algebras $A$.
Book Synopsis Flat Extensions of Positive Moment Matrices: Recursively Generated Relations by : Raúl E. Curto
Download or read book Flat Extensions of Positive Moment Matrices: Recursively Generated Relations written by Raúl E. Curto and published by American Mathematical Soc.. This book was released on 1998 with total page 73 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, the authors develop new computational tests for existence and uniqueness of representing measures $\mu$ in the Truncated Complex Moment Problem: $\gamma {ij}=\int \bar zizj\, d\mu$ $(0\le i+j\le 2n)$. Conditions for the existence of finitely atomic representing measures are expressed in terms of positivity and extension properties of the moment matrix $M(n)(\gamma )$ associated with $\gamma \equiv \gamma {(2n)}$: $\gamma {00}, \dots ,\gamma {0,2n},\dots ,\gamma {2n,0}$, $\gamma {00}>0$. This study includes new conditions for flat (i.e., rank-preserving) extensions $M(n+1)$ of $M(n)\ge 0$; each such extension corresponds to a distinct rank $M(n)$-atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consistency, and normal consistency, the existence problem for minimal representing measures is reduced to the solubility of small systems of multivariable algebraic equations. In a variety of applications, including cases of the quartic moment problem ($n=2$), the text includes explicit contructions of minimal representing measures via the theory of flat extensions. Additional computational texts are used to prove non-existence of representing measures or the non-existence of minimal representing measures. These tests are used to illustrate, in very concrete terms, new phenomena, associated with higher-dimensional moment problems that do not appear in the classical one-dimensional moment problem.
Book Synopsis Homogeneous Integral Table Algebras of Degree Three: A Trilogy by : Harvey I. Blau
Download or read book Homogeneous Integral Table Algebras of Degree Three: A Trilogy written by Harvey I. Blau and published by American Mathematical Soc.. This book was released on 2000 with total page 89 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book features homogeneous integral table algebras of degree three with a faithful real element. The algebras of the title are classified to exact isomorphism; that is, the sets of structure constants which arise from the given basis are completely determined. Other results describe all possible extensions (pre-images), with a faithful element which is not necessarily real, of certain simple homogeneous integral table algebras of degree three. On antisymmetric homogeneous integral table algebras of degree three. This paper determines the homogeneous integral table algebras of degree three in which the given basis has a faithful element and has no nontrivial elements that are either real (symmetric) or linear, and where an additional hypothesis is satisfied.It is shown that all such bases must occur as the set of orbit sums in the complex group algebra of a finite abelian group under the action of a fixed-point-free automorphism of order three. Homogeneous integral table algebras of degree three with no nontrivial linear elements. The algebras of the title which also have a faithful element are determined to exact isomorphism. All of the simple homogeneous integral table algebras of degree three are displayed, and the commutative association schemes in which all the nondiagonal relations have valency three and where some relation defines a connected graph on the underlying set are classified up to algebraic isomorphism.
Book Synopsis Invariants Under Tori of Rings of Differential Operators and Related Topics by : Ian Malcolm Musson
Download or read book Invariants Under Tori of Rings of Differential Operators and Related Topics written by Ian Malcolm Musson and published by American Mathematical Soc.. This book was released on 1998 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: If $G$ is a reductive algebraic group acting rationally on a smooth affine variety $X$, then it is generally believed that $D(X)^G$ has properties very similar to those of enveloping algebras of semisimple Lie algebras. In this book, the authors show that this is indeed the case when $G$ is a torus and $X=k^r\times (k^*)^s$. They give a precise description of the primitive ideals in $D(X)^G$ and study in detail the ring theoretical and homological properties of the minimal primitive quotients of $D(X)^G$. The latter are of the form $B^x=D(X)^G/({\mathfrak g}-\chi({\mathfrak g}))$ where ${\mathfrak g}=\textnormal{Lie}(G)$, $\chi\in {\mathfrak g}^\ast$ and ${\mathfrak g}-\chi({\mathfrak g})$ is the set of all $v-\chi(v)$ with $v\in {\mathfrak g}$. They occur as rings of twisted differential operators on toric varieties. It is also proven that if $G$ is a torus acting rationally on a smooth affine variety, then $D(X[LAMBDA]!/G)$ is a simple ring.
Book Synopsis Rational Homotopical Models and Uniqueness by : Martin Majewski
Download or read book Rational Homotopical Models and Uniqueness written by Martin Majewski and published by American Mathematical Soc.. This book was released on 2000 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main goal of this paper is to prove the following conjecture of Baues and Lemaire: the differential graded Lie Tlgebra associated with the Sullivan model of a space is homotopy equivalent to its Quillen model. In addition we show the same for the cellular Lie algebra model which we build from the simplicial analog of the classical Adams-Hilton model. It turns out that this cellular Lie algebra model is one link in a chain of models connecting the models of Quillen and Sullivan.The key result which makes all this possible is Anick's correspondence between differential graded Lie algebras and Hopf algebras up to homotopy. In addition we show that the Quillen model is a rational homotopical equivalence, and we conclude the same for the other models using our main result. Theconstruction of the three models is given in detail. The background from homotopy theory, differential algebra, and algebra is presented in great generality.
Book Synopsis Hopf Algebras, Polynomial Formal Groups, and Raynaud Orders by : Lindsay Childs
Download or read book Hopf Algebras, Polynomial Formal Groups, and Raynaud Orders written by Lindsay Childs and published by American Mathematical Soc.. This book was released on 1998 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives two new methods for constructing $p$-elementary Hopf algebra orders over the valuation ring $R$ of a local field $K$ containing the $p$-adic rational numbers. One method constructs Hopf orders using isogenies of commutative degree 2 polynomial formal groups of dimension $n$, and is built on a systematic study of such formal group laws. The other method uses an exponential generalization of a 1992 construction of Greither. Both constructions yield Raynaud orders as iterated extensions of rank $p$ Hopf algebras; the exponential method obtains all Raynaud orders whose invariants satisfy a certain $p$-adic condition.
Book Synopsis Categories of Operator Modules (Morita Equivalence and Projective Modules) by : David P. Blecher
Download or read book Categories of Operator Modules (Morita Equivalence and Projective Modules) written by David P. Blecher and published by American Mathematical Soc.. This book was released on 2000 with total page 109 pages. Available in PDF, EPUB and Kindle. Book excerpt: We employ recent advances in the theory of operator spaces, also known as quantized functional analysis, to provide a context in which one can compare categories of modules over operator algebras that are not necessarily self-adjoint. We focus our attention on the category of Hilbert modules over an operator algebra and on the category of operator modules over an operator algebra. The module operations are assumed to be completely bounded - usually, completely contractive. Wedevelop the notion of a Morita context between two operator algebras A and B. This is a system (A,B,{} {A}X {B},{} {B} Y {A},(\cdot,\cdot),[\cdot,\cdot]) consisting of the algebras, two bimodules {A}X {B and {B}Y {A} and pairings (\cdot,\cdot) and [\cdot,\cdot] that induce (complete) isomorphisms betweenthe (balanced) Haagerup tensor products, X \otimes {hB} {} Y and Y \otimes {hA} {} X, and the algebras, A and B, respectively. Thus, formally, a Morita context is the same as that which appears in pure ring theory. The subtleties of the theory lie in the interplay between the pure algebra and the operator space geometry. Our analysis leads to viable notions of projective operator modules and dual operator modules. We show that two C*-algebras are Morita equivalent in our sense if and only ifthey are C*-algebraically strong Morita equivalent, and moreover the equivalence bimodules are the same. The distinctive features of the non-self-adjoint theory are illuminated through a number of examples drawn from complex analysis and the theory of incidence algebras over topological partial orders.Finally, an appendix provides links to the literature that developed since this Memoir was accepted for publication.
Book Synopsis Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem by : Lawrence C. Evans
Download or read book Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem written by Lawrence C. Evans and published by American Mathematical Soc.. This book was released on 1999 with total page 66 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume, the authors demonstrate under some assumptions on $f^+$, $f^-$ that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measure $\mu{^+}=f^+dx$ onto $\mu^-=f^-dy$ can be constructed by studying the $p$-Laplacian equation $- \mathrm{div}(\vert DU_p\vert^{p-2}Du_p)=f^+-f^-$ in the limit as $p\rightarrow\infty$. The idea is to show $u_p\rightarrow u$, where $u$ satisfies $\vert Du\vert\leq 1,-\mathrm{div}(aDu)=f^+-f^-$ for some density $a\geq0$, and then to build a flow by solving a nonautonomous ODE involving $a, Du, f^+$ and $f^-$.
