scholarly journals HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS

2018 ◽  
Vol 6 ◽  
Author(s):  
MATTHIAS KRECK ◽  
HAGGAI TENE
Keyword(s):  
Dimensional Space ◽  
Characteristic Classes ◽  
Geometric Description ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Singular Cohomology ◽  
Cobordism Theory ◽  
Proper Maps ◽  
Definition Of ◽  
Fredholm Maps

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen’s geometric description of cobordism groups for finite-dimensional smooth manifolds [Quillen, ‘Elementary proofs of some results of cobordism theory using steenrod operations’, Adv. Math., 7 (1971)]. Quillen stresses the fact that this construction allows the definition of Gysin maps for ‘oriented’ proper maps. For finite-dimensional manifolds one has a Gysin map in singular cohomology which is based on Poincaré duality, hence it is not clear how to extend it to infinite-dimensional manifolds. But perhaps one can overcome this difficulty by giving a Quillen type description of singular cohomology for Hilbert manifolds. This is what we do in this paper. Besides constructing a general Gysin map, one of our motivations was a geometric construction of equivariant cohomology, which even for a point is the cohomology of the infinite-dimensional space $BG$, which has a Hilbert manifold model. Besides that, we demonstrate the use of such a geometric description of cohomology by several other applications. We give a quick description of characteristic classes of a finite-dimensional vector bundle and apply it to a generalized Steenrod representation problem for Hilbert manifolds and define a notion of a degree of proper oriented Fredholm maps of index $0$.

DUAL FORMULATION OF OPTIMAL PROBLEM FOR CONTINUOUS-TIME SYSTEMS

2005 ◽  
Vol 02 (03) ◽  
pp. 251-258
Author(s):  
HANLIN HE ◽  
QIAN WANG ◽  
XIAOXIN LIAO
Keyword(s):  
Objective Function ◽  
Continuous Time ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Error Response ◽  
Infinite Dimensional ◽  
Dual Formulation ◽  
Finite Dimensional ◽  
Continuous Time Systems ◽  
Time Systems

The dual formulation of the maximal-minimal problem for an objective function of the error response to a fixed input in the continuous-time systems is given by a result of Fenchel dual. This formulation probably changes the original problem in the infinite dimensional space into the maximal problem with some restrained conditions in the finite dimensional space, which can be researched by finite dimensional space theory. When the objective function is given by the norm of the error response, the maximum of the error response or minimum of the error response, the dual formulation for the problems of L1-optimal control, the minimum of maximal error response, and the minimal overshoot etc. can be obtained, which gives a method for studying these problems.

Determination of Hydraulic Permeability Based on the Measurements of Outflow

2015 ◽  
Vol 3 ◽  
pp. 105-116
Author(s):  
Jozef Kačur ◽  
Jozef Minár
Keyword(s):  
Unsaturated Flow ◽  
Dimensional Space ◽  
A Priori ◽  
Hydraulic Permeability ◽  
Infinite Dimensional ◽  
Tuning Parameters ◽  
Finite Dimensional ◽  
Partially Saturated Porous Media ◽  
The Cost

In this paper we present a method for the determination of the hydraulic permeability for flow in partially saturated porous media. The dependence of hydraulic permeability on effective saturation is not assumed to be a member of any specific finite dimensional class of functions (e.g. vanGenuchten-Mualem, Burdin-Mualem, Brook-Corey). Instead, an infinite dimensional space of functions with limited a priori assumptions (e.g. smoothness, monotonicity) is considered. Consequently, we face a more challenging problem compared to the finite-dimensional case, in which only few tuning parameters need to be determined. We consider the case of 1D unsaturated flow and assume that the data are collected at the outflow of the sample. The hydraulic permeability is determined in an iterative way. We minimize the cost functional reflecting the discrepancy between the measured and computed data. In doing so, we use the Gateaux differential to obtain the direction of the descent.

scholarly journals Dagger linear logic for categorical quantum mechanics

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Robin Cockett ◽  
Cole Comfort ◽  
Priyaa Srinivasan
Keyword(s):  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Physics ◽  
Linear Logic ◽  
Infinite Dimensional ◽  
Graphical Calculus ◽  
Finite Dimensional ◽  
Categorical Quantum Mechanics ◽  
Definition Of ◽  
Categorical Semantics

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.

scholarly journals The weak Bernoulli property for matrix Gibbs states

2018 ◽  
Vol 40 (8) ◽  
pp. 2219-2238 ◽  
Author(s):  
MARK PIRAINO
Keyword(s):  
General Result ◽  
Gibbs State ◽  
Dimensional Space ◽  
Gibbs States ◽  
Sufficient Condition ◽  
Infinite Dimensional Space ◽  
Infinite Dimensional ◽  
Ergodic Properties ◽  
Finite Dimensional ◽  
Novel Approach

We study the ergodic properties of a class of measures on $\unicode[STIX]{x1D6F4}^{\mathbb{Z}}$ for which $\unicode[STIX]{x1D707}_{{\mathcal{A}},t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\Vert A_{x_{0}}\cdots A_{x_{n-1}}\Vert ^{t}$, where ${\mathcal{A}}=(A_{0},\ldots ,A_{M-1})$ is a collection of matrices. The measure $\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$ is called a matrix Gibbs state. In particular, we give a sufficient condition for a matrix Gibbs state to have the weak Bernoulli property. We employ a number of techniques to understand these measures, including a novel approach based on Perron–Frobenius theory. We find that when $t$ is an even integer the ergodic properties of $\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$ are readily deduced from finite-dimensional Perron–Frobenius theory. We then consider an extension of this method to $t>0$ using operators on an infinite- dimensional space. Finally, we use a general result of Bradley to prove the main theorem.

scholarly journals DIFFERENCE-DIFFERENTIAL GAME OF CONVERGENCE - EVASION IN HILBERT SPACE, II

2017 ◽  
Vol 20 (10) ◽  
pp. 74-83
Author(s):  
V.L. Pasikov
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Dynamic Game ◽  
Continuous Functions ◽  
Scientific School ◽  
Finite Dimensional Space ◽  
Space Of Continuous Functions ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Right

For conflict operated differential system with delay studying of dynamic game of convergence - evasion relatively functional goal set, now regarding evasion and solution of a problem of existence of alternative in the case under consideration is continued. In the work realization of condition of saddle point relatively to the right part of operated system is not supposed. Earlier similar tasks were set and solved for finite-dimensional space at scientific school of the academicianN.N. Krasovsky. For a case of infinite-dimensional space of continuous functions similar tasks were considered by the author. In the suggested work at theorem proving about convergence - evasion, the norm of Hilbert space is used.

scholarly journals The Lie Group in Infinite Dimension

2011 ◽  
Vol 2011 ◽  
pp. 1-35 ◽  
Author(s):  
V. Tryhuk ◽  
V. Chrastinová ◽  
O. Dlouhý
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Vector Fields ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Symmetries Of Differential Equations ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Infinitesimal Transformations

A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local,C∞smooth) action of a Lie group on infinite-dimensional space (a manifold modelled onℝ∞) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

scholarly journals General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions

2014 ◽  
Vol 2014 ◽  
pp. 1-37 ◽  
Author(s):  
Josef Diblík ◽  
Hana Halfarová
Keyword(s):  
Dimensional Space ◽  
Discrete Systems ◽  
Infinite Dimensional Space ◽  
Initial Interval ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Differential Systems ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Linear Discrete Systems

Planar linear discrete systems with constant coefficients and delaysx(k+1)=Ax(k)+∑l=1n‍Blxl(k-ml)are considered wherek∈ℤ0∞:={0,1,…,∞},m1,m2,…,mnare constant integer delays,0<m1<m2<⋯<mn,A,B1,…,Bnare constant2×2matrices, andx:ℤ-mn∞→ℝ2. It is assumed that the considered system is weakly delayed. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension2(mn+1)is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.

ALMOST FRÉCHET DIFFERENTIABILITY OF LIPSCHITZ MAPPINGS BETWEEN INFINITE-DIMENSIONAL BANACH SPACES

2002 ◽  
Vol 84 (3) ◽  
pp. 711-746 ◽  
Author(s):  
WILLIAM B. JOHNSON ◽  
JORAM LINDENSTRAUSS ◽  
DAVID PREISS ◽  
GIDEON SCHECHTMAN
Keyword(s):  
Banach Spaces ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Lipschitz Mapping ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Lipschitz Mappings ◽  
Finite Dimensional ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results.2000 Mathematical Subject Classification: 46G05, 46T20.

scholarly journals Functional regression on the manifold with contamination

Biometrika ◽  
2020 ◽  
Author(s):  
Zhenhua Lin ◽  
Fang Yao
Keyword(s):  
Nonparametric Regression ◽  
Dimensional Space ◽  
Linear Manifold ◽  
Real Data ◽  
Contamination Level ◽  
Functional Regression ◽  
Dimensional Manifold ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Noisy Measurements

Summary We propose a new method for functional nonparametric regression with a predictor that resides on a finite-dimensional manifold, but is observable only in an infinite-dimensional space. Contamination of the predictor due to discrete or noisy measurements is also accounted for. By using functional local linear manifold smoothing, the proposed estimator enjoys a polynomial rate of convergence that adapts to the intrinsic manifold dimension and the contamination level. This is in contrast to the logarithmic convergence rate in the literature of functional nonparametric regression. We also observe a phase transition phenomenon related to the interplay between the manifold dimension and the contamination level. We demonstrate via simulated and real data examples that the proposed method has favourable numerical performance relative to existing commonly used methods.

scholarly journals The Linear Programming Approach to Reach-Avoid Problems for Markov Decision Processes

2017 ◽  
Vol 60 ◽  
pp. 263-285 ◽  
Author(s):  
Nikolaos Kariotoglou ◽  
Maryam Kamgarpour ◽  
Tyler H. Summers ◽  
John Lygeros
Keyword(s):  
Markov Decision Processes ◽  
Dimensional Space ◽  
Decision Processes ◽  
Linear Program ◽  
Control Synthesis ◽  
Programming Approach ◽  
Finite Dimensional Space ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Markov Decision

One of the most fundamental problems in Markov decision processes is analysis and control synthesis for safety and reachability specifications. We consider the stochastic reach-avoid problem, in which the objective is to synthesize a control policy to maximize the probability of reaching a target set at a given time, while staying in a safe set at all prior times. We characterize the solution to this problem through an infinite dimensional linear program. We then develop a tractable approximation to the infinite dimensional linear program through finite dimensional approximations of the decision space and constraints. For a large class of Markov decision processes modeled by Gaussian mixtures kernels we show that through a proper selection of the finite dimensional space, one can further reduce the computational complexity of the resulting linear program. We validate the proposed method and analyze its potential with numerical case studies.

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