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Let M n + 1 ( n ≥ 2 ) be a simply-connected space form of sectional curvature - κ 2 for some κ ≥ 0 , and I an interval not containing [ - κ , κ ] in its interior. It is known that the domain of a closed immersed hypersurface of M whose principal curvatures lie in I must be diffeomorphic to the n-sphere S n . These hypersurfaces are thus topologically rigid. The purpose of this paper is to show that they are also homotopically rigid. More precisely, for fixed I, the space F of all such hypersurfaces is either empty or weakly homotopy equivalent to the group of orientation-preserving diffeomorphisms of S n . An equivalence assigns to each element of F a suitable modification of its Gauss map. For M not simply-connected, F is the quotient of the corresponding space of hypersurfaces of the universal cover of M by a natural free proper action of the fundamental group of M. [ABSTRACT FROM AUTHOR]

This paper introduced the calculus of mappings on finite product of spaces, allowing different degrees of differentiability in the different factors. This enables us to prove an important feature in the infinite-dimensional Lie theory, the exponential law in generalized setting for locally convex spaces and for manifolds modelled on locally convex spaces. [ABSTRACT FROM AUTHOR]

Abstract For about 25 years, global methods from the calculus of variations have been used to establish the existence of chaotic behavior for some classes of dynamical systems. Like the analytical approaches that were used earlier, these methods require nondegeneracy conditions, but of a weaker nature than their predecessors. Our goal here is study such a nondegeneracy condition that has proved useful in several contexts including some involving partial differential equations, and to show this condition has an equivalent formulation involving stable and unstable manifolds. [ABSTRACT FROM AUTHOR]

We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy—a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity—under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the Γ-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing and on the infinite-dimensional shape manifold of viscous rods. [ABSTRACT FROM AUTHOR]

Representing the conditional independences present in a multivariate random vector via graphs has found widespread use in applications, and such representations are popularly known as graphical models or Markov random fields. These models have many useful properties, but their fundamental attractive feature is their ability to reflect conditional independences between blocks of variables through graph separation, a consequence of the equivalence of the pairwise, local, and global Markov properties demonstrated by Pearl and Paz (1985). Modern-day applications often necessitate working with either an infinite collection of variables (such as in a spatial-temporal field) or approximating a large high-dimensional finite stochastic system with an infinite-dimensional system. However, it is unclear whether the conditional independences present in an infinite-dimensional random vector or stochastic process can still be represented by separation criteria in an infinite graph. In light of the advantages of using graphs as tools to represent stochastic relationships, we undertake in this paper a general study of infinite graphical models. First, we demonstrate that näıve extensions of the assumptions required for the finite case results do not yield equivalence of the Markov properties in the infinite-dimensional setting, thus calling for a more in-depth analysis. To this end, we proceed to derive general conditions which do allow representing the conditional independence in an infinite-dimensional random system by means of graphs, and our results render the result of Pearl and Paz as a special case of a more general phenomenon. We conclude by demonstrating the applicability of our theory through concrete examples of infinite-dimensional graphical models. [ABSTRACT FROM AUTHOR]

Summary: We develop a novel frequency‐based H∞‐control method for a large class of infinite‐dimensional linear time‐invariant systems in transfer function form. A major benefit of our approach is that reduction or identification techniques are not needed, which avoids typical distortions. Our method allows to exploit both state‐space or transfer function models and input/output frequency response data when only such are available. We aim for the design of practically useful H∞‐controllers of any convenient structure and size. We use a nonsmooth trust‐region bundle method to compute arbitrarily structured locally optimal H∞‐controllers for a frequency‐sampled approximation of the underlying infinite‐dimensional H∞‐problem in such a way that (i) exponential stability in closed loop is guaranteed and that (ii) the optimal H∞‐value of the approximation differs from the true infinite‐dimensional value only by a prior user‐specified tolerance. We demonstrate the versatility and practicality of our method on a variety of infinite‐dimensional H∞‐synthesis problems, including distributed and boundary control of partial differential equations, control of dead‐time and delay systems, and using a rich testing set. [ABSTRACT FROM AUTHOR]

We investigate the Oseledets splitting for Banach space-valued random dynamical systems based on the theory of center manifolds. This technique gives us random one-dimensional invariant spaces which turn out to be the Oseledets subspaces under suitable assumptions. We apply these results to a stochastic parabolic evolution equation driven by a fractional Brownian motion. [ABSTRACT FROM AUTHOR]

Consider L a Leibniz algebra additionally graded by an arbitrary set I (set grading). We show that L decomposes as the sum of well-described graded ideals plus (maybe) a certain linear subspace. Under certain conditions, the simplicity of L is characterized in terms of the connection techniques. [ABSTRACT FROM AUTHOR]

We define a diffeology on the Milnor classifying space of a diffeological group G , constructed in a similar fashion to the topological version using an infinite join. Besides obtaining the expected classification theorem for smooth principal bundles, we prove the existence of a diffeological connection on any principal bundle (with mild conditions on the bundles and groups), and apply the theory to some examples, including some infinite-dimensional groups, as well as irrational tori. [ABSTRACT FROM AUTHOR]

Quantum correlations between two free spinless dissipative distinguishable particles (interacting with a thermal bath) are studied analytically using the quantum master equation and tools of quantum information. Bath-induced coherence and correlations in an infinite-dimensional Hilbert space are shown. We show that for temperature T> 0 the time-evolution of the reduced density matrix cannot be written as the direct product of two independent particles. We have found a time-scale that characterizes the time when the bath-induced coherence is maximum before being wiped out by dissipation (purity, relative entropy, spatial dispersion, and mirror correlations are studied). The Wigner function associated to the Wannier lattice (where the dissipative quantum walks move) is studied as an indirect measure of the induced correlations among particles. We have supported the quantum character of the correlations by analyzing the geometric quantum discord. [ABSTRACT FROM AUTHOR]

For a normed infinite-dimensional space, we prove that the family of all locally convex topologies which are compatible with the original norm topology has cardinality greater or equal to $\mathfrak{c}$. [ABSTRACT FROM AUTHOR]

Let ( N , g ) be a Riemannian manifold. Given a compact, connected and oriented submanifold M of N , we define the space of volume preserving embeddings Emb μ ( M , N ) as the set of smooth embeddings f : M ↪ N such that f ⁎ μ f = μ , where μ f (resp. μ ) is the Riemannian volume form on f ( M ) (resp. M ) induced by the ambient metric g (the orientation on f ( M ) being induced by f ). In this article, we use the Nash–Moser inverse function Theorem to show that the set of volume preserving embeddings in Emb μ ( M , N ) whose mean curvature is nowhere vanishing forms a tame Fréchet manifold, and determine explicitly the Euler–Lagrange equations of a natural class of Lagrangians. As an application, we generalize the Euler equations of an incompressible fluid to the case of an “incompressible membrane” of arbitrary dimension moving in N . [ABSTRACT FROM AUTHOR]

In this paper, firstly, we derive some inequalities about the relative entropy for infinite-dimensional quantum systems. Secondly, we propose a new measurement based on the relative entropy of entanglement for infinite-dimensional systems with bounded mean energy, and give a lower bound on this entanglement measure. Lastly, we generalize this measure to multi-partite quantum systems. [ABSTRACT FROM AUTHOR]

Reduction of concentration polarisation is of great value in reverse osmosis membrane systems. Concentration polarisation leads to a reduction in flux, which corresponds to a reduction in separation efficiency. This paper studies an approach by which to reduce concentration polarisation in reverse osmosis channels using a steady, spatially variant slip velocity profile. A method is developed to identify the most effective wall slip velocity profile for increasing the diffusive driving force away from the wall, which in turn increases mass transfer away from the membrane and reduces concentration polarisation. The nonlinear partial differential equations (PDEs) that govern fluid and mass transport behaviours are difficult to solve. In this work, an approximate solution to the nonlinear system is developed using systems of linearised ordinary differential equations (ODEs) to approximate the behaviour of the PDEs and determine the steady-state actuation profile that most effectively increases mass transfer at the wall. This leads to a systematic method for determination of a spatially-varying actuation profile (the most effective slip velocity profile) that decreases concentration polarisation in reverse osmosis membrane channels. [ABSTRACT FROM AUTHOR]

Consider a semiparametric model indexed by a Euclidean parameter of interest and an infinite-dimensional nuisance parameter. In many applications, pseudolikelihood provides a convenient way to infer the parameter of interest, where the nuisance parameter is replaced by a consistent estimator. The purpose of this paper is to establish the asymptotic behaviour of the pseudolikelihood ratio statistic under semiparametric models. In particular, we consider testing the hypothesis that the parameter of interest lies on the boundary of its parameter space. Under regularity conditions, we establish the equivalence between the asymptotic distributions of the pseudolikelihood ratio statistic and a likelihood ratio statistic for a normal mean problem with a misspecified covariance matrix. This result holds when the nuisance parameter is estimated at a rate slower than the usual rate in parametric models. We study three examples in which the asymptotic distributions are shown to be mixtures of chi-squared variables.We conduct simulation studies to examine the finite-sample performance of the pseudolikelihood ratio test. [ABSTRACT FROM AUTHOR]

A collisionless kinetic plasma model may often be cast as an infinite-dimensional noncanonical Hamiltonian system. I show that, when this is the case, the model can be discretized in space and particles while preserving its Hamiltonian structure, thereby producing a finite-dimensional Hamiltonian system that approximates the original kinetic model. I apply the general theory to two example systems: the relativistic Vlasov-Maxwell system with spin and a gyrokinetic Vlasov-Maxwell system. [ABSTRACT FROM AUTHOR]

The problem of finite-dimensional compensator design for the monodomain equations with the FitzHugh-Nagumo model is investigated. Exponential stabilizability and detectability of the linearized infinite-dimensional control system is studied. It is shown that the system is not exactly nullcontrollable but still can be exponentially stabilized by finite-rank input and output operators provided the desired stability margin is small enough. Based on existing results on model order reduction of infinite-dimensional systems, a finite-dimensional compensator is obtained by LQG-balanced truncation. Using partial measurements, the compensator produces a feedback control that is shown to be locally stabilizing for the infinite-dimensional nonlinear control system. Examples motivated by cardiophysiology are used to illustrate these results in a numerical setup. [ABSTRACT FROM AUTHOR]

Our aim is to present a solution to a general linear-quadratic (LQ) problem as well as to a Kalman-Yacubovich-Popov (KYP) problem for infinite-dimensional systems with bounded operators. The results are then applied, via the reciprocal system approach, to the question of solvability of some Lur'e resolving equations arising in the stability theory of infinite-dimensional systems in factor form with unbounded control and observation operators. To be more precise the Lur'e resolving equations determine a Lyapunov functional candidate for some closed-loop feedback systems on the base of some properties of an uncontrolled (open-loop) system. Our results are illustrated in details by an example of a temperature of a rod stabilization automatic control system. [ABSTRACT FROM AUTHOR]

For any N ≥ 2 and α = (α1, ⋯, αN+1) ∈ (0, ∞)N+1, let μ(N)α be the corresponding Dirichlet distribution on Δ(N) := {x = (xi)1≤i≤N ∈ [0, 1]N : |x|1 := Σ1≤i≤N xi ≤ 1}. We prove the Poincaré inequality ... for f ∈ C¹(Δ(N)), and show that the constant 1/αN+1 is sharp. Consequently, the associated diffusion process on Δ(N) converges to μ(N)α in L²(μ(N)α) at the exponentially rate αN+1. The whole spectrum of the generator is also characterized. Moreover, the sharp Poincaré inequality is extended to the infinite-dimensional setting, and the spectral gap of the corresponding discrete model is derived. [ABSTRACT FROM AUTHOR]

We propose an L -version of the Laplacian with respect to a measure on the infinite-dimensional Riemannian manifold. The Dirichlet problem for equations with the proposed Laplacian is solved in a region of the Riemannian manifold from a certain class. [ABSTRACT FROM AUTHOR]

A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to the maximum of the isonormal Gaussian process over this set. Using this formula we compute the first intrinsic volume of some infinite-dimensional convex compact sets including unit balls with respect to Sobolev-type seminorms and ellipsoids in the Hilbert space. We relate the distribution of the random one-dimensional projections of these sets to the distributions S1, S2, C1, C2 studied by Biane, Pitman, Yor [Bull. AMS 38 (2001)]. We show that the k-th intrinsic volume of the set of all functions on [0, 1] which have Lipschitz constant bounded by 1 and which vanish at 0 (respectively, which have vanishing integral) is given by ... This is related to the results of Gao and Vitale [Discrete Comput. Geom. 26 (2001); Elect. Comm. Probab. 8 (2003)], who considered a similar question for functions with a restriction on the total variation instead of the Lipschitz constant. Using the results of Gao and Vitale we give a new proof of the formula for the expected volume of the convex hull of the d-dimensional Brownian motion which is due to Eldan [Elect. J. Probab. 19 (2014)]. Additionally, we prove an analogue of Eldan's result for the Brownian bridge. Similarly, we show that the results on the intrinsic volumes of the Lipschitz balls can be translated into formulae for the expected volumes of zonoids (Aumann integrals) generated by the Brownian motion and the Brownian bridge. Also, these results have discrete versions for Gaussian random walks and bridges. Our proofs exploit Sudakov's and Tsirelson's theorems which establish a connection between the intrinsic volumes and the isonormal Gaussian process. [ABSTRACT FROM AUTHOR]

We introduce and analyze a framework and corresponding method for compressed sensing in infinite dimensions. This extends the existing theory from finite-dimensional vector spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary by demonstrating that existing finite-dimensional techniques are ill suited for solving a number of key problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. A conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. Central to this work is the introduction of two novel concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize an infinite-dimensional problem. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new suboptimal filter by deriving the Ito?s stochastic differential equations of the estimation of higher order central moments, satisfy, and impose some conditions to form a closed system. The essentially infinite-dimensional cubic sensor problem has been investigated in detail numerically to illustrate the reasonableness of the imposed conditions, and the numerical experiments support our discussion. A two-dimensional polynomial filtering problem has also been experimented. [ABSTRACT FROM PUBLISHER]

This paper presents a new approach to studying bilevel programming problems in infinite-dimensional spaces, based on infinite-dimensional duality and evolutionary variational inequalities. The result is applied to the evolutionary emission price problem. In our model, the government chooses the optimal price of pollution emissions, with consideration to firms' response to the price. Moreover, firms choose the optimal quantities of production to maximize their profits, given the price of pollution emissions. Finally, we provide a numerical example to show the feasibility of our approach. [ABSTRACT FROM AUTHOR]

Population managers will often have to deal with problems of meeting multiple goals, for example, keeping at specific levels both the total population and population abundances in given stage-classes of a stratified population. In control engineering, such set-point regulation problems are commonly tackled using multi-input, multi-output proportional and integral (PI) feedback controllers. Building on our recent results for population management with single goals, we develop a PI control approach in a context of multi-objective population management. We show that robust set-point regulation is achieved by using a modified PI controller with saturation and anti-windup elements, both described in the paper, and illustrate the theory with examples. Our results apply more generally to linear control systems with positive state variables, including a class of infinite-dimensional systems, and thus have broader appeal. [ABSTRACT FROM AUTHOR]

We deal with jointly convex generalized Nash equilibrium problems in infinite-dimensional spaces. For their solution, we extend a finite-dimensional optimization approach and design a convergent algorithm in Hilbert space. Then we apply our investigations to a class of multiobjective optimal control problems with control and state constraints that are governed by elliptic partial differential equations. We present a new reformulation as a jointly convex generalized Nash equilibrium problem. We study a finite element approximation of such a multiobjective optimal control problem, and further we prove convergence in appropriate function spaces. Finally, we provide some numerical results that show the effectiveness of our algorithm for multiobjective optimal control problems. [ABSTRACT FROM AUTHOR]

A--martingale density for a given stochastic processis a local-martingalestarting at 1 such that the productis a--martingale. Existence of a--martingale density is equivalent to a classic absence-of-arbitrage property of, and it is invariant if we replace the reference measurewith a locally equivalent measure. Now suppose that there exists a--martingale density for. Can we find another--martingale density forhaving some extra local integrabilityunder? We show that the answer is always positive for one part ofthat we identify, and we show that the complete answer depends in a precise quantitative way on the local integrability of the drift-to-jump ratio of the remaining ‘jumpy’ part of. Our proofs provide in addition new ideas and results in infinite-dimensional spaces. [ABSTRACT FROM PUBLISHER]

We discuss the spectrum phenomenon for Lipschitz functions on the infinite-dimensional torus. Suppose that is a measurable, real-valued, Lipschitz function on the torus . We prove that there exists a number with the following property: For any , there exists a parallel, infinite-dimensional subtorus such that the restriction of the function to the subtorus has an -norm of at most . [ABSTRACT FROM AUTHOR]

We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem. [ABSTRACT FROM AUTHOR]

The additivity of the minimal output entropy and that of the -capacity are known to be equivalent for finite-dimensional irreducibly covariant quantum channels. In this paper, we formulate a list of conditions allowing to establish similar equivalence for infinite-dimensional covariant channels with constrained input. This is then applied to bosonic Gaussian channels with quadratic input constraint to extend the classical capacity results of the recent paper [Giovannetti et al., Commun. Math. Phys. 334(3), 1553-1571 (2015)] to the case where the complex structures associated with the channel and with the constraint operator need not commute. In particular, this implies a multimode generalization of the "threshold condition," obtained for single mode in Schäfer et al. [Phys. Rev. Lett. 111, 030503 (2013)], and the proof of the fact that under this condition the classical "Gaussian capacity" resulting from optimization over only Gaussian inputs is equal to the full classical capacity. Complex structures correspond to different squeezings, each with its own normal modes, vacuum and coherent states, and the gauge. Thus our results apply, e.g., to multimode channels with a squeezed Gaussian noise under the standard input energy constraint, provided the squeezing is not too large as to violate the generalized threshold condition. We also investigate the restrictiveness of the gauge-covariance condition for single- and multimode bosonic Gaussian channels. [ABSTRACT FROM AUTHOR]

The control design for decentralized active damping of large-scale manipulators with hydraulic actuation is considered in a distributed-parameter framework. The concepts of modern light-weight construction enable the production of machines like mobile concrete pumps or maritime crane systems with extended operating range and less static load. However, due to the reduced weight the elasticity of the construction elements has a significant influence on the dynamic behavior of the boom. In this paper, a modular decentralized control strategy is presented and the asymptotic stability of the closed-loop system is rigorously proven in the infinite-dimensional setting. The proposed damping control strategy features a robust behavior since it is independent of the number and pose of the boom segments and of the exact knowledge of the system parameters. At the end, the practical implementation of the control strategy is discussed and validated by means of measurements on an industrial mobile concrete pump with four joints and an operating range of about 40 m. [ABSTRACT FROM AUTHOR]

Matrix-valued functions in the Wiener class on the imaginary line are considered in this note. This class of functions is large enough to be suitable for many applications in systems and control of infinite-dimensional systems. For this class of functions three kinds of factorization are discussed: (right-)standard factorization, canonical factorization, and J -spectral factorization. In particular, we focus on an algorithmic procedure to find a (right-)standard factorization and a J -spectral factorization for matrix-valued functions in the Wiener class under the assumption that such factorizations exists. In practice, the J -spectral factors for irrational functions are usually calculated using rational approximations. We show that approximation using rational functions may be achieved in the Wiener norm. [ABSTRACT FROM AUTHOR]

By using main properties of uniformly distributed sequences of increasing finite sets in infinite-dimensional rectangles in ℝ∞ described in [Real Anal. Exchange 36 (2010/2011), no. 2, 325-340], a new approach for an infinite-dimensional Monte-Carlo integration is introduced and the validity of some infinite-dimensional strong law type theorems are obtained in this paper. In addition, by using properties of uniformly distributed sequences in the unit interval, a new proof of Kolmogorov's strong law of large numbers is obtained which essentially differs from its original proof. [ABSTRACT FROM AUTHOR]

Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems. [ABSTRACT FROM AUTHOR]

Extensions of locally convex topological spaces are considered such that finite cylindrical measures which are not countably additive on their initial domains turn out to be countably additive on the extensions. Extensions of certain transformations of the initial spaces with respect to which the initial measures are invariant or quasi-invariant to the extensions of these spaces are described. Similar questions are considered for differentiable measures. The constructions may find applications in statistical mechanics and quantum field theory. [ABSTRACT FROM AUTHOR]

Connections between quantum anomalies and transformations of pseudomeasures of the type of Feynman pseudomeasures are studied. Mathematical objects related to the notion of the volume element in an infinite-dimensional space considered in the physics literature [1] are discussed. [ABSTRACT FROM AUTHOR]

In this article, we will formulate a mathematical framework that allows us to treat character animations as points on infinite-dimensional Hilbert manifolds. Constructing geodesic paths between animations on those manifolds allows us to derive a distance function to measure similarities of different motions. This approach is derived from the field of geometric shape analysis, where such formalisms have been used to facilitate object recognition tasks. Analogously to the idea of shape spaces, we construct motion spaces consisting of equivalence classes of animations under reparametrizations. Especially cyclic motions can be represented elegantly in this framework. We demonstrate the suitability of this approach in multiple applications in the field of computer animation. First, we show how visual artefacts in cyclic animations can be removed by applying a computationally efficient manifold projection method. We next highlight how geodesic paths can be used to calculate interpolations between various animations in a computationally stable way. Finally, we show how the same mathematical framework can be used to perform cluster analysis on large motion capture databases, which can be used for or as part of motion retrieval problems. [ABSTRACT FROM AUTHOR]

We reformulate the equation characterizing the critical points of the hypersymplectic action functional as solutions of a Hamiltonian system on the iterated loop space. The intent is to gain more insight into dynamics of hyperkähler Floer theory. [ABSTRACT FROM AUTHOR]