Noncompact Spaces Research Articles

On Volterra-type integral equations in noncompact metric space

In the present work, we consider one of the possible generalizations of linear and nonlinear Volterra integral equations of the second kind in the case when the independent variable belongs to an arbitrary noncompact metric space. Sufficient conditions are obtained for the existence of solutions of Volterra-type integral equations in the nonhomogeneous case. Some applications of the obtained results to the integral inequalities are given. MSC:35Q99, 35R35, 65M12, 65M70.

Conley's theorem for dispersive systems

In this article, we study Conley's theorem about the chain recurrence in dynamical systems, that is, the chain recurrent set of continuous map $f$ is the complement of union of $B_{U}(A)-A$, where $A$ is an attractor and $B_{U}(A)$ is a basin of $A$. In this paper, we generalize this theorem to dispersive systems on noncompact spaces.

New examples of non-symmetric Einstein solvmanifolds of negative Ricci curvature

We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In Tamaru (Parabolic subgroups of semisimple Lie groups and einstein solvmanifolds. Math Ann 351(1):51–66, 2011) constructs new attached solvmanifolds, which are submanifolds of the solvmanifolds correspond to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply it to associated solvmanifolds, described in (Ann Global Anal Geom 19(1):75–101, 2001), obtained by modifying the algebraic structure of the solvable Lie algebras corresponding to noncompact symmetric spaces. Our new examples are Einstein solvmanifolds with nilradicals of high nilpotency, which are geometrically distinct from noncompact symmetric spaces and their submanifolds.

The Refined BPS Index from Stable Pair Invariants

A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi-Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a C ∗ action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M-theory reduced on these Calabi-Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local P 1 . We explicitly compute refined invariants in low degree for local P 2 and local P 1 x P 1 and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula. The modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined PT invariants on appropriate geometries to Nekrasov’s partition function and a refinement of Chern-Simons theory on a lens space. We also relate our product formula to wallcrossing.

Equivariant Poincaré–Alexander–Lefschetz duality and the Cohen–Macaulay property

We prove a Poincare-Alexander-Lefschetz duality theorem for rational torus-equivariant cohomology and rational homology manifolds. We allow non-compact and non-orientable spaces. We use this to deduce certain short exact sequences in equivariant cohomology, originally due to Duflot in the differentiable case, from similar, but more general short exact sequences in equivariant homology. A crucial role is played by the Cohen-Macaulayness of relative equivariant cohomology modules arising from the orbit filtration.

Zero Temperature Limits of Gibbs States for Almost-Additive Potentials

This paper is devoted to study ergodic optimisation problems for almost-additive sequences of functions (rather than a fixed potential) defined over countable Markov shifts (that is a non-compact space). Under certain assumptions we prove that any accumulation point of a family of Gibbs equilibrium measures is a maximising measure. Applications are given in the study of the joint spectral radius and to multifractal analysis of Lyapunov exponent of non-conformal maps.

Survey on the role of accelerator modes for anomalous diffusion: the case of the standard map.

We perform an extensive and detailed analysis of the generalized diffusion processes in deterministic area preserving maps with noncompact phase space, exemplified by the standard map, with the special emphasis on understanding the anomalous diffusion arising due to the accelerator modes. The accelerator modes and their immediate neighborhood undergo ballistic transport in phase space, and also the greater vicinity of them is still much affected ("dragged") by them, giving rise to the non-Gaussian (accelerated) diffusion. The systematic approach rests upon the following applications: the GALI method to detect the regular and chaotic regions and thus to describe in detail the structure of the phase space, the description of the momentum distribution in terms of the Lévy stable distributions, and the numerical calculation of the diffusion exponent and of the corresponding diffusion constant. We use this approach to analyze in detail and systematically the standard map at all values of the kick parameter K, up to K = 70. All complex features of the anomalous diffusion are well understood in terms of the role of the accelerator modes, mainly of period 1 at large K ≥ 2π, but also of higher periods (2,3,4,...) at smaller values of K ≤ 2π.

Random-Weighted Sobolev Inequalities on $${\mathbb{R}^d }$$ R d and Application to Hermite Functions

We extend a randomization method, introduced by Shiffman–Zelditch and developed by Burq–Lebeau on compact manifolds for the Laplace operator, to the case of \({\mathbb{R}^d}\) with the harmonic oscillator. We construct measures, thanks to probability laws which satisfies the concentration of measure property, on the support of which we prove optimal-weighted Sobolev estimates on \({\mathbb{R}^d}\). This construction relies on accurate estimates on the spectral function in a non-compact configuration space. As an application, we show that there exists a basis of Hermite functions with good decay properties in \({L^{\infty}(\mathbb{R}^{d})}\), when d ≥ 2.

Curvature of the space of positive Lagrangians

A Lagrangian submanifold in an almost Calabi–Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional curvatures are non-positive. The motivation for our calculation comes from mirror symmetry. Roughly speaking, an exact isotopy class of positive Lagrangians corresponds under mirror symmetry to the space of Hermitian metrics on a holomorphic vector bundle. The latter space is an infinite-dimensional analog of the non-compact symmetric space dual to the unitary group, and thus has non-positive curvature.

The c-function expansion of a basic hypergeometric function associated to root systems

We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems. The basic hypergeometric function in question was constructed as an explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted ane root system is reduced. Its construction was extended to the nonreduced case by the author. It is a meromorphic Weyl group invariant solution of the spectral problem of the Macdonald q-dierence operators. The c-function expansion is its explicit expansion in terms of the basis of the space of meromorphic solutions of the spectral problem consisting of q-analogs of the Harish-Chandra series. We express the expansion coecients in terms of a q-analog of the Harish-Chandra c-function, which is explicitly given as product of q-Gamma functions. The c-function expansion shows that the basic hypergeometric function formally is a q-analog of the HeckmanOpdam hypergeometric function, which in turn specializes to elementary spherical functions on noncompact Riemannian symmetric spaces for special values of the parameters.

End compactifications and general compactifications

We use the insights of Robinson's nonstandard analysis as a powerful tool to extend and simplify the construction of compactifications of regular spaces. In particular, we deal with the Stone-Čech compactification and compactifications formed from topological ends. For the nonstandard extension of a metric space, the monad of a standard point x is the set of all points infinitesimally close to x. Monads of standard points can also be defined for non-metric spaces. The new points of a compactification are formed from equivalence classes of points that are not in the monad of any standard point. Adjoining such points to the original point set allows a better understanding of the relationship between the original space and the set of compactifying points. Our results for end compactifications are established for regular, connected and locally connected spaces. Simple examples of end compactifications are the two point compactification of the real line and the one point compactification of the complex plane. In general, one or more ends form the "far" termination of a non-compact space, while any "hole" in the space corresponds to an end that is "near". Our results on ends extend previous work initiated by Hans Freudenthal on ends understood as equivalence classes of nested sequences of nonempty open sets with compact boundaries. We show, for example, that a product of spaces with at least two non-compact factors has only one end. A brief overview and introduction to nonstandard analysis begins the discussion.

Local gradient estimate for harmonic functions on Finsler manifolds

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorems on noncompact Finsler manifolds with nonnegative Ricci curvature.

Stability properties of some particle filters

Under multiplicative drift and other regularity conditions, it is established that the asymptotic variance associated with a particle filter approximation of the prediction filter is bounded uniformly in time, and the nonasymptotic, relative variance associated with a particle approximation of the normalizing constant is bounded linearly in time. The conditions are demonstrated to hold for some hidden Markov models on noncompact state spaces. The particle stability results are obtained by proving $v$-norm multiplicative stability and exponential moment results for the underlying Feynman–Kac formulas.

An RS-KKM Theorem in FC-Metric Spaces with the Application to Fixed Points

In this paper, RS-KKM mappings are introduced and an RS-KKM theorem is established in noncompact complete FC-metric spaces. As applications, a Ky Fan section theorem, a maximal element theorem and a Fan-Browder type fixed point theorem are obtained in noncompact complete FC-metric spaces. These results unify, improve and generalize some known results in recent literature.

Ky Fan Matching Theorems for MFC Mappings in FC-Metric Spaces with the Application to Minimax Inequalities

In this paper, MFC mappings is introduced and Ky Fan matching theorems for MFC mappings with transfer compactly open values are established in noncompact complete FC-metric spaces. As applications, a Fan-Browder type coincidence theorem and a minimax inequality are obtained. Our results unify, improve and generalize some known results in recent literature.

On classical q-deformations of integrable σ-models

Abstract A procedure is developed for constructing deformations of integrable σ-models which are themselves classically integrable. When applied to the principal chiral model on any compact Lie group F, one recovers the Yang-Baxter σ-model introduced a few years ago by C. Klimčík. In the case of the symmetric space σ-model on F/G we obtain a new one-parameter family of integrable σ-models. The actions of these models correspond to a deformation of the target space geometry and include a torsion term. An interesting feature of the construction is the q-deformation of the symmetry corresponding to left multiplication in the original models, which becomes replaced by a classical q-deformed Poisson-Hopf algebra. Another noteworthy aspect of the deformation in the coset σ-model case is that it interpolates between a compact and a non-compact symmetric space. This is exemplified in the case of the SU(2)/U(1) coset σ-model which interpolates all the way to the SU(1, 1)/U(1) coset σ-model.

Asymptotic Poincaré compactification and finite-time singularities

We provide an extension of the method of asymptotic decompositions of vector fields with finite-time singularities by applying the central extension technique of Poincaré to the dominant part of the vector field at approach to the singularity. This leads to a bundle of fan-out asymptotic systems whose equilibria at infinity govern the dynamics of the asymptotic solutions of the original system. We show how this method can be useful to describe a single-fluid isotropic universe at the time of maximum expansion, and discuss possible relations of our results to structural stability and non-compact phase spaces.

EASY HOLOGRAPHY

It is argued that the role of infinite-dimensional asymptotic symmetry groups in gravity theories are essential for a holographic description of gravity and possibly to a resolution of the black hole information paradox. I present a simple toy model in two-dimensional hyperbolic/anti-de Sitter (AdS) space and describe, by very elementary considerations, how the asymptotic symmetry group is responsible for the entropy area law. Similar results apply also in three-dimensional AdS space. The failure of the approach in higher-dimensional AdS spaces is explained and resolved by considering other asymptotically noncompact homogeneous spaces.

The Entropy of Co-Compact Open Covers

Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space(compactness and metrizability not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: 1) it does not require the space to be compact, and thus generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems, and 2) it is an invariant of topological conjugation, compared to Bowen's entropy that is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system (R, f) defined by f(x) = 2x, the co-compact entropy is zero, while Bowen's entropy for this system is at least log 2. More general, it is found that co-compact entropy is a lower bound of Bowen's entropies, and the proof of this result generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces, too.

Global attractors for semigroup actions

In this paper the notions of global attractor and global uniform attractor are extended to the setting of general semigroup actions. We study the necessary and sufficient conditions for the existence of the global attractor for semigroups acting on noncompact metric spaces. We present conditions for the equivalence between the global attractor and global uniform attractor.