Compact Closure Research Articles

A sufficient condition for a discrete spectrum of the Kirchhoff plate with an infinite peak

Elliptic boundary value problems on domains which have a Lipschitz boundary and a compact closure, in particular when they generate positive self-adjoint operators, have fully discrete spectra. However, if the domain loses the Lipschitz property or compactness, other situations may occur. It is well-known that for the Dirichlet case boundedness is sufficient but not necessary for a discrete spectrum. See the famous paper by Rellich [26] or the more recent contributions by [27, 3]. On the other hand, for the Neumann problem of the Laplace operator there exist numerous examples of bounded domains such that the spectrum gets a non-empty continuous component (see e.g. [7, 17, 18, 28, 11]). The literature on the spectra for the Laplace operator with various boundary conditions is focussed on domains that have a cusp, a finite or infinite peak or horn [3, 11, 13, 8, 14, 12, 6, 4, 15] or even considered a rolled horn [28]. The criteria in [1] and [9] for the embedding H1(Ω) ⊂ L2(Ω) to be compact, show that the Neumann-Laplace problem on a domain Ω with the infinite peak

Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models

In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete “operational” approach, in which the states and measurement outcomes associated with a physical system are represented in terms of what we here call a convex operational model: a certain dual pair of ordered linear spaces–generally, not isomorphic to one another. On the other hand, state spaces for which there is such an isomorphism, which we term weakly self-dual, play an important role in reconstructions of various quantum-information theoretic protocols, including teleportation and ensemble steering. In this paper, we characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol—hence, as a form of classical probabilistic conditioning. In a large class of cases, which includes both the classical and quantum cases, the relevant compact closed categories are degenerate, in the weak sense that every object is its own dual. We characterize the dagger-compactness of such a category (with respect to the natural adjoint) in terms of the existence, for each system, of a symmetric bipartite state, the associated conditioning map of which is an isomorphism.

The truncated K-moment problem for closure of open sets

We consider the truncated K-moment problem when K is the closure of a, not necessarily bounded, open set. We completely characterize the interior of the convex cone of finite sequences that have a representing measure on K. It is the domain of the Legendre–Fenchel transform associated with a certain convex function. And so in this context, detecting whether a sequence is in the interior of this cone reduces to solving a finite-dimensional convex optimization problem. This latter problem is related to maximum-entropy methods for approximating an unknown density from knowing only finitely many of its moments. The proposed approach is essentially geometric and of independent interest, as it also addresses the abstract problem of characterizing the interior of a convex cone C which is the conical hull of a set continuously parametrized by a compact closure of an open set.

Variations on ω-boundedness

Let Open image in new window be a property (or, equivalently, a class) of topological spaces. A space X is called Open image in new window-bounded if every subspace of X with (or in) Open image in new window has compact closure. Thus, countable-bounded has been known as ω-bounded and (σ-compact)-bounded as strongly ω-bounded.

Finite Temperature Formalism for Composite Quantum Particles

This Letter provides the missing part of the newly constructed many-body formalism for composite quantum particles: the introduction of a finite temperature. The finite T formalism we propose deeply relies on the existence of a compact closure relation for the (overcomplete) set of N-composite-particle states. As a first application, we here calculate the energy mean value of the exciton gas outside the condensation regime. We show that carrier exchanges increase its temperature dependence compared to elementary bosons, a signature of the degree-of-freedom increase resulting from the particle composite nature.

Remarks on some basic concepts in the KKM theory

In the KKM theory, some authors adopt the concepts of the compact closure (ccl), compact interior (cint), transfer compactly closed-valued multimap, transfer compactly l.s.c. multimap, and transfer compactly local intersection property, respectively, instead of the closure, interior, closed-valued multimap, l.s.c. multimap, and possession of a finite open cover property. In this paper, we show that such adoption is inappropriate and artificial. In fact, any theorem with a term with “transfer” attached is equivalent to the corresponding one without “transfer”. Moreover, we can invalidate terms with “compactly” attached by giving a finer topology on the underlying space. In such ways, we obtain simpler formulations of KKM type theorems, Fan–Browder type fixed point theorems, and other results in the KKM theory on abstract convex spaces.

On the global attractivity of monotone random dynamical systems

Suppose that ( θ , φ ) (\theta ,\varphi ) is a monotone (order-preserving) random dynamical system (RDS for short) with state space V V , where V V is a real separable Banach space with a normal solid minihedral cone V + V_{+} . It is proved that the unique equilibrium of ( θ , φ ) (\theta ,\varphi ) is globally attractive if every pull-back trajectory has compact closure in V V .

On the solvability of vector fields with real linear coefficients

The following result is proved: for a vector field with real linear coefficients to be locally solvable in R n \mathbb {R}^{n} it is necessary and sufficient that not all its orbits have a compact closure in the complement of the critical set of the vector field.

Certain sequences with compact closure

This paper deals with a question which is stated by quite simple definitions. A sequence { x n } in a space X is called a β-sequence if every subsequence of it has a cluster point in X. The closure of the sequence { x n } means the closure of { x n : n ∈ ω } in X. Here we consider the question when a β-sequence has compact closure. We give several answers to this question.

K*-Metrizable spaces and their applications

In this paper, we introduce and study a new class of generalized metric spaces, which we call k*-metrizable spaces, and suggest various applications of such spaces in topological algebra, functional analysis, and measure theory. By definition, a Hausdorff topological space X is k*-metrizable if X is the image of a metrizable space M under a continuous map f: M → X which has a section s: X → M preserving precompact sets in the sense that the image s(K) of any compact set K ⊂ X has compact closure in X.

Axiomatic Description of Mixed States From Selinger's CPM-construction

We recast Selinger's CPM-construction of mixed states completely positive maps [P. Selinger (2007) Dagger compact closed categories and completely positive maps. In: Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), Electronic Notes in Theoretical Computer Science 170, 139–163. Available from Peter Selinger's homepage] as an axiomatization of maximally mixed states. This axiomatization also guarantees categories of completely positive maps to satisfy the preparation-state agreement axiom of [Coecke, B. (2007) De-linearizing linearity: projective quantum axiomatics from strong compact closure. In: Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), Electronic Notes in Theoretical Computer Science 170, 49–72. E-print: arXiv:quant-ph/0506134], and admits a physical interpretation in terms of purification of mixed states and CPMs. Internal traces, which are crucial in quantum information theory, are the adjoints to these maximally mixed states.

Sharp pointwise estimates for solutions of strongly elliptic second order systems with boundary data from L p

The strongly elliptic system with constant m × m matrix-valued coefficients for a vector-valued functions u = (u 1, … , u m ) in the half-space as well as in a domain with smooth boundary ∂ Ω and compact closure is considered. A representation for the sharp constant in the inequality is obtained, where |·| is the length of a vector in the m-dimensional Euclidean space, , and ‖·‖ p is the L p -norm of the modulus of an m-component vector-valued function, 1 ≤p ≤∞. It is shown that where is a point at ∂ Ω nearest to x ∈ Ω, u is the solution of Dirichlet problem in Ω for the strongly elliptic system with boundary data from , and is the sharp constant in the aforementioned inequality for u in the tangent space to ∂ Ω at . As examples, Lamé and Stokes systems are considered. For instance, in the case of the Stokes system, the explicit formula is derived, where 1 < p < ∞.

Elastodynamics in domains with edges

Let G be a domain in \(\mathbb R^m\) with compact closure \(\overline {G}\), m ≥ 2. The boundary ∂G may contain nonintersecting closed smooth “edges” of various dimensions. In the cylinder {(x, t) : x ∈ G, −∞ < t < +∞}, we study the equations of elastodynamics. The Dirichlet condition (displacements) or the Neumann condition (stresses) is given on the boundary \(\partial G\times \mathbb R\) of the cylinder. The goal is to describe the behavior of solutions near edges.

De-linearizing Linearity: Projective Quantum Axiomatics From Strong Compact Closure

Elaborating on our joint work with Abramsky in [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124] we further unravel the linear structure of Hilbert spaces into several constituents. Some prove to be very crucial for particular features of quantum theory while others obstruct the passage to a formalism which is not saturated with physically insignificant global phases.First we show that the bulk of the required linear structure is purely multiplicative, and arises from the strongly compact closed tensor which, besides providing a variety of notions such as scalars, trace, unitarity, self-adjointness and bipartite projectors [S. Abramsky, B. Coecke, B. (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130, S. Abramsky, B. Coecke, (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124], also provides Hilbert-Schmidt norm, Hilbert-Schmidt inner-product, and in particular, the preparation-state agreement axiom which enables the passage from a formalism of the vector space kind to a rather projective one, as it was intended in the (in)famous Birkhoff & von Neumann paper [G. Birkhoff, J. von Neumann, (1936) The logic of quantum mechanics. Annals of Mathematics 37, 823–843].Next we consider additive types which distribute over the tensor, from which measurements can be build, and the correctness proofs of the protocols discussed in [S. Abramsky, B. Coecke, (2004) A categorical semantics of quantum protocols. Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. Extended version including proofs at arXiv:quant-ph/0402130] carry over to the resulting weaker setting. A full probabilistic calculus is obtained when the trace is moreover linear and satisfies the diagonal axiom, which brings us to a second main result, characterization of the necessary and sufficient additive structure of a both qualitatively and quantitatively effective categorical quantum formalism without redundant global phases. Along the way we show that if in a category a (additive) monoidal tensor distributes over a strongly compact closed tensor, then this category is always enriched in commutative monoids.

Strongly Compact Closed Semantics

These lecture notes survey some joint work with Samson Abramsky as it was presented by me at Mathematical Foundations of Programming Semantics XXI in Birmingham (2005).22I only present part of that talk since some material is already available elsewhere [15]. It concerns a categorical semantics for quantum mechanics introduced in [Abramsky, S. and Coecke, B. (2004) A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS'04), IEEE Computer Science Press. An extended & improved version is available at http://arXiv.org/quant-ph/0402130; Abramsky, S. and Coecke, B. (2005) Abstract physical traces. Theory and Applications of Categories 14, 111–124. Available at www.tac.mta.ca/tac/volumes/14/6/14-06abs.html]. We present this semantics with a particular focus on the connection between categorical diagrams and certain pictures involving typed squares, triangles, diamonds and lines. Along the way we unravel the structural components which come with strong compact closure. We provide pointers to related literature. Keywords: Strong compact closure, semantics, quantum mechanics, quantum logic, quantum informatics.

Theory of Newtonian self-gravitating stationary spherically symmetric systems

We investigate spherically symmetric equilibrium states of the Vlasov–Poisson system, relevant in galactic dynamics. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a region with compact closure. Based on a dynamical systems analysis we derive theorems that guarantee that the steady state solutions have finite mass and compact support.

Dirichlet problems for harmonic maps from regular domains

Given an open set $\Omega$ of compact closure in $\mathbb{R}^m$, the classical Dirichlet problem is to extend a given continuous function $\psi : \partial \Omega \to \mathbb{R}$ to a continuous function $\phi : \overline \Omega \to \mathbb{R}$ such that $\phi$ is harmonic (that is, satisfies the Laplace equation) in $\Omega$. The set $\Omega$ is termed regular if the Dirichlet problem has a (necessarily unique) solution for any continuous boundary function $\psi$. For example, every simply connected planar domain is regular (but may have a 'bad' boundary $\partial\Omega$, for instance, a fractal). In this article it is shown that, when $\Omega$ is regular (in the above sense), every continuous map $\psi$ from $\partial\Omega$ to a simply connected complete Riemannian manifold $(N, h)$ of sectional curvature at most 0 has a unique continuous extension $\phi : \overline\Omega \to N$ which is harmonic in $\Omega$. This is done with $\mathbb{R}^m$ replaced more generally by an $m$-dimensional Riemannian manifold $(M, g)$. The proof relies on the unique solvability of the corresponding variational Dirichlet problem (for any open set $\Omega \Subset M$). And for that, the above target manifold $N$ can be replaced more generally by any simply connected complete geodesic space $Y$ of curvature at most 0 in the sense of A. D. Alexandrov. Assuming that $M$ satisfies the Poincare inequality, we show that, for any map $\psi : M \to Y$ of finite energy in the sense of N. J. Korevaar and R. M. Schoen, there exists a unique map $\phi : M \to Y$ with $\phi = \psi$ on $M \setminus \Omega$ such that $\phi$ minimizes the energy of all maps $M \to Y$ which agree with $\psi$ on $M \setminus \Omega$. If $\Omega$ is regular then $\phi$ is continuous at any point of $\partial\Omega$ at which $\psi$ is continuous. For a Lipschitz (and hence regular) domain $\Omega \Subset M$, existence and uniqueness of the variational solution $\phi$ was obtained by Korevaar and Schoen, and earlier for suitable polyhedral targets $Y$ by Gromov and Schoen. Instead of the Riemannian manifold $M$ our domain space can still more generally be an admissible Riemannian polyhedron (as studied in the recent Cambridge Tract by J. Eells and the present author); the variational solution $\phi$ is then in general only Holder continuous in $\Omega$. The proofs of the stated results of this article rely in part on potential theory relative to the fine topology of H. Cartan.

Some isoperimetric comparison theorems for convex bodies in Riemannian

We prove that the isoperimetric profile of a convex domain Ω with compact closure in a Riemannian manifold (M n+1 , g) satisfies a second-order differential inequality that only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of Q. Regularity properties of the profile and topological consequences on isoperimetric regions arise naturally from this differential point of view. Moreover, by integrating the differential inequality, we obtain sharp comparison theorems: not only can we derive an inequality that should be compared with Levy-Gromov Inequality but we also show that if Ric > n6 on Q, then the profile of Q is bounded from above by the profile of the half-space H n+1 δ in the simply connected space form with constant sectional curvature δ. As a consequence of isoperimetric comparisons we obtain geometric estimations for the volume and the diameter of Ω, and for the first non-zero Neumann eigenvalue for the Laplace operator on Ω.

Déformation localisée de surfaces de Riemann

Let Y be a Riemann surface with compact boundary embedded into a hyperbolic Riemann surface of finite type X. It is proved that the space of deformations D of Y into X is an open subset of the Teichmuller space T (X) of X. Furthermore, D has compact closure if and only if Y is simply connected or isomorphic to a punctured disk, and D = T (X) if and only if the components of X Y are all disks or punctured disks.

Two classes of locally compact sober spaces

We deal with two classes of locally compact sober spaces, namely, the class of locally spectral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSN‐spaces, for short). We prove that locally spectral coherent spaces are precisely the coherent sober spaces with a basis of compact open sets. We also prove that CSN‐spaces are exactly the locally spectral coherent spaces in which every compact open set has a compact closure.