Topology Of Compact Convergence Research Articles

Fundamental Solutions of the Equations of the Oscillation Theory for Anisotropic Elastic Media

The construction of fundamental solutions in R3 for the equations of harmonic vibrations in the theory of elasticity of anisotropic elastic media is carried out. Solutions are constructed in the form of multipole series. Theorems on the convergence of series in the topology of compact convergence in R3/0 are proved. The problems on constructing some singular solutions of the theory of vibrations of an anisotropic body are discussed. The fundamental solution of the oscillation equations for an isotropic medium is obtained in a closed form.

Artificial Intelligence-Assisted Design and Virtual Diagnostic for the Initial Condition of a Storage-Ring-Based Quantum Information System

Developments in Artificial Intelligence (AI) are helping to solve complex physical problems that otherwise may be too computationally demanding to solve using traditional approaches. Universal Approximation Theorems tell us that we can model any physical system if we can approximate the system with some continuous function (i.e., compact convergence topology and algorithmically generated sets of functions, such as the convolutional neural network), whether for an arbitrary depth or arbitrary width neural network. We consider the problem of solving a set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> coupled algebraic equations as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> becomes very large and apply machine learning (ML) to solve this problem for any value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> . The physical problem we are focusing on is to model the equilibrium positions of ions in an ion trap. A storage ring quantum computer could contain well over tens of thousands of ions. Quickly determining the equilibrium positions will be important to minimize the time to target and observe each ion. As each ion serves as a single qubit, this is important for setting and measuring the individual qubit states. The phonon modes from a collection of ions acts as another qubit, useful for gate operations. Measuring the phonon modes, where ions are oscillating around their respective equilibrium positions also means understanding the equilibrium positions very well. Turning all of this into a virtual diagnostic allows real time prediction and comparison to ensure unique definition of each ion.

Approximation of analytical functions by $$k$$ k -positive linear operators in the closed domain

This work treats the problem of convergence for the sequences of linear $$k$$ -positive operators on a space of functions that are analytic in a closed domain. By convergence in this space, we mean a uniform convergence in a closed domain that contains the original domain strictly inside itself, while the linear $$k$$ -positive operators are naturally associated with Faber polynomials related to the considered domain. Until now, this problem has been solved in the space of functions analytic in an open bounded domain with the topology of compact convergence.

Approximation of operators in dual spaces by adjoint operators

Let X and Y be Banach spaces, and let T : Y * → X * be a linear continuous operator. We discuss the possibility of approximation of T by adjoint operators S * : Y * → X * in the topology of compact convergence in L(Y *, X *) and in the topology of pointwise X × Y *−convergence in L(Y *, X *) and also approximation of the operator T *∣ X : X → Y ** by operators from X to Y in the corresponding topologies. Bibliography: 11 titles.

Permutation operators, entanglement entropy, and theXXZ spin chain in the limit

In this paper we develop a new approach to the investigation of the bi-partite entanglemententropy in integrable quantum spin chains. Our method employs the well-known replicatrick, thus taking a replica version of the spin chain model as starting point. At each sitei of this new model we construct an operator which acts as a cyclic permutation among then replicas of the model. Infinite products of give rise to local operators, precursors of branch-point twist fields of quantumfield theory. The entanglement entropy is then expressed in terms of correlationfunctions of such operators. Employing this approach we investigate thevon Neumann and Rényi entropies of a particularly interesting quantum stateoccurring as a limit (in a compact convergence topology) of the antiferromagneticXXZ quantum spin chain. We find that, for large sizes, the entropy scales logarithmically, butnot conformally.

On many-server queues in heavy traffic

We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided.

Closed ideals in topological algebras: A characterization of the topological Φ-algebra C k (x)

Let A be a uniformly closed and locally m-convex Φ-algebra. We obtain internal conditions on A stated in terms of its closed ideals for A to be isomorphic and homeomorphic to Ck(X), the Φ-algebra of all the real continuous functions on a normal topological space X endowed with the compact convergence topology.

Order topologies on l-algebras

We consider different topologies on an l-algebra associated in a natural way with its order, and briefly study the relationship between them. As a consequence we obtain a partial answer to the problem of the characterization of the topological algebra C(X) of real continuous functions on a topological space X, endowed with its compact convergence topology. In fact, we prove that if A is an Archimedean l-algebra and its order topology τ o is locally m-convex, bornological and complete, then there is a realcompact k r -space X such that ( A, τ o ) is isomorphic and homeomorphic with C(X) .

A characterization of the topology of compact convergence on C( X)

Let C( X) be the algebra of all K-valued continuous functions K = R or C on a normal (and T 1) topological space X, and let τ X be the topology of compact convergence on C( X). The following properties are well known: 1. (i) ( C( X), τ X ) is a locally m-convex K-algebra; 2. (ii) the set C( X): 0 ∉ ( K) is open in ( C( X), τ X ) for each compact subset K of X; 0806 118 3. (iii) the only closed ideals of ( C( X), τ X ) are the ideals of the form C( X): ( C) = 0 0806 118 (with C ⊆ X). In this paper we prove that properties (i), (ii) and (iii) characterize the topology τ X .

Expansion of entire functions into Lagrange series

This paper is devoted to the problem of representing entire functions, in spaces described by the order and the type of these functions, by Lagrange series that converge in the natural topology in these spaces; this topology is stronger than the topology of compact convergence.

Generalization of Berg-Dimovski convolution in spaces of analytic functions

In the space H(G) of functions analytic in a ρ-convex region G equipped with the topology of compact convergence, we construct a convolution for the operator Jπ+L where Jρ is the generalized Gel’fond-Leont’ev integration operator and L is a linear continuous functional on H(G). This convolution is a generalization of the well-known Berg-Dimovski convolution. We describe the commutant of the operator Jπ+L in ℋ(G) and obtain the representation of the coefficient multipliers of expansions of analytic functions in the system of Mittag-Leffler functions.

The Composition of Operator-Valued Measurable Functions is Measurable

Given separable Frechet spaces, E, F, and G, let $\mathcal {L}(E,F),\mathcal {L}(F,G)$, and $\mathcal {L}(E,G)$ denote the space of continuous linear operators from E to F , F to G, and E to G, respectively. We topologize these spaces of operators by any one of a family of topologies including the topology of pointwise convergence and the topology of compact convergence. We will show that if $(X,\mathcal {F})$ is any measurable space and both $A:X \to \mathcal {L}(E,F)$ and $B:X \to \mathcal {L}(F,G)$ are Borelian, then the operator composition $BA:X \to \mathcal {L}(E,G)$ is also Borelian. Further, we will give several consequences of this result.

The composition of operator-valued measurable functions is measurable

Given separable Frechet spaces, E, F, and G, let L ( E , F ) , L ( F , G ) \mathcal {L}(E,F),\mathcal {L}(F,G) , and L ( E , G ) \mathcal {L}(E,G) denote the space of continuous linear operators from E to F , F to G, and E to G, respectively. We topologize these spaces of operators by any one of a family of topologies including the topology of pointwise convergence and the topology of compact convergence. We will show that if ( X , F ) (X,\mathcal {F}) is any measurable space and both A : X → L ( E , F ) A:X \to \mathcal {L}(E,F) and B : X → L ( F , G ) B:X \to \mathcal {L}(F,G) are Borelian, then the operator composition B A : X → L ( E , G ) BA:X \to \mathcal {L}(E,G) is also Borelian. Further, we will give several consequences of this result.

Uniform topologies on function spaces and topologies on power sets

A hyperspace construction is shown to yield a theorem, with a one-line proof, one of the corollaries of which is that the compact-open topology coincides, on the set of all continuous functions, with the topology of compact convergence.

Strongly harmonizing operators and strongly harmonizable approximations of continuous random fields on LCA groups

First we introduce a family of strongly harmonizing operators which smooth every suitably weighted continuous random field on an LCA group G into a strongly harmonizable one. Then, by means of these operators, we prove that the set of strongly harmonizable fields whose support is compact and which admit a spectral stochastic density is dense in the set of continuous random fields on G endowed with the compact convergence topology T ( K ). Finally, new sequential approximation properties of such harmonizable fields are derived when T ( K ) is metrizable (e.g. G= R , R k or Z ).

Corrigenda

‘On the global attractivity ina generalized delay-logistic differential equation’In the proof of Theorem 3·1, the function space and its topology are incorrect, The correct function space and its topology with the related modifications are given below, which will replace the contents of the last line of page 188 and the first eight lines of page 189 of the above article.Let B denote the real linear space of all continuous functions defined on [0, ∞)topologized by a separating family of seminorms Pn, whereSince p1 ≤ p2 ≤ p3 ≤ …, the setsform a convex local base for the topology of B. The topology generated by {Vn} is also compatible with that of the metric d, whereThus B becomes a metrizable, Hausdorif, locally convex linear topological space. The strong version of the Tychonoff fixed-point theorem of S. A. Morris and E. S. Noussair (Mat. Čs. 25 (1975), 165–172) states ‘If T is a continuous mapping of a convex subset A of a Hausdorff locally convex linear topological space B into a compact subset of A then T has a fixed point’. In the following, the compactness of the image T(A) ⊂ A will follow from an application of the generalized version of Arzela's theorem found in J. R. Munkres (Topology, A first course, Prentice-Hall, 1975, p. 292). Note that the set A is not closed in B but is convex; we will show below that the image set T(A) is contained in A and the members of T(A) are pointwise bounded by (3·16) and pointwise equicontinuous, from which the sequential compactness of T(A) in the topology of compact convergence will follow by the generalized Arzela's theorem; since the real line is metrizable, such a sequential compactness implies sequential compactness in the compact-open topology of T(A); but the set T(A) is itself metrizable by d with a topology compatible with the compact-open topology. Thus the sequential compactness of T(A) in the compact-open topology implies the compactness of the set T(A) in the compact-open topology. Thus it is sufficient to show the pointwise boundedness and equicontinuity of elements of T(A).

Antisymmetry and analytic structure in the spectrum of a uniform Fr�chet algebra

Let M be a C∞-smooth submanifold of a domain G⊂Cn. Denote by 0(M) the algebra of germs of holomorphic functions on M, that is, each f∈0(M) is holomorphic in some neighbourhood of M, the neighbourhood dependent of f. Now define A(M) as the closure of 0(M) with respect to the topology of compact convergence on M.

The order topology for function lattices and realcompactness

A lattice K(X, Y) of continuous functions on space X is associated to each compactification Y of X. It is shown for K(X, Y) that the order topology is the topology of compact convergence on X if and only if X is realcompact in Y. This result is used to provide a representation of a class of vector lattices with the order topology as lattices of continuous functions with the topology of compact convergence. This class includes every C(X) and all countably universally complete function lattices with 1. It is shown that a choice of K(X, Y) endowed with a natural convergence structure serves as the convergence space completion of V with the relative uniform convergence.

Analytische Struktur in Spektren—ein Zugang über die ∞-dimensionale holomorphie

Under what conditions does the spectrum of a topological C -algebra exhibit C -analytic structure? Some sufficient conditions are known for uniform Banach algebras. In this paper, we treat the case of uniform Fréchet-nuclear algebras (=( FN)-algebras). Assume that the spectrum σA of a (FN)-algebra A is locally compact (under the usual Gelfand topology). Then σA carries, in a natural way, the structure of a (DFN)-analytic space ( DFN = strong dual of a ( FN)-space); we denote it by (σ A, A ) where A is the sheaf of germs of ( DFN)-analytic functions. The notion of ( DFN)-analytic spaces is defined in analogy to the one of Douady's Banach-analytic spaces. A ringed space (X, A ) ist called uniform if the section algebras A (U) are complete under the topology of compact convergence on U, for all open U ⊂ X.

FIA] B − Gruppen und Hypergruppen

The spaceGB of\(\bar B\)-orbits of an [FIA]B− groupG is a commutative hypergroup (=commutative convo in Jewett's terminology). Its space of “hypergroup characters” is a hypergroup itself (see [3]) which can be identified withE (G, B) (the extreme points ofB-invariant positive definite continuous functionsp:G→ℂ withp (e)=1), endowed with the topology of compact convergence. We determine the structure space ofL1 (E(G, B), μPL) whereμPL is the Plancherel measure (which is the “Haar measure” ofE(G, B)) and characterize the convolution algebrasM (E(G, B)) andL1(E(G, B)) via the inverse Fourier-Stieltjes transformation.