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Two examples illustrating properties of discrete measures are given. In the first part of the paper, it is proved that, for any probability measure $$\mu$$ with $$\operatorname{supp}{\mu}=[-1,1]$$ whose logarithmic potential is continuous on $$[-1,1]$$ , there exists a (discrete) measure $$\sigma=\sigma(\mu)$$ with $$\operatorname{supp}{\sigma}=[-1,1]$$ such that the corresponding orthogonal polynomials $$P_n(x;\sigma)=x^n+\dotsb$$ satisfy the condition $$(1/n)\chi(P_n(\,\cdot\,;\sigma))\xrightarrow{*}\mu$$ , $$n\to\infty$$ , where $$\chi(\,\cdot\,)$$ is the measure counting the zeros of a polynomial. The proof of the existence of such a measure $$\sigma$$ is based on properties of weighted Leja points. In the second part, an example of a compact set and a sequence of discrete measures supported on it with a special property is given. Namely, the sequence of measures converges in the $$*$$ -weak topology to the equilibrium measure on the compact set, but the corresponding sequence of logarithmic potentials converges in capacity to the equilibrium potential in no neighborhood of this compact set.

In this paper, we show that the sum of a compact convex subset and a simultaneously $\tau$-strongly proximinal convex subset (resp. simultaneously approximatively $\tau$-compact convex subset) of a Banach space X is simultaneously $\tau$-strongly proximinal (resp. simultaneously approximatively $\tau$-compact ), and the sum of a weakly compact convex subset and a simultaneously approximatively weakly compact convex subset of X is still simultaneously approximatively weakly compact, where $\tau$ is the norm or the weak topology. Moreover, some related results on the sum of simultaneously proximinal subspaces are presented.

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hern\'andez-Guti\'errez. We also conclude that, for any infinite dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous. We generalize some results of Hru\v{s}\'ak, Zamora Avil\'es, and Hern\'andez-Guti\'errez concerning countable dense homogeneous products., Comment: slightly modified and expanded version

Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement *X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group Γ of the locally compact Abelian group G is extended to the set of the above mentioned equivalence classes. A compact topology on is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in .

The objective of this paper is to introduce and demonstrate a robust method for multi-constrained topology optimization. The method is derived by combining the topological sensitivity with the classic augmented Lagrangian formulation. The primary advantages of the proposed method are: (1) it rests on well-established augmented Lagrangian formulation for constrained optimization, (2) the augmented topological level-set can be derived systematically for an arbitrary set of loads and constraints, and (3) the level-set can be updated efficiently. The method is illustrated through numerical experiments.

A topology is said to be strongly useful if every weakly continuous preorder admits a continuous order-preserving function. A strongly useful topology is useful, in the sense that every continuous total preorder admits a continuous utility representation. In this paper, I study the structure of strongly useful topologies. The existence of a natural one-to-one correspondence is proved, between weakly continuous preorders and equivalence classes of families of complete separable systems. In some sense, this result completely clarifies the connections between order theory and topology. Then, I characterize strongly useful topologies and I present a property concerning subspace topologies of strongly useful topological spaces.

We prove quantitative statistical stability results for a large class of small \begin{document}$ C^{0} $\end{document} perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.

We show that the Kantorovich-Rubinstein quasi-metrics d KR and d KR a of Part I extend naturally to various spaces of previsions, and in particular not just the linear previsions (roughly, measures) of Part I. There are natural isomorphisms between the Hoare and Smyth powerdomains, as used in denotational semantics, and spaces of discrete sublinear previsions, and of discrete normalized superlinear previsions, respectively. Turning to the corresponding hyperspaces, namely the same powerdomains, but equipped with the lower Vietoris and upper Vietoris topologies instead, this turns into homeomorphisms with the corresponding space of previsions, equipped with the so-called weak topology. Through these isomorphisms again, the two powerdomains inherit quasi-metrics d H and d Q , respectively, that are reminiscent of the well-known Hausdorff metric. Then we show that the Hoare and Smyth powerdomains of an algebraic complete quasi-metric space are again algebraic complete, with those quasi-metrics, and similarly that the corresponding powerdomains of continuous complete quasi-metric spaces are continuous complete. Furthermore, in the continuous complete case, the d H -Scott topology coincides with the lower Vietoris topology, and the d Q -Scott topology coincides with the upper Vietoris topology.

UDC 519.21 This paper is a survey of Skorohod differentiability of measures on linear spaces, which also gives new proofs of some key results in this area along with some new observations.

Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical Christoffel–Darboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to large-scale systems since the computational complexity is dominated by inverting an N × N Hermitian positive-definite Toeplitz matrix, where N is the number of moments, for which efficient and numerically stable algorithms exist; in particular, the complexity of the approach is independent of the dimension of the underlying state-space. We also show how to compute, from measured data, the spectral projection on a given segment of the unit circle, allowing us to obtain a finite approximation of the operator that explicitly takes into account the point and continuous parts of the spectrum. Finally, we describe a relationship between the proposed method and the so-called Hankel Dynamic Mode Decomposition, providing new insights into the behavior of the eigenvalues of the Hankel DMD operator. A number of numerical examples illustrate the approach, including a study of the spectrum of the lid-driven two-dimensional cavity flow.

In this study, we aim to present some new versions of classical Schauder and Banach fixed point theorems under weak topology in general Banach spaces. We have also another main goal which is to prove that Krasnoselskii theorem ensures in general Banach spaces using these new results. As an application, an illustrative example providing such results is given. Our results extend known results on the issue.

In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer.

In this paper, we establish sufficient conditions on the existence of mild solutions for semilinear impulsive differential inclusions by using Glicksberg-Ky Fan fixed point theorem with weak topology technique. We do not require the compactness of the evolution operator generated by the linear part and of the multivalued nonlinear term. An example is also presented.

The global well-posedness for the KP-Ⅱ equation is established in the anisotropic Sobolev space \begin{document}$ H^{s, 0} $\end{document} for \begin{document}$ s>-\frac{3}{8} $\end{document} . Even though conservation laws are invalid in the Sobolev space with negative index, we explore the asymptotic behavior of the solution by the aid of the \begin{document}$ I $\end{document} -method in which Colliander, Keel, Staffilani, Takaoka, and Tao introduced a series of modified energy terms. Moreover, a-priori estimate of the solution leads to the existence of global attractor for the weakly damped, forced KP-Ⅱ equation in the weak topology of the Sobolev space when \begin{document}$ s>-\frac{1}{8} $\end{document} .

We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Property (co) essentially says that the operation of taking convex combinations of elements of the unit ball is, in a sense, an open map. We show that if a finite dimensional Banach space $X$ has property (co), then for any scattered locally compact Hausdorff space $K$, the space $C_0(K,X)$ of continuous $X$-valued functions vanishing at infinity has property CWO. Several Banach spaces are proved to possess this geometric property; among others: 2-dimensional real spaces, finite dimensional strictly convex spaces, finite dimensional polyhedral spaces, and the complex space $\ell_1^n$. In contrast to this, we provide an example of a $3$-dimensional real Banach space $X$ for which $C_0(K,X)$ fails to have property CWO. We also show that $c_0$-sums of finite dimensional Banach spaces with property (co) have property CWO. In particular, this provides examples of such spaces outside the class of $C_0(K,X)$-spaces., Comment: 22 pages

It is well known that a finite-dimensional linear system cannot be chaotic. In this article, by introducing a weak topology into a two-dimensional Euclidean space, it shows that Li–Yorke chaos can be generated by a linear map, where the weak topology is induced by a linear functional. Some examples of linear systems are presented, some are chaotic while some others regular. Consequently, several open problems are posted.

We show that the space of subprobability measures, equivalently of subprobability continuous valuations, on an algebraic (resp., continuous) complete quasi-metric space is again algebraic (resp., continuous) and complete, when equipped with the Kantorovich-Rubinstein quasi-metrics d KR (unbounded) or d KR a (bounded), themselves asymmetric forms of the well-known Kantorovich-Rubinstein metric. We also show that the d KR -Scott and the d KR a -Scott topologies then coincide with the weak topology. We obtain similar results for spaces of probability measures, equivalently of probability continuous valuations, with the d KR a quasi-metrics, or with the d KR quasi-metric under an additional rootedness assumption.

In this paper, we study the topological structure of the Hahn field F whose elements are functions from the additive abelian group of rational numbers Q to the real numbers field R , with well-ordered support. After reviewing the algebraic and order structures of the field F , we introduce different vector topologies on F that are induced by families of semi-norms, all of which are weaker than the order or valuation topology. We compare those vector topologies and we identify the weakest one which we denote by τ w and whose properties are similar to those of the weak topology on the Levi-Civita field (Shamseddine, 2010). In particular, we state and prove a convergence criterion for power series in F , τ w that is similar to that for power series on the Levi-Civita field in its weak topology (Shamseddine, 2013).

In this manuscript, by means of the technique of measures of weak noncompactness, we establish a generalized form of fixed point theorems for a 2 x 2 block operator matrix involving multivalued maps acting on suitable Banach algebras. The results obtained are then applied to a coupled system of nonlinear integral equations.

We further investigate the weak topology generated by the irreducible unitary representations of a group $G$. A deep result due to Ernest \cite{Ernest1971} and Hughes \cite{Hughes1973} asserts that every weakly compact subset of a locally compact (LC) group $G$ is compact in the LC-topology, generalizing thereby a previous result of Glicksberg \cite{glicks1962} for abelian locally compact (LCA) groups. Here, we first survey some recent findings on the weak topology and establish some new results about the preservation of several compact-like properties when going from the weak topology to the original topology of LC groups. Among others, we deal with the preservation of countably compactness, pseudocompactness and functional boundedness., arXiv admin note: text overlap with arXiv:1704.03438