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In this work, we give three kinds of criteria for weak sets in Orlicz–Bochner sequence spaces l(Φ)(X)$l_{(\Phi)}(X)$ without constraints, conditions posited in each criterion are necessary and sufficient. As an application, we give criteria for weak sets in Orlicz sequence spaces. Well‐known conclusions are exhibited once more, such as Schur's theorem, Banach–Alaoglu's theorem, and the boundedly compact principle of finite dimension space. The results obtained show that the weak compactness may not be extrapolated straightforwardly from X$X$ to l(Φ)(X)$l_{(\Phi)}(X)$, for example, l∞(X)$l_{\infty }(X)$. [ABSTRACT FROM AUTHOR]

In this paper, we compare the Stone–Čech compactification of the space of Radon probability measures on a Tychonoff space , equipped with the weak topology, with the space of Radon probability measures on the Stone–Čech compactification of the space . It is shown that for any noncompact metric space , the compactification does not coincide with . We discuss the case of more general Tychonoff spaces and also the case of the Samuel compactification, for which the coincidence holds. [ABSTRACT FROM AUTHOR]

We establish two new criteria for the closedness of Archimedean cones in countable-dimensional locally convex spaces in terms of projective parallelotopes and projective automorphisms. We also answer some open questions about quasidenseness and quasi-interior. [ABSTRACT FROM AUTHOR]

In this work, we give some criteria of the weakly compact sets and a representation theorem of Riesz's type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function. Finally, as an immediate consequence of the criteria considered in this paper, the criteria of the weakly compact sets of Orlicz sequence spaces are deduced. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the continuity of positive periodic solutions for Lotka–Volterra equations in coefficient functions. It will be proven that these positive periodic solutions are continuous in coefficient functions with respect to weak topologies. [ABSTRACT FROM AUTHOR]

We provide an exhaustive description of the class of locally convex spaces in which all Archimedean cones are closed. We introduce the notion of quasidense set and prove that the above class consists of all finite-dimensional and countable-dimensional spaces whose topological dual is quasidense in the algebraic dual of . [ABSTRACT FROM AUTHOR]

When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study the possible behaviors of persistence, gain, or loss of Krylov solvability under suitable small perturbations of the infinite-dimensional inverse problem – the underlying motivations being the stability or instability of infinite-dimensional Krylov methods under small noise or uncertainties, as well as the possibility to decide a priori whether an infinite-dimensional inverse problem is Krylov solvable by investigating a potentially easier, perturbed problem. [ABSTRACT FROM AUTHOR]

Let E and F be Banach spaces. We present some results about the topology induced on E by the strong operator topology on the space L (L (E , F) , F) of continuous linear operators from L (E , F) into F , and study several classes of homogeneous polynomials from E into F enjoying different types of w EF continuity. [ABSTRACT FROM AUTHOR]

In this paper, we present a simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. We introduce the concept of weak open and closed countable chain condition (WOCCC) relative to a topology, and we then show that a useful topology always satisfies this condition. The most important result in the paper shows that a completely regular topology is useful if and only if it is separable and it satisfies WFOCCC (a stricter version of WOCCC). In this way, we generalize all the previous results concerning useful topologies. We finish the paper by presenting a simple axiomatization of useful topologies under the well-known Souslin Hypothesis. [ABSTRACT FROM AUTHOR]

In this paper, we use a new technique for the treatment of systems based on the advantage of vector-valued norms and of the weak topology. We first present vector versions of the Leray-Schauder alternative and then some Krasnoselskii type fixed point theorems for a sum of two mappings. Applications are given to a system of nonlinear transport equations, and systems of mixed fractional differential equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semigroup, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution implies the existence of a periodic solution. Several results regarding uniqueness and global attractiveness of periodic solutions are also established. The analysis relies on a fixed point theorem of Chow and Hale's type and uses some arguments of weak topology. Our theorems extend in a broad sense some new and classical related results. An application to a transport equation with delay is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we present a new simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. In particular, we show that, for completely regular spaces, a topology is useful, if and only if it is separable, and every isolated chain of open and closed sets is countable. As a specific application to optimization theory, we characterize the continuous representability of all continuous total preorders, which admit both a maximal and a minimal element. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the spectral problem ( y 1 y 2 ) x = ( − 1 2 1 2 λ m − 1 2 λ m 1 2 ) ( y 1 y 2 ) for the periodic generalized Camassa-Holm equations. The first aim is to study the structure of eigenvalues and prove the continuous dependence of eigenvalues on potentials. We prove that as nonlinear functionals of potentials, eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L 1 0 , T. The second aim is to find the estimates of the eigenvalues when the L 1 norm of potentials is given, based on an application of trace formulas. [ABSTRACT FROM AUTHOR]

This paper deals with a new existence theory for periodic solutions to a broad class of evolution equations. We first establish new fixed point theorems for affine maps in locally convex spaces and ordered Banach spaces. Our new fixed point results extend, encompass and complement a number of well-known theorems in the literature, including the famous Chow and Hale fixed point theorem. With these obtained fixed point results, we investigate the existence of periodic solutions for some class of nonhomogeneous linear systems in Banach spaces with lack of compactness. Some illustrative examples are also given. [ABSTRACT FROM AUTHOR]

This paper is concerned with weak pullback mean random attractors for mean random dynamical systems defined in Bochner spaces. We first introduce the concept of weak pullback mean random attractor with respect to the weak topology of reflexive Bochner spaces and then provide a sufficient criterion for existence and uniqueness of such attractors over a complete filtered probability space. As an application, we prove the existence and uniqueness of weak pullback mean random attractors for the stochastic reaction–diffusion equations with nonlinear drift terms as well as nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic reaction–diffusion equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time. [ABSTRACT FROM AUTHOR]

In this paper we show that, in contrast to what happens in a normed space, the weak topology of a finite dimensional asymmetric normed space is not necessarily the same as the topology of the asymmetric norm. We provide a class of finite dimensional asymmetric normed spaces where both topologies coincide. We also prove that the weak topology of an infinite dimensional asymmetric normed space is strictly coarser than the topology of the asymmetric norm. [ABSTRACT FROM AUTHOR]

In this paper, using the technique of measure of weak noncompactness and by removing the convexity hypothesis, we prove some fixed set results for the sum and the product of three multivalued mappings AB + C, acting on Banach algebras satisfying a certain sequential condition in the weak topology setting. In addition, by using a new definition of the multivalued mapping , we get new fixed set theorems for the mappings of the form under some suitable conditions on the operators A, B, and C in Banach algebras. These hybrid theorems improve some recently obtained results by Ben Amar et al. We also apply these results to the theory of self-similarity. [ABSTRACT FROM AUTHOR]

We investigate a probabilistic model for routeing of messages in relay-augmented multihop ad-hoc networks, where each transmitter sends one message to the origin. Given the (random) transmitter locations, we weight the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured in terms of the number of pairs of hops using the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of an optimization of the individual trajectories. In the limit of high spatial density of users, we describe the totality of all the message trajectories in terms of empirical measures. Employing large deviations arguments, we derive a characteristic variational formula for the limiting free energy and analyse the minimizer(s) of the formula, which describe the most likely shapes of the trajectory flow. The empirical measures of the message trajectories well describe the interference, but not the congestion; the latter requires introducing an additional empirical measure. Our results remain valid under replacing the two penalization terms by more general functionals of these two empirical measures. [ABSTRACT FROM AUTHOR]

We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of ℓ 1 . For that purpose, we transfer to general locally compact groups the notion of interpolation ( I 0 ) set, which was defined by Hartman and Ryll-Nardzewsky [24] for locally compact abelian groups. Thus we prove that for every sequence { g n } n < ω in a locally compact group G , then either { g n } n < ω has a weak Cauchy subsequence or contains a subsequence that is an I 0 set. This result is subsequently applied to obtain sufficient conditions for the existence of Sidon sets in a locally compact group G , an old question that remains open since 1974 (see [31] and [19] ). Finally, we show that every locally compact group strongly respects compactness extending thereby a result by Comfort, Trigos-Arrieta, and Wu [13] , who established this property for abelian locally compact groups. [ABSTRACT FROM AUTHOR]

In this paper we discuss some topological properties of the set F(S0, T,z):= {x ∊ X : S0 x ∊ Tx + zg, where T is a nonlinear multi-valued mappings and S0 is a single-valued mappings acting on a Banach space X. This study is based on a new concept, the so called weakly relative demicompactness for nonlinear operators. [ABSTRACT FROM AUTHOR]

A topology τ on a nonempty set X is called a clopen topology provided each member of τ is both open and closed. Given a function f from X to Y, the operator E 7→ f-1( f (E)) is a closure operator on the power set of X whose fixed points are closed subsets corresponding to a clopen topology on X. Conversely, for each clopen topology τ on X, we produce a function f with domain X such that τ = {E ⊆ X : E = f-1( f (E))}. We characterize the clopen topologies on X as those that are weak topologies determined by a surjective function with values in some discrete topological space. Paralleling this result, we show that a topology admits a clopen base if and only if it is a weak topology determined by a family of functions with values in discrete spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the topological structure of solution sets for stochastic evolution inclusions in Hilbert spaces when the semigroup is compact as well as noncompact. It is shown that the solution set is nonempty, compact, and anRδ-set, which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As applications of the obtained results, an example is given. [ABSTRACT FROM PUBLISHER]

For a compact space K we denote by C w ( K ) ( C p ( K ) ) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we discuss the following basic question which seems to be open: Let K and L be infinite compact spaces. Can it happen that C w ( K ) and C p ( L ) are homeomorphic? M. Krupski proved that the above problem has a negative answer when K = L and K is finite-dimensional and metrizable. We extend this result to the class of finite-dimensional Valdivia compact spaces K . [ABSTRACT FROM AUTHOR]

Predictive equivalence in discrete stochastic processes has been applied with great success to identify randomness and structure in statistical physics and chaotic dynamical systems and to inferring hidden Markov models. We examine the conditions under which predictive states can be reliably reconstructed from time-series data, showing that convergence of predictive states can be achieved from empirical samples in the weak topology of measures. Moreover, predictive states may be represented in Hilbert spaces that replicate the weak topology. We mathematically explain how these representations are particularly beneficial when reconstructing high-memory processes and connect them to reproducing kernel Hilbert spaces. • Predictive states reveal internal structure and information-generating mechanisms. • Measure-theoretic and topological foundations of predictive states. • Predictive states can be convergently estimated from empirical data. • Example analyses for finite-memory Markov and hidden Markov processes. • Analyses of infinite-memory hidden semi-Markov and indexed context-free processes. [ABSTRACT FROM AUTHOR]

In this paper we prove that the eigenvalues of a vibrating beam have a strongly continuous dependence on the elastic destructive force, i.e., the eigenvalues, as nonlinear functionals of the elastic destructive force, are continuous in the elastic destructive force with respect to the weak topologies in the Lebesgue spaces L p . In virtue of the minimax characterization for eigenvalues, we prove first the continuity of the lowest eigenvalue and then all the eigenvalues by the induction principle. [ABSTRACT FROM AUTHOR]

In this paper, using the technique of measure of weak noncompacteness, we prove some fixed set results of Krasnosel'skii type for the sum of two multivalued operators in the setting of weak topology, without the assumption of the convexity of their common domain. Applications to the theory of self-similarity are also given. [ABSTRACT FROM AUTHOR]

In this paper we will prove that the eigenvalues of nonhomogeneous hinged vibrating rods have a strongly continuous dependence on weights, i.e., as nonlinear functionals of weights, eigenvalues are continuous in weights with respect to the weak topologies in the Lebesgue spaces L p . [ABSTRACT FROM AUTHOR]

The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach-Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of -convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179-182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and -spaces, but not in , . -convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of -convergence with the Brezis-Lieb lemma and gives a version of the latter without an assumption of convergence a.e. [ABSTRACT FROM AUTHOR]

We establish a compactness result in approach theory which we apply to obtain a generalization of Prokhorov's Theorem for the continuity approach structure. [ABSTRACT FROM AUTHOR]

In this paper, we will prove some fixed point theorems for the sum and the product of nonlinear weakly sequentially continuous operators acting on a WC-Banach algebra. Our results improve and correct some recent results given by Banas and Taoudi, and extend several earlier works using the condition (P). [ABSTRACT FROM AUTHOR]

A definition of qualitative robustness for point estimators in general statistical models is proposed. Some criteria for robustness are established and applied to estimators in parametric, semiparametric, and nonparametric models. In specific nonparametric models, the proposed definition boils down to Hampel robustness. It is also explained how plug-in estimators in certain nonparametric models can be reasonably classified w.r.t. their degrees of robustness. [ABSTRACT FROM AUTHOR]

We describe the topology of any cosmic space and any ℵ 0 -space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ ( E , E ′ ) are ℵ 0 -spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) ℓ 1 and satisfying the Heinrich density condition we prove that ( E , σ ( E , E ′ ) ) is an ℵ 0 -space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain ℓ 1 , then ( E , σ ( E , E ′ ) ) is an ℵ 0 -space if and only if E ′ is separable. The last part of the paper studies the question: Which spaces ( E , σ ( E , E ′ ) ) are ℵ 0 -spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a ( DF )-space whose strong dual E ′ is separable, then ( E , σ ( E , E ′ ) ) is an ℵ 0 -space. Supplementing an old result of Corson we show that, for a Čech-complete Lindelöf space X the following are equivalent: (a) X is Polish, (b) C c ( X ) is cosmic in the weak topology, (c) the weak ⁎ -dual of C c ( X ) is an ℵ 0 -space. [ABSTRACT FROM AUTHOR]

Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a sublattice that is lattice isomorphic to the space of transition kernels. As an application we present a purely analytic proof of Doob's theorem concerning stability of transition semigroups. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with existence results for a nonlinear Hammerstein integral equation inL1spaces. Making use of the De Blasi measure of weak noncompactness, we establish variants of some Krasnosel'skii type theorem involving the weak topology of Banach spaces. [ABSTRACT FROM AUTHOR]

We further investigate the weak topology generated by the irreducible unitary representations of a group G. A deep result due to Ernest [13] and Hughes [22] 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 [19] 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 countable compactness, pseudocompactness and functional boundedness. [ABSTRACT FROM AUTHOR]

Abstract: A method is proposed to prove the global attractor existence for multivalued semiflows with weak continuity properties. An application to the reaction–diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented. [Copyright &y& Elsevier]

Abstract: M. Talagrand showed that, for the Čech–Stone compactification βω of the space of natural numbers ω, the norm and the weak topology generate different Borel structures in the Banach space . We prove that the Borel structures in generated by the weak and the pointwise topology are also different. We also show that in , where , there is no countable family of pointwise Borel sets separating functions from . [Copyright &y& Elsevier]

Abstract: We use Stein’s method to prove a generalization of the Lindeberg–Feller CLT providing an upper and a lower bound for the superior limit of the Kolmogorov distance between a normally distributed random variable and the rowwise sums of a rowwise independent triangular array of random variables which is asymptotically negligible in the sense of Feller. A natural example shows that the upper bound is of optimal order. The lower bound improves a result by Andrew Barbour and Peter Hall. [Copyright &y& Elsevier]

In 1979, Trotman proved, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is (α)-regular. Here we first discuss the Thom transversality theorem for the weak topology and then give a similiar kind of result for the weak topology under very weak hypotheses. Recently, several transversality theorems have been proved for complex manifolds and holomorphic maps. In view of these transversality theorems we also prove a result analogous to Trotman's result in the complex case. [ABSTRACT FROM AUTHOR]

In this paper we consider some open questions concerning ?- convergence in complete CAT(0) metric spaces (i.e. Hadamard spaces). Suppose (X, d) is a Hadamard space such that the sets {z ? X| d(x, z) = d(z, y)} are convex for each x, y ? X. We introduce a so-called half-space topology such that convergence in this topology is equivalent to ?-convergence for any sequence in X. For a major class of Hadamard spaces, our results answer positively open questions nos. 1, 2 and 3 in [W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008) 3689- 3696]. Moreover, we give a new characterization of ?-convergence and a new topology that we call the weak topology via a concept of a dual metric space. The relations between these topologies and the topology which is induced by the distance function have been studied. The paper concludes with some examples. [ABSTRACT FROM AUTHOR]

We give a counterexample to Theorem 3.2 in Toumi and Toumi [The range of an orthomorphism, J. Algebra Appl. 8 (2009) 863-868]. In the process we add a new equivalent statement to those in Theorem 7.48 of Aliprantis and Burkinshaw [ Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, Vol. 105 (American Mathematical Society, 2003)]. [ABSTRACT FROM AUTHOR]

Abstract: For an integrable potential q on the unit interval, let be the zeroth Neumann eigenvalue of the Sturm–Liouville operator with the potential q. In this paper we will solve the minimization problem , where potentials q have mean value zero and norm r. The final result is . The approach is a combination of variational method and limiting process, with the help of continuity results of solutions and eigenvalues of linear equations in potentials and in measures with weak topologies. These extremal values can yield optimal estimates on the zeroth Neumann eigenvalues. [Copyright &y& Elsevier]

Abstract: In this paper we reconsider the basic topological and metric structures on spaces of probability measures and random variables, such as e.g. the weak topology and the total variation metric, replacing them with more intrinsic and richer approach structures. We comprehensibly investigate the relationships among, and basic facts about these structures, and prove that fundamental results, such as e.g. the portmanteau theorem and Prokhorovʼs theorem, can be recaptured in a considerably stronger form in the new setting. [Copyright &y& Elsevier]