Generalization of perfectly continuous,regular set-connected and clopen functions

Summary Noiri in 1984, Dontchev, Ganster and Reilly in recent years and Reilly and Vamanamurthy in 1983 introduced the notion of perfectly continuous, regular set-connected and clopen functions, respectively. The aim of this paper is to introduce the notion of a new class of functions which is called almost clopen functions including the classes of perfectly continuous, regular set-connected and clopen functions. Furthermore, properties of almost clopen functions are obtained and relationships among almost clopenness, perfect continuity, regular set-connectedness, clopenness and almost continuity are investigated.     

On almost clopen continuity

Recently the class of clopen continuous functions between topological spaces has been generalized by the definition of the class of almost clopen continuous functions. The aim of this paper is to reconsider this second class of functions from the perspective of change of topology. Indeed, we show that the concept of almost clopen continuity coincides with the classical notion of continuity provided that suitable changes are made to the topologies of the domain and codomain of the function. We investigate some of the consequences of this situation.     

On the Hencl's notion of absolute continuity

We prove that a slight modification of the notion of α-absolute continuity introduced in [D. Bongiorno, Absolutely continuous functions in , J. Math. Anal. Appl. 303 (2005) 119–134] is equivalent to the notion of n, λ-absolute continuity given by S. Hencl in [S. Hencl, On the notions of absolute continuity for functions of several variables, Fund. Math. 173 (2002) 175–189].     

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Topologies on function spaces

Function spaces play an important role in complex analysis, in the theory of differential equations, in functional analysis and in almost every other branch of modern mathematics. In this paper we give and study the notion of clopen convergence. Also, we study the notion of clopen continuity and define new topologies on function spaces. These results generalize basic results of R. Arens, J. Dugundji and A. Di Concilio (see [1], [4], [2] and [3]).     

几种绝对连续函数的等价性

本文讨论了取值于Banach空间中函数的三种绝对连续相互等价性,得到强绝对 连续与绝对连续等价当且仅当空间是有限维的;绝对连续与弱绝对连续等价当且仅当空 间不含c0.     

On almost rc-Lindelöf sets

In [4], Dlaska introduced the class of almost rc-Lindelöf sets and studied some basic properties of such sets. In this paper, we obtain further results concerning almost rc-Lindelöf sets. We also introduce new concepts to obtain several mapping properties concerning almost rc-Lindelöf sets and almost rc-Lindelöf spaces. The property of being an almost rc-Lindelöf set is invariant under functions which are slightly continuous and weakly θ-irresolute. It is also shown that the property of being an almost rc-Lindelöf space is inverse invariant under functions which are weakly almost open, ω-regular open, and whose fibers are S-sets.     

Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(?), using “local smoothing” estimates. L ²(?) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in $L^{2}(\mathbb{T})In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(ℝ), using “local smoothing” estimates. L ²(ℝ) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in L²(\mathbbT)L^{2}(\mathbb{T}). Our results are in line with previous work on the cubic nonlinear Schr?dinger equation, where Goubet and Molinet (Nonlinear Anal. 71, 317–320, 2009) showed weak continuity in L ²(ℝ) and Molinet (Am. J. Math. 130, 635–683, 2008) showed lack of weak continuity in L²(\mathbbT)L^{2}(\mathbb{T}).     

Almost periodicity and almost automorphy for some evolution equations using Favard’s theory in uniformly convex Banach spaces

In this work, we use an approach due to Favard (Acta Math 51:31–81, 1928) to study the existence of weakly almost periodic and almost automorphic solutions for some evolution equation whose linear part generates a \(C_{0}\)-group satisfying the Favard condition in uniformly convex Banach spaces. When this \(C_{0}\)-group is bounded, which is a condition stronger than Favard’s condition, we prove the equivalence between almost automorphy and weak almost automorphy of solutions.     

A one-parametric class of merit functions for the symmetric cone complementarity problem

In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [C. Kanzow, H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Comput. Optim. Appl. 11 (1998) 227-251] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [C. Kanzow, Y. Yamashita, M. Fukushima, New NCP functions and their properties, J. Optim. Theory Appl. 97 (1997) 115-135] to the SCCP. By exploiting the Cartesian P-properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R₀₂-property. All of these results generalize some recent important works in [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005) 293-327; C.-K. Sim, J. Sun, D. Ralph, A note on the Lipschitz continuity of the gradient of the squared norm of the matrix-valued Fischer-Burmeister function, Math. Program. 107 (2006) 547-553; P. Tseng, Merit function for semidefinite complementarity problems, Math. Program. 83 (1998) 159-185] under a unified framework.     

Existence of minmax points in discontinuous strategic games

In this note we prove the existence of minmax points for strategic form games where the sets of strategies are topological spaces and the payoff functions satisfy conditions weaker than continuity. The employed tools are the class of transfer weakly upper continuous functions and the class of weakly lower pseudocontinuous functions. An example shows that our result is of minimal character.     

Maharam's problem

We solve Maharam's problem [D. Maharam, An algebraic characterization of measure algebras, Ann. Math. 48 (1947) 154–167. [3]], also known as the Control Measure Problem. We construct a non-zero exhaustive submeasure on the algebra of clopen sets of the Cantor set that is not absolutely continuous with respect to a measure. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Fixed point theorems for nonexpansive mappings and Suzuki-generalized nonexpansive mappings on a Banach lattice

We prove the existence of fixed points for multivalued nonexpansive nonself-mappings on a weakly orthogonal reflexive Banach lattice with uniformly monotone norm. Moreover, for single-valued mappings, we extend Betiuk-Pilarska and Prus’s result [A. Betiuk-Pilarska, S. Prus, Banach lattices which are order uniformly noncreasy, J. Math. Anal. Appl. 342 (2008) 1271–1279] on the weak fixed point property to continuous mappings satisfying condition (C) on a w-weakly orthogonal OUNC Banach lattice.     

Excision-preserving cubical approach to the algorithmic computation of the discrete Conley index

We introduce a new approach to the algorithmic computation of the Conley index for continuous maps. We use the technique of splitting an index pair into two layers which is inspired by the work of Mrozek, Reineck and Srzednicki [M. Mrozek, J.F. Reineck, R. Srzednicki, The Conley index over a base, Trans. Amer. Math. Soc. 352 (2000) 4171–4194]. The main advantage of our construction over the approach based directly on the one introduced by Mischaikow, Mrozek and Pilarczyk [K. Mischaikow, M. Mrozek, P. Pilarczyk, Graph approach to the computation of the homology of continuous maps, Found. Comput. Math. 5 (2005) 199–229] is that our cubical sets have the excision property. Moreover, our solution has some advantages in comparison to the approach recently proposed by Mrozek [M. Mrozek, Index pair algorithms, Found. Comput. Math. 6 (2006) 457–493].     

Stability of best rational Chebyshev approximation

In this paper we consider best Chebyshev approximation to continuous functions by generalized rational functions using an optimization theoretical approach introduced in [[5.]]. This general approach includes, in a unified way, usual, weighted, one-sided, unsymmetric, and also more general rational Chebychev approximation problems with side-conditions. We derive various continuity conditions for the optimal value, for the feasible set, and the optimal set of the corresponding optimization problem. From these results we derive conditions for the upper semicontinuity of the metric projection, which include some of the results of Werner [On the rational Tschebyscheff operator, Math. Z. 86 (1964), 317–326] and Cheney and Loeb [On the continuity of rational approximation operators, Arch. Rational Mech. Anal. 21 (1966), 391–401].     

On the history of the study of ideal class groups

This is a survey of a series of results about the class groups of algebraic number fields, with particular emphasis on two articles of Chebotarev [Eine Verallgemeinerung des Minkowski'schen Satzes mit Anwendung auf die Betrachtung der Körperidealklassen, Berichte der wissenschaftlichen Forschungsinstitute in Odessa 1(4) (1924) 17–20; Zur Gruppentheorie des Klassenkörpers, J. Reine Angew. Math. 161 (1929/30) 179–193; corrigendum, ibid. 164 (1931) 196] which seem to be almost forgotten. Their relationship to earlier work on the one hand, and to selected subsequent contributions on the other hand, is discussed. In this way, there emerges an interesting line of development, up to the present day, of results due to Kummer, Hasse, Leopoldt, Iwasawa, and others. More recent work treated here includes results by Cornell and Rosen (1981) and Lemmermeyer (2003) describing the structure of the class group under quite general conditions.     

Asymptotic relations between the Hahn-type polynomials and Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials

It has been shown in Ferreira et al. [Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials, Adv. in Appl. Math. 31(1) (2003) 61–85], López and Temme [Approximations of orthogonal polynomials in terms of Hermite polynomials, Methods Appl. Anal. 6 (1999) 131–146; The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis, J. Comput. Appl. Math. 133 (2001) 623–633] that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic relations. In Ferreira et al. [Limit relations between the Hahn polynomials and the Hermite, Laguerre and Charlier polynomials, submitted for publication] we have established new asymptotic connections between the fourth level and the two lower levels. In this paper, we continue with that program and obtain asymptotic expansions between the fourth level and the third level: we derive 16 asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials. From these expansions, we also derive three new limits between those polynomials. Some numerical experiments show the accuracy of the approximations and, in particular, the accuracy in the approximation of the zeros of those polynomials.     

Frolík decompositions for lattice-ordered groups

Frolík’s theorem says that a homeomorphism from a certain kind of topological space to itself decomposes the space into the clopen set of fixed points together with three clopen sets, each of whose images is disjoint from the original set. Stone’s theorem translates this result to a corresponding theorem about the Riesz space of continuous functions on the topological space. We prove a theorem analogous to that for Riesz spaces in the much more general setting of (possibly noncommutative) lattice-ordered groups and group-endomorphisms. The groups to which our result applies satisfy a weak condition, introduced by Abramovich and Kitover, on the polars; the images of our endomorphisms have a kind of order-density on their polars; the double polars of the images are cardinal summands; and the endomorphisms themselves are disjointness-preserving in both directions. We explain how to extend our result to larger groups to which it does not apply, and, to give additional insight, we provide many examples.     

Some subclasses of the real-valued honorary Baire two functions on ℝ n

Certain subclasses of the class of Baire one real-valued functions have very nice properties, especially concerning their points of continuity and their preservation of connectedness for many connected sets. A Gibson [weakly Gibson] is defined by the requirement that \(f(\overline{U})\subseteq\overline{f(U)}\) for every open [open connected] set U?? ⁿ . It is known that Baire one, Gibson functions are continuous, and that Baire one, weakly Gibson functions have Darboux-like properties in the sense that if U is an open connected set and \(U\subseteq S\subseteq\overline{U}\), then f(S) is an interval. Here we study the situation where the Baire one condition is replaced by honorary Baire two. Distinctly different results are found.     

Global existence of periodic solutions on a simplified BAM neural network model with delays

A simplified n-dimensional BAM neural network model with delays is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu [Wu J. Symmetric functional-differential equations and neural networks with memory. Trans Am Math Soc 1998;350:4799–838], and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [Li MY, Muldowney J. On Bendixson’s criterion. J Differ Equations 1994;106:27–39]. Finally, computer simulations are performed to illustrate the analytical results found.