Excesses, duality gaps and weak compactness

We explore the connection between the concepts ``excess' and ``duality gap' from epigraphical analysis and optimization, and the functional analytic concepts of weak* and weak compactness. We also discuss briefly the connection with R. C. James's ``sup theorem'.

     

The quantitative difference between countable compactness and compactness

We establish here some inequalities between distances of pointwise bounded subsets H of RX to the space of real-valued continuous functions C(X) that allow us to examine the quantitative difference between (pointwise) countable compactness and compactness of H relative to C(X). We prove, amongst other things, that if X is a countably K-determined space the worst distance of the pointwise closure of H to C(X) is at most 5 times the worst distance of the sets of cluster points of sequences in H to C(X): here distance refers to the metric of uniform convergence in RX. We study the quantitative behavior of sequences in H approximating points in . As a particular case we obtain the results known about angelicity for these Cp(X) spaces obtained by Orihuela. We indeed prove our results for spaces C(X,Z) (hence for Banach-valued functions) and we give examples that show when our estimates are sharp.     

Well posedness of nonlinear parabolic systems beyond duality

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial–boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system${?}_{t}u?\mathrm{div}(\nu (|\mathrm{?}u|)\mathrm{?}u)=?\mathrm{div}f$ with a given strictly positive bounded function ν, such that ${\lim }_{k\to \infty }?\nu (k)={\nu }_{\infty }$ and $f\in L^q$ with $q\in (1,\infty )$. The existence, uniqueness and regularity results for $q\ge 2$ are by now standard. However, even if a priori estimates are available, the existence in case $q\in (1,2)$ was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range $q\in (1,\infty )$.Furthermore, our paper includes several new results that may be of independent interest and serve as the starting point for further analysis of more complicated problems. They include a parabolic Lipschitz approximation method in weighted spaces with fine control of the time derivative and a theory for linear parabolic systems with right hand sides belonging to Muckenhoupt weighted $L^q$ spaces.     

Duality in mathematical programming of set functions: On Fenchel duality theorem

The optimization problem of a set function defined on a family a′ of measurable subsets in an atomless finite measure space (X, a, m) is investigated. The generalized Fenchel theorem is formulated and proved in this note.     

The duality between algebras and coalgebras

In this paper we introduce the category of Yang-Baxter structures. We give examples of objects in this category. We construct full and faithful embbedings from the categories of algebra and coalgebra structures to the category of Yang-Baxter structures. Then we give a new duality theorem which extends the duality between finite dimensional algebras and coalgebras.

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Sunto In questo articolo introduciamo la categoria delle strutture di Yang-Baxter. Diamo esempi di oggetti in questa categoria. Costruiamo immersioni piene e fedeli dalle categorie di algebre e coalgebre alla categoria di strutture di Yang-Baxter. Infine diamo un nuovo teorema di dualità che estende la dualità tra algebre e coalgebre di dimensione finita.

     

Fuzzy measures and discreteness

The main result is to show that the space of nonmonotonic fuzzy measures on a measurable space (X,X) with total variation norm is separable if and only if the σ-algebra X is a finite set. Our result is related to fuzzy analysis, functional spaces and discrete mathematics.     

模与余模间的对偶

本文讨论模的对偶余模，把代数的（有限）对偶余代数的有关结论完整地推广到模的对偶余模上．作为这个理论的应用，我们给出一个例子说明Ｈ－模代数Ａ的对偶Ａ°不一定是Ｈ°－余模余代数     

The Fundamental Relations between Geometric Programming Duality, Parametric Programming Duality, and Ordinary Lagrangian Duality

Every formulation of mathematical programming duality (known to the author) for continuous finite-dimensional optimization can easily be viewed as a special case of at least one of the following three formulations: the geometric programming formulation (of the generalized geometric programming type), the parametric programming formulation (of the generalized Rockafellar-perturbation type), and the ordinary Lagrangian formulation (of the generalized Falk type). The relative strengths and weaknesses of these three duality formulations are described herein, as are the fundamental relations between them. As a theoretical application, the basic duality between Fenchel's hypothesis and the existence of recession directions in convex programming is established and then expressed within each of these three duality formulations     

Duality properties between spectra and tilings

Spectra and tilings play an important role in analysis and geometry respectively.The relations between spectra and tilings have bafied the mathematicians for a long time.Many conjectures,such as the Fuglede conjecture,are placed on the establishment of relations between spectra and tilings,although there are no desired results.In the present paper we derive some characteristic properties of spectra and tilings which highlight certain duality properties between them.     

Symplectic duality between complex domains

In this paper after extending the definition of symplectic duality [given in Di Scala and Loi (Adv Math 217:2336–2352, 2008) for bounded symmetric domains] to arbitrary complex domains of \({{\mathbb{C}}^n}\) centered at the origin we generalize some of the results proved in Di Scala and Loi (Adv Math 217:2336–2352, 2008) and Di Scala et al. (Trans Groups 13(2):283–304, 2008) to those domains.     

Symplectic duality between complex domains

In this paper after extending the definition of symplectic duality [given in Di Scala and Loi (Adv Math 217:2336–2352, 2008) for bounded symmetric domains] to arbitrary complex domains of \mathbbCⁿ{{\mathbb{C}}^n} centered at the origin we generalize some of the results proved in Di Scala and Loi (Adv Math 217:2336–2352, 2008) and Di Scala et al. (Trans Groups 13(2):283–304, 2008) to those domains.     

On the duality between sampling and interpolation

We present a simple method based on the stability and duality of the properties of sampling and interpolation, which allows one to substantially simplify the proofs of some classical results.     

On the strong equality between supercompactness and strong compactness

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension ZFC + GCH in which, (a) (preservation) for regular, if is supercompact', then is supercompact' and so that, (b) (equivalence) for regular, is strongly compact' iff is supercompact', except possibly if is a measurable limit of cardinals which are supercompact.