Quasi-regular Dirichlet Forms and L^p-resolvents on Measurable Spaces

We prove that for any semi-Dirichlet form on a measurable Lusin space E there exists a Lusin topology with the given -algebra as the Borel -algebra so that becomes quasi-regular. However one has to enlarge E by a zero set. More generally a corresponding result for arbitrary -resolvents is proven.     

Generalized Continuous Posets and a New Cartesian Closed Category

With every subset selection for posets, there is associated a certain ideal completion . As shown by Erné, such completions help to extend classical results on domains and similar structures in the absence of the required joins. Some results about –predistributive or –precontinuous posets and –continuous functions are summarized and supplemented. In particular, several central results on function spaces in domain theory are extended to the setting of productive closed subset selections. The category FSBP, in which objects are finitely separated and upper bounded posets and arrows are continuous functions between them, is shown to be cartesian closed. This research is supported by the National Natural Science Foundation of China, 10471035.     

Coherence for Product Monoids and their Actions

Let and be two monoids (algebras) in a monoidal category . Further let be a distributive law in the sense of [J. Beck, Lect. Notes Math., 80:119–140, 1969]; naturally yields a monoid . Consider a word in the symbols , , and . The first coherence theorem proved in this paper asserts that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , and . Assume now that an object is endowed with both an -object structure , and an -object structure . Further assume that these two structures are compatible, in the sense that they naturally yield an -object . Let be a word in , , , and , which contains a single instance of , in the rightmost position. The second coherence theorem states that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , , , and .     

Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes

We construct extremal stochastic integrals of a deterministic function with respect to a random Fréchet () sup-measure. The measure is sup-additive rather than additive and is defined over a general measure space , where is a deterministic control measure. The extremal integral is constructed in a way similar to the usual stable integral, but with the maxima replacing the operation of summation. It is well-defined for arbitrary , and the metric metrizes the convergence in probability of the resulting integrals.This approach complements the well-known de Haan's spectral representation of max-stable processes with Fréchet marginals. De Haan's representation can be viewed as the max-stable analog of the LePage series representation of stable processes, whereas the extremal integrals correspond to the usual stable stochastic integrals. We prove that essentially any strictly stable process belongs to the domain of max-stable attraction of an Fréchet, max-stable process. Moreover, we express the corresponding Fréchet processes in terms of extremal stochastic integrals, involving the kernel function of the stable process. The close correspondence between the max-stable and stable frameworks yields new examples of max-stable processes with non-trivial dependence structures.This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grant DMS-0505747 at Boston University.     

The Gold Partition Conjecture

We present the Gold Partition Conjecture which immediately implies the – Conjecture and tight upper bound for sorting. We prove the Gold Partition Conjecture for posets of width two, semiorders and posets containing at most elements. We prove that the fraction of partial orders on an -element set satisfying our conjecture converges to when approaches infinity. We discuss properties of a hypothetical counterexample.     

Weak Barriers in Nonlinear Potential Theory

We characterize regular boundary points for p-harmonic functions using weak barriers. We use this to obtain some consequences on boundary regularity. The results also hold for -harmonic functions under the usual assumptions on , and for Cheeger p-harmonic functions in metric spaces.      

On Approximate Efficiency in Multiobjective Programming

This paper is focused on approximate ( -efficient) solutions of multiobjective mathematical programs. We introduce a new -efficiency concept which extends and unifies different notions of approximate solution defined in the literature. We characterize these -efficient solutions in convex multiobjective programs through approximate solutions of linear scalarizations, which allow us to obtain parametric representations of different -efficiency sets. Several classical -efficiency notions are considered in order to show the concepts introduced and the results obtained.This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.     

From light tails to heavy tails through multiplier

Let X and Y be two independent nonnegative random variables, of which X has a distribution belonging to the class or for some γ ≥ 0 and Y is unbounded. We study how their product XY inherits the tail behavior of X. Under some mild technical assumptions we prove that the distribution of XY belongs to the class or accordingly. Hence, the multiplier Y builds a bridge between light tails and heavy tails.      

The geometric invariants $${\Omega^{n}}$$ of a product of groups

We compute the geometric invariants of a product G × H of groups in terms of and . This gives a sufficient condition in terms of and for a normal subgroup of G × H with abelian quotient to be of type F _n . We give an example involving the direct product of the Baumslag–Solitar group BS_1,2 with itself.      

Universality of Coproducts in Categories of Lax Algebras

Categories of lax -algebras are shown to have pullback-stable coproducts if preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax -algebras, with preserving inverse images. Moreover, we show that any such category of -algebras has a concrete, coproduct preserving functor into the category of topological spaces.     

The \bar \partial -Neumann operator and commutators of the Bergman projection and multiplication operators

We prove that compactness of the canonical solution operator to restricted to (0, 1)-forms with holomorphic coefficients is equivalent to compactness of the commutator defined on the whole L _(0,1)²(Ω), where is the multiplication by and is the orthogonal projection of L _(0,1)²(Ω) to the subspace of (0, 1) forms with holomorphic coefficients. Further we derive a formula for the -Neumann operator restricted to (0, 1) forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by z and . Partially supported by the FWF grant P19147-N13.     

Quasilinearization and curvature of Aleksandrov spaces

We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space is an Aleksandrov domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.      

Boundary Limits for Bounded Quasiregular Mappings

In this paper we establish results on the existence of nontangential limits for weighted -harmonic functions in the weighted Sobolev space , for some q>1 and w in the Muckenhoupt A _q class, where is the unit ball in . These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted -harmonic functions are weak solutions of the partial differential equation

Deep Matrix Algebras of Finite Type

Deep matrix algebras based on a set over a ring are introduced and studied by McCrimmon when has infinite cardinality. Here, we construct a new -module that is faithful for of any cardinality. For a field of arbitrary characteristic and , is studied in depth. The algebra is shown to possess a unique proper non-zero ideal, isomorphic to . This leads to a new and interesting simple algebra, , the quotient of by its unique ideal. Both and the quotient algebra are shown to have centers isomorphic to . Via the new faithful representation, all automorphisms of are shown to be inner in the sense of Definition 18.Presented by D. Passman.     

Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

An algorithm for matching point sets using the l_1 norm

The problem is considered of matching two sets of points in , by translation and rotation. There are many applications, for example in geodesy, computer vision and in the assessment of manufactured parts. When the matching criterion is least squares, there is a well known solution process based on the singular value decomposition of an matrix. Here we consider the use of the norm, which may be more appropriate than least squares in the context of wild points in the data. An algorithm is developed, and is illustrated by some examples for the case .     

Pseudomonotone$${_{\ast}}$$ maps and the cutting plane property

Pseudomonotone maps are a generalization of paramonotone maps which is very closely related to the cutting plane property in variational inequality problems (VIP). In this paper, we first generalize the so-called minimum principle sufficiency and the maximum principle sufficiency for VIP with multivalued maps. Then we show that pseudomonotonicity of the map implies the “maximum principle sufficiency” and, in fact, is equivalent to it in a sense. We then present two applications of pseudomonotone maps. First we show that pseudomonotone maps can be used instead of the much more restricted class of pseudomonotone₊ maps in a cutting plane method. Finally, an application to a proximal point method is given.      

On some types of radical classes

Let be an infinite cardinal. We denote by the collection of all -representable Boolean algebras. Further, let be the collection of all generalized Boolean algebras B such that for each b ∈ B, the interval [0, b] of B belongs to . In this paper we prove that is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized MV-algebras. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-032002. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information.     

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.