Modular Group Algebras with Almost Maximal Lie Nilpotency Indices

Let be a field of positive characteristic and the group algebra of a group . It is known that, if is Lie nilpotent, then its upper (and lower) Lie nilpotency index is at most , where is the order of the commutator subgroup. The authors previously determined those groups for which this index is maximal and here they determine the groups for which it is `almost maximal', that is, it takes the next highest possible value, namely .Presented by V. Dl a b.Dedicated to Professor Vjacheslav Rudko on his 65th birthday.The research was supported by OTKA No. T 037202, No. T 038059 and Italian National Research Project “Group Theory and Application.”     

Symmetric Stable Processes on Unbounded Domains

Let be a symmetric -stable process in , , . We give necessary and sufficient condition under which the expectation of a very general function of the exit time from horns is finite. These domains include the symmetric domains given by increasing functions studied earlier by various authors. Our methods differ from those in earlier papers in that we obtain our results from estimates on the transition densities instead of harmonic measure. Some of this estimates are of independent interest.Supported in part by NSF grant #9700585-DMS and RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP.     

Kernel B-splines and interpolation

This paper applies difference operators to conditionally positive definite kernels in order to generate kernel -splines that have fast decay towards infinity. Interpolation by these new kernels provides better condition of the linear system, while the kernel -spline inherits the approximation orders from its native kernel. We proceed in two different ways: either the kernel -spline is constructed adaptively on the data knot set , or we use a fixed difference scheme and shift its associated kernel -spline around. In the latter case, the kernel -spline so obtained is strictly positive in general. Furthermore, special kernel -splines obtained by hexagonal second finite differences of multiquadrics are studied in more detail. We give suggestions in order to get a consistent improvement of the condition of the interpolation matrix in applications.     

A method of convergence acceleration of some continued fractions

A new method of convergence acceleration is proposed for continued fractions , where and are polynomials in (, ) for sufficiently large. It uses the fact that the modified approximant approaches the continued fraction value, if is sufficiently close to the th tail . Presented method is of iterative character; in each step, by means of an approximation , it produces a new better approximation of the th tail . Formula for is very simple and contains only arithmetical operations. Hence described algorithm is fully rational.     

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 .     

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

On Auslander–Reiten Components and Heller Lattices for Integral Group Rings

Let be a finite group, a complete discrete valuation ring of characteristic zero with residue class field of characteristic , and a block of the group ring . Suppose that is of infinite representation type and is sufficiently large to satisfy certain conditions. Let be the Auslander–Reiten quiver of and a connected component of . In this paper, we show that if contains some Heller lattices then the tree class of the stable part of is . Also, we show that has infinitely many components of type if a defect group of is neither cyclic nor a Klein four group.Presented by Jon Carlson.     

Koszul and Gorenstein Properties for Homogeneous Algebras

The Koszul property was generalized to homogeneous algebras of degree in [5], and related to -complexes. We show that if the -homogeneous algebra is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem to i.e., there is a Poincaré duality between Hochschild homology and cohomology of as for .     

A representation of bivariate extreme value distributions via norms on \mathbb{R}^{2}

It is known that a bivariate extreme value distribution (EVD) with reverse exponential margins can be represented as , , where is a suitable norm on . We prove in this paper the converse implication, i.e., given an arbitrary norm on , , , defines an EVD with reverse exponential margins, if and only if the norm satisfies for the condition . This result is extended to bivariate EVDs with arbitrary margins as well as to extreme value copulas. By identifying an EVD , , with the unit ball corresponding to the generating norm , we obtain a characterization of the class of EVDs in terms of compact and convex subsets of .     

Skew Distributions Generated from Different Families

Following the recent paper by Gupta et al. [8], skew pdfs of the form are generated, where the pdf and the cdf are taken to be different and to come from normal, Student's , Cauchy, Laplace, logistic or the uniform distribution. The properties of the resulting distributions are studied. In particular, expressions for the th moment and the characteristic function are derived. Graphical illustrations are also provided.     

Uniform Estimates of the Fundamental Solution for a Family of Hypoelliptic Operators

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: , in where is the Laplace operator, , and the limit operator is hypoelliptic. It is well known that admits a fundamental solution . Here we establish some a priori estimates uniform in of it, using a modification of the lifting technique of Rothschild and Stein. As a consequence we deduce some a priori estimates uniform in , for solutions of the approximated equation . These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.     

On applicability of the sparse grid method in the worst case setting

Let be bounded linear operators. We provide several sufficient conditions for the validity of the inequality . These results can be applied to error and cost estimates for the sparse grid method.     

Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit

is the category of archimedean -groups with distinguished weak order unit, with -group homomorphisms which preserve unit. This category includes all rings of continuous functions and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in . There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each – object, we define a topology and a convergence . These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject of is dense iff is epically embedded. The topology is , but only sometimes Hausdorff or an -group topology. The convergence is a Hausdorff -group convergence, but only sometimes topological. The associations of to , and to , are functorial. Dedicated to Bernhard Banaschewski for his 80th birthday.     

Nonlinear Parabolic Equations with Regularized Derivatives in Colombeau Algebra

We consider several types of nonlinear parabolic equations with singular like potential and initial data. To prove the existence-uniqueness theorems we employ regularized derivatives. As a framework we use Colombeau space and Colombeau vector space      

Equivalence of $n$-point Gauss–Chebyshev rule and $4n$-point midpoint rule in computing the period of a Lotka–Volterra system

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of restricted on , , respectively. In computing this period numerically, the integral (3.6), which possesses a weak singularity of the square root type at each endpoint of the integration, is an excellent example of using the Gauss–Chebyshev integration rule of the first kind; while the integral (4.7), which is an integral of a smooth periodic function over its period , is an excellent example of using the midpoint rule, but not the trapezoidal rule, suggested by Waldvogel [39, 40], due to a removable singularity of the integrand at , , , , and , respectively. This paper shows, in computing the period of a periodic solution of the Lotka–Volterra system, the -point Gauss–Chebyshev integration rule of the first kind applied to the integral (3.6) becomes the -point midpoint rule to the integral (4.7). Dedicated to R. Bruce Kellogg on the occasion of his 75th birthday.     

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

Homology of planar polygon spaces

In this paper, we study topology of the variety of closed planar n-gons with given side lengths . The moduli space where , encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces as functions of the length vector . We also find sharp upper bounds on the sum of Betti numbers of depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.      

Schur–Weyl Reciprocity between the Quantum Superalgebra and the Iwahori–Hecke Algebra

In this paper, we establish Schur–Weyl reciprocity between the quantum general super Lie algebra and the Iwahori–Hecke algebra . We introduce the sign -permutation representation of on the tensor space of dimensional -graded -vector space . This action commutes with that of derived from the vector representation on . Those two subalgebras of satisfy Schur–Weyl reciprocity. As special cases, we obtain the super case (), and the quantum case (). Hence this result includes both the super case and the quantum case, and unifies those two important cases.Presented by A. Verschoren.     

The modal logic of continuous functions on cantor space

Let be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality and a temporal modality , understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language by interpreting in dynamic topological systems, i.e. ordered pairs , where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. Zhang and Mints have shown that S4C is complete relative to a particular topological space, Cantor space. The current paper produces an alternate proof of the Zhang-Mints result.     

H-matrix theory vs. eigenvalue localization

The eigenvalue localization problem is very closely related to the -matrix theory. The most elegant example of this relation is the equivalence between the Geršgorin theorem and the theorem about nonsingularity of SDD (strictly diagonally dominant) matrices, which is a starting point for further beautiful results in the book of Varga [19]. Furthermore, the corresponding Geršgorin-type theorem is equivalent to the statement that each matrix from a particular subclass of -matrices is nonsingular. Finally, the statement that all eigenvalues of a given matrix belong to minimal Geršgorin set (defined in [19]) is equivalent to the statement that every -matrix is nonsingular. Since minimal Geršgorin set remained unattainable, a lot of different Geršgorin-type areas for eigenvalues has been developed recently. Along with them, a lot of new subclasses of -matrices were obtained. A survey of recent results in both areas, as well as their relationships, will be presented in this paper.This work is partly supported by Republic of Serbia, Ministry of Science and Environmental Protection, grant 144025.