Delta sets for divisors supported in two points

In Duursma and Park (2010) [7], the authors formulate new coset bounds for algebraic geometric codes. The bounds give improved lower bounds for the minimum distance of algebraic geometric codes as well as improved thresholds for algebraic geometric linear secret sharing schemes. The bounds depend on the delta set of a coset and on the choice of a sequence of divisors inside the delta set. In this paper we give general properties of delta sets and we analyze sequences of divisors supported in two points on Hermitian and Suzuki curves.     

Codes associated to the zero-schemes of sections of vector bundles

We consider the algebraic geometric codes associated to the zero-schemes of sections of vector bundles on a smooth projective variety. We give lower bounds for the minimum distances of the codes exploiting the Cayley–Bacharach property of zero-dimensional subschemes.     

Some properties of skew codes over finite fields

After recalling the definition of some codes as modules over skew polynomial rings, whose multiplication is defined by using an endomorphism and a derivation, and some basic facts about them, in the first part of this paper we study some of their main algebraic and geometric properties. Finally, for module skew codes constructed only with an automorphism, we give some BCH type lower bounds for their minimum distance.     

On the Poincaré problem for foliations of general type

Let be a holomorphic foliation of general type on which admits a rational first integral. We provide bounds for the degree of the first integral of just in function of the degree and the birational invariants of and the geometric genus of a generic leaf. Similar bounds for invariant algebraic curves are also obtained and examples are given showing the necessity of the hypothesis. Revised version: 3 July 2001 / Published online: 4 April 2002     

Error Bounds for the Solution Sets of Quadratic Complementarity Problems

In this article, two types of fractional local error bounds for quadratic complementarity problems are established, one is based on the natural residual function and the other on the standard violation measure of the polynomial equalities and inequalities. These fractional local error bounds are given with explicit exponents. A fractional local error bound with an explicit exponent via the natural residual function is new in the tensor/polynomial complementarity problems literature. The other fractional local error bounds take into account the sparsity structures, from both the algebraic and the geometric perspectives, of the third-order tensor in a quadratic complementarity problem. They also have explicit exponents, which improve the literature significantly.     

Refined Noether Normalization Theorem and Sharp Degree Bounds for Dominating Morphisms

Abstract

After the proof of a not very known refinement of the Noether Normalization Theorem, we obtain two sharp degree bounds for the geometric degree of a dominating morphism of irreducible affine algebraic varieties and for the degree of the components of the inverse of an isomorphism of such varieties, generalizing by this way the well-known Gabber bound for automorphisms of affine spaces.     

Distance bounds for algebraic geometric codes

Various methods have been used to obtain improvements of the Goppa lower bound for the minimum distance of an algebraic geometric code. The main methods divide into two categories, and all but a few of the known bounds are special cases of either the Lundell-McCullough floor bound or the Beelen order bound. The exceptions are recent improvements of the floor bound by Güneri, Stichtenoth, and Taskin, and by Duursma and Park, and of the order bound by Duursma and Park, and by Duursma and Kirov. In this paper, we provide short proofs for all floor bounds and most order bounds in the setting of the van Lint and Wilson AB method. Moreover, we formulate unifying theorems for order bounds and formulate the DP and DK order bounds as natural but different generalizations of the Feng-Rao bound for one-point codes.     

A geometric approach to divergent points of higher dimensional Collatz mappings

We define generalized Collatz mappings on free abelian groups of finite rank and study their iteration trajectories. Using geometric arguments we describe cones of points having a divergent trajectory and we deduce lower bounds for the density of the set of divergent points. As an application we give examples which show that the iteration of generalized Collatz mappings on rings of algebraic integers can behave quite differently from the conjectured behavior in \(\mathbb {Z}\).     

Error-Correcting Codes from Higher-Dimensional Varieties

In this paper we use intersection theory to develop methods for obtaining lower bounds on the parameters of algebraic geometric error-correcting codes constructed from varieties of arbitrary dimension. The methods are sufficiently general to encompass many of the codes previously constructed from higher-dimensional varieties, as well as those coming from curves. And still, the bounds obtained are usually as good as the ones previously known (at least of the same order of magnitude with respect to the size of the ground field). Several examples coming from Deligne–Lusztig varieties, complete intersections of Hermitian hyper-surfaces, and from ruled surfaces (or more generally, projective bundles over a curve) are given.     

Lower bounds for computing geometric spanners and approximate shortest paths

We consider the problems of constructing geometric spanners, possibly containing Steiner points, for a set of n input points in d-dimensional space , and constructing spanners and approximate shortest paths among a collection of polygonal obstacles on the plane. The complexities of these problems are shown to be Ω(n log n) in the algebraic computation tree model. Since O(n log n)-time algorithms are known for solving these problems, our lower bounds are tight up to a constant factor.     

Essential Dimension of Algebraic Groups and Integral Representations of Weyl Groups

Upper bounds on the essential dimension of algebraic groups can be found by examining related questions about the integral representation theory of lattices for their Weyl groups. We examine these questions in detail for all simple affine algebraic groups, expanding on work of Lorenz and Reichstein for PGL_n. This results in upper bounds on the essential dimensions of these simple affine algebraic groups which match or improve on the previously known upper bounds.     

Implicit algebraic geometry on categories of universal algebras

Using the notion of an implicit operation on universal algebras, we redefine basic notions of the algebraic geometry of universal algebras. The results obtained for the implicit algebraic geometry imply (as special cases) the known results on the conditional geometric and algebraic geometric comparability of algebras.     

The order bound for general algebraic geometric codes

The order bound gives an in general very good lower bound for the minimum distance of one-point algebraic geometric codes coming from curves. This paper is about a generalization of the order bound to several-point algebraic geometric codes coming from curves.     

Sensitivity analysis of the algebraic Riccati equations

Summary. In this paper, some sharp perturbation bounds for the Hermitian positive semi-definite solution to an algebraic Riccati equation are developed. A further analysis for these bounds is done. This analysis shows that there is, presumably, some intrinsic relation between the sensitivity of the solution to the algebraic Riccati equation and the distance of the spectrum of the closed-loop matrix from the imaginary axis. Received December 16, 1994     

Translation planes and derivation sets

Using ideas from algebraic coding theory, a general notion of aderivation set for a projective plane is introduced. Certain geometric codes are used to locate such sets. These codes also lead to upper bounds for thep-ranks of incidence matrices of translation planes in terms of the dimensions of the associated codes.Dedicated to Professor Tallini on the occasion of his 60 ^th birthdayThis research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

Lower bounds for algebraic complexity of nilpotent associative algebras

Exact algebraic algorithms for calculating the product of two elements of nilpotent associative algebras over fields of characteristic zero are considered (this is a particular case of simultaneous calculation of several multinomials). The complexity of an algebra in this computational model is defined as the number of nonscalar multiplications of an optimal algorithm. Lower bounds for the tensor rank of nilpotent associative algebras (in terms of dimensions of certain subalgebras) are obtained, which give lower bounds for the algebraic complexity of this class of algebras. Examples of reaching these estimates for different dimensions of nilpotent algebras are presented.     

Automorphism groups of compact bordered Klein surfaces with invariant subsets

In this work we get upper bounds for the order of a group of automorphisms of a compact bordered Klein surface S of algebraic genus greater than 1. These bounds depend on the algebraic genus of S and on the cardinals of finite subsets of S which are invariant under the action of the group. We use our results to obtain upper bounds for the order of a group of automorphism whose action on the set of connected components of the boundary of S is not transitive. The bounds obtained this way depend only on the algebraic genus of S. The author is partially supported by the European Network RAAG HPRN-CT-2001-00271 and the Spanish GAAR DGICYT BFM2002-04797.     

Geometric and algebraic multigrid techniques for fluid dynamics problems on unstructured grids

Issues concerning the implementation and practical application of geometric and algebraic multigrid techniques for solving systems of difference equations generated by the finite volume discretization of the Euler and Navier–Stokes equations on unstructured grids are studied. The construction of prolongation and interpolation operators, as well as grid levels of various resolutions, is discussed. The results of the application of geometric and algebraic multigrid techniques for the simulation of inviscid and viscous compressible fluid flows over an airfoil are compared. Numerical results show that geometric methods ensure faster convergence and weakly depend on the method parameters, while the efficiency of algebraic methods considerably depends on the input parameters.