Representability of Hilbert Schemes and Hilbert Stacks of Points

We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space. We also show the algebraicity of the Hilbert stack of points on an algebraic stack and the algebraicity of the Weil restriction of an algebraic stack along a finite flat morphism. For the latter two results, no separation assumptions are necessary.     

The Structure of Normal Algebraic Monoids

We show that any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid. This extends (and builds on) Chevalley's structure theorem for algebraic groups.     

Algebraic quantum hypergroups

An algebraic quantum group is a regular multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup. We define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other. The algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied (in an operator algebraic context) by Chapovsky and Vainerman.We will give some basic examples in order to illustrate different aspects of the theory. In a separate note, we will consider more special cases and more complicated examples. In particular, in that note, we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework.     

A Projected Algebraic Multigrid Method for Linear Complementarity Problems

We present an algebraic version of an iterative multigrid method for obstacle problems, called projected algebraic multigrid (PAMG) here. We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem. This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising, for example, in financial engineering.     

Effective algebraicity

Results of R. Miller in 2009 proved several theorems about algebraic fields and computable categoricity. Also in 2009, A. Frolov, I. Kalimullin, and R. Miller proved some results about the degree spectrum of an algebraic field when viewed as a subfield of its algebraic closure. Here, we show that the same computable categoricity results also hold for finite-branching trees under the predecessor function and for connected, finite-valence, pointed graphs, and we show that the degree spectrum results do not hold for these trees and graphs. We also offer an explanation for why the degree spectrum results distinguish these classes of structures: although all three structures are algebraic structures, the fields are what we call effectively algebraic.     

New representations of algebraic domains and algebraic L-domains via closure systems

Closure systems (spaces) play an important role in characterizing certain ordered structures. In this paper, FinSet-bounded algebraic closure spaces are introduced, and then used to provide a new approach to constructing algebraic domains. Then, a special family of algebraic closure spaces, algebraic L-closure spaces, are used to represent algebraic L-domains. Next, algebraic approximate mappings are defined and serve as the appropriate morphisms between algebraic closure spaces, respectively, algebraic L-closure spaces. On the categorical level, we show that algebraic closure spaces (respectively, algebraic L-closure spaces,) each equipped with algebraic approximate mappings as morphisms, are equivalent to algebraic domains (respectively, algebraic L-domains) with Scott continuous functions as morphisms.

     

Algebraic and real K-theory of Real varieties

An algebraic variety defined over the real numbers has an associated topological space with involution, and algebraic vector bundles give rise to Real vector bundles. We show that the associated map from algebraic K-theory to Atiyah's Real K-theory is, after completion at two, an isomorphism on homotopy groups above the dimension of the variety.     

Geometric realizability of covariant derivative K?hler tensors for almost pseudo-Hermitian and almost para-Hermitian manifolds

The covariant derivative of the K?hler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.     

Algebraic modules and the Auslander-Reiten quiver

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by the direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander-Reiten quiver. For dihedral 2-groups, we also show that there is at most one algebraic module on each component of the (stable) Auslander-Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.     

Relative local equivalence of Engel structures

The conditions to determine germs of Engel structures relative to arbitrary subsets are studied. We show that germs of Engel structures at a point relative to an arbitrary subset are determined by the algebraic restrictions of the Engel structures themselves to the subset, and the projected algebraic restrictions of the derived even-contact structures to the subset. When the subset is a smooth submanifold, algebraic restriction is equivalent to geometric restriction. Even when the subset is a smooth submanifold, we need a new stricter notion, projected algebraic restriction.     

Vector bundles over real algebraic varieties

The object of this paper is to study continuous vector bundles, over real algebraic varieties, admitting an algebraic structure. For large classes of real varieties, we obtain explicit information concerning the Grothendieck group of algebraic vector bundles. We show that in many cases this group is small compared to the corresponding group of continuous vector bundles. These results are used elsewhere to study the geometry of real algebraic varieties.Dedicated to Professor Alexander Grothendieck on the occasion of his 60th birthdaySupported by the NSF Grant DMS-8602672.     

Real intersection points of piecewise algebraic curves

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this work, we present an algorithm for computing the real intersection points of piecewise algebraic curves. It is primarily based on the interval zeros of the univariate interval polynomial in Bernstein form. An illustrative example is provided to show that the proposed algorithm is flexible.     

Maximum algebraic connectivity augmentation is NP-hard

The algebraic connectivity of a graph, which is the second-smallest eigenvalue of the Laplacian of the graph, is a measure of connectivity. We show that the problem of adding a specified number of edges to an input graph to maximize the algebraic connectivity of the augmented graph is NP-hard.     

Additive Hilbert's Theorem 90 in the ring of algebraic integers

For a number field K, we give a complete characterization of algebraic numbers which can be expressed by a difference of two K-conjugate algebraic integers. These turn out to be the algebraic integers whose Galois group contains an element, acting as a cycle on some collection of conjugates which sum to zero. Hence there are no algebraic integers which can be written as a difference of two conjugate algebraic numbers but cannot be written as a difference of two conjugate algebraic integers. A generalization of the construction to a commutative ring is also given. Furthermore, we show that for n ?_ 3 there exist algebraic integers which can be written as a linear form in n K-conjugate algebraic numbers but cannot be written by the same linear form in K-conjugate algebraic integers.     

Curvature Tensors Whose Jacobi or Szabo Operator is Nilpotent on Null Vectors

The authors show that any k-Osserman Lorentzian algebraic curvaturetensor has constant sectional curvature, and give an elementaryproof that any local 2-point homogeneous Lorentzian manifoldhas constant sectional curvature. They also show that a SzabóLorentzian covariant derivative algebraic curvature tensor vanishes.2000 Mathematics Subject Classification 53B20.     

A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function

In this paper we present a refined version of the Newton polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bit-complexity of computing the singular part of a Puiseux expansion by this algorithm, and use a recent quantitative version of Eisenstein's theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial defining the algebraic function.

     

Topologically Algebraic Algebras

We show that a locally convex algebra is topologically algebraic if, and only if, it is algebraic.Thanks are offered to Professor M. Oudadess for many remarks which allowed an improvement of the first version of this paper. We are also indebted to Professor A. Mallios for valuable suggestions.     

From Lie algebras of vector fields to algebraic group actions

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → Der_K(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.     

The universal exponentiable arrow

We show that the essentially algebraic theory of generalized algebraic theories, regarded as a category with finite limits, has a universal exponentiable arrow in the sense that any exponentiable arrow in any category with finite limits is the image of the universal exponentiable arrow by an essentially unique functor.