Generalized Tensor Structures of Kawaguchi Spaces

We prove that, in a generalized Kawaguchi space, there exist intrinsic almost product and almost complex structures associated to a metric of this space. We derive conditions for the integrability of these structures and find compatible affine connections.     

Blaschke structure for a special affine immersion

In this paper we determine a Blaschke structure for affine immersion of Euclidian and Hyperbolic type for plane equi-affine curve. In particular, we consider this structure in the case where the Ricci tensor of affine immersion is constant and give a necessary and sufficient condition for Ricci tensor to be constant.     

Helix splines as an example of affine Tchebycheffian splines

The present paper summarizes the theory of affine Tchebycheffian splines and presents an interesting affine Tchebycheffian free-form scheme, the “helix scheme”. The curve scheme provides exact representations of straight lines, circles and helix curves in an arc length parameterization. The corresponding tensor product surfaces contain helicoidal surfaces, surfaces of revolution and patches on all types of quadrics. We also show an application to the construction of planarC ² motions interpolating a given set of positions. Because the spline curve segments are calculated using a subdivision algorithm, many algorithms, which are of fundamental importance in the B-spline technique, can be applied to helix splines as well. This paper should demonstrate how to create an affine free-form scheme fitting to certain special applications.     

Sur le produit tensoriel des groupes affines

Let A and B be two commutative affine group schemes over a field. There exists an affine group A?B such that Hom(A?B,C)?Bil(A×B,C) for any affine group C. We use technics of the commutative algebraic groups theory, in order to compute these tensor products and to characterize “flat” groups in the unipotent case. The tensor product of commutative affine groups has most properties of the usual tensor product but it is not always associative. As an application we prove a structure theorem of the category of modules over some affine connected prosmooth rings.     

Affine immersion ofn-dimensional manifold intoR n+n(n+1)/2 and affine minimality

We formulate an affine theory of immersions of ann-dimensional manifold into the Euclidean space of dimensionn+n(n+1)/2 and give a characterization of critical immersions relative to the induced volume functional in terms of the affine shape operator.     

Almost Affine Codes

An almost affine code is a code C for which the size of all codes obtained by multiple puncturing of C is a power of the alphabet size. Essentially, almost affine codes are the same as ideal perfect secret haring schemes or partial affine geometries. The present paper explores these interrelations, gives short proofs of known and new results, and derives some properties of the distance distribution of almost affine codes.     

On decomposability of affine partial linear spaces

In the paper we characterize normal subspaces of an affine partial linear space and characterize affine partial linear spaces which can not be represented as the Segre product of some affine partial linear spaces.     

Almost periodic solutions of affine ito equations

We discuss the problem of the existence of almost periodic in distribution solutions of affine stochastic differential equations with almost periodic coefficients. We prove that if the linear part of the affine equation is exponentially stable in mean square then the unique continuous L² -bounded solution of the affine system has the onedimensional distributions almost periodic. An analogous result is shown for the asymptotic almost periodic case     

Affine Properties of Convex Equal-Area Polygons

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous to the six-vertex theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.     

A3型仿射Weyl群最低双边胞腔上的Kazhdan-Lusztig系数

对于Kazhdan-Lusztig多项式Py,w(q),μ(y,w)为它的首项系数(简称KL系数).首项系数在李代数及其表示理论中起着重要的作用.在文章中,W为A3型仿射Weyl群,通过它对应的Hecke代数的性质及其KL基{Cw}的乘积计算,以及不可约模在张量积中的重数公式,给出了A3型仿射Weyl群最低双边胞腔上的KL系数.     

Determinants of the Hypergeometric Period Matrices of an Arrangement and Its Dual

We fix three natural numbers k, n, N, such that n+k+1 = N, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of N hyperplanes in a k-dimensional affine space, the other is an arrangement of N hyperplanes in an n-dimensional affine space. We assign weights α ₁, . . . , α _N to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of k-dimensional hypergeometric integrals and the second is a matrix of n-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler’s gamma function multiplied by a product of exponentials of the weights. Supported in part by NSF grant DMS-0244579.     

Efficient algorithms for solving certain nonconvex programs dealing with the product of two affine fractional functions

Two algorithms for finding a global minimum of the product of two affine fractional functions over a compact convex set and solving linear fractional programs with an additional constraint defined by the product of two affine fractional functions are proposed. The algorithms are based on branch and bound techniques using an adaptive branching operation which takes place in one-dimensional intervals. Results from numerical experiments show that large scale problems can be efficiently solved by the proposed methods.     

Flags of almost affine codes and the two-party wire-tap channel of type II

We describe a two-party wire-tap channel of type II in the framework of almost affine codes. Its cryptological performance is related to some relative profiles of a pair of almost affine codes. These profiles are analogues to relative generalized Hamming weights in the linear case.     

Characters of Simple Bounded Modules over an Untwisted Affine Lie Algebra

After V. Chari and A. Pressley, a simple integrable module with finite-dimensional weight spaces over an affine Lie algebra is either a standard module (highest or lowest weight), in which case its formal character is given by the famous Weyl–Kac formula, or a subquotient of a tensor product of loop modules. In this paper we compute formal characters of generic simple integrable modules of the latter type.     

On solving the Lagrangian dual of integer programs via an incremental approach

The Lagrangian dual of an integer program can be formulated as a min-max problem where the objective function is convex, piecewise affine and, hence, nonsmooth. It is usually tackled by means of subgradient algorithms, or multiplier adjustment techniques, or even more sophisticated nonsmooth optimization methods such as bundle-type algorithms. Recently a new approach to solving unconstrained convex finite min-max problems has been proposed, which has the nice property of working almost independently of the exact evaluation of the objective function at every iterate-point.     

Affine Jacquet functors and Harish-Chandra categories

We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan on the structure of Verma modules in the Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman. The latter provides an extension of the works of Rocha-Caridi and Wallach [A. Rocha-Caridi, N.R. Wallach, Projective modules over infinite dimensional graded Lie algebras, Math. Z. 180 (1982) 151-177] and Deodhar, Gabber and Kac [V. Deodhar, O. Gabber, V. Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. Math. 45 (1982) 92-116] on block decompositions of BGG categories for Kac-Moody algebras. We also derive a compatibility relation between the affine Jacquet functor and the Kazhdan-Lusztig tensor product and apply it to prove that the affine Harish-Chandra category is stable under fusion tensoring with the Kazhdan-Lusztig category. This compatibility will be further applied in studying translation functors for the affine Harish-Chandra category, based on the fusion tensor product.     

Geometry of Affine Warped Product Hypersurfaces

The purpose of this article is the study of warped product manifolds which can be realized either as centroaffine or graph hypersurfaces in some affine space. First, we show that there exist many such realizations. Then we establish general optimal inequalities in terms of the warping function and the Tchebychev vector field for such affine hypersurfaces. We also investigate warped product affine hypersurfaces which verify the equality case of the inequalities. Several applications are also presented.     

Toric fiber products versus Segre products

The toric fiber product is an operation that combines two ideals that are homogeneous with respect to a grading by an affine monoid. The Segre product is a related construction that combines two multigraded rings. The quotient ring by a toric fiber product of two ideals is a subring of the Segre product, but in general this inclusion is strict. We contrast the two constructions and show that any Segre product can be presented as a toric fiber product without changing the involved quotient rings. This allows to apply previous results about toric fiber products to the study of Segre products. We give criteria for the Segre product of two affine toric varieties to be dense in their toric fiber product, and for the map from the Segre product to the toric fiber product to be finite. We give an example that shows that the quotient ring of a toric fiber product of normal ideals need not be normal. In rings with Veronese type gradings, we find examples of toric fiber products that are always Segre products, and we show that iterated toric fiber products of Veronese ideals over Veronese rings are normal.     

Construction of almost integral manifolds of affine control systems

We consider the problem of constructing manifolds of a special form, which are said to be almost integral, lie in the state space of an affine control system, and have the following property: the restriction of the control system to such a manifold is an almost trivial control system defining the part of trajectories of the original system lying on that manifold.     

Affine interval exchange transformation without an isometric model

Contrary to the case of interval exchange transformation, we show that generalized affine interval exchange transformation (affine GIET), with or without flips and admitting dense orbits, may not be conjugated to an isometric GIET. This result is proved by constructing explicitly one such affine GIET. The author was partially supported by FAPESP – Proj. Tematico No. 03/03107-9.