Vector Lattices Associated with Ordered Vector Spaces

Let V be a real, Archimedian ordered, vector space, whose positive cone V ⁺ satisfies V =  V ⁺ – V ⁺. To V we associate a Dedekind complete vector lattice W containing V (by abuse of notation). In the case when V has an order unit the determination of W is already known. Let W₀ ì W{W_0 \subset W} be the vector lattice generated by V. We study W ₀ in the case when the cone C of all positive linear forms on V separates the elements of V. The determination of W ₀ involves the extreme rays of C. We determine the cone of positive linear forms on W ₀ in terms of conical measures on C.     

Zeros of analytic functions,with or without multiplicities

The classical Mason–Stothers theorem deals with nontrivial polynomial solutions to the equation a + b = c. It provides a lower bound on the number of distinct zeros of the polynomial abc in terms of deg a, deg b and deg c. We extend this to general analytic functions living on a reasonable bounded domain W ì \mathbb C{\Omega\subset{\mathbb C}}, rather than on the whole of \mathbb C{{\mathbb C}}. The estimates obtained are sharp, for any Ω, and a generalization of the original result on polynomials can be recovered from them by a limiting argument.     

Manifolds with S 1-category 2 have cyclic fundamental groups

A closed topological n-manifold M ⁿ is of S ¹-category 2 if it can be covered by two open subsets W ₁, W ₂ such that the inclusions W _i → M ⁿ factor homotopically through maps W _i → S ¹. We show that for n > 3 the fundamental group of such an n-manifold is either trivial or infinite cyclic.     

Non-special scrolls with general moduli

Abstract  In this paper we study smooth, non-special scrolls S of degree d, genus g ≥ 0, with general moduli. In particular, we study the scheme of unisecant curves of a given degree on S. Our approach is mostly based on degeneration techniques. Keywords Ruled surfaces, Hilbert schemes of scrolls, Moduli, Embedded degenerations Mathematics Subject Classification (2000) 14J26, 14D06, 14C20, (Secondary) 14H60, 14N10     

Domains with Hyperbolic Orbit Accumulation Boundary Points

Let Ω be a smoothly bounded pseudoconvex domain in ℂ ⁿ satisfying the condition R. Suppose that its Bergman kernel extends to [`(W)]×[`(W)]\overline{\Omega}\times\overline{\Omega} minus the boundary diagonal set as a locally bounded function. In this paper we show that for each hyperbolic orbit accumulation boundary point p, there exists a contraction f∈Aut(Ω) at p. As an application, we show that Ω admits a hyperbolic orbit accumulation boundary point if and only if it is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial and that Ω is of finite D’Angelo type.     

The divergence of Banach space valued random variables on Wiener space

The domain of definition of the divergence operator δ on an abstract Wiener space (W,H,μ) is extended to include W–valued and – valued “integrands”. The main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in δ’s extended domain have divergence zero. These results are then applied to the analysis of quasiinvariant flows induced by W-valued vector fields and, among other results, it turns out that these divergence-free vector fields “are responsible” for generating measure preserving flows. Mathematics Subject Classification (2000): Primary 60H07, Secondary 60H05 An erratum to this article is available at .     

Uniform Regularity in a Wedge and Regularity of Traces of CR Functions

We discuss in Sect. 1 the property of regularity at the boundary of separately holomorphic functions along families of discs and apply, in Sect. 2, to two situations. First, let W\mathcal{W} be a wedge of ℂ ⁿ with C ^ω , generic edge ℰ: a holomorphic function f on W\mathcal{W} has always a generalized (hyperfunction) boundary value bv(f) on ℰ, and this coincides with the collection of the boundary values along the discs which have C ^ω transversal intersection with ℰ. Thus Sect. 1 can be applied and yields the uniform continuity at ℰ of f when bv(f) is (separately) continuous. When W\mathcal{W} is only smooth, an additional property, the temperateness of f at ℰ, characterizes the existence of boundary value bv(f) as a distribution on ℰ. If bv(f) is continuous, this operation is consistent with taking limits along discs (Theorem 2.8). By Sect. 1, this yields again the uniform continuity at ℰ of tempered holomorphic functions with continuous bv. This is the theorem by Rosay (Trans. Am. Math. Soc. 297(1):63–72, 1986), in whose original proof the method of “slicing” by discs is not used.     

An efficient string sorting algorithm for weighing matrices of small weight

In this paper, we demonstrate that the search for weighing matrices of small weights constructed from two circulants can be viewed as a string sorting problem together with a linear time algorithm to locate common strings in two sorted arrays. We also introduce a sparse encoding of the periodic autocorrelation function vector, based on concepts from Algorithmic Information Theory, also known as Kolmogorov complexity, that allows us to speed up the algorithm considerably. Finally, we use these ideas to find new weighing matrices W(2 · n, 9) constructed from two circulants, for many values of n in the range 100 ≤  n ≤  300. These matrices are given here for the first time. We also discuss briefly a connection with Combinatorial Optimization.     

Optimal Prefix-Free Codes for Unequal Letter Costs: Dynamic Programming with the Monge Property

In this paper we discuss the problem of finding optimal prefix-free codes for unequal letter costs, a variation of the classical Huffman coding problem. Our problem consists of finding a minimal cost prefix-free code in which the encoding alphabet consists of unequal cost (length) letters, with lengths α and β. The most efficient algorithm known previously requires O(n^{2 + max(α, β)}) time to construct such a minimal-cost set of n codewords, provided α and β are integers. In this paper we provide an O(n^max(α, β)) time algorithm. Our improvement comes from the use of a more sophisticated modeling of the problem, combined with the observation that the problem possesses a “Monge property” and that the SMAWK algorithm on monotone matrices can therefore be applied.     

Dihedral manifold approximate fibrations over the circle

Consider the cyclic group C ₂ of order two acting by complex-conjugation on the unit circle S ¹. The main result is that a finitely dominated manifold W of dimension > 4 admits a cocompact, free, discontinuous action by the infinite dihedral group D _∞ if and only if W is the infinite cyclic cover of a free C ₂-manifold M such that M admits a C ₂-equivariant manifold approximate fibration to S ¹. The novelty in this setting is the existence of codimension-one, invariant submanifolds of M and W. Along the way, we develop an equivariant sucking principle for orthogonal actions of finite groups on Euclidean space.     

Off-line dynamic maintenance of the width of a planar point set

Agarwal, P.K. and M. Sharir, Off-line dynamic maintenance of the width of a planar point set, Computational Geometry: Theory and Applications 1 (1990) 65-78. In this paper we present an efficient algorithm for the off-line dynamic maintenance of the width of a planar point set in the following restricted case: We are given a real parameter W and a sequence Σ=(σ₁,...,σ_n) of n insert and delete operations on a set S of points in ², initially consisting of n points, and we want to determine whether there is an i such that the width of S the ith operation is less than or equal to W. Our algorithm runs in time O(nlog³n) and uses O(n) space.     

Limit Law of the Local Time for Brox’s Diffusion

We consider Brox’s model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t,m _log t +x)/t, x∈R), where m _log t is the bottom of the deepest valley reached by the process before time t, behaves asymptotically like a process which only depends on W. As a consequence, we get the weak convergence of the local time to a functional of two independent three-dimensional Bessel processes and thus the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space case for which the same questions have been answered recently by Gantert, Peres, and Shi (Ann. Inst. Henri Poincaré, Probab. Stat. 46(2):525–536, 2010).     

Pattern Matching with Swaps

Let a text string T of n symbols and a pattern string P of m symbols from alphabet Σ be given. A swapped version T′ of T is a length n string derived from T by a series of local swaps (i.e., t′_ℓ ← t_ℓ + 1 and t′_ℓ + 1 ← t_ℓ), where each element can participate in no more than one swap. The pattern matching with swaps problem is that of finding all locations i for which there exists a swapped version T′ of T with an exact matching of P in location i of T′. It has been an open problem whether swapped matching can be done in less than O(nm) time. In this paper we show the first algorithm that solves the pattern matching with swaps problem in time o(nm). We present an algorithm whose time complexity is O(nm^1/3 log m log σ) for a general alphabet Σ, where σ = min(m,Σ).     

Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix

The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) can be viewed as a semismooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x, Mx + q) = 0, which is equivalent to the LCP. When M is an M-matrix of order n, the algorithm is known to converge in at most n iterations. We show in this paper that this result no longer holds when M is a P-matrix of order ≥ 3, since then the algorithm may cycle. P-matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the LCP for an arbitrary q. Incidentally, convergence occurs for a P-matrix of order 1 or 2.     

Degree theory on Wiener space

Summary. Let (W, H, μ) be an abstract Wiener space and let Tw  =  w + u (w), where u is an H-valued random variable, be a measurable transformation on W. A Sard type lemma and a degree theorem for this setup are presented and applied to derive existence of solutions to elliptic stochastic partial differential equations. Received: 19 March 1996 / In revised form: 7 January 1997     

A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

The concept of acombinatorial(W; P; U)-geometryfor a Coxeter groupW, a subsetPof its generating involutions and a subgroupUofWwithP  Uyields the combinatorial foundation for a unified treatment of the representation theories of matroids and of even Δ-matroids. The concept of a (W, P)-matroid as introduced by I. M. Gelfand and V. V. Serganova is slightly different, although for many important classes ofWandPone gets the same structures. In the present paper, we extend the concept of the Tutte group of an ordinary matroid to combinatorial (W; P; U)-geometries and suggest two equivalent definitions of a (W; P; U)-matroid with coefficients in a fuzzy ringK. While the first one is more appropriate for many theoretical considerations, the second one has already been used to show that (W; P; U)-matroids with coefficients encompass matroids with coefficients and Δ-matroids with coefficients.     

P-kernels,IC bases and Kazhdan–Lusztig polynomials

In [J. Amer. Math. Soc. 5 (1992) 805–851] Stanley introduced the concept of a P-kernel for any locally finite partially ordered set P. In [Proc. Sympos. Pure Math., Vol. 56, AMS, 1994, pp. 135–148] Du introduced, for any set P, the concept of an IC basis. The purpose of this article is to show that, under some mild hypotheses, these two concepts are equivalent, and to characterize, for a given Coxeter group W, partially ordered by Bruhat order, the W-kernel corresponding to the Kazhdan–Lusztig basis of the Hecke algebra of W. Finally, we show that this W-kernel factorizes as a product of other W-kernels, and that these provide a solution to the Yang–Baxter equations for W.     

Left cells in the weyl group of type 7

By applying an algorithm, we get a representative set of the left cells and all the left cell graphs for the Weyl group W of type E ₇,. From this, we see that the generalized r-invariants characterize all the non-exceptional left cells of W. We give a criterion to check the relation [Ltilde] on the elements in the exceptional left cells of W.     

The degree of the special linear characters of a rank two free group

Given a word W in the free group on 2 letters F ₂, Horowitz showed that the special linear character of W is an integer polynomial in the 3 characters of the basic words of F ₂. Special linear characters are defined via the trace of their representations, and the polynomial character of an arbitrary W can be found by application of certain “trace relations”, which allow one to write the character of a complicated word as a sum of products of the characters of simpler words. Even in the n = 2 case, where the polynomial is uniquely defined, this procedure can be difficult and tedious. In this note, we use the structure of a free group word in F ₂ to compute the degree and the leading monomial of its SL(2,\mathbbC){SL(2,\mathbb{C})} -character without actually computing the full polynomial.