Singular curves bounding two unstable minimal disks

Let A⊆ℝ³ be a convex body and Γ the union of two Jordan curves on ∂A which meet each other at two points with prescribed angles. Then Γ bounds two unstable minimal disks. Received: 21 September 1998 / Revised version: 17 May 1999     

Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form

We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex ‘modulus’ and a complex ‘argument’. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of −1), but the complex phase is multiplied by a different complex root of −1 in the exponential function. We show how to calculate the ‘modulus’ and ‘argument’ from an arbitrary quaternion in Cartesian form.     

Structure of coordinate groups for algebraic sets in partially commutative nilpotent groups

The results obtained deal in algebraic geometry over partially commutative class two nilpotent ℚ-groups, where ℚ is a field of rationals. It is proved that two arbitrary non-Abelian partially commutative class two nilpotent ℚ-groups are geometrically equivalent. A necessary and sufficient condition of being universally geometrically equivalent is specified for two partially commutative class two nilpotent ℚ-groups. Algebraic sets for systems of equations in one variable, as well as for some special systems in several variables, are described. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Translated from Algebra i Logika, Vol. 48, No. 2, pp. 378–399, May–June, 2009.     

Ads Manifolds With Particles and Earthquakes on Singular Surfaces

We prove two related results. The first is an “earthquake theorem” for closed hyperbolic surfaces with cone singularities where the total angle is less than π: any two such metrics in are connected by a unique left earthquake. The second result is that the space of “globally hyperbolic” AdS manifolds with “particles” – cone singularities (of given angle) along time-like lines – is parametrized by the product of two copies of the Teichmüller space with some marked points (corresponding to the cone singularities). The two statements are proved together. F.B. was partially supported by the A.N.R. project GEODYCOS. J.-M.S. was partially supported by the A.N.R. programs RepSurf, 2006-09, ANR-06-BLAN-0311, GeomEinstein, 2006-09, 06-BLAN-0154, and FOG, 2007-10, ANR-07-BLAN-0251-01.     

Schreier type theorems for bicrossed products

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H;G; α; β) is deformed using a combinatorial datum (σ; v; r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → H in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H _α⋈β G ≅H _{α′⋈β′} (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H _α⋈β G ≅ H _{α′⋈β′} G′ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed products of groups are given.     

Cyclic orders: Equivalence and duality

Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomassé’s recent proof of Gallai’s conjecture. We explore this notion further: we prove that two cyclic orders are equivalent if and only if the winding number of every circuit is the same in the two. The proof is short and provides a good characterization and a polynomial algorithm for deciding whether two orders are equivalent. We then derive short proofs of Gallai’s conjecture and a theorem “polar to” the main result of Bessy and Thomassé, using the duality theorem of linear programming, total unimodularity, and the new result on the equivalence of cyclic orders.     

A note on quantum products of Schubert classes in a Grassmannian

Given two Schubert classes σ_λ and σ_μ in the quantum cohomology of a Grassmannian, we construct a partition ν, depending on λ and μ, such that σ_ν appears with coefficient 1 in the lowest (or highest) degree part of the quantum product σ_λ⋆σ_μ. To do this, we show that for any two partitions λ and μ, contained in a k × (n − k) rectangle and such that the 180^∘-rotation of one does not overlap the other, there is a third partition ν, also contained in the rectangle, such that the Littlewood-Richardson number c _λμ ^ν is 1.     

Hamilton cycles in highly connected and expanding graphs

In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on two properties only: one requiring expansion of “small” sets, the other ensuring the existence of an edge between any two disjoint “large” sets. We also discuss applications in positional games, random graphs and extremal graph theory.     

On graphs related to conjugacy classes of groups

LetG be a finite group. Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at the Tel Aviv University under the supervision of Prof. Marcel Herzog.     

To solving problems of algebra for two-parameter matrices. IV

This paper continues the series of publications devoted to surveying and developing methods for solving the following problems for a two-parameter matrix F (λ, μ) of general form: exhausting points of the mixed regular spectrum of F (λ, μ); performing operations on polynomials in two variables (computing the GCD and LCM of a few polynomials, division of polynomials, and factorization); computing a minimal basis of the null-space of polynomial solutions of the matrix F (λ, μ) and separation of its regular kernel; inversion and pseudo in version of polynomial and rational matrices in two variables, and solution of systems of nonlinear algebraic equations in two unknowns. Most of the methods suggested are based on rank factorizations of a two-parameter polynomial matrix and on the method of hereditary pencils. Bibliography: 8 titles.     

Cyclotomic and simplicial matroids

We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid μ_n represented by then ^th roots of unity 1, ζ, ζ², …, ζ^n-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then ^th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of μ_n in the case thatn has two prime factors. First author supported by NSF Postdoctoral Fellowship. Second author supported by NSF grant DMS-0245379.     

On the Borel-Cantelli lemma

A new variant of the “divergent” part of the Borel-Cantelli lemma for events derived from a Markov chain is given. Further two applications are considered. One of the applications refers to the denumerable Markov chain and the second is a new proof of the “strong” theorem corresponding to the “arc sine law”.     

Nine convex sets determine a pentagon with convex sets as vertices

It is proved that if ℱ is a family of nine pairwise disjoint compact convex sets in the plane such that no member of ℱ is contained in the convex hull of the union of two other sets of ℱ, then ℱ has a subfamily ℱ′ with five elements such that no member of ℱ′ is contained in the convex hull of the union of the other sets of ℱ′.     

Neighborhoods of extreme points

An examination of relationship between two neighborhood systems (relative to two linear topologies) of extreme points yields a unified approach to some known and new results, among which are Bessaga-Pełczyński’s theorem on closed bounded convex subsets of separable conjugate Banach spaces and Ryll-Nardzewski’s fixed point theorem. This research was partly supported by the U.S. National Science Foundation.     

Recovering of Damping Coefficients for a System of Coupled Wave Equations with Neumann Boundary Conditions: Uniqueness and Stability

The authors study the inverse problem of recovering damping coefficients for two coupled hyperbolic PDEs with Neumann boundary conditions by means of an additional measurement of Dirichlet boundary traces of the two solutions on a suitable, explicit subportion Γ1 of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on Γ0 = ΓnΓ1, T > 0, the uniqueness and stability of the damping coefficients are established. The proof uses critically the Carleman estimate due to Lasiecka et al. in 2000, together with a convenient tactical route “post-Carleman estimates” suggested by Isakov in 2006.     

Recurrence relations for Tchebycheffian B-splines

Four-term recurrence relations with constant coefficients are derived for a wide class of T chebycheffian B-splines, LB-splines and complex B-splines. Such a relation exists whenever the differential operator defining the underlying “polynomial” space can be factored in two essentially different ways. The four lower order B-splines in the recurrence relation appear in two pairs, each pair corresponding to one of these factorization. It is shown that the two-term recurrence relations for polynomial, trigonometric and hyperbolic B-splines as well as other known two-term recurrence relations are obtained directly from the four-term recurrence relations in a unified and systematic way. The above derivation also yields two different two-term recurrence relations for Green’s functions of these “polynomial” spaces In this context the special examples of exponential functions and rational functions are analyzed in detail.     

Further remarks on <Emphasis Type="Italic">δ</Emphasis>- and <Emphasis Type="Italic">θ</Emphasis>-modifications

In generalizing constructions of N.V. Veličko, the paper starts from two generalized topologies μ and μ′ on a set X and introduces two more generalized topologies gd(μ, μ′) and δ(μ,μ′) with the examination of their properties. Research (partially) supported by Hungarian Foundation for Scientific Research, grant Nos. T 49786, T 046846, K 68398.     

Dominated estimates of convex combinations of commuting isometries

The principal result of this paper is that the convex combination of two positive, invertible, commuting isometries ofL _p(X,F, μ) 1<p<+∞, one of which is periodic, admits a dominated estimate with constantp/p−1. In establishing this, the following analogue of Linderholm’s theorem is obtained: Let σ and ε be two commuting non-singular point transformations of a Lebesgue Space with τ periodic. Then given ε>O, there exists a periodic non-singular point transformation σ′ such that σ′ commutes with τ and μ(x:σ′x≠σx}<ε. Byan approximation argument, the principal result is applied to the convex combination of two isometries ofL _p (0, 1) induced by point transformations of the form τx=x ^k,k>0 to show that such convex combinations admit a dominated estimate with constantp/p−1. Research supported in part by NSF Grant No. GP-7475. A portion of the contents of this paper is based on the author’s doctoral dissertation written under the direction of Professor R. V. Chacon of the University of Minnesota.     

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well. Dedicated to Dick Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—05A19; Secondary—33C45     

A note on intersections of free submonoids of a free monoid

According to a theorem of Tilson [6] any intersection of free submonoids of a free monoid is free. Here we consider intersections of the form {x, y}^* ∩ {u, v}^*, where x, y, u and v are words in a finitely generated free monoid Σ^*, and show that if both the monoids {x, y}^* and {u, v}^* are of the rank two, then the intersection is a free monoid generated either by (at most) two words or by a regular language of the form β₀ + β((γ(1+ δ + ... δ^t))^*ε for some words β⁰, β, γ, δ and ε, and some integer t≥0. An example is given showing that the latter possibility may occur for each t≥0 with nonempty values of the words.