The Wide Partition Conjecture and the Atom Problem in Discrete Tomography

The Wide Partition Conjecture (WPC) was introduced by Chow and B. Taylor as an attempt to prove inductively Rotaʼs Basis Conjecture, and in the simplest case tries to characterize partitions whose Young diagram admits a “Latin” filling. Chow et al. [T. Chow, C. K. Fan, M. Goemans, and J. Vondrak. Wide partitions, latin tableaux, and rotaʼs basis conjecture. Adv. in Appl. Math., 31(2):334–358, 2003] showed how the WPC is related to problems such as edge-list coloring and multi commodity flow. As far as we know, the conjecture remains widely open.We show that the WPC can be formulated using the k-atom problem in Discrete Tomography [C. Dürr, F. Guíñez, and M. Matamala. Reconstructing 3-Colored Grids from Horizontal and Vertical Projections is NP-Hard: A Solution to the 2-Atom Problem in Discrete Tomography. SIAM J Discrete Math, 26(1):330, 2012.]. In this approach, the WPC states that the sequences arising from partitions admit disjoint realizations if and only if any combination of them can be realizable independently. This realizability condition is not sufficient in general. A stronger condition, the saturation condition, was used in [F. Guíñez, M. Matamala, and S. Thomassé. Realizing disjoint degree sequences of span at most two: A tractable discrete tomography problem. Discrete Appl.Math., 159(1):23–30, 2011] to solve instances were the realizability condition fails. We prove that in our case, the saturation condition is satisfied providing the realizability condition does. Moreover, we show that the saturation condition can be obtained from the Langrangean dual of a natural LP formulation of the k-atom problem.     

Induced Projective Modules Over Group von Neumann Algebras

Let G be a group and k a subring of the field of complex numbers. In this paper we study the additive map in reduced K-theory, which is associated with the inclusion of the group algebra kG into the group von Neumann algebra G, and obtain necessary and sufficient conditions for it to be identically zero (or zero modulo torsion). Our results complete the work of Eckmann [Comm. Math. Helvet. 71 (1996), 453–462; Arch. Math. 76 (2001), 241–249] and Schafer [K-theory 19 (2000), 211–217], while reducing to Swan’s theorem on induced representations [Ann. Math. 71 (1960), 552–578], in the case where the group G is finite. (Received: January 2005) Research supported by University of Athens grant Pythagoras 70/3/7298.     

Grassmannian Trilogarithms

In the previous works of the first author, two completely different constructions of single valued Grassmannian trilogarithms were given. One of the constructions, in Math. Res. Lett. 2 (1995), 99–114, is very simple and provides Grassmannian n-logarithms for all n. However its motivic nature is hidden. The other construction in Adv. in Math. 114 (1995), 197–318, is very explicit and motivic, but the generalization for n>4 is not known. In this paper we will compute explicitly the Grassmannian trilogarithm constructed in Math. Res. Lett. 2 (1995), 99–114 and prove that it differs from the motivic Grassmannian trilogarithm by an explicitly given product of logarithms. We also derive some general results about the Grassmannian polylogarithms.     

Local energy decay for linear wave equations with non‐compactly supported initial data

A local energy decay problem is studied to a typical linear wave equation in an exterior domain. For this purpose, we do not assume any compactness of the support on the initial data. This generalizes a previous famous result due to Morawetz (Comm. Pure Appl. Math. 1961; 14 :561–568). In order to prove local energy decay we mainly apply two types of new ideas due to Ikehata–Matsuyama (Sci. Math. Japon. 2002; 55 :33–42) and Todorova–Yordanov (J. Differential Equations 2001; 174 :464). Copyright © 2004 John Wiley & Sons, Ltd.     

Estimates for domains of local invertibility of diffeomorphisms

Using a novel Wintner-type formulation of the classical Peano's existence theorem [Math. Ann. 37 (1890), 182-228], we enhance Wazewski's result on invertibility of maps defined on closed balls [Ann. Soc. Pol. Math. 20 (1947), 81-125] securing the size of the domain of invertibility that agrees with the bounds derived by John [Comm. Pure Appl. Math. 21 (1968), 77-110] and Sotomayor [Z. Angew. Math. Phys. 41 (1990), 306-310].

     

On a Conjectured Formula for Quiver Varieties

In A.S. Buch and W. Fulton [Invent. Math. 135 (1999), 665–687] a formula for the cohomology class of a quiver variety is proved. This formula writes the cohomology class of a quiver variety as a linear combination of products of Schur polynomials. In the same paper it is conjectured that all of the coefficients in this linear combination are non-negative, and given by a generalized Littlewood-Richardson rule, which states that the coefficients count certain sequences of tableaux called factor sequences. In this paper I prove some special cases of this conjecture. I also prove that the general conjecture follows from a stronger but simpler statement, for which substantial computer evidence has been obtained. Finally I will prove a useful criterion for recognizing factor sequences.     

Minimal Submanifolds in a Locally Symmetric and Conformally Flat Space

We study minimal submanifolds in the locally symmetric and conformally flat Riemannian manifold and generalize Yau's result obtained in J. Amer. Math. 97 (1975), 76–100.     

Books and Teaching Aids Received

Bachert , Russel E., Jr ., Hundreds of Ideas for Outdoor Education. Waters , John F., Hatchlings: The Lives of Baby Turtles. Cornett , Charles AND Cornett , Claudia , Reading Math. Greenlaw , James , History of Mathematics—18 Posters.     

More on Kleinewillinghöfer Types of Flat Laguerre Planes

Polster and Steinke [Result. Math., 46 (2004), 103–122] determined the possible Kleinewillingh?fer types of flat Laguerre planes. These types reflect transitivity properties of groups of certain central automorphisms. We exclude three more types from the list given there with respect to Laguerre homotheties. This yields a complete determination of all possible single types with respect to Laguerre homotheties that can occur in flat Laguerre planes. Building on results by M?urer and Hartmann to characterize ovoidal or miquelian Laguerre planes we further characterize certain flat Laguerre planes in terms of their Kleinewillingh?fer types. Received: January 16, 2007. Revised: July 26, 2007.     

Three-dimensional FC Artin Groups are CAT(0)

Building upon earlier work of T. Brady, we construct locally CAT(0) classifying spaces for those Artin groups which are three-dimensional and which satisfy the FC (flag complex) condition. The approach is to verify the ‘link condition’ by applying gluing arguments for CAT(1) spaces and by using the curvature testing techniques of Elder and McCammond [Expositio Math. 11(1) (2002), 143–158].     

On the uniform decay for the Korteweg–de Vries equation with weak damping

The aim of this work is to consider the Korteweg–de Vries equation in a finite interval with a very weak localized dissipation namely the H^?1‐norm. Our main result says that the total energy decays locally uniform at an exponential rate. Our analysis improves earlier works on the subject (Q. Appl. Math. 2002; LX (1):111–129; ESAIM Control Optim. Calculus Variations 2005; 11 (3):473–486) and gives a satisfactory answer to a problem suggested in (Q. Appl. Math. 2002; LX (1):111–129). Copyright © 2007 John Wiley & Sons, Ltd.     

A perturbation result for bounded imaginary powers

We give a new proof of a perturbation result due to J. Prüss and H. Sohr [11]: if an operator A has bounded imaginary powers, then so does A+w (w ≧ 0). Instead of Mellin transform on which the proof in [11] is based, we use the functional calculus for sectorial operators developed in particular by A. McIntosh ([8], [3] and [1]). It turns out that our method gives a more general result than the one used in [11].     

Tame Kernels for Biquadratic Number Fields

Let and , d₁ and d₂ squarefree integers, be an imaginary field and a biquadratic field, respectively. Let S be the set consisting of all infinite primes, all dyadic primes and all finite primes which ramify in E. Suppose the 4-rank of the class group of F is zero and the S-ideal class group of F has odd order, we give the forms of all elements of order ⩽ 2 in K₂O_E and use the Hurrelbrink and Kolster’s method [Hurrelbrink, J. and Kolster, M.: J. reine angew. Math. 499 (1998), 145–188] to obtain the forms of all elements of order 4 in K₂O_E. Received: September 2004     

S. G Mikhlin (1908 – 1990)

Our concern is to find a representation theorem for operators in B(c(X), c(Y)) where X and Y are Banach spaces with Y containing an isomorphic copy of c₀. Cass and Gao [1] obtained a representation theorem that always applies if Y does not contain an isomorphic copy of c₀. Maddox [3], Melvin - Melvin [4], and Robinson [5] consider operators in B(c(X), c(Y)) that are given by matrices. In this paper we show that Cass's and Gao's result in [1] can be extended, when Y contains an isomorphic copy of c₀, to certain operators that we call represent able. In addition, we show that when Y contains an isomorphic copy of co there are always operators that fall outside the scope of our representation theorem. Light is also cast on a theorem given in Maddox [3, Theorem 4.2] and [5, Theorem IV].     

On mappings into ℝ2ℓ

LetM be a compact orientable manifold. We know how to calculateX(M), the Euler characteristic ofM, from a stable mapf: M→R, with information only onS(f), the singular set off. This result was extended to stable maps into the plane by H. Levine [L-2] whenM has dimension 2n, and it is also calculated fromS(f). The purpose of this work is to generalize the above result for maps intoR ^2l, wheren≥l. In this caseS(f) is not a manifold. We use the process of resolution of singularities [L-3] to get a homomorphism having only singularities of codimension 1 and use simmilar technics as in [L-2]. Supported by FAPESP and FINEP.     

On Aki's nonparametric test for symmetry

Summary Aki (1987,Ann. Inst. Statist. Math,39, 457–472) develops a theory of extending the test for symmetry about zero of a continuous distribution functionF. In this paper we discuss the same testing problem in the case where the probabilityF(0) of being negative is unknown, which is assumed to be known in Aki's paper.     

Well‐posedness and asymptotic analysis for a Penrose–Fife type phase field system

In this paper, an asymptotic analysis of the (non‐conserved) Penrose–Fife phase field system for two vanishing time relaxation parameters ε and δ is developed, in analogy with the similar analyses for the phase field model proposed by G. Caginalp (Arch. Rational Mech. Anal. 1986; 92 :205–245), which were carried out by Rossi and Stoth (Adv. Math. Sci. Appl. 2003; 13 :249–271; Quart. Appl. Math. 1995; 53 :695–700). Although formally the singular limits for ε ↓ 0 and for ε and δ ↓ 0 are, respectively, the viscous Cahn–Hilliard equation and the Cahn–Hilliard equation, it turns out that the Penrose–Fife system is indeed a bad approximation for these equations. Therefore, we consider an alternative approximating phase field system, which could be viewed as a generalization of the classical Penrose–Fife phase field system, featuring a double non‐linearity given by two maximal monotone graphs. A well‐posedness result is proved for such a system, and it is shown that the solutions converge to the unique solution of the viscous Cahn–Hilliard equation as ε ↓ 0, and of the Cahn–Hilliard equation as ε ↓ 0 and δ ↓ 0. Copyright © 2004 John Wiley & Sons, Ltd.     

Viability of circle and sphere theorems in potential theory and hydrodynamics via Maxwell's conjecture

In A Treatise on Electricity and Magnetism, Maxwell determines the angles of intersection for which one may use Kelvin's inversion method to obtain the perturbed electric potential upon placing intersecting spherical conductors into a region with a known potential. There are numerous modern applications utilizing this geometric construction in potential theory and hydrodynamics, and generalized circle and sphere theorems play a foundational role in this area of mathematical physics. In his work, Maxwell gives an intuitive argument for obtaining the perturbed potential based on intersecting planar conductors and a spherical inversion, and in this paper we extend his ideas to a full proof using rotational transformations and reflections. In the process, we disprove results in [Proc Lond Math Soc., 1966:3(16)] and [Stud Appl Math., 2001:106(4); Z. Angew. Math. Mech., 2001:81(8)] on boundary value problems in hydrodynamics involving intersecting circles and spheres, and we detail the angles of intersection for which these theorems are viable. Moreover, our proof recovers a special case overlooked by Maxwell for which Kelvin's inversion method may be utilized to obtain full solutions.     

Maximal K3's and Hamiltonicity of 4‐connected claw‐free graphs

Let cl(G) denote Ryjá?ek's closure of a claw‐free graph G. In this article, we prove the following result. Let G be a 4‐connected claw‐free graph. Assume that G[N_G(T)] is cyclically 3‐connected if T is a maximal K₃ in G which is also maximal in cl(G). Then G is hamiltonian. This result is a common generalization of Kaiser et al.'s theorem [J Graph Theory 48(4) (2005), 267–276] and Pfender's theorem [J Graph Theory 49(4) (2005), 262–272]. © 2011 Wiley Periodicals, Inc. J Graph Theory