On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

The (Δ,d,d′, Δ − 1)-problem with applications to computer networks

The (,d, d, – 1)-problem is that of finding large graphs with maximum degree and diameterd such that the subgraphs obtained by deleting any set of up to – 1 vertices have diameterd. In this paper, we deduce upper bounds on the order of such graphs and present some of the largest known ones. We argue that these graphs can be used to construct extremely "robust" networks, and explain why we require this robustness property when designing transputer networks for certain applications. In particular, we investigate the suitability of the odd graphO ₄ as a topology for such networks.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

On a coupled system of abstract differential equations similar to the thermoelasticity equations

The study of a model system of differential equations arising from the dynamical problems of thermo elasticity is continued. The case of shifts of the general type is investigated. We employ the commutant method based on the properties of the operator (A,B)=AB-BA.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 8, pp. 1162–1165, August, 1993.     

Perturbation bounds for theLDL H andLU decompositions

LetA andA+A be Hermitian positive definite matrices. Suppose thatA=LDL ^H and (A+A)=(L+L)(D+D)(L+L)^H are theLDL ^H decompositons ofA andA+A, respectively. In this paper upper bounds on |D| _F and |L| _F are presented. Moreover, perturbation bounds are given for theLU decomposition of a complexn ×n matrix.     

Optimal growth rates for a viscous Hamilton-Jacobi equation

The growth of the L^m-norm, m [1,], of non-negative solutions to the Cauchy problem _t u – u = |u| is studied for non-negative initial data decaying at infinity. More precisely, the function is shown to be bounded from above and from below by positive real numbers. This result indicates an asymptotic behaviour dominated by the hyperbolic Hamilton-Jacobi term of the equation. A one-sided estimate for ln u is also established.     

Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .     

G-Uniformities,LR-Proximities and Hypertopologies

We look closely at the relationships between hit-and-miss and proximal hit-and-miss -topologies, in the setting of proximity spaces. We provide equivalent conditions that force comparisons among proximal hit-and-miss -topologies determined by different proximities. We pay attention to these topologies when consists of the family of all closed balls of a proximity space, and we study their interplay with the Wijsman convergence expressed in proximity spaces. Finally we study the supremum of all Wijsman convergences and of all proximal ball topologies when X is at most regular, and the infimum of all Wijsman convergences when X is at least Tychonoff.     

Isoperimetric inequalities in a boundary value problem on a Riemannian manifold

In this paper the problem u+1=0 in ,u=0 on is considered. Here is a finite domain on a Riemannian manifold and the associated Laplace-Beltrami operator. By means of maximum principles isoperimetric bounds for the maximum ofu and the maximum of the absolute value of the gradient ofu, as well as some related bounds are derived.

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Zusammenfassung Diese Arbeit behandelt das Problem u+1=0 in ,u=0 auf , wobei ein Gebiet auf einer zweidimensionalen Riemann'schen Mannigfaltigkeit ist, und der zugehörige Laplace-Beltrami Operator. Es werden isoperimetrische Schranken für das Maximum vonu und |u| aus gewissen Maximumsprinzipien hergeleitet, sowie einige verwandte Resultate.

     

Gröbner bases and triangulations of the second hypersimplex

The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex (2,n). We present a quadratic Gröbner basis for the associated toric idealK(K _n ). The simplices in the resulting triangulation of (2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK _n . Forn6 the number of distinct initial ideals ofI(K _n ) exceeds the number of regular triangulations of (2,n); more precisely, the secondary polytope of (2,n) equals the state polytope ofI(K _n ) forn5 but not forn6. We also construct a non-regular triangulation of (2,n) forn9. We determine an explicit universal Gröbner basis ofI(K _n ) forn8. Potential applications in combinatorial optimization and random generation of graphs are indicated.Research partially supported by a doctoral fellowship of the National University of Mexico, the National Science Foundation, the David and Lucile Packard Foundation and the U.S. Army Research Office (through ACSyAM/MSI, Cornell).     

On the Drinfeld discriminant function

The discriminant function is a certain rigid analytic modularform defined on Drinfelds upper half-plane . Its absolutevalue may be considered as a function on theassociated Bruhat–Tits tree T. We compare log with the conditionally convergent complex-valued Eisenstein series Edefined on T and thereby obtain results about the growth of and of some related modular forms. We further determine to what extent roots may be extracted of (z)/(nz),regarded as a holomorphic function on . In some cases, this enables us to calculate cuspidal divisor class groups of modular curves.     

The local nature of Δ-coloring and its algorithmic applications

Given a connected graphG=(V, E) with |V|=n and maximum degree such thatG is neither a complete graph nor an odd cycle, Brooks' theorem states thatG can be colored with colors. We generalize this as follows: letG-v be -colored; then,v can be colored by considering the vertices in anO(log n) radius aroundv and by recoloring anO(log n) length augmenting path inside it. Using this, we show that -coloringG is reducible inO(log³ n/log) time to (+1)-vertex coloringG in a distributed model of computation. This leads to fast distributed algorithms and a linear-processorNC algorithm for -coloring.A preliminary version of this paper appeared as part of the paper Improved Distributed Algorithms for Coloring and Network Decomposition Problems, in theProceedings of the ACM Symposium on Theory of Computing pages 581–592, 1992. This research was done when the authors were at the Computer Science Department of Cornell University. The research was supported in part by NSF PYI award CCR-89-96272 with matching funds from UPS and Sun Microsystems.     

Construction of po-groups with quasi-divisors theory

A method is presented making it possible to construct po-groups with a strong theory of quasi-divisors of finite character and with some prescribed properties as subgroups of restricted Hahn groups H(, ), where are finitely atomic root systems. Some examples of these constructions are presented.     

Solution of spectral problems for polynomial matrices

For polynomial matrices of full rank, including matrices of the form A - I and A - B, numerical methods for solving the following problems are suggested: find the divisors of a polynomial matrix whose spectra coincide with the zeros of known divisors of its characteristic polynomial; compute the greatest common divisor of a sequence of polynomial matrices; solve the inverse eigenvalue problem for a polynomial matrix. The methods proposed are based on the W and V factorizations of polynomial matrices. Applications of these methods to the solution of certain algebraic problems are considered. Bibliography: 3 titles._________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 122–138.     

Existence and construction of Δ1-splines of class Ck on a three-directional mesh

Let ^* be the equilateral triangulation of the plane and let ₁ ^* be the equilateral triangle formed by four triangles of ^*. We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ ^*, having a sufficiently high degree n and which are invariant with respect to the group of symmetries of ₁ ^*. Such splines are called ₁ ^*-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁ ^*-splines of class C ^k and minimal degree, but these splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines.     

On a Problem of Karpilovsky

Let G be a finite elementary group. Let ⁿ (G) denote the nth power of the augmentation ideal (G) of the integral group ring G. In this paper, we give an explicit basis of the quotient group Q_n(G) = ⁿ(G)/ⁿ⁺¹ (G) and compute the order of Qⁿ (G).2000 Mathematics Subject Classification: 16S34, 20C05     

Field Extensions and Extensions of Linear Differential Operators

Let K R P be a tower of fields, N be a P-module, and : R N be a K-linear differential operator. The aim of this paper is to investigate whether the operator has an extension to P, i.e. if these exists a differential operator : P N such that |_R = . The results of this paper were published in Russian in Mat. Zametki 30(2) (1981), 237–248.     

Absence of Continuous Spectral Types for Certain Non-Stationary Random Schrödinger Operators

We consider continuum random Schrödinger operators of the type H = – + V₀ + V with a deterministic background potential V₀. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of – + V₀. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (random tube) in arbitrary dimension.submitted 07/04/04, accepted 19/08/04     

Die Dupinschen Zykliden im pseudoeuklidischen Raum

In the pseudoeuclidean space¹ E ³ a surface of degree 3 or 4 is calledDupin-cyclide if there exists a (pseudoeuclidean) torus such that a (pseudoeuclidean) spherical inversion maps upon . If the axis of is respectively space-like, isotropic or time-like is calledDupin-r-,-l- or-z-cyclide. ADupin-cyclide is the envelope of two families of spheres, the caustic surfaces of degenerate in caustic curves and the lines of curvature of constitute two (orthogonal) families of circles.     

Blow-up rates for parabolic systems

Let ⁿ be a bounded domain andB _R be a ball in ⁿ of radiusR. We consider two parabolic systems: ut=u +f(), i= +g(u) in × (0,T) withu=v=0 on × (0,T) andu _t =u, v _t =v inB _r × (0,T) withe/v=f (v), e/v=g(u) onB _R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.