Non-trivial simple poles at negative integers and mass concentration at singularity

Let (X,0) be the germ of a normal space of dimension n+1 and let f be the germ at 0 of a holomorphic function on X. Assume both X and f have an isolated singularity at 0. Denote by J the image of the restriction map , where F is the Milnor fibre of f at 0. We prove that the canonical Hermitian form on , given by poles of order at in the meromorphic extension of , passes to the quotient by J and is non-degenerate on . We show that any non-zero element in J produces a “mass concentration” at the singularity which is related to a simple pole concentrated at for (in a non-na?ve sense). We conclude with an application to the asymptotic expansion of oscillatory integrals , for , when . Received: 28 May 2001 / Published online: 26 April 2002     

Dense Minors In Graphs Of Large Girth

We show that a graph of girth greater than 6 log k+3 and minimum degree at least 3 has a minor of minimum degree greater than k. This is best possible up to a factor of at most 9/4. As a corollary, every graph of girth at least 6 log r+3 log log r+c and minimum degree at least 3 has a K _r minor.     

Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Let A and B be C*-algebras. A linear map T : A → B is said to be a *-homomorphism at an element z ∈ A if ab* = z in A implies T (ab*) = T (a)T (b)* = T (z), and c*d = z in A gives T (c*d) = T (c)*T (d) = T (z). Assuming that A is unital, we prove that every linear map T : A → B which is a *-homomorphism at the unit of A is a Jordan *-homomorphism. If A is simple and infinite, then we establish that a linear map T : A → B is a *-homomorphism if and only if T is a *-homomorphism at the unit of A. For a general unital C*-algebra A and a linear map T : A → B, we prove that T is a *-homomorphism if, and only if, T is a *-homomorphism at 0 and at 1. Actually if p is a non-zero projection in A, and T is a ^?-homomorphism at p and at 1 ? p, then we prove that T is a Jordan *-homomorphism. We also study bounded linear maps that are *-homomorphisms at a unitary element in A.     

Cauchy’s residue theorem for a class of real valued functions

Let [a, b] be an interval in ℝ Rand let F be a real valued function defined at the endpoints of [a, b] and with a certain number of discontinuities within [a, b]. Assuming F to be differentiable on a set [a, b] | E to the derivative f, where E is a subset of [a, b] at whose points F can take values ±∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that KH-vt ∝ _a ^b f = F(b) − F(a), where KH-vt denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate the theory.     

Intersection Divisors of a Canonically Embedded Curve with its Osculating Spaces

We present a new result on the geometry of nonhyperelliptic curves; namely, the intersection divisors of a canonically embedded curve C with its osculating spaces at a point P, not considering the intersection at P, can only vary in dimensions given by the Weierstrass semigroup of the curve C at P. We obtain, under a reasonable geometrical hypothesis, monomial bases for the spaces of higher-order regular differentials. We also give a sufficient condition on the Weierstrass semigroup of C at P in order for this geometrical hypothesis to be true. Finally, we give examples of Weierstrass semigroups satisfying this condition.     

Cycle Cover Ratio of Regular Matroids

A cycle in a matroid is a disjoint union of circuits. This paper proves that every regular matroidM without coloops has a set S of cycles whose union is E(M) such that every element is in at most three of the cycles in S. It follows immediately from this that, on average, each element of M is in at most three members of the cycle cover S. This improves on a 1989 result of Jamshy and Tarsi who proved that there is a cycle cover for which this average is at most 4, and conjectured that a cycle cover exists for which the average is at most 2.     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

On an Extended Lagrange Claim

Lagrange once made a claim having the consequence that a smooth function f has a local minimum at a point if all the directional derivatives of f at that point are nonnegative. That the Lagrange claim is wrong was shown by a counterexample given by Peano. In this note, we show that an extended claim of Lagrange is right. We show that, if all the lower directional derivatives of a locally Lipschitz function f at a point are positive, then f has a strict minimum at that point.     

The Asymptotic Workload Behavior of Two Coupled Queues

We consider a system of two coupled queues Q ₁ and Q ₂. When both queues are backlogged, they are each served at unit rate. However, when one queue empties, the service rate at the other queue increases. Thus, the two queues are coupled through the mechanism for dynamically sharing surplus service capacity. We derive the asymptotic workload behavior at Q ₁ for various scenarios where at least one of the two queues has a heavy-tailed service time distribution. First of all, we consider a situation where the traffic load at Q ₁ is below the nominal unit service rate. We show that if the service time distribution at Q ₁ is heavy-tailed, then the workload behaves exactly as if Q ₁ is served in isolation at a constant rate, which only depends on the service time distribution at Q ₂ through its mean. In addition, we establish that if the service time distribution at Q ₁ is exponential, then the workload distribution is either exponential or semi-exponential, depending on whether the traffic load at Q ₂ exceeds the nominal service rate or not. Next, we focus on a regime where the traffic load at Q ₁ exceeds the nominal service rate, so that Q ₁ relies on the surplus capacity from Q ₂ to maintain stability. In that case, the workload distribution at Q ₁ is determined by the heaviest of the two service time distributions, so that Q ₁ may inherit potentially heavier-tailed characteristics from Q ₂.     

Every Graph of Sufficiently Large Average Degree Contains a <Emphasis Type="Italic">C</Emphasis><Subscript>4</Subscript>-Free Subgraph of Large Average Degree

We prove that for every k there exists d=d(k) such that every graph of average degree at least d contains a subgraph of average degree at least k and girth at least six. This settles a special case of a conjecture of Thomassen.     

Coalescing and annihilating random walk with ‘action at a distance’

We consider a system of particles which perform continuous time random walks onZ ^d . These random walks are independent as long as no two particles are at the same site or adjacent to each other. When a particle jumps from a site x to a sitey and there is already another particle aty or at some neighbory′ ofy, then there is an interaction. In the coalescing model, either the particle which just jumped toy is removed (or, equivalently, coalesces with a particle aty ory′) or all the particles at the sites adjacent toy (other thanx) are removed. In the annihilating random walk, the particle which just jumped toy and one particle aty ory′ annihilate each other. We prove that when the dimensiond is at least 9, then the density of this system is asymptotically equivalent toC/t for some constant C, whose value is explicitly given.     

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.     

Extremal graphs of diameter at most 6 after deleting any vertex

Suppose G is a graph of n vertices and diameter at most d having the property that, after deleting any vertex, the resulting subgraph has diameter at most 6. Then G contains at least max{n, (4n - 8)/3} edges if 4 ≤ d ≤ 6.     

Broadcasting in geometric radio networks

We consider deterministic broadcasting in geometric radio networks (GRN) whose nodes know only a limited part of the network. Nodes of a GRN are situated in the plane and each of them is equipped with a transmitter of some range r. A signal from this node can reach all nodes at distance at most r from it but if a node u is situated within the range of two nodes transmitting simultaneously, then a collision occurs at u and u cannot get any message. Each node knows the part of the network within knowledge radius s from it, i.e., it knows the positions, labels and ranges of all nodes at distance at most s.The aim of this paper is to study the impact of knowledge radius s on the time of deterministic broadcasting in a GRN with n nodes and eccentricity D of the source. Our results show sharp contrasts between the efficiency of broadcasting in geometric radio networks as compared to broadcasting in arbitrary graphs. They also show quantitatively the impact of various types of knowledge available to nodes on broadcasting time in GRN. Efficiency of broadcasting is influenced by knowledge radius, knowledge of individual positions when knowledge radius is zero, and awareness of collisions.     

Forcing highly connected subgraphs

A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex‐degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex‐degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2^k at the vertices and a vertex‐degree of order k log k at the ends which have no k‐connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge‐degree for the ends (which is defined as the maximum number of edge‐disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge‐degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)‐edge‐connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007     

Positive and maximal positive solutions of singular mixed boundary value problem

The paper is concerned with existence results for positive solutions and maximal positive solutions of singular mixed boundary value problems. Nonlinearities h(t;x;y) in differential equations admit a time singularity at t=0 and/or at t=T and a strong singularity at x=0.     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

The Serendipity Family of Finite Elements

We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s−r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r−2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.     

The tangent approximation to one-sided Brownian exit densities

Summary Let p(t) be the density of the first-exit time of a Brownian motion over a one-sided moving boundary, and let p ₁(t) be the density at t of the first-exit time over the tangent to the boundary at t. When is p ₁(t) a good approximation to p(t)? We investigate this question by means of a new integral equation for p(t) which makes possible explicit estimates for the error of the approximation.Work supported by the Deutsche Forschungsgemeinschaft at the Sonderforschungsbereich 123, Universität Heidelberg