An upper bound for a percolation constant

Résumé Dans cette note, nous étudions le temps minimum que prend un fluide pour se répandre à travers un milieu quand les temps nécessaires pour traverser les arcs du milieu sont des variables aléatoires indépendantes les unes des autres et tirées de la distribution uniforme et rectangulaire.     

On a global upper bound for Jensen's inequality

In this paper we shall give a global upper bound for Jensen's inequality without restrictions on the target convex function f. We also introduce a characteristic c(f) i.e. an absolute constant depending only on f, by which the global bound is improved.     

A bound for minimal graphs with a normal at infinity

It is shown that a minimal graph with a normal at infinity is in a-priori bounded vertical distance from its approximating halfcatenoid. This is used to show that the exterior contact angle problem is wellposed under natural geometric conditions on the domain, while the exterior Dirichlet problem can be solvable only for data which satisfy an oscillation bound.This paper was written under the support of the Deutsche Forschungsgemeinschaft while the author was visiting the department of mathematics at Stanford University.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.     

An upper bound for the diameter of a polytope

The distance between two vertices of a polytope is the minimum number of edges in a path joining them. The diameter of a polytope is the greatest distance between two vertices of the polytope. We show that if P is a d-dimensional polytope with n facets, then the diameter of P is at most $\text{1}\text{3}\text{2}{}^{\mathrm{d?3}}\text{(n?d+}\text{5}\text{2}\text{)}$.     

An upper bound for the zero-one knapsack problem and a branch and bound algorithm

A new way of computing the upper bound for the zero-one knapsack problem is presented, substantially improving on Dantzig's approach. A branch and bound algorithm is proposed, based on the above mentioned upper bound and on original backtracking and forward schemes. Extensive computational experiences indicate this new algorithm to be superior to the fastest algorithms known at present.     

An upper bound for the k-domination number of a graph

The k-domination number of a graph G, γk(G), is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k, then γk(G) ≤ kp/(k + 1).     

An upper bound for the minimum rank of a graph

For a graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound is true over for all graphs, and prove a related bound for all zero-nonzero patterns of (not necessarily symmetric) matrices. Most of the results are valid for matrices over any infinite field, but need not be true for matrices over finite fields.     

An upper bound for the path number of a graph

The path number of a graph G, denoted p(G), is the minimum number of edge-disjoint paths covering the edges of G. Lovász has proved that if G has u odd vertices and g even vertices, then p(G) ≤ 1/2 u + g - 1 ≤ n - 1, where n is the total number of vertices of G. This paper clears up an error in Lovász's proof of the above result and uses an extension of his construction to show that p(G) ≤ 1/2 u + [3/4g] ≤ [3/4n].     

An upper bound for the Waring rank of a form

Let \({\phi(n)}\) denote the Euler-totient function. We study the error term of the general k-th Riesz mean of the arithmetical function \({\frac {n}{\phi(n)}}\) for any positive integer \({k \ge 1}\) , namely the error term \({E_k(x)}\) where $${\frac{1}{k!} \sum_{n \leq x} \frac{n}{\phi(n)} \left(1-\frac{n}{x}\right)^k = M_k(x) + E_k(x).}$$ The upper bound for \({| E_k(x)|}\) established here thus improves the earlier known upper bounds for all integers \({k\geq 1}\) .     

An upper bound on the path number of a digraph

A proof is presented of the conjecture of Alspach and Pullman that for any digraph G on n ≥ 4 vertices, the path number of G is at most [$\text{n}{}^2\text{4}$].     

The upper connected geodetic number and forcing connected geodetic number of a graph

For a connected graph G of order p≥2, a set S⊆V(G) is a geodetic set of G if each vertex v∈V(G) lies on an x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A connected geodetic set of G is a geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number of G and is denoted by g_c(G). A connected geodetic set of cardinality g_c(G) is called a g_c-set of G. A connected geodetic set S in a connected graph G is called a minimal connected geodetic set if no proper subset of S is a connected geodetic set of G. The upper connected geodetic number is the maximum cardinality of a minimal connected geodetic set of G. We determine bounds for and determine the same for some special classes of graphs. For positive integers r,d and n≥d+1 with r≤d≤2r, there exists a connected graph G with , and . Also, for any positive integers 2≤a<b≤c, there exists a connected graph G such that g(G)=a, g_c(G)=b and . A subset T of a g_c-set S is called a forcing subset for S if S is the unique g_c-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected geodetic number of S, denoted by f_c(S), is the cardinality of a minimum forcing subset of S. The forcing connected geodetic number of G, denoted by f_c(G), is f_c(G)=min{f_c(S)}, where the minimum is taken over all g_c-sets S in G. It is shown that for every pair a,b of integers with 0≤a≤b−4, there exists a connected graph G such that f_c(G)=a and g_c(G)=b.     

An upper bound for a valence of a face in a parallelohedral tiling

Consider a face-to-face parallelohedral tiling of R^d${\mathbb{R}}^d$ and a (d−k)$(d-k)$-dimensional face F$F$ of the tiling. We prove that the valence of F$F$ (i.e. the number of tiles containing F$F$ as a face) is not greater than 2^k$2^k$. If the tiling is affinely equivalent to a Voronoi tiling for some lattice (the so called Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay k$k$-cell. Yet we emphasize that such an affine equivalence is not assumed in the proof.     

Curvature-free upper bounds for the smallest area of a minimal surface

In this paper we will present two upper estimates for the smallest area of a possibly singular minimal surface in a closed Riemannian manifold Mⁿ with a trivial first homology group. The first upper bound will be in terms of the diameter of Mⁿ, the second estimate will be in terms of the filling radius of a manifold, leading also to the estimate in terms of the volume of Mⁿ. If n = 3 our upper bounds are for the smallest area of a smooth embedded minimal surface. After that we will establish similar upper bounds for the smallest volume of a stationary k-dimensional integral varifold in a closed Riemannian manifold Mⁿ with . The above results are the first results of such nature. Received: October 2004 Revision: May 2005 Accepted: June 2005     

On a generalization of an upper bound for the exponential function

With the introduction of a new parameter n, Kim recently generalized an upper bound for the exponential function that implies the inequality between the arithmetic and geometric means. In this paper, we answer some of Kim's conjectures about the inequalities between Kim's generalized upper bound and the original one. We also see the validity of Kim's generalization for some further negative values of x for the case in which the n is rational with both numerator and denominator odd. The range of its validity for negative x is investigated through the study of the zero distribution of a certain family of quadrinomials.