Let G be a finite abelian group with exponent n. Let s(G) denote the smallest integer l such that every sequence over G of length at least l has a zero-sum subsequence of length n. For p-groups whose exponent is odd and sufficiently large (relative to Davenport’s constant of the group) we obtain an improved upper bound on s(G), which allows to determine s(G) precisely in special cases. Our resu...

Let E be an elliptic curve defined over Fq, the finite field of q elements. We show that for some constant η > 0 depending only on q, there are infinitely many positive integers n such that the exponent of E(Fqn), the group of Fqn-rational points on E, is at most q exp ( −n log logn ) . This is an analogue of a result of R. Schoof on the exponent of the group E(Fp) of Fp-rational points, when a...

We study the average number A (n) per site of the number of different configurations of a branched polymer of n bonds on the Given-Mandelbrot family of fractals using exact real-space renormalization. Different members of the family are characterized by an integer parameter b , 2 < or = b < or = infinity . The fractal dimension varies from log(2) 3 to 2 as b is varied from 2 to infinity. We fin...

Let D = (V ,E) be a primitive digraph. The local exponent of D at a vertex u ∈ V , denoted by exp D (u), is defined to be the least integer k such that there is a directed walk of length k from u to v for each v ∈ V . Let V = {1, 2, . . ., n}. The vertices of V can be ordered so that exp D (1) 6 exp D (2) 6 · · · 6 exp D (n) = γ (D). We define the kth local exponent set En(k) := {expD(k) | D ∈ ...

I develop n-site cluster approximations for a stochastic sandpile in one dimension. A height restriction is imposed to limit the number of states: each site can harbor at most two particles (height zi ≤ 2). (This yields a considerable simplification over the unrestricted case, in which the number of states per site is unbounded.) On the basis of results for n ≤ 11 sites, I estimate the critical...

We study the asymptotic shape of self-avoiding random walks (SAW) on the backbone of the incipient percolation cluster in d-dimensional lattices analytically. It is generally accepted that the configurational averaged probability distribution function for the end-to-end distance r of an N step SAW behaves as a power law for r-->0. In this work, we determine the corresponding exponen...

The exponent of a string is the quotient of its length over its smallest period. The exponent and the period of a string can be computed in time proportional to the string length. We design an algorithm to compute the maximal exponent of all factors of an overlap-free string. Our algorithm runs in lineartime on a fixed-size alphabet, while a naive solution of the question would run in cubic tim...

Scales for perceived egocentric distance produced by three psychophysical methods in four ranges of distances were compared. It was found that (a) the exponents produced by ratio and fractionation methods are in good agreement; (b) the exponents of both these methods were larger than those produced by magnitude estimation; (c) an increase in range of distance was associated with a decrease in e...

Two-dimensional mound coarsening is studied using a continuum model in which the effects of deposition noise have been included but in which coalescence due to mass transfer is assumed to be negligible as expected at low temperature. In agreement with scaling arguments for the case of fluctuation-dominated mound coarsening, we find n = β = 1/3, where n is the mound coarsening exponent and β is ...

Example 1. Set f1 = x d 1 and fi = xi−1 − x d i for i = 2, . . . , n. Then Φ(x) := maxi{|fi(x)|} > 0 for x 6= 0. Let p(t) = (t d , t n−2 , . . . , t). Then limt→0 ||p(t)||/|t| = 1 and Φ(p(t)) = t d . Thus the Lojasiewicz exponent is ≥ d. (In fact it equals d.) This works both over R and C. In the real case set F = ∑ f 2 i . Then degF = 2d, F has an isolated real zero at the origin and the Lojas...