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Brownian sheet and capacity
, 1999
"... Summary. The main goal of this paper is to present an explicit capacity estimate for hitting probabilities of the Brownian sheet. As applications, we determine the escape rates of the Brownian sheet, and also obtain a local intersection equivalence between the Brownian sheet and the additive Brownia ..."
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Cited by 21 (9 self)
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Summary. The main goal of this paper is to present an explicit capacity estimate for hitting probabilities of the Brownian sheet. As applications, we determine the escape rates of the Brownian sheet, and also obtain a local intersection equivalence between the Brownian sheet and the additive Brownian motion. Other applications concern quasiâ€“sure properties in Wiener space.
On the Structure of Dense TriangleFree Graphs
, 1997
"... As a consequence of an early result of Pach we show that every maximal trianglefree graph is either homomorphic with a member of a specific infinite sequence of 3colourable graphs or it contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations ..."
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Cited by 11 (1 self)
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As a consequence of an early result of Pach we show that every maximal trianglefree graph is either homomorphic with a member of a specific infinite sequence of 3colourable graphs or it contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that if a (not necessarily maximal) trianglefree graph of order n has minimum degree ffi n=3 then the graph is either homomorphic with a member of the indicated family or contains the Petersen graph with one edge contracted. As a corollary we get a recent result due to Chen, Jin and Koh. Finally, we show that every trianglefree graph with ffi ? n=3 is either homomorphic with C5 or it contains the Mobius ladder as a subgraph. A major tool is the observation that every trianglefree graph with ffi n=3 has a unique maximal trianglefree supergraph.
Point Sets with Distinct Distances
, 1995
"... For positive integers d and n let fd(n) denote the maximum cardinality of a subset of the n d grid f1; 2; : : : ; ng d with distinct mutual euclidean distances. Improving earlier results of Erdos and Guy, it will be shown that f2(n) c \Delta n 2=3 and, for d 3, that fd(n) cd \Delta n 2=3 ..."
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Cited by 7 (1 self)
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For positive integers d and n let fd(n) denote the maximum cardinality of a subset of the n d grid f1; 2; : : : ; ng d with distinct mutual euclidean distances. Improving earlier results of Erdos and Guy, it will be shown that f2(n) c \Delta n 2=3 and, for d 3, that fd(n) cd \Delta n 2=3 \Delta (ln n) 1=3 , where c; cd ? 0 are constants. Also improvements of lower bounds of Erdos and Alon on the size of Sidonsets in f1 2 ; 2 2 ; : : : ; n 2 g are given. Furthermore, it will be proven that any set of n points in the plane contains a subset with distinct mutual distances of size c1 \Delta n 1=4 , and for point sets in general position, i.e. no three points on a line, of size c2 \Delta n 1=3 with constants c1 ; c2 ? 0. To do so, it will be shown that for n points in R 2 with distinct distances d1 ; d2 ; : : : ; d t , where d i has multiplicity m i , one has P t i=1 m 2 i c \Delta n 3:25 for a positive constant c. If the n points are in general position,...
On The Oracle Complexity Of Factoring Integers
 COMPUTATIONAL COMPLEXITY
, 1996
"... The problem of factoring integers in polynomial time with the help of an (infinitely powerful) oracle who answers arbitrary questions with yes or no is considered. The goal is to minimize the number of oracle questions. Let N be a given composite nbit integer to be factored, where n = dlog 2 ..."
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Cited by 6 (0 self)
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The problem of factoring integers in polynomial time with the help of an (infinitely powerful) oracle who answers arbitrary questions with yes or no is considered. The goal is to minimize the number of oracle questions. Let N be a given composite nbit integer to be factored, where n = dlog 2 Ne. The trivial method of asking for the bits of the smallest prime factor of N requires n/2 questions in the worst case. A nontrivial algorithm of Rivest and Shamir requires only n/3 questions for the special case where N is the product of two n/2bit primes. In this paper, a polynomialtime oracle factoring algorithm for general integers is presented which, for any ffl ? 0, asks at most ffln oracle questions for sufficiently large N , thus solving an open problem posed by Rivest and Shamir. Based on a plausible conjecture related to Lenstra's conjecture on the running time of the elliptic curve factoring algorithm it is shown that the algorithm fails with probability at most N ...