Results 1  10
of
102
Recent work connected with the Kakeya problem
 Prospects in mathematics (Princeton, NJ
, 1996
"... A Kakeya set in R n is a compact set E ⊂ R n containing a unit line segment in every direction, i.e. ∀e ∈ S n−1 ∃x ∈ R n: x + te ∈ E ∀t ∈ [ − 1 1 ..."
Abstract

Cited by 62 (2 self)
 Add to MetaCart
A Kakeya set in R n is a compact set E ⊂ R n containing a unit line segment in every direction, i.e. ∀e ∈ S n−1 ∃x ∈ R n: x + te ∈ E ∀t ∈ [ − 1 1
Some extremal functions in Fourier analysis
 Bull. Amer. Math. Soc
, 1985
"... Abstract. We obtain extremal majorants and minorants of exponential type for a class of even functions on R which includes log x  and x  α, where −1 < α < 1. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As a ..."
Abstract

Cited by 49 (2 self)
 Add to MetaCart
Abstract. We obtain extremal majorants and minorants of exponential type for a class of even functions on R which includes log x  and x  α, where −1 < α < 1. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional HardyLittlewoodSobolev inequalities. A further application provides an ErdösTurántype inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials. 1.
Smoothness and Decay Properties of the Limiting Quicksort Density Function
, 2000
"... Using Fourier analysis, we prove that the limiting distribution of the standardized random number of comparisons used byQuicksort to sort an array of n numbers has an everywhere positive and infinitely differentiable density f, and that each derivative f (k) enjoys superpolynomial decay at ±∞. In pa ..."
Abstract

Cited by 37 (18 self)
 Add to MetaCart
Using Fourier analysis, we prove that the limiting distribution of the standardized random number of comparisons used byQuicksort to sort an array of n numbers has an everywhere positive and infinitely differentiable density f, and that each derivative f (k) enjoys superpolynomial decay at ±∞. In particular, each f (k) is bounded. Our method is sufficiently computational to prove, for example, that f is bounded by 16.
The Kadison–Singer problem in mathematics and engineering
 Proc. Natl. Acad. Sci. USA 103 (2006
, 2006
"... Abstract. We will show that the famous, intractible 1959 KadisonSinger problem in C ∗algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well ..."
Abstract

Cited by 33 (14 self)
 Add to MetaCart
Abstract. We will show that the famous, intractible 1959 KadisonSinger problem in C ∗algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of KadisonSinger. In some areas we will prove what we believe will be the strongest results ever available in the case that KadisonSinger fails. Finally, we will give some directions for constructing a counterexample to KadisonSinger. 1.
On the Distribution of Free Path Lengths for the Periodic Lorentz Gas
"... Consider the domain Z " = fx 2 Rn j dist(x; "Zn) ? "flg; and let the free path length be defined as ..."
Abstract

Cited by 30 (5 self)
 Add to MetaCart
Consider the domain Z " = fx 2 Rn j dist(x; "Zn) ? "flg; and let the free path length be defined as
On the statistical properties of Diffie–Hellman distributions
 MR 2001k:11258 Zbl 0997.11066
"... Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an rth power residue for all small factors of p − 1. The corresponding DiffieHellman (DH) distribution is (ϑ x, ϑ y, ϑ xy) where x, y are randomly chosen from Z ∗ p. A recently formulated assumption is that giv ..."
Abstract

Cited by 29 (10 self)
 Add to MetaCart
Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an rth power residue for all small factors of p − 1. The corresponding DiffieHellman (DH) distribution is (ϑ x, ϑ y, ϑ xy) where x, y are randomly chosen from Z ∗ p. A recently formulated assumption is that given p, ϑ of the above form it is infeasible to distinguish in reasonable time between DH distribution and triples of numbers chosen
On certain exponential sums and the distribution of DiffieHellman triples
 J. London Math. Soc
, 1999
"... Let g be a primitive root modulo a prime p. It is proved that the triples (gx,gy,gxy), x,y�1,…,p�1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε�0 be fixed. Then p−� x,y=� exp0 2πiagx�bgy�cgxy ..."
Abstract

Cited by 26 (14 self)
 Add to MetaCart
Let g be a primitive root modulo a prime p. It is proved that the triples (gx,gy,gxy), x,y�1,…,p�1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε�0 be fixed. Then p−� x,y=� exp0 2πiagx�bgy�cgxy
The Integer Chebyshev Problem
, 1995
"... . We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most n with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval [0; 1] in most detail ..."
Abstract

Cited by 25 (10 self)
 Add to MetaCart
. We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most n with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval [0; 1] in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal "integer Chebyshev" polynomials, showing for example, that on small intevals [0; ffi] and for small degrees d, x d achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys' and others, as to what the "integer transfinite diameter" of [0; 1] should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due t...
Approximating the Limiting Quicksort Distribution
, 2001
"... The limiting distribution of the normalized number of comparisons used by Quicksort to sort an array of n numbers is known to be the unique xed point with zero mean of a certain distributional transformation S. We study the convergence to the limiting distribution of the sequence of distributions o ..."
Abstract

Cited by 24 (5 self)
 Add to MetaCart
The limiting distribution of the normalized number of comparisons used by Quicksort to sort an array of n numbers is known to be the unique xed point with zero mean of a certain distributional transformation S. We study the convergence to the limiting distribution of the sequence of distributions obtained by iterating the transformation S, beginning with a (nearly) arbitrary starting distribution. We demonstrate geometrically fast convergence for various metrics and discuss some implications for numerical calculations of the limiting Quicksort distribution. Finally, we give companion lower bounds which show that the convergence is not faster than geometric. AMS 2000 subject classications. Primary 68W40; secondary 68P10, 60E05, 60E10, 60F05. Key words and phrases. Quicksort, characteristic function, density, moment generating function, sorting algorithm, coupling, Fourier analysis, Kolmogorv{Smirnov distance, total variation distance, integral equation, numerical analysis, d p metric. Date. January 15, 2001; modied August 22, 2001. 1 Research supported by NSF grant DMS{9803780, and by the Acheson J. Duncan Fund for the Advancement of Research in Statistics. 1 1
Harald Cramér and the distribution of prime numbers
 Scandanavian Actuarial J
, 1995
"... “It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the ..."
Abstract

Cited by 20 (1 self)
 Add to MetaCart
“It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the midthirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramér’s ideas have directed and motivated research ever since. One can only fully appreciate the significance of Cramér’s contributions by viewing his work in the appropriate historical context. We shall begin our discussion with the ideas of the ancient Greeks, Euclid and Eratosthenes. Then we leap in time to the nineteenth century, to the computations and heuristics of Legendre and Gauss, the extraordinarily analytic insights of Dirichlet and Riemann, and the crowning glory of these ideas, the proof the “Prime Number Theorem ” by Hadamard and de la Vallée Poussin in 1896. We pick up again in the 1920’s with the questions asked by Hardy and Littlewood,