We study the average number of intersecting points of a given curve with random hyperplanes in an n-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan, this problem is closely linked to finding the a...
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We study the average number of intersecting points of a given curve with random hyperplanes in an n-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan, this problem is closely linked to finding the average number of real zeros of random polynomials. They showed that a real polynomial of degree n has on average 2/pi log n + O(1) real zeros (M. Kac's theorem). This result leads us to the following problem: given a real sequence (alpha (k))(k is an element ofN), study the average 1/N Sigma (N-1)(n=0) rho (f(n)), where rho (f(n)) is the number of real zeros of f(n)(X) = alpha (0) + alpha X-1 + ... + alpha X-n(n). We give theoretical results for the Thue-Morse polynomials and numerical evidence for other polynomials.
We develop two methods for obtaining new lower bounds for the cardinality of covering codes. Both are based on the notion of linear inequality of a code. Indeed, every linear inequality of a code (defined on F-q(n)) a...
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We develop two methods for obtaining new lower bounds for the cardinality of covering codes. Both are based on the notion of linear inequality of a code. Indeed, every linear inequality of a code (defined on F-q(n)) allows to obtain, using a classical formula (inequality (2) below), a lower bound on K-q(n, R), the minimum cardinality of a covering code with radius R. We first show how to get new linear inequalities (providing new lower bounds) from old ones. Then, we prove some formulae that improve on the classical formula (2) for linear inequalities of some given types. Applying both methods to all the classical cases of the literature, we improve on nearly 20% of the best lower bounds on K-q(n, R). (C) 2000 Elsevier Science B.V. All rights reserved.
We investigate the function R(T, sigma), which denotes the error term in the asymptotic formula for integral(0)(T) \ log zeta(sigma + it)\(2)dt, It is shown that R(T, sigma) is uniformly bounded for sigma > 1 and a...
We investigate the function R(T, sigma), which denotes the error term in the asymptotic formula for integral(0)(T) \ log zeta(sigma + it)\(2)dt, It is shown that R(T, sigma) is uniformly bounded for sigma > 1 and almost periodic in the sense of Bohr for fixed sigma > 1;hence R(T, sigma) = Omega(1) when T --> infinity. In case 1/2 < sigma < 1 is fixed we can obtain the bound R(T, sigma) <<(epsilon) T(9-2 sigma)/8+epsilon.
A code is called isodual if it is equivalent to its dual code, and a lattice is called isodual if it is isometric to its dual lattice. In this note, we investigate isodual codes over Z(2k). These codes give rise to is...
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A code is called isodual if it is equivalent to its dual code, and a lattice is called isodual if it is isometric to its dual lattice. In this note, we investigate isodual codes over Z(2k). These codes give rise to isodual lattices;in particular, we construct a 22-dimensional isodual lattice with minimum norm 3 and kissing number 2464.
We study binary codes of length n with covering radius one via their characteristic functions. The covering condition is expressed as a system of linear inequalities. The excesses then have a natural interpretation th...
We study binary codes of length n with covering radius one via their characteristic functions. The covering condition is expressed as a system of linear inequalities. The excesses then have a natural interpretation that makes congruence properties clear. We present new congruences and give several improvements on the lower bounds for K(n, 1) (the minimal cardinality of such a code) given by Zhang (1991, 1992). We study more specifically the cases n = 5 mod 6 and n = 2, 4 mod 6, and get new lower bounds such as K(14, 1) greater than or equal to 1172 and K(20, 1) greater than or equal to 52 456.
We classify n-dimensional pairs of dual lattices by their minimal vectors. This leads to the notion of a ''perfect pair'', a natural enlargement by duality of the usual notion of a perfect lattice, and...
We classify n-dimensional pairs of dual lattices by their minimal vectors. This leads to the notion of a ''perfect pair'', a natural enlargement by duality of the usual notion of a perfect lattice, and we show that there are only finitely many of them in any given dimension. As an application, we obtain a finiteness theorem for pairs of dual lattices which are extremal with respect to the geometrical mean gamma' of their Hermite invariants. (C) 1995 Academic Press, Inc.
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