In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the inhomogeneous exponent of approximation to a generic point in R by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent...

We obtain randomized algorithms for factoring degree n univariate polynomials over Fq requiring O(n1.5+o(1) log q + n1+o(1) logq) bit operations. When log q < n, this is asymptotically faster than the best previous algorithms [J. von zur Gathen and V. Shoup, Comput. Complexity, 2 (1992), pp. 187–224; E. Kaltofen and V. Shoup, Math. Comp., 67 (1998), pp. 1179– 1197]; for log q ≥ n, it matches th...

It is well-known that the maximum exponent that an n-by-n boolean primitive circulant matrix can attain is n − 1. In this paper, we find the maximum exponent attained by n-by-n boolean primitive circulant matrices with constant number of nonzero entries in their generating vector. We also give matrices attaining such exponents. Solving this problem we also solve two equivalent problems: 1) find...

In RSA, the public modulus N = pq is the product of two primes of the same bit-size, the public exponent e and the private exponent d satisfy ed ≡ 1 (mod (p−1)(q−1)). In many applications of RSA, d is chosen to be small. This was cryptanalyzed by Wiener in 1990 who showed that RSA is insecure if d < N. As an alternative, Quisquater and Couvreur proposed the CRT-RSA scheme in the decryption phas...

It is well-known that the maximum exponent that an n-by-n boolean primitive circulant matrix can attain is n− 1. In this paper, we find the maximum exponent that n-by-n boolean primitive circulant matrices with constant number of nonzero entries in its generating vector can attain. We also give matrices attaining such exponents. Solving this problem we also solve two equivalent problems: 1) fin...

We consider directed last passage percolation on $${\mathbb {Z}}^2$$ with exponential times the vertices. A topic of great interest is coupling structure weights geodesics as endpoints are varied spatially and temporally. particular specialization when one considers to points varying in time direction starting from a given initial data. This paper flat condition which corresponds line-to-point ...

We study the one-dimensional discrete quasi-periodic Schrödinger equation −ϕ(n + 1) − ϕ(n − 1) + λV (x + nω)ϕ(n) = Eϕ(n), n ∈ Z We show that for " typical " C 3 potential V , if the coupling constant λ is large, then for most frequencies ω, the Lyapunov exponent is positive for all energies E, and the corresponding eigenfunctions ϕ decay exponentially.

The method of reconstruction for an n-dimensional system from observations is to form vectors of m consecutive observations, which for m > 2n, is generically an embedding. This is Takens’ result. Our analytical examples show that it is possible to obtain spurious Lyapunov exponents that are even larger than the largest Lyapunov exponent of the original system. Therefore, we present examples whe...

We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups U(N),O(N), and Sp(2N). In particular we calculate critical exponents s,ν, and z, corresponding to the energy gap, correlation length, and dynamic exponent, respectively. We also co...

We show that in preferential attachment models with power-law exponent τ ∈ (2, 3) the distance between randomly chosen vertices in the giant component is asymptotically equal to (4 + o(1)) log logN − log(τ−2) , where N denotes the number of nodes. This is twice the value obtained for the configuration model with the same power-law exponent. The extra factor reveals the different structure of ty...