On the Law of The Iterated Logarithm for Rapidly Increasing Subsequences

The usual law of the iterated logarithm states that the partial sums S_n of independent and identically distributed random variables can be normalized by the sequence a_n = √nlog log n, such that limsup_n→∞ S_n/a_n = √2 a.s. As has been pointed out by Gut (1986) the law fails if one considers the limsup along subsequences which increase faster than exponentially. In particular, for very rapidly increasing subsequences {n_k≥1} one has limsup_k→∞ S_nk/a_nk = 0 a.s. In these cases the normalizing constants a_nk have to be replaced by √nk log k to obtain a non-trivial limiting behaviour: limsup_k→∞ S_nk/ √nk log k = √2 a.s. We will present an intelligible argument for this structural change and apply it to related results.     

Difference operators,Green's matrix,sampling theory and applications in signal processing

This paper introduces sampling representations for discrete signals arising from self adjoint difference operators with mixed boundary conditions. The theory of linear operators on finite-dimensional inner product spaces is employed to study the second-order difference operators. We give necessary and sufficient conditions that make the operators self adjoint. The equivalence between the difference operator and a Hermitian Green's matrix is established. Sampling theorems are derived for discrete transforms associated with the difference operator. The results are exhibited via illustrative examples, involving sampling representations for the discrete Hartley transform. Families of discrete fractional Fourier-type transforms are introduced with an application to image encryption.     

Green's function study of a mixed spin-1 and spin-3/2 Heisenberg ferrimagnetic system

The magnetic properties of a mixed spin-1 and spin-3/2 Heisenberg ferrimagnetic system on a square lattice are investigated by using the double-time temperature-dependent Green's function technique. In order to decouple the higher order Green's functions, Anderson and Callen's decoupling and random phase approximations have been used. The nearest- and next-nearest-neighbor interactions and the single-ion anisotropies are considered and their effects on compensation and critical temperature are studied.     

Can a Circulating Light Beam Produce a Time Machine

In a recent paper, Mallett found a solution of the Einstein equations in which closed timelike curves (CTC’s) are present in the empty space outside an infinitely long cylinder of light moving in circular paths around an axis. Here we show that, for physically realistic energy densities, the CTC’s occur at distances from the axis greater than the radius of the visible universe by an immense factor. We then show that Mallett’s solution has a curvature singularity on the axis, even in the case where the intensity of the light vanishes. Thus it is not the solution one would get by starting with Minkowski space and establishing a cylinder of light.     

Multi-Conjugate Adaptive Optics for ELTs: constraints and limitations

We present how enlarging the size of a telescope from the current 10 meter telescope to the future 100 meter Extremely Large Telescopes increases the complexity of a classical or multiconjugate adaptive optics instrument. We point out elements or parameters of the system for which it is critical to propose new ideas as solutions and we study the effect of the increase of the diameter on the point spread function of an MCAO and a Ground Layer AO system. To cite this article: R. Ragazzoni et al., C. R. Physique 6 (2005).     

Efficient Krylov subspace methods for uncertainty quantification in large Bayesian linear inverse problems

Uncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems) and for problems where computation of the square root and inverse of the prior covariance matrix are not feasible. This work exploits Krylov subspace methods to develop and analyze new techniques for large‐scale uncertainty quantification in inverse problems. In this work, we assume that generalized Golub‐Kahan‐based methods have been used to compute an estimate of the solution, and we describe efficient methods to explore the posterior distribution. In particular, we use the generalized Golub‐Kahan bidiagonalization to derive an approximation of the posterior covariance matrix, and we provide theoretical results that quantify the accuracy of the approximate posterior covariance matrix and of the resulting posterior distribution. Then, we describe efficient methods that use the approximation to compute measures of uncertainty, including the Kullback‐Liebler divergence. We present two methods that use the preconditioned Lanczos algorithm to efficiently generate samples from the posterior distribution. Numerical examples from dynamic photoacoustic tomography demonstrate the effectiveness of the described approaches.     

An old conjecture of Erdos-Turán on additive bases

There is a 1941 conjecture of Erdos and Turán on what is now called additive basis that we restate:

Conjecture 0.1(Erdos and Turán). Suppose that is an increasing sequence of integers and

Suppose that

If 0$"> for all , then is unbounded.

Our main purpose is to show that the sequence cannot be bounded by . There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.

     

Influence of the variance of degree distributions on the evolution of cooperation in complex networks

We study how initial network structure affects the evolution of cooperation in a spatial prisoner’s dilemma game. The network structure is characterized by various statistical properties. Among those properties, we focus on the variance of the degree distribution, and inquire how it affects the evolution of cooperation by three methods of imitation. For every method, it was found that a scale-free network does not always promote the evolution of cooperation, and that there exists an appropriate value of the variance, at which cooperation is optimal.     

Values of coefficients of cyclotomic polynomials

Let a(k,n) be the k-th coefficient of the n-th cyclotomic polynomials. In 1987, J. Suzuki proved that . In this paper, we improve this result and prove that for any prime p and any integer l≥1, we have

{a(k,p^ln)∣n,k∈N}=Z.