We construct a full data rate space-time block code over M =2 transmit antennas and T =2 symbol periods, and we prove that it achieves a transmit diversity of 2 over all constellations carved from Z[i] . Further, we optimize the coding gain of the proposed code and then compare it to the Alamouti code. It is shown that the new code outperforms the Alamouti code at low and high SNR when the number of receive antennas N>1. The performance improvement is further enhanced when N or the size of the constellation increases. We relate the problem of space-time diversity gain to algebraic number theory, and the coding gain optimization to the theory of simultaneous Diophantine approximation in the geometry of numbers. We find that the coding gain optimization is equivalent to find irrational numbers "the furthest" from any simultaneous rational approximations.

We consider discrete one-dimensional Schrodinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely ff-continuous spectrum. All these results hold uniformly on the hull generated by a given potential.

Pursuant to the authors' previous chaotic-dynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental constant log 2 = P n2Z + 1=(n2 n ) is not yet known to be 2-normal (i.e. normal to base 2), we are able to establish b-normality (and transcendency) for constants of the form P 1=(nb n ) but with the index n constrained to run over certain subsets of Z + . In this way we demonstrate, for example, that the constant 2;3 = P n=3;3 2 ;3 3 ;::: 1=(n2 n ) is 2-normal. The constants share with ; log 2 and others the property that isolated digits can be directly calculated, but for the new class such computation is extraordinarily rapid. For example, we find that the googol-th (i.e. 10 100 - th) binary bit of 2;3 is 0. We also present a collection of other results -- such as density results and irrationality proofs based on PRNG ideas -- for various special numbers.

Abstract. We obtain a Central Limit Theorem for a general class of additive parameters (costs, observables) associated to three standard Euclidean algorithms, with optimal speed of convergence. We also provide very precise asymptotic estimates and error terms for the mean and variance of such parameters. For costs that are lattice (including the number of steps), we go further and establish a Local Limit Theorem, with optimal speed of convergence. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various other techniques: Dirichlet series, Perron’s formula, quasi-powers theorems, and the saddle-point method. Such dynamical analyses had previously been used to perform the average-case analysis of algorithms. For the present (dynamical) analysis in distribution, we require estimates on transfer operators when a parameter varies along vertical lines in the complex plane. To prove them, we adapt techniques introduced recently by Dolgopyat in the context of continuous-time dynamics [16]. 1.

Extending the classical Legendre's result, we describe all solutions of the inequality |alpha - a/b| < c/b^2 in terms of convergents of continued fraction expansion of alpha. Namely, we show that a/b =(rp_{m+1} +- sp_{m})/(rq_{m+1} +- sq_{m}) for some nonnegative integers m, r, s such that rs < 2c. As an application of this result, we describe a modification of Verheul and van Tilborg variant of Wiener's attack on RSA cryptosystem with small secret exponent.

We obtain new results regarding the precise average-- case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the average-case complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters ---digits and continuants--- that intervene in an entire class of gcd-like algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.

We consider Sturmian sequences and provide an explicit formula for the index of such a sequence in terms of the continued fraction expansion coecients of its slope.

We consider the nonparametric estimation of a periodic function that is observed in additive Gaussian white noise after convolution with a “boxcar,” the indicator function of an interval. This is an idealized model for the problem of recovery of noisy signals and images observed with “motion blur. ” If the length of the boxcar is rational, then certain frequencies are irretreviably lost in the periodic model. We consider the rate of convergence of estimators when the length of the boxcar is irrational, using classical results on approximation of irrationals by continued fractions. A basic question of interest is whether the minimax rate of convergence is slower than for nonperiodic problems with 1/f-like convolution filters. The answer turns out to depend on the type and smoothness of functions being estimated in a manner not seen with “homogeneous ” filters. 1. Introduction. 1.1. Statement of problem and motivation. Suppose that we observe Y(t) for t ∈[−1, 1],whereYisdrawn from an indirect estimation model in Gaussian white noise:

In this paper we give combinatorial proofs of the classification of unori- ented and oriented rational knots based on the now known classifica- tion of alternating knots and the calculus of continued fractions. We also characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination.

We present new O(n 3) algorithms to compute very accurate SVDs of Cauchy matrices, Vandermonde matrices, and related \unit-displacement-rank " matrices. These algorithms compute all the singular values with guaranteed relative accuracy, independent of their dynamic range. In contrast, previous O(n 3) algorithms can potentially lose all relative accuracy in the tiniest singular values.