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An application of (iterated) Bauer-Muir acceleration can give an Ap\'ery-like continued fraction for $\pi$ with irrational coefficients, and much faster convergence. It can be considered a generalized continued fraction with the same matrix…
We show tight lower bounds for the entire trade-off between space and query time for the Approximate Near Neighbor search problem. Our lower bounds hold in a restricted model of computation, which captures all hashing-based approaches. In…
We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the \$n\$th time level…
In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of…
The purpose of [1] was as follows. ?We consider special sets of continuants which occur in applications. For these sets we solve the problem of finding maximal and minimal continuants. There are several methods for finding extremum such as…
We provide completely effective error estimates for Stirling numbers of the first and second kind, denoted by $s(n,m)$ and $S(n,m)$, respectively. These bounds are useful for values of $m \geq n - O(\sqrt{n})$. An application of our Theorem…
This paper concerns the relationships between continued fractions and the geometry of the Stern-Brocot diagram. Each rational number can be expressed as a continued fraction $[a_0; a_1, \ldots, a_n]$ whose terms $a_i$ are integers and are…
We give an exponential upper bound on the probabilitywith which the denominator of the $n$th convergent in the regular continued fraction expansion stays away from the mean $\frac{n\pi^2}{12\log2}$. The exponential rate is best possible,…
We present a new, dynamical way to study powers (that is, repetitions) in Sturmian words based on results from Diophantine approximation theory. As a result, we provide an alternative and shorter proof of a result by Damanik and Lenz…
We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schr\"odinger equation with small potential and the nonlinear Schr\"odinger equation (NLSE) with weak nonlinearity. For the…
We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aimed to compare the efficiency by describing the rate at which the…
A classical theorem in continued fractions due to Serret shows that for any two irrational numbers x and y related by a transformation $\gamma$ in PGL(2,Z) there exist s and t for which the complete quotients x_s and y_t coincide. In this…
We propose and study a generalized continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued…
In this paper a finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation is presented and analyzed. We first propose a new finite difference method to approximate the time fractional derivatives, and…
We study fine-grained error bounds for differentially private algorithms for counting under continual observation. Our main insight is that the matrix mechanism when using lower-triangular matrices can be used in the continual observation…
The objective of this paper is to show (a)=(b)=(c) as rational functions of $T$, $U$ for (a), (b), (c) given by (a) continued fractions of length $2^{n+1}-1$ with explicit partial denominators in $\left\{-T,U^{-1}T\right\}$, (b) truncated…
We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the $n$-th term in a recurrent sequence of suitable…
We construct continued fraction expansions for several families of the Laurent series in $\mathbb{Q}[[t^{-1}]]$. To the best of the author's knowledge, this is the first result of this kind since Gauss derived the continued fraction…
The finite sum of the squares of the Mie coefficients is very useful for addressing problems of classical light scattering. An approximate formula available in the literature, and still in use today, has been developed to determine a priori…
We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give,…