Related papers: Convergence of improper Iwasawa Continued Fraction…
We look at a class of transcendental real numbers xi which, together with their square, satisfy some extremal property of simultaneous approximation by rational numbers with the same denominator. We give a sufficient condition for such a…
Based on the weighted and shifted Gr\"{u}nwald difference (WSGD) operators [24], we further construct the compact finite difference discretizations for the fractional operators. Then the discretization schemes are used to approximate the…
The connection between a Taylor series and a continued-fraction involves a nonlinear relation between the Taylor coefficients $\{ a_n \}$ and the continued-fraction coefficients $\{ b_n \}$. In many instances it turns out that this…
In this paper we generalize some of our results from, `A note on Farey fractions with odd denominators' to subsets of Farey fractions consisting of fractions with denominators not divisible by a given prime. We also investigate the joint…
The Hankel transform of an integer sequence is a much studied and much applied mathematical operation. In this note, we extend the notion in a natural way to sequences of $d$ integer sequences. We explore links to generalized continued…
It is well known that the set of origami constructible numbers is larger than the classical straight-edge and compass constructible numbers. However, the Huzita-Justin-Hatori origami constructible numbers remain algebraic so that the…
We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of…
We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose derivatives are…
We detail the continued fraction expansion of the square root of the general monic quartic polynomial, noting that each line of the expansion corresponds to addition of the divisor at infinity. We analyse the data yielded by the general…
We use sequences which depend on two parameters to define families of ultradifferentiable functions which contain Gevrey classes. It is shown that such families are closed under superposition, and therefore inverse closed as well.…
We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed…
The irreducible decomposition of a unitary representation often contains continuous spectrum when restricted to a non-compact subgroup. The author singles out a nice class of branching problems where each irreducible summand occurs…
We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to…
We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable…
Let $X$ be a real Banach space with a normalized duality mapping uniformly norm-to-weak$^\star$ continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping $J_{\Phi}$ with gauge $\phi$. Let $f$ be…
It is shown that a separated sequence of points in the unit disc of the complex plane is in fact uniformly separated, if there exists a certain intermediate sequence whose separated subsequences are uniformly separated. This property is…
For regular continued fractions (CFs), points with finite expansions are exactly the rationals and, by Lagrange's theorem, points with eventually-periodic expansions are exactly the roots of non-degenerate quadratic equations with integer…
The textbook Newton's iteration is practically inapplicable on solutions of nonlinear systems with singular Jacobians. By a simple modification, a novel extension of Newton's iteration regains its local quadratic convergence toward…
We give a new proof of Tietze Theorem on the convergence of infinite semi-regular continued fractions.
We provide the detailed asymptotic behavior for first-order aggregation models of heterogeneous oscillators. Due to the dissimilarity of natural frequencies, one could expect that all relative distances converge to definite positive value…