On the complexity of extending the convergence region for Traub’s method

The convergence region of Traub’s method for solving equations is small in general. This fact limits its applicability. We locate a more precise region containing the Traub iterations leading to at least as tight Lipschitz constants as before. Our convergence analysis is finer, and obtained without additional conditions. The new theoretical results are tested on numerical examples that illustrate their superiority over earlier results.     

On the complexity of extending the convergence ball of Wang’s method for finding a zero of a derivative

Ball convergence results are very important, since they demonstrate the complexity in choosing initial points for iterative methods. One of the most important problems in the study of iterative methods is to determine the convergence ball. This ball is small in general restricting the choice of initial points. We address this problem in the case of Wang’s method utilized to determine a zero of a derivative. Finding such a zero has many applications in computational fields, especially in function optimization. In particular, we find the convergence ball of Wang’s method using hypotheses up to the second derivative in contrast to earlier studies using hypotheses up to the fourth derivative. This way, we also extend the applicability of Wang’s method. Numerical experiments used to test the convergence criteria complete this study.     

Algorithmic complexity of Greenberg’s conjecture

Archiv der Mathematik - Let k be a totally real number field and p a prime. We show that the “complexity” of Greenberg’s conjecture ( $$\lambda = \mu = 0$$ ) is governed (under...     

On an Analog of the Plan’s Formula

In this paper we obtain an analog of the Plan’s formula, which plays an essential role in obtaining a functional relation for classical Riemann zeta-function.We provide examples of rational functions that satisfy a certain symmetry condition and admit a Maclaurin series expansion with coefficients equal to zero or one.     

On improving the accuracy of Horner’s and Goertzel’s algorithms

It is known that Goertzels algorithm is much less numerically accurate than the Fast Fourier Transform (FFT) (cf. [2]). In order to improve accuracy we propose modifications of both Goertzels and Horners algorithms based on the divide-and-conquer techniques. The proof of the numerical stability of these two modified algorithms is given. The numerical tests in Matlab demonstrate the computational advantages of the proposed modifications. The appendix contains the proof of numerical stability of Goertzels algorithm of polynomial evaluation. AMS subject classification 65F35, 65G50     

On the Sharpness of Mockenhaupt’s Restriction Theorem

We prove that the range of exponents in Mockenhaupt’s restriction theorem for Salem sets (Geom Funct Anal 10:1579–1587, 2000), with the endpoint estimate due to Bak and Seeger (Math Res Lett 18:767–781, 2011), is optimal.     

On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let T _g : [?1, 1] ?? [?1, 1] be the Feigenbaum map. It is well known that T _g has a Cantor-type attractor F and a unique invariant measure ??₀ supported on F. The corresponding unitary operator (U _g ??)(x) = ??(g(x)) has pure point spectrum consisting of eigenvalues ?? _n,r , n ?? 1, 0 ?? r ?? 2 ^n?1 ? 1 with eigenfunctions e _r ⁽ⁿ⁾ (x). Suppose that f ?? C ¹([?1, 1]), f?? is absolutely continuous on [?1, 1] and f?? ?? L ^p ([?1, 1], d??₀), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T _g we prove that S _n (f) ?? 2^?n q ⁿ , as n ?? ??, where the constant q ?? (0, 1) does not depend on f.     

On the proof of Erdős’ inequality

Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality \({\left\| {p'} \right\|_{\left[ { - 1,1} \right]}} \leqslant \frac{1}{2}{\left\| p \right\|_{\left[ { - 1,1} \right]}}\) for a constrained polynomial p of degree at most n, initially claimed by P. Erd?s, which is different from the one in the paper of T.Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval (?1, 1) and establish a new asymptotically sharp inequality.     

On the Moments of the Absorption Time of Kingman’s Coalescent

Simple integral representations are derived for the moments of the absorption time of Kingman’s coalescent Kingman (J Appl Probab 19:27-43, (1982a)). Their computational efficiency versus known representations is established.     

The equational complexity of Lyndon’s algebra

The equational complexity of Lyndon’s nonfinitely based 7-element algebra lies between n − 4 and 2n + 1. This result is based on a new algebraic proof that Lyndon’s algebra is not finitely based. We prove that Lyndon’s algebra is inherently nonfinitely based relative to a rather rich class of algebras. We also show that the variety generated by Lyndon’s algebra contains subdirectly irreducible algebras of all cardinalities except 0, 1, and 4.     

Circular proofs for the Gödel-Löb provability logic

Sequent calculus for the provability logic GL is considered, in which provability is based on the notion of a circular proof. Unlike ordinary derivations, circular proofs are represented by graphs allowed to contain cycles, rather than by finite trees. Using this notion, we obtain a syntactic proof of the Lyndon interpolation property for GL.     

On the equivalence of Lagrange’s axiom to the Lotschnittaxiom

We prove that, in Hilbert’s plane absolute geometry, an axiom used by Lagrange in a proof of the Euclidean parallel postulate in a paper read on 3 February 1806 at the Institut de France, which states that “If a and b are two parallels from P to g, then the reflection of a in b is parallel to g as well”, is equivalent to F. Bachmann’s Lotschnittaxiom, which states that “The perpendiculars on the sides a right angle always intersect.”     

On the complexity of the surrogate dual of 0–1 programming

It is shown that the surrogate dual of a 0–1 programming problem can be solved by 0(m ³) knapsack calls, ifm denotes the number of constraints.

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Zusammenfassung Es wird gezeigt, daß das surrogate duale Problem zu einer linearen Optimierungsaufgabe mit binären Variablen durch 0(m ³) Rucksackprobleme gelöst werden kann. Dabei bezeichnetm die Anzahl der Nebenbedingungen.