Some transcendence results from a harmless irrationality theorem

The arithmetic nature of values of some functions of a single variable, particularly, $\sin{z}$, $\cos{z}$, $\sinh{z}$, $\cosh{z}$, $e^z$, and $\ln{z}$, is a relevant topic in number theory. For instance, all those functions return transcendental values for all non-zero algebraic values of $z$ ($z \ne 1$ in the case of $\ln{z}$). On the other hand, not even an irrationality proof is known for some numbers like $\,e^e$, $\,\pi^e$, $\,\pi^\pi$, $\,\ln{\pi}$, $\,\pi + e\,$ and $\,\pi \, e$, though it is well-known that at least one of the last two numbers is irrational. In this note, I first derive a more general form of this last result, showing that at least one of the sum and product of any two transcendental numbers is transcendental. I then use this to show that, given any complex number $\,t \ne 0, 1/e$, at least two of the numbers $\,\ln{t}$, $\,t + e\,$ and $\,t \, e\,$ are transcendental. I also show that $\,\cosh{z}$, $\sinh{z}\,$ and $\,\tanh{z}\,$ return transcendental values for all $\,z = r \, \ln{t}$, $\,r \in \mathbb{Q}$, $r \ne 0$. Finally, I use a recent algebraic independence result by Nesterenko to show that, for all integer $\,n > 0$, $\,\ln{\pi}\,$ and $\,\sqrt{n} \, \pi\,$ are linearly independent over $\mathbb{Q}$.Comment: 12 pages, no figures. Inclusion of a new theorem (Theor.3). Submitted to Expos. Math. (Feb/07/2014

A rapidly converging Ramanujan-type series for Catalan's constant

In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series, including a classical formula by Ramanujan and a recent formula by Lupas. Interestingly, this suggests that an Ap\'{e}ry-like irrationality proof could be found for this constant.Comment: Improved version of the previous manuscript, with revised text and small corrections. 11 pages, 1 table. Submitted (06/03/2017

On the possible exceptions for the transcendence of the log-gamma function at rational entries

In a recent work [JNT \textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $\,\log{\Gamma(x)} + \log{\Gamma(1-x)}\,$, $x$ being a rational number between $0$ and $1$, is transcendental with at most \emph{one} possible exception, but the proof presented there in that work is \emph{incorrect}. Here in this paper, I point out the mistake they committed and I present a theorem that establishes the transcendence of those numbers with at most \emph{two} possible exceptions. As a consequence, I make use of the reflection property of this function to establish a criteria for the transcendence of $\,\log{\pi}$, a number whose irrationality is not proved yet. This has an interesting consequence for the transcendence of the product $\,\pi \cdot e$, another number whose irrationality remains unproven.Comment: 7 pages, 1 figure. Fully revised and shortened (02/05/2014

A shortcut for evaluating some log integrals from products and limits

In this short paper, I introduce an elementary method for exactly evaluating the definite integrals $\, \int_0^{\pi}{\ln{(\sin{\theta})}\,d\theta}$, $\int_0^{\pi/2}{\ln{(\sin{\theta})}\,d\theta}$, $\int_0^{\pi/2}{\ln{(\cos{\theta})}\,d\theta}$, and $\int_0^{\pi/2}{\ln{(\tan{\theta})}\,d\theta} \,$ in finite terms. The method consists in to manipulate the sums obtained from the logarithm of certain products of trigonometric functions at rational multiples of $\pi$, putting them in the form of Riemann sums. As this method does not involve any search for primitives, it represents a good alternative to more involved integration techniques. As a bonus, I show how to apply the method for easily evaluating $\,\int_0^1{\ln{\Gamma(x)} \, d x}$.Comment: 6 pages, no figures. Revised form. Some small corrections. Submitted to: IJMEST (06/26/2012

Root Finding by High Order Iterative Methods Based on Quadratures

We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with $n+1$ nodes is used the resulting iterative method has convergence order at least $n+2$, starting with the case $n=0$ (which corresponds to the Newton's method)

Numerical Simulations in Two-Dimensional Neural Fields

In the present paper we are concerned with a numerical algorithm for the approximation of the two-dimensional neural field equation with delay. We consider three numerical examples that have been analysed before by other authors and are directly connected with real world applications. The main purposes are 1) to test the performance of the mentioned algorithm, by comparing the numerical results with those obtained by other authors; 2) to analyse with more detail the properties of the solutions and take conclusions about their physical meaning.Comment: This article is closely related to previous publication in Arxiv: http://arxiv.org/abs/1508.07484, 201

Numerical Solution of the Neural Field Equation in the Two-dimensional Case

We are concerned with the numerical solution of a class integro-differential equations, known as Neural Field Equations, which describe the large-scale dynamics of spatially structured networks of neurons. These equations have many applications in Neuroscience and Robotics. We describe a numerical method for the approximation of solutions in the two-dimensional case, including a space-dependent delay in the integrand function. Compared with known algorithms for this type of equation we propose a scheme with higher accuracy in the time discretisation. Since computational efficiency is a key issue in this type of calculations, we use a new method for reducing the complexity of the algorithm. The convergence issues are discussed in detail and a number of numerical examples is presented, which illustrate the performance of the method.Comment: 25 pages, 1 figur

On the convergence rate of the scaled proximal decomposition on the graph of a maximal monotone operator (SPDG) algorithm

Relying on fixed point techniques, Mahey, Oualibouch and Tao introduced the scaled proximal decomposition on the graph of a maximal monotone operator (SPDG) algorithm and analyzed its performance on inclusions for strongly monotone and Lipschitz continuous operators. The SPDG algorithm generalizes the Spingarn's partial inverse method by allowing scaling factors, a key strategy to speed up the convergence of numerical algorithms. In this note, we show that the SPDG algorithm can alternatively be analyzed by means of the original Spingarn's partial inverse framework, tracing back to the 1983 Spingarn's paper. We simply show that under the assumptions considered by Mahey, Oualibouch and Tao, the Spingarn's partial inverse of the underlying maximal monotone operator is strongly monotone, which allows one to employ recent results on the convergence and iteration-complexity of proximal point type methods for strongly monotone operators. By doing this, we additionally obtain a potentially faster convergence for the SPDG algorithm and a more accurate upper bound on the number of iterations needed to achieve prescribed tolerances, specially on ill-conditioned problems

Biases on cosmological parameter estimators from galaxy cluster number counts

Sunyaev-Zel'dovich (SZ) surveys are promising probes of cosmology - in particular for Dark Energy (DE) -, given their ability to find distant clusters and provide estimates for their mass. However, current SZ catalogs contain tens to hundreds of objects and maximum likelihood estimators may present biases for such sample sizes. In this work we use the Monte Carlo approach to determine the presence of bias on cosmological parameter estimators from cluster abundance as a function of the area and depth of the survey, and the number of cosmological parameters fitted. Assuming perfect knowledge of mass and redshift some estimators have non-negligible biases. For example, the bias of $\sigma_8$ corresponds to about $40$ of its statistical error bar when fitted together with $\Omega_c$ and $w_0$. Including a SZ mass-observable relation decreases the relevance of the bias, for the typical sizes of current surveys. The biases become negligible when combining the SZ data with other cosmological probes. However, we show that the biases from SZ estimators do not go away with increasing sample sizes and they may become the dominant source of error for an all sky survey at the South Pole Telescope (SPT) sensitivity. The results of this work validate the use of the current maximum likelihood methods for present SZ surveys, but highlight the need for further studies for upcoming experiments. [abridged]Comment: 27 pages, 5 figures, submitted to JCAP. New discussion on biases for large SZ surveys. Minor text revision and references adde

Some transcendental functions with an empty exceptional set

A transcendental function usually returns transcendental values at algebraic points. The (algebraic) exceptions form the so-called \emph{exceptional set}, as for instance the unitary set $\{0\}$ for the function $f(z) = e^z \,$, according to the Hermite-Lindemann theorem. In this note, we give some explicit examples of transcendental entire functions whose exceptional set are empty.Comment: 8 pages, no figures. Submitted in revised form to KYUNGPOOK Math. J. (July/2012