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

Majority-vote model with heterogeneous agents on square lattice

We study a nonequilibrium model with up-down symmetry and a noise parameter $q$ known as majority-vote model of M.J. Oliveira 1992 with heterogeneous agents on square lattice. By Monte Carlo simulations and finite-size scaling relations the critical exponents $\beta/\nu$, $\gamma/\nu$, and $1/\nu$ and points $q_{c}$ and $U^*$ are obtained. After extensive simulations, we obtain $\beta/\nu=0.35(1)$, $\gamma/\nu=1.23(8)$, and $1/\nu=1.05(5)$. The calculated values of the critical noise parameter and Binder cumulant are $q_{c}=0.1589(4)$ and $U^*=0.604(7)$. Within the error bars, the exponents obey the relation $2\beta/\nu+\gamma/\nu=2$ and the results presented here demonstrate that the majority-vote model heterogeneous agents belongs to a different universality class than the nonequilibrium majority-vote models with homogeneous agents on square lattice.Comment: 9 pages e 8 figures. arXiv admin note: substantial text overlap with arXiv:1306.034

Electronic structure of a graphene superlattice with massive Dirac fermions

We study the electronic and transport properties of a graphene-based superlattice theoretically by using an effective Dirac equation. The superlattice consists of a periodic potential applied on a single-layer graphene deposited on a substrate that opens an energy gap of $2\Delta$ in its electronic structure. We find that extra Dirac points appear in the electronic band structure under certain conditions, so it is possible to close the gap between the conduction and valence minibands. We show that the energy gap $E_g$ can be tuned in the range $0\leq E_g \leq 2\Delta$ by changing the periodic potential. We analyze the low energy electronic structure around the contact points and find that the effective Fermi velocity in very anisotropic and depends on the energy gap. We show that the extra Dirac points obtained here behave differently compared to previously studied systems

Assigning Grammatical Relations with a Back-off Model

This paper presents a corpus-based method to assign grammatical subject/object relations to ambiguous German constructs. It makes use of an unsupervised learning procedure to collect training and test data, and the back-off model to make assignment decisions.Comment: To appear in Proceedings of the Second Conference on Empirical Methods in Natural Language Processing, 7 pages, LaTe

Majority-vote model on Opinion-Dependent Networks

We study a nonequilibrium model with up-down symmetry and a noise parameter $q$ known as majority-vote model of M.J. Oliveira $1992$ on opinion-dependent network or Stauffer-Hohnisch-Pittnauer networks. By Monte Carlo simulations and finite-size scaling relations the critical exponents $\beta/\nu$, $\gamma/\nu$, and $1/\nu$ and points $q_{c}$ and $U^*$ are obtained. After extensive simulations, we obtain $\beta/\nu=0.230(3)$, $\gamma/\nu=0.535(2)$, and $1/\nu=0.475(8)$. The calculated values of the critical noise parameter and Binder cumulant are $q_{c}=0.166(3)$ and $U^*=0.288(3)$. Within the error bars, the exponents obey the relation $2\beta/\nu+\gamma/\nu=1$ and the results presented here demonstrate that the majority-vote model belongs to a different universality class than the equilibrium Ising model on Stauffer-Hohnisch-Pittnauer networks, but to the same class as majority-vote models on some other networks.Comment: 9 figures, accepted for publication in IJMP

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

A shortened recurrence relation for the Bernoulli numbers

In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers $B_{2 n}$, $n$ being any positive integer. This new recurrence seems advantageous in comparison to other known formulae since it allows the computation of both $B_{4 n}$ and $B_{4 n +2}$ from only $B_0, B_2,..., B_{2n}$.Comment: 7 pages, no figures. Submitted to "J. Number Theory" (09/22/2011

Potts model with q states on directed Barabasi-Albert networks

On directed Barabasi-Albert networks with two and seven neighbours selected by each added site, the Ising model with spin S=1/2 was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. However, on these networks the Ising model spin S=1 was seen to show a spontaneous magnetisation. In this model with spin S=1 a first-order phase transition for values of connectivity z=2 and z=7 is well defined. On these same networks the Potts model with q=3 and 8 states is now studied through Monte Carlo simulations. We have obtained also for q=3 and 8 states a first-order phase transition for values of connectivity z=2 and z=7 of the directed Barabasi-Albert network. Theses results are different from the results obtained for same model on two-dimensional lattices, where for q=3 the phase transition is of second order, while for q=8 the phase transition is first-order.Comment: 14 pages including many firgures, for Communications in Computational Physic