Jean Dayantis

Monte Carlo precise determination of the end‐to‐end distribution function of self‐avoiding walks on the simple‐cubic lattice

Chains have been generated on the simple‐cubic lattice to determine, by Monte Carlo simulation, the end‐to‐end distribution function of self‐avoiding walks. The modulus r of the end‐to‐end distribution vector r, the square of this modulus, and the interactions of all orders were recorded for each chain. The Alexandrowicz dimerization procedure has been used to circumvent attrition and thus obtain statistically significant samples of large chains. This made it possible to obtain samples involving 12 000–16 000 chains, within ‘‘windows’’ of width Δρ=0.2, where ρ=r/Nν, N being the number of steps in the walk and ν  the scaling exponent. It was found that the mean value 〈r〉=aNν, with ν=0.5919 and the prefactor a very close (perhaps strictly equal) to unity. The above value of ν is slightly larger than that calculated by Le Guillou and Zinn‐Justin, but in accord with the Wilson e=4−d expansion, where d is the space dimensionality. Also, 〈r 2〉=bN2ν, with b=1.136±0.05. An attempt was made to fit the data to the ...

Statistics of confined self-avoiding walks. Part I. Chain dimensions and concentration profiles

Self-avoiding (SAWs) and random-flight (RFWs) walks of varying number N of steps have been generated inside spheres of varying diameter R, using a random number generator and an ad hoc computer program. The Monte Carlo samples, usually of 100 000 walks, thus obtained, allowed the determination of the following ratios, as a function of N and R: first, the ratio A=(r(N,R))/(r(N, infinity )), where (r(N, R)) stands for the mean (in modulus) end-to-end distance of an N-step confined walk, and (r,(N, infinity )) the same quantity for an N-step non-confined walk; also, the corresponding ratios B, C and D, for the root-mean-square end-to-end distance (r2)12/, the mean radius of gyration (rg), and, finally, the root-mean-square radius of gyration (rg2)12/. If reduced lengths are used, where the reduction length is of the form Nv, v being a scaling exponent, it is found that scaling, i.e. independence of the above ratios with respect to the step number in the walk, is well obeyed. The scaling exponent is equal to 0.592 for SAWs and to 0.500 for RFWs. In order to determine the concentration profiles of end, mid- and overall steps inside the sphere, the last has been divided in a prescribed number of spherical shells, up to 22, of the same thickness, and the number of steps falling inside each shell registered. Again using reduced lengths, it was thus found that all concentration profiles obey scaling, that is, the concentration profile as a function of the reduced distance from the centre of the sphere is defined through a single curve, whatever the value of N. Our results allow a comparison of the parameters for confined SAWs and RFWs.

A search for primes from lesser primes

Abstract The purpose of this paper is to show that there is an algorithm which permits to define new primes from lesser already known primes. The method is based on the definition of “compact” sets of primes, and on the fundamental property, i.e., the fact that the sum or difference of two mutually prime integers have no common decomposition factors with these two mutually prime integers.

Statistics of confined chains. III: Hamiltonian paths

The entropy of self-avoiding walks embedded in a square lattice has been Monte Carlo estimated inside plane squares of various side sizes R. The length of the walks ranged from one to steps, the maximum allowed length, which corresponds to the so-called Hamiltonian paths. It was found that if is the ratio of the occupied over the total number of available lattice sites inside the square, the number of configurations scales to a good approximation as . The limiting curve has then been estimated from the available data, and expressed as a fourth-degree polynomial in . A table is given for Z(1), that is Hamiltonian paths, comparing values obtained from the theoretical relationship given by Orland et al, from the exact enumeration data of Mayer et al, and from the Monte Carlo estimates of the present work.

Generalised arnold numbers and series

Abstract After shortly recalling previous results about Arnold couples of integer numbers and Arnold series obtained using the Arnold-Avez matrix, we proceed to generalize these numbers and series using any arbitrary two dimensional matrix with elements integer numbers. It is found that the set of Arnold series is infinite. In particular, if some parameter to be found in the text has the values minus one or plus one, the well known Fibonacci, and Lucas series appear as particular cases of these generalized Arnold series. Further, these Fibonacci and Lucas series may display successive terms which are all positive, as in the usual series, or all negative, or alternating in algebraic signs.

Complementary results on the search of primes from lesser primes

Abstract In this short note we complete recently published results on the search of primes from lesser primes. The method is based on the concept of “compact” sets of primes and on the “fundamental property”, i.e., the fact that the decomposition in primes of the sum or the difference of two coprime integers never has common decomposition factors with these two coprime integers. The earlier finding is here enhanced, according to which a proportion of primes of about two thirds is to be found in the specified intervals where the result of the algorithm leads either to primes or the product of two primes.