New Sum-Product Estimates for Real and Complex Numbers

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set A of positive real numbers, it is true that |{a+bc+d:a,b,c,d∈A}|≥2|A|2-1.As a consequence of this result, it is also established that |4k-1A(k)|:=|A…A⏟ktimes+⋯+A…A⏟4k-1times|≥|A|k.Later on, it is shown that both of these bounds hold in the case when A is a finite set of complex numbers, although with smaller multiplicative constants. © 2015, Springer Science+Business Media New York

Expanders with superquadratic growth

We prove several expanders with exponent strictly greater than 2. For any finite set A ⊂ ℝ, we prove the following six-variable expander results: (Formula Presented)

The Pecularity of Formation and Extraction of Ionic Associates of Diclofenac, Indomethacin, Ketoprofen and Piroxicam with Basic Dyes

Изучены условия образо- вания и экстракции ионных ассоциатов (ИА) диклофенака, индометацина, кетопрофена и пироксикама с основными красителями и возможности использования их химико-аналитических свойств в аналитической практике. Определены спектрофотометрические и экстракционные характеристики ИА диклофенака, индоме- тацина, кетопрофена и пироксикама с полиметиновыми красителями. The conditions of creation and extraction of IA diclofenac, indomethacin, ketoprofen, and piroxicam with basic dyes were developedand the possibility of using their chemico-analytical properties in analytical practice. The spectrophotometric and extraction characteristics of IA diclofenac, indomethacin, ketoprofen, and piroxicam with polymethyne dyes were determined

Excited Random Walk in One Dimension

We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q=1-p. If the walk hops onto an empty site, there is no bias. For the 1-excited walk on the half-line (one cookie initially at each site), the probability of first returning to the starting point at time t scales as t^{-(2-p)}. Although the average return time to the origin is infinite for all p, the walk eats, on average, only a finite number of cookies until this first return when p<1/2. For the infinite line, the probability distribution for the 1-excited walk has an unusual anomaly at the origin. The positions of the leftmost and rightmost uneaten cookies can be accurately estimated by probabilistic arguments and their corresponding distributions have power-law singularities near the origin. The 2-excited walk on the infinite line exhibits peculiar features in the regime p>3/4, where the walk is transient, including a mean displacement that grows as t^{nu}, with nu>1/2 dependent on p, and a breakdown of scaling for the probability distribution of the walk.Comment: 14 pages, 13 figures, 2-column revtex4 format, for submission to J. Phys.

Universal interface width distributions at the depinning threshold

We compute the probability distribution of the interface width at the depinning threshold, using recent powerful algorithms. It confirms the universality classes found previously. In all cases, the distribution is surprisingly well approximated by a generalized Gaussian theory of independant modes which decay with a characteristic propagator G(q)=1/q^(d+2 zeta); zeta, the roughness exponent, is computed independently. A functional renormalization analysis explains this result and allows to compute the small deviations, i.e. a universal kurtosis ratio, in agreement with numerics. We stress the importance of the Gaussian theory to interpret numerical data and experiments.Comment: 4 pages revtex4. See also the following article cond-mat/030146

Isotropic Transverse XY Chain with Energy- and Magnetization Currents

The ground-state correlations are investigated for an isotropic transverse XY chain which is constrained to carry either a current of magnetization J_M or a current of energy J_E. We find that the effect of nonzero J_M on the large-distance decay of correlations is twofold: i) oscillations are introduced and ii) the amplitude of the power law decay increases with increasing current. The effect of energy current is more complex. Generically, correlations in current carrying states are found to decay faster than in the J_E=0 states, contrary to expectations that correlations are increased by the presence of currents. However, increasing the current, one reaches a special line where the correlations become comparable to those of the J_E=0 states. On this line, the symmetry of the ground state is enhanced and the transverse magnetization vanishes. Further increase of the current destroys the extra symmetry but the transverse magnetization remains at the high-symmetry, zero value.Comment: 7 pages, RevTex, 4 PostScript figure

Theory of periodic swarming of bacteria: application to Proteus mirabilis

The periodic swarming of bacteria is one of the simplest examples for pattern formation produced by the self-organized collective behavior of a large number of organisms. In the spectacular colonies of Proteus mirabilis (the most common species exhibiting this type of growth) a series of concentric rings are developed as the bacteria multiply and swarm following a scenario periodically repeating itself. We have developed a theoretical description for this process in order to get a deeper insight into some of the typical processes governing the phenomena in systems of many interacting living units. All of our theoretical results are in excellent quantitative agreement with the complete set of available observations.Comment: 11 pages, 8 figure