Stability Analysis of Continuous Waves in Nonlocal Random Nonlinear Media

On the basis of the competing cubic-quintic nonlinearity model, stability (instability) of continuous waves in nonlocal random non-Kerr nonlinear media is studied analytically and numerically. Fluctuating media parameters are modeled by the Gaussian white noise. It is shown that for different response functions of a medium nonlocality suppresses, as a rule, both the growth rate peak and bandwidth of instability caused by random parameters. At the same time, for a special form of the response functions there can be an ''anomalous'' subjection of nonlocality to the instability development which leads to further increase of the growth rate. Along with the second-order moments of the modulational amplitude, higher-order moments are taken into account.Comment: This is a contribution to the Proc. of the Seventh Inter. Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Wick Rotation and Fermion Doubling in Noncommutative Geometry

In this paper, we discuss two features of the noncommmutative geometry and spectral action approach to the Standard Model: the fact that the model is inherently Euclidean, and that it requires a quadrupling of the fermionic degrees of freedom. We show how the two issues are intimately related. We give a precise prescription for the Wick rotation from the Euclidean theory to the Lorentzian one, eliminating the extra degrees of freedom. This requires not only projecting out mirror fermions, as has been done so far, and which leads to the correct Pfaffian, but also the elimination of the remaining extra degrees of freedom. The remaining doubling has to be removed in order to recover the correct Fock space of the physical (Lorentzian) theory. In order to get a Spin(1,3) invariant Lorentzian theory from a Spin(4) invariant Euclidean theory such an elimination must be performed after the Wick rotation. Differences between the Euclidean and Lorentzian case are described in detail, in a pedagogical way.Comment: Minor changes. To appear in PRD. 25 page

Determination the different categories of buyers based on the Jaynes’ information principle

Purpose: The article aims to reduce the volume of statistical data, necessary for determination the buyer’s structure. The correct clustering of clients is important for successful activity for both commercial and non-profit organizations. This issue is devoted to a large number of studies. Their main mathematical apparatus is statistical methods. Input data are results of buyer polls. Polls are labor-consuming and quite often annoying buyers. The problem of determination of structure (various categories) of buyers by the mathematical methods demanding a small amount of these polls is relevant. Design/Methodology/Approach: The approach offered in this report based on the Jaynes' information principle (principle of maximum entropy). Jaynes idea is as follows. Let us consider a system in which the conditions cannot be calculated or measured by an experiment. However, each state of the system has a certain measured implication, the average value of which is known (or can be defined), and the average result of these implications is known from the statistical data. Then the most objective are probabilities of states maximizing Shannon’s entropy under restrictions imposed by information about average implications of states. Findings: In this work the task of determination of percentage of buyers for computer shop by the average check is set and solved provided that average checks for each concrete category of buyers are known. Input data for calculation are their average checks. Determination of these values requires much less statistical data, than to directly determine relative number of buyers of various categories. Practical Implications: The results are of particular interest to marketing experts. Originality/Value: The article deals with practical situation when initially there are only three different groups of customers. For this case, the problem of maximizing entropy under given constraints reduced to the problem of finding a solution to a system of three equations, of which only one is nonlinear. This is a completely new result.peer-reviewe