Renormalization of parameters of a soft breakdown of supersymmetry in the regime of strong yukawa coupling within a nonminimal supersymmetric standard model

Within a nonminimal supersymmetric (SuSy) model, the renormalization of trilinear coupling constants A _i(t) for scalar fields and of specific combinations $\mathfrak{M}_i^2 (t)$ of the scalar-particle masses is investigated in the regime of strong Yukawa coupling. The dependence of these parameters on their initial values at the Grand Unification scale disappears as solutions to the renormalization-group equations approach infrared quasifixed points with increasing Y _i(0). In the vicinities of quasifixed points for $\tilde \alpha _{GUT} \ll Y_i (0) \ll 1$ , all solutions A _i(t) and $\mathfrak{M}_i^2 (t)$ are concentrated near some straight lines or planes in the space of parameters of a soft breakdown of supersymmetry. This behavior of the solutions in question is explained by a sufficiently slow disappearance of the A _i(0) and $\mathfrak{M}_i^2 (t)$ dependence of the trilinear coupling constants and combinations of the scalar-particle masses. A method is proposed for deriving equations describing the aforementioned straight lines and planes, and the process of their formation is discussed by considering the example of exact and approximate solutions to the renormalization-group equations within a nonminimal supersymmetric standard model.     

Extremes and student’s <Emphasis Type="Italic">t</Emphasis><Subscript>2</Subscript>-distribution

Some new characterizations of Student’s distribution on two degrees of freedom by regressional properties of maximal order statistics are obtained. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 232–237.     

On families of distributions characterized by certain properties of ordered random variables

In this work, we obtain new characterizations of certain probability distributions by relations with different ordered random variables. Such variables include order statistics, sequential maxima, and records. We consider relations that include not only upper, but also lower record values. The presented ordered objects are based on sequences of independent random variables with a common continuous distribution function. We also investigate equalities in the distribution of sequential maxima exposed by various random shifts. These shifts (one-sided or two-sided) have exponential distributions. Certain theorems and their corollaries present corresponding characterizations of distributions by relations of such a type. In addition, we consider exponentially shifted order statistics such that simple relations among them also characterize certain probability distributions. All of the presented results yield a set of characterizations of various distributions. For particular cases, we present the relations that characterize families of classical exponential and logistic distributions.     

On the Problem of the Optimal Choice of Record Values

Let the independent random variables X1, X2, … have the same continuous distribution function. The upper record values X(1) = X₁ < X(2) < … generated by this sequence of variables, as well as the lower record values x(1) = X₁ > x(2) > …, are considered. It is known that in this situation, the mean value c(n) of the total number of the both types of records among the first n variables X is given by the equality c(n)=2(1+1/2+…+1/n), n = 1, 2, …. The problem considered here is following: how, sequentially obtaining the observed values x₁, x₂, … of variables X and selecting one of them as the initial point, to obtain the maximal mean value e(n) of the considered numbers of records among the rest random variables. It is not possible to come back to rejected elements of the sequence. Some procedures of the optimal choice of the initial element X _r are discussed. The corresponding tables for the values e(n) and differences δ(n)= e(n)–c(n) are presented for different values of n. The value of δ= lim_n→∞δ(n)is also given. In some sense, the considered problem and optimization procedure presented in this paper are quite similar to the classical “secretary problem,” in which the probability of selecting the last record value in the set of independent identically distributed X is maximized.     

Joint distribution of order statistics generated by records

Probability distributions of some order statistics connected with the k th records values are discussed. Some useful representations for the exponential records and the corresponding order statistics are given. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarrow POMI, Vol. 228, 1996, pp. 24–30.     

On the maximal value of the expectation of record numbers

There are n independent identically distributed random variables with a continuous distribution function. The problem considered is, getting sequentially the values of these variables and selecting one of them as an initial point, how we can maximize the expected number of records among the rest of this sequence of random variables (without knowledge of the future values).     

Rate of convergence to the normal law of sums of induced order statistics

One obtains a certain estimate of order 1/n of the rate of convergence to the normal law for induced order statistics. The result includes the multidimensional case.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 108, pp. 45–56, 1981.     

Influence of technological factors on the supramolecular structure and frictional properties of the surface layers of thermoplastics

X-ray and optical diffraction methods have been used to study the changes in supramolecular structure, degree of orientation, and drawratio of the material in the 0–650-µm range for polypropylene samples obtained by molding. Using a microtribometric technique, the influence of these factors on the frictional properties of the polymer have been elucidated.Institute of Mechanics of Metal-Polymer Systems, Academy of Sciences of the Belorussian SSR, Gomel'. Translated from Mekhanika Polimerov, No. 2, pp. 202–206, March–April, 1976.     

Structure determination in "shiftless" solid state NMR of oriented protein samples

An efficient formalism for calculating protein structures from oriented-sample NMR data in the torsion-angle space is presented. Angular anisotropies of the NMR observables are treated by utilizing an irreducible spherical basis of rotations. An intermediate rotational transformation is introduced that greatly speeds up structural fitting by rendering the dependence on the torsion angles Φ and Ψ in a purely diagonal form. Back-calculation of the simulated solid-state NMR spectra of protein G involving 15N chemical shift anisotropy (CSA), and 1H-15N and 1Hα-13Cα dipolar couplings was performed by taking into account non-planarity of the peptide linkages and experimental uncertainty. Even a relatively small (to within 1 ppm) random variation in the CSA values arising from uncertainties in the tensor parameters yields the RMSD's of the back-calculated structures of more than 10 ?. Therefore, the 15N CSA has been substituted with heteronuclear dipolar couplings which are derived from the highly conserved bond lengths and bond angles associated with the amino-acid covalent geometry. Using the additional 13Cα-15N and 13C'-15N dipolar couplings makes it possible to calculate protein structures entirely from "shiftless" solid-state NMR data. With the simulated "experimental" uncertainty of 15 Hz for protein G and 120 Hz for a helical hairpin derived from bacteriorhodopsin, back-calculation of the synthetic dipolar NMR spectra yielded a converged set of solutions. The use of distance restraints dramatically improves structural convergence even if larger experimental uncertainties are assumed.