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We define quantum determinants in quantum matrix algebras related to pairs of compatible braidings. We establish relations between these determinants and the so-called column and row determinants, which are often used in the theory of integrable systems. We also generalize the quantum integrable spin systems using generalized Yangians related to pairs of compatible braidings. We demonstrate that such quantum integrable spin systems are not uniquely determined by the "quantum coordinate ring" of the basic space . For example, the "quantum plane" yields two different integrable systems: rational and trigonometric. [ABSTRACT FROM AUTHOR]

As is known, in the reflection equation (RE) algebra associated with an involutive or Hecke -matrix, the elements (called quantum power sums) are central. Here, is the generating matrix of this algebra, and is the operation of taking the -trace associated with a given -matrix. We consider the problem of whether this is true in certain RE-like algebras depending on a spectral parameter. We mainly study algebras similar to those introduced by Reshetikhin and Semenov-Tian-Shansky (we call them algebras of RS type). These algebras are defined using some current -matrices (i.e., depending on parameters) arising from involutive and Hecke -matrices by so-called Baxterization. In algebras of RS type. we define quantum power sums and show that the lowest quantum power sum is central iff the value of the "charge" in its definition takes a critical value. This critical value depends on the bi-rank of the initial -matrix. Moreover, if the bi-rank is equal to and the charge has a critical value, then all quantum power sums are central. [ABSTRACT FROM AUTHOR]

For a couple of associative algebras we define the notion of their double and give a set of examples. Also, we discuss applications of such doubles to representation theory of certain quantum algebras and to a new type of Noncommutative Geometry. [ABSTRACT FROM AUTHOR]

By generalized Yangians, we mean Yangian-like algebras of two different classes. One class comprises the previously introduced so-called braided Yangians. Braided Yangians have properties similar to those of the reflection equation algebra. Generalized Yangians of the second class, RTT-type Yangians, are defined by the same formulas as the usual Yangians but with other quantum R-matrices. If such an R-matrix is the simplest trigonometric R-matrix, then the corresponding RTT-type Yangian is called a q-Yangian. We claim that each generalized Yangian is a deformation of the commutative algebra Sym(gl(m)[t ]) if the corresponding R-matrix is a deformation of the flip operator. We give the explicit form of the corresponding Poisson brackets. [ABSTRACT FROM AUTHOR]

The development of a Gersevanov Institute procedure is given for analyzing bed settlement under a slab foundation, with consideration of the foundation shape, uneven load, and bed nonuniformity. In the proposed procedure alternative, which has been adapted for the analysis of foundation slabs on a nonuniform rock bed, the load distribution under the slab bottom is taken from the results of a finite-element analysis of a building structure on a base assumed to be absolutely inelastic. Implementation of the analytical procedure is shown using the example of calculating the settlement and deformation of a high-rise building slab, for which a "weakened" area was identified in the bed during surveys. Stochastic simulation of additional "columns" of geological engineering boreholes is carried out to estimate the impact of insufficiently complete information on the properties of the rock mass. The impact of bed nonuniformity on the results of analyzing settlement, slab tilt, and forces within the slab are analyzed. [ABSTRACT FROM AUTHOR]

We obtain a parameterization for power sums of a GL(m|n)-type quantum (super)matrix in terms of its spectral values. [ABSTRACT FROM AUTHOR]

We continue the study of quantum matrix algebras of the GL( m|n) type. We find three alternative forms of the Cayley-Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of “even” and “odd” eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions s λ ( M) that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood-Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring Λ of symmetric functions of countably many variables. [ABSTRACT FROM AUTHOR]

We consider a new method for constructing finite-dimensional irreducible representations of the reflection equation algebra. We construct a series of irreducible representations parameterized by Young diagrams. We calculate the spectra of central elements sk = TrqLk of the reflection equation algebra on “q-symmetric” and “q-antisymmetric” representations. We propose a rule for decomposing the tensor product of representations into irreducible representations. [ABSTRACT FROM AUTHOR]

Let ? be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, g, with the Lie algebra ?. We study one and two parameter quantizations ? h and ? t,h of ? such that the multiplication on the quantized algebra is invariant under action of the Drinfeld–Jimbo quantum group, U h (?). In particular, the algebra ? t,h specializes at h= 0 to a U(?)-invariant ($G$-invariant) quantization, %Ascr; t ,0. We prove that the Poisson bracket corresponding to ? h must be the sum of the so-called r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H 2( M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, $? t,h , corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases. [ABSTRACT FROM AUTHOR]

Partial trisomy 11q due to a sporadic translocation was found in a mentally retarded girl with multiple abnormalities. The proportion of sporadic translocations involved in the total incidence of partial trisomies is discussed. [ABSTRACT FROM AUTHOR]

Two cases of partial 10p trisomy due to a t(10;20)(p12;p12) in two generations of a family are presented. Analysis of 20 known cases of such aberrations confirmed the opinion of Schleiermacher et al. (1974) that partial trisomy 10p is a distinct clinically recognizable entity. The most important diagnostic features of this syndrome are dolichocephaly, prominent forehead, wide open sutures and fontanelles, broad root of the nose, cleft lip and palate, clubfoot, and cystic changes in kidneys. [ABSTRACT FROM AUTHOR]

An analysis of data about the+p9 syndrome revealed that this clinical entity may occur in some different genetic forms. The recurrence risk in cases with familial translocations is due to the type of meiotic segregation, 2:2 or 3:1. It was shown a nonrandomness of involvement of chromosomes 15 and 22 in translocations with chromosome number 9. [ABSTRACT FROM AUTHOR]

We describe two types of Poisson pencils generated by a linear bracket and a quadratic one arising from a classical R-matrix. A quantization scheme is discussed for each. The quantum algebras are represented as the enveloping algebras of 'generalized Lie algebras.' [ABSTRACT FROM AUTHOR]

We describe various ways of constructing Poisson structures associated to Drinfeld-Jimbo R-matrices on homogeneous spaces, research some relations between these structures, and quantize some of them. [ABSTRACT FROM AUTHOR]

This article considers application of nonholonomic filtration associated with the structure of Lie bialgebras to description, in terms of spectral sequences, of classical objects associated with quasi-Hopf algebras, the so-called (quasi)Lie (quasi)bialgebras. [ABSTRACT FROM AUTHOR]

We analyze the relation between the properties of Hecke symmetry (i.e., Hecke type R-matrix) and the algebraic structure of the corresponding reflection equation (RE) algebra. Analogues of the Newton relations and Cayley–Hamilton theorem for the matrix of generators of the RE algebra associated with a finite rank even Hecke symmetry are derived. [ABSTRACT FROM AUTHOR]

Given a simple Lie algebra g, we consider the orbits in g which are of the R-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-called R-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of the R-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions of q-deformed Lie brackets, braided coadjoint vector fields, and tangent vector fields are discussed as well. [ABSTRACT FROM AUTHOR]