Some Remarks on Producing Hopf Algebras

We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to a q-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed. Another example demonstrates that the bialgebra structure of sl_q(2) can be completely determined by requiring the q-oscillator algebra to be its covariant comodule, in analogy with Manin's approach to define SL_q(2) as a symmetry algebra of the bosonic and fermionic quantum planes.Comment: 6 pages, LATEX, no figures, Contribution to the Proceedings of the 4th Colloquium "Quantum Groups and Integrable Systems" (Prague, June 1995

On the oscillation properties of eigenfunctions of Sturm--Liouville problem with singular coefficients

In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\lambda r)y=0$, $(U-1)y^{\vee}+i(U+1)y^{\wedge}=0$, where function $p\in L_{\infty}[0,1]$ is uniformly positive, generalized functions $q,r\in W_2^{-1}[0,1]$ are real-valued and unitary matrix $U\in\mathbb C^{2\times 2}$ is diagonal. The main goal is to prove that well-known (for smooth case) facts about number and distribution of zeros of eigenfunctions hold in general case.Comment: 7 page

Dynamic spin susceptibility in the t-J model

A relaxation-function theory for the dynamic spin susceptibility in the $t$--$J$ model is presented. By a sum-rule-conserving generalized mean-field approximation (GMFA), the two-spin correlation functions of arbitrary range, the staggered magnetization, the uniform static susceptibility, and the antiferromagnetic correlation length are calculated in a wide region of hole doping and temperaturs. A good agreement with available exact diagonalization (ED) data is found. The correlation length is in reasonable agreement with neutron-scattering experiments on La_{2-\delta}Sr_\delta)CuO_4. Going beyond the GMFA, the self-energy is calculated in the mode-coupling approximation. The spin dynamics at arbitrary frequencies and wave vectors is studied for various temperatures and hole doping. At low doping a spin-wave-type behavior is found as in the Heisenberg model, while at higher doping a strong damping caused by hole hopping occurs, and a relaxation-type spin dynamics is observed in agreement with the ED results. The local spin susceptibility and its (\omega/T) scaling behavior are calculated in a reasonable agreement with experimental and ED data.Comment: 13 pages, 14 figure

Every latin hypercube of order 5 has transversals

We prove that for all n>1 every latin n-dimensional cube of order 5 has transversals. We find all 123 paratopy classes of layer-latin cubes of order 5 with no transversals. For each $n\geq 3$ and $q\geq 3$ we construct a (2q-2)-layer latin n-dimensional cuboid with no transversals. Moreover, we find all paratopy classes of nonextendible and noncompletable latin cuboids of order 5.Comment: Supplementary data https://zenodo.org/records/1020402

FIRST DATA ON THE CONCENTRATIONS AND DISTRIBUTION OF NOBLE METALS IN NI-CU SULFIDE ORES OF THE SOUTH MAKSUT DEPOSIT (EAST KAZAKHSTAN)

The magmatic sulfide deposits in the Central Asian orogenic belt are hosted in a series of mafic–ultramafic intrusions in the Maksut zone (E Kazakhstan), the Kalatongke and the Huangshan zones in Xinjiang (NW China) and the Hongqiling zone in NE China. In the Maksut zone there are several intrusions, the best studied from which is the South Maksut intrusion with Cu–Ni–PGE mineralization.The magmatic sulfide deposits in the Central Asian orogenic belt are hosted in a series of mafic–ultramafic intrusions in the Maksut zone (E Kazakhstan), the Kalatongke and the Huangshan zones in Xinjiang (NW China) and the Hongqiling zone in NE China. In the Maksut zone there are several intrusions, the best studied from which is the South Maksut intrusion with Cu–Ni–PGE mineralization