On the relation between two quantum group invariants of 3-cobordisms

We prove in the context of quantum groups at even roots of unity that a Turaev-Viro type invariant of a three-dimensional cobordism M equals the tensor product of the Reshetikhin-Turaev invariants of M and M*, where the latter denotes M with orientation reversed

Integrality of quantum 3-manifold invariants and a rational surgery formula

We prove that the Witten-Reshetikhin-Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates Seifert fibered integral homology spaces and can be used to detect the unkno

A unified quantum SO(3) invariant for rational homology 3-spheres

Given a rational homology 3-sphere M with |H 1 (M,ℤ)|=b and a link L inside M, colored by odd numbers, we construct a unified invariant I M,L belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten-Reshetikhin-Turaev invariants of the pair (M,L). If b=1 and L=∅,I M coincides with Habiro's invariant of integral homology 3-spheres. For b>1, the unified invariant defined by the third author is determined by I M . Important applications are the new Ohtsuki series (perturbative expansions of I M ) dominating quantum SO(3) invariants at roots of unity whose order is not a power of a prime. These series are not known to be determined by the LMO invariant

Integrality of quantum 3-manifold invariants and a rational surgery formula

We prove that the Witten-Reshetikhin-Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates Seifert fibered integral homology spaces and can be used to detect the unkno

RESVERATROL IN KUBAN WINES

Purpose: The main purpose of viticulture is to improve the quality of the grapes, both to a greater extent for ampelotherapy and winemaking, and, to a lesser extent, to onotherapy. Methodology: The article highlights the results of perennial (from 2014) studies of 18 promising technical grape varieties from different zones of the Krasnodar Territory: Anapo-Taman, Central, as well as Amur from the Black Sea zone of the Krasnodar Territory and two control Western European world-famous and most common varieties Merlot and Cabernet-Sauvignon in the same zones. Result: The average values of resveratrol were found in wine materials from the varieties Vladimir and Dmitry (4.7 mg / dm3), Podlesny (3.9 mg / dm3), Saperavi Severny (3.5 mg / dm3), 40 let Octiabria (3.3 mg / dm3), Kurchansky and 40 let Pobedy (3.0 and 2.9 mg / dm3, respectively). On the other hand, as shown by the analysis of wine materials, the Antaris, Varyushkin, Mitsar and Plechistik varieties synthesize a lower content of resveratrol (1.0 and 0.9 mg / dm3, respectively). Applications: This research can be used for the universities, teachers and education students. Novelty/Originality: In this research, the model of resveratrol in Kuban wines is presented in a comprehensive and complete manner

Observables in 3-dimensional quantum gravity and topological invariants

In this paper we report some results on the expectation values of a set of observables introduced for 3-dimensional Riemannian quantum gravity with positive cosmological constant, that is, observables in the Turaev-Viro model. Instead of giving a formal description of the observables, we just formulate the paper by examples. This means that we just show how an idea works with particular cases and give a way to compute 'expectation values' in general by a topological procedure.Comment: 24 pages, 47 figure

ARTIFICIAL BOG OAK

The wood is bog oak – the exclusive material used to create artistic products of high value, which explains the numerous studies in the field of the same material in an industrial environment. The proposed technology of heat treatment allows to obtain wood similar in appearance with wood stained oak and with improved physical and mechanical properties.Представленная работа выполнялась при поддержке гранта Президента Российской Федерации для государственной поддержки молодых российских ученых – докторов наук (МД-5596.2016.8)

Quantum Group Invariant Integrable n-State Vertex Models with Periodic Boundary Conditions

An $U_q(sl(n))$ invariant transfer matrix with periodic boundary conditions is analysed by means of the algebraic nested Bethe ansatz for the case of $q$ being a root of unity. The transfer matrix corresponds to a 2-dimensional vertex model on a torus with topological interaction w.r.t. the 3-dimensional interior of the torus. By means of finite size analysis we find the central charge of the corresponding Virasoro algebra as $c=(n-1) \left[1-n(n+1)/(r(r-1))\right]$.Comment: 19 page