Invariants of knots and links at roots of unity

We present a comprehensive classification of invariants of knots and links associated with irreducible representations of \uqslN{}, when the parameter of quantization $q$ is a root of unity. We demonstrate that, besides the standard HOMFLY-PT invariants, which are associated with representations with highest and lowest weights, non-trivial invariants can be associated only with nilpotent representations with parameters. We define the corresponding invariants and discuss their relations with standard invariants at particular values of parameters.Comment: 26 pages, 12 figure

Explicit examples of DIM constraints for network matrix models

Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.Comment: 46 page

AERATED CONCRETE MICROWAVE REFLECTION AND TRANSMISSION PROPERTIES IN A WET ENVIRONMENT

Excess water content within the building material could greatly alter the interaction of the microwave with the material. In this study aerated concrete (AEROC) block walls (60x60cm) were investigated for their microwave (2.4 GHz) properties under wetting conditions. The spray wetting of the wall was conducted to simulate the environmental processes such as raining or water damage. 50ml/m² water dosage was applied on the surface of the concrete 21 times in 1 min intervals. The results show a noticeable decrease in microwave penetration power through the material after the sample gets enriched with water. As the water content on the surface of the material rises, so does the transmission loss. During the different water content of the aerated concrete, the reflection loss varied from -15.04 dB (dry wall) to -5.03 dB (wet wall’s surface). The transmission loss continues to rise during the entire length of the experiment, from -4.5 dB as a dry sample to -8.3 dB after 441 ml (1035 ml/m² during 23 min) of sprayed water. The variation of reflected microwave power was approximately ten times, which is quite a significant indicator of alteration of microwave propagation. The results demonstrate wetting process as an considerable factor in assessing microwave propagation in near the sources, such as mobile phone base station antennas, industrial microwave heaters and ovens etc

Spectral Duality Between Heisenberg Chain and Gaudin Model

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GL(k) Heisenberg chain is dual to the special reduced k+2-points gl(N) Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.Comment: 36 page

Deviation from Alday-Maldacena Duality For Wavy Circle

Alday-Maldacena conjecture is that the area A_P of the minimal surface in AdS_5 space with a boundary P, located in Euclidean space at infinity of AdS_5, coincides with a double integral D_P along P, the Abelian Wilson average in an auxiliary dual model. The boundary P is a polygon formed by momenta of n external light-like particles in N=4 SYM theory, and in a certain n=infty limit it can be substituted by an arbitrary smooth curve (wavy circle). The Alday-Maldacena conjecture is known to be violated for n>5, when it fails to be supported by the peculiar global conformal invariance, however, the structure of deviations remains obscure. The case of wavy lines can appear more convenient for analysis of these deviations due to the systematic method developed in arXiv:0803.1547 for (perturbative) evaluation of minimal areas, which is not yet available in the presence of angles at finite n. We correct a mistake in that paper and explicitly evaluate the h^2\bar h^2 terms, where the first deviation from the Alday-Maldacena duality arises for the wavy circle.Comment: 23 page

Sequence–structure relationships in yeast mRNAs

It is generally accepted that functionally important RNA structure is more conserved than sequence due to compensatory mutations that may alter the sequence without disrupting the structure. For small RNA molecules sequence–structure relationships are relatively well understood. However, structural bioinformatics of mRNAs is still in its infancy due to a virtual absence of experimental data. This report presents the first quantitative assessment of sequence–structure divergence in the coding regions of mRNA molecules based on recently published transcriptome-wide experimental determination of their base paring patterns. Structural resemblance in paralogous mRNA pairs quickly drops as sequence identity decreases from 100% to 85–90%. Structures of mRNAs sharing sequence identity below roughly 85% are essentially uncorrelated. This outcome is in dramatic contrast to small functional non-coding RNAs where sequence and structure divergence are correlated at very low levels of sequence similarity. The fact that very similar mRNA sequences can have vastly different secondary structures may imply that the particular global shape of base paired elements in coding regions does not play a major role in modulating gene expression and translation efficiency. Apparently, the need to maintain stable three-dimensional structures of encoded proteins places a much higher evolutionary pressure on mRNA sequences than on their RNA structures