On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture

In their recent paper, Alday et al. (Lett Math Phys 91:167–197, 2010) proposed a relation between \({\mathcal{N}=2}\) four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired by the exact form of the instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra \({\mathcal{A}={\sf Vir} \otimes\mathcal{H}}\) which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of \({\mathcal{A}}\). The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called \({Z_{{\sf bif}}}\) appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in Alday et al. (2010). We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin–Ono integrable hierarchy.     

Environmentally benign one-pot multicomponent synthesis of 1,4-dihydropyridine derivatives applying montmorillonite K10 as reusable catalyst

Chemistry of Heterocyclic Compounds - Environmentally benign one-pot multicomponent synthesis of different 1,4-dihydropyridine derivatives using montmorillonite K10 has been achieved. The...     

Computational study of the US stock market evolution: a rank correlation-based network model

This paper presents a computational study of global characteristics of the US stock market using a network-based model referred to as the market graph. The market graph reflects similarity patterns between stock return fluctuations via linking pairs of stocks that exhibit “coordinated” behavior over a specified period of time. We utilized Spearman rank correlation as a measure of similarity between stocks and considered the evolution of the market graph over the recent decade between 2001–2011. The observed market graph characteristics reveal interesting trends in the stock market over time, as well as allow one to use this model to identify cohesive clusters of stocks in the market.     

Simple measure of similarity for the market graph construction

A simple measure of similarity for the construction of the market graph is proposed. The measure is based on the probability of the coincidence of the signs of the stock returns. This measure is robust, has a simple interpretation, is easy to calculate and can be used as measure of similarity between any number of random variables. For the case of pairwise similarity the connection of this measure with the sign correlation of Fechner is noted. The properties of the proposed measure of pairwise similarity in comparison with the classic Pearson correlation are studied. The simple measure of pairwise similarity is applied (in parallel with the classic correlation) for the study of Russian and Swedish market graphs. The new measure of similarity for more than two random variables is introduced and applied to the additional deeper analysis of Russian and Swedish markets. Some interesting phenomena for the cliques and independent sets of the obtained market graphs are observed.     

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to minus n+2 hyperplanes.Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a real n-torus as its fiber. The base accommodates the geometric genus of a hypersurface V. Its homotopy type is a wedge of h^n,o(V) spheres Sⁿ.     

On Separable Schur Rings over Abelian Groups

A finite group is said to be weakly separable if every algebraic isomorphism between two S-rings over this group is induced by a combinatorial isomorphism.We prove that every abelian weakly separable group only belongs to one of several explicitly given families.     

Rectifiable Curves in Proximally Smooth Sets

Set-Valued and Variational Analysis - In this article we study some geometric properties of proximally smooth sets. First, we introduce a modification of the metric projection and prove its...     

Three types of transitions to phase synchronization in coupled chaotic oscillators

We study the effect of noncoherence on the onset of phase synchronization of two coupled chaotic oscillators. Depending on the coherence properties of oscillations characterized by the phase diffusion, three types of transitions to phase synchronization are found. For phase-coherent attractors this transition occurs shortly after one of the zero Lyapunov exponents becomes negative. At rather strong phase diffusion, phase locking manifests a strong degree of generalized synchronization, and occurs only after one positive Lyapunov exponent becomes negative. For intermediate phase diffusion, phase synchronization sets in via an interior crises of the hyperchaotic set.     

加压富氧燃烧模式下SOx/NOx强耦合促进SO3形成的机理研究

加压富氧煤粉燃烧技术由于净发电效率高被认为是最有可能得到应用的燃煤CO2控制技术之一,但受燃烧压力影响,烟气中的酸性气体(SOx/NOx)分压显著提高,导致腐蚀风险加剧,而目前对该问题相关的基础研究很少。本文利用详细反应机理(72组分,428反应),基于实验数据对照,揭示了加压富氧模式下SOx/NOx强耦合促进SO3和NO2形成的协同效应。动力学机理计算结果表明:SOx/NOx的交互作用可促进SO2和NO向SO3和NO2的转化,并且该促进效应在一定的温度和加压条件下十分显著。基于反应路径和敏感性分析,本文最终发展了一个能用于准确预测加压富氧燃烧模式下SO3和NO2在后火焰区形成特性的9步骨架反应机理。