Unoriented HQFT and its underlying algebra

Turaev and Turner introduced a bijection between unoriented topological quantum field theories and extended Frobenius algebras. In this paper, we will show that there exists a bijective correspondence between unoriented (1+1)-dimensional homotopy quantum field theories and extended crossed group algebras.     

The non-perturbative renormalization group in the ordered phase

We study some analytical properties of the solutions of the non-perturbative renormalization group flow equations for a scalar field theory with Z₂ symmetry in the ordered phase, i.e. at temperatures below the critical temperature. The study is made in the framework of the local potential approximation. We show that the required physical discontinuity of the magnetic susceptibility χ(M) at M=±M₀ (M₀ spontaneous magnetization) is reproduced only if the cut-off function which separates high and low energy modes satisfies to some restrictive explicit mathematical conditions; we stress that these conditions are not satisfied by a sharp cut-off in dimensions of space d<4.By generalizing a method proposed earlier by Bonanno and Lacagnina [Nucl. Phys. B 693 (2004) 36] to any kind of cut-off we propose to solve numerically the renormalization group flow equations for the threshold functions rather than for the local potential. It yields an algorithm sufficiently robust and precise to extract universal as well as non-universal quantities from numerical experiments at any temperature, in particular at sub-critical temperatures in the ordered phase. Numerical results obtained for the φ⁴ potential with three different cut-off functions are reported and compared. The data confirm our theoretical predictions concerning the analytical behavior of χ(M) at M=±M₀.Fixed point solutions of the adimensioned renormalization group flow equations are also obtained in the same vein, that is by solving the fixed points equations and the associated eigenvalue problem for the threshold functions rather than for the potential. We report high precision data for the odd and even spectra of critical exponents for different cut-offs obtained in this way.     

Proving AGT conjecture as HS duality: Extension to five dimensions

We extend the proof from Mironov et al. (2011) [25], which interprets the AGT relation as the Hubbard-Stratonovich duality relation to the case of 5d gauge theories. This involves an additional q-deformation. Not surprisingly, the extension turns out to be straightforward: it is enough to substitute all relevant numbers by q-numbers in all the formulas, Dotsenko-Fateev integrals by the Jackson sums and the Jack polynomials by the MacDonald ones. The problem with extra poles in individual Nekrasov functions continues to exist, therefore, such a proof works only for β=1, i.e. for q=t in MacDonald?s notation. For β≠1 the conformal blocks are related in this way to a non-Nekrasov decomposition of the LMNS partition function into a double sum over Young diagrams.     

岩白菜属植物研究的新进展

从植物分布、栽培技术、生物学特性、化学成分以及药理作用5个方面综述了岩白菜研究取得的新进展,同时分析了当前研究中存在的问题,并对岩白菜今后的研究和发展方向进行了展望.     

The lyapunov spectrum and stable manifolds for stochastic linear delay equations

A class of linear stochastic retarded functional differential equations is considered. These equations have diffusion coefficients that do not look into the past. It is shown that the trajectories of such equations form a continuous linear cocycle on the underlying state space. At times greater than the delay the cocycle is almost surely compact. Consequently, using an infinite-dimensional Oseledec multiplicative ergodic theorem of Ruelle, the existence of a countable non-random Lyapunov spectrum is proved. In the hyperbolic case it is shown that the state space admits an almost sure saddle-point splitting which is cocycle-invariant and corresponds to an exponential dichotomy for the stochastic flow     

On the colored Tutte polynomial of a graph of bounded treewidth

We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601-622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39-54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276-290], using a different algorithm based on logical techniques.     

An algorithmic sign-reversing involution for special rim-hook tableaux

Eğecioğlu and Remmel [Linear Multilinear Algebra 26 (1990) 59–84] gave an interpretation for the entries of the inverse Kostka matrix K⁻¹ in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK⁻¹=I but were unable to do the same for the equation K⁻¹K=I. We define an algorithmic sign-reversing involution on rooted special rim-hook tableaux which can be used to prove that the last column of this second product is correct. In addition, following a suggestion of Chow [preprint, math.CO/9712230, 1997] we combine our involution with a result of Gasharov [Discrete Math. 157 (1996) 193–197] to give a combinatorial proof of a special case of the (3+1)-free Conjecture of Stanley and Stembridge [J. Combin. Theory Ser. A 62 (1993) 261–279].     

Dilute solution behaviour of progressively hydrolyzed polyacrylamide in water-N, N dimethylformamide mixtures

The intrinsic viscosities [η’s] of anionic (hydrolyzed; low and high carboxyl content) and nonionic polyacrylamide (unhydrolyzed) were measured in water-N, N dimethylformamide mixtures at various temperatures. Non-polyelectrolyte behavior of low carboxyl content polyacrylamide was observed in mixed solvent system. The plots of [η] vs. solvent composition in a mixed solvent system pass through minima for both high as well as low carboxyl content polymers but through a maximum for nonionic polyacrylamide. Observed minimum for charged polymers may be attributed to the loss of polymer sites available to interact with solvent for H-bonding interaction between neighboring amide and the acid groups. The maximum for nonionic polymer at the particular solvent composition arises for the most powerful cosolvent effect. Existence of two antagonistic effects is apparent in [η] values of nonionic polymer at various temperatures. Huggins constant (K_H) also indicates a significant variation of cosolvency as a function of solvent composition. Activation parameters of viscous flow were calculated using Frenkel-Eyring equation. The volume related parameter and the shape factor were also computed. Shape factor data indicate that polymer molecules are more or less rigid spheres and are not affected by temperature and composition of solvent.