The Remodeling Conjecture and the Faber–Pandharipande Formula

In this note, we prove that the free energies F _g constructed from the Eynard–Orantin topological recursion applied to the curve mirror to ${\mathbb{C}^3}$ reproduce the Faber–Pandharipande formula for genus g Gromov–Witten invariants of ${\mathbb{C}^3}$ . This completes the proof of the remodeling conjecture for ${\mathbb{C}^3}$ .     

Energy–entropy dispersion relation in DNA sequences

For a number of virus- and bacterium genomes we use the concept of block entropy from information theory and compare it with the corresponding configurational energy, defined via the ionization energies of the nucleotides and a hopping term for their interactions in the sense of a tight-binding model. Additionally to the four-letter alphabet of the nucleotides we discuss a reduction to a two-letter alphabet. We find a well defined relation between block entropy and block energy for a not too large block length which can be interpreted as a generalized dispersion relation for all genome sequences. The relation can be used to look for enhanced interactions between virus and bacterium genomes. Well known examples for virus–virus and virus–bacterium interactions are analyzed along this line.     

n-Extended Lorentzian Kac–Moody algebras

Letters in Mathematical Physics - We investigate a class of Kac–Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac–Moody algebras defined by their...     

Abnormal breakdown of Stokes–Einstein relation in liquid aluminium

We present the results of systematic molecular dynamics simulations of pure aluminium melt with a well-accepted embedded atom potential. The structure and dynamics were calculated over a wide temperature range, and the calculated results(including the pair correlation function, self-diffusion coefficient, and viscosity) agree well with the available experimental observations. The calculated data were used to examine the Stokes–Einstein relation(SER). The results indicate that the SER begins to break down at a temperature T_x(~1090 K) which is well above the equilibrium melting point(912.5 K).This high-temperature breakdown is confirmed by the evolution of dynamics heterogeneity, which is characterised by the non-Gaussian parameter α_2(t). The maximum value of α 2(t), α_(2,max), increases at an accelerating rate as the temperature falls below Tx. The development of α_(2,max) was found to be related to the liquid structure change evidenced by local fivefold symmetry. Accordingly, we suggest that this high-temperature breakdown of SER has a structural origin. The results of this study are expected to make researchers reconsider the applicability of SER and promote greater understanding of the relationship between dynamics and structure.     

Kramers–Kronig relation in a Doppler-broadened Λ-type three-level system

We measure the absorption and dispersion in a Doppler-broadened Λ-type three level system by resonant stimulated Raman spectroscopy with homodyne detection. Through studying the dressed state energies of the system, it is found that the absorption and dispersion satisfy the Kramers–Kronig relation. The absorption and dispersion spectra calculated by employing this relation agree well with our experimental observations.     

Drinfeld–Sokolov reduction in quantum algebras: canonical form of generating matrices

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).     

On the relation of Manin’s quantum plane and quantum Clifford algebras

In a recent work we have shown that quantum Clifford algebras — i.e. Clifford algebras of an arbitrary bilinear form — are closely related to the deformed structures asq-spin groups, Hecke algebras,q-Young operators and deformed tensor products. The question to relate Manin’s approach to quantum Clifford algebras is addressed here. Explicit computations using the CLIFFORD Maple package are exhibited. The meaning of non-commutative geometry is reexamined and interpreted in Clifford algebraic terms. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Covariance in the Batalin–Vilkovisky formalism and the Maurer–Cartan equation for curved Lie algebras

We express covariance of the Batalin–Vilkovisky formalism in classical mechanics by means of the Maurer–Cartan equation in a curved Lie superalgebra, defined using the formal variational calculus and Sullivan’s Thom–Whitney construction. We use this framework to construct a Batalin–Vilkovisky canonical transformation identifying the Batalin–Vilkovisky formulation of the spinning particle with an AKSZ field theory.

     

FRT presentation of classical Askey–Wilson algebras

Automorphisms of the infinite-dimensional Onsager algebra are introduced. Certain quotients of the Onsager algebra are formulated using a polynomial in these automorphisms. In the simplest case, the quotient coincides with the classical analog of the Askey–Wilson algebra. In the general case, generalizations of the classical Askey–Wilson algebra are obtained. The corresponding class of solutions of the non-standard classical Yang–Baxter algebra is constructed, from which a generating function of elements in the commutative subalgebra is derived. We provide also another presentation of the Onsager algebra and of the classical Askey–Wilson algebras.

     

Poisson–Nijenhuis structures on quiver path algebras

We introduce a notion of noncommutative Poisson–Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to the study of the Calogero–Moser and Gibbons–Hermsen integrable systems. In the former case, we give a new interpretation of the bihamiltonian reduction performed in Bartocci et al. (Int Math Res Not 2010:279–296, 2010. arXiv:0902.0953).     

A class of Lorentzian Kac–Moody algebras

We consider a natural generalisation of the class of hyperbolic Kac–Moody algebras. We describe in detail the conditions under which these algebras are Lorentzian. We also construct their fundamental weights, and analyse whether they possess a real principal so(1,2) subalgebra. Our class of algebras include the Lorentzian Kac–Moody algebras that have recently been proposed as symmetries of M-theory and the closed bosonic string.     

Fractional Stokes–Einstein relation in TIP5P water at high temperatures

Fractional Stokes–Einstein relation described by D ～(τ/T)~ξ is observed in supercooled water, where D is the diffusion constant, τ the structural relaxation time, T the temperature, and the exponent ξ =τ~(-1). In this work, the Stokes–Einstein relation in TIP5 P water is examined at high temperatures within 400 K–800 K. Our results indicate that the fractional Stokes–Einstein relation is explicitly existent in TIP5P water at high temperatures, demonstrated by the two usually adopted variants of the Stokes–Einstein relation, D ～τ~(-1)τand D ～ T/τ, as well as by D ～ T/η, where η is the shear viscosity. Both D ～τ~(-1)τand D ～ T/τ are crossed at temperature T_x= 510 K. The D ～τ~(-1)τis in a fractional form as D ～τξwith ξ =-2.09 for T ≤ T_xand otherwise ξ =τ~(-1).25. The D ～ T/τ is valid with ξ =τ~(-1).01 for T ≤ T_xbut in a fractional form for T T_x. The Stokes–Einstein relation D ～ T/η is satisfied below T_x = 620 K but in a fractional form above T_x. We propose that the breakdown of D ～ T/η may result from the system entering into the super critical region, the fractional forms of D ～τ~(-1)τand D ～ T/τ are due to the disruption of the hydration shell and the local tetrahedral structure as well as the increase of the shear viscosity.     

Geometric approach to Kac–Moody and Virasoro algebras

In this paper we show the existence of a group acting infinitesimally transitively on the moduli space of pointed-curves and vector bundles (with formal trivialization data) and whose Lie algebra is an algebra of differential operators. The central extension of this Lie algebra induced by the determinant bundle on the Sato Grassmannian is precisely a semidirect product of a Kac–Moody algebra and the Virasoro algebra. As an application of this geometric approach, we give a local Mumford-type formula in terms of the cocycle associated with this central extension. Finally, using the original Mumford formula we show that this local formula is an infinitesimal version of a general relation in the Picard group of the moduli of vector bundles on a family of curves (without any formal trivialization).     

Supercooled liquids analogous fractional Stokes–Einstein relation in NaCl solution above room temperature

The Stokes–Einstein relation D～T/η and its two variants D～τ~(-1) and D～T/τ follow a fractional form in supercooled liquids, where D is the diffusion constant, T the temperature, η the shear viscosity, and τ the structural relaxation time.The fractional Stokes–Einstein relation is proposed to result from the dynamic heterogeneity of supercooled liquids.In this work, by performing molecular dynamics simulations, we show that the analogous fractional form also exists in sodium chloride(NaCl) solutions above room temperature.D～τ~(-1) takes a fractional form within 300–800 K; a crossover is observed in both D～T/τ and D～T/η.Both D～T/τ and D～T/η are valid below the crossover temperature T_x,but take a fractional form for T T_x.Our results indicate that the fractional Stokes–Einstein relation not only exists in supercooled liquids but also exists in NaCl solutions at high enough temperatures far away from the glass transition point.We propose that D～T/η and its two variants should be critically evaluated to test the validity of the Stokes–Einstein relation.     

Homological unimodularity and Calabi–Yau condition for Poisson algebras

In this paper, we show that the twisted Poincaré duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincaré duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi–Yau algebra if the Poisson structure is unimodular.