On The Structure and Representations of the Insertion–Elimination Lie Algebra

We examine the structure of the insertion–elimination Lie algebra on rooted trees introduced in Connes and Kreimer (Ann. Henri Poincar 3(3):411–433, 2002). It possesses a triangular structure , like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a “lowest weight” . We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.      

Vanishing of the Batalin–Vilkovisky Algebra Structure for TCFTs

The purpose of this short note is to show that for topological conformal field theories in dimension two (or closed string field theories) the Batalin–Vilkovisky algebra structure on the homology of the BRST-complex [G] ([Z]) is trivial after a certain element of ghost number 0 is inverted. The important point here is that we do not restrict our considerations to the tree level but incorporate the action of moduli spaces for higher genera. Received: 22 September 1998 / Accepted: 1 March 1999     

The Principal SO(1,2) Subalgebra of a Hyperbolic Kac–Moody Algebra

The analog of the principal SO(3) subalgebra of a finite-dimensional simple Lie algebra can be defined for any hyperbolic Kac–Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact algebra SO(1,2). We exhibit the decomposition of g(A) into representations of SO(1,2). With the exception of the adjoint SO(1,2) algebra itself, all of these representations are unitary. We compute the Casimir eigenvalues; the associated exponents are complex and noninteger.     

The Structure of the Algebra of Observables in the Intermediate Situation of the ∈-Model

We look at the action of the spin-1/2 operatorsof quantum mechanics on the state of an entity in aphysical way, and use this as a guideline to define theoperators of the intermediate situations of a general spin-1/2 measurement model called the-model. Then we test the possible linearity ofthe operators so constructed.     

The Lie–Poisson representation of the nonlinearized eigenvalue problem of the Kac–van Moerbeke hierarchy

The Kac–van Moerbeke hierarchy is studied by a 3×3 discrete eigenvalue problem and the corresponding nonlinearized one an integrable Poisson map with a Lie–Poisson structure is also presented. Moreover, the 2×2 nonlinearized eigenvalue problem associated with the Kac–van Moerbeke hierarchy is proved to be a reduction of the Poisson map on the leaves of the symplectic foliation.     

The Kac Version of the Sherrington–Kirkpatrick Model at High Temperatures

We study the Kac version of the Sherrington–Kirkpatrick (SK) model of a spin glass, i.e., a spin glass with long- but finite-range interaction on and Gaussian mean zero couplings. We prove that for all < 1, the free energy of this model converges to that of the SK model as the range of the interaction tends to infinity. Moreover, we prove that for all temperatures for which the infinite-volume Gibbs state is unique, the free energy scaled by the square root of the volume converges to a Gaussian with variance c _, , where ^–1 is the range of the interaction. Moreover, at least for almost all values of , this variance tends to zero as goes to zero, the value in the SK model. We interpret our finding as a weak indication that at least at high temperatures, the SK model can be seen as a reasonable asymptotic model for lattice spin glasses.     

Tightness of the Ising–Kac Model on the Two-Dimensional Torus

We consider the sequence of Gibbs measures of Ising models with Kac interaction defined on a periodic two-dimensional discrete torus near criticality. Using the convergence of the Glauber dynamic proven by Mourrat and Weber (Commun Pure Appl Math 70:717–812, 2017) and a method by Tsatsoulis and Weber employed in (arXiv:1609.08447 2016), we show tightness for the sequence of Gibbs measures of the Ising–Kac model near criticality and characterise the law of the limit as the \(\Phi ^4_2\) measure on the torus. Our result is very similar to the one obtained by Cassandro et al. (J Stat Phys 78(3):1131–1138, 1995) on \(\mathbb {Z}^2\), but our strategy takes advantage of the dynamic, instead of correlation inequalities. In particular, our result covers the whole critical regime and does not require the large temperature/large mass/small coupling assumption present in earlier results.     

Rigidity of Kac–Schwarz operators

Letters in Mathematical Physics - In his work on the mathematical formulation of 2d quantum gravity Schwarz established a rigidity result for Kac–Schwarz operators for the n-KdV hierarchies....     

Investigation of the Models of Magnetic Dendrimers by the Wang–Landau Method

Physics of the Solid State - The thermodynamic properties of the models of magnetic dendrimers were studied by the Monte Carlo method. The system state density is calculated; the ground state...     

Formulation of the Snell–Descartes Laws in Terms of Geometric Algebra

Russian Physics Journal - A formulation of the laws of reflection and refraction of optical rays by the interface of two isotropic media using the mathematical apparatus of geometric algebra is...     

The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold ${\mathcal {M}}$ , defined by the Lie structure of the vector fields, their action and their module structure over ${C^\infty({\mathcal {M}})}$ , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra ${\Lambda_{R}({\mathcal {M}})}$ , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact ${{\mathcal {M}}}$ ) ${Z}$ which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z  =  i z, z  =  0 and ${z = \hbar}$ , respectively; canonical quantization uniquely follows from such a general geometrical structure. For ${z =\hbar \neq 0}$ , the regular factorial Hilbert space representations of ${\Lambda_{R}({\mathcal{M}})}$ describe quantum mechanics on ${{\mathcal {M}}}$ . For z  =  0, if Diff( ${{\mathcal {M}}}$ ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on ${{\mathcal {M}}}$ .     

Investigations of the Drell–Yan process at the LHC: Results and prospects

Experimental results on the Drell–Yan process obtained with the CMS detector in the first run of the Large Hadron Collider are briefly reviewed. Some original results on this process are presented, and the prospects for its further investigations in the second LHC run are analyzed.     

On the Diffuse Structure of the Toluene–Water Interface

The electric density profile along the normal to the phase interface between aromatic hydrocarbon toluene and water has been studied by X-ray reflectometry using synchrotron radiation. According to the experimental data, the width of the interface under normal conditions is (3.9 ± 0.1) Å. This value is much larger than a theoretical value of (5.7 ± 0.2) Å predicted by the theory of capillary waves with an interphase tension of (36.0 ± 0.1) mN/m. The observed broadening of the interface is attributed to its own diffuse near-surface structure with a width no less than Å, which is about the value previously discussed for (high-molecular-weight saturated hydrocarbon–water) and (1,2-dichloroethane–water) interfaces.     

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.     

The Influence of Polyvinylpyrrolidone on the Structure and Optical Properties of ZnO–MgO Nanocomposites Synthesized by the Polymer–Salt Method

Optics and Spectroscopy - The influence of polyvinylpyrrolidone on the structure and optical properties of ZnO–MgO nanocomposites synthesized by the polymer–salt method is studied. The...