QuantumR matrices related to the spin representations ofB n andD n

We present the explicit form of the trigonometricR matrices related to the spin representations of the simple Lie algebrasX _n=B _n,D _n. We conjecture that one dimensional configuration sums of the corresponding vertex models in the face formulation are the string functions ofX _n ⁽¹⁾ modules.     

Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non polynomial interactions

We consider quantum field theoretical models inn dimensional space-time given by interaction densities which are bounded functions of an ultraviolet cut-off boson field. Using methods of euclidean Markov field theory and of classical statistical mechanics, we construct the infinite volume imaginary and real time Wightman functions as limits of the corresponding quantities for the space cut-off models. In the physical Hilbert space, the space-time translations are represented by strongly continuous unitary groups and the generator of time translationsH is positive and has a unique, simple lowest eigenvalue zero, with eigenvector , which is the unique state invariant under space-time translations. The imaginary time Wightman functions and the infinite volume vacuum energy density are given as analytic functions of the coupling constant. The Wightman functions have cluster properties also with respect to space translations.     

Gauge potentials and bundles over the 4-torus

The construction of principal bundles over a four dimensional torus is considered. The class of groups considered is SU(n)/Z _n , and for this class the Pontrjagin class has even integer values.     

Autonomous models solvable through the full interval method

The most general exclusion single species one dimensional reaction-diffusion models with nearest-neighbor interactions which are both autonomous and can be solved exactly through full interval method are introduced. Using a generating function method, the general solution for F _n , the probability that n consecutive sites be full, is obtained. Some other correlation functions of number operators at nonadjacent sites are also explicitly obtained. It is shown that for a special choice of initial conditions some correlation functions of number operators called full intervals remain uncorrelated.     

Gauge sector statistics of intersecting D‐brane models

In this paper, which is based on the first part of the author's PhD thesis, we review the statistics of the open string sector in T⁶/(ℤ₂ × ℤ₂) orientifold compactifications of type IIA. After an introduction to the orientifold setup we discuss the two different techniques that have been developed, using either a saddle point approximation or a direct computer based method. We explain the two approaches by means of eight‐ and six‐dimensional toy models and present the results for the four‐dimensional models in detail. Special emphasis is placed on models containing phenomenologically interesting gauge groups, in particular those containing a standard model, Pati‐Salam or SU(5) part.     

The dimensional reduction of eleven dimensional supergravity into a product of Robertson-Walker spacetime and the seven sphere

The dimensional reduction of eleven dimensional supergravity is discussed. It is shown that there is no dimensional reduction onto Robertson-Walker space with the asymmetric tensorF giving a realistic fluid. Furthermore it is shown that the ansatz’s for the scale factorR:R=at ⁿ, R=a exp (bt ⁿ), andR=aZ ⁿ, there is no dimensional reduction except the known example of the Freund-Rubin-Englert solution.     

Nonclassical type representations of the q-deformed algebra U′q(son)

The nonstandard q-deformation U_q(so_n) of the universal enveloping algebra U(so _n ) has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional representations of U(so _n ). But U_q(so_n) also has irreducible finite dimensional representations which have no classical analogue. The aim of this paper is to give these representations which are called nonclassical type representations. They are given by explicit formulas for operators of the representations corresponding to the generators of U_q(so_n).     

The problem of chaotic behaviour in a homogeneous arbitrarily dimensional cosmology

The problem of chaotic behaviour in multidimensional mixmaster models is discussed. We show that the chaotic regime disappears in the full class of homogeneous arbitrarily dimensional cosmological models if the dimension of space is n≠3.     

Macdonald polynomials from Sklyanin algebras: A conceptual basis for thep-adics-quantum group connection

We establish a previously conjectured connection betweenp-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which interpolate between the zonal spherical functions of related real andp-adic symmetric spaces. The elliptic quantum algebras underlie theZ _n -Baxter models. We show that in then limit, the Jost function for the scattering offirst level excitations in the 1+1 dimensional field theory model associated to theZ _n -Baxter model coincides with the Harish-Chandra-likec-function constructed from the Macdonald polynomials associated to the root systemA ₁. The partition function of theZ ₂-Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandrac-function, albeit in a less simple way. We relate the two parametersq andt of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular thep-adic regimes in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of q-deforming Euler products.Work supported in part by the NSF: PHY-9000386     

On CP 1 and CP 2 Maps and Weierstrass Representations for Surfaces Immersed into Multi-Dimensional Euclidean Spaces

Abstract

An extension of the classic Enneper–Weierstrass representation for conformally parametrised surfaces in multi-dimensional spaces is presented. This is based on low dimensional CP ¹ and CP ² sigma models which allow the study of the constant mean curvature (CMC) surfaces immersed into Euclidean 3- and 8-dimensional spaces, respectively. Relations of Weierstrass type systems to the equations of these sigma models are established. In particular, it is demonstrated that the generalised Weierstrass representation can admit different CMC-surfaces in ?³ which have globally the same Gauss map. A new procedure for constructing CMC-surfaces in ?ⁿ is presented and illustrated in some explicit examples.     

One-dimensional configuration sums in vertex models and affine Lie algebra characters

We study the local state probabilities of the vertex models in the face formulation associated with the simple Lie algebras X _n =A _n, B _n, C _n, D _n. The corner transfer matrix method expresses them in terms of one-dimensional configuration sums. We show that the latter are the string functions of X _n ⁽¹⁾ modules. We also present similar results for the restricted face models of types B _n ⁽¹⁾, C _n ⁽¹⁾, D _n ⁽¹⁾.     

2-Cocycles and Twisting of Kac Algebras

We describe the twisting construction with the help of 2-cocycles on Hopf–von Neumann and George Kac algebras; we show that twisted Kac algebras are again Kac algebras. Using this construction, we give a wide class of new quantizations of the Heisenberg group and describe several series of non-trivial finite- dimensional -Hopf algebras (Kac algebras) of dimensions 4n and as twisting of finite groups. Received: Received: 21 March 1997 / Accepted: 2 June 1997     

A physical model for dimensional reduction and its effects on the observable parameters of the universe

We construct a physical model to study the effects of dimensional reduction that might have taken place during the inflationary phase of the universe. The model we propose is a (1 + D)-dimensional (D > 3), nonsingular, spatially homogeneous and isotropic Friedmann model. We consider dimensional reduction to take place in a stepwise manner and interpret each step as a phase transition. Independent of the details of the process of dimensional reduction, we impose suitable boundary conditions across the transitions and trace the effects of dimensional reduction to the currently observable parameters of the universe. In order to exhibit the cosmological features of the proposed model, we construct a (1 + 4)-dimensional toy model for both closed and open cases of Friedmann geometries. It is shown that in these models the universe makes transition into the lower dimension when the critical length parameter l _4,3, which signals dimensional reduction, reaches the Planck length in D = 3. The numerical models we present in this paper have the capability of making definite predictions about the cosmological parameters of the universe such as the Hubble parameter, age and density.     

Linear kinematical groups

We prove a theorem which states that in an (n+1)-dimensional space-time (n3) the only linear kinematical groups which are compatible with the isotropy of space are the Lorentz and Galilei groups. The special casesn=1 andn=2 are also briefly discussed.On leave of absence from Instituto di Fisica dell' Universitá, Milano, Italy. A. v. Humboldt fellow.     

Dynamical reduction in multidimensional cosmological models

We present conditions under which there occurs a dynamical dimensional reduction of cosmological models in the form of Bianchi I×(N-3)-dimensional torus filled with matter of the ideal fluid type.     

Electron‐counting rules,three‐dimensional aromaticity,and the boundaries of the Periodic Table

To describe aromaticity of planar and three‐dimensional molecules, different electron counting rules are employed. Here, the relationship between the Hückel 4n + 2 rule and the Hirsch 2(n + 1)² rule is established based on formal approach considering the electrons as objects in an arbitrary n‐dimensional space of states. Two types of three‐dimensional aromaticity, referred to as ‘spherical’ (following the 4n + 2 electron counting rule) and ‘spatial’ (following the 2(n + 1)² electron counting rule) are distinguished. A conclusion concerning the boundaries of the Periodic Table of Elements is made based on the same formal approach that no g‐ (or higher) elements can exist and that possible extension of the Periodic Table beyond the seventh period must be followed by filling of the 8p or inner 6f or 7d levels. Copyright © 2011 John Wiley & Sons, Ltd.     

The Pfaffian line bundle

We analyze the holomorphic Pfaffian line bundle defined over an infinite dimensional isotropic Grassmannian manifold. Using the infinite dimensional relative Pfaffian, we produce a Fock space structure on the space of holomorphic sections of the dual of this bundle. On this Fock space, an explicit and rigorous construction of the spin representations of the loop groupsLO _n is given. We also discuss and prove some facts about the connection between the Pfaffian line bundle over the Grassmannian and the Pfaffian line bundle of a Dirac operator.Supported by a National Science Foundation Graduate Fellowship     

Coherent state operators and n-point invariants for U q ((sl(2))

We write down a complete set of n-point U_q(sl(2)) invariants (using a polynomial basis for the irreducible finite dimensional U _q -modules) that are regular for all nonzero values of the deformation parameter q.     

Theoretical investigation of the phase behaviour of model ternary mixtures containing n-alkanes,perfluoro-n-alkanes,and perfluoroalkylalkane diblock surfactants

We have used the hetero-SAFT-VR approach developed by McCabe and collaborators [Mol. Phys. 104, 571 (2006)] to investigate the phase equilibria of a number of binary and ternary mixtures of n-alkanes, perfluoro-n-alkanes, and perfluoroalkylalkane diblock surfactants. We focused our work on the understanding of the microscopic conditions that control the phase behaviour of these mixtures, with a particular emphasis of the effect on the liquid–liquid separation and the stabilisation of n-alkane + perfluoro-n-alkane mixtures when a diblock surfactant is added. We used very simple molecular models for n-alkanes, and perfluoro-n-alkanes that describe the molecules as chains with tangentially bonded segments with molecular parameters taken from the literature. In the particular case of semifluorinated alkanes or SFA surfactants, we used an hetero-segmented diblock chain model where the parameters for the alkyl and perfluoroalkyl segments taken from the corresponding linear alkanes and perfluoroalkanes, as shown in our previous work [J. Phys. Chem. B 111, 2856 (2007)]. Our goal was to identify the main effects on the phase behaviour when different perfluoroalkylalkane surfactants are added to mixtures of n-alkanes and perfluoro-n-alkanes. We selected the n-heptane + perfluoromethane binary mixture, and studied the changes on the phase behaviour when a symmetric (same number of alkyl and perfluoroalkyl chemical groups) or an asymmetric (different number of alkyl and perfluoroalkyl chemical groups) diblock surfactants is added to the binary mixture. We have obtained the phase diagrams of a wide range of binary and ternary mixtures at different thermodynamic conditions. We have found a variety of interesting behaviours as we modify the alkyl or/and the perfluoroalkyl chain-length of the diblock surfactants: the usual changes in the vapour–liquid phase separation, changes in the type of phase diagrams (typically from type I to type V phase behaviour according to the Scott and Konynenburg classification), azeotropy, and Bancroft points. We noted that the main effect of adding a symmetric or an asymmetric surfactant to the n-heptane + perfluoromethane mixture is to stabilise the system, i.e. to decrease the two-phase (liquid–liquid) immiscibility region of the ternary diagram as the surfactant concentration is increased. This effect becomes larger as the chain length of the surfactant is increased, which is consistent with a higher number of alkyl–alkyl and perfluoroalkyl–perfluoroalkyl favourable interactions in the mixture.