A Simple Cluster Model for the Liquid–Glass Transition

Using the classical distribution-function approach to simple liquids, we estimate the orientational interaction between clusters consisting of a particle and its nearest neighbors. We show that there are density and temperature ranges where the interaction changes sign as a function of the cluster radius. On this basis, the corresponding model of interacting cubic and icosahedral clusters (of the type of a spin glass model) is proposed and solved in the replica-symmetric approximation. We show that the glass order parameter grows continuously on cooling and the replica-symmetry-breaking temperature can be identified with the glass transition temperature. We also show that on cooling a system of particles with a Lennard-Jones interaction, cubic clusters freeze first. The transition temperature for icosahedral clusters is somewhat lower; therefore, the cubic structure of the short-range order is more likely in a Lennard-Jones glass near transition.     

Spin-One <Emphasis Type="Italic">p</Emphasis>-Spin Glass: Exact Solution for Large <Emphasis Type="Italic">p</Emphasis>

We study the low-temperature properties of the p-spin spin glass model in the spin-one (three-state) case for large values of p. We show that the one-step replica symmetry-breaking phase is unstable at a very low temperature, and we calculate the explicit boundary of the stability interval, the Gardner temperature, analytically for large values of p. This temperature for the spin-one model has the same form of dependence on p as in the case of Ising spins (two states). In the one-step replica symmetrybreaking state, a quadrupolar orientational glass coexists with the spin glass and also with a regular quadrupole ordering.     

Groups in NTP<Subscript>2</Subscript>

We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP₂ theory.     

Replica Symmetry Breaking in an Axial Model of Quadrupolar Glass

We investigate the replica symmetry breaking (RSB) in a neighborhood of the instability point of the replica-symmetric solution in the axial quadrupolar glass model. We show that the solution with the first-stage RSB is stable against the subsequent RSB. Although there is no reflection symmetry, the first-stage RSB solution continuously bifurcates from the replica-symmetric one. This implies that our model does not belong to either of the two classes into which spin glasses are usually divided.     

On the modularity of the GL<Subscript>2</Subscript>-twisted spinor L-function

In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke’s converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat’s Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL₂-twisted spinor L-function Z _{G ⊗ h} (s) related to automorphic forms G,h on the symplectic group GSp₂ and GL₂. This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insights into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan’s work on the modularity of the Rankin-Selberg L-series.     

Elasticity of composites with irregularly oriented shape-anisotropic filler particles

A variant of determining the elastic characteristics of composites containing irregularly oriented shape-anisotropic filler particles of two types (short fibers and thin platelets) is considered. The effective elastic constants of the composites are calculated by using the method of orientational averaging of elastic characteristics of isolated transversely isotropic structural elements reinforced with unidirectionally oriented short fibers or coplanarly arranged thin platelets. The superposition of elastic properties of the irregularly oriented structural elements, with account of their orientational distribution in the composite material, is accepted. The calculation results are compared with experimental data for the effective elastic moduli of polymeric composites reinforced with short glass fibers and of polymeric nanocomposites containing the platelet-type particles of organically modified montmorillonite. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 285–300, May–June, 2006.     

The spectrum of the Dirac operator on SU<Subscript>2</Subscript>/Q<Subscript>8</Subscript>

We compute the fundamental Dirac operator for the three-parameter-family of homogeneous Riemannian metrics and the four different spin structures on SU₂/Q₈, where Q₈ denotes the group of quaternions. We deduce its spectrum for the Berger metrics and show the sharpness of Christian Bär’s upper bound for the smallest Dirac eigenvalue in the particular case where SU₂/Q₈ is a homogeneous minimal hypersurface of S ⁴.     

Microscopic theory of superconductivity in MgB<Subscript>2</Subscript>-type systems in a magnetic field: A neighborhood of <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis>2</Subscript>

We calculate the upper critical magnetic field H _c2 in the framework of a microscopic superconductivity theory with two energy bands of different dimensions on the Fermi surface with the cavity topology typical of the compound MgB₂ taken into account (an anisotropic system). We assume an external magnetic field parallel to the crystallographic z axis. We obtain analytic formulas in the low-temperature range (T/T_c ≪ 1) and also near the critical temperature ((T-T_c)/T_c ≪ 1). We compare the temperature dependence of H_c2 for a two-band anisotropic system with that of H _c2 ⁰ corresponding to a two-band isotropic system (with Fermi-surface cavities of the same topology). We determine the role of the band-structure anisotropy, the positive curvature of the upper critical field near the critical temperature, and the important role of the ratio v₁/v₂ of the velocities on the Fermi surface in determining H_c2. We also obtain the values of the parameters Δ₁ and Δ₂ along the line of the critical magnetic field. This paper is dedicated to the 90th birthday of Professor D. N. Zubarev __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 113–128, January, 2008.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-norm packings from function fields

In this paper, we study some packings in a cube, namely, how to pack n points in a cube so as to maximize the minimal distance. The distance is induced by the L₁-norm which is analogous to the Hamming distance in coding theory. Two constructions with reasonable parameters are obtained, by using some results from a function field including divisor class group, narrow ray class group, and so on. We also present some asymptotic results of the two packings.     

A resolvent approach to the stability of essential and critical spectra of perturbed <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroups on Hilbert spaces with applications to transport theory

We give sufficient conditions in terms of resolvents implying the stability of the essential or critical spectra of perturbed C₀-semigroups on Hilbert spaces. We also show how these results apply to transport theory.     

Semi-self-decomposable distributions on Z<Subscript>+</Subscript>

We present a notion of semi-self-decomposability for distributions with support in Z ₊. We show that discrete semi-self-decomposable distributions are infinitely divisible and are characterized by the absolute monotonicity of a specific function. The class of discrete semi-self-decomposable distributions is shown to contain the discrete semistable distributions and the discrete geometric semistable distributions. We identify a proper subclass of semi-self-decomposable distributions that arise as weak limits of subsequences of binomially thinned sums of independent Z ₊-valued random variables. Multiple semi-self-decomposability on Z ₊ is also discussed.     

Maximal Poisson Commutative subalgebras for truncated parabolic subalgebras of maximal index in sl<Subscript>n</Subscript>

We show that several properties of the semisimple algebras carry over to a certain family of parabolic subalgebras of maximal index in sl_n. More precisely we prove an analogue of Kostant's slice theorem [B. Kostant, Amer. J. Math. 85 (1963), 327-404] for these algebras and construct a maximal Poisson commutative subalgebra in the symmetric algebra, following the theory presented in [A.S. Mishchenko and A.T. Fomenko, Math. USSR-Izv. 12 (1978), 371-389]. These results are quite remarkable since these algebras do not admit appropriate sl₂-triples.     

Restrictions of Modular Representations of Special Linear Groups to A<Subscript>1</Subscript>×A<Subscript>1</Subscript> Subgroups

We study the restrictions of infinitesimally irreducible modular representations of algebraic groups of type A_n to subsystem subgroups of type A₁ × A₁. Under some restrictions on the highest weight of a representation we find their composition factors.     

A Note on Polynomially Growing C<Subscript>0</Subscript>-Semigroups

We characterize polynomial growth of a C₀-semigroup in terms of the first power of the resolvent of its generator. We do this for a class of semigroups which includes C₀-semigroups on Hilbert spaces and analytic semigroups on Banach spaces. Furthermore, we characterize polynomial growth for discrete semigroups.     

Lifting elementary embeddings <Emphasis Type="Italic">j</Emphasis>: <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript> → <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript>

We describe a fairly general procedure for preserving I₃ embeddings j: V _λ → V _λ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I₃ embedding, V = HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding V → V”. This improves upon an earlier result in which consistency was established assuming an I₁ embedding.      

Interpolation of operators of weak type (ϕ<Subscript>0</Subscript>, ψ<Subscript>0</Subscript>, ϕ<Subscript>1</Subscript>, ψ<Subscript>1</Subscript>) in Lorentz spaces

We prove theorems on interpolation of quasilinear operators of weak type (ϕ₀, ψ₀, ϕ₀, ψ₁) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and convex functions ϕ₀(t), ψ₀(t), ϕ₁(t), and ψ₁(t). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005.     

Gamma homology,Lie representations and <Emphasis Type="Italic">E</Emphasis><Subscript>∞</Subscript> multiplications

We prove that the stable homotopy of any Γ-module F is the homology of a bicomplex Ξ(F), in which the (q−1)st row is the two-sided bar construction ℬ(Lie^* _q ,Σ _q ,F[q]). This gives a natural homotopical cotangent bicomplex for graded commutative algebras, in a form suitable for use in a new obstruction theory for classifying E _∞ ring structures on spectra. The E _∞ structure on certain Lubin-Tate spectra is a corollary. Oblatum 15-X-2001 & 14-X-2002?Published online: 24 February 2003     

Sturmian morphisms,the braid group <Emphasis Type="Italic">B</Emphasis><Subscript>4</Subscript>, Christoffel words and bases of <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F ₂) of automorphisms of the rank two free group F ₂ and show that it can be realized as a monoid in the group B ₄ of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F ₂ lifting any given basis of the free abelian group Z ². We further give an algorithm allowing to decide whether two elements of F ₂ form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes. Mathematics Subject Classification (2000) 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15