Locally primitive Cayley graphs of finite simple groups

A graph Г is said to be G-locally primitive, where G is a subgroup of automorphisms of Г, if the stabiliser G_a of a vertex α acts primitively on the set Г( α ) of vertices of Г adjacent to α. For a finite non-abelian simple group L and a Cayley subset S of L, suppose that L ⊴ G ⩽ Aut( L), and the Cayley graph Г = Cay ( L, S) is G-locally primitive. In this paper we prove that L is a simple group of Lie type, and either the valency of Г is an add prine divisor of |Out(L)|, orL =PΩ ₈ ⁺ (q) and Г has valency 4. In either cases, it is proved that the full automorphism group of Г is also almost simple with the same socle L.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

Pairwise ?-independence of the sets TiA for a mixing transformation T

If an ergodic automorphism T of a probability space is not partially rigid, then for any numbers a ∈ (0, 1) and ɛ > 0 there exists a set A such that all sets T ⁱ A, i > 0, are pairwise ɛ-independent. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 43, No. 2, pp. 88–91, 2009 Original Russian Text Copyright ? by V. V. Ryzhikov This research was carried out under the Program for Support of Leading Scientific Schools in the Russian Federation (grant no. 6849.2006.1).     

Représentations du groupe des tresses généralisées dans des groupes de Lie

Let Γ be a Lie group.Then any automorphism of the free group of rank n induces a diffeomorphism of Γ ⁿ . We use this remark and a result of P. Vogel to construct linear representations of a certain automorphism group of the algebraic closure of the free group. This automorphism group is closely related to the string link cobordism group.

Received:     

Automorphism groups of graphs with forbidden subgraphs

For a graph Ф letF(Ф) be the class of finite graphs which do not contain an induced subgraph isomorphic to Ф. We show that whenever Ф is not isomorphic to a path on at most 4 vertices or to the complement of such a graph then for every finite groupG there exists a graph ГєF(Ф) such thatG is isomorphic to the automorphism group of Г. For all paths д on at most 4 vertices we determine the class of all automorphism groups of members ofF(д).     

Graphs of actions on R-trees

LetG be a finitely generated group acting on anR-treeT. First assume that the action is free, and minimal (there is no proper invariant subtree), or more generally that it satisfies a certain finiteness condition. Then it may be described as agraph of transitive actions: the action may be recovered from a finite graph, together with additional data; in particular, every vertexv carries an action (G _v, T_v) whose orbits are dense. For the action (G, T), it follows for instance that the closure of any orbit is a discrete union of closed subtrees: it cannot meet a segment in a Cantor set. Now let ℓ be the length function for an arbitrary action ofG. For ɛ>0 small enough, the subgroupG(ɛ)⊂G generated by elementsg withg is independent of ɛ, andG/G(ɛ) is free. Several interpretations are given for the rank ofG/G(ɛ).     

Renormalization and homogenization of fractional diffusion equations with random data

 This paper presents a renormalization and homogenization theory for fractional-in-space or in-time diffusion equations with singular random initial conditions. The spectral representations for the solutions of these equations are provided. Gaussian and non-Gaussian limiting distributions of the renormalized solutions of these equations are then described in terms of multiple stochastic integral representations. Received: 30 May 2000 / Revised version: 9 November 2001 / Published online: 10 September 2002 Mathematics Subject Classification (2000): Primary 62M40, 62M15; Secondary 60H05, 60G60 Key words or phrases: Fractional diffusion equation – Scaling laws – Renormalised solution – Long-range dependence – Non-Gaussian scenario – Mittag-Leffler function – Stable distributions – Bessel potential – Riesz potential     

Automorphism group and representation of a twisted multi-loop algebra

Let G be the complexification of the real Lie algebra so（3） and A = C[t1^±1, t2^±1] be the Lau-ent polynomial algebra with commuting variables. Let L：（t1, t2, 1） = G c .A be the twisted multi-loop Lie algebra. Recently we have studied the universal central extension, derivations and its vertex operator representations. In the present paper we study the automorphism group and bosonic representations ofL（t1, t2, 1）.     

Affinely equivalent complete flat manifolds

Let E _Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E _Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝ ^m-n . As an application we give some estimates of card E _Aff(Γ,G, m).     

Indefinite higher Riesz transforms

Stein’s higher Riesz transforms are translation invariant operators on L ²(R ⁿ ) built from multipliers whose restrictions to the unit sphere are eigenfunctions of the Laplace–Beltrami operators. In this article, generalizing Stein’s higher Riesz transforms, we construct a family of translation invariant operators by using discrete series representations for hyperboloids associated to the indefinite quadratic form of signature (p,q). We prove that these operators extend to L ^r -bounded operators for 1<r<∞ if the parameter of the discrete series representations is generic.     

Representations in L-Spaces on Infinite-Dimensional Symmetric Cones

In this paper we study representations of the automorphism groups of classical infinite-dimensional tube domains. In particular we construct the L²-realization of all unitary highest weight representations, including the vector-valued case. We also find a projective representation of the full identity component of the affine automorphism group of the Hilbert-Schmidt version of the tube domain with trivial cocycle on the subgroup corresponding to the trace class version, but non-trivial on the large group. Finally we show that the operator-valued measures corresponding to the vector valued highest weight representations have densities of a rather weak type with respect to Wishart distributions which makes it possible to determine their “supports.”     

A geometric Jacquet-Langlands correspondence for <Emphasis Type="Italic">U</Emphasis>(2) Shimura varieties

Let G be a unitary group over ℚ, associated to a CM-field F with totally real part F ⁺, with signature (1, 1) at all the archimedean places of F ⁺. Under certain hypotheses on F ⁺, we show that Jacquet-Langlands correspondences between certain automorphic representations of G and representations of a group G′ isomorphic to G except at infinity can be realized in the cohomology of Shimura varieties attached to G and G′.     

Deriving hydrodynamic equations for lattice systems

We study the dynamics of lattice systems in ℤ^d, d ≥ 1. We assume that the initial data are random functions. We introduce the family of initial measures {μ₀^ɛ, ɛ > 0}. The measures μ₀^ɛ are assumed to be locally homogeneous or “slowly changing” under spatial shifts of the order o(ɛ⁻¹ ) and inhomogeneous under shifts of the order ɛ⁻¹ . Moreover, correlations of the measures μ₀^ɛ decrease uniformly in ɛ at large distances. For all τ ∈ ℝ \ 0, r ∈ ℝ^d, and κ > 0, we consider distributions of a random solution at the instants t = τ/ɛ^κ at points close to [r/ɛ] ∈ ℤ^d. Our main goal is to study the asymptotic behavior of these distributions as ɛ → 0 and to derive the limit hydrodynamic equations of the Euler and Navier-Stokes type.     

Homogenization of some parabolic operators with several time scales

The main focus in this paper is on homogenization of the parabolic problem ∂ _t uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ ^r )∇u ^ɛ ) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.     

Automorphisms of groups and of schemes of finite type

We show first that certain automorphism groups of algebraic varieties, and even schemes, are residually finite and virtually torsion free. (A group virtually has a property if some subgroup of finite index has it.) The rest of the paper is devoted to a study of the groups of automorphisms. Aut(Γ) and outer automorphisms Out(Γ) of a finitely generated group Γ, by using the finite-dimensional representations of Γ. This is an old idea (cf. the discussion of Magnus in [11]). In particular the classes of semi-simplen-dimensional representations of Γ are parametrized by an algebraic varietyS _n (Γ) on which Out(Γ) acts. We can apply the above results to this action and sometimes conclude that Out(Γ) is residually finite and virtually torsion free. This is true, for example, when Γ is a free group, or a surface group. In the latter case Out(Γ) is a “mapping class group.” Partially supported by the NSF under Grant MCS 80-05802.     

On maximal proper subgroups of field automorphism groups

Let G be the automorphism group of an extension of algebraically closed fields of characteristic zero of transcendence degree n, 1 ≤ n ≤ ∞. In this paper we

Theta series and trace operators for Г<Subscript>n</Subscript>[q]

We consider the action of suitable trace operators on non homogeneous theta series that are Siegel modular forms for the principal congruence subgroups of the symplectic group of odd levelq: Г _n [q]. This is used for investigating whether modular forms forГ _n [N], withN|q, which are linear combination of such theta series, can be expressed as combination of theta series that are modular forms with respect toГ _n [N].     

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, ${{\rm Sp}(2,\mathbb {R})}Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, or SO(p, 2) with p ≥ 3. The symmetric spaces associated to these G’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of K?hler manifolds and Higgs bundles we study representations of the lattice Г into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU(p, 2) with p ≥ 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) × U(2)), on which it acts cocompactly.     

Eigenvalue distributions from impacts on a ring

We consider the collision dynamics produced by three beads with masses (m ₁, m ₂, m ₃) sliding without friction on a ring, where the masses are scaled so that m ₁ = 1/ɛ, m ₂ = 1, m ₃ = 1 − ɛ, for 0 ⩽ ɛ ⩾ 1. The singular limits ɛ = 0 and ɛ = 1 correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e. 0 < ɛ < 1, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses from continuous to discrete in the two singular limits ɛ → 0 and ɛ → 1 for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit ɛ → 0, the distribution forms Gaussian peaks around the discrete limiting values ± 1, ± i, with variances that scale in power law form as σ ² ∼ αɛ ^β. By contrast, the convergence in the limit ɛ → 1 to the discrete values ±1 is shown to follow a logarithmic power-law σ ² ∼ log(ɛ ^β).     

Nilpotent groups admitting an almost regular automorphism of order four

We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results are proved: (1) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then (a) the subgroup {ie176-1} contains a subgroup of m-bounded index in {ie176-2} which is nilpotent of m-bounded class, and (b) the group G contains a subgroup V of m-bounded index such that the subgroup {ie176-3} is nilpotent of m-bounded class (Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup {ie176-4}, generated by all f(m) th powers of elements in {ie176-5} is nilpotent of class ≤3 (Theorem 2). Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 314–333, May–June, 1996.