Automatic continuity of homomorphisms and fixed points on metric compacta

Let G₂(ℝ) × Sp₆(ℝ) and G₂(ℝ) × F₄(ℝ) be split dual pairs in split E₇(ℝ) and E₈(ℝ), respectively. It is known that the exceptional correspondences for these dual pairs are functorial on the level of infinitesimal characters. In this paper we show that these dual pair correspondences are functorial for the minimal K-types of principal series representations. Gordan Savin’s research is partially supported be NSF Grant no. DMS-0551846     

Categorical Morita Equivalence for Group-Theoretical Categories

A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, ω), where G is a finite group and ω ? H ³(G, k ^×). A group-theoretical and cohomological interpretation of this relation is given. A series of concrete examples of pairs of groups that are categorically Morita equivalent but have nonisomorphic Grothendieck rings are given. In particular, the representation categories of the Drinfeld doubles of the groups in each example are equivalent as braided tensor categories and hence these groups define the same modular data.     

POLYNOMIALS RELATED TO HALL NUMBERS

Let Λ be a finite dimensional algebra of finite representation type over a finite field k. For any modules A, B and Pin mod Λ with P projective, we prove that there exists a polynomial ? ^B _(P)Λ over Z whose evaluation at |E| for any conservative finite field extension E of Λ is the sum of Hall numbers F ^{B E} _{C ^E A ^E} where C ^E runs through isoclasses in mod Λ ^E and P ^E is the projective cover of C ^E . As a consequence of this result and its dual version, Hall polynomials ? ^E _CA exist when C or A is semisimple. As applications of the main result, we obtain the existence of Hall polynomials for Nakayama algebras and some selfinjective algebras.

     

Intertwining operators for L 2(E)

Let (E,Q) be a finite dimensional quadratic vector space over a finite field. For the natural representation -π of the isometry group G of (E,Q) in the space L ²(E) of all complex valued functions on E, we analyse when the intertwining algebra of π is generated by just one averaging operator.     

Elliptic Fibrations and Symplectic Automorphisms on K3 Surfaces

Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice U ³ ⊕ E ₈(?1)² depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.     

A simple characterization of the set ofμ-entropy pairs and applications

We present simple characterizations of the setsE _μ andE _X of measure entropy pairs and topological entropy pairs of a topological dynamical system (X, T) with invariant probability measureμ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relationP forms a residual subset of the setE _X . Finally an example of a minimal point distal dynamical system is constructed for whichE _X ∩(X ₀×X ₀)≠ , whereX ₀ is the denseG _δ subset of distal points inX.     

Co-Point Modules Over Frobenius Koszul Algebras

Let A be a Frobenius Koszul algebra such that its Koszul dual A ^! is a quantum polynomial algebra. Co-point modules over A were defined as dual notion of point modules over A ^! with respect to the Koszul duality. In this article, we will see that various important functors between module categories over A used in representation theory of finite dimensional algebras send co-point modules to co-point modules. As a consequence, we will show that if (E, σ) is a geometric pair associated to A ^!, then the map σ:E → E is an automorphism of the point scheme E of A ^!, so that there is a bijection between isomorphism classes of left point modules over A ^! and those of right point modules over A ^!.     

On a Class of Generalized Vector Quasiequilibrium Problems with Set-Valued Maps

This paper considers the following generalized vector quasiequilibrium problem: find a point (z ₀,x ₀) of a set E×K such that x ₀∈A(z ₀,x ₀) and

Existence Results for Set-Valued Vector Quasiequilibrium Problems

This paper deals with the set-valued vector quasiequilibrium problem of finding a point (z ₀,x ₀) of a set E×K such that (z ₀,x ₀)∈B(z ₀,x ₀)×A(z ₀,x ₀), and, for all η∈A(z ₀,x ₀),

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.     

On the Automorphism Group of a Free Pro-l Meta-abelian Group and an Application to Galois Representations

In § l of this article, we study group-theoretical properties of some automorphism group Ψ^* of the meta-abelian quotient § of a free pro-l group § of rank two, and show that the conjugacy class of some element of order two of Ψ^* is not determined by the action induced on the abelian quotient § of § in the case of § = 2. In § 2 we apply the results to the outer Galois representation § attached to the curve C deleted one point from an elliptic curve E, and give an example that §_c does not factor through the l-adic representation attached to E.     

Uniform harmonic approximation with continuous extension to the boundary

Let Ω be an open set in ℝ ⁿ andE be a relatively closed subset of Ω. Further, letC _e(E) be the collection of real-valued continuous functions onE which extend continuously to the closure ofE in ℝ ⁿ . We characterize those pairs (Ω,E) which have the following property: every function inC _e(E) which is harmonic onE ⁰ can be uniformly approximated onE by functions which are harmonic on Ω and whose restrictions toE belong toC _e(E).     

Some sets obeying harmonic synthesis

LetX be a (not necessarily closed) subspace of the dual spaceB ^* of a separable Banach spaceB. LetX ₁ denote the set of all weak ^* limits of sequences inX. DefineX _a , for every ordinal numbera, by the inductive rule:X _a = (U _{b < a} X _b ) ₁ .There is always a countable ordinala such thatX _a is the weak ^* closure ofX; the first sucha is called theorder ofX inB ^* . LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak ^* dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa. This work was done in Jerusalem, when the second-named author was a visitor at the Institute of Mathematics of the Hebrew University of Jerusalem, with the support of an NSF International Travel Grant and of NSF Grant GP33583.     

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

On Nuclei and Duals of Semifields of Order q 4, q is a Power of 2

The aim of this article is to investigate the nuclei of a semifield A of order q ⁴, q is a power of 2, admitting a four-group E of automorphisms isomorphic to Z ₂ × Z ₂ acting freely on A and having GF(q ²) in its left nucleus. We also study the dual semifield A* of A and show that A and A* are isomorphic.     

Special orthogonal splittings of<Emphasis Type="Italic">L</Emphasis><Stack><Subscript>1</Subscript><Superscript>2<Emphasis Type="Italic">k</Emphasis></Superscript></Stack>

We show that for each positive integerk there is ak×k matrixB with ±1 entries such that puttingE to be the span of the rows of thek×2k matrix [√kI _k,B], thenE,E ^⊥ is a Kashin splitting: TheL ₁ ^2k and theL ₂ ^2k are universally equivalent on bothE andE ^⊥. Moreover, the probability that a random ±1 matrix satisfies the above is exponentially close to 1. Supported by the Israel Science Foundation.     

On mackey convergence in locally convex spaces

After some introductory propositions, we give a dual characterization of those locally convex spaces which satisfy the Mackey convergence condition or the fast convergence condition by means of Schwartz topologies. Making use of the universal Schwartz space (l _∞ ,τ(l _∞ ,l ₁)) we prove some representation theorems for bornological and ultrabornological spaces, that is, every bornological spaceE is a dense subspace of an inductive limit lim indE _a, a∈A, ofseparable Banach spacesE _a, and every Mackey null sequence inE is a null sequence in someE _a. IfE is ultrabornological, thenE can be represented as lim indE _a,a∈A, allE _a separable Banach spaces, such that every fast null sequence inE is a null sequence in someE _a.     

The effect of a surface energy term on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing among functions u:?^d?Ω→?^d, u_∣?Ω=0, and measurable subsets E of Ω. Here f_h⁺, f^? denote quadratic potentials defined on Ω¯×{symmetric d×d matrices}, h is the minimum energy of f_h⁺ and ε(u) is the symmetric gradient of the displacement field u. An equilibrium state û, Ê of J(u,E) is called one‐phase if E=?? or E=Ω, two‐phase otherwise. For two‐phase states, σ∣?E∩Ω∣ measures the effect of the separating surface, and we investigate the way in which the distribution of phases is affected by the choice of the parameters h??, σ>0. Additional results concern the smoothness of two‐phase equilibrium states and the behaviour of inf J(u,E) in the limit σ↓0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non‐zero boundary values. Copyright © 2002 John Wiley & Sons, Ltd.     

Two comments on Dvoretzky’s sphericity theorem

For any two positive integersk, l and anyɛ>0 there exists anN(k, l, ɛ) so that given anyl convex bodiesC ₁, …,C _l symmetric about the origin inE ⁿ withn≧N there exists a subspaceE ^k so that eachC _i intersectsE ^k, or has a projection intoE ^k, in a set which is nearly spherical (asphericity <ɛ). The measure of the totality ofE ^k which intersect a given body inE ⁿ in a nearly ellipsoidal set is considered and an affine invariant measure is introduced for that purpose.