Automorphic Plancherel density theorem

Let F be a totally real field, G a connected reductive group over F, and S a finite set of finite places of F. Assume that G(F ?_? ?) has a discrete series representation. Building upon work of Sauvageot, Serre, Conrey-Duke-Farmer and others, we prove that the S-components of cuspidal automorphic representations of $G\left( {\mathbb{A}_F } \right)$ are equidistributed with respect to the Plancherel measure on the unitary dual of G(F _S ) in an appropriate sense. A few applications are given, such as the limit multiplicity formula for local representations in the global cuspidal spectrum and a quite flexible existence theorem for cuspidal automorphic representations with prescribed local properties. When F is not a totally real field or G(F ?_? ?) has no discrete series, we present a weaker version of the above results.     

Bernstein theorems and transformations of correlation measures in statistical physics

We study the class of endomorphisms of the cone of correlation functions generated by probability measures. We consider algebraic properties of the products (·, ?) and the maps K, K ^?1 which establish relationships between the properties of functions on the configuration space and the properties of the corresponding operators (matrices with Boolean indices): F(γ) → F?(γ) = {F(α?β)}α,β?γ. For the operators F?(γ) and F?(γ), we prove conditions which ensure that these operators are positive definite; the conditions are given in terms of complete or absolute monotonicity properties of the function F(γ).     

Free and almost-free subsemigroups of a free semigroup

A subsemigroup S of a free semigroup F(Σ) is almost-free if there is a free subsemigroupT such that S?T?F(Σ) and T/S is finite. It is shown that it is decidable whether a subsemigroup generated by a regular subset of F(Σ) is almost-free. Sufficient- conditions are given such that if a family F of subsets of F(Σ) satisfies these conditions, then it is undecidable for L∈F whether the subsemigroup generated by L is free and also whether it is almost-free.     

Hyperbolicity of algebras with involution over a given extension

Let F be a field and (A,σ) a central simple F-algebra with involution. Let π(t) be a separable polynomial over F. Let F(π)=F[t]/(π(t)). Tignol considered the question whether the algebra A⊗F(π) is hyperbolic. He introduced the algebra H_π, which is universal for this question. Haile and Tignol determined the structure of the algebra H_π and introduced a certain homomorphic image C_π of H_π. In this paper, we give a new characterization of C_π and introduce a new algebra A_π that classifies the commutative algebras with involution that become hyperbolic over F(π). We determine the structure of A_π and use it to examine some examples.     

Corner boundary layer in nonlinear singularly perturbed elliptic problems

The Dirichlet problem in a rectangle is considered for the elliptic equation ?²Δu = F(u, x, y, ?), where F(u, x, y, ?) is a nonlinear function of u. The method of corner boundary functions is applied to the problem. Assuming that the leading term of the corner part of the asymptotics exists, an asymptotic expansion of the solution is constructed and the remainder is estimated.     

Verma Modules Over Generalized Block Algebras B(b (1), b (2))

ABSTRACT

Given a field F with characteristic zero, a free Abelian group G with rank two, and a total order ? on G which is compatible with the addition, we define Verma modules M([ddot], ?) over the generalized Block algebra B(b ⁽¹⁾, b ⁽²⁾) with b ⁽¹⁾, b ⁽²⁾ ∈ F. The irreducibility of the module M([ddot], ?) is completely determined in this article.     

Transformation and summation formulas for double hypergeometric series

A number of new transformation formulas for double hypergeometric series are presented. The series appearing here are the so-called Kampé de Fériet functions of type F_1:1;2^0:3;4(1,1) and F_0:2;2^1:2;2(1,1). The transformation formulas relate such double series to a single hypergeometric series of ₄F₃(1) type. By specializing certain parameters, a list of new summation formulas for F_0:2;2^1:2;2(1,1) series is obtained. The origin of the results comes from studying symmetries of the 9-j coefficient appearing in quantum theory of angular momentum.     

The minimal dimension of maximal commutative subalgebras of full matrix algebras

Let F be a field and A a maximal commutative subalgebra of the full matrix algebra M_n(F). It is shown that dim A > (2n)^{$\text{2}\text{3}$} ? 1. It is also shown that if the radical of A has cube zero, then dim A ? [3n^{$\text{2}\text{3}$} ? 4], and that this result is best possible for infinitely many natural numbers n.     

Primitive stable representations of free Kleinian groups

In this paper, we give a complete criterion for a discrete faithful representation ρ: F _n →PSL(2, ?) to be primitive stable. This will answer Minsky’s conjectures about geometric conditions on ?³/ρ(F _n ) regarding the primitive stability of ρ.     

On Partially Ordered Semigroups and an Abstract Set-Difference

In this paper we prove the independence of a system of five axioms (S1)–(S5), which was proposed in the book of Pallaschke and Urbański (Pairs of Compact Convex Sets, vol. 548, Kluwer Academic Publishers, Dordrecht, 2002) for partially ordered commutative semigroups. These five axioms (S1)–(S5) are stated in the introduction below. A partially ordered commutative semigroup satisfying these axioms is called a F-semigroup. By the use of a further axiom (S6) we define an abstract difference for the elements of a F-semigroup and prove some basic properties. The most interesting example of a F-semigroup are the nonempty compact convex sets of a topological vector space endowed with the Minkowski sum as operation and the inclusion as partial order. In Section 4 we apply the abstract difference to the problem of minimality of convex fractions. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

Differential operators and BV structures in noncommutative geometry

We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operators on a commutative algebra in such a way that derivations of the commutative algebra are replaced by \mathbbDer(A){\mathbb{D}{\rm er}(A)}, the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A){\mathcal{F}(A)}, a certain ‘Fock space’ associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)){\mathcal{D}(\mathcal{F}(A))} of differential operators is filtered and gr D(F(A)){\mathcal{D}(\mathcal{F}(A))}, the associated graded algebra, is commutative in some ‘wheeled’ sense. The resulting ‘wheeled’ Poisson structure on gr D(F(A)){\mathcal{D}(\mathcal{F}(A))} is closely related to the double Poisson structure on T_A \mathbbDer(A){T_{A} \mathbb{D}{\rm er}(A)} introduced by Van den Bergh. Specifically, we prove that gr D(F(A)) @ F(T_A(\mathbbDer(A)),{\mathcal{D}(\mathcal{F}(A))\cong\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)),} provided the algebra A is smooth. Our construction is based on replacing vector spaces by the new symmetric monoidal category of wheelspaces. The Fock space F(A){\mathcal{F}(A)} is a commutative algebra in this category (a “commutative wheelgebra”) which is a structure closely related to the notion of wheeled PROP. Similarly, we have Lie, Poisson, etc., wheelgebras. In this language, D(F(A)){\mathcal{D}(\mathcal{F}(A))} becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper, we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on \mathbbDer(A){\mathbb{D}{\rm er}(A)} gives rise to a second-order (wheeled) differential operator, a noncommutative analogue of the Batalin-Vilkovisky (BV) operator, that makes F(T_A(\mathbbDer(A))){\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)))} a BV wheelgebra. In the final section, we explain how the wheeled differential operators D(F(A)){\mathcal{D}(\mathcal{F}(A))} produce ordinary differential operators on the varieties of n-dimensional representations of A for all n ≥ 1.     

Nilpotent fusion categories

In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger.     

On free p-superalgebras

In this article we consider several aspects of algebraic combinatorics and combinatorial algebra over fields of prime characteristics. P-super-Radford theorem gives the structure of the free associative algebra over a field of prime characteristic with the new multiplication given by the super shuffle product, we show that this algebra is isomorphic to the reduced free super commutative algebra on s-regular words. We prove the elimination theorem for free partially commutative color Lie p-superalgebras and obtain a Schreier type formula for free Lie p-superalgebras using formal power series techniques.     

Homology of free Lie powers and torsion in groups

LetG be a group that is given by a free presentationG=F/R, and let_γ4 R denote the fourth term of the lower central series of R. We show that ifG has no elements of order 2, then the torsion subgroup of the free central extensionF/[_γ4 R,F] can be identified with the homology groupR _γ6(G, ℤ/2ℤ). This is a consequence of our main result which refers to the homology ofG with coefficients in Lie powers of relation modules.     

On the structure of a relatively free Grassmann algebra

We investigate the multiplicative and T-space structure of the relatively free algebra F ⁽³⁾ with a unity corresponding to the identity [[x ₁ , x ₂], x ₃] = 0 over an infinite field of characteristic p > 0. The highest emphasis is placed on unitary closed T-spaces over a field of characteristic p > 2. We construct a diagram containing all basic T-spaces of the algebra F ⁽³⁾, which form infinite chains of the inclusions. One of the main results is the decomposition of quotient T-spaces connected with F ⁽³⁾ into a direct sum of simple components. Also, the studied T-spaces are commutative subalgebras of F ⁽³⁾; thus, the structure of F(3) and its subalgebras can be described as modules over these commutative algebras. Separately, we consider the specifics of the case p = 2. In the Appendix, we study nonunitary closed T-spaces and the case of a field of zero characteristic.     

Torsion and divisibility in finitely generated commutative semirings

It is conjectured that (additive) divisibility is equivalent to (additive) idempotency in a finitely generated commutative semiring S. In this paper we extend this conjecture to weaker forms of these properties—torsion and almost-divisibility (an element \(a\in S\) is called almost-divisible in S if there is \(b\in \mathbb {N}\cdot a\) such that b is divisible in S by infinitely many primes). We show that a one-generated semiring is almost-divisible if and only if it is torsion. In the case of a free commutative semiring F(X) we characterize those elements \(f\in F(X)\) such that for every epimorphism \(\pi \) of F(X) torsion and almost-divisibility of \(\pi (f)\) are equivalent in \(\pi (F(X))\).     

A personnel assignment problem

The following personnel assignment problem is considered. Let (T, ?) be a linearly ordered set where T is a set (of people), and let (P, ?) be a partially ordered set where P, a set of positions of two types, is of the same cardinality as T. Each person i in T is to be assigned to a position. A feasible assignment of personnel to positions is an embedding of (P, ?) in (T, ?). Given measures of each person's effectiveness in both types of positions, an optimal assignment maximizes the total measure of effectiveness. The general assignment problem is shown to be NP-complete. O(n log n) algorithms for two special cases of the problem are presented.     

Endomorphisms of the Semigroup G2(r) Over Partially Ordered Commutative Rings Without Zero Divisors and with 1/2

Let R be a partially ordered commutative ring without zero divisors and with 1/2. Let G _n (R) be the subsemigroup of GL _n (R) consisting of matrices with nonnegative elements. In the paper, we describe endomorphisms of this semigroup for n = 2.     

Automorphisms of the semigroup of nonnegative invertible matrices of order 2 over rings

In this paper, we describe automorphisms of the semigroup G₂(R) of nonnegative invertible matrices if R is a (not necessarily commutative) partially ordered ring without zero divisors with 1/n for some natural number n?>?1.     

Algebras of holomorphic mappings in (DF)-spaces

Let U be an absolutely convex open subset of a complex barrelled (DF)-space E and let F be a commutative Banach algebra with identity. Let H_b(U, F) be the space of holomorphic mappings from U into F that are bounded on the U-bounded sets and let H_b (U, F) be the space of the holomorphic mappings from U into F that are uniformly weakly continuous on the U-bounded sets, both endowed with the topology τ_b of uniform convergence on the U-bounded sets. The spectra of (H_wu (U, F), τ_b) and (H_b(U, F), τ_b) are studied.