Galois G-covering of quotients of linear categories

In this paper, we introduce the notion of G-liftable ideals, which extends the liftable ideas defined by Assem and Le Meur. We characterize the G-liftable ideals and construct the Galois G-coverings of quotients of categories associated to the G-liftable ideals. In particular, we study the behavior of G-liftable admissible ideals under Galois G-coverings. Furthermore, we show that the ideals generated by finite dimensional projective modules over a locally bounded linear categories are admissible G-liftable ideals. As an application, we provide a reduction technique for dealing with the existence of Serre functors in the stable categories of Gorenstein projective objects.     

Realisability for infinitary intuitionistic set theory

We introduce a realisability semantics for infinitary intuitionistic set theory that is based on Ordinal Turing Machines (OTMs). We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke-Platek set theory are realised. Finally, we use a variant of our notion of realisability to show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible rules of intuitionistic propositional logic.     

Annihilator varieties, adduced representations, Whittaker functionals, and rank for unitary representations of GL(n)

In this paper, we study irreducible unitary representations of ${GL_{n}(\mathbb{R})}$ and prove a number of results. Our first result establishes a precise connection between the annihilator of a representation and the existence of degenerate Whittaker functionals, thereby generalizing results of Kostant, Matumoto and others. Our second result relates the annihilator to the sequence of adduced representations, as defined in this setting by one of the authors. Based on those results, we suggest a new notion of rank of a smooth admissible representation of ${GL_{n}(\mathbb{R})}$ , which for unitarizable representations refines Howe’s notion of rank. Our third result computes the adduced representations for (almost) all irreducible unitary representations in terms of the Vogan classification. We also indicate briefly the analogous results over complex and p-adic fields.     

Nonconvex variational problem with recursive integral functionals in Sobolev spaces: Existence and representation

The purpose of this paper is twofold. First, we present the existence theorem of an optimal trajectory in a nonconvex variational problem with recursive integral functionals by employing the norm-topology of a weighted Sobolev space. We show the continuity of the integral functional and the compactness of the set of admissible trajectories. Second, we show that a recursive integrand is represented by a normal integrand under the conditions guaranteeing the existence of optimal trajectories. We also demonstrate that if the recursive integrand satisfies the convexity conditions, then the normal integrand is a convex function. These results are achieved by the application of the representation theorem in Lp-spaces.     

Asymptotics and integrability properties for matrix coefficients of admissible representations of reductive p-adic groups

We obtain the necessary and sufficient conditions for the pth power integrability of a matrix coefficient of an admissible representation of a reductive p-adic group in terms of the exponents of the matrix coefficient.     

Embedding Problems of Division Algebras

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann, and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.     

Admissible metrics for line bundles on curves and abelian varieties over non-archimedean local fields

Following the approach in the archimedean case, we introduce the notion of admissible metrics for line bundles on curves and abelian varieties over non-archimedean local fields. Several properties of admissible metrics are considered and we show that this approach yields the same notion of admissible metrics over curves as doing harmonic analysis on the reduction graph of the curve. Received: 9 September 2002     

Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles II: Higgs sheaves and admissible structures

We study the basic properties of Higgs sheaves over compact Kähler manifolds and establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we prove some properties of semistable Higgs sheaves.     

Some Finiteness Results in the Representation Theory of Isometry Groups of Regular Trees

Let G denote the isometry group of a regular tree of degree ≥3. The notion of congruence subgroup is introduced and finite generation of the congruence Hecke algebras is proven. Let U be congruence subgroup and (G; U) be the category of smooth representations of G generated by their U-fixed vectors. We also show that this subcategory is closed under taking subquotients. All these results are analogues of well-known results in the case of p-adic groups. It is also shown that the category of admissible representation of G is Noetherian in the sense that every subrepresentation of a finitely generated admissible representation is again finitely generated. Since we want to emphesize the similarities between these groups and p-adic groups, we give the same proofs which also work in the p-adic case whenever possible.     

Complete nearness σ-frames

We show that complete strong nearness σ-frames are exactly the cozero parts of complete separable strong Lindelöf nearness frames. We also relate nearness σ-frames and metric σ-frames and show that every metric σ-frame admits an admissible nearness such that it is complete as a metric σ-frame if and only if it is complete in this admissible nearness.     

Restriction to compact subgroups in the cyclic homology of reductive p-adic groups

Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.     

Binary games in constitutional form and collective choice

In this paper, we define the notion of binary game in constitutional form. For this game, we define a core and give a necessary and sufficient condition for a game to be stable.We define a representation of a collective choice rule by a binary game in constitutional form and characterize those collective choice rules which are representable.We finally introduce the notion of c-social decision function and characterize, as an application of our theorem on stability of binary constitutional games, the collective choice rules which are c-social decision functions.Our representation of a collective choice rule by a binary game in constitutional form is an obvious improvement of the classical representation by a simple game.     

The influence of domain interpretations on computational models

Computational models are usually defined over specific domains. For example, Turing machines are defined over strings, and the recursive functions over the natural numbers. Nevertheless, one often uses one computational model to compute functions over another domain, in which case, one is obliged to employ a representation, mapping elements of one domain into the other. For instance, Turing machines (or modern computers) are understood as computing numerical functions, by interpreting strings as numbers, via a binary or decimal representation, say.We ask: Is the choice of the domain interpretation important? Clearly, complexity is influenced, but does the representation also affect computability? Can it be that the same model computes strictly more functions via one representation than another? We show that the answer is “yes”, and further analyze the influence of domain interpretation on the extensionality of computational models (that is, on the set of functions computed by the model).We introduce the notion of interpretation-completeness for computational models that are basically unaffected by the choice of domain interpretation, and prove that Turing machines and the recursive functions are interpretation-complete, while two-counter machines are incomplete. We continue by examining issues based on model extensionality that are influenced by the domain interpretation. We suggest a notion for comparing computational power of models operating over arbitrary domains, as well as an interpretation of the Church-Turing Thesis over arbitrary domains.     

Metric techniques for convex stationary ergodic Hamiltonians

We adapt the metric approach to the study of stationary ergodic Hamilton?CJacobi equations, for which a notion of admissible random (sub)solution is defined. For any level of the Hamiltonian greater than or equal to a distinguished critical value, we define an intrinsic random semidistance and prove that an asymptotic norm does exist. Taking as source region a suitable class of closed random sets, we show that the Lax formula provides admissible subsolutions. This enables us to relate the degeneracies of the critical stable norm to the existence/nonexistence of exact or approximate critical admissible solutions.     

Derivatives for smooth representations of <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℝ) and <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℂ)

The notion of derivatives for smooth representations of GL(n, ? _p ) was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the “adduced” representation. In this paper we define derivatives of all orders for smooth admissible Fréchet representations of moderate growth. The real case is more problematic than the p-adic case; for example, arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation.In the companion paper [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations.We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS13a].     

Nilpotence and generation in the stable module category

Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG-module M, the tensor M-generation number and the tensor M-ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M.     

Admissibility for nonuniform exponential contractions

We study the relation between the notions of nonuniform exponential stability and admissibility. In particular, using appropriate adapted norms (which can be seen as Lyapunov norms), we show that if any of their associated Lp spaces, with p∈(1,∞], is admissible for a given evolution process, then this process is a nonuniform exponential contraction. We also provide a collection of admissible Banach spaces for any given nonuniform exponential contraction.     

Monotone thematic factorizations of matrix functions

We continue the study of the so-called thematic factorizations of admissible very badly approximable matrix functions. These factorizations were introduced by V.V. Peller and N.J. Young for studying superoptimal approximation by bounded analytic matrix functions. Even though thematic indices associated with a thematic factorization of an admissible very badly approximable matrix function are not uniquely determined by the function itself, R.B. Alexeev and V.V. Peller showed that the thematic indices of any monotone non-increasing thematic factorization of an admissible very badly approximable matrix function are uniquely determined. In this paper, we prove the existence of monotone non-decreasing thematic factorizations for admissible very badly approximable matrix functions. It is also shown that the thematic indices appearing in a monotone non-decreasing thematic factorization are not uniquely determined by the matrix function itself. Furthermore, we show that the monotone non-increasing thematic factorization gives rise to a great number of other thematic factorizations.     

On the cohomology of Banach A∞-module over admissible Banach A∞-algebra

In this paper we are concerned with Banach A_∞-module M over admissible Banach A_∞-algebra A. We give some properties of admissible modules and algebras. We study the cohomology of the complex C_∞(A, M). We show that the vanishing of cohomology of this complex in certain dimensions implies to the existence of the A_∞-module structure.     

A characterization of linear admissible transformations for the m-travelling salesmen problem: A result of Berenguer

Various methods for solving the m-travelling salesmen problem employ transformations of the distance matrix. Such a transformation is called admissible if it preserves or reverses the total ordering of the set of feasible solutions according to criterion value. We show that the only linear admissible transformations are those obtained by adding constants to the rows and columns of a scalar multiple of the distance matrix.