Arbitrary Extensions Research Articles

Spectrally arbitrary pattern extensions

A matrix pattern is often either a sign pattern with entries in {0,+,−} or, more simply, a nonzero pattern with entries in {0,⁎}. A matrix pattern A is spectrally arbitrary if for any choice of a real matrix spectrum, there is a real matrix having the pattern A and the chosen spectrum. We describe a graphical technique, a triangle extension, for constructing spectrally arbitrary patterns out of some known lower order spectrally arbitrary patterns. These methods provide a new way of viewing some known spectrally arbitrary patterns, as well as providing many new families of spectrally arbitrary patterns. We also demonstrate how the technique can be applied to certain inertially arbitrary patterns to obtain larger inertially arbitrary patterns. We then provide an additional extension method for zero–nonzero patterns, patterns with entries in {0,⁎,⊛}.

Observable effects of general new scalar particles

We classify all possible new scalar particles that can have renormalizable linear couplings to Standard Model fields and therefore be singly produced at colliders. We show that this classification exhausts the list of heavy scalar particles that contribute at the tree level to the Standard Model effective Lagrangian to dimension six. We compute this effective Lagrangian for a general scenario with an arbitrary number of new scalar particles and obtain flavor-preserving constraints on their couplings and masses. This completes the tree-level matching of the coefficients of dimension five and six operators in the effective Lagrangian to arbitrary extensions of the Standard Model.

THE WEIGHT PART OF SERRE’S CONJECTURE FOR

AbstractLet $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.

On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups

It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus $g>1$ is bounded by the linear polynomial $12(g-1)$, and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalen

A framework for post-silicon realization of arbitrary instruction extensions on reconfigurable data-paths

In this paper we present a framework for realizing arbitrary instruction set extensions (IE) that are identified post-silicon. The proposed framework has two components viz., an IE synthesis methodology and the architecture of a reconfigurable data-path for realization of the such IEs. The IE synthesis methodology ensures maximal utilization of resources on the reconfigurable data-path. In this context we present the techniques used to realize IEs for applications that demand high throughput or those that must process data streams. The reconfigurable hardware called HyperCell comprises a reconfigurable execution fabric. The fabric is a collection of interconnected compute units. A typical use case of HyperCell is where it acts as a co-processor with a host and accelerates execution of IEs that are defined post-silicon. We demonstrate the effectiveness of our approach by evaluating the performance of some well-known integer kernels that are realized as IEs on HyperCell. Our methodology for realizing IEs through HyperCells permits overlapping of potentially all memory transactions with computations. We show significant improvement in performance for streaming applications over general purpose processor based solutions, by fully pipelining the data-path. (C) 2014 Elsevier B.V. All rights reserved.

Twisted homological stability for extensions and automorphism groups of free nilpotent groups

AbstractWe prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups. The proof presented here uses a general result that applies to arbitrary extensions of groups, and that has other applications as well.

Identifying Frobenius elements in Galois groups

We present a method to determine Frobenius elements in arbitrary Galois extensions of global fields, which may be seen as a generalisation of Euler’s criterion. It is a part of the general question how to compare splitting fields and identify conjugacy classes in Galois groups, which we will discuss as well.

Numerical Solutions of the Spectral Problem for Arbitrary Self-Adjoint Extensions of the One-Dimensional Schrödinger Equation

A numerical algorithm to solve the spectral problem for arbitrary self-adjoint extensions of one-dimensional regular Schrodinger operators is presented. It is shown that the set of all self-adjoint extensions of one-dimensional regular Schrodinger operators is in one-to-one correspondence with the group of unitary operators on the finite-dimensional Hilbert space of boundary data, and they are characterized by a generalized class of boundary conditions that include the well-known Dirichlet, Neumann, Robin, and (quasi-)periodic boundary conditions. The numerical algorithm is based on a nonlocal boundary extension of the finite element method and their convergence is proved. An appropriate basis of boundary functions must be introduced to deal with arbitrary boundary conditions and the conditioning of its computation is analyzed. Some significant numerical experiments are also discussed as well as the comparison with some standard algorithms. In particular it is shown that appropriate perturbations of stand...

On Brauer–Kuroda type relations of S-class numbers in dihedral extensions

Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer–Kuroda relations, as a unit index. Our formula is valid for arbitrary extensions with Galois group D2q and for arbitrary Galois-stable sets of primes S, containing the Archimedean ones. Our results have curious applications to determining the Galois module structure of the units modulo the roots of unity of a D2q-extension from class numbers and S-class numbers. The techniques we use are mainly representation theoretic and we consider the representation theoretic results we obtain to be of independent interest.

Robustness of Equations Under Operational Extensions

Sound behavioral equations on open terms may become unsound after conservative extensions of the underlying operational semantics. Providing criteria under which such equations are preserved is extremely useful; in particular, it can avoid the need to repeat proofs when extending the specified language. This paper investigates preservation of sound equations for several notions of bisimilarity on open terms: closed-instance (ci-)bisimilarity and formal-hypothesis (fh-)bisimilarity, both due to Robert de Simone, and hypothesis-preserving (hp-)bisimilarity, due to Arend Rensink. For both fh-bisimilarity and hp-bisimilarity, we prove that arbitrary sound equations on open terms are preserved by all disjoint extensions which do not add labels. We also define slight variations of fh- and hp-bisimilarity such that all sound equations are preserved by arbitrary disjoint extensions. Finally, we give two sets of syntactic criteria (on equations, resp. operational extensions) and prove each of them to be sufficient for preserving ci-bisimilarity.

Functoriality of the Canonical Fractional Galois Ideal

AbstractThe fractional Galois ideal is a conjectural improvement on the higher Stickelberger ideals defined at negative integers, and is expected to provide non-trivial annihilators for higherK-groups of rings of integers of number fields. In this article, we extend the definition of the fractional Galois ideal to arbitrary (possibly infinite and non-abelian) Galois extensions of number fields under the assumption of Stark's conjectures and prove naturality properties under canonical changes of extension. We discuss applications of this to the construction of ideals in non-commutative Iwasawa algebras.

Consistency of perturbativity and gauge coupling unification

We investigate constraints that the requirements of perturbativity and gauge coupling unification impose on extensions of the standard model and of the minimal supersymmetric standard model. In particular, we discuss the renormalization group running in several supersymmetric left-right symmetric and Pati-Salam models and show how the various scales appearing in these models have to be chosen in order to achieve unification. We find that unification in the considered models occurs typically at scales below ${M}_{\mathrm{B\ensuremath{\llap{\not\;}}}}^{\mathrm{min}}={10}^{16}\text{ }\text{ }\mathrm{GeV}$, implying potential conflicts with the nonobservation of proton decay. We emphasize that extending the particle content of a model in order to push the grand unified theory scale higher or to achieve unification in the first place will very often lead to nonperturbative evolution. We generalize this observation to arbitrary extensions of the standard model and of the minimal supersymmetric standard model and show that the requirement of perturbativity up to ${M}_{\mathrm{B\ensuremath{\llap{\not\;}}}}^{\mathrm{min}}$, if considered a valid guideline for model building, severely limits the particle content of any such model, especially in the supersymmetric case. However, we also discuss several mechanisms to circumvent perturbativity and proton decay issues, for example, in certain classes of extra dimensional models.

Regionale Energieversorgung

Let k be a finite field, a global field or a local non-archimedean field. Let H_1 and H_2 be two split, connected, semisimple algebraic groups defined over k. We prove that if H_1 and H_2 share the same set of maximal k-tori up to k-isomorphism, then the Weyl groups W(H_1) and W(H_2) are isomorphic, and hence the algebraic groups modulo their centers are isomorphic except for a switch of a certain number of factors of type B_n and C_n. We remark that due to a recent result of Philippe Gille, above result holds for fields which admit arbitrary cyclic extensions.

Combinatorial geometries of field extensions

We classify the algebraic combinatorial geometries of arbitrary field extensions of transcendence degree greater than 4 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and Hrushovski in the case of algebraically closed fields. The classification of projective planes in algebraic combinatorial geometries in arbitrary fields of characteristic zero will also be given.

Stably just infinite rings

We study just infinite algebras which remain so upon extension of scalars by arbitrary field extensions. Such rings are called stably just infinite. We show that just infinite rings over algebraically closed fields are stably just infinite provided that the ring is either right noetherian (4.2) or countably generated over a large field (6.4). We give examples to show that, over countable fields, a just infinite algebra which is either affine or non-noetherian need not remain just infinite under extension of scalars. We also give a concrete classification of PI stably just infinite rings (5.5) and give two characterizations of non-PI stably just infinite rings in terms of Martindale's extended center (3.4), (3.5).

Capitulation, ambiguous classes and the cohomology of the units

This paper presents results on both the kernel and cokernel of the S-capitulation map for arbitrary finite Galois extensions K/F of global fields (with Galois group G) and arbitrary finite sets of primes S of F (assumed to contain the archimedean primes in the number field case).

Tate classes and poles of L-functions of twisted quaternionic Shimura surfaces

In this article we generalize a result obtained by Harder, Langlands and Rapoport in the case of Hilbert modular surfaces and we prove in particular the equality between the dimension of the space of Tate classes of twisted quaternionic Shimura surfaces defined over arbitrary solvable extensions of totally real fields and the order of the pole at s = 2 of the zeta functions of these surfaces.

Unitary Groups Acting on Grassmannians Associated with a Quadratic Extension of Fields

Let (V, H) be an anisotropic Hermitian space of finite dimension over the algebraic closure of a real closed field K. We determine the orbits of the group of isometries of (V, H) in the set of K-subspaces of V . Throughout the paper K denotes a real closed field and K its algebraic closure. Then it is well known (see, for example, [4, Chapter 2], [23]; see also [8]) that K = K(i) with i = √−1. Also we let (V,H) be an anisotropic Hermitian space (with respect to the involution underlying the quadratic field extension K/K) of finite dimension n over K. In this context we consider the natural action of the unitary group U = U(V,H) of isometries of (V,H) on the set Xd of all ddimensional K-subspaces of V . The analogous problem where (V,H) is a symplectic space was treated in [1] (for arbitrary quadratic field extensions). It turns out that, in contrast with the symplectic case, there are infinitely many orbits for the action of the unitary group U on Xd. In group theoretic language the stated problem turns into the determination of the double coset spaces of the form (1) GW \G/U, where G = GL (VK) and GW denotes the parabolic subgroup of G stabilizing a member W ∈ Xd (we write VK to indicate that we are regarding V as a vector space over K). The precise structure of double coset spaces involving classical groups is of great interest in applying the classical Rankin-Selberg method for explicit construction of automorphic L-functions, as Garrett [2] and Piatetski-Shapiro and Rallis [6] worked out. 2000 AMS Mathematics Subject Classification. Primary 51N30, 15A21, Secondary 11E39. Received by the editors on October 13, 2003. Copyright c ©2006 Rocky Mountain Mathematics Consortium

Zeta regularized determinants for conic manifolds

We study (relative) zeta regularized determinants of Laplace type operators on compact conic manifolds. We establish gluing formulae for relative zeta regularized determinants. For arbitrary self-adjoint extensions of the Laplace–Beltrami operator, we express the relative ζ-determinants for these as a ratio of the determinants of certain finite matrices. For the self-adjoint extensions corresponding to Dirichlet and Neumann conditions, the formula is particularly simple and elegant.

Maximal tori determining the algebraic groups

Let k be a finite field, a global field, or a local non-archimedean field, and let H 1 and H 2 be split, connected, semisimple algebraic groups over k. We prove that if H 1 and H 2 share the same set of maximal k-tori, up to k-isomorphism, then the Weyl groups W(H 1 ) and W(H 2 ) are isomorphic, and hence the algebraic groups modulo their centers are isomorphic except for a switch of a certain number of factors of type B n and C n . (Due to a recent result of Philippe Gille, this result also holds for fields which admit arbitrary cyclic extensions.).