Ordinary Representations Research Articles

Signal Analysis by Analogy with Text Analysis

In the paper a structural approach to information representation distinguishing from ordinary digital signal representation is discussed. In accordance with structural approach the information representation is computed as the robust signal component composed of elementary carriers of information units. At that it defines the hierarchy of signal partitions into nested signals. The background approaches are pointed out. To explain the essence of structural approach the comparison of signal analysis with text analysis independently of language specificity is used. A method of algorithmic simulation of abilities of some artificial or natural recognition system with account for signal peculiarities is proposed.

UNBOUNDED TAME QUASIREPRESENTATIONS WITH COMMUTATIVE DISCREPANCIES FOR AMENABLE GROUPS

Continuing the study of representations of amenable groups, we discuss one of the simplest cases in which a (not necessarily bounded) Banach space quasirepresentation of an amenable group is close to an ordinary representation of the group in the same space.

Quasirepresentations of amenable groups: Unbounded tame quasirepresentations

Continuing the study of representations of amenable groups, we discuss a model case in which a (not necessarily bounded) Banach space quasirepresentation of an amenable group is close to an ordinary representation of the group in the same space.

Bound states of the spin-orbit coupled ultracold atom in a one-dimensional short-range potential

We solve the bound state problem for the Hamiltonian with the spin-orbit and the Raman coupling included. The Hamiltonian is perturbed by a one-dimensional short-range potential V which describes the impurity scattering. In addition to the bound states obtained by considering weak solutions through the Fourier transform or by solving the eigenvalue equation on a suitable domain directly, it is shown that ordinary point-interaction representations of V lead to spin-orbit induced extra states.

Dihedral Iwasawa theory of nearly ordinary quaternionic automorphic forms

AbstractLetπ(f) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebraBover a totally real fieldFwith associated Galois representationρf. LetKbe a totally complex quadratic extension ofFembedding inB. Using families of CM points on towers of Shimura curves attached toBandK, we construct an Euler system forρf. We prove that it extends top-adic families of Galois representations coming from Hida theory and dihedral ℤdp-extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory.

Conformal blocks of chiral fields in $$ \mathcal{N}=2 $$ SUSY CFT and affine Laumon spaces

We consider the problem of computing N=2 superconformal block functions. We argue that the Kazama-Suzuki coset realization of N=2 superconformal algebra in terms of the affine sl(2) algebra provides relations between N=2 and affine sl(2) conformal blocks. We show that for N=2 chiral fields the corresponding sl(2) construction of the conformal blocks is based on the ordinary highest weight representation. We use an AGT-type correspondence to relate the four-point sl(2) conformal block with Nekrasov's instanton partition functions of a four-dimensional N=2 SU(2) gauge theory in the presence of a surface operator. Since the previous relation proposed by Alday and Tachikawa requires some special modification of the conformal block function, we revisit this problem and find direct correspondence for the four-point conformal block. We thus find an explicit representation for the affine sl(2) four-point conformal block and hence obtain an explicit combinatorial representation for the N=2 chiral four-point conformal block.

Kernels of linear representations of Lie groups, locally compact groups, and pro-Lie Groups

Abstract. For a topological group the intersection of all kernels of ordinary representations is studied. We show that is contained in the center of if is a connected pro-Lie group. The class is determined explicitly if is the class of connected Lie groups or the class of almost-connected Lie groups: in both cases, it consists of all compactly-generated abelian Lie groups. Every compact abelian group and every connected abelian pro-Lie group occurs as for some connected pro-Lie group . However, the dimension of is bounded by the cardinality of the continuum if is locally compact and connected. Examples are given to show that becomes complicated if contains groups with infinitely many connected components.

Companion forms for unitary and symplectic groups

We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp4 over totally real fields.

Spin representations of Weyl groups and the Springer correspondence

We give a common framework for the classification of projective spin irreducible representations of a Weyl group, modeled after the Springer correspondence for ordinary representations.

Real representation spheres and the real monomial Burnside ring

We introduce a restriction morphism, called the Boltje morphism, from a given ordinary representation functor to a given monomial Burnside functor. In the case of a sufficiently large fibre group, this is Robert Boltjeʼs splitting of the linearization morphism. By considering a monomial Lefschetz invariant associated with real representation spheres, we show that, in the case of the real representation ring and the fibre group { ± 1 } , the image of a modulo 2 reduction of the Boltje morphism is contained in a group of units associated with the idempotents of the 2-local Burnside ring. We deduce a relation on the dimensions of the subgroup-fixed subspaces of a real representation.

An analogy between representations of knot groups and Galois groups

We discuss an analogy between the deformations of hyperbolic structures on a knot complement and of certain nearly ordinary Galois representations.

Infinite-Dimensional Representations of 2-Groups

A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called "measurable categories" (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie 2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and sub-intertwiners - features not seen in ordinary group representation theory. We study irreducible and indecomposable representations and intertwiners. We also study "irretractable" representations - another feature not seen in ordinary group representation theory. Finally, we argue that measurable categories equipped with some extra structure deserve to be considered "separable 2-Hilbert spaces", and compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras.

An Approach to Robust Control of the Hopf Bifurcation

The paper illustrates a novel approach to modify the Hopf bifurcation nature via a nonlinear state feedback control, which leaves the equilibrium properties unchanged. This result is achieved by recurring to linear and nonlinear transformations, which lead the system to locally assume the ordinary differential equation representation. Third‐order models are considered, since they can be seen as proper representatives of a larger class of systems. The explicit relationship between the control input and the Hopf bifurcation nature is obtained via a frequency approach, that does not need the computation of the center manifold.

Tensor products of representations up to homotopy

We study the construction of tensor products of representations up to homotopy, which are the $A_\infty$ version of ordinary representations. We provide formulas for the construction of tensor products of representations up to homotopy and of morphisms between them, and show that these formulas give the homotopy category a monoidal structure which is uniquely defined up to equivalence.

Decomposition of reductive regular Prehomogeneous Vector Spaces

Let (G,V) be a regular prehomogeneous vector space (abbreviated to PV), where G is a connected reductive algebraic group over C. If $V= \oplus_{i=0}^{n}V_{i}$ is a decomposition of V into irreducible representations, then, in general, the PV's $(G,V_{i})$ are no longer regular. In this paper we introduce the notion of quasi-irreducible PV (abbreviated to Q-irreducible), and show first that for completely Q-reducible PV's, the Q-isotopic components are intrinsically defined, as in ordinary representation theory. We also show that, in an appropriate sense, any regular PV is a direct sum of quasi-irreducible PV's. Finally we classify the quasi-irreducible PV's of parabolic type.

Le traitement de l’anglais dans certaines entreprises françaises : un globish ou des globishes ?

Despite the current European recommendations in favour of developing linguistic plurality, English in France now occupies a well-established hegemonic position in the world of work. For the last ten years or so, the anglicization of some companies has progressed more and more rapidly and seems, in certain cases, to originate from the fact that these companies want to be noticed, with this issue forming part of their managerial ideology. The insertion of "all-English" at work often takes place at the same time as structural changes motivated by involvement in the globalisation process. It is necessary to analyze these phenomena from a sociolinguistic point of view in connection with the French system representation. This is how, according to ordinary representations, a simplified form of English has come to exist, globish, considered as the most appropriate form of English inside the company. However, there is probably not only one globish but several globishes linked to context and to the speakers using them.

Measuring non-steady-state metabolic fluxes in starch-converting faecal microbiota in vitro

This paper explores human gut bacterial metabolism of starch using a combined analytical and computational modelling approach for metabolite and flux analysis. Non-steady-state isotopic labelling experiments were performed with human faecal microbiota in a well-established in vitro model of the human colon. After culture stabilisation, [U-13C] starch was added and samples were taken at regular intervals. Metabolite concentrations and 13C isotopomeric distributions were measured amongst other things for acetate, propionate and butyrate by mass spectrometry and NMR. The vast majority of metabolic flux analysis methods based on isotopomer analysis published to date are not applicable to metabolic non-steady-state experiments. We therefore developed a new ordinary differential equation-based representation of a metabolic model of human faecal microbiota to determine eleven metabolic parameters that characterised the metabolic flux distribution in the isotope labelling experiment. The feasibility of the model parameter quantification was demonstrated on noisy in silico data using a downhill simplex optimisation, matching simulated labelling patterns of isotopically labelled metabolites with measured metabolite and isotope labelling data. Using the experimental data, we determined an increasing net label influx from starch during the experiment from 94±1 µmol/l/min to 133±3 µmol/l/min. Only about 12% of the total carbon flux from starch reached propionate. Propionate production mainly proceeded via succinate with a small contribution via acrylate. The remaining flux from starch yielded acetate (35%) and butyrate (53%). Interpretation of 13C NMR multiplet signals further revealed that butyrate, valerate and caproate were mainly synthesised via cross-feeding, using acetate as a co-substrate. This study demonstrates for the first time that the experimental design and the analysis of the results by computational modelling allows the determination of time-resolved effects of nutrition on the flux distribution within human faecal microbiota in metabolic non-steady-state.

ELECTROMAGNETISM AND MULTIPLE-VALUED LOOP-DEPENDENT WAVE FUNCTIONALS

We quantize the Maxwell theory in the presence of an electric charge in a "dual" Loop Representation, i.e. a geometric representation of magnetic Faraday's lines. It is found that the theory can be seen as a theory without sources, except for the fact that the wave functional becomes multivalued. This can be seen as the dual counterpart of what occurs in Maxwell theory with a magnetic pole, when it is quantized in the ordinary Loop Representation. The multivaluedness can be seen as a result of the multiply-connectedness of the configuration space of the quantum theory.

On the irreducible representations of the alternating group which remain irreducible in characteristic 𝑝

We consider the problem of which ordinary irreducible representations of the alternating group A n \mathfrak {A}_n remain irreducible modulo a prime p p . We solve this problem for p = 2 p=2 , and present a conjecture for odd p p , which we prove in one direction.

A Decomposition Theorem for Group Representations

In this paper we establish a decomposition theorem for an ordinary representation of a finite group G in any category \({\mathcal C}\) which expresses a suitable irreducible representation of G as the tensor product of two projective ones. The celebrated theorem due to Clifford for a linear representation turns out to be a particular case of it. For that purpose, a definition of projective extension of an ordinary representation of a normal subgroup of G is introduced, as well as a tensor product between two of them.