General Representation Theory Research Articles

Influence of the geometry, relative position and length of sewer plate heat exchangers on their effectiveness

The effect of various dimensions of the shape, relative position and length of channels of plate heat exchangers on the effectiveness of the processes of hydrodynamics and heat transfer are analyzed. Based on the general theory of representations of the boundary layer, it is shown that the length of the entrance section of the hydrodynamic stabilization of flow while creating new and more efficient designs of plate heat exchangers should be accounted for. On the basis of the proposed hypothesis about the qualitative hydrodynamic analogy between the flow streams in plate and mixing heat exchangers, the effect of the relative position of the neighboring channels of plates on the hydrodynamics and the efficiency of heat transfer in a plate heat exchanger is shown. Keywords: plate heat exchangers, heat exchangers, mixing, geometry of channels, the input section of hydrodynamic stabilization of flow, the efficiency of heat transfer, aerodynamics, surface area, angle of attack

First order representations of Fliess models

Fuhrmann’s state-space construction (in its generalized form) is used to obtain a general theory of first order representations of Fliess models defined over an arbitrary noetherian commutative ring. The case of arbitrary linear delay differential equations is involved.

Representation theory of MV-algebras

In this paper we develop a general representation theory for MV-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of MV-algebras and MV-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. We prove that any MV-algebra is isomorphic to the MV-algebra of all global sections of a sheaf of MV-chains on a compact topological space. This result is intimately related to McNaughton’s theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton’s theorem. In spite of the language used in this abstract, we have written this paper in the hope that it can be read by experts in MV-algebras but not in sheaf theory, and conversely.

Generic representation theory of quivers with relations

The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra Λ are explored, with regard to both their geometry and the structure of the modules they encode. Provided that the base field is algebraically closed and has infinite transcendence degree over its prime field, we establish the existence and uniqueness (not up to isomorphism, but in a strong sense to be specified) of a generic module for any irreducible component C , that is, of a module which displays all categorically defined generic properties of the modules parametrized by C ; the crucial point of the existence statement – a priori almost obvious – lies in the description of such a module in a format accessible to representation-theoretic techniques. Our approach to generic modules over path algebras modulo relations, by way of minimal projective resolutions, is largely constructive. It is explicit for large classes of algebras of wild type. We follow with an investigation of the properties of such generic modules in terms of quiver and relations. The sharpest specific results on all fronts are obtained for truncated path algebras, that is, for path algebras of quivers modulo ideals generated by all paths of a fixed length; this class of algebras extends the comparatively thoroughly studied hereditary case, displaying many novel features.

Jogos Didáticos: Um Estudo de Representações Sociais

The present article, drawing on the general theory of social representations, is based on a research developed with a group of undergraduates of a History course at a private University in Sao Paulo. It was the research’s basic intention to identify the constitutive elements of the didactic game’s semantic universe in the interface between cognitive and affective aspects of social representations with the daily needs by questioning the polarization between cognition and emotion. In order to approach such semantic universe, instruments based on methods of compilation, organization, and interrogation were used. Results showed the existence of a complex and unstable organization, operating by means of a contradictory logic (which, although, was not an excluding one), composing the semantic universe of the didactic game.

C. S. Peirce’s Definition of Symbol in §14 of the New List

The main body of this paper consists of a commentary on §14 of Peirce’s 1867 paper “On a New List of Categories,” in which Peirce derives the concept of Symbol within his general theory of representations. Rather than aiming at a comprehensive study of the New List, the paper closely observes how Peirce arrives at his definition of Symbol in the New List by elucidating its main ideas. The paper suggests that Symbol occupies a unique conceptual locus in Peirce’s theory of categories and that his definition of Symbol in the New List is consistent with Peirce’s later formulations of Symbol. The paper includes five brief case studies of Peirce’s formulations of Symbol over the years 1866 - c.1911.

A note on Weil index

Let F be a non-archimedean local field of characteristic 0 and ψ a nontrivial additive character. Weil first defined the Weil index γ(a, ψ) (a ∊ F*) in his famous paper, from which we know that γ(a, ψ)γ(b, ψ) = γ(ab, ψ)γ(1, ψ)(a, b) and γ(a, ψ)4 = (−1, −1), where (a, b) is the Hilbert symbol for F. The Weil index plays an important role in the theory of theta series and in the general representation theory. In this paper, we establish an identity relating the Weil index γ(a, ψ) and the Gauss sum.

Risk Measures and Robust Optimization Problems

These lecture notes give a survey on recent developments in the theory of risk measures. The first part outlines the general representation theory of risk measures in a static one-period setting. In particular, it provides structure theorems for law-invariant risk measures. Examples include Value at Risk, Average Value at Risk, distortion risk measures, and risk measures arising from robust preferences. The second part analyzes risk measures and associated robust optimization problems in the framework of dynamic financial market models. The concept of efficient hedging, as introduced by Föllmer and Leukert[32], is discussed in terms of the more general framework of convex risk measures. The last two sections are devoted to the construction of optimal investment strategies under Knightian uncertainty.

The LQG string—loop quantum gravity quantization of string theory: I. Flat target space

We combine (i) background-independent loop quantum gravity (LQG) quantization techniques, (ii) the mathematically rigorous framework of algebraic quantum field theory (AQFT) and (iii) the theory of integrable systems resulting in the invariant Pohlmeyer charges in order to set up the general representation theory (superselection theory) for the closed bosonic quantum string on flat target space. While we do not solve the, expectedly, rich representation theory completely, we present a, to the best of our knowledge, new, non-trivial solution to the representation problem. This solution exists (1) for any target space dimension, (2) for Minkowski signature of the target space, (3) without tachyons, (4) manifestly ghost free (no negative norm states), (5) without fixing a worldsheet or target space gauge, (6) without (Virasoro) anomalies (zero central charge), (7) while preserving manifest target space Poincaré invariance and (8) without picking up UV divergences. The existence of this stable solution is, on one hand, exciting because it raises the hope that among all the solutions to the representation problem (including fermionic degrees of freedom) we find stable, phenomenologically acceptable ones in lower dimensional target spaces, possibly without supersymmetry, that are much simpler than the solutions that arise via compactification of the standard Fock representation of the string. On the other hand, if such solutions are found, then this would prove that neither a critical dimension (D = 10, 11, 26) nor supersymmetry is a prediction of string theory. Rather, these would be features of the particular Fock representation of current string theory and hence would not be generic. The solution presented in this paper exploits the flatness of the target space in several important ways. In a companion paper, we treat the more complicated case of curved target spaces.

Analytic representation of the square-root operator

In this paper, we use the theory of fractional powers of linear operators to construct a general (analytic) representation theory for the square-root energy operator of relativistic quantum theory, which is valid for all values of the spin. We focus on the spin 1/2 case, considering a few simple yet solvable and physically interesting cases, in order to understand how to interpret the operator. Our general representation is uniquely determined by the Green's function for the corresponding Schrodinger equation. We find that, in general, the operator has a representation as a nonlocal composite of (at least) three singularities. In the standard interpretation, the particle component has two negative parts and one (hard core) positive part, while the antiparticle component has two positive parts and one (hard core) negative part. This effect is confined within a Compton wavelength such that, at the point of singularity, they cancel each other providing a finite result. Furthermore, the operator looks like the identity outside a few Compton wavelengths (cutoff). To our knowledge, this is the first example of a physically relevant operator with these properties.

The bubble algebra: structure of a two-colour Temperley–Lieb Algebra

We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley–Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang–Baxter equations.

A Stratification of Generic Representation Theory and Generalized Schur Algebras

If Fq is the nite eld of characteristic p and order q = p s , let F(Fq) be the category whose objects are functors from nite dimensional Fq{vector spaces to Fq{vector spaces, and with morphisms the natural trans- formations between such functors. We dene an innite lattice of thick subcategories of F(Fq). Our main re- sult then identies various subquotients as categories of modules over products of symmetric groups, via recollement diagrams. Our lattice of thick subcategories is a renemen t of the Eilenberg{MacLane polynomial degree ltration F 0 (Fq) F 1 (Fq) F 2 (Fq) : : : of F(Fq) which has been extensively studied and used in the algebraic K{theory lit- erature. Our main theorem implies a description of F d (Fq)=F d 1 (Fq) that renes and extends earlier results of Pirashvili and others. If q r, one of the subcategories is the Friedlander{Suslin category P r of 'strict polynomial functors of degree r', equivalent to the category of modules over the Schur algebra S(n; r) with n r. Our results can thus also be viewed as rening and extending the classic relationship between S(n; r){modules and r{modules via the Schur functor. In fact, (essentially) all our subcategories are equivalent to categories of modules over various nite dimensional algebras, and our lattice can be interpreted in terms of lattices of idempotent two sided ideals in these generalized Schur algebras. Applications include a simple proof, free of algebraic group theory, of a generalized Steinberg Tensor Product Theorem. This then implies the classic theorem for GLn(Fq), shedding some new light on this classic result. Our tensor product theorem is then used to study when various of our generalized Schur algebras are Morita equivalent. We use two technical tricks which may be of some independent interest. Firstly, we 'twist' by an action of the Galois group Gal(Fq; Fp) to be able to work entirely with vector spaces over the prime eld Fp, making discussions of 'base change' unnecessary. Secondly, we systematically use 'functors with product', a.k.a. lax symmetric monoidal functors, to dene our subcategories.

Locally analytic distributions and 𝑝-adic representation theory, with applications to 𝐺𝐿₂

In this paper we study continuous representations of locally L L -analytic groups G G in locally convex K K -vector spaces, where L L is a finite extension of Q p \mathbb {Q}_p and K K is a spherically complete nonarchimedean extension field of L L . The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of G G , along with interesting new objects such as the action of G G on global sections of equivariant vector bundles on p p -adic symmetric spaces. We introduce a restricted category of such representations that we call “strongly admissible” and we show that, when G G is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of G G . As an application we prove the topological irreducibility of generic members of the p p -adic principal series for G L 2 ( Q p ) GL_2(\mathbb {Q}_p) . Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous K K -valued representations of locally L L -analytic groups.

Algorithms for Fast Convolutions on Motion Groups

In this paper we apply techniques from noncommutative harmonic analysis to the development of fast algorithms for the computation of convolution integrals on motion groups. In particular, we focus on the group of rigid-body motions in 3-space, which is denoted here as SE(3). The general theory of irreducible unitary representations (IURs) of the 3D motion group is described briefly. Using IURs in operator form, we write the Fourier transform of functions on the motion group as an integral over the product space SE(3)×S2. The integral form of the Fourier transform matrix elements allows us to apply fast Fourier transform (FFT) methods developed previously for R3, S2, and SO(3) to speed up considerably the computation of convolutions of functions on SE(3). Such convolutions have been shown to play an important role in a number of engineering disciplines. An algorithm for the fast computation of the Fourier transform is given and its complexity is discussed. The Fourier transform for the 3D “discrete motion group” (semi-direct product of the icosahedral group with the translation group) is also developed and the computational complexity is discussed.

A diagrammatic approach to Hecke algebras of the reflection equation

Hecke algebra deformations of the wreath product Cd Sn arise in solutions to the Yang-Baxter equations with boundary (the reflection equation). We use a simple diagrammatic approach to construct bases for these algebras. We hence introduce quotient algebras suitable for constructing physical solutions to the reflection equation, with a well-defined thermodynamic (large n ) limit. For all d and n we determine the generic representation theory of these algebras, developing a formalism suitable for the analysis of corresponding q -spin-chain Hamiltonian and transfer matrix spectra.

Dual representations of hulls for functions satisfying f (0) = inf f (X\\{0})*

Let Xbe a setW set of extended-real valued functions on Xand 0 an element of Xsuch that w(0) = 0 for all win W. We give a general theory of dual representations, without any extra parameters, of various hulls for extended-real valued functions on X satisfying f(0)= inf f(X\\{0})We use the unifying framework of quasi-convex functions with respect to families of subsets of Xand a condition generalizing the bipolar theorem. Our results contain, as particular cases, some recent results of Thach (1991, 1993) and Rubinov and Glover (1996) on W-pseudo-affine and W-quasi-coaffine hulls.We also give some results in the converse direction. These yield, in particular, that the bipolar theorem is equivalent to a certain property of the lower semi-continuous quasi-convex hulls of functions on locally convex spaces

The Polarity Parameter for Ellipsis Coherence

In this paper I argue that Catalan polarity particles (PPs) si/no, tambe/tampoc, which one finds in some special realizations of ellipsis in Catalan, impose constraints on Discourse Structure. By virtue of their explicit or implicit markedness with respect to the [± neg] feature, I distinguish them as strong/weak PPs. I demonstrate that the polarity carried by these PPs provides a test-bed for discourse coherence, supervising the processing of the preceding discourse in relation to the last state. It is claimed that such PPs inherit in discourse the locality conditions that are present in the sentence-level in terms of comparison discourse relations like PARALLELISM or similarity (signalled by si/tambe) and contrast (signalled by no/tampoc). I propose a formalism within a general discourse representation theory known as Segmented DRT (Asher, 1993).

Partition Algebras are Cellular

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.

Rational cohomology and cohomological stability in generic representation theory

With F q a finite field of characteristic p , let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]( q ) be the category whose objects are functors from finite dimensional F q -vector spaces to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-vector spaces. Friedlander and Suslin have introduced a category [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] of "strict polynomial functors" which has the same relationship to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]( q ) that the category of rational GL m -modules has to the category of GL m ( F q )-modules. Our main theorem says that, for all finite objects F, G ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /], and all s , the natural restriction map from [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] ( F ( k ) , G ( k ) ) to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] ( F, G ) is an isomorphism for all large enough k and q . Here F ( k ) denotes F twisted by the Frobenius k times. This combines with an analogous theorem of Cline, Parshall, Scott, and van der Kallen to show that, for all finite F, G ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /], and all s , evaluation on an m dimensional vector space V m induces an isomorphism from [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="09i" /] ( F, G ) to Ext s GL m ( F q ) ( F ( V m ), F ( V m )) for all large enough m and q . Thus group cohomology of the finite general linear groups has often been identified with MacLane (or Topological Hochschild) cohomology.

The structure of indecomposable injectives in generic representation theory

The structure of indecomposable injectives in generic representation theory