181 research outputs found
An invariant for locally finite dimensional semisimple algebras.
No full textComplete invariants were found for the category of unital direct limits of finite dimensional semisimple complex algebras and the category of unital direct limits of finite dimensional semisimple real algebras by G. A. Elliott ( (E)) and by K. R. Goodearl and D. E. Handelman ( (GH)) respectively. We are naturally led to consider similar complete invariants for other algebras of this type. For other fields, the situation is much more complicated, since the set of division rings containing a field F that is neither real closed nor algebraically closed is infinite (even ignoring the noncommutative ones). So let \Omega=\{D\sb{i}\} be a finite set of finite dimensional division algebras, we shall only study the categories of unital direct limits of finite direct products of matrix algebras involving just this set of division rings. The conjecture of (GH) concerning a proposed complete invariant for direct limit algebras is simplified, and we show that this invariant (essentially a diagram of ordered K\sb0-groups) is complete, establishing the conjecture
An invariant for locally finite dimensional semisimple algebras.
No full textComplete invariants were found for the category of unital direct limits of finite dimensional semisimple complex algebras and the category of unital direct limits of finite dimensional semisimple real algebras by G. A. Elliott ( (E)) and by K. R. Goodearl and D. E. Handelman ( (GH)) respectively. We are naturally led to consider similar complete invariants for other algebras of this type. For other fields, the situation is much more complicated, since the set of division rings containing a field F that is neither real closed nor algebraically closed is infinite (even ignoring the noncommutative ones). So let \Omega=\{D\sb{i}\} be a finite set of finite dimensional division algebras, we shall only study the categories of unital direct limits of finite direct products of matrix algebras involving just this set of division rings. The conjecture of (GH) concerning a proposed complete invariant for direct limit algebras is simplified, and we show that this invariant (essentially a diagram of ordered K\sb0-groups) is complete, establishing the conjecture
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Reappraising regulation: the politics of “regulatory retreat” in the United Kingdom
No full textAs the contributions to this volume ably demonstrate, the current economic crisis is a result of a systemic uncontrollability in the international economy. Yet our understanding of how we have reached the present crisis has generally been framed within a macroeconomics that has consciously and systematically distorted what markets are, how markets work, and the role of regulation therein. Of particular interest is the framing of the idea of regulation in ways that separate states from markets, indeed subsuming the former into the latter within some crude zero-sum conception of power.
In this essay, we offer a critique of the narrow view that the system is merely an inevitable consequence of “underregulation” and that what is needed to “mend” the system is just “better” or more “responsive” regulation. Further, we argue that this framing also raises questions about the utility of seeking to understand the present crisis through the lens of crime.
This chapter is written, at least in terms of its empirical reference points, from a British perspective. We do not claim that it can be read to fit any situation. However, since our focus is on the nature and idea of regulation in general—what regulation is, and can be, and how the idea is represented, distorted, limited, and reframed hegemonically—we hope that the theoretical argument developed here has a more general applicability
Eigenvectors for infinite Markov chains and dimension groups.
No full textThis thesis relates the theory of dimension groups to the study of nonnegative infinite matrices. Given a nonnegative matrix P = Pg,hg,h∈ G (Gamma countable and infinite), we obtain information concerning the nonnegative eigenvectors of P by studying the associated dimension groups and their trace (state) spaces. For a particular class of countable discrete Markov chains, we exhibit affine homeomophisms between nonnegative eigenvector spaces and certain subspaces of related trace spaces. This thesis also establishes some necessary conditions for the weak ergodicity of sequences of 2 x 2 real matrices
A Linearization Technique for Multivariate Polynomials Using Convex Polyhedra Based on Handelman-Krivine's Theorem
No full textNational audienceWe present a new linearization method to over-approximate non-linear multivariate polynomials with convex polyhedra.It is based on Handelman-Krivine's theorem and consists in using products of constraints of a polyhedron to over-approximate a polynomial on this polyhedron. We implemented it together with two other linearization methods that we will not detail in this paper, but that we shall use as comparison. Our implementation in Ocaml generates certificates that can be verified by a trusted checker, certified in Coq, that guarantees the correctness of our linear approximation
Book Review: Siva in the Forest of Pines: An Essay on Sorcery and Self-Knowledge
No full textA review of Siva in the Forest of Pines: An Essay on Sorcery and Self-Knowledge by Don Handelman and David Shulman
Finitely asymptotic properties of powers of characters of compact semisimple Lie groups.
No full textThis thesis has to do with decomposing tensor powers of irreducible representations of compact semisimple Lie groups or their Lie algebras. We will be concerned only with the set of irreducible representations appearing in the decomposition. In particular, we determine whether, for a sufficiently high tensor power, this set is "maximal", in a sense to be made more precise below. We rely on the theory of semisimple Lie algebras, and describe the results in terms of characters. A (reducible) character of a finite dimensional complex semisimple Lie algebra is saturated if for each of its dominant weights, the corresponding irreducible character is a summand in its orthogonal decomposition. A character is eventually saturated if some power is saturated. In the first part of the thesis, we describe all eventually saturated irreducible characters. In particular, it is shown that any irreducible character whose highest weight is in the interior of the Weyl chamber is eventually saturated; if the Lie algebra is simply laced, then all irreducible characters are eventually saturated; if not, then there are irreducible characters (with highest weights on the boundary of the Weyl chamber) that are not eventually saturated. We use the PRV conjecture to derive necessary and sufficient conditions for eventual saturation. These conditions are expressed in terms of cones generated by the weights of the character. The convex hull of the weight diagram, and the weights adjacent to the highest weight along the edges of the convex hull, are described in detail. We show in the second part of the thesis that if the Lie algebra is A\sb{d}, and 5, then for any integer + 1 and any irreducible character , the product \chi\sbsp{\lambda}{n} is saturated. We also establish this result for certain irreducible characters of A\sb{d}, d \ge 5. These results are proved by induction on the rank of the Lie algebra and on the highest weight of the character. The geometry of the convex hull of the set of weights comes in to play here as well, and the dominant faces of this set are described. A reduction result, relating the "restriction to a dominant face" of the decomposition of a product of characters to that of a corresponding product in an algebra of lower rank, is established. The Littlewood-Richardson rule is used to compare products in which the component irreducible characters have highest weights which differ by a small amount. Similar induction arguments are used to describe all the irreducible characters appearing in the decomposition of a product of irreducible characters of A\sb2. Some of the irreducible characters appearing in a product of characters of higher rank algebras can also be determined using this type of induction. The questions considered here arise in the study of product type actions of compact groups. The results on eventual saturation of irreducible characters are useful in computing the equivariant ordered K-theory of certain fixed point C\sp*-algebras under the corresponding group actions
Representation Theory of Compact Inverse Semigroups
No full textW. D. Munn proved that a finite dimensional representation of an inverse semigroup
is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this
thesis will be to give new analytic proof that every finite dimensional representation
of a compact inverse semigroup is equivalent to a ⋆-representation.
The second goal is to parameterize all finite dimensional irreducible representations
of a compact inverse semigroup in terms of maximal subgroups and order
theoretic properties of the idempotent set. As a consequence, we obtain a new and
simpler proof of the following theorem of Shneperman: a compact inverse semigroup
has enough finite dimensional irreducible representations to separate points if and
only if its idempotent set is totally disconnected.
Our last theorem is the following: every norm continuous irreducible ∗-representation
of a compact inverse semigroup on a Hilbert space is finite dimensional
Strong positivity results for polynomials of bounded degree.
No full textGiven a polynomial f possibly having negative coefficients, and a polynomial P having only positive coefficients, consider the problem of determining whether, {\rm for\ some}\ n,\ {\rm the\ product}\ P\sp{n}f\ {\rm has\ no\ negative\ coefficients}.\eqno(*) Generalizations of the positivity problem can be made in several ways. De Angelis (DA1) has dropped the condition that P be positive, requiring instead that \vert P(re\sp{i\theta})\vert0 and Another generalization involves permitting P to vary. That is, to replace P\sp n by a product P\sb1P\sb2P\sb3\cdot\cdot\cdot P\sb n, of positive polynomials. This was first considered in (H3, Appendix C), which gives some results for the case of several variables. The appropriate generalization of the problem is to take a sequence \{P\sb i\} of positive polynomials and to ask whether, for all eligible polynomials f, P\sb{k+1}P\sb{k+2}\cdot\cdot\cdot P\sb{k+n}\cdot f has no negative coefficients.(**) In (BH) the problem (**) is studied extensively for the case of polynomials in a single variable x. The sequence \{P\sb i\} of polynomials is said to be strongly positive if (**) holds for all f for which f\vert\sb{(0,\infty)} is strictly positive. It is frequently convenient to permit the polynomials P\sb i and f to be Laurent polynomials; that is, polynomials admitting both positive and negative powers of the indeterminate x. Before studying strong positivity, it is important to be familiar with what is called the fluctuation of a Laurent polynomial. If P=\sum\sb ip\sb ix\sp i, then the fluctuation of P is the number \sum\sb i\vert p\sb i-p\sb{1+1}\vert\over \sum\sb ip\sb i. It measures the differences between adjacent coefficients of P as a fraction of the total mass of P. A necessary condition for a sequence \{P\sb j\} to be strongly positive is that {\cal F}(P\sb{k+1}P\sb{k+2}\cdot\cdot\cdot P\sb{k+n})\to 0,\quad {\rm as}\ n\to\infty,\ {\rm for\ every} k.\eqno(+) We end chapter 5 by giving a few circumstances in which we can guarantee strong positivity even though \sum\sb i{\cal P}(P\sb i) is finite. These are generally cases in which lockings into products will readily yield divergent persistence. We also note that these cases tend to require that the dominant coefficients have relatively prime spacing. Failing this, the sequence is often not even zero-fluctuation sequence. An underlying topic which surfaces at various points throughout the thesis is strong unimodality of a Laurent polynomial (also known as log-concavity). In chapter 7 we prove a generalization of a theorem due to Odlyzko and Richmond (OR). They proved that under suitable conditions on the positioning of the nonzero coefficients of P, all large powers P\sp n will be strongly unimodal. Our generalization proves the strong unimodality of all products P\sb1P\sb2\cdot\cdot\cdot P\sb n for n sufficiently large, subject to appropriate conditions on the sequence \{P\sb i\}. In chapter 6 we prove another strong unimodality result, this time for sums of products or linear polynomials. The result is phrased as strong unimodality of various cross-sections of a two-dimensional triangular grid of coefficients. Results of this type come into play when seeking to estimate coefficients of powers of certain univariate polynomials. (Abstract shortened by UMI.
Representation Theory of Compact Inverse Semigroups
No full textW. D. Munn proved that a finite dimensional representation of an inverse semigroup
is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this
thesis will be to give new analytic proof that every finite dimensional representation
of a compact inverse semigroup is equivalent to a ⋆-representation.
The second goal is to parameterize all finite dimensional irreducible representations
of a compact inverse semigroup in terms of maximal subgroups and order
theoretic properties of the idempotent set. As a consequence, we obtain a new and
simpler proof of the following theorem of Shneperman: a compact inverse semigroup
has enough finite dimensional irreducible representations to separate points if and
only if its idempotent set is totally disconnected.
Our last theorem is the following: every norm continuous irreducible ∗-representation
of a compact inverse semigroup on a Hilbert space is finite dimensional
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