Set Of Equivalence Classes Research Articles

QUANTUM KINEMATICS OF THE CHERN-SIMONS FIELD WITH MATTER IN THE LOOP REPRESENTATION

The scalar field coupled to the Abelian Chern-Simons gauge field is quantized in the loop representation. The physical space of states is found to be labeled by equivalence classes of sets of open paths. The relations among them are studied, showing that the long range interaction provided by the gauge field gives rise to a topological interaction among the paths which is explicitly computed.

II 1 Factors, Their Bimodules and Hypergroups

In this paper, we introduce a notion that we call a hypergroup; this notion captures the natural algebraic structure possessed by the set of equivalence classes of irreducible bifinite bimodules over a II 1 factor. After developing some basic facts concerning bimodules over II 1 factors, we discuss abstract hypergroups. To make contact with the problem of what numbers can arise as index-values of subfactors of a given II 1 factor with trivial relative commutant, we define the notion of a dimension function on a hypergroup, and prove that every finite hypergroup admits a unique dimension function, we then give some nontrivial examples of hypergroups, some of which are related to the Jones subfactors of index 4cos 2 π/(2n+1). In the last section, we study the hypergroup invariant corresponding to a bifinite module, which is used, among other things, to obtain a transparent proof of a strengthened version of what Ocneanu terms 'the crossed-product remembering the group'.

Undecidability and initial segments of the (r.e.) tt-degrees

A notion of reducibility ≤r between sets is specified by giving a set of procedures for computing one set from another. We say that a set A is r-reducible to a set B, A ≤rB, if one of the procedures applied to B gives A. Associated with any such reducibility notion is the structure of r-degrees, the equivalence classes of sets with respect to this reducibility, with the induced ordering. The most general notion of a computable reducibility is that of Turing, ≤T. Here we say that A ≤TB if there is a Turing machine φe which, when equipped with an oracle for B, computes A: φeB = A. Such Turing degree computations are characterized by the phenomenon that only during the computation itself do we discover which questions about B need to be answered to compute A(x). In contrast, for nearly all other computable reducibilities the set of questions needed is given in advance by a recursive procedure. Perhaps the most common example of such a procedure is many-one reducibility, ≤m: A ≤mB if there is a recursive function f such that x ∈ A ⇔ f(x) ∈ B.Reducibilities with the property that the output, A(x), is determined by the answers that B gives to a set of questions calculated recursively from x are said to be of tabular type. The most general tabular reducibility is called truth-table reducibility, ≤tt. The procedures [e] associated with this reducibility are specified by a recursive function f (= {e}) which, for each x, gives a set of n questions about the oracle and, for each of the possible 2n sets of answers, gives the corresponding output. As usual this defines A ≤ttB as “there is an e such that [e]B = A”. It is with this notion of reducibility and the associated tt-degrees that we shall be concerned in this paper. Basic information on several such strong reducibilities can be found in Rogers [28]. For more information we recommend the survey articles by Odifreddi [25] and Degtev [2] as well as Odifreddi's book [26].

Bounds for the number of moduli for irregular varieties of general type

LetX be a minimal surface of general type and M (X) the set of equivalence classes of complex structures on the differentiable manifold underlyingX; denoting byMx the dimension of M(X) at [X], the point corresponding to the complex structure ofX we consider the problem of finding an upper bound forMx in terms of the basic numerical invariants ofX. The main result is the Castelunovo's bound:Mx≤Pg(X)+2q(X) for certain irregular surfaces. We also generalize the above bound to an arbitrary dimension.

Formal deformations of the associative algebra of smooth matrices

There is a canonical map from the set of equivalence classes of formal deformations of the associative algebra of smooth functions on a manifold and that of the associative algebra of smooth matrices of any given dimension. Using the so-called Morita equivalence for Hochschild Cohomology, one shows that this mapping is a bijection.

A cohomological interpretation of the reflexive Brauer group

Throughout this note R will be a normal affine domain, i.e., R is the coordinate ring of a normal afftne variety X, over an algebraically closed field of characteristic zero. Let K denote the field of fractions of R and let C be a central simple algebra over K. A subring A of C is said to be reflexive Azumaya algebra over R iff: (I) the center of n is equal to R, (2) n is f.g. and reflexive as R-module, (3) A,, is an Azumaya algebra over the discrete valuation ring R, for every height one prime p of R. Two reflexive Azumaya algebras (in possibly distinct central simple K-algebras) are said to be equivalent if there exist Eg. reflexive R-modules M and N such that A 0; End,(M) = r@X End,(N) as R-algebras, where the modified tensor-product 0; is defined to be (-OR-)**. The set of equivalence classes of reflexive Azumaya algebras equipped with the modified tensor product as a multiplication rule is a group, the so- called reflexive Brauer group ,!l(R) of R or of the corresponding normal afftne variety X Whereas the natural morphism Br( R) --+ Br(K)

Sections boréliennes pour les feuilletages mesurés

Borelian sections for measured foliations have been defined. A sufficient condition for the existence of borelian sections in the measured foliations is given and it is also proved that the measure on the set of equivalence classes obtained via Whitehead's relation is a dense probability measure.

Software reliability modeling (tutorial)

There are a number of views as to what software reliability is and how it should be quantified. Some people believe that this measure should be binary in nature so that an imperfect program would have zero reliability while a perfect one would have a reliability value of one. This view parallels that of program proving whereby the program is either correct or incorrect. Others, however, feel that software reliability should be defined as the relative frequency of the times that the program works as intended by the user. This view is similar to that taken in testing where a percentage of the successful ewes is used as a measure of program quality. According to the latter viewpoint, software reliability is a probabilistic measure and can be defined as follows: Let F be a class of faults, defined arbitrarily, and T be a measure of relevant time, the units of which are dictated by the application at hand. Then the reliability of the software package with respect to the class of faults F and with respect to the metric T, is the probability that no fault of the class occurs during the execution of the program for a prespecified period of relevant time. A number of models have been proposed during the past fifteen years to estimate software reliability and several other performance measures. These are based mainly on the failure history of software and can be classified according to the nature of the failure process studied as indicated below. Times Between Failures Models: In this class of models the process under study is the time between failures. The most common approach is to assume that the time between, say, the (i-1)st and ith failures, follows a distribution whose parameters depend on the number of faults remaining in the program during this interval. Failure Count Models: The interest of this class of models is in the number of faults or failures in specified time intervals rather than in times between failures. The failure counts are assumed to follow a known stochastic process with a time dependent discrete or continuous failure rate. Fault Seeding Models: The basic approach in this class of models is to “seed” a known number of faults in a program which is assumed to have an unknown number of indigenous faults. Input Domain Based Models: The basic approach taken here is to generate a set of test cases from an input distribution which is assumed to be representative of the operational usage of the program. Because of the difficulty in obtaining this distribution, the input domain is partitioned into a set of equivalence classes, each of which is usually associated with a program path. In this tutorial we discuss the key models from the above classes and the related issues of parametric estimation, unification of models, Bayesian interpretation, validation and comparison of models, and determination of optimum release time.

Weak containment and tensor products of group representations. II

Let G be a locally compact group and G the set of equivalence classes of irreducible unitary representations of G. (~, equipped with the Jacobson topology, is called the dual space of G. In this paper, we continue our study of problems related to supports in G of tensor products of representations of G. It is well known that ifn and ~ are finite dimensional representations of G and ~g denotes the conjugate representation of Q, then the tensor product n| contains the trivial one-dimensional representation 1G if and only ifg and Q have a common subrepresentation. Can this be generalized to the case, where n and ~ are arbitrary and containment is replaced by weak containment? This question was first discussed by Fell [6] and has been taken up in [4] and [21]. The object of Sect. 1 is to add some further results concerning this problem. Our main results are the following. We show that, for a polynomially growing discrete group G and any two unitary representations n and ~ of G, 1, is weakly contained in nQff (i.e. 1~ ~suppn| ) if and only if suppnnsupp04:0. This extends the corresponding result for locally nilpotent discrete groups in [21]. If G is a motion group and n e (~, we give a criterion on n to satisfy that laesuppn| ~ is equivalent to ff=n for all ~Ed.

Composition and genera of norm-type forms

M. Kneser has recently discovered a way to define a composition of binary quadratic forms in general [5]. His basic idea can be described as expanding the structure to include a specified action of a ring of similitudes. This approach avoids the traditional problem of orienting the forms, since the proper equivalences can be defined simply as the isometries that preserve the action of the similitude ring. But more is true: when we view his idea in this way, we can extend it to norm-type forms of higher degrees. Besides throwing a new light on the quadratic case, this extension reveals a natural concept of underlying the genus fields already known in number theory. Fix a base ring R (commutative with unit). If P is a free i?-module, then a of degree m supported by P is of course a homogeneous polynomial ƒ of degree m in the coordinates on P; technically, this means that ƒ is an element of the symmetric power S(P*), and in this version we can (and do) extend the definition to projective P of finite rank. Carrying over the usual terminology for quadratic forms, we call (P, ƒ) primitive if ƒ is not identically zero modulo any maximal ideal of R. Now fix an extension C of R, and assume that C is projective of rank m as an i?-module. A form of type C/R will be a pair (P, ƒ) where (1) P is an invertible C-module (and hence projective of rank m over R), (2) ƒ is a primitive form of degree m on the i?-module P, and (3) there is a formal identity f(cp) = N(c)f(p), where N is the norm from CtoR. Two such forms are equivalent if there is a form isometry preserving the C-module structure. Let F(C/R) be the set of equivalence classes. If R —• S is any ring homomorphism, then ® H S induces a map F(C/R) —• F(C ® S/S). There is always at least one form of type C/R, the trivial form (C, iV), and in a sense this is the basic one:

Representations of Well-Founded Preference Orders

A preference order, or linear preorder, on a set X is a binary relation which is transitive, reflexive and total. This preorder partitions the set X into equivalence classes of the form . The natural relation induced by on the set of equivalence classes is a linear order. A well-founded preference order, or prewellordering, will similarly induce a well-ordering. A representation or Paretian utility function of a preference order is an order-preserving map f from X into the R of real numbers (provided with the standard ordering). Mathematicians and economists have studied the problem of obtaining continuous or measurable representations of suitably defined preference orders [4, 7]. Parametrized versions of this problem have also been studied [1, 7, 8]. Given a continuum of preference orders which vary in some reasonable sense with a parameter t, one would like to obtain a continuum of representations which similarly vary with t.

On the category of homotopy pairs

A category is described whose objects are pairs in the sense of Eckmann and Hilton. The set of morphisms π( f, g) from f to g is the set of equivalence classes (under a certain equivalence relation) of triples (φ, ψ, h 1), where φ and ψ form together with f and g a homotopy-commutative diagram and h 1 is a homotopy from φ f to gψ, π( f, g) depends only on the homotopy classes of f and g and π n(α,β) = π(ƒ nf,g) is a group for n⩾1. The groups fit into a ‘cylinder-web’ diagram with multiple exact sequences that can enable their computation provided that all relevant ‘absolute’ homotopy groups π n ( X, Y) are known and information regarding composition of absolute classes is available. For the Hopf classes η and v the groups π n ( η, η) ( n⩾1), π n ( η, v) (1⩽ n⩽5) and π n ( η 3, η 3) ( n = 1, 2) are computed.

On Definability of Nonmeasurable Sets

In [1], Solovay constructed a model of ZFC in which every set of reals in OD (R) is Lebesgue measurable. Here we construct a model in which every equivalence class of sets of reals modulo null sets that is in OD(R) consists of Lebesgue measurable sets. This result immediately implies Solovay's, since the equivalence class of any set of reals in OD (R) is itself OD (R). As a consequence, one cannot provably explicitly define a nonmeasurable set modulo null sets within ZFC. We do not know whether this holds in the model Solovay uses (where an inaccessible cardinal is collapsed to ω 1). Instead, our model is a generic extension of his model.In the model we construct, a somewhat stronger statement holds: every set in OD(R) of sets of reals which has < 2C inequivalent elements modulo null sets, consists entirely of Lebesgue measurable sets of reals.

An algorithm for computation of translation invariant groups and the subgroup lattice

For given integern>2 the translation sets of permutations on 1, 2, ...,n−1 form a set of equivalence classes on the symmetric group ℙ n-1. To each translation set there corresponds one step-cycle. A group consisting of a union of disjoint translation sets is called a translation invariant group. In this paper, we want to describe an algorithm for a systematic search for all translation invariant groups in ℙ n-1.

A new construction of the real numbers

There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.

Some rate-<tex>p/(p+1)</tex>quasi-cyclic codes (Corresp.)

Some optimal rate- \frac{1}{2} quasi-cyclic codes found by Chert et aL [3] are grouped into a small set of equivalence classes with the assistance of a computer. The weight distribution of a selection of these rate- \frac{1}{2} codes is tabulated. In addition, a list of new optimal rate- \frac{2}{3} quasi-cyclic codes of lengths up to 54 is presented.

Degrees of computational complexity

We consider a measure Φ of computational complexity. The measure Φ determinesa binary relation on the recursive functions; F is no harder to compute than G iff for every index g of G there is an index f of F such that for nearly all x , the difficulty of f at x (as measured by Φ ) is no more than the difficulty of g at x . The corresponding symmetric relation is an equivalence relation, and the set of equivalence classes (the degrees of complexity) is partially ordered. In this paper we give a simple proof of a result of McCreight: An arbitrary countable partial ordering can be embedded in this ordering of degrees of complexity.

Singular Integrals on Ultraspherical Series

One of the main uses of harmonic analysis on the sphere is to discover new theorems about series of ultraspherical (Gegenbauer) polynomials. In this paper, we will construct singular integral operators from scalar functions on the sphere to vector functions. These operators when restricted to zonal functions give Lp-bounded (1 < p < ∞ ) operators on ultraspherical series.We will use [7, Chapter 9] as our main reference. Let G denote a compact group, with identity e, and Ĝ its dual, the set of equivalence classes of continuous irreducible unitary representations of G.

A note on homotopy equivalences

Given a homotopy equivalence f : X → Y f:X \to Y , a homotopy inverse g of f, and a homotopy H : X × I → X H:X \times I \to X from g ∘ f g \circ f to 1 X {1_X} . We show that there is a homotopy K : Y × I → Y K:Y \times I \to Y from f ∘ g f \circ g to 1 Y {1_Y} such that f ∘ H ≃ K ∘ ( f × 1 I ) rel X × ∂ I f \circ H \simeq K \circ (f \times {1_I})\,{\text {rel}}\,X \times \partial I and H ∘ ( g × 1 I ) ≃ g ∘ K rel Y × ∂ I H \circ (g \times {1_I}) \simeq g \circ K\,{\text {rel}}\,Y \times \partial I .

Determinacy theory for the Livšic moments problem

The Livtc moments problem as formulated by Gil de Lamadrid [6] is presented in terms of a given vector space F of complex functions, called test functions, defined on R = (-00, + co) and a given pseudo-inner product (nonnegative sesquilinear form) on F. The problem is to find a positive measure p on R which gives an integral representation of the pseudo-inner product. This problem includes the classical Hamburger moments problem and the Bochner theorem on positive definite functions on R. Livgic [IO] and other Russian authors [l, 81 have limited their discussion of this problem to the existence of a solution. On the other hand, even for the Hamburger problem, the solution is not in general unique. This has led to the classical determinacy theory for the Hamburger problem [l, 17, 181. This theory is concerned with the classification of solutions, the study of the structure of the set of solutions, the characterization of those problems which are determinate (have a unique solution), and the study of the supports of solutions. Recently, Gil de Lamadrid [6] initiated the study of the determinacy theory for the LivSic problem. His method, as that of LivZic, was based on Naimark’s theory of transcendental extensions (extensions to larger spaces) of a symmetric operator [12, 13, 14, 151. U 1 aimark proved the existence of transcendental extensions of a symmetric operator and analyzed the set of all such extensions. He pointed to the application of his theory to the moments problem by establishing a one-to-one correspondence between the set of solutions of the Hamburger problem and the set of equivalence classes of minimal self-adjoint extensions of the operator of multiplication by the independent variable which appears naturally in the problem. Later, LivEic used Naimark’s existence theorem to establish the existence of a solution of a