Showing papers on "Equivalence class published in 1982"

Distributions of Maximal Invariants Using Quotient Measures

Abstract: This paper demonstrates the use of proper actions and quotient measures in representations of non-central distributions of maximal invariants.

Constructing measures for path integrals

Abstract: The overcompleteness of the coherent states for the Heisenberg–Weyl group implies that many different integral kernels can be used to represent the same operator. Within such an equivalence class we construct an integral kernel to represent the quantum‐mechanical evolution operator for certain dynamical systems in the form of a path integral that involves genuine (Wiener) measures on continuous phase‐space paths. To achieve this goal it is necessary to employ an expression for the classical action different from the usual one.

The equivalence classes of the Vasil'ev codes of length 15

Abstract: In this paper we determine the equivalence classes of the perfect Vasilev codes of length 15: There exist 19 non equivalent Vasil'ev codes (including the Hamming code). If we restrict the equivalence transformations to permutations of coordinates we get 64 different Vasil'ev codes.

The nonrealizability of the quotient a//a2 of the steenrod algebra

Abstract: It is proved that there is no spectrum whose Z2-cohomol- ogy is the quotient of the Steenrod algebra by the left ideal generated by Sqt, Sq2, and Sq4. This quotient arises as a summand in H*MO (8>.

A variable-order Markov chain for coding of speech spectra

Abstract: We present a method that reduces the bit rate of a low rate LPC vocoder by modelling the sequence of quantized spectra by a Markov chain. To minimize the bit rate, one would want to use a high-order chain. Unfortunately, a high-order chain would require an inordinate amount of data for training. We describe in this paper the use of a variable order Markov chain that maximizes the effective use of a given amount of speech data. To reduce the number of states of a high-order chain, we define an equivalence relation on the states, i.e., "similar" states are grouped together in an equivalence class and a single conditional distribution is associated with the equivalence class. We introduce two equivalence relations. In the first, called variable order Markov chain, the equivalence classes represent the most probable states of any order. In the second method, called variable resolution, the equivalence class is obtained by decreasing the quantization accuracy in representing a spectrum that belongs to a more remote past. For an LPC vocoder with 64 possible spectra (using 6-bit vector quantization), the second method is superior to the first and decreases the entropy from 6 bits to 4 bits per spectrum with 256 equivalence classes.

Equivalence classes of functions on finite sets

Abstract: By using Polya's theorem of enumeration and de Bruijn's generalization of Polya's theorem, we obtain the numbers of various weak equivalence classes of functions in RD relative to permutation groups G and H where RD is the set of all functions from a finite set D to a finite set R, G acts on D and H acts on R. We present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn's theorem, i.e., to determine which functions belong to the same equivalence class. We also use our algorithm to construct the family of non-isomorphic fm-graphs relative to a given group.

Approximate topology on ()

Abstract: Let A be a C*-algebra and Rep(A) the set of all nondegenerate representations of A. We define a new topology on Rep(A) and study the relations with the point weak topology of Rep(A). 1. htroduction Throughout the paper, let A be a nonzero C*-algebra, and let P(A), A and Prim(A) denote, respectively, the pure states, the unitary equivalence classes of all nonzero irreducible representations and the set of all primitive ideals of A. As usual, we consider Prim(A) as a topological space with the Jacobson topology [3, p. 70]. The spectrum A is topologized with the inverse image of the Jacobson topology under the canonical surjection ca1: A -+ Prim(A), sending the unitary equivalence class [7r] of a unitary representation 7r to the kernel ker(ir) of 7r. Thus a, is a continuous open surjection. Prim(A) is a To-space [3, p. 70] and not Hausdorff in general. On the other hand, the spectrum A is not even a T0-space in the general situation [3, p. 71]. We regard P(A) as the topological space relativised from the weak* topology o(A*,A) on the norm dual space A* of A. For any f E P(A), let lrf be the irreducible representation of A associated with f, under the Gelfand-Naimark-Segal construction. The mapping a2: f -4 [7rf ] is an open and continuous surjection [3, p. 79], but it is many-to-one, [3, p. 54]. Let Rep(A) denote the set of all nondegenerate representations of A on nonzero Hilbert spaces. We will consider the weak topology TW on Rep(A), which is essentially the same as the topology on Rep(A: H) of M. Takesaki [10, p. 376] or the strong topology of L. T. Gardner [5, p. 445]. Let Irr(A) be the set of all nonzero irreducible representations of A. Let be the approximate equivalence in Rep(A) [1, 6]. In the main theorem (Theorem), the set of equivalence classes in Rep(A) under I, equipped with the quotient topology Tw, will be shown to be homeomorphic with Prim(A). But our principal concern of this paper is to introduce a new topology T on Rep(A), called the approximate topology, which will induce a Hausdorff topology on Prim(A) stronger than the Jacobson topology. This result is also contained in the Theorem. 2. The approximate topology For brevity, we write X for Rep(A). If ir E X, then we denote by H, the representation space of ir. For any two ir, p E X, we say that they are approximately equivalent, denoted by r p, if there is a net {Ui} of unitary operators Ui: Hp -* Hr such that f1Ui*7r(a)U -p(a)f l l0, for every a E A Received by the editors July 27, 1981. 1980 Mathematics Subject Classification. Primary 46L05; Secondary 46K10.

Dominates on Equivalence Classes of Semigroup Operations.

Abstract: : Initially the problem was to study the dominates relation on a collection of semigroup operations called triangular norms This led to an equivalent problem -- studying subadditivity of certain semigroup operations defined on the non-negative reals The setting was later generalized to include both problems and to bring essentials of the problem into sharper focus In the generalization, a partially ordered set S was endowed with the collection, Op(s), of all semigroup operations which had the same identity, e, and were non-decreasing in place The dominates relation was defined on Op(S) The collection, Map(S), of order-preserving bijections from S to S map e to itself was used to partition Op(S) into equivalence classes -- two objects being placed in the same class if they were isomorphic via some member of Map(S) Dominates restricted to any equivalence class in via some member of Map(S) Dominates restricted to any equivalence class in Op(S) was shown to to exhibit a certain homogeniety relative to composition of elements in Map(S) Transitivity of dominates on an equivalence class was shown to be equivalent to an appropriate subset of Map(S) being alegbraically closed under composition The equivalence classes determined by continuous triangular norms were characterized in terms of ordinal sums of semigroups (Author)