5,452 research outputs found

    Relations on FP-Soft Sets Applied to Decision Making Problems

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    In this work, we first define relations on the fuzzy parametrized soft sets and study their properties. We also give a decision making method based on these relations. In approximate reasoning, relations on the fuzzy parametrized soft sets have shown to be of a primordial importance. Finally, the method is successfully applied to a problems that contain uncertainties.Comment: soft application

    On Maximum Contention-Free Interleavers and Permutation Polynomials over Integer Rings

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    An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Contention-free interleavers have been recently shown to be suitable for parallel decoding of turbo codes. In this correspondence, it is shown that permutation polynomials generate maximum contention-free interleavers, i.e., every factor of the interleaver length becomes a possible degree of parallel processing of the decoder. Further, it is shown by computer simulations that turbo codes using these interleavers perform very well for the 3rd Generation Partnership Project (3GPP) standard.Comment: 13 pages, 2 figures, submitted as a correspondence to the IEEE Transactions on Information Theory, revised versio

    What Is the Validity Domain of Einstein’s Equations? Distributional Solutions over Singularities and Topological Links in Geometrodynamics

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    The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which the former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions

    Tensor products and regularity properties of Cuntz semigroups

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    The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra AA, its (concrete) Cuntz semigroup Cu(A)Cu(A) is an object in the category CuCu of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter CuCu-semigroups. We establish the existence of tensor products in the category CuCu and study the basic properties of this construction. We show that CuCu is a symmetric, monoidal category and relate Cu(A⊗B)Cu(A\otimes B) with Cu(A)⊗CuCu(B)Cu(A)\otimes_{Cu}Cu(B) for certain classes of C*-algebras. As a main tool for our approach we introduce the category WW of pre-completed Cuntz semigroups. We show that CuCu is a full, reflective subcategory of WW. One can then easily deduce properties of CuCu from respective properties of WW, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in WW are much easier since the objects are purely algebraic. We also develop a theory of CuCu-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a CuCu-semiring. We give explicit characterizations of CuCu-semimodules over such CuCu-semirings. For instance, we show that a CuCu-semigroup SS tensorially absorbs the CuCu-semiring of the Jiang-Su algebra if and only if SS is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.Comment: 195 pages; revised version; several proofs streamlined; some results corrected, in particular added 5.2.3-5.2.

    Further Results on Quadratic Permutation Polynomial-Based Interleavers for Turbo Codes

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    An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Also, the recently proposed quadratic permutation polynomial (QPP) based interleavers by Sun and Takeshita (IEEE Trans. Inf. Theory, Jan. 2005) provide excellent performance for short-to-medium block lengths, and have been selected for the 3GPP LTE standard. In this work, we derive some upper bounds on the best achievable minimum distance dmin of QPP-based conventional binary turbo codes (with tailbiting termination, or dual termination when the interleaver length N is sufficiently large) that are tight for larger block sizes. In particular, we show that the minimum distance is at most 2(2^{\nu +1}+9), independent of the interleaver length, when the QPP has a QPP inverse, where {\nu} is the degree of the primitive feedback and monic feedforward polynomials. However, allowing the QPP to have a larger degree inverse may give strictly larger minimum distances (and lower multiplicities). In particular, we provide several QPPs with an inverse degree of at least three for some of the 3GPP LTE interleaver lengths giving a dmin with the 3GPP LTE constituent encoders which is strictly larger than 50. For instance, we have found a QPP for N=6016 which gives an estimated dmin of 57. Furthermore, we provide the exact minimum distance and the corresponding multiplicity for all 3GPP LTE turbo codes (with dual termination) which shows that the best minimum distance is 51. Finally, we compute the best achievable minimum distance with QPP interleavers for all 3GPP LTE interleaver lengths N <= 4096, and compare the minimum distance with the one we get when using the 3GPP LTE polynomials.Comment: Submitted to IEEE Trans. Inf. Theor

    Convex and Concave Soft Sets and Some Properties

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    In this study, after given the definition of soft sets and their basic operations we define convex soft sets which is an important concept for operation research, optimization and related problems. Then, we define concave soft sets and give some properties for the concave sets. For these, we will use definition and properties of convex-concave fuzzy sets in literature. We also give different some properties for the convex and concave soft sets
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