Arithmetic Mean

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Jumat Sulaiman - One of the best experts on this subject based on the ideXlab platform.

Yan-kuen Wu - One of the best experts on this subject based on the ideXlab platform.

  • On the Max-quasi-Arithmetic Mean Powers of a Fuzzy Matrix
    2009 International Joint Conference on Computational Sciences and Optimization, 2009
    Co-Authors: Yan-kuen Wu
    Abstract:

    Since Thomason's paper in 1977 showing that the max-min powers of a fuzzy matrix either converge or oscillate with a finite period, many different algebraic operations are employed to explore the limiting behavior of powers of a fuzzy matrix, such as max-min/max-product/max-Archimedean t-norm/max-t-norm/max-Arithmetic Mean operations. In this article, we consider the max-quasi-Arithmetic Mean powers of a fuzzy matrix which is an extensive case of the max-Arithmetic Mean, max-root power Mean and max-convex Mean. We also show that the powers of such fuzzy matrices are always convergent. Some numerical examples are provided to show the situation of convergence.

  • CSO (2) - On the Max-quasi-Arithmetic Mean Powers of a Fuzzy Matrix
    2009 International Joint Conference on Computational Sciences and Optimization, 2009
    Co-Authors: Yan-kuen Wu
    Abstract:

    Since Thomason's paper in 1977 showing that the max-min powers of a fuzzy matrix either converge or oscillate with a finite period, many different algebraic operations are employed to explore the limiting behavior of powers of a fuzzy matrix, such as max-min/max-product/max-Archimedean t-norm/max-t-norm/max-Arithmetic Mean operations. In this article, we consider the max-quasi-Arithmetic Mean powers of a fuzzy matrix which is an extensive case of the max-Arithmetic Mean, max-root power Mean and max-convex Mean. We also show that the powers of such fuzzy matrices are always convergent. Some numerical examples are provided to show the situation of convergence.

  • Convergence of powers for a fuzzy matrix with convex combination of max-min and max-Arithmetic Mean operations
    Information Sciences, 2009
    Co-Authors: Yan-kuen Wu
    Abstract:

    Fuzzy matrices have been proposed to represent fuzzy relations on finite universes. Since Thomason's paper in 1977 showing that powers of a max-min fuzzy matrix either converge or oscillate with a finite period, conditions for limiting behavior of powers of a fuzzy matrix have been studied. It turns out that the limiting behavior depends on the algebraic operations employed, which usually in the literature includes max-min/max-product/max-Archimedean t-norm/max t-norm/max-Arithmetic Mean operations, respectively. In this paper, we consider the powers of a fuzzy matrix with convex combination of max-min and max-Arithmetic Mean operations. We show that the powers of such a fuzzy matrix are always convergent.

  • convergence of max Arithmetic Mean powers of a fuzzy matrix
    Fuzzy Sets and Systems, 2007
    Co-Authors: Yan-kuen Wu
    Abstract:

    Fuzzy matrices provide convenient representations for fuzzy relations on finite universes. In the literature, the powers of a fuzzy matrix with max-min/max-product/max-Archimedean t-norm compositions have been studied. It turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved. In this paper, the max-Arithmetic Mean composition is considered for the fuzzy relations. We show that the max-Arithmetic Mean powers of a fuzzy matrix always are convergent.

Kwang-cheng Chen - One of the best experts on this subject based on the ideXlab platform.

  • Data extraction via histogram and Arithmetic Mean queries: Fundamental limits and algorithms
    2016 IEEE International Symposium on Information Theory (ISIT), 2016
    Co-Authors: I-hsiang Wang, Shao-lun Huang, Kwang-cheng Chen
    Abstract:

    The problems of extracting information from a data set via histogram queries or Arithmetic Mean queries are considered. We first show that the fundamental limit on the number of histogram queries, m, so that the entire data set of size n can be extracted losslessly, is m = Θ(n/log n), sub-linear in the size of the data set. For proving the lower bound (converse), we use standard arguments based on simple counting. For proving the upper bound (achievability), we proposed two query mechanisms. The first mechanism is random sampling, where in each query, the items to be included in the queried subset are uniformly randomly selected. With random sampling, it is shown that the entire data set can be extracted with vanishing error probability using Ω(n/log n) queries. The second one is a non-adaptive deterministic algorithm. With this algorithm, it is shown that the entire data set can be extracted exactly (no error) using Ω(n/log n) queries. We then extend the results to Arithmetic Mean queries, and show that for data sets taking values in a real-valued finite Arithmetic progression, the fundamental limit on the number of Arithmetic Mean queries to extract the entire data set is also Θ(n/log n).

  • ISIT - Data extraction via histogram and Arithmetic Mean queries: Fundamental limits and algorithms
    2016 IEEE International Symposium on Information Theory (ISIT), 2016
    Co-Authors: I-hsiang Wang, Shao-lun Huang, Kwang-cheng Chen
    Abstract:

    The problems of extracting information from a data set via histogram queries or Arithmetic Mean queries are considered. We first show that the fundamental limit on the number of histogram queries, m, so that the entire data set of size n can be extracted losslessly, is m = Θ(n/log n), sub-linear in the size of the data set. For proving the lower bound (converse), we use standard arguments based on simple counting. For proving the upper bound (achievability), we proposed two query mechanisms. The first mechanism is random sampling, where in each query, the items to be included in the queried subset are uniformly randomly selected. With random sampling, it is shown that the entire data set can be extracted with vanishing error probability using Ω(n/log n) queries. The second one is a non-adaptive deterministic algorithm. With this algorithm, it is shown that the entire data set can be extracted exactly (no error) using Ω(n/log n) queries. We then extend the results to Arithmetic Mean queries, and show that for data sets taking values in a real-valued finite Arithmetic progression, the fundamental limit on the number of Arithmetic Mean queries to extract the entire data set is also Θ(n/log n).

Mohana Sundaram Muthuvalu - One of the best experts on this subject based on the ideXlab platform.

Gary Weiss - One of the best experts on this subject based on the ideXlab platform.

  • majorization and Arithmetic Mean ideals
    arXiv: Functional Analysis, 2012
    Co-Authors: Victor Kaftal, Gary Weiss
    Abstract:

    Following "An infinite dimensional Schur-Horn theorem and majorization theory", Journal of Functional Analysis 259 (2010) 3115-3162, this paper further studies majorization for infinite sequences. It extends to the infinite case classical results on "intermediate sequences" for finite sequence majorization. These and other infinite majorization properties are then linked to notions of infinite convexity and invariance properties under various classes of substochastic matrices to characterize Arithmetic Mean closed operator ideals and Arithmetic Mean at infinity closed operator ideals.

  • B(H) lattices, density and Arithmetic Mean ideals
    Houston Journal of Mathematics, 2011
    Co-Authors: Victor Kaftal, Gary Weiss
    Abstract:

    This paper studies lattice properties of operator ideals in B(H) and their appli- cations to the Arithmetic Mean ideals which were introduced in (10) and further studied in the project (14)-(17) of which this paper is a part. It is proved that the lattices of all principal ideals, of principal ideals with a generator that satisfies the �1/2-condition, of Arithmetic Mean stable principal ideals (i.e., those with an am-regular generator), and of Arithmetic Mean at infinity stable principal ideals (i.e., those with an am-∞ regular generator) are all both upper and lower dense in the lattice of general ideals. This Means that between any ideal and an ideal (nested above or below respectively) in one of these sublattices, lies another ideal in that sublattice. Among the applications: a principal ideal is am-stable (and similarly for am-∞ stable principal ideals) if and only if any (or equivalently, all) of its first order Arithmetic Mean ideals are am-stable, such as its am-interior, am-closure and others. A principal ideal I is am-stable (and similarly for am-∞ stable principal ideals) if and only if it satisfies any (equivalently, all) of the first order equality cancellation properties, e.g, Ja = Ia ⇒ J = I. These cancellation properties can fail even for am-stable countably generated ideals. It is proven that while the inclusion cancellation Ja ⊃ Ia implies J ⊃ I does not hold in general, even for I am-stable and principal, there is always a largest ideal b I for which Ja ⊃ Ia ⇒ J ⊃ b I. Furthermore, if I = (�) is principal, then b I is principal as ' for the harmonic sequence !, 0 < p < 1, and 1/p − 1/p ' = 1.

  • a survey on the interplay between Arithmetic Mean ideals traces lattices of operator ideals and an infinite schur horn majorization theorem
    arXiv: Functional Analysis, 2007
    Co-Authors: Victor Kaftal, Gary Weiss
    Abstract:

    The main result in (23) on the structure of commutators showed that Arithmetic Means play an important role in the study of operator ideals. In this survey we present the notions of Arithmetic Mean ideals and Arithmetic Mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sub- lattice structure of ideals, Arithmetic Mean ideal cancellation properties of first and second order, and softness properties - a term that we introduced, but a notion that is ubiquitous in operator ideals. Arithmetic Mean closure of ideals leads us to investigate majorization for infinite sequences and this in turn leads us to an infinite Schur-Horn majorization theorem that extends recent work by Arveson and Kadison to nonsummable sequences. This survey covers the material announced towards the beginning of the project in PNAS- US (33) and then expanded and developed in a series of papers (34)-(38).

  • soft ideals and Arithmetic Mean ideals
    arXiv: Functional Analysis, 2007
    Co-Authors: Victor Kaftal, Gary Weiss
    Abstract:

    This article investigates the soft-interior and the soft-cover of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the Arithmetic Mean operations is essential for the study of the multiplicity of traces (see arXiv:0707.3169v1 [math.FA]). Many classical ideals are "soft", i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic Mean (am) operations were proven to be intrinsic to the theory of operator ideals by the work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators and Arithmetic Mean operations at infinity were studied in arXiv:0707.3169v1 [math.FA]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-infinity interior of an ideal.

  • b h lattices density and Arithmetic Mean ideals
    arXiv: Functional Analysis, 2007
    Co-Authors: Victor Kaftal, Gary Weiss
    Abstract:

    This part of a multi-paper project studies the lattice properties of the Arithmetic Mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and Wodzicki. We prove: the lattices of all principal ideals, of Arithmetic Mean or Arithmetic Mean at infinity stable principal ideals or of principal ideals with a generator that satisfies the Delta_1/2 condition, are all both upper and lower dense in the lattice of general ideals. That is, between any ideal and an ideal (nested above or below respectively) in one of these sublattices, lies another ideal in that sublattice. Among the applications: a principal ideal I is am-stable (I = I_a) if and only if any of its first order Arithmetic Mean ideals are am-stable if and only if the ideal satisfies the first order equality cancellation property: J_a = I_a implies J = I. We show that this cancellation property can fail even for am-stable countably generated ideals. Similar results hold for Arithmetic Mean at infinity ideals. Inclusion cancellations do not hold in general even for principal ideals, but for every ideal I there is a largest ideal I^ for which J_a contains I_a implies that J contains I^. When I is principal, I^ too is principal. We show that I=I^ is a strictly stronger property than am-stability. For example, for I the p < 1 power of the principal ideal J generated by diag {1/n}, I^ is the q power of J where 1/p-1/q = 1.