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Mrinal Kanti Roychowdhury - One of the best experts on this subject based on the ideXlab platform.

  • Local Dimensions and Quantization Dimensions in Dynamical Systems
    The Journal of Geometric Analysis, 2020
    Co-Authors: Mrinal Kanti Roychowdhury, Bilel Selmi
    Abstract:

    Let $$\mu $$ μ be a Borel Probability Measure generated by a hyperbolic recurrent iterated function system defined on a nonempty compact subset of $$\mathbb R^k$$ R k . We study the Hausdorff and the packing dimensions, and the quantization dimensions of $$\mu $$ μ with respect to the geometric mean error. The results establish the connections with various dimensions of the Measure $$\mu $$ μ and generalize many known results about local dimensions and quantization dimensions of Measures.

  • Optimal quantization for the Cantor distribution generated by infinite similutudes
    Israel Journal of Mathematics, 2019
    Co-Authors: Mrinal Kanti Roychowdhury
    Abstract:

    Let P be a Borel Probability Measure on ℝ generated by an infinite system of similarity mappings { S _ j : j ∈ ℕ} such that $$P=\Sigma_{j=1}^{\infty}\frac{1}{2^{j}}P\circ{S}_j^{-1}$$ P = Σ j = 1 ∞ 1 2 j P ∘ S j − 1 , where for each j ∈ ℕ and x ∈ ℝ, $$S_j(x)=\frac{1}{3^j}x+1-\frac{1}{3^{j-1}}$$ S j ( x ) = 1 3 j x + 1 − 1 3 j − 1 . Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f _1, f _2 : ℝ → ℝ such that f _1( x ) = 1/3 x and f _2( x ) = 1/3 x + 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings { S _ j : j ∈ ℕ} associated with the Probability vector $$(\frac{1}{2},\frac{1}{{{2^2}}},...)$$ ( 1 2 , 1 2 2 , ... ) , for all n ∈ ℕ, we determine the optimal sets of n -means and the n th quantization errors for the infinite self-similar Measure P . The technique obtained in this paper can be utilized to determine the optimal sets of n -means and the n th quantization errors for more general infinite self-similar Measures.

  • Uniform distributions on curves and quantization
    arXiv: Probability, 2018
    Co-Authors: Joseph Rosenblatt, Mrinal Kanti Roychowdhury
    Abstract:

    The basic goal of quantization for Probability distribution is to reduce the number of values, which is typically uncountable, describing a Probability distribution to some finite set and thus to make an approximation of a continuous Probability distribution by a discrete distribution. It has broad application in signal processing, and data compression. In this paper, first we have defined the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of $n$-means, and the $n$th quantization errors for different values of $n$ with respect to the uniform distributions defined on the curves. In each case, we have further calculated the quantization dimension and show that it is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number, which supports the well-known result of Bucklew and Wise (1982), which says that for a Borel Probability Measure $P$ with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number.

  • least upper bound of the exact formula for optimal quantization of some uniform cantor distributions
    Discrete & Continuous Dynamical Systems - A2018 Volume 38 Pages 4555-4570, 2018
    Co-Authors: Mrinal Kanti Roychowdhury
    Abstract:

    The quantization scheme in Probability theory deals with finding a best approximation of a given Probability distribution by a Probability distribution that is supported on finitely many points. Let \begin{document} $P$\end{document} be a Borel Probability Measure on \begin{document} $\mathbb R$\end{document} such that \begin{document} $P = \frac 12 P\circ S_1^{-1}+\frac 12 P\circ S_2^{-1},$\end{document} where \begin{document} $S_1$\end{document} and \begin{document} $S_2$\end{document} are two contractive similarity mappings given by \begin{document} $S_1(x) = rx$\end{document} and \begin{document} $S_2(x) = rx+1-r$\end{document} for \begin{document} $0 and \begin{document} $x∈ \mathbb R$\end{document} . Then, \begin{document} $P$\end{document} is supported on the Cantor set generated by \begin{document} $S_1$\end{document} and \begin{document} $S_2$\end{document} . The case \begin{document} $r = \frac 13$\end{document} was treated by Graf and Luschgy who gave an exact formula for the unique optimal quantization of the Cantor distribution \begin{document} $P$\end{document} (Math. Nachr., 183 (1997), 113-133). In this paper, we compute the precise range of \begin{document} $r$\end{document} -values to which Graf-Luschgy formula extends.

  • Canonical sequences of optimal quantization for condensation Measures
    arXiv: Dynamical Systems, 2017
    Co-Authors: Dogan Comez, Mrinal Kanti Roychowdhury
    Abstract:

    Let $P:=\frac 1 3 P\circ S_1^{-1}+\frac 13 P\circ S_2^{-1}+\frac 13\nu$, where $S_1(x)=\frac 15 x$, $S_2(x)=\frac 1 5 x+\frac 45$ for all $x\in \mathbb R$, and $\nu$ be a Borel Probability Measure on $\mathbb R$ with compact support. Such a Measure $P$ is called a condensation Measure, or an an inhomogeneous self-similar Measure, associated with the condensation system $(\{S_1, S_2\}, (\frac 13, \frac 13, \frac 13), \nu)$. Let $D(\mu)$ denote the quantization dimension of a Measure $\mu$ if it exists. Let $\kappa$ be the unique number such that $(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}+(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}=1$. In this paper, we have considered four different self-similar Measures $\nu:=\nu_1, \nu_2, \nu_3, \nu_4$ satisfying $D(\nu_1)>\kappa$, $D(\nu_2) \kappa$, and $D(\nu_4)=\kappa$. For each Measure $\nu$ we show that there exist two sequences $a(n)$ and $F(n)$, which we call as canonical sequences. With the help of the canonical sequences, we obtain a closed formula to determine the optimal sets of $F(n)$-means and $F(n)$th quantization errors for the condensation Measure $P$ for each $\nu$. Then, we show that for each Measure $\nu$ the quantization dimension $D(P)$ of the condensation Measure $P$ exists, and satisfies: $D(P)=\max\{\kappa, D(\nu)\}$. Moreover, we show that for $D(\nu_1)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients are finite, positive and unequal; on the other hand, for $\nu=\nu_2, \nu_3, \nu_4$, the $D(P)$-dimensional lower quantization coefficient is infinity. This shows that for $D(\nu)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients can be either finite, positive and unequal, or it can be infinity.

Trubee Davison - One of the best experts on this subject based on the ideXlab platform.

  • unitary representations of the baumslag solitar group on the cantor set
    Journal of Fourier Analysis and Applications, 2020
    Co-Authors: Trubee Davison
    Abstract:

    The Cantor Set supports a Borel Probability Measure known as the Hutchinson Measure which satisfies a well known fixed point relationship (Hutchinson in Indiana University Math J 30(5):713–747, 1981). Previously it has been shown by Jorgensen and Dutkay that the Cantor set can be extended to an inflated Cantor set, $${\mathcal {R}}$$, on a subset of the real line, which supports an extended Hutchinson Measure $${\bar{\mu }}$$ (Dutkay and Jorgensen in Rev. Mat. Iberoamericana 22(1):131–180, 2006). Unitary dilation and translation operators can be defined on $$L^2({\mathcal {R}}, {\bar{\mu }})$$ which satisfy the Baumslag–Solitar group relation, and give rise to a filtration of the Hilbert space $$L^2({\mathcal {R}}, {\bar{\mu }})$$ called a multi-resolution analysis (Dutkay and Jorgensen 2006). The low pass filter function corresponding to this construction can be used to produce a Measure, m, on a compact topological group called the 3-solenoid, denoted $${\mathcal {S}}_3$$ (Dutkay in Trans Am Math Soc 358(12):5271–5291, 2006). The Hilbert space $$L^2({\mathcal {S}}_3, m)$$ also admits a unitary representation of the Baumslag–Solitar group, and there exists a generalized Fourier transform between $$L^2({\mathcal {R}}, {\bar{\mu }})$$ and $$L^2({\mathcal {S}}_3,m)$$ (Dutkay 2006). In this paper, we build off of Jorgensen and Dutkay’s work to show that the unitary operators on $$L^2({\mathcal {S}}_3,m)$$ mentioned above are related to each other via a family of partial isometries, which satisfy properties resembling the Cuntz relations.

  • A Positive Operator-Valued Measure for an Iterated Function System
    Acta Applicandae Mathematicae, 2018
    Co-Authors: Trubee Davison
    Abstract:

    Given an iterated function system (IFS) on a complete and separable metric space  Y $Y$ , there exists a unique compact subset X ⊆ Y $X \subseteq Y$ satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel Probability Measure, called the Hutchinson Measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson Measure to a projection-valued Measure, under the assumption that the IFS does not have essential overlap (Jorgensen in Adv. Appl. Math. 34(3):561–590, 2005 ; Operator Theory, Operator Algebras, and Applications, pp. 13–26, 2006 ). In previous work, we developed an alternative approach to proving the existence of this projection-valued Measure (Davison in Acta Appl. Math. 140(1):11–22, 2015 ; Acta Appl. Math. 140(1):23–25, 2015 ; Generalizing the Kantorovich metric to projection-valued Measures: with an application to iterated function systems, 2015 ). The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in Jorgenson et al. (J. Math. Phys. 48(8):083511, 35, 2007 ). We build off their work to generalize the Hutchinson Measure to a positive-operator valued Measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued Measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R.F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory (Werner in J. Quantum Inf. Comput. 4(6):546–562, 2004 ). We conclude with a discussion of Naimark’s dilation theorem with respect to this positive operator-valued Measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric.

  • a positive operator valued Measure for an iterated function system
    arXiv: Functional Analysis, 2016
    Co-Authors: Trubee Davison
    Abstract:

    Given an iterated function system (IFS) on a complete and separable metric space $Y$, there exists a unique compact subset $X \subseteq Y$ satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel Probability Measure, called the Hutchinson Measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson Measure to a projection-valued Measure, under the assumption that the IFS does not have essential overlap. In previous work, we developed an alternative approach to proving the existence of this projection-valued Measure. The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in. We build off their work to generalize the Hutchinson Measure to a positive-operator valued Measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued Measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R.F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory. We conclude with a discussion of Naimark's dilation theorem with respect to this positive operator-valued Measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric.

Zhiren Wang - One of the best experts on this subject based on the ideXlab platform.

  • Measure complexity and mobius disjointness
    Advances in Mathematics, 2019
    Co-Authors: Wen Huang, Zhiren Wang
    Abstract:

    Abstract In this paper, the notion of Measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's Mobius disjointness conjecture holds for any system for which every invariant Borel Probability Measure has sub-polynomial Measure complexity. We then apply this result to a number of situations, including certain systems whose invariant Measure don't all have discrete spectrum.

Wen Huang - One of the best experts on this subject based on the ideXlab platform.

  • a minimal distal map on the torus with sub exponential Measure complexity
    Ergodic Theory and Dynamical Systems, 2020
    Co-Authors: Wen Huang
    Abstract:

    In this paper the notion of sub-exponential Measure complexity for an invariant Borel Probability Measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential Measure complexity is constructed.

  • Measure complexity and mobius disjointness
    Advances in Mathematics, 2019
    Co-Authors: Wen Huang, Zhiren Wang
    Abstract:

    Abstract In this paper, the notion of Measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's Mobius disjointness conjecture holds for any system for which every invariant Borel Probability Measure has sub-polynomial Measure complexity. We then apply this result to a number of situations, including certain systems whose invariant Measure don't all have discrete spectrum.

Ugolini Sara - One of the best experts on this subject based on the ideXlab platform.

  • Toward a Probability theory for product logic: states, integral representation and reasoning
    'Elsevier BV', 2018
    Co-Authors: Flaminio Tommaso, Godo Lluis, Ugolini Sara
    Abstract:

    The aim of this paper is to extend Probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive Probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel Probability Measure. Furthermore, the relation between states and Measures is shown to be one-one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the Measure is a state in the above sense.Comment: 27 pages, 1 figur

  • Towards a Probability theory for product logic: States, integral representation and reasoning
    'Elsevier BV', 2018
    Co-Authors: Flaminio Tommaso, Godo Lluis, Ugolini Sara
    Abstract:

    The aim of this paper is to extend Probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive Probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel Probability Measure. Furthermore, the relation between states and Measures is shown to be one–one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the Measure is a state in the above sense. © 2017 Elsevier Inc.The authors acknowledge partial support by the SYSMICS project ( EU H2020-MSCA-RISE-2015 Project 689176). Also, Flaminio and Godo acknowledge partial support by the FEDER/MINECO project TIN2015-71799-C2-1-PPeer Reviewe