A Markov-Chain-Based Model for Group Message Distribution in Connected Networks

The authors introduce a stochastic Markov-chain-based model for the recursive distribution of a message from the source node to all remaining nodes. This recursive message distribution process produces a spanning tree topology over the connected network of nodes. The model has five input parameters: (1) the number of nodes n in the group, (2) the maximum number of child nodes, (3) the number of sub-message components needed to transfer a single message, (4) the probability p1 that two adjacent nodes in a network initiate a connection (edge) in the spanning tree, and (5) the probability p2 that each sub-message component is transferred correctly between nodes. The authors derive a closed-form expression for the expected group message distribution time, measured in discrete-time epochs, that is verified via Monte Carlo simulations. Since both the closed-form formulas and the Monte Carlo simulations are computationally intensive for networks with a large number of nodes n, this paper derives a reliable approximate formula for the expected distribution time for networks as large as n = 1000.

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Noniterative Three-Dimensional Location Estimation Using Azimuth and Elevation Measurements at Multiple Locations

Journal of Sensors ◽

10.1155/2019/3743475 ◽

2019 ◽

Vol 2019 ◽

pp. 1-13

Author(s):

Ji Woong Paik ◽

Joon-Ho Lee

Keyword(s):

Closed Form ◽

Three Dimensional ◽

Location Estimation ◽

Closed Form Expression ◽

Sensor Location ◽

Form Expression ◽

Emitter Location ◽

Initial Location ◽

Computationally Intensive ◽

Location Estimate

Closed-form expression of three-dimensional emitter location estimation using azimuth and elevation measurements at multiple locations is presented in this paper. The three-dimensional location estimate is obtained from three-dimensional sensor locations and the azimuth and elevation measurements at each sensor location. Since the formulation is not iterative, it is not computationally intensive and does not need initial location estimate. Numerical results are presented to show the validity of the proposed scheme.

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Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽

10.1109/access.2021.3096419 ◽

2021 ◽

pp. 1-1

Author(s):

Yassine Zouaoui ◽

Larbi Talbi ◽

Khelifa Hettak ◽

Naresh K. Darimireddy

Keyword(s):

Closed Form ◽

Closed Form Expression ◽

Form Expression

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Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

ACM SIGMETRICS Performance Evaluation Review ◽

10.1145/3453953.3453974 ◽

2021 ◽

Vol 48 (3) ◽

pp. 91-96

Author(s):

Shigeo Shioda

Keyword(s):

Distribution Function ◽

Probability Density Function ◽

Closed Form ◽

Probability Density ◽

Infinite Number ◽

Stable Distribution ◽

Random Variable ◽

Cauchy Distribution ◽

Closed Form Expression ◽

Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

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Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

Journal of High Energy Physics ◽

10.1007/jhep06(2021)063 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Vivek Kumar Singh ◽

Rama Mishra ◽

P. Ramadevi

Keyword(s):

Closed Form ◽

Topological Strings ◽

Chebyshev Polynomials ◽

Fibonacci Numbers ◽

Infinite Family ◽

Closed Form Expression ◽

Two Dimensional ◽

Laurent Polynomials ◽

Form Expression ◽

Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

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