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The Generalized pascal triangle and the Matrix Variate Jensen-Logistic Distribution

COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2015年第13期44卷 2738-2752页

作者： Caro-Lopera, Francisco J. Gonzalez-Farias, Graciela Balakrishnan, N. Univ Medellin Dept Ciencias Basicas Medellin Colombia Ctr Invest Matemat Monterrey Mexico McMaster Univ Dept Math & Stat Hamilton ON Canada

This article defines the so called Generalized Matrix Variate Jensen-Logistic distribution. The relevant applications of this class of distributions in Configuration Shape Theory consist of a more efficient computation, supported by the corresponding inference. This demands the solution of two important problems: (1) the development of analytical and efficient formulae for their k-th derivatives and (2) the use of the derivatives to transform the configuration density into a polynomial density under some special matrix Kummer relation, indexed in this case by the Jensen-Logistic kernel. In this article, we solve these problems by deriving a simple formula for the k-th derivative of the density function, avoiding the usual partition theory framework and using a generalization of pascal triangles. Then we apply the results by obtaining the associated Jensen-Logistic Kummer relations and the configuration polynomial density in the setting of Statistical Shape Theory.

关键词： Jensen-Logistic distribution pascal triangle Statistical shape theory Zonal polynomials Generalized Kummer relations

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Deep Convolutional Neural Network for Active Intrusion Detection and Protect data from Passive Intrusion by pascal triangle

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WIRELESS PERSONAL COMMUNICATIONS 2024年 1-14页

作者： Alqahtani, Abdulrahman Saad Univ Bisha Coll Comp & Informat Technol Bisha Saudi Arabia

Active and passive intrusion are the two types of intrusion. The active intrusion attempts to modify the data and the passive intrusion observes the data and replica. This intrusion creates big damage and congestion in the network. Deep learning techniques have been extensively used to advance intrusion detection systems, which can rapidly and effectively identify and classify intrusions at various levels of networks. These technologies are capable of properly and swiftly identifying threats. Networks, however, require a sophisticated security solution due to the frequent emergence and evolution of hostile threats. Publicly accessible intrusion databases must be updated often due to the intricacy of attacks and constantly varying detection. To solve these issues, Deep Convolutional Neural Network for Active intrusion detection and Protect data from Passive Intrusion by pascal triangle is introduced. This work proposes a convolutional neural network (CNN) based network intrusion detection model with five convolutional layers. Furthermore, this approach uses the pascal triangle method, making the dummy route for protecting the data from passive intrusion. This model is evaluated for binary and multiclass classification using the CICIDS2018 dataset, a publicly available dataset comprising 80 statistical features. The dataset is preprocessed by data transformation and numerical standardization techniques. To evaluate the performance of the suggested system, experiments are carried out. The research findings show that the proposed CNN performs better at detecting multiclass categorization than existing intrusion detection methods, with average accuracy, precision, recall, and F1-score values of 99.16, 99.20, 99.63, and 98.76%, respectively.

关键词： Deep learning Intrusion detection Convolutional neural network pascal triangle CICIDS2018 Multiclass classification Accuracy

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Numerical problems with the pascal triangle in moment computation

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2016年 306卷 53-68页

作者： Kautsky, Jaroslav Flusser, Jan Flinders Univ South Australia CSEM GPO Box 2100 Adelaide SA 5001 Australia Acad Sci Czech Republic Inst Informat Theory & Automat Vodarenskou Vezi 4 CR-18208 Prague 8 Czech Republic

Moments are important characteristics of digital signals and images and are commonly used for their description and classification. When calculating the moments and their derived functions numerically, we face, among other numerical problems studied in the literature, certain instabilities which are connected with the properties of pascal triangle. The pascal triangle appears in moment computation in various forms whenever we have to deal with binomial powers. In this paper, we investigate the reasons for these instabilities in three particular cases central moments, complex moments, and moment blur invariants. While in the first two cases this phenomenon is tolerable, in the third one it causes serious numerical problems. We analyze these problems and show that they can be partially overcome by choosing an appropriate polynomial basis. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Stable calculations Polynomials Moments pascal triangle Orthogonal moments Moment invariants

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Multiperfect numbers on lines of the pascal triangle

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JOURNAL OF NUMBER THEORY 2009年第5期129卷 1136-1148页

作者： Luca, Florian Luis Varona, Juan Univ La Rioja Dpto Matemat & Computac Logrono 26004 Spain Univ Nacl Autonoma Mexico Inst Matemat Morelia 58089 Michoacan Mexico

A number n is said to be multiperfect (or multiply perfect) if n divides its sum of divisors sigma(n). In this paper, we study the multiperfect numbers on straight lines through the pascal triangle. Except for the lines parallel to the edges, we show that all other lines through the pascal triangle contain at most finitely many multiperfect numbers. We also study the distribution of the numbers sigma(n)/n whenever the positive integer n ranges through the binomial coefficients on a fixed line through the pascal triangle. (C) 2008 Elsevier Inc. All rights reserved.

关键词： Multiperfect Multiply perfect Perfect Sum of divisors pascal triangle Binomial coefficients Catalan numbers

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THE NUMBER OF PATHS IN BOUNDARY RESTRICTED pascal triangle

THE NUMBER OF PATHS IN BOUNDARY RESTRICTED PASCAL TRIANGLE

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Information Theory and Applications Workshop

作者： Kobayashi, Kingo Sato, Hajime Hoshi, Mamoru Univ Electrocommun Tokyo Japan Senshu Univ Tokyo Japan

ISBN: (纸本)9781509025299

In this talk, we provide several forms of the generator for counting the number of paths in the boundary restricted pascal triangle. In the section 1, we consider the problem of single boundary. In the section 2, we proceed the discussion into the problem of two boundaries. Specifically, we give three seemingly different forms to count the number of paths. If time permits, we will give the proof of the equivalence between them. The study treated in the paper is a continuation of papers [2, 3].

关键词： random walk symbolic method pascal triangle Dyck path Fibonacci polynomial Chebyshev polynomial of the second kind

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The pascal triangle (1654), the Reed-Muller-Fourier Transform (1992), and the Discrete pascal Transform (2005) 46

The Pascal triangle (1654), the Reed-Muller-Fourier Transfor...

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46th IEEE International Symposium on Multiple-Valued Logic (ISMVL)

作者： Moraga, Claudio Stankovic, Radomir Stankovic, Milena TU Dortmund Univ Fac Comp Sci Chair 1 D-44227 Dortmund Germany Univ Nis Fac Elect Engn Nish Serbia

ISBN: (纸本)9781467394888

This paper makes a theoretical comparative analysis of the Reed-Muller-Fourier Transform, pascal matrices based on the pascal triangle, and the Discrete pascal Transform. The Reed-Muller-Fourier Transform was not originated by a pascal matrix, however it happens to show a strong family resemblance with it, sharing several basic properties. Its area of application is the multiple-valued switching theory, mainly to obtain polynomial expressions from the value vector of multiple-valued functions. The Discrete pascal Transform was introduced over a decade later, based on an ad hoc modification of a pascal matrix, for applications on picture processing. It is however shown that a Discrete pascal Transform of size p, taken modulo p equals the special Reed-Muller-Fourier Transform for the same p and n = 1. The Sierpinski fractal is close related to the pascal matrix. Data structures based on the Sierpinski triangle have been successfully used to solve special problems in switching theory. Some of them will be addressed in the paper.

关键词： pascal triangle pascal matrices Reed-Muller-Fourier Transform Discrete pascal Transform Fractals

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Distribution of the extreme values of the number of ones in Boolean analogues of the pascal triangle

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DISCRETE MATHEMATICS AND APPLICATIONS 2017年第3期27卷 149-176页

作者： Malyshev, Fedor M. Russian Acad Sci Steklov Math Inst Moscow Russia

The paper is concerned with estimating the number xi of ones in triangular arrays consisting of elements of the field GF (2) which are defined by the bottom row of s elements. The elements of each higher row are obtained (as in pascal triangles) by the summation of pairs of elements from the corresponding lower row. It is shown that there exists a monotone unbounded sequence 0 = k(0) < k(1) < k(2) < ... of rational numbers such that, for any k > 0, for sufficiently large.. the admissible values of xi which are smaller than.. or larger than s (s + 1)/3 - sk/3 are concentrated in neighbourhoods of points k(i)s and s(s + 1)/3 - sk(i)/3, i >= 0. The resulting estimates of the neighbourhoods are functions of i for each i >= 0 and do not depend on s. The distributions of the numbers of triangles with values xi in these neighbourhoods depend only on the residues of s with respect to moduli that depend on i >= 0.

关键词： pascal triangle (0-1)-matrix extreme combinatorial configuration

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A multiple-scale pascal polynomial triangle solving elliptic equations and inverse Cauchy problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2016年 62卷 35-43页

作者： Liu, Chein-Shan Kuo, Chung-Lun Natl Taiwan Univ Dept Civil Engn Taipei 10617 Taiwan

The polynomial expansion method is a useful tool to solve partial differential equations (PDEs). However, the researchers seldom use it as a major medium to solve PDEs due to its highly ill-conditioned behavior. We propose a single-scale and a multiple-scale pascal triangle formulations to solve the linear elliptic PDEs in a simply connected domain equipped with complex boundary shape. For the former method a constant parameter R-0 is required, while in the latter one all introduced scales are automatically determined by the collocation points. Then we use the multiple-scale method to solve the inverse Cauchy problems, which is very accurate and very stable against large noise to 20%. Numerical results confirm the validity of the present multiple-scale pascal polynomial expansion method. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Multi-scale polynomial expansion pascal triangle Elliptic PDEs Inverse Cauchy problems

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Fabrication of pascal-triangle Lattice of Proteins by Inducing Ligand Strategy

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ANGEWANDTE CHEMIE-INTERNATIONAL EDITION 2020年第24期59卷 9617-9623页

作者： Liu, Rongying Kochovski, Zdravko Li, Long Yin, Yue-wen Yang, Jing Yang, Guang Tao, Guoqing Xu, Anqiu Zhang, Ensong Ding, Hong-ming Lu, Yan Chen, Guosong Jiang, Ming Fudan Univ State Key Lab Mol Engn Polymers Shanghai 200433 Peoples R China Fudan Univ Dept Macromol Sci Shanghai 200433 Peoples R China Helmholtz Zentrum Berlin Mat & Energie Inst Electrochem Energy Storage D-14109 Berlin Germany Soochow Univ Sch Phys Sci & Technol Ctr Soft Condensed Matter Phys & Interdisciplinar Suzhou 215006 Peoples R China Univ Potsdam Inst Chem D-14476 Potsdam Germany Fudan Univ Multiscale Res Inst Complex Syst Shanghai 200433 Peoples R China

A protein pascal triangle has been constructed as new type of supramolecular architecture by using the inducing ligand strategy that we previously developed for protein assemblies. Although mathematical studies on this famous geometry have a long history, no work on such pascal triangles fabricated from native proteins has been reported so far due to their structural complexity. In this work, by carefully tuning the specific interactions between the native protein building block WGA and the inducing ligand R-SL, a 2D pascal-triangle lattice with three types of triangular voids has been assembled. Moreover, a 3D crystal structure was obtained based on the 2D pascal triangles. The distinctive carbohydrate binding sites of WGA and the intralayer as well as interlayer dimerization of RhB was the key to facilitate nanofabrication in solution. This strategy may be applied to prepare and explore various sophisticated assemblies based on native proteins.

关键词： inducing ligand pascal triangle self-assembly wheat germ agglutinin

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Inequality-To-Inequality Zhang Equivalency(ZE) for Integrals of 0,1,2,3,...,Times Illustrating pascal triangle

Inequality-To-Inequality Zhang Equivalency(ZE) for Integrals...

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第34届中国控制与决策会议

作者： Yunong Zhang Wuyi Yang Zhenyu Li School of Computer Science and Engineering Sun Yat-sen University Guangdong Key Laboratory of Modern Control Technology School of Artificial Intelligence Sun Yat-sen University

By applying equivalency,it is easier to solve original problems at different aspects and *** equivalency(ZE) has been originally proposed for equalities(i.e.,equations),which means that one can solve original equations at different ***,in order to satisfy the requirements of practical applications,inequality ZE has been proposed,which supplements and develops the concept of *** this study,the concept of inequality ZE is *** order-n inequality ZE is proposed and proved by mathematical *** applying inequality ZE,the differentorder ZE formulas of 0,1,2,3,…,times integral inequalities are *** the process of applying inequality ZE,we find that the coefficients of inequality-to-inequality ZE formulas are consistent with the rows of pascal ***,some theoretics are proposed to reveal the *** the help of pascal triangle,we can quickly construct any order inequality ZE formula to solve various problems.

关键词： Zhang equivalency(ZE) inequality to inequality pascal triangle mathematical induction

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