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On the asymptotic spectral distribution of increasing size matrices: test functions, spectral clustering, and asymptotic estimates of outliers

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LINEAR ALGEBRA AND ITS APPLICATIONS 2024年 697卷 615-638页

作者： Bianchi, Davide Garoni, Carlo Sun Yat Sen Univ Sch Math Zhuhai Zhuhai 519082 Peoples R China Univ Roma Tor Vergata Dept Math I-00133 Rome Italy

Let {A(n)}(n) be a sequence of square matrices such that size(A(n)) = d(n) -> infinity as n -> infinity. We say that {A(n)}(n) has an asymptotic spectral distribution described by a Lebesgue measurable function f: D subset of R-k -> C if, for every continuous function F: C -> C with bounded support, lim(n ->infinity) 1/d(n) Sigma(dn)(i=1) F(lambda(i)(A(n))) = 1/mu(k)(D) integral F-D(f(x))dx, where mu(k) is the Lebesgue measure in R-k and lambda(1)(A(n)),..., lambda(dn)(A(n)) are the eigenvalues of A(n). In the last decades, the asymptotic spectral distribution of increasing size matrices has become a subject of investigation by several authors. Special attention has been devoted to matrices A(n) arising from the discretization of differential equations, whose size d(n) diverges to infinity along with the mesh-fineness parameter n. However, despite the popularity that the topic has reached nowadays, the role of the so-called test functions F appearing in the above limit relation has been inexplicably neglected so far. In particular, a natural question such as "Which is the largest set of test functions F for which the above limit reis satisfied?" is still unanswered. In the present paper, we provide a definitive answer to this question by identifying the largest set of test functions F for which the above limit relation is satisfied. We also present some applications of this result to the analysis of spectral clustering, including new asymptotic estimates on the number of outliers. A special attention is devoted to the so-called "inner outliers" of Hermitian block Toeplitz matrices. We conclude the paper with an interpretation of the main result in the context of the vague convergence of probability measures. (c) 2024 The Author(s). Published by Elsevier Inc.

关键词： Asymptotic spectral and singular value distribution test functions Spectral clustering Outliers Hermitian block Toeplitz matrices Vague convergence of probability measures Urysohn's lemma and Tietze's extension theorem

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FACTORIZATION OF functions IN THE SCHUR-AGLER CLASS RELATED TO test functions

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2024年第9期152卷 3991-4001页

作者： Bhowmik, Mainak Kumar, Poornendu Indian Inst Sci Dept Math Bangalore 560012 India Univ Manitoba Dept Math Winnipeg MB R3T 2N2 Canada

We provide necessary and sufficient conditions for operator-valued functions on arbitrary sets associated with a collection of test functions to have factorizations in several situations.

关键词： Colligation extreme points factorization realization formula Schur- Agler class test functions

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A Review of Benchmark and test functions for Global Optimization Algorithms and Metaheuristics

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WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL STATISTICS 2025年第2期17卷

作者： Naser, M. Z. Al-Bashiti, Mohammad Khaled Tapeh, Arash Teymori Gharah Naser, Ahmad Kodur, Venkatesh Hawileeh, Rami Abdalla, Jamal Khodadadi, Nima Gandomi, Amir H. Eslamlou, Armin Dadras Clemson Univ Clemson SC 29634 USA Univ Manitoba Winnipeg MB Canada Michigan State Univ E Lansing MI USA Amer Univ Sharjah Sharjah U Arab Emirates Univ Miami Coral Gables FL USA Univ Technol Sydney Sydney NSW Australia Obuda Univ Budapest Hungary Khazar Univ Baku Azerbaijan Clemson Univ Glenn Dept Civil Engn Clemson SC USA

Benchmarking in optimization is a critical step in evaluating the performance, robustness, and scalability of machine learning algorithms and metaheuristics. While trends in benchmark design continue to evolve, synthetic functions remain vital for fundamental stress tests and theoretical evaluations. As several benchmark and test functions have been developed and derived over the past decades, little attention has been given to classifying such test functions and the rationale behind their usage. From this lens, this paper reviews and categorizes a broad range of functions often employed in assessing optimizers and metaheuristics. More specifically, we classify test functions based on modality, dimensionality, separability, smoothness, constraints, and noise characteristics to offer a broad view that aids in selecting appropriate benchmarks for various algorithmic challenges. Then, this review also discusses in detail the 25 most commonly used functions in the open literature and proposes two new, highly dimensional, dynamic, and challenging functions that could be used for testing new algorithms. Finally, this review identifies gaps in current benchmarking practices and directions for future research, as well as suggests best practices and guidelines.

关键词： artificial intelligence optimization test functions

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Archimedean Godement-Jacquet zeta integrals and test functions

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JOURNAL OF NUMBER THEORY 2018年 191卷 396-426页

作者： Lin, Bingchen Sichuan Univ Sch Math Chengdu 610065 Sichuan Peoples R China

We uniformly find suitable functions, which are called test functions, in the space of joint harmonics of Schrodinger model such that the Godement-Jacquet zeta integrals are equal to the principal L-functions for GL(n)(R). We give the explicit expressions for test functions by using Langlands parameters and harmonic functions. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Zeta integrals Principal L-functions Theta correspondence Joint harmonics Lowest K-types test functions

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The exponential law for spaces of test functions and diffeomorphism groups

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INDAGATIONES MATHEMATICAE-NEW SERIES 2016年第1期27卷 225-265页

作者： Kriegl, Andreas Michor, Peter W. Rainer, Armin Univ Vienna Fak Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria

We prove the exponential law A(E x F, G) congruent to A(E, A(F, G)) (bornological isomorphism) for the following classes A of test functions: B (globally bounded derivatives), W-infinity,W-P (globally p-integrable derivatives), S (Schwartz space), 1) (compact support), B-[M] (globally Denjoy-Carleman), W-[M],W-P (Sobolev-Denjoy-Carleman), S-[L]([M]) (Gelfand-Shilov), and D-[M] (Denjoy-Carleman with compact support). Here E, F, G are convenient vector spaces which are finite dimensional in the cases of D, W-infinity,W-P, D-[M], and W-[M],W-P. Moreover, M = (M-k) is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms Diff B, Diff W-infinity,W-P, Diff S, and Diff D are C-infinity Lie groups, and that Diff B-{M}, Diff W-{M},W-P, Diff S-{L}({M}), and Diff D-{M}, for non-quasianalytic M, are C-{M} Lie groups, where Diff A = {Id +f : f is an element of A(R-n , R-n), inf(x is an element of Rn) det (IIn + df (x)) > 0}. We also discuss stability under composition. (C) 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

关键词： Convenient setting Exponential law test functions Sobolev functions Denjoy-Carleman classes Gelfand-Shilov classes

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A novel adaptive genetic algorithm for global optimization of mathematical test functions and real-world problems

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ENGINEERING SCIENCE AND TECHNOLOGY-AN INTERNATIONAL JOURNAL-JESTECH 2016年第4期19卷 2002-2021页

作者： Mahmoodabadi, M. J. Nemati, A. R. Sirjan Univ Technol Dept Mech Engn Sirjan Iran

Genetic algorithm (GA) is a population-based stochastic optimization technique that has two major problems, i.e. low convergence speed and falling down in local optimum points. This paper introduces an adaptive genetic algorithm (AGA) consisting of new crossover and mutation operators to handle these drawbacks. The crossover operator is based on a combination of the traditional crossover mechanism and the particle swarm optimization (PSO) operator. The proposed mutation operator intelligently uses sliding mode control (SMC) to escape from local minimums and converges to the global optimum. The performance of the proposed genetic algorithm is challenged by using twenty well-known test functions. The comparison of the obtained numerical results with those of the other optimization algorithms reported in literature demonstrates the superiority of the proposed algorithm in finding the global optimum points. At the end, the proposed method is employed to estimate the oil demand in Iran based on socio-economic indicators and using linear and exponential forms as a real-world problem that shows the AGA's effectiveness. (C) 2016 Karabuk University. Publishing services by Elsevier B.V.

关键词： Adaptive genetic algorithm Particle swarm optimization Sliding mode control test functions Oil demand estimation

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Chaotic Aquila Optimization algorithm for solving global optimization and engineering problems

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ALEXANDRIA ENGINEERING JOURNAL 2024年 108卷 135-157页

作者： Gopi, S. Mohapatra, Prabhujit Vellore Inst Technol Sch Adv Sci Dept Math Vellore 632014 Tamil Nadu India

The Aquila Optimization (AO) algorithm is a newly established swarm-based method that mimics the hunting behavior of Aquila birds in nature. However, in complex optimization problems, the AO has shown a slow convergence rate and gets stuck in the local optimal region throughout the optimization process. To overcome this problem, a hybrid with AO and twelve chaotic maps has been proposed to adjust its main parameter. This new mechanism, namely the Chaotic Aquila Optimization (CAO) algorithm, is employed with chaotic maps with the AO algorithm. The proposed chaotic AO (CAO) approach takes seriously a variety of chaotic maps while setting the main AO parameter, which helps in managing exploration and exploitation. To validate the performance of the CAO algorithm, estimates for CEC 2005 and CEC 2022 test functions and the first chaotic map results are compared with the AO algorithm to select the best results of the CAO algorithm, and then CAO results are compared with nine popular optimization algorithms such as FFA, AVOA, MGO, AGTO, SSA, GWO, MVO, SCA, TSA, and AO. Moreover, statistical analyses such as the Wilcoxon rank-sum test and the t-test are performed to analyze the significant difference between the proposed CAO and other algorithms. Furthermore, the proposed CAO has been employed to solve six real-world engineering problems. The results demonstrate the CAO's superiority and capability over other algorithms in solving complex optimization problems. The results demonstrate that CAO achieved outstanding performance and effectiveness in solving an extensive variety optimization problems.

关键词： Aquila optimization algorithm Chaotic maps Chaotic Aquila optimization algorithm test functions Real-life problems

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Mixed solutions and multiple solitary wave solutions for a (3+1)-dimensional nonlinear system

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NONLINEAR DYNAMICS 2024年第23期112卷 21343-21354页

作者： Feng, Qing-Jiang Zhang, Guo-Qing Univ Shanghai Sci & Technol Coll Sci Shanghai 200093 Peoples R China Kaili Univ Coll Sci Kaili 556011 Guizhou Peoples R China

The dynamic behavior of the mixed solution and the multiple solitary wave solutions in a (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli type equation is discussed. Based on the Hirota bilinear method, the lump-kink solution is constructed by introducing a new test function, and the existence conditions for the lump-kink solution are provided. The lump-kink solutions describe two types of interaction phenomena: the lump and the kink move head-on and collide elastically, and the lump chases the kink and collides elastically. In addition, the lump can pass through the kink wave, and the waveform and velocity of these localized waves remain unchanged after the collision. What is more, the innovation of this paper is that the multiple periodic-lump solutions are derived by setting some new test functions. Compared with the single period-lump solution in previous research, the multiple periodic-lump solutions exhibit richer dynamic behavior. Finally, the motion of two types of solitary waves on a periodic wave background is studied. Therefore, studying the elastic collision of lump-kink waves and the interaction of multiple periodic lump waves can help people understand nonlinear systems and solve nonlinear problems in nature.

关键词： Hirota bilinear method test functions Elastic collision Lump wave

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Lump Solution, Breather Soliton and More Soliton Solutions for a (2+1)-Dimensional Generalized Benjamin-Ono Equation

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QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 2023年第2期22卷 1-16页

作者： Ma, Hongcai Yue, Shupan Gao, Yidan Deng, Aiping Donghua Univ Dept Appl Math Shanghai 201620 Peoples R China

Our work is to discuss some new exact solutions of the (2+1)-dimensional generalized Benjamin-Ono (gBO) equation, which describes small amplitude long waves on the surface of shallow water. After careful consideration, lump, breather soliton and new solitons solutions of gBO equation are gained by using the Hirota bilinear method, the test function and the idea of the homogeneous balance, which have not been studied yet. By improving the expression of the test function, the quadratic function with two squares is increased to the quadratic function with three squares, and various lump solutions are obtained. Combining trigonometric function with hyperbolic function, breather soliton is derived. Obviously, sometimes the test function form of tanh-coth and tan-cot methods are so simple that it can't get the desired result. Utilizing the improved tanh-coth and tan-cot methods, whose solutions depend on the parameter n, we can get more soliton solutions. Finally, by determining different parameters, one can draw the three-dimensional plots and density plots at different times. By observing these figures, we analyze the dynamic behavior of (2+1)-dimensional gBO equation in detail. These results can help us to understand nonlinear systems better.

关键词： test functions Lump solution Breather soliton More soliton solutions The improved tanh-coth and tan-cot methods

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Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams

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MECCANICA 2014年第6期49卷 1469-1477页

作者： Sarkar, Korak Ganguli, Ranjan Indian Inst Sci Dept Aerosp Engn Bangalore 560012 Karnataka India

In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.

关键词： Rotating beams Timoshenko beams Free vibration test functions

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