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Journal articles on the topic 'Minimal nonnegative solution'

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Author: Grafiati

Published: 10 March 2023

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1

Guo, Pei-Chang. "Newton-Shamanskii Method for a Quadratic Matrix Equation Arising in Quasi-Birth-Death Problems." East Asian Journal on Applied Mathematics 4, no. 4 (November 2014): 386–95. http://dx.doi.org/10.4208/eajam.040914.301014a.

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AbstractIn order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.

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2

Guo, Pei-Chang. "A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/1-Type Markov Chains." Mathematical Problems in Engineering 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/4018239.

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For the nonlinear matrix equations arising in the analysis of M/G/1-type and GI/M/1-type Markov chains, the minimal nonnegative solution G or R can be found by Newton-like methods. We prove monotone convergence results for the Newton-Shamanskii iteration for this class of equations. Starting with zero initial guess or some other suitable initial guess, the Newton-Shamanskii iteration provides a monotonically increasing sequence of nonnegative matrices converging to the minimal nonnegative solution. A Schur decomposition method is used to accelerate the Newton-Shamanskii iteration. Numerical examples illustrate the effectiveness of the Newton-Shamanskii iteration.

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3

Garić-Demirović, M., M. R. S. Kulenović, and M. Nurkanović. "Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation." Scientific World Journal 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/210846.

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We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the formxn+1=xn-12/(axn2+bxnxn-1+cxn-12),n=0,1,2,…,where the parameters a, b, and c are positive numbers and the initial conditionsx-1andx0are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.

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4

Ivanov, Ivan G. "ITERATIVE COMPUTING THE MINIMAL SOLUTION OF THE COUPLED NONLINEAR MATRIX EQUATIONS IN TERMS OF NONNEGATIVE MATRICES." Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application 12, no. 1-2 (2020): 226–37. http://dx.doi.org/10.56082/annalsarscimath.2020.1-2.226.

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We investigate a set of nonlinear matrix equations with nonnegative matrix coefficients which has arisen in applied sciences. There are papers where the minimal nonnegative solution of the set of nonlinear matrix equations is computed applying the different procedures. The alternate linear implicit method and its modifications have intensively investigated because they have simple computational scheme. We construct a new decoupled modification of the alternate linear implicit procedure to compute the minimal nonnegative solution of the considered set of equations. The convergence properties of the proposed iteration are derived and a sufficient condition for convergence is derived. The performance of the proposed algorithm is illustrated on several numerical examples. On the basis of the experiments we derive conclusions for applicability of the computational schemes.

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5

Hammoudi, Alaaeddine, Oana Iosifescu, and Martial Bernoux. "Mathematical analysis of a spatially distributed soil carbon dynamics model." Analysis and Applications 15, no. 06 (August 2, 2017): 771–93. http://dx.doi.org/10.1142/s0219530516500081.

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The aim of this paper is to study the mathematical properties of a new model of soil carbon dynamics which is a reaction–diffusion–advection system with a quadratic reaction term. This is a spatial version of Modeling Organic changes by Micro-Organisms of Soil model, recently introduced by M. Pansu and his group. We show here that for any nonnegative initial condition, there exists a unique nonnegative weak solution. Moreover, if we assume time periodicity of model entries, taking into account seasonal effects, we prove existence of a minimal and a maximal periodic weak solution. In a particular case, these two solutions coincide and they become a global attractor of any bounded solution of the periodic system.

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6

Zhang, Xuemei, Xiaozhong Yang, and Meiqiang Feng. "Minimal Nonnegative Solution of Nonlinear Impulsive Differential Equations on Infinite Interval." Boundary Value Problems 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/684542.

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7

Ivanshin, Pyotr. "Functions of Minimal Norm with the Given Set of Fourier Coefficients." Mathematics 7, no. 7 (July 20, 2019): 651. http://dx.doi.org/10.3390/math7070651.

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We prove the existence and uniqueness of the solution of the problem of the minimum norm function ∥ · ∥ ∞ with a given set of initial coefficients of the trigonometric Fourier series c j , j = 0 , 1 , … , 2 n . Then, we prove the existence and uniqueness of the solution of the nonnegative function problem with a given set of coefficients of the trigonometric Fourier series c j , j = 1 , … , 2 n for the norm ∥ · ∥ 1 .

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8

Guo, Chun-Hua. "A note on the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation." Linear Algebra and its Applications 357, no. 1-3 (December 2002): 299–302. http://dx.doi.org/10.1016/s0024-3795(02)00431-7.

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9

Zhang, Lihong, Bashir Ahmad, and Guotao Wang. "Monotone iterative method for a class of nonlinear fractional differential equations on unbounded domains in Banach spaces." Filomat 31, no. 5 (2017): 1331–38. http://dx.doi.org/10.2298/fil1705331z.

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In this paper, we investigate the existence of minimal nonnegative solution for a class of nonlinear fractional integro-differential equations on semi-infinite intervals in Banach spaces by applying the cone theory and the monotone iterative technique. An example is given for the illustration of main results.

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10

Miyajima, Shinya. "Fast verified computation for the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation." Computational and Applied Mathematics 37, no. 4 (February 13, 2018): 4599–610. http://dx.doi.org/10.1007/s40314-018-0590-x.

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11

He, Qi-Ming, and Marcel F. Neuts. "On the convergence and limits of certain matrix sequences arising in quasi-birth-and-death Markov chains." Journal of Applied Probability 38, no. 2 (June 2001): 519–41. http://dx.doi.org/10.1239/jap/996986760.

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We study the convergence of certain matrix sequences that arise in quasi-birth-and-death (QBD) Markov chains and we identify their limits. In particular, we focus on a sequence of matrices whose elements are absorption probabilities into some boundary states of the QBD. We prove that, under certain technical conditions, that sequence converges. Its limit is either the minimal nonnegative solution G of the standard nonlinear matrix equation, or it is a stochastic solution that can be explicitly expressed in terms of G. Similar results are obtained relative to the standard matrix R that arises in the matrix-geometric solution of the QBD. We present numerical examples that clarify some of the technical issues of interest.

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12

He, Qi-Ming, and Marcel F. Neuts. "On the convergence and limits of certain matrix sequences arising in quasi-birth-and-death Markov chains." Journal of Applied Probability 38, no. 02 (June 2001): 519–41. http://dx.doi.org/10.1017/s0021900200020015.

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We study the convergence of certain matrix sequences that arise in quasi-birth-and-death (QBD) Markov chains and we identify their limits. In particular, we focus on a sequence of matrices whose elements are absorption probabilities into some boundary states of the QBD. We prove that, under certain technical conditions, that sequence converges. Its limit is either the minimal nonnegative solution G of the standard nonlinear matrix equation, or it is a stochastic solution that can be explicitly expressed in terms of G. Similar results are obtained relative to the standard matrix R that arises in the matrix-geometric solution of the QBD. We present numerical examples that clarify some of the technical issues of interest.

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13

Seo, Sang-hyup, Jong-Hyeon Seo, and Hyun-Min Kim. "Convergence of a modified Newton method for a matrix polynomial equation arising in stochastic problem." Electronic Journal of Linear Algebra 34 (February 21, 2018): 500–513. http://dx.doi.org/10.13001/1081-3810.3762.

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The Newton iteration is considered for a matrix polynomial equation which arises in stochastic problem. In this paper, it is shown that the elementwise minimal nonnegative solution of the matrix polynomial equation can be obtained using Newton's method if the equation satisfies the sufficient condition, and the convergence rate of the iteration is quadratic if the solution is simple. Moreover, it is shown that the convergence rate is at least linear if the solution is non-simple, but a modified Newton method whose iteration number is less than the pure Newton iteration number can be applied. Finally, numerical experiments are given to compare the effectiveness of the modified Newton method and the standard Newton method.

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14

Litovchenko, V. A., and G. M. Unguryan. "Parabolic systems of Shilov-type with coefficients of bounded smoothness and nonnegative genus." Carpathian Mathematical Publications 9, no. 1 (June 8, 2017): 72–85. http://dx.doi.org/10.15330/cmp.9.1.72-85.

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The Shilov-type parabolic systems are parabolically stable systems for changing its coefficients unlike of parabolic systems by Petrovskii. That's why the modern theory of the Cauchy problem for class by Shilov-type systems is developing abreast how the theory of the systems with constant or time-dependent coefficients alone. Building the theory of the Cauchy problem for systems with variable coefficients is actually today. A new class of linear parabolic systems with partial derivatives to the first order by the time $t$ with variable coefficients that includes a class of the Shilov-type systems with time-dependent coefficients and non-negative genus is considered in this work. A main part of differential expression concerning space variable $x$ of each such system is parabolic (by Shilov) expression. Coefficients of this expression are time-dependent, but coefficients of a group of younger members may depend also a space variable. We built the fundamental solution of the Cauchy problem for systems from this class by the method of sequential approximations. Conditions of minimal smoothness on coefficients of the systems by variable $x$ are founded, the smoothness of solution is investigated and estimates of derivatives of this solution are obtained. These results are important for investigating of the correct solution of the Cauchy problem for this systems in different functional spaces, obtaining forms of description of the solution of this problem and its properties.

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15

Liu, Yuanyuan. "Additive Functionals for Discrete-Time Markov Chains with Applications to Birth-Death Processes." Journal of Applied Probability 48, no. 4 (December 2011): 925–37. http://dx.doi.org/10.1239/jap/1324046010.

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In this paper we are interested in bounding or calculating the additive functionals of the first return time on a set for discrete-time Markov chains on a countable state space, which is motivated by investigating ergodic theory and central limit theorems. To do so, we introduce the theory of the minimal nonnegative solution. This theory combined with some other techniques is proved useful for investigating the additive functionals. This method is used to study the functionals for discrete-time birth-death processes, and the polynomial convergence and a central limit theorem are derived.

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16

Liu, Yuanyuan. "Additive Functionals for Discrete-Time Markov Chains with Applications to Birth-Death Processes." Journal of Applied Probability 48, no. 04 (December 2011): 925–37. http://dx.doi.org/10.1017/s0021900200008536.

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In this paper we are interested in bounding or calculating the additive functionals of the first return time on a set for discrete-time Markov chains on a countable state space, which is motivated by investigating ergodic theory and central limit theorems. To do so, we introduce the theory of the minimal nonnegative solution. This theory combined with some other techniques is proved useful for investigating the additive functionals. This method is used to study the functionals for discrete-time birth-death processes, and the polynomial convergence and a central limit theorem are derived.

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17

Hautphenne, Sophie. "Extinction Probabilities of Supercritical Decomposable Branching Processes." Journal of Applied Probability 49, no. 3 (September 2012): 639–51. http://dx.doi.org/10.1239/jap/1346955323.

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We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

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18

Hautphenne, Sophie. "Extinction Probabilities of Supercritical Decomposable Branching Processes." Journal of Applied Probability 49, no. 03 (September 2012): 639–51. http://dx.doi.org/10.1017/s0021900200009438.

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We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

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19

Fu, Zufeng, and Daoyun Xu. "Uniquely Satisfiable d-Regular (k,s)-SAT Instances." Entropy 22, no. 5 (May 19, 2020): 569. http://dx.doi.org/10.3390/e22050569.

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Unique k-SAT is the promised version of k-SAT where the given formula has 0 or 1 solution and is proved to be as difficult as the general k-SAT. For any k ≥ 3 , s ≥ f ( k , d ) and ( s + d ) / 2 > k − 1 , a parsimonious reduction from k-CNF to d-regular (k,s)-CNF is given. Here regular (k,s)-CNF is a subclass of CNF, where each clause of the formula has exactly k distinct variables, and each variable occurs in exactly s clauses. A d-regular (k,s)-CNF formula is a regular (k,s)-CNF formula, in which the absolute value of the difference between positive and negative occurrences of every variable is at most a nonnegative integer d. We prove that for all k ≥ 3 , f ( k , d ) ≤ u ( k , d ) + 1 and f ( k , d + 1 ) ≤ u ( k , d ) . The critical function f ( k , d ) is the maximal value of s, such that every d-regular (k,s)-CNF formula is satisfiable. In this study, u ( k , d ) denotes the minimal value of s such that there exists a uniquely satisfiable d-regular (k,s)-CNF formula. We further show that for s ≥ f ( k , d ) + 1 and ( s + d ) / 2 > k − 1 , there exists a uniquely satisfiable d-regular ( k , s + 1 ) -CNF formula. Moreover, for k ≥ 7 , we have that u ( k , d ) ≤ f ( k , d ) + 1 .

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20

Nikolopouls, P. V., and V. P. Sreedharan. "An algorithm for computing nonnegative minimal norm solutions." Numerical Functional Analysis and Optimization 15, no. 1-2 (January 1994): 87–103. http://dx.doi.org/10.1080/01630569408816552.

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21

Guo, Dajun. "Minimal nonnegative solutions for nth order integro-differential equations in Banach spaces." Applied Mathematics and Computation 113, no. 1 (July 2000): 55–65. http://dx.doi.org/10.1016/s0096-3003(99)00069-7.

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22

Guo, Dajun. "Terminal value problems of impulsive integro-differential equations in Banach spaces." Journal of Applied Mathematics and Stochastic Analysis 10, no. 1 (January 1, 1997): 71–78. http://dx.doi.org/10.1155/s1048953397000075.

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This paper uses cone theory and the monotone iterative technique to investigate the existence of minimal nonnegative solutions of terminal value problems for first order nonlinear impulsive integro-differential equations of mixed type in a Banach space.

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23

Bai, Zhong-Zhi, Xiao-Xia Guo, and Shu-Fang Xu. "Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations." Numerical Linear Algebra with Applications 13, no. 8 (2006): 655–74. http://dx.doi.org/10.1002/nla.500.

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24

Evéquoz, Gilles, and Tolga Yeşil. "Dual ground state solutions for the critical nonlinear Helmholtz equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 3 (January 30, 2019): 1155–86. http://dx.doi.org/10.1017/prm.2018.103.

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AbstractUsing a dual variational approach, we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation $$-\Delta u-k^2u = Q(x) \vert u \vert ^{2^*-2}u,\quad u\in W^{2,2^*}({\open R}^{N})$$for N ⩾ 4, where 2* : = 2N/(N − 2). The coefficient $Q \in L^{\infty }({\open R}^{N}){\setminus }\{0\}$ is assumed to be nonnegative, asymptotically periodic and to satisfy a flatness condition at one of its maximum points. The solutions obtained are so-called dual ground states, that is, solutions arising from critical points of the dual functional with the property of having minimal energy among all nontrivial critical points. Moreover, we show that no dual ground state exists for N = 3.

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25

Ho, Ky, and Inbo Sim. "On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth." Advances in Nonlinear Analysis 12, no. 1 (September 3, 2022): 182–209. http://dx.doi.org/10.1515/anona-2022-0269.

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Abstract In this article, we study the existence of multiple solutions to a generalized p ( ⋅ ) p\left(\cdot ) -Laplace equation with two parameters involving critical growth. More precisely, we give sufficient “local” conditions, which mean that growths between the main operator and nonlinear term are locally assumed for p ( ⋅ ) p\left(\cdot ) -sublinear, p ( ⋅ ) p\left(\cdot ) -superlinear, and sandwich-type cases. Compared to constant exponent problems (e.g., p p -Laplacian and ( p , q ) \left(p,q) -Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the mountain pass theorem for p ( ⋅ ) p\left(\cdot ) -sublinear and p ( ⋅ ) p\left(\cdot ) -superlinear cases and constructing critical values defined by a minimax argument in the genus theory for sandwich-type case. Moreover, we also obtain a nontrivial nonnegative solution for sandwich-type case changing the role of parameters. Our work is a generalization of several existing works in the literature.

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26

Mel'nikov, A. K. "ABOUT CHOOSING THE METHOD OF EXACT APPROXIMATIONS OF DISCRETE STATISTICS PROBABILITY DISTRIBUTIONS." Vestnik komp'iuternykh i informatsionnykh tekhnologii, no. 204 (June 2021): 39–48. http://dx.doi.org/10.14489/vkit.2021.06.pp.039-048.

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In the paper, we consider the methods of exact approximations of statistics probabilities distribution. As the exact approximations, we consider ∆-exact distributions. The difference between the ∆-exact distributions and the exact approximations does not exceed a predefined arbitrary small value ∆ that defines the accuracy of the approximations. Besides, we consider the methods of the first and second multiplicity, which use statistic characteristics of samples. The first multiplicity method is based on the properties of the components of the first multiplicity vector, which are nonnegative integer solutions of a linear equation. The linear equation relates the alphabet sign frequency and the sample size. The second multiplicity method is based on the solution of a system of linear equations. The linear equations of the system relate the sample size and the alphabet cardinality with the number of the alphabet signs that have equal frequency in the sample. For the considered methods of exact approximations, we give expressions to estimate the computational complexity of exact approximations of distributions for any sample parameters. To provide the approximations accuracy of 10–5, and the computing resource with the performance of 1018 operations per second, we calculated the sample parameters. For these samples, we can calculate the exact approximations of distributions, using the considered methods, the available computing resource, and the declared accuracy. We formed the parameter regions for the samples, and the exact approximations of distributions can be calculated for these samples with the help of various methods. We compared the regions themselves and with the so-called region of uncertainty, which is limited from above not more than 5-fold excess of the sample size over the alphabet cardinality. On the base of the comparison of the parameter regions of the samples, which are suitable for calculation of the exact approximations of the distributions, we compared their calculation methods. It is shown that owing to the second multiplicity method, we can make calculations for all values of the alphabet cardinality from 2 to 256. In contrast to the second multiplicity method, the first multiplicity method does not allow calculations for the alphabet cardinality over 73. The parameter region of the samples, which are suitable for calculation of the limit approximations of the distributions by the second multiplicity method, contains the complete parameter region of the samples, suitable for calculation of the limit approximations of the distributions by the first multiplicity method, and exceeds it more than in 52 times. Owing to the comparison of the methods of exact approximations, it is proved that if we have the same computing resource, we can calculate the exact approximations with the help of the second multiplicity method for a greater number of samples with the increased parameters in comparison with the first multiplicity method. Hence, to calculate the exact approximations of statistics probability distributions, we choose the second multiplicity method. Practical significance of the research is possibility of calculation of the maximal values of the sample parameters. The current technological level of computer systems allows calculation of the exact approximations of the distributions for these values, which provide the minimal loss of criteria efficiency in comparison with the limit approximations used for the sample parameters. The scientific novelty of the research is the comparative analysis of the methods of exact approximations of distributions for calculation of distributions for the sample parameters, which do not allow calculation of the exact distributions due to their high computational complexity.

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27

Rüschendorf, Ludger. "On stochastic recursive equations of sum and max type." Journal of Applied Probability 43, no. 3 (September 2006): 687–703. http://dx.doi.org/10.1239/jap/1158784939.

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In this paper we consider stochastic recursive equations of sum type, , and of max type, , where Ai, bi, and b are random, (Xi) are independent, identically distributed copies of X, and denotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimal Ls-metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.

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28

Kuang, Juhong, Weiyi Chen, and Zhiming Guo. "Periodic solutions with prescribed minimal period for second order even Hamiltonian systems." Communications on Pure & Applied Analysis, 2021, 0. http://dx.doi.org/10.3934/cpaa.2021166.

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<p style='text-indent:20px;'>In this paper, we develop a new method to study Rabinowitz's conjecture on the existence of periodic solutions with prescribed minimal period for second order even Hamiltonian system without any convexity assumptions. Specifically, we first study the associated homogenous Dirichlet boundary value problems for the discretization of the Hamiltonian system with given step length and obtain a sequence of nonnegative solutions corresponding to different step lengths by using discrete variational methods. Then, using the sequence of nonnegative solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, by utilizing the limit function of convergent subsequence and the symmetry of the potential, we will obtain the desired periodic solution. In particular, we prove Rabinowitz's conjecture in the case when the potential satisfies a certain symmetric assumption. Moreover, our main result greatly improves the related results in the literature in the case where <inline-formula><tex-math id="M1">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>.</p>

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29

Miyajima, Shinya. "Fast enclosure for the minimal nonnegative solution to the nonsymmetric T-Riccati equation." Calcolo 59, no. 3 (July 19, 2022). http://dx.doi.org/10.1007/s10092-022-00475-4.

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30

Chacha, Chacha Stephen, and Hyun-Min Kim. "Elementwise Minimal Nonnegative Solutions for a Class of Nonlinear Matrix Equations." East Asian Journal on Applied Mathematics 9, no. 4 (2019). http://dx.doi.org/10.4208/eajam.300518.120119.

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