E. Feireisl, M. Lukáčová-Medvid’ová, Simon Schneider, Bangwei She
Applying the concept of S-convergence, based on averaging in the spirit of Strong Law of Large Numbers, the vanishing viscosity solutions of the Euler system are studied. We show how to efficiently compute a viscosity solution of the Euler system as the S-limit of numerical solutions obtained by the viscosity finite volume method. Theoretical results are illustrated by numerical simulations of the Kelvin–Helmholtz instability problem.
{"title":"Approximating viscosity solutions of the Euler system","authors":"E. Feireisl, M. Lukáčová-Medvid’ová, Simon Schneider, Bangwei She","doi":"10.1090/mcom/3738","DOIUrl":"https://doi.org/10.1090/mcom/3738","url":null,"abstract":"Applying the concept of S-convergence, based on averaging in the spirit of Strong Law of Large Numbers, the vanishing viscosity solutions of the Euler system are studied. We show how to efficiently compute a viscosity solution of the Euler system as the S-limit of numerical solutions obtained by the viscosity finite volume method. Theoretical results are illustrated by numerical simulations of the Kelvin–Helmholtz instability problem.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87656411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $zeta_K(s)$ of $K$. More precisely, we show that for $T geq 1$, $$ Big| N_K (T) - frac{T}{pi} log Big( d_K Big( frac{T}{2pi e}Big)^{n_K}Big)Big| le 0.228 (log d_K + n_K log T) + 23.108 n_K + 4.520, $$ which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.
{"title":"Counting zeros of Dedekind zeta functions","authors":"Elchin Hasanalizade, Quanli Shen, PENG-JIE Wong","doi":"10.1090/MCOM/3665","DOIUrl":"https://doi.org/10.1090/MCOM/3665","url":null,"abstract":"Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $zeta_K(s)$ of $K$. More precisely, we show that for $T geq 1$, $$ Big| N_K (T) - frac{T}{pi} log Big( d_K Big( frac{T}{2pi e}Big)^{n_K}Big)Big| \u0000le 0.228 (log d_K + n_K log T) + 23.108 n_K + 4.520, $$ which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75603577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The transitive simultaneous conjugacy problem asks whether there exists a permutation $tau in S_n$ such that $b_j = tau^{-1} a_j tau$ holds for all $j = 1,2, ldots, d$, where $a_1, a_2, ldots, a_d$ and $b_1, b_2, ldots, b_d$ are given sequences of $d$ permutations in $S_n$, each of which generates a transitive subgroup of $S_n$. As from mid 70' it has been known that the problem can be solved in $O(dn^2)$ time. An algorithm with running time $O(dn log(dn))$, proposed in late 80', does not work correctly on all input data. In this paper we solve the transitive simultaneous conjugacy problem in $O(n^2 log d / log n + dnlog n)$ time and $O(n^{3/ 2} + dn)$ space. Experimental evaluation on random instances shows that the expected running time of our algorithm is considerably better, perhaps even nearly linear in $n$ at given $d$.
传递同时共轭问题询问是否存在一个排列$tau in S_n$，使得$b_j = tau^{-1} a_j tau$对所有$j = 1,2, ldots, d$都成立，其中$a_1, a_2, ldots, a_d$和$b_1, b_2, ldots, b_d$是$S_n$中$d$排列的序列，每一个都生成$S_n$的传递子群。从70年代中期开始，人们就知道这个问题可以在$O(dn^2)$时间内解决。80年代末提出的运行时间为$O(dn log(dn))$的算法不能正确处理所有输入数据。本文解决了$O(n^2 log d / log n + dnlog n)$时间和$O(n^{3/ 2} + dn)$空间上的传递联立共轭问题。在随机实例上的实验评估表明，我们的算法的预期运行时间相当好，在给定$d$的情况下，$n$甚至可能接近线性。
{"title":"The simultaneous conjugacy problem in the symmetric group","authors":"A. Brodnik, A. Malnic, Rok Požar","doi":"10.1090/MCOM/3637","DOIUrl":"https://doi.org/10.1090/MCOM/3637","url":null,"abstract":"The transitive simultaneous conjugacy problem asks whether there exists a permutation $tau in S_n$ such that $b_j = tau^{-1} a_j tau$ holds for all $j = 1,2, ldots, d$, where $a_1, a_2, ldots, a_d$ and $b_1, b_2, ldots, b_d$ are given sequences of $d$ permutations in $S_n$, each of which generates a transitive subgroup of $S_n$. As from mid 70' it has been known that the problem can be solved in $O(dn^2)$ time. An algorithm with running time $O(dn log(dn))$, proposed in late 80', does not work correctly on all input data. In this paper we solve the transitive simultaneous conjugacy problem in $O(n^2 log d / log n + dnlog n)$ time and $O(n^{3/ 2} + dn)$ space. Experimental evaluation on random instances shows that the expected running time of our algorithm is considerably better, perhaps even nearly linear in $n$ at given $d$.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75514541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The “Schur-Siegel-Smyth trace problem” is a famous open problem that has existed for nearly 100 years. To study this problem with the known methods, we need to find all totally positive algebraic integers with small trace. In this work, on the basis of the classical algorithm, we construct a new type of explicit auxiliary functions related to Chebyshev polynomials to give better bounds for S k S_k , and reduce sharply the computing time. We are then able to push the computation to degree 15 15 and prove that there is no such totally positive algebraic integer with absolute trace 1.8 1.8 . As an application, we improve the lower bound for the absolute trace of totally positive algebraic integers to 1.793145 ⋯ 1.793145cdots .
“Schur-Siegel-Smyth迹问题”是一个存在了近100年的著名开放问题。为了用已知的方法研究这个问题，我们需要找到所有带小迹的全正代数整数。本文在经典算法的基础上，构造了一种新的与Chebyshev多项式相关的显式辅助函数，给出了S k S_k更好的界，大大减少了计算时间。然后，我们能够将计算推到15次15次，并证明不存在绝对迹为1.8 1.8的完全正代数整数。作为一个应用，我们将完全正代数整数的绝对迹的下界改进为1.793145⋯1.793145cdots。
{"title":"Totally positive algebraic integers with small trace","authors":"Congjie Wang, Jie Wu, Qiang Wu","doi":"10.1090/MCOM/3636","DOIUrl":"https://doi.org/10.1090/MCOM/3636","url":null,"abstract":"The “Schur-Siegel-Smyth trace problem” is a famous open problem that has existed for nearly 100 years. To study this problem with the known methods, we need to find all totally positive algebraic integers with small trace. In this work, on the basis of the classical algorithm, we construct a new type of explicit auxiliary functions related to Chebyshev polynomials to give better bounds for \u0000\u0000 \u0000 \u0000 S\u0000 k\u0000 \u0000 S_k\u0000 \u0000\u0000, and reduce sharply the computing time. We are then able to push the computation to degree \u0000\u0000 \u0000 15\u0000 15\u0000 \u0000\u0000 and prove that there is no such totally positive algebraic integer with absolute trace \u0000\u0000 \u0000 1.8\u0000 1.8\u0000 \u0000\u0000. As an application, we improve the lower bound for the absolute trace of totally positive algebraic integers to \u0000\u0000 \u0000 \u0000 1.793145\u0000 ⋯\u0000 \u0000 1.793145cdots\u0000 \u0000\u0000.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87424571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For the Hodge–Laplace equation in finite element exterior calculus, we introduce several families of discontinuous Galerkin methods in the extended Galerkin framework. For contractible domains, this framework utilizes seven fields and provides a unifying inf-sup analysis with respect to all discretization and penalty parameters. It is shown that the proposed methods can be hybridized as a reduced two-field formulation.
{"title":"An Extended Galerkin analysis in finite element exterior calculus","authors":"Q. Hong, Yuwen Li, Jinchao Xu","doi":"10.1090/mcom/3707","DOIUrl":"https://doi.org/10.1090/mcom/3707","url":null,"abstract":"For the Hodge–Laplace equation in finite element exterior calculus, we introduce several families of discontinuous Galerkin methods in the extended Galerkin framework. For contractible domains, this framework utilizes seven fields and provides a unifying inf-sup analysis with respect to all discretization and penalty parameters. It is shown that the proposed methods can be hybridized as a reduced two-field formulation.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90665922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dow Drake, Jay Gopalakrishnan, J. Schöberl, C. Wintersteiger
Finite element methods for symmetric linear hyperbolic systems using unstructured advancing fronts (satisfying a causality condition) are considered in this work. Convergence results and error bounds are obtained for mapped tent pitching schemes made with standard discontinuous Galerkin discretizations for spatial approximation on mapped tents. Techniques to study semidiscretization on mapped tents, design fully discrete schemes, prove local error bounds, prove stability on spacetime fronts, and bound error propagated through unstructured layers are developed.
{"title":"Convergence analysis of some tent-based schemes for linear hyperbolic systems","authors":"Dow Drake, Jay Gopalakrishnan, J. Schöberl, C. Wintersteiger","doi":"10.1090/mcom/3686","DOIUrl":"https://doi.org/10.1090/mcom/3686","url":null,"abstract":"Finite element methods for symmetric linear hyperbolic systems using unstructured advancing fronts (satisfying a causality condition) are considered in this work. Convergence results and error bounds are obtained for mapped tent pitching schemes made with standard discontinuous Galerkin discretizations for spatial approximation on mapped tents. Techniques to study semidiscretization on mapped tents, design fully discrete schemes, prove local error bounds, prove stability on spacetime fronts, and bound error propagated through unstructured layers are developed.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73465800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Blanes, F. Casas, P. Chartier, A. Escorihuela-Tomàs
We analyze composition methods with complex coefficients exhibiting the so-called “symmetry-conjugate” pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when projected on the real axis and we compare them with the usual left-right palindromic compositions. New schemes within this family up to order 8 are proposed and their efficiency is tested on several examples. Our analysis shows that higherorder schemes are more efficient even when time step sizes are relatively large. AMS numbers: 65L05, 65P10, 37M15
{"title":"On symmetric-conjugate composition methods in the numerical integration of differential equations","authors":"S. Blanes, F. Casas, P. Chartier, A. Escorihuela-Tomàs","doi":"10.1090/mcom/3715","DOIUrl":"https://doi.org/10.1090/mcom/3715","url":null,"abstract":"We analyze composition methods with complex coefficients exhibiting the so-called “symmetry-conjugate” pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when projected on the real axis and we compare them with the usual left-right palindromic compositions. New schemes within this family up to order 8 are proposed and their efficiency is tested on several examples. Our analysis shows that higherorder schemes are more efficient even when time step sizes are relatively large. AMS numbers: 65L05, 65P10, 37M15","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77077712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, based on a domain decomposition (DD) method, we shall propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from the edge element approximation of the Maxwell eigenvalue problem. In order to eliminate the components in orthogonal complement space of the eigenvalue, we shall solve a parallel preconditioned system and a Helmholtz projection system together in fine space. After one coarse space correction in each iteration and minimizing the Rayleigh quotient in a small dimensional Davidson space, we finally get the error reduction of this two-level PHJD method as γ = c(H)(1 − C δ H2 ), where C is a constant independent of the mesh size h and the diameter of subdomains H , δ is the overlapping size among the subdomains, and c(H) decreasing as H → 0, which means the greater the number of subdomains, the better the convergence rate. Numerical results supporting our theory shall be given.
本文基于域分解(DD)方法，提出了一种有效的两级预条件Helmholtz-Jacobi-Davidson (PHJD)方法，用于求解由Maxwell特征值问题的边元近似引起的代数特征值问题。为了消去特征值在正交补空间中的分量，我们将在精细空间中求解一个平行预条件系统和一个亥姆霍兹投影系统。在每个迭代和最小化一个粗空间校正后的瑞利商小维戴维森空间,我们最后得到的错误减少二级PHJD方法γ= c (H)(1−cδH2), c是一个恒定的独立的筛孔尺寸H和子域的直径H,δ是子域之间的重叠的大小,和c (H)降低H→0时,这意味着更大的子域的数量,收敛速度就越好。将给出支持我们理论的数值结果。
{"title":"A Two-Level Preconditioned Helmholtz-Jacobi-Davidson Method for the Maxwell Eigenvalue Problem","authors":"Qigang Liang, Xuejun Xu","doi":"10.1090/mcom/3702","DOIUrl":"https://doi.org/10.1090/mcom/3702","url":null,"abstract":"In this paper, based on a domain decomposition (DD) method, we shall propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from the edge element approximation of the Maxwell eigenvalue problem. In order to eliminate the components in orthogonal complement space of the eigenvalue, we shall solve a parallel preconditioned system and a Helmholtz projection system together in fine space. After one coarse space correction in each iteration and minimizing the Rayleigh quotient in a small dimensional Davidson space, we finally get the error reduction of this two-level PHJD method as γ = c(H)(1 − C δ H2 ), where C is a constant independent of the mesh size h and the diameter of subdomains H , δ is the overlapping size among the subdomains, and c(H) decreasing as H → 0, which means the greater the number of subdomains, the better the convergence rate. Numerical results supporting our theory shall be given.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77132226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by the desire to compute complete Lyapunov functions for nonlinear dynamical systems, we develop a general theory of discretizing a certain type of continuous minimization problems with differential inequality constraints. The resulting discretized problems are quadratic optimization problems, for which there exist e fficient solution algorithms, and we show that their unique solutions converge strongly in appropriate Sobolev spaces to the unique solution of the original continuous problem. We develop the theory and present examples of our approach, where we compute complete Lyapunov functions for nonlinear dynamical systems. A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is divided into the chain-recurrent set, where the complete Lyapunov function is constant along solutions, and the part characterizing the gradient-like flow, where the complete Lyapunov function is strictly decreasing along solutions. We propose a new method to compute a complete Lyapunov function as the solution of a quadratic minimization problem, for which no information about the chain-recurrent set is required. The solutions to the discretized problems, which can be solved using quadratic programming, converge to the complete Lyapunov function.
{"title":"Minimization with differential inequality constraints applied to complete Lyapunov functions","authors":"P. Giesl, C. Argáez, S. Hafstein, H. Wendland","doi":"10.1090/mcom/3629","DOIUrl":"https://doi.org/10.1090/mcom/3629","url":null,"abstract":"Motivated by the desire to compute complete Lyapunov functions for nonlinear dynamical systems, we develop a general theory of discretizing a certain type of continuous minimization problems with differential inequality constraints. The resulting discretized problems are quadratic optimization problems, for which there exist e fficient solution algorithms, and we show that their unique solutions converge strongly in appropriate Sobolev spaces to the unique solution of the original continuous problem. \u0000We develop the theory and present examples of our approach, where we compute complete Lyapunov functions for nonlinear dynamical systems. \u0000A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is divided into the chain-recurrent set, where the complete Lyapunov function is constant along solutions, and the part characterizing the gradient-like flow, where the complete Lyapunov function is strictly decreasing along solutions. We propose a new method to compute a complete Lyapunov function as the solution of a quadratic minimization problem, for which no information \u0000about the chain-recurrent set is required. The solutions to the discretized problems, which can be solved using quadratic programming, converge to the complete Lyapunov function.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73667987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1971, Tutte wrote in an article that"it is tempting to conjecture that every 3-connected bipartite cubic graph is hamiltonian". Motivated by this remark, Horton constructed a counterexample on 96 vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on 50 vertices which was discovered independently by Georges and Kelmans. In this article we show that there is no smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic graphs in the literature have cyclic 4-cuts -- even if they have girth 6 -- it is natural to ask whether this is a necessary prerequisite. In this article we answer this question in the negative and give a construction of an infinite family of non-hamiltonian cyclically 5-connected bipartite cubic graphs. In 1969, Barnette gave a weaker version of the conjecture stating that 3-connected planar bipartite cubic graphs are hamiltonian. We show that Barnette's conjecture is true up to at least 90 vertices. We also report that a search of small non-hamiltonian 3-connected bipartite cubic graphs did not find any with genus less than 4.
{"title":"The Minimality of the Georges-Kelmans Graph","authors":"G. Brinkmann, Jan Goedgebeur, B. McKay","doi":"10.1090/mcom/3701","DOIUrl":"https://doi.org/10.1090/mcom/3701","url":null,"abstract":"In 1971, Tutte wrote in an article that\"it is tempting to conjecture that every 3-connected bipartite cubic graph is hamiltonian\". Motivated by this remark, Horton constructed a counterexample on 96 vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on 50 vertices which was discovered independently by Georges and Kelmans. In this article we show that there is no smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic graphs in the literature have cyclic 4-cuts -- even if they have girth 6 -- it is natural to ask whether this is a necessary prerequisite. In this article we answer this question in the negative and give a construction of an infinite family of non-hamiltonian cyclically 5-connected bipartite cubic graphs. In 1969, Barnette gave a weaker version of the conjecture stating that 3-connected planar bipartite cubic graphs are hamiltonian. We show that Barnette's conjecture is true up to at least 90 vertices. We also report that a search of small non-hamiltonian 3-connected bipartite cubic graphs did not find any with genus less than 4.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90659108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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