Pub Date : 2021-05-10DOI: 10.4236/AJCM.2021.112010
D. Cacuci, R. Fang
This work extends to fourth-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 7477 imprecisely known (uncertain) model parameters which have nonzero values. These parameters are as follows: 180 microscopic total cross sections; 7101 microscopic scattering sections; 60 microscopic fission cross sections; 60 parameters that characterize the average number of neutrons per fission; 60 parameters that characterize the fission spectrum; 10 parameters that characterize the fission source; and 6 parameters that characterize the isotope number densities. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 7477 first-order and 27,956,503 second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters. These works showed that largest response sensitivities involve the total microscopic cross sections, which motivated the recent computation of all of the (180)3 third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. It turned out that some of these 3rd-order cross sections were far larger than the corresponding 2nd-order ones, thereby having the largest impact on the uncertainties induced in the PERP benchmark’s response. This finding has motivated the development of the original 4th-order formulas presented in this work, which are valid not only for the PERP benchmark but can also be used for computing the 4th-order sensitivities of response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of the largest fourth-order sensitivities of the PERP benchmark’s response to the total microscopic cross section and use them for a pioneering fourth-order uncertainty analysis of the PERP benchmark’s response.
{"title":"Fourth-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: II. Mathematical Expressions and CPU-Time Comparisons for Computing 4th-Order Sensitivities","authors":"D. Cacuci, R. Fang","doi":"10.4236/AJCM.2021.112010","DOIUrl":"https://doi.org/10.4236/AJCM.2021.112010","url":null,"abstract":"This work extends to fourth-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 7477 imprecisely known (uncertain) model parameters which have nonzero values. These parameters are as follows: 180 microscopic total cross sections; 7101 microscopic scattering sections; 60 microscopic fission cross sections; 60 parameters that characterize the average number of neutrons per fission; 60 parameters that characterize the fission spectrum; 10 parameters that characterize the fission source; and 6 parameters that characterize the isotope number densities. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 7477 first-order and 27,956,503 second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters. These works showed that largest response sensitivities involve the total microscopic cross sections, which motivated the recent computation of all of the (180)3 third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. It turned out that some of these 3rd-order cross sections were far larger than the corresponding 2nd-order ones, thereby having the largest impact on the uncertainties induced in the PERP benchmark’s response. This finding has motivated the development of the original 4th-order formulas presented in this work, which are valid not only for the PERP benchmark but can also be used for computing the 4th-order sensitivities of response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of the largest fourth-order sensitivities of the PERP benchmark’s response to the total microscopic cross section and use them for a pioneering fourth-order uncertainty analysis of the PERP benchmark’s response.","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"11 1","pages":"133-156"},"PeriodicalIF":0.0,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46003421","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}
Pub Date : 2021-03-01DOI: 10.4236/AJCM.2021.111002
Yoshihiro Tanaka, Mitsuru Togashi
There are a few studies that focus on solution methods for finding a Nash equilibrium of zero-sum games. We discuss the use of Karmarkar’s interior point method to solve the Nash equilibrium problems of a zero-sum game, and prove that it is theoretically a polynomial time algorithm. We implement the Karmarkar method, and a preliminary computational result shows that it performs well for zero-sum games. We also mention an affine scaling method that would help us compute Nash equilibria of general zero-sum games effectively.
{"title":"Polynomial Time Method for Solving Nash Equilibria of Zero-Sum Games","authors":"Yoshihiro Tanaka, Mitsuru Togashi","doi":"10.4236/AJCM.2021.111002","DOIUrl":"https://doi.org/10.4236/AJCM.2021.111002","url":null,"abstract":"There are a few studies that focus on solution methods for finding a Nash equilibrium of zero-sum games. We discuss the use of Karmarkar’s interior point method to solve the Nash equilibrium problems of a zero-sum game, and prove that it is theoretically a polynomial time algorithm. We implement the Karmarkar method, and a preliminary computational result shows that it performs well for zero-sum games. We also mention an affine scaling method that would help us compute Nash equilibria of general zero-sum games effectively.","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"11 1","pages":"23-30"},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49124492","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}
Pub Date : 2021-03-01DOI: 10.4236/AJCM.2021.111005
Yongzhou
In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and L1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.
{"title":"Convergence and Superconvergence of Fully Discrete Finite Element for Time Fractional Optimal Control Problems","authors":"Yongzhou","doi":"10.4236/AJCM.2021.111005","DOIUrl":"https://doi.org/10.4236/AJCM.2021.111005","url":null,"abstract":"In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and L1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"11 1","pages":"53-63"},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41399410","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}
Pub Date : 2021-03-01DOI: 10.4236/AJCM.2021.111003
G. Vergara-Hermosilla
In this work we develop necessary and sufficient conditions for describing the family of anti-Hurwitz polynomials, introduced by Vergara-Hermosilla et al. in [1]. Specifically, we studied a dual version of the Theorem of Routh-Hurwitz and present explicit criteria for polynomials of low order and derivatives. Another contribution of this work is establishing a dual version of the Hermite-Biehler Theorem. To this aim, we give extensions of the boundary crossing Theorems and a zero exclusion principle for anti-Hurwitz polynomials.
{"title":"On a Dual to the Properties of Hurwitz Polynomials I","authors":"G. Vergara-Hermosilla","doi":"10.4236/AJCM.2021.111003","DOIUrl":"https://doi.org/10.4236/AJCM.2021.111003","url":null,"abstract":"In this work we develop necessary and sufficient conditions for describing the family of anti-Hurwitz polynomials, introduced by Vergara-Hermosilla et al. in [1]. Specifically, we studied a dual version of the Theorem of Routh-Hurwitz and present explicit criteria for polynomials of low order and derivatives. Another contribution of this work is establishing a dual version of the Hermite-Biehler Theorem. To this aim, we give extensions of the boundary crossing Theorems and a zero exclusion principle for anti-Hurwitz polynomials.","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"11 1","pages":"31-41"},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47570444","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}
Pub Date : 2021-03-01DOI: 10.4236/AJCM.2021.111004
H. Sarafian
In search of nonlinear oscillations, we envision a 3D elliptic curva-ture-dependent nonuniform charge distribution to creating an electric field along the symmetry axis causing a massive point-like charged particle placed on the symmetry axis to oscillate in a delayed/hesitant nonlinear mode. The charge distribution is a 3D twisted line creating nontrivial electric field causing an unexpected oscillation that is non-orthodox defying the common sense. Calculation of this research flavored investigation is entirely based on utilities accompanied with Computer Algebra Systems (CAS) especially Mathematic [1]. The characteristics of the delayed oscillations in addition to embodying classic graphics displaying the time-dependent kinematic quantities are augmented including various phase diagrams signifying the nonlinear oscillations. The output of our investigation is compared to nonlinear non-delayed oscillations revealing fresh insight. For comprehensive understanding of the hesitant oscillator a simulation program is crafted clarifying visually the scenario on hand.
{"title":"Nonlinear Electrostatic “Hesitant” Oscillator","authors":"H. Sarafian","doi":"10.4236/AJCM.2021.111004","DOIUrl":"https://doi.org/10.4236/AJCM.2021.111004","url":null,"abstract":"In search of nonlinear oscillations, we envision a 3D elliptic curva-ture-dependent nonuniform charge distribution to creating an electric field along the symmetry axis causing a massive point-like charged particle placed on the symmetry axis to oscillate in a delayed/hesitant nonlinear mode. The charge distribution is a 3D twisted line creating nontrivial electric field causing an unexpected oscillation that is non-orthodox defying the common sense. Calculation of this research flavored investigation is entirely based on utilities accompanied with Computer Algebra Systems (CAS) especially Mathematic [1]. The characteristics of the delayed oscillations in addition to embodying classic graphics displaying the time-dependent kinematic quantities are augmented including various phase diagrams signifying the nonlinear oscillations. The output of our investigation is compared to nonlinear non-delayed oscillations revealing fresh insight. For comprehensive understanding of the hesitant oscillator a simulation program is crafted clarifying visually the scenario on hand.","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"11 1","pages":"42-52"},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70509294","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}
Pub Date : 2021-03-01DOI: 10.4236/AJCM.2021.111001
Vardges Melkonian
In this paper we develop modeling techniques for a social partitioning problem. Different social interaction regulations are imposed during pandemics to prevent the spread of diseases. We suggest partitioning a set of company employees as an effective way to curb the spread, and use integer programming techniques to model it. The goal of the model is to maximize the number of direct interactions between employees who are essential for company’s work subject to the constraint that all employees should be partitioned into components of no more than a certain size implied by the regulations. Then we further develop the basic model to take into account different restrictions and provisions. We also give heuristics for solving the problem. Our computational results include sensitivity analysis on some of the models and analysis of the heuristic performance.
{"title":"Mathematical Models for a Social Partitioning Problem","authors":"Vardges Melkonian","doi":"10.4236/AJCM.2021.111001","DOIUrl":"https://doi.org/10.4236/AJCM.2021.111001","url":null,"abstract":"In this paper we develop modeling techniques for a social partitioning problem. Different social interaction regulations are imposed during pandemics to prevent the spread of diseases. We suggest partitioning a set of company employees as an effective way to curb the spread, and use integer programming techniques to model it. The goal of the model is to maximize the number of direct interactions between employees who are essential for company’s work subject to the constraint that all employees should be partitioned into components of no more than a certain size implied by the regulations. Then we further develop the basic model to take into account different restrictions and provisions. We also give heuristics for solving the problem. Our computational results include sensitivity analysis on some of the models and analysis of the heuristic performance.","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45174678","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}
Pub Date : 2021-03-01DOI: 10.4236/AJCM.2021.111006
H. Sarafian
It is customary to apply Newton’s cooling as the standard model investigating the temperature profile of a hot substance exposed to a cool ambient. The rate of change of temperature in Newton’s model is simplistically related to linear-temperature difference of the two e.g. [1]. In our research flavored investigation, we consider a fresh model, cooling that depends to the difference of temperature-squared conducive to similar results. Utilizing a Computer Algebra System (CAS), especially Mathematica [2] we show the equivalency of the two.
{"title":"Alternate Cooling Model vs Newton’s Cooling","authors":"H. Sarafian","doi":"10.4236/AJCM.2021.111006","DOIUrl":"https://doi.org/10.4236/AJCM.2021.111006","url":null,"abstract":"It is customary to apply Newton’s cooling as the standard model investigating the temperature profile of a hot substance exposed to a cool ambient. The rate of change of temperature in Newton’s model is simplistically related to linear-temperature difference of the two e.g. [1]. In our research flavored investigation, we consider a fresh model, cooling that depends to the difference of temperature-squared conducive to similar results. Utilizing a Computer Algebra System (CAS), especially Mathematica [2] we show the equivalency of the two.","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"11 1","pages":"64-69"},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44051368","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}
Pub Date : 2021-01-01DOI: 10.4236/ajcm.2021.112009
D. Cacuci, R. Fang
{"title":"Fourth-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: I. Mathematical Expressions and CPU-Time Comparisons for Computing 1st-, 2nd- and 3rd-Order Sensitivities","authors":"D. Cacuci, R. Fang","doi":"10.4236/ajcm.2021.112009","DOIUrl":"https://doi.org/10.4236/ajcm.2021.112009","url":null,"abstract":"","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70509364","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}
Pub Date : 2021-01-01DOI: 10.4236/ajcm.2021.114017
H. Sarafian
Nucleons are fermions with intrinsic spins exhibiting dipole character. Di-pole-dipole interaction via their dipole moments is the key feature quantify-ing the short-range nucleonics interaction in two-body physics. For a pair of interacting dipoles, the energy of a pair is the quantity of interest. The same is true for chemical polar molecules. For both cases, derivation of energy almost exclusively is carried out vectorially [1]. Although uncommon the interacting energy can be derived algebraically too. For the latter Taylor, expansion is applied [2]. The given expression although appears to be correct it is incom-plete. In our report, by applying Taylor’s expansion up to the 4th order and utilizing a Computer Algebra System we formulate the missing terms. Our report highlights the impact of correcting missing terms by giving two expli-cit examples.
{"title":"Higher-Order Corrections to Algebraic Derivation of Electric Dipole-Dipole Interaction","authors":"H. Sarafian","doi":"10.4236/ajcm.2021.114017","DOIUrl":"https://doi.org/10.4236/ajcm.2021.114017","url":null,"abstract":"Nucleons are fermions with intrinsic spins exhibiting dipole character. Di-pole-dipole interaction via their dipole moments is the key feature quantify-ing the short-range nucleonics interaction in two-body physics. For a pair of interacting dipoles, the energy of a pair is the quantity of interest. The same is true for chemical polar molecules. For both cases, derivation of energy almost exclusively is carried out vectorially [1]. Although uncommon the interacting energy can be derived algebraically too. For the latter Taylor, expansion is applied [2]. The given expression although appears to be correct it is incom-plete. In our report, by applying Taylor’s expansion up to the 4th order and utilizing a Computer Algebra System we formulate the missing terms. Our report highlights the impact of correcting missing terms by giving two expli-cit examples.","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70509428","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}
Pub Date : 2021-01-01DOI: 10.4236/ajcm.2021.114019
Joachim Moussounda Mouanda
We prove that every matrix ( ) k n F M ∈ is associated with the smallest positive integer ( ) 1 d F ≠ such that ( ) d F F ∞ is always bigger than the sum of the operator norms of the Fourier coefficients of F. We establish some inequalities for matrices of complex polynomials. In application, we show that von Neumann’s inequality holds up to the constant 2 n for matrices of the algebra ( ) k n M .
我们证明了每一个矩阵()k n F M∈都与最小的正整数()1d F≠相关联，使得()d F F∞总是大于F的傅里叶系数的算子范数之和。在应用中，我们证明了von Neumann不等式对于代数()k n M的矩阵保持常数2n。
{"title":"On Von Neumann’s Inequality for Matrices of Complex Polynomials","authors":"Joachim Moussounda Mouanda","doi":"10.4236/ajcm.2021.114019","DOIUrl":"https://doi.org/10.4236/ajcm.2021.114019","url":null,"abstract":"We prove that every matrix ( ) k n F M ∈ is associated with the smallest positive integer ( ) 1 d F ≠ such that ( ) d F F ∞ is always bigger than the sum of the operator norms of the Fourier coefficients of F. We establish some inequalities for matrices of complex polynomials. In application, we show that von Neumann’s inequality holds up to the constant 2 n for matrices of the algebra ( ) k n M .","PeriodicalId":64456,"journal":{"name":"美国计算数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70509487","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：