Solving Algebraic Problems with Geometry Diagrams(APGDs)poses a significant challenge in artificial intelligence due to the complex and diverse geometric relations among geometric objects.Problems typically involve bo...Solving Algebraic Problems with Geometry Diagrams(APGDs)poses a significant challenge in artificial intelligence due to the complex and diverse geometric relations among geometric objects.Problems typically involve both textual descriptions and geometry diagrams,requiring a joint understanding of these modalities.Although considerable progress has been made in solving math word problems,research on solving APGDs still cannot discover implicit geometry knowledge for solving APGDs,which limits their ability to effectively solve problems.In this study,a systematic and modular three-phase scheme is proposed to design an algorithm for solving APGDs that involve textual and diagrammatic information.The three-phase scheme begins with the application of the statetransformer paradigm,modeling the problem-solving process and effectively representing the intermediate states and transformations during the process.Next,a generalized APGD-solving approach is introduced to effectively extract geometric knowledge from the problem’s textual descriptions and diagrams.Finally,a specific algorithm is designed focusing on diagram understanding,which utilizes the vectorized syntax-semantics model to extract basic geometric relations from the diagram.A method for generating derived relations,which are essential for solving APGDs,is also introduced.Experiments on real-world datasets,including geometry calculation problems and shaded area problems,demonstrate that the proposed diagram understanding method significantly improves problem-solving accuracy compared to methods relying solely on simple diagram parsing.展开更多
A qualia role-based entity-dependency graph(EDG)is proposed to represent and extract quantity relations for solving algebra story problems stated in Chinese.Traditional neural solvers use end-to-end models to translat...A qualia role-based entity-dependency graph(EDG)is proposed to represent and extract quantity relations for solving algebra story problems stated in Chinese.Traditional neural solvers use end-to-end models to translate problem texts into math expressions,which lack quantity relation acquisition in sophisticated scenarios.To address the problem,the proposed method leverages EDG to represent quantity relations hidden in qualia roles of math objects.Algorithms were designed for EDG generation and quantity relation extraction for solving algebra story problems.Experimental result shows that the proposedmethod achieved an average accuracy of 82.2%on quantity relation extraction compared to 74.5%of baseline method.Another prompt learning result shows a 5%increase obtained in problem solving by injecting the extracted quantity relations into the baseline neural solvers.展开更多
The Landau problem on non-commutative quantum mechanics is studied, where the Heisenberg algebra and the Landau energy levels as well as the non-commutative angular momentum are constructed in detail in non-commutativ...The Landau problem on non-commutative quantum mechanics is studied, where the Heisenberg algebra and the Landau energy levels as well as the non-commutative angular momentum are constructed in detail in non-commutative space and non-commutative phase space respectively.展开更多
This research paper evaluated and investigated the effectiveness of the comprehensible technique (CT) introduced by the researcher as an alternative solution to Algebra’s linear equations in solving consecutive numbe...This research paper evaluated and investigated the effectiveness of the comprehensible technique (CT) introduced by the researcher as an alternative solution to Algebra’s linear equations in solving consecutive number problems. The CT is a proposed easy and efficient way of solving number problems which aimed to help the learners reduce their difficulties in using variables since the technique only uses basic arithmetic and mental Mathematics. Evaluation research was utilized in this study where 20 people, from various ages and professions, were chosen purposively as respondents of the study. The study found out that CT was very effective for the respondents especially in unlocking difficulties in solving consecutive number problems. Moreover, according to them, the CT is easy to use, it is engaging and enjoyable, direct and simple and can be utilized as shortcut way of solving consecutive number problems when competing in Math quiz bees or taking board examinations where time is of the essence.展开更多
Let A be a factor von Neumann algebra and Ф be a nonlinear surjective map from A onto itself. We prove that, if Ф satisfies that Ф（A）Ф（B） - Ф（B）Ф（A）＊ -- AB - BA＊ for all A, B ∈ A, then there exist a l...Let A be a factor von Neumann algebra and Ф be a nonlinear surjective map from A onto itself. We prove that, if Ф satisfies that Ф（A）Ф（B） - Ф（B）Ф（A）＊ -- AB - BA＊ for all A, B ∈ A, then there exist a linear bijective map ψA →A satisfying ψ（A）ψ（B） - ψ（B）ψ（A）＊ = AB - BA＊ for A, B ∈ A and a real functional h on A with h（0） -= 0 such that Ф（A） = ψ（A） ＋ h（A）I for every A ∈ A. In particular, if .4 is a type I factor, then, Ф（A） = cA ＋ h（A）I for every A ∈ .4, where c = ±1.展开更多
We present new sufficient conditions on the solvability and numericalmethods for the following multiplicative inverse eigenvalue problem: Given an n × nreal matrix A and n real numbers λ1 , λ2 , . . . ,λn, fin...We present new sufficient conditions on the solvability and numericalmethods for the following multiplicative inverse eigenvalue problem: Given an n × nreal matrix A and n real numbers λ1 , λ2 , . . . ,λn, find n real numbers c1 , c2 , . . . , cn suchthat the matrix diag(c1, c2,..., cn)A has eigenvalues λ1, λ2,..., λn.展开更多
We envision utilizing the versatility of a Computer Algebra System, specifically Mathematica to explore designing physics problems. As a focused project, we consider for instance a thermo-mechanical-physics problem sh...We envision utilizing the versatility of a Computer Algebra System, specifically Mathematica to explore designing physics problems. As a focused project, we consider for instance a thermo-mechanical-physics problem showing its development from the ground up. Following the objectives of this investigation first by applying the fundamentals of physics principles we solve the problem symbolically. Applying the solution we investigate the sensitivities of the quantities of interest for various scenarios generating feasible numeric parameters. Although a physics problem is investigated, the proposed methodology may as well be applied to other scientific fields. The codes needed for this particular project are included enabling the interested reader to duplicate the results, extend and modify them as needed to explore various extended scenarios.展开更多
The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity proble...The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.展开更多
This research develops a solution method for project scheduling represented by a max-plus-linear (MPL) form. Max-plus-linear representation is an approach to model and analyze a class of discrete-event systems, in whi...This research develops a solution method for project scheduling represented by a max-plus-linear (MPL) form. Max-plus-linear representation is an approach to model and analyze a class of discrete-event systems, in which the behavior of a target system is represented by linear equations in max-plus algebra. Several types of MPL equations can be reduced to a constraint satisfaction problem (CSP) for mixed integer programming. The resulting formulation is flexible and easy-to-use for project scheduling;for example, we can obtain the earliest output times, latest task-starting times, and latest input times using an MPL form. We also develop a key method for identifying critical tasks under the framework of CSP. The developed methods are validated through a numerical example.展开更多
Around 1945, Alfred Tarski proposed several questions concerning the elementary theory of non-abelian free groups. These remained open for 60 years until they were proved by O. Kharlampovich and A. Myasnikov and indep...Around 1945, Alfred Tarski proposed several questions concerning the elementary theory of non-abelian free groups. These remained open for 60 years until they were proved by O. Kharlampovich and A. Myasnikov and independently by Z. Sela. The proofs, by both sets of authors, were monumental and involved the development of several new areas of infinite group theory. In this paper we explain precisely the Tarski problems and what has been actually proved. We then discuss the history of the solution as well as the components of the proof. We then provide the basic strategy for the proof. We finish this paper with a brief discussion of elementary free groups.展开更多
In this paper, an algorithm based on a shifted inverse power iteration for computing generalized eigenvalues with corresponding eigenvectors of a large scale sparse symmetric positive definite matrix pencil is present...In this paper, an algorithm based on a shifted inverse power iteration for computing generalized eigenvalues with corresponding eigenvectors of a large scale sparse symmetric positive definite matrix pencil is presented. It converges globally with a cubic asymptotic convergence rate, preserves sparsity of the original matrices and is fully parallelizable. The algebraic multilevel itera-tion method (AMLI) is used to improve the efficiency when symmetric positive definite linear equa-tions need to be solved.展开更多
The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . T...The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .展开更多
基金supported by the National Natural Science Foundation of China(No.61977029)the Fundamental Research Funds for the Central Universities,CCNU(No.3110120001).
文摘Solving Algebraic Problems with Geometry Diagrams(APGDs)poses a significant challenge in artificial intelligence due to the complex and diverse geometric relations among geometric objects.Problems typically involve both textual descriptions and geometry diagrams,requiring a joint understanding of these modalities.Although considerable progress has been made in solving math word problems,research on solving APGDs still cannot discover implicit geometry knowledge for solving APGDs,which limits their ability to effectively solve problems.In this study,a systematic and modular three-phase scheme is proposed to design an algorithm for solving APGDs that involve textual and diagrammatic information.The three-phase scheme begins with the application of the statetransformer paradigm,modeling the problem-solving process and effectively representing the intermediate states and transformations during the process.Next,a generalized APGD-solving approach is introduced to effectively extract geometric knowledge from the problem’s textual descriptions and diagrams.Finally,a specific algorithm is designed focusing on diagram understanding,which utilizes the vectorized syntax-semantics model to extract basic geometric relations from the diagram.A method for generating derived relations,which are essential for solving APGDs,is also introduced.Experiments on real-world datasets,including geometry calculation problems and shaded area problems,demonstrate that the proposed diagram understanding method significantly improves problem-solving accuracy compared to methods relying solely on simple diagram parsing.
基金supported by the National Natural Science Foundation of China (Nos.62177024,62007014)the Humanities and Social Sciences Youth Fund of the Ministry of Education (No.20YJC880024)+1 种基金China Post Doctoral Science Foundation (No.2019M652678)the Fundamental Research Funds for the Central Universities (No.CCNU20ZT019).
文摘A qualia role-based entity-dependency graph(EDG)is proposed to represent and extract quantity relations for solving algebra story problems stated in Chinese.Traditional neural solvers use end-to-end models to translate problem texts into math expressions,which lack quantity relation acquisition in sophisticated scenarios.To address the problem,the proposed method leverages EDG to represent quantity relations hidden in qualia roles of math objects.Algorithms were designed for EDG generation and quantity relation extraction for solving algebra story problems.Experimental result shows that the proposedmethod achieved an average accuracy of 82.2%on quantity relation extraction compared to 74.5%of baseline method.Another prompt learning result shows a 5%increase obtained in problem solving by injecting the extracted quantity relations into the baseline neural solvers.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 90303003 and 10575026） and the Natural Science Foundation of Zhejiang Province, China （Grant No M103042）.
文摘The Landau problem on non-commutative quantum mechanics is studied, where the Heisenberg algebra and the Landau energy levels as well as the non-commutative angular momentum are constructed in detail in non-commutative space and non-commutative phase space respectively.
文摘This research paper evaluated and investigated the effectiveness of the comprehensible technique (CT) introduced by the researcher as an alternative solution to Algebra’s linear equations in solving consecutive number problems. The CT is a proposed easy and efficient way of solving number problems which aimed to help the learners reduce their difficulties in using variables since the technique only uses basic arithmetic and mental Mathematics. Evaluation research was utilized in this study where 20 people, from various ages and professions, were chosen purposively as respondents of the study. The study found out that CT was very effective for the respondents especially in unlocking difficulties in solving consecutive number problems. Moreover, according to them, the CT is easy to use, it is engaging and enjoyable, direct and simple and can be utilized as shortcut way of solving consecutive number problems when competing in Math quiz bees or taking board examinations where time is of the essence.
基金supported by National Natural Science Foundation of China(10871111)the Specialized Research Fund for Doctoral Program of Higher Education(200800030059)(to Cui)Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education,Science and Technology(NRF-2009-0070788)(to Park)
文摘Let A be a factor von Neumann algebra and Ф be a nonlinear surjective map from A onto itself. We prove that, if Ф satisfies that Ф（A）Ф（B） - Ф（B）Ф（A）＊ -- AB - BA＊ for all A, B ∈ A, then there exist a linear bijective map ψA →A satisfying ψ（A）ψ（B） - ψ（B）ψ（A）＊ = AB - BA＊ for A, B ∈ A and a real functional h on A with h（0） -= 0 such that Ф（A） = ψ（A） ＋ h（A）I for every A ∈ A. In particular, if .4 is a type I factor, then, Ф（A） = cA ＋ h（A）I for every A ∈ .4, where c = ±1.
文摘We present new sufficient conditions on the solvability and numericalmethods for the following multiplicative inverse eigenvalue problem: Given an n × nreal matrix A and n real numbers λ1 , λ2 , . . . ,λn, find n real numbers c1 , c2 , . . . , cn suchthat the matrix diag(c1, c2,..., cn)A has eigenvalues λ1, λ2,..., λn.
文摘We envision utilizing the versatility of a Computer Algebra System, specifically Mathematica to explore designing physics problems. As a focused project, we consider for instance a thermo-mechanical-physics problem showing its development from the ground up. Following the objectives of this investigation first by applying the fundamentals of physics principles we solve the problem symbolically. Applying the solution we investigate the sensitivities of the quantities of interest for various scenarios generating feasible numeric parameters. Although a physics problem is investigated, the proposed methodology may as well be applied to other scientific fields. The codes needed for this particular project are included enabling the interested reader to duplicate the results, extend and modify them as needed to explore various extended scenarios.
基金Supported by the Funds of Ministry of Education of China for PhD (20020141013)the NNSF of China (10471015).
文摘The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.
文摘This research develops a solution method for project scheduling represented by a max-plus-linear (MPL) form. Max-plus-linear representation is an approach to model and analyze a class of discrete-event systems, in which the behavior of a target system is represented by linear equations in max-plus algebra. Several types of MPL equations can be reduced to a constraint satisfaction problem (CSP) for mixed integer programming. The resulting formulation is flexible and easy-to-use for project scheduling;for example, we can obtain the earliest output times, latest task-starting times, and latest input times using an MPL form. We also develop a key method for identifying critical tasks under the framework of CSP. The developed methods are validated through a numerical example.
文摘Around 1945, Alfred Tarski proposed several questions concerning the elementary theory of non-abelian free groups. These remained open for 60 years until they were proved by O. Kharlampovich and A. Myasnikov and independently by Z. Sela. The proofs, by both sets of authors, were monumental and involved the development of several new areas of infinite group theory. In this paper we explain precisely the Tarski problems and what has been actually proved. We then discuss the history of the solution as well as the components of the proof. We then provide the basic strategy for the proof. We finish this paper with a brief discussion of elementary free groups.
文摘In this paper, an algorithm based on a shifted inverse power iteration for computing generalized eigenvalues with corresponding eigenvectors of a large scale sparse symmetric positive definite matrix pencil is presented. It converges globally with a cubic asymptotic convergence rate, preserves sparsity of the original matrices and is fully parallelizable. The algebraic multilevel itera-tion method (AMLI) is used to improve the efficiency when symmetric positive definite linear equa-tions need to be solved.
文摘The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .
