This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the incl...This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture.展开更多
The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(A,B;E(G))with | A |=mand | B |=nwhich does not contain H as a subgraph. Whenmin{ m,n }>2t...The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(A,B;E(G))with | A |=mand | B |=nwhich does not contain H as a subgraph. Whenmin{ m,n }>2t, the problem of determining the value of ex(m,n;Km−t,n−t)has been solved by Balbuena et al. in 2007, whose proof focuses on the structural analysis of bipartite graphs. In this paper, we provide a new proof on the value of ex(m,n;Km−t,n−t)by virtue of algebra method with the tool of adjacency matrices of bipartite graphs, which is inspired by the method using { 0,1 }-matrices due to Zarankiewicz [Problem P 101. Colloquium Mathematicum, 2(1951), 301].展开更多
The purpose of this note is to establish a general representation of Hankel matrices of Bell numbers and the convoluted Bell numbers. As a special case, the results of Aigner are extended.
A method for ranking complementary judgment matrixes with traspezoidal fuzzy numbers based on Hausdorff metric distance and fuzzy compromise decision approach is proposed. With regard to fuzzy number complementary jud...A method for ranking complementary judgment matrixes with traspezoidal fuzzy numbers based on Hausdorff metric distance and fuzzy compromise decision approach is proposed. With regard to fuzzy number complementary judgment matrixes given by a decider group whose members have various weights, the expert＇s information was aggregated first by means of simple weight average（SWA） method and Bonissone calculational method. Hence a matrix including all the experts＇ preference information was got. Then the matrix＇ column members were added up and the fuzzy evaluation values of the alternatives were got. Lastly, the Hausdorff metric distance and fuzzy compromise decision approach were used to rank the fuzzy evaluation values and then the ranking values of all the alternatives were got. Because exact numbers and triangular fuzzy numbers could all be transformed into trapezoidal fuzzy numbers, the method developed can rank complementary judgment matrixes with trapezoidal fuzzy numbers, triangular fuzzy numbers and exact numbers as well. An illustrative example is also given to verify the developed method and to demonstrate its feasibility and practicality.展开更多
We present componentwise condition numbers for the problems of MoorePenrose generalized matrix inversion and linear least squares. Also, the condition numbers for these condition numbers are given.
In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp m...In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp matrix equations. Then the fuzzy approximate solution of the fuzzy matrix equation is obtained by solving two crisp matrix equations. The existence condition of the strong LR fuzzy solution to the fuzzy matrix equation is also discussed. Some examples are given to illustrate the proposed method. Our results enrich the fuzzy linear systems theory.展开更多
The authors study the generation of matrices with complex entries belonging to some matrix groups, mainly those that are defined by a scalar product space. These matrices have useful applications in quantum mechanical...The authors study the generation of matrices with complex entries belonging to some matrix groups, mainly those that are defined by a scalar product space. These matrices have useful applications in quantum mechanical problems and complex control problems. In this work, the authors try to generate matrices such that： （1） the condition number of these types of matrices is controlled and （2） The algorithm used to generate these matrices preserves their structure.展开更多
为了减小低快拍数和低信噪比下采样协方差矩阵误差,并降低其运算复杂度,提出了一种基于实数化的均匀圆阵采样协方差矩阵重构方法。针对均匀圆阵的特点,通过组建特殊的基向量,构成特殊的重构矩阵。通过将采样协方差矩阵实数化,进一步降...为了减小低快拍数和低信噪比下采样协方差矩阵误差,并降低其运算复杂度,提出了一种基于实数化的均匀圆阵采样协方差矩阵重构方法。针对均匀圆阵的特点,通过组建特殊的基向量,构成特殊的重构矩阵。通过将采样协方差矩阵实数化,进一步降低了重构矩阵的复杂度。考虑到多通道不一致性对重构矩阵的影响,引入0位校正算法,提高了重构方法的稳健性。最后应用重构后的协方差矩阵进行子空间类波达方向估计(direction of arrival,DOA)。实验仿真证明,该特殊重构矩阵在实数化下与原矩阵重构能力相同;当快拍数为100、信噪比为0 dB时,双信源分辨力较重构前由74%提高到95%以上;理论重构运算复杂度降低到原来的53.99%。展开更多
This study employs mathematical modeling to analyze the impact of active immigrants on Foot and Mouth Disease (FMD) transmission dynamics. We calculate the reproduction number (R<sub>0</sub>) using the nex...This study employs mathematical modeling to analyze the impact of active immigrants on Foot and Mouth Disease (FMD) transmission dynamics. We calculate the reproduction number (R<sub>0</sub>) using the next-generation matrix approach. Applying the Routh-Hurwitz Criterion, we establish that the Disease-Free Equilibrium (DFE) point achieves local asymptotic stability when R<sub>0</sub> α<sub>1</sub> and α<sub>2</sub>) are closely associated with reduced susceptibility in animal populations, underscoring the link between immigrants and susceptibility. Furthermore, our findings emphasize the interplay of disease introduction with population response and adaptation, particularly involving incoming infectious immigrants. Swift interventions are vital due to the limited potential for disease establishment and rapid susceptibility decline. This study offers crucial insights into the complexities of FMD transmission with active immigrants, informing effective disease management strategies.展开更多
文摘This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture.
文摘The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(A,B;E(G))with | A |=mand | B |=nwhich does not contain H as a subgraph. Whenmin{ m,n }>2t, the problem of determining the value of ex(m,n;Km−t,n−t)has been solved by Balbuena et al. in 2007, whose proof focuses on the structural analysis of bipartite graphs. In this paper, we provide a new proof on the value of ex(m,n;Km−t,n−t)by virtue of algebra method with the tool of adjacency matrices of bipartite graphs, which is inspired by the method using { 0,1 }-matrices due to Zarankiewicz [Problem P 101. Colloquium Mathematicum, 2(1951), 301].
文摘The purpose of this note is to establish a general representation of Hankel matrices of Bell numbers and the convoluted Bell numbers. As a special case, the results of Aigner are extended.
文摘A method for ranking complementary judgment matrixes with traspezoidal fuzzy numbers based on Hausdorff metric distance and fuzzy compromise decision approach is proposed. With regard to fuzzy number complementary judgment matrixes given by a decider group whose members have various weights, the expert＇s information was aggregated first by means of simple weight average（SWA） method and Bonissone calculational method. Hence a matrix including all the experts＇ preference information was got. Then the matrix＇ column members were added up and the fuzzy evaluation values of the alternatives were got. Lastly, the Hausdorff metric distance and fuzzy compromise decision approach were used to rank the fuzzy evaluation values and then the ranking values of all the alternatives were got. Because exact numbers and triangular fuzzy numbers could all be transformed into trapezoidal fuzzy numbers, the method developed can rank complementary judgment matrixes with trapezoidal fuzzy numbers, triangular fuzzy numbers and exact numbers as well. An illustrative example is also given to verify the developed method and to demonstrate its feasibility and practicality.
基金the NSF of China under grant 10471027 and Shanghai Education Commission.
文摘We present componentwise condition numbers for the problems of MoorePenrose generalized matrix inversion and linear least squares. Also, the condition numbers for these condition numbers are given.
文摘In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp matrix equations. Then the fuzzy approximate solution of the fuzzy matrix equation is obtained by solving two crisp matrix equations. The existence condition of the strong LR fuzzy solution to the fuzzy matrix equation is also discussed. Some examples are given to illustrate the proposed method. Our results enrich the fuzzy linear systems theory.
文摘The authors study the generation of matrices with complex entries belonging to some matrix groups, mainly those that are defined by a scalar product space. These matrices have useful applications in quantum mechanical problems and complex control problems. In this work, the authors try to generate matrices such that： （1） the condition number of these types of matrices is controlled and （2） The algorithm used to generate these matrices preserves their structure.
文摘为了减小低快拍数和低信噪比下采样协方差矩阵误差,并降低其运算复杂度,提出了一种基于实数化的均匀圆阵采样协方差矩阵重构方法。针对均匀圆阵的特点,通过组建特殊的基向量,构成特殊的重构矩阵。通过将采样协方差矩阵实数化,进一步降低了重构矩阵的复杂度。考虑到多通道不一致性对重构矩阵的影响,引入0位校正算法,提高了重构方法的稳健性。最后应用重构后的协方差矩阵进行子空间类波达方向估计(direction of arrival,DOA)。实验仿真证明,该特殊重构矩阵在实数化下与原矩阵重构能力相同;当快拍数为100、信噪比为0 dB时,双信源分辨力较重构前由74%提高到95%以上;理论重构运算复杂度降低到原来的53.99%。
文摘This study employs mathematical modeling to analyze the impact of active immigrants on Foot and Mouth Disease (FMD) transmission dynamics. We calculate the reproduction number (R<sub>0</sub>) using the next-generation matrix approach. Applying the Routh-Hurwitz Criterion, we establish that the Disease-Free Equilibrium (DFE) point achieves local asymptotic stability when R<sub>0</sub> α<sub>1</sub> and α<sub>2</sub>) are closely associated with reduced susceptibility in animal populations, underscoring the link between immigrants and susceptibility. Furthermore, our findings emphasize the interplay of disease introduction with population response and adaptation, particularly involving incoming infectious immigrants. Swift interventions are vital due to the limited potential for disease establishment and rapid susceptibility decline. This study offers crucial insights into the complexities of FMD transmission with active immigrants, informing effective disease management strategies.
