Applications of shuffle product to restricted decomposition formulas for multiple zeta values

In this paper we obtain a recursive formula for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values (MZVs). The first formula generalizes the decomposition formula of Euler and is similar to the restricted formula of Eie and Wei for MZVs with one strings of 1's. The second formula generalizes the previous results to the product of two MZVs with one and two strings of 1's respectively.Comment: 11 page

(E)-2-Meth­oxy-N′-(4-methoxy­benzyl­idene)benzohydrazide

The mol­ecule of the title compound, C16H16N2O3, displays an E configuration about the C=N bond. The dihedral angle between the two benzene rings is 99.0 (2)°. In the crystal structure, mol­ecules are linked through inter­molecular N—H⋯O hydrogen bonds, forming chains running along the b axis

(E)-4-Chloro-N′-(5-hydr­oxy-2-nitro­benzyl­idene)benzohydrazide

The title compound, C14H10ClN3O4, was synthesized by the reaction of 5-hydr­oxy-2-nitro­benzaldehyde with an equimolar quantity of 4-chloro­benzohydrazide in methanol. The mol­ecule displays an E configuration about the C=N bond. The dihedral angle between the two benzene rings is 3.9 (2)°. In the crystal structure, mol­ecules are linked through inter­molecular N—H⋯O and O—H⋯O hydrogen bonds, forming chains running along the b axis

(E)-3-Bromo-N′-(2,4-dichloro­benzyl­idene)benzohydrazide

The title compound, C14H9BrCl2N2O, was synthesized by the reaction of 2,4-dichloro­benzaldehyde with an equimolar quantity of 3-bromo­benzohydrazide in methanol. The mol­ecule displays an E configuration about the C=N bond. The dihedral angle between the two benzene rings is 5.3 (2)°. In the crystal structure, mol­ecules are linked through inter­molecular N—H⋯O and C—H⋯O hydrogen bonds, forming chains running along the c axis

(E)-N′-(2-Chloro­benzyl­idene)-2-methoxy­benzohydrazide

The mol­ecule of the title compound, C15H13ClN2O2, displays an E configuration about the C=N bond. The dihedral angle between the two benzene rings is 77.1 (2)°. In the crystal structure, mol­ecules are linked through inter­molecular N—H⋯O hydrogen bonds, forming chains running along the b axis

(E)-4-Chloro-N′-(2-chloro­benzyl­idene)benzohydrazide

The title compound, C14H10Cl2N2O, was synthesized by the reaction of 2-chloro­benzaldehyde with an equimolar quantity of 4-chloro­benzohydrazide in methanol. The mol­ecule displays an E configuration about the C=N bond. The dihedral angle between the two benzene rings is 8.6 (2)°. In the crystal structure, mol­ecules are linked through inter­molecular N—H⋯O hydrogen bonds, forming chains running along the c axis

(E)-3-Bromo-N′-(4-methoxy­benzyl­idene)benzohydrazide methanol solvate

The title compound, C15H13BrN2O2·CH3OH, was synthesized by the reaction of 4-methoxy­benzaldehyde with an equimolar quantity of 3-bromo­benzohydrazide in methanol. The benzohydrazide mol­ecule displays an E configuration about the C=N bond. The dihedral angle between the two benzene rings is 4.0 (2)°. The benzohydrazide and methanol mol­ecules are linked into a chain propagating along the b axis by O—H⋯O, O—H⋯N, N—H⋯O and C—H⋯O hydrogen bonds

A note on the growth factor in Gaussian elimination for generalized Higham matrices

The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite and $\mathrm{i}=\sqrt{-1}$ is the imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth factor in Gaussian elimination is less than 3. In this paper, based on the previous results, a new bound of the growth factor is obtained by using the maximum of the condition numbers of matrixes B and C for the generalized Higham matrix A, which strengthens this bound to 2 and proves the Higham's conjecture.Comment: 8 pages, 2 figures; Submitted to MOC on Dec. 22 201

A similarity-based cooperative co-evolutionary algorithm for dynamic interval multi-objective optimization problems

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Dynamic interval multi-objective optimization problems (DI-MOPs) are very common in real-world applications. However, there are few evolutionary algorithms that are suitable for tackling DI-MOPs up to date. A framework of dynamic interval multi-objective cooperative co-evolutionary optimization based on the interval similarity is presented in this paper to handle DI-MOPs. In the framework, a strategy for decomposing decision variables is first proposed, through which all the decision variables are divided into two groups according to the interval similarity between each decision variable and interval parameters. Following that, two sub-populations are utilized to cooperatively optimize decision variables in the two groups. Furthermore, two response strategies, rgb0.00,0.00,0.00i.e., a strategy based on the change intensity and a random mutation strategy, are employed to rapidly track the changing Pareto front of the optimization problem. The proposed algorithm is applied to eight benchmark optimization instances rgb0.00,0.00,0.00as well as a multi-period portfolio selection problem and compared with five state-of-the-art evolutionary algorithms. The experimental results reveal that the proposed algorithm is very competitive on most optimization instances