In this paper we provide an explicit minimal projective bimodule resolution for some weakly symmetric algebras with radical cube zero. Then by using this resolution we compute the dimension of its Hochschild cohomology groups.

In this paper we give explicit projective bimodule resolutions for algebras in a class of some special biserial algebras, which contains a cluster-tilted algebra of type ${\mathbb{D}}_4$. As a main result we completely determine the dimensions of the Hochschild cohomology groups for these algebras.

In this paper, we present inexact sequential quadratically constrained quadratic programming (SQCQP) methods with nonmonotone line searches for the convex programming problem. Kato, Narushima and Yabe proposed the SQCQP method whose subproblem is solved inexactly. Their inexact SQCQP method uses a monotone line search strategy. To reduce the number of merit function evaluations and to accept the unit step size easier, we apply the nonmonotone line search to their inexact SQCQP method. We present the algorithms of the inexact SQCQP method with the nonmonotone line searches and prove their global and superlinear convergence properties. Moreover, we give some numerical experiments to investigate the numerical performance of our proposed methods.

For a $T$-variate density function, the present article defines the quasi-symmetry of order and the marginal symmetry of order $k$, and gives the theorem that the density function is $T$-variate permutation symmetric if and only if it is quasi-symmetric and marginal symmetric of order $k$. The theorem is illustrated for the multivariate normal density function.

We continue our recent work on the problem of testing the equality of two normal mean vectors when the data have two-step monotone pattern missing observations. This paper extends the two-sample problem in our previous paper to the $k$-sample problem. Under the assumption that the population covariance matrices are equal, we obtain the likelihood ratio test statistic for testing the hypothesis $H_0:{\mu }^{(1)}={\mu }^{(2)}=\cdots ={\mu }^{(k)}$ against $H_1$ : at least two ${\mu }^{(i)}$s are unequal. Then, we provide Hotelling’s $T^2$ type statistic for testing any two mean vectors and propose the approximate upper percentile of this statistic. The accuracy of the approximation is investigated by Monte Carlo simulation.

A mapping $f:V(G)\to \{0,1,2\}$ is called a 3-product cordial labeling if $|v_f(i)-v_f(j)|\le 1$ and $|e_f(i)-e_f(j)|\le 1$ for any $i,j\in \{0,1,2\}$, where $v_f(i)$ denotes the number of vertices labeled with $i,e_f(i)$ denotes the number of edges $xy$ with $f(x)f(y)\equiv i(mod3)$. A graph with a 3-product cordial labeling is called a 3-product cordial graph. In this paper, we investigate the 3-product cordial behaviour for some standard graphs.