In this paper we study the pseudo-projective curvature tensor on warped product manifolds. We obtain some significant results of the pseudo-projective curvature tensor on warped product manifolds in terms of its base and fiber manifolds. Moreover, we derive some interesting results which describe the geometry of base and fiber manifolds for a pseudo-projectively at warped product manifold. Lastly, we study the pseudo-projective curvature tensor on generalized Robertson-Walker space-times and standard static space-times.

Any net of an isotetrahedron I (a tetrahedron with all congruent four faces) and a rectangle dihedra RD satisfies the Conway criterion. Does the converse proposition hold? If so, are there practical algorithms for it? The difficult part of the proof of it comes down to the following problem [5], [6]: Find the practical algorithm for folding a parallelogramic strip into I or RD. The process of how to cover a thin rectangular board by a long tape without making gaps or overlaps has been known as a folklore among natives in various places globally [1]. By generalizing the known folklore foldings, this paper gives the practical algorithms to obtain all isotetrahedra and rectangle dihedra into which a strip can be folded. As a result of it, it is also proved that there is no other way to fold a given strip into rectangle dihedra other than the known two types of folklore foldings.

Tate-Hochschild cohomology of an algebra is a generalization of ordinary Hochschild cohomology, which is defined on positive and negative degrees and has a ring structure. Our purpose of this paper is to study the eventual periodicity of an algebra by using the Tate-Hochschild cohomology ring. First, we deal with eventually periodic algebras and show that they are not necessarily Gorenstein algebras. Secondly, we characterize the eventual periodicity of a Gorenstein algebra as the existence of an invertible homogeneous element of the Tate-Hochschild cohomology ring of the algebra, which is our main result. Finally, we use tensor algebras to establish a way of constructing eventually periodic Gorenstein algebras.

The marginal homogeneity (MH) model is well-known for analyzing ordinal square contingency tables. This study proposes a non-inferiority marginal symmetry (NiMS) model, which has a different marginal symmetry structure than the MH model. In the NiMS model, the probability of an observation falling in row category $i$ or below and column category $i$ or above is equal to the probability of an observation falling in row category $i$ or above and column category $i$ or below. Additionally, two kinds of extended NiMS models are proposed. These extended NiMS models constantly hold when the NiMS model holds. However, the converse is not necessarily true. This study examines what a model should be necessary, in addition to the extended NiMS model, to satisfy the NiMS model.

For square contingency tables, the measure to represent the degree of departure from the partial symmetry model was proposed. It is necessary to estimate the measure because it is constructed of unknown parameters. Although many studies consider using the plug-in estimator to estimate the measure, the bias of the plug-in estimator is large when the sample size is not so large. In this study, we consider to estimate the measure when the sample size is not so large. This paper presents the improved approximate unbiased estimators of the measure which are obtained using the second-order term in Taylor series expansion. Some simulation studies show the performances of proposed estimators for finite sample cases.

In this paper, by utilising the Riesz functional calculus in a Banach algebra $\mathcal{B}$, we provide some norm inequalities for the generalized commutator $$f(y)z-zf(x)$$ where $x,$ $y,$ $z\in \mathcal{B}$ and $f$ is an analytic function for which the elements $f(y)$ and $f(x)$ exist. Some examples for the resolvent and exponential functions are also given.

In this paper, we give the estimate of the solutions to Schrödinger equation with vector potential on modulation spaces. We assume that vector potential $\mathbf{a}(t,x)$ is first degree polynomial with respect to $x$, and it corresponds to constant magnetic field or time-dependent magnetic field for physics.

For square contingency tables with ordered categories, indexes that represent the degree of departure from the marginal homogeneity (MH) model have been proposed. There are two types of directionalities of departure from MH. The existing indexes, however, can analyze either the degree of departure from MH or the directionality but not both. To address this issue, this study proposes a two-dimensional index, which combines the existing indexes, can simultaneously analyze both the degree and directionality of departure from MH. This study also evaluates the usefulness of the proposed index for visually comparing degrees of departure from MH in several square contingency tables using confidence region.