We improve some known inequalities describing the mutual relation between small transfinite dimension and transfinite dimension $D$ in separable metrizable spaces. We also estimate small transfinite dimension of a product with a finite-dimensional factor, generalizing the results due to Luxemburg.

In this paper we show that periodic Takahashi 3-manifolds are cyclic coverings of the connected sum of two lens spaces (possibly cyclic coverings of $\bm{S}^{3}$), branched over knots. When the base space is a $3$-sphere, we prove that the associated branching set is a two-bridge knot of genus one, and we determine its type. Moreover, a geometric cyclic presentation for the fundamental groups of these manifolds is obtained in several interesting cases, including the ones corresponding to the branched cyclic coverings of $\bm{S}^{3}$.

We give a necessary and sufficient condition for the existence of spacelike minimal surfaces in $4$-dimensional Lorentzian space forms, which is a generalization of the Ricci condition for minimal surfaces in $3$-dimensional Riemannian space forms.

We classify the real hypersurfaces in a non-flat quaternionic space form satisfying several conditions on their Weingarten endomorphism. Firstly, we study on the maximal quaternionic distribution of the real hypersurface a relationship between a certain metric tensor and the restriction of the metric of the ambient manifold. Secondly, we consider some formulae relating the Weingarten endomorphism to the curvature operator.

The classical obstruction to minimal isometric immersions into Euclidean space is $Ric \geq 0$. In this article we construct examples of Riemannian manifolds with $Ric \lt 0$ which don't admit any minimal isometric immersion into Euclidean space for any codimension, by applying Chen invariants.

We consider a weakly hyperbolic operator with constant coefficients. We shall derive a priori estimates for it and by applying the estimate we prove local existence of the solution of semi-linear Cauchy problem and investigate the propagation of analyticity of the solutions.

Geometrical and structural properties are proved for a class of even-dimensional manifolds which are equiped with a $\mathscr{T}$-parallel connection.

A space $X$ is called (discretely) star-Lindelöf if for every open cover $\mathscr{U}$ of $X$, there exists a (discrete closed) countable subset $B$ of $X$ such that $St(B, \mathscr{U}) = X$. We investigate the relationship between these spaces and $\omega_{1}$-compact spaces, and also study topological properties of discretely star-Lindelöf spaces.

We give an example of non-local pseudo-differential operators which do not have the transmission property. The symbol of this operator is given by the composition of a complete Bernstein function by a quadratic form. The study is motivated by the boundary problem of non-local operators.

A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity. A rigid space can be constructed so as to admit compact operators [14]. This paper proves that the rigid space admitting compact operators constructed in [14] can be modified to be an $AR$, and hence is homeomorphic to the Hilbert space $\ell_{2}$.

Introducing a fictitious time evolution in a random matrix model having Gaussian entries, we prove that the empirical distribution of the scaled eigenvalues of the random matrix converges in probability to the Wigner's semi-circle law.

The index forms of minimal anti invariant submanifolds in Sasakian and Kähler manifolds are obtained. We give lower bounds for the index of these submanifolds in terms of their intrinsic quantities. Certain stability of the minimal submanifolds is also considered, which is related to eigenspaces of the Laplacian.