We deal with equations on post-composition of holomorphic functions on the open unit disk under the condition that quotients of de Branges-Rovnyak kernels are positive semi-definite.

There are a variety of characterizations in classical orthogonal polynomials. First of all, by explicit expressions, secondly by generating functions, thirdly as polynomial solutions of differential equations. There also exist other ways. Needless to say, these definitions are equivalent one another.

In the case of the disk polynomials, a similar situation occurs. However, there are few studies that refer to relations with their generating function. The purpose of this paper is to show by a heuristic method that the partial differential equations of second order which the disk polynomials satisfy are derived from their generating function.

Inequalities for positive linear maps of positive selfadjoint operators in Hilbert spaces via some recent reverses of Young's inequality are given. Operator and vector inequalities involving the weighted operator geometric mean are also obtained. Reverses of the celebrated Ando's inequality are provided.

We observe the curvature of four-dimensional almost Hermitian manifolds satisfying each of the three conditions by Gray. Further, we apply the obtained results to the integrability of certain classes of four-dimensional almost Kähler manifolds.

The modal logic KC4 is the smallest normal modal logic K with the additional axiom $\Box\Box A \mathbin{\supset} \Box A$ called C4, which claims density of the accessible relation. In this article, a modified subformula property for this logic is shown by devising an appropriate sequent calculus. The finite model property and decidability of this logic are easy corollaries.

It is peculiar that a modified subformula for this logic is defined for a set of formulas but not for a single formula.

In this paper, we consider a kind of discrete surfaces in the three-dimensional Euclidean space called a regular pentagon ring. It is a discrete surface obtained by attaching a finite number of pairwise congruent regular pentagons along their edges such that the closed polygonal line on the surface, which connects the midpoints of those edges with line segments, is a trivial knot.

In the main theorem, we will show that if a regular pentagon ring is planar, it can be folded in one regular pentagon.