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.
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.
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.
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.
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 Kahler manifolds.
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.
In the first and fourth authors' paper in 2017, it was shown that there exists a BSE-algebra of type I isomorphic to no C*-algebras, which solved negatively a question posed by the fourth author and O. Hatori. However, this result suggests a further investigation of commutative Banach algebra of type I. In the first part of the paper, we classify type I algebras into six families by means of BSE, BED, and Tauberian. It is shown that a Banach algebra of type I is isomorphic to a Segal algebra in some commutative C*-algebra if and only if it is Tauberian. In the second part, we give concrete examples of type I algebras to show that all of six families mentioned above are nonempty.
Let $\mathbb{P}^1(\overline{\mathbb{Q}})$ be the projective line over $\overline{\mathbb{Q}}$ and $H$ the Weil height on $\mathbb{P}^1(\overline{\mathbb{Q}})$. A classical result in algebraic number theory, so called Kronecker's theorem, states that $H(1,x)=1$ if and only if $x\in\overline{\mathbb{Q}}$ is 0 or a root of unity. In [4], Talamanca introduced some height functions on $M_n(\overline{\mathbb{Q}})$. The purpose of this paper is to show analogues of Kronecker's theorem for these heights: We determine height one matrices relative to these heights.
The invariant theory of finite groups can connect the coding theory to the number theory. In this paper, under this conformity, we obtain the minimal generators of the rings of E-polynomials constructed from the groups related to $\mathbb{Z}_4$-codes. In addition, we determine the generators of the invariant rings appearing by E-Polynomials and complete weight enumerators of Type II $\mathbb{Z}_4$-codes.
Tanabe [5] gives a geometrical proof of the existence of $\lim\frac{p(1,t)-1}{t}$ for a natural norm $p$ on ${\mathbb R}^2$. Following his idea, a shorter proof is given.
Let H(𝔻) be the linear space of all analytic functions on the open unit disc 𝔻 and H^p(𝔻) the Hardy space on 𝔻 . The characterization of complex linear isometries on 𝒮^p={ f∈ H(𝔻):f'∈ H^p(𝔻) } was given for 1 ≤ p < ∞ by Novinger and Oberlin in 1985. Here, we characterize surjective, not necessarily linear, isometries on 𝒮^∞ .
Let $k$ be a field of characteristic $p \geq 0$ and $A = k[x_0, x_1, x_2, \ldots]$ the polynomial ring in countably many variables over $k$. We construct a rational higher $k$-derivation on $A$ whose kernel is not the kernel of any higher $k$-derivation on $A$. This example extends [5, Example 4].
Let $X$ be a log del Pezzo surface of rank one. In [8], the first author determined the possible singularity type of $X$ when $X$ contains the affine plane as a Zariski open subset. In this paper, we prove that, if $X$ contains a non-cyclic quotient singular point and its singularity type is one of the list of [8, Appendix C], then it contains the affine plane as a Zariski open subset.
Let $S$ be a $\mathbb Q$-homology projective plane, $C$ a rational unicuspidal curve on $S^0 = S - \operatorname{Sing} S$ and $C'$ the proper transform of $C$ with respect to the minimal embedded resolution of $C$. We prove that $S^0 - C$ is affine ruled if and only if $C'^2 \geq -1$ and determine the pairs $(S,C)$ when $\overline{\kappa}(S^0 -C) = -\infty$ and $C'^2 \leq -2$.
In this paper we introduce the concept of quadratic quantum $f$-divergence measure for a continuos function $f$ defined on the positive semi-axis of real numbers, the invertible matrix $T$ and matrix $V$ by $$\mathcal{S}_{f}\left( V,T\right) :=\mathrm{tr}\left[ \left\vert T^{\ast }\right\vert ^{2}f\left( \left\vert VT^{-1}\right\vert ^{2}\right) \right].$$ Some fundamental inequalities for this quantum $f$-divergence in the case of convex functions are established. Applications for particular quantum divergence measures of interest are also provided.
We consider an asymptotic version of $\alpha$-$\psi$ contractive mappings. We show the existence and uniqueness of fixed points. Caccioppoli's fixed point theorem is deduced from main results in this paper. Moreover, we discuss an asymptotic version of mappings related with $(c)$-comaprison functions.
We produce the family of Calabi-Yau hypersurfaces $X_{n}$ of $(\mathbb{P}^{1})^{n+1}$ in higher dimension whose inertia group contains non commutative free groups. This is completely different from Takahashi's result \cite{ta98} for Calabi-Yau hypersurfaces $M_{n}$ of $\mathbb{P}^{n+1}$.
In this research, we propose computational methods to evaluate scalarizing functions, which are defined via set-relations. In recent years, many theoretical results of the scalarizing functions for sets have been published. The aim of this paper is to show that each value of the scalarizing functions can be computed and to introduce computational algorithms of them for convex polytopes in a finite dimensional space.
Real hypersurfaces satisfying the condition $\phi l = l \phi$, $(l = R( . , \xi)\xi)$, have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex hyperbolic plane $\mathbb{C}H^{2}$ satisfying a generalization of $\phi l = l \phi$ under an additional restriction on a specific function.
Due to Rentschler, Miyanishi and Kojima, the invariant ring for a G_a-action on the affine plane over an arbitrary field is generated by one coordinate. In this note, we give a new short proof for this result using the automorphism theorem of Jung and van der Kulk.
