We give a list of nonisolated hypersurface singularities of which normalizations are the rational triple point singularities of surfaces and construct their minimal resolution graphs by a method introduced by Oka for isolated complete intersection singularities. We also show that nonisolated forms of rational triple point singularities and their normalizations are both Newton non-degenerate.

Let $X$ be a completely regular Hausdorff space. We denote by $C(X,A)$ the algebra of all continuous functions on $X$ with values in a complex commutative unital Banach algebra $A$. Let $C_{b}(X,A)$ be its subalgebra consisting of all bounded continuous functions and endowed with the uniform norm. In this paper, we give conditions equivalent to the density of a natural continuous image of $X\times \mathcal {M}(A)$ in the maximal ideal space of $C_{b}(X,A)$.

In this paper, we find the group of automorphisms (up to finite index) for K3 surfaces in a class of Wehler K3 surfaces with Picard number $4$. In doing so, we demonstrate a variety of techniques, both general and ad hoc, that can be used to find the group of automorphisms of a K3 surface, particularly those with small Picard number.

Let $A$ be a local Artinian Gorenstein algebra with maximal ideal $\fM $, \[P_A(z) := \sum _{p=0}^{\infty } (\tor _p^A(k,k))z^p \] its Poicar\'{e} series. We prove that $P_A(z)$ is rational if either $\dim _k({\fM ^2/\fM ^3}) \leq 4 $ and $ \dim _k(A) \leq 16,$ or there exist $m\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $ m$ for $n\in [2,c]$ and equal to $1$ for $n =c+1$. The results are obtained due to a decomposition of the apolar ideal $\Ann (F)$ when $F=G+H$ and $G$ and $H$ belong to polynomial rings in different variables.

In this short note, we give a proof of the Riemann hypothesis for the Goss $v$-adic zeta function $\zeta _{v}(s)$, when $v$ is a prime of $\mathbb {F}_{q}[t]$ of degree one.

This paper deals with the problem of finding two new closed formulae for linearization coefficients of two special nonsymmetric cases for Jacobi polynomials $P^{(\alpha ,\beta )}_n(x)$ corresponding to the parameters' values $\beta =-\alpha =\pm 1/2$. From these two formulae, the linearization coefficients of the products of Chebyshev polynomials of the third and fourth kinds are established. Based on using algorithmic methods, such as the algorithms by Zeilberger, Petkovsek and Van-Hoeij, and two certain Whipple's transformations, six new closed formulae for summing certain terminating hyper\-geometric functions of unit argument are given.

In this paper, we deal with comparison theorems for the first eigenvalue of the Schr\"{o}dinger operator, and we present some criteria for the compactness of a Riemannian manifold in terms of the eigenfunctions of its Laplacian. Firstly, we establish a comparison theorem for the first Dirichlet eigenvalue $\mu _1^D (B (p,r))$ of a given Schr\"{o}dinger operator. We then prove that, for the space form $ M_K^n$ with constant sectional curvature $K$, the first eigenvalue of the Laplacian operator $\lambda _1 (M_K^n)$ is greater than the limit of the corresponding first Dirichlet eigenvalue $\lambda _1^D(B_K(p,r))$. Based on these, we present a characterization of a compact gradient shrinking Ricci soliton locally being an $n$-dim space form by the first eigenfunctions of the Laplacian operator, which gives a generalization of an interesting result by Cheng \cite {4} from 2-dim to $n$-dim. This result also gives a partial proof of a conjecture by Hamilton \cite {7} that a compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. Finally, we derive a criterion of the compactness of manifolds, which gives a partial proof of another conjecture by Hamilton \cite {6} that, if a complete Riemannian 3-manifold $(M^3, g)$ satisfies the Ricci pinching condition $Rc \ge \varepsilon Rg$, where $R > 0$ and $\varepsilon $ is a positive constant, then it is compact. In fact, our result is also true for general $n$-dim manifolds.

This paper determines the inertia groups (isotropy groups) of the points of a toric Deligne-Mumford stack $[Z/G]$ (considered over the category of smooth manifolds) that is realized from a quotient construction using a stacky fan or stacky polytope. The computation provides an explicit correspondence between certain geometric and combinatorial data. In particular, we obtain a computation of the connected component of the identity element $G_0 \subset G$ and the component group $G/G_0$ in terms of the underlying stacky fan, enabling us to characterize the toric DM stacks which are global quotients. As another application, we obtain a characterization of those stacky polytopes that yield stacks equivalent to weighted projective stacks and, more generally, to \textit {`fake' weighted projective stacks}. Finally, we illustrate our results in detail in the special case of \textit {labeled sheared simplices}, where explicit computations can be made in terms of the facet labels.

It is well known that, if the ring has acc on essential right ideals, then for every quasi-continuous module over the ring, the finite exchange property implies the full exchange property. In this paper, we obtain the former implication for the generalizations of quasi-continuous modules over a ring with acc on right annhilators of elements of the module. Moreover, we focus on direct sums and direct summands of weak $C_{12} $ modules i.e., modules with the property that every semisimple submodule can be essentially embedded in a direct summand. To this end, we prove that since weak $ C_{12} $ is closed under direct sums. Amongst other results, we provide several counterexamples including the tangent bundle of a real sphere of odd dimension over its coordinate ring for the open problem of whether weak $ C_{12}$ implies the $ C_{12} $ condition.

For arbitrary real closed fields $R$, we study the structure of the space $M(R(y))$ of $\mathbb{R}$-places of the rational function field in one variable over $R$ and determine its dimension to be 1. We determine small subbases for its topology and discuss a suitable metric in the metrizable case. In the case of non-archimedean $R$, we exhibit the rich variety of homeomorphisms of subspaces that can be found in such spaces.

Dorff et al.~\cite {do and no} proved that the harmonic convolutions of the standard right half-plane mapping $F_0=H_0+\overline {G}_0$ (where $H_0+G_0=z/(1-z)$ and $G_0'=-zH_0'$) and mappings $f_\beta =h_\beta +\overline {g}_{\beta }$ (where $f_\beta $ are obtained by shearing of analytic vertical strip mappings with dilatation $e^{i\theta }z^n$, $n=1,2$, $\theta \in \mathbb {R}$) are in $S_H^0$ and are convex in the direction of the real axis. In this paper, by using Cohn's rule, we generalize this result by replacing the standard right half-plane mapping $F_0$ with a family of right half-plane mappings $F_a=H_a+\overline {G}_a$ (with $H_a+G_a=z/(1-z)$ and $G'_a/H'_a= {(a-z)}/{(1-az)}$, $a\in (-1,1)$) and including the cases $n=3$ and $n=4$ (in addition to $n=1$ and $n=2$) for dilatations of $f_\beta $.

We shall present an elementary approach to extremal decompositions of (quantum) covariance matrices determined by densities. We give a new proof on former results and provide a sharp estimate of the ranks of the densities that appear in the decomposition theorem.

This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a ``boundary'' divisor, and we prove general tropical versions of the WDVV, respectively, topological recursion equations (under some assumptions). As a direct application, we prove that, for the toric varieties $\PP ^1$, $\PP ^2$, $\PP ^1 \times \PP ^1$ and with $\Psi $-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $\PP ^2$ in \cite {MR08}). Our approach uses tropical intersection theory and unifies and simplifies some parts of the existing tropical enumerative geometry (for rational curves).

Let $X_0$ be a smooth hypersurface (assumed not to be generic) in projective space $\mathbf {P}^n$, $n\geq 4$, over complex numbers, and $C_0$ a smooth rational curve on $X_0$. We are interested in deformations of the pair $C_0$ and $X_0$. In this paper, we prove that, if the first order deformations of the pair exist along each deformation of the hypersurface $X_0$, then $\deg (C_0)$ cannot be in the range \[ \bigg ( m\frac {2\deg (X_0)+1}{\deg (X_0)+1}, \frac {2+m(n-2)}{2n-\deg (X_0)-1}\bigg ), \] where $m$ is any non negative integer less than \[ \dim (H^0(\mathcal {O}_{\mathbf {P}^n}(1))|_{C_0} )-1. \]

In this paper, a generalization of Ramanujan's cubic transformation, in the form of an inequality, is proved for zero-balanced Gaussian hypergeometric function $F(a,b;a+b;x)$, $a,b>0$.