Volume 34, Issue 1

For four wide classes of topological rings R\mathfrak{R}, we show that all flat left R\mathfrak{R}-contramodules have projective covers if and only if all flat left R\mathfrak{R}-contramodules are projective if and only if all left R\mathfrak{R}-contramodules have projective covers if and only if all descending chains of cyclic discrete right R\mathfrak{R}-modules terminate if and only if all the discrete quotient rings of R\mathfrak{R} are left perfect. Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhoods of zero formed by open two-sided ideals such that either the ring is commutative, or it has a countable base of neighborhoods of zero, or it has only a finite number of semisimple discrete quotient rings. The fourth class consists of all the topological rings with a base of neighborhoods of zero formed by open right ideals which have a closed two-sided ideal with certain properties such that the quotient ring is a topological product of rings from the previous three classes. The key technique on which the proofs are based is the contramodule Nakayama lemma for topologically T-nilpotent ideals.

In this paper, we consider ℤr{\mathbb{Z}^{r}}-graded modules on the Cl⁡(X){\operatorname{Cl}(X)}-graded Cox ring ℂ⁢[x1,…,xr]{\mathbb{C}[x_{1},\ldots,x_{r}]} of a smooth complete toric variety X . Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive ℤs+r+2{\mathbb{Z}^{s+r+2}}-graded modules over any non-standard bigraded polynomial ring ℂ⁢[x0,…,xs,y0,…,yr]\mathbb{C}[x_{0},\ldots,x_{s},\allowbreak y_{0},\ldots,y_{r}]. In this case, we give sharp bounds for the multigraded regularity index of their multigraded Hilbert function, and a method to compute their Hilbert polynomial.

The 4⁢N{4N}-carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4⁢N{4N}-carpet F , let {Fn}n≥0{\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂nFn=F{\bigcap_{n}F_{n}=F}. On each Fn{F_{n}}, let ℰ⁢(u,v)=∫FN∇⁡u⋅∇⁡v⁢d⁢x{\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and un{u_{n}} be the unique harmonic function on Fn{F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ=ρ⁢(N)>1{\rho=\rho(N)>1} such that ℰ⁢(un,un)⁢ρn{\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n . Such estimates have implications for the existence and scaling properties of Brownian motion on F .

In the present paper, we study the Newton polygons for the L -functions of n -variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top-dimensional Dwork cohomology. Using Wan’s decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for the L -function of F¯⁢(λ¯,x):=∑i=1nxiai+λ¯⁢∏i=1nxi-1,\bar{F}(\bar{\lambda},x):=\sum_{i=1}^{n}x_{i}^{a_{i}}+\bar{\lambda}\prod_{i=1}% ^{n}x_{i}^{-1}, with a1,…,an{a_{1},\ldots,a_{n}} being pairwise coprime for n≥2{n\geq 2}.

We show that locally compact abelian p -groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

Let 𝔛{\mathfrak{X}} be a homogeneous tree and let ℒ{\mathcal{L}} be the Laplace operator on 𝔛{\mathfrak{X}}. In this paper, we address problems of the following form: Suppose that {fk}k∈ℤ{\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛{\mathfrak{X}} such that for all k∈ℤ{k\in\mathbb{Z}} one has ℒ⁢fk=A⁢fk+1{\mathcal{L}f_{k}=Af_{k+1}} and ∥fk∥≤M{\lVert f_{k}\rVert\leq M} for some constants A∈ℂ{A\in\mathbb{C}}, M>0{M>0} and a suitable norm ∥⋅∥{\lVert\,\cdot\,\rVert}. From this hypothesis, we try to infer that f0{f_{0}}, and hence every fk{f_{k}}, is an eigenfunction of ℒ{\mathcal{L}}. Moreover, we express f0{f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛{\mathfrak{X}}.

This paper is concerned with Hankel determinants for starlike and convex functions related to modified sigmoid functions. Sharp bounds are given for second and third Hankel determinants.

We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.

The purpose of this paper is to establish a necessary and sufficient condition for the boundedness of general product singular integral operators introduced by Han, Li and Lin [Y. Han, J. Li and C.-C. Lin, Criterion of the L2L^{2} boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 2016, 3, 845–907] on the multiparameter Lipschitz spaces of homogeneous type M~=M1×⋯×Mn{\tilde{M}=M_{1}\times\cdots\times M_{n}}. Each factor space Mi{M_{i}}, 1≤i≤n{1\leq i\leq n}, is a space of homogeneous type in the sense of Coifman and Weiss. These operators generalize those studied by Journé on the Euclidean space and include operators studied by Nagel and Stein on Carnot–Carathéodory spaces. The main tool used here is the discrete Littlewood–Paley–Stein theory and almost orthogonality together with a density argument for the product Lipschitz spaces in the weak sense.

We develop the Gompf fiber connected sum operation for symplectic orbifolds. We use it to construct a symplectic 4-orbifold with b1=0{b_{1}=0} and containing symplectic surfaces of genus 1 and 2 that are disjoint and span the rational homology. This is used in turn to construct a K-contact Smale–Barden manifold with specified 2-homology that satisfies the known topological constraints with sharper estimates than the examples constructed previously. The manifold can be chosen spin or non-spin.

In this paper, we present a new approach to obtaining the off-diagonal upper estimate of the heat kernel for any regular Dirichlet form without a killing part on the doubling space. One of the novelties is that we have obtained the weighted L2{L^{2}}-norm estimate of the survival function 1-PtB⁢1B{1-P_{t}^{B}1_{B}} for any metric ball B , which yields a nice tail estimate of the heat semigroup associated with the Dirichlet form. The parabolic L2{L^{2}} mean-value inequality is borrowed to use.