Compactness for immersions of prescribed Gaussian curvature II: geometric aspects

We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used to solve the Plateau problem for immersions of constant (and prescribed) Gaussian curvature in general Hadamard manifolds.     

Uncountable categoricity for gross models

A model is said to be gross if all infinite definable sets in have the same cardinality (as ). We prove that if for some uncountable , has a unique gross model of cardinality , then for any uncountable , has a unique gross model of cardinality .

     

Uncountable Homogeneous Partial Orders

A partially ordered set (P, ≤) is called k‐homogeneous if any isomorphism between k‐element subsets extends to an automorphism of (P, ≤). Assuming the set‐theoretic assumption ⋄(ϰ₁), it is shown that for each k, there exist partially ordered sets of size ϰ₁ which embed each countable partial order and are k‐homogeneous, but not (k + 1)‐homogeneous. This is impossible in the countable case for k ≥ 4.     

Uncountable Cofinalities of Permutation Groups

A sufficient criterion is found for certain permutation groupsG to have uncountable strong cofinality, that is, they cannotbe expressed as the union of a countable, ascending chain (H_i)_i     

Compactness methods in the theory of homogenization II: Equations in non-divergence form

We prove C^{0, α}, C^{1, α} and C^{1, 1} a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form Σ,^j=1a^{i j}(x/?)δ²/δx_iδx_j. The constants in these estimates are independent of the small parameter ?, and hence our results imply strengthened versions of the classical averaging theorems. These results generalize to a wide class of linear operators in non-divergence form, including equations with lower-order terms and nonuniformly oscillating coefficients, as well as to certain nonlinear problems, which we discuss in the last section.     

Uncountable Strongly Surjective Linear Orders

Order - We call a linear order L strongly surjective if whenever K is a suborder of L then there is a surjective f : L → K so that x ≤ y implies f(x) ≤ f(y). We prove various...     

不可数多个GO-空间乘积的k-仿紧性

证明了不可数多个GO-空间的乘积是k-仿紧空间(相似文献   

Uncountable critical points for congruence lattices

The critical point between two classes \({{\mathcal K}}\) and \({{\mathcal L}}\) of algebras is the cardinality of the smallest semilattice isomorphic to the semilattice of compact congruences of some algebra in \({{\mathcal K}}\), but not in \({{\mathcal L}}\). Our paper is devoted to the problem of determining the critical point between two finitely generated congruence-distributive varieties. For a homomorphism \({\varphi: S \rightarrow T}\) of \({(0, \vee)}\)-semilattices and an automorphism \({\tau}\) of T, we introduce the concept of a \({\tau}\)-symmetric lifting of \({\varphi}\). We use it to prove a criterion which ensures that the critical point between two finitely generated congruence-distributive varieties is less or equal to \({\aleph_{1}}\). We illustrate the criterion by constructing two new examples with the critical point exactly \({\aleph_{1}}\).     

A Function with Locally Uncountable Rotation Set

The rotation set, , of a Lebesgue measurable real valued function on the circle is the set of R for which converges as n for almost every x. Z. L. Buczolich has constructed a non-integrable function whose rotation set contains at least countably many irrational rotations. In this paper we construct a function whose rotation set has uncountable intersection with each non-empty open subset of R.     

不分明化拓扑中近似紧性和几乎紧性

在不分明化拓扑空间中,从正则开集出发引入了近似紧性和几乎紧性的概念,并且给出了它们的一些性质.这些概念的结合有助于我们对不分明化拓扑的研究。     

L-Smooth紧

在模糊格上讨论Smooth拓扑空间的Smooth紧性,它定位于每一个模糊集,并兼顾整个Smooth拓扑空间。     

Uncountable Saturated Structures have the Small Index Property

We prove the following theorem. Let m be an uncountable saturatedstructure of cardinality = ^< and assume that G is a subgroupof Aut (m) whose index is less than or equal to . Then thereexists a subset A of cardinality strictly less than such thatevery automorphism of m leaving A pointwise fixed is in G.     

Uncountable families of vertex-transitive graphs of finite degree

We prove that the set of vertex-transitive graphs of finite degree is uncountably large.     

Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: existence

We apply the compactness results obtained in the first part of this work, to prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation

$$-\triangle u + V \left(\left|x\right|\right) u = g \left(\left|x\right|, u\right) \quad {\rm in} \Omega \subseteq \mathbb{R}^{N},\,N \geq 3,$$

where \({\Omega}\) is a radial domain (bounded or unbounded) and u satisfies u =  0 on \({\partial\Omega}\) if \({\Omega \neq\mathbb{R}^{N}}\) and \({u \rightarrow 0}\) as \({\left|x\right| \rightarrow \infty}\) if \({\Omega}\) is unbounded. The potential V may be vanishing or unbounded at zero or at infinity and the nonlinearity g may be superlinear or sublinear. If g is sublinear, the case with a forcing term \({g\left(\left|\cdot\right|, 0\right) \neq 0}\) is also considered. Our results allow to deal with V and g exhibiting behaviours at zero or at infinity which are new in the literature and, when \({\Omega = \mathbb{R}^{N}}\), do not need to be compatible with each other.

     

Construction of Some Uncountable 2-Arc-Transitive Bipartite Graphs

We give various constructions of uncountable arc-transitive bipartite graphs employing techniques from partial orders, starting with the cycle-free case, but generalizing to cases where this may be violated. This work was supported by a grant from the EPSRC.     

三维时滞微分系统的不可列个周期解

本文对一类三维时滞微分方程系统给出了存在无穷不可列个周期解的条件，并对有关文献中的一些不足之处作了讨论．     

On Uncountable Unions and Intersections of Measurable Sets

We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a good additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are considered in detail, and it is shown that the Lebesgue measure and the Baire property differ essentially in this aspect. Another difference between measure and category is illustrated in the case of some uncountable intersections of sets of full measure (comeager sets, respectively). We also discuss a topological form of the Vitali covering theorem, in connection with the Baire property of uncountable unions of certain sets.