On uniform estimate in Calabi-Yau theorem

We show that the uniform estimate in the Calabi-Yau theorem easily follows from the local stability of the complex Monge-Ampère equation.     

On uniform estimate in Calabi-Yau theorem

We show that the uniform estimate in the Calabi-Yau theorem easily follows from the local stability of the complex Monge-Ampere equation.     

On a uniform estimate for the quaternionic Calabi problem

We establish a C ⁰ a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampère type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version of the Gauduchon theorem.     

A uniform Birkhoff theorem

Birkhoff’s HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra B satisfies all equations that hold in an algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a (possibly infinite) direct power of A. The former statement is equivalent to the existence of a natural map sending term functions of the algebra A to those of B—the natural clone homomorphism. The study of continuity properties of natural clone homomorphisms has been initiated recently by Bodirsky and Pinsker for locally oligomorphic algebras.Revisiting the argument of Bodirsky and Pinsker, we show that for any algebra B in the variety generated by an algebra A, the induced natural clone homomorphism is uniformly continuous if and only if every finitely generated subalgebra of B is a homomorphic image of a subalgebra of a finite power of A. Based on this observation, we study the question as to when Cauchy continuity of natural clone homomorphisms implies uniform continuity. We introduce the class of almost locally finite algebras, which encompasses all locally oligomorphic as well as all locally finite algebras, and show that, in case A is almost locally finite, then the considered natural homomorphism is uniformly continuous if (and only if) it is Cauchy-continuous. In particular, this provides a locally finite counterpart of the result by Bodirsky and Pinsker. Along the way, we also discuss some peculiarities of oligomorphic permutation groups on uncountable sets.     

On generalized Calabi-Yau nilmanifolds

We consider two possible definitions generalizing the notion of Calabi-Yau manifolds, and we describe some examples of these structures. Moreover, we prove a classification theorem for four and six complex dimensional nilmanifolds admitting an invariant generalized Calabi-Yau structure.     

On an estimate of Cranston and McConnell for elliptic diffusions in uniform domains

Summary We show that ifD ⁿ ,n3,n3, is a bounded uniform domain, then the lifetime of the Doobh-paths inD for elliptic diffusions in divergence form is finite. This result holds for any bounded domainD in the plane.Research supported by a Bantrell Fellowship     

A uniform estimate for rough paths

It is well known that for two p  -rough paths, if their first ⌊p⌋$\lfloor p\rfloor$ levels of iterated integrals are close in p  -variation sense, then all levels of their iterated integrals are close. In this paper, we prove that a similar result holds for the paths provided the first ⌊p⌋$\lfloor p\rfloor$ terms are close in a ‘uniform’ sense. The estimate is explicit, dimension free, and only involves the p  -variation of two paths and the ‘uniform’ distance between the first ⌊p⌋$\lfloor p\rfloor$ terms. Applications include estimation of the difference of the signatures of two uniformly close paths (Lyons and Xu, 2011 [6]), and convergence rates for Gaussian rough paths (Riedel and Xu, 2012 [7]).     

Decidable fan theorem and uniform continuity theorem with continuous moduli

The uniform continuity theorem $(\mathsf{UCT})$ states that every pointwise continuous real-valued function on the unit interval is uniformly continuous. In constructive mathematics, $\mathsf{UCT}$ is strictly stronger than the decidable fan theorem $(\mathsf{DFT})$, but Loeb [17] has shown that the two principles become equivalent by encoding continuous real-valued functions as type-one functions. However, the precise relation between such type-one functions and continuous real-valued functions (usually described as type-two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus of a continuous real-valued function on [0, 1], and show that real-valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that $\mathsf{DFT}$ is equivalent to the statement that every pointwise continuous real-valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that $\mathsf{DFT}$ is equivalent to a similar principle for real-valued functions on the Cantor space ${\{0,1\}}^{\mathbb{N}}$. These results extend Berger's [2] characterisation of $\mathsf{DFT}$ for integer-valued functions on ${\{0,1\}}^{\mathbb{N}}$ and unify some characterisations of $\mathsf{DFT}$ in terms of functions having continuous moduli.     

Topological uniform descent and Weyl type theorem

The generalized Weyl’s theorem holds for a Banach space operator T if and only if T or T^∗ has the single valued extension property in the complement of the Weyl spectrum (or B-Weyl spectrum) and T has topological uniform descent at all λ which are isolated eigenvalues of T. Also, we show that the generalized Weyl’s theorem holds for analytically paranormal operators.     

A Skorohod representation theorem for uniform distance

Let??? _n be a probability measure on the Borel ??-field on D[0, 1] with respect to Skorohod distance, n ?? 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X _n such that X _n ~ ?? _n for all n ?? 0 and ||X _n ? X ₀|| ?? 0 in probability, where ||·|| is the sup-norm. Such conditions do not require??? ₀ separable under ||·||. Applications to exchangeable empirical processes and to pure jump processes are given as well.