A Banach–Stone Theorem for Uniformly Continuous Functions

 In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U(X) of all uniformly continuous real functions on X.     

A Banach–Stone Theorem for Uniformly Continuous Functions

 In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U(X) of all uniformly continuous real functions on X. (Received 3 March 2000; in revised form 29 June 2000)     

Strong Convergence Theorems for Perturbed Maximal Monotone Operators in Banach Spaces

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Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process

Journal of Theoretical Probability - We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein–Uhlenbeck process, providing the foundation for the...     

Cannon–Thurston maps for hyperbolic free group extensions

This paper gives a detailed analysis of the Cannon–Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E_Γ of F, and the main theorem of Dowdall–Taylor [DT14] states that in this situation E_Γ is hyperbolic if and only if Γ is purely atoroidal.     

Nonlinear Calderón–Zygmund inequalities for maps

Being motivated by the problem of deducing \(\mathsf {L}^{p}\)-bounds on the second fundamental form of an isometric immersion from \(\mathsf {L}^{p}\)-bounds on its mean curvature vector field, we prove a nonlinear Calderón–Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.     

Asymptotics for Szegö polynomial zeros

The Wiener-Levinson method and algorithm, formulated here in terms of Szegö polynomials _n ( _N,I ;z) orthogonal on the unit circle, is used to find unknown frequencies _j from anN-sample of a discrete time signal consisting of the superposition of sinusoidal waves with frequencies ₁,...,₁. In a recent paper the authors (and W.J. Thron) have shown that zerosz(j, n, N, I) of _n ( _N,I ;z) converge asN to the critical points ,j=1, 2,...,I, providednn ₀ (I)=2I+L, whereL is 0 or 1. The present paper gives results on the convergence of zerosz(j, n, N, I) to some of the for the case in whichnn ₀ (I), wheren is the degree of _n ( _N,I ;z).Research supported in part by the United States Educational Foundation in Norway (Fulbright Grant), the Norwegian Research Council (NAVF) and the US National Science Foundation under Grant No. DMS-9103141.     

A Banach–Zarecki Theorem for functions with values in Banach spaces

A Banach–Zarecki Theorem for a Banach space-valued function  \(F : [0,1] \rightarrow X\) with compact range is presented. We define the strong absolute continuity ( \(sAC_{||.||_{F}}\) ) and the bounded variation ( \(BV_{||.||_{F}}\) ) of \(F\) with respect to the Minkowski functional \(||.||_{F}\) associated to the closed absolutely convex hull \(C_{F}\) of \(F([0,1])\) . It is proved that \(F\) is \(sAC_{||.||_{F}}\) if and only if \(F\) is \(BV_{||.||_{F}}\) , weak continuous on \([0,1]\) and satisfies the weak property \((N)\) .     

Rigidity Theorems for Einstein–Thorpe Metrics Dedicated to my parents

We will prove that every Einstein–Thorpe metric on T ⁸ must be flat and that on compact oriented hyperbolic manifolds of dimension 8, every Einstein–Thorpe metric is a hyperbolic metric up to rescalings and diffeomorphisms.     

Addition–Deletion Theorems for factorizations of Orlik–Solomon algebras and nice arrangements

We study the notion of a nice partition or factorization of a hyperplane arrangement due to Terao from the early 1990s. The principal aim of this note is an analogue of Terao’s celebrated addition–deletion theorem for free arrangements for the class of nice arrangements. This is a natural setting for the stronger property of an inductive factorization of a hyperplane arrangement by Jambu and Paris.In addition, we show that supersolvable arrangements are inductively factored and that inductively factored arrangements are inductively free. Combined with our addition–deletion theorem this leads to the concept of an induction table for inductive factorizations.Finally, we prove that the notions of factored and inductively factored arrangements are compatible with the product construction for arrangements.     

On quasimöbius maps in real Banach spaces

Suppose that E and E′ denote real Banach spaces with dimension at least 2, that D $ \subseteq $ E and D′ $ \subseteq $ E′ are domains, that f: D → D′ is an (M,C)-CQH homeomorphism, and that D is uniform. The aim of this paper is to prove that D′ is a uniform domain if and only if f extends to a homeomorphism $\overline f :\overline D \to {\overline D ^\prime }$ and $\overline f $ is η-QM relative to ?D. This result shows that the answer to one of the open problems raised by Väisälä from 1991 is affirmative.     

Chermak–Delgado Lattice Extension Theorems

If G is a finite group with subgroup H, then the Chermak–Delgado measure of H (in G) is defined as |H||C _G (H)|. The Chermak–Delgado lattice of G, denoted 𝒞𝒟(G), is the set of all subgroups with maximal Chermak–Delgado measure; this set is a moduar sublattice within the subgroup lattice of G. In this paper we provide an example of a p-group P, for any prime p, where 𝒞𝒟(P) is lattice isomorphic to 2 copies of ?₂ (a quasiantichain of width 2) that are adjoined maximum-to-minimum. We introduce terminology to describe this structure, called a 2-string of 2-diamonds, and we also give two constructions for generalizing the example. The first generalization results in a p-group with Chermak–Delgado lattice that, for any positive integers n and l, is a 2l-string of n-dimensional cubes adjoined maximum-to-minimum and the second generalization gives a construction for a p-group with Chermak–Delgado lattice that is a 2l-string of ? _p+1 (quasiantichains, each of width p + 1) adjoined maximum-to-minimum.     

Radon–Nikodym Theorems for Nonnegative Forms,Measures and Representable Functionals

The aim of this paper is to establish two Radon–Nikodym-type theorems for nonnegative Hermitian forms defined on a real or complex vector space and to apply these results to provide some known Radon–Nikodym-type theorems of the theory of representable positive functionals, \(\sigma \)-additive and finitely additive measures.     

Erdős–Szekeres-Type Theorems for Segments and Noncrossing Convex Sets

A family of convex sets is said to be in convex position, if none of its members is contained in the convex hull of the others. It is proved that there is a function N(n) with the following property. If is a family of at least N(n) plane convex sets with nonempty interiors, such that any two members of have at most two boundary points in common and any three are in convex position, then has n members in convex position. This result generalizes a theorem of T. Bisztriczky and G. Fejes Tóth. The statement does not remain true, if two members of may share four boundary points. This follows from the fact that there exist infinitely many straight-line segments such that any three are in convex position, but no four are. However, there is a function M(n) such that every family of at least M(n) segments, any four of which are in convex position, has n members in convex position.