The Hillam–Thron Theorem in Higher Dimensions

Let D be a disc in the complex plane, and let s _n be a sequence of Möbius maps each of which maps D into itself. In 1965 Hillam and Thron proved (essentially) that if the points s _n (), n=12,..., lie in a compact subset of D then the functions s ₁^...^s _n converge locally uniformly to a constant on D. They then made applications to continued fractions. In this paper, the corresponding results are proved in all dimensions.     

The Brezis–Browder Theorem in a general Banach space

During the 1970s Brezis and Browder presented a now classical characterization of maximal monotonicity of monotone linear relations in reflexive spaces. In this paper, we extend (and refine) their result to a general Banach space. We also provide an affirmative answer to a problem posed by Phelps and Simons.     

The Loomis–Sikorski Theorem revisited

We prove, constructively, that the Loomis–Sikorski Theorem for σ-complete Boolean algebras follows from a representation theorem for Archimedean vector lattices and a constructive representation of Boolean algebras as spaces of Carathéodory place functions. We also prove a constructive subdirect product representation theorem for arbitrary partially ordered vector spaces. Received August 10, 2006; accepted in final form May 30, 2007.     

The Cauchy–Kowalewski Theorem in the Space of Pseudo Q-holomorphic Functions

In this work, we prove the Cauchy–Kowalewski theorem for the initial-value problem

$$\begin{aligned} \frac{\partial w}{\partial t}= & {} Lw \\ w(0,z)= & {} w_{0}(z) \end{aligned}$$

where

$$\begin{aligned} Lw:= & {} E_{0}(t,z)\frac{\partial }{\partial \overline{\phi }}\left( \frac{ d_{E}w}{dz}\right) +F_{0}(t,z)\overline{\left( \frac{\partial }{\partial \overline{\phi }}\left( \frac{d_{E}w}{dz}\right) \right) }+C_{0}(t,z)\frac{ d_{E}w}{dz} \\&+G_{0}(t,z)\overline{\left( \frac{d_{E}w}{dz}\right) } +A_{0}(t,z)w+B_{0}(t,z)\overline{w}+D_{0}(t,z) \end{aligned}$$

in the space \(P_{D}\left( E\right) \) of Pseudo Q-holomorphic functions.

     

A Refinement of The Fatou–Naim–Doob Theorem

We introduce the notion of harmonic thin sets and establish a refinement of the Fatou–Naim–Doob Theorem in the axiomatic system of Brelot, with an added assumptiom which is fufilled for the classical Laplace operator in ⁿ. We verify this assumption for the Poisson–Szegö integrals on the unit ball in ⁿ as well as the Weinstein equation on a halfspace in ⁿ. Thus, a stronger version of the Fatou–Naim–Doob Theorem is given in those cases. The assumption is expected to be verified for a large class of second order elliptic differential operators. The application of our result to sets of determination (for harmonic functions), introduced by Beurling [2], Dahlberg [9], Bonsall [3], and Hayman and Lyons [16], will follow in a separate paper.     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

Infinitely many universally tight torsion free contact structures with vanishing Ozsváth–Szabó contact invariants

Ozsváth–Szabó contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsváth–Szabó invariant vanishes. We also discuss the relation between these invariants and an invariant on T³ and construct other examples of new phenomena in Heegaard–Floer theory. Along the way, we prove two conjectures of K. Honda, W. Kazez and G. Matić about their contact topological quantum field theory. Almost all the proofs in this paper rely on their gluing theorem for sutured contact invariants.     

The Proof of a Theorem in DC Problem

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The FRT-type Theorem for the Hom–Long Equation

In this article, we introduce the concepts of Hom–Long equation and Hom–Long dimodule. We discuss how Hom–Long dimodules are connected to the Hom–Long equation. As the main result, we study the FRT-type theorem for the Hom–Long equation.     

The Brauer–Ostrowski Theorem for Matrices of Operators

The classical Brauer-Ostrowski Theorem gives a localization of the spectrum of a matrix by a union of Cassini ovals. In this paper we prove a corresponding result for operator matrices.     

On differential–algebraic equations in infinite dimensions

We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.     

Generalized Ginzburg–Landau equations in high dimensions

In this work, we study critical points of the generalized Ginzburg–Landau equations in dimensions \(n\ge 3\) which satisfy a suitable energy bound, but are not necessarily energy-minimizers. When the parameter in the equations tend to zero, such solutions are shown to converge to singular n-harmonic maps into spheres, and the convergence is strong away from a finite set consisting (1) of the infinite energy singularities of the limiting map, and (2) of points where bubbling off of finite energy n-harmonic maps could take place. The latter case is specific to dimensions \({>}2\). We also exhibit a criticality condition satisfied by the limiting n-harmonic maps which constrains the location of the infinite energy singularities. Finally we construct an example of non-minimizing solutions to the generalized Ginzburg–Landau equations satisfying our assumptions.     

The Erdős–Ko–Rado Theorem for Integer Sequences

n , q, t we determine the maximum number of integer sequences which satisfy for , and any two sequences agree in at least t positions. The result gives an affirmative answer to a conjecture of Frankl and Füredi. Received: January 26, 1998     

The Atiyah–Bott–Lefschetz Theorem for Manifolds with Conical Singularities

We establish an Atiyah–Bott–Lefschetz formula for elliptic operators on manifolds with conical singular points.     

The Paley–Wiener Theorem and Limits of Symmetric Spaces

We extend the Paley–Wiener theorem for Riemannian symmetric spaces to an important class of infinite-dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.     

The Schur–Weyl Graph and Thoma’s Theorem

Functional Analysis and Its Applications - We define a graded graph, called the Schur–Weyl graph, which arises naturally when one considers simultaneously the RSK algorithm and the classical...     

Note On The Utility Of A Theorem Of Leindler–Meir–Totik

Our purpose is to generalize and to extend a theorem of S. Sharma and S. K. Varma [15] concerning the order of approximation by Abel means in the Lipschitz norm. The proof is basically based on a simple extension of a general theorem of L. Leindler, A. Meir and V. Totik [6] related to approximation by finite summability methods.     

The Homotopy Fixed Point Theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory

We settle two conjectures for computing higher Grothendieck–Witt groups (also known as Hermitian K-groups) of noetherian schemes X, under some mild conditions. It is shown that the comparison map from the Hermitian K-theory of X to the homotopy fixed points of K  -theory under the natural Z/2$\mathbb{Z}/2$-action is a 2-adic equivalence. We also prove that the mod 2^ν$2^{\nu }$ comparison map between the Hermitian K-theory of X and its étale version is an isomorphism on homotopy groups in the same range as for the Quillen–Lichtenbaum conjecture in K-theory. Applications compute higher Grothendieck–Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.