On the Lusternik-Schnirelmann category of Stiefel manifolds

We determine the Lusternik-Schnirelmann category of real Stiefel manifolds Vn,k and quaternionic Stiefel manifolds Xn,k for n?2k which is equal to the cup-length of the mod 2 cohomology of Vn,k and the integer cohomology of Xn,k, respectively.     

A quantum cup-length estimate for symplectic fixed points

A new lower bound for the number of fixed points of Hamiltonian automorphisms of closed symplectic manifolds (M,ω) is established. The new estimate extends the previously known estimates to the class of weakly monotone symplectic manifolds. We prove for arbitrary closed symplectic manifolds with rational symplectic class that the cup-length estimate holds true if the Hofer energy of the Hamiltonian automorphism is sufficiently small. For arbitrary energy and on weakly monotone symplectic manifolds we define an analogon to the cup-length based on the quantum cohomology ring of (M,ω) providing a quantum cup-length estimate. Oblatum 12-IX-1997     

The Conley index, cup-length and bifurcation

A module structure of the cohomology Conley index is used to define a relative cup-length. This invariant is applied then to prove a multiplicity theorem for periodic solutions to Hamiltonian systems.     

Evolution of conformal maps with quasiconformal extensions

We study one-parameter curves on the universal Teichmüller space T and on the homogeneous space M=DiffS¹/RotS¹ embedded into T. As a result, we deduce evolution equations for conformal maps that admit quasiconformal extensions and, in particular, such that the associated quasidisks are bounded by smooth Jordan curves. This approach allows us to understand the Laplacian growth (Hele-Shaw problem) as a flow in the Teichmüller space.     

Approximation of analytic functions by Legendre functions

We will solve the inhomogeneous Legendre’s differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate Legendre function.     

A Remark on Fine Differentiability

In [7], B. Fuglede has proved that finely holomorphic functions on a finely open subset U of the complex plane C are finely locally extendable to usual continuously differentiable functions. We shall adopt B. Fuglede’s approach to show that the same remains true even for functions which have only finely continuous fine differential on U. In higher dimensions, an analogous result may be obtained and the result can be applied to finely monogenic functions which were introduced recently as a higher dimensional analogue of finely holomorphic functions. I acknowledge the financial support from the grant GA 201/05/2117. This work is also a part of the research plan MSM 0021620839, which is financed by the Ministry of Education of the Czech Republic.     

Nonvanishing of Koecher–Maass series attached to Siegel cusp forms

We prove a nonvanishing result for Koecher–Maass series attached to Siegel cusp forms of weight k and degree n   in certain strips on the complex plane. When n=2$n=2$, we prove such a result for forms orthogonal to the space of the Saito–Kurokawa lifts ‘up to finitely many exceptions’, in bounded regions.     

Cellularity of twisted semigroup algebras

In this paper, the cellularity of twisted semigroup algebras over an integral domain is investigated by introducing the concept of cellular twisted semigroup algebras of type JH. Partition algebras, Brauer algebras and Temperley-Lieb algebras all are examples of cellular twisted semigroup algebras of type JH. Our main result shows that the twisted semigroup algebra of a regular semigroup is cellular of type JH with respect to an involution on the twisted semigroup algebra if and only if the twisted group algebras of certain maximal subgroups are cellular algebras. Here we do not assume that the involution of the twisted semigroup algebra induces an involution of the semigroup itself. Moreover, for a twisted semigroup algebra, we do not require that the twisting decomposes essentially into a constant part and an invertible part, or takes values in the group of units in the ground ring. Note that trivially twisted semigroup algebras are the usual semigroup algebras. So, our results extend not only a recent result of East, but also some results of Wilcox.     

Compact embeddings and proper mappings

Sufficient conditions are given to assert that a C¹-proper mapping defined on a compact embedding between Banach spaces over K has a fixed point. The proof of the result is based upon continuation methods.     

Continuous embeddings and continuation methods

Sufficient conditions are given to assert that a compact mapping defined on a continuous embedding between Hilbert spaces over K$\mathbb{K}$ has a fixed point. The proof of the result is based upon continuation methods.     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

Path regularity and explicit convergence rate for BSDE with truncated quadratic growth

We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang’s path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and then derive an analogous result for qgBSDE.     

The structure of Valdivia compact lines

We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The conjecture asserted that a compact line is Valdivia compact if its weight does not exceed ℵ₁, every point of uncountable character is isolated from one side and every closed first countable subspace is metrizable. It turns out that the last condition is not sufficient. On the other hand, we show that the conjecture is valid if the closure of the set of points of uncountable character is scattered. This improves an earlier result of the first author.     

Approximation of analytic functions by Hermite functions

We solve the inhomogeneous Hermite equation and apply this result to estimate the error bound occurring when any analytic function is approximated by an appropriate Hermite function.     

A λ-lemma for foliations

We show that a C⁰ codimension one foliation with C¹ leaves F of a closed manifold is minimal if there are a foliation G transverse to F, and a diffeomorphism f preserving both foliations, such that every leaf of F intersects every leaf of G and f expands G. We use this result to study of Anosov actions on closed manifolds.     

Universal Teichmüller space and BMO

We first give some new characterizations on BMOA–Teichmüller space and various characterizations on VMOA–Teichmüller space as well. In particular, we prove that a quasisymmetric conformal welding h$h$ corresponds to an asymptotically smooth curve in the sense of Pommerenke (1978) [32] precisely when h$h$ is absolutely continuous with logh^′∈VMO$log{h}^{\prime }\in \text{VMO}$. We then show that these BMO–Teichmüller spaces have natural complex structures.     

Formal Hopf algebra theory I: Hopf modules for pseudomonoids

In this article we develop some of the basic constructions of the theory of Hopf algebras in the context of autonomous pseudomonoids in monoidal bicategories. We concentrate on the notion of Hopf modules. We study the existence and the internalisation of this notion, called the Hopf module construction. Our main result is the equivalence between the existence of a left dualization for A (i.e., A is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case a Hopf module construction for A always exists. We recover from the general theory developed here results on coquasi-Hopf algebras.     

On the existence of local strong solutions for the Navier-Stokes equations in completely general domains

There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω⊆R³, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the L^q-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution u∈L⁸(0,T;L⁴(Ω)) in some interval [0,T), 0<T≤∞, with u(0)=u₀, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u₀∈D(A^1/4) is strictly stronger and therefore not optimal.     

Optimal recovery from a finite set in Banach spaces of entire functions

We develop an approach to the problem of optimal recovery of continuous linear functionals in Banach spaces through information on a finite number of given functionals. The results obtained are applied to the problem of the best analytic continuation from a finite set in the complex space Cn, n?1, for classes of entire functions of exponential type which belong to the space Lp, 1<p<∞, on the real subspace of Cn. These latter are known as Wiener classes.     

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne–Simpson problem.