A-stable Runge–Kutta methods for semilinear evolution equations

We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale. Under the same assumptions, we prove that a class of implicit, A-stable Runge–Kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale. Under the additional assumption that the linear part of the evolution equation is normal or sectorial, we prove full order convergence of the semidiscretization in time for initial data on open sets. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrödinger equation.     

Localization,completion and detecting equivariant maps on skeletons

In the category of equivariant spaces with base point, we prove the injectivity of the induced map between homotopy sets under some conditions. We study some relations between the localization and the completion. By using these results, we characterize continuous maps which are homotopic on skeletons, and obtain a generalization of the theory of phantom maps.     

Observations on quasi-uniform products

We prove that any product of quotient maps in the category of quasi-uniform spaces and quasi-uniformly continuous maps is a quotient map. We also show that a quasi-uniformly continuous map from a product of quasi-uniform spaces into a quasi-pseudometric T₀-space depends on countably many coordinates.Furthermore we characterize those quasi-uniformities that are unique in their quasi-proximity class and prove that this property is preserved under arbitrary products in the category of quasi-uniform spaces.     

A refinement of Stone duality to skew Boolean algebras

We establish two theorems that refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem, we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of étale spaces over locally compact Boolean spaces whose morphisms are étale space cohomomorphisms over continuous proper maps. In the second theorem, we prove that the category of left-handed skew Boolean -algebras whose morphisms are proper skew Boolean -algebra homomorphisms is equivalent to the category of étale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective étale space cohomomorphisms over continuous proper maps.     

On Maps Of Bounded p-Variation With p>1

This paper addresses properties of maps of bounded p-variation (p>1) in the sense of N. Wiener, which are defined on a subset of the real line and take values in metric or normed spaces. We prove the structural theorem for these maps and study their continuity properties. We obtain the existence of a Hölder continuous path of minimal p-variation between two points and establish the compactness theorem relative to the p-variation, which is an analog of the well-known Helly selection principle in the theory of functions of bounded variation. We prove that the space of maps of bounded p-variation with values in a Banach space is also a Banach space. We give an example of a Hölder continuous of exponent 0<<1 set-valued map with no continuous selection. In the case p=1 we show that a compact absolutely continuous set-valued map from the compact interval into subsets of a Banach space admits an absolutely continuous selection.     

On the complexity of expansive measures

We prove that the existence of positively expansive measures for continuous maps on compact metric spaces implies the existence of e 0 and a sequence of(m, e)-separated sets whose cardinalities go to infinite as m →∞. We then prove that maps exhibiting such a constant e and the positively expansive maps share some properties.     

Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.     

A duality of generalized metric spaces

We develop a duality theory for Lawvere?s generalized metric spaces that extends the Lawson duality for continuous dcpos and open filter reflecting maps: we prove that the category of relatively cocomplete and continuous [0,∞]-categories considered with open filter reflecting maps is self-dual.     

Ordered group-valued probability,positive operators,and integral representations

Probability maps are additive and normalised maps taking values in the unit interval of a lattice ordered Abelian group. They appear in theory of affine representations and they are also a semantic counterpart of Hájek’s probability logic. In this paper we obtain a correspondence between probability maps and positive operators of certain Riesz spaces, which extends the well-known representation theorem of real-valued MV-algebraic states by positive linear functionals. When the codomain algebra contains all continuous functions, the set of all probability maps is convex, and we prove that its extreme points coincide with homomorphisms. We also show that probability maps can be viewed as a collection of states indexed by maximal ideals of a codomain algebra, and we characterise this collection in special cases.     

Preference orders and continuous representations

In this paper we prove some general theorems on the existence of continuous order-preserving functions on topological spaces with a continuous preorder. We use the concepts of network and netweight to prove new continuous representation theorems and we establish our main results for topological spaces that are countable unions of subspaces. Some results in the literature on path-connected, locally connected and separably connected spaces are shown to be consequences of the general theorems proved in the paper. Finally, we prove a continuous representation theorem for hereditarily separable spaces.     

Dimension zero at all scales

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale dimension. We show that in all categories a space has dimension zero if and only if it is equivalent to an ultrametric space. Also, 0-dimensional spaces are characterized by means of retractions to subspaces. There is a universal zero-dimensional space in all categories. In the Lipschitz Category spaces of dimension zero are characterized by means of extensions of maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is coarsely equivalent to a direct sum of cyclic groups. We construct uncountably many examples of coarsely inequivalent ultrametric spaces.     

Cheeger Type Sobolev Spaces for Metric Space Targets

In this paper, we consider the natural generalization of Cheeger type Sobolev spaces to maps into a metric space. We solve Dirichlet problem for CAT(0)-space targets, and obtain some results about the relation between Cheeger type Sobolev spaces for maps into a Banach space and those for maps into a subset of that Banach space. We also prove the minimality of upper pointwise Lipschitz constant functions for locally Lipschitz maps into an Alexandrov space of curvature bounded above.     

General logical metatheorems for functional analysis

In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Hölder-Lipschitz, uniformly continuous, bounded and weakly quasi-nonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

     

Fredholm theory and transversality for noncompact pseudoholomorphic mapsin symplectizations

We study pseudoholomorphic maps from a punctured Riemann surface into the symplectization of a contact manifold. A Fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the Euler characteristic of the Riemann surface and the asymptotics data given by the periodic solutions of the Reeb vector field associated to the contact form. The transversality results establish the existence of additional structure for these spaces. To be more precise, we prove that these spaces are generically smooth manifolds, and therefore their virtual dimension coincides with their actual dimension. © 2004 Wiley Periodicals, Inc.     

Entropy formula and continuity of entropy for piecewise expanding maps

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.     

Zig-zag chains and metric equivalences between ultrametric spaces

We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (bi-uniform equivalences). Using a combinatoric approach we consider every ultrametric space as the end space of a chain and prove that all these equivalences can be characterized by the existence of a common zig-zag chain.     

Some generalizations of Fedorchuk duality theorem—I

Generalizing duality theorem of V.V. Fedorchuk [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. Mat. Zh. 14 (5) (1973) 1088-1099; English translation: Siberian Math. J. 14 (1973) 759-767 (1974)], we prove Stone-type duality theorems for the following four categories: the objects of all of them are the locally compact Hausdorff spaces, and their morphisms are, respectively, the continuous skeletal maps, the quasi-open perfect maps, the open maps, the open perfect maps. In particular, a Stone-type duality theorem for the category of compact Hausdorff spaces and open maps is obtained. Some equivalence theorems for these four categories are stated as well; two of them generalize the Fedorchuk equivalence theorem [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. Mat. Zh. 14 (5) (1973) 1088-1099; English translation: Siberian Math. J. 14 (1973) 759-767 (1974)].     

On the Libera operator

We show that the Libera operator, L, on some spaces of analytic functions is a continuous extension of the conjugate of the Cesàro operator. Results on L acting on various spaces are obtained. In particular, L maps the Bloch space into BMOA. We also prove some results on the best approximation by polynomials in Hardy and Bergman spaces.     

R(L)型诱导空间及连通性

本文研究了R（L）-型诱导空间，利用R（L）-值下半连续函数，讨论了R（L）型诱导空间的基本性质，得到了R（L）型诱导空间保持笛卡积和连通性，并证明了诱导映射和生成映射是可交换的。     

Expansive measures for flows

We extend the concept of expansive measure [2] from homeomorphism to flows. We prove for continuous flows on compact spaces that every expansive measure has no singularities in the support, is aperiodic, is expansive with respect to time-T maps (but not conversely), remains expansive under topological equivalence, vanishes along the orbits and is natural under suspensions. We apply these properties to prove that there are no expansive flows (in the sense of [26]) of any closed surface.