Topological dynamics of transformations induced on the space of probability measures

LetT be a continuous transformation of a compact metric spaceX. T induces in a natural way a transformationT _M on the spaceM (X) of probability measures onX, and a transformationT _K on the spaceK (X) of closed subsets ofX. This note investigates which of the topological properties ofT∶X→X (like distality, transitivity, mixing property etc. ...) are “inherited” byT _M∶M (X)→M (X) andT _K∶K (X)→K (X).     

Sulla unicità della soluzione di alcuni problemi di controllo ottimo in spazi di Orlicz

Let(E)z _xy+A(x, y)z_x+B(x, y)z_y+C(x, y)z=U(x, y) a.e. in Δ=]0,a[×]0,b[, be a distributed parameter control process. We study the uniqueness of some optimal control problems (minimum distance and minimum effort problems) relative to (E). We take the Orlicz spaceL ^M(Δ) as the «permanent control space, whereas both the «initial control» space and the «target» space are subspaces of the Orlicz-Sobolev spaceW ^1,M (]0,a[)×W ^1,M (]0,b[).     

Zur Möbiusgeometrie der Berührkreisscharen

Making use of an earlier result of the theory of stripes in three-dimensional conformal spaceM ₃ we obtain a moving frame and derivational formulas for one-parameter families of tangent circles inM ₃. Avoiding invariant parameters we can set up a bijection of the tangent circles into the osculating circles that preserves the duple ratio. The following section deals with loxodromes on Dupin cyclides. Finally with the aid of a modified stereographic projection we show how to get the Frenet formulas of the euclidian theory of spacecurves from the derivational formulas for tangent circles inM ₃.     

Constrained best approximation in Hilbert space

In this paper we study the characterization of the solution to the extremal problem inf{‖x‖x ∈C ∩M}, wherex is in a Hilbert spaceH, C is a convex cone, andM is a translate of a subspace ofH determined by interpolation conditions. We introduce a simple geometric property called the “conical hull intersection property” that provides a unifying framework for most of the basic results in the subject of optimal constrained approximation. Our approach naturally lends itself to considering the data cone as opposed to the constraint cone. A nice characterization of the solution occurs, for example, if the data vector associated withM is an interior point of the data cone.     

Hyperbolic imbedding and spaces of continuous extensions of holomorphic maps

LetX be a complex subspace of a complex spaceY. We show that hyperbolic imbeddedness ofX inY is characterized by relative compactness in the compact-open topology of certain spaces of continuous extensions of holomorphic maps from the punctured diskD* toX and fromM -A toX whereM is a complex manifold andA is a divisor onM with normal crossings. We apply these characterizations to obtain generalizations and extensions of theorems of Kobayashi, Kiernan, Kwack, Noguchi and Vitali forD and for higher dimensions. Relative compactness ofX inY is not assumed.     

On the distortion required for embedding finite metric spaces into normed spaces

We investigate the minimum dimensionk such that anyn-point metric spaceM can beD-embedded into somek-dimensional normed spaceX (possibly depending onM), that is, there exists a mappingf: M→X with $$\frac{1}{D}dist_M (x,y) \leqslant \left| {f(x) - f(y)} \right| \leqslant dist_M (x,y) for any$$ Extending a technique of Arias-de-Reyna and Rodríguez-Piazza, we prove that, for any fixedD≥1,k≥c(D)n ^1/2D for somec(D)>0. For aD-embedding of alln-point metric spaces into the samek-dimensional normed spaceX we find an upper boundk≤12Dn ^1/[(D+1)/2]lnn (using thel _∞ ^k space forX), and a lower bound showing that the exponent ofn cannot be decreased at least forD?[1,7)∪[9,11), thus the exponent is in fact a jumping function of the (continuously varied) parameterD.     

Extension of positive functionals on certain ordered spaces

An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL. A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism. These properties were introduced by Schmidt in [5]. In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM. An analogous theorem is derived for Riesz spaces with extension property (E2).     

Infinite-dimensional 4-manifold of finite cohomological dimension under the continuum hypothesis

Under the assumption of the continuum hypothesis, a differentiable 4-manifoldM of dimension dimM=∞ and cohomological dimension cA—dimM=4 is constructed. The spaceM is perfectly normal and hereditarily separable. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 664–670, November, 1999.     

Basic sequences and norming subspaces in non-quasi-reflexive Banach spaces

A Banach spaceX is non-quasi-reflexive (i.e. dimX ^**/X=∞) if and only if it contains a basic sequence spanning a non-quasi-reflexive subspace. In fact, this basic sequence can be chosen to be non-k-boundedly complete for allk. A basic sequence which is non-k-shrinking for allk exists inX if and only ifX ^* contains a norming subspace of infinite codimension. This need not occur even ifX is non-quasi-reflexive. Every norming subspace ofX ^* has finite codimension if and only if for every normingM inX ^*, everyM-closedY inX,M∩Y ^T is norming overX/Y. This solves a problem due to Schäffer [19].     

Finite dimensional imbeddings of harmonic spaces

For a noncompact harmonic manifoldM we establish finite dimensionality of the eigensubspacesV _γ generated by radial eigenfunctions of the form coshr+c. As a consequence, for such harmonic manifolds, we give an isometric imbedding ofM into (V _γ,B), whereB is a nondegenerate symmetric bilinear indefinite form onV _γ (analogous to the imbedding of the real hyperbolic spaceH ⁿ into ? ⁿ⁺¹ with the indefinite formQ(x,x)=?x ₀ ² +Σx _i ² ). This imbedding is minimal in a ‘sphere’ in (V _γ,B). Finally we give certain conditions under whichM is symmetric.     

Orlicz spaces,spline systems,and brownian motion

There are three results proved in this paper. The first one characterizes the Hölder classes in Orlicz spaces by the coefficients of the orthogonal spline expansions of the Franklin type. The second one gives a sharp estimate for the correlation of two random variables obtained as a composition of two Borel functions with the components of a given two-dimensional Gaussian vector. The third one is obtained with the help of the first two and it states that the Wiener measure is concentrated on the Banach space of Hölder functions with exponent 1/2 but in the norm of the Orlicz spaceL _M ^* withM(t)=expt(t ²)?1.     

Topological and bornological characterisations of ideals in von Neumann algebras: II

SupposeM is a von Neumann algebra on a Hilbert spaceH andI is any norm closed ideal inM. We extend to this setting the well known fact that the compact operators on a Hilbert space are precisely those whose restrictions to the closed unit ball are weak to norm continuous.     

An existence theorem for the complementarity problem

LetC be a pointed, solid, closed and convex cone in then-dimensional Euclidean spaceE ⁿ ,C* its polar cone,M:C→E ⁿ a map, andq a vector inE ⁿ . The complementarity problem (q|M) overC is that of finding a solution to the system $$(q|M) x \varepsilon C, M(x) + q \varepsilon C{^*} , \left\langle {x, M(x) + q} \right\rangle = 0.$$ It is shown that, ifM is continuous and positively homogeneous of some degree onC, and if (q|M) has a unique solution (namely,x=0) forq=0 and for someq=q ₀ ∈ intC*, then it has a solution for allq ∈E ⁿ .     

The classification of ϕ-symmetric Sasakian manifolds

A -symmetric spaceM is a complete connected regular Sasakian manifold, that fibers over an Hermitian symmetric spaceN, so that the geodesic involutions ofN lift to define global (involutive) automorphisms of the Sasakian structure onM. In the present paper the complete classification of -symmetric spaces is obtained. The groups of automorphisms of the Sasakian structures and the groups of isometries of the underlying Riemannian metrics are determined. As a corollary, the Sasakian space forms are also determined.     

The structure of hemigroups

A hemigroup is a continuous binary operation on a spaceM which satisfies (xy)(zy)=xz. The structure of these and their relationship with semigroups is described.     

Some isomorphically polyhedral orlicz sequence spaces

A Banach space is polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is known that a polyhedral Banach space has a separable dual and isc ₀-saturated, i.e., each closed infinite dimensional subspace contains an isomorph ofc ₀. In this paper, we show that the Orlicz sequence spaceh _M is isomorphic to a polyhedral Banach space if lim _t→0 M(Kt)/M(t)=∞ for someK<∞. We also construct an Orlicz sequence spaceh _M which isc ₀-saturated, but which is not isomorphic to any polyhedral Banach space. This shows that beingc ₀-saturated and having a separable dual are not sufficient for a Banach space to be isomorphic to a polyhedral Banach space.