Weak Similarities of Finite Ultrametric and Semimetric Spaces

Weak similarities form a special class of mappings between semimetric spaces. Two semimetric spaces X and Y are weakly similar if there exists a weak similarity Φ: X → Y. We find a structural characteristic of finite ultrametric spaces for which the isomorphism of its representing trees implies a weak similarity of the spaces. We also find conditions under which the Hasse diagrams of balleans of finite semimetric spaces are isomorphic.     

Random difference equations with subexponential innovations

We consider the random difference equations S = _d (X + S)Y and T = _d X + TY, where = _d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.     

Holomorphic mappings of once-holed tori

Let T be the space of marked once-holed tori and Y₀ be a Riemann surface with marked handle. We investigate geometric properties of the set T_a[Y₀] of X ∈ T that allow holomorphic mappings of X into Y₀. We also examine the set T_c[Y₀] of marked once-holed tori conformally embedded into Y₀. It turns out that T_a[Y₀] and T_c[Y₀] have several properties in common. Our basic tool is a new notion, called a handle condition.     

Relatively computably enumerable reals

A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and \({X \not\leq_T Y}\). A real X is relatively simple and above if there is a real Y < _T X such that X is c.e.(Y) and there is no infinite set \({Z \subseteq \overline{X}}\) such that Z is c.e.(Y). We prove that every nonempty \({\Pi^0_1}\) class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above.     

Generic elements in isometry groups of Polish ultrametric spaces

This paper presents a study of generic elements in full isometry groups of Polish ultrametric spaces. We obtain a complete characterization of Polish ultrametric spaces X whose isometry group Iso(X) has a neighborhood basis at the identity consisting of open subgroups with ample generics. It also gives a characterization of the existence of an open subgroup in Iso(X) with a comeager conjugacy class.We also study the transfinite sequence defined by the projection of a Polish ultrametric space X on the ultrametric space of orbits of X under the action of Iso(X).     

The Morse minimal system is nearly continuously Kakutani equivalent to the binary odometer

Ergodic homeomorphisms T and S of Polish probability spaces X and Y are evenly Kakutani equivalent if there is an orbit equivalence ?: X ₀ → Y ₀ between full measure subsets of X and Y such that, for some A ? X ₀ of positive measure, ? restricts to a measurable isomorphism of the induced systems T _A and S _?(A). The study of even Kakutani equivalence dates back to the seventies, and it is well known that any two zero-entropy loosely Bernoulli systems are evenly Kakutani equivalent. But even Kakutani equivalence is a purely measurable relation, while systems such as the Morse minimal system are both measurable and topological.Recently del Junco, Rudolph and Weiss studied a new relation, called nearly continuous Kakutani equivalence. A nearly continuous Kakutani equivalence is an even Kakutani equivalence where also X ₀ and Y ₀ are invariant G _δ sets, A is within measure zero of both open and closed, and ? is a homeomorphism from X ₀ to Y ₀. It is known that nearly continuous Kakutani equivalence is strictly stronger than even Kakutani equivalence, and nearly continuous Kakutani equivalence is the natural strengthening of even Kakutani equivalence to the nearly continuous category—the category of maps that are continuous after sets of measure zero are removed. In this paper, we show that the Morse minimal substitution system is nearly continuously Kakutani equivalent to the binary odometer.     

On Linear Preservers of Two-Sided Gut-Majorization On <Emphasis Type="Bold">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis></Subscript>

For X, Y ∈ M_n,m it is said that X is gut-majorized by Y, and we write X ?_gutY, if there exists an n-by-n upper triangular g-row stochastic matrix R such that X = RY. Define the relation ~_gut as follows. X ~_gutY if X is gut-majorized by Y and Y is gut-majorized by X. The (strong) linear preservers of ?_gut on ?ⁿ and strong linear preservers of this relation on M_n,m have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ~_gut on ?ⁿ and M_n,m.     

On isomorphisms of some Köthe function <Emphasis Type="Italic">F</Emphasis>-spaces

We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (Ω _X ; Σ _X , µ _X ) and (Ω _Y ; Σ _Y ; µ _Y ), respectively, with absolute continuous norms are isomorphic and have the property

$\mathop {\lim }\limits_{\mu (A) \to 0} \left\| {\mu (A)^{ - 1} 1_A } \right\| = 0$

(for µ = µ _X and µ = µ _Y , respectively) then the measure spaces (Ω _X ; Σ _X ; µ _X ) and (Ω _Y ; Σ _Y ; µ _Y ) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L _p (µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces having no nonzero separable quotient space.

     

Khinchin’s inequality,Dunford-Pettis and compact operators on the space <Emphasis Type="Italic">C</Emphasis>([0, 1], <Emphasis Type="Italic">X</Emphasis>)

We prove that if X, Y are Banach spaces, Ω a compact Hausdorff space and U:C(Ω, X) → Y is a bounded linear operator, and if U is a Dunford-Pettis operator the range of the representing measure G(Σ) ? DP(X, Y) is an uniformly Dunford-Pettis family of operators and ∥G∥ is continuous at Ø. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space C([0, 1], X) with values in c ₀ or l _p, (1 ≤ p < ∞) be Dunford-Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.     

Banach–Stone Theorems for maps preserving common zeros

Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F) a Banach–Stone map if it has the form T f (y) =  S _y (f (h(y))) for a family of linear operators S _y : E → F, \({y \in Y}\) , and a function h: Y → X. In this paper, we consider maps having the property:

$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z}) $

where Z(f) =  {f =  0}. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including C ^∞), as Banach–Stone maps. In particular, we confirm a conjecture of Ercan and Önal: Suppose that X and Y are realcompact spaces and E and F are Hausdorff topological vector lattices (respectively, C ^*-algebras). Let T: C(X, E) → C(Y, F) be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that

$ Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. $

Then X is homeomorphic to Y and E is lattice isomorphic (respectively, C ^*-isomorphic) to F. Some results concerning the continuity of T are also obtained.

     

On the stability of tangent bundle on double coverings

Let Y be a smooth projective surface defined over an algebraically closed field k with char k≠2, and let π : X →Y be a double covering branched along a smooth divisor. We show that y_X is stable with respect to π~*H if the tangent bundle y_Y is semi-stable with respect to some ample line bundle H on Y.     

Applications of balanced pairs

Let (X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to (X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y (resp. Y-coresolution dimension of X) is finite, then the bounded homotopy category of Y (resp. X) is contained in that of X (resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.     

Deviation of elements of a Banach space from a system of subspaces

We prove that if X is a real Banach space, Y ₁ ? Y ₂ ? ... is a sequence of strictly embedded closed linear subspaces of X, and d ₁ ≥ d ₂ ≥ ... is a nonincreasing sequence converging to zero, then there exists an element x ∈ X such that the distance ρ(x, Y _n ) from x to Y _n satisfies the inequalities d _n ≤ ρ(x, Y _n ) ≤ 8d _n for n = 1, 2, ....     

Spaces with large relative extent

In this paper, we prove the following statements: (1) For every regular uncountable cardinal κ, there exist a Tychonoff space X and Y a subspace of X such that Y is both relatively absolute star-Lindelöf and relative property (a) in X and e(Y, X) ? κ, but Y is not strongly relative star-Lindelöf in X and X is not star-Lindelöf. (2) There exist a Tychonoff space X and a subspace Y of X such that Y is strongly relative star-Lindelöf in X (hence, relative star-Lindelöf), but Y is not absolutely relative star-Lindelöf in X.     

A conditional limit theorem for a bivariate representation of a univariate random variable and conditional extreme values

We first consider a real random variable X represented through a random pair (R,T) and a deterministic function u as X = R?u(T). Under quite weak assumptions we prove a limit theorem for (R,T) given X>x, as x tends to infinity. The novelty of our paper is to show that this theorem for the representation of the univariate random variable X permits us to obtain in an elegant manner conditional limit theorems for random pairs (X,Y) = R?(u(T),v(T)) given that X is large. Our approach allows to deduce new results as well as to recover under considerably weaker assumptions results obtained previously in the literature. Consequently, it provides a better understanding and systematization of limit statements for the conditional extreme value models.     

On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ? X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω ₁-monolithic compact space X, if C _p (X)is star countable then it is Lindelöf.     

Functional envelope of Cantor spaces

Given a topological dynamical system(X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of(X, T) is the system(S(X), FT), where FT is defined by FT(?) = T ? ? for any ? ∈ S(X). We show that(1) If(Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(2) If(S(Σ), F_σ) is transitive then it is Devaney chaos, where(Σ, σ) is a subshift of finite type;(3) If(Σ, T) has shadowing property, then(SU(Σ), FT) has shadowing property,where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(4) If(X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T : X → X is continuous, then(SU(X), FT) is sensitive;(5) If Σ is a closed subset of a Cantor set with infinite points and T : Σ→Σ is positively expansive then the entropy ent U(FT) of the functional envelope of(Σ, T) is infinity.     

Uniform Homeomorphisms of Unit Spheres and Property H of Lebesgue–Bochner Function Spaces

Assume that the unit spheres of Banach spaces X and Y are uniformly homeomorphic.Then we prove that all unit spheres of the Lebesgue–Bochner function spaces L_p(μ, X) and L_q(μ, Y)are mutually uniformly homeomorphic where 1 ≤ p, q ∞. As its application, we show that if a Banach space X has Property H introduced by Kasparov and Yu, then the space L_p(μ, X), 1 ≤ p ∞,also has Property H.     

More on stability of almost surjective <Emphasis Type="Italic">ε</Emphasis>-isometries of Banach spaces

Let X and Y be two Banach spaces, and f: X → Y be a standard ε-isometry for some ε ≥ 0. In this paper, by using a recent theorem established by Cheng et al. (2013–2015), we show a sufficient condition guaranteeing the following sharp stability inequality of f: There is a surjective linear operator T: Y → X of norm one so that

$$\left\| {Tf(x) - x} \right\| \leqslant 2\varepsilon , for all x \in X.$$

As its application, we prove the following statements are equivalent for a standard ε-isometry f: X → Y:

1. (i)
   lim inf _t→∞ dist(ty, f(X))/|t| < 1/2, for all y ∈ S _Y ;
    
2. (ii)
   \(\tau(f)\equiv sup_{y\epsilon S_{Y}}\) lim inf _t→∞dist(ty, f(X))/|t| = 0;
    
3. (iii)
   there is a surjective linear isometry U: X → Y so that

   $$\left\| {f(x) - Ux} \right\| \leqslant 2\varepsilon , for all x \in X.$$
    

This gives an affirmative answer to a question proposed by Vestfrid (2004, 2015).     

On the Gomory–Hu Inequality

It was proved by R. Gomory and T. Hu in 1961 that, for every finite nonempty ultrametric space (X, d), the inequaliy \( \left| {\mathrm{Sp}(X)} \right|\leq \left| X \right|-1 \), where Sp(X) = {d(x, y) : x, y ∈ X, x ≠ y} , holds. We characterize the spaces X for which the equality is attained by the structural properties of some graphs and show that the set of isometric types of such X is dense in the Gromov–Hausdorff space of the isometric types of compact ultrametric spaces.