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.     

Unbounded norm topology beyond normed lattices

In this paper, we generalize the concept of unbounded norm (un) convergence: let X be a normed lattice and Y a vector lattice such that X is an order dense ideal in Y; we say that a net \((y_\alpha )\) un-converges to y in Y with respect to X if \(\bigl |\bigl ||y_\alpha -y|\wedge x\bigr |\bigr |\rightarrow 0\) for every \(x\in X_+\). We extend several known results about un-convergence and un-topology to this new setting. We consider the special case when Y is the universal completion of X. If \(Y=L_0(\mu )\), the space of all \(\mu \)-measurable functions, and X is an order continuous Banach function space in Y, then the un-convergence on Y agrees with the convergence in measure. If X is atomic and order complete and \(Y=\mathbb R^A\) then the un-convergence on Y agrees with the coordinate-wise convergence.     

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.     

Tail behavior of the product of two dependent random variables with applications to risk theory

Let the random vector (X,Y) follow a bivariate Sarmanov distribution, where X is real-valued and Y is nonnegative. In this paper we investigate the impact of such a dependence structure between X and Y on the tail behavior of their product Z?=?XY. When X has a regularly varying tail, we establish an asymptotic formula, which extends Breiman’s theorem. Based on the obtained result, we consider a discrete-time insurance risk model with dependent insurance and financial risks, and derive the asymptotic and uniformly asymptotic behavior for the (in)finite-time ruin probabilities.     

On two questions of the theory of retracts

We establish that condition (Γ) on brick decomposition is indecomposable. This answers K. Borsuk’s question [1]. We prove that there exist metric spaces X and Y and a point (a, b) ∈ X × Y such that (a, b) is an r-point of the product X × Y; moreover, a is not an r-point of X. This answers A. Kosinski’s question [2].     

Embeddings of quotient division algebras of rings of differential operators

Let k be an algebraically closed field of characteristic zero, let X and Y be smooth irreducible algebraic curves over k, and let D(X) and D(Y) denote respectively the quotient division rings of the ring of differential operators of X and Y. We show that if there is a k-algebra embedding of D(X) into D(Y), then the genus of X must be less than or equal to the genus of Y, answering a question of the first-named author and Smoktunowicz.     

Large sublattices in subsets of Banach lattices

A classical problem (initially studied by N. Kalton and A. Wilansky) concerns finding closed infinite dimensional subspaces of X / Y, where Y is a subspace of a Banach space X. We study the Banach lattice analogue of this question. For a Banach lattice X, we prove that X / Y contains a closed infinite dimensional sublattice under the following conditions: either (i) Y is a closed infinite codimensional subspace of X, and X is either order continuous or a C(K) space, where K is a compact subset of \({\mathbb {R}}^n\); or (ii) Y is the range of a compact operator.     

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.     

Asymptotics for the finite-time ruin probability in a discrete-time risk model with dependent insurance and financial risks<Superscript>*</Superscript>

We consider a discrete-time risk model with insurance and financial risks. Within period i ≥ 1, the real-valued net insurance loss caused by claims is the insurance risk, denoted by X _i , and the positive stochastic discount factor over the same time period is the financial risk, denoted by Y _i . Assume that {(X, Y), (X _i , Y _i ), i ≥ 1} form a sequence of independent identically distributed random vectors. In this paper, we investigate a discrete-time risk model allowing a dependence structure between the two risks. When (X, Y ) follows a bivariate Sarmanov distribution and the distribution of the insurance risk belongs to the class ?(γ) for some γ > 0, we derive the asymptotics for the finite-time ruin probability of this discrete-time risk model.     

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.     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.     

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.     

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.

     

On the Dimension of Preimages of Certain Paracompact Spaces

It is proved that if X is a normal space which admits a closed fiberwise strongly zero-dimensional continuous map onto a stratifiable space Y in a certain class (an S-space), then IndX = dimX. This equality also holds if Y is a paracompact σ-space and ind Y = 0. It is shown that any closed network of a closed interval or the real line is an S-network. A simple proof of the Kateˇ tov–Morita inequality for paracompact σ-spaces (and, hence, for stratifiable spaces) is given.     

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.     

On the product of relatively weakly almost Lindelöf subsets

The purpose of this note is to show that there exist two Tychonoff spaces X, Y, a subset A of X and a subset B of Y such that A is weakly almost Lindelöf in X and B is weakly almost Lindelöf in Y, but A × B is not weakly almost Lindelöf in X × Y.     

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.     

From Isomorphic Rooted Trees to Isometric Ultrametric Spaces

For every finite ultrametric space X we can put in correspondence its representing tree T_X. We found conditions under which the isomorphism of representing trees T_X and T_Y implies the isometricity of ultrametric spaces X and Y having the same range of distances.     

On operators which attain their norm

The following problem is considered. LetX andY be Banach spaces. Are those operators fromX toY which attain their norm on the unit cell ofX, norm dense in the space of all operators fromX toY? It is proved that this is always the case ifX is reflexive. In general the answer is negative and it depends on some convexity and smoothness properties of the unit cells inX andY. As an application a refinement of the Krein-Milman theorem and Mazur’s theorem concerning the density of smooth points, in the case of weakly compact sets in a separable space, is obtained.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.