Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Let θ ∈ (0, 1), λ ∈ [0, 1) and p, p ₀, p ₁ ∈ (1,∞] be such that (1 ? θ)/p ₀ + θ/p ₁ = 1/p, and let φ, φ₀, φ₁ be some admissible functions such that φ, φ₀ ^p/p0 and φ₁ ^p/p1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L _φ0 ^p0),λ (X), L _φ1 ^{p1), λ} (X), θ> of two generalized grand Morrey spaces on a quasi-metric measure space X is the generalized grand Morrey space L _φ ^p),λ (X). Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.     

Note: A conjecture on partitions of groups

We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A _n A _n ^?1 ) = |G| for each n ∈ ω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.     

Low distortion embeddings of some metric graphs into Banach spaces

We give a simple example of a countable metric graph M such that M Lipschitz embeds with distortion strictly less than 2 into a Banach space X only if X contains an isomorphic copy of l ₁. Further we show that, for each ordinal α < ω ₁, the space C([0, ω ^α ]) does not Lipschitz embed into C(K) with distortion strictly less than 2 unless K ^(α) ≠ 0. Also \(C\left( {\left[ {0,{\omega ^{{\omega ^\alpha }}}} \right]} \right)\) does not Lipschitz embed into a Banach space X with distortion strictly less than 2 unless Sz(X) ≥ ω ^α+1.     

The SL(<Emphasis Type="Italic">V</Emphasis>)-ample cone of product of flag varieties

Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set ω = (\(\vec d\)(1), ..., \(\vec d\)(m)) of sequences of positive integers, denote by L _ω the ample line bundle corresponding to the polarization on the product X = Π _i=1 ^m Flag(V, \(\vec n\)(i)) of flag varieties of type \(\vec n\)(i) determined by ω. We study the SL(V)-linearization of the diagonal action of SL(V) on X with respect to L _ω. We give a sufficient and necessary condition on ω such that X ^ss (L _ω) ≠ \(\not 0\) (resp., X ^s (L _ω) ≠ \(\not 0\)). As a consequence, we characterize the SL(V)-ample cone (for the diagonal action of SL(V) on X), which turns out to be a polyhedral convex cone.     

A sharp eigenvalue theorem for fractional elliptic equations

The invisibility graph I(X) of a set X ? R ^d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number ω(I(X)), the chromatic number χ(I(X)) and the convexity number γ(X), which is the minimum number of convex subsets of X that cover X.We settle a conjecture of Matou?ek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.We also find sets X in R⁵ with χ(X) = 2, but γ(X) arbitrarily large.     

A limit theorem for continuous selectors

Any (measurable) function K from Rⁿ to R defines an operator K acting on random variables X by K(X) = K(X₁,..., X_n), where the X_j are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x₁, x₂,..., x_n) ∈ {x₁, x₂,..., x_n}. For each such continuous selector H (except for projections onto a single coordinate) there is a unique point ω_H in the interval (0, 1) so that, for any random variable X, the iterates H^(N) acting on X converge in distribution as N → ∞ to the ωH-quantile of X.     

bmo _ρ(ω) Spaces and Riesz transforms associated to Schrdinger operators

We introduce the BMO-type space bmoρ(ω) and establish the duality between h_ρ~1(ω) and bmo _ρ(ω),where ω∈A_1~(ρ, ∞)(R~n) and ω's locally behave as Muckenhoupt's weights but actually include them. We also give the Fefferman-Stein type decomposition of bmoρ(ω) with respect to Riesz transforms associated to Schrdinger operator L, where L =-? + V is a Schrdinger operator on R~n(n≥3) and V is a non-negative function satisfying the reverse Hlder inequality.     

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.     

Domains of definition of Monge-Ampère operators on compact Kähler manifolds

Let (X, ω) be a compact Kähler manifold. We introduce and study the largest set DMA(X, ω) of ω-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set PSH(X, ω) of all ω-psh functions. We prove that certain twisted Monge-Ampère operators are well defined for all ω-psh functions. As a consequence, any ω-psh function with slightly attenuated singularities has finite weighted Monge-Ampère energy.     

Remarks on discretely absolutely star-Lindelöf spaces

In this paper, we prove the following statements(1) There exists a Hausdorff Lindelöf space X such that the Alexandroff duplicate A(X) of X is not discretely absolutely star-Lindelöf.(2) If X is a regular Lindelöf space, then A(X) is discretely absolutely star-Lindelöf.(3) If X is a normal discretely star-Lindelöf space with e(X) < ω ₁, then A(X) is discretely absolutely star-Lindelöf.     

Symmetric function kernels and sweeping of measures

This is a potential theoretic study of balayage (sweeping) of a positive Radon measure ω on a locally compact (Hausdorff) space X onto a closed, or, more generally, a quasiclosed set A ? X (that is, a set which can be approximated in outer capacity by closed sets). The setting is that of potentials with respect to a suitable symmetric function kernel G: X × X → [0,+∞]. We consider energy capacity, not as a set function, but as a functional, acting on positive numerical functions on X. The finiteness of the upper capacity of the function 1 _A Gω is sufficient for the possibility of the sweeping in question (1 _A denoting the indicator function of A and Gω the G-potential of ω).     

General Hausdorff functions,and the notion of one-sided measure and dimension

The main facts about Hausdorff and packing measures and dimensions of a Borel set E are revisited, using determining set functions \(\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)\), where \(\mathcal{B}_E\) is the family of all balls centred on E and α is a real parameter. With mild assumptions on φ_α, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set V in ? ^D , these notions are used to introduce the interior and exterior measures and dimensions of any Borel subset of ?V. We stress that these dimensions depend on the choice of φ_α. Two determining functions are considered, φ_α(B)=Vol _D (B∩V)diam(B)^α-D and φ_α(B)=Vol _D (B∩V)^α/D , where Vol _D denotes the D-dimensional volume.     

On the sectionwise connectedness of a contingent

Let X be a real normed space and let f: ? → X be a continuous mapping. Let T _f (t ₀) be the contingent of the graph G(f) at a point (t ₀, f(t ₀)) and let S ⁺ ? (0,∞) × X be the “right” unit hemisphere centered at (0, 0 _X ). We show that

1. 1.
   If dimX < ∞ and the dilation D(f, t ₀) of f at t ₀ is finite then T _f (t ₀) ∩ S ⁺ is compact and connected. The result holds for \(T_f (t_0 ) \cap \overline {S^ + } \) even with infinite dilation in the case f: [0,∞) → X.
    
2. 2.
   If dimX = ∞, then, given any compact set F ? S ⁺, there exists a Lipschitz mapping f: ? → X such that T _f (t ₀) ∩ S ⁺ = F.
    
3. 3.
   But if a closed set F ? S ⁺ has cardinality greater than that of the continuum then the relation T _f (t ₀) ∩ S ⁺ = F does not hold for any Lipschitz f: ? → X.
    

     

The quasi-triangular structures over Hom-<Emphasis Type="Italic">ω</Emphasis>-smash coproduct Hopf algebras

Let (C, α) and (H, β) be Hom-bialgebras and ω: C ?H → H ?C a linear map. We introduce a Hom-ω-smash coproduct (C _ω ? H, γ) and give necessary and sufficient conditions for (C _ω ? H, γ) to be a Hom-bialgebra. We study the quasi-triangular structures over (C _ω ? H, γ) and show the necessary and sufficient conditions for (C _ω ? H, γ, R) to be a quasi-triangular Hom-Hopf algebra. As applications of our results, we introduce the concept of D(H)* and construct quasi-triangular structures over D(H)*.     

Generalized varieties of sums of powers

Let X ? P^N be an irreducible, non-degenerate variety. The generalized variety of sums of powers V S P_H^X(h) of X is the closure in the Hilbert scheme Hilb_h (X) of the locus parametrizing collections of points {x₁,..., x_h} such that the (h -1)-plane >x₁,..., x_h> passes through a fixed general point p ∈ P^N. When X = V_dⁿ is a Veronese variety we recover the classical variety of sums of powers V S P(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V S P_H^X(h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of V S P_H^X(h) are inherited from the birational geometry of variety X itself.     

Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X_1 and X_2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann(CR) manifolds of dimensions 2m-1 and 2n-1 in C~(m+1)and C~(n+1), respectively. We introduce the ThomSebastiani sum X = X_1 ⊕X_2which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in C~(m+n+2). Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1for all n 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X_1 ⊕ X_2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly,we show that if X = X_1 ⊕ X_2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X_1 and X_2 provided that X_2 admits a transversal holomorphic S~1-action.     

A note on transitively <Emphasis Type="Italic">D</Emphasis>-spaces

In this note, we show that if for any transitive neighborhood assignment φ for X there is a point-countable refinement ? such that for any non-closed subset A of X there is some V ∈ ? such that |V ∩ A| ? ω, then X is transitively D. As a corollary, if X is a sequential space and has a point-countable wcs*-network then X is transitively D, and hence if X is a Hausdorff k-space and has a point-countable k-network, then X is transitively D. We prove that if X is a countably compact sequential space and has a pointcountable wcs*-network, then X is compact. We point out that every discretely Lindelöf space is transitively D. Let (X, τ) be a space and let (X, ?) be a butterfly space over (X, τ). If (X, τ) is Fréchet and has a point-countable wcs*-network (or is a hereditarily meta-Lindelöf space), then (X, ?) is a transitively D-space.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

First-order and monadic properties of highly sparse random graphs

A random graph is said to obey the (monadic) zero–one k-law if, for any property expressed by a first-order formula (a second-order monadic formula) with a quantifier depth of at most k, the probability of the graph having this property tends to either zero or one. It is well known that the random graph G(n, n^–α) obeys the (monadic) zero–one k-law for any k ∈ ? and any rational α > 1 other than 1 + 1/m (for any positive integer m). It is also well known that the random graph does not obey both k-laws for the other rational positive α and sufficiently large k. In this paper, we obtain lower and upper bounds on the largest at which both zero–one k-laws hold for α = 1 + 1/m.     

Limiting law results for a class of conditional mode estimates for functional stationary ergodic data

The main purpose of the present work is to establish the functional asymptotic normality of a class of kernel conditional mode estimates when functional stationary ergodic data are considered. More precisely, consider a random variable (X,Z) taking values in some semi-metric abstract space E × F. For a real function φ defined on F and for each x ∈ E, we consider the conditional mode, say ?_φ(x), of the real random variable φ(Z) given the event “X = x”. While estimating the conditional mode function by Θ?_φ,n(x), using the kernel-type estimator, we establish the limiting law of the family of processes {Θ?_φ(x) - Θ_φ(x)} (suitably normalized) over Vapnik–Chervonenkis class C of functions φ. Beyond ergodicity, no other assumption is imposed on the data. This paper extends the scope of some previous results established under mixing condition for a fixed function φ. From this result, the asymptotic normality of a class of predictors is derived and confidence bands are constructed. Finally, a general notion of bootstrapped conditional mode constructed by exchangeably weighting samples is presented. The usefulness of this result will be illustrated in the construction of confidence bands.