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For a non-empty compact set $E$ in a proper subdomain $\Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $\Omega$ by $d(E)$ and $d(E,\partial\Omega),$ respectively. The quantity $d(E)/d(E,\partial\Omega)$ is invariant under similarities and plays an important role in Geometric Function Theory. In the present paper, when $\Omega$ has the hyperbolic distance $h_\Omega(z,w),$ we consider the infimum $\kappa(\Omega)$ of the quantity $h_\Omega(E)/\log(1+d(E)/d(E,\partial\Omega))$ over compact subsets $E$ of $\Omega$ with at least two points, where $h_\Omega(E)$ stands for the hyperbolic diameter of the set $E.$ We denote the upper half-plane by $\mathbb{H}$. Our main results claim that $\kappa(\Omega)$ is positive if and only if the boundary of $\Omega$ is uniformly perfect and that the inequality $\kappa(\Omega)\leq\kappa(\mathbb{H})$ holds for all $\Omega,$ where equality holds precisely when $\Omega$ is convex., Comment: 21 pages, 2 figures

We establish sufficient conditions for the almost compactness of partially integral operators in $ L_{p} $ .

Assuming the Continuum Hypothesis (CH), we prove that there exists a perfectly normal compact topological space $ Z $ and a countable set $ E\subset Z $ such that $ Z\setminus E $ does not condense onto any compact set. The space $ Z $ enables us to answer in the negative (under CH) the following problem of Ponomarev: Is each perfectly normal compact set an $ a $ -space? We also prove that the product of $ a $ -spaces need not be an $ a $ -space.

An F-compactum or a Fedorchuk compactum is a Hausdorff compact space that admits decomposition into a special well-ordered inverse system with fully closed neighboring projections. We prove that the square of Aleksandroff’s “double arrow” space is not an F-compactum of countable spectral height. Using this, we demonstrate the impossibility of representing the Helly space as the inverse limit of a countable system of resolutions with metrizable fibers. This gives a negative answer to a question posed by Watson in 1992.

We prove an analog of the Dodds–Fremlin–Wickstead Theorem on compact domination for homogeneous orthogonally additive polynomials in Banach lattices. The proof is based on linearization of the polynomials which was established earlier by the author.

We consider the extension of an orthogonally additive operator from a lateral ideal and a lateral band to the whole space. We prove in particular that every orthogonally additive operator, extended from a lateral band of an order complete vector lattice, preserves lateral continuity, narrowness, compactness, and disjointness preservation. These results involve the strengthening of a recent theorem about narrow orthogonally additive operators in vector lattices.

Using a new approach, we show that, for any ideal space X with nonempty regular part, the maximal function operator MB constructed from an arbitrary quasidensity differential basis B is not compact if considered in a pair of weighted spaces (Xw, Xv) generated by X. For special differential bases that include convex quasidensity bases, we prove that MB is not compact in a pair of weighted spaces (Xw, Xv) generated by an arbitrary ideal space X. An example is given of a quasidensity differential basis such that the maximal function operator constructed from this basis is compact in (L∞, L∞).

We introduce the notion of variative seminormal functor F and prove that, for each of these functors and every compact space X, the normality of the space F(X) X implies that the character of X is countable. Thus, we obtain a generalization of the Arkhangel’skiĭ-Kombarov theorem of 1990 on the countability of the character of a compact space which is normal outside the diagonal. Under the assumption of Jensen’s principle, we prove that the above assertion fails for finite nonvariative functors.

For systems of ordinary differential equations, we obtain some sufficient conditions of existence of an optimal control in terms of their right-hand sides and the cost functional with the use of compactness methods. In the theorems proven, either the time interval is finite and a solution belongs to some domain or the time interval coincides with the semiaxis.

Let α ∈ (0, 1). Consider the Riemann-Liouville fractional operator of the form $f \to T_\alpha f(x): = v(x)\int\limits_0^x {\frac{{f(y)u(y)dy}} {{(x - y)^{1 - \alpha } }}} ,x > 0, $ with locally integrable weight functions u and v. We find criteria for the L p → L q -boundedness and compactness of T α when 0 1/α under the condition that u monotonely decreases on ℝ+:= [0,∞). The dual versions of this result are given.

Let X be a Banach space and let T: X → X be a power bounded linear operator. Put X0 = {x ∈ X ∣ Tnx → 0}. Assume given a compact set K ⊂ X such that lim infn→∞ρ{Tnx, K} ≤ η < 1 for every x ∈ X, ∥x∥ ≤ 1. If \(\eta < \tfrac{1} {2} \), then codim X0 < ∞. This is true in X reflexive for \(\eta \in [\tfrac{1} {2},1) \), but fails in the general case.

Let X be a real normed space and let f: ℝ → X be a continuous mapping. Let Tf(t0) be the contingent of the graph G(f) at a point (t0, f(t0)) and let S+ ⊂ (0,∞) × X be the “right” unit hemisphere centered at (0, 0X). We show that 1. If dimX < ∞ and the dilation D(f, t0) of f at t0 is finite then Tf(t0) ∩ 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. If dimX = ∞, then, given any compact set F ⊂ S+, there exists a Lipschitz mapping f: ℝ → X such that Tf(t0) ∩ S+ = F. 3. But if a closed set F ⊂ S+ has cardinality greater than that of the continuum then the relation Tf(t0) ∩ S+ = F does not hold for any Lipschitz f: ℝ → X.

Considering the integral operators with nonnegative kernels and variable integration limits, we obtain criteria of boundedness and compactness in weighted Lebesgue spaces under some conditions on the kernels that are weaker than those studied before.

We calculate the principal term of the asymptotics of the Kolmogorov ɛ-entropy of some compact set of periodic infinitely differentiable functions that is continuously embedded into the space of continuous periodic functions.

Let φ1 and φ2 be holomorphic self-maps of the unit polydisk \(\mathbb{D}^N\) , and let u1 and u2 be holomorphic functions on \(\mathbb{D}^N\) . We characterize the boundedness and compactness of the difference of weighted composition operators Wφ1, u1 and Wφ2, u2 from the weighted Bergman space \(A_{\vec \alpha }^p\) , 0 −1, j = 1,..., N, to the weighted-type space Hυ∞ of holomorphic functions on the unit polydisk \(\mathbb{D}^N\) in terms of inducing symbols φ1, φ2, u1, and u2.

Many articles deal with large deviation principles (LDPs) (see [1-4] for instance and the references in [3,4]), studying mainly the LDP for the sums of random elements or for various stochastic models and dynamical systems. For a sequence of random elements in a metric space, in studying LDPs it turns out natural to introduce the concepts of the local LDP and extended LDP. They enable us to state and prove LDP-type statements in those cases when the usual LDP (cf. [3,4]) fails (see [5,6] and Section 6 of this article). We obtain conditions for the fulfillment of the extended LDP in metric spaces. The main among these conditions is the fulfillment of the local LDP. The latter is usually much simpler to prove than the extended LDP.

In ℂn we consider an algebraic surface Y and a finite collection of hypersurfaces Si. Froissart’s theorem states that if Y and Si are in general position in the projective compactification of ℂn together with the hyperplane at infinity then for the homologies of Y \∪ Si we have a special decomposition in terms of the homology of Y and all possible intersections of Si in Y. We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of ℂn in which Y and Si are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in Y leaving invariant all hypersurfaces Y ∩ Sk with the exception of one Y ∩ Si, which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface Y the complement of a surface in a compact toric variety to a collection of hypersurfaces in it.

A seminormal functor kF enjoys the Katěetov property (K-property) if for every compact set X the hereditary normality of kF(X) implies the metrizability of X. We prove that every seminormal functor of finite degree n>3 enjoys the K-property. On assuming the continuum hypothesis (CH) we characterize the weight preserving seminormal functors with the K-property. We also prove that the nonmetrizable compact set constructed in [1] on assuming CH is a universal counterexample for the K-property in the class of weight preserving seminormal functors.

Let \( \mathbb{B} \) be the unit ball in ℂn and let H(\( \mathbb{B} \)) be the space of all holomorphic functions on \( \mathbb{B} \). We introduce the following integral-type operator on H(\( \mathbb{B} \)): $$ I_\phi ^g (f)(z) = \int\limits_0^1 {\operatorname{Re} f(\phi (tz))g(tz)\frac{{dt}} {t}} ,z \in \mathbb{B}, $$ where g e H(\( \mathbb{B} \)), g(0) = 0, and φ is a holomorphic self-map of \( \mathbb{B} \). Under study are the boundedness and compactness of the operator from the mixed norm space H(p, q, ϕ)(\( \mathbb{B} \)) to the Bloch-type space ℬµ(\( \mathbb{B} \)).

Under study are the sets in ℝn (NED sets) each of which does not affect the conformal capacity of any condenser with connected plates disjoint from this set. These sets are removable singularities of quasiconformal mappings, which explains our interest in them. For compact sets on a hyperplane we obtain a geometric criterion of the NED property; we point out a simple sufficient condition for an NED set in terms of the connected attainability of its points from its complement in the hyperplane. For compact sets on a hypersphere we obtain a criterion for an NED set in terms of the reduced module at a pair of points in its complement. We establish that a compact set on a hypersphere S, removable for the capacity in at least one spherical ring concentric with S and containing S, is an NED set.

We study the compactness of the Hardy-Littlewood operator on several spaces of harmonic functions on the unit ball in ℝn such as: a-Bloch, weighted Hardy, weighted Bergman, Besov, BMOp, and Dirichlet spaces.

Assuming the continuum hypothesis we construct an example of a nonmetrizable compact set X with the following properties

We introduce some nested classes of Volterra type integral operators. For the operators of these classes we establish criteria for boundedness and compactness in Lebesgue spaces.

We study the following integral type operator $$T_g (f)(z) = \int\limits_0^{z_{} } { \cdots \int\limits_0^{z_n } {f(\zeta _1 , \ldots ,\zeta _n )} g(\zeta _1 , \ldots ,\zeta _n )d\zeta _1 , \ldots ,\zeta _n } $$ in the space of analytic functions on the unit polydisk U n in the complex vector space ℂn. We show that the operator is bounded in the mixed norm space , with p, q ∈ [1, ∞) and α = (α1, …, αn), such that αj > −1, for every j = 1, …, n, if and only if $$\sup _{z \in U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| < \infty $$ . Also, we prove that the operator is compact if and only if $$\lim _{z \to \partial U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| = 0$$ .

Using the Fourier-Laplace transform for functionals, we describe the duals of some spaces of the infinitely differentiable functions given on convex compact sets or convex domains in ℝN and such that the growth of their derivatives is determined by weight sequences of a general form.

We establish a criterion for existence of an L2-limit and a weak L2-limit of a polyharmonic function on a regular analytic boundary of a bounded plane domain.

We study a new cardinal-valued invariant ndw(X) (calling it the nd-weight of X) of a topological space which is defined as the least upper bound of the weights of nowhere dense subsets of X. The main result is the proof of the inequality hl(X)≤ndw(X) for compact sets without isolated points ((hl is the hereditary Lindelof number). This inequality implies that a compact space without isolated points of countable nd-weight is completely normal. Assuming the continuum hypothesis, we construct an example of a nonmetrizable compact space of countable nd-weight without isolated points.

The notion is introduced of quasientropy solution to the equations of an isothermal gas in Lagrangian variables. Compactness of bounded quasientropy solutions is established. Reduction of Young measures to Dirac measures bases on group analysis for the entropy wave equation.

We consider various definitions of a pseudocompact mapping and the basic properties of pseudocompact mappings. Moreover, we consider the definition of countable compactness of a continuous mapping and study the properties of a countably compact mapping similar to the corresponding properties for countably compact spaces and also the interrelation between countable compactness and pseudocompactness of mappings. We also extend the notions of local bicompactness and k-space to continuous mappings.

We study the homeomorphic embeddings of a compact set K, a union of nondegenerate continua, into \(\overline {\mathbb{R}} ^n\) which preserve the conformal moduli of all condensers whose plates are continua in K. Using a result by V. N. Dubinin together with the estimates for the conformal moduli of infinitesimal condensers, we prove that Belinskii's conjecture (that such a mapping extends to a Mobius automorphism of the whole space \(\overline {\mathbb{R}} ^n2\)), corroborated by the author in 1990 for n=2 is also valid for n>2 if the compact set in question is regular at some collection of (n+2) points. This essentially strengthens the previous result of the author (1992) in which regularity was required at each point of the compact set.

This addendum to [1] completely characterizes the boundedness and compactness of a recently introduced integral type operator from the space of bounded holomorphic functions H∞(\( \mathbb{D}^n \)) on the unit polydisk \( \mathbb{D}^n \) to the mixed norm space Open image in new window with p, q ∈ [1,∞) and α = (α1, ..., αn) such that αj > −1 for every j = 1, ..., n. We show that the operator is bounded if and only if it is compact and if and only if g ∈ Open image in new window , where \( \vec q \) = (q, ..., q).

M. F. Prokhorova UDC 513.83 In the article [1], E. Gordon defined the predicate "to be standard relative to t': zstt r 3't~o : (Ffin(~)) & (t E dome) & (z E ~(t)) (here F fin(~) means that ~ is a function whose values are finite sets) and introduced the natural notions of t-infinitesimal, t-illimited number, and t-limited real number. For example, a t-infinitesimal is defined as a number whose modulus is less than every positive t-standard number (this notion is a particular case of the notion of superinfinitesimal or 7r-monad [2]). It was demonstrated in [1] that, for some nonstandard natural number N, not all points of the interval [0, 1] are N-nearstandard (i.e., there exists z E [0, 1] such that there is no N-standard number N-infinitely close to x). The following question arises: is it possible to choose a nonstandard natural number N so that each point of the interval [0, 1] has an N-standard part, i.e., is N-nearstandard. If the answer to this question were positive then we could construct an external function similar to but "more detailed" than the standard part map. Unfortunately, as is shown below, the answer to this question is negative. Moreover, it remains negative in a more general case when we replace the set of natural numbers with an arbitrary set of nonmeasurable cardinality and the interval with an arbitrary Hausdorff space other than a rare compact set [3]. The theorems below are also of interest from the viewpoint of constructing the propositions of bounded internal set theory, BIST, of [4] which are equivalent to the conjecture of existence of mea- surable cardinals [5] (this conjecture is unprovable in ZFC). Let (X, r) be a topological space (r is the family of open sets). We call t X-appropriate if all points of X are t-nearstandard (i.e., for each x E X there is a t-standard y E X such that x is contained in every t-standard neighborhood of y). It is clear that such t must be bounded (i.e., t lies in some standard set). We define the complexity of a bounded t to be the cardinal complt = min{cardT : st(T) & t e T}.

UDC 513.88 It the article we study sublinear functionals that are defined on the space of Bochner integrable functions and possess the properties of decomposition and scalar compactness. We give an integral representation for the functionals. Such functionals are used in proving theorems on existence of continuous selections for a family of lower semicontinuous multivalued mappings with decomposable closed nonconvex values in the space of Bochner integrable functions. The article relates to the studies [1-3] and others. 1. We present basic definitions, notation, and a series of auxiliary results. Throughout what follows, X is a separable Banach space with norm H" ]l, T is a Hausdorff compact space with positive Radon measure/z and a-algebra ~ of/,-measurable subsets, and R stands for the real axis. We denote by L1 (T, X) the Banach space of equivalence classes of Bochner integrable functions u: T ---* X, with the norm I[u(.)HL1 = fJ[u(t)Hd#. T By X t and L ~ (T, X) we mean the topological dual spaces of X and L1 (T, X) with the strong (norm) topologies; (x, xl~ and (u(-), u'(.)) are the canonical bilinear forms establishing duality between X and X' and between LI(T, X) and UI(T , X) respectively. For the norm of X ~ we use the same notation as for that of X. The spaces X' and LI(T,X ) endowed with weak a(X~,X) - and a( L~ (T,X), Ll(T,X) )-topologies will be denoted by a-X' and a-L](T,X) (see [4, p. 6831). A function v : T ~ X I is called scalarly measurable if, given x E X, the scalar function (x, v(t)) is measurable. A function v : T ~ X ~ is called weakly measurable if, for every e > 0, there is a closed set Te C T, p(T \ Te)

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As Suss showed [I], if the projections of two convex compact sets in R ~ (n> 2) on any hyperplane can be superposed by parallel transport, then the sets themselves can be superposed in Rn by parallel transport. Earlier in [1-4] this assertion was generalized to cases in which the compact sets are not convex Or in which their projections can be superposed by means of motions rather than parallel transport. Here we are concerned with weakening yet another condition of Suss's lemma, namely, we shall assume that the information on projections is known only for part of the projection directions a situation that arises often in inverse problems of integral geometry and tomography [5-7].

The following question was posed in Rudin [I]: is it true that any nontrivial closed subalgebra of the algebra of all functions continuous on an open ball, which is invariant under the group of analytic automorphisms of the ball in C n, is the algebra of all analytic functions (or their complex conjugates)? The corresponding result was obtained previously by the author, but the proof was only published in preprint [2]. In the present paper we give an improved version of this proof and we get some similar results.

INTRODUCTION Two model equations and the problems for them, which are not within the framework of the well-known theory of parabolic equations, are considered in this paper. In the first equation the coefficient of the second derivative, ~(ux), which depends on the gradient, becomes infinite. The need to study such equations is indicated in the section of unsolved problems in [i] in connection with problems of subsurface hydrodynamics. The second problem relates to the well-known problem of nonuniformly parabolic equations when the growth with respect to the gradient of the coefficient of the second derivative, m(Ux), is not polynomial. The theorem of compactness for nonlinear sets from [3] and the lemmas in the appendixes, which generalize some already well known, are used to overcome the difficulties that arise in the proof of a passage to the limit.