On images of Borel measures under Borel mappings

Let and be metric spaces. We show that the tight images of a (fixed) tight Borel probability measure on , under all Borel mappings , form a closed set in the space of tight Borel probability measures on with the weak-topology. In contrast, the set of images of under all continuous mappings from to may not be closed. We also characterize completely the set of tight images of under Borel mappings. For example, if is non-atomic, then all tight Borel probability measures on can be obtained as images of , and as a matter of fact, one can always choose the corresponding Borel mapping to be of Baire class 2.

     

On separable supports of Borel measures

Some properties of Borel measures with separable supports are considered. In particular, it is proved that any -finite Borel measure on a Suslin line has a separable support, and from this fact it is deduced, using the continuum hypothesis, that any Suslin line contains a Luzin subspace with the cardinality of the continuum.     

Classes of interval graphs under expanding length restrictions

Let C(α) denote the finite interval graphs representable as intersection graphs of closed real intervals with lengths in [1, α]. The points of increase for C are the rational α ≥ 1. The set D(α) = [∩_β>αC(β)]\C(α) of graphs that appear as soon as we go past α is characterized up to isomorphism on the basis of finite sets E(α) of irreducible graphs for each rational α. With α = p/q and p and q relatively prime, ∣E(α)∣ is computed for all (p,q) with q ? 2 and p = q + 1. When q = 1, E(p) contains only the bipartite star K¹, p_+2. A lowr bound on ∣E(α)∣ is given for all rational α.     

Borel measures with a density on a compact semi-algebraic set

Let ${\mathbf{K} \subset \mathbb{R}^n}$ be a compact basic semi-algebraic set. We provide a necessary and sufficient condition (with no à priori bounding parameter) for a real sequence y = (y _α), ${\alpha \in \mathbb{N}^n}$ , to have a finite representing Borel measure absolutely continuous w.r.t. the Lebesgue measure on K, and with a density in ${\cap_{p \geq 1} L_p(\mathbf{K})}$ . With an additional condition involving a bounding parameter, the condition is necessary and sufficient for the existence of a density in L _∞(K). Moreover, nonexistence of such a density can be detected by solving finitely many of a hierarchy of semidefinite programs. In particular, if the semidefinite program at step d of the hierarchy has no solution, then the sequence cannot have a representing measure on K with a density in L _p (K) for any p ≥ 2d.     

Borel complexity of the space of probability measures

Using a technique developed by Louveau and Saint Raymond, we find the complexity of the space of probability measures in the Borel hierarchy: if is any non-Polish Borel subspace of a Polish space, then , the space of probability Borel measures on with the weak topology, is always true , where is the least ordinal such that is .

     

Hausdorff measures of different dimensions are not Borel isomorphic

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H ^s ) and (ℝ, B, H ^t ) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H ^d is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d. We also prove this statement in a more general form: If A ⊂ ℝⁿ is Borel and ƒ: A → ℝ^m is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A). Partially supported by the Hungarian Scientific Research Fund grant no. T 49786.     

On soft mappings of the unit ball of Borel measures

The main result of this paper consists of two theorems. One of them asserts that the functor U _τ takes the 0-soft mappings between spaces of weight ≤ω ₁ and Polish spaces to soft mappings. The other theorem, which is a corollary of the first one, asserts that the functor U _τ takes the AE(0)-spaces of weight ≤ω ₁ to AE-spaces. These theorems are proved under Martin’s axiom MA(ω ₁). The results cannot be extended to spaces of weight ≥ω ₂. For spaces of weight ω ₁, these results cannot be obtained without additional set-theoretic assumptions. Thus, the question as to whether the space is an absolute extensor cannot be answered in ZFC. The main result cannot be transferred to the functor U _R of the unit ball of Radon measures. Indeed, the space is not real-compact and, therefore, . __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 41–54, 2003.     

Classes of Carleson-type measures generated by capacities

We introduce classes of measures in the half-space generated by Riesz, or Bessel, or Besov capacities in , and give a geometric characterization of these as Carleson-type measures. G. Dafni was supported in part by NSERC, the CRM, Barcelona, and the CRM, Montreal. G. E. Karadzhov was supported in part by Memorial and Concordia Universities, NSERC, and the CRM, Montreal. J. Xiao was supported in part by NSERC of Canada and Dean of Science Start-up Funds of Memorial University.     

Borel Subalgebras Redux with Examples from Algebraic and Quantum Groups

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J. 45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.     

Rigid versus unique determination of protein structures with geometric buildup

We introduce a geometric buildup approach to the distance geometry problem in protein modeling, and discuss the necessary and sufficient conditions on the distances for rigid or unique determination of a protein structure. We describe a new buildup algorithm for determining protein structures rigidly instead of uniquely. The algorithm requires even fewer distance constraints than the general buildup algorithm. We present the test results from applying the algorithm to determining the protein structures with varying degrees of availability of the distances, and show that the new development increases the modeling ability of the geometric buildup method even more while retaining much of the computational feasibility of the method.     

Certain strict topologies on the space of regular Borel measures on locally compact groups

Let G   denote a locally compact Hausdorff group and M(G)$M(G)$ be the space of all bounded complex-valued regular Borel measures on G  . In this paper, we define two strict topologies on M(G)$M(G)$ and study various properties of these topologies such as metrizability, barrelledness and completeness. We also determine the dual space of M(G)$M(G)$ and consider various continuity properties for the convolution product on M(G)$M(G)$ under these topologies.     

Ideals with bases of unbounded Borel complexity

We present several naturally defined σ‐ideals which have Borel bases but, unlike for the classical examples, these ideals are not of bounded Borel complexity. We investigate set‐theoretic properties of such σ‐ideals.     

On unique determination of domains in Euclidean spaces

The paper is devoted to two new directions in developing the classical geometric subjects related to studying the problem of unique determination of closed convex surfaces by their intrinsic metrics. The first of these directions is the study of unique determination of domains (i.e., open connected sets) in Euclidean spaces by relative metrics of the boundaries of these domains. It appeared about 25–30 years ago and was developed owing to the efforts of Russian scientists. The first part of the paper (Secs. 3–7) contains an overview of the results referring to this direction. The foundations of the second direction are presented in the second part of the paper, i.e., in Sec. 8, for the first time. This direction is closely related with the first one and consists of studying the problem of unique determination of conformal type. The main result of the section is the theorem on the unique determination of bounded convex domains by relative conformal moduli of their boundary conductors. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 22, Geometry, 2007.     

Stability analysis of fractional-order complex-valued neural networks with time delays

In this paper, we consider the problem of stability analysis of fractional-order complex-valued Hopfield neural networks with time delays, which have been extensively investigated. Moreover, the fractional-order complex-valued Hopfield neural networks with hub structure and time delays are studied, and two types of fractional-order complex-valued Hopfield neural networks with different ring structures and time delays are also discussed. Some sufficient conditions are derived by using stability theorem of linear fractional-order systems to ensure the stability of the considered systems with hub structure. In addition, some sufficient conditions for the stability of considered systems with different ring structures are also obtained. Finally, three numerical examples are given to illustrate the effectiveness of our theoretical results.     

Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

We study the inheritance of properties of free backward propagators associated with transition probability functions by backward Feynman-Kac propagators corresponding to functions and time-dependent measures from non-autonomous Kato classes. The inheritance of the following properties is discussed: the strong continuity of backward propagators on the space , the -smoothing property of backward propagators, and various generalizations of the Feller property. We also prove that a propagator on a Banach space is strongly continuous if and only if it is separately strongly continuous and locally uniformly bounded.