Convolutions of cantor measures without resonance

Denote by µ _a the distribution of the random sum \((1 - a)\sum\nolimits_{j = 0}^\infty {{w_j}{a^j}} \), where P(ω _j = 0) = P(ω _j = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the measure µ _a is supported on C _a , the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length (1 ? 2a), and iterating this process inductively on each of the remaining intervals. We investigate the convolutions µ _a * (µ _b ° S _λ ^?1 ), where S _λ (x) = λx is a rescaling map. We prove that if the ratio log b/ log a is irrational and λ ≠ 0, then

$D({\mu _a} * ({\mu _b} \circ S_\lambda ^{ - 1})) = \min ({\dim _H}({C_a}) + {\dim _H}({C_b}),1)$

, where D denotes any of correlation, Hausdorff or packing dimension of a measure.

We also show that, perhaps surprisingly, for uncountably many values of λ the convolution µ_1/4* (µ_1/3 ° S _λ ^?1 ) is a singular measure, although dim _H (C _1/4) + dim _H (C _1/3) > 1 and log(1/3)/ log(1/4) is irrational.     

On a question of A. E. Nussbaum on measurability of families of closed linear operators in a Hilbert space

The purpose of this note is to answer a question A. E. Nussbaum formulated in 1964 about the possible equivalence between weak measurability of a family of densely defined, closed operators {T(t)} _t∈ℝ in a separable complex Hilbert space H\mathcal{H} on one hand, and the notion of measurability of the 2 × 2 operator-valued matrix of projections {(P(Γ(T(t))) _j,k )_1⩽j,k⩽2} _t∈ℝ onto the graph Γ(T(t)) of T(t) on the other, in the negative.     

Lead Finder docking and virtual screening evaluation with Astex and DUD test sets

Lead Finder is a molecular docking software. Sampling uses an original implementation of the genetic algorithm that involves a number of additional optimization procedures. Lead Finder's scoring functions employ a set of semi-empiric molecular mechanics functionals that have been parameterized independently for docking, binding energy predictions and rank-ordering for virtual screening. Sampling and scoring both utilize a staged approach, moving from fast but less accurate algorithm versions to computationally more intensive but more accurate versions. Lead Finder includes tools for the preparation of full atom protein and ligand models. In this exercise, Lead Finder achieved 72.9% docking success rate on the Astex test set when the original author-prepared full atom models were used, and 74.1% success rate when the structures were prepared by Lead Finder. The major cause of docking failures were scoring errors resulting from the use of imperfect solvation models. In many cases, docking errors could be corrected by the proper protonation and the use of correct cyclic conformations of ligands. In virtual screening experiments on the DUD test set the early enrichment factor of several tens was achieved on average. However, the area under the ROC curve ("AUC ROC") ranged from 0.70 to 0.74 depending on the screening protocol used, and the separation from the null model was not perfect-0.12-0.15 units of AUC ROC. We assume that effective virtual screening in the whole range of enrichment curve and not just at the early enrichment stages requires more accurate solvation modeling and accounting for the protein backbone flexibility.     

Induced Super‐Halogen Behavior of Metal Moieties in Halogen‐Doped Clusters: LinI(‐) and AlnI(‐), n = 13, 1, 2, 3.

ChemInform is a weekly Abstracting Service, delivering concise information at a glance that was extracted from about 200 leading journals. To access a ChemInform Abstract, please click on HTML or PDF.     

On a possible empirical meaning of meets and joins for quantum propositions

Following up an idea of Jauch, an empirical basis is worked out for the meets (and joins) of quantum propositions through approximate measurement to an arbitrary degree of accuracy of special yes-no effects constructed from chains of yes-no measurements. This is done with a view to interpreting the meets (and joins) as Dedekind's cuts in analogy with irrationals.     

Mott's cloud-chamber theory made explicit and the relative-collapse interpretation of quantum mechanics thus obtained

It is pointed out that the concept of a shiftable split between object and subject with a well-defined subject, introduced and utilized in two preceding articles of the author, is present in Mott's historical cloud-chamber measurement theory. The crucial question of the occurrence of the subject events (the main constituents of the subject) is also given a tacit answer in Mott's text. When Mott's theory is made sufficiently explicit, the object-subject complementarity principle (an elaborated form of Bohr's macroscopic complementarity principle) emerges. The (individual-system) relative-collapse postulate, which appears as a natural completion of Born's (ensemble) postulate, takes form in it. The proposed interpretation is compared with the many-worlds and the modal approaches, which share with the former the idea of nondestruction of coherence (in contrast to interpretations that contain the idea of an absolute, i.e., observer-independent collapse, which destroys coherence irrevocably).     

A structural study of copper(II) carboxylates: Crystal structure and physical characterisation of [Cu2(2-bromopropanoato)4(caffeine)2]

Based on the bond-valence sum model the continual transition between the bidentate and monodentate bonding mode of carboxylato group and minimum of Cu–O bond lengths were estimated. The dependence of Cu⋯Cu separation on Cu–N bond length in binuclear Cu(II) carboxylates with caffeine were derived and fitted to the observed data. The crystal structure of [Cu₂(2-bromopropanoato)₄(caffeine)₂] has been determined by X-ray crystallography. The structure consists of centrosymmetric binuclear units where the two Cu atoms are coordinated by four disordered bridging 2-bromopropionates and two caffeine ligands at the apices of a bicapped square prism. Both 2-bromopropionates show disorder of their –Br and –CH₃ substituents. Stacking π·π interactions between the adjacent caffeine molecules link the complex units in 1-D networks. The binuclear structure of the studied compound is consistent with magnetic data and EPR spectrum.     

Instantons and mirror K3 surfaces

The instanton moduli space of a real 4-dimensional torus is an 8-dimensional Calabi-Yau manifold. Associated to this Calabi-Yau manifold are two (singular) K3 surfaces, one a quotient, the other a submanifold of the moduli space; both carry a natural Calabi-Yau metric. They are curiously related in much the same way as special examples of complex 3-dimensional mirror manifolds; however, in our case the mirror is present in the form of instanton moduli.     

First example of the cascade acylation/IMDAV/ene reaction sequence,leading to N-arylbenzo[f]isoindole-4-carboxylic acids possessing anti-viral activity

The reaction between readily accessible N-aryl-3-phenylallylamines and maleic anhydride led to unexpected products – polysubstituted hydrogenated benzo[f]isoindole-4-carboxylic acids. This transformation proceeds through a previously unknown sequence of steps: N-acylation of the allylamine with maleic anhydride, intramolecular Diels-Alder reaction of the vinylarene in the intermediate N-maleamide, and Alder-ene reaction of the products of the previous two steps. Selected benzo[f]isoindoles displayed antiviral activity.