Dimensions of some fractals defined via the semigroup generated by 2 and 3

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σ _m ={0, ...,m?1}^? that are invariant under multiplication by integers. The results apply to the sets {x∈Σ _m :? k, x _k x _2k ... x _nk =0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.     

Semi-invariant Matrices over Finite Groups

The semi-center of an artinian semisimple module-algebra over a finite group G can be described using the projective representations of G. In particular, the semi-center of the endomorphism ring of an irreducible projective representation over an algebraically closed field has a structure of a twisted group algebra. The following group-theoretic result is deduced: the center of a group of central type embeds into the group of its linear characters.     

Polymer and biopolymer mediated self-assembly of gold nanoparticles

Gold nanoparticle-polymer composites are versatile and diverse functional materials, with applications in optical, electronic and sensing devices. This tutorial review focuses on the use of polymers to control the assembly of gold nanoparticles. Examples of synthetic polymers and biopolymers are provided, as well as applications of the composite materials in sensing and memory devices.     

Fluctuation of a planar Brownian loop capturing a large area

We consider a planar Brownian loop that is run for a time and conditioned on the event that its range encloses the unusually high area of , with being large. The conditioned process, denoted by , was proposed by Senya Shlosman as a model for the fluctuation of a phase boundary. We study the deviation of the range of from a circle of radius . This deviation is measured by the inradius and outradius , which are the maximal radius of a disk enclosed by the range of , and the minimal radius of a disk that contains this range. We prove that, in a typical realization of the conditioned measure, each of these quantities differs from by at most .

     

Effect of metal cations on polydiacetylene Langmuir films

Polydiacetylene (PDA) Langmuir films (LFs) are a unique class of materials that couple a highly aligned conjugated backbone with tailorable pendant side groups and terminal functionalities. The films exhibit chromatic transitions from monomer to blue polymer and finally to a red phase that can be activated optically, thermally, chemically, and mechanically. The properties of PDA LFs are strongly affected by the presence of metal cations in the aqueous subphase of the film due to their interaction with the carboxylic head groups of the polymer. In the present study the influence of divalent cadmium, barium, copper, and lead cations on the structural, morphological, and optical properties of PDA LFs was investigated by means of surface pressure-molecular area (π-A) isotherms, atomic force microscopy, optical absorbance, and Raman spectroscopy. The threshold concentrations for the influence of metal cations on the film structure, stability, and phase transformation were determined by π-A analyses. It was found that each of the investigated cations has a unique influence on the properties of PDA LFs. Cadmium cations induce moderate phase transition kinetics with reduced domain size and fragmented morphology. Barium cations contribute to stabilization of the PDA blue phase and enhanced linear strand morphology. On the other hand, copper cations enhance rapid formation of the PDA red phase and cause fragmented morphology of the film, while the presence of lead cations results in severe perturbation of the film with only a small area of the film able to be effectively polymerized. The influence of the metal cations is correlated with the solubility product (K(sp)), association strength, and ionic-covalent bond nature between the metal cations and the PDA carboxylic head groups.     

Comparative study of the photoprotolytic reactions of D-luciferin and oxyluciferin

Optical steady-state and time-resolved spectroscopic methods were used to study the photoprotolytic reaction of oxyluciferin, the active bioluminescence chromophore of the firefly's luciferase-catalyzed reaction. We found that like D-luciferin, the substrate of the firefly bioluminescence reaction, oxyluciferin is a photoacid with pK(a)* value of ～0.5, whereas the excited-state proton transfer (ESPT) rate coefficient is 2.2 × 10(10) s(-1), which is somewhat slower than that of D-luciferin. The kinetic isotope effect (KIE) on the fluorescence decay of oxyluciferin is 2.5 ± 0.1, the same value as that of D-luciferin. Both chromophores undergo fluorescence quenching in solutions with a pH value below 3.     

Mesoscopic stoner instability in metallic nanoparticles revealed by shot noise

We study sequential tunneling through a metallic nanoparticle close to the Stoner instability coupled to parallel magnetized electrodes. Increasing the bias voltage successively opens transport channels associated with excitations of the nanoparticle's total spin. For the current this leads just to a steplike increase. The Fano factor, in contrast, shows oscillations between large super-Poissonian and sub-Poissonian values as a function of bias voltage. We explain the enhanced Fano factor in terms of generalized random-telegraph noise and propose the shot noise as a convenient tool to probe the mesoscopic Stoner instability.     

Self-similar sets of zero Hausdorff measure and positive packing measure

We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑ ₀ ⁸ a _n 4⁻ⁿ a _n 4⁻ⁿ with digits witha _n ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections. Research of Y. Peres was partially supported by NSF grant #DMS-9803597. Research of K. Simon was supported in part by the OTKA foundation grant F019099. Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics at The Hebrew University of Jerusalem.     

Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p ₂>p ₁>p _c , each infinite p ₂-cluster contains an infinite p ₁-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θ _v (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p _c . All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group. Received: 22 December 1997 / Revised version: 1 July 1998     

Toward extracting $$\gamma $$ from $$B\rightarrow DK$$ without binning

SABRE (Sodium iodide with Active Background REjection) is a direct detection dark matter experiment based on arrays of radio-pure NaI(Tl) crystals. The experiment aims at achieving an ultra-low background rate and its primary goal is to confirm or refute the results from the DAMA/LIBRA experiment. The SABRE Proof-of-Principle phase was carried out in 2020–2021 at the Gran Sasso National Laboratory (LNGS), in Italy. The next phase consists of two full-scale experiments: SABRE South at the Stawell Underground Physics Laboratory, in Australia, and SABRE North at LNGS. This paper focuses on SABRE South and presents a detailed simulation of the detector, which is used to characterise the background for dark matter searches including DAMA/LIBRA-like modulation. We estimate an overall background of 0.72 cpd/kg/\(\hbox {keV}_{\hbox {{ee}}}\) in the energy range 1–6 \(\hbox {keV}_{\hbox {{ee}}}\) primarily due to radioactive contamination in the crystals. Given this level of background and considering that the SABRE South has a target mass of 50 kg, we expect to exclude (confirm) DAMA/LIBRA modulation at \(4~(5)\sigma \) within 2.5 years of data taking.