On the Hausdorff dimension of fibres

It is well known that there are planar sets of Hausdorff dimension greater than 1 which are graphs of functions, i.e., all their vertical fibres consist of 1 point. We show this phenomenon does not occur for sets constructed in a certain “regular” fashion. Specifically, we consider sets obtained by partitioning a square into 4 subsquares, discarding 1 of them and repeating this on each of the 3 remaining squares, etc.; then almost all vertical fibres of a set so obtained have Hausdorff dimension at least 1/2. Sharp bounds on the dimensions of sets of exceptional fibres are presented. Partially supported by a grant from the Landau Centre for Mathematical Analysis.     

Points of increase for random walks

Say that a sequenceS ₀, ..., S_n has a (global) point of increase atk ifS _k is maximal amongS ₀, ..., S_k and minimal amongS _k, ..., S_n. We give an elementary proof that ann-step symmetric random walk on the line has a (global) point of increase with probability comparable to 1/logn. (No moment assumptions are needed.) This implies the classical fact, due to Dvoretzky, Erdős and Kakutani (1961), that Brownian motion has no points of increase. Research partially supported by NSF grant # DMS-9404391.     

Symmetry and “Magic” Numbers or from the pythagoreans to Eugene Wigner

I trace the evolution of the scientific world view in general and of the variational approach specifically, including transformation groups I then review the impact of these ideas at the atomic, nuclear and particle levels.     

Application of Banach limits to the study of sets of integers

The Kamae and Mendes France version of the Van der Corput equidistribution theorem is extended further to summability methods different from Cesàro summability and groups different from the circle. The theorem is shown to follow naturally from consideration of Banach limits and spectral theory.     

CHN,QCD, and\overline {SA} (4,R)

CHN≓ (1966)was an algebraic algorithm which reproduced and extended the predictions of the non-interacting≓ quark model in the asymptotic high-energy region. It wus formulated within the conceptual framework of on- mass- shell physics and of the complex angular-momentum plane. Prior to the advent of the standard model, it was reinterpreted in terms of the Melosh transformation relating current≓ to constituent≓ quarks. It is now lied up to the QCD paradigm.Also on leave from Center for Particle Physics, University of Texas, Austin, Texas 78712.     

World spinors—Construction and some applications

The existence of a topological double-covering for the GL(n, R) and diffeomorphism groups is reviewed. These groups do not have finite-dimensional faithful representations. An explicit construction and the classification of all (n, R), n=3,4 unitary irreducible representations is presented. Infinite-component spinorial and tensorial fields, “manifields”, are introduced. Particle content of the ladder manifields, as given by the (3, R) “little” group, is determined. The manifields are lifted to the corresponding world spinorial and tensorial manifields by making use of generalized infinite-component frame fields. World manifields transform w.r.t. corresponding representations, which are constructed explicitly. Supported in part by the Science Foundation (Belgrade).     

Exploring long-range proton conduction,the conduction mechanism and inner hydration state of protein biopolymers

Proteins are the main proton mediators in various biological proton circuits. Using proteins for the formation of long-range proton conductors is offering a bioinspired approach for proton conductive polymers. One of the main challenges in the field of proton conductors is to explore the local environment within the polymers, along with deciphering the conduction mechanism. Here, we show that the protonic conductivity across a protein-based biopolymer can be hindered using straightforward chemical modifications, targeting carboxylate- or amine-terminated residues of the protein, as well as exploring the effect of surface hydrophobicity on proton conduction. We further use the natural tryptophan residue as a local fluorescent probe for the inner local hydration state of the protein surface and its tendency to form hydrogen bonds with nearby water molecules, along with the dynamicity of the process. Our electrical and spectroscopic measurements of the different chemically-modified protein materials as well as the material at different water–aprotic solvent mixtures result in our fundamental understanding of the proton mediators within the material and gaining important insights on the proton conduction mechanism. Our biopolymer can be used as an attractive platform for the study of bio-related protonic circuits as well as a proton conducting biopolymer for various applications, such as protonic transistors, ionic transducers and fuel cells.

Post formation modification of protein-based materials can attenuate the proton conduction efficiency resulting from change in conduction mechanism, charge carrier mobility, carrier concentrations and inner hydration layer.     

Effect of hot acid etching on the mechanical strength of ground YAG laser elements

High-power pumped Nd:YAG elements may exceed their tensile strength under high thermally induced stress. Providing extra strength to such rods is essential for their employment in high-power lasers. The tensile strength of YAG elements was increased by chemical etching in concentrated phosphoric acid. The highest tensile strength was achieved by etching of fine-ground YAG components: an average for slabs, and for rods, which are 3.6 times and 5 times higher than those of non-etched elements, respectively. The measurements were carried out by four-point flexure strength test. We have established a dependency among the micro-roughness of YAG elements, the surface morphology obtained by etching, and the tensile strength: the tensile strength of the etched element improves for finer after-etch surface texture, which is obtained for finer initial micro-roughness.

To assure the withstanding of Nd:YAG rods under high thermal gradients, a new approach was employed, namely, increasing the pump-power applied to the Nd:YAG rod till fracture. Our results show an increase by more than 2.7 times in tensile strength of etched Nd:YAG rods as compared to standard commercial rods, which corresponds to a thermal loading of excess of 434 W/cm.     

Boolean constant degree functions on the slice are juntas

We show that a Boolean degree $d$ function on the “slice” $\left(\genfrac{}{}{0ex}{}{\left[n\right]}{k}\right)?\{(x_1,\dots ,x_n)\in \{0,1\}:{\sum }_{i=1}^n{x}_i=k\}$ is a junta (depends on a constant number $\gamma \left(d\right)$ of coordinates), assuming that $k,n?k$ are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube ${\{0,1\}}^n$. Moreover, we show that the maximum number of coordinates that a Boolean degree $d$ function can depend on is the same on the slice and on the hypercube.     

Trees and Markov Convexity

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors. The research of the first author was conducted while he was at U. C. Berkeley and the Institute for Advanced Study.