Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝ ^d , let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C ₊², then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C ₊³) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K. Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).     

Violation of distribution symmetry in statistical evaluation of absolute enantioselective synthesis

Enantiomeric excesses obtained in absolute enantioselective synthesis by chiral autocatalysis (Soai-reaction) were statistically analyzed. Two sets of parallel experiments, which were performed under chemically different conditions, are available. One group contains 37, while the other contains 84 preparative results. The former group shows some interesting tendencies but does not give conclusive statistical results. The sample of 84 parallel experiments, providing 39 R- and 45 S-excesses have shown that these data represent two distinct, non-symmetric sets with different non-Gaussian distributions. Clear S preference was found.     

Hausdorff Dimension for Randomly Perturbed Self Affine Attractors

In this paper we shall consider a self-affine iterated function system in , d ≥ 2, where we allow a small random translation at each application of the contractions. We compute the dimension of a typical attractor of the resulting random iterated function system, complementing a famous deterministic result of Falconer, which necessarily requires restrictions on the norms of the contraction. However, our result has the advantage that we do not need to impose any additional assumptions on the norms. This is of benefit in practical applications, where such perturbations would correspond to the effect of random noise. We also give analogous results for the dimension of ergodic measures (in terms of their Lyapunov dimension). Finally, we apply our method to a problem originating in the theory of fractal image compression. Research of Jordan and Pollicott was supported by the EPSRC and the research of Simon was supported by an EU-Marie Curie grant and the OTKA Foundation #T42496.     

Determination of the thermal energy and its distribution in peptides

The mean thermal energy and its distribution has been calculated for various peptides. The table, figures, and analytical expressions provided in the paper can be used to determine the mean thermal energy and its distribution in peptides in a very simple way, without having to use complex mathematics. Accuracy is ∼±6%. The same expressions can be used for most organic compounds with an estimated accuracy of ∼±10%. Data for W(CO)₆, the “thermometer molecule,” are also given.     

M?ssbauer spectroscopy in heterogeneous catalysis: A brief overview and recent study on paramagnetic features detected in catalytic centres of Fe-ZSM-5

A short retrospection is given on the development of M?ssbauer spectroscopy and on the early period of its application to heterogeneous catalysis in Hungary. Besides the historical view, we present results of recent studies of a latest application on a nanosystem. Namely, catalytic centres in Fe-ZSM-5 are studied by in situ spectroscopy, the results are correlated with parallel activity measurements in the decomposition of N₂O. Changes in paramagnetic relaxation features are interpreted as the effects of Fe²⁺/Fe³⁺ interconversions. From the analysis, the dominance of the mononuclear species is suggested at lower temperature (620 K). The dinuclear Fe⁽³⁺⁾ _framework-O-Fe⁽²⁺⁾ _{extra-framework} species play probably more expressed role at higher temperatures (∼ 770 K) in the decompositon of nitrous oxide.     

Comparison of Fe-AlPILC and Fe-ZSM-5 catalysts used for degradation of methomyl

Catalytic performances of Fe-AlPILC (14 wt.% Fe) and Fe-ZSM-5 (5 wt.% Fe) catalysts are compared in the wet oxidative degradation of methomyl. Fe-ZSM-5 exhibits outstanding whereas Fe-AlPILC shows only mediocre activity. Positions of iron are analysed in the two catalysts by Mössbauer spectroscopy. Iron is in highly dispersed state in Fe-AlPILC whereas in the other case a hematite/ZSM-5 composite is formed. The catalytic activity is attributed to iron located and stabilized in ionic dispersion.     

The Dimension of Projections of Fractal Percolations

Fractal percolation or Mandelbrot percolation is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by $E\subset\mathbb{R}^{2}$ a typical realization of the fractal percolation on the plane,

• If $\dim_{\rm H}E<1$ then for all lines ? the orthogonal projection E _? of E to ? has the same Hausdorff dimension as E,
• If $\dim_{\rm H}E>1$ then for any smooth real valued function f which is strictly increasing in both coordinates, the image f(E) contains an interval.

The second statement is quite interesting considering the fact that E is almost surely a Cantor set (a random dust) for a large part of the parameter domain, see Chayes et al. (Probab. Theory Relat. Fields, 77(3):307–324, 1988). Finally, we solve a related problem about the existence of an interval in the algebraic sum of d≥2 one-dimensional fractal percolations.     

Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space

The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. In particular, we prove that if C(n) denotes the largest number of touching pairs in a packing of n>1 congruent balls in Euclidean 3-space, then $0.695<\frac{6n-C(n)}{n^{\frac{2}{3}}}< \sqrt[3]{486}=7.862\dots$ for all $n=\frac{k(2k^{2}+1)}{3}$ with k??2.