An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension

The famous 1960's construction of Golod and Shafarevich yields infinite dimensional nil, but not nilpotent, algebras. However, these algebras have exponential growth. Here, we construct an infinite dimensional nil, but not locally nilpotent, algebra which has polynomially bounded growth.

     

Isospectrality and galois projective geometries

We construct a series of pairs of domains in the plane and pairs of surfaces with boundary that are isospectral but not isometric. The construction is based on the existence of finite transformation groups that are spectrally equivalent but not isomorphic. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 660–664, May, 1998. This research was partially supported by the Russian Foundation for Basic Research and by INTAS under joint grant No. 96-01-00043.     

Dimension Theory and Nonstable K 1

This article provides an introduction to A. Bak's theory of group-valued functors on categories with structure and dimension and applies this theory to the algebraic K-theory functors nonstable K ₁ and K ₂.     

On classical analogues of free entropy dimension

We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on Rn. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.     

Some refinements of Dade's projective conjecture

In this paper a further refinement of Dade's projective conjecture, due to Boltje, is presented. This new statement includes ideas first published by Isaacs and Navarro as well as the recent contractibility version of Alperin's conjecture introduced by Boltje. Leaning heavily on the work of Robinson, weaker forms of the conjecture are proved in the case of p-solvable groups.     

Size distribution of fractured areas in one-dimensional systems

We study a one-dimensional model for fracture, identifying fractured areas with intervals on which a stress field exceeds a threshold value. When is a diffusion process, the cumulative numberN(l) of fractured areas whose length is greater thanl obeys a power lawCl ^–p asl0 with probability one. The exponentp and the constantC are determined. The exponentp agrees with the Hausdorff dimension of the end points of fractured areas, i.e., ^–1(). Even if is self-similar with parameterH>0, i.e.,(cx)– is equivalent toc ^H {(x)–} for anyc>0, the exponentp does not depend solely onH;p=H, where(0, 1/H) is another parameter characterizing. Non-diffusion processes are given whereN(l) does not follow a power law.     

Pattern formation in diffusion-limited reactions

The conditions for macroscopic segregation ofA andB in a steady-stateA+B 0 reaction are studied in infinite systems. Segregation occurs in one and two dimensions and is marginal ford=3. We note the dependence of these results on the precise experimental conditions assumed in the theory. We also note the difference between these results and our earlier ones for finite systems where the critical dimension isd=2.     

On the critical behavior of the magnetization in high-dimensional Ising models

We derive rigorously general results on the critical behavior of the magnetization in Ising models, as a function of the temperature and the external field. For the nearest-neighbor models it is shown that ind4 dimensions the magnetization is continuous atT _c and its critical exponents take the classical values=3 and=1/2, with possible logarithmic corrections atd=4. The continuity, and other explicit bounds, formally extend tod>3 1/2. Other systems to which the results apply include long-range models ind=1 dimension, with 1/|x–y| couplings, for which 2/(–1) replacesd in the above summary. The results are obtained by means of differential inequalities derived here using the random current representation, which is discussed in detail for the case of a nonvanishing magnetic field.Research supported in part by NSF grant PHY-8301493 A02, and by a John S. Guggenheim Foundation fellowship (M.A.).     

Characterization of Titania/Silica Gel by Means of Low-Pressure Nitrogen Adsorption

Adsorbents synthesized by grafting of titania onto mesoporous silica gel surfaces at different temperatures were studied by means of nitrogen adsorption–desorption and water desorption. The pore size distribution f(R_p) of titania/silica gel depends on the titania concentration (C_TiO2) and the temperature of titania synthesis. Nonuniformity of TiO₂ phase is maximal at a low C_TiO2 value (3.2 wt.% anatase deposited at 473 K), and two peaks of the fractal dimension distribution f(D) are observed at such a concentration of titania, but at larger C_TiO2 values, only one f(D) peak is seen. More ordered filling of pores and adsorption sites by nitrogen, reflecting in the shape of adsorption energy distributions f(E) at different pressures of adsorbate, is observed for adsorbent with titania (rutile+anatase) grafted on silica gel at a higher temperature (673 K).     

On the classification of geometric codes by polynomial functions

It is shown how to represent algebraically all functions that have a zero sum on all -dimensional subspaces ofPG(n,q) or ofAG(n,q). In this way one can calculate the dimensions of related codes, or one can represent interesting sets of points by functions.