Folding degrees of azurins and pseudoazurins. Implications for structure and function

A quantitative measure of the degree of folding of azurins and pseudoazurins has been made. We have found that the reduction potential of azurins and pseudoazurins is a function of the contribution to the degree of folding of His117, a key amino acid in electron transfer which is directly bonded to copper in these proteins. The folding degree of His117 explains 95% of the variance in the experimental values of the reduction potential of azurins and pseudoazurins. The change in the folding degree of this amino acid influences several geometric parameters of the main backbones of these proteins. Among them, the angle formed between N(His117)...Cu...S(Cys112), which plays an important role in electron transport, but not the N(His117)...Cu distance, shows some non-linear correlation with the reduction potential of azurins and pseudoazurins. However, it is only able to explain less than 75% in the variance of the reduction potential of these proteins instead of the 95% explained by the folding degree of His117.     

From Pappus’ Theorem to the Twisted Cubic

We discuss a classical result in planar projective geometry known as Steiners theorem involving 12 interlocking applications of Pappus theorem. We prove this result using three dimensional projective geometry then uncover the dynamics of this construction and relate them to the geometry of the twisted cubic.Mathematics Subject Classification (2000). Primary 51N15.     

Observables in a noncommutative approach to the unification of quanta and gravity: a finite model

We further develop a noncommutative model unifying quantum mechanics and general relativity proposed in Gen. Rel. Grav. (36, 111–126 (2004)). Generalized symmetries of the model are defined by a groupoid given by the action of a finite group on a space E. The geometry of the model is constructed in terms of suitable (noncommutative) algebras on . We investigate observables of the model, especially its position and momentum observables. This is not a trivial thing since the model is based on a noncommutative geometry and has strong nonlocal properties. We show that, in the position representation of the model, the position observable is a coderivation of a corresponding coalgebra, coparallelly to the well-known fact that the momentum observable is a derivation of the algebra. We also study the momentum representation of the model. It turns out that, in the case of the algebra of smooth, quickly decreasing functions on , the model in its quantum sector is nonlocal, i.e., there are no nontrivial coderivations of the corresponding coalgebra, whereas in its gravity sector such coderivations do exist. They are investigated.This revised version was published online in April 2005. The publishing date was inserted.     

On d-Dimensional Dual Hyperovals

d-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.     

Croissance lineaire et géométrie bornée

We show that on a noncompact manifold which has finite topology at infinity, there exists a Riemannian metric with bounded geometry and linear growth-type.     

Metric and isoperimetric problems in symplectic geometry

Our first result is a reduction inequality for the displacement energy. We apply it to establish some new results relating symplectic capacities and the volume of a Lagrangian submanifold in a number of different settings. In particular, we prove that a Lagrange submanifold always bounds a holomorphic disc of area less than , where is some universal constant. We also explain how the Alexandroff-Bakelman-Pucci inequality is a special case of the above inequalities. Our inequality on displacement of reductions is also applied to yield a relation between length of billiard trajectories and volume of the domain. Two simple results concerning isoperimetric inequalities for convex domains and the closure of the symplectic group for the norm are included.
     

Criteria for -ampleness

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a -ample divisor, where is an automorphism of a projective scheme . Many open questions regarding -ample divisors have remained.

We derive a relatively simple necessary and sufficient condition for a divisor on to be -ample. As a consequence, we show right and left -ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms yield a -ample divisor.

     

Skein modules and the noncommutative torus

We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cylinder over a torus, and give a straightforward computation of the -th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.

     

“Magic Relation” Between the Structures of Coexisting Phases at a First-Order Phase Transition in a Hard Sphere System

In the statistical geometry of a hard sphere system of any number of dimensions, V _o and S _o, the so-called available space and the area of the interface between the available and unavailable space, respectively, can be used as surrogates for chemical potential and pressure. It is shown exactly that, if a first-order transition occurs, the relation dV _o/dS _o=–/2D, where is the diameter of a sphere and D is the dimensionality of the system, must hold for densities in the phase coexistence region. This relation is remarkable in that –/2D is the ratio of the volume to the surface area of a sphere. Also, it is shown that it is possible for the system to have two successive first-order transitions, but that the occurrence of a continuous transition (even in two dimensions) is unlikely. It is argued that this unlikelihood is substantially strengthened by the absence of temperature (except as a trivial factor) as a variable in hard-sphere systems. This suggests that the findings of the KTHNY theory, recent simulations, and colloid experiments (specialized to sticky hard disks) can be extended to true hard disks. The fundamental physics underlying the magic relation is yet to be exposed. The author continues to search for the underlying reason and hopes that the present paper will stimulate others to join the search.     

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi-Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer-Severi varieties and, in the quaternion case, we show how to obtain some Brauer-Severi varieties from G-Hilbert schemes of P¹-bundles.