Meyer's concept of quasicrystal and quasiregular sets

This paper relates two mathematical concepts of long-range order of a set of atoms , each of which is based on restrictions on the set of interatomic distances –. A set in ⁿ is aMeyer set if is a Delone set and there is a finite setF such that . Meyer proposed that such sets include the possible structures of quasicrystals. He obtained a structure theory for such sets, which reformulates results that he obtained in harmonic analysis around 1970, and which relates these sets to cut-and-project sets. In geometric crystallography V.I. Galiulin introduced the concept ofquasiregular set, which is a set such that both and – are Delone sets. This paper shows that these two concepts are identical.     

Nonnegative Radix Representations for the Orthant

Let be a nonnegative real matrix which is expanding, i.e. with all eigenvalues , and suppose that is an integer. Let consist of exactly nonnegative vectors in . We classify all pairs such that every in the orthant has at least one radix expansion in base using digits in . The matrix must be a diagonal matrix times a permutation matrix. In addition must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set can be diagonally scaled to lie in . The proofs generalize a method of Odlyzko, previously used to classify the one--dimensional case.

     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

COMPARATIVE PHOTOCHEMICAL ANALYSIS OF HIGHLY PURIFIED 124 KILODALTON OAT and RYE PHYTOCHROMES in vitro

Abstract A direct comparison of the photochemical interconversions between red (P_r-) and far-red (P_fr-) absorbing forms of highly-purified 124 kDa oat and rye phytochromes under identical experimental conditions was performed. In two different buffer systems at 5°C, the quantum yields for the P_r to P_tr and P_fr to P_r phototransformations under constant red and far-red illumination, φ _r and φ_fr respectively, were determined to be 0.152-0.154 and 0.060-0.065 for oat preparations and 0.172-0.174 and 0.074-0.078 for rye preparations. These values as well as the wavelength dependence of the photoequilibrium produced under continuous illumination throughout the visible and near-ultraviolet spectrum were based on the absorption spectra of the two phytochrome preparations and revised molar absorption coefficients. The molar absorption coefficients were estimated by quantitative amino acid analysis and shown to be identical for the two monocot phytochromes (i.e. 132 mM ⁻¹ cm⁻¹ at the red absorption maximum for the P_r form). Because these measurements were performed under identical experimental conditions, including buffer, temperature, light fluence rate, and instrumentation, the differences observed must reflect structural features inherent to the two different monocotyledonous phytochromes.     

Biliverdin reduction by cyanobacterial phycocyanobilin:ferredoxin oxidoreductase (PcyA) proceeds via linear tetrapyrrole radical intermediates

Cyanobacterial phycocyanobilin:ferredoxin oxidoreductase (PcyA) catalyzes the four electron reduction of biliverdin IXalpha (BV) to phycocyanobilin, a key step in the biosynthesis of the linear tetrapyrrole (bilin) prosthetic groups of cyanobacterial phytochromes and the light-harvesting phycobiliproteins. Using an anaerobic assay protocol, optically detected bilin-protein intermediates, produced during the PcyA catalytic cycle, were shown to correlate well with the appearance and decay of an isotropic g approximately 2 EPR signal measured at low temperature. Absorption spectral simulations of biliverdin XIIIalpha reduction support a mechanism involving direct electron transfers from ferredoxin to protonated bilin:PcyA complexes.     

Apollonian circle packings: number theory

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: . Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element −n as a class number, and give an exact formula for it. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. We present evidence suggesting that the set of integer radii of curvatures that appear in an integral Apollonian circle packing has positive density, and in fact represents all sufficiently large integers not excluded by congruence conditions. Finally, we discuss asymptotic properties of the set of curvatures obtained as the packing is recursively constructed from a root quadruple.     

Single-suit two-person card play

We consider a two-person constant sum perfect information game, which we call theEnd Play Game, which arises from an abstraction of simple end play positions in card games of the whist family, including bridge. This game was described in 1929 by Emanuel Lasker, the mathematician and world chess champion, who called itwhistette. The game uses a deck of cards that consists of a single totally ordered suit of 2n cards. To begin play the deck is divided into two handsA andB ofn cards each, held by players Left and Right, and one player is designated as having thelead. The player on lead chooses one of his cards, and the other player after seeing this card selects one of his own to play. The player with the higher card wins a “trick” and obtains the lead. The cards in the trick are removed from each hand, and play then continues until all cards are exhausted. Each player strives to maximize his trick total, and thevalue of the game to each player is the number of tricks he takes. Despite its simple appearance, this game is quite complicated, and finding an optimal strategy seems difficult. This paper derives basic properties of the game, gives some criteria under which one hand is guaranteed to be better than another, and determines the optimal strategies and value functions for the game in several special cases.     

The d-Step Conjecture and Gaussian Elimination

The d-step conjecture is one of the fundamental open problems concerning the structure of convex polytopes. Let Δ (d,n) denote the maximum diameter of a graph of a d-polytope that has n facets. The d-step conjecture Δ (d,2d) = d is proved equivalent to the following statement: For each ``general position' real matrix M there are two matrices drawn from a finite group matrices isomorphic to the symmetric group on d letters, such that has the Gaussian elimination factorization L ^-1 U in which L and U are lower triangular and upper triangular matrices, respectively, that have positive nontriangular elements. If #(M) is the number of pairs giving a positive L ^-1 U factorization, then #(M) equals the number of d-step paths between two vertices of an associated Dantzig figure. One consequence is that #(M)≤ d!. Numerical experiments all satisfied #(M) ≥ 2 ^d-1 , including examples attaining equality for 3 ≤ d ≤ 15. The inequality #(M) ≥ 2 ^d-1 is proved for d=3. For d≥ 4, examples with #(M) =2 ^d-1 exhibit a large variety of combinatorial types of associated Dantzig figures. These experiments and other evidence suggest that the d-step conjecture may be true in all dimensions, in the strong form #(M) ≥ 2 ^d-1 . Received April 10, 1995, and in revised form August 23, 1995.