Powers of rationals modulo 1 and rational base number systems

A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite right transducer. Every real number has at least one such expansion and a countable infinite number of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions. The results that we derive are pertinent on their own and also as they relate to other problems in combinatorics and number theory. A first example is a new interpretation and expansion of the constant K(p) from the so-called “Josephus problem.” More important, these expansions in the base allow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers. Work partially supported by the CNRS/JSPS contract 13 569, and by the “ACI Nouvelles Interfaces des Mathématiques”, contract 04 312.     

Generalized radix representations and dynamical systems. IV

For r = (r₁,…, r_d) ∈ ?^d the mapping τ_r:?^d →?^d given byτ_r(a₁,…,a_d) = (a₂, …, a_d,−⌊r₁a₁+…+ r_da_d⌋)where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ?^d there exists an integer k > 0 with τ_r^k(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).     

Ultrafast optical technologies for large-capacity TDM/WDM photonic networks

As the Internet data traffic increases explosively, the optical networks will evolve into "photonic networks", in which "wavelength" label is used to route large-capacity optical signals at optical path cross connect (OPXCs) nodes. Further development of ultrafast all-optical processing technologies enables us to further upgrade the photonic networks into those with much more flexibility and higher transparency, where ultrafast optical signals of any bit rates (2.5, 10, 40, 100, 160 G/ch) and frame/modulation formats (GbE, 10 GbE, SDH, OTN/ NRZ, RZ, CS-RZ, RZ-DPSK, etc.) are processed and transmitted all-optically without optical-electrical-optical (O-E-O) conversion from end-to-end.     

Diastereomeric selective effects of double-stranded peptides conjugated with -Phe-Phe- residue for growth inhibition and permeability of Ca(2+) on A431, src(ts)NRK, and A549 cells proliferation

Our aim in this study is to elucidate the correlations between inhibition and chirality, especially, diastereomer, against cell proliferation of double-stranded peptides. In previous studies, we reported on the design, synthesis, and chemical properties on a series of novel double-stranded peptides, (y-AA-x-AA)(2)-spacer(S) (AA=amino acid, S=spacer, symbols x and y represent L- or D-forms, and (y-, x-) as represent of the symbol) conjugated with -y-AA-x-AA- and -z-AA-y-AA-x-AA- sequences to a spacer of carbon number n. The inhibition of A431 and src(ts)NRK cells growth by four diastereomers of the N(1),N(12)-bis(y-Phe-x-Phe)dodecanediamines (n=12) increased in the following order: (L-, L-)<(D-, D-)<(L-, D-)<(D-, L-). A similar trend was seen in the order for the activity of (y-AA-x-AA)(2)-spacer(S) with a spacer of carbon number n=2, 3, 4, 5, 6, and 12 against the cell growth inhibition. To understand the mechanism of diasteromer selective cell growth inhibition, the correlations between chirality and cell growth inhibition were investigated from the measurement of the changes in cytosolic Ca(2+) concentration (=[Ca(2+)](c)) of A431 cells. Although less active N(1),N(12)-bis(L-Phe-L-Phe)dodecanediamine increases cytosolic [Ca(2+)](c), more active diasteromers, N(1),N(12)-bis(L-Phe-D-Phe)dodecanediamine and N(1),N(12)-bis(D-Phe-L-Phe)dodecanediamine, decrease cytosolic [Ca(2+)](c) in A431 cell. This study provides diastereomeric effected new insights - this controls the polarity of double-stranded peptides - into the control of tumor cell proliferation and design of the uptake by penetration through the cell membrane of drugs.     

Efficient implementation of three-dimensional reference interaction site model self-consistent-field method: application to solvatochromic shift calculations

The authors present an implementation of the three-dimensional reference interaction site model self-consistent-field (3D-RISM-SCF) method. First, they introduce a robust and efficient algorithm for solving the 3D-RISM equation. The algorithm is a hybrid of the Newton-Raphson and Picard methods. The Jacobian matrix is analytically expressed in a computationally useful form. Second, they discuss the solute-solvent electrostatic interaction. For the solute to solvent route, the electrostatic potential (ESP) map on a 3D grid is constructed directly from the electron density. The charge fitting procedure is not required to determine the ESP. For the solvent to solute route, the ESP acting on the solute molecule is derived from the solvent charge distribution obtained by solving the 3D-RISM equation. Matrix elements of the solute-solvent interaction are evaluated by the direct numerical integration. A remarkable reduction in the computational time is observed in both routes. Finally, the authors implement the first derivatives of the free energy with respect to the solute nuclear coordinates. They apply the present method to "solute" water and formaldehyde in aqueous solvent using the simple point charge model, and the results are compared with those from other methods: the six-dimensional molecular Ornstein-Zernike SCF, the one-dimensional site-site RISM-SCF, and the polarizable continuum model. The authors also calculate the solvatochromic shifts of acetone, benzonitrile, and nitrobenzene using the present method and compare them with the experimental and other theoretical results.     

Chiral spin pairing in helical magnets

A concept of chiral spin pairing is introduced to describe a vector-chiral liquid-crystal order in frustrated spin systems. It is found that the chiral spin pairing is induced by the coupling to phonons through the Dzyaloshinskii-Moriya interaction and the four-spin exchange interaction of the Coulomb origin under the edge-sharing network of magnetic and ligand ions. This produces two successive second-order phase transitions upon cooling: an O(2) chiral spin nematic, i.e., spin cholesteric, order appears with an either parity, and then the O(2) symmetry is broken to yield a helical magnetic order. Possible candidate materials are also discussed as new multiferroic systems.