The way of Diophantus: Some clarifications on Diophantus' method of solution

In the introduction of the Arithmetica Diophantus says that in order to solve arithmetical problems one has to “follow the way he (Diophantus) will show.” The present paper has a threefold objective. Firstly, the meaning of this sentence is discussed, the conclusion being that Diophantus had elaborated a program for handling various arithmetical problems. Secondly, it is claimed that what is analyzed in the introduction is definitions of several terms, the exhibition of their symbolism, the way one may operate with them, but, most significantly, the main stages of the program itself. And thirdly, it is argued that Diophantus' intention in the Arithmetica is to show the way the stages of his program should be practically applied in various arithmetical problems.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

The Sheffer group and the Riordan group

We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs.     

Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hènon maps. For this class of systems we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Hènon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism. 2000 Mathematics Subject Classification. Primary: 37C45, 37D20, 37D35, Secondary: 37A35, 37E30     

Roots,iterations and logarithms of formal automorphisms

In this paper it is proved that having a logarithm is equivalent to having roots of arbitrary order in the group of automorphisms of a formal power series ring, and in algebraic subgroups too     

Long-Term Changes in a Heat Exchanger Steel

The steel ASTM A213 P22 is used for superheater outlet header in power plants. During duty cycles lasting over tens of years the temperature is of the order of 545°C and the pressure 125 atm. The microscopic changes in these steels are hard to analyze under working conditions, but they are believed to be responsible for the appearance of creeps in such devices. Investigation of the microscopic modifications will help to predict future failures due to creeps, increasing the reliability and saving a lot of money.We report the investigation of such changes in ASTM A213 P22 steel in use for 30 years as a Superheater header at Haifa power station. The analysis has been carried out using Mössbauer Spectroscopy, X-ray diffraction and optical microscopy. As a result it seems that the morphology and phase change of the carbide phases M₃C/M₇C₃ and M₂₃C₆ are the most prominent changes leading to the material failures.     

Optical spectroscopy of Yb/Er codoped NaY(WO4)2 crystal

Erbium and ytterbium codoped double tungstates NaY(WO₄)₂ crystals were prepared by using Czochralski (CZ) pulling method. The absorption spectra in the region 290-2000 nm have been recorded at room temperature. The Judd-Ofelt theory was applied to the measured values of absorption line strengths to evaluate the spontaneous emission probabilities and stimulated emission cross sections of Er³⁺ ions in NaY(WO₄)₂ crystals. Intensive green and red lights were measured when the sample were pumped by a 974 nm laser diode (LD), especially, the intensities of green upconversion luminescence are very strong. The mechanism of energy transfer from Yb³⁺ to Er³⁺ ions was analyzed. Energy transfer and nonradiative relaxation played an important role in the upconversion process. Photoexcited luminescence experiments are also fulfilled to help analyzing the transit processes of the energy levels.     

Structural studies of Cr-doped mullite derived from single-phase precursors

Cr-doped mullites were prepared from single-phase precursors containing up to 9.60 wt% Cr₂O₃ using a sol-gel technique followed by thermal treatment. Particle induced X-ray emission spectroscopy and X-ray powder diffraction were used to characterize the samples. Mullites were orthorhombic, space group Pbam. Cr doping caused the increase of unit-cell parameters. Strongest expansion was noticed along c-axis followed by a and b (Δc/c=0.089, Δa/a=0.061, Δb/b=0.045% per mole Cr₂O₃). A second phase, namely θ-(Al,Cr)₂O₃, was revealed by XRD in the sample containing 9.60 wt% Cr₂O₃. The structure of mullites was refined by the Rietveld method, location of Cr³⁺ was performed by the EPR spectroscopy. At low chromium doping level (Cr₂O₃ content less than ∼5 wt%) Cr³⁺ ions were substituted for Al³⁺ in the AlO₆ octahedra of the mullite structure (M1 site). For higher doping level, Cr³⁺ ions were additionally substituted for Al³⁺ in the AlO₆ octahedra of the second phase [θ-(Al,Cr)₂O₃ at 1400 °C, or α-(Al,Cr)₂O₃ at 1600 °C] which segregated in the system. Substitution of Cr³⁺ for Al³⁺ on M1 site in the mullite structure resulted in increase of average distances in (M1)O₆ octahedron and decrease of average distances in T*O₄ tetrahedron, while average distances in TO₄ tetrahedron stayed almost constant.     

Numerical study of charge ordering in NaxCoO2

A simple microscopic model of charge ordering in the Na_xCoO₂ system is presented. The model takes into account the interplane interactions between the ordered Na ions and d electrons from the CoO₂ layers as well as the nearest-neighbor intraplane Coulomb interactions between d electrons. It is shown that a driving force of charge ordering in the CoO₂ layers is the interplane interaction that alone is able to describe various types of inhomogeneous charge ordering (e.g., the striped phases) as well as to predict correctly the conducting properties of the system.     

Asymptotic expansions in sequential estimation for the first-order random coefficient autoregressive model: Regenerative approach

Edgeworth expansions for the distribution of a sequential least squares estimator in the random coefficient autoregressive (RCA) model are derived. The regenerative approach to second-order asymptotic analysis of Markov-type statistical models is developed.