Structure of 4,4′-butane-(1,4,7-three-p-tolylsulphonyl-1,4,7-three amine) diphthalonitrile

Abstract

The crystal structure of the title compound, C₄₁ H₃₅ N₇ O₆ S₃ was determined as monoclinic by single crystal X-Ray diffraction technique. The molecular structure was identified by IR, ¹H-NMR, ¹³C-NMR and elemental analysis. The crystal parameters of this compound are as follows: monoclinic P 2 1/n, a = 12.694(2) Å, b = 26.204(2) Å, c = 13.005(2) Å, β = 102.95(2)°, V = 4216.02(1) Å.³, Z = 4, Dx = 1.289 g/cm³, F(000) = 1704, λ (MoKα) = 0.71070 Å, μ = 0.2 mm^?1. The structure was solved by SHELXS-97 and refined by SHELXL-97. R = 0.06 for 3178 observed reflections with I > 2σ (I).     

Anomalous field dependence of muon Knight shift and line width in the singlet ground state compound PrCu2

We report on ZF and TF‐studies of PrCu₂ above the induced Jahn–Teller transition at 7.3 K. Generally a two‐component signal is found, one showing inhomogeneous the other one homogeneous, temperature dependent line broadening. In ZF the former component is well represented by a Gaussian Kubo–Toyabe function with \varDelta \simeq 6.5\,μs^-1 at 7.5 K, corresponding to a field width of 76 G. This is about 30 times larger than what is calculated to arise from the ¹⁴¹Pr‐nuclear dipole moments alone, pointing to strong hyperfine enhanced features. TF‐field scans at 12 K revealed that the enhancement is suppressed in external fields exceeding 1 kG. In parallel the Knight shift drops from very large values well above 10% at 100 G to shifts of the order of 1% above 1 kG. A scaling of the Knight shifts with the corresponding relaxation rates seems to imply that the strange field dependence below 1 kG is associated with the magnetic susceptibility of the muons’s nearest neighbour Pr³⁺‐ions, a result for which we have no explanation yet to offer. This revised version was published online in July 2006 with corrections to the Cover Date.     

Heating mechanism in relation to the laser machining process

Laser workpiece interaction mechanism is an important phenomenon which will assist in the development of laser machining systems. The interaction mechanism is generally complicated and depends on the laser and workpiece properties. In the present study a mathematical analysis for the laser material removal by evaporation and radial ejection of liquid is carried out. In the analysis the time unsteady problem is solved and nucleation explosions are predicted.     

Analytical Approach to Fractional Zakharov-Kuznetsov Equations by He＇s Homotopy Perturbation Method

The aim of this paper is to obtain the approximate analytical solution of a fractional Zakharov-Kuznetsov equation by using homotopy perturbation method （HPM）. The fractional derivatives are described in the Caputo sense. Several examples are given and the results are compared to exact solutions. The results reveal that the method is very effective and simple.     

Analytical Approach to Space- and Time-Fractional Burgers Equations

A scheme is developed to study numerical solution of the space- and time-fractional Burgers equations under initial conditions by the homotopy analysis method. The fractional derivatives are considered in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.     

Quivers of finite mutation type and skew-symmetric matrices

Quivers of finite mutation type are certain directed graphs that first arised in Fomin-Zelevinsky’s theory of cluster algebras. It has been observed that these quivers are also closely related with different areas of mathematics. In fact, main examples of finite mutation type quivers are the quivers associated with triangulations of surfaces. In this paper, we study structural properties of finite mutation type quivers in relation with the corresponding skew-symmetric matrices. We obtain a characterization of finite mutation type quivers that are associated with triangulations of surfaces and give a new numerical invariant for their mutation classes.     

Expected gain–loss pricing and hedging of contingent claims in incomplete markets by linear programming

We analyze the problem of pricing and hedging contingent claims in the multi-period, discrete time, discrete state case using the concept of a “λ gain–loss ratio opportunity”. Pricing results somewhat different from, but reminiscent of, the arbitrage pricing theorems of mathematical finance are obtained. Our analysis provides tighter price bounds on the contingent claim in an incomplete market, which may converge to a unique price for a specific value of a gain–loss preference parameter imposed by the market while the hedging policies may be different for different sides of the same trade. The results are obtained in the simpler framework of stochastic linear programming in a multi-period setting, and have the appealing feature of being very simple to derive and to articulate even for the non-specialist. They also extend to markets with transaction costs.     

Application of the Exp-function method for nonlinear differential-difference equations

In this paper, we established abundant travelling wave solutions for some nonlinear differential-difference equations. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics.     

Unified scaling law in the coherent noise model

The waiting time distribution between successive events and the unified scaling law is studied using the coherent noise model. It is shown that, although this model generates uncorrelated event sizes and does not exhibit criticality, it still provides the unified scaling law. We argue the role of characteristic kink observed in the unified scaling law and the meaning of the parameter C$C$ used to fix the peak of the kink to unity. Our results indicate that the parameter C$C$ is indeed a physical quantity localizing the end of the linear tendency in the scaling law, which corresponds to the completion of the dominance of correlated events in time.