A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain

Barros-Neto and Gelfand (Duke Math. J. 98 (3) (1999) 465; Duke Math. J. 117 (2) (2003) 561) constructed for the Tricomi operator on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the Tricomi operator. We give for the Tricomi-type operator a fundamental solution relative to an arbitrary point of Rⁿ⁺¹ with the support in the region t?0, where the operator is hyperbolic. Our key observation is that the fundamental solution for the Tricomi-type operator can be written like an integral of the distributions generated by the fundamental solution of the Cauchy problem for the wave equation. The application of that fundamental solution to the L^p-L^q estimate for the forced Tricomi-type equation is given as well.     

Large time behavior of the heat kernel

In this paper we study the large time behavior of the (minimal) heat kernel k_P^M(x,y,t) of a general time-independent parabolic operator Lu=u_t+P(x,∂_x)u which is defined on a noncompact manifold M. More precisely, we prove that

     

Well-posedness for a scalar conservation law with singular nonconservative source

We consider the Cauchy problem for the 2×2 strictly hyperbolic system

     

Existence theory for nonclassical entropy solutions of scalar conservation laws

We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.Received: June 3, 2002; revised: November 12, 2002     

Explicit exact solutions for a new generalized Hamiltonian amplitude equation with nonlinear terms of any order

Making use of a proper transformation and a generalized ansatz, we consider a new generalized Hamiltonian amplitude equation with nonlinear terms of any order, iu_x + u_tt + (|u|^p + |u|^2p)u + u_xt = 0. As a result, many explicit exact solutions, which include kink-shaped soliton solutions, bell-shaped soliton solutions, periodic wave solutions, the combined formal solitary wave solutions and rational solutions, are obtained.Received: April 4, 2002     

Wave front shape for a constrained cylindrically anisotropic elastic solid

The two-dimensional wave front shape caused by a point impulse excitation in a cylindrically anisotropic elastic solid is considered. The elastic parameters of the solid are constrained such that E = G This constraint allows the parametric equations of the wave front to be expressed exactly in terms of elementary transcendental functions. The precise location of double and cusp points on the front is treated in detail. Time histories of several wave front patterns are presented and an interesting feature of the front is generalized to the unconstrained solid.     

Self-organized Monte Carlo localization of critical point via linear filtering

Self-organized Monte Carlo simulations are suggested. Their essence is artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by biased random walk in temperature space. The action of walker is driven by feedback utilizing the linear filtering recursion based on the instantaneous estimates of Binder cumulants. The simulation for 2d Ising model demonstrates that the mean temperature typical for the steady noncanonical equilibrium regime properly approximates the true critical temperature. The estimates of the critical Binder cumulants and critical exponents are also discussed.     

Chiral polymers containing a carbohydrate moiety: synthesis

This paper reports the preparation of poly(acetal-ethers) by two alternative synthetic pathways (alkaline and acid-catalyzed conditions). Polycondensations of methyl 2,3:4,6-di-O-salicylidene-alpha-D-mannopyranoside (3) (mixtures of endo-H and exo-H dioxolan-2-yl-diastereomers) with 1,4-dibromobutane (4) (method I a and I b) were performed in solvents (DMF, butyl acetate/DMF, DMSO) and were catalyzed by K2CO3/KI or KOH. A similar polymer (6) was formed by the reaction of methyl alpha-D-mannopyranoside (1) and 1,4-bis(2-formylphenoxy)butane (7), catalyzed by p-toluenesulfonic acid (method II). Regardless of conversion or initial comonomer feed ratios, the composition of the polycondensates depended on the reaction conditions, leading to the formation of macrocyclic [1 + 1] (5) and [2 + 2] compounds, which were macromolecules with diverse molecular weights and optical properties. The regioselective polycondensation was examined by 1H NMR spectroscopy of selected polymers. In the case of 5-membered cyclic acetal units, mixtures of the endo-H and exo-H dioxolan-2-yl system, diastereomers were formed in the polymer chain. The macrocycles and linear oligomers were identified by NMR and electrospray mass spectrometry (ESI-MS). Thermodynamically controlled reactions for making macrocycles as well as oligomers in the absence of templates are also discussed.     

Hardness and phase analysis of IN 718 deformed at high strain rate

Specimens of the nickel base alloy IN 718 deformed at high strain rate (~10 s^–1), as realized in a screw press, have higher strength than parts forged conventionally in a hydraulic press. Microstructure analyses in light and transmission electron microscopes reveal the precipitation of very small Ni₃Nb particles (-phase) to be the reason for the increased hardness. Several processing routes are discussed and analyzed in relation to the TTT-diagram of IN 718.