Some functional equations originating from number theory

We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.     

Enhanced Binding in Non-Relativistic QED

 We consider a spinless particle coupled to a photon field and prove that even if the Schr?dinger operator p ² +V does not have eigenvalues the system can have a ground state. We describe the coupling by means of the Pauli-Fierz Hamiltonian and our result holds in the case where the coupling constant α is small. Received: 7 January 2002 / Accepted: 8 November 2002 Published online: 13 January 2003     

The focusing problem for the Eikonal equation

We study the focusing problem for the eikonal equation¶¶ ?_tu=| ?u| ², \partial _{t}u=\left| \nabla u\right| ^{2}, ¶¶i.e., the initial value problem in which the support of the initial datum is outside some compact set in R^d \mathbf{R}^{d} . The hole in the support will be filled in finite time and we are interested in the asymptotics of the hole as it closes. We show that in the radially symmetric case there are self-similar asymptotics, while in the absence of radial symmetry essentially any convex final shape is possible. However in R² \mathbf{R}^2 , for generic initial data the asymptotic shape will be either a vanishing triangle or the region between two parabolas moving in opposite directions (a closing eye). We compare these results with the known results for the porous medium pressure equation which approaches the eikonal equation in the limit as m? 1 m\rightarrow 1 .     

An existence result for a quadrature surface free boundary problem

The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.     

Microwave Assisted Conversion of Oximes and Semicarbazones to Carbonyl Compounds Using Benzyltriphenylphosphonium Peroxymonosulfate

Summary.  Benzyltriphenylphosphonium peroxymonosulfate in the presence of catalytic amounts of bismuth chloride was found to be an efficient and mild reagent for the oxidative cleavage of oximes and semicarbazones to the corresponding carbonyl compounds under microwave irradiation. Corresponding author. E-mail: haji@cc.iut.ac.ir Received March 4, 2002; accepted (revised) April 8, 2002     

Rückbezüge des Mathematikunterrichts und der Methodik des Mathematikunterrichts in der DDR auf historische Vorausentwicklungen

The author discusses at some examples exemplarily, how the pedagogy of mathematics education in the GDR from curricular and educational/psychological knowledge from the time before 1945 has developed scientifically.     

Dependence of high electric-field-induced strain on the composition and orientation of Pb(Mg1/3Nb2/3)O3-PbTiO3 crystals

Electric-field-induced strain behavior of (1−x)Pb(Mg_1/3Nb_2/3)O₃-xPbTiO₃ (PMNT) crystals with different orientations and compositions was investigated for use as electromechanical actuators. Crystallographically, high strains with low hysteresis were achieved for 〈001〉 oriented rhombohedral crystals (29%≤x≤31%) near a morphotropic phase boundary, rather than 〈110〉 and 〈111〉. Domain instability could explain inferior strain levels and large hysteresis for 〈110〉 and 〈111〉 oriented crystals. Ultrahigh strain levels up to 1.8% could be achieved for 〈001〉 oriented PMNT crystals, being related to an E-field induced phase transition. −2 kV/cm negative E-field can be applied to PMNT ferroelectric material with low hysteresis. High strain with low hysteresis makes PMNT crystals promising candidates for high performance solid-state actuators.