LP-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen II

This paper deals with the global classical solvability of the Cauchy problem for nonlinear wave equations of higher order of the following form where denotes the Laplacian, a real constant, m a positive integer, and f the nonlinearity. Using results of the first part [6] concerning estimations for the kernel of the associated integral equation a priori estimates for certain L^P-norms of the local regular solution are established for higher space dimensions and for a class of nonlinearities satisfying certain growth and positivity conditions. This leads to completely regular solutions by means of the Sobolev embedding theorems.     

Der quadratfreie Kern der Eulerschen φ-Funktion

It is shown by a simple induction that any square-free natural number is the square-free kernel of Euler's -function of infinitely many natural numbers. Furthermore every square-free natural number >1 is the square-free kernel of Euler's -function of infinitely many square-numbers.     

Striktionseigenschaften von Regelscharen imE n

Let ={e(u)|uI} be a one-parameter family of straight lines forming a ruledC ^r-2-surface E ⁿ (n2,r1) without singular generatorse(u) (uI). As a synopsis, a generalization and an improvement of various results already known about the strictional properties of ruled surfaces E ⁿ (especially in the casen=3) the author demonstrates a uniform geometrical way of defining and uniquely obtaining thestriction point S(u) and theparameter of distribution d(u) of a generatore(u) under the minimal assumptions thate(u)E ⁿ (n2) be noncylindrical andr1. Other methods of obtainingS(u) andd(u) are discussed in comparison, and special strictional properties ofskew ruled surfaces E ⁿ are proved.

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Herrn Prof. Dr. H. R. Müller zum 65. Geburtstag     

A variational proof of the Gauss-Bonnet formula

The classical Gauss-Bonnet formula has the form I(g_ij)=2, where I(g_ij) represents a sum of three terms each of which depends on the metric tensor g_ij. It is shown that the first variation I of I(g_ij) with respect to the metric g_ij vanishes and that for the Euclidean metric _ij we have I(_ij)=2. From this the formula I(g_ij)=2 follows. In the process, explicit expressions are obtained for the first variation of each of the three terms which comprise I(g_ij). Furthermore, a general expression for the first variation of a multiple integral whose integrand is a scalar density depending on the metric tensor g_ij and its derivatives up to the second order is obtained with the aid of results of Rund [1].     

Mean value estimates in lattice point theory

LetQ(u) be a positive definite quadratic form inr2 variables with a real symmetric coefficient matrix of determinantD. Given a real vectorb with 0b _j <1, forx>0 letA(x) be the number of lattice points in the ellipsoidQ(u+b)x, letV(x) be the volume of this ellipsoid andP(x)=A(x)–V(x). Let . By introduction of a parameter we shall show how the treatment of estimates onP(x) and onM(x) can be unified.     

Lexicographic quasiconcave multiobjective programming

Summary Letf _i :A R ben real-valued objective functions on a convex setA -K ^m ,K:=R orC, n, mN. Letg: A R ⁿ be defined by , where for eachxA, (i ₁ (x), ..., i _n (x)) is a permutation of (1, ...,n) such that . In this paper we treat the problem of findingx ^*A such that , wherel-max denotes the lexicographic maximum. If the f_i's are strongly quasiconcave we can reduce the problem stepwise until finally it is in the form of a scalar programming problem. Further, we consider conditions for the existence and uniqueness of a solution and discuss the relationship of the problem to the vector maximum (i.e. Pareto) and maxmin (i.e. Chebychev) problems.

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Zusammenfassung f _i :AR seienn reellwertige Zielfunktionen über einer konvexen MengeA-K ^m ,K:=R oderC, n, mN. g:AR ⁿ sei definiert durch , wobei für jedesxA (i ₁ (x), ... i _n (x)) eine Permutation von (1, ...,n) derart ist, daß Wir betrachten das Problem, einx ^*A so zu finden, daß , wobeil-max das lexikographische Maximum bedeute. Falls dief _i stark quasikonkav sind, läßt sich das Problem stufenweise reduzieren, bis es schließlich die Gestalt eines skalaren Optimierungsproblems annimmt. Wir geben Existenz- und Eindeutigkeitsbedingungen an und besprechen Zusammenhänge mit dem Vektormaximumproblem (d.h. Pareto-Optimierung) und dem Maxmin-Problem (d.h. Tschebyscheff-Optimierung).

     

Effect of take-up speed on physical properties and permeation performance of cellulose acetate hollow fibers

Cellulose acetate (CA) ultrafiltration hollow fibers were spun via the dry-jet wet spinning technique. The effect of the take-up speed on the mechanical properties, morphology, thermal properties, pure water permeation, retention, and surface characterization of hollow fiber membranes were investigated. Both the inner and outer diameters of the hollow fiber decreased with the increase of take-up speed. Macrovoids were observed on the inner surface of the drawn hollow fibers. The d-space decreased with the increase of the take-up speed. The ultimate tensile stress (UTS) increased and the breaking elongation decreased with the increase of take-up speed. The permeation performance was measured. The hydraulic permeability increased and the retention decreased slightly with the increase of the take-up speed. The surface roughness increased with the increase of the take-up speed. The thermal analysis results showed that the endothermic peak shifts to the higher temperature region and coefficient of thermal expansion (CTE) decrease for a higher take-up speed.     

Biosorption of americium-241 by Saccharomyces cerevisiae

The biosorption of radionuclide ²⁴¹Am from solution by Saccharomyces cerevisiae (S. cerevisiae), and the effects of experimental conditions on the adsorption were investigated. The preliminary results showed thatS. cerevisiae is a very efficient biosorbent. An average of more than 99% of the total ²⁴¹Am could be removed by S. cerevisiae of 2.1 g/l (dry weight) from ²⁴¹Am solutions of 17.54–4386.0 mg/l (2.22 MBq/l–555 MBq/l) with adsorption capacities of 7.45–1880.0 mg/g biomass (dry weight) (0.94 MBq/g–237.9 MBq/g). The adsorption equilibrium was achieved within 1 hour and the optimum pH ranged 1–3. No significant differences on ²⁴¹Am adsorption were observed at 10–45 °C, or in solutions containing Au³⁺ or Ag⁺, even 2000 times above ²⁴¹Am concentration. The relationship between concentrations and adsorption capacities of ²⁴¹Am indicated the biosorption process should be described by the Freundlich adsorption isotherm.     

Rational design of an “all-in-one” phototheranostic

We report here porphodilactol derivatives and their corresponding metal complexes. These systems show promise as “all-in-one” phototheranostics and are predicated on a design strategy that involves controlling the relationship between intersystem crossing (ISC) and photothermal conversion efficiency following photoexcitation. The requisite balance was achieved by tuning the aromaticity of these porphyrinoid derivatives and forming complexes with one of two lanthanide cations, namely Gd³⁺ and Lu³⁺. The net result led to a metalloporphodilactol system, Gd-trans-2, with seemingly optimal ISC efficiency, photothermal conversion efficiency and fluorescence properties, as well as good chemical stability. Encapsulation of Gd-trans-2 within mesoporous silica nanoparticles (MSN) allowed its evaluation for tumour diagnosis and therapy. It was found to be effective as an “all-in-one” phototheranostic that allowed for NIR fluorescence/photoacoustic dual-modal imaging while providing an excellent combined PTT/PDT therapeutic efficacy in vitro and in vivo in 4T1-tumour-bearing mice.

We report here porphodilactol derivatives and their corresponding metal complexes as “all-in-one” phototheranostics by controlling the relationship between intersystem crossing (ISC) and photothermal conversion efficiency following photoexcitation.     

Synthesis and near infrared luminescence of a tetrametallic Zn2Yb2 architecture from a trinuclear Zn3L2 Schiff base complex

Near infrared luminescence is observed in tetrametallic [Zn2Yb2L2(mu-OH)2Cl4].2MeCN which is obtained from the Zn3 Schiff-base complex [Zn3L2(NO3)2].MeOH, (H2L =N,N'-bis(5-bromo-3-methoxysalicylidene)propylene-1,3-diamine).