Positivstellensätze for differentiable functions

We present a canonical proof of both the strict and weak Positivstellensatz for rings of differentiable and smooth functions. Our construction is explicit, preserves definability in expansions of the real field, and it works in definably complete expansions of real closed fields as well as for real-valued functions on Banach spaces.     

Positivstellens?tze for real function algebras

We look for algebraic certificates of positivity for functions which are not necessarily polynomial functions. Similar questions were examined earlier by Lasserre and Putinar [Positivity and optimization for semi-algebraic functions (to appear), Proposition 1] and by Putinar [A Striktpositivestellensatz for measurable functions (corrected version) (to appear), Theorem 2.1]. We explain how these results can be understood as results on hidden positivity: The required positivity of the functions implies their positivity when considered as polynomials on the real variety of the respective algebra of functions. This variety is however not directly visible in general. We show how algebras and quadratic modules with this hidden positivity property can be constructed. We can then use known results, for example Jacobi’s representation theorem (Jacobi in Math Z 237:259–273, 2001, Theorem 4), or the Krivine-Stengle Positivstellensatz (Marshall in Positive polynomials and sums of squares. Mathematical Surveys and Monographs 146, 2008, page 25), to obtain certificates of positivity relative to a quadratic module of an algebra of real-valued functions. Our results go beyond the results of Lasserre and Putinar, for example when dealing with non-continuous functions. The conditions are also easier to check. We explain the application of our result to various sorts of real finitely generated algebras of semialgebraic functions. The emphasis is on the case where the quadratic module is also finitely generated. Our results also have application to optimization of real-valued functions, using the semidefinite programming relaxation methods pioneered by Lasserre [SIAM J Optim 11(3): 796–817, 2001; Lasserre in Moments, positive polynomials and their applications. Imperial College Press, London, 2009; Lasserre and Putinar in Positivity and optimization for semi-algebraic functions (to appear); Marshall in Positive polynomials and sums of squares. Mathematical Surveys and Monographs 146, 2008, page 25].     

Verallgemeinerte Primzahlsätze

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Zwei Determinantensätze

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Einige Sätze

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Integralsätze der Flächentheorie

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Fundamentalsätze der Wahrscheinlichkeitsrechnung

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Schliessungssätze in Laguerre-Ebenen

For Laguerre planes Artzy [1] showed that a 4-point Pascal theorem on an oval leads to a configuration , consisting of six points and five regular circles, which is equivalent Miquel's theorem. We represent some similar incidence assumptions which are again equivalent to in each Laguerre plane. Besides, a uniform denotation to characterize different kinds of miquelian theorems in Benz planes is suggested.

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Gewidmet Herrn Professor Benz zum 60. Geburtstag     

Punctured combinatorial Nullstellensätze

In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g ₁,g ₂, …, g _n , that are the product of linear factors. We then prove a punctured version which states, for simple zeros, that if f vanishes at nearly all, but not all, of the common zeros of g ₁(X ₁), …,g _n (X _n ) then every residue of f modulo the ideal generated by g ₁, …, g _n , has a large degree.     

Einige arithmetische Sätze

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Fixpunktsätze in uniformen Räumen

Ohne ZusammenfassungHerrnJosef Lense zum 70. Geburtstag am 28. Oktober 1960 gewidmet     

A unified ansätze for the solution of nonlinear differential equations

The aim of the paper is to propose a generalized ansätze for constructing exact solutions to nonlinear ordinary differential equations. This unified transformation is manipulated to acquire analytical solutions that are general solutions of simpler linear or nonlinear systems of ordinary differential equations that are either integrable or possess special solutions. The method is implemented to obtain several families of traveling wave solutions for a class of nonlinear evolution equations and for higher order wave equations of KdV type (I).     

Fundamentalsätze der quasielliptischen Differentialgeometrie

It is shown that in the quasielliptic space there exist two different sets of respectively quasielliptic congruent curves with the same natural equations =(s), =(s). On the curves of one set the sense of increasing arc length s corresponds with the positive orientation, to say with the sense of increasing x¹/x⁰; on curves of the other set the orientation of s is negativ. Further we prove that the fundamental quantities of surface theory as defined in [3] cannot always characterize an unique parametrized surface apart from quasielliptic motions. In the exceptional case one Christoffel symbol must be given too.