On mappings preserving orthogonality of non-singular vectors

Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then, \(\dot V: = \{ z \in V|q(z) \ne 0\} \) is the set of non-singular vectors ofV, and forx, y ∈ \(\dot V\) , ?(x, y) ?f(x, y) ²/(q(x) · q(y)) is theq-measure of (x, y), where ?(x,y)=0 means thatx, y are orthogonal. For an arbitrary mapping \(\sigma :\dot V \to \dot V\) we consider the functional equations $$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$ and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N?{0, 1, 2} ∧ ∣F∣ > 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F?{0} such thatF x ^σ =F x ^ξ ?x ∈ \(\dot V\) andf(x ^ξ,y ^ξ)=λ · (f(x, y))^ρ ?x, y ∈V. Moreover, (II) implies ρ =id _F ∧q(x ^ξ) = λ ·q(x) ?x ∈ \(\dot V\) , and (III) implies ρ=id _F ∧ λ ∈ {1,?1} ∧x ^σ ∈ {x ^ξ, ?x ^ξ} ?x ∈ \(\dot V\) . Other results obtained in this paper include the cases dimV = 2 resp. dimV ?N resp. ∣F∣ = 3.     

Preconditioning a class of fourth order problems by operator splitting

We develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space. We consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two (discrete) linear second order elliptic operators, and exploit this property in designing the preconditioners. The underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available. We propose symmetric and non-symmetric preconditioners, along with theory and numerical experiments. They both document crucial properties of the preconditioners as well as their practical performance. It is important to note that we neither need H ^s -regularity, s > 1, of the continuous problem nor quasi-uniform grids.     

On non-semisplit extensions,tensor products and exactness of groupC *-algebras

Summary We show the existence of a block diagonal extensionB of the suspensionS(A) of the reduced groupC ^*-algebraA = C _r ^* (SL ₂()), such that there is only oneC ^*-norm on the algebraic tensor productB ^op B, butB is not nuclear (even not exact). Thus the class of exactC ^*-algebras is not closed under extensions.The existence comes from a new established tensorial duality between the weak expectation property (WEP) of Lance and the local variant (LLP) of the lifting property.We characterize the local lifting property of separable unitalC ^*-algebrasA as follows:A has the local lifting property if and only if Ext (S(A)) is a group, whereS(A) is the suspension ofA.If moreoverA is the quotient algebra of aC ^*-algebra withWEP (for brevity:A isQWEP) but does not satisfyLLP then there exists a quasidiagonal extensionB of the suspensionS(A) by the compact operators such that on the algebraic tensor productB ^op B there is only oneC ^*-norm.The question if everyC ^*-algebra isQWEP remains open, but we obtain the following results onQWEP: AC ^*-algebraC isQWEP if and only ifC ^** isQWEP. A von NeumannII ₁-factorN with separable predualN _* isQWEP if and only ifN is a von Neumann subfactor of the ultrapower of the hyperfiniteII ₁-factor. IfG is a maximally almost periodic discrete non-amenable group with Haagerup's Herz-Schur multiplier constant _G =1 then the universal groupC ^*-algebraC ^*(G) is not exact but the reduced groupC ^*-albegraC _r ^* (G) is exact and isQWEP but does not satisfyWEP andLLP.We study functiorial properties of the classes ofC ^*-algebras satisfyingWEP, LLP resp. beingQWEP.As applications we obtain some unexpected relations between some open questions onC ^*-algebras.Oblatum 13-IV-92Work partially supported by DFG     

Kazhdan constants and the dual space topology

Supported by NATO collaborative research grant CRG 900029     

Indefinite Eigenvalue Problems with Several Singular Points and Turning Points

We consider a general class of eigenvalue problems with two-point boundary conditions on a finite interval generated by a differential equation with an indefinite weight function which has several zeros and/or poles. As a basic result we derive asymptotic estimates for a special fundamental system of solutions of the corresponding differential equation and determine the asymptotic distribution of the eigenvalues. Finally we prove the uniform convergence of eigenfunction expansions for some class of functions f.     

Sequence spaces and inverse of an infinite matrix

We give here some properties of the sets _α(uΔ) generalizing the space of generalized difference sequencesl ^∞(uΔ). Then we study spaces related to the sets of sequences that are strongly convergent or strongly bounded. Next we define from the sets of spaces that are (N,q) summable or bounded the sets of spaces that are (N,q)α-bounded orr-bounded. Then we give some properties of these spaces using Banach space of the forms _α.     

Weak spectral synthesis in commutative Banach algebras. II

Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). Continuing our investigation in [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008) 987-1002], we establish various results on intersections and unions of weak spectral sets and weak Ditkin sets in Δ(A). As an important example, the algebra of n-times continuously differentiable functions is studied in detail. In addition, we prove a theorem on spectral synthesis for projective tensor products of commutative Banach algebras which applies to Fourier algebras of locally compact groups.     

Using Multibody Systems for the Investigation of Dynamic Aberrations in High Precision Optics

The coupling of multibody system dynamics and optical simulations by using ray tracing through moving lens systems will be summarized for both rigid and flexible lens systems. Furthermore, a method will be introduced for efficient simulations in the time domain. This method provides a direct integration of the optical simulation into the equations of motion of a multibody system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fuzzy-Based Analysis of a Hill-Type Muscle Model

Among the most challenging problems in present day mechanics is the realistic simulation of the human kinetic behavior which requires correct modeling of the muscles – the human actuators. We are, therefore, implementing and validating a 1-D massless Hill-type muscle model in the program Neweul-M². As a part of this effort, various anatomical and physiological parameters need to be chosen prior to the simulation. These parameters are always subject to natural variability or scatter. Therefore, it is important to consider those parameters as uncertain fuzzy parameters and then determine the sensitivity measures of the parameters involved in various simulation scenarios using the program Famous. The applied sensitivity analysis quantifies the influence of individual parameters or subsets of parameters on the output quantity of interest. Thus, this analysis identifies those parameters with large influence on the simulation response of muscles which, therefore, needs to be chosen very carefully. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)