Form sums of nonnegative selfadjoint operators

Summary The sum of two unbounded nonnegative selfadjoint operators is a nonnegative operator which is not necessarily densely defined. In general its selfadjoint extensions exist in the sense of linear relations (multivalued operators). One of its nonnegative selfadjoint extensions is constructed via the form sum associated with A and B. Its relations to the Friedrichs and Krein--von Neumann extensions of A+Bare investigated. For this purpose, the one-to-one correspondence between densely defined closed semibounded forms and semibounded selfadjoint operators is extended to the case of nondensely defined semibounded forms by replacing semibounded selfadjoint operators by semibounded selfadjoint relations. In particular, the inequality between two closed nonnegative forms is shown to be equivalent to a similar inequality between the corresponding nonnegative selfadjoint relations.</o:p>     

Semi-dualizing modules and related Gorenstein homological dimensions

A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom_A(C,C)?A in the derived category D(A).We show how each such module gives rise to three new homological dimensions which we call C-Gorenstein projective, C-Gorenstein injective, and C-Gorenstein flat dimension, and investigate the properties of these dimensions.     

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832     

Linear Fractional Transformations of Nevanlinna Functions Associated with a Nonnegative Operator

In the present paper a subclass of scalar Nevanlinna functions is studied, which coincides with the class of Weyl functions associated to a nonnegative symmetric operator of defect one in a Hilbert space. This class consists of all Nevanlinna functions that are holomorphic on (?∞, 0) and all those Nevanlinna functions that have one negative pole a and are injective on ${(-\infty, a)\,\cup\, (a, 0)}$ . These functions are characterized via integral representations and special attention is paid to linear fractional transformations which arise in extension and spectral problems of symmetric and selfadjoint operators.     

Confluences of agendas: Emigrant mathematicians in transit in Denmark, 1933–1945

The present paper analyses the confluence of agendas held by Danish mathematicians and German refugees from Nazi oppression as they unfolded and shaped the mathematical milieu in Copenhagen during the 1930s. It does so by outlining the initiatives to aid emigrant intellectuals in Denmark and contextualises the few mathematicians who would be aided. For most of those, Denmark would be only a transit on the route to more permanent immigration, mainly in the US. Thus, their time in Copenhagen would exert only temporary influence over Danish mathematics; but as it will be argued, the impacts of their transit would be more durable both for the emigrants and for the Danish mathematical milieu. It is thus argued that the influx of emigrant mathematicians helped develop the institutional conditions of mathematics in Copenhagen in important ways that simultaneously bolstered the international outlook of Danish mathematicians. These confluences of agendas became particularly important for Danish mathematics after the war, when the networks developed during the 1930s could be drawn upon.     

Gender differences in the mathematics achievements of German primary school students: results from a German large-scale study

In Germany, national standards for mathematics for the end of primary school were established in 2004. In the present study, data were collected to evaluate these standards, and were used to compare the mathematical skills of girls and boys. Many studies have shown that gender differences are strongest at the highest levels of education. The findings from primary school are less consistent. Thus, in our study we analyzed achievement differences in a sample of approximately 10,000 third and fourth graders, representative of the German elementary school population. Gender-specific competencies were compared in the different content domains, both for the general mathematical competence, and for the cognitive levels of the tasks. Overall, boys outperformed girls, but substantial variation was found between the content domains and general mathematical achievement. Differences were higher in grade three than in grade four. The proportion of boys in the classroom did not appear to affect the individual level of performance. Analysis of the items on which boys or girls clearly outperformed each other reproduced a pattern of specific item characteristics predicting gender bias consistent with those reported in previous studies in other countries.     

Synthesis of 2-aryl-4H-pyrano[2,3-b]pyridin-4-ones by a one-pot deprotection-cyclization reaction

An efficient synthesis of 2-aryl-4H-pyrano[2,3-b]pyridine-4-ones is reported, using a one-pot, two step process in the presence of pyridinium hydrochloride. The methodology is compatible with a series of functional groups useful for the synthesis of second generation analogs, as part of our SAR program. In addition, the method proved to be scalable (>100 g), allowing for efficient synthesis of material to support animal studies.     

Fundamental aspects of NOx adsorption on BaO

The nature of NO₂ and nitrite/nitrate pairs adsorbed on BaO has been studied within the density functional theory. Size effects are investigated by comparing results for the BaO molecule, BaO clusters [(BaO)_x, x = 4, 6, 9, 12] and BaO(1 0 0). The adsorption energies show weak dependence on size, which is a manifestation of the fast size convergence of the BaO electronic structure and the local character of the NO₂ adsorption bond. Nitrite/nitrate pair formation is associated with a significant energy gain. For BaO(1 0 0), the stability of the pair is insensitive to the nitrite-nitrate separation, a finding that demonstrates a surface mediated non-local mechanism of molecular pair formation on oxide surfaces. The results have implications for the understanding of NO_x storage and reduction catalysts.     

DEFORMATIONS OF HYPERCOMPLEX STRUCTURES ASSOCIATED TO HEISENBERG GROUPS

Let X be a compact quotient of the product of the real Heisenberggroup H_4m+1 of dimension 4m + 1 and the three-dimensional realEuclidean space R³. A left-invariant hypercomplex structureon H_4m+1 x R³ descends onto the compact quotient X. The spaceX is a hyperholomorphic fibration of 4-tori over a 4m-torus.We calculate the parameter space and obstructions to deformationsof this hypercomplex structure on X. Using our calculations,we show that all small deformations generate invariant hypercomplexstructures on X but not all of them arise from deformationsof the lattice. This is in contrast to the deformations on the4m-torus.     

Lifetime measurements for metastable He like ions

We present work in progress on the development of a new technique for measurements of μs lifetimes of metastable excited states of He like ions. The key element in this new technique is an electrostatic storage device for ions with keV energies. This revised version was published online in August 2006 with corrections to the Cover Date.