A characterization of cardinalsκ such that 2λ = 2κ wheneverκ ≦λ < 2κ

We characterize cardinalsκ such that 2^λ = 2^κ wheneverκ ≦λ < 2^κ using ideals in small algebras of sets satisfying certain completeness and saturation conditions. Research of this author was supported by an NSF Grant.     

Embedded eigenvalues of Sturm Liouville operators

In this work we study the behavior of embedded eigenvalues of Sturm-Liouville problems in the half axis under local perturbations. When the derivative of the spectral function is strictly positive, we prove that the embedded eigenvalues either disappear or remain fixed. In this case we show that local perturbations cannot add eigenvalues in the continuous spectrum. If the condition on the spectral function is removed then a local perturbation can add infinitely many eigenvalues.     

A note on a result by M. Bognár concerning the Hahn--Mazurkiewicz Theorem

Summary In this note we use Theorem 2.4 in<span lang=EN-US style='font-size:10.0pt;mso-ansi-language:EN-US'>[1]to give a very short proof of a recent result due to M. Bognár (Theorem A in [3]). We also prove that Bognár's result is in fact equivalent to the classical Hahn-Mazurkiewicz Theorem. Finally we give generalizations in the non-compact setting.     

Finitely additive integration and local integration with respect to upper integrals

Summary We study the integration theory for general integral metrics when restricted to upper integrals q, finding improvements in the relation between the classes of the q-integrable and the q_l-integrable functions. We give new results and notions which lead to the desirable characterizations of q-integrable functions as q_l-integrable f with q(|f|) < ∞, and of q_l-integrable functions via the integrability of their upper truncations, under natural conditions which are fulfilled in most finitely additive integration theories.     

On-line arbitrarily vertex decomposable trees

A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by τ. An on-line version of the problem of characterizing arbitrarily vertex decomposable trees is completely solved here.     

Physics, mind, society: Back and forth

The life course of the physicist and biologist George Feher may be seen as an epitome of science of the second half of the 20th century and the beginning of the 21st century. Feher, a native of Slovakia, barely escaped Nazism and communism and became a scientist in the USA. The Nazi concentration camps and the communist gulags have become a symbol of the 20th century. This symbol stands here to pose a question: How the two totalitarian systems, fraught with irrationality, may have arisen and thrived in parallel with an unprecedented expansion of science, the paragon of rationalism? The question has become even more urgent in the 21st century. The Ground Zero, an empty spot left after the collapse of the twin towers of the World Trade Center on 11 September 2001, has become the symbol of the entrance of humankind into the new millennium. We can do much, but we understand too little about who we are and what we are doing — this is a message that the two symbols convey about the precarious stage of our evolution. The second message concerns the role of artifacts, specifically scientific instruments, in the advancement of science. Human cultural evolution has been steadily progressing, in a form of a ratchet, only because artifacts have been continually evolving. Contrary to the common Popperian wisdom, the demarcation in science may not consist in the amenability to theoretical falsification, but rather in the amenability to instrumental grasping. Scientific instruments have empowered humans for impressive feats of manipulation with Nature and themselves. Knowledge arising in the course of autonomous evolution of artifacts may surpass the horizon of human understanding and grasping. New knowledge may still be power, but no longer the power of humans. We may need a revision of some fundamental ideas of European thought. Our understanding of the human mind may entirely reshape our comprehension of the nature of physical knowledge, and vice versa.     

Trovacenchemie. 13 [1] Weg und Irrweg zu Di([5]trovacenyl)keton

Trovacene Chemistry. 13 [1] On Being Lead and Mislead in the Synthesis of Di([5]trovacenyl)ketone Di([5]trovacenyl)ketone ( 3¨ ) has been prepared from lithio‐[5]trovacene and dimethylcarbamoyl chloride and studied by X‐ray diffraction, cyclic voltammetry, magnetic susceptometry and EPR spectroscopy. Slight variation of the synthetic protocol affords an unusual trinuclear complex 5 ? that is also fully characterized. Spin‐spin exchange interaction is smaller in 3¨ than in 1,1‐di([5]trovacenylethene ( 2¨ ), which differs from 3¨ by a replacement of O for CH₂. The novel trinuclear complex 5 ? , however, displays exchange coupling very similar to that observed in 1‐methoxy‐2,5,6‐tri([5]trovacenyl)benzene ( 7 ? ). In both cases, an unsymmetrical triangular arrangement of S = 1/2 centers is encountered. These findings are discussed in terms of the nature of the respective spacers.     

Eigenvalue ratios of vibrating strings

Summary New results for the eigenvalue ratios of vibrating strings are presented partially in connection with previous results concerning Schr?dinger operators.     

Sequencing games without initial order

In this note we study uncertainty sequencing situations, i.e., one-machine sequencing situations in which no initial order is specified. We associate cooperative games with these sequencing situations, study their core, and provide links with the classic sequencing games introduced by Curiel et al. (Eur J Oper Res 40:344–351, 1989). Moreover, we propose and characterize two simple cost allocation rules for uncertainty sequencing situations with equal processing times.