Projective geometry with Clifford algebra

Projective geometry is formulated in the language of geometric algebra, a unified mathematical language based on Clifford algebra. This closes the gap between algebraic and synthetic approaches to projective geometry and facilitates connections with the rest of mathematics.This work was partially supported by NSF grant MSM-8645151.     

Arguments for and against Engesser's buckling formulas

Summary To account for shear in buckling pioblems, one uses either the approach of Engesser or the socalled modified approach, proposed by Haringx in connection with helical springs. Both are based on onedimensional reasoning. While most authors seem to prefer the second method, Nänni [4] has shown by means of the three-dimensional theory of elasticity that, for bars, the first one is superior. The present paper is based on a more fundamental one-dimensional approach and confirms Nänni's and hence Engesser's results. It also shows that, together with shear, shortening of the bar must be taken into account, and it improves Engesser's formulas accordingly. It finally shows why Haringx' approach is correct for springs.

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Argumente für und gegen Engessers Knickformeln

Übersicht Um bei Knickproblemen den Schub zu berücksichtigen, verwendet man entweder das Engessersche oder das sogenannte modifizierte Verfahren, das von Haringx zur Behandlung von Schraubenfedern vorgeschlagen worden ist. Während die meisten Autoren die zweite Methode vorzuziehen scheinen, hat Nänni [4] mittels der dreidimensionalen Elastizitätstheorie gezeigt, daß bei Stäben die erste überlegen ist. Die vorliegende Arbeit stützt sich auf ein fundamentaleres eindimensionales Verfahren und bestätigt Nännis und damit Engessers Ergebnisse. Sie zeigt auch, daß mit dem Schub zusammen die Verkürzung des Stabes zu berücksichtigen ist, und sie verbessert Engessers Formeln dementsprechend. Schließlich zeigt sie, weshalb das Verfahren von Haringx bei Federn korrekt ist.

     

Simple polytopes without small separators

We show that by cutting off the vertices and then the edges of neighborly cubical polytopes, one obtains simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least Ω(n/log^3/2 n). This disproves a conjecture by Kalai from 1991/2004.     

On biquadratic fields that admit unit power integral basis

We consider biquadratic number fields whose maximal orders have power integral bases consisting of units. We prove an effective and efficient criteria to decide whether the maximal order of a biquadratic field has a unit power integral basis or not. In particular we can determine all trivial biquadratic fields whose maximal orders have a unit power integral basis.     

Controllable and reversible inversion of the electronic structure in nickel N-confused porphyrin: a case when MCD matters

Nickel N-confused tetraphenylporphyrin, 1, and nickel 2-N-methyl-N-confused tetraphenylporphyrin, 1-Me, exhibit unusual sign-reversed (positive-to-negative intensities in ascending energy) MCD spectra in the Q-type band region, suggesting a rare ΔHOMO < ΔLUMO relationship between π and π* MOs in the porphyrin core. Simple and reversible deprotonation of the external NH proton in 1 dramatically changes the electronic structure of the porphyrin core into the ΔHOMO > ΔLUMO combination characteristic for the meso-(tetraaryl)porphyrins. DFT, time-dependent DFT, and semiempirical ZINDO/S calculations on 1, 1-Me, and 1(-) confirm the experimental finding and successfully explain the MCD pattern in the target compounds.     

Über Phenolalkohole

Ohne Zusammenfassung     

Über Phenolalkohole

Ohne Zusammenfassung