From Bolzano‐Weierstraß to Arzelà‐Ascoli

We show how one can obtain solutions to the Arzelà‐Ascoli theorem using suitable applications of the Bolzano‐Weierstraß principle. With this, we can apply the results from 10 and obtain a classification of the strength of instances of the Arzelà‐Ascoli theorem and a variant of it. Let be the statement that each equicontinuous sequence of functions contains a subsequence that converges uniformly with the rate and let be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that is instance‐wise equivalent, over , to the Bolzano‐Weierstraß principle and that is instance‐wise equivalent, over , to , and thus to the strong cohesive principle (). Moreover, we show that over the principles , and are equivalent.     

Calabi-Yau 4-folds and toric fibrations

We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi-Yau 4-folds. We find 914 164 weights with degree d ≤ 150 whose maximal Newton polyhedra are reflexive and 525 572 weights with degree d ≤ 4000 that give rise to weighted projective spaces such that the polynomial defining a hypersurface of trivial canonical class is transversal. We compute all Hodge numbers, using Batyrev's formulas (derived by toric methods) for the first and Vafa's formulas (obtained by counting of Ramond ground states in N = 2 LG models) for the latter class, checking their consistency for the 109 308 weights in the overlap. Fibrations of k-folds, including the elliptic case, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is a subpolyhedron of that corresponding to the k-fold, whereas the fan determining the base is a linear projection of the fan corresponding to the k-fold.     

Geometry of kinematicK-loops

AK-loop is called kinematic, if a further condition (K7) is valid. Such a loop (L, ⊕) can be provided in a natural way with a left and right structureL andG such that (L,L) and (L,G) become incidence (linear) spaces. For (L,L) andt∈L, each left translationt ⁺:L→L;x→b⊕x is a collineation and (L,G) can be turned in an incidence space with parallelism (L,G, ‖). Examples of kinematicK-loops are given for which the corresponding automorphisms δ_a,b are either the identity or fixed point free.     

A remark on polar geometry

Dedicated to ProfessorDr. Walter Benz on the occasion of his sixtieth birthday     

Analysis of dynamic pattern formation in nonlinear Fabry-Perot resonators

We present a detailed analysis of transverse effects and pattern formation in bistable optical elements. The system we investigate consists of a Fabry-Perot resonator for the optical feedback element with a nematic liquid-crystal cell used as an optically nonlinear intracavity medium. On illumination with a cw-laser beam, the system causes the beam to break up into several individual spots, passing through several transitions before finally reaching a stationary state. We devise a theoretical model which is used as the basis for numerical simulations of the system. The simulation results are in good agreement with experiment. Finally, we characterize the principal instability of the system using a linear stability analysis of the theoretical model.     

Bifurcations and subharmonic resonances in multi-degree-of-freedom panel models

This paper describes the nonlinear, postcritical behavior of parametrically excited, shallow, cylindrical panels, which are modeled with two or four degrees of freedom. The analysis shows complicated dynamic behavior. Stable, periodic motions coexist with the trivial solution for very small values of the excitation amplitude. Moreover, a stable, chaotic attractor could be found coexisting with the trivial solution.

---

Sommario Si studia il comportamento postcritico nonlineare di pannelli cilindrici ribassati, soggetti ad eccitazione parametrica e modellati con due o quattro gradi di libertà. L'analisi evidenzia un comportamento dinamico complesso. Moti periodici stabili coesistono con la soluzione banale per valori molto piccoli dell'ampiezza dell'eccitazione. Un attrattore caotico stabile coesiste altresì con tale soluzione per alcuni valori della frequenza dell'eccitazione.

     

On the Cayley-Bacharach Property

The Cayley-Bacharach Property (CBP), which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations and explicit algorithms for checking the CBP directly, via the canonical module, and in combination with the property of being a locally Gorenstein ring. Moreover, we characterize strict Gorenstein rings by the CBP and the symmetry of their affine Hilbert function, as well as by the strict CBP and the last difference of their affine Hilbert function.     

Double fluorescence lifetimes of metal-free phthalocyanine in a mixed-solvent Shpol'skii matrix at 4·2 K

We report identical double fluorescence lifetimes of τ = 3·7 and 6·3 ns for the 0-0 fluorescences of each of the two tautomeric species of metal-free phthalocyanine (H₂Pc) in a mixed solvent matrix of α-chloronaphthalene and n-octane at 4·2K. Such double lifetime feature has not been reported previously for this molecule nor, to our knowledge, for any of the related phthalocyanines and porphyrins.     

Kinetic lattice gas model: Langmuir,Ising and interaction kinetics

The kinetic lattice gas model is formulated properly to account for adsorption, desorption, and diffusion at surfaces. We examine three choices for the transition probabilities in the master equation, which we term Langmuir, Ising and interaction kinetics, and show how they lead to different sticking coefficients and desorption rates.