Improving the Fundamental Theorem of Algebra

Theorems 2 and 3 establish the minimum algebraic conditions necessary for a field to be algebraically closed, and they can therefore be said to “optimize” the Fundamental Theorem of Algebra. But each specific“degree implication” is a first-order consequence of the axioms for fields, and could have been discovered two centuries ago; the existence of these finitary relationships appears to have been unsuspected by practically everyone, with one important exception.     

High-frequency magnetoimpedance in nanocomposites

The transmission of millimeter-range electromagnetic waves (30–50 GHz) through a magnetic nanocomposite thin film exhibiting tunnel magnetoresistance (TMR) is calculated. The relative change of transmission coefficient in an applied magnetic field due to the magnetorefractive effect is approximately linear with TMR and strongly depends on nanocomposite resistivity and film thickness. The obtained results are in a good agreement with experiment.     

Singlet versus triplet dynamics of β-carotene studied by quantum control spectroscopy

Biomolecules very often present complex energy deactivation networks with overlapping electronic absorption bands, making their study a difficult task. This can be especially true in transient absorption spectroscopy when signals from bleach, excited state absorption and stimulated emission contribute to the signal. However, quantum control spectroscopy can be used to discriminate specific electronic states of interest by applying specifically designed laser pulses. Recently, we have shown the control of energy flow in bacterial light-harvesting using shaped pump pulses in the visible and the selective population of pathways in carotenoids using an additional depletion pulse in the transient absorption technique. Here, we apply a closed-loop optimization approach to β-carotene using a spatial light modulator to decipher the energy flow network after a multiphoton excitation with a shaped ultrashort pulse in the near-IR. After excitation, two overlapping bands were detected and identified as the S₁ state and the first triplet state T₁. Using the transient absorption signal at a specific probe delay as feedback, the triplet signal could be optimized over the singlet contribution.     

Analysis of Decomposition Algorithms with Benders Cuts for p-Median Problem

In this paper the algorithms for solving the p-median problem based on the Benders decomposition are investigated. A family of problems hard for solving with such algorithms is constructed and then generalized to a special NP-hard case of the p-median problem. It is shown that the effectiveness of the considered algorithms depends on the choice of the optimal values of the dual variables used in Benders cuts. In particular, the depth of the cuts can be equal to one.     

New Non-Linear Inequalities for Flag-Vectors of 4-Polytopes

In this paper we prove four new (infinite) lists of quadratic inequalities, and four cubic inequalities, for the flag f-vectors of 4-polytopes. These extend and supplement the only four currently known non-linear inequalities, which were proved by Bayer in 1987. The new lists of inequalities for flag f-vectors yield new lists of inequalities for f-vectors of 4-polytopes. Using the latter, we managed to improve an estimate discovered by Hoppner and Ziegler concerning upper bounds of f₁ in terms of f₀ and f₃.     

Ueber ein neues Urankolloid

Ohne Zusammenfassung     

Heat polynomial analogues for equations with higher order time derivatives

We generalize the heat polynomials for the heat equation to more general partial differential equations, of higher order with respect to both the time variable and the space variables. Whereas the heat equation requires only one family of polynomials, for an equation of the ?th order with respect to time we introduce ? families of polynomials. These families correspond to the ? initial conditions specified by the Cauchy problem.     

The convex hull property and topology of hypersurfaces with nonnegative curvature

We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certain required conditions. This gives a convex hull property, dual to the classical one for surfaces with nonpositive curvature. A version of this result in the nonsmooth category is obtained as well. We show that our boundary conditions determine the topology of the surface up to at most two choices. The proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing procedure.