Computing Dynamic Output Feedback Laws

The pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed-loop system has a desired set of eigenvalues. Converting this problem into a question of enumerative geometry, efficient numerical homotopy algorithms to solve this problem for general Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While dynamic feedback laws offer a wider range of use, the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before. In this paper we present symbolic-numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine. We report on numerical experiments with our publicly available software and illustrate its application on various control problems from the literature.Comment: 20 pages, 3 figures; the software described in this paper is publicly available via http://www.math.uic.edu/~jan/download.htm

On Stimulated Radiation of Black Holes

The Unruh's thermal state in the vicinity of the event horizon of the black hole provides conditions where impinging particles can radiate other particles. The subsequent decays may eventually lead to observable radiation of photons and neutrinos induced even by massive particles with gravitational interaction only. The hadronic particles will induce $\sim 30$ MeV gamma radiation from $\pi^{0}$ decays.Comment: 4 pages, 1 figur

Integrable Systems in n-dimensional Riemannian Geometry

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an (n)-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair in (\orth{n+1}) with the vector modified Korteweg-De Vries equation (vmKDV) \vk{t}= \vk{xxx}+\fr32 ||\vk{}||^2 \vk{x} as integrability condition. We indicate that other integrable vector evolution equations can be found by using a different Ansatz on the form of the Lax pair. We obtain these results by using the {\em natural} or {\em parallel} frame and we show how this can be gauged by a generalized Hasimoto transformation to the (usual) {\em Fren{\^e}t} frame. If one chooses the curvature to be zero, as is usual in the context of integrable systems, then one loses information unless one works in the natural frame

H\"older-type inequalities and their applications to concentration and correlation bounds

Let $Y_v, v\in V,$ be $[0,1]$-valued random variables having a dependency graph $G=(V,E)$. We show that $$\mathbb{E}\left[\prod_{v\in V} Y_{v} \right] \leq \prod_{v\in V} \left\{ \mathbb{E}\left[Y_v^{\frac{\chi_b}{b}}\right] \right\}^{\frac{b}{\chi_b}},$$ where $\chi_b$ is the $b$-fold chromatic number of $G$. This inequality may be seen as a dependency-graph analogue of a generalised H\"older inequality, due to Helmut Finner. Additionally, we provide applications of H\"older-type inequalities to concentration and correlation bounds for sums of weakly dependent random variables.Comment: 15 page

Synthesis of Hydrophobically and Electrostatically Modified Polyacrylamides and Their Catalytic Effects on the Unimolecular Decarboxylation of 6-Nitrobenzisoxazole-3-carboxylate Anion

A series of hydrophobically and electrostatically modified polyacrylamides (Copol(AM-C12)) has been synthesized by radical-initiated copolymerization of acrylamide with n-dodecylmethyldiallylammonium bromide as the hydrophobe in aqueous solution using ammonium persulfate as the initiator. The formation of hydrophobic microdomains of the copolymers was revealed by large hypsochromic shifts of the longwavelength absorption band of the solvatochromic probe Methyl Orange, noncovalently bound to the macromolecule. It was found that the microdomains formed by these copolymers in aqueous solution are more hydrophobic than those of the cationic polysoaps poly(alkylmethyldiallylammonium halides) containing the same n-dodecyl groups as the side chains as a result of the reduced electrostatic repulsions at the periphery of the microdomains. The reduced cationic character of the copolymers Copol(AM-C12) most likely also accounts for the observation that the anionic dye Methyl Orange does not induce microdomain formation in aqueous solution. The effect of the hydrophobically and electrostatically modified polyacrylamides on the unimolecular decarboxylation of 6-nitrobenzisoxazole-3-carboxylate anion (6-NBIC) has been investigated in aqueous solutions at pH 11.3 and 30 °C. It is suggested that the relatively modest catalytic effects induced by Copol(AM-C12) should be ascribed to hydrogen-bond stabilization of the initial state by NH groups in the macromolecules. The decarboxylation rates of 6-NBIC at binding sites in hydrophobic microdomains increase with increasing n-dodecyl group content in the copolymers.