Morse Theory and Infinite Families¶of Harmonic Maps Between Spheres

Existence of an infinite sequence of harmonic maps between spheres of certain dimensions was proven by Bizoń and Chmaj. This sequence shares many features of the Bartnik–McKinnon sequence of solutions to the Einstein–Yang–Mills equations as well as sequences of solutions that have arisen in other physical models. We apply Morse theoretic methods to prove existence of the harmonic map sequence and to prove certain index and convergence properties of this sequence. In addition, we generalize the result of Bizoń and Chmaj to produce infinite sequences of harmonic maps not previously known. The key features “responsible” for the existence and properties of the sequence are thereby seen to be the presence of a reflection (ℤ₂) symmetry and the existence of a singular harmonic map of infinite index which is invariant under this symmetry. Received: 10 December 1999 / Accepted: 7 July 2000     

Estimation of optimal backward perturbation bounds for the linear least squares problem

In this paper a method of estimating the optimal backward perturbation bound for the linear least squares problem is presented. In contrast with the optimal bound, which requires a singular value decomposition, this method is better suited for practical use on large problems since it requiresO(mn) operations. The method presented involves the computation of a strict lower bound for the spectral norm and a strict upper bound for the Frobenius norm which gives a gap in which the optimal bounds for the spectral and the Frobenius norm must be. Numerical tests are performed showing that this method produces an efficient estimate of the optimal backward perturbation bound.     

<Emphasis Type="Italic">M</Emphasis>-slenderness

Analogues of Nunke’s theorem are proved which characterize variants of slenderness. For a bounded monotone subgroup M of ? ^ω , a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G. It is consistent relative to ordinary set theory (ZFC) that if M ≠ ? ^ω is an unbounded monotone subgroup of ? ^ω , then a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G.     

High-flip-angle slice-selective parallel RF transmission with 8 channels at 7 T

At high magnetic field, B(1)(+) non-uniformity causes undesired inhomogeneity in SNR and image contrast. Parallel RF transmission using tailored 3D k-space trajectory design has been shown to correct for this problem and produce highly uniform in-plane magnetization with good slice selection profile within a relatively short excitation duration. However, at large flip angles the excitation k-space based design method fails. Consequently, several large-flip-angle parallel transmission designs have recently been suggested. In this work, we propose and demonstrate a large-flip-angle parallel excitation design for 90 degrees and 180 degrees spin-echo slice-selective excitations that mitigate severe B(1)(+) inhomogeneity. The method was validated on an 8-channel transmit array at 7T using a water phantom with B(1)(+) inhomogeneity similar to that seen in human brain in vivo. Slice-selective excitations with parallel RF systems offer means to implement conventional high-flip excitation sequences without a severe pulse-duration penalty, even at very high B(0) field strengths where large B(1)(+) inhomogeneity is present.     

Local Wick Polynomials and Time Ordered Products¶of Quantum Fields in Curved Spacetime

In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and K?hler, and by Brunetti and Fredenhagen, but they did not impose any “locality” or “covariance” condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an “extended Wick polynomial algebra”– large enough to contain the Wick polynomials and their time ordered products – by generalizing a construction of Dütsch and Fredenhagen to curved spacetime. We then define the notion of a local, covariant quantum field, and seek a definition of local Wick polynomials and their time ordered products as local, covariant quantum fields. We introduce a new notion of the scaling behavior of a local, covariant quantum field, and impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements – such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigation. Received: 27 March 2001 / Accepted: 6 June 2001     

Inversion problem including a long-ranged Coulomb potential

In this note we extend the WKB inverse problem which involved a short-ranged potential to include a combined long and short-ranged potential.