<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.     

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     

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.     

On particle creation by black holes

Hawking's analysis of particle creation by black holes is extended by explicitly obtaining the expression for the quantum mechanical state vector ψ which results from particle creation starting from the vacuum during gravitational collapse. (Hawking calculated only the expected number of particles in each mode for this state.) We first discuss the quantum field theory of a Hermitian scalar field in an external potential or in a curved but asymptotically flat spacetime with no horizon present. In agreement with previously known results, we find that we are led to a unique quantum scattering theory which is completely well behaved mathematically provided a certain condition is satisfied by the operators which describe the scattering of classical positive frequency solutions. In terms of these operators we derive the expression for the state vector describing particle creation from the vacuum, and we prove that S-matrix is unitary. Making the necessary modification for the case when a horizon is present, we apply this theory for a massless Hermitian scalar field to get the state vector describing the steady state emission at late times for particle creation during gravitational collapse to a Schwarzschild black hole. There is some ambiguity in the theory in this case arising from freedom involved in defining what one means by “positive frequency” at the future event horizon. However, it is proven that the expression for the density matrix formed from ψ describing the emission of particles to infinity is independent of this choice, and thus unambiguous predictions for the results of all possible measurements at infinity are obtained. We find that the state vector describing particle creation from the vacuum decomposes into a simple product of state vectors for each individual mode. The density matrix describing emission of particles to infinity by this particle creation process is found to be identical to that of black body emission. Thus, black hole emission agrees in complete detail (i.e., not only in expected number of particles) with black body emission.     

Existence of the S-matrix in quantum field theory in curved space-time

The existence of the S-matrix is proven for particle creation by an external gravitational field of compact support. No infrared divergences occur even for massless quantum fields; in particular, a localized gravitational field always produces a finite expected total number of particles. The results of this paper apply to both boson and fermion fields as well as to more general linear, external potential interactions.     

Die Aufnahme der Äthylendiamintetraessigsäure-Komplexe von Zink,Cadmium, Zirkon und Niob durch Anionenaustauscher

Zusammenfassung Die radioaktiven Indikatoren⁶⁵Zn,^115mCd,⁹⁵Zr und⁹⁵Nb wurden verwendet, um das Austauschverhalten der Äthylendiamintetraessigsäure-Komplexe dieser Elemente an Säulen des stark basischen Harzes Dowex 1 X4 zu untersuchen. Die Messung der Verteilungskoeffizienten zwischen nitrat-beladenem Austauscher und ÄDTA-haltigen Nitratlösungen, bzw. zwischen ÄDTA-beladenem Austauscher und ÄDTA-Lösungen ermöglicht Aussagen über die Zusammensetzung der Komplexe, die für den Austauschvorgang bestimmend sind. Zn bildet im untersuchten pH-Bereich 3,0–8,8 einen negativen zweiwertigen Komplex, der durch den Austauscher stärker aufgenommen wird als der Bleikomplex. An nitrat-beladenen Austauschersäulen wird die Elutionsgeschwindigkeit sehr stark durch den ÄDTA-Gehalt der Elutionslösungen beeinflußt, sobald das Verhältnis [NO ₃ ^– ]:[H₂Y^2–] unter den Wert 100 absinkt. Der Cadmiumkomplex wird durch den nitrat-beladenen Austauscher im pH-Bereich 5,0–10,0 als zweiwertiger Komplex aufgenommen, jedoch schwächer als der Zinkkomplex. Auch an komplexonbeladenen Austauschersäulen wird der Aufnahmevorgang des Zn und des Cd durch ihre zweiwertigen Komplexe bestimmt, wobei Cd bedeutend schwächer aufgenommen wird als Zn. Zr wird im untersuchten pH_Bereich 5,0–7,5 als zweiwertiger Zirkonylkomplex aufgenommen, unterhalb pH=3,0 ist ein einwertiger Komplex, wohl von der Zusammensetzung ZrH₃Y ₂ ^– , für die Aufnahme maßgeblich. Nb wird als Radiokolloid von der Austauschersäule festgehalten. Der Vergleich der Aufnahme der Komplexe ZnY^2–, PbY^2– und CdY^2– läßt schließen, daß die Größe der Komplexbildungskonstante, ähnlich wie die Ladungszahl, nicht den entscheidenden Faktor für die Stärke der Aufnahme in den Austauscher bildet.Die Möglichkeiten radiochemischer Trennungen von Zn, Cd, Pb, Bi einerseits und von Zr–Nb andererseits werden aufgezeigt.Mit 5 Abbildungen