Phase transitions and positive definite couplings

The method of infrared bounds is extended to a large class of nearest neighbour interactions in classical spin systems. Temperature controlled bounds on fluctuations follow whenever the coupling function is a positive definite kernel. Existence of phase transitions is demonstrated for the RP ^Nmodel for d3.     

On combinatorial and projective geometry

Cross ratios constitute an important tool in classical projective geometry. Using the theory of Tutte groups as discussed in [6] it will be shown in this note that the concept of cross ratios extends naturally to combinatorial geometries or matroids. From a thorough study of these cross ratios which, among other observations, includes a new matroid theoretic version and proof of the Pappos theorem, it will be deduced that for any projective space M= _n (K) of dimension n2 of M over some skewfield K the inner Tutte group is isomorphic to the commutator factor group K ^*/[K ^*, K ^*] of K ^*K{0}. This shows not only that in case M= _n (K) our matroidal cross ratios are nothing but the classical ones. It can also be used to correlate orientations of the matroid M= _n (K) with the orderings of K. And it implies that Dieudonné's (non-commutative) determinants which, by Dieudonné's definition, take their values in K ^*/[K ^*, K ^*] as well, can be viewed as a special case of a determinant construction which works for just every combinatorial geometry.Research supported by the DFG (Deutsche Forschungsgemeinschaft).     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

Periodic solutions of nonlinear differential equations involving the method of scalar nonlinearities

Summary The recently developed method of scalar nonlinearities is applied to establish a new type of existence proof for periodic solutions of nonlinear differential equations. It is proved that given a periodic solution of a certain linear differential equation whose coefficients are subject to some nonlinear constraint, a nonlinear differential equation, which is closely related to the linear one, has a periodic solution (of the same period) as well. While, in general, the nonlinear equation will not be explicitly resolvable, the linear equation (with constraint) will allow for explicitly given solutions.The proof is carried out by constructing a homotopy (between appropriately chosen integral operators) and is based on Leray-Schauder theory. Thus, an essential hypothesis is the a-priori boundedness of certain intermediate problems. The very definition of the homotopy, which seems to be unprecedented in the literature, bears resemblance with the introduction of Dirac's-function.The theory is applied to Duffing's equation, resulting in an abstract existence statement as well as the explicit construction of numerically tractable intermediate problems.     

An instanton-invariant for 3-manifolds

To an oriented closed 3-dimensional manifoldM withH ₁(M, )=0, we assign a ₈-graded homology groupI _*(M) whose Euler characteristic is twice Casson's invariant. The definition uses a construction on the space of instantons onM×.     

WKB properties of the time-dependent Schrödinger system

It is shown that the time-dependent WKB expansion highlights some of the hidden properties of the Schrödinger equation and forms a natural bridge between that equation and the functional integral formulation of quantum mechanics. In particular it is shown that the leading (zero- and first-order in ) terms in the WKB expansion are essentially classical, and the relationship of this result to the classical nature of the WKB partition function, and of the anomalies in quantum field theory, is discussed.     

Energetic ion bombardment of Ag surfaces by C<Stack><Subscript>60</Subscript><Superscript>+</Superscript></Stack> and Ga<Superscript>+</Superscript> projectiles

The ion bombardment-induced release of particles from a metal surface is investigated using energetic fullerene cluster ions as projectiles. The total sputter yield as well as partial yields of neutral and charged monomers and clusters leaving the surface are measured and compared with corresponding data obtained with atomic projectile ions of similar impact kinetic energy. It is found that all yields are enhanced by about one order of magnitude under bombardment with the C60+ cluster projectiles compared with Ga+ ions. In contrast, the electronic excitation processes determining the secondary ion formation probability are unaffected. The kinetic energy spectra of sputtered particles exhibit characteristic differences which reflect the largely different nature of the sputtering process for both types of projectiles. In particular, it is found that under C60+ impact (1) the energy spectrum of sputtered atoms peaks at significantly lower kinetic energies than for Ga+ bombardment and (2) the velocity spectra of monomers and dimers are virtually identical, a finding which is in pronounced contrast to all published data obtained for atomic projectiles. The experimental findings are in reasonable agreement with recent molecular dynamics simulations.