Discrete differential calculus, graphs, topologies and gauge theory

Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a `Hasse diagram' determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two-point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an `internal' discrete space ({\`a} la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, also a `symmetric lattice' is studied which (in a certain continuum limit) turns out to be related to a `noncommutative differential calculus' on manifolds.Comment: 36 pages, revised version, appendix adde

Monte Carlo simulations of copolymers at homopolymer interfaces: Interfacial structure as a function of the copolymer density

By means of extensive Monte Carlo simulations of the bond fluctuation model, we study the effect of adding AB diblock copolymers on the properties of an interface between demixed homopolymer phases. The parameters are chosen such that the homopolymers are strongly segregated, and the whole range of copolymer concentrations in the two phase coexistence region is scanned. We compare the ``mushroom'' regime, in which copolymers are diluted and do not interact with each other, with the ``wet brush'' regime, where copolymers overlap and stretch, but are still swollen by the homopolymers. A ``dry brush'' regime is never entered for our choice of chain lengths. ``Intrinsic'' profiles are calculated using a block analysis method introduced by us in earlier work. We discuss density profiles, orientational profiles and contact number profiles. In general, the features of the profiles are similar at all copolymer concentrations, however, the profiles in the concentrated regime are much broader than in the dilute regime. The results compare well with self-consistent field calculations.Comment: to appear in J. Chem. Phy

Evidence for complex order parameter in La_{1.83}Sr_{0.17}CuO_4

The in-plane magnetic field penetration depth (\lambda_{ab}) in single-crystal La_{1.83}Sr_{0.17}CuO_4 was investigated by means of the muon-spin rotation (\muSR) technique. The temperature dependence of \lambda^{-2}_{ab} has an inflection point around 10-15K, suggesting the presence of two superconducting gaps: a large gap (\Delta_1^d) with d-wave and a small gap (\Delta_2^s) with s-wave symmetry. The zero-temperature values of the gaps at \mu_0H=0.02T were found to be \Delta_1^d(0)=8.2(2)meV and \Delta_2^s(0)=1.57(8)meV.Comment: 5 pages, 3 figure

Interaction-induced Renormalization of Andreev Reflection

We analyze the charge transport between a one-dimensional weakly interacting electron gas and a superconductor within the scaling approach in the basis of scattering states. We derive the renormalization group equations, which fully account for the intrinsic energy dependence due to Andreev reflection. A strong renormalization of the corresponding reflection phase is predicted even for a perfectly transparent metal-superconductor interface. The interaction-induced suppression of the Andreev conductance is shown to be highly sensitive to the normal state resistance, providing a possible explanation of experiments with carbon-nanotube/superconductor junctions by Morpurgo et al. [Science 286, 263 (2001)].Comment: 4 pages, 2 figure

Two-center resonant photo ionization

Photoionization of an atom $A$, in the presence of a neighboring atom $B$, can proceed via resonant excitation of $B$ with subsequent energy transfer to $A$ through two-center electron-electron correlation. We demonstrate that this two-center mechanism can strongly outperform direct photoionization at nanometer internuclear distances and possesses characteristic features in its time development and the spectrum of emitted electrons.Comment: 4 pages, 3 figure

The impact of delivery risk on optimal production and futures hedging

Multiple delivery specifications exist on nearly all commodity futures contracts. Sellers are typically allowed to choose among several grades of the underlying commodity. On the delivery day, the futures price converges to the spot price of the cheapest-to-deliver grade rather than to that of the par-delivery grade of the commodity. This imposes an additional delivery risk on hedgers. This paper derives the optimal production and futures hedging strategy for a risk-averse competitive firm in the presence of delivery risk. We show that, depending on its relative valuation, the delivery option may induce the firm to produce more than in the absence of delivery risk. If delivery risk is additively related to commodity price risk, the firm will under-hedge its exposure to commodity price risk. If delivery risk is multiplicatively related to commodity price risk, the firm will under- or over-hedge this exposure. For constant relative risk aversion, this is illustrated by a numerical example.delivery risk, futures, risk management, production

Restricted Export Flexibility and Risk Management with Options and Futures

This paper examines the production, export and risk management decisions of a risk-averse competitive firm under exchange rate risk. The firm is export flexible in allocating its output to either the domestic market or a foreign market after observing the exchange rate. Export flexibility is restricted by certain minimum sales requirements that are due to long-term considerations. Currency options are sufficient to derive a separation result under restricted export flexibility. Under fairly priced currency futures and options, full hedging with both instruments is optimal. Introducing fairly-priced currency options stimulates production provided that the currency futures market is unbiased.restricted export flexibility, risk management, currency futures, currency options