A Local Functional Calculus and Related Results on the Single-Valued Extension Property

We study the local functional calculus of an operator T having the single-valued extension property. We consider a vector f(T, v) for an analytic function f on a neighborhood of the local spectrum of a vector v with respect to T and show that the local spectrum of v and the local spectrum of f(T, v)are equal with the possible exception of points of the local spectrum of v that are zeros of f, that is, we show that s_T \sigma_{T} (v) is equal to s_T \sigma_{T} (f(T,v)) union the set of zeros of f on s_T \sigma_{T} (v). This local functional calculus extends the Riesz functional calculus for operators. For an analytic function f on a neighborhood of s \sigma (T), we use the above mentioned proposition to obtain proofs of the results that if T has the single-valued extension property, then f(T) also has the single-valued extension property, and conversely if f is not constant on each connected component of a neighborhood of s \sigma (T) and f(T) has the singlevalued extension property, then T also does.     

Water incorporation into a micellar-lamellar phase (dodecyl maltoside-water)

Many alkylated sugars are surfactants, and show lamellar phases which prove to be identical to those of more conventional surfactants. Most, if not all, previous investigations focus on the behaviour of these compounds in pure form. Dodecyl maltoside shows the same lamellar phase as when dry, but on the addition of water the layer spacing expands, and on addition of more water, a hexagonal phase forms. The area per head group increases substantially with water addition, indicating that the water is incorporated into the layers, possibly at the head groups. We suggest that the hexagonal phase is the system's answer to the frustration caused by a large head group area compared with that of the tails.     

Supersymmetry and the Parisi-Sourlas dimensional reduction: A rigorous proof

Functional integrals that are formally related to the average correlation functions of a classical field theory in the presence of random external sources are given a rigorous meaning. Their dimensional reduction to the Schwinger functions of the corresponding quantum field theory in two fewer dimensions is proven. This is done by reexpressing those functional integrals as expectations of a supersymmetric field theory. The Parisi-Sourlas dimensional reduction of a supersymmetric field theory to a usual quantum field theory in two fewer dimensions is proven.Partially supported by the NSF under grant MCS-8301889Partially supported by FAPESP