We prove several vanishing theorems for a class of generalized elliptic genera on foliated manifolds, by using classical equivariant index theory. The main techniques are the use of the Jacobi theta-functions and the construction of a new class of elliptic operators associated to foliations.

Injecting a single hole into a one-dimensional Heisenberg spin chain is probably the simplest case of doping a Mott insulator. The motion of such a single hole will generally induce a many-body phase shift, which can be identified by an exact sign structure of the model known as the phase string. We show that the sign structure is nontrivial even in this simplest problem, which is responsible for the essential properties of Mott physics. We find that the characteristic momentum structure, the Luttinger liquid behavior, and the quantum phase interference of the hole under a periodic boundary condition, can all be attributed to it. We use the density matrix renormalization group (DMRG) numerical simulation to make a comparative study of the t-J chain and a model in which the sign structure is switched off. We further show that the key DMRG results can be reproduced by a variational wave function with incorporating the correct sign structure. Physical implications of the sign structure for doped Mott insulators in general are also discussed.

One of the theorems of Bengt Stolt’s article “On the Diophantine Equation$u$^{2}−$Dv$^{2}=4$N$” is not quite correct in its entirety. A counter-example will be given to show this. A modification of the theorem which he was trying to prove will be given for certain special cases.

Shape optimization aims to optimize an objective function by changing the shape of the computational domain. In recent years, shape optimization has received considerable attentions. On the theoretical side there are several publications dealing with the existence of solution and the sensitivity analysis of the problem; see e.g., [6] and references therein.

Askewing'method is shown to effectively reduce the order of bias of locally parametric estimators, and at the same time retain positivity properties. The technique involves first calculating the usual locally parametric approximation in the neighbourhood of a point x'that is a short distance from the place x where the we wish to estimate the density, and then evaluating this approximation at x. By way of comparison, the usual locally parametric approach takes x'= x. In our construction, x'-x depends in a very simple way on the bandwidth and the kernel, and not at all on the unknown density. Using skewing in this simple form reduces the order of bias from the square to the cube of bandwidth; and taking the average of two estimators computed in this way further reduces bias, to the fourth power of bandwidth. On the other hand, variance increases only by at most a moderate constant factor.