Dedicated to the memory of Professor Katsutoshi Takahashi

In this paper, we introduce p-hyponormal tuples in the sense of D. Xia. Furthermore we extend Putnam’s inequality to these tuples and show an equivalence relation of two spectra.

Consider a group presentation: $$\hat{[Pscr]}\tfrm{=<\tfbf{x};}\tfbf{r}\tfrm{>}$$. Here x is a set and r is a set of non-empty, cyclically reduced words on the alphabet x ∪ x−1 (where x−1 is a set in one-to-one correspondence x[harr]x−1 with x). We assume throughout that $\hat{[Pscr]}$ is finite. Let $\hat{F}$ be the free group on x (thus $\hat{F}$ consists of free equivalence classes [W] of word on x∪x−1), and let N be the normal closure of {[R] : R∈r} in $\hat{F}$. Then the group G=G($\hat{[Pscr]}$) defined by $\hat{[Pscr]}$ is $\hat{F}\tfrm{/}N$. We will write W1 =GW2 if [W1]N=[W2]N.

A finite group G can be represented as a group of automorphisms of a compact Riemann surface, that is, G acts on a Riemann surface. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts (possibly reversing orientation).

A group G is said to be a minimal non-FC group, if G contains an infinite conjugacy class, while every proper subgroup of G merely has finite conjugacy classes. The structure of imperfect minimal non-FC groups is quite well-understood. These groups are in particular locally finite. At the other end of the spectrum, a perfect locally finite minimal non-FC group must be a p-group. And it has been an open question for quite a while now, whether such groups exist or not.

When a thin layer of normal (non-superconducting) material is placed between layers of superconducting material, a superconducting-normal-superconducting junction is formed. This paper considers a model for the junction based on the Ginzburg–Landau equations as the thickness of the normal layer tends to zero. The model is first derived formally by averaging the unknown variables in the normal layer. Rigorous convergence is then established, as well as an estimate for the order of convergence. Numerical results are shown for one-dimensional junctions.

In the small diffusion limit ε→0, metastable dynamics is studied for the generalized Burgers problem

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Here u=u(x, t) and f(u) is smooth, convex, and satisfies f(0)=f′(0)=0. The choice f(u)=u2/2 has been shown previously to arise in connection with the physical problem of upward flame-front propagation in a vertical channel in a particular parameter regime. In this context, the shape y=y(x, t) of the flame-front interface satisfies u=−yx. For this problem, it is shown that the principal eigenvalue associated with the linearization around an equilibrium solution corresponding to a parabolic-shaped flame-front interface is exponentially small. This exponentially small eigenvalue then leads to a metastable behaviour for the time-dependent problem. This behaviour is studied quantitatively by deriving an asymptotic ordinary differential equation characterizing the slow motion of the tip location of a parabolic-shaped interface. Similar metastability results are obtained for more general f(u). These asymptotic results are shown to compare very favourably with full numerical computations.