In this article, we give equations involving homology classes given by the self-intersection of singularity sets of fold maps. As an application, we give an obstruction to existence of fold maps up to cobordism.

We consider the Pauli–Fierz model, which describes a particle (an electron) coupled to the quantized electromagnetic field and limit the number of photons to less than 2. By computing the resolvent explicitly, we located the spectrum of the Hamiltonian mass. Our results do not depend on the coupling constant $e$ nor on the infrared cutoff parameter $R$.

An irreducible criterion of the imaginary Verma modules for the twisted affine Nappi-Witten Lie algebra $\widetilde{H}_{4}[\sigma]$ is determined. For a (reducible) imaginary Verma module, explicit characterizations of the maximal submodule, imaginary vacuum vectors, and the composition structure are also given.

We consider a nonnegative solution $u$ of the Cauchy problem for a quasilinear parabolic equation $u_t=\Delta u^m+\mu(x)u^p$ with the initial data $u_0(x)\,(\not\equiv 0)$ satisfying $\| \tilde\mu u_0\|_{L^{\infty}(\mathbf R^N)}<\infty$, where nonnegative function $\mu(x)$ satisfies some condition and $\tilde\mu=\mu^{1/(p-1)}$. We give sufficient conditions on $u_0$ for a weighted solution $\tilde\mu u$ to blow up at space infinity and for a direction $\psi \in \mathbf S^{N-1}$ to be a blow-up direction of $\tilde\mu u$. We also show that such a weighted solution $\tilde\mu u$ blows up completely at the blow-up time of $\tilde\mu u$.

We consider nonlinear resistive networks. The equivalence of the Liouville property, the Khas'minskii condition and the weak maximum principle for operators of Laplacian with potential is proved, and a number of criteria for these properties are given. The parabolicity of networks is also discussed.

Suppose that an ideal $J$ of $C^\infty$ functions on an open subset of $\mathbf{R}^2$ is a Łojasiewicz ideal. We describe the set of $C^\infty$ functions vanishing on the zeros of $J$ explicitly using $J$ in an open neighborhood of each point in zeros of $J$, it can be obtained by taking real radical and closure starting from $J$ repeatedly for a finite number of times. This gives an another affirmative answer to Bochnak's conjecture in dimension 2, which is first done by Risler.