We call a section of an elliptic surface to be everywhere integral if it is disjoint from the zero-section. The set of everywhere integral sections of an elliptic surface is a finite set under a mild condition. We pose the basic problem about this set when the base curve is P¹. In the case of a rational elliptic surface, we obtain a complete answer, described in terms of the root lattice E₈ and its roots. Our results are related to some problems in Gröbner basis, Mordell-Weil lattices and deformation of singularities, which have served as the motivation and idea of proof as well.

We prove the main theorem on the structure of everywhere integral sections on a rational elliptic surface, which is formulated in the first part of the paper with the same title [18]. A few examples are given to illustrate it, and some open questions in the case of higher arithmetic genus will be discussed.

We show that under the heat-bath Glauber dynamics for the ferromagnetic Ising model on a finite graph, the single spin expectation at a fixed time starting at the all-up configuration is not necessarily an increasing function of coupling constants.

In this paper, we consider a family of cubic fields $\{K_m\}_{m\geq4}$ associated to the irreducible cubic polynomials $P_m(x)=x^3-mx^2-(m+1)x-1,\,\,\,(m\geq4).$ We prove that there are infinitely many $\{K_m\}_{m\geq4}$'s whose class numbers are divisible by a given integer n. From this, we find that there are infinitely many non-normal totally real cubic fields with class number divisible by any given integer n.

We present a different perspective on Nagumo's uniqueness theorem and its various generalizations. This allows us to improve these generalizations.

Using the Mehler kernel, we give an existence result of the Cauchy Dirichlet problem for the Hermite heat equation with homogeneous Dirichlet boundary conditions and continuous and bounded Cauchy data vanishing at x=0.