In this paper, we study perturbations of the $d$-dimensional Ising model for $d\geq 2$, including long range ones to which the Pirogov-Sinai theory is not applicable. We show that the uniqueness of the equilibrium state of the Ising model at high temperature and the coexistence of equilibrium states at low temperature are preserved by spin-flip symmetric perturbations.

Let $G$ be a finite $p$-solvable group and $N$ a normal subgroup of $G$. Suppose that $B$ is a $p$-block of $G$ with defect group $D$ such that $ |D|>|D\cap N|$. Given $\mu \in \mathrm{Irr}(N),$ we show that the set of height zero characters in $\mathrm{Irr}(B)$ that lie over $\mu $ is either empty or contains two or more elements.

We give a method to construct a critical Schrödinger form from the subcritical Schrödinger form by subtracting a suitable positive potential. The method enables us to obtain optimal Hardy-type inequalities.

In this paper we study spectral estimates for perturbations of the Landau Hamiltonian by a pseudo-differential operator with non-smooth Weyl symbol in a modulation space. We obtain an upper bound for the counting function of the eigenvalues in a spectral gap.

Following Getzler's idea from the geometric viewpoint as to symbol calculus on a spin manifold, we introduce a new symbol calculus of $H$-pseudodifferential operators on a contact Riemannian manifold with contact distribution $H$. An explicit formula for the top grading part of the symbol of the composite of $H$-differential operators is presented.

For a certain class of power series, infinite products, and Lambert type series, we establish a necessary and sufficient condition for the infinite set consisting of their values, as well as their derivatives of any order at any algebraic points except their poles and zeroes, to be algebraically independent. As its corollary, we construct an example of an infinite family of entire functions of two variables with the following property: Their values and their partial derivatives of any order at any distinct algebraic points with nonzero components are algebraically independent.

Shimizu introduced a region crossing change unknotting operation for knot diagrams. As extensions, two integral region choice problems were proposed and the existences of solutions of the problems were shown for all non-trivial knot diagrams by Ahara and Suzuki, and Harada. We relate both integral region choice problems with an Alexander numbering for regions of a link diagram,and give alternative proofs of the existences of solutions for knot diagrams. We also discuss the problems on link diagrams. For each of the problems on the diagram of a two-component link, we give a necessary and sufficient condition that there exists a solution.

We study variational problems for integral invariants, which are defined as integrations of invariant functions of the second fundamental form, of a smooth map between pseudo- Riemannian manifolds. We derive the first variational formulae for integral invariants defined from invariant homogeneous polynomials of degree two. Among these integral invariants, we show that the Euler-Lagrange equation of the Chern-Federer energy functional is reduced to a second order PDE. Then we give some examples of Chern-Federer submanifolds in Riemannian space forms.

We analyze the structure of spinor coordinates on resolutions of Gorenstein ideals of codimension four. As an application we produce a family of such ideals with seven generators which are not specializations of the Kustin-Miller model.

Minhyong Kim introduced arithmetic Chern-Simons invariants for totally imaginary number fields as arithmetic analogues of the Chern-Simons invariants for 3-manifolds. In this paper, we extend Kim's definition to any number field, by using the modified étale cohomology groups and fundamental groups which take real primes into account. We then show explicit formulas of mod 2 arithmetic Dijkgraaf-Witten invariants for real quadratic fields $\mathbb{Q} (\sqrt{p_1 p_2 \cdots p_r})$, where $p_i$'s are distinct prime numbers congruent to 1 mod 4, in terms of the Legendre symbols of $p_i$'s. We also show topological analogues of our formulas for 3-manifolds.