We obtain general upper bounds of the sizes and the numbers of Jordan blocks for the eigenvalues λ ≠ 1 in the monodromies at infinity of polynomial maps.

In this paper, we provide a new class of reconstructible finite graphs. We show the following theorem: Let *k* be a positive integer number. Let Γ be a finite graph with at least 3 vertices. Suppose that Γ satisfies the following two conditions: (i) for any two distinct vertices *w,w*′ ∈ *V*(Γ), [*w,w*′] ∈ *E*(Γ) ⬄ *N*(*w*)-{*w*′} ≇ *N*(*s*) for any vertex *s* ∈ *V*(Γ); (ii) there exists a vertex *v* ∈ *V*(Γ) of degree *k* such that for any *k*-vertices *v*_{1}, *v*_{2}, …, *v*_{k} ∈ *V*(Γ)-{*v*}, there exists a vertex *u* ∈ *V*(Γ) such that *St*^{2}(*u*,Γ) ⋂ {*v*, *v*_{1}, *v*_{2}, …, *v*_{k}} = ∅, where *N*(*w*) is the full subgraph of Γ whose vertex set is {*v* ∈ *V*(Γ)|[*w,v*] ∈ *E*(Γ)} and *St*^{2}(*u*,Γ) = ⋂ {*St*(*w*,Γ)| *w* ∈ *V*(*St*(*u*,Γ))}. Then the graph Γ is reconstructible. We also provide some applications and examples.

We give a bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex by applying unprojection theory. From this we derive a bound for the Betti numbers of iterated stellar subdivisions of the boundary complex of a simplex. The bound depends only on the number of subdivisions, and we construct examples which prove that it is sharp.

We prove that an admissible normal function over a surface and the zero section simultaneously extend to sections of a log Néron model. This gives a new proof of the surface base case of the algebraicity of zero loci of admissible normal functions.

We prove that a group *G* with exactly three classes of nonnormal proper subgroups of the same non-prime-power order is nonsolvable if and only if *G* ≃ *A*_{5}, and a group *G* with exactly four classes of nonnormal proper subgroups of the same non-prime-power order is nonsolvable if and only if *G* ≃ *PSL*(2,7) or *PSL*(2,8). Moreover, we prove that any group *G* with at most nine classes of nonnormal nontrivial subgroups of the same order is always solvable except for *G* ≃ *A*_{5}, *PSL*_{2}(7) or *SL*_{2}(5).

Let *p* be a prime number. Let *K* be an abelian number field with *p* ∤ [*K* : ℚ] and ζ_{p} ∈ *K*, *K*_{∞}/*K* the cyclotomic ℤ_{p}-extension, and *K*_{n} the *n*th layer with *K*_{0} = *K*. Let $\mathcal U$_{n} be the group of semi-local principal units of *K*_{n} at the prime *p*, and $\mathcal U$_{n}^{(1)} the elements *u* of $\mathcal U$_{n} satisfying the congruence *u* ≣ 1 modulo ζ_{p} - 1. The Galois module structure of $\mathcal U$_{n} is well understood. The purpose of this paper is to determine the Galois module structure of $\mathcal U$_{n}^{(1)}.

We consider an abstract model which describes an interaction of non-relativistic particles with a Bose field. We show that the essential self-adjointness of the generalized spin-boson Hamiltonian with a quadratic boson interaction for all coupling constant and the Hamiltonian is self-adjoint if it is bounded from below under some conditions.

The aim of this paper is to give the Schwartz kernel theorem for the space of the tempered distributions on the Heisenberg group.

It is well known that, at each point of a 4-dimensional Einstein Riemannian manifold (*M,g*), the tangent space admits at least one so-called Singer-Thorpe basis with respect to the curvature tensor *R* at *p*. K. Sekigawa put the question "how many" Singer-Thorpe bases exist for a fixed curvature tensor *R*. Here we work only with algebraic structures ($\mathbb{V}$, ⟨,⟩, *R*), where ⟨,⟩ is a positive scalar product and *R* is an algebraic curvature tensor (in the sense of P. Gilkey) which satisfies the Einstein property. We give a partial answer to the Sekigawa problem and we state a reasonable conjecture for the general case. Moreover, we solve completely a modified problem: how many there are orthonormal bases which are Singer-Thorpe bases simultaneously for a natural 5-dimensional family of Einstein curvature tensors *R*. The answer is given by what we call "the universal Singer-Thorpe group" and we show that it is a finite group with 2304 elements.