We discuss the simple constituents of Specht module $S^{\lambda}$ for the symmetric group $S_n$ defined over the field of $p$ elements. We firstly give an easier proof to the result in [6] which asserts that there exists a simple constituent of $S^{\lambda}$ with the shape of ``a branch'' of $\lambda$ (Theorem 3.3), and secondly give a sufficient condition for $\lambda$ to have a particular type branch as a constituent (Proposition 3.4).

The classical theorem of Shub and Sullivan states that a sequence of local fixed point indices of iterations of a $C^1$ self-map of $\mathbb{R}^m$ is periodic. The paper generalizes this result to a wider class of maps.

Given a fibered link, consider the characteristic polynomial of the monodromy restricted to first homology. This generalizes the notion of the Alexander polynomial of a knot. We define a construction, called iterated plumbing, to create a sequence of fibered links from a given one. The resulting sequence of characteristic polynomials for these links has the same form as those arising in work of Salem and Boyd in their study of distributions of Salem and P-V numbers. From this we deduce information about the asymptotic behavior of the large roots of the generalized Alexander polynomials, and define a new poset structure for Salem fibered links.

The Hermite constant $\gamma_n(D)$ of a quaternion skew field $D$ over a global field is defined and studied. We obtain an upper bound of $\gamma_{n}(D)$. In the case that the base field is a number field, we introduce the notion of quaternionic Humbert forms over $D$. Then $\gamma_{n}(D)$ is characterized as a critical value of the Hermite invariants for $n$-ary quaternionic Humbert forms. We extend Voronoï's theorem on extreme forms to quaternionic Humbert forms.

The paper deals with locally free sheaves $\mathcal{F}_{p,q}$ on $\mathbb{P}^{2}$ obtained from a morphism $\mathbb{P}^{1}\times\mathbb{P}^{1}\to\mathbb{P}^{2}$. Bases of $\mathrm{H}^{i}(\mathbb{P}^{2},\mathcal{F}_{p,q})$ are explicitly given in terms of elements of certain local cohomology modules, which built up canonically a complex for computing cohomology modules of locally free sheaves on $\mathbb{P}^{2}$.

The Brown-Peterson cohomology for skeleta of the classifying space of the group $\mathbb{Z}_{4}\times\mathbb{Z}_{4}$ is analyzed in order to describe obstructions to the motion planning problem for a particle moving in a 4-torsion lens space. We discuss the relationship of this situation to the Euclidean immersion problem for $2^{e}$-torsion lens spaces, and the way this leads to an alternative approach to the classical immersion problem for real projective spaces.

A curve that can be parametrized by polynomials is called a polynomial curve. It is well-known that a polynomial curve has only one place at infinity. Let $C$ be a curve with one place at infinity. Sathaye presented the following question raised by Abhyankar: Is there a polynomial curve associated with the semigroup generated by pole orders of $C$ at infinity? In this paper, we give a negative answer to this question using Gröbner basis computation.

By using Morton's inequality we study the canonical genus of a Whitehead double of a knot. We show that the crossing number of a 2-bridge knot coincides with the canonical genus of its Whitehead double.

It is shown that the symmetric behaviour of certain class of knots can be realized by their polynomial representations. We prove that every strongly invertible knot (open) can be represented by a polynomial embedding $t\mapsto (f(t),g(t),h(t))$ of $\mathbb{R}$ in $\mathbb{R}^{3}$ where among the polynomials $f(t)$, $g(t)$ and $h(t)$ two of them are odd polynomials and one is an even polynomial. We also prove that a subclass of strongly negative amphicheiral knots can be represented by a polynomial embedding $t\mapsto(f(t),g(t),h(t))$ of $\mathbb{R}$ in $\mathbb{R}^{3}$ where all three polynomials $f(t)$, $g(t)$ and $h(t)$ are odd polynomials.

We derive a holomorphic spinor representation formula for spacelike surfaces of constant mean curvature 1 in de Sitter 3-space, and use it to construct examples of spacelike catenoids and trinoids with constant mean curvature 1.

For the elastic wave, there are waves of different modes and a remarkable phenomenon called ``mode-conversion'' which causes serious difficulties in the analysis of singularities of the scattering kernel. In the present paper, by considering the case of a non back-scattering, we examine singularities of the scattering kernel for the elastic wave equation in the case of mode-conversion.

The purpose of this paper is to give an explicit statement with respect to a numerical algorithm for finding balanced metrics, which has already been pointed out by Donaldson [3].

The isovariant Borsuk-Ulam theorem provides nonexistence results on isovariant maps between representations. In this paper we shall deal with the existence problem of isovariant maps as a converse to the isovariant Borsuk-Ulam theorem, and show that the converse holds for representations of an abelian $p$-group or a cyclic groups of order $p^{n}q^{m}$ or $pqr$, where $p,q,r$ are distinct primes.

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-reaching generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope. In this case we retrieve the familiar picture from toric geometry: the equivariant cohomology is the face ring of the nerve simplicial complex and the ordinary cohomology is obtained by factoring out certain linear forms. In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, the equivariant cohomology is still isomorphic to the face ring of an appropriate simplicial poset.