In this note, we will make a remark on finite simple groups with abelian Sylow $p$-subgroups using the Classification Theorem of the Finite Simple Groups.

It is shown that, for any coaction $\alpha$ of a locally compact group $K$ on a properly infinite von Neumann algebra $A$ and a closed subgroup $H$ of $K$, $\alpha$ is cocycle conjugate to a coaction which comes from a coaction of $H$ if and only if the dual action $\widehat{\alpha}$ is induced by an action of $H$. We also include applications of the result concerning almost periodic coactions and the ranges of 1-cocycles on measured equivalence relations.

This paper is concerned with giving explicitly the invariant density for a class of rational transformations from the real line $\mathbf{R}$ into itself. We proved that the invariant density can be written in terms of the fixed point $z_{0}$ in $\mathbf{C} \setminus \mathbf{R}$ or in terms of the periodic point $z_{0}$ in $\mathbf{C} \setminus \mathbf{R}$ with period 2. The explicit form of the density allows us to obtain the ergodic properties of the transformation $R$.

A link is said to be {\it strongly $n$-trivial} if there exists a diagram such that one can choose $n+1$ crossing points with the property that changing crossings on any $0 < m \le n+1$ points of these $n+1$ points yields a trivial link. It is shown that for a positive integer $n$ the components of a strongly $n$-trivial link admit mutually disjoint Seifert surfaces.

The purpose of this paper is to prove that we can inductively construct all finite-dimensional irreducible nilpotent modules of type 1 by using the Schnizer homomorphisms for quantum algebras at roots of unity of type $A_n$, $B_n$, $C_n$, $D_n$, or $G_2$.

In this note we construct explicitly a family of nilpotent polylogarithmic extensions unramified outside three points of the field $\mathbf{C}(z)$ of rational functions of one variable over $\mathbf{C}$. We show how to use these extensions to calculate explicitly $l$-adic polylogarithms introduced in [5].

We show that on a $p$-adic Lie group, any normal semistable measure has a unique semistable embedding. This, in particular, implies the uniqueness of semistable embedding of any (operator-)semistable measure on a finite dimensional $p$-adic vector space. We compare two classes of probability measures on a unipotent $p$-adic algebraic group: the class of semistable measures and that of measures whose domain of semistable attraction is nonempty.

Let $X$ be a compact Hausdorff space and $C(X)$ the Banach algebra of all complex-valued continuous functions on $X$. We consider the following property of $C(X)$: for each $f \in C(X)$ there exist a $g \in C(X)$ and positive integers $p$ and $q$ such that $p$ does not divide $q$ and $f^{q} = g^{p}$. When $X$ is locally connected, we give a necessary and sufficient condition for $C(X)$ to have this property. We also give a characterization of a first-countable compact Hausdorff space $X$ for which $C(X)$ has the property above. As a corollary, we prove that if $X$ is locally connected, or first-countable, then $C(X)$ has the property above if and only if $C(X)$ is algebraically closed.

In this article, we introduce a formula for the Reidemeister-Turaev torsion $\tau^{\varphi}(M, [\mathcal{V}], \mathfrak{o}_M)$ of an arbitrary closed 3-manifold $M$ equipped with a Spin$^c$ structure $[\mathcal{V}]$. As a CW-structure of $M$ needed in the process of the computation, we adopt the one induced from a Heegaard splitting which is compatible, via the concept of flow-spine, with a given Spin$^c$ structure.

Let $M$ be a real hypersurface with almost contact metric structure $(\phi ,\xi ,\eta ,g)$ in a nonflat complex space form $M_{n}(c)$. We denote by $S$ be the Ricci tensor of $M$. In the present paper we investigate real hypersurfaces with constant scalar curvature of $M_{n}(c)$ whose structure Jacobi operator $R_{\xi}$ commute with both $\phi$ and $S$. We characterize Hopf hypersurfaces of $M_{n}(c)$.

In this paper we treat the intersection of fixed point subgroups by the involutive automorphisms of exceptional Lie group $G = F_4, E_6, E_7$. We shall find involutive automorphisms of $G$ such that the connected component of the intersection of those fixed point subgroups coincides with the maximal torus of $G$.

We show a radius sphere theorem for a compact Riemannian manifold whose radial curvature at the base point is bounded from below by that of a 2-sphere of revolution. The diameter sphere theorem is expanded to a wide class of metrics.

Recently, under the influence of Elkies' conjecture [2], the optimal recursive towers of algebraic function fields (of one variable) over the finite fields with square cardinality are studied [3, 5, 6]. In this paper, we define the limit $$ \lim_{i \to \infty} \text{(the number of places of degree $n$ in $F_{i}/\mathbf{F}_{q}$)/(genus of $F_{i}$)} $$ of a tower $F_{0} \subseteq F_{1} \subseteq F_{2} \subseteq \cdots$ over the finite field $\F_{q}$. Using this limit, we prove that all the proper constant field extensions of all the optimal towers over the finite fields with square cardinality are not optimal, and we show a simple criterion whether a tower is optimal or not. Moreover, we give many new recursive towers of finite ramification type.

In this work a precise condition for the singularity of a circuit distance power $C_n^{(d)}$ is derived. Namely, either $n$ and $d$ are not relatively prime or the order of 2 in $d+1$ is strictly smaller than in $n$. It is also shown that the simple eigenvalues of circuit distance powers are contained in $\{-2,0,2d\}$, generalizing a well-known result for circuits. Further, the nullity of $C_n^{(d)}$ is calculated.

The Turaev-Viro invariants are topological invariants of closed 3-manifolds. In this paper, we give a formula of the Turaev-Viro invariants of all orientable closed Seifert fibered manifolds. Our formula is based on a new construction of special spines of all orientable closed Seifert fibered manifolds and the``gluing lemma'' of topological quantum field theory. By using our formula, we get sufficient conditions of coincidence of the Turaev-Viro invariants of orientable closed Seifert fibered manifolds.

The aim of this paper is to extend the domain of polygamma function from the set of complex numbers to the set of bicomplex numbers. We also discuss integral representation, recurrence relation, multiplication formula and reflection formula for this function.

For any given integers $d_0, d_1 \ge 1$, let $\mathcal{F}$ be the family of polynomial maps $f$ such that $f$ has a fixed point at the origin, and moreover has just two distinct critical points 1 and $c_f\neq 1$ of multiplicies $d_0$ and $d_1$, respectively. For the central hyperbolic component $\mathcal{H}$ of $\mathcal{F}$, a monodromy map on $\mathcal{H}$ is obtained by Branner-Hubbard deformations. We show that for any given $\lambda$ with $0 < |\lambda| < 1$ and for any given integer $n \ge 1$, the monodromy map transitively acts on the family of all polynomial maps $f\in \mathcal{H}$ with $f^{\prime}(0) = \lambda$ and $f^{\circ n}(c_f) = 1$.

We shall consider IBVP to a nonlinear equation of suspended string with uniform density to which a nonlinear time-independent outer force works. We shall show the existence of time-global weak solutions of IBVP. To prove our result we shall use the function spaces defined by [Ya1], and apply the method due to Sattinger [Sat] based on the potential-well and the Galerkin method.