A fibering theorem for 3-manifolds

We generalize a result of Moon on the fibering of certain 3-manifolds over the circle. Our main theorem is the following: Let $M$ be a closed 3-manifold. Suppose that $G=\pi_1(M)$ contains a finitely generated group $U$ of infinite index in $G$ which contains a non-trivial subnormal subgroup $N\neq \mathbb{Z}$ of $G$, and suppose that $N$ has a composition series of length $n$ in which at least $n-1$ terms are finitely generated. Suppose that $N$ intersects nontrivially the fundamental groups of the splitting tori given by the Geometrization Theorem and that the intersections of $N$ with the fundamental groups of the geometric pieces are non-trivial and not isomorphic to $\mathbb{Z}$. Then, $M$ has a finite cover which is a bundle over $\mathbb{S}$ with fiber a compact surface $F$ such that $\pi_1(F)$ and $U$ are commensurable.

COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.

Representations induced from the Zelevinsky segment and discrete series in the half-integral case

Let {G_{n}} denote either the group {\mathrm{SO}(2n+1,F)} or {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form {\langle\Delta\rangle\rtimes\sigma}, where {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of {G_{n}}. We determine the composition series of {\langle\Delta\rangle\rtimes\sigma} in the case when {\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.

On the odd order composition factors of finite linear groups

In this paper, we study the product of orders of composition factors of odd order in a composition series of a finite linear group. First we generalize a result by Manz and Wolf about the order of solvable linear groups of odd order. Then we use this result to find bounds for the product of orders of composition factors of odd order in a composition series of a finite linear group.

Anion Distribution, Structural Distortion, and Symmetry-Driven Optical Band Gap Bowing of Mixed Halide Cs2SnX6 Vacancy Ordered Double Perovskites

Mixed anion compounds in the Fm-3m vacancy ordered perovskite structure were synthesised and characterised experimentally and computationally with a focus on compounds where A = Cs+. Pure anion Cs₂SnX₆ compounds were formed with X = Cl, Br and I using a room temperature solution phase method. Mixed anion compounds were formed as solid solutions of Cs₂SnCl₆ and Cs₂SnBr₆ and a second series from Cs₂SnBr₆ and Cs₂SnI₆. Single phase structures formed across the entirety of both composition series, with no evidence of long range anion ordering observed by diffraction. A distortion of the cubic A2BX6 structure was identified in which the spacing of the BX6 octahedra changes to accommodate the A site cation without reduction of overall symmetry. Optical band gap values varied with anion composition between 4.89 eV in Cs₂SnCl₆to 1.35 eV in Cs₂SnI₆, but proved highly non-linear with changes in composition. In mixed halide compounds it was found that lower energy optical transitions appeared that were not present in the pure halide compounds, and this could be attributed to lowering of the local symmetry within the tin halide octahedra. The electronic structure was characterised by photoemission spectroscopy, and Raman spectroscopy revealed vibrational modes in the mixed halide compounds that could be assigned to particular mixed halide octahedra. This analysis was used to determine the distribution of octahedra types in mixed anion compounds, which was found to be consistent with a near-random distribution of halide anions throughout the structure, although some deviations from random halide distribution were noted in mixed iodide-bromide compounds, where the larger iodide anions preferentially adopted trans configurations.