We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a result, direct sums of special linear Lie algebras $\sll(2) \oplus \sll(m_1) \oplus \cdots \oplus \sll(m_k)$ admit left invariant flat complex projective structures if the equality $4 + m_1^2 + \cdots + m_k^2 -k - 4 m_1 m_2 \cdots m_k = 0$ holds. These contain $\sll(2)$, $\sll(2) \oplus \sll(3)$, $\sll(2) \oplus \sll(3) \oplus \sll(11)$ for example.

For each integer $n$, an $n$-folding curve is obtained by folding $n$ times a strip of paper in two, possibly up or down, and unfolding it with right angles. Generalizing the usual notion of infinite folding curve, we define complete folding curves as the curves without endpoint which are unions of increasing sequences of $n$-folding curves for $n$ integer.

We prove that there exists a standard way to extend any complete folding curve into a covering of $R^2$ by disjoint such curves, which satisfies the local isomorphism property introduced to investigate aperiodic tiling systems. This covering contains at most six curves.

In this paper, we consider the second-order nonlinear neutral functional dynamic equation

$\left( p(t)\left( \left[ y(t)+r(t)y(\tau (t))\right] ^{\Delta }\right) ^{\gamma }\right) ^{\Delta }+f(t,y(\delta (t)))=0$

on a time scale $\mathbb{T}$ and establish some new sufficient conditions for oscillation. Our results improve oscillation results for neutral delay dynamic equations on time scales and are new when $\delta (t)>t$ and/or $% 0<\gamma <1.$ Furthermore our results can be applied on the time scales $% \mathbb{T=}h\mathbb{T}$, for $h>0$, $\mathbb{T=}q^{\mathbb{N}}=\{t:t=q^{k}$% \textbf{, }$k\in \mathbb{N}$, $q>1$, $\mathbb{T=N}^{2}=\{t^{2}:t\in \mathbb{N% }\},$ $\mathbb{T}_{2}\mathbb{=}\{\sqrt{n}:n\in \mathbb{N}_{0}\},$ $\mathbb{T}% _{3}\mathbb{=}\{\sqrt[3]{n}:n\in \mathbb{N}_{0}\},$ and when $\mathbb{T=T}% _{n}=\{t_{n}:n\in \mathbb{N}_{0}\}$ where $\{t_{n}\}$ is the set of harmonic numbers, etc.

We study the monoid generated by certain Zariski-van Kampen generators in the positive homogeneous presented fundamental group of the complement of the logarithmic free divisor, called the type $\mathrm{B_{ii}}$ in the list by Sekiguchi. Although the monoid is cancellative, it turns out that the monoid is not Gaussian and, hence, is neither Garside nor Artin. Nevertheless, we show that the monoid carries certain particular elements similar to the fundamental elements in Artin monoid. Hence, we can solve the word problem and the conjugacy problem in the monoid and determine the center of it and the explicit form of the growth function for it. As a result, we can also solve the word problem and the conjugacy problem in the fundamental group, and determine the center of it (Theorem 5.8).

We give a mathematical model of $n$-membered ringed hydrocarbon molecules, and study the topology of a configuration space $C_{n}$ of the model. Under the bond angle conditions required for ringed molecules, we prove that $C_{n}$ is homeomorphic to $(n-4)$-dimensional sphere $S^{n-4}$ when $n = 5, 6, 7$. This result gives an appropriate explanation of the configuration space of $n$-membered ringed hydrocarbon molecules when $n = 5, 6$.

The aim of this paper is to define a homology theory for racks with finite rank $N$ and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in S. Nelson, Link invariants from finite racks, arXiv:0808.0029. For this purpose, we prove that $N$-degenerate chains form a sub-complex of the classical complex defining rack homology. If a rack has rack rank $N=1$ then it is a quandle and our homology theory coincides with the CKJLS homology theory. Nontrivial cocycles are used to define invariants of knots and examples of calculations for classical knots with up to $8$ crossings and classical links with up to $7$ crossings are provided.