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## Hopf Fibrations and their Quaternion Interpretations I

Before I begin discussing the Hopf fibration of the 3-spbhere, one of the simplest yet deeply profound example of a non-trivial fiber bundle, I’d like to recall the definition of a fiber bundle.

Let $E,B,F$ represent the entire space, base space and the fiber respectively where $E,B$ are connected. If $f:E \mapsto B$ is a continuous surjection onto the base space, then the structure $(E,B,F,f)$ is said to be a fiber bundle if for every $x\in E$, there exists a neighborhood $U \subset B$ of $f(x)$ such that there exists a homeomorphism $\psi:f^{-1}(U) \mapsto U \times F$ such that $\psi \circ \pi_{U}=f$.

What this basically means is that locally, the fiber bundle looks like the product $B \times F$ but globally, it may have different topological properties.

A trivial fiber bundle is a fiber bundle which in which the total space is $E=B \times F$. In fact, any fiber bundle over a contractible CW Complex is trivial.

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## CW Complex Structure of Real Projective Space and Kusner’s Parametrization of Boy’s surface

The real projective space represented by $\mathbb{RP}^{n}$ is the space obtained from $\mathbb{R}^{n+1}$ under the equivalence relation $x \sim kx$ $\forall x \in \mathbb{R}^{n+1}$. Basically, $\mathbb{RP}^{n}$ is the set of lines which passed through the origin in $\mathbb{R}^{n+1}$. It can also be understood by identifying antipodal points(points which lie on opposite ends of the diameter) of a unit $n$-sphere,$S^{n}$.

One very basic yet deeply interesting example of these spaces is $\mathbb{RP}^{2}$, known as the real projective plane. While $\mathbb{RP}^{0}$ is a points and $\mathbb{RP}^{1}$ is homeomorphic to a circle with infinity, the real projective plane turns out to be far more interesting indeed. It can’t be embedded in $\mathbb{R}^{3}$ and its immersion/s such as Boy’s surface, Roman surface and Cross Cap have far stranger structures than a mere circle as in the case of $\mathbb{RP}^{2}$. In fact, I will delineate some of the calculations involved in the Kusner-Bryant 3-dimensional parametrization of Boy’s surface. It’s a little fascinating how much complexity can be added to mathematical structures upon generalization especially in the case of the projective space which I believe have a remarkably simple and ‘non-threatening’ definition.

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# Nets/Moore-Smith Sequences

Sequences are common objects in the field of topology. Often, sequences can help identify continuous functions, limit points and compact spaces in metric spaces.

Moore-Smith sequences(or nets) are essentially a generalization of the sequence for an arbitrary topological space and we can see that many foundational theorems of general topology can be stated in terms of nets.

So first, let’s recall what a partial order and direct set is:

A partial order is an order realtion $\leq$ satisfying:-

• $\alpha \leq \alpha$
• If $\alpha \leq \beta$ and $\beta \leq \alpha$, then $\alpha=\beta$.
• If $\alpha \leq \beta,\beta \leq \gamma$, then $\alpha \leq \gamma$.

A directed set $I$ is a set with a partial order relation $\leq$ such that for any $\alpha,\beta \in I$,there exists $\gamma$ such that $\alpha \leq \gamma,\beta \leq \gamma$.

It’s best to think of a directed set as a sort of analogue to an indexing set.

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## Connection between a symmetric linear transformation and the unit sphere

There is an interesting correspondence among the quadratic form of a symmetric linear transformation $T:V \mapsto V$on a real Euclidean space,the extreme values of the sphere and the eigenvectors of $T$

Let $Q(x)=(T(x),x)$ be the quadratic form associated with a symmetric linar transformation which maps $V$ into itself,then the set of elements $u$ in V satisfying $\langle u,u \rangle=1$ is called the unit sphere of $V$

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## The irrationality of e

This is the post excerpt.

The irrationality of e can be proved using the infinite series expansion.

$e=\sum_{k=0}^{\infty} \frac{1}{k!}$

Assume that e is rational.i.e $e=\frac{p}{q}$ where $p \in \mathbb{Z}$ and $q \in \mathbb{Z}-{0}$.

## Using Legendre Polynomials in irrationality proofs and establishing irrationality measures

The Legendre polynomials are a set of polynomial solutions to the Legendre’s Differential equation:

$(1-x^{2})y^{''}-2xy^{'}+n(n+1)y=0$

The edifice of most irrationality arguments use the fact the repeated integration by parts can be used on generating functions involving the polynomial.Another useful property is that these polynomials have integer coefficients.

Now,I’ll present a few important examples which display this property.

## Inequalities related to the Gamma Function and related functions

There are numerous inequalities related to the classical gamma function($\Gamma (x)$) ,incomplete gamma function($\Gamma (a;x)$) and the psi function(also called the di-gamma function which is simply the derivative of the logarithm of the gamma function).

One of the simplest bounds for the function $\Gamma (1+a)$ where $0 \leq a \leq 1$ is:

$2^{a-1} \leq \Gamma (1+a) \leq 1$.

A stronger inequality can be given by

$e^{(x-1)(1- \gamma)} \leq \Gamma(1+a) \leq 1$ where $\gamma$ is the Euler-Mascheroni constant.

(1)

with equality on the lower bound iff $x=1$