My first ever riding of the LaTeX bicycle @TeXtip #Theorem #GroupTheory
During a recession period, when you have almost nothing to do throughout the working days, it could be a nice idea to fill your spare time with science.
In pure mathematics and theoretical physics, in particular, all you need is a good Internet connection and some original resources to learn from them. A Nobel prize winner, Gerard ‘t Hooft, put it this way: "…the costs of becoming a theoretical physicist should not exceed much the price of a computer with internet connection, a printer, and lots of paper and pens."

I well remember a friend of mine in the math department of Sharif University of Technology back in 1990s who is now in MSU used to write his homework and  research papers via LaTex but then I was more focused on humanities and didn’t need to learn this coding environment.

Given the fact that I am a first time user I was wondering that if someone could help me to locate the proper file for installation and also the basic help stuff for a beginner. Very soon, however, I learned that LaTeX is an open and free coding language. And that there is a jungle of downloadable files here and there!

Yesterday I checked out one of them and tried my best to see if I can install a workable option. I just wanted to write a document with some math and equations.
Apparently this software is based on a sort of modular package. You download and install the basic version and the software will get updated via Internet. Also, found this resource which is indeed a helpful guide for beginners like me.

As a first exercise today morning I coded via LaTeX a proof of Maschke’s Theorem which is an elegant result in modular representation of finite groups. Learning to do LaTeX was very much like my first time of riding a bicycle. It took a few hours of painful frustrations but the outcome is quite enjoyable and stylish.

Below is the code and the picture is the output for those interested.

\documentclass[12pt,a4paper]{article}

\usepackage{amsmath}

\usepackage{amsfonts}

\begin{document}

\title{Proof of Maschke Theorem}

\author {@VictorScenario First Exercise in \LaTeX{}}

\maketitle

Let G be a finite group, $F=\mathbb{R}$ or $F=\mathbb{C}$, and let V be an FG-module. If U is an FG-submodule of V, then there is an FG-submodule W of V such that

$V= U \oplus W.$

\textit{Proof:}

Choose any subspace $W_0$ of V such that $V=U \oplus W_0$.

Take a basis $(v_1, \dots , v_m)$ of U, extend it to a basis $(v_1, \dots, v_n)$ of V, and let $$W_0=sp(v_{m+1}, \dots ,v_n)$$.

For $v \in V$, we have $v=u+w$ for unique vectors $u \in U$ and $w \in W_0$, we define $\phi : V \to V$ by setting $v\phi=u$.

Clearly $\phi$ is a projection of V with kernel $W_0$ and image U. Define $\vartheta:V \to V$ by

$v\vartheta=\frac {1}{|G|} \sum_{g \in G} vg \phi g^{-1} \quad (v \in V).$

Note that $\vartheta$ is an endomorphism of V and $Im \vartheta \subseteq U$. For $v \in V$ and $x \in G$,

$(vx) \vartheta=\frac{1}{|G|}\sum_{g \in G}(vx)g \phi g^{-1}.$

Suppose that $h=xg$. Thus

$(vx) \vartheta=\frac {1}{|G|}\sum_{h \in G}vh \phi h^{-1}x =\left(\frac {1}{|G|}\sum_{h \in G} vh \phi h^{-1}\right) x =(v \vartheta)x.$

Hence $\vartheta$ is an FG-homomorphism. For $u \in U, g \in G \Rightarrow ug \in U \Rightarrow (ug) \phi = ug$. Therefore

$u \vartheta=\frac {1}{|G|} \sum_{ g \in G} ug \phi g^{-1}=\frac {1}{|G|} \sum_{g \in G} (ug)g^{-1}=\frac {1}{|G|} \sum_{g \in G} u=u.$

Let $v \in V$. Then $v \vartheta \in U \Rightarrow (v \vartheta) \vartheta = v \vartheta \Rightarrow \vartheta^{2}=\vartheta$. Hence $\vartheta$ is a projection.

Also $Im \vartheta = U$. Now if we let $W=Ker \vartheta$ $\Rightarrow$ W is an FG-submodule of V and $V=U \oplus W$. \

\textit{Source of the Proof: Gordon James, Martin Liebeck, (2001), Representations and Characters of Groups, Second Edition, Cambridge University Press.}

\end{document}

Benefit of an overcrowded planet for the future of science

In science 2.0 my favorite is crowdsourcing astronomy. Imagine that you have 1 billion of people helping scientists to explore and make sense of the vastness of the cosmos.

This gives you a why and justification about should we encourage even more Earth population, that is approaching 10 billion and beyond it.

Clearly more population means that there will be more working brains to pave the way for the next breakthrough in science given the fact that there are billions of stars and galaxies over there to be dealt with.

Every person assigned to one star or one galaxy under the supervision of a great star scientist promises some unimagined revolutions.

The power of anticipation or self-fulfilling prophecy? #ISIS #Iraq #USA #Sunni #Shia #Caliphate

December 2004

WASHINGTON DC—-“The tenuous peace inside Iraq that America had stitched together so laboriously came undone with the sudden re-igniting of the Sunni insurgency; the insurgents proclaimed themselves the true Caliphate and battled anew both Shia and the American garrisons.”

Source:

Pages 83-92, Mapping the Global Future, Report of the National Intelligence Council’s 2020 Project

June 2014

BAGHDAD—“The Sunni Islamist militant group whose three-week blitz through northern Iraq has nearly upended the country’s fragile unity announced itself as a new Islamist “caliphate” on Sunday, unilaterally declaring statehood and demanding allegiance from other Islamist groups.”

Source:

The Wall Street Journal

The #science of what transcends #death
Our world, as been shown by modern science, is at a fundamental level based on quantum mechanics.
Earlier I started a technical discussion in the physics forum about the relationship between observable facts and their existence. As it turns out there is still a live and ongoing debate among scientists on how to interpret the reality or existence of quantum states which are essential tools or mathematical objects for calculations in quantum mechanics.
Nonetheless, in science, existence is tightly, reciprocally, and closely related to the observable outcome of measurements. That is to say what cannot be observed for science does not exist. It is not anything at all, so to speak. Science can only approve or disprove characteristics of outcomes of what is already observable, of something.
To be observable is the limit of science. So you cannot disprove what is by definition not observable, not a thing, except perhaps in fantasy tales, using the scientific method.
Soul, spirit, or whatever you like to call what transcends death all fall beyond that limit or boundary and I would say that a scientist should better take “an agnostic position” with respect to such notions that emerge from human creative intellect.
The question of change recognition in the future
Most futurists, including me, are enthusiasts about change and further evolution. Some favor it radical, rapid and revolutionary while others would prefer a gradual, elaborate, and slow shift away from status quo. None of us can possibly question or cast doubt on the notion of change per se. Some people get the notion of the need to change but simply do not agree or accept its nature or associated process or are not committed to it.
In quantum mechanics, in a stationary state, simultaneous recognition of to change and being constant won’t pose any serious problem, change of state vector occurs in an abstract complex numbers space but the real observable facts of energy are and remain constant regardless.

But the lesson learned from quantum mechanics says that, on an ontological level,  perhaps even if we are dealing with the same issue, say the future of women in the Middle East, it might be the case that in a particular mindset the change itself is not recognized at all let alone to be accepted or agreed and acted upon.

Now the question is that if we are supposed not to recognize change, remove it from our vocabulary, at least for a limited yet sometimes powerful rich group of people, and observe the world within and around us as simply constant then what a futurist has to to offer or talk about?
For them our shared evolution means that we should conclude once and for all time that what we observe right now will be projected indefinitely into the future.
However, it appears to me that sometime through the evolution of our civilization we need to face a huge survival challenge, that is on an existential level, between those who recognize change and those who simply don’t. Because simultaneous existence of both could pose some serious problems to future of the planet Earth itself.
Pattern of simultaneously to change and be constant

In quantum mechanics the evolution of a system is a tricky notion, it could mean that the particle remains at the same energy state indefinitely into the future, but mathematically it’s not constant and therefore changing, but in fact it is constant and not changing at all!

http://en.wikipedia.org/wiki/Stationary_state

Isn’t the history of our civilization obeying the same pattern of simultaneously changing and constant:

1914 - 2014: One hundred years time lapse http://vimeo.com/96057108

Please don’t knock on the door we will know when you have arrived

Berlin artist Julian Oliver has written a simple script that finds and detects Google Glass on the local network and kick them off.

Tagged as `humor´ too, because I always lol at the word Glassholes :D

Deep #insight about hidden #dimensions of situations #math #management

When you walk on the ground you sense it to be flat. But if go up and see the ground from a large enough distance above, you will note that you were indeed walking on the surface of a sphere, on the globe of Earth. So as we call it in math a two-sphere is embedded in a three dimensional space. But how can we be sure that our 3-dimensional world might be itself a three-sphere embedded in a four dimensional universe?

The first time I did my exercise in mathematics of differentiable manifolds I felt absolutely thrilled because the equations were crystal clear, showing that our three dimensional space could be itself just a surface of a sphere, a three-sphere, embedded inside a four dimensional universe. The picture shown here in this post is my handwritten calculations. A one page note in which I for the first time learned how to induce a flat 4-dimensional metric on the three-sphere. In other words, how to pull back a flat metric in four dimensions onto the surface of points in four dimensions at unit distance from origin.

This deep insight inspired and helped me to accept it as an essential mental habit to keep a very open mind to accept the possibility of hidden dimensions to any problem or situation I face, those dimensions beyond my direct sense making and recognition, indicating that what I presume to know for sure could be just a surface of a much larger big picture.

Unlike math and physics in which you can demonstrate and convince yourself analytically of the possibility of extra dimensions without even experiencing them using just one page of paper and pencil, in other disciplines such as leadership and management it could be enormously difficult if not impossible to do just that. However, we need to know that sometimes perhaps we are wrong, we are just scratching the surface of a much larger set of factors and dimensions that are relevant to our choices, decision, and actions, yet hidden from our direct sense making and experience.