A Perfect World.

Lev, one of the most gifted graduates of The Academy of Unlimited Arts, with personal interests in Perpetual Omnipotence, decided to focus on universe creation in his postgraduate pursuits. The world he planned to make this time was meant to be unlike any other - a dynamic cosmos full of sentient and intelligent beings. The idea was to make it a perfect world; no pain, no sorrow, no longing, no broken hearts. No death. In short - ideal.

After all the world-making only one thing remained to be done - finding a fitting name for this wonderful creation. Lev was happy and content having finally settled on an appropriately adequate one - a perfect name. Upon completion of his cosmic project, Lev attached a plaque to its Luminiferous Aether, and on it an engraving in a particularly baroque font "A place with no dreams".

Beginning of the End

Elijah had been seeing Barbara for almost a month by then. The first few times they had met for coffee, drawn towards each other with a youthful curiosity and suppressed desire, days passed like hours. They relished in this new unexplored space of companionship. Immersed in philosophical conversations about life, exchanging views about what seemed most important, they managed, it seemed, to create their private realm outside of time which did not conform to the usual laws governing the universe. Before Elijah could realize, their quite lengthy acquaintance, for they had by then known each other for over a year, within no more than a few weeks evolved to a state which was an enchanted and exciting platform full of endless possibilities. Both quite shy, and careful not to spoil what grew out of what seemed like hours in their personal timeless world, enjoyed this curious state full of promise yet free from promises. Aside from Barbara’s cheerful, witty and energetic yet controlled demeanor which captivated with its reliably suitable degree of spontaneity, there was one other particular quality which Elijah admired. Unlike anyone he had met before, Barbara wholeheartedly engaged herself into any issue at hand, and with a particular and genuinely considerate approach when immersed in conversation with a colleague, or a friend. "She makes me feel present" was, Elijah felt the best characterization of her caring, engaging and non-selfish nature.
Barbara had a quintessentially feminine and beautiful Rubenesque figure, which sadly did not reflect her desired self image. This bitter dissonance between love expressed and actual self-love, was to become a blemish which would not only encroach into the blissful realm the two friends had conjured, but eventually mercilessly annihilate it.
The first cracks appeared during one of their many outings through the city’s major bookstores. Whilst strolling through the forest of towering bookshelves inside the city’s largest ‘used books’ store, and admiring the beautiful publications of Shakespeare’s works from the beginning of the 20th century, Elijah captured by an almost juvenile spontaneity with a trace of romantic intention, which he wouldn't deny, turned to Barbara and whispered among the colonnade of world’s major literary works: “Close our eyes, and let me lead you into a place within this forest where you will open them again only after you have blindly picked some book – one which you’ll promise to read”. Barbara agreed promptly – and soon, they were silently dancing through this symbolic storage room of millions of stories, fictional and actual lives, hopes and dreams. Upon stopping among some particularly tall bookshelves Elijah delicately twirled Barbara twice and lead her carefully as she blindly reached out for a book - this mysterious book, which they both anticipated to contain a magical spell which would embody everything that led to its emergence from the dormant and dusty forest. It would also bring a promise, a guide, or clue on this treasure hunt for true love. Barbara hesitated a little before allowing her fingers tentatively feel the spines of the books at shoulder level. She pulled a dusty book out, and after what appeared as having glanced at the title she silently turned with a jerk of the neck to Elijah who, not having seen the title himself only puzzled in horror at Barbara’s cold, narrowed eyes piercing him with a deadly and stern gaze out of the frame of a stone cold and fierce grimace which momentarily disfigured her face. She pushed the book into his chest, said “very funny” and marched off with an impatient glance at her watch. Elijah stood there in utter bewilderment at what had just happened, and his eyes widened as he read “Low fat cooking” - a title screaming at him with its colorful and boldface font from the dusty cover of the book.

Chance's Revenge

Joseph was known to occasionally bewilder random strangers, mostly those unfortunate enough to travel on the same train as him, by unexpectedly turning to them and reciting some lengthy and arbitrary Latin sentence. This exercise in buffoonery was mostly intended to evoke a laugh from his friends who shared his appreciation for the hilarity of the kaleidoscopic mixture of puzzlement, amusement and horror - all inevitably provided by his victim's faces.

Yesterday morning Joseph, up to his usual mischief did just that - while conversing with his typically extroverted group he suddenly turned around without warning and uttered a preselected Latin phrase to the young woman sitting behind him - "You are the most beautiful girl I have ever seen in my not so quite short life". This time something odd happened however. Whilst uttering this provocative sentence he slowed down half way through and what began as a nonchalant and lively baritone quietened to a quivering whisper, for at that very moment Joseph realized that what he was saying was true. The jest backfired, and Chance took her revenge on him.

N>1 roots of primes are irrational

The question came up in a conversation recently, and although it seemed intuitively true, I couldn't find a proof online. Many of you reading this are probably familar with the proof for the irrationality of √2. Well, this is a more general version for all n integer roots of any prime.

Theorem

Proof

Euclid's first theorem states that for a, b ∈ Z and any prime p:

p|ab  →   p|a  or  p|b 

An immediate corollary of it is, for a, 1 ≤ n ∈ Z and any prime p:

p|aⁿ   →   p|a

Since

p|aⁿ = aa^n-1    →  p|a  or  p|a^n-1

but,

p|a^n-1 = aa^n-2   →   p|a  or  p|a^n-2

:

p|a^n-(n-2) = a² = aa   →   p|a  or  p|a

→  p|a  or  p|a  or  p|a  or... p|a  ( n times )   →   p|a

* * *

Now suppose for contradiction that

∃a,b,2≤n∈ Z, p∈P (p^1/n = a / b), where a nad b have no common factor.

p^1/n = a / b   →   p =  aⁿ / bⁿ 

→   pbⁿ =  aⁿ  →   p|aⁿ   by def. of divides, since bⁿ∈ Z

→  p|a by Corollary to Euclid's first theorem.

Hence a = pk ,  k∈ Z by def. of p|a.        (1)

But  p^1/n = a / b   →   a = p^1/nb = pk   →   pbⁿ = pⁿkⁿ 

→ bⁿ = p( p^n-2kⁿ )  →   p|bⁿ   by def. of divides, since p^n-2kⁿ∈ Z

→ p|b by Corollary to Euclid's first theorem.

Hence b = pm ,  m∈ Z by def. of p|b.     (2)

Statements (1) and (2) jointly contradict the statement that "a and b have no common factor".

Hence ∀a,b,2≤n∈ Z, p∈P (p^1/n ≠ a / b)  q.e.d.

UNSAFE PROFANATION

An artist doesn't need a lecture on the value of art or the creative process in general. It is a blessing of the muses, a charitable gesture of serendipity for which all artists yearn like junkies. Why is that? Because both the creative process and inspiration are what an artist's soul feeds on and is defined by, which makes them sacred. They're sacred gifts whose value largely springs from their ephemeral and inconsistent nature due to the muses' capriciousness.

Therefore it seems natural that to a genuine apprentice of the muses the acquisition of a respect for the value of this gift and the recognition of its instantiations in the form of "perceivable artworks" is inevitable. It follows that the role of an art school should be both to aid the apprentice of the muses to evoke the longing for this gift, and nurture a respect for its instantiations.

It has come to my attention that during a recent social function at the Queensland College of Art (Griffith University), students have been physically tampering with an unfinished sculpture of one of their absent colleagues. This included climbing on the sculpture en masse, and ultimately soiling it with sand and what appears to be alcohol. Whereas such behavior should be generally labeled as disrespectful or thoughtless in the least, within the artistic community it is nothing short of a profanation.

One may be tempted to endorse a more sympathetic stance, and claim that the students deserve a measure of consideration due to their sheer ignorance. As a proof of their naivety serves the fact that by publishing their photos of the function's activities on Facebook, they did not care to exclude those of the sculpture being tampered with. To the contrary, judging by the number of photographs depicting the activity in question, it seems that it was one of the evening's highlights. Imagine the fury of young "Phidias" upon discovering that the profanation of the progeny of his inspiration served as a vehicle of base amusement to a herd of intoxicated morons.

Pleading ignorance, may get the students off the hook, but the story does not end there. A senior  lecturer and a PhD student were present at the function, as evidenced by the published photographs, which makes them passive participants. What is one to say of the ignorance of those senior figures of an artistic and academic institution? Why didn't they react? Did they succumb to a momentary primitive group think? Were they so overwhelmed by the libations as to be completely oblivious of the surrounding goings on? Or maybe their formal statuses are merely empty labels masking equally ignorant individuals? Surely all those alternatives seem equally unthinkable!

The senior academics are denied a plea of ignorance. If that was granted, what would that imply about the school (QCA) which they represent? After all it would be dreadful to permit the thought that the "Art" in QCA is also just an empty label. So let's give the school the benefit of the doubt and suppose there does exist a genuine spirit of art within its walls. But that would seem inconsistent with the described incident!

What went wrong? Consider this. I'm quite sure that once the story surfaces, the highest ranking bureaucrats of Griffith University Inc. who are more concerned with avoiding liability suits, and making a profit then actually focusing on educating the youths would be appalled that such misconduct had occurred: "damaging a piece of art without taking the appropriate safety measures? The photos clearly depict students climbing the sculpture without adequate headgear and goggles! This is an outrage!".

A lesson from Bach

This anecdote has been attributed to the biography of young Johann Sebastian Bach.
One evening, Johann was playing his latest sonata to a German baroness at her estate. When he had finished, the baroness exclaimed with a bewildered delight: “Ah! Maestro! It was wonderful!… But what did you mean by it?" Bach promptly approached the keyboard again, and replayed the sonata. When he had finished, he turned to the baroness and explained: “That, dear madam, is what I meant.”

Tangibility of the transcendental

I was doing a bit of reading about transcendental numbers today, and had an idea of constructing my own well defined transcendental. I was inspired by the Liouville constant.

Anyway, my idea was to construct a number which is a base 10 concatonation of the digits of consecutive prime numbers:
m_p =  0.2357911131719232931...
or the natural numbers:
m_N = 0.12345678910111213...
The construction proceeded as follows. First we define the function lambda:

Now we can express the numbers as infinite sums:

Where p_n is the n'th prime.

Initially I thought this to be an exercise in creative procrastination, since firstly the idea is dead simple, and secondly who would be silly enough in going to the trouble of formulating such numbers. Surely only a fresh graduate with too much time on his hands.

To my surprise the idea behind those numbers has been already formulated by not one, but a few mathematicians. My m_N is in fact a constant previously formulated by D. G. Champernowne. And it turns out that m_p is already known as the
Copeland-Erdos constant:

Notice the striking similarity in the formulation - I have formulated m_p completly oblivious to the Copeland-Edros formulation. The similarity lies in the fact that my lambda function acts a lot like the floor of base 10 logarithm of y, plus 1. In fact the relation is the following:

Which makes the sums in the exponents of the definitions of m_p and the Copeland-Edros constant equal, since for each i we subtract 1 i.e. we subtract 1, k times, we need to add k back to the sum.

Well, I guess I feel better knowing that I'm in good company deeming quite obscure and prima facie pointless ideas valuable. I knew that both m_N and m_p are irrational, but now I also know that m_N is in fact transcendental (mission accomplished!), and hence don't have to provide a proof, which i feel would be a herculean task.