Infinite #WinterThoughts

What is time?

tumblr_ojgi5s19hs1qbdfz5o1_540

 

Only propositions have sense; only in the nexus of a proposition does a name have meaning.

~ Ludwig Wittgenstein, Tractatus Logico-Philosophicus (1921), 3.3

We live surrounded by symbols. In this city, where you and I dream, love, walk and invent new causes to believe, infinity lives through their immortality.

The ghosts have names, some secrets, as yet unrevealed. They have left for us so many traces of their own dreams: Viktoria Hill, the Iron Cross, the Blue Angel, abandoned airfields, hideous ruins, and for each one we can discover them, silent, ever so present, braving the flow of time, as ice covering the Spree.

The lakes are now frozen, the air carries the scents of wood and coal fires, perhaps the lingering sounds of ancient wars. So, you and I, my love, we walk with the Dead, from time to time, listening to their calm voices, evoking infinity.

Picture: The season of fallen leaves. © 2017 Irina Urumova

 

Zeta #AtoZAprilChallenge

image001

 

The Zeta function ζ(s) is defined as above, for example :

ζ(2) = 1/12 + 1/22 + 1/32 … = π2 / 6.

This converges for s>1 (i.e. tends towards a limit), but diverges for anything else. The Riemann Zeta function extends this range to allow us to compute the result for any complex number. The Riemann Hypothesis claims that the ‘zeros’ or ‘roots’ of this extended function, i.e. solutions for which ζ(s) = 0, have the form 1/2 + ai, i.e. where the real part of the complex number is always 1/2 (as well as certain ‘trivial’ roots which don’t have this form).

Unsolved! Proposed by Bernhard Riemann in 1859. This is one of the 7 ‘Millennium Prize’ problems, for which there is a $1m reward.

Get cracking!

There’s a number of consequences if this is true, but perhaps the most important is that it reveals the distribution of the prime numbers. The Prime Number Theorem allowed us to estimate the number of primes up to a given number. If armed with all the roots of the Riemann Zeta function, then we can work out the exact number!

Fight #TheDailyPost

Write a new post in response to today’s one-word prompt.

 

tumblr_m1s0pb2rvd1qb30dwo1_500

 

It took me a long time, to understand who you were, and how it was that you came to us now, reminding me of a nearly all forgotten past. Sarah and you are one, even if herself did not see that when you first came in our lives. It is a complicated tale for us humans to fully apprehend – and yes, I know you are as human as us, only ahead of us, the being we will one day become. Sarah’s happy, for me and for herself. We are reconciled with you, Melissa, and I am reconciled with my lost youth.

Of course I cannot follow all the mathematics, and even less the physics, although Gabrielle’s spent a fair time explaining the transforms to us. Sarah is a much better mathematician, and she does understand quantum physics far more than I do. You and her had a good time discussing the reasons for Lagrangian logic, or we would say, mechanics. Old Newton must be turning round in his grave…

As you recall I am an incorrigible romantic: watching the two of you, in Gabrielle’s old house, laughing and juggling with those exquisite slides, I kept dreaming. How similar you two are, and how beautiful. Gabrielle said I had nothing to fear: neither she nor you are pretending to be extraordinary, merely living at a level of complexity slightly away from us, but still it leaves us plenty of space and time to enjoy ourselves, with you. Sarah has bought into the idea that I am now able to visit you, Melissa, in Gabrielle’s world, and that does not involve any risk to my body. Still it is a little difficult for me to accept that simple reality: what travels are quantum of information, to use our archaic description, and this avoids the quantum electrodynamics limits of old very gracefully. So, for now, I have given up deciphering the equations, I just enjoy listening to you, the sound of your voice, the warmth you and Sarah have brought to my life. As a writer I am very privileged.

But will I be able to tell our story? That is without betraying the sweet secret: Melissa is immortal.

Originally posted as “What I see”

#AtoZChallenge2015: Orthogonal

Orthogonal projectionThe word orthogonal conjures up memories of wonderful maths classes, many years ago, as well as more obscure readings, much later, of absconse topological subjects. Geometry was one of the great pleasures of my youth: yes, we are all different!

I quote from: Barile, Margherita. “Orthogonal.” From MathWorld–A Wolfram Web Resource, created by Eric W. Weissteinhttp://mathworld.wolfram.com/Orthogonal.html

“In elementary geometry, orthogonal is the same as perpendicular. Two lines or curves are orthogonal if they are perpendicular at their point of intersection. Two vectors v and w of the real plane R^2 or the real space R^3 are orthogonal iff their dot product v·w=0. This condition has been exploited to define orthogonality in the more abstract context of the n-dimensional real space R^n.

More generally, two elements v and w of an inner product space E are called orthogonal if the inner product of v and w is 0. Two subspaces V and W of E are called orthogonal if every element of V is orthogonal to every element of W. The same definitions can be applied to any symmetric or differential k-form and to any Hermitian form.”

For those of my readers so inclined, Wikipedia has an interesting article on matrices!

“In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e.

Q^\mathrm{T} Q = Q Q^\mathrm{T} = I,

where I is the identity matrix.

This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:

Q^\mathrm{T}=Q^{-1}, \,

An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q*) and therefore normal (Q*Q = QQ*) in the reals. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. In other words, it is a unitary transformation.

The set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.

The complex analogue of an orthogonal matrix is a unitary matrix.”

Image: orthogonal projection at http://english.rejbrand.se/algosim/visualisation.asp?id=orthogonal_projection

#AToZAprilChallenge: Genius

Salvatore Bruno Wikipedia: “A genius is a person who displays exceptional intellectual ability, creativity, or originality, typically to a degree that is associated with the achievement of an unprecedented leap of insight. This may refer to a particular aspect of an individual, or the individual in his or her entirety; to a scholar in many subjects (e.g. Gottfried Wilhelm Leibniz or Leonardo da Vinci)[1] or a scholar in a single subject (e.g., Albert Einstein or Charles Darwin). There is no scientifically precise definition of genius, and the question of whether the notion itself has any real meaning has long been a subject of debate.”

Genius came into English from C14, in its main Latin sense  – from genius – a guardian spirit. It was extended to mean ‘a characteristic disposition or quality’ from C16, as still in ‘every man has his genius’ (Johnson, 1780), and ‘barbarous and violent genius of the age’ (Hume, 1754). The development towards the dominate modern meaning of ‘extraordinary ability’ is complex; it occurred, interactively, in both English and French, and later German. It seems to have been originally connected with the idea of ‘spirit’ through the notion of ‘inspiration’… This sense is always close to the developing sense of Creative… A good test case is ‘the English genius for compromise’. (Keywords)

#AtoZAprilChallenge: Fiction

You can't stop me “Fiction has the interesting double sense of a kind of Imaginative Literature and of pure (sometimes deliberately deceptive) invention… A general use, ranging between a consciously formed hypothesis (‘mathematical fictions’ 1579) and an artificial  and questionable assumption (‘of his own fiction’), was equally common. Fictitious, from C17, ranged from this to the sense of deceptive invention; the literary use required the later fictional. The popularity of novels led to a curious C20 back-formation, in library and book-trade use, in non-fiction (at time made equivalent to ‘serious’ reading…)

Novel, now so nearly synonymous with fiction, has its own interesting history. The two senses now indicated by the noun (prose fiction) and the adjective (new, innovating, whence novelty) represent different branches of development from Latin novus – new. Until C18 novel, as a noun, carried both senses: (i) a tale; (ii) what we now call, with the same sense, news…

(In) Fielding: – ‘What novel’s this? – Faith! It may be a pleasant one to you.’

It was from this range of senses that novelist meant successively any kind of innovator (C17), a newsmonger (C18) and a writer of prose fiction.” (Keywords)

#WritersWednesday: His Hero is Marcel

Time Line of the Universe

Time Line of the Universe Credit: NASA/WMAP Science Team Source: Original version: File:CMB Timeline75.jpg

It goes for colours, type-faces, places, objects, smiles, books… The human spirit is attracted, inspired, by “things”, in a fashion that appears random to the observer (“tastes and colours…” goes the French saying). But it isn’t. There are reasons for everything, and randomness is often a metaphor for “we can’t explain this”.

Julian is attracted by – universes. Worlds, galaxies, star systems… Or should I write “multiverses”: the existence of multiple universes that rarely intersect, merely coexist, and, mostly, in ignorance of each other? He knows, has read about, that most physicists, mathematicians, philosophers, are generally skeptical about the concept. Generally, but sometimes not. And Julian is attracted by those writers who are less than skeptical, the party of the “cosmic inflation”, and its far away consequences. Julian believes in the Two Moons of Huraki Murakami: he too has seen them…

Sarah, who’s a far better mathematician than her husband, is willing to discuss strings theory and other quantum wonders, and let him indulge in his quest. He too is after the “Ultimate Nature of Reality” [*]. I do understand, and she does, that Julian seeks his inspiration from serious subjects: history, science, philosophy, the “thinking” authors of weird and wonderful stories.

So it goes for time: our Julian is obsessed by it. His hero is, of course, Marcel Proust, and he’s often written about Marcel, and written him into his stories, as himself or as his little prisoner. I am fascinated by this, as it links to his other obsessions, his writing style, and, finally, his love for both Sarah and Melissa, the two women in his life, the inspiration for his writing. There are reasons to believe that, for Julian, his friend Melissa is a reincarnation of the docile Prisoner, dear to Marcel, his Albertine…

But Sarah has another theory: Julian wishes to be Albertine, someone’s property, or, to be precise, his wife’s. So that Melissa maybe Julian, in the end, just in another “universe”. This intrigues me too, as often Melissa has told me she wished to be Julian, to live in his skin. Poor soul. What I keep to myself, for now, is that Melissa has also claimed to be Sarah, to “merge” with her.

Sarah, Albertine, Odette, Julian, Melissa, Swann? Julian is “à la recherche”, in this universe, or, as necessary, in another. Which writer is not?

[*] “Our Mathematical Universe: My Quest for the Ultimate Nature of Reality”, by Max Tegmark, was reviewed by Brian Rotman in The Guardian of February 1, 2014.

#AtoZChallenge: April 5, 2013 ~ Euler and “e”

Euler and e

Leonhard Euler Leonhard Euler is a towering  figure of Mathematics and Physics in the 18th century, and one of the greatest mathematicians of all times.  Born in 15 April 1707 in Basel (Schweiz, Switzerland) Euler’s legacy includes “e” the Euler number, with pi one of the fundamental constants of mathematics, and volumes on infinitesimal calculus, geometry, algebra and number theory. Euler lived in Saint-Petersburg, where he died on 18 September 1783, and in Berlin.  Students of mathematics the world over owe him the Euler’s Identity:

euler described as “the most beautiful mathematical formula ever”.