Newton on pi

Newton had proposed variations on a formula for
approximating pi. At pi/6, the sine function is at .5.

0! is 1

The desmos on-line shows the correct decimals with the last rounded; above
for 30 iterations.

Basel P.

The Basel Problem: sum of inverse squares on consecutive numbers.
This comes to (pi^2)/6, and this precise number was the work of Euler.
Basel was the home town of the Bernoullis, who first grappled with
the problem. Johann Bernoulli tutored Euler in mathematics.

This is quite interesting because it is precisely finding and proving this
that led Euler to first propose the Zeta function; and its infamous descendant
for us, the Rieman Zeta function on the complex plane.

One can see easily enough that for exponents 0 and 1, the series is infinite.
The problem with squares and higher is to find the convergence formula.

                                            source: Gerard Villemin

                                                             *     *     *
Went to Wikipedia to help me through a proof for the Basel problem. The
starred solution was in Russian! Below, in machine translation to English:

https://www.desmos.com/calculator/yu7h2cqcnl

SuperZ

Noteworthy

It was a more Liberals and more Greens EU election.

The UK Brexit Party's close ally from Italy - the Five Star mouvement - did
not do well. Will the UK  lead the Direct Democracy Group in the new EU !?😂

Congratulations to all for an interesting election!

                                               *     *     *

France's Animal Rights Party didn't elect anyone, but they got votes:

Superfluous

So this afternoon I got curious about what Riemann actually
said with respect to his famous hypothesis. Actually, it was
a curiously stupid statement: in all probability they (the zeroes)
are all real, but  after a few weak attemps, a fully rigorous
proof was not really necessary, because superfluous to his purpose...

Well, NOT. We currently have armies of mathematicians with computers
pumping out billions of Riemann zeroes, and hoping for a way out
of the nightmare. Reimann himself, on his death, left behind cases of
computational papers that his wife had burned; man was a fire
hazard and no computational slouch.

This is not the first occurence of the phenomenon. Fermat's last theorem
( a^n + b^n = c^n has no solution for n > 2), was found in the margin of a
notebook, with the note he just didn't have room for the proof; a ridiculous
assertion, as attested by his son. Gauss - not one to miss the party - left us with
this gem: "If others would but reflect on mathematical truths as deeply and 
continuously as I have, they would make my discoveries."

And as Abel said of Gauchy, (who defned the notions of limit and convergence):
"Cauchy is mad and there is nothing that can be done about him, although, right now, 
he is the only one who knows how mathematics should be done."

So there we have it, running smalltalk about mathematics from its greats. Andrew
Wiles ending up developing new mathematical tools based on modularity to prove
Fermat; but that won't work here. The issue seems to be time, not space.

Je dis ça, je n'ai rien dit...

https://www.mscs.dal.ca/~kgardner/History.html

Daily Math

The zeta function on real values for s; on Graph:

Below: complex zeta, with s = a + bi:

The critical line (showing cos at 1/2) itself moves, to accomodate the function!?

http://math-it.org/Mathematik/Riemann/RiemannApplet.html

                                                   *     *     *

Every turn through the origin is an infamous 'zero'. It doesn't happen

for all real values; indeed there seems to be only one...

https://www.geogebra.org/m/T47QECTP

Hey Hayders!

The Briggs-Myers test of personality is what people use to
introduce themselves on Tinder. I won't give anything away, but I share
a type with Newton, Sir Isaac.

https://www.betterhelp.com/advice/general/you-can-take-the-myers-briggs-personality-test-free-on-these-websites/

I am also including the cover picture from my account
with My Fitness Pal. Today marks 365 days of weight
management (aka no snacking, no junk) with sterling
results. Looking back, I did have that psycho-rigid look
one sees on Sir Isaac paintings at the beginning; I knew
'hangry' and 'hungbored' moments lay ahead.

I mention all this because I have been dismayed by the
distressing amount of scutiny public figures get on the net.
Even the New York Times - America's Grey Lady - went so
far this week as to headline that the UK was 'on the brink
of Boris Johnson' for PM  'and Chaos', while the Economist
announced the man was a huge gamble like that guzzler, Winston
Churchill, had been.

Give a guy an honest break: attending Balliol College, Oxford is not,
prima facie, a 'turn-off'. Balliol is the school which trains England's
politicians: he is the real deal, the house wine. And while on the subject,
I was surprised by how E. Macron navigated through the 'gilets jaunes'
crisis. He really is dead-center for French politics, 'En Marche' says it all.
The May-Junker imbroglio on Brexit happened slightly to the cautious
right of that. To watch: the results of the European elections, coming today!

Veils

Sometimes, math feels like a veil lifting...

https://youtu.be/AAir4vcxRPU

                                               *     *     *

Algebraic

The Euler identity in mathematics is easy enough
to use: on the polar complex plane, one multiplies
the radius of one, adds so many degrees - rads -
to the angle measure. But what does the expression
e^i*pi mean. Not always helpful is that the historical
progression of concepts - and proofs - can be confusing.

One thing to keep in mind is the particuliarity of i, (-1)^.5.
9^.5 = 3
9^-.5 = 1/3
9^i = 9^((-1)^.5)
The only way to escape i is by squaring it!!

Below, seeing how the series noted e can be viewed in terms
of sin and cos. Historically, the Taylor series came within Euler's lifetime
in 1710 but is based on calculus. Taylor adjudicated Newton at the
Royal Society!?

Having made it there, one more step gets one to e^(i*pi). The takeaway
on all this: the expression does have meaning but it is about e!! (The
Donald Trump of mathematical constants!) So the uses
of cos and sin might be unexpected, counter-intuitive for many. It
is algebraic...

https://youtu.be/F_0yfvm0UoU

*     *     *

                                                                       *     *      *

Google gave me the above numbers for 'evaluate 9^i'.

One starts by solving for e^(i θ)= 9^i

The angle is some 54 degrees...

From e

I should go shopping more often: my morning math
problem is no longer an issue.😐

So indeed, 3^x and e^(x*ln3) are one and the same curve.
Both clock in at 1 for (x=0), as does e itself, but are otherwise
different from the authentic e. One has effectively moved on
from e.

Using e notation is still the standard in Science, to make calculations
easier.

And e is still 1 and all of its divisors. Like pi, it is an irrational constant, but

unlike pi, it is computationally ideal rather than geometrically so.

All the terms in this series are positive!

Away

It's the day before the European Election, and I'm armchair-
peeking again. This has nothing to do with me, but then it does.
From a good light, Europe defines progress, and perhaps progressivism:
what it is possible to achieve in this historical moment. In a bad light,
is it not state gangsterism, our price and conditions or none.

On the elections front, it pays to be rude. Given proportional  representation -
one votes for a list with candidates actually running down the list - there
will be an inevitable push to popular and outrageous stances. Or - why not -
adding good-looking blondes to the list at the last minute. For the national parties,
it's clever to be crude, and play one's European arm into position. Actual issues
be overlooked, everybody is Lady Gaga bu-sy!

I have read (and translated) many a rant by frustrated academics about the
untractable machine of European policy-makers who govern from the Dark Side.
Yeah, the Center (economic liberal) -Right (Christian/Social Democrats). A salad of
ever-consensual policies! Stay strong, Europe! President Trump is coming in June...

 I remain optimistic about the whole thing because the system is open. Below, a
YouTube video that has gone largely un-commented. Blows me away:

Digital P.

How do I feel about digital populism? I like milkshakes
too much to throw any, but I would be weary of voting
on emotion, which seems to be the essence of the thing
for both the UK Brexit Party and Italy's Five Star Movement,
which is a close ally and EU partner.

It was also inevitable that an evolving EU would eventually
throw shade on 'archaic' national political institutions. Couldn't
seem to see that coming. At least they are not victims of fake news
from foreign powers; they control their own story arc!

https://www.spiegel.de/international/europe/interview-with-brexit-party-
leader-nigel-farage-a-1267728.html

https://www.theguardian.com/politics/2019/may/21/brexit-party-nigel-
farage-italy-digital-populists-five-star-movement

                                      *     *     *

Parties arguing for a Europe of independents (economy, the military), as opposed to a
federalist view, by country:

https://www.lemonde.fr/les-decodeurs/article/2019/05/21/28-pays-56-listes-et-autant-
de-nuances-la-carte-d-europe-des-souverainistes_5465227_4355770.html

The PNT

Number theory does have an accepted formula for the
number of primes up to or equal to - noted pi(x) - a certain
number. It was first guessed at by Gauss (at 16) and worked
on by Riemann and Chebechev, to be finally proven by
Hadamard and Vallée Poussin independently in 1896.

Here it is:

Ignoring 2 and 3, we can have fun using x/ln(x)!

For up to or equal to 100, we get:

Not bad; there are 25 and we predicted 22...

One could also look at for 150 minus for 100...

It's a play day!

https://de.wikipedia.org/wiki/Primzahlsatz

http://www-history.mcs.st-and.ac.uk/Biographies/Chebyshev.html

The RZF

Euler's Identity (1737):

Riemann Zeta Function (1859):

source: Gérard Villemin

             Wikipedia

                                                                   *     *     *
The Tchebychev Linkage 👩

By Van helsing - self-made largely based on "How to draw a straight line, by A.B. Kempe, B.A.", [1] and [2], CC BY 2.5, Link

Circles

Been putting my math files in order. Good
to know...

How to output a circle with the 3 Graph function types:
f(x) functions take x values in and output y. One has to
go to Pythagoras to get the circle.
The complex plane takes a parameter t, in rads, and assigns values
to both x and y.
The polar plane takes integers as the modulus (length), and t as the
argument (angle) of a rotating arm. You get what you ask for!

https://www.mathsisfun.com/sets/domain-range-codomain.html

Who

Really enjoyed this documentary, yesterday. Second guessing
long-departed historical figures is always hazardous: what was
he thinking? Did he/she believe his own line? Henry VIII did insist
on being addressed as 'Your Majesty', perhaps as a reminder he
was creating history...

The business about his obesity in later life was handled diplomatically.
He was injured and had no way of burning what he was eating! (From his
armour measurements, we know he went from  200 pounds as a young man,
but was up to 300 at his death.) At least no one is fat-shaming the poor man
although his lifetime coincided with the development of the sugar trade from
the New World (16th century). In any event, a fierce man!

This is a PBS documentary, so an American approach to story telling. I have
seen BBC versions where the accent was on wanting a male heir at all cost.
Still, this nasty habit of 'disposing' of his exes remains mysterious to me.
Who would do such a thing!?



                                            *     *     *

400 pounds, actually. And mistreatment to make him loose his mind...

Dream

I had a very odd dream last night. I was at the
Mall, only the inside had turned into a giant mathematical
Cartesian grid. I sat down at a point table, and found myself
opposite a dark-haired man trying to explain some math to me; I
could hardly hear him. Then he was replaced by a blond man with
a puffy hairstyle, who took off his shirt. This was clearly some
sort of signal, but I couldn't remember what it meant. I
looked over to the right and my mother was sitting alone, only
this was a very fragile and bent version of my mother. I said:
'This is impossible, you must be 99 years old because I am close
to seventy'.  Indeed my mother has been dead for many years.
She laughed and said: 'You'll never change!' and slowly
got up to leave. I thought to myself: 'I wonder if this is what death
is like...' I woke up.

So this morning I read on the Net about the death of Grumpy Cat, the beloved
 diminutive female cat, at age seven. RIP, you are well remembered, Grumpy.

MEPs

Libération is holding a day of debate on the issues
facing Europe this Saturday, May 18, in Strasbourgh.
Below, an interesting piece on the current mood in
Poland:

https://www.liberation.fr/evenements-libe/2019/05/15/pologne-du-
communisme-au-nationalisme-une-transition-par-les-mots_1726908

source: Libération

author: Grazyna Plebanek, writer

translation: doxa-louise

The Strasbourg Forum

Poland’s shift from Communism to Nationalism, in words

Grazyna Plebanek is a writer, explaining the hardened political discourse

in Poland, which has accompanied the rise of Nationalism. She will be

a participant in Saturday’s May 18 Libération Forum in Strasbourg.

...The Forum agenda will touch on ideas, values, culture...

The language of political discourse in Poland under communism was pure propaganda. 

The Communist Party held a monopoly on, apart from national symbols, words such as ‘homeland’, ‘people’, ‘honor’ and ‘blood’. At the time, everything published, be it articles or literary works, had to pass the censorship bureau. After the first free elections, June 4 1989, the language of officialdom became free of propaganda. One of he expressions most used by our fitrst free government was ‘turn over a new page’, to symbolize renewal. It is also a description of our relationship with the Communist past, which was done through the organization of the round table of 1989, and not by the execution of the leadership, as was the case in Romania.

The introduction thirty years ago of Capitalism - which has taken a fierce turn - changed how we speak. Words and expressions reminiscent of homo economicus entered the language: ‘to tighten one’s belt’, ‘a project’, ‘an effort’, etc. Since then, we Poles have tired of the pressure to consume ever more, of the admonition to an ostentatious lifestyle. The less fortunate are also frustrated because they could not afford it. A certain distance to capitalism has taken hold, and the Western Way of Life has come to be seen as something forced on us from the outside.

This shift was noticed by the Law and Justice Party, which won the election after eight years of government by Civic Platform Party. L&J played on the frustration of the Polish electorate, on their complexes toward Western Europe and their fear of Russia. All that as a wave of Nationalism was starting to be felt in Europe.

L&J laid down a pompous way of speaking akin to the patriotic values of the XIXth Century, a time when Poland was split among three neighboring Empires: ‘getting our dignity back’, ‘holding up one’s head’, ‘sovereign’. The party championed family values all the while opposing the ‘stranger’. The persona of the immigrant is that of a barbarian arriving ‘with his parasites’ ready to ‘rape our women’. This constitutes going backwards mentally and using to one’s ends the language of patriarchy. ‘Our women’: objectified women, deprived of will, passive and weak, waiting for men to act.

The vocabulary of Communist propaganda has also made a comeback. Lech Walesa has been discredited in public opinion, accused of a past as an ‘agent’ of security services under Communism. Donald Tusk, now an ‘agent for Germany at the behest of Angela Merkel’. Women who opposed L&J were singled out as ‘the worst kinds of citizens’ or ‘traitors’.

Donald Tusk recently made a speech at the University of Warsaw May3, (my Alma Mater)marking the anniversary of the Polish constitution - the fist democratic constitution in Europe. He recommended using an inclusive rather than exclusive way of speaking. ‘It is time to use and instead of or’ he said. But the reaction from certain quarters was stunning. Tusk has been so demonized by L&J that many no longer listen to what he is saying, but only who he might be speaking for: him, the despised ‘agent’. All this only goes to show to what extent the national myth created by L&J - including  conspiracy allegations surrounding the crash of the Presidential pane in 2010 - has come to divide the nation.

Even more troubling, public television -now a propaganda tool of L&J - has been showing the pronouncements of Donald Tusk to pictures of Hitler and Stalin. And all this is happening after the assasination of the mayor of Gdansk, a few months ago, in this hatemongering atmosphere now prevalent in Poland.

Many are currently uniting their voices  to combat this language. In his speech, Donald Tusk used humor to deal with L&J’s approach. What else can be done? One can refuse to enter the spiral of hate discourse and up the level of discussion with arguments on concrete subjects. One can help civil society come together by promoting groups such as the Wielka Orkiesttra Swiatecznej Pomocy, the national charity organization. One can ask people to participate in democracy and not only demand it. And one can point to the role of independent journalism and fact-checking by bringing to light any progress of disinformation.                               

                                             *     *     *

                                           source: Wikipedia

The European Parliament is often governed from the center-left;

interestingly because of Britain with 24 sitting as Eurosceptics, 

and 18 (+1) as Conservatives out of their 71 MEPs in the last 5-year

session. In fact they held the balance of power, as I read it!!

                                        *     *     *

http://www.europarl.europa.eu/unitedkingdom/en/european-elections/uk_meps.html

                                       *     *     *

A first with this election, citizens from any European
country - if they are residents - can join a candidate list.
Thus, Yavis Varoufakis - formerly a Greek minister - is
running in Germany; Nicolas Barnier, son of the EU negotiator,
is running in Belgium... Not that many, but some. France
even has a party that argues that nationality should be at
the European level, giving access to medical cards and social
services anywhere in Europe. Something to watch for in
the future!

http://www.lefigaro.fr/elections/europeennes/elections-europeennes-ces-candidats-
qui-se-presentent-hors-de-leurs-frontieres-20190516

M. Hero

Woke up in a good mood this morning, thinking about
mathematics. Surely because of all that tinkering yesterday, trying
to visualize how the complex plane redraws a simple
cartesian one.

Reading about how Bernard Riemann, who came to believe conventional
planes were restrictive. There could be many spaces with different
metrics, one might move around on. Or that could transform one
into another. Voilà Topology, a new branch of mathematics.

A manifold is a topological space that resembles Euclidean space near each point. 
More precisely, each point of an n-dimensional manifold has a neighborhood that 
is homeomorphic to the Euclidean space of dimension n.Wikipedia

We are in the world of that beleaguered mathematical hero, the small bug
on the sphere, who is perhaps encountering strange phenomena...

                                          *     *     *
Below, the well-known Möbius strip, a long-standing object of fascination.
One can travel the lenth of the strip on both sides, without lifting a finger.

Verified this for myself by making one with a black line on one side.

But if one travelled one, how would one know the distance travelled.
One would need markers on the strip (maybe construct one with lined
paper). Even so, there are difficulties.

Let us consider an application, using what in French is referred to
metro-boulot-dodo: the daily grind of the urban worker one might
translate to tube-cube-snooze. A simple work day would involve metro
to and metro back, but repeated forays from the metro to shop, browse,
have a cupa...etc do add up. A bit like twisting the M-strip and yanking.
Because eventually there are two round trips, overground and underground!

                                                 *     *     *



                                               *     *     *

https://www.quora.com/Is-a-mobius-strip-flat

Point Perfect

Worked on getting a point on a polar circle on Graph, last night. Ended up with

different frames of reference on the one graph...

*     *     *

This morning, on macOS Grapher. The first is the example for Global vs Local

parametric equations on the complex plane. The second is straight entries on 

Polar Parametric. 👍

*     *     *

Below, today's problem. How to replicate that right-hand figure:

https://www.amq.math.ca/wp-content/uploads/bulletin/vol57/no1/05-Riemann.pdf

Mapping a squared complex polar point amounts to squaring the modulus value

and doubling the angle.

A conformal map is fancier: it conserves angles but not orientation as one goes

from one type of map to another. Our model is a conformal maping!!

http://cmgreen.io/2015/12/24/the-ahlfors-iteration.html

http://www-users.math.umn.edu/~%20olver/ln_/cml.pdf

https://www.pacifict.com/Examples/Example48.html