Κυριακή, 14 Μαΐου 2017

Βέλτιστη γωνία παρατήρησης εικόνας

(117) Βέλτιστη γωνία παρατήρησης εικόνας - mathematica.gr



Βέλτιστη γωνία παρατήρησης εικόνας

Τις ημέρες αυτές πλήθος κόσμου επισκέπτεται την εκκλησία, προσκυνά και κάποιοι παρατηρούν τις εικόνες ή τις τοιχογραφίες.
Ας υποθέσουμε ότι μία εικόνα, μήκους 1 μέτρου, βρίσκεται (το κάτω μέρος της) σε ύψος 2 μέτρα από το έδαφος και ένας προσκυνητής, του οποίου τα μάτια βρίσκονται σε ύψος 1,8 μέτρα από το έδαφος παρατηρεί την εικόνα.
Να βρεθεί η θέση του παρατηρητή, ώστε η γωνία παρατήρησης της εικόνας ACB (γωνία άνω μέρους, οφθαλμών, κάτω μέρους) να γίνει μέγιστη.

Σάββατο, 29 Απριλίου 2017

Where's the maths in beer? | plus.maths.org - Τα Μαθηματικά της Μπύρας!

Where's the maths in beer? | plus.maths.org










This article is based on one of Budd's Gresham College lectures. Watch the full lecture here, and see here to find out more about this free, public lecture series.
There's actually quite a lot of maths in beer. To convince you of this, here are three beer-related maths stories.



Blowing bubbles

Let's start with the head. One of the key features of a pint of
Guinness is the wonderful creamy foam head. This is in contrast to the
much smaller head that we find on a pint of bitter beer. For the
manufacturers of beer to get both types of head involves a lot of
science and maths. The foam in the head of a pint of bitter is made of
networks of bubbles of carbon dioxide separated by thin films of the
beer itself, with surface tension giving the strength to the thin walls
surrounding each bubble. The walls of these bubbles move as a result of
surface tension, with smaller bubbles moving faster as they have a
higher curvature. This results in the smaller bubbles being "eaten" by
the larger ones in a process called Oswald ripening.
Basically small bubbles shrink and large bubbles grow, leading to a
coarse foam made up of large bubbles only. Eventually the liquid drains
from the large bubbles and they pop, and the foam disappears.



The remarkable mathematician John von Neumann,
who was (amongst many other achievements) responsible for the
development of the modern electronic computer, devised an equation in
1952 which explained the patterns that we see in such cellular
structures in two dimensions. In 2007 this was extended to three
dimensions by a group of mathematicians in Princeton interested in the
applications of maths to beer. It's a hard life!



Why the widget?

Guinness
Guinness has a lovely creamy head. Image: PDPhoto.org.


Another group of mathematicians, appropriately from Limerick in
Ireland, have made a study of the foam on a pint of Guinness. This is
much creamier than the foam on a pint of bitter. The reason is that
whilst the foam on bitter is made up of air bubbles, the foam on a pint
of Guinness is made up of nitrogen. This gas diffuses a hundred times
slower in air than carbon dioxide, meaning that the bubbles are smaller
and the foam is much more stable and creamier.



The nitrogen needs to be introduced into the Guinness when it is
poured. In a pub this is achieved by having a separate pipe, linked to a
nitrogen supply, which supplies the nitrogen at the same time as the
beer is served from the barrel.

For many years Guinness in cans did not have a head. However, this problem was solved by the introduction of a widget,
which is a nitrogen container in the can, and which releases precisely
the right amount of nitrogen when the can is opened. This process must
be very carefully controlled, and a lot of careful design work is
required to make the widget work well. The whole process was analysed by
(it appears!) the whole of the applied mathematics department at
Limerick, and described in the charmingly titled paper The initiation of Guinness. Notably the same group has now done a complete analysis of the mathematics of making coffee.


Why statisticians couldn't organise a piss up in a brewery

In the year 2005 the British Science Festival
came to Dublin. At that time I had the honour of being the president of
the maths section of the British Science Association, which was
organising the festival, and had the responsibility of devising a
mathematics programme for the event.



Bitter
William Sealy Gosset (1876 - 1937).


One of our plans was to have a mathematics visit to the Guinness
Breweries in Dublin. Obviously there are many reasons why we might want
to visit a brewery, but why should mathematicians want to go there, and
why should they want to go to Guinness? The answer to both of these
questions lies in the person of William Gosset (pictured) who was the chief statistician at Guinness in the first part of the 20th Century.

Guinness
was in many ways ahead of its time in the production and quality
control that it applied to its product (as well as the way that it was
advertised). Gosset was employed in part to ensure that the Guinness
stout was of a consistent quality. This was done by making careful
measurements of a sample of the product and using these to assess both
its general quality and its variability. This was, at the time, a
difficult problem in statistics. To solve it Gosset devised a new
statistical test to compare the measurements. This worked extremely well and made a very real difference
to improving the quality of Guinness stout. Gosset felt it important to
publish this test, but was reluctant to disclose his identity and that
of his employer. Instead it was published in the journal Biometrika, in 1908, under the anonymous name of "Student". Ever since this test has been known as Student's t-test. It plays a central role in testing and maintaining the quality of food and drink all over the world.



So, let's get back to the British Science Festival. Having decided to
go to Guinness we set up a sub-committee to organise the trip to it
during the science festival, in part to celebrate the invention there of
the t-test and its contribution to modern statistics. Clearly such a
trip should include a reception and a drink of a pint of Guinness.
Unfortunately, through no one's fault, it wasn't possible in the end to
do this. It was only after the event that we realised we could be
accused of being unable to organise a piss up in a brewery.



Three mathematicians walk into a pub...

I will finish this article with a bad story/joke about mathematicians
and drinking. You have to concentrate a bit to get the joke.



Three mathematicians go into a pub and the bar tender asks, "Does anyone want a lager"?



The first mathematician pauses for thought, and then says, "I don't know".
The second mathematician likewise says, "I don't know".
Finally the third mathematician says, "No!"



So the bar tender ask, "Does everyone want a bitter then?"



The first mathematician pauses for thought, and then again says, "I don't know".
The second mathematician likewise says, "I don't know".
Finally the third mathematician says, "Yes!"



So they all have a bitter.




About this article

This article is adapted from one of Budd's Gresham College lectures. See here to find out more about this free, public lecture series.


Chris Budd
Chris Budd.


Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the Institute of Mathematics and its Applications, Chair of Mathematics for the Royal Institution and an honorary fellow of the British Science Association. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.



He has co-written the popular mathematics book Mathematics Galore!, published by Oxford University Press, with C. Sangwin, and features in the book 50 Visions of Mathematics ed. Sam Parc.

Παρασκευή, 14 Απριλίου 2017

Παράγοντες με σταθερό άθροισμα

Αν σε ένα γινόμενο, οι παράγοντές του έχουν σταθερό άθροισμα, τότε το γινόμενο γίνεται μέγιστο, όταν οι παράγοντές του είναι ίσοι μεταξύ τους.

Μπορείτε να το αποδείξετε;

Δοκιμάστε πρώτα με γινόμενο δύο παραγόντων.

Τομές κώνου (ή κωνικές τομές)

Θα τις μελετήσουμε στη Β΄Λυκείου κατεύθυνση:

Φωτογραφία του Constantinos Kyventidis.

Τρίτη, 21 Μαρτίου 2017

Βραβείο Άμπελ στον Yves Meyer - Thales + Friends

Βραβείο Άμπελ στον Yves Meyer - Thales + Friends



Αναρτήθηκε σε 21 Μαρτίου, 2017 κατηγορία: Ειδήσεις | Tags: , ,


Στον Γάλλο μαθηματικό Yves Meyer από την École Normale Supérieure Paris-Saclay, θα απονεμηθεί, σε ειδική τελετή που θα γίνει στο Όσλο στις 23 Μαΐου, το Βραβείο Άμπελ
για το 2017. Η βράβευση, που ανακοινώθηκε σήμερα από τη Νορβηγική
Ακαδημία Επιστημών και Γραμμάτων, τιμά τη σημαντική προσφορά του
77χρονου μαθηματικού στην ανάπτυξη της μαθηματικής θεωρίας των
κυματιδίων.

Πρόκειται για το κορυφαίο διεθνές βραβείο για τη μαθηματική επιστήμη,
που απονέμεται κάθε χρόνο από τον Βασιλιά της Νορβηγίας σε έναν ή
περισσότερους μαθηματικούς με σπουδαία συνεισφορά στην επιστήμη. Το
βραβείο φέρει το όνομα του Νορβηγού μαθηματικού Νιλς Χένρικ Άμπελ (1802-1829) και συνοδεύεται από χρηματικό έπαθλο ύψους 675.000 ευρώ.

Ο Yves Meyer χαρακτηρίζεται στην ανακοίνωση της Ακαδημίας πρωτοπόρος
οραματιστής στη σύγχρονη ανάπτυξη μιας θεωρίας που συνδυάζει τα
μαθηματικά, την πληροφορική και την υπολογιστική επιστήμη.

Η ανάλυση κυματιδίων έχει αξιοποιηθεί, άλλωστε, σε μια ευρεία κλίμακα
πεδίων που ξεκινά από την υπολογιστική αρμονική ανάλυση και τη συμπίεση
δεδομένων και εκτείνεται σε τομείς όπως είναι η μείωση θορύβου, η
ιατρική απεικόνιση, η αρχειοθέτηση, ο ψηφιακός κινηματογράφος, η
αποσυνέλιξη των εικόνων που λαμβάνουμε από το διαστημικό τηλεσκόπιο
Hubble, αλλά και η πρόσφατη ανίχνευση βαρυτικών κυμάτων (LIGO) που
δημιουργήθηκε από τη σύγκρουση ανάμεσα σε δύο μαύρες τρύπες.

Ο Yves Meyer μνημονεύεται, επίσης, για τον εντυπωσιακό τρόπο με τον
οποίο αξιοποιεί στο έργο του θεωρητικούς τομείς των μαθηματικών,
προκειμένου να αναπτύξει πρακτικά εργαλεία που εφαρμόζονται στην
Επιστήμη των Υπολογιστών και την Πληροφορική. Το παράδειγμά του
αναδεικνύει, όπως σημειώνεται στην ανακοίνωση της Ακαδημίας, τον
ισχυρισμό που υποστηρίζει ότι η εργασία στα καθαρά μαθηματικά έχει πολύ
συχνά χρησιμότητα σε πρακτικές εφαρμογές του … πραγματικού κόσμου.

Σάββατο, 4 Μαρτίου 2017

Βράβευση Μαθητών Ευκλείδη

Βράβευση μαθητών για το διαγωνισμό "Ευκλείδης" 2017

ΕΛΛΗΝΙΚΗ ΜΑΘΗΜΑΤΙΚΗ ΕΤΑΙΕΙΑ
77ΟΣ ΠΑΝΕΛΛΗΝΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ
ΣΤΑ ΜΑΘΗΜΑΤΙΚΑ «Ο ΕΥΚΛΕΙΔΗΣ»
ΒΡΑΒΕΥΣΗ
ΚΥΡΙΑΚΗ 5 MAΡTΙΟΥ 2017
Αμφιθέατρο Αντωνιάδου, Οικονομικό Πανεπιστημίου Αθηνών και ώρα 12.00 π.μ. 
 
Από το σχολείο μας (Πρότυπο ΓΕΛ και Γυμνάσιο Ευαγγελικής Σχολής ΣΜύρνης): 
Β ΓΥΜΝΑΣΙΟΥ
ΕΜΜΑΝΟΥΗΛ
ΔΗΜΗΤΡΙΟΣ
ΠΕΙΡΑΜΑΤΙΚΟ ΓΥΜΝΑΣΙΟ ΕΥΑΓΓΕΛΙΚΗΣ ΣΧΟΛΗΣ ΣΜΥΡΝΗΣ
Α
ΜΠΑΛΑΤΟΣ
ΔΗΜΗΤΡΙΟΣ
ΠΕΙΡΑΜΑΤΙΚΟ ΓΥΜΝΑΣΙΟ ΕΥΑΓΓΕΛΙΚΗΣ ΣΧΟΛΗΣ ΣΜΥΡΝΗΣ
Β
ΚΑΡΒΟΥΝΗΣ
ΟΡΕΣΤΗΣ
ΠΕΙΡΑΜΑΤΙΚΟ ΓΥΜΝΑΣΙΟ ΕΥΑΓΓΕΛΙΚΗΣ ΣΧΟΛΗΣ ΣΜΥΡΝΗΣ
Γ
Γ ΓΥΜΝΑΣΙΟΥ
ΜΠΕΚΙΑΡΗΣ
ΔΗΜΗΤΡΙΟΣ
ΠΕΙΡΑΜΑΤΙΚΟ ΓΥΜΝΑΣΙΟ ΕΥΑΓΓΕΛΙΚΗΣ ΣΧΟΛΗΣ ΣΜΥΡΝΗΣ
Β
ΚΑΝΤΙΑΝΗΣ
ΘΕΟΔΩΡΟΣ
ΠΕΙΡΑΜΑΤΙΚΟ ΓΥΜΝΑΣΙΟ ΕΥΑΓΓΕΛΙΚΗΣ ΣΧΟΛΗΣ ΣΜΥΡΝΗΣ
Γ
Γ ΛΥΚΕΙΟΥ
ΜΕΛΑΣ
ΔΗΜΗΤΡΙΟΣ ΧΡΥΣΟΒΑΛΑΝΤΗΣ
ΣΧΟΛΗ ΜΩΡΑΪΤΗ
A
ΣΗΜΑΝΤΗΡΗΣ
ΙΩΑΝΝΗΣ
1ο ΓΕΝΙΚΟ ΛΥΚΕΙΟ ΓΛΥΦΑΔΑΣ
A
ΓΑΛΑΝΗΣ
ΧΡΗΣΤΟΣ
ΠΕΙΡΑΜΑΤΙΚΟ ΛΥΚΕΙΟ ΕΥΑΓΓΕΛΙΚΗΣ ΣΧΟΛΗΣ ΣΜΥΡΝΗΣ
Β

Κατασκευασιμα πολύγωνα

https://m.youtube.com/watch?v=TAHczLeIUTc

Παρασκευή, 3 Φεβρουαρίου 2017

Η ιστορία του μηδενός.

The History of Zero: How Ancient Mesopotamia Invented the Mathematical Concept of Nought and Ancient India Gave It Symbolic Form – Brain Pickings



The History of Zero: How Ancient Mesopotamia Invented the Mathematical Concept of Nought and Ancient India Gave It Symbolic Form

“If you look at zero you see nothing; but look through it and you will see the world.”

The History of Zero: How Ancient Mesopotamia Invented the Mathematical Concept of Nought and Ancient India Gave It Symbolic Form

If
the ancient Arab world had closed its gates to foreign travelers, we
would have no medicine, no astronomy, and no mathematics — at least not
as we know them today.

Central to humanity’s quest to grasp the nature of the universe and
make sense of our own existence is zero, which began in Mesopotamia and
spurred one of the most significant paradigm shifts in human
consciousness — a concept first invented (or perhaps discovered) in
pre-Arab Sumer, modern-day Iraq, and later given symbolic form in
ancient India. This twining of meaning and symbol not only shaped
mathematics, which underlies our best models of reality, but became
woven into the very fabric of human life, from the works of Shakespeare,
who famously winked at zero in King Lear by calling it “an O without a figure,” to the invention of the bit that gave us the 1s and 0s underpinning my ability to type these words and your ability to read them on this screen.

Mathematician Robert Kaplan chronicles nought’s revolutionary journey in The Nothing That Is: A Natural History of Zero (public library).
It is, in a sense, an archetypal story of scientific discovery, wherein
an abstract concept derived from the observed laws of nature is named
and given symbolic form. But it is also a kind of cross-cultural fairy
tale that romances reason across time and space



Art by Paul Rand from Little 1 by Ann Rand, a vintage concept book about the numbers

Kaplan writes:

If you look at zero you see nothing; but look through it
and you will see the world. For zero brings into focus the great,
organic sprawl of mathematics, and mathematics in turn the complex
nature of things. From counting to calculating, from estimating the odds
to knowing exactly when the tides in our affairs will crest, the
shining tools of mathematics let us follow the tacking course everything
takes through everything else – and all of their parts swing on the
smallest of pivots, zero

With these mental devices we make visible the hidden laws controlling
the objects around us in their cycles and swerves. Even the mind itself
is mirrored in mathematics, its endless reflections now confusing, now
clarifying insight.

[…]

As we follow the meanderings of zero’s symbols and meanings we’ll see
along with it the making and doing of mathematics — by humans, for
humans. No god gave it to us. Its muse speaks only to those who ardently
pursue her.
With an eye to the eternal question of whether mathematics is discovered or invented — a question famously debated by Kurt Gödel and the Vienna Circle — Kaplan observes:

The disquieting question of whether zero is out there or a
fiction will call up the perennial puzzle of whether we invent or
discover the way of things, hence the yet deeper issue of where we are
in the hierarchy. Are we creatures or creators, less than – or only a
little less than — the angels in our power to appraise?


Art by Shel Silverstein from The Missing Piece Meets the Big O

Like all transformative inventions, zero began with necessity — the
necessity for counting without getting bemired in the inelegance of
increasingly large numbers. Kaplan writes:

Zero began its career as two wedges pressed into a wet
lump of clay, in the days when a superb piece of mental engineering gave
us the art of counting.

[…]

The story begins some 5,000 years ago with the Sumerians, those
lively people who settled in Mesopotamia (part of what is now Iraq).
When you read, on one of their clay tablets, this exchange between
father and son: “Where did you go?” “Nowhere.” “Then why are you late?”,
you realize that 5,000 years are like an evening gone.

The Sumerians counted by 1s and 10s but also by 60s. This may seem
bizarre until you recall that we do too, using 60 for minutes in an hour
(and 6 × 60 = 360 for degrees in a circle). Worse, we also count by 12
when it comes to months in a year, 7 for days in a week, 24 for hours in
a day and 16 for ounces in a pound or a pint. Up until 1971 the British
counted their pennies in heaps of 12 to a shilling but heaps of 20
shillings to a pound.

Tug on each of these different systems and you’ll unravel a history
of customs and compromises, showing what you thought was quirky to be
the most natural thing in the world. In the case of the Sumerians, a
60-base (sexagesimal) system most likely sprang from their dealings with
another culture whose system of weights — and hence of monetary value —
differed from their own.
Having to reconcile the decimal and sexagesimal counting systems was a
source of growing confusion for the Sumerians, who wrote by pressing
the tip of a hollow reed to create circles and semi-circles onto wet
clay tablets solidified by baking. The reed eventually became a
three-sided stylus, which made triangular cuneiform marks at varying
angles to designate different numbers, amounts, and concepts. Kaplan
demonstrates what the Sumerian numerical system looked like by 2000 BCE:



This cumbersome system lasted for thousands of years, until someone
at some point between the sixth and third centuries BCE came up with a
way to wedge accounting columns apart, effectively symbolizing “nothing
in this column” — and so the concept of, if not the symbol for, zero was
born. Kaplan writes:

In a tablet unearthed at Kish (dating from perhaps as far
back as 700 BC), the scribe wrote his zeroes with three hooks, rather
than two slanted wedges, as if they were thirties; and another scribe at
about the same time made his with only one, so that they are
indistinguishable from his tens. Carelessness? Or does this variety tell
us that we are very near the earliest uses of the separation sign as
zero, its meaning and form having yet to settle in?
But zero almost perished with the civilization that first imagined
it. The story follows history’s arrow from Mesopotamia to ancient
Greece, where the necessity of zero awakens anew. Kaplan turns to
Archimedes and his system for naming large numbers, “myriad” being the
largest of the Greek names for numbers, connoting 10,000. With his
notion of orders of large numbers, the great Greek polymath
came within inches of inventing the concept of powers, but he gave us
something even more important — as Kaplan puts it, he showed us “how to
think as concretely as we can about the very large, giving us a way of
building up to it in stages rather than letting our thoughts diffuse in
the face of immensity, so that we will be able to distinguish even such
magnitudes as these from the infinite.”



“Archimedes Thoughtful” by Domenico Fetti, 1620

This concept of the infinite in a sense contoured the need for naming
its mirror-image counterpart: nothingness. (Negative numbers were still
a long way away.) And yet the Greeks had no word for zero, though they
clearly recognized its spectral presence. Kaplan writes:

Haven’t we all an ancient sense that for something to
exist it must have a name? Many a child refuses to accept the argument
that the numbers go on forever (just add one to any candidate for the
last) because names run out. For them a googol — 1 with 100 zeroes after
it — is a large and living friend, as is a googolplex (10 to the googol
power, in an Archimedean spirit).

[…]

By not using zero, but naming instead his myriad myriads, orders and
periods, Archimedes has given a constructive vitality to this vastness —
putting it just that much nearer our reach, if not our grasp.
Ordinarily, we know that naming is what gives meaning to existence. But names are given to things, and zero is not a thing — it is, in fact, a no-thing. Kaplan contemplates the paradox:

Names belong to things, but zero belongs to nothing. It
counts the totality of what isn’t there. By this reasoning it must be
everywhere with regard to this and that: with regard, for instance, to
the number of humming-birds in that bowl with seven — or now six —
apples. Then what does zero name? It looks like a smaller version of
Gertrude Stein’s Oakland, having no there there.
Zero, still an unnamed figment of the mathematical imagination,
continued its odyssey around the ancient world before it was given a
name. After Babylon and Greece, it landed in India. The first surviving
written appearance of zero as a symbol appeared there on a stone tablet
dated 876 AD, inscribed with the measurements of a garden: 270 by 50,
written as “27°” and “5°.” Kaplan notes that the same tiny zero appears
on copper plates dating back to three centuries earlier, but because
forgeries ran rampant in the eleventh century, their authenticity can’t
be ascertained. He writes:

We can try pushing back the beginnings of zero in India
before 876, if you are willing to strain your eyes to make out dim
figures in a bright haze. Why trouble to do this? Because every story,
like every dream, has a deep point, where all that is said sounds
oracular, all that is seen, an omen. Interpretations seethe around these
images like froth in a cauldron. This deep point for us is the cleft
between the ancient world around the Mediterranean and the ancient world
of India.
But if zero were to have a high priest in ancient India, it would
undoubtedly be the mathematician and astronomer Āryabhata, whose
identity is shrouded in as much mystery as Shakespeare’s. Nonetheless,
his legacy — whether he was indeed one person or many — is an indelible
part of zero’s story.



Āryabhata (art by K. Ganesh Acharya)

Kaplan writes:

Āryabhata wanted a concise way to store (not calculate
with) large numbers, and hit on a strange scheme. If we hadn’t yet our
positional notation, where the 8 in 9,871 means 800 because it stands in
the hundreds place, we might have come up with writing it this way:
9T8H7Te1, where T stands for ‘thousand’, H for “hundred” and Te for
“ten” (in fact, this is how we usually pronounce our numbers, and how
monetary amounts have been expressed: £3.4s.2d). Āryabhata did something
of this sort, only one degree more abstract.

He made up nonsense words whose syllables stood for digits in places,
the digits being given by consonants, the places by the nine vowels in
Sanskrit. Since the first three vowels are a, i and u, if you wanted to
write 386 in his system (he wrote this as 6, then 8, then 3) you would
want the sixth consonant, c, followed by a (showing that c was in the
units place), the eighth consonant, j, followed by i, then the third
consonant, g, followed by u: CAJIGU. The problem is that this system
gives only 9 possible places, and being an astronomer, he had need of
many more. His baroque solution was to double his system to 18 places by
using the same nine vowels twice each: a, a, i, i, u, u and so on; and
breaking the consonants up into two groups, using those from the first
for the odd numbered places, those from the second for the even. So he
would actually have written 386 this way: CASAGI (c being the sixth
consonant of the first group, s in effect the eighth of the second
group, g the third of the first group)…

There is clearly no zero in this system — but interestingly enough,
in explaining it Āryabhata says: “The nine vowels are to be used in two
nines of places” — and his word for “place” is “kha”. This kha
later becomes one of the commonest Indian words for zero. It is as if we
had here a slow-motion picture of an idea evolving: the shift from a
“named” to a purely positional notation, from an empty place where a
digit can lodge to “the empty number”: a number in its own right, that
nudged other numbers along into their places.
Kaplan reflects on the multicultural intellectual heritage encircling the concept of zero:

While having a symbol for zero matters, having the notion
matters more, and whether this came from the Babylonians directly or
through the Greeks, what is hanging in the balance here in India is the
character this notion will take: will it be the idea of the absence of
any number — or the idea of a number for such absence? Is it to be the
mark of the empty, or the empty mark? The first keeps it estranged from
numbers, merely part of the landscape through which they move; the
second puts it on a par with them.
In the remainder of the fascinating and lyrical The Nothing That Is,
Kaplan goes on to explore how various other cultures, from the Mayans
to the Romans, contributed to the trans-civilizational mosaic that is
zero as it made its way to modern mathematics, and examines its profound
impact on everything from philosophy to literature to his own domain of
mathematics. Complement it with this Victorian love letter to mathematics and the illustrated story of how the Persian polymath Ibn Sina revolutionized modern science.

MacTutor History of Mathematics - Εξαιρετικός ιστότοπος για την Ιστορία των Μαθηματικών.

MacTutor History of Mathematics

Παρασκευή, 20 Ιανουαρίου 2017

Θερινά μαθήματα στην Οξφόρδη - mathematica.gr

(7) Θερινά μαθήματα στην Οξφόρδη - mathematica.gr



Έλαβα από το Πανεπιστήμιο της Οξφόρδης ανακοίνωση για Θερινά μαθήματα Μαθηματικών σε ταλαντούχους μαθητές Λυκείου.

Ακολουθεί η ανακοίνωση στα Αγγλικά. Δεν την μεταφράζω γιατί έτσι και αλλιώς αυτοί που
θα ενδιαφερθούν για τα μαθήματα, πρέπει να γνωρίζουν Αγγλικά, πέρα από τα άριστα Μαθηματικά.

Περισσότερες λεπτομέρειες στην ιστοσελίδα που παραπέμπουν,

http://www.promys-europe.org/programme/application

Μαζί με την αίτηση πρέπει ο ενδιαφερόμενος να στείλει, μεταξύ άλλων, και τις λύσεις του σε μια σειρά ωραίων προβλημάτων.

Θα επισυνάψω τα προβλήματα (κάντε υπομονή γιατί έχω αργή σύνδεση και όποτε ανεβάζω ή κατεβάζω επισυναπτόμενο, το ... μετανιώνω.)

Προσοχή όμως, ΔΕΝ ΠΡΕΠΕΙ ΝΑ ΒΑΛΟΥΜΕ ΛΥΣΕΙΣ ΕΔΩ. Τις λύσεις πρέπει να τις βρει μόνος του όποιος
ενδιαφέρεται να κάνει αίτηση και εμείς ΔΕΝ ΕΧΟΥΜΕ ΣΚΟΠΟ ΝΑ ΠΑΡΑΚΑΜΨΟΥΜΕ τους οργανωτές.
(Αν ήταν έτσι, θα έβαζα δικές μου λύσεις, αφού τις έλυσα έτσι και αλλιώς).

Τις ασκήσεις τις βάζω μόνο για θέαση και για το ενδιαφέρον που παρουσιάζουν.



Call for Applications for PROMYS Europe 2017

PROMYS
Europe, a challenging six-week residential summer programme at the
University of Oxford, is seeking pre-university students from across
Europe who show unusual readiness to think deeply about mathematics. I
am writing to ask for your assistance in spreading the word to
potentially interested applicants and their teachers.

PROMYS
Europe is designed to encourage mathematically ambitious students who
are at least 16 to explore the creative world of mathematics.
Participants tackle fundamental mathematical questions within a richly
stimulating and supportive community of fellow first-year students,
returning students, undergraduate counsellors, research mentors,
faculty, and visiting mathematicians.

First-year students focus
primarily on a series of very challenging problem sets, daily lectures,
and exploration projects in Number Theory. There will also be a
programme of talks by guest mathematicians and the counsellors, on a
wide range of mathematical subjects, as well as courses aimed primarily
at students who are returning to PROMYS Europe for a second or third
time.

PROMYS Europe is a partnership of Wadham College and the
Mathematical Institute at the University of Oxford, the Clay Mathematics
Institute, and PROMYS (Program in Mathematics for Young Scientists,
founded in Boston in 1989).

The programme is dedicated to the
principle that no one should be unable to attend for financial reasons.
Most of the cost is covered by the partnership and by generous donations
from supporters. In addition, full and partial needs-based financial
aid is available, which can cover the fee and travel costs, in part or
in full.

The application and application problem set are
available on the PROMYS Europe website. The deadline for first-year
student applications is 19 March. PROMYS Europe 2017 will run from 9
July to 19 August at the University of Oxford.

We hope you will
spread the word to potentially interested applicants and their teachers
by circulating the attached flyer and/or encouraging them to visit the
website.


Thank you for helping us reach out to students who
would benefit from this opportunity. If you wish to be removed from our
mailing list, please unsubscribe.


Glenn Stevens
Director of PROMYS Europe


PROMYS Europe Board:
Professor Henry Cohn (Principal Researcher at Microsoft Research, and Adjunct Professor of Mathematics at MIT)
Dr Vicky Neale ( Whitehead Lecturer in the Mathematical Institute and at Balliol College, University of Oxford)
Professor Glenn Stevens ( Director, Program in Mathematics for Young Scientists (PROMYS), and
Professor of Mathematics, Boston University) and
Professor Nicholas Woodhouse (President of the Clay Mathematics Institute and Professor of Mathematics,
University of Oxford and Fellow of Wadham College)