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.