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<h1 class="card-title">
We now have math to describe black holes.
</h1>
<div class="card-text-lead">
A new set of equations can precisely describe the reflections of
the Universe that appear in the warped light around a black
hole.
<p>
The proximity of each reflection is dependent on the angle of
observation with respect to the black hole, and the rate of
the black hole's spin, according to a mathematical solution
worked out by physics student Albert Sneppen of the Niels Bohr
Institute in Denmark.This is really cool, absolutely, but it's
not just really cool. It also potentially gives us a new tool
for probing the gravitational environment around these extreme
objects.
</p>
<p>
"There is something fantastically beautiful in now
understanding why the images repeat themselves in such an
elegant way," Sneppen said. "On top of that, it provides new
opportunities to test our understanding of gravity and black
holes."
</p>
<p>
If there's one thing that black holes are famous for, it's
their extreme gravity. Specifically that, beyond a certain
radius, the fastest achievable velocity in the Universe, that
of light in a vacuum, is insufficient to achieve escape
velocity.
</p>
<p>
That point of no return is the event horizon – defined by
what's called the Schwarszchild radius – and it's the reason
why we say that not even light can escape from a black hole's
gravity.
</p>
<p>
Just outside the black hole's event horizon, however, the
environment is also seriously wack. The gravitational field is
so powerful that the curvature of space-time is almost
circular.
</p>
<p>
Any photons entering this space will, naturally, have to
follow this curvature. This means that, from our perspective,
the path of the light appears to be warped and bent.
</p>
<p>
At the very inner edge of this space, just outside the event
horizon, we can see what is called a photon ring, where
photons travel in orbit around the black hole multiple times
before either falling towards the black hole, or escaping into
space.
</p>
<p>
This means that the light from distant objects behind the
black hole can be magnified, distorted and 'reflected' several
times. We refer to this as a gravitational lens; the effect
can also be seen in other contexts, and is a useful tool for
studying the Universe.
</p>
<p>
So we've known about the effect for some time, and scientists
had figured out that the closer you look towards the black
hole, the more reflections you see of distant objects.
</p>
<p>
To get from one image to the next image, you needed to look
about 500 times closer to the black hole's optical edge, or
the exponential function of two pi (e2π), but why this was the
case was difficult to mathematically describe.
</p>
<p>
Sneppen's approach was to reformulate the light trajectory,
and quantify its linear stability, using second order
differential equations. He found not only did his solution
mathematically describe why the images repeat at distances of
e2π, but that it could work for a rotating black hole - and
that repeat distance is dependent on spin."It turns out that
when it rotates really fast, you no longer have to get closer
to the black hole by a factor of 500, but significantly less,"
Sneppen said. "In fact, each image is now only 50, or five, or
even down to just two times closer to the edge of the black
hole."
</p>
<p>
In practice, this is going to be difficult to observe, at
least any time soon - just look at the intense amount of work
that went into the unresolved imaging of the ring of light
around supermassive black hole Pōwehi (M87*).
</p>
<p>
Theoretically, however, there should be infinite rings of
light around a black hole. Since we have imaged the shadow of
a supermassive black hole once, it's hopefully only a matter
of time before we're able to obtain better images, and there
are already plans for imaging a photon ring.
</p>
<p>
One day, the infinite images close to a black hole could be a
tool for studying not just the physics of black hole
space-time, but the objects behind them - repeated in infinite
reflections in orbital perpetuity.
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<h1 class="card-title">
The turing machine should run forever-unless maths is wrong.
</h1>
<div class="card-text-lead">
One hundred and fifty years of mathematics will be proved wrong
if a new computer program stops running. Thankfully, it’s
unlikely to happen, but the code behind it is testing the limits
of the mathematical realm.
<p>
The program is a simulated Turing machine, a mathematical
model of computation created by codebreaker Alan Turing. In
1936, he showed that the actions of any computer algorithm can
be mimicked by a simple machine that reads and writes 0s and
1s on an infinitely long tape by working through a set of
states, or instructions. The more complex the algorithm, the
more states the machine requires.
</p>
<p>
Now Scott Aaronson and Adam Yedidia of the Massachusetts
Institute of Technology have created three Turing machines
with behaviour that is entwined in deep questions of
mathematics. This includes the proof of the 150-year-old
Riemann hypothesis – thought to govern the patterns of prime
numbers.
</p>
<p>
Turing’s machines have long been used to probe such questions.
Their origins lie in a series of philosophical revelations
that rocked the mathematical world in the 1930s. First, Kurt
Gödel proved that some mathematical statements can never be
proved true or false – they are undecidable. He essentially
created a mathematical version of the sentence “This sentence
is false”: a logical brain-twister that contradicts itself.
</p>
<p>No proof of everything</p>
<p>
Gödel’s assertion has a get-out clause. If you change the base
assumptions on which proofs are built – the axioms – you can
render a problem decidable. But that will still leave other
problems that are undecidable. That means there are no axioms
that let you prove everything.
</p>
<p>
Following Gödel’s arguments, Turing proved that there must be
some Turing machines whose behaviour cannot be predicted under
the standard axioms – known as Zermelo-Fraenkel set theory
with the axiom of choice (C), or more snappily, ZFC –
underpinning most of mathematics. But we didn’t know how
complex they would have to be.
</p>
<p>
Now, Yedidia and Aaronson have created a Turing machine with
7918 states that has this property. And they’ve named it “Z”.
</p>
<p>
“We tried to make it concrete, and say how many states does it
take before you get into this abyss of unprovability?” says
Aaronson.
</p>
<p>
The pair simulated Z on a computer, but it is small enough
that it could theoretically be built as a physical device,
says Terence Tao of the University of California, Los Angeles.
“If one were then to turn such a physical machine on, what we
believe would happen would be that it would run indefinitely,”
he says, assuming you ignore physical wear and tear or energy
requirements.
</p>
<p>No end in sight</p>
<p>
Z is designed to loop through its 7918 instructions forever,
but if it did eventually stop, it would prove ZFC
inconsistent. Mathematicians wouldn’t be too panicked, though
– they could simply shift to a slightly stronger set of
axioms. Such axioms already exist, and could be used to prove
the behaviour of Z, but there is little to be gained in doing
so because there will always be a Turing machine to exceed any
axiom.
</p>
<p>
“One can think of any given axiom system as being like a
computer with a certain limited amount of memory or processing
power,” says Tao. “One could switch to a computer with even
more storage, but no matter how large an amount of storage
space the computer has, there will still exist some tasks that
are beyond its ability.”
</p>
<p>
But Aaronson and Yedidia have created two other machines that
might give mathematicians more pause. These will stop only if
two famous mathematical problems, long believed to be true,
are actually false. These are Goldbach’s conjecture, which
states that every even whole number greater than 2 is the sum
of two prime numbers, and the Riemann hypothesis, which says
that all prime numbers follow a certain pattern. The latter
forms the basis for parts of modern number theory, and
disproving it would be a major, if unlikely, upset.
</p>
<p>Practical benefits</p>
<p>
Practically, the pair have no intention of running their
Turing machines indefinitely in an attempt to prove these
problems wrong. It’s not a particularly efficient way to
attack that problem, says Lance Fortnow of the Georgia
Institute of Technology in Atlanta.
</p>
<p>
Expressing mathematical problems as Turing machines has a
different practical benefit: it helps to work out how complex
they are. The Goldbach machine has 4888 states, the Riemann
one has 5372, while Z has 7918, suggesting the ZFC problem is
the most complex of the three. “That would match most people’s
intuitions about these sorts of things,” Aaronson says.
</p>
<p>
Yedidia has placed his code online, and mathematicians may now
compete to reduce the size of these Turing machines, pushing
them to the limit. Already a commenter on Aaronson’s blog
claims to have created a 31-state Goldbach machine, although
the pair have yet to verify this.
</p>
<p>
Fortnow says the actual size of the Turing machines are
irrelevant. “The paper tells us that we can have very
compressed descriptions that already go beyond the power of
ZFC, but even if they were more compressed, it wouldn’t give
us much pause about the foundations of math,” he says.
</p>
<p>
But Aaronson says shrinking Z further could say something
interesting about the limits of the foundations of maths –
something Gödel and Turing are likely to have wanted to know.
“They might have said ‘that’s nice, but can you get 800
states? What about 80 states?’” Aaronson says. “I would like
to know if there is a 10-state machine whose behaviour is
independent of ZFC.”
</p>
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<h1 class="card-title">The mathematics behind poker</h1>
<div class="card-text-lead">
<p>
What's the difference between an average poker player and a
successful one? Some say that it's mainly down to luck and how
the cards fall in any given situation. Others say it's down to
strategy and their successful implementation as well as the
ability to bluff with confidence.
</p>
<p>
What a lot of people fail to mention is that poker is a game
based on mathematics, specifically probability and that is the
basis of every successful strategy. Luck and strategy on their
own are useless if you have little idea on the future
possibilities of the flop, the turn or the river. Just look at
what happened when man came up against machine.
</p>
<p>
In this article we'll show you how basic mathematics can
improve your poker game.
</p>
<p>What is probability?</p>
<p>
The strand of mathematics that concerns poker is probability,
which is the likely outcome of a certain event. In layman's
terms probability can best be explained by the toss of a coin
- when a coin is tossed, there are two possible outcomes,
heads or tails.
</p>
<p>
Therefore the likelihood of the coin landing on one outcome is
1 in 2 or 50%. In Poker the probabilities are harder to
calculate. In any given poker game the cards will be dealt
from a deck of 52 cards with 4 different suits.
</p>
<p>
The probability of getting a King in is therefore 4 in 52 or
7.7% and then the chances of getting a second King is 3 in 51
as one card is already missing from the deck, leaving you with
a 5.9% chance.
</p>
<p>
To work out your odds of receiving a pocket pair of Kings you
have to multiply the probabilities of receiving each card so
in this instance you would do the following sum;
</p>
<p>(4/52) x (3/51) =</p>
<p>(12/2652) =</p>
<p>(1/221) =</p>
<p>0.45%</p>
<p>
That means that your chances of landing pocket Kings are
incredibly low and that on average you should receive these
cards once in 221 hands. Don't worry though, you don't need to
get your calculator out and do these sums at the table, you
simply need to know the basic probabilities of receiving
certain hands.
</p>
<p>What are pot odds?</p>
<p>
In basic terms, pot odds are the ratio of the size of the pot
in play to the cost of your next potential move. Pot odds are
used by players to compare the chances of winning a hand with
a future card, so they can estimate the value of the call.
</p>
<p>
Pot odds are usually expressed in ratios as they are easier to
ascertain than percentages, but for ease and accessibility
percentages are used by TV broadcasters.
</p>
<p>How to work out your poker odd</p>
<p>
Working out your pot odds requires a bit of practice, once
you've got the basics sorted it will come as second nature to
you at the poker table.
</p>
<p>
To work out your pot odds you'll firstly need to calculate
your 'outs'. These are quite simply the cards that will help
you improve your hand and make it better than your opponents.
</p>
<p>
Take this situation for example, you've been dealt the queen
and nine of hearts, and the dealer lays out the ace of hearts,
the king of hearts and the seven and four of spades. That
means there are 9 hearts left in the deck, and you'll need
just one of them to appear on the river for you to win.
</p>
<p>
From the cards that we can see, there are 46 cards remaining
at play, meaning there are 37 cards that will see you lose and
9 that will see you win. In simple ratio terms that means you
are 4 times as likely to lose as you are to win, leaving you
with a 20% chance of success.
</p>
<p>
You might think that this seems quite complex and difficult to
implement mid game, but it really isn't. The best thing you
can do is enlist yourself in some soft or practice poker games
and give it a go until you get your head around it.
</p>
<p>
Working out your pre-flop odds is a little more complex and
will require a bit more skill, but again it's eminently
achievable. Take a look at some of the key things to look out
for below.
</p>
<p>
Premium hands: As discussed earlier in the article, the
chances of getting pocket picture pairs are incredibly low. So
it's best not to base your game on the chances of receiving
these hands, but if you do pull them your chances of success
will be considerably higher than your opponents.
</p>
<p>
The river flush: If you're just one card short of a full flush
after the flop, your chances of drawing a full flop on the
river are 34.97%, meaning you can be fairly confident of
winning the hand.
</p>
<p>
Suits: Some players will tell you that playing any two cards
because they're suited is a great tactic to employ. These
players haven't worked out the odds, they're simply telling
you about their anecdotal experiences. Playing two suited
hands only improves your chances of winning by a measly 2.5%
</p>
<p>
The river odds: By the time the game reaches the river, your
chances of making a pair increase by around 50%.
</p>
<p>
The better pair: On the occasion that two pairs go head to
head, the higher pair wins roughly 80% of the time. So if
you're holding queens, you might feel fairly confident, but be
wary, if your opponent raises and re-raises, the likelihood is
they're holding aces or kings
</p>
<p>
Its race time: A coin-flip or race as some players call it, is
simply a pair against two overcards because they each win
about half of the time. If overcards are suited, the pair will
win around 54% of the time, if they're not then it increases
to 57% of the time.
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