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https://plus.maths.org/content
enTricky teacups
https://plus.maths.org/content/tricky-teacups
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/tea_cups_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="max-width: 300px;"><img src="/content/sites/plus.maths.org/files/articles/2017/teacups/tea_cups.png" alt="Tea cups" width="300" height="258" />
<p></p>
</div>
<!-- Image taken from the science festival poster -->
<p>One of our favourite problems from our sister site <a href="http://nrich.mathsorg">NRICH</a> are the <A href="http://nrich.maths.org/7397">teacups</a>. Imagine you have four sets of cups and saucers. One set is red, one is blue, one is green and one is yellow. In each set there are four cups and four saucers, given you a total of sixteen cups and sixteen saucers all together. This means that there are a total of sixteen different cup-and-saucer combinations, which you
can arrange in a square 4x4 grid. </p>
<p>Here's the challenge. Can you arrange them so that: </p>
<ul>
<li> In any row there is only one cup of each colour</li>
<li> In any row there is only one saucer of each colour;</li>
<li>In any column there is only one cup of each colour</li>
<li>In any column there is only one saucer of each colour</li>
<li>Any cup-and-saucer combination (for example red cup, blue saucer) appears only once in the grid.</li>
</ul>
<div class="leftimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2017/teacups/grid.png" alt="Grid" width="250" height="247" />
<p>Can you arrange cups and saucers in the grid in a way that satisfies the rules? (You don't have to start with a blue cup and blue saucer in the top left corner.)</p>
</div>
<!-- Image from fotolia and made by MF -->
<p>In case you haven't got four sets of cups and saucers at home, we have made
<a href="/content/sites/plus.maths.org/files/articles/2017/teacups/cups_and_saucers.pdf">these paper templates</a> for you to cut out cups and saucers and play. Alternatively, if you can make it to this Saturday's Cambridge Science Festival <a href="https://www.sciencefestival.cam.ac.uk/events/maths-public-open-day">Maths Public Open day</a> there's a set of real cups and saucers for you to play with.</p>
<p>We love this challenge because it's a lot trickier than you might at first think. It also has an interesting history.</p>
<p>You may have heard about <em>Latin squares</em> before: they are arrangements of symbols (for example numbers) in a square grid so that every symbol occurs exactly once in every row and every column of the grid. The solution to a sudoku puzzle forms a Latin square on a 9x9 grid in which every number from 1 to 9 occurs exactly once in each row and column.</p>
<div class="rightimage" style="width: 183px;"><img src="/content/sites/plus.maths.org/files/articles/2017/teacups/sudoku.jpg" alt="Sudoku" width="183" height="188" />
<p>A completed sudoku grid forms a Latin square.</p>
</div>
<!-- Image from fotolia.com -->
<p>Our tea cups are slightly more challenging. Once you have solved the problem, the grid of cups and saucers form a Graeco-Latin square, in which there are two objects in every little square of the grid. Generally, mathematicians use symbols rather than cups and saucers when they construct such squares. For example, instead of cups and saucers you could have the numbers 1 to 4 and the letters A to D. A little square would then contain a pair made up of a letter and a number, such as A1 or D3. A Graeco-Latin square is an arrangement of symbols in a square grid, which obeys the following rules:</p>
<ul><li>Each square of the grid contains a pair of symbols, one from each of two kinds (one letter, one number).</li><li>
In every column of the grid, every symbol occurs exactly once. And the same is true for each row. (Each number, and each letter, exactly once in every column and every row).</li><li>
Every possible pair of objects occurs exactly once in the entire grid.</li></ul>
<p>Graeco-Latin squares were studied extensively by <A href="/content/five-eulers-best-0">Leonhard Euler</a>, one of the most prolific mathematicians of all time, at the end of the 18th century. Euler was inspired by a puzzle called the <em>36 officers problem</em>. Imagine there's a war and you're in command of an army that consists of six regiments, each containing six officers of six different ranks. Can you arrange the officers in a 6x6 grid so that each row and each column of the square holds only one officer from each regiment and only one officer from each rank? In this case, the two "symbols" in each little square are the rank and regiment of an officer. The difference to the teacup puzzle is that here we are dealing with a 6x6 grid, rather than a 4x4 grid.</p>
<p>Euler gave a lot of thought to the 36 officers problem, and eventually <a
href="http://eulerarchive.maa.org/docs/originals/E530_intro.pdf">concluded that</a>, "We have to admit that such an arrangement is impossible, though we can't give a rigorous demonstration of this". A proof that the problem is impossible to solve didn't come along until nearly 130 years later, when the French mathematician <a
href="https://en.wikipedia.org/wiki/Gaston_Tarry">Gaston Terry</a> provided it in 1901. </p>
<p>So what about our teacup puzzle? As you may already have found out, a solution does exist in this case. It would be cruel to present you with an unsolvable problem! In fact, there are a total of 1152 solutions. Once you have found one solution, you can immediately generate others: simply rotate or reflect the arrangement and the result is also a solution. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/teacups/saucers.JPG" alt="Saucer symmetry" width="427" height="428" />
<p style="max-width: 427px;">These solutions (involving only the saucers, so they form a Latin square
— we don't want to give the whole solution away) look different, but
they are related through rotations: rotate the first solution by 90 degrees
clockwise and you get the second (top right), another 90 degree
rotation gives the third solution (bottom left), and another gives the
fourth (bottom right).</p>
</div>
<!-- Image made by MF -->
<p>If you consider solutions related by such symmetries to be the same, because essentially they are, then the number of solutions shrinks to 144. The idea that problems and their solutions can exhibit symmetries is an important one in mathematics and theoretical physics, and often provides a powerful tool, not only to solve problems, but also to sniff out new lines of inquiry. </p>
<p>What if we are dealing with five sets of cups and saucers, or six, seven, or any other positive number? In that case we would be looking for a 5x5, 6x6, 7x7, or, more generally, an <em>n</em>x<em>n</em> Graeco-Latin square. Euler showed that <em>n</em>x<em>n</em> Graeco-Latin squares exist for all odd numbers <em>n</em> and for all numbers <em>n</em> that are divisible by 4. The number <em>n</em>=6 doesn't belong to either of these classes, and as we have seen with the 36 officer problem above, a 6x6 Graeco-Latin square doesn't exist. Euler believed that the same was true for any even number <em>n</em> that is not divisible by 4. In other words, that there is no <em>n</em>x<em>n</em> Graeco-Latin square for n=6, n=10, n=14, etc. So did many other people, until Euler was proved wrong in 1960. The mathematicians <a href="https://en.wikipedia.org/wiki/Raj_Chandra_Bose"> Raj Chandra Bose</a>, <a href="https://en.wikipedia.org/wiki/Sharadchandra_Shankar_Shrikhande">Sharadchandra Shankar Shrikhande</a> and <a href="https://en.wikipedia.org/wiki/E._T._Parker">E. T. Parker</a> enlisted the help of computers to prove that Graeco-Latin squares exist for all <em>n</em> except 2 and 6.</p>
<p>But Euler's work on Graeco-Latin squares wasn't a failure. As one would expect from a great mathematician, he persevered, and even though his initial conclusion was wrong, he came up with interesting and important new ideas in the process of tackling the problem. The 36 officer problem inspired Euler to contribute important work to an area of maths called <a href="/content/tags/combinatorics"><em>combinatorics</em></a>
which, as you may have guessed, is about combining objects in ways that satisfy particular constraints. It has many applications in the real world - from
designing schedules, such as school timetables or sports tournaments, to
detecting patterns, for example in DNA sequences or large amounts of data.</p>
</div></div></div>Fri, 24 Mar 2017 15:18:10 +0000mf3446818 at https://plus.maths.org/contenthttps://plus.maths.org/content/tricky-teacups#commentsWavelets catch Abel Prize
https://plus.maths.org/content/wavelets-catch-abel-prize
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/meyers_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/Abel17/meyers.jpg" alt="Sine wave" width="350" height="532" />
<p>Yves Meyer. Photo: B. Eymann/Académie des Sciences.</p>
</div>
<p>This year's <a href="http://www.abelprize.no">Abel Prize</a> has been awarded to Yves Meyer for the
development of an incredibly powerful mathematical tool: <em>wavelet theory</em>. The theory enables you to break
all sorts of different types of information into simpler
components which are easier to analyse, process and store — which is why it finds
applications in a huge range of areas, from medical imaging to the detection of
gravitational waves.</p>
<p> The inspiration for Meyer's work came, not from within
mathematics, but from the oil industry. In the 1980s the French
engineer <a href="https://en.wikipedia.org/wiki/Jean_Morlet">Jean Morlet</a> wondered how to best use seismic data to find
oil: if you send vibrations into the ground, how can you use their
echo to figure out where the oil is hiding? Morlet, together with the physicist <a href="https://en.wikipedia.org/wiki/Alex_Grossmann">Alex Grossmann</a>, came up with a way
of analysing the signals, and also coined the term "wavelet", but the
oil industry wasn't interested. Morlet's method wasn't used, but it
was published in a scientific journal.</p>
<div class="leftimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/Abel17/sine.png" alt="Sine wave" width="350" height="119" />
<p>This is a sine wave. It extends indefinitely to the left and right. Since sine waves are related to cosine, this could also be seen as a representation of a cosine wave.</p>
</div><!-- image made by MF -->
<p>Meyer came across Morlet's results and spotted their
potential. Mathematicians and engineers already had a powerful tool
for analysing and processing certain types of information. <em>Fourier
analysis</em>, as it's called, is best explained using sound as an
example. The sound of the middle A on a tuning fork is represented by
a perfect <em>sine wave</em>, such as the one seen on the left. Other sounds, like
that of a violin playing the same note, are more complicated. However,
it turns out that any periodic sound, in fact any type of periodic
<em>signal</em>, can be decomposed into a sum of
sine and cosine waves of different frequencies. This enables you to do a whole range of things.
For example, it enables you to manipulate the different frequencies of
a sound individually, or to "clean it up" by removing interfering
noices. (You can read more about <a href="/content/tags/fourier-analysis">Fourier analysis</a> on <em>Plus</em>.)</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/carola/Fourier_transform.gif"/><p style="max-width:300px"><p>
The function <em>f</em> varies in time –
representing a sound wave. The Fourier transform process takes <em>f</em> and decomposes it
into its constituent sine waves, with particular frequencies and
amplitudes. The Fourier transform is represented as spikes in the frequency domain, the height of the spike showing the amplitude of the wave of that frequency.</p></div>
<!-- movie in public domain - -->
<p>Fourier analysis is a versatile tool. It can also be used to
analyse and process images and other types of
information. However, it does come with a limitation: because the
basic components — the sine and cosine waves — are periodic,
Fourier analysis works best
for repeating signals. It's not so good on non-periodic signals, which
contain blips and spikes and all sorts of
other irregular features. An endlessly undulating wave is no good at localising a unique
feature such as a spike.
Unfortunately for Fourier analysis, most
real-life phenomena, from the sound of speech to seismic data,
fall in the non-periodic category. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/Abel17/speech.png"/><p style="max-width:433px">
<<p>This is waveform comes from human speech. It has regularities, but it's not periodic.</p></div>
<!-- Image made by MF - -->
<p>This is where wavelet theory comes in. The idea is to start, not
with a sine or cosine wave that's defined over all of the infinite
number line, but with a piece of an oscillating function defined over
just an interval. That's the <em>mother wavelet</em>. You can then form
<em>daughter wavelets</em> by squashing the mother wavelet to a
smaller interval (thereby increasing its frequency), expanding it over
a larger interval (thereby reducing its frequency), or just shifting
it along. A signal, such as the sound of human speech, is then
expressed as a combination of such a system of wavelets. Such a
decomposition enables you to capture repeating information in the
signal, but it also allows you to zoom in on local irregularities, such as spikes, using a
sequence of increasingly contracted versions of the mother
wavelet. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/Abel17/meyer.jpg"/><p style="max-width:264px"><p>
<p>This is an example of two versions of a wavelet. A squashed version of the wavelet (above) has a higher frequency than the original version (below).</p></div>
<!-- image concocted by MF - -->
<p> To store such a decomposition of a signal you only need the information
describing the original mother wavelet, as well as the contributions
of the various daughter wavelets: these are enough to put the original
signal back together again. This makes
wavelets a particularly useful tool for compression of
information. The FBI, for example, uses wavelets to store information
of fingerprints which, due to their high level of detail, would
otherwise take up a huge amount of storage space. </p>
<p>The origins of wavelet theory go quite a long way back. The
mathematician <a href="https://en.wikipedia.org/wiki/Alfréd_Haar">Alfréd Haar</a> already discovered a version of a
wavelet over a hundred years ago. But it was Meyer who gave the theory
solid mathematical foundations. He came up with families of wavelets
that exhibit the mathematical properties that are necessary for the
processing and analysis of signals and developed the general
mathematical framework of the theory.</p>
<p> <a href="https://en.wikipedia.org/wiki/Stéphane_Mallat">Stéphane Mallat</a>, who
collaborated with Meyer on wavelet theory, calls him a
"visionary" whose work doesn't belong to any one area such as pure
maths, applied maths, or computer science — it can only be labelled
"amazing". Apart from the prestige of the Abel Prize, awarded since
2003 to recognise "contributions of extraordinary depth and
influence to the mathematical sciences", Meyer can now enjoy the
765,000 Euros that come as prize money.</p>
<hr />
<h3>Further reading</h3>
<p>The <a href="http://www.abelprize.no/c69461/seksjon/vis.html?tid=69535">Abel
Prize website</a> has some lovely and accessible descriptions of
Meyer's work. Another good description of wavelets can be found in <a
href="http://www.pacm.princeton.edu/~ingrid/parlez-vous%20wavelets.pdf">this</a>
article from <em>What's happening in the mathematical sciences?</em></p></div></div></div>Thu, 23 Mar 2017 15:33:35 +0000mf3446817 at https://plus.maths.org/contenthttps://plus.maths.org/content/wavelets-catch-abel-prize#commentsContextuality — The most quantum thing
https://plus.maths.org/content/contextuality-most-quantum-thing
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/puzzle-icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Brendan Foster</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>What happens when we're not looking? What happens when we do
look? These two questions are the heart of the mystery of quantum
mechanics.</p>
<div style="float:right; margin-left: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/ZJAnixX8T4U?rel=0" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;">Watch <a href="http://trin-hosts.trin.cam.ac.uk/fellows/butterfield/">Jeremy Butterfield</a>, philosopher of physics at the University of Cambridge, explain contextuality.</p>
</div>
<p>By looking we really mean observing: for example, observing an experiment involving the smallest building blocks of matter, such as electrons. Quantum mechanics tells us that while we are not looking, such particles behave very differently from when we are looking. Before an observation is made, the particles do not occupy specific places in space and they do not travel at well-defined speeds. It's only when we make an observation, that particles assume specific locations and speeds (see <a href="/content/watch-and-learn">here</a> for more details). So quantum mechanics tells us that something special happens when an observation
is made. But it doesn't tell us why or how that
happens. It doesn't tell us what counts as an observation or an observer.</p>
<p>Quantum mechanics doesn't give us the whole story of what happens in
between observations. It does tell us what happens to the mathematical object describing a system, the <em>wave function</em>,
during those times. But what about the electrons and photons, and all the
other stuff that makes up the world? What exactly is that stuff doing when
we are not observing it?</p>
<p>Although quantum mechanics doesn't give us the answers to these questions, we do know something about the answers, if they exist. We know that
they must be different from the answers people had before quantum
mechanics, in the days of Newton and classical physics. The biggest difference may be something called <em>contextuality</em> — a part
of the complicated relationship between observers and observations. It
may be the most quantum thing about quantum mechanics.</p>
<h3>The meaning of contextuality</h3>
<p>
<p>"Context" means all the details that surround an event. Context gives us
more information about what happened and how it happened: what else
happened at the same time, or just before, or just after.</p>
<div class="leftimage" style="max-width: 283px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/lama.jpg" alt="llama" width="424" height="300" />
<p>Not what you'd expect to find in your garage.</p>
</div>
<!-- Image from fotolia -->
<p>I asked physicist <a href=" http://www.anselm.edu/Academics/Majors-and-Departments/Physics/Faculty/Ian-Durham.htm">Ian Durham</a> of St. Anselm College, in New Hampshire
USA, to summarise it in a sentence. He said, "When you go to your garage,
you get a car; you don't get a llama." Is a llama in your parking space unusual? Most likely very unusual if you
live in a city like London. But maybe not if you live on a ranch in the Andes
Mountains. The context — where you live — adds meaning to the basic
event — a llama in your garage.</p>
<p>Contextuality in physics describes how or whether the details of an
observation affect what is observed. We mean that the results of
measurements can depend on how we made the measurement, or what
combination of measurements we chose to do.</p>
<p>In physics this applies to measurements of quantities such an electron's <em><a href="https://en.wikipedia.org/wiki/Spin_(physics)">spin</a></em>: the answer you get for a particular direction of spin may vary depending on what you choose to measure alongside it. To see what this would mean if it applied to things we are familiar with, imagine measuring a person's height. If contextuality applied in this context, you might get a different value for height if you measured the person's weight along side it, than you would get if you measured the person's shoe size alongside it.</p>
<p>Is this weird? Maybe. It's weird if you think an
electron's spin (or a person's height) had a preset value before you measured it, and you think
the whole point of a measurement is to give you a true report of that value.
You would expect to get that one answer when you make the
measurement.</p>
<h3>Definite theories</h3>
<p>Durham suggested another way to picture contextuality. Imagine you're
putting together a jigsaw puzzle. Somehow the box got lost, so you don't
know what the picture is supposed to be. You put the puzzle together
starting from the outer edges, and you find a picture of a cat. Then you
break the puzzle up and put it together again, now starting from the inner
pieces. It's the same pieces to the same puzzle, but this time you find the
picture is — a llama.</p>
<div class="rightimage" style="max-width: 354px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/puzzle.jpg" alt="puzzle" width="354" height="275" />
<p>How is reality pieced together?</p>
</div>
<!-- Image from fotolia -->
<p>This is weird if you think the story of the world should have a prefixed
image, like a normal jigsaw puzzle. But does it? Or is the story of the world
made up as it goes along, and changed by our choices?</p>
<p>Quantum mechanics doesn't tell us what electrons are doing when we
are not observing those values. The electrons and other particles live
secret, unknowable lives, as far as we can predict. A theory that tells us
more might give us a complete picture of what electrons are doing at all
times. It would also tell us the values of things we can measure like
momentum or spin, even when we are not trying to measure them.</p>
<p>Classical Newtonian mechanics is a theory that has these features.
Classical particles are like rocks. They have concrete positions and speeds.
They have a real story about what they are doing when we don't look at
them. Experiments show us the true values of those things.</p>
<p>Philosophers of science call this feature <em>value definiteness</em>, because the
measurable things have definite values. So what about quantum
mechanics? Could there be a "definite" theory underneath quantum
mechanics? A theory that would give us a real story about the electrons,
and that delivers the same predictions as quantum mechanics for
experiments?</p>
<h3>The Kochen-Specker theorem and contextuality</h3>
<p>An argument called the Kochen–Specker theorem tells us what we can
and can't have. The theorem is named for mathematicians Simon Kochen
and Ernst Specker, who published their work in 1967. The physicist John
Bell found a similar result around the same time and various other people have
come up with simpler versions of the proof since then. Here's what it tells
us.</p>
<p>Imagine you have invented a new theory to explain what quantum
particles are really doing, even when we are not watching them. Your
theory delivers the same predications as quantum mechanics for
observations. But your theory has an added feature — it gives us a concrete
story about the world. Electrons have definite speeds and spins and so on.
Their speeds and spins have definite values that exist even when we are not
trying to detect them. When we do try to detect them, your theory predicts
that tests will show us those values that the speed and spin had just before
we did the test. In other words, you theory has value definiteness. (An example of such a theory is <em>pilot wave theory</em>, you can find out more <a href="/content/riding-pilot-wave">here</a>.)</p>
<div class="leftimage" style="max-width: 300px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/sphere3.png" alt="Sphere with axes" width="300" height="264" />
<p>The Kochen-Specker theorem can be proved by a relatively simple argument involving colouring arrows within a sphere. See <a href="/content/kochen-specker-theorem">here</a> to find out more.</p>
</div>
<!-- Image made by MF -->
<p>The Kochen–Specker theorem tells us that your theory must be
contextual. The values that your electron has for its speed and spin, those
values must somehow or sometimes depend on how you try to measure
them.</p>
<p>It may be tricky to give a clear explanation of how contextual effects
could actually happen without upsetting those preset, definite values, or
without changing what sort of things can have preset values. The preset values are not just
"the spin of an electron". They must be things like "the spin as measured in this particular
detector, while these other particular things are also measured," and so
on. We seem to lose the point that we are trying to save – that there is
something real in the world that exists outside of our particular
observations.</p>
<p>There is an alternative – you can try to change your theory to avoid
contextuality. The Kochen–Specker theorem says you must then give up
value definiteness. The way you choose to measure things won't effect the
values you get in the experiments, but those values didn't exist before you
did the measurements! You can't say the electron had that speed and that
spin until after you ran it into the detector and measured those values.
Before the detector, we cannot say what is happening in the world.</p>
<p>This non-contextual option takes us back to the starting point —
quantum mechanics as a mysterious theory that tells us nothing about
what happens when we do not observe the world. Some are bothered by
this option, but for some people, this option is not a problem. This is just
quantum mechanics' message.</p>
<p>(Incidentally, the Kochen-Specker theorem has a surprisingly straightforward proof, involving colouring a sphere with two different colours — you can find out more <a href="/content/kochen-specker-theorem">here</a>.)</p>
<h3>Beyond quantum mechanics</h3>
<div class="rightimage" style="max-width: 284px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/camera.jpg" alt="camera" width="284" height="300" />
<p>Despite the various research efforts, the role of the observer in physics is still being debated.</p>
</div>
<!-- Image from fotolia -->
<p>Researchers continue to think about contextuality and to argue about
whether we should want to avoid it. They have adapted the Kochen–Specker theorem to clarify the meaning of contextuality and to come up
with different ways to talk about it.</p>
<p>An influential result came in 2004, when physicist <a href=" https://www.perimeterinstitute.ca/people/robert-spekkens">Rob Spekkens</a> from
Canada's Perimeter Institute found a way to think about contextuality in
theories besides quantum mechanics. His work makes it possible to ask if we can use a different theory in order to avoid contextuality, and what other weirdness we have to accept to avoid it.
</p>
<p>Some people are looking for creative ways around the Kochen–Specker
theorem. <a href=" https://www.chapman.edu/our-faculty/matt-leifer">Matt Leifer</a> of Chapman University in California told me he hopes
we can avoid contextuality through <em><a href="http://fqxi.org/grants/large/awardees/view/__details/2015/leifer">retrocausality</a></em> — that is by allowing
future events to have an effect on the past. But even allowing that, he is
not sure that we can get away from all appearances of contextuality. It may
be a very deep part of the real world. He describes it as a sort of "unifying
principle" for many other quantum things, like <em>quantum tunnelling</em>. If you want to
avoid contextuality, you will still have to explain these things.
</p><p>So is contextuality weird? Is a contextual world a problem? The answer
for now is maybe – it all depends on how you look at it.</p>
<hr>
<h3>About this article</h3>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/brendan.jpg" alt="Brendan" width="250" height="245" />
<p></p>
</div>
<p>Brendan Foster is a writer, musician, and sometime physicist in Baltimore, Maryland, USA. He is also science programs coordinator for the <a href="http://fqxi.org/home">Foundational Questions Institute</a>, and cohost of the FQXi podcast, with Zeeya Merali and (sometimes) Puffy the Cat.</p>
<p><em>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-context-everything">here</a> to see more articles and videos about contextuality. </em></p>
<div class="leftimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div></div></div></div>Tue, 21 Mar 2017 16:49:20 +0000mf3446777 at https://plus.maths.org/contenthttps://plus.maths.org/content/contextuality-most-quantum-thing#commentsThe quantum reality paradox
https://plus.maths.org/content/quantum-reality-paradox
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/sphere_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Kate Becker</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p><em>This article first appeared on the <a href="http://www.fqxi.org/community">FQXi community website</a>. FQXi are our partners in our <a href="/content/whos-watching-context-everything">Who's watching? The physics of observers project</a>, within which we explore quantum contextuality. Click <a href="/content/whos-watching-context-everything">here</a> to find out more about contextuality. </em></p>
<hr/>
<p>Is the concept of an omniscient God compatible with the laws of physics? In the 1960s, Swiss mathematician <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Specker.html">Ernst Specker</a> set out to find the answer. His investigation led him, along with American mathematician <a href="https://www.math.princeton.edu/directory/simon-kochen">Simon Kochen</a>, to codify one of the strangest rules of quantum reality: <A href="/content/contextuality-most-quantum-thing">contextuality</a>.</p>
<p>To understand the oddness of contextually, imagine that choosing to step on the scale and measure your weight could change your eye color. That's the everyday equivalent of what seems to happen when we decide how to measure the quantum properties of a particle.</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/guhne_small.jpg" alt="magic" width="350" height="263" />
<p>Jan-Åke Larsson, Adán Cabello, Mile Gu, Karoline Wiesner, & Otfried Gühne
at a quantum contextuality workshop in Siegen, September 2014.
Image: Alex Monrás</p></div>
<p> Physicists <a href="http://www.physik.uni-siegen.de/tqo/members/guehne/">Otfried Gühne</a>, at the University of Siegen, in Germany, <a href="http://personal.us.es/adan/home.htm">Adán Cabello</a>, at the University of Seville, in Spain, and <a href="http://people.isy.liu.se/jalar/">Jan-Åke Larsson</a>, at the University of Linköping, in Sweden, are investigating this surprising brand of quantum weirdness. The team is looking to the future to see if quantum contextuality could one day drive more powerful computation. They are also digging back into the past to learn about the religious quest that led to the concept of contextually.</p>
<h3>Quantum coin flipping</h3>
<p>Quantum theory often feels like a piling-on of the bizarre. We're told, for instance, that the simple act of making a measurement can change a particle's state in an unpredictable manner. But contextuality is a fresh violation of our intuition about how the world is supposed to work. To illustrate just how counterintuitive it is, Larsson offers the example of a coin flip. Imagine taking three coins from your pocket—a penny, a nickel, and a dime, for example—and flipping any two of them. Say that you choose to flip the penny and the dime. The outcome of the dime flip shouldn't depend on whether you flipped it along with the penny or along with the nickel—and, of course, in the everyday world, it does not.</p>
<p>Now transpose this scenario to the quantum world. Instead of pulling two of three coins from your pocket, imagine that you have a particle on which you can make two of three possible measurements. The measurements are all of the same property, a quantity called the <em>magnetic quadrupole moment</em> (which is equivalent to the value of the particle's spin, squared). The quadrupole moment is three-dimensional, and you can choose two of three axes along which to measure it: the x-axis, the y-axis, and a third which is 45 degrees away from the y-axis but still perpendicular to the x-axis. Just like the coin toss, you don't expect that your choice to measure along the second or third axes should affect the result you get for the measurement along x-axis. Yet, in the quantum world, it does. This strange link between the choice of one measurement and the result of a second is called contextually.</p>
<p>Though physicists had suspected that quantum contextually might hold since the early days of the field, the idea wasn't formally described until 1967, when Kochen and Specker devised a mathematically elegant proof. The duo wanted to see if a more intuitive deterministic picture of reality—such as the one preferred by Einstein—could explain the outcome of particle experiments. In this view, particles do not pick their properties in an instant, at the point of measurement, apparently on a whim, as standard quantum theory asserts. Instead, the particles contain "hidden variables" that determine the outcome of any future measurements in a predictable manner. These hidden variables provide a complicated set of instructions, even pre-programming particles to return different responses to the same measurement, depending on the order in which measurements are carried out—thus mimicking contextuality. But crucially, since experimenters have no access to these hidden variables, to them, the results appear to be mysterious and are impossible to predict before the experiment is performed.</p>
<p>With their theorem, Kochen and Specker showed that such "non-contextual" theories that invoke hidden variables cannot explain the outcome of quantum measurements without hitting a paradox. The mathematicians achieved this by considering how such information might be stored within a single particle, and proved that there's simply no way to encode these instructions so that you cover every possible different result that could be seen in an experiment. These theories are fundamentally at odds with quantum mechanics.</p>
<p>But to test the <a href="/content/kochen-specker-theorem">Kochen-Specker theorem</a>, physicists would have to make a series of simultaneous (or nearly simultaneous) measurements on a single particle. This experiment was out of reach when the theorem was first published because, at the time, measuring the state of a particle demolished it, preventing further tests. "It's a very beautiful result, but it's a mathematical one," says Cabello. With no way to test the theorem experimentally, it languished for almost 40 years.</p>
<h3>Non-demolition measurements</h3>
<p>Finally, around 2003, a new delicate technique for making "non-demolition measurements" enabled physicists to test the theorem for the first time. It allowed physicists to preserve the state of an ion in a trap, even after measuring it. In 2009, Cabello and his colleagues performed the first lab experiment affirming a simplified form of the Kochen-Specker theorem. A more rigorous test, also by Cabello and colleagues, following exactly the measurements that Kochen and Specker laid out, came in 2013.</p>
<div class="leftimage" style="max-width: 360px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/guhne_ks.jpg" alt="magic" width="360" height="360" />
<p>The Kochen Specker theorem considers a number of directions along
which a particle's properties can be fixed before measurement.
Image: Jan-Åke Larsson</p></div>
<p>"Vindication" was Cabello's first reaction, he recalls. But this was only the beginning. "Now I realise they provided the ground needed for the next set of questions: why quantum theory is as contextual as it is? What is quantum theory really telling us about the world?"</p>
<p>To answer those questions, Gühne, Cabello, and Larsson must address the gap between ideal experiments, in which measurements are made simultaneously, and real ones, in which the measurements are made in sequence. "When you do these measurements, you can't perform them ideally. That's a fact of life," says Larsson. "There can be delays—will be delays—between the detector that measures one property and the one that measures the other." What inaccuracies might follow from these delays? That is something that the researchers are now trying to figure out.</p>
<p>Bizarre as it seems, contextuality may also provide insight into why quantum computers could be so powerful at solving certain classes of problems. Despite decades of progress in building quantum computers, physicists still disagree about exactly which quantum effect gives them an edge. Last year, a team of researchers from Ireland and Canada <a href="http://www.nature.com/nature/journal/v510/n7505/full/nature13460.html">argued in <em>Nature</em></a> that contextually is the critical ingredient driving error correction in quantum computers. But some argue that contextually may even be the source of the speed-up that would make <a href="/content/quantum-computers-get-real">quantum computers outperform classical machines</a>. (See also, <em><a href="/content/quantum-context">Quantum in context</a></em>.) The connection between the power of quantum computers and contextuality "is almost one-to-one," says Cabello. "You have this power because you have this magic."</p>
<p>The practical benefits of contextuality are still up for discussion among researchers who have been meeting regularly at a series of FQXi-funded workshops in Spain, Germany, Singapore, and Sweden. "Definitely we had a heated debate on contextually," says FQXi member Pawel Kurzynski, a physicist at the Centre for Quantum Technologies (CQT) at the National University of Singapore, who participated in the Spanish workshop, and co-organised the Singapore meeting.
</p>
<p>The Swedish meeting, which took place in August 2015, emphasised the philosophy and history of field. Cabello has recorded about a dozen video interviews with key contextuality researchers so far, including Kochen and Specker (Specker died in 2011). "You start filming, and there's always some point at which you capture something special," says Cabello.</p>
<h3>God's omniscience?</h3>
<p>It was through these interviews that Cabello discovered that Specker wanted to study quantum reality—whether it was indeterministic, as standard quantum theory asserts, or could instead be described by a deterministic hidden-variables theory—in part, to better understand deep religious questions. Could God be omniscient, if reality itself is not set until it is observed by humans?</p>
<p>Cabello believes that contextuality does not force physicists to give up a belief that there is real world out there. But perhaps it is not as concrete as some might wish. "We do believe in a reality," says Cabello. "An electron has an electric charge, for instance." But other properties of the quantum system have no deeper reality; they are created by the act of measurement, he says.</p>
<p>Gühne, Cabello, Larsson and their colleagues underscore that view with new research uncovering flaws in those interpretations of quantum mechanics that assume that the quantum properties we measure are reflections of some intrinsic reality that exists independently of the act of measuring. Their conclusion comes from an unlikely source: thermodynamics, the same branch of physics that describes classical systems like steam engines. They found that, under certain assumptions, if quantum probabilities are determined by intrinsic properties, a quantum system could release infinite heat with each successive measurement. This rules out such models. "Ultimately," they write, "our work indicates that the long-standing question, 'Do the outcomes of experiments on quantum systems correspond to intrinsic properties?' is not purely metaphysical."</p>
<p>The team now aims to create a web archive of their contextulaity video interviews, for both experts and the general public. "Many scientists are not really interested in history of science because they are absorbed in their daily research," says FQXi member <a href="http://www.quantumlah.org/people/dagomir">Dagomir Kaszlikowski</a>, a contextually expert also at CQT. But, he adds—possibly with an intentional pun—knowing the historical "context" of the theory's origins is crucial, to avoid misunderstanding what it is really about.</p></div></div></div>Tue, 21 Mar 2017 15:34:26 +0000mf3446781 at https://plus.maths.org/contenthttps://plus.maths.org/content/quantum-reality-paradox#commentsWho's watching: Context is everything
https://plus.maths.org/content/whos-watching-context-everything
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/puzzle-icon_0.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div><p>Quantum mechanics appears to say that the world at its smallest scales is fuzzy. Little particles, such as electrons, don't have precise locations in space and they don't travel at well-defined speeds, for example. It's only when we look, that is, when we make a measurement, that reality somehow "snaps" into place and particles are found sitting at well-defined places and moving with well-defined speeds. The exact values of properties like location and speed appear to be chosen at random.</p>
<div style="float:right; margin-left: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/ZJAnixX8T4U?rel=0" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;"><a href="http://trin-hosts.trin.cam.ac.uk/fellows/butterfield/">Jeremy Butterfield</a>, philosopher of physics at the University of Cambridge, explains contextuality.</p>
</div>
<p>The predictions of quantum mechanics have been tested endlessly in experiments and they hold true, but could it be that the theory simply isn't complete? Could there be <em>hidden variables</em> — some extra information — which, if we include them, give us a theory that isn't random or fuzzy? The answer is yes, but not without a price. Such a theory will always exhibit something called <em>contextuality</em>: the outcome of a measurement will be heavily distorted by your experimental set-up, so try as you might, you can never be an impartial observer. The following articles and video explore this concept of contextuality and related topics.</p>
<p><em>These articles and videos are part of our <a href="/content/whos-watching-physics-observers">Who's watching: The physics of observers</a> project, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with the Foundational Questions Institute</a>. </em>
</p>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/puzzle-icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/contextuality-most-quantum-thing">Contextuality: The most quantum thing</a> — What exactly is contextually and how do we know that we can't get away from it? </p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/wave-icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/riding-pilot-wave">Riding the pilot wave</a> — Pilot wave theory is an extension of quantum mechanics that isn't random or fuzzy: there's a sharply defined reality that's there even when we're not looking. However, pilot wave theory is contextual. Here is a quick introduction.</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/pens_icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/kochen-specker-theorem">Colouring by numbers: The Kochen-Specker theorem</a> — A quick look at the theorem that delivers contextuality. Its proof can be thought of as a colouring problem!</p></div>
<a name="fqxi"></a>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; "><p>
The following articles first appeared on the <a href="http://www.fqxi.org/community">FQXi</a> website.</p>
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/5/18_sep_2015_-_1235/kandk_magic_icon.jpg?1442576108" alt="" width="100" height="100" /> </div>
<p><a href="/content/quantum-context">Quantum in context</a> — Contextuality could provide the magic needed for quantum computation, and perhaps even open the door to time travel.</p>
<br class="brclear" />
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/sphere_icon.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/quantum-reality-paradox">The quantum reality paradox</a> — What does contextually mean for real-life measurements and what does it have to do with religious questions?</p>
</div></div></div></div>Tue, 21 Mar 2017 15:20:46 +0000mf3446780 at https://plus.maths.org/contenthttps://plus.maths.org/content/whos-watching-context-everything#commentsRiding the pilot wave
https://plus.maths.org/content/riding-pilot-wave
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/wave-icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Is the Moon still there when you're not looking at it? You can't know because you'd have to look at it to check, but you probably think that it is. Why should the existence of something as neutral as the Moon depend on where we are looking?</p>
<div class="rightimage" style="width: 400px;"><img src="/content/sites/plus.maths.org/files/packages/2016/schrodingerscat/cat_image.jpg" alt="Cat" width="400" height="287" />
<p>The idea that a particle does not have a well-defined position in space but can be in several places at once can be translated into <a href="/content/who-killed-schrodingers-cat">Schrödinger's famous thought experiment</a> involving a cat that is simultaneously dead and alive.</p>
</div><!-- image from postcard created by Charles Trevelyan -->
<p> Yet, the question, which was
posed for illustrative purposes by Albert Einstein, highlights one of
the major problems with quantum mechanics. While the theory doesn't apply to large things such as the Moon, it seems to suggest that
reality, at the smallest scales, is fuzzy. According to the standard interpretation of quantum mechanics, particles, such as electrons or protons, don't have
precise locations in space, they don't travel at
well-defined speeds, and they don't have precise values for other physical quantities either. Only when we look — when we make a measurement — does a
particle "assume" a location or speed through some mysterious mechanisms nobody understands. (See <a href="/content/watch-and-learn">here</a> to find out more about this problem of measurement.)</p>
<p>What's also strange is that
nature appears fundamentally random. The location or speed a particle assumes when we measure it appears as if picked out of a hat. To use Einstein's famous phrase, it seems that God plays dice with nature. </p>
<p>Physicists believe that nature behaves in this strange way because of a mathematical object called the <em>wave function</em>. Just like Newton's laws of motion describe the behaviour of macroscopic objects, so the wave function is believed to describe the behaviour of tiny particles. You can use the wave function to calculate the probabilities of possible outcomes of an experiment, such as the particle appearing at point A or point B. Its predictions have been tested to a higher degree of accuracy than any other theory in physics and appear to be correct. But if you take the wave function as gospel, then you have to accept that, in general, particles really don't have well-defined values for properties like position or speed. The wave function simply doesn't speak that kind of language.</p>
<p>But why should we take the wave function as gospel? It does make correct predictions, so it's clearly onto something, but perhaps it's not all there is.
Could we perhaps add something to quantum mechanics, to create a new theory which isn't random and gives us more of a clue as to
what particles are doing when we are not looking?</p>
<h3>Ride the wave</h3>
<p>The answer is yes. An example is <em>pilot wave
theory</em>, first proposed, and then abandoned, by one of the originators of quantum
mechanics, <a href="https://en.wikipedia.org/wiki/Louis_de_Broglie">Louis de Broglie</a>, and later developed further by <a href="https://en.wikipedia.org/wiki/David_Bohm">David Bohm</a>. De Broglie insisted that particles as we normally think of them do exist. They travel through space along well-defined paths and at every moment in time have a precise location, speed and so on. Those paths, however, are not what we'd expect them to be from ordinary physics. The particles are guided, not by Newton's laws of motion, but by the wave function from quantum mechanics. "[The wave function] has another role, apart from the practical role of spitting
out probabilities of outcomes of measurements," explains <a href="http://trin-hosts.trin.cam.ac.uk/fellows/butterfield/">Jeremy Butterfield</a>, a philosopher of physics at the University of Cambridge. "It also tells the individual particle where to go. It guides it. It's like a pilot of a
big ocean liner." Hence the name "pilot wave theory".</p>
<div class="leftimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/wave_0.jpg" alt="Surfer" width="350" height="234" />
<p>A surfer's path is guided by the wave he or she is riding. According to pilot wave theory, a particle's path is also guided by a wave.</p>
</div><!-- image from postcard created by Charles Trevelyan -->
<p>In contrast to ordinary quantum mechanics, pilot wave theory doesn't involve any fundamental randomness at all: particles don't just turn up in unpredictable places when we make a measurement. However, calculating their exact paths using pilot wave theory is a bit like predicting the outcome of a horse race. There are so many factors involved — the form of each horse, the quality of the turf, the jockeys' mood — you really have no chance in hell of getting it right. People resort to probabilities instead and pilot wave theory allows us to do the same. We might not be able to predict for certain where a particle will be when we measure it, but at least we can work out what chance it's got of being found at a particular spot. In both cases, the uncertainty involved is not down to nature being fundamentally random, but to our ignorance of the many factors involved.</p>
<p>This kind of use of probabilities to deal with our ignorance wasn't a new thing in physics when de Broglie first developed his theory. When people first studied the behaviour of gases in the 19th century, they soon realised they had no chance of tracking the motion of each individual molecule, and resorted to statistics instead, ending up with a theory now known as <em>statistical mechanics</em>. </p>
<p> Luckily (or rather, by de Broglie's ingenious design) the probabilities given by pilot wave theory match those given by quantum mechanics — so since quantum mechanics agrees with experiments, pilot wave theory does too. The crucial difference is that in ordinary quantum mechanics, the uncertainty is due to nature being fundamentally random, while in pilot wave theory it's due to our ignorance of all the details involved, just like in statistical mechanics.</p>
<h3>Who's looking?</h3>
<p>All this sounds remarkably sane, but it wouldn't be quantum physics if all the weirdness had gone away. And it hasn't. The theory says that any act of measurement interferes with the system you're measuring so much, it changes it drastically. "When you trundle on some massive [measuring] apparatus, which is billions of times bigger than your tiny [particle], you radically disturb it," explains Butterfield. The outcome of a measurement depends sensitively on the details of your measuring apparatus, and in particular, it depends on which quantities you decide to measure at the same time. And because the disturbance is so radical, that outcome is liable to tell you very little about what the system was actually doing before you looked. "In this view measurement is distorting, it's disturbing, it's not truth-telling — it's fibbing to you," says Butterfield. Even though the theory admits a definite reality that exists when we are not looking, you'll never catch that reality unawares. Physicists call this phenomenon <em>quantum contextuality</em> (you can find out more about contextuality <a href="/content/contextuality-most-quantum-thing">here</a>).</p>
<p>Crucially, this also means that certain properties can't be simultaneously measured to any desired degree of accuracy. The position and the momentum of a particle are an example: measuring the position disturbs the particle in a way that makes it hard to measure momentum and vice versa. This awkward relationship between position and momentum is summed up in Heisenberg's famous <em>uncertainty principle</em> from ordinary quantum mechanics. However, in ordinary quantum mechanics the principle is regarded as a fundamental uncertainty of nature, while in pilot wave theory it's down to in principle limitations of measurement.</p>
<div style="float:right; margin-left: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/WIyTZDHuarQ" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;">In this video Derek Muller shows how silicone oil droplets bouncing on a pool of oil come with their own "pilot waves" and can replicated quantum effects such as the double slit experiment. </p>
</div>
<p>Contextuality is also to blame for the most shocking aspect of pilot wave theory: "spooky action at a distance" to quote Einstein again. Several particles may be guided by the same wave function, and there are no geographical bounds to how far these particles can move apart. But because the guiding wave is the same, a measurement that affects a particle on Earth can also instantaneous affect another particle several hundred light years away.
This weird effect sat uncomfortably with many of the originators of quantum mechanics, including Einstein, but since then it has been proved that quantum physics simply can't get away from it. And the phenomenon has also been verified in experiments: it's what physicists call <em>quantum entanglement</em>. </p>
<p>Pilot wave theory isn't the mainstream version of quantum mechanics. That's partly because it's hard to reconcile with Einstein's special relativity. It's only in the last twenty to thirty years that it has found slightly wider acceptance among physicists. A special boost came from a discovery involving droplets of silicone oil: when bouncing on a puddle of oil, these little droplets produce their own pilot waves and replicate the behaviour produced by pilot wave theory. They are guided by their pilot waves and, even though they themselves aren't quantum objects, replicate quantum phenomena. For example, they can create the same wave-like interference patterns that electrons produce in the famous <em><a href="/content/physics-minute-double-slit-experiment-0">double slit experiment</a></em>. The video above explains how.</p>
<p>So what's right, pilot wave theory, orthodox quantum mechanics, or something else? Nobody knows and it doesn't seem likely we'll find out any time soon. To quote Nobel laureate <a href="https://web2.ph.utexas.edu/~weintech/weinberg.html">Steven Weinberg</a>, <a href="http://www.nybooks.com/articles/2017/01/19/trouble-with-quantum-mechanics/">writing in the New York Review of Books</a>, "Regarding [...] the future of quantum mechanics, I have to echo Viola in <em>Twelfth Night</em>: 'O time, thou must untangle this, not I.'" </p>
<hr>
<h3>About this article</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. She interviewed Jeremy Butterfield in Cambridge in February 2017. </p>
<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p><em>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-context-everything">here</a> to see more articles and videos about contextuality. </em></p>
</div></div></div>Tue, 21 Mar 2017 14:01:28 +0000mf3446776 at https://plus.maths.org/contenthttps://plus.maths.org/content/riding-pilot-wave#commentsFace fusion
https://plus.maths.org/content/face-fusion
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/obama_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Would you like a computer game avatar that looks just like you? Or enhance your features with those of someone incredibly beautiful? If yes, then a bit of maths will come in handy.</p>
<div class="rightimage" style="max-width: 248px;"><img src="/content/sites/plus.maths.org/files/news/2017/Facefusion/obama_fused.jpg" alt="Face fusion" width="248" height="243" />
<p>Who's this? <br/> (Original image by <a href="https://commons.wikimedia.org/wiki/File:Official_portrait_of_Barack_Obama.jpg">Pete Souza</a>, <a href="https://creativecommons.org/licenses/by/3.0/deed.en">CC BY 3.0</a>. Image fusion by Martin Benning, Michael Möller, Raz Z. Nossek, Martin Burger, Daniel Cremers, Guy Gilboa, and Carola-Bibiane Schönlieb.)</p>
</div>
<!-- Image used by permission -->
<p>At this year's <a href="https://www.sciencefestival.cam.ac.uk">Cambridge Science Festival</a> mathematicians from the <a href="http://www.damtp.cam.ac.uk/research/cia/">Cambridge Image Analysis Group</a> will present software that performs
<em>face fusion</em>: if you enter an image of your face and an image of someone else's face, the software will merge the two, resulting in a strange combination of both. The idea behind it is familiar from music, when different tracks of a song are mixed together. A sound can be decomposed into its component frequencies, and you can then adjust those frequencies separately using an equaliser (see <a href="/content/career-interview-audio-software-engineer">this article</a> to find out more). In the case of the images, it's not sound frequencies that are isolated, but structures within an image that display a particular level of detail. For example, the wrinkles on someone's face display a particular level of detail (hopefully a very fine one). If you can isolate structures with that level of detail and adjust them, you can enhance or diminish the wrinkles. Alternatively, you could replace them with the wrinkles you've isolated on someone else's face — that's the basic idea behind our face fusion. And it can all be done mathematically.</p>
<p>At the heart of the face fusion process lies a type of <em>diffusion equation</em>. The most famous diffusion equation is the <em>heat equation</em>, which describes how heat spreads (diffuses) through a material. For example when you hold one end of a metal rod held into a fire the heat will gradually spread towards your hand. When it comes to image analysis and image processing, we can use diffusion equations, not to spread heat around, but to spread information from the various pixels that make up the image. The diffusion equation used here is more complicated than the ordinary heat equation, and it preserves the larger features of the image, such as the overall shape of the face. It is known as the total variation (TV) flow and it is represented by a <em>differential equation</em>.</p>
<p>The result of applying the TV flow is a blurring of the image: as you can see in figure 1, more and more details are removed as you move through the image sequence, with the image becoming more and more blurred.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/Facefusion/figure1.png" alt="Blurring of image." width="600" height="170" /><p style="max-width: 600px;">Figure 1: Example of the TV flow blurring an image. (Underlying image by <a href="https://commons.wikimedia.org/wiki/File:Yang_Liwei.jpg">Dyor</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>. Image processing by Chris Irving.)</p>
</div>
<!-- Image used by permission -->
<p>Since the TV flow is a mathematical object, you can also use mathematics to observe exactly how the flow changes over time. That is, you can see how structures of a particular level of detail (fine or coarse) respond to it — this gives you a way of getting hold of these structures, and a mathematical tool called the TV transform can be used to separate them out. Figure 2 shows a sequence of images containing structures of decreasing level of detail, fine on the left to coarse on the right.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/Facefusion/Figure2.png" alt="Blurring of image." width="600" height="169" /><p style="max-width: 600px;">Figure 2: The TV transform separating out structures of varying level of detail, from fine to coarse. (Underlying image by <a href="https://commons.wikimedia.org/wiki/File:Yang_Liwei.jpg">Dyor</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>. Image processing by Chris Irving.)</p>
</div>
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<p>Now that these components have been separated out, you can manipulate them individually, just as an equaliser manipulates individual frequencies in a song. You can also reconstruct the original picture (or the original picture without wrinkles) simply by "adding" the various components (or all the components except the ones containing wrinkles). The mathematics behind the TV transform was inspired by the mathematics behind sound processing and has resulted in a powerful tool.</p>
<p>We're now ready for some face fusion. Suppose you'd like to enhance (or perhaps spoil?) Barack Obama's face with some of the features of Ronald Reagan. You use the mathematics to remove a lot of the structure from Obama's face, so that you are left with a "background" image of him. Depending on what kind of effect you'd like to achieve, you might also decide to leave some of Obama's features, such as the eyes, mouth and hair, unchanged: this will ensure that Obama is still recognisable, even after Reaganification. Using the TV transform you then separate the fine and medium scale structures from Reagan's face, such as his wrinkles and the puffy cheeks. Then, after carefully aligning the images, you superimpose those features on Obama's background image. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/Facefusion/Oba%2Ca.jpg" alt="Obama/Reagan" width="600" height="292" /><p style="max-width: 600px;">Figure 3: Image of Obama (left) and the image equipped with some of Reagan's features (right). (Original image by <a href="https://commons.wikimedia.org/wiki/File:Official_portrait_of_Barack_Obama.jpg">Pete Souza</a>, <a href="https://creativecommons.org/licenses/by/3.0/deed.en">CC BY 3.0</a>. Image fusion by Martin Benning, Michael Möller, Raz Z. Nossek, Martin Burger, Daniel Cremers, Guy Gilboa, and Carola-Bibiane Schönlieb.)</p>
</div>
<!-- Image used by permission -->
<p>Similarly, you can start with a background Reagan and superimpose Obama's fine and medium scale features. Which one of the two is better looking, Ronald Obama or Barack Reagan, is up to you to decide.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/Facefusion/figure4.png" alt="Obama/Reagan" width="600" height="296" /><p style="max-width: 600px;">Figure 4: Image of Reagan (left) and the image equipped with some of Obama's features (right). (Image fusion by Martin Benning, Michael Möller, Raz Z. Nossek, Martin Burger, Daniel Cremers, Guy Gilboa, and Carola-Bibiane Schönlieb.)
</p>
</div>
<!-- Image used by permission -->
<p>The mathematics developed for the face fusion software is likely to find a range of applications, for example in the computer gaming industry and eventually perhaps also in the movies. Next time you're impressed by some particularly spectacular computer generated imagery, give a thought to mathematics! And if you can, head down to the <a href="http://www.cms.cam.ac.uk">Centre for Mathematical Sciences</a> on March 25th to play with the software at the Cambridge Science Festival <a href="https://www.sciencefestival.cam.ac.uk/events/maths-public-open-day">Maths Public Open day</a>.</p>
<p><em>You can read more about image processing using diffusion equations in <a href="/content/restoring-profanity">this</a> <em>Plus</em> article.</em></p>
<hr/><h3>About this article</h3><p>
We'd like to thank <a href="http://www.damtp.cam.ac.uk/people/mb941/">Martin Benning</a>, Leverhulme Early Career Research Fellow at the University of Cambridge, for his help in putting together this news story.</p>
</div></div></div>Tue, 21 Mar 2017 13:03:09 +0000mf3446816 at https://plus.maths.org/contenthttps://plus.maths.org/content/face-fusion#commentsColouring by numbers: The Kochen-Specker theorem
https://plus.maths.org/content/kochen-specker-theorem
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/pens_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger and Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>We have seen in <a href="/content/contextuality-most-quantum-thing ">a previous article</a> that quantum mechanics is a strange theory. Taken at face value, the theory says that particles, such as electrons, don't have well-defined values for properties such as their position or speed, until you make a measurement. In some mysterious way that nobody understands, the particles "assume" the definite values only when you measure them. </p>
<div class="rightimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/pens.jpg" alt="Pens" width="350" height="234" />
<p>It's nice that a sophisticated theorem that's relevant to quantum physics can be reduced to a simple colouring problem.</p>
</div><!-- image from fotoila.com -->
<p>So perhaps the theory just isn't complete? Perhaps there is another, as yet undiscovered theory, which tells us exactly what particles are doing even when we are not watching them? The theory
would have to give the same predictions as quantum mechanics for
observations because those predictions have been verified in experiments more than any other theory. But unlike ordinary quantum mechanics, let's imagine our new theory does allow for particles to have
definite locations, speeds and spins and so on, even when we are not
trying to detect them. </p>
<p>This sounds like a plausible idea and indeed such theories have been developed — see <a href="/content/riding-pilot-wave">here</a> for an example. However, a result called the <em>Kochen–Specker theorem</em> tells us that things can't ever be as straightforward as we might hope. It says that such a new theory must be
<em>contextual</em>. The values you measure for a particle's properties — speed, position, spin, etc — those
values must somehow or sometimes depend on how you try to measure
them. If this sounds confusing, then you might want to read <a href="/content/contextuality-most-quantum-thing">this</a> article to find out more about the idea of contextuality. Here we look at the Kochen-Specker theorem in more detail and even give you an idea of the proof, which is refreshingly simple.</p>
<h3>Getting in a spin</h3>
<p>
The Kochen-Specker theorem was proved by the mathematicians <a href="https://en.wikipedia.org/wiki/Simon_B._Kochen">Simon Kochen</a> and <a href="https://en.wikipedia.org/wiki/Ernst_Specker">Ernst Specker</a> in 1967, so it celebrates its 50th birthday this year. "The theorem makes the idea of contextuality very vivid, without bothering your head with all these misty and controversial issues about the interpretations of quantum theory," explains <a href="http://trin-hosts.trin.cam.ac.uk/fellows/butterfield/">Jeremy Butterfield</a>, a philosopher of physics at the University of Cambridge. "That's because it articulates contextuality as a problem of colouring."</p>
<div class="leftimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/spin_family.jpg" alt="Spin family" width="350" height="253" />
<p>Elementary particles with different values of spin are represented in the work <em>Spin Family</em> (2009) by <a href="http://www.JulianVossAndreae.com/">Julian Voss-Andreae</a>. (<a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>)</p>
</div><!-- image from fotoila.com -->
<p>Let's start with some basics. Quantum particles have a property called <em>spin</em>: it's a little bit akin to what you'd think of as the "spin" of an object that is rotating on an axis, but it's more subtle than that. We don't need to go into the details here, suffice to say that when you measure the component of a particle's spin in a particular direction in space, you will only ever get one of a very limited number of values. For example, for <em>spin 1 particles</em> the spin can only take the values 1, −1 or 0. Quantum mechanics predicts that a spin 1 particle will have any one of these values with equal probability (i.e., the spin value will be 1 roughly a third of the time, 0 a third of the time and −1 a third of the time). </p>
<p>
As spin 1 particles have spin of either 1, −1 or 0, the <em>squared component </em> of spin in any direction can only be 1 (=1<sup>2</sup> =(-1)<sup>2</sup>) or 0 (=0<sup>2</sup>). The spin in that direction is 1, −1, or 0 with probability 1/3 for each of these values. After you've squared the spin there's a 2/3 probability of the squared spin being 1, and a 1/3 probability of the squared spin being 0, because −1 and 1 have been lumped together.
</p>
<p>Now quantum mechanics predicts a deceptively straightforward result: <em>If you measure the squared spin of a spin 1 particle in three perpendicular directions, you will always get the answers 1, 0, 1 in some order. </em>(Although you can't simultaneously measure the spin of a particle in different directions — they form incompatible pairs in Heisenberg's uncertainty principle — you can simultaneously measure the squared component of spin of a particle in three perpendicular directions.)
</p>
<h3>Colouring by numbers</h3>
<p>Now here comes the colouring problem. Imagine a sphere and a triple of perpendicular axes running through its centre. Your task is to colour one of these axes red and the other two blue. But here is the catch: you need to assign the two colours to <em>every</em> triple of perpendicular axes in the entire sphere, but in a coherent way. Each axis must be either red or blue, but not both. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/2spheres.jpg" alt="sphere with axes" width="500" height="237" />
<p style="max-width: 500px;">Two triples of perpendicular axes. One axis of each triple must be coloured red, the other two blue.</p>
</div>
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<p>You can see that this is a difficult problem. Let's single out an axis that has already been coloured red, and let's assume it's the vertical one (if it isn't vertical, we simply rotate the sphere so that it is). Now every axis in the equatorial plane (and there are infinitely many of them) is perpendicular to the vertical one, and since the vertical axis is red, each of them needs to be coloured blue. "For one red I now have a vast infinity of blues," says Butterfield. "But didn't I want to have roughly a third of the sphere red and roughly two thirds blue? It looks like the blues are bound to swamp the reds when we notice that the entire equator will have to be blue."</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/sphere3.png" alt="Sphere with axes" width="300" height="264" />
<p style="max-width: 300px;">If the vertical axis is coloured red, then all axes in the equatorial plane must be coloured blue.</p>
</div>
<!-- Image made by MF -->
<p> Kochen and Specker showed that it is indeed impossible to colour all axes either red or blue so that in every triple of perpendicular axes exactly one is red. You don't actually need to consider infinitely many axes to prove their result: a specific collection of only 33 is enough (see <a href="/content/proof-kochen-specker-theorem">here</a> for the details). When you try to colour them according to the rules, you'll eventually find an axis that needs to be painted red because it's part of a triple in which the other two are blue, but that also needs to be coloured blue because it's part of a triple in which there's already a red axis.</p>
<h3>Context is everything</h3>
<p>What does this mean for our measurements of spin 1 particles? Remember we are thinking of a theory in which particles have well-defined values for their spin (and therefore also for their squared spin in all directions). Let's represent the outcomes of a measurement of square spin in a triple of perpendicular directions by a triple of perpendicular axes in our sphere picture. As we mentioned above, these outcomes always give a 0 in one of the directions, and 1s in the other two direction. The 0 outcome is represented by a red axis and the other two outcomes are represented by the two blue axes in the triple. </p>
<p>Since we know that the colouring problem is impossible to solve, we know that there is at least one awkward axis: one that should be painted red if you consider it as part of one triple (call it triple 1), but should be painted blue if you consider it part of another triple (call it triple 2). This axis then corresponds to a direction in which the squared spin will be measured as 0 if you measure it as part of triple 1 of directions, and in which the squared spin will be measured as 1 if you measure it as part of triple 2 of directions.
In other words, the value of the squared spin you see when you measure the system in the awkward direction depends on what other two directions you choose to measure squared spin in!
"Kochen and Specker force on you the idea that the value you get depends on the triple, and therefore there is contextuality," says Butterfield.</p>
<p>You can't get away from the cold truth demonstrated by the Kochen-Specker theorem. Either you accept that a particle doesn't have a definite preset value for its spin before you measure it (that's the ordinary interpretation of quantum mechanics) or you accept that the preset value depends on what other measurements you decide to make at the same time. Both options are strange — take your pick!</p>
<hr>
<h3>About this article</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> and <a href="/content/people/index.html#rachel">Rachel Thomas</a> are Editors of <em>Plus</em>. They interviewed Jeremy Butterfield in Cambridge in February 2017. </p>
<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p><em>This article is in part based on a <a href="/content/john-conway-discovering-free-will-part-ii">2011 article by Rachel Thomas</a>, based on an interview with the mathematician John Conway. It is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-context-everything">here</a> to see more articles and videos about contextuality. </em></p>
</div></div></div>Tue, 21 Mar 2017 10:37:44 +0000mf3446778 at https://plus.maths.org/contenthttps://plus.maths.org/content/kochen-specker-theorem#commentsMaths in a minute: Linear regression
https://plus.maths.org/content/maths-minute-linear-regression
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/regression_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Wim Hordijk</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>A linear regression tries to estimate a linear relationship that best fits a given set of data. For example, we might want to find out how the number of tropical storms has changed over the years. In this case, we can plot the number of storms against time. The linear regression will find the straight line that best fits the plotted data, and calculate several statistics indicating how well the line fits the data and whether the slope of the line is significantly different from zero (i.e., whether there is a real trend or not). (See <a href="/content/citizen-science-public-databases-and-bit-maths">this article</a> for more on the tropical storm example.)</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/earthquakes/Hurricanes_cropped.jpg" alt="Storms" width="600" height="185" /><p> Yearly tropical storms. The blue line indicates the result of a linear regression on the number of storms over time.
</p></div>
<!-- image provided by author -->
<p>In general, when you perform a linear regression, a dependent variable (say <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0001.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" />, the number of tropical storms in our example) is assumed to be linearly dependent on one or more explanatory variables (say <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0002.png" alt="$x_1$" style="vertical-align:-2px;
width:15px;
height:9px" class="math gen" />, <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0003.png" alt="$x_2$" style="vertical-align:-2px;
width:15px;
height:9px" class="math gen" />, etc, up to <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0004.png" alt="$x_ n$" style="vertical-align:-2px;
width:17px;
height:9px" class="math gen" />). The general equation for the linear relationship is </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0005.png" alt="\[ y = a_0+a_1x_1+a_2x_2+ ...+ a_ nx_ n. \]" style="width:249px;
height:14px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Given a set of observed values for the dependent variable <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0001.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> and corresponding explanatory variables <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0002.png" alt="$x_1$" style="vertical-align:-2px;
width:15px;
height:9px" class="math gen" /> to <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0004.png" alt="$x_ n$" style="vertical-align:-2px;
width:17px;
height:9px" class="math gen" />, the linear regression then estimates values for the model parameters <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0006.png" alt="$a_0$" style="vertical-align:-2px;
width:15px;
height:9px" class="math gen" /> to <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0007.png" alt="$a_ n$" style="vertical-align:-2px;
width:17px;
height:9px" class="math gen" /> such that the total <em>error</em> (i.e., the differences between the observed values <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0001.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> and the model predicted values <img src="/MI/03b6a3cfa4b7cfbec147948711ddadc5/images/img-0008.png" alt="$\hat{y} = a_0+a_1x_1+a_2x_2+ ... +a_ nx_ n$" style="vertical-align:-3px;
width:243px;
height:16px" class="math gen" />) is minimised. </p>
<p>For example, in the case of tropical storms the dependent variable is the number of storms each year (<em>hurricanes</em>), and the (single) explanatory variable is time (<em>year</em>). For a set of 18 observations from the <a href="http://www.nhc.noaa.gov/data/#monthly">Monthly Storm Reports</a>, one for each year from 1999 to 2016, a linear regression results in the following estimated model:</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7f373f8a429c99f3ddb55905c3629a21/images/img-0001.png" alt="\[ hurricanes = -637.9+0.3323 \times year. \]" style="width:277px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>In this data set, the observed number of hurricanes for the year 2001 is 29. However, the model predicted value for this year is <img src="/MI/7f373f8a429c99f3ddb55905c3629a21/images/img-0002.png" alt="$-637.9+0.3323\times 2001 = 27.03$" style="vertical-align:-1px;
width:228px;
height:14px" class="math gen" />. In other words, the error is <img src="/MI/7f373f8a429c99f3ddb55905c3629a21/images/img-0003.png" alt="$29-27.03 = 1.97$" style="vertical-align:0px;
width:126px;
height:13px" class="math gen" />. In the model estimation, the values of the parameters were chosen in such a way that the total error (summed over all years) is minimised (in reality, a linear regression actually minimises the sum of <em>squared</em> errors). </p>
<div class="rightimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/earthquakes/regression.jpg" alt="A graph" width="350" height="272" />
<p>Linear regression can help you spot trends in your data.</p>
</div>
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<p>A statistic that is often used to indicate how well the estimated model fits the given data is the <em>coefficient of determination</em>, denoted <img src="/MI/dde067e7d9bb5522b1954594c2c20272/images/img-0001.png" alt="$R^2$" style="vertical-align:0px;
width:18px;
height:14px" class="math gen" />, which measures the proportion of the <a href="https://en.wikipedia.org/wiki/Variance">variance</a> in the dependent variable that is actually explained by the explanatory variable(s). This value is usually on a scale from zero to one, with zero indicating no explanatory value (i.e., complete unpredictability) and one indicating full explanatory value (i.e., complete predictability). The <img src="/MI/a6eb914ba6513aa6e65e935749028880/images/img-0001.png" alt="$R^2$" style="vertical-align:0px;
width:18px;
height:14px" class="math gen" /> value in the regression performed here is <img src="/MI/a6eb914ba6513aa6e65e935749028880/images/img-0002.png" alt="$R^2=0.11$" style="vertical-align:0px;
width:70px;
height:14px" class="math gen" />, which is low. This indicates that the estimated models have very little explanatory power, and that the data is mostly random rather than having a linear dependence on time.</p>
<p>Finally, a linear regression analysis also tests the <em>null hypothesis</em> that the slope of the regression line is zero (i.e., that there is actually no dependence of the dependent variable on the explanatory variables). The statistic calculated to decide whether to accept or reject this null hypothesis is the <em>p</em>-value. This statistic (a value between zero and one) indicates the probability that the data is the way it is under the assumption that there is no dependence. Another way to put this is to say that the <em>p</em>-value indicates the probability of making a mistake when rejecting the null hypothesis (i.e., the probability of rejecting a true hypothesis).</p>
<p>A standard threshold (or <em>significance level</em>) used for the <em>p</em>-value is 0.01, or a 1% probability of rejecting a true hypothesis. This means that any <em>p</em>-value that is larger than 0.01 is not considered enough statistical evidence to reject the null hypothesis. Sometimes a more "forgiving" significance level of 0.05 (or 5%) is used, but the main idea is that the higher the <em>p</em>-value of the regression, the more likely it is that the slope of the linear model is not significantly different from zero. (You can find out more about <em>p</em>-values and significance levels in <a href="/content/what-are-sigma-levels-0">this</a> article.)</p>
<h3>Do it yourself</h3>
<p>
If you'd like to perform your own linear regression, you might want to use the <a href="https://www.r-project.org"><font style="font-family:Courier New;" >R</font> program</a>. For example, for the hurricane data (after extracting the number of storms for each year from the public database), the regression can be performed with the following R commands:</p>
<p><font style="font-family:Courier New;" >year<-c(1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016)</br>
hurricanes<-c(20,32,29,24,29,27,42,25,23,32,26,26,30,36,31,28,30,36)</br>
myModel<-lm(hurricanes ~ year)</br>
summary(myModel)</br>
predict(myModel)</font></p>
<p>The <font style="font-family:Courier New;" >summary</font> command prints out a summary of the linear regression, including the estimated values for the model parameters, the
<img src="/MI/dde067e7d9bb5522b1954594c2c20272/images/img-0001.png" alt="$R^2$" style="vertical-align:0px;
width:18px;
height:14px" class="math gen" />
value and the <em>p</em>-value, and several other statistics. The <font style="font-family:Courier New;" >predict</font> command provides the model predicted values (with which the blue line in the figure above was plotted).</p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2016/altruism/wim.jpg" alt="Wim Hordijk" width="200" height="200" /></div>
<p>Wim Hordijk is a computer scientist currently on a fellowship at the Konrad Lorenz Institute in Klosterneuburg, Austria. He has worked on many research and computing projects all over the world, mostly focusing on questions related to evolution and the origin of life. More information about his research can be found on his <a href="http://www.worldwidewanderings.net">website</a>.</p></div></div></div>Mon, 20 Mar 2017 16:14:03 +0000mf3446798 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-linear-regression#commentsCitizen science: Facts or fake news?
https://plus.maths.org/content/citizen-science-public-databases-and-bit-maths
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/volcanoe_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Wim Hordijk</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In this day and age of the internet, where anyone can post anything, it is often difficult to know what is true and what is not. One person claims one thing, while another states the exact opposite. Who to believe among all this (sometimes deliberate) confusion?</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/earthquakes/volcano.jpg" alt="A volcano" width="350" height="248" />
<p>Have volcanic eruptions become more frequent? Find out with a little bit of maths.</p>
</div>
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<p>The upside, also thanks to the internet, is that you don't need to be a professional scientist to find out at least some of the truth for yourself. With the increasing availability of public online databases and easy-to-use software, "citizen science" can go a long way at countering unsubstantiated claims.</p>
<p>Take, for example, the confusion surrounding the topic of recent earth activity, in particular earthquakes, tropical storms, and volcanic eruptions. Many claims are going around that all of these have drastically increased over the past several years. However, few of these claims include supporting evidence, or if they do the statistics are often presented in a biased or distorted way to make the claims seem true.</p>
<p>In this article I will present some simple statistical analyses to show how any person with a computer and an internet connection can decide for themselves whether to believe such claims or not. All it takes is some public databases and a little bit of basic maths...</p>
<h3>Earthquakes</h3>
<p>To start with earthquakes, probably the most complete and accurate database is maintained by the US Geological Survey (USGS). <a href="https://earthquake.usgs.gov/earthquakes/search/">This database</a> is publicly available on the internet, and data can be downloaded for any range of magnitudes, time period, or geographic region.</p>
<p>The downloaded data can then be opened and analysed in any spreadsheet program, such as <a href="http://www.openoffice.org/product/calc.html">OpenOffice Calc</a>. Even better software for doing statistics is the <a href="https://www.r-project.org"><font style="font-family:Courier New;" >R</font> program</a>. This software requires a little more effort to learn how to use it properly, but it provides very nice plotting facilities for visualising your data and statistical results. Alternatively, you can use <a href="http://www.gnuplot.info">gnuplot</a> for making quick and easy plots. All these programs are available for free and run on all platforms (Windows, MacOS, and Linux). I used a combination of <font style="font-family:Courier New;" >R</font> for doing the statistics and gnuplot for plotting the results.</p>
<p>From the USGS database I downloaded all earthquakes of magnitude 5.0 or larger (M5+), from 1 January 1996 until 31 December 2016, worldwide. The reason I chose M5+ is that those are potentially damaging. Then I used the <font style="font-family:Courier New;" >R</font> program to group these into monthly counts (i.e., counting the total number of earthquakes in each calendar month), and to calculate a 12-month running average. A running average is an average that is calculated over a subsequence (or "window") of the data, and then "sliding" this window along the entire data sequence. In this case, the running average is calculated over the past 12 months, counting backwards from the current month while moving forward in time. For example, the running average for the month of April 2014 is the average over the 12-month window starting from May 2013 and ending at April 2014.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/earthquakes/Earthquakes.jpg" alt="Figure 1" width="600" height="464" /><p>Figure 1: Monthly number of earthquakes. The total number of earthquakes per month since January 1996 (blue bars), and a 12-month running average (red line).</p></div>
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<p>The results are shown in figure 1, where the blue bars indicate the monthly counts and the red line the 12-month running average. Various observations can be made from this plot. First, and most obvious, there is the large spike in March 2011. This is due to the M9.0 earthquake off the coast of Japan on 11 March 2011 (which triggered a tsunami that caused widespread damage and casualties). This extremely violent earthquake caused many large aftershocks, generating the unusually high number (just over 700) of M5+ earthquakes for that month. Since this event is an exception, the monthly count for March 2011 was not included in the running average calculations.</p>
<p>The second striking observation is that, even though there seems to have been an increasing trend in number of earthquakes from early 2000 onwards (indicated by the red line), after the March 2011 event there was actually a clear drop in the number of earthquakes, which seems to have leveled off since then. In fact, the number of earthquakes in the most recent months is very similar to what is was more than 10 years ago. Perhaps the tectonic pressure that had been building up over the years before 2011 was released sufficiently by the large M9.0 earthquake that everything has "calmed down" again. This gradual increase followed by a significant drop is even more clearly visible when the earthquakes are grouped into yearly counts, as is shown in Figure 2.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/earthquakes/yearly_M5.jpg" alt="Figure 2" width="600" height="464" /><p>Figure 2: Yearly number of earthquakes. The total number of earthquakes per year since 1996 (blue bars). Note that the vertical axis starts at 1000 instead of zero, so the increase looks more exagerated than in Figure 1 (one of those little tricks to "manipulate" statistics).</p></div>
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<p>So, in conclusion, even though there indeed seems to have been a clear increase in the number of M5+ earthquakes since around 2000, this number dropped significantly after the March 2011 event, and is now back at the level it was more than 10 years ago. Moreover, this initial increase is completely absent when considering only M6+ earthquakes. If you don't believe this, you now know how to verify (or refute) this for yourself, rather than simply accepting what anyone else claims!</p>
<h3>Tropical storms</h3>
<p>Next up are hurricanes. Data on tropical storms in both the Atlantic and eastern Pacific can be obtained from the <em>National Hurricane Center</em>. <a href="http://www.nhc.noaa.gov/data/#monthly">Monthly storm reports</a> since 1999 are available under the <em>Tropical Cyclone Monthly Summary Archive</em> section. In particular, the final report of each year contains a full summary of all storms for that particular year, including data on maximum sustained winds (in MPH). Storms with maximum sustained winds of at least 40MPH (which are considered to be tropical storms or worse) are included in the analysis here.</p>
<p>Figure 3 presents the results of collecting this data, showing the total number of storms in each year (Atlantic and eastern Pacific combined; purple line), their average strength (in MPH; green line), and the maximum observed strength in that year (in MPH; red line). Although there was a record-breaking storm with maximum sustained winds of more than 200MPH in 2015 (Hurricane Patricia which developed off the coast of Mexico in October 2015), there is no clear trend visible in the data, let alone an obvious increase.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/earthquakes/Hurricanes.jpg" alt="Figure 3" width="600" height="464" /><p>Figure 3: Yearly tropical storms. The total number, average strength, and maximum strength of tropical storms per year since 1999, combined over the Atlantic and eastern Pacific. The blue line indicates the result of a linear regression on the number of storms over time.
</p></div>
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<p>To test this lack of a trend statistically, a <em><a href="/content/maths-minute-linear-regression">linear regression</a></em> can be performed on the data. Briefly, in this context a linear regression tries to fit a straight line to the given data, and calculates several statistics indicating how well the line fits the data and whether the slope of the line is significantly different from zero (i.e., whether there is a real trend or not). See <a href="/content/maths-minute-linear-regression">here</a> for a more detailed explanation of this type of statistical analysis and how to interpret the statistics (such as <em>R</em><sup>2</sup> and <em>p</em>-values).</p>
<p>Performing a linear regression on the number of tropical storms against time (years), the fit is very poor:
<em>R</em><sup>2</sup>=0.11. The result of this regression is shown in Figure 3 by the blue line, but it explains only just over 10% of the variance in the data. Moreover, the <em>p</em>-value for testing the null hypothesis that the slope of the regression is equal to zero is 0.172, so there is far from enough statistical evidence to reject this hypothesis, even at a more "forgiving" significance level of 0.05 (or 5%). In other words, there is no statistical justification to claim that there is an increasing trend in the number of tropical storms over the past (almost) 20 years.</p>
<h3>Volcanic eruptions</h3>
<p>Finally, we'll have a look at volcanoes. An <a href="http://volcano.si.edu/search_eruption.cfm">online database</a> of volcanic eruptions is provided by the Smithsonian Institution. I downloaded data for the years 1976-2016, which is presented as yearly counts in Figure 4.
</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/earthquakes/Eruptions.jpg" alt="Figure 4" width="600" height="464" /><p>Figure 4: Yearly volcanic eruptions. The total number of volcanic eruptions per year since 1976, worldwide (red line). The blue line indicates the result of a linear regression on the number of eruptions over time.
</p></div>
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<p>Performing a linear regression on the number of eruptions against time, the fit is again very poor:
<em>R</em><sup>2</sup>=0.12. However, in this case the linear relationship, represented by the blue line in Figure 4, does seem to have a clear positive slope. The <em>p</em>-value for testing the null hypothesis that the slope of the regression is equal to zero is 0.024. This value is somewhat in the grey area, but still above the standard significance level of 0.01. This, together with the poor fit of the linear model, and the fact that the number of eruptions over the past eight years is (on average) lower than in the previous eight-year period (as is clear from Figure 4), suggests that there is little statistical evidence for an obvious increase in the number of volcanic eruptions over the past 40 years.</p>
<h3>Some afterthoughts...</h3>
<p>The above analyses of publicly available data does not provide much statistical support for claims of a significant recent increase in seismic, atmospheric, or volcanic activity on our planet. Even though there was a clear increase in M5+ earthquakes in the build-up to March 2011, this was followed by a significant decrease since then. Furthermore, any clear trend (in either direction) is completely absent in the number of M6+ earthquakes and tropical storms. The number of volcanic eruptions does show a slight positive trend, but this can hardly be considered statistically significant.</p>
<div class="rightimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/earthquakes/storm.jpg" alt="A storm" width="350" height="247" />
<p>The more people are affected by a storm, the more likely it is to be recorded and reported.</p>
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<p>So, how come such claims exist (and persist) in the first place? Or, if they are somehow true after all, why is there an apparent lack of statistical evidence? I can imagine several possible answers to these questions.</p>
<ul><li><em>Lack of longer-term data</em>: One could argue that to see real trends, we need data even further back in time. However, given that the majority of claims about an increase in earth activity seem to specifically emphasise recent years (2012 seems to be a particularly popular date), data for the past two to four decades should still clearly show the claimed increase.</li><li>
<em>Incomplete or unreliable data</em>: Related to this, older data might not be reliable or complete enough. For example, on the USGS website it is clearly stated that there were no machine-readable earthquake records available for the years before 1980. This older data literally had to be scanned in from paper documents. In other words, possibly increasing trends over longer time scales may simply be an artefact of missing data.</li><li>
<em>Increase in population size</em>: The world's population has increased exponentially over the past few decades. As a consequence, urban areas have grown rapidly, both in size and number. Even if the frequency and magnitude of natural disasters like earthquakes, hurricanes, and volcanic eruptions has remained the same, the average number of affected people per event will inevitably have increased, perhaps giving the illusion that things have become worse, or more frequent.</li><li>
<em>Increase in news reporting</em>: On a similar note, news reporting has increased drastically as well over the past years or decades. Anytime anything happens anywhere, we hear or read about it in the news. However, this increase in the number and extent of news reports on natural incidents does not necessary reflect an actual increase in such events, but may also simply give the illusion of an increased frequency.</li></ul>
<p>I hope the simple analyses in this article have provided some examples of how we can all be "citizen scientists" in pursuit of truth. With the increasing availability of online public databases and free software, it has become possible for anyone with access to a computer and the internet to find out for themselves whether certain claims are true or not. And of course a little bit of basic mathematics is very helpful too. As the famous evolutionary biologist J. B. S. Haldane once said: "An ounce of algebra is worth a ton of verbal argument."</p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2016/altruism/wim.jpg" alt="Wim Hordijk" width="200" height="200" /></div>
<p>Wim Hordijk is a computer scientist currently on a fellowship at the Konrad Lorenz Institute in Klosterneuburg, Austria. He has worked on many research and computing projects all over the world, mostly focusing on questions related to evolution and the origin of life. More information about his research can be found on his <a href="http://www.worldwidewanderings.net">website</a>.</p>
</div></div></div>Mon, 20 Mar 2017 15:48:07 +0000mf3446797 at https://plus.maths.org/contenthttps://plus.maths.org/content/citizen-science-public-databases-and-bit-maths#comments