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Theorem of the week

This result has a very uninformative name, I’m afraid, and it doesn’t identify the result uniquely, but it is the standard name so I’d better stick with it.

I’m going to write this post on the assumption that you know about Fermat’s little theorem and Euler’s criterion.  Fortunately, those links take you to places on this blog where you can read everything you need to know, and then you can come back here.

Ready?  Are you sitting comfortably?  Then we’ll begin.

Let’s fix an odd prime $latex p$, and some number $latex a$ that is coprime to $latex p$.  We’d like to know whether $latex a$ is a quadratic residue modulo $latex p$.  Putting that another way, we’d like to find the value of the Legendre symbol $latex \genfrac{(}{)}{}{}{a}{p}$.  Helpfully, Euler‘s criterion tells us that $latex \genfrac{(}{)}{}{}{a}{p} \equiv a^{\frac{p-1}{2}} \pmod{p}$.  I say “helpfully”, but it rather…

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When the Counting Gets Tough, the Tough Count on Mathematics: Fibonacci numbers.

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“Fibonacci Identities with Matrices: one matrix identity generates 4 identities for the Fibonacci numbers. As a bonus, we obtain a property of divisibility of Fibonacci numbers.”

Read more about Alexander Bogomolny’s blog post Fibonacci Identities with Matrices at Cut the Knot

It is arguable that the work of Gauss has a greater impact on our daily lives than the magnificent creations of Beethoven.

Read more at The Two Cultures.

Hydrobates

The music did seem to have a positive effect on the synchronization of lectures. Unfortunately it was not always there – for instance it was not there before my talk – and it seems to have been getting less and less. One good thing is that the name tags, as well as showing the usual information have the first name (or nickname) printed in large letters at the top. I find that this can be very useful for recognizing people after only having met them fleetingly.

The plenary talk of Claire Tomlin yesterday was about the HER2 receptor which plays an important role in breast cancer. It is connected to transcription factors in the nucleus by a signalling network containing two main pathways. One of these includes the MAP kinase cascade while another passes through the substance Akt. Excessive activity of this type of signalling can be reduced by a…

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Hydrobates

On Tuesday I will travel to Knoxville for the annual meeting of the Society for Mathematical Biology. On Wednesday I will give a talk there about my work on the NFAT signalling pathway. The programme of the conference is very dense: apart from the times when there are plenary talks there are seven sessions in parallel. My usual tactics at conferences of this type is to choose whole sessions to attend rather than individual talks. Anything else is usually frustrating due to the poor synchronization of the talks in different sessions. Maybe it will be better in this case. It is planned to have music to mark the breaks between talks which will be heard in all the rooms. This could overcome any lack of discipline imposed by the chairs of the individual sessions. Since all the rooms are in one building and, to judge by their numbering, close…

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Dr. Keith Devlin, a mathematician at Stanford University, will be launching his first free online math course, Introduction to Mathematical Thinking. It’s now scheduled to start on September 17, and the registration page just went live on Coursera, the Stanford spin-off MOOC platform now offering online courses from a number of the nation’s best universities.

MOOCtalk

A real-time chronicle of a seasoned professor embarking on his first massively open online course.

I’ve been pretty quiet on this blog since launching it on May 5.

Partly that is due to summer vacation and the start of great cycling weather. But a lot of my time got swallowed up planning and developing my fall MOOC. It’s now scheduled to start on September 17, and the registration page just went live on Coursera, the Stanford spin-off MOOC platform now offering online courses from a number of the nation’s best universities.

All my Stanford colleagues who gave courses in the first round earlier this year reported how much time it takes to create such a course, no matter how long you have been teaching at university level. Knowing that you won’t be in the same room as the students, where there is ongoing interaction and constant, instant feedback…

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I just purchased two tickets to see The Dark Knight Rises! TDKR Considering my wife and I have not been to a movie theater since our daughter was born 16 months ago, we are both very excited to see this movie on the big screen!

Click here to watch a 13 minute featurette.

Now for some fun math. Here is what you get when you type bat-insignia in WolframAlpha:

To generate the bat-insignia in Sage use the following code:

x,y = var('x,y')
f1 = ((x/7)^2*sqrt((abs(abs(x)-3))/(abs(x)-3))+(y/3)^2*sqrt((abs(y+3*sqrt(33)/7)/(y+3*sqrt(33)/7)))-1)
f2 = abs(x/2)-(3*sqrt(33)-7)/112*x^2-3+sqrt(1-(abs(abs(x)-2)-1)^2)-y
f3 = 9*sqrt(abs((abs(x)-1)*(abs(x)-0.75))/((1-abs(x))*(abs(x)-0.75)))-8*abs(x)-y
f4 = -y+3*abs(x)+0.75*sqrt(abs((abs(x)-0.75)*(abs(x)-0.5))/(-(abs(x)-0.75)*(abs(x)-0.5)))
f5 = 2.25*sqrt(abs((x-0.5)*(x+0.5))/(-(x-0.5)*(x+0.5)))-y
f6 = 6*sqrt(10)/7+(1.5-0.5*abs(x))*sqrt(abs(abs(x)-1)/(abs(x)-1))-6*sqrt(10)/14*sqrt(4-(abs(x)-1)^2)-y
f=[f1,f2,f3,f4,f5,f6]
num = 2000
sum([implicit_plot(g,(x,-8,8),(y,-5,5),plot_points=num)for g in f])

Statistical Significance.

Reading graphs can be tricky.

Mini-polymath4 discussion thread.

Mini-polymath4 begins in three hours.

Another rant by one of my favorite bloggers, mathematician Cathy O’Neil (mathbabe). This is really a must read IMO and it illustrates why it is important to understand when statistics are being misused.

Another good post on the misuse of statistics is “If correlation doesn’t imply causation, then what does?
by Michael Nielsen

Check them both out!

mathbabe

In the pharmaceutical industry, where companies are making enormous bets with huge money and people’s lives, it makes sense that there are conflicting interests. The companies, who are in charge of testing their drugs for safety and for successful treatment, tend to want to emphasize the good and ignore the bad.

That’s why they are expected to describe beforehand how they are planning to do the tests. It stands to reason that, if they did a thousand tests and then only reported on the best ones, the public would get a biased view of the safety of their products.

For some reason, though, this standard doesn’t seem to be universally followed, and lying with statistics seems to be okay.

The newest example comes from Merck (see Pharmalot article here), which changed its statistical methods on testing Vioxx for Alzheimer’s patients from an intent-to-treat analysis to an on-treatment analysis even though their stipulated…

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