A Few Interesting Answers to Some
Hard Questions
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Continuum Hypothesis
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Ordinal Numbers
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Odd Perfect Numbers
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N vs. NP - resolved!
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I think developing an “elevator speech” is important, and
I encourage all students to do this before they
graduate.
-Judy Walker
In general, it's good to give people an idea that we're still
working. Math isn't done. I don't play the role of a
monk sitting in my office, keeping the store of all
mathematical knowledge safe to pass on to the next
generation. New technology like GPS or Google
requires new mathematical thought and progress. You
can't just look up how to do it in a book.
-Tom Marley
Operator Theory
What I usually do is discuss the idea of transformation via the idea of
electronic communication: think of changing a signal such as your voice
into a digital format. The sound waves are sampled and then converted
into sequences of 0s and 1s. This is an example of a linear
transformation. I then tell people that I don't study these transformations
themselves, but I study collections of things like them, which have certain
algebraic properties.
-David Pitts
One of the big ideas of 20th century science was quantum mechanics, and
the key thing quantum mechanics found was that there's a fundamental
limit to how well you can measure some things (speed vs position).
Physicists found it comes down to the fact that in ordinary arithmetic
when you multiply two numbers, it doesn't matter what order you do it in,
but the quantum view builds a model of algebra where a times b and b
times a need not be the same. Now the operator theory I study doesn't
work with quantum theory directly, but looks at the algebra you get with
families of these non-commuting variables together. So operator theory
really just works to develop other parts of mathematics with the quantum
mechanical perspective.
-John Orr
Beatles Unknown "A Hard Day's Night" Chord
Mystery Solved Using Fourier Transform
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http://www.youtube.com/watch?v=cD4TAgdS_Xw
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“instantly recognizable twang”
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For 40 years, no one knew what chord George Harrison was
playing on his 12-string Rickenbacker.
Jason Brown of Dalhousie University applied Fourier Transforms
to decompose the sound into its original frequencies, and
parse the notes.
Lennon played a six-string, Paul had his bass, Ringo was
drums... none of them quite fit.
In the end, Brown discovered that there had to be a piano playing
an F note, which was impossible to play with the other notes
on the guitar.
Science 2.0, October 2008, www.science20.com
Diff Eq and PDEs
Differential equations arise in many different areas of the physical sciences
and engineering and also in diverse subjects such as medicine,
psychology, economics, operations research to mention just a few. As
simple examples, differential equations arise in the theory of electric
circuits, gravitational equilibrium of a star, learning of tasks, vibrating
strings, heat flow, wave motion, and so on.
-Lynn Erbe
In the sciences and engineering, mathematical models are developed to aid
in the understanding of physical phenomena. These models often yield
an equation that contains some derivatives of an unknown function.
Such an equation is called a differential equation. Two examples of
models developed in calculus are the free fall of a body and the decay of
a radioactive substance.
-"Fundamentals of Differential Equations"
Visual Effects in the Movies
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Fluid Animation: water, fire, air, smoke, snow, even fur or
cloth
You choose how, and at what rate, particles are emitted in
your space, and you control their speed. Then you
determine what kind of forces or acceleration they should
be subject to. (gravity field)
Fountain, Smoke, Sand
The 30 second water sphere scene in Harry Potter took six
months to simulate.
Trade-off between accuracy (implementing the complex
mathematical equations describing each physical
process) and efficiency.
+Plus Magazine, June 2009, plus.maths.org
Math model may decrease phantom
traffic jams
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Some traffic jams have no apparent cause –
no accident, no stalled vehicle, no
construction.
High Traffic Density
Traffic jams can be modeled as a selfsustaining wave, similar to detonation
waves produced by explosions.
Jamitons
MSNBC, June 2009
Coding Theory
There are many examples of instances in which information needs to
be transmitted reliably across a channel --- satellite pictures from
outer space, cellular phone conversations, email, etc.; even data
storage (on a hard disk or a flash drive or cd, for example) can be
thought of in this way. No matter how careful we are, errors are
bound to occur. It is the goal of coding theory to find ways of
adding redundancy to the information so that these errors can be
detected and efficiently corrected.
-Judy Walker
Detecting Altered Photos
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Consider the shadows...
Tampering with an image
leaves statistical traces
in the file.
If something is removed,
part of the background
will be copied, and the
two parts of the file will
be identical.
AMS Math Moments, 2006
Mathematician Receives Grammy
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Dr. Kevin Short of the University of New Hampshire won a
Grammy award (historical category) in 2008 for his
restoration of a 1949 wire recording of a live Woody
Guthrie concert.
It is the only known recording of Guthrie performing in
concert.
It was recorded by a Rutgers college student onto spools of
wire.
Once it was carefully transferred to a digital formal, Short
used Chaotic Compression Technology to fill in the
broken signal.
Signal Processing: cell phone downloads, better hearing
aids, more sensitive explosive materials detectors
ScienceNews, 2008
Math Education
Most simply, mathematics education research is inquiry by
carefully developed research methods aimed at providing
evidence about the nature and relationships of many
mathematics learning and teaching phenomena. It seeks to
clarify the phenomena, illuminate them, explain how they are
related to other phenomena, and how this may be related to
undergraduate mathematics course organization and
teaching.
-“Mathematics Education Research: A Guide for the Research
Mathematician” by McNight2, Magid, & Murphy
Gut Instinct's Surprising Role in Math
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Two distinct number systems
Approximate number system vs genuine
computation
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Emphasize the power of the ballpark figure
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Fermi Problems
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Test
The New York Times, September 2008
Truro Zoning Decision
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There was a vote on how soon hotels can be
converted to condominiums, and the measure
needed 2/3 vote to pass.
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The vote was 136 to 70.
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The town accountant computed 206 x .66 = 135.96.
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“But using .6666 — a more accurate version of twothirds — the affirmative vote needed to be 137
instead of 136, according to an anonymous caller
to town hall and to the Times.”
Cape Cod Times, April 2009
Functional Integration
You're doing Calculus but instead of using points on the real
line (or complex plane), your points are functions. So you're
integrating functions of functions (called functionals). How to
explain Calculus to someone who's never had it? Calculus
makes hard problems easy. It is useful for finding max/mins
in real applications.
-Dave Skoug
Do dogs know Calculus?
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Professor Tim Pennings of Hope College had
an idea while playing fetch with his Welsh
Corgi, Elvis, near Lake Michigan.
Elvis could run faster than he could swim.
Was Elvis's path optimal for retrieving the
ball as quickly as possible?
MAA - The College Mathematics Journal, May 2003
Let r denote the running
speed, and s the swimming
speed. Also, let's call z the
total distance from A to B
(through D). Finally, we will
denote the time it takes to get
to the ball as T(y).
Then...
is minimized by
On average, Elvis ran at r = 6.40 m/s and swam at s = 0.910
m/s, so from before, we get a relationship of y = 0.144x.
Pennings ran 35 “throw and fetch” trials with Elvis, measuring
the values of x and y.
Commutative Algebra and Algebraic Geometry
If I want to explain very briefly what I do, I say that I work with
different number systems and finding solutions to systems of
equations in these number systems. If I have more time or a
chalkboard, I like to describe the geometric connections: you can
think of a line in 3-space as the intersection of two planes. What
about a surface in 4-space? Is it the intersection of two 3dimensional objects?
-Tom Marley
The motivating goal in Algebraic Geometry is to understand solution
sets of systems of polynomial equations in several variables.
Algebraic Geometry thus subsumes linear algebra, where the goal
is to understand solutions sets of systems of linear equations in
several variables, but non-linear equations turn out to be much
harder to deal with.
-Brian Harbourne
Mathematicians to thank for great graphics
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100 powerful supercomputers perform geometrical, algebraic and
calculus-based calculations to animate Pixar's characters.
Tony DeRose:
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Without mathematics, we wouldn't have these visually rich
environments, and visually rich characters.
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You didn't see any water in Toy Story, whereas by the time
we got to Finding Nemo, we had the computer
techniques that were needed to create all the splash
effects.
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I remember as a mathematics student thinking, 'Well,
where am I ever going to use simultaneous equations?'
And I find myself using them every day, all the time now.
The supercomputers run 24 hours a day, seven days a week, and
it still takes them five to six hours to render a single frame
lasting 1/24th of a second. So, for every second of film, it takes
the computer six days.
Science Daily, May 2007
Math Biology
Whenever I say Math Bio, the response is usually a surprise that biology
uses mathematics! So, I try to talk about the large amounts of data
collection, using Ecology as an example, and get them to realize these
are things they already knew Biologists did, and so using mathematics
should be obvious. Then I explain that as a math biologist, I try to put
together all of this vast amount of data and use it to create equations that
will be able to make predictions about the future of the ecosystem. They
usually think then that I'm a biology statistician, but it's hard to get across
the idea of a model (as opposed to analyzing data from things that have
already happened). There are a few examples I use to help with the
concept of modeling: pest outbreaks, global climate change, and
managing a fishery.
-David Logan
YouTube Usage Decoded
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Why are certain videos on YouTube watched
millions of times while 90 percent of the
contributions find only the odd viewer?
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Top Videos of All Time
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Herd-like behavior
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Tracked viewer statistics for 5 million videos over
two years
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“Junk”, “Viral”, and “Quality” videos
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Recognizing potential blockbusters early
Science Daily, December 2008
Math Neuroscience
A central challenge in neuroscience is to understand how
neurons represent and process information about the world.
Our brain is constantly constructing and updating a
representation of the world around us, largely from sensory
inputs (visual, auditory, tactile, etc.). How is this information
represented in the brain? This requires understanding how
neurons - and, more generally, populations of neurons
working together in neural networks - encode and transform
information that comes in through the senses.
-Carina Curto
Improving Computer Language
Recognition
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With most current programs, if you speak just a little too
quickly or slowly, if your pronunciation isn’t clear, or if
there is background noise, the system often fails to
work properly
The brain classifies the various signals from the
smallest, fast-changing components (single sound
units like 'e' or 'u') up to big, slow-changing elements
(the topic).
Predicting the next speech sound
Science Daily, August 2009
For emaxlpe...
Arocdnicg to rsceearch at Cmabrigde
Uinervtisy, it deosn’t mttaer in waht oredr the
ltteers in a wrod are, the olny iprmoatnt tihng
is taht the frist and lsat ltteer are in the rghit
pcale. The rset can be a toatl mses and you
can sitll raed it wouthit pobelrm. Tihs is
buseace the huamn mnid deos not raed
ervey lteter by istlef, but the wrod as a wlohe.
Discrete Mathematics
Usually I say that I study the structure of networks, using the
specific example of the Internet and computer networks.
-Stephen Hartke
Combinatorics is a fancy word for counting. Combinatorics is
concerned with determining the number of logical
possibilities of some event without necessarily listing all the
particular outcomes. One can often perform calculations
involving probabilities simply by counting the possible
outcomes. Combinatorics often requires counting the
number of rearrangements or groupings of a set of objects.
-MathCounts
Betting on March Madness
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64 games determine the winner
So a randomly chosen bracket has a 1 in 264
chance of being completely correct.
If every person on Earth could fill out a
bracket every second, then it would take
them roughly one century to fill out all
possibilities.
Science Daily, March 2006
Groups, Semigroups, & Topology
When you have an object and you bend it a little bit (or a lot!), the
object hasn't really inherently changed; topology is about
understanding what it is about the object that hasn't changed. As
an example, what makes a ball a ball, and not a doughnut? Knot
theory is really the topology of knotted circles; what is different
about your shoes when they are tied as opposed to when they
aren't? How many different ways can you tie your shoes? Since
lots of physical objects can be modeled as knotted objects
(molecules, elementary particles="strings"), the concepts and
techniques of knot theory can be applied to many physical
systems.
-Mark Brittenham
Geometric Map of the Internet
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The researchers at San Diego Supercomputer Center and
Cooperative Association for Internet Data Analysis at UC - San
Diego have created the first geometric "atlas" of the Internet as
part of a project to prevent our most ubiquitous form of
communication from collapsing within the next decade or so.
They discovered a latent hyperbolic, or negatively curved, space
hidden beneath the Internet's topology, leading them to devise
a method to create an Internet map using hyperbolic geometry.
Such a map would lead to a more robust Internet routing
architecture because it simplifies path-finding throughout the
network.
Internet Black Holes
Science Daily, September 2010
Other Cool Applications of Math
For people with very little background in math (say, someone who
took high school math 25 years ago) I don't really try to say
"what's operator theory." Instead I try to explain what pure math is
and how it's different from grade school arithmetic. So I might talk
about something from graph theory and say “thats what pure
mathematicians do, and thats how what they do connects to the
modern world.”
-John Orr
People always like real world applications (they never ask about
applications of fine arts, but somehow math requires applications,
rather than simply appreciating the beauty of the ideas).
-David Pitts
Gallons per mile
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Which improvement will save more gas over
the course of a year: changing from 34
mpg to 50mpg, or from 18 to 28mpg?
How about 12mpg to 15mpg vs 30mpg to
60mpg?
Manufacturers should list efficiency in terms
of gallons per 10,000 miles driven.
Duke University News, June 2008
Congressional Districts
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Two person fair division: “I cut. You choose.”
Theoretically possible to double actual
percentage
Each party divides their part as they would like.
A party with 40% of the vote could get 80% of
the seats on their half of the state. If the other
party received all the votes on the other half,
than the original party will have 40% of the
seats.
Science News, February 2009
Basketball Free Throws
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Rick Barry
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NBA Career Free Throw Percentage of 89.3%
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In 1978-79, it hit a high of 94.7%.
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The higher the arc, the larger the window is, so
there's more room for error.
The optimum angle for the shot is 45 degrees plus half the angle from the top of the player’s
hand to the rim.
Discover Magazine, August 2008
The Donovan Index
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What is the most successful US pro-sports city?
Start by dividing, for each franchise, the sum of the
number of teams in the league during each year
that franchise won a championship
(championship points) by the number of seasons
the team has existed.
For example, the Phillies get 30 championship
points for their World Series title in 2009 and 26
for their 1980 title. They've been in existence for
102 years, so that's an index of 0.55.
Doesn't adjust for population
Wall Street Journal, The Numbers Guy, February 2009
SPF
Dermatologists say this is just a numbers game that confuses customers.
SPF
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SPF measures how much the product shields the
sun’s shorter-wave UVB rays, which can cause
sunburn.
If adequately applied, sunscreens with sky-high
SPFs offer slightly better protection against
lobster-red burns than an SPF 30.
In 2007, the FDA proposed capping SPF at 50+.
Calculated by comparing the time needed for a
person to burn unprotected with how long it takes
for that person to burn wearing sunscreen
Half the sunscreen = square root of the protection
The New York Times, May 2009
Lewis Carroll
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A cup contains 50 spoonfuls of brandy, and
another contains 50 spoonfuls of water. A
spoonful of brandy is taken from the first
cup and mixed into the second cup. Then a
spoonful of the mixture is taken from the
second cup and mixed into the first. Is
there more or less brandy in the second
cup than there is water in the first cup?
Final Random Facts
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th
The 46 Mersenne Prime
42,643,801
(2
-1) was one of
TIME's best inventions
of 2008.
In 2009, the Wall Street
Journal ranked
Mathematician as the
best job, and
Lumberjack as the
worst.
2 42,643,801
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