Mathematicians trace source of Rogers-Ramanujan identities, find algebraic gold

Mathematicians have found a framework
for the celebrated Rogers-Ramanujan
identities and their arithmetic
properties, solving another long-
standing mystery stemming from the work of
Indian math genius Srinivasa Ramanujan.
The findings, by mathematicians at Emory
University and the University of Queensland,
yield a treasure trove of algebraic numbers and
formulas to access them.

"Algebraic numbers are among the first
numbers you encounter in mathematics," says
Ken Ono, a number theorist at Emory "And yet,
it's surprisingly difficult to find functions that
return them as values in a uniform and
systematic way."

Ono is the co-author of the new findings,
along with S. Ole Warnaar of the University of
Queensland and Michael Griffin, an Emory
graduate student.

Ono announced the findings in April as a
plenary speaker at the Applications of
Automorphic Forms in Number Theory and
Combinatorics conference at Louisiana State
University. He will also present them as a
plenary speaker at the 2015 Joint Mathematics
Meetings, the largest mathematics meeting in
the world, set for January in San Antonio.
Warnaar, Griffin and others will give additional
talks on the findings during an invited special
session to accompany Ono's plenary address.
The most famous algebraic number of all is the
golden ratio, also known by the Greek letter
phi.

Many great works of architecture and art,
such as the Parthenon, are said to embody the
pleasing proportions of the golden ratio, which
is also seen in beautiful forms in nature.

Mathematicians, artists and scientists, from
ancient times to today have pondered the
qualities of phi, which is approximately equal
to 1.618, although its digits just keep on going,
with no apparent pattern.

"People studied the golden ratio before there
was a real theory of algebra," Ono says. "It
was a kind of prototype for algebraic
numbers."

Although no other algebraic units are as
famous as the golden ratio, they are of central
importance to algebra. "A fundamental
problem in mathematics is to find functions
whose values are always algebraic numbers,"
Ono says. "The famous Swiss mathematician
Leonhard Euler made some progress on this
problem in the 18th century. His theory of
continued fractions, where one successively
divides numbers in a systematic way, produces
some very special algebraic numbers like the
golden ratio. But his theory cannot produce
algebraic numbers which go beyond the stuff
of the quadratic formula that one encounters
in high school algebra."

Ramanujan, however, could produce such
numbers, and he made it look easy.
"Ramanujan has a very special, almost mythic,
status in mathematics," says Edward Frenkel, a
mathematician at the University of California,
Berkeley. "He had a sort of Midas touch that
seemed to magically turn everything into gold."
And the Rogers-Ramanujan identities are
considered among Ramanujan's greatest
legacies, adds Frenkel, a leading expert on the
identities.
"They are two of the most remarkable and
important results in the theory of q-series, or
special functions," says Warnaar, who began
studying the Rogers-Ramanujan identities
shortly after he encountered them while
working on his PhD in statistical mechanics
about 20 years ago.

Although originally discovered by L. J. Rogers
in 1894, the identities became famous through
the work of Ramanujan, who was largely self-
taught and worked instinctively.
In 1913, Ramanujan sent a letter from his
native India to the British mathematician G. H.
Hardy that included the two identities that
Rogers discovered and a third formula that
showed these identities are essentially modular
functions and their quotient has the special
property that its singular values are algebraic
integral units. That result came to be known as
the Rogers-Ramanujan continued fraction.
Hardy was astonished when he saw the
formulas. "I had never seen anything in the
least like this before," Hardy wrote. "A single
look at them is enough to show they could
only be written down by a mathematician of
the highest class. They must be true because
no one would have the imagination to invent
them."

"Ramanujan seemed to produce this result out
of thin air," Ono says.
Ramanujan died in 1920 before he could
explain how he conjured up the formulas.

"They have been cited hundreds of times by
mathematicians," Ono says. "They are used in
statistical mathematics, conformal field theory
and number theory. And yet no one knew
whether Ramanujan just stumbled onto the
power of these two identities or whether they
were fragments of a larger theory."
For nearly a century, many great
mathematicians have worked on solving the
mystery of where Ramanujan's formulas came
from and why they should be true.
Ono uses the analogy of going for a walk in a
creek bed and discovering a piece of gold. Had
Ramanujan accidentally found a random
nugget? Or was he drawn to that area because
he knew of a rich seam of gold nearby?

Warnaar was among those who pondered these
questions. "Just like digging for gold, in
mathematics it's not always obvious where to
look for a solution," he says. "It takes time
and effort, with no guarantee of success, but it
helps if you develop a lot of intuition about
where to look."
Finally, after 15 years of focusing almost
entirely on the Rogers-Ramanujan identities,

Warnaar found a way to embed them into a
much larger class of similar identities using
something known as representation theory.
"Ole found the mother lode of identities," Ono
says.
When Ono saw Warnaar's work posted last
November on arXiv.org, a mathematics-physics
archive, his eyes lit up.

"It just clicked," Ono recalls. "Ole found this
huge vein of gold, and we then figured out a
way to mine the gold. We went to work and
showed how to come full circle and make use
of the formulas. Now we can extract infinitely
many functions whose values are these
beautiful algebraic numbers."
"Historically, the Rogers-Ramanujan identities
have tantalized mathematicians," says George
Andrews, a mathematician at Penn State and
another top authority on the identities.

"They
have played an almost magical role in many
areas of math, statistical mechanics and
physics."

The collaboration of Warnaar, Ono and Griffin
"has given us a big picture of the general
setting for these identities, and deepened our
theoretical understanding for many of the
breakthroughs in this area of mathematics
during the past 100 years,"

Andrews says.

"They've given us a whole new set of tools to
be able to attack new problems."

"It's incredibly exciting to solve any problem
related to Ramanujan, he's such an important
figure in mathematics," Warnaar says. "Now
we can move on to more questions that we
don't understand. Math is limitless, and that's
fantastic."

The physics of curly hair: Researchers develop first detailed model for a 3-D strand of curly hair

Summary:

February 12, 2014
Massachusetts Institute of
Technology
The heroes and villains in animated films tend
to be on opposite ends of the moral spectrum.
But they're often similar in their hair, which is
usually extremely rigid or -- if it moves at all
-- is straight and swings to and fro. It's rare to
see an animated character with bouncy, curly
hair, since computer animators don't have a
simple mathematical means for describing it.
But now, researchers have developed the first
detailed model for a 3-D strand of curly hair.

FULL RESEARCH

The heroes and villains in animated films
tend to be on opposite ends of the moral
spectrum. But they're often similar in
their hair, which is usually extremely rigid
or -- if it moves at all -- is straight and swings
to and fro. It's rare to see an animated
character with bouncy, curly hair, since
computer animators don't have a simple
mathematical means for describing it.
However, change may be coming soon to a
theater near you: In a paper appearing in the
Feb. 13 issue of Physical Review Letters ,
researchers at MIT and the Université Pierre et
Marie Curie in Paris provide the first detailed
model for the 3-D shape of a strand of curly
hair.

This work could have applications in the
computer animation film industry, but it also
could be used by engineers to predict the
curve that long steel pipes, tubing, and cable
develop after being coiled around a spool for
transport. In the field, these materials often
act like a stubborn garden hose whose intrinsic
curves make it behave in unpredictable ways.

In engineering terminology, these items -- and
hair -- are all examples of a slender, flexible
rod.
Co-authors on the paper are Pedro Reis, an
assistant professor in MIT's Department of Civil
and Environmental Engineering and Department
of Mechanical Engineering; Basile Audoly and
Arnaud Lazarus, of the Université Pierre et
Marie Curie; and former MIT graduate student
James Miller, who is now a research associate
at Schlumberger-Doll Research. Miller worked
on this project as part of his doctoral thesis
research and is lead author of the paper.
"Our work doesn't deal with the collisions of
all the hairs on a head, which is a very
important effect for animators to control a
hairstyle," Reis says. "But it characterizes all
the different degrees of curliness of a hair and
describes mathematically how the properties
of the curl change along the arc length of a
hair."

When Reis set out to investigate the natural
curvature in flexible rods, he wasn't thinking
of hair. But as he studied several small flexible,
curved segments of tubing suspended from a
structure in his lab, he realized they weren't so
different from strands of curly hair hanging on
a head. That's when he contacted Audoly, who
had previously developed a theory to explain
the 2-D shape of human hair.
Using lab experimentation, computer
simulation, and theory -- "the perfect triangle
of science," Reis says -- the team identified the
main parameters for curly hair and simplified
them into two dimensionless parameters for
curvature (relating to the ratio of curvature
and length) and weight (relating to the ratio of
weight and stiffness). Given curvature, length,
weight, and stiffness, their model will predict
the shape of a hair, steel pipe, or Internet
cable suspended under its own weight.
As a strand of hair curls up from the bottom,
its 2-D hook grows larger until it reaches a
point where it becomes unstable under its own
weight and falls out of plane to become a 3-D
helix. Reis and co-authors describe the 3-D
curl as a localized helix, where only a portion
of the strand is curled, or a global helix, if the
curliness extends the entire length up to the
head.

A curl can change phase -- from 2-D to 3-D
local helix to 3-D global helix, and back again
-- if its parameters change. Because a strand
of hair is weighted from the bottom by gravity,
the top of the strand has more weight under it
than the tip, which has none. Thus, if the
weight on a hair is too great for its innate
curliness, the curl will fail and become either
straight or helical, depending on the strand's
length and stiffness.
For the curvature study, Miller created
flexible, thin rods using molds as small as a
bottle of Tabasco sauce and as large as the
columns in MIT's Lobby 7 (about a meter in
diameter). He injected a rubber-like material
inside hollow flexible tubing wrapped around
these molds. Once the rubber material cured
and the tubing was cut away, Miller and Reis
had flexible polyvinyl thin rods whose natural
curvature was based on the size of the object
around which they had been wrapped.

The researchers' use of dimensionless numbers
to describe innate curvature means the
equation will hold true at all scales. Even with
lengths measured in kilometers, the steel
piping used by the oil industry is flexible
enough to be spooled. "We think of steel pipes
as being nice and straight but usually at some
point they're getting wrapped around
something," Miller says. "And at large
dimensions, they're so flexible that it's like you
and I dealing with a limp spaghetti noodle."
"The mathematician [Leonhard] Euler first
derived the equation for a slender elastic body
-- like a hair strand -- in 1744," Audoly says.
"Even though the equations are well-known,
they have no explicit solution and, as a result,
it is challenging to connect these equations
with real shapes."
"The fact that I am bald and worked on this
problem for several years became a nice
running joke in our lab," Reis says. "But joking
aside, for me the importance of the work is
being able to take the intrinsic natural
curvature of rods into account for this class of
problems, which can dramatically affect their
mechanical behavior. Curvature can delay
undesirable instability that happens at higher
loads or torsion, and this is an effect that
engineers need to be able to understand and
predict."

NET / JRF IN MATHEMATICS

Council of Scientific and
Industrial Research
Human Resource
Development Group
Examination Unit

CSIR-UGC (NET) EXAM FOR AWARD
OF JUNIOR RESEARCH FELLOWSHIP
AND ELIGIBILITY FOR LECTURERSHIP

MATHEMATICAL SCIENCES

EXAM SCHEME
TIME: 3
HOURS
MAXIMUM M ARKS: 200

CSIR-UGC (NET) Exam for Award of
Junior Research Fellowship and
Eligibility for Lecturership shall be a
Single Paper Test having Multiple Choice
Questions (MCQs). The question paper
shall be divided in three parts.

Part 'A'

This part shall carry 20
questions pertaining to
General Science,
Quantitative Reasoning &
Analysis and Research
Aptitude. The candidates
shall be required to answer
any 15 questions. Each
question shall be of two
marks. The total marks
allocated to this section
shall be 30 out of 200.

Part 'B'

This part shall contain 40
Multiple Choice Questions
(MCQs) generally covering
the topics given in the
syllabus. A candidate shall
be required to answer any
25 questions. Each question
shall be of three marks. The
total marks allocated to this
section shall be 75 out of
200.

Part 'C'

This part shall contain 60
questions that are designed
to test a candidate's
knowledge of scientific
concepts and/or application
of the scientific concepts.
The questions shall be of
analytical nature where a
candidate is expected to
apply the scientific
knowledge to arrive at the
solution to the given
scientific problem. The
questions in this part shall
have multiple correct
options. Credit in a question
shall be given only on
identification of ALL the
correct options. No credit
shall be allowed in a
question if any incorrect
option is marked as correct
answer. No partial credit is
allowed. A candidate shall
be required to answer any
20 questions. Each question
shall be of 4.75 marks. The
total marks allocated to this
section shall be 95 out of
200.

· For Part ‘A’ and ‘B’
there will be Negative
marking @25% for each
wrong answer. No Negative
marking for Part ‘C’.

· To enable the
candidates to go through the
questions, the question
paper booklet shall be
distributed 15 minutes
before the scheduled time of
the exam. The Answer sheet
shall be distributed at the
scheduled time of the exam.
· On completion of the
exam i.e. at the scheduled
closing time of the exam, the
candidates shall be allowed
to carry the Question Paper
Booklet. No candidate is
allowed to carry the
Question Paper Booklet in
case he/she chooses to leave
the test before the scheduled
closing time.

· Model Question Paper
is available on HRDG
website www.csirhrdg.res.in

Basic importent topic

18, 2013 — When early elementary math
teachers ask students to explain their problem-
solving strategies and then tailor instruction to
address specific gaps in their understanding,
students learn significantly more than those
taught using a more traditional approach. This
was the conclusion of a yearlong study of
nearly 5,000 kindergarten and first-grade
students conducted by researchers at Florida
State University.
The researchers found that “formative
assessment,” or the use of ongoing evaluation
of student understanding to inform targeted
instruction, increased students’ mastery of
foundational math concepts that are known to
be essential to later achievement in
mathematics and science.
Their results corroborated those of two earlier
pilot projects indicating that implementation of
the Mathematics Formative Assessment System
(MFAS) can markedly improve academic
performance in mathematics. The findings
further suggested that MFAS may help close
the gender gap that often develops by third
grade.
“The results of the most recent study
conducted in schools across Florida are
exciting,” said Laura Lang, principal
investigator who directed development and
testing of MFAS. “The randomized field trial
showed that students in K-3 classes where
teachers used MFAS were well ahead of other
students taught by teachers using more
traditional approaches. As one of the
elementary principals of a participating school
put it, MFAS is a real ‘game changer’ in terms
of student engagement and success in math.”
MFAS was created through the efforts of
researchers at the Florida Center for Research
in Science, Technology, Engineering and
Mathematics (FCR–STEM) who received $2.9
million in competitively awarded grant funds
from the Florida Department of Education’s
Race to the Top program to pursue the
project. MFAS is fully aligned with the Common
Core State Standards adopted in Florida and
many other states.
The randomized field trial was conducted in
partnership with 31 schools and 301 teachers
in three Florida districts across the state —
one urban, one suburban and one rural.
Schools were randomly assigned to either the
MFAS treatment group or to a group that used
a more typical approach to math instruction.
Comparing average annual gains in math on
nationally normed tests to the results, learning
was accelerated when teachers integrated MFAS
in their day-to-day instruction.
“In kindergarten, we can infer that students
learned at a rate equivalent to an extra six
weeks of instruction,” Lang said. “In first
grade, the gains were even greater — two
months of extra instruction. It was as if we
extended the school year without actually
adding any more days to it.”
In constructing MFAS, Lang and her team drew
upon research demonstrating that the learning
of mathematics is facilitated when teachers
gain deeper insights into what their students
already know and are able to do as well as
what students do not know and are unable to
do. Teachers gather these insights through
careful observation and by engaging students
in discussions of their mathematical thinking.
“Formative assessment is a process, not a
test,” Lang said, “and feedback is a key
element.”
The approach enables teachers to address each
child’s instructional needs. Teachers can avoid
holding back those who are ready to advance,
while efficiently helping those who are
struggling. This contrasts sharply with current
practice in many elementary classrooms.
“Based on our classroom observations over the
past four years, teachers typically rely heavily
on a math textbook to guide the planning of
day-to-day instruction and often provide
students feedback only on whether their
answers are correct,” Lang said. “Teachers
integrating formative assessment in instruction
not only ask students to do math tasks but
also to explain their reasoning and to justify
their solutions. As a result, teachers are better
equipped to identify misconceptions,
determine gaps in understanding and adjust
their instruction accordingly.”
Students play a key role in the formative
assessment process. MFAS actively engages
students, encouraging them to monitor and
regulate their own learning. Students also
evaluate each other’s work and provide
productive feedback, working as a team.
MFAS also has potential long-term effects on
closing the gender gap in mathematics, Lang
said. Studies show that even though both boys
and girls enter school with a fundamental
number sense, by the third grade boys tend to
do better in mathematics.
The results of a pilot study conducted in
second- and third-grade classrooms suggest
that, in classrooms where MFAS was used, by
third grade the girls showed no statistically
significant difference in mathematics
achievement from boys, according to Mark
LaVenia, methodologist on the MFAS team.
However, in classrooms with more
conventional instruction, girls continued to lag
behind boys in math achievement.

Story Source:

The above story is based on materials provided
by Florida State University, via Newswise.

Unlocking Biology With Math

Oct. 7, 2013 — Scientists at USC have created
a mathematical model that explains and
predicts the biological process that creates
antibody diversity -- the phenomenon that
keeps us healthy by generating robust immune
systems through hypermutation.
The work is a collaboration between Myron
Goodman, professor of biological sciences and
chemistry at the USC Dornsife College of
Letters, Arts and Sciences; and Chi Mak,
professor of chemistry at USC Dornsife.
"To me, it was the holy grail," Goodman said.
"We can now predict the motion of a key
enzyme that initiates hypermutations in
immunoglobulin (Ig) genes."

Goodman first described the process that
creates antibody diversity two years ago. In
short, an enzyme called "activation-induced
deoxycytidine deaminase" (or AID) moves up
and down single-stranded DNA that encodes
the pattern for antibodies and sporadically
alters the strand by converting one nitrogen
base to another, which is called "deamination."

The change creates DNA with a different
pattern -- a mutation.
These mutations, which AID creates a million-
fold times more often than would otherwise
occur, generate antibodies of all different sorts
-- giving you protection against germs that
your body hasn't even seen yet.
"It's why when I sneeze, you don't die,"
Goodman said.
In studying the seemingly random motion of
AID up and down DNA, Goodman wanted to
understand why it moved how it did, and why
it deaminated in some places much more than
others.

"We looked at the raw data and asked what the
enzyme was doing to create that," Goodman
said. He and his team were able to develop
statistical models whose probabilities roughly
matched the data well, and were even able to
trace individual enzymes visually and watch
them work.

But they were all just
approximations, albeit reasonable ones.
Collaborating with Mak, however, offered
something better: a rigorous mathematical
model that describes the enzyme's motion and
interaction with the DNA and an algorithm for
directly reading out AID's dynamics from the
mutation patterns.
At the time, Mak was working on the
mathematics of quantum mechanics. Using
similar techniques, Mak was able to help
generate the model, which has been shown
through testing to be accurate.

"Mathematics is the universal language behind
physical science, but its central role in
interpreting biology is just beginning to be
recognized," Mak said. Goodman and Mak
collaborated on the research with Phuong
Pham, assistant research professor, and Samir
Afif, a graduate student at USC Dornsife. An
article on their work, which will appear in
print in the Journal of Biological Chemistry on
October 11, was selected by the journal as a
"paper of the week."

Next, the team will generalize the
mathematical model to study the "real life"
action of AID as it initiates mutations during
the transcription of Ig variable and constant
regions, which is the process needed to
generate immunodiversity in human B-cells.

Non-Traditional Mathematics Curriculum Results in Higher Standardized Test Scores

Sept16, 2013 — For many years, studies have
shown that American students score
significantly lower than students worldwide in
mathematics achievement, ranking 25 th among
34 countries. Now, researchers from
theUniversity of Missouri have found high
school students in the United States achieve
higher scores on a standardized mathematics
test if they study from a curriculum known as
integrated mathematics.
James Tarr, a professor in the MU College of
Education, and Doug Grouws, a professor
emeritus from MU, studied more than 3,000
high school students around the country to
determine whether there is a difference in
achievement when students study from an
integrated mathematics program or a more
traditional curriculum. Integrated mathematics
is a curriculum that combines several
mathematic topics, such as algebra, geometry
and statistics, into single courses. Many
countries that currently perform higher than
the U.S. in mathematics achievement use a
more integrated curriculum. Traditional U.S.
mathematics curricula typically organize the
content into year-long courses, so that a 9 th
grade student may take Algebra I, followed by
Geometry, followed by Algebra II before a pre-
Calculus course.
Tarr and Grouws found that students who
studied from an integrated mathematics
program scored significantly higher on
standardized tests administered to all
participating students, after controlling for
many teacher and student attributes. Tarr says
these findings may challenge some long-
standing views on mathematics education in
the U.S.
"Many educators in America have strong views
that a more traditional approach to math
education is the best way to educate high
school students," Tarr said. "Results of our
study simply do not support such impassioned
views, especially when discussing high-
achieving students. We found students with
higher prior achievement scores benefitted
more from the integrated mathematics
program than students who studied from the
traditional curriculum."

Tarr and Grouws' papers, which were recently
published in the Journal for Research in
Mathematics Education, come from a three-
year study measuring educational outcomes
for students studying from different types of
mathematics curricula. Tarr says improving
American mathematics education is vital for
the future of the country
.
"Many countries that the U.S. competes with
economically are outpacing us in many fields,
particularly in mathematics and science," Tarr
said. "It is crucial that we re-evaluate our
school mathematics curricula and how it is
implemented if we hope to remain competitive
on a global stage."
Tarr and Grouws' longitudinal study is funded
by grant of more than $2 million from the
National Science Foundation.

Story Source:

The above story is based on materials provided
by University of Missouri-Columbia.

Journal Reference:

1. James E. Tarr, Douglas A. Grouws, Óscar
Chávez, and Victor M. Soria. The Effects of
Content Organization and Curriculum
Implementation on Students’ Mathematics
Learning in Second-Year High School
Courses .
Journal for Research in Mathematics.

Photo maths

Some maths articles :  only image,  just like topic.

Arresting Model Stops Cars

Sep. 5, 2013 — Researchers in China have
developed a mathematical model that could
help engineers design a flexible vehicle-arrest
system for stopping cars involved in criminal
activity or terrorism, such as suspect car
bombers attempting break through a check
point, without wrecking the car or killing the
occupants.

Writing in a forthcoming issue of the
International Journal of Vehicle Design, Pak Kin
Wong and colleagues in the Department of
Electromechanical Engineering at the
University of Macau, in Taipa, Macao, explain
how common vehicle-arrest systems used by
law enforcement, the military and in anti-
terrorism activities, usually cause serious
damage to the vehicle and maim or kill the
occupants. A more positive system for bringing
a car chase to a halt or stopping a car-bomber
in their tracks is needed if perpetrators,
witnesses and evidence are to be protected.

A flexible system would increase the stopping
distance of a vehicle involved in criminal or
terrorist activity and allow its kinetic energy to
be dissipated without the complete destruction
of the vehicle as otherwise occurs with solid,
immovable barriers and equipment currently
used. The team's mathematical model of
vehicle arrest with different flexible materials
and designs bears up to theoretical and
experimental scrutiny and offers engineers a
new set of variables to embed in their design
program in the development of new, effect
vehicle arrest systems. Moreover, the system
could allow the design of an "intelligent"
vehicle-arrest system for roadblocks and
checkpoints that could respond differently
depending on vehicle speed and type and allow
for greater control in bringing a vehicle to a
stop

Story Source:

The above story is based on materials provided
by Inderscience Publishers , via EurekAlert!, a
service of AAAS.
And ( science daily magazine ) .

Note: Materials may be edited for content and
length. For further information, please contact
the source cited above.

Journal Reference:

1. Pak Kin Wong et al. Modelling and testing of
arresting process in flexible vehicle
arresting systems. Int. J. Vehicle Design ,
2013, 64, 1-25

Generosity Leads to Evolutionary Success, Biologists Show

Sep. 2, 2013 — With new insights into the
classical game theory match-up known as the
"Prisoner's Dilemma," University of
Pennsylvania biologists offer a mathematically
based explanation for why cooperation and
generosity have evolved in nature.
Their work builds upon the seminal findings of
economist John Nash, who advanced the field
of game theory in the 1950s, as well as those
of computational biologist William Press and
physicist-mathematician Freeman Dyson, who
last year identified a new class of strategies for
succeeding in the Prisoner's Dilemma.
Postdoctoral researcher Alexander J. Stewart
and associate professor Joshua B. Plotkin, both
of Penn's Department of Biology in the School
of Arts and Sciences, examined the outcome of
the Prisoner's Dilemma as played repeatedly by
a large, evolving population of players. While
other researchers have previously suggested
that cooperative strategies can be successful in
such a scenario, Stewart and Plotkin offer
mathematical proof that the only strategies
that succeed in the long term are generous
ones. They report their findings in the
Proceedings of the National Academy of
Sciences the week of Sept. 2.
"Ever since Darwin," Plotkin said, "biologists
have been puzzled about why there is so much
apparent cooperation, and even flat-out
generosity and altruism, in nature. The
literature on game theory has worked to
explain why generosity arises. Our paper
provides such an explanation for why we see
so much generosity in front of us."
The Prisoner's Dilemma is a way of studying
how individuals choose whether or not to
cooperate. In the game, if both players
cooperate, they both receive a payoff. If one
cooperates and the other does not, the
cooperating player receives the smallest
possible payoff, and the defecting player the
largest. If both players do not cooperate, they
receive a payoff, but it is less than what they
would gain if both had cooperated. In other
words, it pays to cooperate, but it can pay
even more to be selfish.
In the Iterated Prisoner's Dilemma, two players
repeatedly face off against one another and can
employ different strategies to beat their
opponent. In 2012, Press and Dyson "shocked
the world of game theory," Plotkin said, by
identifying a group of strategies for playing
this version of the game. They called this class
of approaches "zero determinant" strategies
because the score of one player is related
linearly to the other. What's more, they
focused on a subset of zero determinant
approaches they deemed to be extortion
strategies. If a player employed an extortion
strategy against an unwitting opponent, that
player could force the opponent into receiving
a lower score or payoff.
Stewart and Plotkin became intrigued with this
finding, and last year wrote a commentary in
PNAS about the Press and Dyson work. They
began to explore a different approach to the
Prisoner's Dilemma. Instead of a head-to-head
competition, they envisioned a population of
players matching up against one another, as
might occur in a human or animal society in
nature. The most successful players would get
to "reproduce" more, passing on their
strategies to the next generation of players.
It quickly became clear to the Penn biologists
that extortion strategies wouldn't do well if
played within a large, evolving population
because an extortion strategy doesn't succeed
if played against itself.
"The fact that there are extortion strategies
immediately suggests that, at the other end of
the scale, there might also be generous
strategies," Stewart said. "You might think
being generous would be a stupid thing to do,
and it is if there are only two players in the
game, but, if there are many players and they
all play generously, they all benefit from each
other's generosity."

In generous strategies, which are essentially
the opposite of extortion strategies, players
tend to cooperate with their opponents, but, if
they don't, they suffer more than their
opponents do over the long term.
"Forgiveness" is also a feature of these
strategies. A player who encounters a defector
may punish the defector a bit but after a time
may cooperate with the defector again.
Stewart noticed the first of these generous
approaches among the zero determinant
strategies that Press and Dyson had defined.

After simulating how some generous strategies
would fare in an evolving population, he and
Plotkin crafted a mathematical proof showing
that, not only can generous strategies succeed
in the evolutionary version of the Prisoner's
Dilemma, in fact these are the only approaches
that resist defectors over the long term.
"Our paper shows that no selfish strategies will
succeed in evolution," Plotkin said. "The only
strategies that are evolutionarily robust are
generous ones."
The discovery, while abstract, helps explain the
presence of generosity in nature, an inclination
that can sometimes seem counter to the
Darwinian notion of survival of the fittest.

"When people act generously they feel it is
almost instinctual, and indeed a large
literature in evolutionary psychology shows
that people derive happiness from being
generous," Plotkin said. "It's not just in
humans. Of course social insects behave this
way, but even bacteria and viruses share gene
products and behave in ways that can't be
described as anything but generous."
"We find that in evolution, a population that
encourages cooperation does well," Stewart
said. "To maintain cooperation over the long
term, it is best to be generous."

Story Source:

The above story is based on materials provided
by University of Pennsylvania , via
EurekAlert!, a service of AAAS.

How Vegetation Competes for Rainfall in Dry Regions

Aug. 30, 2013 — The greater the plant density
in a given area, the greater the amount of
rainwater that seeps into the ground. This is
due to a higher presence of dense roots and
organic matter in the soil. Since water is a
limited resource in many dry ecosystems, such
as semi-arid environments and semi-deserts,
there is a benefit to vegetation to adapt by
forming closer networks with little space
between plants.

Hence, vegetation in semi-arid environments
(or regions with low rainfall) self-organizes into
patterns or "bands." The pattern formation
occurs where stripes of vegetation run parallel
to the contours of a hill, and are interlaid with
stripes of bare ground. Banded vegetation is
common where there is low rainfall. In a paper
published last month in the SIAM Journal on
Applied Mathematics, author Jonathan A.
Sherratt uses a mathematical model to
determine the levels of precipitation within
which such pattern formation occurs.

"Vegetation patterns are a common feature in
semi-arid environments, occurring in Africa,
Australia and North America," explains
Sherratt. "Field studies of these ecosystems are
extremely difficult because of their remoteness
and physical harshness; moreover there are no
laboratory replicates. Therefore mathematical
modeling has the potential to be an extremely
valuable tool, enabling prediction of how
pattern vegetation will respond to changes in
external conditions."
Several mathematical models have attempted
to address banded vegetation in semi-arid
environments, of which the oldest and most
established is a system of partial differential
equations, called the Klausmeier model.
The Klausmeier model is based on a water
redistribution hypothesis, which assumes that
rain falling on bare ground infiltrates only
slightly; most of it runs downhill in the
direction of the next vegetation band. It is
here that rain water seeps into the soil and
promotes growth of new foliage. This implies
that moisture levels are higher on the uphill
edge of the bands. Hence, as plants compete
for water, bands move uphill with each
generation. This uphill migration of bands
occurs as new vegetation grows upslope of the
bands and old vegetation dies on the
downslope edge.

In this paper, the author uses the Klausmeier
model, which is a system of reaction-diffusion-
advection equations, to determine the critical
rainfall level needed for pattern formation
based on a variety of ecological parameters,
such as rainfall, evaporation, plant uptake,
downhill flow, and plant loss. He also
investigates the uphill migration speeds of the
bands. "My research focuses on the way in
which patterns change as annual rainfall varies.
In particular, I predict an abrupt shift in
pattern formation as rainfall is decreased,
which dramatically affects ecosystems," says
Sherratt. "The mathematical analysis enables
me to derive a formula for the minimum level
of annual rainfall for which banded vegetation
is viable; below this, there is a transition to
complete desert."

The model has value in making resource
decisions and addressing environmental
concerns. "Since many semi-arid regions with
banded vegetation are used for grazing and/or
timber, this prediction has significant
implications for land management," Sherratt
says. "Another issue for which mathematical
modeling can be of value is the resilience of
patterned vegetation to environmental change.

This type of conclusion raises the possibility of
using mathematical models as an early warning
system that catastrophic changes in the
ecosystem are imminent, enabling appropriate
action (such as reduced grazing)."
The simplicity of the model allows the author
to make detailed predictions, but more
realistic models are required to further this
work. "All mathematical models are a
compromise between the complexity needed to
adequately reflect real-world phenomena, and
the simplicity that enables the application of
mathematical methods.

My paper concerns a
relatively simple model for vegetation
patterning, and I have been able to exploit this
simplicity to obtain detailed mathematical
predictions," explains Sherratt. "A number of
other researchers have proposed more realistic
(and more complex) models, and
corresponding study of these models is an
important area for future work. The
mathematical challenges are considerable, but
the rewards would be great, with the potential
to predict things such as critical levels of
annual rainfall with a high degree of
quantitative accuracy."

Story Source:

The above story is based on materials provided
by Society for Industrial and Applied
Mathematics.
And ( sciencedaily magzine  ).

Journal Reference:

1. Jonathan A. Sherratt. Pattern Solutions of the
Klausmeier Model for Banded Vegetation in
Semiarid Environments V: The Transition
from Patterns to Desert. SIAM Journal on
Applied Mathematics, 2013; 73 (4): 1347 DOI:
10.1137/120899510

Maths cartoons

See maths cartoons photos.

Do Girls Really Experience More Math Anxiety?

Aug. 27, 2013 — Girls report more math
anxiety on general survey measures but are
not actually more anxious during math classes
and exams, according to new research
forthcoming in Psychological Science , a journal
of the Association for Psychological Science.

Existing research suggests that females are
more anxious when it comes to mathematics
than their male peers, despite similar levels of
achievement. But education researchers
Thomas Götz and Madeleine Bieg of the
University of Konstanz and the Thurgau
University of Teacher Education and colleagues
identified a critical limitation of previous
studies examining math anxiety: They asked
students to describe more generalized
perceptions of mathematics anxiety, rather
than assessing anxiety during actual math
classes and exams.

To address this limitation, the researchers
conducted two studies in which they collected
data from approximately 700 students from
grades 5 to 11. In the first study, they
compared students' responses on two different
measures: A questionnaire measuring anxiety
about math tests, and their real-time self-
reports of anxiety directly before and during a
math exam. In the second study, they
compared questionnaire measures of math
anxiety with repeated real-time assessments
obtained during math classes via mobile
devices.

Findings from the two studies replicated prior
research and existing gender stereotypes,
showing that girls reported more math anxiety
than boys on generalized assessments, despite
similar math achievement.
However, the data obtained during math exams
and classes revealed that girls did not
experience more anxiety than boys in real-life
settings.
The data further suggest that lower self-
reported competence in mathematics may
underlie the discrepancy between the levels of
anxiety reported by girls in the two settings.

The researchers note that general
questionnaires may allow inaccurate beliefs
about math ability to negatively bias girls'
assessments of their math abilities and
exacerbate their math anxiety.
According to Götz, Bieg, and colleagues, these
results suggest that stereotyped beliefs
regarding math ability, rather than actual
ability or anxiety differences, may be largely
responsible for women not choosing to pursue
careers in math-intensive domains.
Co-authors include Oliver Ludtke of Humboldt
University Berlin (Germany), Reinhard Pekrun
of the University of Munich (Germany), and
Nathan C. Hall of McGill University (Canada).
This research was supported by grants from
the German Research Foundation to the fourth
author (Project for the Analysis of Learning
and Achievement in Mathematics Grants PE
320/11-1, PE 320/11-2, PE 320/11-3, and PE
320/11-4).

Story Source:

The above story is based on materials provided
by Association for Psychological Science .
And ( sciencedaily magazine ).

Matroid Theory: Mathematician Solves 40-Year-Old Problem

Aug. 28, 2013 — A team of mathematicians
has solved a problem first posed more than 40
years ago that has confounded modern
mathematicians, until now.
Professor Jim Geelen of the University of
Waterloo and his colleagues, Professor Bert
Gerards of Centrum Wiskunde & Informatica
and the University of Maastricht in the
Netherlands, and Professor Geoff Whittle of
Victoria University of Wellington in New
Zealand have proved the famous Rota's
Conjecture.

The three men worked for almost 15 years to
solve this problem posed by the famous
mathematician and philosopher Gian-Carlo
Rota in 1970. Earlier this year, in Waterloo,
the trio completed the final step in their epic
project.
Rota's Conjecture relates to a specialized area
of mathematics known as matroid theory, a
modern form of geometry, which was
pioneered by the mathematician Bill Tutte.
The theory investigates the embedding of
abstract geometric structures, or matroids,
into concrete geometric frameworks - namely,
projective geometries over a given finite field.

The conjecture is that, for each finite field,
there is a finite set of obstructions preventing
such a realization. The conjecture was posed by
Rota at the International Congress of
Mathematics in 1970, serendipitously, one
week before Geelen was born.
"For me the most rewarding part of the
research project has been the collaboration
with Bert and Geoff. We work together about
three times a year typically for periods of
three weeks either here in Waterloo or in New
Zealand or the Netherlands," said Professor
Geelen. "Those visits are intense; we sit in a
room together, all day every day, in front of a
whiteboard. The discussion can be very lively
at times, while at other times, when we are
stuck, we might sit there for two hours without
saying a word; each just thinking about ways
to overcome the particular obstacle."
In 1999, Geelen, Gerards and Whittle joined
forces to work on Rota's Conjecture as well as
generalizing the famous Graph Minor Theory
developed by Robertson and Seymour to
matroids.

Last year they completed their Matroid Minor
Theory which gives deep insights into the
structure of matroids. The proof of Rota's
Conjecture relies on the full power of that
theory and, in addition, required
groundbreaking new results on matroid
connectivity.

According to the trio, the real hard work only
just began when early this year they started
writing up the results of their work. The Graph
Minors Theory itself filled more than 600
journal pages and the Matroid Minors Theory
is set to be at least as long. The team expects
that it will take them at least three years to
complete the writing.
Jim Geelen is a Professor in the Department of
Combinatorics and Optimization at the
University of Waterloo and holds a Canada
Research Chair.

Story Source:

The above story is based on materials provided
by University of Waterloo .

And ( science daily magazine ).

New Energy Model Offers Transparency to Let Others Replicate Findings

Aug. 27, 2013 — Computer models are used to
inform policy decisions about energy, but
existing models are generally "black boxes"
that don't show how they work, making it
impossible for anyone to replicate their
findings. Researchers from North Carolina
State University have developed a new open-
source model and are sharing the data they
put into it, to allow anyone to check their
work -- an important advance given the
environmental and economic impact of energy
policy decisions.

"Most models show you the math behind how
they work, but don't share the source code
that is supposed to implement that math -- so
you can't tell how faithful the model is to the
mathematics," says Dr. Joseph DeCarolis, an
assistant professor of civil, construction and
environmental engineering at NC State and co-
author of a paper on the new model. "And the
people utilizing existing models often don't
share the data they use. So, in effect, you can't
check their work.

"That's a problem, because the results of those
models are informing policy decisions with
billions of dollars on the line."
The new open-source model, called Temoa, is
an energy economy optimization (EEO) model.
EEO models are computer models that inform
policy and industry decisions by offering
insights into how energy costs and production
are likely to change over time.

For example,

EEO models could be used to identify strategies
that would drive down energy costs and
reduce greenhouse gas emissions over the next
10, 20 or 30 years.
DeCarolis's team designed Temoa to be flexible,
allowing users to look at any timeframe and on
any scale, from a global model to a model of a
single city. They also designed Temoa to be
more rigorous than existing models when it
comes to addressing uncertainty. Specifically,
they plan to combine multiple forms of
analysis to give policymakers more information
on the potential impact of specific policy
alternatives.

Story Source:

The above story is based on materials provided
by North Carolina State University.
And ( sciencedaily magzine )

Journal Reference:

1. Kevin Hunter, Sarat Sreepathi, Joseph F.
DeCarolis. Modeling for insight using Tools
for Energy Model Optimization and Analysis
(Temoa) . Energy Economics, 2013; 40: 339
DOI: 10.1016/j.eneco.2013.07.014

Fractions Gain Traction With Real- Life Models

Aug. 27, 2013 — If 3 is greater than 2, then ⅓
must be bigger than ½ -- right? Wrong. As
thousands of students head back to school
next week, many will use exactly that kind of
thinking when faced with fractions for the first
time. New research from Concordia University
shows that for children to understand math,
teachers must constantly make the connection
between abstract numbers and real world
examples.

Helena Osana, associate professor in
Concordia's Department of Education, and PhD
candidate Nicole Pitsolantis put this theory to
the test in a classroom of fifth and sixth
graders. Their findings -- published in the
professional journal Teaching Children
Mathematics, as well as in the British Journal
of Educational Psychology -- show students
understand math much more clearly when
teachers use pictures and concrete models to
demonstrate what fractions actually mean.
Those connections are even stronger when the
model is personally meaningful to the
students. Write out '¾' on the blackboard and
the concept is not so clear. Show kids ¾ of a
shoelace or talk about running ⅓of the way to
school and suddenly they get it.
Although teachers already use models when
talking about fractions -- for instance, to show
a picture of a pie with slices eaten -- they
often put them away too quickly. To prove that
the constant use of models made a bigger
impact, Osana and Pitsolantis tried teaching
with models for only part of the lesson and
then the entire lesson.
They found that students showed much greater
understanding when the models were
continually present. "Our study shows teachers
should not only include pictures and models
while teaching fractions, but also have them
side by side throughout the class while
continually making clear connections between
the concepts and the models," says Osana.

The lessons produced by this research have the
potential to go beyond the classroom. "This is
something not only useful for teachers but
also for parents," says Osana. "Because
children are studying fractions, parents think
they're not able to help. But parents can have
positive effect on learning too. Something as
simple as writing '¾' out on a piece of paper,
then demonstrating what it means to use ¾ of
a cup of sugar, or filling up the gas tank until
it reaches the ⅔ mark, then writing and
pointing to the numbers '⅔ ,' can really go a
long way towards demystifying math," she
says.
Pitsolantis, who also teaches fourth and fifth
grade math at Lower Canada College in
Montreal, says that the depth of
misunderstanding that happens when models
are abandoned surprised her. She and Osana
are now testing out how teachers can
successfully incorporate concrete models into
other grades.

Story Source:

The above story is based on materials provided
by Concordia University .

And ( sciencedaily magazine ).

Formulae----1

Maths photo

Spatial Training Boosts Math Skills

June 25, 2013 — Training young children in
spatial reasoning can improve their math
performance, according to a groundbreaking
study from Michigan State University
education scholars.

The researchers trained 6- to 8-year-olds in
mental rotation, a spatial ability, and found
their scores on addition and subtraction
problems improved significantly. The mental
rotation training involved imagining how two
halves of an object would come together to
make a whole, when the halves have been
turned at an angle.

Past research has found a link between spatial
reasoning and math, but the MSU study is the
first to provide direct evidence of a causal
connection -- that when children are trained in
one ability, improvement is seen in the other.

The findings will be published in a forthcoming
issue of the Journal of Cognition and
Development.
Kelly Mix, professor of educational psychology,
said the findings suggest spatial training
"primes" the brain to better tackle calculation
problems. Mix authored the study with Yi-Ling
Cheng, a doctoral student in MSU's College of
Education.

"What's shocking is that we saw these
improvements in math performance after
giving the students just one 20-minute training
session in spatial ability," Mix said. "Imagine if
the training had been six weeks."
Understanding the connection between spatial
ability and math, she said, is especially
important in the early elementary grades
because many studies indicate early
intervention is critical for closing achievement
gaps in math.

Spatial ability is important for success in many
fields, from architecture to engineering to
meteorology, according to a Johns Hopkins
University paper. An astronomer must visualize
the structure of the solar system and the
motions of the objects in it, for example, while
a radiologist must be able to interpret the
image on an X-ray.
Some education experts have called for
including spatial reasoning in the elementary
math curriculum. But there are many forms of
spatial ability and Mix said it's important to
first figure out how each of them may or may
not relate to the various math disciplines.

To that end, Mix is leading a larger study that
tests elementary students on different forms of
spatial ability and math performance.
Mix's research into spatial ability and math is
funded by two grants totaling $2.8 million
from the Institute of Education Sciences, the
research arm of the U.S. Department of
Education.

Story Source:

The above story is based on materials provided
by Michigan State University .
Note: Materials may be edited for content and
length. For further information, please contact
the source cited above.

Journal Reference:

1. Yi Ling Cheng, Kelly S. Mix. Spatial Training
Improves Children's Mathematics Ability .
Journal of Cognition and Development , 2012; :
120919075341007 DOI:
10.1080/15248372.2012.725186

Stabilizing Aircraft During Takeoff and Landing Using Math

Aug. 20, 2013 — One of the lesser known
concerns about commercial aircraft is their
stability on the ground during taxiing, takeoff,
and landing. During these processes, planes
must maintain stability under various operating
conditions. However, in some situations, the
aircraft landing gear displays unwanted
oscillations, which are referred to as shimmy
oscillations.

In a paper published last month in the SIAM
Journal on Applied Dynamical Systems, authors
Chris Howcroft, Bernd Krauskopf, Mark
Lowenberg, and Simon Neild study the
dynamics of aircraft landing gear using
nonlinear models. The dynamics of landing
gear shimmy and the wheel-ground interaction
are fundamentally nonlinear.

"Shimmy oscillations of aircraft landing gear
have long been a problem, and their
prediction and prevention remains an ongoing
challenge in landing gear design," explains
author Chris Howcroft. "The issue is that a
landing gear may display the desired behavior
during ground take-off/landing manoeuvres
over several hundred or so flights, but then
suddenly oscillate given just the right -- or
rather, the wrong -- conditions."

Fortunately, mathematical models provide
cost-effective ways to study the dynamics of
the main landing gear (MLG) and determine the
types of oscillations that may occur under
different conditions. "The work we conducted
clarifies under which conditions shimmy
oscillations can be encountered in the MLG of
a representative midsize passenger aircraft. We
identified different types of shimmy
oscillations and showed where they occur,"
says Howcroft.

"Having the right mathematical model is really
the key," he adds. "Actual testing is extremely
expensive; however, nonlinear analysis
methods are very well suited to identifying
these hard-to-find dynamics. They may also be
employed to determine the shimmy
characteristics before the aircraft has actually
been built."

The model can provide insights not only into
aircraft operation, but also design features,
and can aid in adjusting both for optimum
stability.
Aircraft landing gear supports the weight of
the aircraft during landing and ground
operations. In addition to their wheels, landing
gear also have shock absorbing equipment or
"shock struts" as well as brakes, retraction
mechanisms, controls, and structural entities
that attach the gear to the aircraft.
The model in the paper takes into account tire-
contact dynamics and the orientation of the
side-stay, the part of the aircraft that supports
the shock strut. It characterizes the motion of
the system in terms of dynamics of the MLG,
which are expressed as three degrees of
freedom: rotation about the main strut, and
in-plane and out-of-plane motion with respect
to the plane of main strut and side-stay. After
determining the dynamics for the simplest
geometric case, where the side-stay is
perpendicular to the direction of travel, the
authors use the model to study different side-
stay orientations.

"For the specific case of MLG, we developed a
nonlinear and fully parameterised model that
allowed us to map out how its dynamics
depends on operational parameters, such as
aircraft velocity and loading, and design
parameters describing the geometry of the
landing gear," explains Howcroft. "In contrast
to the more traditional approach of performing
large numbers of simulations, this was
achieved by employing advanced tools from
dynamical systems that track solutions and
stability changes in parameters directly."

Moreover, other parameters could be
incorporated into the model further down the
road, such as runway conditions or tire
pressure, or physical effects such as the
dynamics of the shock absorber.

"Future directions of this research will focus
on the incorporation and assessment of
additional nonlinear effects," says Howcroft.
"For example, mechanical joints loosen over
the lifespan of an aircraft landing gear, and
this may have a dramatic effect on dynamic
performance, service life and maintenance
requirements of the landing gear."

"Nonlinear modeling and analysis are now
being introduced as tools into industrial
practice, via the recent development of a
MATLAB toolbox," says author Simon Neild.

"This is exciting to see, and has allowed our
group to tackle not only the problem of
landing gear shimmy, but of aircraft ground
manoeuvres and airliner loss-of-control in
flight."
Author Bernd Krauskopf adds.

"This project is
part of a larger research effort in collaboration
with Airbus into aircraft ground dynamics via
the bifurcation analysis of nonlinear models.
Related work concerns shimmy in nose landing
gears and its interplay with the dynamics of
the fuselage."
Future work would integrate many different
aspects into a unifying model.

"Ultimately our
current research is moving towards the
integration of landing gear and airframe into
an overall model that allows us to create a full
dynamic picture of aircraft ground dynamics,"
says author Mark Lowenbergm.

Story Source:

The above story is based on materials provided
by Society for Industrial and Applied
Mathematics. And  ( sciencedaily magazine  )

Note: Materials may be edited for content and
length. For further information, please contact
the source cited above.

Journal Reference:

1. Chris Howcroft, Bernd Krauskopf, Mark H.
Lowenberg, Simon A. Neild. Influence of
Variable Side-Stay Geometry on the Shimmy
Dynamics of an Aircraft Dual-Wheel Main
Landing Gear . SIAM Journal on Applied
Dynamical Systems , 2013; 12 (3): 1181 DOI:
10.1137/120887643

"Math in Daily Life"

When you buy a car, follow a recipe, or
decorate your home, you're using math
principles. People have been using these same
principles for thousands of years, across
countries and continents. Whether you're
sailing a boat off the coast of Japan or building
a house in Peru, you're using math to get
things done.

How can math be so universal?

First,
human beings didn't invent math
concepts; we discovered them. Also, the
language of math is numbers, not English
or German or Russian. If we are well versed in
this language of numbers, it can help us make
important decisions and perform everyday
tasks. Math can help us to shop wisely, buy the
right insurance, remodel a home within a
budget, understand population growth, or even
bet on the horse with the best chance of
winning the race.