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.