Category: Mathematics Page 6 of 8

Mathematical Restoration of Renaissance Masterpieces

By Olivia Zhu

On September 22, 2014

In Computers/Technology, Engineering, Mathematics, Visualization

The Ghissi Masterpiece, missing the ninth panel

By Olivia Zhu

Ninth panel of the Ghissi masterpiece, as reconstructed by Charlotte Caspers

What do Renaissance masterpieces and modern medical images have in common?

The same mathematical technique, “oriented elongated filters,” originally developed to detect blood vessels in medical images can actually be used to detect cracks in digital images of antiquated Renaissance paintings.

On September 19, Henry Yan, Rowena Gan, and Ethan Levine, three undergraduate students at Duke, presented their work on oriented elongated filters and many other techniques to the Math Department. Yan, Gan, and Levine performed summer research to detect and correct cracks in the digitized Ghissi masterpiece, an altarpiece done by 14-century Italian painter Francescuccio di Cecco Ghissi. The altarpiece originally consisted of nine panels, but one was lost in the annals of history and has been recently reconstructed by artist and art historian Charlotte Caspers.

The role of the three undergrads was to digitally rejuvenate the panels of the Ghissi masterpiece, which had faded and accumulated cracks in paint layers because of weathering factors like pressure and temperature. Using various mathematical analysis techniques based in Matlab, including oriented elongated filters, linear combinations of 2-D

Henry Yan’s K-SVD analysis to detect cracks in the image at left

Gaussian kernels (which essentially create directional filters), K-SVD (which updates atoms to better fit an image), and multi-scale top-hat (which extracts small elements and details from an image), the research group created a “crack map,” which they overlaid on the original image.

Then they instructed the computer to fill in the cracks with the colors directly adjacent to the cracks, thereby creating a smoother, crack-free image—this method is called inpainting.

In the future, Yan, Gan, and Levine hope to optimize the procedures they have already developed to accomplish color remapping to digitally age or refurbish images so that they look contemporary to their historical period, and to digitally restore gilding, the presence of gold leaf on paintings.

Math junior flips for 'bit flips'

By Karl Bates

On August 11, 2014

In Engineering, Mathematics, Students

By Ashley Mooney

Paul Ziquan Yang is using mathematical techniques to eliminate errors in computer hardware.

Over the summer, the rising junior math major worked with Robert Calderbank, Charles S. Sydnor professor of computer science, as part of the PRUV Fellowship program, a six-week mentorship sponsored by the Department of Mathematics. Yang now plans to continue to work with Calderbank in the fall and may turn the project into a senior thesis.

Paul Z. Yang’s summer fellowship in math wasn’t all work and no play.

“Professor Calderbank has been a great source of encouragement and inspiration,” Yang said. “I used to be an engineering major at Pratt, and he was also the one who helped me make my mind to transfer to Trinity and focus on math because that suits me better. I have learned much more from him than just math.”

Yang, from Beijing, China, is studying coding theory, and focusing on how to repair incorrectly stored data. In a computer’s hardware, all information is stored in the form of bits—binary values of zero or one.

Robert Calderbank, director of the Information Initiative at Duke

Within computers, something called a bit flip occasionally occurs where a value of one is replaced with zero or vice versa. In small amounts, these errors are harmless, but when they accumulate they can actually prevent machines from running correctly.

“Our aim is to add redundant information to a fixed list of binary bits so that we can detect the error and possibly correct it,” Yang said. “It’s sort of like cryptography but the aim is different.”

Cryptography is the study and use of mathematical techniques to secure communications and data in the presence of third parties. While cryptography is used to enforce the security of information, Yang employs similar mathematical methods to fix binary coding errors.

Yang noted that he is very excited about his research. “My favorite part is to see the interaction of various branches of math, and how research can connect these branches, even if they seem unrelated at first.”

His classes have provided him with both a theoretical and applied background for his research. His coursework has also trained him in the ways of thinking necessary to develop research questions.

Outside of his classes and research, Yang enjoys spending time with his friends and reading. He said he has also started taking tennis lessons for the first time, and is enjoying the sport more than he expected.

Yang said his long-term plan is eventually to become a professor. After graduation, he is planning on getting a Ph.D. in math, but is still unsure of what he would like to focus on in the field

“The research done at an undergraduate level doesn’t necessarily determine your subject at a graduate level,” Yang said. “I’ll see what interests me more over the next two years of courses.”

Teachers Look to 'Alice' for Help

By Karl Bates

On July 18, 2014

In Computers/Technology, Guest Post, Mathematics, Science Communication & Education

Guest Post by Leah Montgomery, NC Central University

With technology and computer science among the fastest growing fields of study today, it’s a wonder there are so few computer science classes in public middle and high schools.

Florida teacher Chari Distler’s message to a Duke classroom full of her middle and high school teaching colleagues was a promising one: They can get a new generation of kids interested in computer science.

School teachers from all over the country learned programming at Duke this summer.

All they have to do is follow Alice.

Alice is a 3D virtual worlds programming environment that offers an easy way to create animations for games and storytelling. Since 2008, Duke Professor Susan Rodger has led a two-week summer program training teachers to use Alice to help promote computer literacy among young students.

“What we’re trying to do is teach middle school and high school teachers, in all disciplines, how to program and then help them to integrate it into their discipline,” said Rodger. “The teachers will then expose students to what computer science is. The idea is that if they know what it is then they might choose it as a career when they go to college.”

Distler attended her first Adventures in Alice Programming session at Duke two years ago and returned this week to advise this year’s class on how she implemented the program in her classes.

She said one of her students from North Broward Preparatory School won second place in the annual Alice contest for his animated 45-second video titled “From Rags to Riches.”

Audrey Toney, an instructional coach for teachers in the North Carolina New Schools network, said she learned about Alice through a teacher who wanted to add programming to her curriculum.

“It gives students computational thinking and critical thinking and offers another way to present other than PowerPoint and Prezi,” said Toney.

Toney wants to challenge her professional development students to use Alice to replicate a design of a robotic arm that will lift and unload boxes. The program will allow students to budget money, price the cost of parts and code the robot’s movements.

During the first week of the workshop, teachers get familiar with the Alice software through interactive activities. Teachers created worlds with flying dragons, flipping princesses and annoyed Garfields.

The teachers worked together on learning Alice programming. (Les Todd, Duke Photography)

In week two, teachers learned about the use of 3-D imaging in the classroom at the Duke Immersive Virtual Environment (DiVE). The teachers also started creating their own Alice-based lesson plans this week. New Jersey high school teacher Kenneth McCarthy said he found his inspiration in the Sunday paper.

“I was thumbing through the Sunday paper and saw Garfield,” said McCarthy, who teaches algebra two and a beginner programming class . “It just looked like something that could be easily used with Alice.”

McCarthy is familiar with Alice, having used the program last year when his students participated in the Hour of Code, an initiative that challenges students and teachers to learn programming in one hour.

“I think the traditional thought was that you have to know algebra two (and other higher mathematics) to learn this, but Alice can be used in elementary schools,” said McCarthy.

Rising Duke senior Samantha Huerta was a workshop assistant for Susan Rodger for nine weeks this summer, helping develop workshop materials and finding ways to integrate computer science into math and other subjects.

“I wasn’t exposed to any type of computer science growing up,” said Huerta. “This is a field that isn’t going to go away, and we need to have more diversity. As a female Latina, I am a double minority and it is my hope to continue researching and bringing diversity to this field.”

Math and Comp Sci Junior Studies Fruit Flies

By Karl Bates

On July 9, 2014

In Animals, Biology, Computers/Technology, Mathematics, Students

By Ashley Mooney

Dorsal closure is a stage in fruitfly embryonic development that is used to study wound-healing.

Roger Zou, a computer science and math major from Solon, Ohio, is working on creating more efficient ways to study wound-healing in fruit flies. It turns out that the way fruit flies heal actually has implications for how mammals heal too.

The junior is developing computational methods that can more accurately quantify cellular properties of fruit flies. As fruit fly embryos develop, he tracks cells through space and time to learn more about a process called dorsal closure. It’s a developmental stage that is similar to wound healing, where a gap in the embryo’s epithelium—which is like its skin—is closed by the coordinated effort of different types of cells. (see movie below)

Roger Zou is a junior spending the summer in Dan Kiehart’s lab.

“It’s fun to study the morphological forces because it’s not entirely understood how organisms develop,” Zou said.

In his analysis, Zou uses a laser under a microscope to make cuts on areas of the fly embryos. After making cuts, Zou uses computational methods to measure the wound healing.

Beyond collecting such data, Zou is developing a computer program that analyzes images from the microscope more accurately.

Zou has worked in Biology Professor Daniel Kiehart’s lab since his freshman year. His project was originally a component of a graduate student’s dissertation, but after she graduated, he continued some aspects of her research.

His project has been funded by the Dean’s Summer Research Fellowship for two consecutive summers. He also has done several independent study projects. Although Zou is planning on publishing his research this summer, he will likely use the data eventually to do a senior thesis.

Several of Zou’s math and computer science classes have given him a background in the techniques needed to use a computer to analyze large sets of image data, he said.

“My favorite thing about my research is the ability to learn new things independently,” Zou said. “[Kiehart] is very good at leading me in the right direction but allowing me to be very independent and I think because of that I’ve been able to learn a lot more and learn from my mistakes.”

Outside of his research, Zou is a teaching assistant for the computer science class Data Structures and Algorithms. He also tutors Duke students in organic chemistry and middle school children in math through the America Reads*America Counts program. And he also does web development for The Chronicle, Duke University’s independent student newspaper.

After graduating, Zou said he hopes to pursue a PhD in either computational biology or computer science or maybe go for a combined MD-PhD program. No matter which program he chooses, Zou said he wants to continue doing research.

[youtube http://www.youtube.com/watch?v=Yk-O_W1Wqbc?rel=0]

Student's Summer is a Deep Dive into Deep Math

By Karl Bates

On July 2, 2014

In Mathematics, Students

By Ashley Mooney

Math and computer science student Will Victor is spending his summer immersed in Albert Einstein’s theory of general relativity and learning to conduct mathematics research.

Will Victor is a rising senior at Duke who’s spending the summer studying math — for fun!

Victor, from Aiken, South Carolina, is working with math and physics Professor Hubert Bray to explore how geometry relates to general relativity. General relativity explains how gravity affects space and time — pulling on them to create curvature we can both predict and measure. Think of how a world map must have curved latitude and longitude lines so that the representation of distances between spots on the curved surface of the globe are correct.

Victor is studying a class of objects called manifolds. It’s a field of math that describes the surface of complicated structures — anything from landmasses to a misshapen wad of clay — and it has practical applications in areas of mathematical physics such as black holes, dark matter and dark energy, the phenomena that Victor’s mentor studies.

“There’s a lot to be said about how fun it is to really know what the boundaries of the field are and to think about how we might attack answering the questions of what we don’t know,” Victor said.

This summer, he’s reading mathematical research and doing practice problems to learn the material. “I work problems every day to prepare my mind for research,” Victor said. He also works with Bray’s graduate students to see how they are tackling their own research questions.

The rising senior is part of the PRUV Fellowship program, a 6-week research mentorship sponsored by the Department of Mathematics. After the summer, he will participate in independent studies and eventually write a senior thesis on his research or pursue graduation with distinction.

Victor said his math classes have been essential to preparing him for research by helping him solve math problems, learn theorems, improve proof techniques and get exposure to different types of problems.

“Having a depth and breadth of mathematical knowledge is essential to being able to conduct research in the field,” Victor said. “Learning mathematics consists of acquiring both a formal language and a toolbox of argumentative frameworks which allow a student to speak precisely about theoretical problems.”

Victor noted that the classes he has taken in algebraic structures and classical analysis have provided the framework for talking about any type of modern mathematical question.

Beyond his research, Victor is involved with the selective living group Round Table and a member of the juggling club. He also works with the Community Empowerment Fund, an organization that works with and fosters relationships with the working poor in Durham and Chapel Hill.

“Our goal is to empower people to eventually obtain stable housing and stable employment,” Victor said. “We provide an accountability source for them, so we meet and talk weekly.”

After graduation, Victor says he hopes to go to graduate school in math and continue doing math research. “I enjoy the field I’m in, but I haven’t seen enough math to necessarily know this is what I’m most interested in,” Victor said. “I want to get exposed to more before deciding what I will specialize in.”

Duke Researchers Cited for Their Influence

By Karl Bates

On June 13, 2014

In Behavior/Psychology, Business/Economics, Cardiology, Chemistry, Environment/Sustainability, Faculty, Global Health, Mathematics, Medicine, Physics

We are the champions, my friend.

By Karl Leif Bates

A new compilation of the world’s most-cited scientists just released by Thomson Reuters (our friends from March Madness), shows that 32 Duke researchers are in the top one percent of their fields.

There are 3215 most-cited scientists on the list, so perhaps that makes Duke the one percent of the one percent?

Most-cited means a particular paper has been named frequently in the references by other papers in that field.

And that “is a measure of gross influence that often correlates well with community perceptions of research leaders within a field,” Thomson Reuters says. The database company admits their study methodology does favor senior authors who have had their papers out there longer, but there are quite a few younger Duke researchers in this list too.

From the Medical Center, the tops in citations in clinical medicine are cardiologists Eric Peterson, Robert Califf, Christopher Granger, and Eric Magnus Ohman. Michael Pencina, a biostatistician at the Duke Clinical Research Institute, is also most-cited in clinical medicine.

Perhaps not surprisingly, Nobel laureate, biochemist, and father of the G-protein coupled receptor Robert Lefkowitz made the list in pharmacology and toxicology.

Barton Haynes and David Montefiori of the Duke Human Vaccine Institute are listed in the microbiology category.

Medical School basic scientist Bryan Cullen of Molecular Genetics and Microbiology was cited in microbiology.

In psychiatry/psychology, A. John Rush, the vice dean for clinical research at Duke-NUS School of Medicine in Singapore, made the list, as did Richard Keefe, Joseph McEvoy of psychiatry and Avshalom Caspi, and Terrie Moffitt of Psychology & Neuroscience in Arts & Sciences.

Also from Trinity College of Arts and Sciences, Ahmad Hariri and HonaLee Harrington of Psychology & Neuroscience also made the list in psychiatry/psychology. Benjamin Wiley was oft-cited in Chemistry, James Berger and Ingrid Daubechies in mathematics, and plant biologists Philip Benfey, Xinnian Dong and Tai-Ping Sun in the category of plant and animal science.

Sanford School of Public Policy Dean Kelly Brownell is on the list in general social sciences, along with Arts & Sciences sociologist James Moody and nutrition researcher Mary Story of community and family medicine and the Duke Global Health Institute.

Nicholas School of the Environment researchers Robert Jackson and Heather Stapleton were cited the environment/ecology category.

From the Pratt School of Engineering, David R. Smith was cited in the physics category and Jennifer West in materials science.

The economics and business category includes Dan Ariely along with his Fuqua School of Business colleagues Campbell Harvey and Arts & Sciences economist Tim Bollerslev.

The Thomson Reuters analysis is based on their Web of Science database. This is the first time it has been done since 2001, when there were 45 Duke names on the list (including five that appeared again this time), but the methodology has changed somewhat.

UPDATE – There’s now a full PDF report from Thomson Reuters for download – http://sciencewatch.com/sites/sw/files/sw-article/media/worlds-most-influential-scientific-minds-2014.pdf

Women in Statistics Starts Here

By Karl Bates

On May 14, 2014

In Mathematics

By Karl Leif Bates

Duke statisticians formed a key variable in launching the first-ever “Women in Statistics” conference that kicks off Thursday, May 15 in Cary.

Duke statistician Dalene Stangl helped launch the first-ever Women in Statistics conference.

The meeting has attracted more than 300 attendees from around the country, including women from both academics and industry, said co-chair Dalene Stangl, a professor of the practice and associate chair and director of undergraduate studies in Duke’s department of statistics.

Rather than hearing the latest on Beyesian analysis, the group is going to focus on networking, mentoring, career advice and leadership, Stangl said.

The meeting was inspired by a conference called Women in Computer Science that has grown to more than 4,000 attendees. “We’re trying to replicate that,” said Stangl, who is chair of the American Statistical Association’s committee on women.

The inclusion of industry speakers from pharmaceuticals, retail, and government agencies as well as academics is deliberate Stangl said. “Many of the professional societies are organized around that academic role, but industry is a great option for many women.”

In addition to Stangl, Duke speakers will include Mine Cetinkaya-Rundel and Merlise Clyde (who built the conference web site).

Attendees seem to be a healthy mix of novice and experienced, she added. “If we’re missing a group, it’s probably in the middle, ” she said. The conference runs through Saturday.

UPDATE – Read Harvard post-doc Stephanie Hicks’ posts from the conference. May 20 and May 14.

Grad Student Solves 30-Year-Old Physics Problem

By Erin Weeks

On March 27, 2014

In Faculty, Mathematics, Physics, Students

By Erin Weeks

Sometimes an age-old question just needs a fresh set of eyes.

That was the case in Duke’s physics department, where a graduate student and professor recently resolved a calculating dilemma that has vexed computational physicists for decades.

Emilie Huffman, second year PhD student in physics

Emilie Huffman is a second-year PhD student from Charlotte, North Carolina. Last spring she began working with Shailesh Chandrasekharan, an associate professor and the director of graduate studies in physics, on what’s known as a sign problem.

Chandrasekharan is a theoretical nuclear and particle physicist who specializes in solving sign problems, which arise when one uses certain computational algorithms to calculate the behavior of large numbers of particles called fermions.

“Almost all the matter we know of are made with fermions,” Chandrasekharan said. “As building blocks of matter, it’s very important to be able to do calculations with them.”

But calculations of such complexity get tricky, and sign problems make it easy for wrong results to surface.

“It’s a very broad problem that affects almost all fields of physics involving quantum mechanics with strong correlations, where Monte Carlo methods are essential to perform calculations,” Chandrasekharan said.

Some in the field have simply moved on since the 1980s, leaving interesting questions plagued by sign problems unexplored. Other scientists have found workarounds and approximations. Very few, including Chandrasekharan, have tried to figure out solutions through the years. Huffman began work to expand on one of her advisor’s solutions, involving a grouping concept called fermion bags, and apply them to a new class of problems.

“She finally figured out a nice formula,” Chandrasekharan said. “Although the formula is quite simple and elegant, I couldn’t guess it.”

“In physics, often there’s a truth, and if you’re hitting on the right truth, everything starts falling into place.” Chandrasekharan says that’s what happened when he began applying Huffman’s formula to a class of problems.

Their paper appeared recently in the journal Physical Review B’s Rapid Communications.

“Now that I have a solution, I can begin to apply it,” Huffman said. Starting with condensed matter physics, Huffman plans to apply her solution to various questions that have been stymied by sign problems. “I can use this solution to study properties of graphene,” she said, referring to the single-layer carbon that has been touted as the strongest material in the world. Many puzzles remain in the field, especially involving multi-layer graphene sheets.

Wherever she turns her attention next, it’s clear Huffman has a promising career ahead.

Citation: “Solution to sign problems in half-filled spin-polarized electronic systems,” Emilie Huffman and Shailesh Chandrasekharan. Physical Review B Rapid Communications, March 12, 2014. DOI: 10.1103/PhysRevB.89.111101.

Jane Austen and Game Theory

By Olivia Zhu

On March 25, 2014

In Behavior/Psychology, Lecture, Mathematics, Science Communication & Education

Attendees played Regency Era card games involving game theory before the talk

By Olivia Zhu

“It is a great deal better to choose than to be chosen.” –Jane Austen, in Emma.

Jane Austen — novelist, romantic, and social critic — can now add another title to her repertoire: game theorist.

This role has been bestowed upon her by Michael Chwe, a game theorist in the Department of Political Science at UCLA and author of the book Jane Austen, Game Theorist. Chwe claims that Austen acts as a social scientist by setting up a theoretical framework for game theory in her novels. In his talk to a lively crowd well-versed in Austen’s works on March 25^th, Chwe explained Austen’s uncanny emphasis on choice, preference, and strategic thinking.

Chwe’s illustration of Jane’s choices and commensurability analysis in Pride and Prejudice

According to Chwe, Austen does not attribute actions to random variables, but rather to careful consideration of all alternatives. For example, Fanny Price in Mansfield Park chooses to refuse Henry Crawford’s offer of marriage after weighing her options; she does so entirely out of personal preference. Similarly, a major tenet in game theory is that the individual chooses what she wants to do without much consideration past her own wishes. Chwe said that Austen places a criticism on game theory here, when Fanny’s uncle, Sir Thomas, chastises Fanny being selfish instead of marrying Henry for the family’s financial security.

Chwe also introduced the game theory concept commensurability, in which negative factors are literally subtracted from positive factors in a decision to produce a single number of utility. He stated that Austen’s language, including phrases such as “finely checkered” happiness, “two

Chwe’s playful histogram of Elizabeth Bennet’s quantification of emotion.

pleasures, however unlike in kind,” and “on the whole, no cause to repine,” clearly illustrate Austen’s intent to quantify emotions for commensurability.

Finally, Chwe pointed out the bounty of strategic thinking, another element of game theory, present in Austen’s novels. Austen does not portray calculation as unnatural or cold, he says. She mentions the word “scheme” 126 times, “contrive” 54 times, “foresight” 49 times, and “calculate” 41 times. Her strong, female characters often pride themselves on their ability to anticipate others’ actions.

Chwe concluded that though there is no direct evidence that Austen infused game theory into her novels, she clearly explores the concept of choice in her work.

Sign Up For Datafest 2014 to Work on Mystery Big Data

By Erin Weeks

On March 5, 2014

In Computers/Technology, Mathematics, Science Communication & Education, Students, Visualization

Heads up Duke undergrads and graduate students — here’s an opportunity to hang out in the beautifully renovated Gross Hall, get creative with your friends using big data and compete for cash prizes and statistics fame.

Datafest, a data analysis competition that started at UCLA, is in its third year in the Triangle. Every year, a mystery client provides a dataset that teams can analyze, tinker with and visualize however they’d like over the course of a weekend. Think hackathon, but for data junkies.

“The datasets are bigger and more complex than what you’ll see in a classroom, but they’re of general interest,” said organizer Mine Çetinkaya-Rundel, an assistant professor of the practice in the Duke statistics department. “We want to encourage students from all levels.”

Last year’s mystery client was online dating website eHarmony (you can read about it here), and teams investigated everything from heightism to Myers-Briggs personality matches in online dating. In 2012, the dataset came from Kiva, the microlending site.

This year’s dataset provider will be revealed on the first day of Datafest. Sign up ends ~~this Friday, March 7~~, Monday, March 10, so assemble your team and register here!