Nick Higham on beautiful equations

The Most Beautiful Equations in Applied Mathematics

By Nick Higham

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From p. 171 of PCAM,
typeset in all its splendour in the Lucida Bright font.

The BBC Earth website has just published a selection of short articles on beautiful mathematical equations and is asking readers to vote for their favourite.

I wondered if we had included these equations in The Princeton Companion to
Applied Mathematics
(PCAM), specifically in Part III: Equations, Laws, and Functions of Applied Mathematics. We had indeed included the ones most
relevant to applied mathematics. Here are those equations, with links to the
BBC articles.

  • The wave equation (which quotes PCAM author Ian Stewart). PCAM has a short
    article by Paul Martin of the same title (III.31), and the wave equation
    appears throughout the book.
  • Einstein’s field equation. PCAM has a 2-page article Einstein’s Field
    Equations
    (note the plural), by Malcolm MacCallum (article III.10).
  • The Euler-Lagrange equation. PCAM article III.12 by Paul Glendinning is about
    these equations, and more appears in other articles, especially The
    Calculus of Variations
    (IV.6), by Irene Fonseca and Giovanni Leoni.
  • The Dirac equation. A 3-page PCAM article by Mark Dennis (III.9) describes
    this equation and its quantum mechanics roots.
  • The logistic map. PCAM article The logistic equation (III.19), by Paul
    Glendinning treats this equation, in both differential and difference forms.
    It occurs in several places in the book.
  • Bayes’ theorem. This theorem appears in the PCAM article Bayesian Inference in Applied Mathematics (V.11), by Des Higham, and in other articles employing
    Bayesian methods.

A natural equation is: Are there other worthy equations that are the
subject of articles in Part III of PCAM that have not been included in the BBC
list? Yes! Here are some examples (assuming that only single equations are
allowed, which rules out the Cauchy-Riemann equations, for example).

  • The Black-Scholes equation.
  • The diffusion (or heat) equation.
  • Laplace’s equation.
  • The Riccati equation.
  • Schrödinger’s equation.

Higham jacketThis article is cross posted to Nick Higham’s blog.

Check out the Princeton Companion to Applied Math here.

Solving last week’s L.A. Math challenge

LA MathWe’re back with the conclusion to last week’s LA Math challenge, The Case of the Vanishing Greenbacks, (taken from chapter 2 of the book). After the conclusion of the story, we’ll talk a little more with the author, Jim Stein. Don’t forget to check out the fantastic trailer for LA Math here.

Forty‑eight hours later I was bleary‑eyed from lack of sleep. I had made no discernible progress. As far as I could tell, both Stevens and Blaisdell were completely on the up‑and‑up.   Either I was losing my touch, or one (or both) of them were wasting their talents, doctoring books for penny‑ante amounts.   Then I remembered the envelope Pete had sealed. Maybe he’d actually seen something that I hadn’t.

I went over to the main house, to find Pete hunkered down happily watching a baseball game. I waited for a commercial break, and then managed to get his attention.

“I’m ready to take a look in the envelope, Pete.”

“Have you figured out who the guilty party is?”

“Frankly, no. To be honest, it’s got me stumped.” I moved to the mantel and unsealed the envelope. The writing was on the other side of the piece of paper. I turned it over. The name Pete had written on it was “Garrett Ryan and the City Council”!

I nearly dropped the piece of paper. Whatever I had been expecting, it certainly wasn’t this. “What in heaven’s name makes you think Ryan and the City Council embezzled the money, Pete?”

“I didn’t say I thought they did. I just think they’re responsible for the missing funds.”

I shook my head. “I don’t get it. How can they be responsible for the missing funds if they didn’t embezzle them?”

“They’re probably just guilty of innumeracy. It’s pretty common.”

“I give up. What’s innumeracy?”

“Innumeracy is the arithmetical equivalent of illiteracy. In this instance, it consists of failing to realize how percentages behave.” A pitching change was taking place, so Pete turned back to me. “An increase in 20% of the tax base will not compensate for a reduction of 20% in each individual’s taxes.   Percentages involve multiplication and division, not addition and subtraction. A gain of 20 dollars will compensate for a loss of 20 dollars, but that’s because you’re dealing with adding and subtracting. It’s not the same with percentages, because the base upon which you figure the percentages varies from calculation to calculation.”

“You may be right, Pete, but how can we tell?”

Pete grabbed a calculator. “Didn’t you say that each faction was out $198,000?”

I checked my figures. “Yeah, that’s the amount.”

Pete punched a few numbers into the calculator. “Call Ryan and see if there were 99,000 taxpayers in the last census. If there were, I’ll show you where the money went.”

I got on the phone to Ryan the next morning. He confirmed that the tax base in the previous census was indeed 99,000. I told Pete that it looked like he had been right, but I wanted to see the numbers to prove it.

Pete got out a piece of paper. “I think you can see where the money went if you simply do a little multiplication. The taxes collected in the previous census were $100 for each of 99,000 individuals, or $9,900,000. An increase of 20% in the population results in 118,800 individuals. If each pays $80 (that’s the 20% reduction from $100), the total taxes collected will be $9,504,000, or $396,000 less than was collected after the previous census. Half of $396,000 is $198,000.”

I was convinced. “There are going to be some awfully red faces down in Linda Vista. I’d like to see the press conference when they finally announce it.” I went back to the guesthouse, called Allen, and filled him in. He was delighted, and said that the check would be in the mail.   As I’ve said before, when Allen says it, he means it. Another advantage of having Allen make the arrangements is that I didn’t have to worry about collecting the fee, which is something I’ve never been very good at.

I wondered exactly how they were going to break the news to the citizens of Linda Vista that they had to pony up another $396,000, but as it was only about $3.34 per taxpayer I didn’t think they’d have too much trouble. Thanks to a combination of Ryan’s frugality and population increase, the tax assessment would still be lower than it was after the previous census, and how many government agencies do you know that actually reduce taxes? I quickly calculated that if they assessed everyone $3.42 they could not only cover the shortage, but Allen’s fee as well. I considered suggesting it to Ryan, but then I thought that Ryan probably wasn’t real interested in hearing from someone who had made him look like a bungler.

My conscience was bothering me, and I don’t like that. I thought about it, and finally came up with a compromise I found acceptable. I went back to the main house.

Pete was watching another baseball game. The Dodgers fouled up an attempted squeeze into an inning‑ending double play. Pete groaned. “It could be a long season,” he sighed.

“It’s early in the year.” I handed him a piece of paper. “Maybe this will console you.”

“What’s this?” He was examining my check for $1,750. “Your rent’s paid up.”

“It’s not for the rent, Pete. It’s your share of my fee.”

“Fee? What fee?”

“That embezzling case in Orange County. It was worth $3,500 to me to come up with the correct answer. I feel you’re entitled to half of it. You crunched the numbers, but I had the contacts and did the legwork.”

Pete looked at the check. “It seems like a lot of money for very little work. Tell you what. I’ll take $250, and credit the rest towards your rent.”

A landlord with a conscience! Maybe I should notify the Guinness Book of Records. “Seems more than fair to me.”

Pete tucked the check in the pocket of his shirt. “Tell me, Freddy, is it always this easy, doing investigations?”

I summoned up a wry laugh. “You’ve got to be kidding. So far, I’ve asked you two questions that just turned out to be right down your alley. I’ve sometimes spent months on a case, and come up dry. That can make the bottom line look pretty sick. What’s it like in your line of work?”

“I don’t really have a line of work. I have this house and some money in the bank. I can rent out the guesthouse and make enough to live on. People know I’m pretty good at certain problems, and sometimes they hire me. If it looks like it might be interesting, I’ll work on it.” He paused. “Of course, if they offer me a ridiculous amount of money, I’ll work on it even if it’s not interesting. Hey, we’re in a recession.”

“I’ll keep that in mind.”   I turned to leave the room. Pete’s voice stopped me.

“Haven’t you forgotten something?”

I turned around. “I give up. What?”

“We had a bet. You owe me five bucks.”

I fished a five out of my wallet and handed it over. He nodded with satisfaction as he stuffed it in the same pocket as the check, and then turned his attention back to the game.

What made you include this particular idea in the book?

JS: The story features one of the most common misunderstandings about percentages.  There are innumerable mistakes made because people assume that percentages work the same way as regular quantities.  But they don’t — if a store lowers the cost of an item by 30% and then by another 20%, the cost of the item hasn’t been lowered by 50% — although many people make the mistake of assuming that it has.  I’m hoping that the story is sufficiently memorable that if a reader is confronted by a 30% discount followed by a 20% discount, they’ll think “Wasn’t there something like that in The Case of the Vanishing Greenbacks?

There are 14 stories in the book, and each features a mathematical point, injected into the story in a similar fashion as the one above.  I think the stories are fun to read, and if someone reads the book and remembers just a few of the points, well, I’ve done a whole lot better than when I was teaching liberal arts math the way it is usually done.

James D. Stein is emeritus professor in the Department of Mathematics at California State University, Long Beach. His books include LA Math, Cosmic Numbers (Basic) and How Math Explains the World (Smithsonian).

An interview with Nicholas Higham on The Princeton Companion to Applied Mathematics

Higham jacket

We are excited to be running a series of posts on applied mathematics by Nicholas Higham over the next few weeks. Higham is editor of The Princeton Companion to Applied Mathematics, new this month. Recently he took the time to answer some questions about the book, and where the field is headed. Read his popular first post on color in mathematics here.

What is Applied Mathematics?

NH: There isn’t any generally agreed definition, but I rather like Lord Rayleigh’s comment that applied mathematics is about using mathematics to solve real-world problems “neither seeking nor avoiding mathematical difficulties”. This means that in applied mathematics we don’t go out of our way to consider special cases that will never arise in practice, but equally we do not sidestep genuine difficulties.

What is the purpose of The Companion?

NH: The Companion is intended to give an overview of the main areas of applied mathematics, to give examples of particular problems and connections with other areas, and to explain what applied mathematicians do—which includes writing, teaching, and promoting mathematics as well as studying the subject. The coverage of the book is not meant to be exhaustive, but it is certainly very broad and I hope that everyone from undergraduate students and mathematically interested lay readers to professionals in mathematics and related subjects will find it useful.

What is an example of something aspect of applied mathematics that you’ve learned while editing the book?

NH: Applied mathematics is a big subject and so there are many articles on topics outside my particular areas of expertise. A good example concerns applications of computational fluid dynamics in sport. An article by Nicola Parolini and Alfio Quarteroni describes the mathematical modeling of yachts for the America’s cup. The designer wishes to minimize water resistance on the hull and maximize the thrust produced by the sails. Numerical computations allow designs to be simulated without building and testing them. The article also describes mathematical modeling of the hi-tech swimsuits that are now banned from competition. The model enables the benefit of the suits on race times to be estimated.

The Companion is about 1000 pages. How would advise people to read the book.

NH: The book has a logical structure, with eight parts ranging from introductory material in Part I, the main areas of applied mathematics in Part IV (the longest part), through to broader essays in the final part. It is a good idea to start by reading some of the articles in Part I, especially if you are less familiar with the subject. But a perfectly sensible alternative approach is to select articles of interest from the table of contents, read them, and follow cross-references. Or, you can just choose a random article and start reading—or simply follow interesting index entries! We worked very hard on the cross-references and index so an unstructured approach to reading should lead you around the book and allow you to discover a lot of relevant material.

What was the hardest thing about editing The Companion?

NH: The hardest aspect of the project was ensuring that it was completed in a reasonable time-frame. With 165 authors it’s hard to keep track of everything and to to ensure that drafts, revisions, and corrected proofs are delivered on time.

How much of the book did you write?

NH: I wrote about 100 of the 1000 pages. This was great fun, but it was some of the hardest writing I’ve done. The reason is partly that I was sometimes writing about topics that I don’t normally write about. But it was also because Companion articles are quite different from the papers I’m used to writing: they should have a minimal number of equations and formal statements of theorems, lots of diagrams and illustrations, and no citations (just Further Reading at the end of the article).

How did you choose the cover?

NH: We considered many different ideas. But after a lot of thought we settled on the motor boat picture, which captures the important topics of fluid mechanics, waves, and ocean, all of which are covered in the book in a number of articles.

What is the most unexpected article?

NH: Perhaps the article Mediated Mathematics: Representations of Mathematics in Popular Culture and Why These Matter by sociologist of education Heather Mendick. She discusses the way mathematics is represented in numerous TV shows and films.

What would you be doing if you hadn’t become a mathematician?

NH: I’d be playing a Hammond B3 organ in a jazz or blues band. I’m a keen musician and played keyboards semi-professionally for many years, starting in my teens.

How did you go about organizing the book?

NH: I recruited five Associate Editors with expertise in different areas and we met and planned out the eight parts of the book and the articles, along with a list of authors to invite. We looked for authors who are leading international experts in their field and are at the same time excellent expositors. Signing up the 165 authors was quite a long process. We were able to find authors for almost every article, so just a very small number had to be dropped. In some cases the authors suggested changes of content or emphasis that we were happy to agree with.

What range of readers is The Companion aimed at?

NH: The target audience for The Companion is very broad. It includes mathematicians at undergraduate level or above, students, researchers, and professionals in other subjects who use mathematics, and mathematically interested lay readers. Some articles will also be accessible to students studying mathematics at pre-university level.

Why not just seek information online? Why is there a need for a book?

NH: When Princeton University Press asked me to edit The Companion they told me that reference books still have great value. Many people have trouble navigating the vast amount of information available online and so the need for carefully curated thematic reference works, written by high calibre authors, is as great as ever. So PUP’s experience is that print is definitely not dead, and indeed my own experience is that I have many books in PDF form on my computer, but if I want to read them seriously I use a hard copy.

How have you ensured that the book will not go out of date quickly?

NH: This was a major consideration. This was a five and a half year project and we wanted to make sure that the book will still be relevant 10, 20, or 50 years from now. To do that we were careful to choose articles on topics that have proven long-term value and are not likely to be of short-term interest. This is not to say that we don’t cover some relatively new, hot topics. For example, there are articles on compressed sensing (recovering sparse, high-dimensional data from a small number of indirect measurements) and on cloaking (hiding an object from an observer who is using electromagnetic, or other, forms of imaging, as in Harry Potter or Romulan space ships in Star Trek), both of which are areas that have grown tremendously in the last decade.

What sort of overview of applied mathematics does the book give?

NH: Applied mathematics is a huge subject, so we cannot cover everything in 1000 pages. We have tried to include the main areas of research as well as key underlying concepts, key equations, function and laws, as well as lots of example of applied mathematics problems. The examples range from the flight of a golf ball, to robotics, to ranking web pages. We also cover the use of applied mathematics in other disciplines such as engineering, biology, computer science, and physics. Indeed the book also has a significant mathematical physics component.

Where is the field going?

NH: Prior to the 20th century, applied mathematics was driven by problems in astronomy and mechanics. In the 20th century physics became the main driver, with other areas such as biology, chemistry, economics, engineering, and medicine also providing many challenging mathematical problems from the 1950s onwards. With the massive and still growing amounts of data available to us in today’s digital society information, in its many guises, will be an increasingly important influence on applied mathematics in the 21st century.

To what extent does The Companion discuss the history of applied mathematics?

NH: We have an excellent 25-page article in Part I titled The History of Applied Mathematics by historians of mathematics June Barrow-Green and Reinhard Siegmund-Schultze. Many articles contain historical information and anecdotes. So while The Companion looks to the future it also gives an appreciation of the history of the subject.

How do you see the connections between applied mathematics and other disciplines developing?

NH: Applied mathematics is becoming ever more interdisciplinary. Many articles in The Companion illustrate this. For example,

  • various facets of imaging feature in several articles, including those on compressed sensing, medical imaging, radar, and airport baggage screening,
  • the article on max-plus algebras shows how what may seem like an esoteric area of pure mathematics has applications to all kinds of scheduling processes,
  • the article on the spread of infectious diseases shows the value of mathematical models in epidemiology,
  • several articles show how mathematics can be used to help understand the earth’s weather and climate, focusing on topics such as weather prediction, tsunamis, and sea ice.

What are you thoughts on the role of computing in applied mathematics?

NH: Computation has been a growing aspect of applied mathematics ever since the first stored program computer was invented here in Manchester. More and more it is the case that numerical computations and simulations are used in tandem with, or even in place of, the classical analysis that relies just on pen and paper. What I find particularly interesting is that while the needs of mathematics and of science in general have, naturally, influenced the development of computers and programming languages, there have been influences in the other direction. For example, the notation for the ceiling and floor functions that map a real number to the next larger or smaller integer, respectively, was first introduced in the programming language APL.

Of course numerical computations are expressed in terms of algorithms, and algorithms are ubiquitous in applied mathematics, and strongly represented in the book.

Do you have any views on ensuring the correctness of work in applied mathematics?

NH: As the problems we solve become every more complicated, and the computations we perform run for longer and longer, questions about the correctness of our results become more important. Applied mathematicians have always been good at estimating answers, perhaps by an asymptotic analysis, so we usually know roughly what the answer should look like and we may be able to spot gross errors. Several particular aspects of treating correctness are covered in The Companion.

Uncertainty quantification is about understanding how uncertainties in the data of a problem affect the solution. It’s particularly important because often we don’t know the problem data exactly—for example, in analyzing groundwater flow we don’t know the exact structure of what lies under the ground and so have to make statistical assumptions, and we want to know how these impact the computed flows.

A different aspect of correctness concerns the reproducibility of our computations and treats issues such as whether another scientist can reproduce our results and whether a computation on a high-performance computer will produce exactly the same answer when the computation is repeated.

All of these issues are covered in multiple articles in the book.

Nicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester. Mark R. Dennis is professor of theoretical physics at the University of Bristol. Paul Glendinning is professor of applied mathematics at The University of Manchester. Paul A. Martin is professor of applied mathematics at the Colorado School of Mines. Fadil Santosa is professor of mathematics at the University of Minnesota. Jared Tanner is professor of the mathematics of information at the University of Oxford.

Nicholas Higham on Mathematics in Color

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We are excited to be running a series of posts on applied mathematics by Nicholas Higham over the next few weeks. Higham is editor of The Princeton Companion to Applied Mathematics, which is out this month. A slightly longer version of this post on color in mathematics can be found on Higham’s blog, and it has been cross posted at John Cook’s blog, The Endeavour. —PUP Blog Editor

Color is a fascinating subject. Important early contributions to our understanding of it came from physicists and mathematicians such as Newton, Young, Grassmann, Maxwell, and Helmholtz. Today, the science of color measurement and description is well established and we rely on it in our daily lives, from when we view images on a computer screen to when we order paint, wallpaper, or a car, of a specified color.

For practical purposes color space, as perceived by humans, is three-dimensional, because our retinas have three different types of cones, which have peak sensitivities at wavelengths corresponding roughly to red, green, and blue. It’s therefore possible to use linear algebra in three dimensions to analyze various aspects of color.

Metamerism

A good example of the use of linear algebra is to understand metamerism, which is the phenomenon whereby two objects can appear to have the same color but are actually giving off light having different spectral decompositions. This is something we are usually unaware of, but it is welcome in that color output systems (such as televisions and computer monitors) rely on it.

Mathematically, the response of the cones on the retina to light can be modeled as a matrix-vector product Af, where A is a 3-by-n matrix and f is an n-vector that contains samples of the spectral distribution of the light hitting the retina. The parameter n is a discretization parameter that is typically about 80 in practice. Metamerism corresponds to the fact that Af_1 = Af_2 is possible for different vectors f_1 and f_2. This equation is equivalent to saying that Ag = 0 for a nonzero vector g =f_1-f_2, or, in other words, that a matrix with fewer rows than columns has a nontrivial null space.

Metamerism is not always welcome. If you have ever printed your photographs on an inkjet printer you may have observed that a print that looked fine when viewed indoors under tungsten lighting can have a color cast when viewed in daylight.

LAB Space: Separating Color from Luminosity

In digital imaging the term channel refers to the grayscale image representing the values of the pixels in one of the coordinates, most often R, G, or B (for red, green, and blue) in an RGB image. It is sometimes said that an image has ten channels. The number ten is arrived at by combining coordinates from the representation of an image in three different color spaces. RGB supplies three channels, a space called LAB (pronounced “ell-A-B”) provides another three channels, and the last four channels are from CMYK (cyan, magenta, yellow, black), the color space in which all printing is done.

LAB is a rather esoteric color space that separates luminosity (or lightness, the L coordinate) from color (the A and B coordinates). In recent years photographers have realized that LAB can be very useful for image manipulations, allowing certain things to be done much more easily than in RGB. This usage is an example of a technique used all the time by mathematicians: if we can’t solve a problem in a given form then we transform it into another representation of the problem that we can solve.

As an example of the power of LAB space, consider this image of aeroplanes at Schiphol airport.

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Original image.

Suppose that KLM are considering changing their livery from blue to pink. How can the image be edited to illustrate how the new livery would look? “Painting in” the new color over the old using the brush tool in image editing software would be a painstaking task (note the many windows to paint around and the darker blue in the shadow area under the tail). The next image was produced in
just a few seconds.

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Image converted to LAB space and A channel flipped.

How was it done? The image was converted from RGB to LAB space (which is a nonlinear transformation) and then the coordinates of the A channel were replaced by their negatives. Why did this work? The A channel represents color on a green–magenta axis (and the B channel on a blue–yellow axis). Apart from the blue fuselage, most pixels have a small A component, so reversing the sign of this component doesn’t make much difference to them. But for the blue, which has a negative A component, this flipping of the A channel adds just enough magenta to make the planes pink.

You may recall from earlier this year the infamous photo of a dress that generated a huge amount of interest on the web because some viewers perceived the dress as being blue and black while others saw it as white and gold. A recent paper What Can We Learn from a Dress with Ambiguous Colors? analyzes both the photo and the original dress using LAB coordinates. One reason for using LAB in this context is its device independence, which contrasts with RGB, for which the coordinates have no universally agreed meaning.

Higham jacketThe Princeton Companion to Applied Mathematics

Nicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester, and editor of The Princeton Companion to Applied Mathematics. His article Color Spaces and Digital Imaging in The Princeton Companion to Applied Mathematics gives an introduction to the mathematics of color and the representation and manipulation of digital images. In particular, it emphasizes the role of linear algebra in modeling color and gives more detail on LAB space.

 

 

Five places you didn’t expect to encounter applied math

You don’t need to step into a classroom to have a run-in with mathematics. Professionals from a range of backgrounds — engineering, economics, physics, biology, computer science — use mathematics every day. To celebrate the publication of the much-anticipated Princeton Companion to Applied Mathematics, edited by Nicholas J. Higham, we’re thinking about all of the unique places and situations where applied mathematics is at work. Here is a list of just a few, compiled with a little help from our numerically inclined friends.

On the Golf Course

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Golf involves mathematics, and not just when keeping score. The flight of your golf ball is affected by how air interacts with the surface of the ball. Did you know that the dimples in golf balls have a purpose, one with a mathematical explanation? Douglas N. Arnold, professor of Mathematics at the University of Minnesota, tells us more:

In the middle of the nineteenth century, when rubber golf balls were introduced, golfers noticed that old scuffed golf balls traveled farther than new smooth balls, although no one could explain this unintuitive behavior. This eventually gave rise to the modern dimpled golf ball. Along the way a great deal was learned about aerodynamics and its mathematical modeling. Hundreds of different dimple patterns have been devised, marketed, and patented. However, even today the optimal dimple pattern lies beyond our reach, and its discovery remains a tough challenge for applied mathematics and computational science.

Check out Dr. Arnold’s entry, “The Flight of a Golf Ball,” where he explains why golf ball dimples are an important part of your Saturday morning tee time.

On Wall Street

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Wall Street is all about the numbers. Whether modeling the risk of a single stock or mapping the complex interactions that make up the world’s financial structure, mathematics helps the financial sector to study and evaluate systemic risk.

“The complexity, unpredictability, and evolving nature of financial markets continues to provide an enormous challenge to mathematicians, engineers, and economists in identifying, analyzing, and quantifying the issues and risks they pose,” write Dr. René A. Carmona and Dr. Ronnie Sircar of Princeton University.

In their entry, “Financial Mathematics,” Dr. Carmona and Dr. Sircar discuss how the finance industry uses mathematics. They also examine the role of mathematics in understanding and regulating financial markets in light of the financial crisis of 2008.

On Your Phone’s Weather App

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Do you check the 10-day forecast during the weekend before a big outdoor event, fingers crossed for clear skies and no rain? There’s math behind that “chance of thunderstorms” prediction. NWP [numerical weather prediction] helps meteorologists to predict weather patterns for more than a week ahead. Better numerical schemes are partially responsible for moving us forward from the weather prediction methods of fifty years ago.

In his article “Numerical Weather Prediction,” Peter Lynch presents the mathematical principles of NWP and illustrates the process by considering some specific models and their application to practical forecasting. Dr. Lynch describes the many conditions that can be better predicted using NWP:

NWP models are used to generate special guidance for the marine community. Predicted winds are used to drive wave models, which predict sea and swell heights and periods. Prediction of road ice is performed by specially designed models that use forecasts of temperature, humidity, precipitation, cloudiness, and other parameters to estimate the conditions on the road surface. Trajectories are easily derived from limited-area models. These are vital for modeling pollution drift, for nuclear fallout, smoke from forest fires, and so on. Aviation benefits significantly from NWP guidance, which provides warnings of hazards such as lightning, icing, and clear-air turbulence.

In the Airport Security Line

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On your next trip through airport security, take a look at the x-ray machine. Once an object, like your suitcase, is scanned, the image can be viewed from multiple angles by a security officer. Threat detection software can also be used to locate problematic items. There is math at work here too.

W. R. B. Lionheart, professor of Applied Mathematics at the University of Manchester, explains this technology in his entry “Airport Baggage Screening with X-Ray Tomography.”

While Researching Your Next Vacation

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Getting ready for your first vacation of the fall? Buying tickets, making dinner reservations, researching tourist attractions — what did we do without the internet? Or rather, what did we do before the organized internet?

When the internet was still in its early stages, search engines were not as advanced as they are today, and webpage results were ranked by simple rules. Searching for “New York sightseeing” may have led you to the page where the search term appears the most, instead of a page with the most useful information. Today, search engines use a more advanced method for ranking web pages: grouping pages into authority pages, which have many links to them, and hub pages, which point to many authorities. The catch is that these terms depend on one another. How does this work? In the Princeton Companion to Applied Mathematics, editor Nicholas Higham explains the mathematics behind webpage ranking.

Looking for more examples of math in the world? Check out this video from SIAM, where SIAM conference attendees are asked how they use math in their work. Math really is all around us.

New and Forthcoming Titles in Mathematical Sciences

We invite you to see what is new and forthcoming in our 2011 Mathematical Sciences catalog.

View the catalog online at:
http://press.princeton.edu/catalogs/math2011.pdf

From Computational Aspects of Modular Forms to Loving and Hating Mathematics, you’ll find something you want to read. The catalog is full of great books by great authors. Professors, make sure to check out pages 6-10 for course books. And if you’re in New Orleans for the Joint Mathematics Meeting, please stop by booth no. 509 to say hello and browse the books.

You can also learn about new math books by joining our e-mail list at:
http://press.princeton.edu/subscribe/