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# The star for favourites gives the same tool tip text regardless of state [duplicate]

Regardless the state of a favourite star, the tool tip text is:

"Click again to undo".

When deselected, the tool tip text should read

"Click to mark as a favourite".

This follows Steve Krug's "Don't Make me think" principle.

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## marked as duplicate by Oded♦Feb 10 '14 at 15:04

+1 Nice find! I hadn't noticed that myself either. –  Jonathan Sampson Feb 11 '10 at 18:08
The same apply to the accepted answer check-box. Click to set this answer as your accepted answer; click again to toggle. –  heavyd Feb 11 '10 at 19:31
I just KNEW that star was the spawn of evil. –  squillman Feb 11 '10 at 21:07
@squillman: upvote for mentioning the "spawn of evil" :P –  Shaharyar Feb 11 '10 at 21:54
The tooltip on the unseleted star reads "This is a favorite question (click again to undo)". I guess the "Click again to undo" part is fine but saying "This is a favorite question" looks wrong. –  Alexandre Jasmin Jun 30 '11 at 19:28

Talking about this, I guess the part where it tells you (click again to undo) is fair, as it's just a for your information part. But the part where it says, This is a favorite question, that should be a typo, since it is not a favorite question.

Nice catch by you, but team should place a try block and remove this! And update it to something like this

Click to mark it as a favorite question (Click again to undo)

Where the part inside the bracket would be taken as a for your information part.

That would be more like it when its not a favorite question and you need to favorite it.

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I think the message is clear enough.

It says: "This is a favorite question (click again to undo)"

I don't think this needs to be explained any further, I understand it.

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But if I haven't marked it as a favorite, it is not a favorite, therefore the tool tip is a dirty liar. Also, if I haven't done anything, there is nothing to undo. This tool tip is completely out of control! –  raven Feb 11 '10 at 19:15


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# Toeplitz matrix

In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix:

${\displaystyle {\begin{bmatrix}a&b&c&d&e\\f&a&b&c&d\\g&f&a&b&c\\h&g&f&a&b\\i&h&g&f&a\end{bmatrix}}.}$

Any n×n matrix A of the form

${\displaystyle A={\begin{bmatrix}a_{0}&a_{-1}&a_{-2}&\cdots &\cdots &a_{-(n-1)}\\a_{1}&a_{0}&a_{-1}&\ddots &&\vdots \\a_{2}&a_{1}&\ddots &\ddots &\ddots &\vdots \\\vdots &\ddots &\ddots &\ddots &a_{-1}&a_{-2}\\\vdots &&\ddots &a_{1}&a_{0}&a_{-1}\\a_{n-1}&\cdots &\cdots &a_{2}&a_{1}&a_{0}\end{bmatrix}}}$

is a Toeplitz matrix. If the i,j element of A is denoted Ai,j, then we have

${\displaystyle A_{i,j}=A_{i+1,j+1}=a_{i-j}.\ }$

A Toeplitz matrix is not necessarily square.

## Solving a Toeplitz system

A matrix equation of the form

${\displaystyle Ax=b\ }$

is called a Toeplitz system if A is a Toeplitz matrix. If A is an ${\displaystyle n\times n}$  Toeplitz matrix, then the system has only 2n−1 degrees of freedom, rather than n2. We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case.

Toeplitz systems can be solved by the Levinson algorithm in Θ(n2) time.[1] Variants of this algorithm have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems).[2] The algorithm can also be used to find the determinant of a Toeplitz matrix in O(n2) time.[3]

A Toeplitz matrix can also be decomposed (i.e. factored) in O(n2) time.[4] The Bareiss algorithm for an LU decomposition is stable.[5] An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant.

Algorithms that are asymptotically faster than those of Bareiss and Levinson have been described in the literature, but their accuracy cannot be relied upon.[6][7][8][9]

## General properties

• A Toeplitz matrix may be defined as a matrix A where Ai,j = ci−j, for constants c1−ncn−1. The set of n×n Toeplitz matrices is a subspace of the vector space of n×n matrices under matrix addition and scalar multiplication.
• Two Toeplitz matrices may be added in O(n) time and multiplied in O(n2) time.
• A positive semi-definite n×n Toeplitz matrix ${\displaystyle A}$  of rank r < n can be uniquely factored as
${\displaystyle A=\sum _{k=1}^{r}d_{i}v(f_{k})v(f_{k})^{\operatorname {H} }=VDV^{\operatorname {H} }}$
where ${\displaystyle D=\mathrm {diag} (d_{1},\ldots ,d_{r})}$  is an r×r positive definite diagonal matrix, ${\displaystyle V=[v(f_{1}),\ldots ,v(f_{r})]}$  is an n×r Vandermonde matrix such that the columns are ${\displaystyle v(f)=[1,e^{i2\pi f},\ldots ,e^{i2\pi (n-1)f}]^{\operatorname {T} }}$ . Here ${\displaystyle i={\sqrt {-1}}}$  and ${\displaystyle f_{k}\in [0,1]}$  is normalized frequency, and ${\displaystyle V^{\operatorname {H} }}$  is the Hermitian transpose of ${\displaystyle V}$ . If the rank r = n, then the Vandermonde decomposition is not unique.[10]
• For symmetric Toeplitz matrices, there is the decomposition
${\displaystyle {\frac {1}{a_{0}}}A=GG^{\operatorname {T} }-(G-I)(G-I)^{\operatorname {T} }}$
where ${\displaystyle G}$  is the lower triangular part of ${\displaystyle {\frac {1}{a_{0}}}A}$ .
• The inverse of a nonsingular symmetric Toeplitz matrix has the representation
${\displaystyle A^{-1}={\frac {1}{\alpha _{0}}}(BB^{\operatorname {T} }-CC^{\operatorname {T} })}$
where ${\displaystyle B}$  and ${\displaystyle C}$  are lower triangular Toeplitz matrices and ${\displaystyle C}$  is a strictly lower triangular matrix.[11]

## Discrete convolution

The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. For example, the convolution of ${\displaystyle h}$  and ${\displaystyle x}$  can be formulated as:

${\displaystyle y=h\ast x={\begin{bmatrix}h_{1}&0&\cdots &0&0\\h_{2}&h_{1}&&\vdots &\vdots \\h_{3}&h_{2}&\cdots &0&0\\\vdots &h_{3}&\cdots &h_{1}&0\\h_{m-1}&\vdots &\ddots &h_{2}&h_{1}\\h_{m}&h_{m-1}&&\vdots &h_{2}\\0&h_{m}&\ddots &h_{m-2}&\vdots \\0&0&\cdots &h_{m-1}&h_{m-2}\\\vdots &\vdots &&h_{m}&h_{m-1}\\0&0&0&\cdots &h_{m}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\vdots \\x_{n}\end{bmatrix}}}$
${\displaystyle y^{T}={\begin{bmatrix}h_{1}&h_{2}&h_{3}&\cdots &h_{m-1}&h_{m}\end{bmatrix}}{\begin{bmatrix}x_{1}&x_{2}&x_{3}&\cdots &x_{n}&0&0&0&\cdots &0\\0&x_{1}&x_{2}&x_{3}&\cdots &x_{n}&0&0&\cdots &0\\0&0&x_{1}&x_{2}&x_{3}&\ldots &x_{n}&0&\cdots &0\\\vdots &&\vdots &\vdots &\vdots &&\vdots &\vdots &&\vdots \\0&\cdots &0&0&x_{1}&\cdots &x_{n-2}&x_{n-1}&x_{n}&0\\0&\cdots &0&0&0&x_{1}&\cdots &x_{n-2}&x_{n-1}&x_{n}\end{bmatrix}}.}$

This approach can be extended to compute autocorrelation, cross-correlation, moving average etc.

## Infinite Toeplitz matrix

A bi-infinite Toeplitz matrix (i.e. entries indexed by ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$ ) ${\displaystyle A}$  induces a linear operator on ${\displaystyle \ell ^{2}}$ .

${\displaystyle A={\begin{bmatrix}&\vdots &\vdots &\vdots &\vdots \\\cdots &a_{0}&a_{-1}&a_{-2}&a_{-3}&\cdots \\\cdots &a_{1}&a_{0}&a_{-1}&a_{-2}&\cdots \\\cdots &a_{2}&a_{1}&a_{0}&a_{-1}&\cdots \\\cdots &a_{3}&a_{2}&a_{1}&a_{0}&\cdots \\&\vdots &\vdots &\vdots &\vdots \end{bmatrix}}.}$

The induced operator is bounded if and only if the coefficients of the Toeplitz matrix ${\displaystyle A}$  are the Fourier coefficients of some essentially bounded function ${\displaystyle f}$ .

In such cases, ${\displaystyle f}$  is called the symbol of the Toeplitz matrix ${\displaystyle A}$ , and the spectral norm of the Toeplitz matrix ${\displaystyle A}$  coincides with the ${\displaystyle L^{\infty }}$  norm of its symbol. The proof is easy to establish and can be found as Theorem 1.1 in the google book link: [12]

• Circulant matrix, a Toeplitz matrix with the additional property that ${\displaystyle a_{i}=a_{i+n}}$

format: plain


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# How do you find the area of a rectangle that is 7.5 cm long and 4 cm wide?

$30$ sq. cm.
Formula for area of a rectangle is $A = l \times w$
$\textcolor{w h i t e}{\text{XXX}}$where $A$ is the area, $l$ is the length, and $w$ is the width
$A = 7.5 \text{ cm" xx 4 "cm}$
color(white)("X")=30 " cm"^2


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# What's Happening in the Mathematical Sciences

What's Happening in the Mathematical Sciences is a series of publications that report on the latest mathematical research in a way that both mathematicians and non-mathematicians can appreciate.

Order What's Happening in the Mathematical Sciences, Volume 12 on the AMS Bookstore

### Fifty Ways to Beat a Virus (Part 1)

In 2020, the world for the first time in a century confronted a global pandemic that would claim millions of lives. While students were sent home and campuses closed, many mathematicians found an opportunity to join the fight against COVID-19. Even simple differential equation models can teach us important lessons about the exponential growth of a new epidemic and the importance of threshold behavior. Using more elaborate (and realistic) models, two mathematical modeling groups, in Texas and Illinois, had a profound and positive effect on the management of the epidemic by local and state authorities during the first waves.

### Fifty Ways to Beat a Virus (Part 2)

Continuing the previous chapter, Part 2 discusses the problems confronted by mathematicians and epidemiologists in the later part of 2020 and in early 2021. How could universities re-open safely? How does the uncontrolled spread of an epidemic in prisons affect the surrounding community, and what can be done about it? And the biggest question: could vaccination bring the epidemic under control? Even though the coronavirus kept throwing surprises at us, mathematicians did a surprisingly good job of developing strategies, giving realistic answers and highlighting the main reasons for uncertainty.

### Fifty Ways to Beat a Virus (Part 3)

Another important front in the battle against COVID-19 was to understand how the infection progresses within the human body. Why do some people have life-threatening symptoms, while others have none at all? An online group called the “Immune Gals” highlighted the delayed release of interferon as a characteristic marker of severe cases. Another mathematician used the tools of graph theory to identify parts of the viral RNA that are especially vulnerable to attack by drugs or gene therapy. And a third group adapted a machine-learning language model to detect escape variants of coronavirus. In effect, they taught the computer to “speak virus.”

### Descartes' Homework

A homework problem assigned in 1643 by René Descartes to his star pupil, Princess Elizabeth of Bohemia, backfired when Descartes himself couldn’t solve it. But he redeemed himself by solving a special case that still fascinates geometers and number theorists. Descartes’ theorem and its generalizations have revealed startling symmetries in infinite foams (called Apollonian packings) that are created by squeezing circles as tightly as possible into the spaces created by other circles. New Apollonian and Apollonian-like patterns continue to be discovered in the plane, in 3-space, and in higher dimensions all the way up to dimension 20.

### Square Pegs and Squiggly Holes

A century-old “folklore problem” in topology asks whether any continuous loop in the plane that does not cross itself must contain four points that form a perfect square. This is known as the Square Peg Conjecture. Recently, a team of two geometers proved something even better: if the curve is smooth (i.e., it has no corners or cusps) then it contains rectangles of all conceivable shapes, and also quadrilaterals of any shape that will fit snugly in a circle. Their proof ingeniously weaves together 2000-year-old theorems of Euclidean geometry with less than 20-year-old theorems of symplectic geometry.

### Dancing on the Edge of the Impossible

The ultimate dream of many number theorists would be an algorithm that inputs a polynomial equation and outputs all the rational solutions to that equation. For equations with integer coefficients in one variable, it’s easy. For equations with nine or more variables, it’s impossible. But equations in two variables are right on the edge. Faltings’ theorem, proved in 1983, guarantees (for most such equations) that there are only finitely many solutions… but it doesn’t say whether there are four or four million. Number theorists are now inching closer to an “effective” Faltings theorem that would tell them when they can stop looking for more solutions.

### A Climate for Math

Unprecedented heat waves. Orange skies. Ten-thousand-year floods. When the weather gets weirder than ever before, whom are you going to call? A climatologist, sure. But you might want to talk with a mathematician, too. This chapter explains how mathematical techniques like extreme value analysis, differential equations, changepoint analysis, and machine learning can give us some idea of what has happened already and what will happen next in climate change.

What is the friendliest research community in mathematics? You could make a good case for zero-forcing. This mash-up of linear algebra and graph theory has been in existence for less than 20 years, but has attracted a large following, with a sizeable proportion of young mathematicians and people from diverse academic and social backgrounds. Zero-forcing is a way to turn hard problems about eigenvalues of matrices into easy games played on graphs (i.e., dot-and-line networks). Or vice versa, it’s a way to turn simple graph games into challenging math problems. Applications to quantum control and electrical and civil engineering are the icing on the cake.

Order What's Happening in the Mathematical Sciences, Volume 11 on the AMS Bookstore

### Gerrymandering: Mathematics on Trial

In the last ten years, mathematicians have developed much more sophisticated methods to detect partisan gerrymandering (i.e., the redrawing of election districts to favor one political party). Unfamiliar words like “outliers” and “efficiency gap” were heard in the US Supreme Court. Though the court eventually withheld its endorsement of any such methods, awareness of the issue among mathematicians reached an all-time high.

### The Calculus of Calories

Dieting comes down to a simple equation: Energy stored = energy intake – energy expenditure. If only it were so easy! Mathematicians have developed new algorithms and smartphone apps to help you take the weight off. But they have also made surprising and sobering discoveries about the ways in which your body will fight to put the weight back on.

### When Black Holes Collide

One of the boldest predictions of Einstein’s theory of general relativity was the possibility of ripples in spacetime, called gravitational waves. The biggest roadblock to detecting them was figuring out what these waves should “sound like.” In 2015, astrophysicists finally heard the mathematically predicted chirp of two black holes colliding, a billion light-years away and a billion years ago.

### The Shape of Data

A new hybrid of statistics and topology, called topological data analysis, uses tools like persistent homology to uncover hidden geometric structures in data, such as loops and bubbles. Applications have run the gamut from figuring out how rat brains encode spatial information to finding optimal molecular structures for porous molecules called zeolites.

### The "Set®" Game Has Met Its Match

A popular card game leads to an unexpectedly challenging problem on arithmetic progressions in finite vector spaces, known as the Cap Set Conjecture. In 2016, mathematicians discovered an unexpectedly simple solution. A very rough analog is that it you don’t need too many stars in the sky before you’ll start seeing “three-in-a-row” patterns like the belt of Orion. (Four in a row, though, is another story!)

### The Mathematics of Commuting

Bike-sharing systems (like New York’s CitiBike) and ridesharing services (like Uber and Lyft) have led to an explosion of interest in optimization problems. How do you move bikes efficiently from hubs with too many to hubs with too few? How much could traffic congestion be reduced if people shared Uber rides? And what would be the effect of a door-to-door solution that combined bike sharing with ride sharing?

### Expanding Horizons

Expanding graphs, or expanders, have multiple personalities. They are networks that are hard to cut in two; networks in which information propagates rapidly; and networks with a large “spectral gap.” In recent years their applications have become even more numerous than their personalities. This chapter explains how they are used in coding, knot theory, and nearest-neighbor search, plus a surprising generalization to hypergraphs.

### Tubing through Hyperspace

In 2012, geometers in rapid succession solved three open problems concerning optimal geometries of tori (inner tubes). The Willmore Conjecture identified the torus with least bending energy; the Lawson Conjecture identified the torus in the hypersphere (a sphere in four-dimensional space) with the least surface area; and the Pinkall-Sterling Conjecture classified all of the tori in the hypersphere that minimize area subject to a volume constraint. Tubing has never been so much fun!

### Tsunamis: Learning from Math, Learning from the Past

The great Japanese tsunami of 2011 killed more than 15,000 people in a country that had been better prepared for tsunamis than any other in the world. The tragedy highlighted gaps in our scientific understanding of this hugely destructive natural phenomenon, and even more so in the public’s understanding. It also pointed out some ways in which mathematical models got it right, but not fast enough.

### Today's Forecast: Ten Percent Chance of Burglary

An interdisciplinary team at UCLA discovered how to adapt a mathematical model developed for earthquake prediction to identify likely crime “hot spots.” Two field tests of their software were resounding successes, and predictive policing was recognized in Time magazine and other media outlets as one of the top scientific discoveries of 2011.

### Topologists Cross Four Off "Bucket List"

Three decades ago, three-dimensional topology seemed like a wild, untamed jungle of disparate examples. Then William Thurston proposed a series of conjectures that brought some order to the chaos. In 2012, just months before Thurston’s death, the Virtual Haken Conjecture and Virtual Fibering Conjecture were finally proved, showing that almost all three-dimensional manifolds are descended from templates constructed in an elementary fashion.

### Mathematicians Do the Twist

Rubik’s cube, the mathematician’s favorite toy, continues to attract new fans and inspire new research. While “speedcubers” developed new algorithms (and manual dexterity) to solve the cube in less than 10 seconds, mathematicians proved that an omniscient being could always solve the classic, 3-by-3-by-3 cube in 20 moves or less.

### The Right Epidemic at the Right Time

In 2009 the world experienced its first flu pandemic in forty years. Fortunately it turned out much milder than the three great pandemics of the twentieth century, but it provided an ideal opportunity to test a variety of mathematical simulations in real time. Conclusions: The simulations worked pretty well, and communications between modelers and field workers were excellent, but the late delivery of vaccine would have been a fiasco in a worse epidemic.

### Thinking Topically

Topic modeling is a new statistical technique named after its ability to identify topics (such as genetics or climate change) in a large body of documents. While still in its early days, it has proved hugely popular in the fields of “digital humanities” and it might enable social-networking websites to respond automatically and anonymously to cyber-bullying.

### Thinking Tropically

Blessed (or perhaps cursed?) with a catchy name, tropical geometry enables mathematicians to solve difficult problems in classical algebraic geometry by making simple combinatorial models that look a lot like stick figures. This new type of geometry also has surprising applications to string theory in physics, evolutionary trees in biology, and the scheduling of trains.

Unfortunately, Volume 9 is no longer available.

### Accounting for Taste

Netflix, a movie rental company, offered a million-dollar prize for a computer algorithm that could significantly upgrade the company's ability to predict its customers' likes and dislikes. The wildly successful competition upset some conventional wisdom in the field of machine learning and vividly demonstrated the power of "crowdsourcing."

### A Brave New Symplectic World

Thirty years ago, Alan Weinstein conjectured that certain kinds of dynamical systems with two degrees of freedom, such as a pendulum that is free to stretch as well as swing, always have periodic (repeating) solutions. At the time, no one had any idea how to prove his conjecture. But that was before "symplectic camels" and a "crazy" homology theory based on the equations for a magnetic monopole. Conclusion: Periodic orbits exist. (Magnetic monopoles still don't.)

### Mathematics and the Financial Crisis

In the first decade of the 2000s, investors embraced credit derivatives: a clever, formula-based method to parcel out the risk of subprime loans and profit from a booming housing market. But they forgot--or chose to ignore--that a mathematical model is only as good as its assumptions.

### The Ultimate Billiard Shot

In an inside-out version of billiards that you can't quite fit into your garage (or even the Milky Way), two groups of researchers find different ways to sink the cue ball into a pocket at infinity.

### SimPatient

Mathematical models of patient populations have begun to supplement or replace clinical trials in cases where the trials would be difficult or impossible to perform. In 2009, a public health panel used six breast cancer models to arrive at a controversial recommendation that women aged 40-49 should no longer be advised to have an annual mammogram.

### Instant Randomness

How long does it take to mix milk in a coffee cup, neutrons in an atomic reactor, atoms in a gas, or electron spins in a magnet? In the Ising model of magnetism, mathematicians have calculated a unique cutoff time when the system abruptly goes from essentially unmixed to almost completely mixed. The same behavior is expected for other systems described by the theory of Markov chains.

### In Search of Quantum Chaos

A classical billiard ball, on a table with curved, "dispersing" sides, travels on a chaotic trajectory that essentially randomizes the ball's position over the long term. However, at low energies, quantum billiard balls are not chaotic. In a tour de force combining pure number theory with physics, mathematicians proved that quantum chaos does emerge at high energies.

### 3-D Surprises

Even in the twenty-first century, mathematics can still reveal new phenomena in ordinary three-dimensional space. Item 1: a stunningly efficient way to pack tetrahedra. Item 2: the Gömböc, a homogeneous body that automatically rights (and wrongs) itself.

### As One Heroic Age Ends, a New One Begins

John Milnor's discovery of "exotic spheres" in 1956 ushered in a new era of high-dimensional topology, with powerful new tools like framed cobordism theory, stable homotopy theory, and surgery. But one question, called the Kervaire Invariant One problem, remained stubbornly unanswered for more than 40 years, until three topologists found "a shortcut to Mount Everest."

Unfortunately, Volume 8 is no longer available.

Order What's Happening in the Mathematical Sciences, Volume 7 on the AMS Bookstore

### A New Twist in Knot Theory

( 846 KB)

Two seemingly unrelated kinds of knots--modular knots and Lorenz knots--turn out to be the same. For number theorists, the payoff is a new way to look at an old concept, the modular surface. For dynamical systems theorists, it's a new way to understand the beginnings of chaos.

### Error-term Roulette and the Sato-Tate Conjecture

Fifteen years ago, Andrew Wiles proved Fermat's Last Theorem, thereby solving the most famous problem in mathematics. The aftershocks are still being felt. A team of number theorists, including Wiles collaborator, Richard Taylor, adapts Wiles' machinery to polish off another celebrated unsolved problem in number theory.

### The Fifty-one Percent Solution

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Persi Diaconis, Susan Holmes and Richard Montgomery tell you how to get the odds in your favor the next time you flip a coin. And no, you don't have to cheat.

### Dominos, Anyone?

An innocent puzzle about tiling a checkerboard with dominos leads, ultimately, to a new model of random surfaces. The theory is both simple enough to be exactly solvable, and complex enough to include phase transitions between "solid," "liquid," and "gaseous" states--a combination that has never occurred previously in statistical physics.

### Not Seeing Is Believing

New "metamaterials" may soon bring invisibility cloaks--and inaudibility cloaks, too--out of the realm of fantasy and into reality. Unlike Harry Potter's cloak, these devices will work strictly on the principles of mathematics, not magic.

### Getting with the (Mori) Program

In algebraic geometry, the Mori Minimal Model Program is an ambitious effort to match the geometry of surfaces defined by polynomial equations to their algebra. After 100 years of carrying out the program one dimension at a time, algebraic geometers race to the top of the ladder.

### The Book that Time Couldn't Erase

More than 1000 years ago, an unknown scribe copied some of Archimedes' works onto parchment. But 200 years later, another scribe erased them and wrote a prayer book over the top. Over the last ten years, high-tech imaging methods have penetrated through the damage done by centuries of neglect and abuse to reveal what Archimedes, the greatest mathematician of ancient Greece, really wrote.

### Charting a 248-dimensional World

Racing against time and the incurable illness of one of their colleagues, a team of twenty group theorists puts together a detailed survey of "the most beautiful structure in mathematics"-- the exceptional Lie group, E8.

### Compressed Sensing Makes Every Pixel Count

( 878 KB)

Proving that more is not always better, mathematicians show that "sparse" or "compressible" signals, such as digital photographs and cell phone messages, can be reconstructed from many fewer measurements than engineers previously supposed. As an extreme proof of principle, engineers design the world's first single-pixel camera.

Order What's Happening in the Mathematical Sciences, Volume 6 on the AMS Bookstore

See links to Cipra and Mackenzie talking about the subject of each chapter.

### First of Seven Millennium Problems Nears Completion

by Barry Cipra ( 1 MB)
hear the podcast interview with the author

In 2002, Grigory Perelman announced a solution to the Poincaré Conjecture, a problem in topology selected in 2000 as one of the seven leading math challenges of the millennium. After more than three years of scrutiny, mathematicians are cautiously accepting his proof, which uses a geometric partial differential equation called "Ricci flow," first studied by Richard Hamilton.

### Classifying Hyperbolic Manifolds--All's Well that Ends Well

by Dana Mackenzie ( 1.62 MB)
hear the podcast interview with the author

While three-dimensional topologists awaited a verdict on the Poincaré Conjecture (which deals with closed manifolds), several of them solved a suite of long-standing conjectures about the ends of open manifolds.

### Digits of Pi

by Barry Cipra ( 317 KB)
hear the podcast interview with the author

Pi lovers rejoice! You can now stick your finger into pi and pluck out any digit (say, the trillionth one) without having to compute all the preceding digits. The only catch is that you have to count in sixteens.

### Combinatoricists Solve a Venn-erable Problem

by Barry Cipra ( 415 KB)
hear the podcast interview with the author

A question posed by an undergraduate student--can a Venn diagram be rotationally symmetric?--leads to some beautiful "doilies," intricate mathematics, and (after forty years) a solution found by another student.

### New Insights into Prime Numbers

by Barry Cipra ( hear the podcast interview with the author)

numbers surprised number theorists. A team of Indian mathematicians, including two students, discovers the first polynomial-time algorithm for testing to see if a number is prime. American, Turkish, and Hungarian mathematicians collaborate to show that small gaps between consecutive primes occur much more often than anyone had previously proved. And another international collaboration finds evenly spaced (but non-consecutive) prime sequences of any desired length.

### From Rubik's Cube to Quadratic Number Fields ... and Beyond

by Dana Mackenzie ( hear the podcast interview with the author)

Manjul Bhargava started with a simple idea of putting numbers into a box. What came out of the box, eventually, was a whole new way to combine and to count objects in algebraic number theory.

### Vortices in the Navier-Stokes Equations

by Barry Cipra ( hear the podcast interview with the author)

From Jupiter's Great Red Spot to the eddies in a stream, vortices are a familiar feature of fluid flow. Mathematicians have gained new insights into the formation and long-term survival of vortices in both two- and three-dimensional fluids.

### Fluid Dynamics Explains Mysteries of Insect Motion

by Dana Mackenzie ( 735 KB)
hear the podcast interview with the author

How do water striders move on a nearly frictionless surface? How do dragonflies hover? These and other conundrums of biology can be explained by the mathematics of the Navier-Stokes equations.

### Brownian Motion, Phase Transitions, and Conformal Maps

by Dana Mackenzie ( hear the podcast interview with the author)

A new random process called Schramm-Loewner evolution turns out to be a good model for a variety of physical phenomena, from the random jitters of air molecules to phase transitions of a magnetic material. The key mathematical property of these systems, proven in some cases and still conjectural in others, is conformal invariance.

### Smooth(ed) Moves

by Barry Cipra ( hear the podcast interview with the author)

Some algorithms of computer science, such as the simplex algorithm for solving linear programming problems, work better than they are supposed to. A new measure of complexity, called smoothed analysis, shows why the standard worst-case scenarios are so misleading.

Order What's Happening in the Mathematical Sciences, Volume 5 on the AMS Bookstore

### New Heights for Number Theory

Progress proceeds apace in the post-Fermat world of elliptic curves and modular forms.

### A Mathematical Twist to Protein Folding

Powerful statistical methods are helping researchers elucidate the three-dimensional structure of life's most important molecules.

### Nothing to Sphere But Sphere Itself

A centuries old problem--the Kepler conjecture--has yielded to new insights and some dogged computation.

### Finite Math

Is the universe finite? Observations of the cosmic microwave background and a new mathematical algorithm may provide an answer.

### The Mathematics of Traffic Jams

Computer models are helping researchers understand where traffic jams come from--and maybe what to do about them.

### Rewriting History

Plimpton 322 is one of mathematicians' favorite cuneiform tablets. But what did it mean to the scribe who composed it?

### It's a Small, Big, Small, Big World

Researchers have found a short distance from theory to applications in the study of small world networks.

### A Celestial Pas de Trois

New methods have revealed a multitude of solutions to an old problem in celestial mechanics: the orbital motion of three bodies.

### Think and Grow Rich

The Clay Mathematics Institute has singled out seven important problems in mathematics, with a \$1 million dollar prize for each.

### Ising on the Cake

A new theorem helps explain why statistical physics has had such a hard time with one of its central problems.

Order What's Happening in the Mathematical Sciences, Volume 4 on the AMS Bookstore

### A Blue-Letter Day for Computer Chess

Deep Blue's victory over Garry Kasparov is the end of a long road in computer chess, but the mathematical study of "perfect" play in combinatorial games has an even longer ways to go.

### Proof by Example: A Mathematician's Mathematician

Paul Erdős, the "Johnny Appleseed" of mathematics, inspired hundreds of colleagues and left a rich legacy of mathematical problems--and solutions.

### Computers Take Algebraic Geometry Back to Its Roots

Computer power has brought algorithmic questions in algebraic geometry back to the fore.

### As Easy as EQP

An automated theorem prover succeeeds in settling a decades-old conjecture in symbolic logic. What's left for human minds to do?

### Beetlemania: Chaos in Ecology

A collaboration between mathematicians and biologists has led to the first experimental evidence for chaotic dynamics in a population.

### From Wired to Weird

Mathematical discoveries are shaping research in a potentially revolutionary kind of computing, based on principles of quantum mechanics.

### Tales from the Cryptosystem

A breakthrough in the theory of computational complexity has implications for cryptographic systems with "guaranteed" security.

### But Is It Math?

Mathematics and art have more in commmon than is commonly supposed. Two twentieth-century artists, M. C. Escher and Marcel Duchamp, used mathematics as an inspiration for works of art.

### Mathematical Discovery (by Henri Poincaré)

Henri Poincaré's thoughts on thinking, written near the beginning of the twentieth century, are well worth repeating at century's end.

Order What's Happening in the Mathematical Sciences, Volume 3 on the AMS Bookstore

### Fermat's Theorem--At Last!

Andrew Wiles has completed the astonishing tour de force that resolves the most famous problem in mathematics. His proof is the realization of a boyhood dream.

### A Tale of Two Theories

A breakthrough in theoretical physics has simplified a notoriously difficult theory in 4-dimensional geometry, and given mathematicians a lot to think about.

### Computer Science Discovers DNA

Will computers of the future be bio-engineered? It's a possibility.

### The Gentle Art of Control

Modern technology relies on mathematical control theory to keep things on an even keel. How do the equations know what to do?

### Computational Fluid Dynamics--Verging on Turbulence

Mathematical techniques, faster computers, and better algorithms are gaining ground in the study of complex fluid flows. For some researchers, computtational turbulence is literally a pipe dream.

### Cellular Automata Offer New Look on Life, the Universe, and Everything

The continuously increasing power of computers has enabled researchers to take a discrete look at the world. Theorists seek to explain the complex patterns that are often seen.

### Are Group Theorists Simple-Minded?

Researchers are working hard to simplify one of the most complicated proofs in mathematical history--the classification of simple groups.

### The Secret Life of Large Numbers

A computational challenge in number theory has been met, considerably sooner than the 20,000 years it was expected to take.

### In Math We Trust

A theorem about multivariate integration may find a home on Wall Street. You could call it a get-rich-quick scheme.

### "A Truly Remarkable Proof"

The announcement last year of a proof of Fermat's Last Theorem stunned the mathematical world. Andrew Wiles's proof, though currently incomplete, has nonetheless drawn rave reviews.

### From Knot to Unknot

What's the quickest way to untie a knot? Researchers have untangled a good part of the answer.

### New Wave Mathematics

Will compact waves cruise the information superhighways of the future? In theory, it's possible.

### Mathematical Insights for Medical Imaging

A team of mathematicians, computer scientists, and engineers has designed a new medical imaging technology based on the safe application of electric current

### Parlez-vous Wavelets?

Mathematicians and scientists are rapidly learning to speak a new language. The results are making a big splash.

### Random Algorithms Leave Little to Chance

Computer scientists will do anything to avoid bottlenecks and speed up computations. But gamble on the results? You bet!

### Soap Solution

Undergraduate students at a summer mathematics research program have found some slick answers to some old problems about the geometry of soap bubbles.

### Straightening Out Nonlinear Codes

A complicated class of error-correcting codes has suddenly gotten much easier to use.

### Quite easily Done

A combinatorial problem, long thought to be difficult, has finally been solved--with surprising ease.

### (Vector) Field of Dreams

A clever construction "pulls the plug" on a 40-year old conjecture about the topology of vector fields.

Unfortunately, Volume 2 is no longer available.

### Equations Come to Life in Mathematical Biology

Mathematicians are working with biologists to delve into some of the most challenging problems in biology today, from understanding the human immune system to "computing" the human heart.

### New Computer Insights from "Transparent" Proofs

Can a computer be trusted when it produces a proof so long and complicated that no human can check the details? Theorists have cooked up a new way to tell whether or not a computer proof is right.

### You Can't Always Hear the Shape of a Drum

Can you hear the shape of a drum? is a famous problem that asks if two drums that look different can make the same sound. After decades of head-scratching, mathematicians have come up with the answer.

### Environmentally Sound Mathematics

Mathematicians have been teaming up with scientists to work on solving environmental problems, from ocean modeling to dealing with hazardous waste.

### Disproving the Obvious in Higher Dimensions

Intuition about our three-dimensional world can be surprisingly misleading when it comes to higher dimensions, as two recent results in geometry show.

### Collaboration Closes in on Closed Geodesics

An unusual blend of differential geometry and dynamical systems has led to an important theoretical result about the number of closed "geodesic" curves on distorted spheres.

### Crystal Clear Computations

Growing crystals--on a computer? Mathematicians are helping materials scientists to better understand the nature of crystals, while picking up some challenging mathematical problems along the way.

### Camp Geometry

A group of talented and inquisitive undergraduates "camped out" last summer at the Geometry Center. Using sophisticated computer graphics and their own imaginations, they came up with some fascinating mathematics.

### Number Theorists Uncover a Slew of Prime Imposters

Strange as it may sound, there are composite numbers that "masquerade" as primes. A group of mathematicians trying to hunt down these prime imposters ended up proving there are infinitely many of them.

### Map-Coloring Theorists Look at New Worlds

How many colors are needed to distinguish neighboring colors on a map? The famous Four Color Theorem notwithstanding, this is a challenging problem in graph theory--especially when your maps aren't flat.

Unfortunately, Volume 1 is no longer available.


text:
## Open Lumify

I worked for a “data discovery and analytics platform” company1 for a couple of years, and got to see first-hand the power of graph-based datastores for analysis, hypothesis-testing and pattern discovery. It truly was an example of intelligence amplification and allowed the analysts I worked with to work more effectively than ever.

Since that day I’ve had a desire to map things out, to build a kind of mega-mindmap2 to really understand some of the more complex and complicated situations — ancient and modern history, the middle east, US politics, the powerful people behind every conspiracy theory imaginable…

So I was quite happy to see that someone somewhere had started an opensource graph-based analysis suite of a similar sort. Unfortunately by the time I found it, it had become abandonware, moving from open- to (assumed) closed-source by the IP owner, who was then bought by another company, then again, renamed, ad absurdum infinitum. I grabbed the last opensource I could find and have started on the long journey to revive it.

So far I’ve updated the build (Maven! Eek) to work with Java 9+ and refactored everything to theorg.openlumify package. It builds, it runs … but I can’t login.

If only there were more hours in the night or day!

## Some plans

In no particular order …

• get it running
• build containers
• make it work in Kubernetes


text:
class galpy.potential.AdiabaticContractionWrapperPotential(amp=1.0, pot=None, baryonpot=None, method='cautun', f_bar=0.157, rmin=None, rmax=50.0, ro=None, vo=None)[source]

AdiabaticContractionWrapperPotential: Wrapper to adiabatically contract a DM halo in response to the growth of a baryonic component. Use for example as:

dm= AdiabaticContractionWrapperPotential(pot=MWPotential2014[2],baryonpot=MWPotential2014[:2])

to contract the dark-matter halo in MWPotential2014 according to the baryon distribution within it. The basic physics of the adiabatic contraction is that a fraction f_bar of the mass in the original potential pot cools adiabatically to form a baryonic component baryonpot; this wrapper computes the resulting dark-matter potential using different approximations in the literature.

__init__(amp=1.0, pot=None, baryonpot=None, method='cautun', f_bar=0.157, rmin=None, rmax=50.0, ro=None, vo=None)[source]

NAME:

__init__

PURPOSE:

INPUT:

amp - amplitude to be applied to the potential (default: 1.)

pot - Potential instance or list thereof representing the density that is adiabatically contracted

baryonpot - Potential instance or list thereof representing the density of baryons whose growth causes the contraction

method= (‘cautun’) Type of adiabatic-contraction formula:

f_bar= (0.157) universal baryon fraction; if None, calculated from pot and baryonpot assuming that at rmax the halo contains the universal baryon fraction; leave this at the default value unless you know what you are doing

rmin= (None) minimum radius to consider (default: rmax/2500; don’t set this to zero)

rmax= (50.) maximum radius to consider (can be Quantity)

ro, vo= standard unit-conversion parameters

OUTPUT:

(none)

HISTORY:

2021-03-21 - Started based on Marius Cautun’s code - Bovy (UofT)


text:
Consider a differentiable function f having a domain of all positive real numbers, and for which...

Question:

Consider a differentiable function f having a domain of all positive real numbers, and for which it is known that {eq}f'(x)= (4 - x)x^{-3} {/eq} for x > 0.

a. Find all intervals on which the graph of f is concave down. Justify answer.

b. Given that f(1)= 2 determine f(x).

Concavity intervals and inflection point

If we have the first derivative, and we need the concave down intervals, we find the second derivative and make a second derivative test, remember: a Function with one independent variable has inflection point where its second derivative is zero, also if the second derivative is positive the function is concave up and if the second derivative is negative, the function is concave down.

The first derivative is: {eq}\displaystyle \ f'(x)= (4 - x)x^{-3} \\ {/eq}

The second derivative is: {eq}\displaystyle \ f''(x)= 2\,{x}^{-3}-12\,{x}^{-4} \\ \displaystyle \ f''(x) = 2\,{\frac {x-6}{{x}^{4}}} \\ {/eq}

Inflection point at {eq}\displaystyle \ f''(x) =0 \; \Rightarrow \; 0= 2\,{\frac {x-6}{{x}^{4}}} \; \Leftrightarrow \; x= 6 \\ {/eq} so, the inflection point is: {eq}(6 , f(6)) {/eq}

Second derivative test:

{eq}\begin{array} \; \text{Interval} \; & { 0 < x < 6 } & { 6 < x < \infty } \\ \hline Test \space{} value & { x = 2 } & { x = 9 } \\ Test & { f''( 2 ) = - < 0 } & { f''( 9 ) = + > 0 } \\ Conclusion & { concave \; downward } & { concave \; upward } \\ \end{array} \\ \therefore \boxed { \text{ concave upward interval is} \; \ ( 6 , \infty )} \\ \therefore \boxed { \text{ concave downward interval is} \; \ ( 0 , 6 )} \\ {/eq}

a. Therefore, the interval on which the graph of f is concave down is {eq}( 0 , 6 ) \\ {/eq}

b. Given that f(1)= 2

{eq}\ f(x) {/eq} is

{eq}\displaystyle \int \ f'(x) \;dx = \int (4 - x)x^{-3} \; dx \\ \displaystyle \ f(x) = {x}^{-1}-2\,{x}^{-2} + C \\ {/eq}

Now, with the given point: {eq}(1,2 ) \\ \displaystyle 2= {(1)}^{-1}-2\,{(1)}^{-2}+C \; \Rightarrow \; C= 3 \\ {/eq}

Therefore, the function is:

{eq}\displaystyle \ f(x) = {x}^{-1}-2\,{x}^{-2}+3 \\ {/eq}

Graph


text:
scroll identifier for mobile
main-content

This practical guide provides comprehensive information on PIV. The third edition extends many aspects of Particle image Velocimetry, in particular the tomographic PIV method, high-velocity PIV, Micro-PIV, and accuracy assessment.

In this book, relevant theoretical background information directly support the practical aspects associated with the planning, performance and understanding of experiments employing the PIV technique. It is primarily intended for engineers, scientists and students, who already have some basic knowledge of fluid mechanics and non-intrusive optical measurement techniques. It shall guide researchers and engineers to design and perform their experiment successfully without requiring them to first become specialists in the field. Nonetheless many of the basic properties of PIV are provided as they must be well understood before a correct interpretation of the results is possible.

### Chapter 1. Introduction

The introduction starts with a short summary of the historical background of PIV. Next, the fundamental principles of PIV are briefly explained in an easily understandable manner, so that the reader already becomes aware of the main advantages, problems and limitations of PIV, before starting to read the detailed explanations in the main part of the book. Several major technological milestones of PIV, achieved during the last decades, resulted in its wide-spread use within different areas today. This fact is illustrated taking examples from applications for fundamental research in turbulent flows and for industrial research in large test facilities.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 2. Physical and Technical Background

The choice of an appropriate tracer is of paramount importance for the success of a PIV experiment. This chapter guides the choice by giving a description of the physical mechanisms governing the motion of tracer particles in the flow. The discussion covers seeding particle generation techniques and their supply into the flow facility from water experiments to aerodynamics and compressible reactive flows. The properties of HFSB tracers are given a specific attention, given their recent introduction for PIV experiments at large scale. The light scattering properties of the tracer particles are of equal importance. The chapter presents the most used physical models to predict the amount of light scattered by the tracers. The fundamental properties of Lasers as the most used devices to illuminate the seeded flow are treated. The discussion includes the properties of emitted light pulses, low- and high-repetition rate systems, light transmission optics and methods of light sheet formation. The growing number of experiments with volume illumination justifies an expanded discussion on the working principles of LED illumination as an alternative approach to Lasers. Digital imaging systems needed to record the scattered light intensity from the particles are described from their working principle, electronic architecture and operating modes. The most relevant differences between CCD and CMOS imagers are explained guiding the choice to the most suited imager for a given experiment. The discussion includes the most recent developments in this domain with the sCMOS architecture.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 3. Recording Techniques for PIV

Based on the physical concepts of electronic imaging introduced in the previous chapter, this chapter focusses on the image sensor technology in the context of PIV image recording. Loosely based on the historical development of CCD image sensor technology, the chapter describes the advancements such as on-chip intermediate storage, asynchronous triggering and the lens-on-chip technique that were instrumental in making the “double-shutter” PIV camera possible, that has been the workhorse throughout the PIV community for the past two decades. Details such as the synchronization of these cameras with the light source are covered. Driven by the consumer market, CMOS imaging is nowadays increasingly replacing the CCD sensor in PIV-suitable cameras, in particular, through the introduction of low noise imagers such as the scientific CMOS. Beyond this, high-speed CMOS imager allow the extension of PIV technique to capture temporally highly resolved velocity data at rates in excess of 10,000 recordings per second.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 4. Mathematical Background of Statistical PIV Evaluation

In this chapter, a simplified mathematical model of the recording and subsequent statistical evaluation of PIV images will be presented. For this purpose the mathematical representation of the particle image locations, the image intensity field and the mean value, the auto-correlation, and the variance of a single exposure recording are described. Then, we analyze the cross-correlation of two frames of singly exposed recordings and expand the theory for the evaluation of doubly exposed recordings.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 5. Image Evaluation Methods for PIV

This chapter covers extensively the methods used to determine the flow velocity starting from the recordings of particle images. After an introduction to the concept of spatial correlation and Fourier methods, an overview of the different PIV evaluation methods is given. Ample discussions devoted to explain the details of the discrete spatial correlation operator in use for PIV interrogation. The main features associated to the FFT implementation (aliasing, displacement range limit and bias error) are discussed. Methods that enhance the correlation signal either in terms of robustness or of accuracy are surveyed. The discussion of ensemble correlation techniques and the use of single-pixel correlation in micro-PIV and macroscopic experiments is a novel addition to the present edition. A detailed description is given of the standard image interrogation based on multigrid image deformation, where the advantages in the treatment of complex flows are discussed as well as the issues in terms of resolution and numerical stability. Another new feature introduced in this chapter is the discussion of the recent developments of algorithms in use for PIV time series as obtained by high-speed PIV systems. Namely, the algorithms to perform Multi frame-PIV, Pyramid Correlation and Fluid Trajectory Correlation and Ensemble Evaluation are treated. Furthermore, a new section that discusses the methods used for individual particle tracking is introduced. The discussion describes the working principles of PTV for planar PIV. The potential of the latter techniques in terms of spatial resolution as well as their limits of applicability in terms of image density are presented.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 6. PIV Uncertainty and Measurement Accuracy

The chapter starts with an overview of common PIV measurement error contributions. Then important parameters for the optimization of PIV measurements are discussed and the significance of the dynamic velocity range and dynamic spatial range is outlined. Thereafter, the concept of measurement error is introduced and the error propagation essentials are discussed. The sensitivity of the measurement uncertainty on the particle image size, particle image density, background noise, particle image shift, out-of-plane motion, light-sheet mismatch, displacement gradients and streamline curvature is discussed in detail. The chapter finishes with strategies to optimize PIV uncertainties and outlines the main implications of the uncertainty analysis for multi-camera recording systems. The aim of this chapter is to familiarize the reader with various sources and sensitivities of PIV measurement uncertainty which will be instructive for optimizing PIV measurements in practice.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 7. Post-processing of PIV Data

While the previous chapter have dealt with recording and evaluation of PIV images, the extracted data require further post-processing in the context of data validation and further data reduction to retrieve fluid mechanical relevant information. This chapter introduces a variety of validation schemes that operate either globally or locally on the data along with methods of data interpolation to fill in data gaps in both space and time. The validated data can then be subjected to differentiation to, for instance, extract gradient information such as vorticity fields. Issues and errors arising through applying differentials to the finitely-spaced data grid are discussed and illustrated. Alternatively, the velocity data can be integrated to retrieve streamlines, body forces or even pressure fields.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 8. Stereoscopic PIV

By extending the “classical” single camera PIV implementation with a second camera the planar light sheet can be imaged stereoscopically from two directions allowing the recovery of the velocity vector normal to the light sheet. This chapter on stereo-PIV introduces two primary implementations of stereoscopic imaging along with the concept of Scheimpflug imaging of obliquely viewed light sheet planes. The stereoscopic reconstruction can follow a variety of approaches but collectively rely on some sort of calibration to recover 3-C velocity data from two separate 2-C vector maps. The error introduced through camera misalignment is discussed along with possible mitigation strategies. The application of the stereo-PIV technique in environments with index-of-refraction changes such as in water facilities requires specific imaging arrangements. The chapter closes with a list of recommendations for successful implementation of the stereo-PIV technique.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 9. Techniques for 3D-PIV

This chapter initially provides the reader with an extensive survey of the many methods available to measure the flow velocity in three-dimensional problems. Afterwards, the chapter devotes three main sections to the most common or emerging methods: Tomographic PIV, 3D-PTV techniques and Shake-the-Box. Tomographic PIV is considered the technique of choice at present, and is extensively discussed. Aspects cover hardware components, requirements for illumination of a volume and techniques to increase particles visibility. A discussion is given of the effects of number of cameras and their configuration to deal with densely seeded experiments, where the phenomenon of ghost particles dominates the experimental errors. The theoretical background is given about the techniques used for tomographic reconstruction of 3D particle intensity distribution, followed by a critical evaluation of the accuracy of reconstruction. The discussion of Tomographic PIV closes with the description of the most recent algorithms based on multi-exposure reconstruction and time-resolved data analysis. The second part of the chapter is a new addition to the book and it deals with the fundamental principles of 3D-PTV. Particle detection, triangulation and pairing are the most important operations to perform a successful 3D-PTV evaluation. Hybrid methods based on tomographic reconstruction and individual particle tracking are discussed. The section concludes suggesting the working range of PTV techniques in 3D experiments. The Shake-the-Box technique is an emerging method that performs Lagrangian particle tracking with high potential in terms of computation speed and for the high accuracy of particle motion estimation. Its rapid diffusion among research laboratories justifies a detailed description of its working principles, main features and characteristic performance. The section elaborates on the concepts of Iterative Particle Reconstruction, Optical Transfer Function calibration and Data Assimilation to restore the results on a Cartesian mesh. Algorithms for image sequences as well as for four-frame recordings are detailed.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 10. Micro-PIV

The chapter begins with a motivation of microfluidic flow analysis and summarizes the main diagnostic tools commonly used for flow measurements in microscopic systems. Thereafter, the typical implementation of 2D planar micro-PIV is presented, followed by a short historical background of significant development steps since 1993. Next, the imaging of volume-illuminated small particles is discussed and the essentials of three-dimensional diffraction pattern are outlined. The concept of depth-of-field and depth-of-correlation are introduced and the problem of particle visibility is discussed in detail. The second half of the chapter focuses on 3D micro-PIV and micro-PTV techniques. First, scanning, stereoscopic and tomographic micro-PIV recording techniques are presented. Thereafter, the confocal scanning microscopy and defocusing techniques are discussed. Finally, the 3D astigmatism PTV technique is outlined in detail and the strength of the technique for 3D time resolved flow analysis in micro-scale systems is demonstrated.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 11. Applications: Boundary Layers

The following two experiments have been performed in the second half of the 1990s’ in the DLR low turbulence wind tunnel (TUG), which is of an Eiffel type. Screens in the settling chamber and a high contraction ratio of 15:1 lead to a low turbulence level in the test section (cross section $$0.3 \times 1.5\,\text {m}^2$$). The basic turbulence level in the test section of the TUG of $$Tu= 0.06\%$$ (measured by means of a hot wire) allows the investigation of acoustically exited transition from laminar to turbulent flow as well as turbulent boundary layers that develop in the relatively long test section. The flow was seeded in the settling chamber upstream of the screens used to reduce the turbulence of the wind tunnel flow.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 12. Applications: Transonic Flows

Solid surfaces in like actuators, fluid mechanical models and surrounding walls can influence the fluid flow and/or can be deformed or displaced by it. The knowledge of the actual surface shape and location is therefore important for many fluid mechanical investigations. In compressible fluids density variations is another important fluid parameter besides velocity. Both, the measurement of density gradients in a flow and the detection of the deformation and position of solid surfaces in contact with the fluid can be easily obtained based on PIV imaging hard- and software. The correlation based procedures for deformation, displacement, and strain analysis have been developed and applied more and more frequently during the past decade. The most common method, the deformation measurement by Digital Image Correlation (DIC) is described first together with examples of applications. The later section describes the theory of the Background-Oriented Schlieren Technique (BOS), which determines density gradients without using any sophisticated optical equipment. Practical aspect of the technique are addressed by the description of its application to a helicopter in hovering flight and to the transonic flow behind a cylinder.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 13. Applications: Helicopter Aerodynamics

In this chapter a number of applications of the PIV technique will be described, contributed by leading PIV experts from different research establishments and universities worldwide. Primarily, the objective of presenting these applications is to show how the PIV technique has spread out to the most different research areas. However, it is of even higher importance to gain the reader access to a wide variety of ideas for PIV measurements by presenting many different applications in fundamental or industrial research. For each experiment the most important parameters of the object under investigation, of the illumination and recording setup, etc. will be given. These data together with the hints and tricks briefly described and the references to further, more detailed, literature may be useful for the reader when trying to solve problems of his own application.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 14. Applications: Aeroacoustic and Pressure Measurements

In the research field of aeroacoustics, fluctuations in turbulent flows which are part of a sound generating process can be identified by means of the so-called causality correlation technique [5,6,29]. In this approach, the coefficient matrix resulting from the calculated correlation coefficients between the acoustic pressure fluctuations in the far field and velocity fluctuations in a turbulent flow field is analysed. The temporal resolution of the coefficient matrix is set by the sampling rate of the pressure signal in the far field. The spatial resolution is determined by the spatial sampling rate of the measured flow quantity (Table  11.22).
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 15. Applications: Flows at Different Temperatures

These experimental investigations of flows by means of PIV have been carried out in 1996 by DLR in cooperation with the Center of Applied Space Technology and Microgravity (ZARM), University of Bremen, in order to complete their numerical simulations and LDV measurements.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 16. Applications: Micro PIV

This section discusses various applications and techniques used in $$\mu \text {PIV}$$. The applications include very high spatial resolution measurements of pressure-driven flow in a rectangular capillary and measurements in a challenging toroidal vortex.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 17. Applications: Stereo PIV and Multiplane Stereo PIV

The various methods of image reconstruction and calibration as described in Sect. 8.1 were applied in the measurement of the unsteady vortex ring flow field in 1995. Figure 11.125 outlines a vortex ring generator having a simple construction with very reproducible flow characteristics. The vortex ring is generated by discharging a bank of electrolytic capacitors ($$60\,000\,\upmu \text {F}$$) through a pair of loudspeakers which are mounted facing inward on to two sides of a wooden box. By forcing the loudspeaker membranes inward, air is impulsively forced out of a cylindrical, sharpened nozzle (inner diameter $$=34.7\,\text {mm}$$) on the top of the box. The shear layer formed at the tip of the nozzle then rolls up into a vortex ring and separates from the nozzle as the membranes move back to their equilibrium positions due to the decay in supply voltage. As long as the charging voltage is kept constant, the formation of the vortex ring will be very reproducible. The generator also has a seeding pipe with a check valve allowing the interior of the box and ultimately the core of the vortex ring to be seeded.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 18. Applications: Volumetric Flow Measurements

The transition to turbulence in circular jets is a complex three-dimensional process that requires the use of 3D-PIV and temporal resolution for the full understanding the dynamical behavior of coherent vortices. Jets are used in a multitude of engineering systems and their study often regards the process heat and mass transfer.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans

### Chapter 19. Related Techniques

As noted in Sect.1.3, PIV developed from Laser Speckle Interferometry. Therefore, one of the early names for this technique was ‘Laser Speckle Velocimetry’ before ‘Particle Image Velocimetry’ was established. The Laser Speckle Interferometry (or Laser Speckle Photography) was mainly developed for the determination of displacement and strain in engineering structures. The laser speckles are created due to random interference of scattered light from an optically rough surface illuminated by coherent light.
Markus Raffel, Christian E. Willert, Fulvio Scarano, Christian J. Kähler, Steven T. Wereley, Jürgen Kompenhans


text:
## positively

What is $F(c)$ if $c$ is a positively oriented circle?

Go to the list of words starting with: a b c d e f g h i j k l m n o p q r s t u v w y z


text:
# PDE, inhomogeneous diffusion equation

1. Feb 16, 2012

### fluidistic

1. The problem statement, all variables and given/known data
Mathews and Walker problem 8-2 (page 253):
Assume that the neutron density n inside $U_{235}$ obeys the differential equation $\nabla ^2 n+\lambda n =\frac{1}{\kappa } \frac{\partial n }{\partial t}$ (n=0 on surface).
a)Find the critical radius $R_0$ such that the neutron density insdie a $U_{235}$ sphere of radius $R_0$ or greater is unstable and increases exponentially with time.

b)Suppose two hemispheres, each just barely stable, are brought together to form a sphere. This sphere is unstable, worth n ~ $e^t/\tau$. Find the "time-constant" tau of the resulting explosion.

2. Relevant equations

3. The attempt at a solution
I tried to solve this PDE via a Laplace transform, but now I realize I do not think it is well appropriated due to the nabla operator.
Applying Laplace transform gave me $\nabla ^2 F(n)+\lambda F(n)=\frac{1}{\kappa } [n(t=0)+sF(n)]$. Also I do not know the initial condition n(t=0).

I guess I'll have to try another method and that it might involve taking the Laplacian in spherical coordinates, I'm open to use any method to solve this PDE. I'd like to know whether I'm right in thinking that it's not solvable using a Laplace transform and what other method could I use.
Thank you very much in advance.

2. Feb 16, 2012

### nlsherrill

Sorry, I really wish I could help but I don't think I know. This sounds like such a cool problem. What book is this question in?

I am assuming the neutron is to be considered a sphere? I think spherical coordinates would be your best bet, but I have not had experience with PDE's in spherical... Isn't the laplace transform usually used for solving PDE's on semi infinite boundaries?

3. Feb 16, 2012

### fluidistic

The book is from Mathews and Walker -Mathematical methods of physics-. I think we do not consider a single neutron, but a sphere (nucleous of uranium) with a "nicely behaved" neutron density function.
I don't really know about the Laplace transform.
Yeah it looks like I'll have to take the Laplacian in spherical coordinates but... I'm unsure what method to use. A kind of separation of variables maybe.

4. Feb 16, 2012

### Mindscrape

Hmm, yeah, in my experience, every PDE is really it's own beast that you just have to try different methods on. I think separation of variables should work out fine if the density is constant, is it?

5. Feb 17, 2012

### nlsherrill

I think maybe try the separation of variables for spherical polar coordinates and see what you get...it will surely involve the tough to deal with bessel functions though

6. Feb 17, 2012

### fluidistic

Ok guys I'll try separation of variables. But the neutron density isn't constant. It vanishes at the boundaries of the sphere (its surface). And inside the sphere I have no idea how it is, probably very complicated. It's the function that satisfies the PDE.

7. Feb 17, 2012

### Mindscrape

My bad, I meant lambda whatever it is... I'm in EM mode. x-(

8. Feb 17, 2012

### fluidistic

No problem.
Indeed this method looks promising. I'm not math ninja though and I think I have to do some tricks.
So I indeed took the Laplacian in spherical coordinates. After simplifications I reached $$\frac{1}{R}\left (\frac{2}{r} R'+R'' \right )+\frac{1}{\Theta r^2 } \left ( \cot (\theta ) \Theta ' + \Theta '' \right )+\frac{\Psi ''}{\Psi r^2 \sin ^2 \theta }+\lambda =\frac{T'}{\kappa T}$$.
Where $T(t)$, $R(r)$, $\Psi (\phi )$ and $\Theta (\theta )$. I assumed $n (r, \theta, \phi , t)$ to be the product of the 4 functions.

After a few considerations (like a function depending only on positon is equal to a function depending only on time for all positions+time must be that these functions are constants. Also, if I multiply the eq. by $r^2 \sin \theta$ or by $r^2$ I get other conditions leading me to what follows), I get that
(1)$\frac{1}{\Psi } \Psi ''=C_1$. I could solve this equation but there will be 3 cases, depending upon the sign of $C_1$ and if it's worth 0.
(2)$T'=\kappa T C_2 \Rightarrow T(t)= \tilde C_2 e ^{\kappa t}$ and $C_2=\frac{T'}{\kappa t }$.
(3)$C_3=R''+\frac{2R'}{r}+R(\lambda - C_2 )$.
(4)$C_4= \Theta ''+\Theta '+\frac{C_1 \Theta }{\sin \theta }$.
Now it's "only" a matter of solving all these equations and then multiply each solutions together to form the general solution to the PDE.

9. Feb 17, 2012

### Mindscrape

Yep, I think you're good so far. You just have to look up the solutions to the ODEs. They should be pretty similar to simply solving the helmholtz equation in spherical coordinates.

10. Feb 17, 2012

### sgd37

what you have there is a modified schrodinger equation i.e. it doesn't give oscillating solutions but you can solve it just the same way

11. Feb 18, 2012

### fluidistic

Ok I've followed a book for Hemlholtz equation (Mathews and Walker page 228) but it's not that easy to me.
I realize I shouldn't have calculated the derivatives in the Laplacian. So I've redone all the algebra keeping terms like in the book. But there's still a difference from the Helmholtz equation $\nabla ^2 X +k^2 X=0$. In my case the equation isn't homogeneous but equal to a constant that I called $C_1$.
$$\frac{1}{Rr^2}\frac{1}{dr}(r^2R')+\frac{1}{r^2 \sin \theta }\frac{d}{d\theta } (\sin (\theta )\Theta )+\frac{1}{r^2 \sin ^2 \theta }\frac{\Psi'}{\Psi }+\lambda=\frac{1}{\kappa} \frac{T'}{T}=C_1$$.
So that $T(t)=Ae^{c_1\kappa t }$. (I made a small mistake in my last post I think).
I also get that $\frac{\Psi '}{\Psi}=\text{constant}=-m^2 \Rightarrow \Psi (\phi )=Be^{im\phi }+Ce^{-im\phi}$. This equation is the same for Helmholtz equation. I do not really know why the book chose the constant as "$-m^2$", probably for convenience in the algebra. Nor do I know why it considered only a negative constant instead of getting 3 cases (positive, negative and equal to 0).
The next equation is $\frac{1}{\sin \theta} \frac{1}{d\theta }(\sin (\theta ) \Theta ' )-\frac{m^2}{\sin ^2 \theta }\Theta = \text{constant}\cdot \Theta=-l(l+1)\Theta$. Which is the same as in Helmholtz equation.
Lastly the radial equation is $\frac{1}{r^2}\frac{d}{dr} (r^2R')+\left [ \lambda -C_1-\frac{l(l+1)}{r^2} \right ]R=0$. Which differs slightly from Helmholtz equation. Indeed, I have $\lambda-C_1$ instead of $k^2$.
The book then gives the solution to the theta equation, $P_l^m (x)$, $Q_l^m(x)$. Where $x=\cos \theta$.
While for the radial equation, adjusting the given result but substituting their k by $\sqrt {\lambda - C_1}$, I get $R=\frac{J_{l+1/2}(\sqrt {\lambda -C_1}r)}{\sqrt r}$ and $\frac{Y_{l+1/2}(\sqrt {\lambda -C_1}r)}{\sqrt r}$. I do not know however if I have to take $\pm \sqrt {\lambda -C_1}$ or just the positive square root, etc.
So to answer to part a) I think I must NOT only look into the radial equation solution. For a particular value of $r=R_0$, when I multiply all functions together, I should get that the solution diverges when t tends to infinity.
Unfortunately multiplying all these functions together will create a too complicated function to analyze...

Last edited: Feb 18, 2012
12. Feb 19, 2012

### sgd37

m varies from +l to -l so you have all three considerations in your solution unless l is half integer of course

13. Jun 24, 2012

### fluidistic

I'm back at this problem. I found some extra help, basically telling me to use the symmetry of the problem.
So I tried to redo all the exercise from start.
I'll use separation of variables and due to the symmetry of the problem I'll assume a solution of the form $n(r,t)=R(r)T(t)$.
My first doubt comes as soon as I take the Laplacian.
If I take the Laplacian in spherical coordinates, the PDE simplifies to $\frac{2R'}{R}+r\frac{R''}{R}+\lambda = \frac{1}{\kappa } \frac{T'}{T}$. This gives me 2 DE's in which one isn't that simple.
However if I use the logic that the Laplacian only acts over the space coordinates (in this case r, and not on t), then the PDE simplifies to $\frac{R''}{R}+\lambda = \frac{1}{\kappa } \frac{T'}{T}$. This gives rise to 2 "simple" DE's.
Why on Earth I get 2 different DE's to solve whether I consider the Laplacian in spherical coordinates or if I consider it only acts over r?
Which way is the correct one and why does the other way fail?
So I've kept following the easiest way (in other words the way of $\nabla ^2 (RT)=T R''$.)
First DE to solve: $\frac{1}{\kappa} \frac{T'}{T}=m$, with solution $T(t)=Ae^{m \kappa t}$.
The second DE to solve: $\frac{R''}{R}+\lambda =m$. Note that m is the constant of separation of the method of separation of variables used in the problem.
Here I used logic and I assumed that only the case $m>\lambda$ holds true. Because if $m<\lambda$ I'd get a periodic solution in r, which I don't think to be true and if $m=0$ I'd get the constant solution equal to 0 which is also to be disgarted.
Hence $R(r)=Be^{r\sqrt{m-\lambda }} + Ce^{-r\sqrt {m-\lambda}}$. Here I guess it's safe to assume $C=0$ so that $n(r,t)$ remains finite when $r=0$.
Thus $R(r)=Be^{r\sqrt {m-\lambda}}$.
Now I apply the boundary condition that $n(\tilde r , t )=0$ where $\tilde r$ is the radius of the sphere. But this implies that $B=0$ and thus $n(r,t)=0$ for all r and all t's. Which is obviously wrong.
Can someone point me out my error(s) and/or help me with this problem?

14. Jun 26, 2012

### Mute

You have to use the 3d Laplacian. Your "simple" case amounted to using the 1d Laplacian, which is incorrect. You really have a 3d problem, you are just assuming that the solution you are seeking is the spherically symmetric solution, such that the angular derivatives in the Laplacian vanish, which leaves you with only the radial term from the 3d Laplacian. This symmetry does not allow you to treat the problem as if it were just a 1d problem. Suppose you were a masochist and wanted to try to solve this problem in cartesian coordinates: you would have to keep all three derivatives. What you did when you only used $\partial^2/\partial r^2$ would amount to only keeping the $\partial^2/\partial x^2$ term in the cartesian Laplacian, for instance.

15. Jun 26, 2012

### EulersFormula

This problem is very similar to the Shrodinger equation. I recommend taking a look at it. I also wouldn't throw away solutions that suggest the neutron density to vary harmonically with distance. This is not uncommon.

16. Jul 1, 2012

### fluidistic

Ok thank you very much guys, now it gets clearer to me.
So basically the 2 DE's to solve are:
(1): $\frac{T'}{\kappa T }=m$.
(2): $\frac{2R'}{rR}+\frac{R''}{R}+\lambda =m$.
The solution to (1) is I said already, $T(t)=T_0e^{\kappa m t}$.
I'm now trying to figure out the solution to (2). I tried the method of Frobenius, I proposed a solution of the form $R(r)=\sum _{n=0} ^{\infty } a_n r^{n+c}$. Deriving with respect to r twice and plugging in the DE (2) I reached the secular equation $c(c+1)=0$ so that $c=0$ or $c=-1$. Since the 2 roots differ by an integer, I can "only" reach 1 type of solution of the form I assumed, not the 2. I'd have to work out the second form of the solution by variation of parameters once I'm done with $R_1 (r)$.
So that I took $c=-1$ (no choice), this gave me the recurrence relation $a_n = - \frac{a_{n-2} (\lambda - m )}{n ^2 -n}$. Where $a_1$ is arbitrary and $a_0$ too (except that it can't be 0).
For convenience I chose $a_0=1$ but even though I calculated the first terms ($a_2$, $a_4$ and $a_6$) I don't really see a pattern to express the general form of $a_n$ independently of the previous $a_{n-p}$'s. Also, should I choose $a_1=0$ for convenience so that all odd terms would be worth 0?
By the way this might be similar to the hydrogen atom in a way yes, but the method to solve the DE isn't the same I guess because one usually "factors out" the behavior the DE when r tends to infinity (in the hydrogen atom case) while here I'm only interested in solving the PDE inside the region of a sphere of radius a or $\tilde r$; so that it's not really the same path to take. I can be wrong of course.
Any idea on how to solve the DE (2)?

17. Jul 1, 2012

### Staff: Mentor

The solution is expressible as Bessel functions. See Eqn. 9.1.52 in Abramowitz and Stegan.

18. Jul 1, 2012

### fluidistic

Thanks a lot for the reference (it's a free ebook).
So it states that the solution to the DE $w''-\frac{2\nu -1 }{z}w'+\lambda ^2 w =0$ is $w =z^{\nu } C_{\nu} (\lambda z )$. Where C could be any linear combination of the Bessel functions of any kind.
My problem is... if I compare the DE to solve (2) or a slightly simplified but equivalent to DE: (3): $R''+ \frac{2}{r}R'+R(\lambda - m )=0$, one finds that $\nu =-\frac{1}{2}$. So clearly nu isn't the smaller roots of the indicial equation as I thought.
What is very strange is that Wolfram alpha gives a totally different answer which seems to me unrelated to the Bessel function (http://www.wolframalpha.com/input/?i=y''+(2/x)y'+y*a=0) and it seems to confirm that $c=-1$ is the smaller root of the indicial equation.
According to the solution in the book of Abramowitz and Stegun the solution would be of the form of $\frac{C_{-\frac{1}{2}}(r \sqrt {\lambda -m})}{\sqrt r}$.
I'm totally confused.

19. Jul 1, 2012

### Mute

For half integer orders the Bessel functions are basically just combinations of signs and cosines divided by some power of x.

See http://en.wikipedia.org/wiki/Spherical_Bessel_function#Spherical_Bessel_functions:_jn.2C_yn

You should be able to mash the solution from Abramowitz and Stegun into looking like the one wolfram alpha gave you.


text:
Lecture 12: An Introduction to Arithmetic Functions

# Summary

Today we talked a little more about the Euler $\phi$ function, then moved on to talk about arithmetic functions more generally. We saw a few examples of arithmetic functions, but we spent the majority of our time discussing the properties these functions have — especially when they respect multiplication.

# Summing Up Euler's Phi

As a final remark on the Euler $\phi$ function, we give the following

Lemma: For any integer n, we have

$\displaystyle \sum_{d\mid n}\phi(d) = n$

Proof: We start the proof by partitioning the integers between 1 and n into groups based on their gcds. Namely, we define

(1)
\begin{align} S_d = \{a: 1 \leq a \leq n \mbox{ and }(a,n) = d\}. \end{align}

Since every integer between 1 and n belongs to exactly one set $S_d$, we know that

(2)
\begin{align} \sum_{d \mid n} |S_d| = n. \end{align}

Now we notice that elements a of the set $S_d$ must have the property $(\frac{a}{d},\frac{n}{d}) = 1$ — after all, once we divide through by the gcd of two elements, these quotients are relatively prime to each other. Hence every element of $S_d$ corresponds to an element between 1 and $\frac{n}{d}$ which is relatively prime to $\frac{n}{d}$. How many such elements are there? Exactly $\phi\left(\frac{n}{d}\right)$; after all, that's exactly what Euler's $\phi$ function counts! Hence we have

(3)
\begin{align} n = \sum_{d \mid n} |S_d| = \sum_{d \mid n} \phi\left(\frac{n}{d}\right). \end{align}

To finish off the proof, we notice that as d runs through divisors of n, so too does the quantity $\frac{n}{d}$. Hence the final sum in our expression is actually the same as $\sum_{d \mid n} \phi(d)$ — it is merely adding the sum numbers up in the opposite order. $\square$

#### Example

In class we verified this by calculating the value of $\phi$ on divisors of 12. We found

(4)
\begin{align} \phi(1) + \phi(2) + \phi(3) + \phi(4) + \phi(6) + \phi(12) = 1 + 1 + 2 + 2 + 2 + 4 = 12. \end{align}

We also verified that for each divisor d, the set $S_d$ had $\phi\left(\frac{12}{d}\right)$ elements. For instance, $S_3 = \{3,9\}$, and $\phi\left(\frac{12}{3}\right) = \phi(4) = 2.$ $\square$

# Arithmetic functions

Having played with Euler's $\phi$ function for a while, we're now ready to think about more functions on the integers. We start with a basic

Definition: A function whose domain is the positive integers is called an arithmetic function.

There are, of course, loads and loads of arithmetic functions. For instance, your old favorite $f(x) = x^2$ becomes an arithmetic function when you only evaluate it on integers. Typically, though, number theorists are more interested in the functions which are connected to number-theoretic properties. For instance, here are a smattering of arithmetic functions which number theorists get really excited about

Function Name What it does
$\phi$ Counts congruence classes relatively prime to n
$\nu$ Counts the number of divisors
$\sigma$ Computes the sum of divisors
$\sigma_k$ Computes the sum of kth powers of divisors

We'll eventually talk about all these functions, but first we want to discuss arithmetic functions more generally.

One of the important properties that nearly all the arithmetic functions we'll deal with share is that they somehow respect multplication, by which I mean one can often evaluate an arithmetic function for a composite number by evaluating it on primes (or powers of primes).

Definition: An arithmetic function f is called multiplicative if $f(nm) = f(n)f(m)$ whenever $(n,m) = 1$. An arithmetic function g is called completely multiplicative if $g(nm) = g(n)g(m)$ for all pairs of integers n and m.

We have already seen that $\phi$ is a multiplicative function. The benefit of multiplicativity is the following

Lemma: If f is an arithmetic function and $n = p_1^{a_1}\cdots p_k^{a_k}$, then

$f(n) = f(p_1^{a_1})\cdots f(p_k^{a_k})$.

If g is completely multiplicative, then we have

$g(n) = g(p_1)^{a_1}\cdots g(p_k)^{a_k}$

To prove either of these results, just use induction and the definition of multiplicativity (or complete multiplicativity).

## An Interesting property of multiplicative functions

We'll finish our discussion of arithmetic functions in much the same way we finished our discussion of Euler's $\phi$ function.

Theorem: If f is a multiplicative function, then

F(n) = \sum_{d \mid n} f(n)

is also a multiplicative function.

In order to prove this result, we'll need the following

Lemma: If $(m,n) = 1$ and d is a divisor of mn, then there exists divisors $d_1$ of n and $d_2$ of m which are relatively prime to each other and whose product is d. Hence divisors of mn corresponds to pairs of divisors $d_1,d_2$ of n and m (respectively).

We didn't prove this result in class, and I won't prove it here. This is good practice for the exam, so try proving it yourself!

Proof of Theorem: If we want to show that F is arithmetic, then we need to show that for any relatively prime integers n and m, we have

(5)
$$F(nm) = F(n)F(m).$$

With this as our goal, we'll go ahead and try to compute $F(nm)$.

First, notice that by definition we have

(6)
\begin{align} F(nm) = \sum_{d\mid n}f(d). \end{align}

Now we can use our lemma above to express each divisor d of nm as a product of (relatively prime) divisors of n and m:

(7)
\begin{align} \sum_{d\mid n}f(d) = {\sum_{d_1\mid n}}_{d_2 \mid m}f(d_1d_2). \end{align}

Now since f is arithmetic, we know that $f(d_1d_2) = f(d_1)f(d_2)$. Hence, after reexpressing as a double sum, we get

(8)
\begin{align} {\sum_{d_1\mid n}}_{d_2 \mid m}f(d_1d_2) = \sum_{d_1 \mid n}\sum_{d_2 \mid m} f(d_1)f(d_2). \end{align}

Finally, notice that the quantity $f(d_1)$ is not dependent on $d_2$, and hence we can pull this quantity out of the second sum. This leaves us with

(9)
\begin{align} \sum_{d_1 \mid n}\sum_{d_2 \mid m} f(d_1)f(d_2)= \sum_{d_1 \mid n}f(d_1)\sum_{d_2 \mid m} f(d_2). \end{align}

But this is just $F(n)F(m)$, and so we've proven

(10)
$$F(nm) = F(n)F(m).$$

$\square$

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