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1. CMB 2011 (vol 56 pp. 272)

Cheng, Lixin; Luo, Zhenghua; Zhou, Yu
 On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate In this note, we first give a characterization of super weakly compact convex sets of a Banach space $X$: a closed bounded convex set $K\subset X$ is super weakly compact if and only if there exists a $w^*$ lower semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly FrÃ©chet differentiable on each bounded set of $X^*$. Then we present a representation theorem for the dual of the semigroup $\textrm{swcc}(X)$ consisting of all the nonempty super weakly compact convex sets of the space $X$. Keywords:super weakly compact set, dual of normed semigroup, uniform FrÃ©chet differentiability, representationCategories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50

2. CMB 2008 (vol 51 pp. 205)

Duda, Jakub
 On GÃ¢teaux Differentiability of Pointwise Lipschitz Mappings We prove that for every function $f\from X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is G\^ateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f\from X\to\R$ cone monotone, $g\from X\to\R$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Fr\'echet differentiable. Keywords:GÃ¢teaux differentiable function, Radon-NikodÃ½m property, differentiability of Lipschitz functions, pointwise-Lipschitz functions, cone mononotone functionsCategories:46G05, 46T20


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One of the key defining features of Tezos is to include the protocol upgrades in the protocol itself to create what has been dubbed as self-amending crypto-ledger. The upgrades are submitted through the blockchain, but before they get deployed, they have to pass multiple stages of votes, that also take place on the blockchain. In this post, we'll look at what encompasses the process of building a protocol upgrade.

## Before you begin...

Say you have an idea for protocol change and you have the capabilities to implement it. Great! But there's of course more to a successful protocol upgrade than just writing the code that shouldn't be overlooked. Consider other teams (currently Nomadic Labs and us, Cryptium Labs) working on evolving the protocol and get in touch with them or open an issue on the main Tezos repository. This generates discussion around the changes you'd like to propose and you'll get some feedback. After all, your proposal will have to convince the stakeholders to vote for it, so it's good to put your design to the test early on. Also, it may be possible to bundle your changes together with other proposals, instead of competing with them (only one upgrade can pass the proposal stage at the time) and possibly get the upgrade deployed sooner.

## Development prerequisites

The obvious part should be to learn OCaml. You'll want to be comfortable with both reading and writing OCaml. There are numerous resources available to help along the way:

The Tezos documentation covers many topics you'll want to be familiar with, especially in the developer section. Both the data encoding and the error monad libraries are used extensively. For protocol upgrade itself it's essential to read the protocol Alpha section and how the protocol is sandboxed (in the protocol, you can only use modules and interfaces that are exposed from lib_protocol_environment and while this might seem to be limiting, there are good reasons for it). You may gain better insight by also looking at the code as you read the documentation.

Make sure you can build from sources. For development, instead of make build-deps invoke make build-dev-deps to install Merlin for code completion and more.

If you're not already familiar with the Tezos client, you can explore the many available commands and RPC interface from within the sandbox mode (a local private network with configurable number of nodes), which is also very useful for quickly experimenting with the changes you've applied to the protocol.

## Protocol code overview

For protocol upgrade, you can typically start your branch from master. Your proposal code starts off the protocol Alpha in src/proto_alpha/lib_protocol, which will contain the code of the latest active protocol (and indeed will have the code of your proposal after it's successfully voted and injected). This is the green eye of the octopus, described in more detail in the aforementioned protocol Alpha section.

In brief,

• Note that the TEZOS_PROTOCOL file defines the linking order of the contained modules and its grouping aids in understanding how the code is structured
• The Alpha_context module re-defines abstract modules typically by glueing together concrete *_repr (representation) and *_storage modules. For example, module Vote is comprised of Vote_repr and Vote_storage. Vote_repr contains data definitions and their encoding while Vote_storage exposes accessors to key-value stores
• All the storage accessors are defined in the Storage module using the many variants of storage signatures from Storage_sigs together with storage builders from Storage_functors
• Much of the protocol logic is built using the Alpha_context abstractions in Apply (protocol's main entry points), Baking and protocol Amendment modules
• Script_* modules deal with Michelson smart contracts, whose code is translated to an intermediate representation to be interpreted when a contract code is being run
• Finally, the *_services modules define the protocol RPC API

The Tezos documentation goes into more detail and we recommend that you get comfortable with it. We hope that this will give you a good introductory overview of the codebase and how to navigate it. If you have questions or get stuck on something, don't hesitate to reach out for help!

In the next part, we'll look at how to handle protocol upgrades that introduce structural changes, that require a data migration to bridge the changes from a previous protocol to a new one.

Written by Tomáš Zemanovič, core developer and researcher at Metastate.

Metastate has ceased its activities on the 31st of January 2021. The team members have joined a new company and working on a new project. If you're interested in distributed systems research and engineering and systems engineering in Rust, checkout the open positions at Heliax.

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# Using polar coordinates to evaluate a multivariable limit

#### Jimmy25

1. The problem statement, all variables and given/known data

When you substitute polar coordinates into a multivariable limit, do you treat theda as a constant when evaluating? (I know how to use polar coordinates to evaluate a limit but haven't learned what they are yet)

2. Relevant equations

3. The attempt at a solution

Related Calculus and Beyond Homework News on Phys.org

#### fzero

Homework Helper
Gold Member
1. The problem statement, all variables and given/known data

When you substitute polar coordinates into a multivariable limit, do you treat theda as a constant when evaluating? (I know how to use polar coordinates to evaluate a limit but haven't learned what they are yet)
Generally you wouldn't treat $$\theta$$ as a constant because the most important concept in taking a multivariable limit is that the limit doesn't exist if it depends on the path you take to the point.

As an example, consider

$$\lim_{(x,y)\rightarrow (0,0)} \frac{x^2y}{x^4+y^2} = \lim_{r->0} \frac{r\cos^2\theta\sin\theta}{r^2 \cos^4\theta + \sin^2\theta} = \lim_{r->0} \frac{r \cos^2\theta}{\sin\theta}.$$

If we just take $$r\rightarrow 0$$, we find that this vanishes. However, if we also take $$\theta\rightarrow 0$$ at the same rate as r, we find $$\cos^2(0)=1$$. Therefore the limit does not exist.

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## Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds

Vsevolod Shevchishin.
We introduce the secondary Stiefel-Whitney class $\tilde w_2$ of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class $\tilde w_2$ vanishes. Using this, we give a detailed description of the combinatorial structure of the diffeotopy group of ruled symplectic 4-manifolds $X$, either minimal or blown-up, and its action on the homology and homotopy groups $H_2(X,\zz)$, $\pi_1(X)$, and $\pi_2(X)$.


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# Mathematics of superhorizon

1. Mar 20, 2014

### the_pulp

Hi, Im reading Tasi lectures on cosmology and in a lot of places states that certain magnitudes (example, comoving curvature R) work in someway (example "does not evolve") outside the horizon k<aH, where k is some fourier frequency of the magnitude we may be talking about. I understand the general idea (outside the horizon things cannot affect what happens inside the horizon -or something like that-. k is a frequency so 1/k > (aH)^-1 -that is wavelength > hubble radius should behave in a special way-) but:

1 i dont see how that intuitive condition transform in "for k<aH" in mathematical language. I get what it tries to say but I cant see the full symmetry between the words and the math
2 what "does not evolve" mean in this context (or whatever other condition is defined on any other magnitude)
3 why does it have to mean that? I insist, i get the idea that things otuaide my horizon cant affect me, my question is how is that idea translated to mathematical language.

I dont know if the question is clear, i hope you can help me! Thanks in advance for your usual useful help!

2. Mar 21, 2014

### bapowell

1) The wavenumber $k$ is comoving, meaning that it does not change as the universe evolves. This is helpful for placing a label on each Fourier mode that "stays with" the mode even though it is stretching. The physical wavenumber, $k/a$, does evolve with the expansion. Now, if $k/a$ is the physical wavenumber, then $\lambda_{\rm phys} = a/\lambda$ is the physical wavelength of the mode. Note that it increases $\propto a$. Meanwhile, the inverse Hubble parameter is the horizon size, $H^{-1} = d_H$. So, rewrite the condition as $$d_H < \lambda_{\rm phys}$$ and it should be clear that the condition k<aH is the same as saying that the physical wavelength of the mode is larger than the horizon.

2) The amplitude of the mode is constant.

3) It's not so much saying that things outside the horizon cannot affect things within. It's saying that this thing -- the Fourier mode -- does not evolve (at least not in a Hubble time). You've got a single fluctuations with a wavelength larger than the causal horizon, or equivalently, a thing with a frequency smaller than the age of the universe. Such a fluctuation is by all accounts frozen.

3. Mar 22, 2014

### the_pulp

Ok, thank you very much. I kept on reading and I have a lot of more doubts so I will keep on posting!

Thanks again.


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# 39  Solving Maxwell's Equations: Green’s Functions, Jefimenko

## The Story So Far...

We have established that the magnetic and electric fields can be expressed in terms of a vector potential $\stackrel{\to }{A}$ and a scalar potential $\phi$

$\stackrel{\to }{B}=\stackrel{\to }{\nabla }×\stackrel{\to }{A},$

$\stackrel{\to }{E}=-\stackrel{\to }{\nabla }\phi -\frac{\partial \stackrel{\to }{A}}{\partial t},$

and that Maxwell's equations are equivalent to linear differential equations for these potentials, having as source terms the charge and current distributions:

$\begin{array}{l}{\nabla }^{2}\phi -\frac{1}{{c}^{2}}\frac{{\partial }^{2}\phi }{\partial {t}^{2}}=-\rho /{\epsilon }_{0},\\ {\nabla }^{2}\stackrel{\to }{A}-\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{A}}{\partial {t}^{2}}=-{\mu }_{0}\stackrel{\to }{j}.\end{array}$

Here the potentials have been chosen to satisfy the Lorenz gauge condition $\stackrel{\to }{\nabla }\cdot \stackrel{\to }{A}+\left(1/{c}^{2}\right)\partial \phi /\partial t=0,$ (this can always be done).

## Green's Function for a Single-Frequency Point Source

Clearly these equations for $\phi ,\stackrel{\to }{A}$ have plane wave solutions in empty space.  But we need a full solution, including regions with nonzero charge and current densities.

The basic strategy is to Fourier transform with respect to time only, then solve the resulting spatial equation for a particular frequency.  This is of course an important case in the real world: an oscillating charge/current is an antenna, the signal is basically a single frequency with information transmitted by modulating the amplitude, the frequency, or digitizing.

Since the equations are linear, any solution can then be found by a superposition of single-frequency terms.

Writing the Fourier transform (with respect to time only) pair

$\begin{array}{l}\phi \left(\stackrel{\to }{r},t\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\phi \left(\stackrel{\to }{r},\omega \right){e}^{-i\omega t}d\omega \\ \phi \left(\stackrel{\to }{r},\omega \right)=\underset{-\infty }{\overset{\infty }{\int }}\phi \left(\stackrel{\to }{r},t\right){e}^{i\omega t}dt\\ \end{array}$

the spatial differential equation for $\phi \left(\stackrel{\to }{r},\omega \right)$ is the Fourier transform of

${\nabla }^{2}\phi -\frac{1}{{c}^{2}}\frac{{\partial }^{2}\phi }{\partial {t}^{2}}=-\rho /{\epsilon }_{0},$

that is,

$\left({\nabla }^{2}+{k}^{2}\right)\phi \left(\stackrel{\to }{r},\omega \right)=-\rho \left(\stackrel{\to }{r},\omega \right)/{\epsilon }_{0}.$

So, for a charge distribution oscillating at $\omega ,$ the actual time-dependent potential is the real part of $\phi \left(\stackrel{\to }{r},\omega \right){e}^{-i\omega t}.$  The time-independent function $\phi \left(\stackrel{\to }{r},\omega \right)$ will be complex, because there will almost always be phase differences between the oscillations at different points in space.

As we discovered in electrostatics, the way to solve equations of this type is to introduce Green’s functions.  Recall that the Green's function is essentially the inverse of a differential operator, meaning$—$purely formally$—$that we want to "solve" the above equation by writing

$\phi \left(\stackrel{\to }{r},\omega \right)=-\frac{1}{\left({\nabla }^{2}+{k}^{2}\right)}\rho \left(\stackrel{\to }{r},\omega \right)/{\epsilon }_{0}.$

Again, we have to figure out how to interpret this formal expression.

Since this is a linear theory, we only have to solve the equation for a point charge, $\rho \left(\stackrel{\to }{r}\right)=\delta \left(\stackrel{\to }{r}\right),$ then we can get the general result by integrating our solution over the actual charge distribution.

That's where the Green's function comes in$—$recall it's defined by (with appropriate boundary conditions)

$\left({\nabla }^{2}+{k}^{2}\right){G}_{k}\left(\stackrel{\to }{r},{\stackrel{\to }{r}}^{\prime }\right)=-4\pi \delta \left(\stackrel{\to }{r}-{\stackrel{\to }{r}}^{\prime }\right).$

For now, we'll take the case of charges and currents in otherwise empty space, so no finite boundaries.

With no boundaries, ${G}_{k}$ is a function of $\stackrel{\to }{R}=\stackrel{\to }{r}-{\stackrel{\to }{r}}^{\prime },$ and in fact only of $\left|\stackrel{\to }{R}\right|=R$, since there is no preferred direction.

Now, from electrostatics, Poisson’s equation ${\nabla }^{2}\psi =-4\pi \delta \left(\stackrel{\to }{r}-{\stackrel{\to }{r}}^{\prime }\right)$ has the solution

$\psi \left(\stackrel{\to }{r}-{\stackrel{\to }{r}}^{\prime }\right)=\frac{1}{\left|\stackrel{\to }{r}-{\stackrel{\to }{r}}^{\prime }\right|}=\frac{1}{R}.$

Our Green’s function ${G}_{k}\left(R\right)$ must look a lot like this in the limit $R\to 0$, because close enough to the origin the ${k}^{2}$ becomes negligible: think of the delta function on the right-hand side as the limit of a very sharp peak, the rapid change in slope ensures that, close in, the ${\nabla }^{2}$ is far more important than the ${k}^{2}.$

So we try a solution of the form ${G}_{k}\left(R\right)=u\left(R\right)/R,$ with $u\left(R\right)\to 1$ as $R\to 0.$  The differential equation becomes

${d}^{2}u\left(R\right)/d{R}^{2}=-{k}^{2}u\left(R\right),$

and the solutions are

${G}_{k}^{\left(±\right)}\left(R\right)=\frac{{e}^{±ikR}}{R}.$

Remember this solution is for a particular time Fourier component $\omega =ck$:  to go back from frequency to time, recall $\phi \left(\stackrel{\to }{r},t\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}\phi \left(\stackrel{\to }{r},\omega \right){e}^{-i\omega t}d\omega$, and since our solution has a single frequency component, to find the time-dependent single-frequency Green's function we just multiply by ${e}^{-i\omega t}$, so

${G}_{k}^{\left(±\right)}\left(R,t\right)=\frac{{e}^{±ikR-i\omega t}}{R}=\frac{{e}^{±i\left(\omega /c\right)R-i\omega t}}{R}.$

This describes spherical waves: the choice of sign ${G}_{k}{}^{\left(+\right)}\left(R,t\right)$ represents a wave going out from the origin. The negative sign represents a wave going into the origin.  When light is scattered by a molecule, the ingoing light can be represented in terms of ingoing spherical waves, so the minus sign is relevant initially, the subsequent outgoing wave of course has the positive sign.

## Green's Function for a Point Source in Both Space and Time

To get the full picture of potentials generated by time dependent charge and current distributions, we need the Green’s function with a delta function source in both space and time:

$\left({\nabla }^{2}-\frac{1}{{c}^{2}}\frac{{\partial }^{2}}{\partial {t}^{2}}\right)G\left(\stackrel{\to }{r},t;{\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)=-4\pi \delta \left(\stackrel{\to }{r}-{\stackrel{\to }{r}}^{\prime }\right)\delta \left(t-{t}^{\prime }\right).$

(This corresponds to charge appearing for a moment at the origin$—$not a physical situation by itself, but the potential from moving charges can be constructed from an integral over such functions.)

Again, we'll take the case of no finite boundaries, so

$G\left(\stackrel{\to }{r},t;{\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)=G\left(R,\tau \right),\text{ }\tau =t-{t}^{\prime },$

The equation is solved by noting that the delta function pulse in time has a Fourier transform in which all frequencies appear equally:

$\delta \left(\tau \right)=\underset{-\infty }{\overset{\infty }{\int }}\frac{d\omega }{2\pi }\text{\hspace{0.17em}}{e}^{-i\omega \tau },$

and therefore (with no finite boundaries) we can just integrate over the contributions from each frequency, using the single-frequency Green's function we found above,

${G}^{\left(±\right)}\left(R,\tau \right)=\underset{-\infty }{\overset{\infty }{\int }}\frac{d\omega }{2\pi }{G}_{k=\omega /c}{}^{\left(±\right)}\left(R,\tau \right)=\underset{-\infty }{\overset{\infty }{\int }}\frac{d\omega }{2\pi }\frac{{e}^{±ikR-i\omega \tau }}{R},\text{ }k=\omega /c.$

The integral over $\omega$ is the delta function, that is,

${G}^{\left(±\right)}\left(R,\tau \right)=\frac{1}{R}\delta \left(\tau \mp \frac{R}{c}\right).$

The choice ${G}^{\left(+\right)}\left(R,\tau \right)=\frac{1}{R}\delta \left(\tau -\frac{R}{c}\right)$ is called the retarded Green’s function: the scalar potential equation ${\nabla }^{2}\phi -\frac{1}{{c}^{2}}\frac{{\partial }^{2}\phi }{\partial {t}^{2}}=-\rho /{\epsilon }_{0}$ is formally solved with this Green’s function to give

$\phi \left(\stackrel{\to }{r},t\right)=\frac{1}{{\epsilon }_{0}}\int d{\stackrel{\to }{r}}^{\prime }d{t}^{\prime }{G}^{\left(+\right)}\left(\stackrel{\to }{r},t;{\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)\rho \left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)$

and, with the simple form of the Green’s function we’ve found, this can be written

$\phi \left(\stackrel{\to }{r},t\right)=\frac{1}{4\pi {\epsilon }_{0}}\int d{\stackrel{\to }{r}}^{\prime }\frac{1}{\left|\stackrel{\to }{r}-{\stackrel{\to }{r}}^{\prime }\right|}\rho {\left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)}_{\text{ret}}$

where the “ret” for retarded means at the time that a light signal emitted would reach $\stackrel{\to }{r}$

at time $t.$

So the solutions to the wave equations

$\begin{array}{l}{\nabla }^{2}\phi -\frac{1}{{c}^{2}}\frac{{\partial }^{2}\phi }{\partial {t}^{2}}=-\rho /{\epsilon }_{0},\\ {\nabla }^{2}\stackrel{\to }{A}-\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{A}}{\partial {t}^{2}}=-{\mu }_{0}\stackrel{\to }{j}.\end{array}$

are

$\begin{array}{l}\phi \left(\stackrel{\to }{r},t\right)=\frac{1}{4\pi {\epsilon }_{0}}\int {d}^{3}{r}^{\prime }\frac{{\left[\rho \left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)\right]}_{\text{ret}}}{R},\\ \stackrel{\to }{A}\left(\stackrel{\to }{r},t\right)=\frac{{\mu }_{0}}{4\pi }\int {d}^{3}{r}^{\prime }\frac{{\left[\stackrel{\to }{J}\left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)\right]}_{\text{ret}}}{R}.\end{array}$

Recall that we are working in the Lorenz gauge,

$\stackrel{\to }{\nabla }\cdot \stackrel{\to }{A}+\frac{1}{{c}^{2}}\frac{\partial \phi }{\partial t}=0.$

So, given the time and space charge and current distribution, we can find the scalar and vector potentials.

Recall that in the static case, we found similar expressions for the electric field as an integral over the charge distribution

$\stackrel{\to }{E}\left(\stackrel{\to }{r}\right)=\frac{1}{4\pi {\epsilon }_{0}}\int {d}^{3}{r}^{\prime }\frac{\rho \left({\stackrel{\to }{r}}^{\prime }\right)\stackrel{^}{\stackrel{\to }{R}}}{{R}^{2}}$

and the Biot-Savart law for the magnetic field in terms of the current distribution.  You might think that for the time-dependent case, these would generalize as the potentials did, by just putting in the retarded times.  But they don't: essentially, because the fields come from differentiating the potentials, and differentiating a retarded time corresponding to a moving source has spatial contributions.  It can get very messy.

## *Deriving General Equations for the Fields from any Charge and Current Distribution

The general equations for the fields in terms of sources are cumbersome, and not often used, real problems are usually simpler.  In Jackson and Griffiths, the general equations are called Jefimenko's equations, from Jefimenko's 1966 text, although they were in fact first written down in 1912 by Schott, a student of J.J. Thomson.

The real point of going over these equations is to show explicitly that Maxwell's equations can be solved to give a complete description of the electric and magnetic fields generated by any time-dependent flow of electric charge.

Also, they make a conceptual/linguistic point: we write down Maxwell’s equations, then we usually say “a changing electric field generates a magnetic field”, and vice versa. But strictly speaking, this is not what happens, it’s an invalid “cause and effect” sequence.  If there is a changing electric field at some space time point, it’s because at a previous light-separated point (meaning retarded point, in the sense used above) some charge was moving, so at that point there was a current, and that’s where the magnetic field is coming from. Perhaps we should say “a changing electric field implies a magnetic field” but in fact we’ll probably continue with the sloppy language, it works fine$—$as long as we get the equations right.

Here's how Jackson derives the general equations. He focusses on deriving equations with the wave-like form on the left-hand side, for the components of the electric and magnetic fields.  You can get these equations directly from Maxwell's equations, using the potentials in intermediate steps, but ending with source terms (meaning the right-hand side of the equation) that are just charge density, current density and their derivatives.

To construct the equations for ${\stackrel{\to }{\nabla }}^{2}\stackrel{\to }{E}-\left(1/{c}^{2}\right)\stackrel{¨}{\stackrel{\to }{E}}$ and similarly for $\stackrel{\to }{B}$, we use

$\stackrel{\to }{E}=-\stackrel{\to }{\nabla }\phi -\stackrel{˙}{\stackrel{\to }{A}},\text{ }\stackrel{\to }{B}=\stackrel{\to }{\nabla }×\stackrel{\to }{A}$

to find

${\nabla }^{2}\stackrel{\to }{E}-\left(1/{c}^{2}\right)\stackrel{¨}{\stackrel{\to }{E}}=-\stackrel{\to }{\nabla }\left({\nabla }^{2}\phi \right)-{\nabla }^{2}\stackrel{˙}{\stackrel{\to }{A}}+\left(1/{c}^{2}\right)\left(\stackrel{\to }{\nabla }\stackrel{¨}{\phi }+\stackrel{⃛}{\stackrel{\to }{A}}\right),$

but we're in the Lorenz gauge, so ${\nabla }^{2}\stackrel{\to }{A}-\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{A}}{\partial {t}^{2}}=-{\mu }_{0}\stackrel{\to }{j},$ the $\stackrel{\to }{A}$ terms on the right hand side add to ${\mu }_{0}\stackrel{˙}{\stackrel{\to }{j}}$, then from ${\nabla }^{2}\phi -\frac{1}{{c}^{2}}\frac{{\partial }^{2}\phi }{\partial {t}^{2}}=-\rho /{\epsilon }_{0}$ the $\phi$ terms add to $\stackrel{\to }{\nabla }\rho /{\epsilon }_{0}.$

Therefore,

${\nabla }^{2}\stackrel{\to }{E}-\left(1/{c}^{2}\right)\stackrel{¨}{\stackrel{\to }{E}}=-\frac{1}{{\epsilon }_{0}}\left(-\stackrel{\to }{\nabla }\rho -\frac{1}{{c}^{2}}\frac{\partial \stackrel{\to }{j}}{\partial t}\right).$

The corresponding equation for the magnetic field,

${\nabla }^{2}\stackrel{\to }{B}-\frac{1}{{c}^{2}}\frac{{\partial }^{2}\stackrel{\to }{B}}{\partial {t}^{2}}=-{\mu }_{0}\stackrel{\to }{\nabla }×\stackrel{\to }{j},$

follows immediately on applying $\stackrel{\to }{\nabla }×$ to the equation for $\stackrel{\to }{A}.$

We can solve these wave-type equations using the Green's function, just as we did for the potentials, to get

$\begin{array}{l}\stackrel{\to }{E}\left(\stackrel{\to }{r},t\right)=\frac{1}{4\pi {\epsilon }_{0}}\int {d}^{3}{r}^{\prime }\frac{1}{R}{\left[-{\stackrel{\to }{\nabla }}^{\prime }\rho -\frac{1}{{c}^{2}}\frac{\partial \stackrel{\to }{j}}{\partial {t}^{\prime }}\right]}_{\text{ret}},\\ B\left(\stackrel{\to }{r},t\right)=\frac{{\mu }_{0}}{4\pi }{\int {d}^{3}{r}^{\prime }\frac{1}{R}\left[{\stackrel{\to }{\nabla }}^{\prime }×\stackrel{\to }{j}\right]}_{\text{ret}}.\end{array}$

But this is where things get complicated!  Take that first term, ${\stackrel{\to }{\nabla }}^{\prime }\rho .$ We're differentiating the charge density $\rho \left({\stackrel{\to }{r}}^{\prime }\right)$ with respect to ${\stackrel{\to }{r}}^{\prime }$, but it's inside the retarded bracket: so, we're finding the difference in $\rho$ between ${\stackrel{\to }{r}}^{\prime }+d{\stackrel{\to }{r}}^{\prime }$ and ${\stackrel{\to }{r}}^{\prime }$, but if these two points are at different distances from the point $\stackrel{\to }{r}$ at which we're finding the field (left-hand side of above equation) then the "ret" requirement will mean that in finding this derivative we're making these measurements at slightly different times, so, if $\rho$ is also varying in time, that gives a contribution.  To separate out this effect, imagine $\rho \left({\stackrel{\to }{r}}^{\prime }\right)$ is uniform in space over some region, but increasing in time (for example, a charged gas that's being compressed).  For this gas,  $\stackrel{\to }{\nabla }\rho$ is constant in the region, so $\rho \left({\stackrel{\to }{r}}^{\prime }+d{\stackrel{\to }{r}}^{\prime }\right)-\rho \left({\stackrel{\to }{r}}^{\prime }\right)=0$ if they're measured simultaneously, but inside the ${\left[\right]}_{\text{ret}}$ they're not: if ${\stackrel{\to }{r}}^{\prime }+d{\stackrel{\to }{r}}^{\prime }$ is a distance $dR$ radially further out than ${\stackrel{\to }{r}}^{\prime },$ we'll be sensing it a time $dt=dR/c$ earlier, so will find a contribution $-\stackrel{˙}{\rho }/c$ in the radial direction to the gradient. That is,

${\left[{\stackrel{\to }{\nabla }}^{\prime }\rho \right]}_{\text{ret}}={\stackrel{\to }{\nabla }}^{\prime }{\left[\rho \right]}_{\text{ret}}-\stackrel{^}{\stackrel{\to }{R}}\stackrel{˙}{\rho }/c.$

Another way of seeing this, following Jackson, is to write ${\left[f\left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)\right]}_{\text{ret}}=f\left({\stackrel{\to }{r}}^{\prime },t-R/c\right)$ then realize that any differentiation with respect to ${\stackrel{\to }{r}}^{\prime }$ must include a term from differentiating $R=\left|\stackrel{\to }{r}-{\stackrel{\to }{r}}^{\prime }\right|.$ Thus.

$\begin{array}{c}{\left[{\stackrel{\to }{\nabla }}^{\prime }×\stackrel{\to }{j}\right]}_{\text{ret}}={\stackrel{\to }{\nabla }}^{\prime }×{\left[\stackrel{\to }{j}\right]}_{\text{ret}}+{\left[\frac{\partial \stackrel{\to }{j}}{\partial {t}^{\prime }}\right]}_{\text{ret}}×{\stackrel{\to }{\nabla }}^{\prime }\left(t-R/c\right)\\ ={\stackrel{\to }{\nabla }}^{\prime }×{\left[\stackrel{\to }{j}\right]}_{\text{ret}}+\frac{1}{c}{\left[\frac{\partial \stackrel{\to }{j}}{\partial {t}^{\prime }}\right]}_{\text{ret}}×\stackrel{^}{\stackrel{\to }{R}}.\end{array}$

Differentiation with respect to ${t}^{\prime }$ is not a problem, since it involves sequential sampling at the same point ${\stackrel{\to }{r}}^{\prime }.$

Putting these expressions into the formulas for the fields gives

$\begin{array}{l}\stackrel{\to }{E}\left(\stackrel{\to }{r},t\right)=\frac{1}{4\pi {\epsilon }_{0}}\int {d}^{3}{r}^{\prime }\left\{\frac{\stackrel{^}{\stackrel{\to }{R}}}{{R}^{2}}{\left[\rho \left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)\right]}_{\text{ret}}+\frac{\stackrel{^}{\stackrel{\to }{R}}}{cR}{\left[\frac{\partial \rho \left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)}{\partial {t}^{\prime }}\right]}_{\text{ret}}-\frac{1}{{c}^{2}R}{\left[\frac{\partial \stackrel{\to }{j}\left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)}{\partial {t}^{\prime }}\right]}_{\text{ret}}\right\}\\ B\left(\stackrel{\to }{r},t\right)=\frac{{\mu }_{0}}{4\pi }\int {d}^{3}{r}^{\prime }\left\{{\left[\stackrel{\to }{j}\left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)\right]}_{\text{ret}}×\frac{\stackrel{^}{\stackrel{\to }{R}}}{{R}^{2}}+{\left[\frac{\partial \stackrel{\to }{j}\left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)}{\partial {t}^{\prime }}\right]}_{\text{ret}}×\frac{\stackrel{^}{\stackrel{\to }{R}}}{cR}\right\}.\end{array}$

To check that these equations make sense, look first at the static limit: they yield the Coulomb electrostatic law, and the Biot-Savart magnetic law.  Now note that there are terms decreasing with distance as $1/R$ and others as $1/{R}^{2}$, the latter dominate close in.  These "near field" terms are in fact the same as the static terms, except for the time delay correction.

Assuming the charge and current distribution is confined to a finite volume, far away these $1/{R}^{2}$ terms become negligible.  Notice that the remaining $1/R$ terms only come from changing charge and current densities.  They are electromagnetic radiation. But they don’t have quite the right form: as we'll see in more detail later (and as$—$hopefully$—$you already know) in an electromagnetic wave, the electric field, the magnetic field and the direction of propagation are all perpendicular to each other.  The above expression for the magnetic field is perpendicular to the direction of propagation, but the second term in the electric field is along that direction.  It turns out that this term is in fact canceled by a contribution form the third term, but this is a tricky calculation (see Kirk Donaldson, or the Fourier treatment in Panofsky and Phillips).

(Note:  Actually, the field components parallel to the direction of propagation, although they are decreasing by a factor $1/R$  relative to the perpendicular components, play one important role: as we'll discuss later, a system can radiate angular momentum (atoms making a transition usually do!) and thinking of that in terms of the Poynting vector it is immediately evident we need that parallel component.  The $1/R$ is compensated in angular momentum by the $\stackrel{\to }{R}×\stackrel{\to }{p}$ term.)

The bottom line is that the electric far field is

${\stackrel{\to }{E}}_{\text{radiation}}\left(\stackrel{\to }{r},t\right)=\frac{1}{4\pi {\epsilon }_{0}}\int {d}^{3}{r}^{\prime }\frac{1}{{c}^{2}R}\left\{{\left[\frac{\partial \stackrel{\to }{j}\left({\stackrel{\to }{r}}^{\prime },{t}^{\prime }\right)}{\partial {t}^{\prime }}\right]}_{\text{ret}}×\stackrel{^}{\stackrel{\to }{R}}\right\}×\stackrel{^}{\stackrel{\to }{R}},$

and since ${\epsilon }_{0}{c}^{2}=1/{\mu }_{0},$ this matches the magnetic field vector.

Griffiths gets the same equations by a slightly different route: he begins by writing the fields in terms of the potentials, then feeds in the expressions for the potentials in terms of the Green's functions, then changes the source terms back to fields.

Remarkably, Feynman gives the results in Volume I of his Lectures, but just to say how wonderful nature is$—$he doesn't derive it.

Kirk McDonald gives a full analysis of why, despite appearances, the far field has the standard wave form, with no component of the electric field in the direction of propagation.  This was done in terms of Fourier transforms by Panofsky and Phillips.


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Diskretnyi Analiz i Issledovanie Operatsii
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 Diskretn. Anal. Issled. Oper., Ser. 2, 2001, Volume 8, Issue 1, Pages 70–87 (Mi da239)

The problem of locating rectangular plants with minimal cost for the connecting network

A. V. Panyukov

South Ural State University

Abstract: We present a method for the hierarchical decomposition of the problem of locating rectangular plants with minimal cost for their connecting network into an optimal ordering problem (the upper level) and two problems of the construction of an optimal flow (the lower level). We obtain the following results: (1) we find necessary and sufficient conditions for the local extremum and give an algorithm for constructing locally optimal solutions; (2) for large-scale problems, we present a solution algorithm based on random search, heuristics, and the decomposition method proposed; (3) for the search for the global extremum, we present an algorithm that is based on the branch and bound method.

Full text: PDF file (1828 kB)

Bibliographic databases:
UDC: 519.854.2
Revised: 22.11.2000

Citation: A. V. Panyukov, “The problem of locating rectangular plants with minimal cost for the connecting network”, Diskretn. Anal. Issled. Oper., Ser. 2, 8:1 (2001), 70–87

Citation in format AMSBIB
\Bibitem{Pan01} \by A.~V.~Panyukov \paper The problem of locating rectangular plants with minimal cost for the connecting network \jour Diskretn. Anal. Issled. Oper., Ser.~2 \yr 2001 \vol 8 \issue 1 \pages 70--87 \mathnet{http://mi.mathnet.ru/da239} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1858423} \zmath{https://zbmath.org/?q=an:1075.90509}

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This publication is cited in the following articles:
1. G. G. Zabudsky, N. S. Veremchuk, “An algorithm for approximate solution to the Weber problem on a line with forbidden gaps”, J. Appl. Industr. Math., 10:1 (2016), 136–144


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QUESTION

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Complete the following reaction:(balance and states)
If no reaction occurs write "NR"

LiOH (aq) + HF (aq) $\longrightarrow$


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You probably want to set sigma to be a fraction of the radius, rather than a fixed value. Sigma controls the actual shape of the filter, while radius just controls how far out the filter gets cut off (as Gaussians truly have infinite radius, so we have to cut them off somewhere to use them in practice). With sigma set to 0.5, you're making a filter that's ...

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The units of the NDF are tricky. For whatever it's worth, Heitz's convention of defining it relative to a 1 m² reference geometric surface is unusual, and although I can see why he would want to define it that way for conceptual simplicity, it does not really match how NDFs are used in practice. I definitely had a head-tilt moment when I first read that ...

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In truth, there is no mathematical maximum value for $\alpha$. As you noted, microfacet slope is unbounded, so in principle you could have arbitrarily large slope values and hence arbitrarily large $\alpha$. There's nothing wrong with that—the mathematics of the microfacet model keeps working fine. As a practical matter, beyond a certain point you don't ...

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The book distinguishes a spectral power distribution from a spectral response curve because they are not the same, they are adjoint. The thing about adjoints that makes them a little tricky to wrap your head around is that they typically have the same representation in software, but keeping them distinct is extremely important in computer graphics. Another ...

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Shadow rays that hit a glass surface or similar are stopped because there's no direct path from the light source to the shading point—there's a piece of glass in the way. for perfectly specular transmission/reflection, you can continue tracing the shadow ray because there's only one possible outgoing vector for a given incoming vector That won't work, ...

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In your example of a BVH cell partitioned along the x axis, if the ray is going left to right (increasing x), then the cell with the lower x comes first; but if it's going right to left (decreasing x), then the cell with the higher x comes first. So, you can look at the sign of the x component of the direction vector to determine which cell to test first.

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The summation doesn't include the BxDF that was picked for sampling. Look again at this line: if (bxdfs[i] != bxdf && bxdfs[i]->MatchesFlags(type)) Here bxdf is the one that was sampled earlier, so when it iterates to that one it skips it.

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# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Minimal actions of homeomorphism groups

### Tom 198 / 2008

Fundamenta Mathematicae 198 (2008), 191-215 MSC: Primary 37B05; Secondary 54H15, 22F50. DOI: 10.4064/fm198-3-1

#### Streszczenie

Let $X$ be a closed manifold of dimension $2$ or higher or the Hilbert cube. Following Uspenskij one can consider the action of ${\rm Homeo}(X)$ equipped with the compact-open topology on ${\mit\Phi}\subset 2^{2^{X}}$, the space of maximal chains in $2^{X}$, equipped with the Vietoris topology. We show that if one restricts the action to $M\subset {\mit\Phi}$, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of ${\rm Homeo}(X)$ on $U_{{{\rm Homeo}(X)}}$, the universal minimal space of ${\rm Homeo}(X)$, is not transitive (improving a result of Uspenskij). Additionally for $X$ as above with ${\rm dim}(X)\geq 3$ we characterize all the minimal subspaces of $V(M)$, the space of closed subsets of $M$, and show that $M$ is the only minimal subspace of ${\mit\Phi}$. For ${\rm dim}(X)\geq 3$, we also show that $(M,{\rm Homeo}(X))$ is strongly proximal.

#### Autorzy

• Yonatan GutmanInstitute of Mathematics
The Hebrew University
Jerusalem, Israel
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## Current carrying wires

Each of the current carrying wires in the figure is long andsraight adn carries the current I either ino or out of the page, asshown.

What is the direction of the net magnetic field produced bythese three wires at the center of the triangle?

Answer should be in the form:  θ=      relative to the positivex-axis


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### Author Topic: Embedded LaTeX formatting available! [instructions]  (Read 1490 times)

#### Daniel

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##### Embedded LaTeX formatting available! [instructions]
« on: December 19, 2013, 10:30:18 PM »
Dedicated enough to write code for your own formatting? Then LaTeX is for you. And more than that, now you can use it on the forum!

This will enable you to insert mathematical formulas and other formatting that isn't possible using just the normal BBCODE of the forum. I'm not sure I'd recommend it for posting a whole article here, but I hope it will be useful!

$\lim_{n \to \infty}\sum_{k=1}^n \frac{1}{k^2}= \frac{\pi^2}{6}$
 $\lim_{n \to \infty}\sum_{k=1}^n \frac{1}{k^2}= \frac{\pi^2}{6}$(And note the "LaTeX" button above the message box.)


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Find the sum of the given series, or show that the series diverges (possibly to infinity or negative infinity). Exercises

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Find the sum of the given series, or show that the series diverges (possibly to infinity or negative infinity). Exercises $11-14$ are telescoping series and should be done by partial fractions as suggested in Example 3 in this section. $\sum_{n=1}^{\infty} \frac{1}{(3 n-2)(3 n+1)}=\frac{1}{1 \times 4}+\frac{1}{4 \times 7}+\frac{1}{7 \times 10}+\cdots$

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text:
## Electronic Journal of Probability

### Parabolic Harnack inequality and local limit theorem for percolation clusters

#### Abstract

We consider the random walk on supercritical percolation clusters in $\mathbb{Z}^d$. Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function.

#### Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 1, 1-26.

Dates
Accepted: 7 January 2009
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819462

Digital Object Identifier
doi:10.1214/EJP.v14-587

Mathematical Reviews number (MathSciNet)
MR2471657

Zentralblatt MATH identifier
1192.60107

Rights

#### Citation

Hambly, Ben; Barlow, Martin. Parabolic Harnack inequality and local limit theorem for percolation clusters. Electron. J. Probab. 14 (2009), paper no. 1, 1--26. doi:10.1214/EJP.v14-587. https://projecteuclid.org/euclid.ejp/1464819462

#### References

• P. Antal, A. Pisztora. On the chemical distance for supercritical bernoulli percolation. Ann. Probab. 24 (1996), 1036–1048.
• M.T. Barlow. Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004), 3024–3084.
• M.T. Barlow, R.F. Bass, Z.-Q. Chen, M. Kassmann. Non-local Dirichlet Forms and Symmetric Jump Processes. To appear Trans. Amer. Math. Soc.
• M.T. Barlow, R.F. Bass, T. Kumagai. Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps. To appear Math. Zeitschrift.
• R.F. Bass. On Aronsen's upper bounds for heat kernels. Bull. London Math. Soc. 34 (2002), 415–419.
• I. Benjamini, R. Lyons, O. Schramm. Percolation perturbations in potential theory and random walks. In: Random walks and discrete potential theory (Cortona, 1997), 56–84, Sympos. Math., XXXIX, Cambridge Univ. Press, Cambridge, 1999.
• N. Berger, M. Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Rel. Fields 137 (2007), 83–120.
• N. Berger, M. Biskup, C.E. Hoffman, G. Kozma. Anomalous heat-kernel decay for random walk amoung bounded random conductances. Ann. Inst. Henri Poincar'e. 44 (2008), 374-392.
• M. Biskup, T.M. Prescott. Functional CLT for random walk among bounded random conductances. Electron. J. Probab. 12 (2007), 1323–1348
• R.M Blumenthal and R.K. Getoor. Markov Processes and Potential Theory. Academic Press, Reading, MA, 1968.
• O. Couronn'e, R.J. Messikh. Surface order large deviations for 2D FK-percolation and Potts models. Stoch. Proc. Appl. 113 (2004), 81–99.
• P.G. de Gennes. La percolation: un concept unificateur. La Recherche 7 (1976), 919–927.
• T. Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Math. Iberoamericana 15 (1999), 181–232.
• A. De Masi, P.A. Ferrari, S. Goldstein, W.D. Wick. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 (1989), 787–855.
• J.-D. Deuschel, A. Pisztora. Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 (1996), 467–482.
• E.B. Fabes, D.W. Stroock. A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash. Arch. Mech. Rat. Anal. 96 (1986), 327–338.
• W. Feller. An introduction to probability theory and its applications. Vol. II}. 2nd ed. Wiley, New York-London-Sydney, 1971.
• G.R. Grimmett. Percolation. (2nd edition). Springer, 1999.
• A. Maritan. About diffusion processes in disordered systems. J. Phys. A: Math. Gen. 21 (1988) 859–863.
• P. Mathieu, E. Remy. Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 (2004), 100–128.
• P. Mathieu, A. Piatnitski. Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), 2287–2307.
• P. Mathieu. Quenched invariance principles for random walks with random conductances. J. of Stat. Phys., 130, (2008) 1025-1046.
• J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958), 931–954.
• L. Saloff-Coste. Aspects of Sobolev-type inequalities. Cambridge Univ. Press 2002.
• V. Sidoravicius and A.-S. Sznitman. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Rel. Fields 129 (2004), 219–244.


text:
• # question_answer One card is drawn from a well-shuffled deck of 52 cards. The probability of drawing an ace is ___. A)  $\frac{1}{12}$                     B) $\frac{1}{13}$                      C)  $\frac{1}{50}$                                 D)  $\frac{3}{10}$

Total number of cards = 52 Total number of aces present in a deck of cards =4 $\therefore$  Probability of drawing an ace $=\frac{4}{52}=\frac{1}{13}$


text:
## Pythagorean triples (x^2 – 1, y^2 – 1, z^2 – 1)

To find a Pythagorean triangle with sides,

$x^2 - 1, \; y^2 - 1, \; z^2 - 1$

where   $x, y, z$   are integers.

Here are two examples,


text:
## Monday, 3 October 2011

### The (in)consistency of PA and consensus in mathematics

(Cross-posted at NewAPPS)

Last week, the foundations of mathematics community was shaken by yet another claim of the inconsistency of Peano Arithmetic (PA). This time, it was put forward by Edward Nelson, professor of mathematics in Princeton, who claimed to have found a proof of the inconsistency of PA. A few months ago, quite some stir had been caused when Fields medalist V. Voevodsky seemed to be saying that the consistency of PA was an open question; but Nelson’s claim was much more radical; he claimed to have proved that PA was outright inconsistent! (Here is a great post by Jeff Ketland with a crash-course on PA and a discussion of ways in which it might be inconsistent.)

Nelson announced his results on the FOM mailing list on September 26th 2011, providing links to two formulations of the proof: one in book form and one in short-summary form. Very quickly, a few math-oriented blogs had posts up on the topic; we all wanted to understand the outlines of Nelson’s purported proof, and most of us bet all our money on the possibility that there must be a mistake in the proof. External evidence strongly suggests that PA is consistent, in particular in that so many robust mathematical results would have to be revised if PA were inconsistent (not to mention several proofs of the consistency of arithmetic in alternative systems, such as Gentzen’s proofs -- see here).

Indeed, it did not take long for someone to find an apparent loophole in Nelson’s purported proof, and not just someone: math prodigy and Fields medalist Terence Tao (UCLA), who is considered by many as the most brilliant mathematician currently in activity. The venue in which Tao first formulated his reservations was somewhat original: on the G+ thread opened by John Baez on the topic. (So those who dismiss social networks as a pure waste of time have at least one occurrence of actual top-notch science being done in a social network to worry about!) At the same time, Daniel Tausk, a professor of mathematics at the University of São Paulo, had identified the same mistake in Nelson’s argument, and alerted Nelson of the problem in private communication. Judging from his replies (which can be read in the FOM archive as well as in The n-Category Café post), Nelson did not immediately understand the objection, but within a few days consensus had emerged that the mistake identified by Tao and Tausk was irreparable. Then, on October 1st, Nelson graciously acknowledged the correctness of the objections and withdrew the claim of having a proof of the inconsistency of PA. As has been remarked by many commentators, it is certainly a sign of mathematical greatness to acknowledge one’s own mistakes so promptly!

While it must have taken true mathematical insight to see what was wrong with the purported proof, once the mistake was spotted, it was actually not overly difficult to understand – if not in all its technical details, which presuppose a good understanding of Chaitin’s theorem, at least in its basic idea: it was based on the unwarranted assumption that the hierarchy of sub-theories built for the proof is going to map neatly into the hierarchy of the Kolmogorov complexity of each of these sub-theories. In the game of mathematics, tacitly relying on assumptions is not a good move, even less so when it turns out that the assumption is false.

From the point of view of the social practice of mathematics, this episode raises a number of interesting questions, which can be framed from the point of view of Jody Azzouni’s insightful analysis of how and why mathematics is unique as a social practice. Azzouni argues that mathematics is one of the most homogeneous groups of human practices around, and in particular one which seems particularly favorable for the emergence of consensus. Indeed, mathematicians virtually always agree on whether a purported proof is or is not a valid mathematical proof; it may take a while for enough members of the community to understand its details (as with Deolalikar’s purported proof of $P \not= NP$ last year, but this was mostly because the latter was rather poorly formulated), but past this initial period, consensus virtually always emerges. (To be clear, there can be disagreement on whether a proof is elegant, interesting etc., but not as to its correctness and incorrectness.) This remains a very puzzling feature of mathematical practice, which at least to some people suggests that Platonism is the only plausible philosophical account of mathematical truth.

The swiftness with which consensus emerged as to what was wrong with Nelson’s purported proof is yet another example of the strong tendency towards consensus in mathematical practice – and arguably, it cannot be fully explained as a merely socially imposed kind of consensus, due to homogeneous ‘indoctrination’ by means of mathematical education. That it only took a few days for consensus to emerge naturally also depends crucially on the technologies of information dissemination currently at our disposal – mailing lists, home-pages, blogs, social networks; but more generally, the episode has been a dramatic but classical occurrence of a well known pattern in mathematics as a social practice.

The comparison with another dramatic claim, put forward for the first time more than 80 years ago, may prove to be informative. Indeed, the reception of Gödel’s incompleteness results (which has been extensively studied by Dawson and Mancosu, among others) also illustrates the emergence of a consensus, but this time it favored the highly surprising claim being put forward; in last week’s episode, by contrast, the highly surprising claim has been refuted.

In September 1930, Gödel made the first public announcement of his incompleteness result. In first instance, he had proved only the first incompleteness theorem, and as is well known, from the first theorem the second incompleteness theorem was proved independently by Gödel himself and by John von Neumann. But von Neumann was one of the few who immediately understood the result (Bernays and Carnap seem to have struggled); in fact, he may have been the first to see the far-reaching consequences it had. In particular, while Gödel was still rather cautions regarding the impact of his results for Hilbert’s program, von Neumann immediately drew the right conclusions. However, quite a few mathematicians were so taken aback by Gödel’s results that they were convinced that there must have been a mistake somewhere in the proof; just as in last week’s episode, many of them set out to find out the ‘loophole’ in Gödel’s proof, but on that occasion to no avail. As narrated in this wonderful paper by P. Mancosu, the news of Gödel’s results quickly spread, but in the absence of the WWW or similar platforms, many people did not have immediate access to off-prints of the paper where the results were presented. For example, before seeing the actual results in detail, mathematician Leon Chwistek suspected that Gödel’s proof would be based on misunderstandings, but once he read the actual paper, he stood corrected:

In my last letter I have raised some doubts concerning Dr Gödel’ s work that have completely disappeared after a more attentive study of the problem. I had at first thought that there was a tacit introduction of non-predicative functions which makes the use of a rigorous symbolical procedure impossible and I even feared the possibility of an antinomy. Now I see that this is out of the question. The method is truly wonderful and it fits pure type theory thoroughly. (Letter to Kaufmann, in Mancosu 1999, 44)

Both the consensus that eventually emerged concerning the correctness of Gödel’s surprising results, and the consensus swiftly established as on what was wrong with Nelson’s purported proof last week are examples of the amazing and quite unique tendency to converge towards consensus in mathematical practice. In many senses, the most pressing task for the philosophy of mathematics, practice-based or otherwise, is to formulate a satisfactory account of this almost eerie fact about mathematics.

1. It would be interesting to also contrast these two episodes with the recent P != NP "proof" that got some press...

2. @Seamus, it would indeed, but I'm afraid it's not going to be done by me... I haven't followed the P != NP controversy from very close last year, and the topic is too far removed from the areas I feel confident I'll be saying something sensible on.

3. I suspect that history (ie people) tend to remember episodes where the surprising suggestion turns out to be true better than when the surprising claim turns out to be false.

4. It's amazing how someone can come along and in a matter of minutes falsify a proof which must have taken Nelson hundreds of man hours to construct.

And should the original claim have been less bold and more carefully checked before publishing to avoid humiliation and publicity?

5. Great post! As a mathematician, I find the relative ease of consensus wonderful and mysterious.

Maybe I'm reading too much into your words, but it seems like you would argue that consensus in mathematics is both easier and swifter than consensus in other sciences like, say, physics. Yet the eventual arrival at consensus is an important feature of physics as well. (For the sake of argument, let's ignore those speculative parts of physics that haven't yet made contact with experiment, and focus on a subfield where theory and experiment see constant interplay, such as condensed matter.)

It is understandable that consensus in physics would take longer than mathematics: it is an experimental science, and experiments take time. But are there any qualitative differences in consensus in physics, once it is reached, from consensus in mathematics?

6. "who is considered by many as the most brilliant mathematician currently in activity."

No serious professional mathematician would accept such a description of any mathematician.

7. ""who is considered by many as the most brilliant mathematician currently in activity."

No serious professional mathematician would accept such a description of any mathematician. "

Funny. I thought Bernoulli believed such a thing about Newton.

8. With regard to you final paragraph on the "most pressing task for the philosophy of mathematics", I can sympathise with that thought, and yet a huge amount of effort has already been devoted to it. Alongside the justification of formalisms by more orthodox philosophy of mathematics, on the practice-based side we have, say, Ken Manders on the control afforded by Euclidean diagrammatic reasoning in 'The Euclidean Diagram' (1995) and Colin McLarty in Voir Dire in the case of mathematical progress describing the advent of shared means of rigorous expression in the twentieth century.

To my mind, an equally or more pressing task is to understand the kinds of slowly emerging consensus that occur as to what directions a field of mathematics should take. With the very light constraints afforded by logical correctness or empirical adequacy, there's some explaining to do.

9. @John,
Indeed, it seems to me that there is a stronger tendency towards consensus in math than in, say, physics, but that's ultimately an empirical claim of course (the 'sociology' of each of these fields). And it's precisely the fact that math is not an experimental science that makes consensus even more puzzling: there isn't anything like, you know, the physical world, to keep mathematical practice in check!

10. @David,
I guess I should have said "the most pressing task for the philosophy of mathematics ... REMAINS to formulate a satisfactory account...". You are absolutely right that there has been a lot of work on the topic, but to my mind it is still essentially an open question.

The higher level question that you mention, i.e. the directions a field of mathematics should take, is definitely also very interesting, but certainly just as hard or even harder! :)

11. What exactly is it that you find so mysterious about the strong tendency towards consensus? A naive answer might be that we have a clear notion of what a proof is, and therefore a watertight test that we can apply to claims of mathematical correctness. To the objection that fully detailed and rigorous proofs are never in practice written out, I would answer that mathematicians are trained to know how to take any part of a mathematical write-up and, if it is correct, expand it into something more formal and detailed -- a process that can be iterated if necessary. Experts in different areas get a feel for what is likely to be true in their area, so if an incorrect proof appears, and is of a sufficiently important result, they will usually (but not always) know which part needs to be clarified. Surprisingly often, the mistake is a simple unjustified assertion that the author mistakenly thinks is obvious and a more sceptical reader doesn't find obvious at all.

I'm curious to have a more detailed formulation of what the real philosophical problem is here. What is unsatisfactory or incomplete about the account (or some modest expansion of it) that I've just given?

12. What about the objection that the proof itself requires a proof and so on ad infinitum. Why is that objection not justified?

13. So, accepting Tim Gowers' description of mathematical practice as I do, perhaps there's a question as to why it was possible for the various branches of mathematics to reach the condition he describes. The advent of today's standards of rigour came piecemeal, some branches taking much longer than others. And you can see reactions to the perceived threat of losing the control that it affords in Jaffe and Quinn's concerns about the influx of research by physicists such as Edward Witten.

That ultimately just about all of mathematics can be rendered in a single and, so far as we can tell, consistent system, ZFC, must tell us something. But the kinds of ways of knowing that there's something fishy in a proposed proof operate at a higher level than that. You can find out quite a lot about this intuition from the Tricki. Category theory also provides a higher-level support.

14. I sometimes say that philosophy is about 'questioning the obvious'. So while the phenomenon of consensus in mathematics may appear to be fairly unproblematic, especially for those who experience it from within (i.e. professional mathematicians), if you think about mathematics as a body of human practices, I'd say it's quite astonishing. As Azzouni argues in the paper I link to above, in terms of institutionalized human practices, mathematics is quite unique in terms of its tendency towards consensus.

One can of course argue that this is a product of clear and precise standards, which are transmitted and perpetuated through education. But the comparison with other fields of scientific inquiry can be illuminating here. Take biology, for example: even though (almost) everybody agrees that evolution is the right framework, there is a lot of disagreement on the details: gradalism vs. non-gradalism, adaptationism vs. by-product accounts etc. It's not obvious in which ways education in biology would not be as conducive to consensus as education in math, and yet the situations seem very different.

And to say that we have a clear, rigorous notion of what a proof is is only the beginning of an answer anyway. As suggested above by David, the emergence of the current standards of what counts as a correct, rigorous proof was a long and winding process. Moreover, while these standards seem widely and unanimously endorsed on the level of practices, i.e. tacitly, a theoretical/philosophical account of what a mathematical proof is is still largely an open question. Typically, a mathematician can tell whether a given purported proof is a 'good proof' or not (as in the case discussed above), but she or he would be hard-pressed to offer a definition of what a mathematical proof is ("I know one when I see one.")

I realize that these questions may sound pedantic and unnecessary from the point of view of mathematical practice, but that's just philosophy, questioning the obvious...

1. Catarina offers an interesting contrast between the achievement of consensus in mathematics and in empirical science, namely in biology and evolutionary theory. Education in biology and education in mathematics prepares both sets of researchers for speaking the same ‘language’, i.e., mastering the existing concepts and theories, and accepting the forms of argument, reasoning, experiment and proof in the respective discipline. However, at the last hurdle, in evaluating new alternative sub-theories, there are major differences in achieving consensus in the two fields. Why is this? Although there may be minor differences in educational style (mathematics assumes theory consensus, biology perhaps does not) is not the real difference in the disciplines themselves? Biology attempts to account for one world – the empirical world of living things we share. In other words, the truth test is fitting with this one world as best we can (subject to all the constraints indicated by Kuhn). Nevertheless, until theories are falsified or otherwise discarded they remain in contention. Mathematics develops theories based on ideas, concepts, assumptions, problems, etc., and ultimately there is no final competition between different theories. Provided they are consistent they may coexist. We have Euclidean and non-Euclidean geometries. There is no contradiction, no competition, no rivalry unless we apply them as applied theories to the empirical world. And then they are just another part of empirical science. The latter strives for an intended model. Mathematical theories only need to satisfy some particular model, out of the infinitely many possible. So consensus is easier to achieve in mathematics than biology. Which is not to say that the process of proving is easier in mathematics than that of conducting empirical tests in biology. It is just that if your derivation is agreed to be correct there is no further final test of truth, irrespective of the epistemological grounds of that agreement.

2. True. But the puzzling thing is precisely this bit:

"It is just that if your derivation is agreed to be correct there is no further final test of truth, irrespective of the epistemological grounds of that agreement."

That is, the consensus on proofs/derivations being correct.

15. As pointed out earlier, consensus in mathematics is a relatively recent phenomenon, basically dating back to the foundational work in the early 20th century. Before then, one had debates about use of infinitesimals, implicit use of the axiom of choice (before this axiom was explicitly formulated), use of complex analytic methods instead of "elementary" methods, etc.

In my view, one of the main reasons we were able to build such solid foundations to mathematics was that mathematics had been maturing for over 2000 years as a cumulatively developing subject, and that by the early 20th century we had finally reached the point where we could build proper foundations. There are other disciplines that are as old as mathematics (philosophy, for instance), but the cumulative development process started later. (Mathematics, for instance, still uses the classical theorems of Greek geometry, but much of Aristotlean physics or metaphysics has now been discarded by their modern counterparts.) Physics, for instance, has been developing more or less cumulatively since Galileo, and it is conceivable that in a century or two they may have a consensus foundation that is as solid as the foundations of mathematics are today. (They're not quite there yet, though.)

So perhaps the deeper question is why mathematics started developing in a cumulative fashion much earlier than most other disciplines.

16. here's a draft mini-paper on convergence in philosophy and why there isn't more of it, by comparison to mathematics, physics, and so on. there's also some data here. of course the foundations of mathematics is intermediate between regular mathematics and philosophy in some respects, so it's not surprising that convergence on the foundations of mathematics is slower than convergence on regular mathematics and faster than convergence in philosophy. it would be nice to pin down the relevant dimensions of difference.

17. It's important to recognise that advances in mathematics are largely brought about by conceptual innovation rather than by grounding in some or other foundational language. Nineteenth century projective geometry texts may need a little tightening up in terms of rigour from our perspective, but far more importantly we don't need to stand in such awe of the fact that theorems and their proofs can be dualised by the simple expedient of exchanging a few terms, 'line' and 'point', 'passes through' and 'lies on', etc., once we understand the role of the outer automorphism of the group SL(3, R). To understand this we need to understand a range of conceptual advances by Felix Klein and others.

I could go on and on here. It was a conceptual necessity for Poincaré to introduce early methods of algebraic topology in the 1890s. In doing so he shone a light on earlier work and transformed the mathematics of the twentieth century.

This isn't to underplay the importance of foundational work, but innovations in the axiomatic method have been as important for allowing conceptual clarification and advance as for the provision of security. How mathematics affords the opportunity for radical changes in its conceptual outlook while preserving a tightly integrated network of theories is the interesting qustion for me.

18. Good points, @Terry. They clearly illustrate my conviction that the philosophical discussion of these issues must be thoroughly historically-informed if it is to go anywhere.

I would just like to clarify on what level I identify a tendency for overwhelming consensus in mathematics: it is mostly on the level of recognizing a given proof as correct or incorrect. Of course one can disagree on whether certain assumptions used along the proof are legitimate or not (e.g. the axiom of choice or what have you), but granting the assumptions, in most cases there seems to be consensus as to whether a proof is correct or not. As suggested by David Chalmers above, it is not so surprising that there should be more divergence of opinions in discussions on the foundations of mathematics, where certain assumptions may be questioned or accepted on grounds other than proofs, than in what he calls 'regular mathematics'.

19. Well, I think the consensus amongst mathematicians is what we created mathematics for. most mathematicians I know have this craving for irrefutable arguments, for always wanting to dispute what has been said by others until no doubt remains (lovely put in cartoon form here http://abstrusegoose.com/353), so in some ways is really not surprising that the consensus is there. sure, it relies on the cumulative nature of the subject. in fact, what is disturbing is when it is *not* there, as is the case of foundations, of the debate between constructive vs classical logic. the need for certainty has turned many mathematicians into constructivists, it seems to me...

20. William wrote:

"What about the objection that the proof itself requires a proof and so on ad infinitum. Why is that objection not justified?"

No mathematicians believe that objection, so even if it were true, it wouldn't affect the way they arrive at a consensus. I.e., we hardly ever mathematicians object to someone's proof claiming the proof needs a proof... although I can imagine someone demanding program verification for a computer-based proof of the 4-color theorem.

21. "hardly ever" -> "hardly ever see"

22. I think Valeria hit the nail on the head: mathematicians crave consensus.

If any sort of argument is of the sort that it only convinces 50% of mathematicians, we'll either say it's "not mathematics", or discuss, polish and/or demolish the argument until convinces either 99% of mathematicians or just 1%. (Example: Cantor's proofs.)

If someone doesn't play the game according to the usual rules, we'll make up a new game and say they're playing that game instead, thus eliminating potential controversy. (Example: intuitionistic mathematics.)

Finally, we reward people who quickly admit their errors, instead of fighting on endlessly. We say they're smart, not wimps. (Example: Edward Nelson.) People who fight on endlessly are labelled crackpots and excluded from the community. (Examples: too numerous to list here.)

23. Well yes, mathematicians crave consensus, but then so do a lot of people. I crave consensus in philosophy (especially if it converges on my views), but I don't expect to find it. The question is why are mathematicians able to satisfy that craving. How have they been able to develop robustly stable practices? How is it possible to sense problems in a proof without dipping down to the machine code of its fully formal rendition.

It would be fascinating to have a transcript of the thought processes which allowed Terry Tao to see the problem with Nelson's proof.

24. Great points, @david! I agree entirely.

25. Dear David,

Actually, Nelson's proof was relatively easy to understand, in part because he took the trouble to write out a short outline which make clear the general strategy of proof while omitting most of the technical details (though it was ambiguous at one very crucial juncture), and also because I had already previously thought about the surprise examination (or unexpected hanging) paradox and the Kritchman-Raz argument (see the last section of http://terrytao.wordpress.com/2011/05/19/epistemic-logic-temporal-epistemic-logic-and-the-blue-eyed-islander-puzzle-lower-bound/ ).

From the outline one could already see that the main idea was to adapt the Kritchman-Raz argument to the theory Q_0^*, which "almost" proved its own consistency in that it contained a hierarchy of theories Q_1, Q_2, Q_3, ..., each of which could prove the consistency of its predecessor.

Now, I did not at the time fully understand the definition of Q_0^*, nor was I fully aware of the Hilbert-Ackermann result which guaranteed this chain of consistency results, but I was willing to accept the existence of such a hierarchy of theories. (I've since read up a bit on these topics, though.) The question was then, given such an abstract hierarchy, whether one could use the arguments of Chaitin and Kritchman-Raz to establish the inconsistency of at least one of these theories.

These arguments were simple enough (they were basically formalisations of the Berry paradox and surprise examination paradox respectively) that I could then try to do that directly, without any further assistance from the outline. And, indeed, when I attempted to do this, I did at first seem to obtain a contradiction (much as paradoxes such as the Berry or surprise examination paradoxes also lead to absurdity if one reasons somewhat carelessly using informal naive argument). So I could see where Nelson was coming from; but then I spent some time trying to expand out my arguments in detail to find the error. The key, as I found out, was to specify exactly what proof verifier would be used for the Chaitin portion of the argument (this was an issue that was left ambiguous in Nelson's outline), and in particular whether it would accept proofs of unbounded complexity or not. Since Nelson wanted to keep all proofs at bounded complexity, I used a proof verifier that enforced such a bound, and eventually worked out that this could not be done while keeping the length of the Chaitin machine constant; this was the objection that I raised in my first few comments. However, after Nelson responded, it became clear that he was using an unrestricted proof verifier, and this led to a different problem, namely that the proofs produced by Chaitin's argument were of unbounded complexity. So there was not a single "flaw" in the argument, but rather there were two separate flaws, one of which was relevant to one interpretation of the argument, and the other of which was relevant to an alternate interpretation of the argument.

Aside from this one ambiguity, though, the outline was quite clear. Certainly there have been other manuscripts claiming major results that were much more difficult to adjudicate because they were written so badly that there were multiple ambiguities or inaccuracies in the exposition, and any high-level perspective on the argument was obscured.

26. Dear Terry, thanks very much for that. For me it's intriguing how high-level perspectives, phrased in a natural mathematical language, can be sufficiently clear that they allow for the requisite checks to be made. I wonder if in the passage you describe in the second half of the penultimate paragraph, your reasoning is assisted by familiarity with similar potential problems, perhaps arising in very different parts of mathematics.

In my book I discuss the claims that automated 'analogical' reasoning allowed the production of a proof of Heine-Borel in two dimensions, from a proof of the one-dimensional case. My point was that even in this simple case, the program, which is operating syntactically, requires a large amount of human assistance.

Put in 'ordinary' language, one would suppose a closed interval with a cover and no finite subcover. Then take half of the interval which can't be finitely covered. Iterate this process. There is a point in the intersection of these nested intervals. It must belong to an element of the cover, and far enough along the sequence the nested intervals must be contained inside that element. Contradiction.

The one worry for the human trying to extend this to two dimensions is to make sure that one can go far enough along the sequence so that each dimension of the nested rectangles fits inside the given element of the cover. The computer needs a big hand just here.

Now some truly analogical reasoning would come into play if the human recognises from similar experience that an attempt to work an infinite-dimensional variant of the proof will not work as we cannot be assured that there's an upper bound to distances along the sequence for each of the dimensions to become small enough.

Would I be right in thinking that

27. In this specific case, all the mathematicians involved relied for their work on a specific highly developed theory- let me call it a module- and none was prepared to bear the cognitive cost of re-writing the entire module which is why there could be a quick resolution. Surely this happens all the time in other disciplines as well? Indeed the less 'rational' or alethic the subject area the lower the cognitive pay-off for rewriting entire modules so we might find even faster resolution, without even the pause for critical thought. A lesser mind might have taken much longer to concede the point.

Still, suppose there was a big cognitive pay-off, currently available, for entirely rewriting the Chatin theorem 'module' or rethinking Kolmogorov complexity- it may be that some on or other of those involved might have taken that tack.

Surely an 'in-built opponent' is a feature of all social communication? It weighs down most heavily where the cognitive cost of re-writing modules are high or the reward is miniscule? I suppose statements about fashion or syntax or what is considered politically correct, have this feature and thus in most social sub-sets there is going to be very quick recantation simply because the cost greatly outweighs any possible benefit.

At this point, I'd like to introduce a notion of ontologically dysphoria- the feeling of being in the wrong Universe, the intuition that the Cartesian duality of mind and body points to something more troubling, bizarre or tragic. It may be that there is a sort of genotypal canalisation towards a widespread feeling of this sort- perhaps, rather than a malaise attributable to the welgeist, ontological dysphoria is the driver for the necessary-but-not-too-much preference diversity needed to drive trade, but also communication and the elaboration of Knowledge systems.

It may make a sort of collocational sense for us to agree that Math represents a limit case of one sort and philosophy, with its distinctions without differences, as the limit case of its opposite.

Godel and Von Neumann, both Theists on their death beds, who agreed on so much in the way of mathematics yet had ontological dysphorias of opposite tropism. It may be that the latter type of 'madness'- or stark solipsistic discontinuity- is as important a driver for breakthroughs in Maths as the great powers of 'reason' both possessed which enabled the Von Neumann to grasp Godel's result, perhaps, more thoroughly and more quickly than he had done himself.


text:
# quatdivide

Divide quaternion by another quaternion

## Syntax

n = quatdivide(q,r)

## Description

n = quatdivide(q,r) calculates the result of quaternion division, n, for two given quaternions, q and r. Inputs q and r can each be either an m-by-4 matrix containing m quaternions, or a single 1-by-4 quaternion. n returns an m-by-4 matrix of quaternion quotients. Each element of q and r must be a real number. Additionally, q and r have their scalar number as the first column.

The quaternions have the form of

$q={q}_{0}+i{q}_{1}+j{q}_{2}+k{q}_{3}$

and

$r={r}_{0}+i{r}_{1}+j{r}_{2}+k{r}_{3}$

The resulting quaternion from the division has the form of

$t=\frac{q}{r}={t}_{0}+i{t}_{1}+j{t}_{2}+k{t}_{3}$

where

$\begin{array}{l}{t}_{0}=\frac{\left({r}_{0}{q}_{0}+{r}_{1}{q}_{1}+{r}_{2}{q}_{2}+{r}_{3}{q}_{3}\right)}{{r}_{0}^{2}+{r}_{1}^{2}+{r}_{2}^{2}+{r}_{3}^{2}}\\ {t}_{1}=\frac{\left({r}_{0}{q}_{1}-{r}_{1}{q}_{0}-{r}_{2}{q}_{3}+{r}_{3}{q}_{2}\right)}{{r}_{0}^{2}+{r}_{1}^{2}+{r}_{2}^{2}+{r}_{3}^{2}}\\ {t}_{2}=\frac{\left({r}_{0}{q}_{2}+{r}_{1}{q}_{3}-{r}_{2}{q}_{0}-{r}_{3}{q}_{1}\right)}{{r}_{0}^{2}+{r}_{1}^{2}+{r}_{2}^{2}+{r}_{3}^{2}}\\ {t}_{3}=\frac{\left({r}_{0}{q}_{3}-{r}_{1}{q}_{2}+{r}_{2}{q}_{1}-{r}_{3}{q}_{0}\right)}{{r}_{0}^{2}+{r}_{1}^{2}+{r}_{2}^{2}+{r}_{3}^{2}}\end{array}$

## Examples

Determine the division of two 1-by-4 quaternions:

q = [1 0 1 0];
r = [1 0.5 0.5 0.75];
d = quatdivide(q, r)

d =

0.7273    0.1212    0.2424   -0.6061

Determine the division of a 2-by-4 quaternion by a 1-by-4 quaternion:

q = [1 0 1 0; 2 1 0.1 0.1];
r = [1 0.5 0.5 0.75];
d = quatdivide(q, r)

d =

0.7273    0.1212    0.2424   -0.6061
1.2727    0.0121   -0.7758   -0.4606

## References

[1] Stevens, Brian L., Frank L. Lewis, Aircraft Control and Simulation, Wiley–Interscience, 2nd Edition.


text:
The Community for Technology Leaders
Issue No. 03 - March (2014 vol. 26)
ISSN: 1041-4347
pp: 595-607
Hao Wang , The University of Hong Kong, Hong Kong
Yilun Cai , The University of Hong Kong, Hong Kong
Yin Yang , Advanced Digital Sciences Center, Singapore
Shiming Zhang , Noah's Ark Lab, Hong Kong
Nikos Mamoulis , The University of Hong Kong, Hong Kong
ABSTRACT
This paper studies the problem of finding objects with durable quality over time in historical time series databases. For example, a sociologist may be interested in the top 10 web search terms during the period of some historical events; the police may seek for vehicles that move close to a suspect 70 percent of the time during a certain time period and so on. Durable top-$(k)$ (DTop-$(k)$) and nearest neighbor ($({\rm D}k{\rm NN})$) queries can be viewed as natural extensions of the standard snapshot top-$(k)$ and NN queries to timestamped sequences of values or locations. Although their snapshot counterparts have been studied extensively, to our knowledge, there is little prior work that addresses this new class of durable queries. Existing methods for DTop-$(k)$ processing either apply trivial solutions, or rely on domain-specific properties. Motivated by this, we propose efficient and scalable algorithms for the DTop-$(k)$ and $({\rm D}k{\rm NN})$ queries, based on novel indexing and query evaluation techniques. Our experiments show that the proposed algorithms outperform previous and baseline solutions by a wide margin.
INDEX TERMS
Time series analysis, Indexing, Trajectory, Knowledge discovery, Data engineering, Search problems,spatiotemporal databases, Durable query, time series, historical data
CITATION
Hao Wang, Yilun Cai, Yin Yang, Shiming Zhang, Nikos Mamoulis, "Durable Queries over Historical Time Series", IEEE Transactions on Knowledge & Data Engineering, vol. 26, no. , pp. 595-607, March 2014, doi:10.1109/TKDE.2013.10
CITATIONS
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