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# Mean independent and correlated variables

Given two real random variables $$X$$ and $$Y$$, we say that:

1. $$X$$ and $$Y$$ are independent if the events $$\{X \le x\}$$ and $$\{Y \le y\}$$ are independent for any $$x,y$$,
2. $$X$$ is mean independent from $$Y$$ if its conditional mean $$E(Y | X=x)$$ equals its (unconditional) mean $$E(Y)$$ for all $$x$$ such that the probability that $$X = x$$ is not zero,
3. $$X$$ and $$Y$$ are uncorrelated if $$\mathbb{E}(XY)=\mathbb{E}(X)\mathbb{E}(Y)$$.

Assuming the necessary integrability hypothesis, we have the implications $$\ 1. \implies 2. \implies 3.$$.
The $$2^{\mbox{nd}}$$ implication follows from the law of iterated expectations: $$\mathbb{E}(XY) = \mathbb{E}\big(\mathbb{E}(XY|Y)\big) = \mathbb{E}\big(\mathbb{E}(X|Y)Y\big) = \mathbb{E}(X)\mathbb{E}(Y)$$

Yet none of the reciprocals of these two implications are true.

## Mean-independence without independence

Let $$\theta \sim \mbox{Unif}(0,2\pi)$$, and $$(X,Y)=\big(\cos(\theta),\sin(\theta)\big)$$.

Then for all $$y \in [-1,1]$$, conditionally to $$Y=y$$, $$X$$ follows a uniform distribution on $$\{-\sqrt{1-y^2},\sqrt{1-y^2}\}$$, so: $\mathbb{E}(X|Y=y)=0=\mathbb{E}(X).$ Likewise, we have $$\mathbb{E}(Y|X) = 0$$.

Yet $$X$$ and $$Y$$ are not independent. Indeed, $$\mathbb{P}(X>0.75)>0$$  and $$\mathbb{P}(Y>0.75) > 0$$, but  $$\mathbb{P}(X>0.75, Y>0.75) = 0$$ because $$X^2+Y^2 = 1$$ and $$0.75^2 + 0.75^2 > 1$$.

## Uncorrelation without mean-independence

A simple counterexample is $$(X,Y)$$ uniformly distributed on the vertices of a regular polygon centered on the origin, not symmetric with respect to either axis.

For example, let $$(X, Y)$$ have uniform distribution with values in $\big\{(1,3), (-3,1), (-1,-3), (3,-1)\big\}.$

Then $$\mathbb{E}(XY) = 0$$ and $$\mathbb{E}(X)=\mathbb{E}(Y)=0$$, so $$X$$ and $$Y$$ are uncorrelated.

Yet $$\mathbb{E}(X|Y=1) = -3$$, $$\mathbb{E}(X|Y=3)=1$$ so we don’t have $$\mathbb{E}(X|Y) = \mathbb{E}(X)$$. Likewise, we don’t have $$\mathbb{E}(Y|X) = \mathbb{E}(Y)$$.

# Determinacy of random variables

The question of the determinacy (or uniqueness) in the moment problem consists in finding whether the moments of a real-valued random variable determine uniquely its distribution. If we assume the random variable to be a.s. bounded, uniqueness is a consequence of Weierstrass approximation theorem.

Given the moments, the distribution need not be unique for unbounded random variables. Carleman’s condition states that for two positive random variables $$X, Y$$ with the same finite moments for all orders, if $$\sum\limits_{n \ge 1} \frac{1}{\sqrt[2n]{\mathbb{E}(X^n)}} = +\infty$$, then $$X$$ and $$Y$$ have the same distribution. In this article we describe random variables with different laws but sharing the same moments, on $$\mathbb R_+$$ and $$\mathbb N$$.

### Continuous case on $$\mathbb{R}_+$$

In the article a non-zero function orthogonal to all polynomials, we described a function $$f$$ orthogonal to all polynomials in the sense that $\forall k \ge 0,\ \displaystyle{\int_0^{+\infty}} x^k f(x)dx = 0 \tag{O}.$

This function was $$f(u) = \sin\big(u^{\frac{1}{4}}\big)e^{-u^{\frac{1}{4}}}$$. This inspires us to define $$U$$ and $$V$$ with values in $$\mathbb R^+$$ by: $\begin{cases} f_U(u) &= \frac{1}{24}e^{-\sqrt[4]{u}}\\ f_V(u) &= \frac{1}{24}e^{-\sqrt[4]{u}} \big( 1 + \sin(\sqrt[4]{u})\big) \end{cases}$

Both functions are positive. Since $$f$$ is orthogonal to the constant map equal to one and $$\displaystyle{\int_0^{+\infty}} f_U = \displaystyle{\int_0^{+\infty}} f_V = 1$$, they are indeed densities. One can verify that $$U$$ and $$V$$ have moments of all orders and $$\mathbb{E}(U^k) = \mathbb{E}(V^k)$$ for all $$k \in \mathbb N$$ according to orthogonality relation $$(\mathrm O)$$ above.

### Discrete case on $$\mathbb N$$

In this section we define two random variables $$X$$ and $$Y$$ with values in $$\mathbb N$$ having the same moments. Let’s take an integer $$q \ge 2$$ and set for all $$n \in \mathbb{N}$$: $\begin{cases} \mathbb{P}(X=q^n) &=e^{-q}q^n \cdot \frac{1}{n!} \\ \mathbb{P}(Y=q^n) &= e^{-q}q^n\left(\frac{1}{n!} + \frac{(-1)^n}{(q-1)(q^2-1)\cdot\cdot\cdot (q^n-1)}\right) \end{cases}$

Both quantities are positive and for any $$k \ge 0$$, $$\mathbb{P}(X=q^n)$$ and $$\mathbb{P}(Y=q^n) = O_{n \to \infty}\left(\frac{1}{q^{kn}}\right)$$. We are going to prove that for all $$k \ge 1$$, $$u_k = \sum \limits_{n=0}^{+\infty} \frac{(-1)^n q^{kn}}{(q-1)(q^2-1)\cdot\cdot\cdot (q^n-1)}$$ is equal to $$0$$.

Continue reading Determinacy of random variables

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## Spread the Word and Run a Chance to Win a Bundle of Goodies from Embedded World

February 21, 2019

If you do, please consider paying close attention for the next few days to the EmbeddedRelated Twitter account and to my personal Linkedin account (feel free to connect).  This is where I will be posting lots of updates about how the EmbeddedRelated.tv live streaming experience is going at Embedded World.

The most successful this live broadcasting experience will be, the better the chances that I will be able to do it...

## Launch of EmbeddedRelated.tv

February 21, 2019

With the upcoming Embedded Word just around the corner, I am very excited to launch the EmbeddedRelated.tv platform.

This is where you will find the schedule for all the live broadcasts that I will be doing from Embedded World next week.  Please note that the schedule will be evolving constantly, even during the show, so I suggest your refresh the page often.  For instance, I am still unsure if I will be able to do the 'opening of the doors' broadcast as...

## Live Streaming from Embedded World!

February 12, 2019

For those of you who won't be attending Embedded World this year, I will try to be your eyes and ears by video streaming live from the show floor.

I am not talking improvised streaming from a phone, but real, high quality HD streaming with a high-end camera and a device that will bond three internet connections (one wifi and two cellular) to ensure a steady, and hopefully reliable, stream. All this to hopefully give those of you who cannot be there in person a virtual...

February 5, 2019

## Embedded Programming Video Course Teaches RTOS

January 20, 2019

If you'd like to understand how a Real-Time Operating System (RTOS) really works, here is a free video course for you:

RTOS part-1: In this first lesson on RTOS you will see how to extend the foreground/background architecture from the previous lesson, so that you can have multiple background loops running seemingly simultaneously.:

RTOS part-2: In this second lesson on RTOS you will see how to automate the context switch process. Specifically, in this lesson, you will start building...

## The Hardest Bug I Never Solved

I agreed to four hours.

Four hours to help hunt down and kill a bug. A terrible malicious bug that was eating away at this project, wreaking havoc upon the foundations of a critical feature, and draining time randomly from every one of eight firmware engineers on this project. Quite honestly, I can’t remember the last time it took more than an hour or two for me to locate, isolate, and fix a firmware bug. Surely I could help find and solve this issue within four...

## Wye Delta Tee Pi: Observations on Three-Terminal Networks

December 23, 2018

Today I’m going to talk a little bit about three-terminal linear passive networks. These generally come in two flavors, wye and delta.

Why Wye?

The town of Why, Arizona has a strange name that comes from the shape of the original road junction of Arizona State Highways 85 and 86, which was shaped like the letter Y. This is no longer the case, because the state highway department reconfigured the intersection

## The Least Interesting Circuit in the World

It does nothing, most of the time.

It cannot compute pi. It won’t oscillate. It doesn’t light up.

Often it makes other circuits stop working.

It is… the least interesting circuit in the world.

What is it?

About 25 years ago, I took a digital computer architecture course, and we were each given use of an ugly briefcase containing a bunch of solderless breadboards and a power supply and switches and LEDs — and a bunch of

## Servo Troubleshooting notes

October 4, 2018

Servo Troubleshooting

Ostendorff

Forcing a servo to rotate when off can damage it.

Malfunction:Servo makes a grinding noise or acts erratic

Probable Cause

Corrective Action

Source

Examine gears for broken teeth

Replace gears

General

Malfunction:Servo Jitters

Probable Cause

Corrective Action

Source

Dirty potentiometer

...

## Linear Feedback Shift Registers for the Uninitiated, Part XVIII: Primitive Polynomial Generation

Last time we figured out how to reverse-engineer parameters of an unknown CRC computation by providing sample inputs and analyzing the corresponding outputs. One of the things we discovered was that the polynomial $x^{16} + x^{12} + x^5 + 1$ used in the 16-bit X.25 CRC is not primitive — which just means that all the nonzero elements in the corresponding quotient ring can’t be generated by powers of $x$, and therefore the corresponding 16-bit LFSR with taps in bits 0, 5,...

## VHDL tutorial - A practical example - part 1 - Hardware

May 18, 20111 comment

In previous posts I described some simple VHDL examples.  This time let's try something a little more complex. This is part one of a multiple part article.  This is intended to be a detailed description of one of several initial designs that I developed for a client.  This design never made it into a product, but a similar design was used and is currently being produced.  As a considerable amount of work was put into this effort, I decided to share this design...

## The habitat of hardware bugs

The Moscow apartment which little me called home was also home to many other creatures, from smallish cockroaches to biggish rats. But of course we rarely met them face to face. Evolution has weeded out those animals imprudent enough to crash your dinner. However, when we moved a cupboard one time, we had the pleasure to meet a few hundreds of fabulously evolved cockroaches.

In this sense, logical bugs aren't different from actual insects. You won't find...

## Ten Little Algorithms, Part 4: Topological Sort

July 5, 20151 comment

Other articles in this series:

Today we’re going to take a break from my usual focus on signal processing or numerical algorithms, and focus on...

## The Least Interesting Circuit in the World

It does nothing, most of the time.

It cannot compute pi. It won’t oscillate. It doesn’t light up.

Often it makes other circuits stop working.

It is… the least interesting circuit in the world.

What is it?

About 25 years ago, I took a digital computer architecture course, and we were each given use of an ugly briefcase containing a bunch of solderless breadboards and a power supply and switches and LEDs — and a bunch of

## Important Programming Concepts (Even on Embedded Systems) Part II: Immutability

September 14, 20141 comment

Other articles in this series:

This article will discuss immutability, and some of its variations in the topic of functional programming.

There are a whole series of benefits to using program variables that… well, that aren’t actually variable, but instead are immutable. The impact of...

## Coding Step 1 - Hello World and Makefiles

Articles in this series:

Step 0 discussed how to install GCC and the make utility with the expectation of writing and compiling your first C program. In this article, I discuss how to use those tools we installed last time. Specifically, how to use GCC to compile a C program and...

## From bare-metal to RTOS: 5 Reasons to use an RTOS

Developers can come up with amazing and convoluted reasons to not use an RTOS. I have heard excuses ranging from they are too expensive (despite open source solutions) all the way to they aren’t efficient and use too much memory. In some circumstances some excuses are justified but there are many reasons why a developer should look to an RTOS to help with their real-time scheduling needs.

From bare-metal to RTOS Quick Links
• Part 1:

## Slew Rate Limiters: Nonlinear and Proud of It!

October 6, 2014

I first learned about slew rate limits when I was in college. Usually the subject comes up when talking about the nonideal behavior of op-amps. In order for the op-amp output to swing up and down quickly, it has to charge up an internal capacitor with a transistor circuit that’s limited in its current capability. So the slew rate limit $\frac{dV}{dt} = \frac{I_{\rm max}}{C}$. And as long as the amplitude and frequency aren’t too high, you won’t notice it. But try to...

## C Programming Techniques: Function Call Inlining

Introduction

Abstraction is a key to manage software systems as they increase in size and complexity. As shown in a previous post, abstraction requires a developper to clearly define a software interface for both data and functions, and eventually hide the underlying implementation.When using the C language, the interface is often exposed in a header '.h' file, while the implementation is put in one or more  corresponding '.c' files.

First, separating an interface from its...

## Another 10 Circuit Components You Should Know

October 30, 20131 comment

It's that time again to review all the oddball goodies available in electronic components. These are things you should have in your bag of tricks when you need to design a circuit board. If you read my previous posts and were looking forward to more, this article's for you!

1. Bus switches

I can't believe I haven't mentioned bus switches before. What is a bus switch?

There are lots of different options for switches:

• mechanical switch / relay: All purpose, two...


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# Reactivity and selectivity in free radical halogenation of alkanes

I read about the free radical halogenation of alkanes with $$\ce{X2}$$ and I have the following questions:

1. Given that the first step is the homolytic breaking of the $$\ce{X-X}$$ bond, and the homolytic bond dissociation enthalpies of halogens (page 23, 31, and 34 of the PDF - I think it does not have data on $$\ce{I-I}$$ - but I'll use values from my textbook) are in the order $$\ce{I_2 < F_2 < Br_2 < Cl_2}$$, why is the order of reactivity of halogens $$\ce{F_2 > Cl_2 > Br_2 > I_2}$$?
2. My book said that bromine is very selective when reacting with alkanes - it will prefer a tertiary carbon greatly over primary or secondary ones. Chlorine is less selective and fluorine is not selective at all - it's happy to react with all types of carbons. Why is this so?

PS: I am using the textbook 'Organic Chemistry For JEE' by Solomons, Fryhle, and Snyder.

• – Loong May 14 '18 at 7:44
• Your order for bond dissociation energies correct (DOI: 10.1039/TF9625800633) – Gaurang Tandon May 14 '18 at 9:15
• – user55119 Oct 19 '18 at 23:51


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# 5 Essential SAT No Brainers You Need Most

As Pavel described in a recent post, the ultimate guide to the SAT requires a steady march along the Path to Competence. That being said, there are a few SAT strategies that are so essential – so crucial to success – that, if ignored, might torpedo the advances you’ve made towards doing the best you can.

Here’s our list of five must-do SAT strategies. While some of them seem obvious, they are all hugely important. Think of this as your to-do list in the months and weeks before the SAT!

1.  Be familiar with the test.

In order to understand how to succeed on the SAT, you need to, well, know how the SAT works. This means understanding how much time is in each section, how the test is scored, and how many breaks there are. The best way to prepare for this is to read the College Board’s description of the test structure. Also, try taking a few practice tests to figure out the timing. If you’d like to take a Math section practice tests with Lemma,

Time management is an essential part of success in all endeavors, but especially the SAT! If I told you that I got every answer right on my SAT you’d be pretty impressed…But, what if I added the point that I managed my time poorly and only answered half the questions… I bet you’d think of my score a little differently!

Lot’s of students struggle with running out of time while taking the test. (I did). The best technique for dealing with this problem is to skip a problem that is tripping you up. Finish all the questions that you can first before circling back to the more difficult ones. Just be very careful with filling in the bubbles correctly when you start going out of order!

3.  Condition yourself

It’s important to acclimatize yourself to specific conditions and to use memory techniques when studying. Popular memory techniques range from somewhat intuitive to totally bizarre. Some people even advise chewing the same type of gum while studying and taking the test, or even wearing the same clothes!

A perhaps more reasonable suggestion is that you study the same way you take the test. Doing one Problem of the Day with a service like ours (or the myriad of others) is not bad for your test preparation. However, it is very important that you try to simulate the test as best you can. Try to find a quiet room to study and practice, avoid playing music in the background, and turn your phone off for a little. The best way to condition yourself is to try to simulate the actual conditions you’ll be taking the test in!

4.  Guessing when you don’t know the answer

Ah guessing… one of the most often discussed test-taking techniques. But for good reason! You aren’t penalized for wrong answers on the SAT, so if you don’t know the exact answer to a question, first try to eliminate answers you know are not options. Check out this example:

QuestionAt Lemma, the cost $$C$$, in US Dollars, of producing $$n$$ Chalkboard Robots is expressed by $$C=12n+261$$. Lemma sells these robots for the all-too-low price of \$15. Lemma makes a profit when the total income generated from our robots is greater than the cost of production $$C$$. Which of the following inequalities defines all possible values for $$n$$ where Lemma makes a profit?

1. $$n<87$$
2. $$n>261$$
3. $$C>87$$
4. $$n>87$$

No need to do all the work on this one- unless you want to. The point is that you recognize that option c is out of the question because the question isn’t asking anything about Cost $$C$$, but how many robots we gotta huck before making some dough!

If you have absolutely no idea on where to even begin on a question, just guess. Like I said, you aren’t penalized, so just go for it!

5.  Don’t Cram and get some sleep!

The social science backs the intellectual ridiculousness of cramming. The problem with crammers is that sometimes they do generate results. I’m sure you have someone in your life who you swear only studies for a test the day beforehand and always gets good grades. Maybe you’re that student! But the SAT isn’t a reading quiz. Well, it kinda is…But it’s a really big reading quiz.

The best technique is steady practice in the months before the test and a good night’s sleep the night before. DO NOT stay up all night cramming the even-b quadratic equation into your head your overall performance will suffer as a result!

To Wrap Up:

If you follow these 5 easy no-brainers, you’ll be well on your way to success on the SAT. If you combine these strategies with following the Path to Competence, you’ll be a superstar. Hard work and diligence are the ultimate tools for beating the SAT!


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# every positive monotone increasing function necessarily the derivative of a convex function?

I was trying to find out if every positive (do we need this?) monotone increasing function $$f :\mathbb{R}_{+}\to\mathbb{R}_{+}$$ was necessarily the derivative of a convex function. I have found this answer that seems related but I am not so sure if this is the good approach. Do I have to verify that for $$n$$ points $$x_1,\ldots, x_n$$ I have $$\langle x_0-x_1,f(x_0)\rangle+\ldots+\langle x_{n-1}-x_n,f(x_{n-1})\rangle+\langle x_{n}-x_0,f(x_n)\rangle\ge 0$$? (for every $$f$$ as above)

• Any function $f :\mathbb{R}_{+}\to\mathbb{R}_{+}$ is necessarily positive – Henry Apr 13 at 14:43
• Hint: A function with increasing derivative is convex. – Giorgos Giapitzakis Apr 13 at 14:45
• It slightly depends on your definition of derivative. $f(x) =x$ for $0 < x < 1$; $f(1)=3$; $f(x)=x+10$ for $x>1$ is a monotone increasing function but not obviously the derivative of its integral – Henry Apr 13 at 14:47
• you don't need positive. Check - $x^2$ has negative derivative for $x<0$, but you may want to add continuous to the requirement? – Rahul Madhavan Apr 13 at 14:48

If $$F: I \to \Bbb R$$ is convex and differentiable on an interval $$I \subset \Bbb R$$ then $$F'$$ is increasing and continuous, see for example Continuity of derivative of convex function.
Conversely, if $$f: I \to \Bbb R$$ is increasing and continuous then $$\tag{*} F(x) = \int_a^x f(t) \, dt$$ (for some $$a \in I$$) is convex with $$F'=f$$.
If $$f$$ is only increasing but not necessarily continuous then we can still define $$F$$ via $$(*)$$ because monotone functions are Lebesgue integrable. Then $$F' = f$$ a.e. in $$I$$ and $$F$$ is convex: For $$x < y < z$$ is $$\frac{F(y)-F(x)}{y-x} = \frac{1}{y-x} \int_x^y f(t) \, dt = \int_0^1 f(x + s(y-x)) \, ds \\ \le f(y) \le \int_0^1 f(y + s(z-y)) \, ds = \frac{F(z)-F(y)}{z-y} \, .$$
• If $f$ is continuous I understand that $F$ is convex, since $F'=f$ and $f$ is increasing. But what is the argument in the case $F'=f$ a.e.? – roi_saumon Apr 13 at 20:04
• Thank you. Is there a typo $f'\to f$? – roi_saumon Apr 13 at 23:22
• @roi_saumon: Yes, it should be $f$ (or $F'$). Hopefully correct now. – Martin R Apr 14 at 4:52


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[BUG] getting a French error message when using Prime in German language
02-23-2014, 04:45 PM
Post: #1
 Stefan Member Posts: 87 Joined: Dec 2013
[BUG] getting a French error message when using Prime in German language
If language is set to german and I enter the following:
$$\int_0^2 \left( \begin{cases}-2-x & -2 \leq x < -1\\ \sin\left(\frac{\pi}{2} x\right) & -1 \leq x < 1 \\ 2-x & 1 \leq x < 2 \end{cases} \right) \cdot \sin\left(n \pi x \right) \rm{d}x$$

I get the error message in french: "Aucun cas ne s'applique". (English version is: "No case applies")

Btw: I entered the big piecewiese expression first in the plot mode (with capital X). I then copied it to the CAS where I had to replace every capital X with a lowercase x. I tried to do this with "subst()", but this fails because X is intialised to zero. Is there a better way than replacing every X manually to do this?
@HP: Please get rid of this capital/non capital variable crazyness.
02-23-2014, 05:01 PM
Post: #2
 Tugdual Senior Member Posts: 751 Joined: Dec 2013
RE: [BUG] getting a French error message when using Prime in German language
It is because only the french were supposed to fail...
Seriously I have most of the error messages in English while I chose French in the settings.
02-23-2014, 09:55 PM (This post was last modified: 02-23-2014 09:55 PM by Joe Horn.)
Post: #3
 Joe Horn Senior Member Posts: 1,611 Joined: Dec 2013
RE: [BUG] getting a French error message when using Prime in German language
(02-23-2014 04:45 PM)Stefan Wrote:  ... I then copied it to the CAS where I had to replace every capital X with a lowercase x. I tried to do this with "subst()", but this fails because X is intialised to zero. Is there a better way than replacing every X manually to do this?

Quoting the original and the replacement does the trick. Case-sensitive example in CAS:

subst('X^3+2*X+3','X=x') --> x^3+2*x+3, as desired.
Notice that both arguments of subst are quoted.

<0|ɸ|0>
-Joe-
02-23-2014, 10:26 PM (This post was last modified: 02-23-2014 10:26 PM by eried.)
Post: #4
 eried Senior Member Posts: 741 Joined: Dec 2013
RE: [BUG] getting a French error message when using Prime in German language
There is a lot of typos even in the english version.

Just check: http://www.hpmuseum.org/forum/thread-494.html in MS Word. Every couple of lines there is a typo in the help. Now think on the fact that the other languages are probably derived from that english version...
I personally use the device in english because spanish is full of weird abbreviations and bad translations in every single screen.

My website: erwin.ried.cl
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3 changed according to first comment

I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here.

I've heard that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter is the same sort of object as an ultrafilterextender, since whereas an ultrafilter on an uncountable $\kappa$ is in $V_{\kappa + 2}$, an extender in general is an element of a much higher level $V_\alpha$. But that an extender captures all the information an ultrafilter has about elementary embeddings. Or so I think.

What I am trying to prove is a precise statement of this:

Lemma: given a non-principal $\kappa$-complete ultrafilter on an uncountable $\kappa$, and $j: V \rightarrow M \cong Ult(V,U)$, there is an extender $E$ such that $j_E : V \rightarrow M$ and also $j_E = j$. ($j_E$ is the embedding built from $E$, that is, $j_E$ elementarily maps $V$ into the direct limit of a direct system, one whose elements are ultraproducts of $V$, and whose maps are appropriately defined to commute with the maps from $V$ to these ultraproducts.)

I've tried to take the extender $F$ of a short length such that $F_a = U$. Then all the maps of the direct system are the identity, and the direct limit will be our original $M$, and $j_E = j$. But I'm not convinced this is a genuine extender in that is satisfies one of the (all equivalent?) definitions of extenders.

Another approach is to take $j_U$, and take the embedding $E_{j_U}$ derived from it, and then try to show $j_{E_{j_U}} = j$. This will in fact be an extender, but I can't show whether the target model $M_E$ agrees with $M$.

Any ideas or suggestions?

2 fixed wording; deleted 8 characters in body; edited title

# HowisanAn Extender is a Generalization of an Ultrafilter?

I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here.

I've heard it said that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter is the same sort of object as an ultrafilter, since whereas an ultrafilter on an uncountable $\kappa$ is in $V_{\kappa + 2}$, an extender in general is an element of a much higher level $V_\alpha$. But that an extender captures all the information an ultrafilter has about elementary embeddings. Or so I think.

What I am trying to prove is a precise statement of this:

Lemma: given a non-principal $\kappa$-complete ultrafilter on an uncountable $\kappa$, and $j: V \rightarrow M \cong Ult(V,U)$, there is an extender $E$ such that $j_E : V \rightarrow M$ and also $j_E = j$. ($j_E$ is the embedding built from $E$, that is, $j_E$ elementarily maps $V$ into the direct limit of a direct system, one whose elements are ultraproducts of $V$, and whose maps are appropriately defined to commute with the maps from $V$ to these ultraproducts.)

One approach

I've tried is to take the extender $F$ of a short length such that $F_a = U$. I can then see that Then all the maps of the direct system are the identity, and that the direct limit will be our original $M$, and that $j_E = j$. But I'm not sure whether convinced this is a genuine extender in that is satisfies all the conditions one of an extender, at the moment I'm a bit lost on (all the conditions an extender has to satisfyequivalent?) definitions of extenders.

Another approach is to take $j_U$, and take the embedding $E_{j_U}$ derived from it, and then try to show $j_{E_{j_U}} = j$. This will in fact be an extender, but I'm not sure I can't show whether the target model $M_E$ agrees with $M$.

Any ideas on either approachor suggestions?Thanks!

1

# How is an Extender a Generalization of an Ultrafilter?

I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here.

I've heard it said that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter is the same sort of object as an ultrafilter, since whereas an ultrafilter on an uncountable $\kappa$ is in $V_{\kappa + 2}$, an extender in general is an element of a much higher level $V_\alpha$. But that an extender captures all the information an ultrafilter has about elementary embeddings. Or so I think.

What I am trying to prove is a precise statement of this:

Lemma: given a non-principal $\kappa$-complete ultrafilter on an uncountable $\kappa$, and $j: V \rightarrow M \cong Ult(V,U)$, there is an extender $E$ such that $j_E : V \rightarrow M$ and also $j_E = j$. ($j_E$ is the embedding built from $E$, that is, $j_E$ elementarily maps $V$ into the direct limit of a direct system, one whose elements are ultraproducts of $V$, and whose maps are appropriately defined to commute with the maps from $V$ to these ultraproducts.)

One approach I've tried is to take the extender $F$ of a short length such that $F_a = U$. I can then see that all the maps of the direct system are the identity, and that the direct limit will be our original $M$, and that $j_E = j$. But I'm not sure whether this satisfies all the conditions of an extender, at the moment I'm a bit lost on all the conditions an extender has to satisfy.

Another approach is to take $j_U$, and take the embedding $E_{j_U}$ derived from it, and then try to show $j_{E_{j_U}} = j$. This will in fact be an extender, but I'm not sure whether the target model $M_E$ agrees with $M$.

Any ideas on either approach? Thanks!


text:
#### typo: flase #mnic

##### August 25, 2014

Everybody leaves typos while typing. However, typos while writing code end up in bugs. Bugs open doors to vulnerabilities.

When I talk about www.mnic.mu or www.gov.mu, many people lash out that they are not connected to the MNIC database therefore irrelevant. Chill out bambinos! No need to get excited. Showing you petty mistakes in your VISIBLE projects is just to make you realize your recklessness. As for the MNIC database or your intranet or whatever, you never published anything, so I am not commenting on those.

The following is a code extract from the video page on the MNIC website.

hdwebplayer({
id       : 'player_11',
swf      : 'http://www.mnic.mu/modules/hdwebplayer_1.6/player/player.swf',
width    : '545',
height   : '307',
video    : 'http://www.mnic.mu/images/video/11.mp4',
autoStart: 'flase',
});

What would autoStart: ‘flase’, do? Yeah, yeah! I am not gonna make a big issue out of this, it should have been autoStart: ‘false’,. It’s just about being cautious while writing code for Government Projects. Showing same carelessness while working with our biometric stuffs will land you in deep soup some day (^^,) …

To add, the typo repeats itself throughout the page revealing a notorious copy & paste habit.


text:
?

Free Version
Moderate

# Thermochemistry

CHEM-MMEEA3

The heats of formation of $N_{2}O_{5}(g)$ and $NO_{2}(g)$ are $11.297$ and $33.85\space kJ/mol$, respectively at $298\space K$.

What is the heat of reaction at $298\space K$ for the reaction below?

$$2N_{2}O_{5}(g) \rightarrow 4NO_{2}(g) + O_{2}(g)$$

A

Cannot be determined based on the information given.

B

$+ 11.297\space kJ/mol$

C

$-22.55\space kJ/mol$

D

$+ 112.8 \space kJ/mole$


text:
0

# What numbers between 10 and 20 have multiples of 26?

Wiki User

2017-09-20 19:23:35

There are no multiples of 26 in that range.

Wiki User

2017-09-20 19:23:35
Study guides

21 cards

## What is the prime factorization of 34

➡
See all cards
4.86
14 Reviews

Wiki User

2017-07-08 00:57:15

None. The multiples of 26 are simply the number 26, multiplied by some integer. Thus, the first multiples are:0, 26, 52, ...

Wiki User

2017-08-02 06:31:12

None of them. Positive multiples of 26 must be greater than 26.


text:
# Shell Method Problem - about y axis - # 5

#### Jason76

$$\displaystyle y = 3x^{4}, y = 0, x = 2$$

about $$\displaystyle x = 4$$ so revolves about y axis

Find limit of integration

$$\displaystyle 3x^{4} = 0$$

$$\displaystyle x = 0$$

$$\displaystyle V = 2\pi \int_{0}^{2} (x)(3x^{4}) dx$$

$$\displaystyle V = 2\pi \int_{0}^{2} 3x^{5}) dx$$

$$\displaystyle V = (2\pi) \dfrac{3x^{6}}{6}$$ evaluated at 0 and 2

$$\displaystyle V = (2\pi) \dfrac{x^{6}}{2}$$ evaluated at 0 and 2

$$\displaystyle V = (2\pi) [[\dfrac{(2)^{6}}{2}] - [0]] = 32 * 2\pi = 64 \pi$$ - wrong on homework ?

#### chiro

MHF Helper
Hey Jason76.

Since it is about the x = 4 axis you will need to shift the function to get the right answer.

This means instead of looking 3x^4 you also need to look at (3x^4 - 4) since you are shifting the function by four units.

Shell integration - Wikipedia, the free encyclopedia

1 person

#### Jason76

If were dealing with y axis then it's [function - (shifted value)]

If were dealing with the x axis, then what's the strategy? The article doesn't explain too well.

#### chiro

MHF Helper
You shift the y value instead.

The difference between rotation around x and y is that you have a function of the other variable. That's basically it.

If you have say y = f(x) and need x as a function of y then you find the inverse function.

So instead of using say x - a you use y - a for the shift.

1 person

#### Jason76

So instead of using say x - a you use y - a for the shift.

It's

$$\displaystyle a - x$$

and

$$\displaystyle a - y$$

when dealing with shifting.

Last edited:

#### Jason76

Would you use the same strategy (a - x), (a - y), regarding revolving around lines (not the y or x axis), with the disk and washer method?


text:
gs_split_by_tree(x)

gs_split_by_channels(x)

gs_check_redundant_nodes(x)

gs_remove_redundant_nodes(x, toRemove)

gs_remove_redundant_channels(gs)

gs_update_channels(gs, map, all = TRUE)

gh_pop_move(gh, node, to)

gs_pop_set_visibility(x, y, FALSE)

## Details

In order to merge multiple GatingSets into single GatingSetList, the gating trees and channel names must be consistent. These functions help removing the discrepancies and standardize the GatingSets so that they are mergable.

gs_split_by_tree splits the GatingSets into groups based on the gating tree structures.

gs_split_by_channels split GatingSets into groups based on their flow channels.

gs_check_redundant_nodes returns the terminal(or leaf) nodes that makes the gating trees to be different among GatingSets and thus can be considered to remove as redundant nodes.

gs_remove_redundant_nodes removes the terminal(or leaf) nodes that are detected as redundant by gs_check_redundant_nodes.

gs_remove_redundant_channels remove the redundant channels that are not used by any gate defined in the GatingSet.

gs_update_channels modifies the channel names in place. (Usually used to standardize the channels among GatingSets due to the letter case discrepancies or typo).

gh_pop_move inserts a dummy gate to the GatingSet. Is is useful trick to deal with the extra non-leaf node in some GatingSets that can not be simply removed by gs_remove_redundant_nodes

gs_pop_set_visibility hide a node/gate in a GatingSet. It is useful to deal with the non-leaf node that causes the tree structure discrepancy.


text:
# American Institute of Mathematical Sciences

2011, 2011(Special): 1351-1357. doi: 10.3934/proc.2011.2011.1351

## Sharp blow-up for semilinear wave equations with non-compactly supported data

 1 Department of Complex and Intelligent Systems, Faculty of Systems Information Science, Future University Hakodate, 116-2 Kamedanakano-cho, Hakodate, Hokkaido 041-8655, Japan 2 Department of Mathematics, College of Science and Technology, Nihon University, Chiyodaku Kanda Surugadai 1-8, Tokyo, 101-8308, Japan 3 The 4th year of undergraduate, Department of Complex Systems, School of Systems Information Science, Future University Hakodate, 116-2 Kamedanakano-cho, Hakodate, Hokkaido 041-8655, Japan

Received  July 2010 Revised  August 2010 Published  October 2011

This paper corrects Asakura's observation on semilinear wave equations with non-compactly supported data by showing a sharp blow-up theorem for classical solutions. We know that there is no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak, in spite of nite propagation speed of the linear wave. Our theorem clari es the final criterion on such a phenomenon.
Citation: Hiroyuki Takamura, Hiroshi Uesaka, Kyouhei Wakasa. Sharp blow-up for semilinear wave equations with non-compactly supported data. Conference Publications, 2011, 2011 (Special) : 1351-1357. doi: 10.3934/proc.2011.2011.1351
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Impact Factor:


text:
(FUNDAMENTAL AND APPLIED MATHEMATICS)

1999, VOLUME 5, NUMBER 2, PAGES 411-416

## On singularity of solution to inverse problems of spectral analysis expressed with equations of mathematical physics

V. V. Dubrovsky
L. V. Smirnova

Abstract

View as HTML     View as gif image    View as LaTeX source

 The inverse problem for the Laplacian under the Robin's boundary conditions is considered. We prove the following{\bfseries Theorem.} If $q_p$, $p=1,2$, are real twice continuously differentiable functions on $\bar{\Omega}$ and there exists a subsequence $i_k$ of positive integers such that $\|v_{i_k}(q_p)\|_{L_2(S)}\leq \mathrm{const} |\lambda_{i_k}|^{\beta}$, where $v_i(q_p)$ are orthonormal eigenfunctions of the operator $-\Delta+q$ in the case of Robin's boundary conditions with the eigenvalues $\lambda_i$, $i\in\mathbb N$, and $0\leq\beta<4^{-1}$ then there exists an infinite subsequence $i_{k_{l_m}}$ of positive integers such that the conditions $$\lambda_i(q_1)=\lambda_i(q_2),\ \ i\neq i_{k_{l_m}},\quad v_i(q_1)|_S=v_i(q_2)|_S,\ \ i\neq i_{k_{l_m}},$$ imply $q_1=q_2$.

All articles are published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/99/992/99204t.htm


text:
## How To Fix Cannot Determine Size Of Graphic Latex Kile (Solved)

Home > No Bounding > Cannot Determine Size Of Graphic Latex Kile

# Cannot Determine Size Of Graphic Latex Kile

## Contents

An easy calculus inequality that I can't prove What is a unifier? Invoking QuickBuild from the main program interface should now trigger PdfLatex. –Michael Palmer Jan 11 '12 at 17:22 | show 1 more comment 1 Answer 1 active oldest votes up vote Was there no income tax before 1913 in the United States? Not the answer you're looking for? http://inviewsoftware.com/no-bounding/cannot-determine-size-of-graphic-in-png-latex.html

Incidentally, you can change this in TexShop from the "Typeset" menu. graphics errors png share|improve this question edited Jan 22 '13 at 15:55 Martin Schröder 11.2k43196 asked May 8 '11 at 11:56 user5371 possible duplicate of: tex.stackexchange.com/questions/11193/… (untagged) –Ciro Santilli Why is using let inside a for loop so slow on Chrome? Use pdftex.

## Latex Error Cannot Determine Size Of Graphic In (no Boundingbox)

Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count). This is my pillow First skills to learn for mountaineering "PermitRootLogin no" in sshd config doesn't prevent su - What are 'hacker fares' at a flight search-engine? How do I handle this? share|improve this answer answered Jan 22 '13 at 15:38 Stephan Lukasczyk 1,70521127 add a comment| Not the answer you're looking for?

I simply added \usepackage[dvipdfm]{graphicx} \usepackage{bmpsize} to my file, and it worked perfectly ! Sample code that doesn't work: \begin{figure}[htb] \begin{center} \leavevmode \includegraphics[width=0.8\textwidth]{graph.png} \end{center} \end{figure} If I have to manually set some measurements of the image, please tell me how I can find out those It probably has to do with what compiler you are using. ! Latex Error: Option Clash For Package Graphicx. But in order for that to work I had to comment out the following 2 packages: %\usepackage[dvips]{epsfig} %\usepackage[dvips]{graphicx} ...and simply use package graphicx: \usepackage{graphicx} It seems [dvips] is problematic when used

pdf latex eps share|improve this question edited May 3 '12 at 13:05 Mr Lister 25.3k85481 asked Apr 8 '09 at 21:29 euphoria83 5,631164862 closed as off-topic by Dave Jarvis, joran, Fraser, No Bounding Box Error In Latex A lot of the ACM journals and conferences actually take word files, which is even worse. –Uri Apr 8 '09 at 22:01 | show 7 more comments up vote 25 down You need to use pdflatex for this. http://tex.stackexchange.com/questions/94869/error-including-a-png-cannot-determine-size-of-graphic This one and converting the pdf to .eps or .ps all make the error go away, but then there is just a big blank space where the image is supposed to

Interconnectivity Teenage daughter refusing to go to school Is adding the ‘tbl’ prefix to table names really a problem? Latex No Bounding Box Jpg When I use pdftex, the command: \includegraphics[height=60mm, width=100mm]{number2.png} works fine, but when I use "Tex and Ghostscript", I get the same error as you, about not being able to get the Converted the .eps file to a .pdf file using the perl program epstopdf 2. You are however able to state the natural size of the images using natwidth and natheight which will make latex compile without error.

1. Go to miktex-setting, tab "root", activate "show miktex roots", it will show you the path to UserConfig (not UserData as I wrote by mistake in the previous comment).
2. On 1941 Dec 7, could Japan have destroyed the Panama Canal instead of Pearl Harbor in a surprise attack?
3. By changing its extension from .JPG to .jpg (yes, just capital to lower) the error went away. \$ pdflatex --verion pdflatex: unrecognized option '--verion' This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live

## No Bounding Box Error In Latex

I have followed suggestions I found in the archives and done the following: 1. http://tex.stackexchange.com/questions/98679/latex-error-cannot-determine-size-of-graphic-in-title-png-no-boundingbox I could not find it here and on the internet. –euphoria83 Apr 8 '09 at 21:56 1 @euphoria: Which journal are you publishing to? Latex Error Cannot Determine Size Of Graphic In (no Boundingbox) For a better animation of the solution from NDSolve How to deal with a coworker that writes software to give him job security instead of solving problems? Latex No Bounding Box Pdf It is often better to take the JPEG and convert it into a PDF (on a mac) or EPS (on a PC).

How are the functions used in cryptographic hash functions chosen? http://inviewsoftware.com/no-bounding/cannot-determine-size-of-graphic-png-latex.html File: title.png Graphic file (type eps) Does anyone know how to fix this problem? LaTeX Error: Cannot determine size of graphic in xfig/ch1/doubsq.pdf (no Boun dingBox). Topics Active topics Unanswered topics News and Articles Posts Unread posts New posts Your posts Other LaTeX forums TeXwelt (deutsch) goLaTeX (deutsch) TeXnique (français) Sidebar Topics Active topics Unanswered topics News Natwidth Latex

more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed On 1941 Dec 7, could Japan have destroyed the Panama Canal instead of Pearl Harbor in a surprise attack? I'm working on debian using emacs graphics errors png share|improve this question edited Jan 22 '13 at 15:51 Martin Schröder 11.2k43196 asked Jan 22 '13 at 15:02 user24726 41112 marked as check over here Count trailing truths Am I interrupting my husband's parenting?

pgfmathparse basic usage Existence proof of Lorentz transformation from lightlike to lightlike vectors How did early mathematicians make it without Set theory? Latex Error File Dvipdfm Def Not Found In the past I have done this with \epsfig but from reading archives of this mailing list --- this doesn't work for pdflatex. If not, please edit your question here to explain why so that people can better focus their attention to help you. –Martin Schröder Jan 22 '13 at 15:54 add a comment|

## How to tar.gz many similar-size files into multiple archives with a size limit First skills to learn for mountaineering Prove that the following statements for a ring R are equivalent: Tank-Fighting

So, you only need to open the file in Inkscape and then SaveAs and select eps format. It just depends whether you're using LaTeX or pdfLaTeX as to which you'd choose. –Will Robertson Apr 9 '09 at 2:14 2 Should we migrate this (and other tex questions) This is how I start: \documentclass[a4paper,12pt,titlepage]{report} \usepackage{fontenc} \usepackage[italian]{babel} \usepackage[utf8x]{inputenc} \usepackage{graphicx} \begin{document} . . . \begin{figure}[htbp] \begin{center} \includegraphics{mol.png} \end{center} \label{fig:uno} \end{figure} . . . \end{document} and this is the error report: ERROR: Convert Png To Eps Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the

Tried to do pdflatex on my file (under linux) With \usepackage{graphicx} I get the error: ! GraphicsConvertor on a mac will do that for you easily. How to tar.gz many similar-size files into multiple archives with a size limit Draw some mountain peaks Wien's oscillator - amplitude stabilization with Zeners (loop's gain) Removal of negative numbers from this content Select QuickBuild from the list on the left.

more hot questions lang-tex about us tour help blog chat data legal privacy policy work here advertising info mobile contact us feedback Technology Life / Arts Culture / Recreation Science Other First skills to learn for mountaineering Draw some mountain peaks This is my pillow n-dimensional circles! I eventually just decided to let the file contain everything. –Uri Apr 8 '09 at 22:02 add a comment| up vote 2 down vote On a Mac (pdftex) I managed to The following one worked for me: \usepackage[dvipdfmx]{graphicx} \usepackage{bmpsize} –zeroos May 12 '15 at 20:56 Just to add: specificying natwidth and natheight also solved a problem I had with jpgs

I am using the following syntax for including the images : \begin{figure}[!h] \begin{center} \scalebox{0.45}{ \includegraphics{tree.jpg}} \end{center} \caption{\small A sample} \label{tree} \end{figure} What should I do ? Join them; it only takes a minute: Sign up Error including image in Latex [closed] up vote 29 down vote favorite 6 I am getting the following error while compiling my Done: \begin{center} \includegraphics[width=140mm]{xfig/ch1/doubsq.pdf} \end{center} in my document. 4. Type H for immediate help. ...


text:
# PMRWindow plugin¶

The PMRWindow plugin offers an interface to PMR. By default, it lists all the published exposures found in PMR:

The list can be filtered. For example, to enter Noble as a filter gives:

You can right click on any of the exposures to view them in PMR, copy their URL to the clipboard, or make a local copy of them (see the PMR Workspaces window for the latter). Alternatively, double click on any of the exposures to reveal their contents. For example, to double click on Noble, 1962 gives:

From there, double click on either noble_1962.cellml or noble_1962.sedml to open them in OpenCOR. You can also right click on any of them to view them in PMR or copy their URL to the clipboard.


text:
alex kuang

professional yak shaver

One sentiment I see from people coming from dynamically typed languages (python, ruby) or even Java and C is a general dismissiveness about static typing and compilers. At best, the sentiment is “Oh, well it makes sure my input is an Int, or a Float, or a Bool… That’s cool I guess, but I can do that with TDD”. At worst, static typing is seen as something to fight against – shackles that limit our creativity and bar us from what we really want to do and the beautiful programs that we could write if only the type-checker got out of our way.

Personally, I think of the type-checker (and by extension the compiler, really) not only as a free suite of rigorous tests built from first principles, but as a friend that is nice enough to correct me, the silly human, when I think I’m making sense but I’m really not. Sure, sometimes that friend is a bit dense (ahem Java, ahem) and can’t quite understand what I’m trying to say, but in the case of a language like Scala I find that the compiler is right more often than not. In fact, I had a whole giant wall of text geared up to talk about the value in using Option over null, in using Either instead of exceptions, in capturing values without using Stringly-Typed data… But then Li Haoyi beat me to it with another addition to his wonderful Strategic Scala Style series: Practical Type Safety. I highly recommend reading it (and the rest of the Strategic Scala Style series) before coming back.

Still here? That post covers a lot of what I wanted to say, but I wanted to put some extra emphasis on one particular topic:

ADTs (Algebraic Data Types) are so, so good. You can take a lot of features away in Scala and I could probably get by, but ADTs – Or at least, the closest Scala approximation – are on the short list that you’d have to pry out of my cold dead hands… Along with higher order functions and pattern matching, probably.

So, a quick review… What are Algebraic Data Types? ADTs are so named because their structure can be described in two operations: Product, and Sum.

Product Types

Product types are present in one form or another in the vast majority of mainstream programming languages. People may know it as a struct in C, or a record, or a tuple, or a case class in Scala. It’s essentially a way of mashing multiple types together into one type. The reason it is called a product type is because the cardinality of the type (i.e., the set of all possible values for it), is the product of the cardinality of the type’s constituents. Some quick examples:

Sum Types

Where product types express “this and that and the other thing”, sum types (also commonly known as union types) express “this or that or the other thing”. Sum types are so named because the cardinality of a sum type is the sum of the cardinality of the type’s consituents. Unfortunately in Scala 2.x, there isn’t direct support for sum types, but they can be roughly approximated with sealed trait. Some examples:

One cheerful note is that proper union type support was announced for Dotty at ScalaDays 2016, so hurrah! *confetti*

So What?

At first glance, ADTs might seem simplistic – kinda like fancied up named tuples. But they are deceptively powerful, because combining them effectively allows you to encode invariants about your program’s logic and state in the type system, and therefore leverage the compiler to keep you from violating those invariants and writing buggy software.

Which sounds like a bunch of abstract nonsense, so it’s time for a concrete example!

Modeling Real Life: Wrangling User Lists

Working in the marketing space, it is very common to deal with lists of users. It can be a list of users from a mailing list, a list of website visitors, or a list of people who have downloaded your white paper. In any case, it is a list of users that you have in your possession, and you want to follow them around the internet and serve them ads. Such a list of users might have attributes like an id and a human-readable name. Since most folks aren’t in the business of building exchanges, it might also have a downstream platform to target (e.g., AdWords, or Facebook). It might also have some different states that keep track of whether the list’s meta-data has been sent to the downstream platform, or whether it’s been archived or soft-deleted and when that state transition happened.

So one reasonable implementation at a data structure for such a user list might look something like:

At first glance, this seems like pretty reasonable code. The various states and platforms are enumerated, and it seems to carry all the information we want with decent naming and so on. At the very least it’s better than carting around a tab-separated string, or something like val list: (Long, String, ListStatus, Long, Option[Long], Option[String]).

But there’s still something a bit off here. In particular, look around the optional fields. With this current model, it is possible to construct an internally inconsistent list:

The example above is internally inconsistent because the list’s active status indicates that it should be onboarded to the downstream platform, but for some reason we do not have a downstream id. There are other “weird” cases like this that are possible but do not make sense semantically – for example, archivedTime = Some(1) with status = Pending.

This data structure is almost trivially simple, but there area already a good number of things that can go wrong, especially once we take outside input and possibly complex business logic into account. And here there’s nothing stopping us from constructing these degenerate cases other than our ability to keep everything in our heads (spotty even at the best of times) and read the code really really carefully every time we work with this particular data structure (good luck).

Another point to consider is what this does to the code that works with the data:

The above code is a conceptually simple method that takes users from our list and sends them to a downstream platform, e.g. Facebook. It technically does what we want, but there is a lot of clutter introduced by the management of the status. The block under list.downstreamId.getOrElse is particularly bad, since it basically amounts to saying “well we don’t think we’re wrong, but we technically could be wrong, so we have to sprinkle this boilerplate-ish error handling into our business logic”. Apparently, this sort of thing is not that uncommon.

One possible remedy is to refactor UserList to encode some of the constraints into the type:

This lets us rewrite the above method as:

At this point, one might protest, “That just makes you push the status validation somewhere else! The first implementation was just bad practice, you could easily have written this simple version with the old type by refactoring to def onboardUsersToList(downstreamId: String, users: Seq[User]).” Yes, this is perfectly true. The validation still has to take place somewhere since the code will presumably be interacting with the outside world. The difference here is that the former implementation can only enforce cleanliness and correctness with malleable things like documentation and best practice guidelines, whereas the latter implementation enforces it by simply refusing to compile until you fix it. The latter implementation also reduces the amount of “unforced” errors since it is impossible to construct an ActiveList object without a downstreamId, whereas it was possible to accidentally create a UserList(status = Active, downstreamId = None) previously.

Stepping back

Hopefully the above example demonstrated a reasonable real-world case where ADTs can be really useful. ADTs seem really basic – and in the vast world of type hackery they are only a starting point – but you can do some surprisingly powerful things with these fundamental building blocks.

There is one last point I want to make. As a professional programmer (i.e., employed by a business to write code that ostensibly generates some amount of profit), the goal of all this is not to encode the world at the type level. The goal is to reduce complexity and mental overhead. There are some insane things possible once you go down the rabbit hole of type hackery – for example, type-level quicksort. This is all great fun and by no means a bad thing, but at some point it’s important to step back and think, “Right, somebody else actually has to read, maintain, and modify this code at some point. Maybe this should not go into production.”

This is especially important to keep in mind in a language like Scala, which provides us with plentiful amounts of power to complicate things and shoot ourselves in the foot if we’re not careful. Or, in other words:


text:
## IACR News item: 13 August 2019

###### Karim Eldefrawy, Vitor Pereira
ePrint Report
Secure Multi-Party Computation (MPC) enables a group of $n$ distrusting parties to jointly compute a function using private inputs. MPC guarantees correctness of computation and confidentiality of inputs if no more than a threshold $t$ of the parties are corrupted. Proactive MPC (PMPC) addresses the stronger threat model of a mobile adversary that controls a changing set of parties (but only up to $t$ at any instant), and may eventually corrupt all $n$ parties over a long time.

This paper takes a first stab at developing high-assurance implementations of (P)MPC. We present high-assurance implementations of (P)MPC protocols. We formalize in EasyCrypt, a tool-assisted framework for building high-confidence cryptographic proofs, several abstract and reusable variations of secret sharing and of (P)MPC protocols building on top of them. Using those, we prove a series of abstract theorems for the proactive setting. We implement and perform computer-checked security proofs of concrete instantiations of all required (abstract) protocols in EasyCrypt.

We also develop a new tool-chain to extract high-assurance executable implementations of protocols implemented and verified in EasyCrypt that uses Why3 as an intermediate tool. Using this new tool, we were able to extract code from our (P)MPC formalizations and to conduct a practical evaluation of it by comparing the performance obtained using our approach to the performance of manually implemented versions using Python-based Charm framework for prototyping cryptographic schemes. We argue that the small overhead of our high-assurance executables is a reasonable price to pay for the increased confidence about their correctness and security.

Additional news items may be found on the IACR news page.


text:
1,962
12
Essay, 18 pages (4500 words)

# Anomalous behavior of highly active helical swimmers

Advertising We'll write a high-quality original custom paper on Anomalous behavior of highly active helical swimmers just for you with a 15% discount for the 1st order

## 1 Introduction

Much of the microscopic world is dominated by living organisms whose sustenance is strongly dependent on their ability to move in fluidic environments. The importance of motility is observed in bacteria’s search for food or spermatozoa’s struggle to reach the ovum. However, the small size of the microorganisms results in a large surface (

$\propto frictionalforce$

) to volume (

$\propto inertialforce$

) ratio (

$\sim {10}^{4}$

). At such low Reynolds numbers, the role of inertia is entirely negligible, which has resulted in the evolution of novel swimming strategies adopted by the microorganisms to overcome the fluidic drag [ 1 ]. A particular strategy that cannot lead to motility at microscopic scales is reciprocal motion. As Purcell [ 2 ] describes Scallop theorem in his “ Life at low Reynolds number”; “…I change my body into a certain shape and then I go back to the original shape by going through the sequence in reverse. At low Reynolds number, everything reverses just fine … So, if the animal tries to swim by a reciprocal motion, it can’t go anywhere”. This kind of motion in bulk Newtonian medium cannot result in net displacement of an individual swimmer.

Indeed, one way to work around [ 3 ] the Scallop theorem is to adopt non-reciprocal swimming strategies. For instance, many bacteria swim by rotating chiral (helical) flagella which leads to translation [ 4 ]. Similar strategies have been used to make synthetic swimmers at small scales, where the rigid magnetic helices propel in the desired direction due to the rotation induced by a rotating magnetic field [ 5 , 6 ]. Flexible filaments subjected to whip-like beating pattern can result in net motion as well [ 7 , 8 ]. Apart from these strategies to break the back and forth symmetry, the physical environment can also result in the violation of the Scallop theorem. For instance, two bodies exhibiting reciprocal actuation with a phase difference exploit the unsteady hydrodynamic flows [ 9 ] created by each other to achieve collective locomotion. Alternately, non-Newtonian properties, e. g., shear thinning [ 10 ] or viscoelastic nature of the surrounding medium can aid in the locomotion of reciprocal swimmers. The fluid elastic stresses [ 11 ] or difference in forward and backward strokes of reciprocal swimming [ 12 , 13 ], can be the source of motion in a non-linear viscoelastic environment.

In this article, we examine a system of artificial reciprocal swimmers built on the experimental platform of rigid magnetic helical nano-propellers, which move under the influence of a rotating magnetic field [ 5 ]. As shown extensively before, this is a versatile, multifunctional bio-manipulation [ 14 16 ], delivery [ 17 , 18 ] and sensing system [ 19 21 ]. Typically, this is a driven system where the sense (clockwise, CW or counterclockwise, CCW) and the plane of rotation of the magnetic field determines the direction of propulsion depending on the chirality of the helix [ 22 ]. This is unlike the other self-propelled synthetic swimmers, where the direction of non-interacting motile objects are independent of each other [ 23 , 24 ]. The helical propulsion system can be changed from driven to active [ 25 ] system by simply changing the magnetic drive from rotating to oscillating field configuration, where the plane of oscillation is perpendicular to the plane (within the microfluidic chamber) where the propellers are experimentally maneuvered. As a result, the orientation of the propellers are independent of the external field, and they undergo back and forth motion, brought about by the CW-CCW rotation of the helices. As shown previously [ 26 , 27 ], a subtle counterplay between the magnetic drive and weight asymmetry of the helices governs the rotational dynamics of the propellers. Note that the time average displacement of a reciprocal helical swimmer is zero in the absence of any diffusion. However, in the presence of thermal fluctuations in the surrounding fluid, these objects can show considerable diffusion compared to their passive counterparts [ 26 , 28 ], which is caused by the orientational fluctuation of the swimmer during the back and forth motion. In summary, the experimental system of reciprocal propellers is a zero torque, zero force active system and the activity can be regulated via frequency of the external magnetic field. The activity of this system can be tuned easily by changing the magnetic drive parameters, and the experimental studies can be performed over long duration which make them an attractive potential system to study the emergence of various interesting active matter properties [ 29 , 30 ] such as the roles of confinement [ 31 , 32 ], surface effects [ 33 , 34 ], and collective interaction [ 35 ].

In this study, we will be specifically discussing the enhancement in diffusivity of the reciprocal swimmers as a function of frequency and the parameter- “ degree of randomness”, described later. In our previous studies [ 25 , 26 ], the enhancement of diffusivity at lower frequencies of actuation was investigated, where the diffusivity increased by 78% compared to the passively diffusing structures. As we show here, new dynamical behavior of the swimmers emerge at higher activity levels (increase in diffusivity up to 300%), including the role of the surface of the fluidic chamber, that was neglected before.

## 2 Results and Discussion

The system of swimmers used here consists of helical structures made of silica and a magnetic material that are fabricated via Glancing Angle Deposition (GLAD) method [ 36 ]. These helices are grown on a substrate patterned with an array of pillars having a diameter of 200 nm. The helical body is made of silica with magnetic material Iron and Cobalt incorporated at one end [ 37 ] of the structure. These 3-micron length propellers are later magnetized along their short axis to have a permanent magnetic moment [ 38 ], as shown in Figure 1B . The swimmers are sonicated in deionized water and laid down on a substrate to image them via SEM, as shown in Figure 1A . The swimmers are suspended in deionized water in a microfluidic device of thickness 20–25 μm, and within few minutes, the swimmers settle down close to the bottom surface, approximately 1

$\sim$

2 microns above the bottom glass surface. For magnetic actuation, a triaxial Helmholtz coil is used where the three pair of coils in xyz direction are powered by amplifiers, thus producing a spatially uniform magnetic field at the center of the coil. The microfluidic chamber is placed at the center of triaxial coil coinciding with the imaging plane of the microscope. A rotating magnetic field due to the X, Y, Z coils drives the swimmers and can be controlled effectively via the strength and frequency of the field. By driving the helices along a specific direction in the xy plane, we find a linear relationship between the translational speeds and the frequencies of the rotating field from where the hydrodynamic pitch of the propellers is estimated to be

${p}_{h}$

= 600 nm. Note the linear relationship is only valid at low frequencies where the drag is not larger than the maximum available magnetic torque. For reciprocal actuation, a sinusoidal signal is fed only to the pair of z-coils resulting in an oscillating field in the z -direction. It is to be noted that the sample is placed in the xy imaging plane, whereas the external magnetic field is perpendicular to this imaging plane. Hence, the permanent magnetic moment vector along the short axis of the swimmer follows the external oscillating magnetic field and turns the propeller clockwise (CW) and counterclockwise (CCW) alternately. This would also result in the propeller moving forward and backwards by a distance of

${p}_{h}/2$

in body frame. Note, the swimmer is subjected to thermal fluctuations in the xy plane, and hence the directionality is not governed by any external force.

FIGURE 1

(A). SEM image of the helical swimmers.(B). A dc-field magnetizes the swimmer with permanent magnetic moment along the short axis.(C). The schematic of a triaxial Helmholtz coil, which enables a uniform magnetic field at the center of the coil. The swimmers are placed in the sample chamber at the center of the coil.(D). The oscillating magnetic field in the z -direction causes the back-forth motion of the propeller.

Previously, in a theoretical work, Lauga [ 28 ] discusses the case of reciprocal swimmers. He found that for time scales larger than the rotational diffusion, swimmers show significant enhancement in diffusivity compared to Brownian motion. The swimmer under such reciprocal actuation experiences enhanced fluctuations with effective diffusion constant

${D}_{rec}$

which is always greater than the Brownian diffusion constant

${D}_{{k}_{B}T}$

. This effect is dependent on the time period of reciprocal actuation, and the orientational diffusion time. In case of Mandal et al [ 26 ], the predicted enhancement in diffusivity is experimentally realized with a system of helical swimmers. They discuss the detailed dynamics of the swimming where the role of asymmetric weight distribution along the propellers’ body is explained. Their numerical simulation took into account that the swimmers are subjected to a sinusoidal drive and the effect of a parameter q , namely degree of randomness; discussed in the following section. However, both these approaches rely on modeling of the swimmer in bulk. While for the experiments reported in this article, we have observed anomalous rise of D for the same swimmer at higher frequencies for which the present theories do not provide an explanation. We believe this needs to be addressed by incorporating an effect of surface, which has not been accounted for in any of the theories.

The effect of enhancement is shown in the two trajectories in Figure 2A corresponding to a propeller under external actuation and another undergoing Brownian motion, tracked using a 50x magnification objective lens and a camera. Both the trajectories are plotted for 120 s, and one of the swimmers was subjected to 80 Gauss of magnetic field strength oscillating at 10 Hz. The corresponding Figure 2B shows the mean squared displacement curves for both the propellers, where one (red) shows an increased diffusivity compared to Brownian propeller (blue). The

${D}_{rec}$

being

$2. 76×{10}^{-13}{m}^{2}{s}^{-1}$

and

${D}_{{k}_{B}T}$

being

$1. 5×{10}^{-13}{m}^{2}{s}^{-1}$

, the

$\text{Δ}D$

is around

$1. 26×{10}^{-13}{m}^{2}{s}^{-1}$

. Ideally, the swimmers should follow a sequence of CW and CCW turns, resulting in a perfect reciprocal sequence (e. g., see Figure 2C ). However, experimentally it has been observed that swimmers do not show a perfect reciprocal sequence. For instance, as shown in Figure 2D , the sequence of CW-CCW has been interrupted twice (red arrows) and this is due to the rotational diffusion around the long axis of the propeller which introduces a degree of randomness [ 26 ], “ q ”. In this case, the q takes the value of 2/10. The behavior of the propellers at lower frequencies and the importance of q was discussed in the prior work [ 26 ]. Here we extend this to much higher frequencies and point out a new, distinct effect where the value of q is found to depend on the actuation frequency for certain swimmers. Note the degree of randomness q has a direct effect on the enhanced diffusivity of the swimmers.

FIGURE 2

(A). The typical trajectory is comparing the Brownian and an active propeller. The propeller (red curve) is performing reciprocal motion as a response to an external field of amplitude 80G and frequency 10 Hz.(B). Comparison of enhancement in diffusivity for the Brownian and active propeller.(C). The enhancement of diffusivity is closely related to the ‘ q’ parameter, which is a ratio of broken reciprocity. The table shows a perfect sequence of clockwise-counterclockwize rotations about the long axis of the propeller.(D). The schematic indicates two instances (red arrows) where the sequence is broken, giving a q value of 2/10.

As shown in Figure 3A , we performed numerical simulations to investigate the relation between the frequency of the magnetic field, degree of randomness and

$\text{Δ}D$

. The

$\text{Δ}D$

always increases with frequency (activity), which is larger for higher values of q . Physically, this dependence on q can be understood as a larger step size of the back and forth motion and therefore, more significant enhancement of the diffusivity. We also show experimental results of

$\text{Δ}D$

as a function of frequency, and this is shown in Figure 3B for two propellers, P1 and P2. The critical point to note in Figure 3B is the difference between the P1 (blue) and P2 (red) propellers. While the

$\text{Δ}D$

for P1 increases monotonically with frequency, it is clear that the values of q for this propeller lies between 0 and 0. 1, without any apparent dependence of q on the applied frequency. The swimmer named P2 appears to have a changing q with frequency.

FIGURE 3

(A). Simulations are carried out to delineate the effect of ‘ q’ on the enhancement of diffusivity. As the frequency of the field is increased, the enhancement increases too. However, the introduction of q allows for much higher activity even at the same frequency.(B). Enhancement of two sample swimmers is plotted onto the simulated results. It is evident that the P1 (red) and the P2 (blue) swimmers behave differently.

Before delving into the discussion regarding this dependence of q on frequency, we will discuss a way to determine this degree of randomness experimentally. Indeed, one way is to image and analyze the rotational dynamics of the swimmer, but this may not be possible to perform for experiments where turning of the swimmer may be difficult to observe, e. g. when the helix dimensions are smaller. Here, we show that the q value can be directly deduced from the time domain Fourier space of the body frame displacements along the long axis of the swimmers. The positions of a swimmer in the lab frame can be tracked from the video. The algorithm detected the propeller as a rod, and its orientation with respect to the x -axis of the lab frame was recorded. It is necessary to record the video at a sampling rate equal to or more than 2 f , f being the frequency of actuation. The displacement value along long axis of the propeller typically represents the step size in half a cycle of actuation. This provides us with a series of alternately positive and negative displacements corresponding to the back-forth motion. Typically, a Fourier transform of such a series must result in a peak at the actuating frequency, where the amplitude of the peak is inversely related to the q . This is expected because, with an increase in q , the periodicity is disrupted and results in a decrease in the peak amplitude. The fast Fourier Transform was implemented in MATLAB for a sequence of

$\text{Δ}x$

displacements of total length T.

$y ( k )=∑ t = 1 T Δ x ( t ) e (− 2 π i )( t − 1 )( k − 1 )/ T ,( 1 )$

In Figure 4A , the simulation shows a peak at the actuating frequency, here 20 Hz for total time T = 100 s and the variation of peak amplitude with q . However, Figure 4A represents an ideal case where the sampling rate of data (here the images per second) has been assumed to be constant.

FIGURE 4

(A). The simulation where the Fourier space is plotted for different ‘ q’ values. It shows a prominent peak at the actuating frequency and variation of Fourier amplitude and width of the peak as a function of ‘ q’.(B). The simulated Fourier space is plotted (red) for an actuation frequency of 40hz and q = 0. 2. The other curve (blue) indicates the same curve but incorporating errors arising due to video capture. Though the peak widens, the area under both the curves remains the same.(C). Comparing the P1 and P2 swimmers at a lower frequency (5 hz) and higher frequency (40 hz).(D). The surface mediated effects might influence the dynamics at lower and higher frequency of actuation. There is a possibility that at higher frequencies, the propeller might not have enough time to ‘ see’ the surface.

On the other hand, during the recording of the experiments, the camera might introduce errors, such as a possibility of dropping of frames or variation in the frame rate. We simulate a condition where if

${F}_{s}$

is the rate per second at which images are captured, δ is the error introduced as the fluctuation in the frame rate. A

$±$

10% probability of frame rate being

${F}_{s}$

$±$

2 is considered using randomly drawn numbers from uniformly distributed set between [0, 1]. The results as shown in Figure 4B , the Fourier peak grew wider and shorter than the ideal case. However, the area under the curve remained same in both the cases; highlighting a vital observation that despite the shortcomings in imaging, Fourier space of the displacements is an excellent way to estimate q .

We believe the anomalous behavior of propeller P2 which exhibits a variation in q with frequency (as shown in Figure 3B ) can be explained by taking into account the role of surface of the microfluidic chamber. As observed in previous study [ 26 ], two main factors affect the dynamics of the helical swimmers. First is the asymmetric distribution of weight along the body of the swimmer. Second is the role of noise around the long axis which is manifested in rotational diffusion around the long axis, i. e., a type of wobbling [ 38 40 ] motion superimposed on the back and forth rotation about the long axis Figure 4D is a schematic of the phenomenon we propose, which could explain the anomalous behavior of swimmer P2. During the experiments, the swimmers (due to gravity) settle down at a distance of 1–2 microns away from the surface. Now, the idea relies on the fact that the swimmers while being rotated by the applied magnetic drive, can move along the vertical direction when their moments are not perfectly perpendicular to their long axis; thus, resulting in a periodic motion along the vertical direction. The sedimentation time (approximately at a rate of

$9×{10}^{-6}m/s$

) for a typical distance of 1 micron is about 0. 11 s. In one limit, when the magnetic field frequency is lower than 10 Hz, the swimmer would be going up and down (vertically) while rotating about its own axis due to the reciprocal drive. At lower frequencies, as shown in Figure 4D , in one cycle of oscillation, half of the swimmer approaches the surface whereas the other half is fairly unaffected. This prevents the swimmer from being affected by surface uniformly across the body length. In the other limit, when the frequency is higher, the swimmer would remain in the same z-plane. Thus, at higher frequencies, swimmers are affected by the surface uniformly throughout the body length. So, during the event of turning, the balance of torques due to gravity and magnetic moment is also affected by presence of surface uniformly across the body. This is evident from Figure 3B where after 10 Hz, the two propellers seem to follow different trends; with P2 showing variation in q .

## 3 Conclusion

We present experimental and numerical study of magnetically driven helical swimmers at higher magnetic drive than what has been applied before. As explained in previous papers, under oscillating magnetic fields, these swimmers show back and forth motion due to interplay of magnetic torque and weight asymmetry. Their diffusivities can be significantly higher than the passively diffusing (not driven) structures due to the role played by orientational diffusion. The reciprocal sequence can be imperfect, determined by a parameter called degree of randomness, which results in even higher diffusivity. Here, we have shown a new method to determine the degree of randomness experimentally and have shown this parameter can be affected by the presence of the surface. This has resulted in anomalous increase in diffusivity in certain swimmers. It will be interesting to check how a collection of swimmers behave in dense suspensions where they interact with each other through hydrodynamic forces.

## Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

## Author Contributions

AG and GP developed the idea. GP performed the experiments and numerical simulations. Both the authors discussed extensively during this research and prepared the manuscript together. AG supervised the work.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

The authors would like to thank Pranay Mandal for helpful discussions and contributions in the building of experimental set-up. We gratefully acknowledge the Micro and Nano Characterization Facility (MNCF, CeNSE) and National Nano Fabrication Center (NNFC, CeNSE) at IISc. We thank SERB for funding this research. We also acknowledge funding from MHRD, MeitY and DST Nano Mission for supporting the facilities at CeNSE.

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text:
# How do you integrate ln(x+sqrt(x^2+1))?

Jun 5, 2016

By parts.

#### Explanation:

You can integrate it by parts with the rule

$\setminus \int f ' \left(x\right) g \left(x\right) \mathrm{dx} = f \left(x\right) g \left(x\right) - \setminus \int f \left(x\right) g ' \left(x\right)$

where we assume that

$f ' \left(x\right) = 1$ and $g \left(x\right) = \ln \left(x + \sqrt{{x}^{2} + 1}\right)$

consequently

$f \left(x\right) = x$ and $g ' \left(x\right) = \frac{1}{\sqrt{{x}^{2} + 1}}$.

The integral is then

$\setminus \int \ln \left(x + \sqrt{{x}^{2} + 1}\right) \mathrm{dx}$

$= x \ln \left(x + \sqrt{{x}^{2} + 1}\right) - \setminus \int \frac{x}{\sqrt{{x}^{2} + 1}} \mathrm{dx}$

$= x \ln \left(x + \sqrt{{x}^{2} + 1}\right) - \sqrt{{x}^{2} + 1} + C$.


text:
# Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I have a few questions on an application of the Weyl character formula.

To start with we work with the $\mathbb{Q}$ version of Hamilton's quaternions and consider the maximal order $\mathfrak{O} = \mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}\frac{1+i+j+k}{2}$.

Now consider the group:

$\Gamma_1 = \left\{A\in M_2(\mathfrak{O}^{\times})\,|\,A = \left( \begin{array}{ccc} a & 0 \\ 0 & b \end{array} \right) \text{ or } A = \left( \begin{array}{ccc} 0 & a \\ b & 0 \end{array} \right) \right\}$

This is a subgroup of the Lie group Sp$(2)$ and is very simple to calculate.

I wish to use the Weyl character formula to evaluate $\sum_{\gamma\in\Gamma_1} \text{tr}(\rho_v(\gamma))$, where $\rho_v$ is the irreducible representation of Sp$(2)$ of highest weight $v(L_1+L_2)$ (using standard notation for $L_1,L_2$).

Now my plan (in order to be able to use the Weyl character formula to get actual traces and not just formal characters) is to shift focus from Sp$(2)$ into Sp$(4,\mathbb{C})\cap$ U$(4)$. There is an isomorphism between the two that behaves as follows on elements of $\Gamma_1$:

$\left(\begin{array}{ccc} a+bi+cj+dk & 0 \\ 0 & \alpha+\beta i+\gamma j + \delta k \end{array}\right) \longmapsto \left( \begin{array}{ccc} a+bi & 0 & c+di & 0\\ 0 & \alpha+\beta i & 0 & \gamma + \delta i \\ -c+di & 0 & a-bi & 0\\ 0 & -\gamma+\delta i & 0 & \alpha - \beta i \end{array} \right)$

$\left(\begin{array}{ccc} 0 & a+bi+cj+dk \\ \alpha+\beta i+\gamma j + \delta k & 0 \end{array}\right)\longmapsto\left( \begin{array}{ccc} 0 & a+bi & 0 & c+di \\ \alpha+\beta i & 0 & \gamma + \delta i & 0\\ 0 & -c+di & 0 & a-bi \\ -\gamma+\delta i & 0 & \alpha - \beta i & 0 \end{array} \right)$

My first question is whether the character of the rep $\rho_v$ of Sp$(2)$ is the same as the character of the similarly defined "$\rho_v$" on the Lie group Sp$(4,\mathbb{C})$ but restricted to Sp$(4,\mathbb{C}) \cap$ U$(4)$? The restriction is important here in order to get a rep on a compact Lie group (hence allowing me to guarantee conjugation into maximal torus to always be possible).

If so then I know that the maximal torus in Sp$(4,\mathbb{C}) \cap$ U$(4)$ consists of matrices of the form:

$\left( \begin{array}{ccc} x & 0 & 0 & 0\\ 0 & y & 0 & 0 \\ 0 & 0 & x^{-1} & 0\\ 0 & 0 & 0 & y^{-1} \end{array} \right)$

where $x,y$ are complex numbers of modulus $1$.

So, when I use the Weyl character formula for the Lie algebra $\mathfrak{sp}_4(\mathbb{C})$ I get the following large expression for the character:

$\frac{e^{(v+2)L_1 + (v+1)L_2} - e^{(v+2)L_1 - (v+1)L_2} - e^{-(v+2)L_1 + (v+1)L_2} + e^{-(v+2)L_1 - (v+1)L_2} - e^{(v+1)L_1 + (v+2)L_2} + e^{-(v+1)L_1 + (v+2)L_2} + e^{(v+1)L_1 - (v+2)L_2} - e^{-(v+1)L_1 - (v+2)L_2}}{e^{2L_1 + L_2}(1-e^{-2L_1})(2-e^{-2L_2})(1-e^{L_2-L_1})(1-e^{-L_1-L_2})}$

The rest of my question is as follows. First, is this correct? Also may I shift this into this formula for the character on the Lie group Sp$(4,\mathbb{C}) \cap$ U$(4)$ (where $T$ is an element of the maximal torus as above):

$\chi(T) = \frac{x^{2v+4}y^{2v+3}-x^{2v+4}y-y^{2v+3}+y-x^{2v+3}y^{2v+4}+xy^{2v+4}+x^{2v+3}-x}{x^vy^v(x^2-1)(y^2-1)(xy-1)(x-y)}$

I got this by using the Lie algebra version above and rewriting multiplicatively, then rearranging.

If I am correct so far then can this be simplified in any way?

-


text:
## Atto: Matrix editor

atto_matrix
Maintained by Geoffrey Rowland
Moodle Atto Editor plugin for TeX/MathJax Matrices and Arrays.
91 sites
5 fans

Moodle Atto Editor plugin for TeX/MathJax Matrices and Arrays.

Edit matrices and arrays in Moodle 2.7, or later, using the MathJax or TeX filter. Includes templates for 1xn 2xn, 3xn and mxn matrices and a range of brackets and braces.

Requirements:

1. The MathJax filter to be enabled (recommended) or the TeX filter enabled (or, in Moodle 2.8, both)

Don't forget to also enable the appropriate filter in the relevant course:

2. The matrix editor button (matrix icon) to be added to the Atto editor toolbar:

Edit Administration > Site administration > Plugins > Text editors > Atto HTML editor > Atto toolbar settings > Toolbar config, to give, for example:

insert = equation, matrix, charmap, table, clear

Immediately after installation and configuration you may need to clear your Moodle server caches:

and local browser cache for correct display.

### Awards

• Sat, 3 Oct 2015, 10:31 PM

Hi. Thanks for sharing this plugin with the community. The plugins team is currently busy with other development tasks for the upcoming Moodle 3.0 release, so please expect delays in the review and approval process. Thanks for your understanding and patience, we will do our best to get to reviewing your plugin soon.

• Mon, 5 Oct 2015, 5:12 PM
Unsure how to deal with the line-length warnings and errors for settings.php. Effectively, settings.php is acting like a lang file.
Will attend to the others.
• Mon, 5 Oct 2015, 5:26 PM
On inspection, am also unsure of how to deal with other warnings and errors. Code structure is similar/identical to (derived from) existing atto_equation and similar Atto plugins,
• Thu, 15 Oct 2015, 5:35 PM

No worries Geoffrey. I checked the generated warnings and they do not seem valid in many cases. It is indeed ok to format the long strings in your settings.php this way. What seems to be valid is the extra @param line in the docblock for your atto_matrix_strings_for_js() as that function does not actually accept any parameters. Other warnings seem to be raised because of using the reserved keyword in the Based on @package .... You should be able to get rid of them by simply removing the @package from there, if you want.

What caught my eye is that you are including \$OUTPUT->header() in your ajax.php response. I think it should be removed and all you want to return is the filter_text() output. I believe the header() does not produce any content in this case anyway, right?

This is really nice UI widget and front-end for the filters. Good idea, nice design and well implemented. I am happy to approve it now. You are cleared to land, welcome to the Plugins directory!

• Wed, 7 Dec 2016, 8:43 PM