text:
# Definite double integral of sin(y²)

I’ll be honest: I had no idea how to do this at first, but then I thought that it was probably easier than I could imagine. What most likely blocks you when you see this is the function inside the integral, which, as we saw here, is a special function, meaning its antiderivative is not an elementary function. This is the only way to express the result:

The function S is not a function we know, it could have also been A, K, G.

But the fact that the result is just S(x) could actually be good for the resolution of our double integral. Let’s see what we can do:

First of all, we calculate the inner integral, which is with respect to y. Let’s first treat it as an indefinite integral:

This is exactly what we have just seen, the only difference being the ‘y‘;

Now we can plug in the bounds of integration, x and 1:

At this point you may wonder what S(1) is. This is what I asked myself too, but then I thought that maybe there was no need to evaluate it, so I just kept going.

This is the result of the inner integral, and now we have to integrate it with respect to x (this is the outer integral with bounds 0 and 1):

Since it is a difference, we can apply linearity and obtain two integrals:

Inside the first integral we have a constant, because there is not the variable x, therefore

and so

Now, what on earth is the integral of S(x)? Again, despite how it looks, it’s actually very simple using integration by parts, the exact same way we found the integral of ln(x).

It would make no sense to let dv be S(x), since that way we would have to integrate it, but that’s exactly what we are trying to do; instead, we can let u=S(x) so that we can take the derivative: if the integral of sin(x²) is S(x), the derivative of this last function is the function inside the integral. Therefore,

As you probably know, the result is uv-int(vdu);

Remember the bounds, since it’s a definite integral. We can plug them in the first part:

and so

But wait: if you remember, we had

and what we’ve just found is this integral, so we can replace it with the result:

As you can see, S(1)-S(1)=0. How amazing is this?

At this point we are left with this integral, which is actually very easy to evaluate performing a u-substitution:

let’s let u=x², so that its derivative will cancel out with the x in front of sin:

Don’t forget that the integral is with respect to x, which means the bounds are values of x. Since we are performing the u-substitution we need to find the bounds of u; since u=x², u1 (lower bound)=x1²=0, and u2 (upper bound)=x2²=1.

The x‘s cancelled out and the 1/2 was brought to the front of the integral (linearity). This is one of the “fundamental” integrals: the antiderivative of sin(x) is -cos(x).

If you set your calculator to rad you’ll get that cos(1) is approximately 0.54, therefore the final result is (-0.54+1)/2=0.46/2=0.23.

Overall,

or, if you prefer,

That’s it! I hope you liked this post and that you found it interesting! If you did, give it a like and subscribe to stay updated!

Join 30 other subscribers.


text:
# Storage on Palmetto

## Overview

Teaching: 15 min
Exercises: 0 min
Questions
• How and where can I store my files?

Objectives
• home directory, scratch space

Every Palmetto user gets 100 GB of storage space; this storage is backed up at the end of every day, and the backup is kept for 42 days. So if you accidentally delete a file that was created more than a day ago, we might be able to restore it. This storage is called home directory.

To see how much space you have left in your home directory, please type:

checkquota

Since most of you are new users of Palmetto, you should be using very little storage at the moment.

When you log into Palmetto, you end up in your home directory. To see which directory you are in, type

pwd

…which stands for “print working directory”. It should give you something like

/home/<your Palmetto username>

For ever Palmetto user, their scratch space is located in /scratch1/<username> folder. You can access it with the cd (“change directory”) command:

cd /scratch1/<your Palmetto username>

To go back to your home directory, you can do

cd /home/<your Palmetto username>

There is also a shortcut; to go to your home directory, you can simply type

cd

Here, I will not go into details about Linux commands. Some of you have taken our Linux workshop. There are many online tutorials, my favourite is this one. Please spend some time getting familiar with going between the directories, as well as with copying, moving, and deleting files.

format: plain


text:
## Algebra 1

Published by Prentice Hall

# Chapter 1 - Foundations for Algebra - Chapter Review - 1-7 The Distributive Property: 82

#### Answer

$y-3$

#### Work Step by Step

The distributive property states that for any real numbers $a, b$, and $c$, $a(b+c) = a\cdot b + a\cdot c$ Use the formula above to obtain: $=(6y-5y)-3 \\=(6-5)y - 3 \\=1(y)-3 \\=y-3$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Simple (but wrong) argument for the generality of positive beta-functions

In the introduction (page 5) of Supersymmetry and String Theory: Beyond the Standard Model by Michael Dine (Amazon, Google), he says

(Traditionally it was known that) the interactions of particles typically became stronger as the energies and momentum transfers grew. This is the case, for example, in quantum electrodynamics, and a simple quantum mechanical argument, based on unitarity and relativity, would seem to suggest it is general.

Of course, he then goes on to talk about Yang-Mills theory and the discovery of negative beta-functions and asymptotic freedom. But it is the mention of the simple but wrong argument that caught my attention.

So, does anyone know what this simple argument is? And how is it wrong?

-
I think this is very appropriate here, and in fact this is exactly the kind of question I'd like to see more of (advanced but not research; should have a well-defined answer). But... I don't know the answer. –  wsc Jan 13 '12 at 14:42
@Ron: Thanks for the comments Ron. I think I might put down a bounty on this question to try to get a more explicit answer. –  Simon Feb 19 '12 at 9:44
@Simon: I'll do it. –  Ron Maimon Feb 20 '12 at 13:21
I wrote to Michael Dine. It's an argument about the spectral representation but not what anyone has said so far. He says he'll forward the details a few days from now. –  Mitchell Porter Feb 21 '12 at 5:40
Ron: Thanks for pointing your rep points on the line (I was going to wait until the weekend). I hope that @Mitchell gets a reply soon. –  Simon Feb 25 '12 at 5:11

Michael Dine's response, quoted with permission:

I now have to think back, but the argument in QED is based on the spectral representation ("Kallen-Lehman representation"). The argument purports to show that the wave function renormalization for the photon is less than one (this you can find, for example, in the old textbook of Bjorken and Drell, second volume; it also can be inferred from the discussion of the spectral function in Peskin and Schroder). This is enough, in gauge theories, to show that the coupling gets stronger at short distances. The problem is that the spectral function argument assumes unitarity, which is not manifest in a covariant treatment of the gauge theory (and not meaningful for off-shell quantities). In non-covariant gauges, unitarity is manifest, but not Lorentz invariance, so the photon (gluon) renormalization is more complicated. In particular, the Coulomb part of the gluon ($A^0$) is not a normal propagating field.

-
Thanks Mitchell, I forgot about this until Manishearth reminded me in the comments above. It would be nice if the answer was more detailed, but I'm happy with Michael Dine's response. (Your bounty will arrive in ~24 hours) –  Simon Apr 11 '12 at 7:04

This is a temporary answer in order to store the generous bounty that Ron offered. When a proper answer to this question is given, I will transfer the 500 rep points (assign an equal bounty) to that answer.

Going by the totalitarian principle of quantum mechanics / quantum field theory, since this move is not explicitly forbidden, it must be compulsory.

-
What an answer! Full bounty! –  Ron Maimon Feb 27 '12 at 14:47

I may be wrong, but the following remark at http://alumnus.caltech.edu/~callaway/trivpurs.pdf (PHYSICS REPORTS 167, No. 5 (1988) 241—320) may be relevant: "Then [2.1] it follows that if $[x, y] = \triangle_+(y-x)$ possesses properties implied by the Garding—Wightman axioms, then... the associated field theory is a generalized free field, i.e., it is a trivial theory." Reference [2.1] there is B. Simon, The $P(\varphi)^2$ Euclidean (Quantum) Field Theory (Princeton, 1974).

By the way, you can just ask the author, M. Dine. Sometimes asking the author is the only way to sort out what (s)he wrote:-). I remember I found a book containing a result I had recently obtained myself, but without any proof. I e-mailed one of the two authors of the book and asked for the relevant reference. It took me a couple of months, but eventually he advised me that they meant something different from the result that I obtained, so my result was new:-)

-
This is not an accurate statement of the remark. The theorem is that if the commutator is equal to the free commutator than the theory is free. This is both a trivial result (made to sound nontrivial) and unrelated to the beta function properties. The rigorous literature is superficially impressive in this field, but all of it is essentially less than worthless. But no downvote--- I appreciate the sincere effort. –  Ron Maimon Feb 21 '12 at 1:05
@Ron Maimon: Could you please explain that? Do you think the statement in the Callaway's article is wrong? Because I don't see how the assumption in the statement is "if the commutator is equal to the free commutator". As for relevance to the beta function properties, I guess if the beta function is positive, the charge is screened, and the theory is non-interacting. –  akhmeteli Feb 21 '12 at 3:01
Apologies, I wrote the first comment without reading the paper you linked. The paper is not rigorous nonsense, it's a good review of triviality in scalar field theory, with a rigorous subsection. But the result you quote is just saying that if a theory obeys Wicks theorem then it's a free field theory. The result I thought you were talking about is a somewhat deeper: if the two point function is exactly free, than the theory is free. Neither have anything obvious to do with the beta-function, but they are positive spectral weight results, and so they are Kallan-isms. –  Ron Maimon Feb 21 '12 at 15:04
just because the beta-function is positive doesn't mean that the theory is trivial. The beta-function could be positive with a strong coupling fixed point, or even a fixed point at relatively weak coupling. Not every growing coupling has to blow up to infinity. Its a different question. But at least now I know what you were thinking, it makes sense why you would give this answer. –  Ron Maimon Feb 21 '12 at 15:06


text:
Is there a website similar to FlightRadar24, but showing general aviation?

As most engine driven aircrafts have at least a mode C transponder, is there a site that displays such traffic as well?

• A lot of gliders are also visible on FR24, since many use an ADS-B-like system called FLARM. – Sanchises Sep 15 '15 at 15:41
• Checkout adsbexchange.com – Steve Kuo Mar 28 '18 at 18:21

Yes, and no.
FlightAware (and I believe FlightRadar24) will show General Aviation aircraft (I panned around a bit and found N802AF, a Pilatus, buzzing around over Bethlehem, Pennsylvania just now).

The catch is most of these systems are fed with ATC data, and generally only pick up flights that are either on an instrument flight plan, or which have picked up flight following & been entered into the ATC system by the controller who set up the strip.

Most light GA aircraft are flying around VFR (transponder code 1200), and don't show up on these flight tracking sites. Similarly if you're assigned a "local" transponder code rather than a system-wide one you typically won't show up on the tracking sites either (this is often the case when you're flying VFR and go through Approach/Departure for Class C or Class B transitions/arrivals).

As more of the fleet is equipped with ADS-B this coverage may improve, as anyone can set up an ADS-B receiver and skim the traffic data. That data is currently only broadcast when a "participating aircraft" (ADS-B Out equipped) is in the area though...

• If I get flight following for a portion of my (GA VFR) flight, am I able to access history data somehow online once I'm on the ground? – CJBS Mar 21 '14 at 6:46
• Scrap that - seems to be available here: flightaware.com/live/flight/N00001 (where N00001 is the tail number). Fascinating! Most recent flight is listed, and it says Registered users (registering is free and easy!) view 4 months history. I might just do that. – CJBS Mar 21 '14 at 6:49
• These systems use crowd sourced ADS-B data, in which case Mode-S equipped GA aircraft showup. – Steve Kuo Mar 28 '18 at 19:48
• @CJBS I was an airborne systems test engineer in upstate NY for several years. When we would operate circuits VFR in and out of the Class C and generally would request flight following. Our track never appeared on FR24 or FA because we were assigned a local xponder. However, when the plane would come from Utica via VFR it would transition through two ATC centers and receive a big boy xponder and show up on FA/FR24. So I guess you need IFR, ADS-B, or to transit multiple ATC centers to show up. – user28387 Mar 28 '18 at 20:52

TL;DR

To properly track GA aircraft, one needs to operate professional equipment or an ATC datastream, both which are not available to most websites.

Flight radar 24 shows GA traffic in some occasions, but in general it cannot track these aircraft.

FR24 works primarily on ADS-B, which is a surveillance technology that transmits aircraft information including position. Strictly speaking, ADS-B messages are not responses since they are unsolicited transmissions.

General aviation equipped with only a Mode A/C (/S) transponder will not transmit ADS-B data. Mode A/C radar replies do not contain position information encoded in them so even if the receiver receives these responses, the position cannot be determined from them.

Only if a message is received by 4 receivers which are carefully synchronized in time, it is possible to determine the position that the message was sent from by analyzing the times of arrival at each receiver. The travel distance of the signal to each receiver is different and even though the signal travels at the speed of light, the time difference of arrival is measurable. Using the 4 times of arrival the position can be revealed using a process called multilateration.

FR24 is using this method for aircraft that transmit Mode S replies but for Mode A/C it is difficult. Firstly there is no aircraft identifier the A/C messages. If a lot of aircraft are flying with the same Mode A code (e.g. 1200 VFR) the arrival times quickly become ambiguous. To add to the difficulty there is no difference between the Mode A and Mode C message formats, so you might easily be comparing the time of a Mode A message from one receiver to a Mode C message from another. Professional multilateration systems used for ATC make use of transmission antenna's that interrogate the aircraft transponders (like radar) to remove any ambiguities. This is not allowed for hobby systems.

Additionally FR24 works on 1090 MHz ADS-B messages. In the USA, an additional ADS-B technology is in use on 978 MHz (UAT). So even if GA is ADS-B equipped, in the USA they might not be detected by FR24.

So to properly track GA aircraft, one needs professional equipment or an ATC datastream, both which are not available to most websites.

The FAA provides 5 minute delayed radar track of aircraft, but only of those aircraft that fly on an IFR flight plan are included. Also aircraft owners can opt-out of that system for privacy reasons.

There are some websites that show GA movement. One that I like is the monitoring website for residents living near Amsterdam airport Schiphol. The website is fed by ATC data, with a delay of about 1 minute. It shows all traffic except military.

• Wow. I learned a lot from this. – masonk Apr 5 '15 at 10:34

FlightAware picks up GA traffic, but only if it is on a flight plan (which sometimes includes VFR aircraft receiving flight following) or if the aircraft is ADS-B Out equipped and within their ADS-B coverage areas.

Either way, you're not currently going to get a full picture of the environment from any source - not in the way that ATC does, at least. Public sources aren't going to show the many, many aircraft not squawking unique codes.

• Do you know if FlightAware picks up/shows all ADS-B flights, or do they still filter VFR (1200) traffic? (I'm dragging my feet on ADS-B, for want of a GPS I love, rather than just one I can tolerate, so I'm not up on how good the coverage is for those aircraft...) – voretaq7 Jan 7 '14 at 19:19
• @voretaq7 Their FAQ addresses that... they were hoping to show all flights by end of 2013. – egid Jan 7 '14 at 19:21
• ah... that whoosh I heard was the deadline sailing by then! :-) – voretaq7 Jan 7 '14 at 19:27

You are right @Force the information presented on flightradar24.com are based on transponder outputs and completed with information stored in a database. Unfortunately a "simple" mode C or S is not sufficient for this. Flightradar24 gets its data not from ATC radars but from private suporters who run a receiver for so called ADS-B outputs from the aircraft. These are more advanced digital information then only the squawk, registration/callsign and altitude of an aircraft used by airliners. But more and more private owners use this technology and that's why you can look foreward to track more and more G/A traffic on flightradar24.com.

I know that flightaware.com also uses this data and additionally uses data provided by the FAA (with a five minute delay) as well. But you won't be able to track any aircraft with a VFR flight plan - unless he is ADS-B equipped - becuase in the US these flight plans are only used for SAR.

A little additional remark: If you observe the German airspace at flightradar24.com you'll sometimes find some microlights on the 'scope' - sometimes I think these aircraft are 100kg aircraft and 250kg avionics.

The remarkable Skylines project offers tracking based on data transmitted from your smartphone, e.g. using the great and free XCSoar software. Quite glider-centric, though.

• Oh good to know. I actually know the guy who did a lot of XCSoar's recoding :) – Force Jan 7 '14 at 21:19
• @Force thank you very much for your efforts! I also updated the Skylines URL. – yankeekilo Sep 6 '19 at 14:30


text:
# Math Help - Formula to work out tax payable

1. ## Formula to work out tax payable

Hi, I'm trying to work on a formula that can calculate the tax payable given that we have different tax rates for different thresholds. So for example, let's say we have the following tax table.

 Taxable Income Marginal Tax Rate $0 -$10,000 10% $10,0001 -$20,000 15%

So, if our income is $x, is there some formula I could use to calculate the tax payable. I know that this can be very easily calculated once we have$x. Although is there a formula as a function of $x that could calculate the tax payable? I was initially thinking of using matrices, although I'm not too sure how that would work. Any suggestions would be appreciated. Thanks 2. ## Re: Formula to work out tax payable If I am interpreting "marginal rate" correctly, it is very simple: if I (income) is less than$10000, the tax is .1I or I/10.

If I is larger than or equal to $10000, the tax is$1000+ .15(I- 10000).
That "$1000" is, of course the tax on the first$10000 income, and then you apply the 15% rate to the remainder: if you earn $45000, you pay 10% of the first$10000, or $1000, plus 15% of 45000- 10000= 25000 so you pay .15(25000)=$3750 on the rest for a total tax of \$4750.


text:
## Copy Date/Time to Windows Clipboard with C++

December 20, 2010 at 12:44 (c++, windows)

I was unable to find a suitable program to quickly generate a timestamp in the format I desired, so I decided to write one using C++, specifically MinGW’s g++ v4.5.0 along with <windows.h>. This is my first real experience with Windows-specific programming other than a few hello worlds, so it was instructive as well as frustrating. It took considerable time to discover and familiarise myself with SetClipboardData(). Eventually I was able to modify the code in this thread (which didn’t compile as-is) and obtain a working clipboard prototype very similar to this, which I didn’t find until afterward. After that, it was mainly a matter of understanding the functions in <ctime>.

The format I was after is this:

2010-12-20 07:44:19 UTC-5

However, the code can of course be easily modified for various other formats. Here is the code:

## Simple Asterisk Pattern with Recursion

October 18, 2010 at 14:23 (code golf, java)

Background: I introduced code golf in a previous post.

I recently came across an easy but fun problem posted on Yahoo! Answers. It could make for a good code golf exercise. I will write the problem statement in a purposely vague manner, but clear enough to make the goal obvious; specifics can be added according to any particular contest or testing environment.

Using only recursion and no loops, print a triangular pattern of asterisks according to the following example:

Input:

1
3

Output:

*
*
**
***
**
*

## Reflexive Assignment in Java

October 11, 2010 at 06:54 (code golf, java)

Background: Code golf is a game in which the goal is to solve programming problems using as few keystrokes as possible. It has its roots in Perl golf, and anyone with golfing experience knows that Perl is particularly well suited to such challenges. J and K, as well as the more esoteric GolfScript, are also known for combining terse syntax with powerful features. Obfuscation is a related topic.

In this post, I will discuss a snippet of Java code one might not normally think to write. I understand that “Java code golf” is practically an oxymoron, since it seems the goal of Java is to write code using as many keystrokes as possible, but let us overlook this point for now.

Suppose we want to increment a variable by an integer from standard input, but only if that integer is positive. We could use this code fragment:

## SPOJ TEST in LOLPython

October 2, 2010 at 06:07 (esoteric language, tutorial)

Background: I introduced SPOJ in a previous post.

If you haven’t heard, LOLCODE is a powerful programming language used mainly by cats. It is deplorable that such a beautiful and natural language has not received wider support thus far. To help remedy the situation, I decided to code a solution to TEST using LOLPython, which is derived from LOLCODE and is just as elegant. Of course the program cannot be submitted because neither LOLCODE nor LOLPython is currently supported by SPOJ. But we must have faith that one day such blatant discrimination against cats and their programming languages will end, and we must prepare for that glorious day!

I can haz code?

WHILE I CUTE?
MENUMBR CAN HAZ NUMBR raw_input WIT OK!
IZ MENUMBR KINDA LIKE 42?
KTHXBYE

Save a kitten – ask for LOLCODE support in your local programming competition (and syntax highlighter) today!

## Java Custom addSlashes()

October 1, 2010 at 06:56 (java, javascript, tutorial, web dev)

Java regexes can get pretty clunky due to the dual use of the backslash character, as is commonly known. Although there’s not a lot to it, below is some code to add slashes for these characters: backslash, line feed, carriage return, null, single quote. It is used in a web application to format JSON strings on the server before passing to the client, where the objects are then processed by JavaScript/Ext JS. Maybe someone will find it useful!

public class StringUtil {
public static String addSlashes(String s) {
s = s.replaceAll("\\\\", "\\\\\\\\");
s = s.replaceAll("\\n", "\\\\n");
s = s.replaceAll("\\r", "\\\\r");
s = s.replaceAll("\\00", "\\\\0");
s = s.replaceAll("'", "\\\\'");
return s;
}
}

Incidentally, there are other places where the following JavaScript replacement is necessary:

s = s.replace(/"/g, '"');

## Frobenius Numbers – Round Robin Algorithm

September 20, 2010 at 14:50 (algorithm, java, math.NT)

Frobenius numbers are solutions to the coin problem. Let $\displaystyle 0 be coin denominations; what is the smallest sum of money that cannot be obtained using these coins? More formally, define the Frobenius number $\displaystyle g(a_1,\,\ldots\,,a_n)$ as the greatest number that is not a linear combination $\sum_{i=1}^n x_ia_i$ with $\displaystyle 0 \le x_i \in \mathbb{Z}$. The Frobenius number exists if and only if $\displaystyle a_1 > 1$ and $\displaystyle \gcd(a_1,\,\ldots\,,a_n)=1$. A special case of Frobenius numbers involves the interestingly named McNugget numbers, and there is a well-known formula when $\displaystyle n=2$ given by $\displaystyle g(a_1,a_2)=a_1a_2-a_1-a_2$ sometimes known as the Chicken McNugget Theorem.

## SPOJ, ideone, and TEST

September 14, 2010 at 05:53 (computer language, tutorial)

Background: I introduced SPOJ in a previous post.

ideone is an excellent tool that allows you to write and execute code in many languages, all online. It’s basically a pastebin with added functionality, making it an online mini-IDE. It is based on the SPOJ engine but supports some languages that SPOJ does not.

I’m mainly writing this post to report a useful discovery regarding ideone and the first SPOJ problem, TEST. If you happen to know a language like JavaScript and would like to submit a solution to TEST but can’t figure out how I/O is supposed to work, there is an easy solution through ideone. Simply go there, set the language to say JavaScript (rhino) and use the link to insert sample that appears above the textarea. A basic solution to TEST in the given language will magically appear, complete with sample input. This seems to work for just about any language on the list (even Whitespace!), and could come in handy when there is no corresponding thread in the SPOJ forums.

## How To: SPOJ TEST Using JAR File

September 12, 2010 at 07:39 (java, tutorial)

Background: Sphere Online Judge (SPOJ) is an online judge system accepting a wide variety of programming languages. It’s a great way to test your skills as a programmer, and can be quite addictive if you like a good challenge.

While browsing the SPOJ forums, I did not find a thread detailing how to successfully submit a JAR for the first problem, TEST. So, here’s a quick guide. I’m using Windows, but there should be little difference with other operating systems since everything is done through the command line. I will assume you already have the JDK installed and paths set up such that if you type java, javac, or jar from the command line the appropriate programs will be executed.

## Mac Newlines Can Cause PHP Headaches

August 31, 2010 at 21:47 (php, web dev)

While working on a project, somehow the majority of my PHP files acquired Mac style \r newline characters rather than DOS/Windows \r\n or UNIX \n. I don’t know exactly how this happened, but the \r newlines were perpetuated by Notepad++, which identified the files as Macintosh and displayed them transparently. The problem is if I take this code:

<?php
echo 'hello';
?>

and upload it to the Linux project server (on Go Daddy), it becomes:

<?phpecho 'hello';?>

which of course causes an error. It took a while to trace the problem to its source, mainly because I had no idea such newline characters existed. So now you know, and knowing is half the battle.

## Undocumented Ext JS PagingToolbar Method

August 22, 2010 at 07:01 (extjs, javascript)

Background: I’ve mentioned Ext JS in a previous post and am still using version 3.2.1.

After another fun-filled trip through ext-all-debug.js, I’ve discovered a simple yet hidden way to load an arbitrary page number in a grid that uses a PagingToolbar (Ext.PagingToolbar). Simply call yourGridPanel.getBottomToolbar().doLoad(start), where start is the row index starting from 0. So for example if you set pageSize = 10, then to display the first page call doLoad(0), second page call doLoad(10), etc. Note that when choosing a non-multiple of pageSize, the page number gets rounded up, so for example doLoad(1) would display items 2-11 (if you have 11 or more items) and report the page number as 2.

And of course, as the title of this post implies, this method is absent from the official API documentation (as of the time of this post) despite its being a public and very useful method.


text:
# Maximize $\int_{0}^{1} \mathbb{E} |X-\mathbb{P}(X\ge u)|^3\,\mathrm du$ over $[0,1]$ valued $X$.

Find the maximal value of

$$\int_{0}^{1} \mathbb{E} |X-\mathbb{P}(X\ge u)|^3\,\mathrm du$$

over $$[0,1]$$ valued random variables $$X$$.

I believe that optimal $$X$$ takes exactly two different values, but I have a hard time trying to actually prove it. I will be glad for any help or insight.

I also wonder if a pair $$(X,\mathbb{P}[X\ge U])$$, where $$U\sim\mathcal{U}[0,1]$$, is a more widely known or studied.

• why the power 3? what is the motivation behind ? Dec 28 '19 at 20:16
• in this case, you could perhaps mention the solution for k=2 in your question; this could be of interest to solve the other cases. Dec 28 '19 at 20:42
• I understand you're asking for general $X$, but if we restrict to binary-valued (indicator) $X$, the max still happens at $P(X=1)=\frac12$ for $k=3$, but more interestingly for $k=4$ the max(es) now happen at $P(X=1)=\frac12 \pm {1 \over 2\sqrt{3}}$ (assuming I didn't make any mistake with math and typing things into wolfram alpha.) Dec 28 '19 at 23:29
• One can easily show that the maximizer is always a discrete random variable with at most $C(k)$ values with some explicit $C(k)$. Unfortunately, as of now, it is of little practical value since I can only show that $C(3)\le 66$, say, and I see no reason why the number of values should not grow to infinity as $k\to\infty$. Would you be interested in some upper bounds that are only not too far from the true maxima for large $k$ as well? Jan 5 '20 at 3:45
• @fedja - I would be highly interested in $C(3)\le 66$ bound. I have been trying to find any such bound by myself and did not succeed. I think it would be quite eyeopening to see it. Jan 5 '20 at 11:06

The equivalent reformulation of the problem is to maximize $$\mathbb E|X-Y|^k$$ under the condition that $$X,Y$$ are independent and the functions $$F_X(u)=\mathbb P\{X\ge u\}$$ and $$F_Y(u)=\mathbb P\{Y\ge u\}$$ are inverse to each other in the sense that their graphs are symmetric (as usual, we connect the jumps in the graph by vertical intervals and consider vertical intervals as jumps).
Now I want to show two things: the upper bound $$\frac 2{k+1}$$ (which should be compared with the example $$X=\frac{k+1}{k+2}\chi_ A, \mathbb P(A)=\frac{k+1}{k+2}$$ giving the value $$\frac 2{k+2}(\frac{k+1}{k+2})^{k+1}\ge \frac 2{e(k+2)}$$) and the claim that the maximizer exists and is a discrete random variable with at most $$C(k)$$ values (though I'll not try to squeeze the best $$C(k)$$ the method can give: it is still too large for practical purposes even when $$k=3$$).
The first claim is rather straightforward: $$\mathbb E|X-Y|^k=\int_0^1 kt^{k-1}(\mathbb P\{X>Y+t\}+\mathbb P\{Y>X+t\})dt.$$ Now define $$s\in[0,1-t]$$ by $$F_X(s+t)=s$$. Notice that if $$X>Y+t$$, then either $$X>s+t$$, the probability of which is $$F_X(s+t)=s$$ or $$Y, the probability of which is $$1-F_Y(s)=1-F_X^{-1}(s)=1-s-t$$, so we always have $$\mathbb P\{X>Y+t\}\le \mathbb P\{X>s+t\}+\mathbb P\{Y and the same inequality holds for the other probability whence the integral in question is at most $$2\int_0^1 kt^{k-1}(1-t)dt=\frac 2{k+1}.$$
The second claim is a bit longer to prove. We fix $$m$$ and consider all discrete random variables with at most $$m$$ values. This is a compact set and we have a continuous functional, so the maximum over that family is attained for some $$X$$ with, say, $$M$$ values. If we bound $$M$$ independently of $$m$$, we will be done. Suppose that $$M$$ is large. Then the graph of $$F_X$$ is a descending stairway with many steps. In particular, there is a value $$x$$ of $$X$$ that is neither $$0$$, nor $$1$$, and the associated probability $$p_{x}=\mathbb P\{X=x\}\le\frac 1{M-2}$$ is very small. Let's try to move $$x$$ to a close value $$x'$$ without changing the probabilities. Our functional then can be written as $$L(x')+\sum_y p_x q_y|x'-y|^k+p_x[|x'-y_+|^k-|x'-y_-|^k](x'-x)$$ where $$L(x')$$ is linear in $$x'$$ and $$y_+-y_-=p_x$$ (i.e., $$y_{\pm}$$ are the levels of the top and the bottom of the step on the stairway graph of $$F_X$$ whose vertical side is $$p_x$$). Taking the second derivative in $$x'$$ at $$x'=x$$ and dividing by $$p_x$$, we get $$k(k-1)\sum_y q_y|x-y|^{k-2}\\+2k[|x-y_+|^{k-1}{\rm sgn}(x-y_+)- |x-y_-|^{k-1}{\rm sgn}(x-y_-)]\le 0$$ (otherwise our configuration cannot be a maximizer). Since the second derivative $$k(k-1)|x-y|^{k-2}$$ is a convex function of $$y$$ for integer $$k\ge 2$$, we can replace the increment of the first derivative by the half-sum of the second derivatives at the endpoints times $$y_+-y_-=p_x$$ and derive the inequality $$\sum_y q_y|x-y|^{k-2}\le p_x[|x-y_+|^{k-2}+|x-y_-|^{k-2}]\tag{*}$$ This should actually hold for every $$x\ne 0,1$$ and I feel that there must be a more efficient way to use $$(*)$$ than the one I'm following below but, at the very least, the RHS is $$\le 2p_x$$, so if $$p_x$$ is very small, then $$Y$$ is concentrated near $$x$$ and, thereby, near its mean $$EY$$. Similarly, choosing small $$q_y$$ with $$y\ne 0,1$$, we see that $$X$$ is concentrated near $$EX$$. But $$EX=EY$$ (both represent the area under the stairway), so in this case $$X-Y$$ is concentrated near $$0$$ and $$E|X-Y|^k$$ is very small. However, we have a maximizer, so we have an a priori lower bound on that expectation for every fixed $$k$$, which tells us that $$M$$ cannot be too large, finishing the argument.
• @StanTuwim Yes, but the problem is that the minimum may occur at an end and the sequence of $p$'s and $q$'s may be unimodal with maximum near the middle or increasing all the way to the other end, so the desired inequality never occurs for $p_x$ or $q_y$. Jan 6 '20 at 12:31


text:
# Inequality conjecture for ordered numbers

Consider the following inequality.

Let $$x_1>x_2>...>x_n>0$$ be some positive numbers. Then

$$\sum_{i=1}^n x_i+\sum_{i=1}^n\sum_{j\neq i} \frac{x_ix_j}{x_i+x_j}\leq M\left [\sum_{i=1}^n i\cdot x_i\right]$$

It is not hard to show that the bound holds for M=2 (by adding and subtracting $$(i-1)x_i$$ to the LHS and using $$\frac{x_i-x_j}{x_i+x_j}\leq 1$$). However, after some experiments I am convinced it also holds for M=4/3, but I have no clue how to demonstrate such a bound. Any ideas or advice would be greatly appreciated!

For $$x_1=\ldots =x_n=1$$ (or, as you required strict order, approximately for $$x_i=1-i\epsilon$$), you want $$M\left[n+n\cdot(n-1)\cdot \frac 12\right]\le 1+2+\ldots +n$$ As $$1+2+\ldots +n=\frac{n(n+1)}2$$, this implies $$M\le 1$$.

• But if inequality holds with $x_i=1$ it does imply that it holds for all choices of x (but I claim it does hold for all ordered x if M=4/3) – Simba_loo May 16 at 21:32
• Apologies, M should've been on the RHS – Simba_loo May 16 at 21:57

This is true for $$M=1$$ and it is the smallest possible constant. To check that $$M\geq 1$$ plug $$x_i\rightarrow1$$ as Hagen suggested. For $$M=1$$ we have:

$$\sum\limits_{i=1}^n x_i+\sum\sum \frac{x_ix_j}{x_i+x_j}\leq\sum\limits_{i=1}^n x_i+\sum\sum \frac{(x_i+x_j)^2}{4(x_i+x_j)}=\sum\limits_{i=1}^n x_i+\frac{2(n-1)}{4}\sum\limits_{i=1}^n x_i =\frac{n+1}{2}\sum\limits_{i=1}^n x_i\leq \sum\limits_{i=1}^n ix_i$$

The last one is true because of rearrangement inequality. For any permutations of $$\{b_i\}=1,2..n$$ we have

$$\sum\limits_{i=1}^n b_{k_i}x_i\leq \sum\limits_{i=1}^n ix_i$$

Adding all possible such inequalities we get:

$$(n-1)!\sum\limits_{i=1}^n \left(\sum\limits_{j=1}^nj\right)x_i\leq n!\sum\limits_{i=1}^n ix_i$$

$$\frac{n+1}{2}\sum\limits_{i=1}^n x_i\leq \sum\limits_{i=1}^n ix_i$$ and we are done

• Thanks for the response, but it doesn't hold for M=1! Try n=2, x_1 = 2, x_2 = 1, LHS = 1+2+2/3+2/3 = 13/3, RHS = 2+1*2 = 12/3 The bounds you use though might be useful, I'll try playing around with them. – Simba_loo May 17 at 12:32
• I think the proof doesn't work because you don't account for the fact that in there are two $x_ix_j$ terms, e.g. for n=2 you get $LHS = x_1+x_2 + \frac{2x_i x_j}{x_i+x_j}$ – Simba_loo May 17 at 12:42


text:
1.6k views

Which one of the following is the tightest upper bound that represents the number of swaps required to sort $n$ numbers using selection sort?

1. $O(\log n$)
2. $O(n$)
3. $O(n \log n$)
4. $O(n^{2}$)
edited | 1.6k views
0
this question is asked 3 times in gate 2006 , 2009 , and  2013

In selection max you can do is $n$ swaps..selecting the smallest element from all the elements and replacing it correct position so $O(n)$
edited by
+27
(n-1) swaps to be precise
+1
I should have written it with big oh notation
0
okay :)
+1
Word swap is important!
0

But the time complexity of Selection Sort is O(n2).. There are two loops, one inside the other

0
The question asks only for the number of swaps.
0

Ok. Thanks Sir... The book from where I read has given wrong algorithm, saying it to be Selection Sort.. As per that algorithm, number of swaps are also O(n2).

+11
Then you should never open that book again
0
ha ha ha :-D
0
for(i=0;i<n-1;i++)
{
min_idx=i;
for(j=i+i;j<n;j++)
{
if(a[j]<a[min_idx])
min_idx=j;
}
swap(a[i],a[min_idx]); // only 1 swap each iteration for a total of n-1 iterations
}

0

@Ayush Upadhyaya  In this algorithm , the first loop should be

for(i=0;i<n-2;i++)

as  for the lase element we don't need any comparison.

0

@Ayush Upadhyaya typing mistake at j=i+1

edit i want  to say that it should be i+1 but typed as i+i

0

@Nandkishor3939-$i \lt n-1$ will ensure last value at which loop runs is $n-2$ only.

@Gurdeep Saini-Why j should not be $i+1$ ? :O

0
https://www.geeksforgeeks.org/selection-sort/

1
2


text:
Welcome to the Monastery PerlMonks

by Cody Pendant (Prior)
 on Jun 01, 2004 at 22:17 UTC Need Help??

Here's my code:
$str = 'msxsl C:\path\foo.xml C:\path\bar.xsl -o C:\path\output.html'; $str;
[download]
Now, if I run that with backticks, I get the expected result. The transform happens and there's no error. If I deliberately break it:
$str = 'msxsl C:\path\foo.xml C:\BAD-path\bar.xsl -o C:\path\output.ht +ml'; $str;
[download]
Now, if I run it, I need to trap the error:
Error occurred while loading document 'C:\BAD-path\bar.xsl'.

Code:   0x800c0006
The system cannot locate the object specified.
[download]

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text:
# Surviving Re-entry in only a space suit

Because who needs a space craft?

Picture this:
While performing a routine spacewalk, outside your small, 1- man space pod, a fault in the propulsion system controls causes the pod to explode, launching you into a low orbit of the Earth.

Now with a limited air supply, and only your own space suit & MMU (Manned Manuvering Unit) at your disposal, you are tasked with re-entering Earth's atmosphere, in a survivable manner. (Please note that chance of survival does not need to be 100%).

The Question:
Is it scientifically possible to survive re-entry in this manner, and if so how? If not, and if it would be possible if the scenario were to be modified, please describe how it could be modified in order to make it possible.

Note: Your MMU does not need to be of the modern NASA variety. The story is set a short time in the future, so the design can be slightly more advanced (but still within reason for an MMU).

Edit: Originally, I had planned for the MMU to provide a wireless transceiver (allowing the astronaut to calculate his re-entry angle based on GPS signals?), a set of manuvering thrusters (set his re-entry vector & slow him down to prevent a crash - if possible), and possibly the material needed to construct a rudimentary heat-shield (if such material would feasibly be used in construction of an MMU).

• Would you be orbiting the Earth six or seven km / second? If so, no. – Mikey Jan 23 '16 at 3:20
• You can be orbiting at any feasible speed. Plus the thrusters on your MMU could be used to alter your speed (to a realistic extent) if necessary. – Chef Cyanide Jan 23 '16 at 3:30
• They might as well say, "You're gonna die Jeff". – Xandar The Zenon Jan 23 '16 at 3:37
• Check out this: en.wikipedia.org/wiki/MOOSE – AMACB Jan 23 '16 at 3:55
• I assumed your MMU to be something like a near-future EVA setup, in which case, absolutely not. Further to this, I would expect your destroyed space pod would have the only chance of things like a parachute? I'm not trying to be contrary, because I think it's a good question (+1), it just seems like a dead end to me. The answers will be good here though. Maybe define what's included with the MMU that's not included with the 1-Man Space Pod? – Mikey Jan 23 '16 at 5:42

There are many problems to overcome:

According to NASA, most meteors of a radius of 80 feet burn up in the atmosphere so that they cause little to no damage. So, we need to somehow cool off our poor guy so that he isn't a pile of ashes by the time he hits (or floats, if he was ashes) to the ground. Add a very potent coolant system or heat shield.

Now that we've solved burning up, we still have to consider landing. Terminal velocity at sea-level for an upright human is about 200 mph. Most meteors burn up around 30 miles, so our friend would have to decelerate in a "mere" 30 miles, either requiring a reverse propulsion system to boost him upward, or a parachute. The former would also require some kind of stabilization system, such as gyroscopes, to avoid flipping over. The problem with a upward thrust is that gravity will push down when you are not at terminal velocity, so you would have to supply a continuous thrust of around 9.9 m/s^2 for about 10 minutes to slow him down to land safely (which is impractical), so you would need a parachute. To be safe, we also include a upward-accelerating jetpack to ensure a safe landing.

All in all, your chances for survival are fairly slim, but it could be possible to survive such a fall.

• Thank you for such a practical analysis of the problem (+1). Out of curiosity, do you think it would be feasible for our astronaut to re-configure his MMU in order to provide the thrust which you mentioned? – Chef Cyanide Jan 24 '16 at 14:29
• Unlikely if he is floating around in space. Even if he was inside a space station, I doubt he could figure out how to even if it was possible. – AMACB Jan 24 '16 at 16:03
• We can dispense with the parachute. He's got quite a range of places to land and presumably good guidance as to the deorbit burn and where to aim. His target should be a place that lacks things sticking up and has just received a huge snowfall. Extremely dangerous but potentially survivable. – Loren Pechtel Jan 24 '16 at 17:48
• Pretty good summary of the issues, but you missed the first one: getting into re-entry in the first place. Unless the hapless voyager can alter his orbit enough to re-enter, he will just orbit forever. Not every orbit that intersects the upper atmosphere is a re-entry: if you come in too shallow, the atmospheric shock just bounces you back up! – Securiger Nov 14 '18 at 11:37
• (And the delta V needed for this manoeuvre is vastly more than real world EVA kits can provide: for example, the MMU does a maximum of 23.4 m/s. when you need km/s.) – Securiger Nov 14 '18 at 11:46

If this sort of scenario was considered likely, the MMU might incorporate a "ballute".

This is a low mass substitute for a reentry heat shield, and has the added bonus of being an air bag for your final landing. Real life use includes being used to retard bombs (so the bomber can escape the blast) and this has been proposed for real life space missions. The movie 2010 has the Russian spacecraft using a ballet to aerobrake in the orbit of Jupiter, so the technique is versatile.

Now this might not be a common thing to add to an MMU, since while lighter than a heat shield, the mass of the fabric and gas bottles to inflate the Ballute will still be considerable, and make manoeuvring in space slow and unwieldy (you will also require far more fuel for your MMU to work). It would be like suggesting you always carry a parachute and a backpack full of survival gear for both desert and arctic conditions, on the off chance you fall out of an airplane or window.

For your scenario, if the construction shack can't send a rescue pod out quickly enough to pick up the unfortunate astronaut, they might be able to launch a "reentry pack" with prepackaged ballute, inflation gear, extra oxygen, survival kit and a powerful solid fuel rocket motor. It intercepts the astronaut, who rapidly straps it on (or the robotic arms on the device grab him and pull him aboard), then it orients itself, the solid fuel motor fires to begin reentry and the ballet inflates. For simplicity, the device might actually be designed to land on water so the ballute becomes an improvised life raft as well (using it as an airy bag on land might bounce you face first into the desert floor....).

Happy landings

• While still not quite as epic a solution as I had originally been hoping for, your solution is both realistic and practical (assuming I can find a way for our unfortunate astronaut to acquire one of these ballute systems). I will certainly consider it if I can't find a way to achieve re-entry using the materials that are already in the astronaut's possession (+1). Thank you very much! – Chef Cyanide Jan 24 '16 at 14:25
• No problem. Don't stand too close to windows, though ;-) – Thucydides Jan 25 '16 at 1:55


text:
# C 0000013 mmmv folder structure for ID labeled artifacts t1

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

The artifacts are labeled with an artifact_ID that consists of only characters that are part of the mmmv_ASCII_subset_t1. Artifacts are placed in folders that have the artifact_ID as the folder name (hereafter artifact_folder). Minimum artifact_ID length is one character and the maximum artifact_ID length is determined by the filesystem(usually 255 on Linux). The artifact_folders are leaves of a tree, where the root node is a folder named "artifacts_root", which shares a parent folder with a folder named "header", which contains text-files that each contain exactly one line of text, which is never an empty string and does not contain any spaces, tabulation characters, linebreaks. An example:

├── artifacts_root
├── i_n_of_characters_per_intermittent_vertex.txt
├── i_n_of_edges_between_the_artifacts_root_and_a_leaf.txt
├── s_tree_format_version.txt
└── s_tree_instance_ID.txt

All of the files that reside at the folder named "header" and have the "txt" file extension (hereafter: text-files) are in the UTF-8 format. The text-files may include the UTF-8 byte order mark (BOM, File:2021 01 20 The byte-order mark BOM in HTML ux-generated files.pdf) independent of the other limits set to them by the current specification. Each of the text-files in the folder named "header" is a field of a record (hereafter: tree-record), where the extensionless text-file name matches with a field name, which can also be seen as a name of a variable.

The text-files in the folder named "header" that start with a prefix "i_" contain a non-negative base 10 whole number. Leading zeros are allowed. For example, the number 72 might be written as 0000072. With the exception of the optional presence of the UTF-8 BOM all of the characters at files "s_tree_format_version.txt" and "s_tree_instance_ID.txt" must be part of the mmmv_ASCII_subset_t1.

The path length from the root of the tree to the leaf of the tree is a constant that is written in i_n_of_edges_between_the_artifacts_root_and_a_leaf. All of the folders that are vertices of the tree, except leaves and the artifacts_root, have exactly i_n_of_characters_per_intermittent_vertex number of characters in their name. All of the folders that are vertices have names that consist of only characters that are part of the mmmv_ASCII_subset_t1.

# Implementation Notes

This specification does not require that the full path of the folder artifact_root consists of only characters that are part of the mmmv_ASCII_subset_t1, but from practicality point of view this extra restriction might be imposed by the application that uses this specification.


text:
# Equivalence Classes Output

I understand that an equivalence relation is a set that is reflexsive, symmetric, and transitive. I don't quite understand equivalence classes though.

For example: What would the equivalence class be of the equivalence relation {(0.0),(1,1),(2,2),(3,3)} if the (original?) set is {0,1,2,3}?

Consider an equivalence relation on the set $S=\{1,2,3,4,5,6\}$ which is given by $$R=\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(1,3),(1,6),(6,1),(6,3),(3,1),(3,6),(2,4),(4,2)\}.$$ You can verify that $R$ is an equivalence relation on $S$. Now, we define an equivalence class of $s\in S$ as $$[s]=\{x\in S : x\ R\ s\}$$ So, what elements in $S$ are related to $6?$ They are $\{1,3,6\}.$ Using above notation we have $$[6]=\{1,3,6\}$$ The set $\{1,3,6\}$ is called an equivalence class of $6$. I'll leave the rest to you to see if you can find all the other equivalence classes.
The equivalence relation is a set of pairs of equivalent things. In your example each element is equivalent to only itself so the equivalence classes are $\{0\},\{1\},\{2\},\{3\}$. If you add $\{(1,0),(0,1)\}$ (you need both pairs to have symmetry) to your equivalence relation then you'll have classes $\{0,1\},\{2\},\{3\}$. Generally your initial set will break up into a disjoint union of equivalence classes whenever you have an equivalence relation.
• So if I have a set {(0.0),(1,1),(1,2),(2,1),(2,2),(3,3)} the equivalence class would be {{0},{1,2},{3}}? – Evan Bloemer May 1 '15 at 0:39


text:
# A Comparative statistics of (trivariate) random event

Tags:
1. Feb 10, 2017

### estebanox

Problem: I'm interested in studying the probability of an event involving a random vector.

Specifically, I'm interested in
(∂/∂a)Pr[X>( (Y-a)/Z )]

Where "a" is a non-random parameter and the random vector {X,Y,Z} is distributed Normal( µ, Σ)
for µ={0,0,0}
and Σ= {{1, 0.5, 0.5}, {0.5, 1, 0}, {0.5, 0, 1}}

What I know: Simulations tell me that the above partial derivative is non-monotonic, i.e. increasing the value of "a" first increases the probability of the event, and then decreases it. Attached is a simulation for a∈[0,3].

I can see that this is likely coming from the fact that Z takes value along the real line, so the effect of "a" flips with the sign of Z.

Question: I struggle to understand exactly what is going on here from a distributional point of view. What determines the point of inflection? I don't need an analytical solution – just some intuition.

#### Attached Files:

• ###### Screen Shot 2017-02-09 at 09.00.48.png
File size:
8.7 KB
Views:
33
Last edited: Feb 10, 2017
2. Feb 10, 2017

### Stephen Tashi

What's your definition of the relation "$>$" when we are comparing two vectors ?

3. Feb 10, 2017

### estebanox

Each of X, Y and Z are random variables. I refer to random vector when talking about (X,Y,Z)

4. Feb 10, 2017

### Stephen Tashi

Then I don't understand the notation:
Does this indicate we are considering 3 different situations for the standard deviations or $3^3$ different situations ?

Did you mean "for $a\in [0,3]$" ?

5. Feb 10, 2017

### estebanox

Oh, yes, typo! I'll fix it. Thanks.

6. Feb 10, 2017

### estebanox

The vector μ and the matrix Σ refer to the parameters of the joint distribution of (X,Y,Z). The simulation is fixing all of these parameters, and tracing the probability for different values of "a".

7. Feb 10, 2017

### Stephen Tashi

I suggest we start by seeing if we can get intuition for a much simpler situation!

Let $Y$ and $Z$ each be uniformly distributed on the interval [-1,1]. Let $x$ be a number instead of a random variable. How does $P(x > (Y- a)/Z$ vary with $a$ ?. Maybe we need to consider two cases: $x< 0$ and $x > 0$.

It might be simpler to ask about $P(x < (Y-a)/Z)$ since that's a question about a cumulative distribution.


text:
Tags

If you’re just jumping into this blog, it will be very important that you read the previous posts in order to understand this one

There are elements or SLNs (species-level food web networks) in S that cannot exist because they possess topologies inconsistent with observed guild species richnesses. We define those SLNs as having probabilities equal to zero, and are therefore interested in the subsets of F, which are those comprising SLNs of probability greater than zero. The arguments on ensemble composition in the earlier post conclude that the members of F cannot be easily enumerated, but the probability of any specific SLN can be determined and if exceeding zero, could therefore have been an actual food web of the paleocommunity. Let species $x_{i} (x_{i}\in G_{i})$ have $r_{i}^{x}$ in-links, or prey, and topology $t_{i}^{x}$. The probability of $t_{i}^{x}$, constrained by the metanetwork topology and guild species-richnesses, is given by the multinomial probability
$p(t_{i}^{x}) = \frac{r_{i}^{x}!} {a_{i1}k_{1}!a_{i2}k_{2}!\ldots a_{i\vert U\vert}k_{\vert U\vert}!} p_{1}^{a_{i1}k_{1}}p_{2}^{a_{i2}k_{2}}p_{\vert U\vert}^{a_{i\vert U\vert}k_{\vert U\vert}}$
where $k_{n}$ is the number of links from species in guild $G_{n}$ to $x_{i}$, $\sum_{n=1}^{\vert U\vert}k_{n}=r_{i}^{x}$ , and $p_{n}$ is the probability that a link, drawn randomly between $x_{i}$ and any other species, will connect to a species in $G_{n}$. Factorial zero is defined conventionally as equal to one. $p_{n}$ is estimated empirically from the data as
$p_{n} = \frac{a_{in}\vert G_{n}\vert}{\sum_{j=1}^{\vert U\vert}a_{ij}\vert G_{j}\vert}$
The multinomial formula calculates the number of ways in which r links can be arranged among the guilds with $k_{1}$ assigned to $G_{1}$, $k_{2}$ to $G_{2}$, and so on, and then multiplies this number by the product of the probabilities, which is equal to the probability of drawing $k_{1}$ links between $x_{i}$ and $G_{1}$, and $k_{2}$ links between $x_{i}$ and $G_{2}$, and so on. In order for $p(t_{i}^{x}$ to be greater than zero, no \$k_{n} can exceed the number of species in $G_{n}$. The probability of any SLN may now be defined formally as
$p(t_{i}^{x}) = \frac{r_{i}^{x}!} {a_{i1}k_{1}!a_{i2}k_{2}!\ldots a_{i\vert U\vert}k_{\vert U\vert}!} p_{1}^{a_{i1}k_{1}}p_{2}^{a_{i2}k_{2}}p_{\vert U\vert}^{a_{i\vert U\vert}k_{\vert U\vert}} \qquad \mathrm{if}\quad 0\leq k_{n}\leq G_{n} \forall n$

It now becomes immediately obvious that among SLNs of probability greater than zero, some are of greater probability than others. Examine the SLNs in the figure, arranged in order of descending probability. p1 is 0.36, p2 is 0.16 and p3 is zero. The most probable food web is therefore one where the predatory species, with two in-links, preys exclusively on species in G2. The third SLN is of probability zero since the metanetwork (earlier post) forbids links between G3 and G4. Most food webs are of course significantly more complicated than the example used here, but the probability of a SLN is simply the product of the topological probabilities of all species in the food web given the metanetwork,
$p(\mathrm{SLN}) = \prod_{i=1}^{\vert U\vert}\prod_{n=1}^{\vert G_{n}\vert} p(t_{i}^{n})$
Finally, although the set of possible SLNs cannot be delimited analytically, the preceding formula may be used as the basis of a likelihood model, facilitating Markov Chain Monte Carlo exploration of (S,F,P).


text:
# Content and measure of point sets ...

by Marian Hortense Covington

Written in English

## Subjects:

• Set theory.,
• Number theory.
The Physical Object
Pagination58 leaves,
Number of Pages58
ID Numbers
Open LibraryOL16882540M

The Theory of Sets of Points Cambridge, It ha involved an amount of labour which would seem to the average reader quite out of proportion to the size of the book. yet I can scarcely hope that the mode of presentation will appeal equally to all mathematicians. Addition Theorem for the content of sets of intervals; Content of a.   The application lets you plan, create, launch, promote, and measure any type of content using six integrated modules — consult, create, collaborate, calendar, promote, and measure. Content Launch also offers complete workflow tools and distribution to 15 content platforms. Contently. The first three volumes cover point set topology, measure spaces, Hilbert and Banach spaces, distribution theory, the Fourier transform, and complex analysis. A Comprehensive Course in Analysis; 5 volume set (includes The Ultimate Companion to A Comprehensive Course in Analysis). "The Point Is: Making Sense of Birth, Death, and Everything in Between," by Lee Eisenberg, came along in my life at a perfect time. I recently celebrated a birthday (#62) and had that little nagging thought about end-of-life, what-have-I-accomplished, what-to-do-next concerns. Mr/5.

Milwaukee uses a nylon blade coating with this tape measure, which effectively reduces the amount of wear and tear on the blade. This coating also helps account for why this tape measure feels so good to use and for the sound it makes. The blade pulls out and retracts easily, and at no point does it feel flimsy/5. Durrett since it is free. Just imo, measure theory is highly overrated. It is good to know what measure 0 sets are, LP spaces, different modes of convergence (convergence in measure, pointwise a.e, L1, distribution), Radon Nikodym tells you when a random variable has a pdf, different convergence theorem is useful to know, ect. In print, the optimal point size for body text is 10–12 point. On the web, the optimal size is 15–25 pixels. Though 12 point has become the default size in digital word processing—and also the basis of many institutional document-formatting rules—that’s mostly due to the typewriter tradition. In the process of writing a review of a book called The Spec Manual, I can’t find material that I know I wrote at one point or another so I can link to the interest of expediency, I’ve created this glossary entry. POM Codes are used to specify the measuring points of a garment or product. If you measure garments, you will need to have codes to indicate the location you measured at.

In an online and social media world, measurement is the key to success If you can measure your key business relationships, you can improve them. Even though relationships are "fuzzy and intangible," they can be measured and managed-with powerful results. Measure What Matters explains simple, step-by-step procedures for measuring customers, social media reputation, influence 5/5(1).   The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers , , . How we measure 'reads' A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text.

## Content and measure of point sets ... by Marian Hortense Covington Download PDF EPUB FB2

The measure μ does not necessarily coincide with the content λ on compact sets, However it does if λ is regular in the sense that for any compact C, λ(C) is the inf of λ(D) for compact sets D containing C in their interiors.

See also. Minkowski content; References. Halmos. Measure here means Lebesgue Measure (there are others) and measure generalizes the "length" or "size" of a set.

It allows us to measure sets which we couldn't do otherwise. We still can't measure EVERY set out there on the real number line but we can measure a lot more than we could before. The book covers much more than the challenge of measuring intangibles.

It also addresses the links with estimating, measurement, uncertainty, and risk management. It considers how to think about measurement - confidence intervals, the value of the measurement and why you have decided you need to measure; It is a book that make you think/5().

The chapter presents the graphs of two typical Lipschitz functions. Rectifiable sets can have countably many rectifiable pieces, perhaps connected by countably many tubes and handles and perhaps with all points in R n as limit points.

From the point of view of measure. Praise for the second edition of How to Measure Anything: Finding the Value of “Intangibles” in Business “How to Measure Anything was already my favorite book (just ahead of Hubbard’s second book, The Failure of Risk Management) and one I actively promote to my students and the Second Edition, improving on the already exquisite first edition, is an achievement of its by: A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞).

Nonzero finite measures are analogous to probability measures in the sense that any finite measure μ is proportional to the probability measure ().A measure μ is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. It would be difficult to measure the added space since it is small especially at small sizes.

So to do this, type is measured from baseline to baseline giving us the size and added space. A gauge with sets of parallel lines at equal distances is used and lined up with. Publisher Summary. This chapter discusses the Lebesgue measure of linear sets. It describes the conditions that need to be satisfied in the case of the Lebesgue measure: (1) it must always be nonnegative; (2) it must coincide with the measure for open sets defined; and (3) the measure of a countable number of disjoint measurable sets must be equal to the sum of their measures.

Integration over Jordan measurable sets. In one variable there is really only one type of reasonable set to integrate over: an interval. In several variables we have many very simple sets we might want to integrate over and these cannot be described so easily.

Let $$S \subset {\mathbb{R}}^n$$ be a bounded Jordan measurable set. An Introduction to Measure Theory. Terence Tao. This is a preliminary version of the book An Introduction to Measure Theory published by the American Mathematical Society (AMS). This preliminary version is made available with the permission of the AMS and may not be changed, edited, or reposted at any other website without explicit written.

4 CONTENT 1 Measures ˙-Algebras and Measures Measure Determining Classes Lebesgue Measure CarathØodory™s Theorem Existence of Linear MeasureFile Size: KB. The Interval Angle is advertised to measure nothing more than, nor less than, a difference in intervallic content of the pitch-class sets involved, but this, we claim, it does very well.

As an aside, we did investigate an ANGLE-like measure which used trichordal embeddings as a basis, and its formula is many times more difficult and involved. Introductory remarks. Topology is about open sets. The characterizing property of a continuous function f is that the inverse image f −1 (G) of an open set G is open.

Measure theory is about measurable sets. The characterizing property of a measurable function f is that the inverse image f −1 (A) of any measurable set is measurable. In topology, one axiomatizes the notion of ‘open set.

Listening Practice Set 1. Read the script. Give yourself 10 minutes to complete this practice set. Narrator: Listen to a conversation between a male student and a female librarian.

Student: Hi. I'm new here I, uh, couldn't come to the student orientation—and I'm wondering if you can give me a few quick pointers about the library. I'd. Despite the overlap with the book of Rudin [17] there are some di er-ences in exposition and content.

A small expository di erence is that in Chapter 1 measurable functions are de ned in terms of pre-images of (Borel) measurable sets rather than pre-images of open sets. The Lebesgue measure. I cannot do a calculated column for the point totals since it is multiple rows so I am running into a problem with getting the count.

Maybe there is a way to summarize the total points in another table (one row per employee) and jsut add a column for the Bins. You can use Monitor Profiler to measure and set the white point (RGB) settings that result in the desired white point.

The white point defines the exact white that the monitor displays. Warm White (K) - Also known as D50, is a yellowish white, recommended for image reproduction on CRT monitors. K is also the recommended viewing light.

GET THE RIGHT SIZE. RIGHT AWAY. If you can usually buy your clothes off the rack, chances are we'll have your size in our extensive stock -- and we'll be able to ship your Point Blank Body Armor the same day.

Measuring Made Easy: Use the sizing chart and diagrams below as a guide. Content marketing isn’t at the point where it can measure how people feel about your content.

Or, at least, I’m not aware of anyone doing it successfully. (In theory, it should be possible – the rise of chatbots and on-site surveys might give you the ability to ask questions about how people : Benyamin Elias.

Sets of Measure Zero. Deﬁnition: A subset A ⊂ R has measure 0 if inf A⊂∪In X ‘(I n) = 0 where {I n} is a ﬁnite or countable collection of open intervals and so A has measure 0.

Sets of Measure Zero. To see that it has measure 0, notice that A n = [2n k=1 B n,k, B n,k = [aFile Size: KB. Thanks for contributing an answer to Mathematics Stack Exchange.

Please be sure to answer the question. Provide details and share your research. But avoid Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. reference point Although rod and level equipment is familiar to most engineers, the requirements for obtaining a profile measure that is valid for computing roughness are much different than for laying out a road.

You must take elevation measures at close intervals of a foot or less. The individual height measures must be accurate to mmFile Size: KB. Hi, I'm creating a point in time report which requires a number of Dynamic components.

I have the majority of them in the report but I seem to have fallen at the last hurdel. In the Screen shot below, I would like the last column 'TEST days' measure to; Sum "Balance" if "Age Category" = "   This textbook presents classical set theory in an intuitive but concrete allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors.

Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. "Henry is a particular sort of pig" and in an effort to record the proper place for everything on the farm he sets out to create a map.

A great story about representing a 3-D environment in a 2-D map and ground views and bird's eye views. Mathematics. Find Books at the Right Level There’s more than one way to use Lexile measures to find the right books for engaging students and encouraging reading.

Using a student’s Lexile reading measure as a starting point, it’s easy to find books that provide the just-right amount. A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text.

Content marketing is a strategic marketing approach focused on creating and distributing valuable, relevant, and consistent content to attract and retain a clearly defined audience — and, ultimately, to drive profitable customer action. Instead of pitching your products or services, you are providing truly relevant and useful content to your.

An Alternate Approach to the Measure of a Set of Real Numbers Seminar on the History and Exploration of Math Problems (S.H.E.M.P.) Think of the open sets as a starting point. We know that a countable union of open sets is open and a ﬁnite intersection of open sets is open.

So to create a new collection of sets based on open sets, we couldFile Size: KB. Green sets up this book seemingly intent on figuring out a better way to measure success in counterinsurgency.

To begin, he establishes the futility of using metrics such as the number of enemy that US forces kill (a callback to the notorious “body count” figures of the Vietnam War), development projects that US forces complete, or.

In this blog post, my goal is to explain the four available options when it comes to configuring SharePoint Document Library.

Each of the four methods below is unique and selecting a particular option depends on a number of variables like the types of documents to be stored in SharePoint Document Library, how well they need to be organized, etc.We are interested principally in sets of harmonic measure zero; under the above hypotheses, such sets can be defined as follows: The point set α is of harmonic measure zero with respect toGif the image point setα x has (linear) measure zero, i.e., provided α x can be covered by a sequence of arcs of arbitrarily small total : Rolf Nevanlinna.Read and learn for free about the following article: Choosing the "best" measure of center If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains * and * are unblocked.


text:
1

### AIPMT 2007

Which of the following oxidation states are the most characteristic for lead and tin respectively?
A
+2, +4
B
+4, +4
C
+2, +2
D
+4, +2

## Explanation

The tendency to from +2 ionic state increases on moving down the group due to inert pair effect. Most characteristic oxidation state for lead and tin are +2, +4 respectively.
2

### AIPMT 2006

In which of the following molecules are all the bonds are not equal?
A
NF3
B
ClF3
C
BF3
D
AlF3

## Explanation

The Cl - F (Cl - Fb) bond length is equal to 1.60 $\mathop A\limits^o$ while each of two axial Cl - F (Cl - Fa) bond length is equal to 1.70 $\mathop A\limits^o$.
3

### AIPMT 2006

Which one of the following orders is not in accordance with the property stated against it?
A
F2 > Cl2 > Br2 > I2 : Bond dissociation energy
B
F2 > Cl2 > Br2 > I2 : Oxidising power
C
HI > HBr > HCl > HF : Acidic property in water
D
F2 > Cl2 > Br2 > I2 : Electronegativity.

## Explanation

Bond dissociation energy is kcal/mol of F - F, Cl - Cl, Br - Br and I - I are 38, 57, 45.5 and 35.6 respectively.

Bond dissociation energy of fluorine is less because of its small size and repulsion between electrons of two atoms.
4

### AIPMT 2006

The electronegativity difference berween N and F is greater than that between N and H yet the dipole moment of NH3 (1.5D) is larger than that of NF3 (0.2 D). This is because
A
in NH3 the atomic dipole and bond dipole are in the opposite directions whereas in NF3 these are in the same direction
B
in NH3 as well as in NF3 the atomic dipole and bond dipole are in the same direction
C
in NH3 the atomic dipole and bond dipole are in the same direction whereas in NF3 these are in opposite directions
D
in NH3 as well as in NF3 the atomic dipole and bond dipole are in opposite directions.

## Explanation

The dipole moment of NF3 is 0.24 D and of NH3 is 1.48 D. In NF3, the dipole moment vector are aligned in opposite direction to that of the dipole moment vector of lone pair on N–atom which partly cancel out. On the other hand, N–H bonds in NH3 are in the same direction of dipole moment of the lone pair on N-atom which adds up the new resultant dipole moment.

### EXAM MAP

NEET

#### Graduate Aptitude Test in Engineering

GATE CE GATE ECE GATE ME GATE IN GATE EE GATE CSE GATE PI


text:
6.1: Definition

You can think of a PRF as a way to achieve random access to a very long [pseudo]random string. That long string is also the truth table of the function $$F(k,·)$$. A distinguisher is allowed to query this function, and the results should look as though they came from a random function — a function whose truth table was chosen uniformly at random. This perspective is the source of the name “pseudorandom function;” it’s an object that (when the seed is chosen uniformly) is indistinguishable from a random function, to polynomial-time computations.

Under this perspective, we can consider a function with any input output length, not just with a single bit of output. Consider the set ? of functions from $$\{0,1\}^{in}$$ to $$\{0,1\}^{out}$$. If F is a good PRF, then when instantiated with a uniformly chosen $$k$$, the function $$F(k,·)$$ should look just like a function chosen uniformly from the set $$?$$, when given query access to the function.

We can therefore formalize security for PRFs by defining two libraries that provide an interface to query a function on arbitrary inputs. The two libraries differ in whether the responses are determined by evaluating a PRF (with randomly chosen seed) or by a evaluating a function chosen truly at random.

For technical reasons, it is important that the libraries for the PRF definition run in polynomial time. This means we must be careful about the library that provides access to a truly random function. We cannot explicitly sample the entire truth table of a random function upfront — the truth table could be exponential in size. Instead, outputs of the function are sampled on demand and cached for later in an associative array. The formal definition looks like this:

Definition $$\PageIndex{1}$$: PRF Security

Let $$F : \{0,1\}^{\lambda} \times \{0,1\}^{in} \rightarrow \{0,1\}^{out}$$ be a deterministic function. We say that $$F$$ is a secure pseudorandom function (PRF) if $$\mathscr{L}^F_{\text{prf-real}} ≋ \mathscr{L}^F_{\text{prf-rand}}$$, where:

Discussion

• We measure the security of a PRF by comparing its outputs to those of a “random function.” The idea of a “random function” is tricky. We do not mean a function that gives different outputs when called twice on the same input. Instead, we mean a function whose truth table is chosen uniformly at random, after which the truth table is absolutely fixed. Calling the function twice on the same input will result in the same output.

But due to the technical limitations mentioned above, we don’t actually write the code of $$\mathscr{L}_{\text{prf-rand}}$$ in this way. The truth table is too large to sample and store explicitly. Rather, we have to sample the values appearing in the truth table on-demand. But we see that once they are chosen, the variable $$T$$ ensures that they remain fixed.

We see also that in $$\mathscr{L}_{\text{prf-real}}$$, the function $$F$$ is deterministic and the key $$k$$ is static between calls to QUERY. So calling QUERY twice on the same input gives the same answer.

• If the input length in is large (e.g., if $$in = \lambda$$), then the calling program will not be able to query the function on all inputs $$x \in \{0,1\}^{in}$$. In fact, it can only query on a negligible fraction of the input space $$\{0,1\}^{in}$$!
• Suppose we know that the calling program is guaranteed to never call the QUERY subroutine on the same x twice. Then there is no need for $$\mathscr{L}_{\text{prf-rand}}$$ to keep track of $$T$$. The library could just as well forget the value that it gave in response to $$x$$ since we have a guarantee that it will never be requested again. In that case, we can considerably simplify the code of $$\mathscr{L}_{\text{prf-rand}}$$ as follows:

$\underline{\text{QUERY}(x\in\space\in\{0,1\}^{in}):}\\z\space\leftarrow\{0,1\}^{out}\\\text{return}\space z\nonumber$

Instantiations

In implementations of cryptographic systems, “provable security” typically starts only above the level of PRFs. In other words, we typically build a cryptographic system and prove its security under the assumption that we have used a secure PRF as one of the building blocks. PRFs are a kind of “ground truth” of applied cryptography.

By contrast, the algorithms that we use in practice as PRFs are not proven to be secure PRFs based on any simpler assumption. Rather, these algorithms are subjected to intense scrutiny by cryptographers, they are shown to resist all known classes of attacks, etc. All of these things taken together serve as evidence supporting the assumption that these algorithms are secure PRFs.

An example of a conjectured PRF is the Advanced Encryption Standard (AES). It is typically classified as a block cipher, which refers to a PRF with some extra properties that will be discussed in the next chapter. In particular, AES can be used with a security parameter of $$\lambda \in \{128,192,256\}$$, and has input/output length $$in = out = 128$$. The amount of scrutiny applied to AES since it was first published in 1998 is considerable. To date, it is not known to be vulnerable to any severe attacks. The best known attacks on AES are only a tiny bit faster than brute-force, and still vastly beyond the realm of feasible computation.


text:
Precalculus (6th Edition)

$x=5$
RECALL: (1) The slope-intercept form of a line's equation is $y=mx+b$ where $m$=slope and $(0, b)$ is the line's y-intercept. (2) A line with an undefined slope is vertical, the equation of which is of the form $x=h$ where $h$=x-intercept. All points on the line have an x-coordinate of $h$. The given line has an undefined slope therefore its equation is of the form $x=h$. The line contains the point $(5, 1)$. The x-coordinate of this point is $5$. Thus, the line's equation is $x=5$. The equation of the line cannot be written in slope-intercept form.


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# Calculus-performing mechanical calculator

A clip from the Discovery Channel's Dirty Jobs program on tanneries demonstrates the workings of a calculus-performing mechanical calculator that measures the surface-area of irregularly shaped hides with a fascinating and clever set of gears, calipers and ratchets.

Dirty Jobs - Tannery Mechanical Surface Integrator (Thanks, Dad!)

## Game theory and bad behavior on Wall Street —

An opinion piece by Chris Arnade on the asymmetry in pay (money for profits, flat for losses), which he describes "the engine behind many of Wall Street’s mistakes" That asymmetry "rewards short-term gains without regard to long-term consequences," Chris writes in a new guest blog at Scientific American. "The results? The over-reliance on excessive leverage, banks that are loaded with opaque financial products, and trading models that are flawed." [Scientific American Blog Network] Xeni

# The world's largest prime number — visualized

Philip Bump took the recently discovered 17-million-digit prime number and, six digits at a time, converted it into RGB colors. This is the result.

## Neil deGrasse Tyson on pi and other constants —

Both the Bible and the Indiana State Legislature have tried to redefine pi to equal something much more simple than 3.14159265358979323846264338327950 ...

## 86.54% liked this —

Science blogger Matt Springer analyzes the surprisingly fascinating math behind Reddit upvotes.

# Probability theory for programmers

Jeremy Kun, a mathematics PhD student at the University of Illinois in Chicago, has posted a wonderful primer on probability theory for programmers on his blog. It's a subject vital to machine learning and data-mining, and it's at the heart of much of the stuff going on with Big Data. His primer is lucid and easy to follow, even for math ignoramuses like me.

For instance, suppose our probability space is $\Omega = \left \{ 1, 2, 3, 4, 5, 6 \right \}$ and $f$ is defined by setting $f(x) = 1/6$ for all $x \in \Omega$ (here the “experiment” is rolling a single die). Then we are likely interested in more exquisite kinds of outcomes; instead of asking the probability that the outcome is 4, we might ask what is the probability that the outcome is even? This event would be the subset $\left \{ 2, 4, 6 \right \}$, and if any of these are the outcome of the experiment, the event is said to occur. In this case we would expect the probability of the die roll being even to be 1/2 (but we have not yet formalized why this is the case).

As a quick exercise, the reader should formulate a two-dice experiment in terms of sets. What would the probability space consist of as a set? What would the probability mass function look like? What are some interesting events one might consider (if playing a game of craps)?

(Image: Dice, a Creative Commons Attribution (2.0) image from artbystevejohnson's photostream)

# Math Fleet: command a space squadron and defend planet Earth with the power of math

Kevin Kelly told me about Phil Scale's new iOS application to teach kids arithmetic. It's called Math Fleet and it sounds great. (Phil created Kevin's Asia Grace photobook app, which is also wonderful.)

Phil says:

I've been an independent iOS app developer for four years, and my wife, Jennifer, and I work together from our home in Austin creating games and educational apps. Our newest app is called Math Fleet, an action game set in space where players must use quick math skills to save Earth from invasion, all while dodging asteroids and battling enemy star fighters.

The inspiration for the game came from our sons, Jack, Luke and Dylan (ages 7, 6 and 3) for whom I've downloaded and tried many educational apps and games. At the beginning of the summer we were all home together and I was brainstorming the next game, which I knew I wanted to set in space. I had my sons Jack and Luke playing a math game I had downloaded to earn playing time for the other games they really wanted to play. We all agreed though that the math game I had them playing, which cost me \$5, wasn't a very good game and there really wasn't very much math in it. I knew I could write a better game and we started talking about what we would do differently, and in that moment, we decided our mission was to create not just a better game, but the most awesome action math game out there, and that Jack and Luke would be in the driver's seat guiding how the game would take shape.

It became the ultimate geek dad summer for me as I fully committed myself to making their ideas a reality, and some of their ideas were pretty challenging to implement. Such as Luke's idea that the user pilot multiple ships, customizable and upgradable -- the eventual foundation of the game; Or the Patrol Sector concept drawn up by my 7-year-old, Jack.

Throughout the eight month journey, as I coded, they tested prototypes of controls, menus, action sequences, effects, space weapons, and ships. They told me what they liked and what they didn't like, what they'd do differently, and in many cases they would contribute key design concepts delivered as crayon drawings or Lego models. I learned more about game design by watching them, listening, and discussing ideas than I learned from writing my previous two games.

Finally, after eight months and 300,000 lines of code, Math Fleet is officially released, and is hopefully everything we set out to create to create an exciting math game that kids actually want to play, while also challenging their minds by combining fast problem solving with the stress and distraction of piloting a space fleet.

Math Fleet

# Crypto and Bletchley Park podcast from BBC's Infinite Monkey Cage

BBC Radio 4's great math and science show "The Infinite Monkey Cage" did a great (and very funny) episode on crypto and Bletchley Park, with Robin Ince, Brian Cox, Dave Gorman, Simon Singh and Dr Sue Black.

(via Schneier)

## MoMath, more problems —

Here's an awesome activity for anybody who happens to be in New York City. Next week, on December 15th, The National Museum of Mathematics (MoMath) will open at a location near the Flatiron Building. Opening weekend festivities (and the museum, itself) look really cool.

# Amazing, invisible work that goes on when you click an HTTPS link

Jeff Moser has a clear, fascinating enumeration of all the incredible math stuff that happens between a server and your browser when you click on an HTTPS link and open a secure connection to a remote end. It's one of the most important (and least understood) parts of the technical functioning of the Internet.

People sometimes wonder if math has any relevance to programming. Certificates give a very practical example of applied math. Amazon's certificate tells us that we should use the RSA algorithm to check the signature. RSA was created in the 1970's by MIT professors Ron *R*ivest, Adi *S*hamir, and Len *A*dleman who found a clever way to combine ideas spanning 2000 years of math development to come up with a beautifully simple algorithm:

You pick two huge prime numbers "p" and "q." Multiply them to get "n = p*q." Next, you pick a small public exponent "e" which is the "encryption exponent" and a specially crafted inverse of "e" called "d" as the "decryption exponent." You then make "n" and "e" public and keep "d" as secret as you possibly can and then throw away "p" and "q" (or keep them as secret as "d"). It's really important to remember that "e" and "d" are inverses of each other.

Now, if you have some message, you just need to interpret its bytes as a number "M." If you want to "encrypt" a message to create a "ciphertext", you'd calculate:

C ≡ Me (mod n)

This means that you multiply "M" by itself "e" times. The "mod n" means that we only take the remainder (e.g. "modulus") when dividing by "n." For example, 11 AM + 3 hours ≡ 2 (PM) (mod 12 hours). The recipient knows "d" which allows them to invert the message to recover the original message:

Cd ≡ (Me)d ≡ Me*d ≡ M1 ≡ M (mod n)

# Free Coursera Calculus course with hand-drawn animated materials

Robert Ghrist from University of Pennsylvania wrote in to tell us about his new, free Coursera course in single-variable Calculus, which starts on Jan 7. Calculus is one of those amazing, chewy, challenging branches of math, and Ghrist's hand-drawn teaching materials look really engaging.

Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include:

the introduction and use of Taylor series and approximations from the beginning;

* a novel synthesis of discrete and continuous forms of Calculus;

* an emphasis on the conceptual over the computational; and

* a clear, entertaining, unified approach.

Calculus: Single Variable (Thanks, Robert!)

# Tallest possible Lego tower height calculated

The good folks on the most-excellent BBC Radio/Open University statistical literacy programme More or Less decided to answer a year-old Reddit argument about how many Lego bricks can be vertically stacked before the bottom one collapses.

They got the OU's Dr Ian Johnston to stress-test a 2X2 Lego in a hydraulic testing machine, increasing the pressure to some 4,000 Newtons, at which point the brick basically melted. Based on this, they calculated the maximum weight a 2X2 brick could bear, and thus the maximum height of a Lego tower:

The average maximum force the bricks can stand is 4,240N. That's equivalent to a mass of 432kg (950lbs). If you divide that by the mass of a single brick, which is 1.152g, then you get the grand total of bricks a single piece of Lego could support: 375,000.

So, 375,000 bricks towering 3.5km (2.17 miles) high is what it would take to break a Lego brick.

"That's taller than the highest mountain in Spain. It's significantly higher than Mount Olympus [tallest mountain in Greece], and it's the typical height at which people ski in the Alps," Ian Johnston says.

"So if the Greek gods wanted to build a new temple on Mount Olympus, and Mount Olympus wasn't available, they could just - but no more - do it with Lego bricks. As long as they don't jump up and down too much."

# Some planets are harder to leave than others

At his Psychology Today blog, Michael Chorost delves into a question about exoplanets that I've not really thought much about before — how easy they would be to leave.

Many of the potentially habitable exoplanets that we've found — the ones we call "Earth-like" — are actually a lot bigger than Earth. That fact has an effect — both on how actually habitable those planets would be for us humans and how easily any native civilizations that developed could slip the surly bonds of gravity and make it to outer space.

The good news, says Chorost is that the change in surface gravity wouldn't be as large as you might guess, even for planets much bigger than Earth. The bad news: Even a relatively small increase in surface gravity can mean a big increase in how fast a rocket would have to be going in order to leave the planet. It starts with one equation — SG=M/R^2.

Let’s try it with [exoplanet] HD 40307g, using data from the Habitable Exoplanet Catalog. Mass, 8.2 Earths. Radius, 2.4 times that of Earth. That gets you a surface gravity of 1.42 times Earth.

... it’s amazingly easy to imagine a super-Earth with a comfortable gravity. If a planet had eight Earth masses and 2.83 times the radius, its surface gravity would be exactly 1g. This is the “Fictional Planet” at the bottom of the table. Fictional Planet would be huge by Earth standards, with a circumference of 70,400 miles and an area eight times larger.

Does that mean we could land and take off with exactly the same technology we use here, assuming the atmosphere is similar? Actually, no. Another blogger, who who goes by the moniker SpaceColonizer, pointed out that Fictional Planet has a higher escape velocity than Earth. Put simply, escape velocity is how fast you have to go away from a planet to ensure that gravity can never bring you back. For Earth, escape velocity is about 25,000 miles per hour. Fictional Planet has an escape velocity 68% higher. That’s 42,000 miles per hour.

Read the full story at Psychology Today blogs

Thanks to Apollo 18, who also helped with the math for Chorost's post.

## Gifts for the space fans in your life —

If yesterday's BoingBoing Gift Guide didn't give you enough holiday ideas, Popular Science has a collection of gifts for aspiring rocket scientists. Includes meteorite jewelry, a scarf printed with a pattern inspired by measurement systems, and some natty blazers designed by NASA.

# Make a green bean matherole! (And other math-based Thanksgiving treats)

Vi Hart is Khan Academy's professional mathemusician. (Yeah, I KNOW, right?) And, this year, she's making the most delightfully nerdy Thanksgiving dinner ever.

It begins with green bean matherole, topped with fried Borromean onion rings. But, besides the fact that it's finished with crispy, delicious hyperbolic geometry, what makes the matherole a matherole?

Vectors. Like the rings, vectors are part of geometry. They've got a magnitude (think: size of the green bean) and they've got a direction (think: which way the green bean is pointing). Most importantly, a single vector can be part of a field of vectors. And that, my friends, is an excellent starting point for a 9 x 13 pan full of beans.


text:
Willkommen, schön sind Sie da!

# Geometry, Algebra, Number Theory, and Their Information Technology Applications

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## Beschreibung

This volume contains proceedings of two conferences held in Toronto (Canada) and Kozhikode (India) in 2016 in honor of the 60th birthday of Professor Kumar Murty. The meetings were focused on several aspects of number theory:

• The theory of automorphic forms and their associated L-functions
• Arithmetic geometry, with special emphasis on algebraic cycles, Shimura varieties, and explicit methods in the theory of abelian varieties
• The emerging applications of number theory in information technology

Kumar Murty has been a substantial influence in these topics, and the two conferences were aimed at honoring his many contributions to number theory, arithmetic geometry, and information technology.

Contains research articles in applications in number theory in information technology with emphasis on cryptographic applications of abelian varieties

Features research articles on explicit methods in abelian varieties, whose explicit constructions will be essential for applications in IT

Addresses questions such as twin rime conjectures with an overview of recent developments in Sieve theory

Klappentext

This volume contains proceedings of two conferences held in Toronto (Canada) and Kozhikode (India) in 2016 in honor of the 60th birthday of Professor Kumar Murty. The meetings were focused on several aspects of number theory:

The theory of automorphic forms and their associated L-functions
Arithmetic geometry, with special emphasis on algebraic cycles, Shimura varieties, and explicit methods in the theory of abelian varieties
The emerging applications of number theory in information technology

Kumar Murty has been a substantial influence in these topics, and the two conferences were aimed at honoring his many contributions to number theory, arithmetic geometry, and information technology.

Inhalt
Overview of the work of Kumar Murty (A. Akbary, S. Gun, M. Ram Murty).- On the average value of a function of the residual index (A. Akbary, A. T. Felix).- Applications of the square sieve to a conjecture of Lang and Trotter for a pair of elliptic curves over the rationals (S. Baier, V. M. Patankar).- $R$-group and multiplicity in restriction for unitary principal series of $GSpin$ and $Spin$ (D. Ban, K. Choiy, D. Goldberg).- The $2$-Class Tower of $\mathbb{Q}(\sqrt{-5460})$ (N. Boston, J. Wang).- On the bad reduction of certain $U(2,1)$ Shimura varieties (E. de Shalit, E. Goren).- Density modulo 1 of a sequence associated to a multiplicative function evaluated at polynomial arguments (J.-M. Deshouillers, M. Nasiri-Zare).- Uniqueness Results for a class of $L$-functions (A. Dixit).- Quadratic periods of meromorphic forms on punctured Riemann surfaces (P. Eskandari).- On the local coefficients matrix for coverings of $\SL_2$ (F. Gao, F. Shahidi, D. Szpruch).- Eisenstein series of weight one, $q$-averages of $0$-logarithm and periods of elliptic curves (D. R. Grayson, D. Ramakrishnan).- On zeros of certain cusp Forms of integral weight for full modular group (M. Manickam, Sandeep E. M.).- A note on Burgess bound (R. Munshi).- A smooth Selberg sieve and applications (M. Ram Murty, A. Vatwani).- Explicit arithmetic on abelian varieties (V. Kumar Murty, P. Sastry).- Derived categories of moduli spaces of vector bundles on curves II (M. S. Narasimhan).- Representations of an integer by some quaternary and octonary quadratic forms (B. Ramakrishnan, B. Sahu, A. K. Singh).- A topological realization of the congruence subgroup kernel (J. Scherk).- Fine Selmer groups and isogeny invariance (R. Sujatha, M. Witte).- Distribution of a subset of non-residues modulo $p$ (by R. Thangadurai, V. Kumar).- On solving a generalized Chinese remainder theorem in the presence of remainder errors (G. Xu).- Endomorphism algebras of abelian varieties with special reference to superelliptic Jacobians Y. G. Zarhin).

## Produktinformationen

 Titel: Geometry, Algebra, Number Theory, and Their Information Technology Applications Untertitel: Toronto, Canada, June, 2016, and Kozhikode, India, August, 2016 Editor: EAN: 9783030073466 ISBN: 3030073467 Format: Kartonierter Einband Herausgeber: Springer International Publishing Genre: Mathematik Anzahl Seiten: 536 Gewicht: 803g Größe: H235mm x B155mm x T28mm Jahr: 2019 Auflage: Softcover reprint of the original 1st ed. 2018

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