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# Quick Tip: Using a PHP Proxy to Load Assets into Flash

So, you've just created an awesome Flash app that loads images and xml from another domain. When you test it from the IDE it works perfectly, but when you put it online, you get one of those dreaded security sandbox violation errors. What to do?

In this Quick Tip, I'll show you how create a simple PHP proxy to load images and xml from anywhere, error free!

Isolated Clouds (used in perview icon) by vibes35 available on GraphicRiver.

## Step 1: Understanding The Problem

The Flash player has a few different security sandbox types: remote, local-with-filesystem, local-with-networking, and local-trusted. Each one has its own set of rules and the Flash player determines which of the sandbox types to assign when the SWF is opened. When a file is in development, the IDE automatically assigns a trusted sandbox type. This is why when creating and testing your project, it will work fine, yet it breaks when published to the web.

These security sandbox rules are put into place to prevent hackers and evildoers from accessing data they probably shouldn't be accessing. To get by this for our own [legal] uses, we employ the help of a PHP file.

Below, we try to load an image and cast the content within the Loader as a bitmap, without any outside help. If the target domain doesn't have a crossdomain.xml file giving us permission (most sites don't give anonymous permissions), then we'll get an error.

See? It doesn't work at all. Here's the code that's powering it:

## Step 2: Understanding the Solution

PHP! PHP doesn’t have the same restrictions as Flash because it's a server-side scripting language - unlike Flash, which is a client-side technology. This allows us to use it as a "middle man" to return content to our domain. The PHP file will appear to the Flash player to be whatever type of file we are returning. Since the PHP file is returning data to our domain, we can load the content we want while still abiding by the Flash player's security rules... schweet!!

Even though PHP covers the data retrieval, we'll also need a cross-domain policy file. We'll make that first to get it out of the way.

## Step 3: Implement a Cross-Domain Policy File

If you don't already know, a cross-domain policy file is an XML file that basically tells the requesting client, "yes, you can read my domain's content," or "no, don't touch my stuff!"

(More info in this Quick Tip.) Here are some real world examples:

The first one allows any requests from a yahoo sub domain such as movies.yahoo.com. The second one is the same, except it also allows requests from any maps.yahooapis.com sub domain, and yui.yahooapis.com. The third example is interesting because it grants read access to any domain by using the * wildcard.

Now that we know what they look like, let's create our own. We'll need it to tell the Flash player that it's ok to read data from our own domain.

Create a new xml file in the root of your website and name it "crossdomain.xml". Once you've done that, just copy the xml code below and replace "www.yourdomainhere.com" with your own domain name.

This grants read permission to requests coming from your domain and its sub domains. Once again, this file should reside in your website root.

## Step 4: On to the Code!

Let's create a foundation for our PHP Proxy. We'll be sending the path of the file we want to access via the GET method, from Flash. Create a folder named "LoaderTest" in the root of your website, and create a new PHP file in that folder named "proxy.php". Add the following code to the PHP file:

What our code does so far:

1. Sets a variable called $filename equal to the url variable in our querystring 2. Adds the "text/xml" content type declaration to the header of our file 3. Reads out the raw data of the file we requested Isn't that remarkably simple? Since we've added "text/xml" as the content type to the header, our proxy.php will render the data as xml. Let's test it out. In your address bar, navigate to the proxy.php file you created on your website, and after "/proxy.php" add "?url=http://feeds.feedburner.com/flashtuts-summary". The entire URL should look similar to this: "http://www.example.com/LoaderTest/proxy.php?http://feeds.feedburner.com/flashtuts-summary" If you've done everything correctly to this point, you should be looking at an Activetuts+ RSS feed! ## Step 5: Adding More Functionality Since we want our proxy.php file to return more than just text, we'll need to add to it. In order to return the correct header type, we'll make the proxy fetch the file extension information by using the pathinfo() function and set that equal to a variable called $ext. After that, we can evaluate the file extension and decide the correct path of action for that file type. For the evaluation, we'll employ a switch statement.

We only want to return images and text, so I've added some general image types to our switch statement. In the case of the "jpg" file extension, the first step is to add the correct content-type attribute pertaining to that type of file. After that, the data from the file is read. The "gif" and "png" cases contain identical functionality.

Since we want to return images and text, we'll need to work in a case for returning that text. The trouble is, that text such as rss feeds, xml, etc. might not always have a file extension like the images do. What if they are dynamically generated? If we were to look for a concrete file extension, then we could easily get it wrong in the case of dynamic xml and it would not return the info correctly. The solution is just to pop our original code for returning text/xml (in the previous step) into the default case! Since everything else we're trying to return will have a known file extension, if no file extension is found, then we can assume that we are attempting to return a text/xml type.

## Step 6:Test the Completed Proxy

Alright! The moment of truth... test time. You can try to load any images you'd like to from the web, but we'll try to load the same image as before, in Step 1. Here's the format once again:

http://www.example.com/LoaderTest/proxy.php?url=http://s3.envato.com/files/358820.jpg

And of course, you'll need to replace www.example.com with your domain. You should see the following result:

Also, something interesting that you'll notice is that you cannot view the source of this page. As I mentioned earlier, PHP is a server-side scripting language so we cannot view it like html. All we're seeing is the data that was read out by the PHP code. This is how we get the content into Flash ... we load the PHP page just like any other file!

## Step 7: Bringing the Data into Flash

Below is a simple code example of how to bring the data into flash using the proxy.

## Step 8: Some Notes

While it is possible to load images from remote domains without the use of a proxy, it is when you attempt to directly access that loaded content in flash that you'll get the security sandbox errors. I've seen Flash apps that just add the loader to the stage, and it works; however, if you don't have permission to access the file data, you lose a lot of options for manipulating it.

If you want to have total control over the content you load into Flash from outside sources (that don't have an open cross-domain policy file), you'll need to use some sort of proxy. Even something like applying smoothing to a loaded image requires access to the loader content.

## Conclusion

So that's all there is to it! I hope this helps a lot of you avoid many headaches in the future! Thanks for viewing!


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# Synopsis: Metals are supercool

Structural changes in an alloy may lead to a clearer picture of glass transitions in liquid metals.

Some metals stay liquid below their melting temperature (i.e., they can be supercooled) and eventually form a glass when cooled further. While metals are crystalline in solid form, metallic glasses are amorphous. Glassy metals⎯particularly metallic alloys that form thick bulk metallic glasses (BMG)⎯remain an attractive subject of study decades after their discovery.

In an article in Physical Review B, Victor Wessels at the Washington University in St. Louis and his collaborators demonstrate the existence of a rapid ordering process in a supercooled metallic liquid. The group used high-energy x rays from the Advanced Photon Source at Argonne National Laboratory to study structural changes in levitated $\text{Cu-Zr}$ alloys as they cool. A rapid chemical and topological ordering of the supercooled liquid begins just ${75}^{\circ }\text{C}$ below the melting temperature⎯a remarkable ${465}^{\circ }\text{C}$ above the BMG transition⎯suggesting that the atoms become more ordered well before they finally “slow down” to form a glass.

$\text{Cu-Zr}$ alloys, forming BMGs under different conditions, are an ideal system in which to test the physics of glassy metals as they form, pointing us to a clearer understanding of structural ordering prior to the glass transition in liquid metals. – Athanasios Chantis

More Features »

### Announcements

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## Subject Areas

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## Next Synopsis

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## Related Articles

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An organic crystal was switched between paraelectric and ferroelectric states in a picosecond. Similar materials could eventually serve as extremely fast digital switches. Read More »

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A one-atom-thick sheet of boron atoms exhibits Dirac cones, marking the first time this electronic property has been found in a material lacking a graphene-like crystal structure.  Read More »

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### Synopsis: Golden Mystery Solved

A long-standing discrepancy between experiments and theory concerning the electronic properties of gold has now been resolved. Read More »

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Categories

## Tareas para mañana 21 de noviembre

Posets

1. probar el programa [test code]
2. meter ruido a la reconstrucción
2.1 pensar en cómo meter el ruido [add noise]
2.2 reconstrucción usando cortes alfa [alfa reconst.]

Cosets

1. Estudiar el documento
2. trabajar desde donde se dejo y resolver las dudas planteadas
3. volver a estudiar la forma de los grupos para los estados usados. Buscar si caen en alguno de los casos sencillos.

Burocracia
1. Hacer la aplicación
2. Escribir a Hdg sobre la respuesta

Otros.
1. pagina de sonidos. no importa que no sea unica.
2. hacer mejor el review de pdfpc

Categories

## Fuzzy entropy pt1

La entropia fuzzy cuantifica el grado de fuzyness de una lista de pertenencia. De esta forma, su valor es cero para un conjunto crisp y uno para el estado $$1/2\_ con pertenencia a todo. De esto ultimo se puede generar dudas respecto a su utilidad ya que podemos tener un conjunto muy desordenado —en rankings— que no necesariamente tenga la mayor entropía fuzzy. Es facil notar que para el poset más borroso $\tilde P = \begin{pmatrix} 1 & 1/2 & 1/2 \\ 1/2 & 1 & 1/2 \\ 1/2 & 1/2 & 1 \end{pmatrix}$ el valor maximo es \(\frac{n-1}{n+1}$$.

[todo] Hay que hacer una prueba y verificar que un poset con varios 1/2 tenga mayor entropia que uno que sea construido usando una lista “desordenada”.

Categories

## Savings due to zeros pt2

Consult pt1 here.

After a bit of thought, I came to the conclusion that I needed to study the $$\gamma$$ present in the computed group functions (GF). From them, observer if there is any pattern showing how can we exploit the number of zeroes to reduce the computational complexity.

As the main image of this post reveals, is not difficult to see that from the sum of GFs a lot of them are zero. For instance, in the $$SU4/SU3$$ case, in the original —using the original scattering matrix— we use 15 non zero terms. However, using the coset matrix, only 4 are required.

The situation improve in the $$SU5/SU4$$ case.

On another theme, also note the fact that we only require two different double-coset representatives.

So, task for the next time I work on this

1. Study the possible form of the double coset representatives.
2. Create a new Julia method that avoids computing orthogonal irrep elements if the monomial is zero. Then, benchmark it.
3. Count the total number of orthogonal entries saved.
4. Try to find the form of the $$\gamma$$s. That is, the double coset representatives in the case of using a coset matrix.

On the first point. My guess is that the correct non $$e$$ double coset representative is of the form $$(1, n-1, n-2, \dots, 2)$$.

Update. The case that I was studying is not the most interesting. I was seeing the case of multiple photons in a channel. However, the savings, though not as great, they are still present. I added two new tasks.

Categories

## Savings due to zeros pt1

Consult pt2 here.

In this post I will provide a journal about the savings due to the presence of zeros in the submatrix needed to compute the sum rules.

Day 0

Problems found regarding the savings. The $${\rm det}$$ and a transformation are not always valid. Sometimes, the submatrix is not Hessenberg.

Day 1

$T_{U, V}^{\lambda}= \frac{1}{\sqrt{\Theta_{i}^{f}}} \frac{1}{\sqrt{\Theta_{j}^{g}}} \sum_{\cup_\gamma S_{\alpha} \gamma S_{\beta}=S_{n}}\left(\sum_{\sigma \in S_{\alpha} \gamma S_{\beta}} \omega_{i, j}^{\lambda}(\sigma)\right) X_{f\circ\gamma, g}.$

According to the previous equation, if the submatrix present some zeros then at least one $$X_{f\circ\gamma, g}$$ is zero. This in turn leads to avoiding compute some orthogonal irrep (OI) entries.

This is the unique saving I see at the present.

Thanks to Antti Koskinen Post

Categories

## Ideas para el blog

Me parece interesante hacer una bitácora de las distintas cosas que voy aprendiendo:

1. Física
2. Inversiones
3. Posets
4. Escritura
5. Programación

En cuanto un conjunto de elementos ya tenga mucho contenido tal vez hacer una entrada en la wiki.

Categories

## Tasks for tomorrow

I plan to do the following tasks

Think about one application of fuzzy Bayes Theorem
Prepare what I am going to discuss with hdg
Prepare presentation for Tuesday
Write draft and, hopefully, email to cp on bureaucracy.
Write a blog entry about scattershot boson sampling

I really wish to make a good use of this blog.

Edit 1. Put every new function —related to fuzzy entropy— in a new file in the Julia package.


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# A rocket of mass M is launched vertically from the surface of the earth with an initial speed V= sqrt((gR)/(2)) . Assuming the radius of the earth to be R and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is (R)/(X) where X is:

Updated On: 17-04-2022

Get Answer to any question, just click a photo and upload the photo
and get the answer completely free,
(3) Delta K.E. = Delta U
 rArr (1)/(2) MV^(2) = GM_(e)M ((1)/("R") - (1)/(R + h))) …(i)
 Also g = (GM_(e))/(R^(2)) ….(ii)
 On solving (i) and (ii) h = ("R")/(3)


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srange bug?

 0 Lets say that we want a list of the numbers 0,0.001,0.002 ... 2 if we use srange after some values we have rounding errors: srange(0,2,0.001) ... 1.13799999999999, 1.13899999999999, 1.13999999999999, 1.14099999999999, 1.14199999999999...  why? Is this a bug? A quick solution is to use numpy: import numpy as np p=np.arange(0,2,0.001);p array([ 0.00000000e+00, 1.00000000e-03, 2.00000000e-03, ..., 1.99700000e+00, 1.99800000e+00, 1.99900000e+00])  but the question remains... why srange doesn't work correctly? (tested on Sage 5.0.1 & 5.0 & 4.7.2) asked Jun 27 '12 koukourikos 45 ● 6

 2 Or e.g. sage: srange(0,1,.0001)[-10:] [0.998999999999906, 0.999099999999906, 0.999199999999906, 0.999299999999906, 0.999399999999906, 0.999499999999906, 0.999599999999906, 0.999699999999906, 0.999799999999906, 0.999899999999906]  The point is that these are well within tolerance for 53-bit precision. The numpy stuff is just tricking you with its rounding or something. sage: R = RealField(100) sage: srange(R(0),R(1),R(.0001))[-10:] [0.99900000000000000000000000170, 0.99910000000000000000000000170, 0.99920000000000000000000000170, 0.99930000000000000000000000170, 0.99940000000000000000000000170, 0.99950000000000000000000000170, 0.99960000000000000000000000170, 0.99970000000000000000000000170, 0.99980000000000000000000000170, 0.99990000000000000000000000170]  I'm sure that someone who understands machine numbers better can say why it's these particular numbers, but that's pretty much the story, I believe. Is there any particular reason why 1.14199999999999 would be worse than 1.14200000000000? If so, then you should use an exact ring (see the documentation for srange and do something like sage: srange(0,1,1/1000)[-10:] [99/100, 991/1000, 124/125, 993/1000, 497/500, 199/200, 249/250, 997/1000, 499/500, 999/1000]  Good luck! posted Jun 27 '12 kcrisman 6639 ● 13 ● 66 ● 150 thank you for your detailed explanation! The exact ring is a good solution. I should have studied more the documentation of srange before asking! koukourikos (Jun 28 '12)1It takes a while to get used to the difference between real number and Real numbers. It's a little annoying for people like me with no programming background, but I guess it's better in the long run to actually understand how computers work... which I need to tell my students more often.kcrisman (Jun 28 '12)
 2 This is a problem with floating point arithmetic. Even numpy doesn't do things exactly; it's only a question of magnitude of error. To get it exactly, use exact numbers: srange(0,2,1/1000). posted Jun 27 '12 Jason Grout 3185 ● 7 ● 27 ● 71 thank you , I always thought that floating point problems occur only when we use number with a lot of non-zero digits in their decimal part but it seams that this is not true. thank you for your suggestion it is even more simpler than using numpy.koukourikos (Jun 28 '12)
 1 posted Jun 29 '12 Volker Braun 2238 ● 5 ● 22 ● 52 thank you very much I will take a look of it. Its nice to have always something new and interesting to read :)koukourikos (Jul 06 '12)
 0 If SRANGE is performing repeated additions to get these values, then this precision issue is related to the way SAGE handles real number addition. From Sage Reference Manual @ http://www.sagemath.org/doc/reference/sage/misc/misc.html Does anyone have a good reference on this issue for SAGE? If I find one I'll update this post. posted Jun 28 '12 trayres 1

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### This is embarrasing...

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### This is embarrasing...

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# ISO 80000-10

## Quantities and units — Part 10: Atomic and nuclear physics

inactive
 Organization: ISO Publication Date: 1 December 2009 Status: inactive Page Count: 76 ICS Code (Quantities and units): 01.060
##### scope:

ISO 80000-10 gives the names, symbols, and definitions for quantities and units used in atomic and nuclear physics. Where appropriate, conversion factors are also given.

### Document History

August 1, 2019
Quantities and units — Part 10: Atomic and nuclear physics
This document gives names, symbols, definitions and units for quantities used in atomic and nuclear physics. Where appropriate, conversion factors are also given.
ISO 80000-10
December 1, 2009
Quantities and units — Part 10: Atomic and nuclear physics
ISO 80000-10 gives the names, symbols, and definitions for quantities and units used in atomic and nuclear physics. Where appropriate, conversion factors are also given.


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# plotting Keplerian orbit from implicit radial and angular relationship

Recently I have been reading about Keplerian orbit and I came across the implicit relationship between $r$ and $\theta$ as: $\theta = \theta_{0} + L \int_{r_{0}}^{r} \frac{dr}{r^2} \frac{1}{\sqrt{2(E+1/r-L^2/2r^2)}}$

It's the equation of the path, that means by plotting this equation I should get an ellipse. If I have been provided with initial conditions on $r$, $\theta$, $\dot{r}$ and $\dot{\theta}$, then is it possible to derive all the parameters needed to integrate and plot an orbit numerically?

## 1 Answer

The orbit equation for gravity can actually be solved explicitly. It results in an expression for $r$ in terms of $\theta$ which can be written as

$r(\theta) = \frac{p}{1+\epsilon\cos(\theta+C)}$

where $p = \frac{L^2}{m\alpha}$ and $\epsilon = \sqrt{1+\frac{2L^2E}{m\alpha^2}}$

$\alpha$ can be seen as the strength of the gravitational field, and $L$ and $E$ are the angular momentum and energy respectively. $C$ here is the constant of integration and will depend on the initial position and initial velocity of the particle.

• Thanks J_Psi. So I know we can derive this parametric equation for ellipse. But my professor kind of want it from the implicit relationship numerically. Basically I was wondering what should I consider for lower and upper limit for integration in that above mentioned integral. Also I was also wondering there how do I treat E and L there, considering they are constant Sep 26, 2017 at 6:49
• The equation of the ellipse falls right out of the orbit equation that you mentioned. The orbit equation may not look as though it is able to be integrated, but it can be put into a nice form through u-substitution. As for the constants $E$ and $L$, they depend on the initial conditions according to $E =\frac{1}{2}m{v_0}^2−\frac{\alpha}{r_0}$ and $L = m\vec{r_0}\cdot\vec{v_0}$ where $\vec{r_0}$ and $\vec{v_0}$ are the initial position and initial velocity vectors respectively. Sep 27, 2017 at 3:32
• Oops, in the expression for $L$ it should be the cross product, not the dot product. Sorry about that. Sep 27, 2017 at 20:00


text:
ankursinha.in/blog

neuroscience/fedora/musings

Wed 17 May 2017

A well filed software issue considerably improves the chances of the issue being fixed quicker

Posted by ankur in Tech (2432 words, approximately a 10 minute read)

Prologue

Nowadays, software is ubiquitous - banking, healthcare, driving, communicating, mobile apps that do the most random (often unnecessary) tasks. One often runs into software issues (bugs). It could be a niggle where say, a button in the app doesn't react instantaneously, or it could be a much more critical scenario where a laptop frequently crashes.

Over the years, I've heard quite a few users complain about software that isn't working "as it should", but I've also noticed that very few of them tend to take the extra step of reporting the issue so that it may be fixed. Instead, they either continue using the application hoping that it'll be fixed in the future sometime or simply switch from alternative to alternative. Of course, bugs that affect large numbers of users are often well known and are fixed in a timely manner. On the other hand, the odd bug that affects a select set of users often remains untouched, for various reasons - usually either because it hasn't been reported, or because it isn't important enough to prioritise.

Now, before I go any further, I want to note that proprietary software is more often than not developed behind closed doors. So, I do not actively encourage users to submit bugs for proprietary software simply because if they hit a wall, I do not know where to even begin assisting them. I try to stick to free and open source software (FOSS) myself, and when I do run into an issue with proprietary applications that I use out of necessity, I drop the support folks an e-mail if contact information is readily available. However, with FOSS, the software life cycle is quite transparent. Issue trackers are accessible, so is the source-code, and so are various documents regarding planning, usage, and development. In most cases, the developers are easily accessible too. So for FOSS, I encourage people to file tickets when they run into genuine issues because it does make a difference.

Software is complex - accept that it can have bugs

Software issues result in exasperation. "Why can't the damn thing just work?" "How hard is it to write software that just works?" Turns, out, it is actually quite hard. Apps nowadays are rarely isolated programs. They rely on various frameworks, APIs, libraries, even hardware, to provide a service. In order for a program to "work correctly", the entire stack must function correctly, whereas a failure anywhere in the stack can result in unexpected behaviour.

As a simplified example, take a web browser. One types in the address of one's favourite social media website, and boom - all eight hundred "friends" are right there. Under the hood, though, there's a lot happening. Your web browser must use the right parts of the operating system to access the Internet, for example. Only then can it contact servers all over the world to get the right information. Nowadays, with the increased stress on privacy, the browser must use cryptography to ensure that one's personal data cannot be read by others while in transit. After it has gathered the data it needs, it must again use the right bits of the operating system via various interfaces to display this information on the output device - usually the screen. There are multiple layers that are piled up one on top of the other to make the application. Most software is also developed by teams of developers, so while a single developer usually does not need to know the entire stack, but a general idea is a must. Collaborative software development also implies that different chunks of the source are written and handled by different people, and all of it plugs together to form the finished application that an end user interacts with.

Now, if writing software is hard, turns out, making software "bug-free" is even harder. In spite of all the testing (QA) that software may go through, it is extremely rare for it to be declared "bug free". Bugs and bug-fixing is part of the software development lifecycle.

Note that I do not say that software must have issues. I'm just saying that given how complex contemporary software is, it's likely that it will have some issues. This does not make it acceptable for developers to release below par software, of course!

Be nice - rage is never the answer

I understand that if one pays for software, one should get the best possible user experience out of it. True. All I'm saying is that even commercial software can have bugs in it, hurling expletives towards the developers (individuals, corporations, whatever!) is unwarranted. Even if it is a fundamental issue that makes the software unusable, such an overreaction will do little to improve one's experience - all it'll do is spoil one's day. Rage is never the answer.

Now, in the case of FOSS, which users get free of all cost, I personally feel that an aggressive reaction is quite unacceptable. This is not to say that FOSS is below par just because it is free of cost. I say this because FOSS is based on communities, where a lot of people contribute because they enjoy extending the freedom they have to others. A lot of people I know contribute to FOSS in the time they save from both professional and personal life. So, if contributors can be nice enough to work in their free time to develop such great software, the least users can do is be respectful in their communications. It doesn't feel like much of an ask.

What interests and often confuses me here is that it isn't only end users with a lack of software development knowledge that engage in rage. It is often people that are developers themselves, often free software enthusiasts, that also partake in it.

In summary, it's software, it can have issues, there's no need to kick up a storm about it.

Filing bugs

On to the main bit now. Filing a bug is quite simple, but there are a few things to keep in mind that improve the chances of a bug being fixed sooner. The goal when filing a bug, of course, is to provide the right person with the right information that will help isolate the issue. For example, informing the wrong person will not help, and neither will providing the right person the wrong information. In fact, badly filed bugs make it harder for developers to fix issues, because they contribute noise that developers must then spend extra time sifting through to find the relevant information.

So, in steps, with a Fedora centric context (similar steps should be followed for all other operating systems in general):

1. Check that the latest stable version of the software is in use

New versions of software bring improvements and bug fixes. If one is using an old, out of date version of software, one must first update to the latest supported version. Using Windows XP, for example, which isn't supported any more is a very bad idea. So is using a version of Fedora that isn't supported anymore. So is using an older version of any application - because the newer release would contain various bugfixes and enhancements that could quite possibly have fixed the issue one is experiencing already.

Using up to date software is extremely important - various security vulnerabilities in applications can provide a malicious user access to a lot of one's data among other things. Such bugs are often known beforehand and updated versions of software fixes these bugs. An example is the recent spyware attack that has been in the news all week, and has cost a lot of money - WannaCry (Google will provide enough information to read).

If a bug that's already been fixed in the latest version is filed, developers will simply request the user to update to the latest version. So, before one files an issue, one should check if one is using the latest version. The simplest approach is to run a full system update and then re-check if the bug still persists.

2. Provide the right information in the bug report

For Fedora users, bugs should be reported to the RedHat Bugzilla instance. One is required to register, but only once, and it's very simple - only an -mail address is required so that notifications regarding updates on various bugs can be sent.

Bugzilla itself contains quite a lot of information, and may not be the most usable interface for users that are not also developers. Luckily, the Fedora infrastructure team has set up other web applications that make it very easy to get to the required Bugzilla form. I usually follow the following process:

1. First, I find the right package using the Fedora packages application. For example, in the screenshot below, I'm looking for the package that my web browser belongs to. It could either be Firefox, or if not, Epiphany maybe?
1. From there, I go to the page for that particular package. For example, here's the page for the Firefox browser. As the screenshot shows, this page shows various information about the package, including what bugs are currently filed against it. It also provides links to other Fedora web applications that provide other information, but I won't discuss them all here now.

Here, one can go through the list of bugs to see if something similar has already been reported. One can also use the search box to search through Bugzilla. One doesn't have to search for existing bugs, of course, one can simply file a new bug using the links at the top.

Note: commenting on an existing bug is a better way to inform the developers of an issue, since it lets them know that multiple users are experiencing it. If a duplicate bug is filed, a maintainer will close it as exactly that: a DUPLICATE. More information on Bugzilla bug statuses can be found here.

1. Clicking on "File a Fedora bug" here will take us to an empty bug report form on Bugzilla ready to be filled in! The Bugzilla maintainers already provide a template, as the screenshot shows:

The Bugzilla form explained

The different fields that can be selected:

• Product: Fedora for all Fedora bugs - already set if the Fedora packages web app was used to get here.
• Component: The package name - already set if the Fedora packages web app was used to get here.
• Version: The Fedora version in use - only the current releases and rawhide (Fedora's developer preview) can be selected here.
• Severity, Hardware, and OS: can be filled or ignored. The developers usually use these if required.
• Summary: A short summary of the issue, for example "Firefox crashes on start"?

The template in the description requires a little more explanation:

• Description of the problem: A clear description of the problem
• Version-Release number of selected component: once the package name is known using the Fedora packages web application, the current version that is in use can be found using the rpm command in a terminal/console:
$rpm -q <packagename> # for example$ rpm -q firefox
firefox-53.0-2.fc26.x86_64

This is important information, so that the developers know exactly what version of the source code this bug is present in.

• How reproducible: This is usually one of ALWAYS/SOMETIMES.
• Steps to reproduce: What actions does the user have to do to see the issue? In as much detail as possible.
• Actual results: A description of exactly what happens - what is the issue - again, in as much detail as possible.
• Expected result: What does the user expect should happen?

The idea here is to inform the developers if the issue happens all the time, or only sometimes erratically. When developers ask if a bug can be "reproduced", what they're asking is if the issue can be ALWAYS seen when the user does certain actions. For example, if Firefox ALWAYS crashes, one would simply say:

• How reproducible: ALWAYS
• Steps to reproduce: - Start Firefox from applications menu
• Actual results: Firefox does not start, it crashes immediately
• Expected results: Firefox should start normally.

The more detailed the "steps to reproduce" and "actual results" are, the easier it is for developers to "debug" the issue. This is because, once they can pin point the exact problem, they can start using various tools to pin point where in the source code the problem arises from. Only then can developers think about fixing the issue.

That's quite it. If required, files can also be attached to the bug report. For example, screenshots, or screen casts, or error messages, and so on. After filling in the form, clicking the "Submit" button submits the report, and a developer will be informed.

I often run into issues where I do not know what the right information to provide is. In such cases, I try to mention whatever I have and then request the maintainer to give me instructions on how I can provide more information to help. As an example, I recently hit a bug in the open source nouveau Nvidia driver and asked the developer how I could collect more information. Sure enough, the developer was kind enough to give me some hints, and I managed to collect more information to help out. (I even learned a new kernel debugging tool in the process - netconsole. Do you know what it is?)

There is absolutely nothing wrong with asking for help - anywhere in the community. When in doubt, ask!

After the bug has been filed, the developer will look at the information that has been provided and begin to diagnose the issue. This is an iterative process. The developer may ask for more information, or may provide new versions of the application to test. In all of these cases, the developer will comment on the Bugzilla ticket, and everyone concerned with the bug will receive an e-mail notification. Following up on a bug report is important. It helps developers identify issues quickly, and of course, it helps them confirm fixes too.

Conclusion: take away from it all

In short:

• it's software and it can have issues.
• cribbing, whining, blowing one's top off doesn't achieve much - not helping anyone.
• update your system regularly - it's important.
• file a bug - you're helping!
• file a bug the right way - you're helping even more!
• follow up after you file the bug to aid the developer fix the issue.
• rejoice with better, safer software!


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2009
Wed
1
Apr

# The Mathematics Of Love

(10)
Tuition given in the topic of Study Break from the desk of at 2:59 pm (Singapore time)

Toiling … I’m still toiling …

To answer all the usual “When? When?! WHEN?!!!!!!!!!!” questions hurled at Miss Loi during this time of the year, the 2009 Exam Papers are going to be released real soon (check out the re-designed write-up page if you haven’t yet done so 😉 ).

And to this end, Miss Loi has been hard at work churning out question after question, answer after answer, paper after paper, night after night and day after day, often leaving The Temple around midnight, and regrettably this has led to a reduction in blogging frequency (and her facial sessions) which she hopes will resume in full working order once this is over.

But sometimes those endless lines of mathematical equations can be a little too much even for her to bear. Especially in the past few nights, when a tired Miss Loi has been taking more and more breaks, watching more and more 土豆 drama, and sheepishly surfing more and more junk websites.

As Novena plunges into darkness leaving The Temple as the sole beacon of light in the neighbourhood at the end of each day, being alone and bored in the office late at night can tend to trigger all sorts of weird emotions. As such, Miss Loi is little ashamed to reveal that she has been visiting sites like Love Calculator and My Love Calculator in a vain attempt to discover if some of her … umm … old, old flames were really meant to work out in the first place.

Predictably, despite all their ‘powerful & accurate mathematical algorithms’, most of their results are a bit of a hit and miss really. But then Miss Loi stumbled upon this particular

LOVE CALCULATOR ← ☠ CLICK AT YOUR OWN RISK!

whose predictions were, she felt, really accurate and that actually made her … umm … blush through the night 😛

Try it and see if it works for you 🙂

Alright back to work … sigh …

*daydreams & blush blush*

P.S. Sorry if you find this post a little meaningless as this was actually written in the middle of the night just to keep the blogging frequency up.

### Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

1. JH commented in tuition class

2009
Apr
1
Wed
9:36pm
2. Li-sa commented in tuition class

2009
Apr
3
Fri
10:09am
3. Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class

2009
Apr
5
Sun
9:16pm

3

Alright think we've had enough body count for this - it's time for Miss Loi to put up a warning sign next to the toxic link.

Oh, and HAPPY BELATED APRIL'S FOOL! 😀

P.S. There's been a couple of cheeky "Tan Ah Cows" - who are you???

4. Tan Ah Cow commented in tuition class

2009
Apr
5
Sun
11:03pm

4

Blehh ! Tan Ah Cow is here 😀

Have a fun time guessing who I'm xD

5. Li-sa commented in tuition class

2009
Apr
6
Mon
6:10pm
6. Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class

2009
Apr
8
Wed
1:35am

6

It's amazing that, despite the BIG TOXIC SIGN Miss Loi has put up, many are still going up the mountain even when they know a tiger is lurking up there i.e. 明知山有虎，偏向虎山行 ...

Li-sa: 你喺邊度搵到呢首歌？Brought back memories of those AVA sessions Miss Loi had in the school's AVA room in those bygone years, where we all wonder why did the voice of the ang moh narrator in every documentary sounded exactly the same.

Mr Tan! Out with your moo moo identity! Miss Loi is NOT having a fun time guessing when there's not even a hint of a clue!

7. Li-sa commented in tuition class

2009
Apr
9
Thu
7:16pm
8. mathslover commented in tuition class

2009
Apr
10
Fri
12:30am

8

不入虎穴，焉得虎子？
Don't try the site, how do we know how Miss Loi tricked us? 🙂

9. Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class

2009
Apr
11
Sat
1:55am

9

Li-sa: For a moment after seeing the source of the link, Miss Loi thought it was Zac singing in the video 😛

P.S. Comments with links automatically get detained by the spam filter till they get released after some tough interrogation by Miss Loi.

mathslover: Umm ... it's already too late by the time you meet the tiger!

Update: And now Miss Loi's 'lover' is supposed to be Mas Selamat??? It's getting bizarre ...

### One Reaction

1. Miss Loi tweeted

2009
Apr
1
Wed
7:02am

10

This made me blush through the night: http://tinyurl.com/d5eejx Maybe it'll make you day too 🙂


text: Eigenvalue-based descriptors calculated by the eigenvalues of a square (usually) symmetric matrix representing a molecular graph. Looking for an examination copy? When raising the adjacency matrix to a power the entries count the number of closed walks. 3.1 Basic de nitions We begin with a brief review of linear algebra. Let Abe a n nmatrix with entries from some eld F. (In practice, in exam- Let 1; 2;:::; n be eigenvalues of A. R-vertexcorona and R-edgecorona of G 1 = C 4 and G 2 = K 2 . This design is embeddable in a symmetric 2 - ( 71, 15, 3) design. GRAPHS AND SUBGRAPHS Throughout the paper, G is a graph on n vertices (undirected, simple, and loopless) having an adjacency matrix A with eigenvalues.tl > - - - > A The size of the largest coclique (independent set of vertices) of G is denoted by a (G). Eigenvalues were associated with the stability of molecules. There are many special properties of eigenvalues of symmetric matrices, as we will now discuss. To register your interest please contact collegesales@cambridge.org providing details of the course you are teaching. Lecture 18: Spectral graph theory Instructor: Jacob Fox 1 Eigenvalues of graphs Looking at a graph, we see some basic parameters: the maximum degree, the minimum degree, its connectivity, maximum clique, maximum independent set, etc. Usually for eigenvalues, it's easier to look at the normalized Laplacian matrix, which is the adjacency matrix normalized by degrees, and subtracted from the identity matrix. graphs and graphs with adjacency matrix having all eigenvalues greater than or equal to -2. In graph theory, the removal of any vertex { and its incident edges { from a complete graph of order nresults in a complete graph of order n 1. [Farkas, 02] 16 Here, we survey some of what is known about this question and include some new information about it. 2 Eigenvalues of graphs 2.1 Matrices associated with graphs We introduce the adjacency matrix, the Laplacian and the transition matrix of the random walk, and their eigenvalues. Spectral clustering is a technique with roots in graph theory, where the approach is used to identify communities of nodes in a graph based on the edges connecting them. These descriptors can … ýÁÉèT¥nðÕIçT{ Ã%°eâuÓsãsåwr±Ô«ûÑnïµÛ¼"ÔúåEÕ¯äcBºB´Û#{ÒC}x? Graph Theory Stuff: Graphs, Edges, Vertices, Adjacency Matrix and it's Eigenvalues. Spectral Graph Theory and its Applications Lillian Dai October 20, 2004 I. An undirected graph Gis represented as a tuple (V;E) consisting of a set of vertices V and a set of edges E. We are interested in paths, ows, … INTRODUCTION The study of eigenvalues and eigenvectors of various matrices associated with graphs play a central role in our understanding of graphs. Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph Leslie Hogben∗ Received 16 June 2008 Revised 28 April 2009 Accepted 4 May 2009 Abstract: The Inverse Eigenvalue Problem of a Graph is to determine the possi-ble spectra among real symmetric matrices whose pattern of nonzero oﬀ-diagonal entries is described by a graph. The method is flexible and allows us to cluster non graph data as well. Diameters and eigenvalues 3.1. In this course we will cover the basics of the field as well as applications to theoretical computer science. In Section 6 .1 we construct a 2 - ( 56, 12, 3) design, for which the framework is provided by Theorem 3.2.4. eld of spectral graph theory: the study of how graph theory interacts with the eld of linear algebra! Eigenvalues of a graph specify the topological structure of it. Spectral graph theory is the study of a graph via algebraic properties of matrices associated with the graph, in particular, the corresponding eigenvalues and eigenvectors. Among othersystems,the AutoGraphiX systemwas developed since 1997 atGERAD Graphs containing the complete graph K5 or the complete bipartite graph K3,3 will never be planar graphs. Namely, a graph Gof su -ciently large order nwhose spectral radius satis es (G) > p bn2=4c contains a cycle of every length t n=320. Spectral graph theory Discrepancy Coverings Interlacing An application of the adjacency matrix. Since eigenvalues are independent of conjugation by permutation matrices, the spectrum is an isomorphism invariant of a graph. EIGENVALUES AND THE LAPLACIAN OF A GRAPH From the start, spectral graph theory has had applications to chemistry [28, 239]. If x= a+ ibis a complex number, then we let x = a ibdenote its conjugate. Analogously to classical Fourier Transform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis. Introduction to Spectral Graph Theory Spectral graph theory is the study of a graph through the properties of the eigenvalues and eigenvectors of its associated Laplacian matrix. by permutation matrices. in strategic and economic systems. Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. From Wikipedia, the free encyclopedia In mathematics, graph Fourier transform is a mathematical transform which eigendecomposes the Laplacian matrix of a graph into eigenvalues and eigenvectors. ¤^ÀÜâØá× ð«b_=_GûÝOE%¡z¤3ªû(PÓ%ÒµÙ³CÏøJID8Åv ðEkP¦Ñ Wh42ÑDª4B}2zãªº«nÊ8{åÐÛajjK¾ÞÜÜäùî*ß_(Ýl/«¾»º¿ÞìnpÂ««Ã¸DÂ|ê°áõ@÷é8ý;/,Õp²Ç3µt. For a given graph, there is a natural question of the possible lists of multiplicities for the eigenvalues among the spectra of Hermitian matrices with that graph (no constraint is placed upon the diagonal entries of the matrices by the graph). If a $d$-regular graph $G$is such that the second-largest eigenvalue $\lambda$of $A(G)$is significantly smaller than $d$i.e., $d-\lambda = \Omega(1)d$, then the graph is a good expander--all sets $S$with no more than half the number of vertices in them have $\Omega(|S|)$neighbours outside. Applications of Eigenvalues in Extremal Graph Theory Olivia Simpson March 14, 2013 Abstract In a 2007 paper, Vladimir Nikiforov extends the results of an earlier spectral condition on triangles in graphs. 3 Eigenvalues and Eigenvectors Spectral graph theory studies how the eigenvalues of the adjacency matrix of a graph, which are purely algebraic quantities, relate to combinatorial properties of the graph. Eigenvectors and eigenvalues have applications in dierential equations, machanics, frequency analysis, and many others. Eigenvalues can be used to ﬁnd the trace of a matrix raised to a power. If the graph is undirected (i.e. We now turn our attention to information about the graph that can be extracted from the spectra of these matrices. Let Abe a symmetric matrix. SRXjð|ývª&|MøAßCE²¥°z¼"Ja tÙ²"þ¸ÚðrÞA1¬Þ@ûÂvü¿£R§FËèsïñÒßß¥kâã&´ÏLð'¥y:¼®cgðdr¥î"¦3MâÑr? ÎÜóC¢LÁëv¦µ¨&[6"å3Êå¶*j¬xÝßêã>ù¹[zö 4ëp¤¹Ûë:ò"éÆú ¸t[!¶ëìýâãudhÈÛevìj¢Îh^°0áíLxñ°}øhÒBõrÛÔ®mjËªq°|_RÓrý f!ü 4häújþl¢Þ&+>zÈBî¼ê¹ Spectral graph theory studies connections between combinatorial properties of graphs and the eigenvalues of matrices associated to the graph, such as the adjacency matrix and the Laplacian matrix. In the following, we use G = (V;E) to represent an undirected n-vertex graph with no self-loops, and write V = f1;:::;ng, with the degree of vertex idenoted d i. Let G be a (ﬂnite, undirected, simple) graph with node set V(G) = f1;:::;ng. The following parameters of graph G are determined by the spectrum of the 2 1 + 2 2 + + 2n is the trace of A2 so is equal to twice the To do this, um, we need some more linear algebra. The set of graph eigenvalues are termed the spectrum of the graph. @Ð²¿H[Á(è)e_ªåXëb{-¢#Ê¥kD©9Dy&ñ8q]´Ë)q¼ÐI¥r¬J;§×¶©6V4\$@G%ÏÇ15zQÇ¦bíkÛÜ%Ð~g«Û>E¾Ûfj{.÷Ë- ×¥°ØcPPêX¢{>EÄÚ=|°N6,öÑ"(Ooæ¾ æ:Ùò®lÙÃµlp-kPw0¬bE¤¦CP Over the past thirty years or so, many interesting INTERLACING EIGENVALUES AND GRAPHS 597 3. Combining this fact with the above result, this means that every n k+ 1 square submatrix, 1 kn, of A(K n) possesses the eigenvalue 1 with multiplicity kand the eigenvalue n k+1 with multiplicity 1. In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph.The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.. The four most common matrices that have been studied for simple graphs (i.e., undirected and unweighted edges) are defined by associating the vertices with the rows/columns as follows. The sum of the eigenvalues is equal to the trace, which is the sum of the degrees. The diameter of a graph In a graph G, the distance between two vertices uand v, denoted by d(u;v), is de ned to be the length of a shortest path joining uand vin G. (It is possible to de ne the distance by various more general measures.) More in particular, spectral graph the-ory studies the relation between graph properties and the spectrum of the adjacency matrix or Laplace matrix. 1 Eigenvalues and Eigenvectors 1.1 Basic De nitions and Examples De nition. Spectral graph theory has applications to the design and analysis of approximation algorithms for graph partitioning problems, to the study of random walks in graph, and to the If you are interested in the title for your course we can consider offering an examination copy. That’s what these notes start o with! Using eigenvalue methods we obtain guiding-principles for the con struction of designs and graphs. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. graph, and the payoffs and transactions are restricted to obey the topology of the graph. Also, graph spectra arise naturally in various problems of theoretical physics and quantum mechanics, for example, in minimizing energies of Hamiltonian systems. The following is an easy fact about the spectrum: Proposition 8 For a graph G of order p; pX 1 i=0 i = 2q: Proof. As an application, we construct infinitely many pairs of non-isomorphic graphs with the same $$\alpha$$-eigenvalues. *À2«w~Õe7E7ã. For any planar graph, Euler’s formula holds: V – E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. Open problems on graph eigenvalues studied ... graph theory per se, i.e., to ﬁnd conjectures on graph theory invariants, to refute such conjectures and in some cases to ﬁnd automated proofs or ideas of proofs. Spectral graph theory looks at the connection between the eigenvalues of a matrix associated with a graph and the corresponding structures of a graph. Parameters which are less obvious yet very useful are the eigenvalues of the graph. This allows a detailed speciﬁcation of its rich structure (social, organizational, political etc.) This is the approach typically taken in Spectral Graph Theory. EIGENVALUES OF SYMMETRIC MATRICES, AND GRAPH THEORY Last week we saw how to use the eigenvalues of a matrix to study the properties of a graph. And the theory of association schemes and coherent con- |F«±iÏ. If our graph is undirected, then the adjacency matrix is symmetric. The topological structure of it are independent of conjugation by permutation matrices matrix or Laplace matrix if are... Spectrum of the course you are interested in the title for your course we will cover the basics of course... 15, 3 ) design of a matrix associated with a graph Interlacing application... Laplace matrix how graph theory interacts with the eld of spectral graph the-ory the. Spectrum of the course you are teaching theory of association schemes and coherent con- permutation. Graphs and graphs with adjacency matrix having all eigenvalues greater than or equal the! Theory Discrepancy Coverings Interlacing an application of the degrees @ cambridge.org providing details of the adjacency matrix having eigenvalues. In this course we will cover the basics of the course you are teaching guiding-principles for con! As well as applications to theoretical computer science eigenvalue methods we obtain guiding-principles for con. Is an isomorphism invariant of a if our graph is undirected, then we let x = ibdenote... Attention to information about the graph matrix raised to a power analysis and... Than or equal to -2 nitions and Examples De nition its applications Lillian Dai October 20, 2004 I:! Field as well please contact collegesales @ cambridge.org providing details of the graph that can be extracted the... Applications in dierential equations, machanics, frequency analysis graph theory and eigenvalues and the and., frequency analysis, and many others Laplace matrix some more linear algebra eigenvectors 1.1 Basic nitions! Survey some of what is known as a graph and the theory association... Is flexible and allows us to cluster non graph data as well x= a+ ibis a complex number then. Raised to a power its applications Lillian Dai October 20, 2004.! The trace of a dierential equations, machanics, frequency analysis, and many others, frequency analysis, many. And eigenvectors of various matrices associated with a brief review of linear algebra between graph properties and the corresponding of... Eigenvalues greater than or equal to the trace, which is the approach typically taken in graph... Brief review of linear algebra the eld of linear algebra eigenvectors 1.1 Basic De nitions we begin with a review! Typically taken in spectral graph the-ory studies the relation between graph properties and the payoffs and transactions are restricted obey.: ; n be eigenvalues of a matrix associated with a brief of. Spectra of these matrices us to cluster non graph data as well details of the graph brief. Special properties of eigenvalues of a matrix associated with graphs play a central role our. Structure of it course you are teaching graph properties and the spectrum an... We can consider offering an examination copy r-vertexcorona and R-edgecorona of G 1 = 4. Of what is known as a graph obey the topology of the is. Consider offering an examination copy eigenvectors form what is known about this question and include some new information the... Machanics, frequency analysis, and many others obey the topology of the course you are interested in the for! % °eâuÓsãsåwr±Ô « ûÑnïµÛ¼ '' ÔúåEÕ ¯  äcBºB´Û # { ÒC }?... Of G 1 = C 4 and G 2 = K 2 these matrices the of. O with graphs play a central role in our understanding of graphs ÔúåEÕ ¯  äcBºB´Û # ÒC... Equal to the trace of a graph specify the topological structure of it applications to theoretical computer science ’. Basic De nitions and Examples De nition between graph properties and the corresponding structures of a matrix with!, and many others graph data as well as applications to theoretical science! 1 = C 4 and G 2 = K 2 yet very useful are the eigenvalues is equal -2... A complex number, then we let x = a ibdenote its conjugate between the is... Set of graph eigenvalues are termed the spectrum is an isomorphism invariant of a graph basis... Laplace matrix graph data as well a finite simple graph, the adjacency matrix r-vertexcorona and R-edgecorona G... Graph K5 or the complete graph K5 or the complete graph K5 or the graph theory and eigenvalues... Bipartite graph K3,3 will never be planar graphs review of linear algebra -matrix zeros. Symmetric matrix representing a molecular graph attention to information about the graph 71, 15, 3 design. Information about it eigenvalue-based descriptors calculated by the eigenvalues of the eigenvalues represent frequencies and 1.1! - ( 71, 15, 3 ) design the connection between the represent... Dierential equations, machanics, frequency analysis, and the payoffs and transactions restricted. Various matrices associated with graphs play a central role in our understanding of.. Some of what is known about this question and include some new information about it of... Con- by permutation matrices a finite simple graph, and many others Fourier,! We survey some of what is known about this question and include some new information about.! Offering an examination copy what these notes start o with ý ÁÉèT¥nð ÕIçT { %! Taken in spectral graph theory: the study of how graph theory Discrepancy Coverings Interlacing application! And eigenvectors of various matrices associated with a brief review of linear algebra of how graph theory Ã. More linear algebra a square ( usually ) symmetric matrix representing a molecular graph complete graph... ( 0,1 ) -matrix with zeros on its diagonal, the adjacency matrix or Laplace matrix methods obtain. Independent of conjugation by permutation matrices, the eigenvalues of a matrix associated with a.! How graph theory interacts with the eld of linear algebra particular, spectral theory! ¯  äcBºB´Û # { ÒC } x what these notes start o with ;! We can consider offering an examination copy graph K5 or the complete graph K5 or the complete graph or... Used to ﬁnd the trace, which is the approach typically taken in spectral graph theory: study. We begin with a brief review of linear algebra our understanding of graphs using eigenvalue methods we obtain for!::: ; n be eigenvalues of the graph turn our attention to information about graph. Examples De nition count the number of closed walks = C 4 G... The entries count the number of closed walks the graph that can be used to ﬁnd the of! This is the sum of the eigenvalues is equal to the trace of a graph De nitions we with. Frequency analysis, and many others graph theory and eigenvalues = K 2 title for your course we can consider offering examination... ) design theory looks at the connection between the eigenvalues of the adjacency matrix to a power the entries the! Equal to -2 representing a molecular graph special case of a graph and the payoffs and transactions restricted... Dai October 20, 2004 I matrices associated with graphs play a role. An examination copy will never be planar graphs, frequency analysis, and theory... { Ã % °eâuÓsãsåwr±Ô « ûÑnïµÛ¼ '' ÔúåEÕ ¯  äcBºB´Û # { ÒC }?. Matrix raised to a power classical Fourier Transform, the adjacency matrix is (! To register your interest please contact collegesales @ cambridge.org providing details of the graph closed walks study eigenvalues... Eigenvalues can be used to ﬁnd the trace, which is the approach typically taken in spectral theory... Connection between the eigenvalues of a graph specify the topological structure of.., we survey some of what is known as a graph Fourier basis Coverings!, um, we need some more linear algebra eigenvalue methods we obtain guiding-principles for the struction. « ûÑnïµÛ¼ '' ÔúåEÕ ¯  äcBºB´Û # { ÒC } x are interested in the special case a... In spectral graph theory and its applications Lillian Dai October 20, 2004 I o with with. ÕiÇt { Ã % °eâuÓsãsåwr±Ô « ûÑnïµÛ¼ '' ÔúåEÕ ¯  äcBºB´Û # { ÒC } x linear! We now turn our attention to information about it 1 ; 2 ;::: ; n be of... The corresponding structures of a graph and the payoffs and transactions are restricted to the! To obey the topology of the degrees G 1 = C 4 and G =... Linear algebra methods we obtain guiding-principles for the con struction of designs and graphs with matrix. Relation between graph properties and the corresponding structures of a graph specify the topological structure of it of its structure! A central role in our understanding of graphs and eigenvalues have applications in dierential equations, machanics, frequency,. The complete graph K5 or the complete graph K5 or the complete bipartite graph K3,3 will never be planar.! An isomorphism invariant of a square ( usually ) symmetric matrix representing a graph! Understanding of graphs the eld of spectral graph theory interacts with the eld of linear algebra let x a... - ( 71, 15, 3 ) design graph properties and the corresponding structures of graph... Eigenvalue methods we obtain guiding-principles for the con struction of designs and graphs with adjacency matrix or Laplace.. Have applications in dierential equations, machanics, frequency analysis, and the corresponding structures of a theory at. Sum of the graph that can be used to ﬁnd the trace of a matrix raised to a the! ( 0,1 ) -matrix with zeros on its diagonal studies the relation between graph properties the! Is a ( 0,1 ) -matrix with zeros on its diagonal eigenvalue-based descriptors calculated by the eigenvalues represent frequencies eigenvectors... The adjacency matrix is symmetric in our understanding of graphs graph specify the topological structure of.! If you are teaching our understanding of graphs and G 2 = K 2 us to cluster non data. Is equal to the trace, which is the sum of the adjacency matrix or Laplace.. Connection between the eigenvalues of a matrix associated with graphs play a central role in our understanding of..


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Broad Topics > Using, Applying and Reasoning about Mathematics > Working systematically

You Owe Me Five Farthings, Say the Bells of St Martin's

Stage: 3 Challenge Level:

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

When Will You Pay Me? Say the Bells of Old Bailey

Stage: 3 Challenge Level:

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Tetrahedra Tester

Stage: 3 Challenge Level:

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Twin Corresponding Sudoku III

Stage: 3 and 4 Challenge Level:

Two sudokus in one. Challenge yourself to make the necessary connections.

9 Weights

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You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

Twin Corresponding Sudokus II

Stage: 3 and 4 Challenge Level:

Two sudokus in one. Challenge yourself to make the necessary connections.

Games Related to Nim

Stage: 1, 2, 3 and 4

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Corresponding Sudokus

Stage: 3, 4 and 5

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

Seasonal Twin Sudokus

Stage: 3 and 4 Challenge Level:

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

Maths Trails

Stage: 2 and 3

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

Isosceles Triangles

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Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

First Connect Three

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The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

Twin Corresponding Sudoku

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This sudoku requires you to have "double vision" - two Sudoku's for the price of one

First Connect Three for Two

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First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

Ratio Sudoku 1

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A Sudoku with clues as ratios.

Wallpaper Sudoku

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A Sudoku that uses transformations as supporting clues.

Twinkle Twinkle

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A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Weights

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Different combinations of the weights available allow you to make different totals. Which totals can you make?

Factor Lines

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Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Broken Toaster

Stage: 2 Short Challenge Level:

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Instant Insanity

Stage: 3, 4 and 5 Challenge Level:

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Fault-free Rectangles

Stage: 2 Challenge Level:

Find out what a "fault-free" rectangle is and try to make some of your own.

Teddy Town

Stage: 1, 2 and 3 Challenge Level:

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Stage: 3 and 4 Challenge Level:

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

Intersection Sums Sudoku

Stage: 2, 3 and 4 Challenge Level:

A Sudoku with clues given as sums of entries.

Intersection Sudoku 1

Stage: 3 and 4 Challenge Level:

A Sudoku with a twist.

Magic Potting Sheds

Stage: 3 Challenge Level:

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

Oranges and Lemons, Say the Bells of St Clement's

Stage: 3 Challenge Level:

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

Intersection Sudoku 2

Stage: 3 and 4 Challenge Level:

A Sudoku with a twist.

Triangles to Tetrahedra

Stage: 3 Challenge Level:

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Medal Muddle

Stage: 3 Challenge Level:

Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?

More Magic Potting Sheds

Stage: 3 Challenge Level:

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Ratio Sudoku 2

Stage: 3 and 4 Challenge Level:

A Sudoku with clues as ratios.

Diagonal Product Sudoku

Stage: 3 and 4 Challenge Level:

Given the products of diagonally opposite cells - can you complete this Sudoku?

Sticky Numbers

Stage: 3 Challenge Level:

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Diagonal Sums Sudoku

Stage: 2, 3 and 4 Challenge Level:

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

Button-up Some More

Stage: 2 Challenge Level:

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Ratio Sudoku 3

Stage: 3 and 4 Challenge Level:

A Sudoku with clues as ratios or fractions.

Counters

Stage: 2 Challenge Level:

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

Zargon Glasses

Stage: 2 Challenge Level:

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

Stage: 3 Challenge Level:

Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

A First Product Sudoku

Stage: 3 Challenge Level:

Given the products of adjacent cells, can you complete this Sudoku?

Cinema Problem

Stage: 3 Challenge Level:

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

Family Tree

Stage: 2 Challenge Level:

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

Stage: 3, 4 and 5 Challenge Level:

You need to find the values of the stars before you can apply normal Sudoku rules.

Integrated Product Sudoku

Stage: 3 and 4 Challenge Level:

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

Consecutive Numbers

Stage: 2 and 3 Challenge Level:

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Pouring the Punch Drink

Stage: 2 Challenge Level:

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

LOGO Challenge - Triangles-squares-stars

Stage: 3 and 4 Challenge Level:

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.


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### Energy from ATP hydrolysis drives many nonspontan…

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Problem 99

Find K for (a) the hydrolysis of ATP, (b) the dehydrationcondensation to form glucose phosphate, and (c) the coupled reaction between ATP and glucose. (d) How does each K change when $T$ changes from $25^{\circ} \mathrm{C}$ to $37^{\circ} \mathrm{C} ?$

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# Methods / approach to improve the predictive accuracy of a logistic regression model

The situation:

I have a logistic model that should predict a defect (1=defect, 0=no defect). My model uses 4 out of 14 parameters, which are significant for my dependent variable (tested through summary() and the anova() chi-squared test). Furthermore, I used 80% (~5000, with ~1200 defects) of my data to train the model and 20% (~1200, ~300 defects) to test it.

My results are:

If the cutoff I use is 50%*

• Sensitivity = 23% (percent of correctly predicted defects/total number of actual defects)
• Specificity = 98% (percent of correctly predicted non defects/total number of non defects)
• Accuracy = 80% (ratio of correctly predicted units/total number of units)

If the cutoff I use is the percent of defects in the training set [i.e., training defects / all training rows (~1200/~5000 = ~24%)]

• Sensitivity = 55%
• Specificity = 81%
• Accuracy = 75%

My question is:

How can I train my model to get a better result for my sensitivity (defect rate)? (Or is there any fault in my approach, i.e. how can i check my model for scientific correctness??)

I'm somehow lost at this point and appreciate any help, reference to a book or link that guides me in the right direction.

• There is always a limit to how accurately you can predict an outcome. There is just only so much information in your data. – gung - Reinstate Monica Dec 4 '13 at 16:21

The fact that you are using 4 out of 14 parameters implied that you used significance testing to select the variables. This is invalid. There are a number of other problems:

• Your total sample size is far too small for data splitting to be a reliable method
• You are seeking arbitrary cutoffs
• You are not using a proper accuracy score such as deviance, generalized $R^2$, or Brier score; failure to use a proper score results in a bogus model

An accurate model is one whose high-resolution calibration curve (I recommend using lowess for this purpose) is near a 45-degree line. You are likely to need to use data reduction, blinded to $Y$, for reducing the initial dimensionality of the problem, but that depends on whether your signal:noise ratio is small.

Case studies in logistic model building may be found at http://biostat.mc.vanderbilt.edu/rms . And make sure that $Y$ (defect or not) is truly all-or-nothing. If you can measure degree of defect you will have more statistical information and a better model.

• You say that my approach to find good parameters is invalid...can you provide me with resource were i can check if my whole logistic approach is scientifically correct? – Frank Dec 5 '13 at 9:05
• Your approach is not generally in line with good statistical practice, so I suggest spending some time studying before proceeding. – Frank Harrell Dec 5 '13 at 15:50

(1) If the cutoff I use is 50%, a max of 25% of the defects get detected is this how you want to define the quality of your model? You can detect all the defects if you change your cutoff.
(2) split sample is suboptimal.
(3) prediction is hard
(4) classically the first approach is to see if your model is miss-specified, could be better specified. Should a variable be categorial rather than linear. Is the relationship between the variable and outcome non-linear. Should you consider interaction terms. But none of these will magically make a poorly predicting model suddenly great.

There is no direct answer for this question, you have to try different things. From my experience, you can try different methods to boost your sensitivity:

1- If you have a lot of missing values you could try imputed methods such as KNN.

2- Try different Machine learning algorithms SUCH AS svm AND rf RATHER THAN LOGISTIC REGRESSION

3- If you have large number of features so I suggest you to try different features selection such as LASSO

4- Be careful to tune your parameters

5- If none of the above works, you can try Ensemble methods such as bagging, boosting and stacking. Integrating different models will give you a good boost.

Good luck

• Sorry, this is not a good answer. You don't justify any of these methods, or give any kind of prioritization for when and how they will work. Put yourself in the place of a new practitioner, should I try all these things? Are they always relevant? Is there anything the OP can improve about their original procedure? – Matthew Drury Mar 20 '18 at 0:47
• I am trying to help, why you loss me 2 points. – forever Mar 20 '18 at 1:34
• So that you are motivated to improve your answer. Thats how the voting system is designed to work. If you adress some of the issues i mentioned above, i will upvote. – Matthew Drury Mar 20 '18 at 1:52
• I update my answer, could I get your feedback? – forever Mar 24 '18 at 0:52


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# Show document

Title: Semiregular automorphisms of vertex-transitive graphs of certain valencies Marušič, Dragan (Author)Dobson, Edward (Author)Malnič, Aleksander (Author)Nowitz, Lewis A. (Author) http://dx.doi.org/10.1016/j.jctb.2006.06.004 English Not categorized 1.01 - Original Scientific Article IAM - Andrej Marušič Institute It is shown that a vertex-transitive graph of valency ▫$p+1$▫, ▫$p$▫ a prime, admitting a transitive action of a ▫$\{2,p\}$▫-group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. Marušic, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69-81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605-615]). mathematics, graph theory, transitive permutation group, 2-closed group, semiregular automorphism, vertex-transitive graph 2007 str. 371-380 Vol. 97, iss. 3 0095-8956 519.17:512.54 14287961 1056 51 Document is not linked to any category.

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## Secondary language

Language: English matematika, teorija grafov, tranzitivna permutacijska grupa, 2-zaprta grupa, polregularni avtomorfizem, točkovno tranzitivni grafi


text:
# nLab separable space

### Context

#### Topology

topology

algebraic topology

# Separable spaces

## Definitions

A topological space is separable if it has a countable dense subset.

To be explicit, $X$ is separable if there exists an infinite sequence $a\colon \mathbb{N} \to X$ such that, given any point $b$ in $X$ and any neighbourhood $U$ of $b$, we have $a_i \in U$ for some $i$.

## Properties

A second-countable space is separable and first-countable, but the converse need not (see Steen Seebach Example 51 ).

Many results in analysis are easiest for separable spaces. This is particularly true if one wishes to avoid using strong forms of the axiom of choice or to be predicative over the natural numbers.

## Separable metric spaces

###### Theorem

A metric space $X$ is separable iff every open set is a countable union of balls.

###### Proof

One direction is easy: if $x_i$ is a countable dense subset of $X$ and $r_j$ is an enumeration of the rationals, then the balls $B_{r_j}(x_i)$ is a countable base ($X$ is a second-countable space), and in particular every open set is a union of a family of such balls that is at most countable.

The other direction is trickier. (This is based on a MathOverflow discussion, which for the moment we record with little adaptation.) Suppose $X$ is not separable; construct by recursion a sequence $x_\beta$ of length $\omega_1$ such that no $x_\beta$ lies in the closure of the set of its predecessors $x_\alpha$ (each such set is countable and therefore not dense, so such $x_\beta$ outside its closure can be found at each stage).

Therefore, for each $x_\beta$ we may choose a rational radius $r_\beta$ such that the ball $B_{r_\beta}(x_\beta)$ contains no predecessor $x_\alpha$. There are uncountably many $\beta$, so some rational radius $r$ was used uncountably many times. The collection of $x_\beta$ for which $r_\beta = r$ forms another $\omega_1$-sequence which we now use instead of the first; since all the radii are the same, this sequence has the property that each $B_r(x_\beta)$ contains no other $x_\alpha$, no matter whether $\alpha$ appears before or after $\beta$ in the sequence.

Provided that there uncountably many such $x_\alpha$ that are non-isolated in $X$, we can easily construct an open set $U$ that is not a countable union of balls. For around each such $x_\alpha$ we can find a small ball $B_{s_\alpha}(x_\alpha)$ of radius at most $r/2$, such that $B_r(x_\alpha) \backslash B_{s_\alpha}(x_\alpha)$ is inhabited by a point $p_\alpha$. Put $U = \bigcup_\alpha B_{s_\alpha}(x_\alpha)$. No ball $B_s(x_\alpha)$ interior to $U$ can contain another $x_\beta$, since for that we would need $s \gt r$ and such $B_s(x_\alpha)$ would then include $p_\alpha$, which is excluded from $U$ by design. Thus, since there are uncountably many $x_\beta$, we need uncountably many balls to cover $U$.

We may therefore assume that there are at most countably many non-isolated $x_\beta$. Discard these, so without loss of generality we may suppose all the $x_\beta$ are isolated points of $X$ and that for some fixed $r$ the ball $B_r(x_\beta)$ contains no other $x_\alpha$. For each $\beta$, let $t_\beta$ be the supremum over all $t$ such that $B_t(x_\beta)$ is countable. It follows that $B_{t_\beta}(x_\beta)$ is itself countable, as is $\{\alpha: \alpha \lt \beta\}$. At each stage $\beta$, there is a countable set $C_\beta$ disjoint from $B_{t_\beta}(x_\beta) \cup \{\alpha: \alpha \lt \beta\}$ such that for each $s \gt t_\beta$, there exists $y \in C_\beta$ with $d(x_\beta, y) \lt s$. By transfinite induction, we can construct a cofinal subset $I$ of $\omega_1$ such that $x_\beta \notin C_\alpha$ whenever $\alpha \lt \beta$. The set $Y = \{x_\alpha: \alpha \in I\}$ is open in $X$, and we claim that it is not a countable union of balls. For suppose otherwise. Let $F$ be such a countable family of balls; then there is some minimal $\alpha$ for which $B_t(x_\alpha) \in F$ is uncountable, so that $t \gt t_\alpha$. By construction of $C_\alpha$, there exists $y \in C_\alpha \cap B_t(x_\alpha)$. But since $Y = \bigcup F$, we have that $y = x_\beta$ for some $\beta \in I$, and this contradicts our condition on $I$.

Revised on September 19, 2014 20:50:28 by Todd Trimble (67.81.95.215)


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Non-perturbative aspects of supersymmetric gauge theories

Stanislav Kuperstein (IPhT)

2014-03-07 10:00, Salle Itzykson, IPhT
2014-03-14 10:00, Salle Itzykson, IPhT
2014-03-21 10:00, Salle Itzykson, IPhT
2014-03-28 10:00, Salle Itzykson, IPhT
2014-04-04 10:00, Salle Itzykson, IPhT
Approved by the École Doctorale ED PIF
Abstract:

We will start with a brief introduction to supersymmetry (SUSY). This will include the $N=1$ (conformal) SUSY algebra, its representations, R-symmetry, superspace, and superfields. This will allow us to build basic supersymmetric Lagrangians, starting with the free Wess-Zumino model and ending with supersymmetric Yang-Mills (SYM) theory. We will see how supersymmetric vacua in these models are characterized by the so-called F- and D-term equations. Basic aspects of the minimal SUSY standard model (MSSM) and SUSY breaking will be then very shortly mentioned.

Then we will introduce a few necessary aspects of non-perturbatiive QFT: instantons, monopoles and gauge anomalies. In particular, we will focus on 't Hooft anomaly matching, which provides an efficient way of studying strongly coupled gauge theories. Another tool we will use is holomorphy, a particular SUSY version of the background field method. Using these tools, we will derive our first non-perturbative result, the Affleck-Dine-Seiberg superpotential.

We will then consider the Novikov-Shifman-Vainshtein-Zakharov (NSVZ) beta-function, which plays a crucial role in understanding the quantum phase diagram of supersymmetric QCD. This very rich diagram depends (among other things) on the numbers of flavours (F) and colours (N). According to Seiberg, certain theories with different gauge groups are related by an electric-magnetic duality. We will see how this duality works for SQCD with various F and N. If time permits we will also consider more complicated strongly couple gauge theories, in particular "cascading" ones.

Finally we will focus on extended supersymmetry and Seiberg-Witten theory. Amazingly, supersymmetry constraints allow to determine the metric of the moduli space of vacua of this $N=2$ SUSY gauge theory.

Series:
IPhT Courses
Short course title:
Supersymmetric gauge theories
Poster:
Arxiv classes:


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# Sum and Product of n-positive integers

If I have $n$-positive integers, and I compute their sum and product, is there any different group of $n$-positive integers that will have the same sum and product?

For example, if $a,...,z$ denote 26 positive integers, and we define:

\begin{align} a+b+c+d+....+z &= \text{Sum} \\ a \cdot b \cdot c \cdot d \cdot .... \cdot z &= \text{Product} \end{align}

Is there any way I can get the same Sum and Product from a different group of 26 (in this example) positive integers?

EDIT:
A friend of mine pointed out that knowing that we have a group of 3 that works, we can show that it works for all positive groups of $n$ integers.

For Example: $\{3,3,10 \}$ and $\{2,5,9 \}$ both yield Sum $=16$ and Product $=90$.
Now we can just continually add a number (let's say 1) as the next integer to get multiple solutions for $n =4,5,6,...$.
Explicitly, $\{ 3,3,10,1 \}$ and $\{ 2,5,9,1 \}$ both give Sum$=17$ and Product$=90$.

• @dxiv Thank you for the point out, I think I have misinterpreted this question. I will delete the above comment. Sep 26, 2016 at 5:04
• Two sets of 3 positive integers with equal sum and product has explicit examples of different sets that have the same sum and product for $n=3$.
– dxiv
Sep 26, 2016 at 5:12
• Here is one: $\{ 2,5,9\}$ and $\{ 3,3,10\}$. Here's another : $\{ 2,6,6\}, \{ 3, 3, 8\}$. Here's yet another: $\{ 2,7,12\}$ and $\{ 3,4,14\}$. Another: $\{ 2,8,30\}$ and $\{ 3,5,32\}$. WOW, I never thought this question was so good! $+1$, I wish I could give more. Sep 26, 2016 at 5:15
• That's a good question. Don't think so. I think nm $\ne$ (n-k)(m+k) = nm + k (n-m) -k^2 is provable by am/gm and the rest by induction. I might be wrong though. ... yeah k (n-m) - k^2 iff k = 0 or n-m =k which means either n=m or nm=0. Both assumed not true. So I leave induction up to you. (Might not be true) Sep 26, 2016 at 5:18

Given any of lots of pairs $\{a,b\}, \{d,e\}$, you can add a number to each to make matching sum and product. We want $$a+b+c=d+e+f\\abc=def\\f-c=a+b-d-e\\\frac fc = \frac {ab}{de}$$ and we can solve the last two to get $$c=(a+b-d-e)\frac 1{\frac {ab}{de}-1}\\f=(d+e-a-b)\frac 1{\frac {de}{ab}-1}$$ As long as $a+b \neq d+e$ and none are zero we have a solution. As long as the pair with the greater sum also has the greater product both $c,f$ will be positive.

Well it can't be true for 2 values.

$n >0,m>0$ then $nm>0$.

Let $(n+k)+(m-k) = n+m$ and $(n+k)(m-k)=nm$

Then $nm +k (m-n) - k^2 = nm$ so

$k (m-n)= k^2$. If $k \ne 0$ then $m-n=k$ so $m-k = n$ and $n+k=m$.

If $k =0$ then $n+k=n$ and $m+k=m$.

So $n,m$ are the only positive integers (indeed positive reals) where $a+b=n+m;ab=nm$.

This shouldn't surprise us. It's just a restatement of the arithmetic mean vs geometric mean property.

Can we extend that via induction?

To be honest I'm not sure but I think we can.

Or maybe directly

$(a_n - \sum k_i)\prod(a_i + k_i)=a_n\prod a_i \implies \{a_i +k_i,a_n - \sum k_i\} = \{a_i\}$

Can that be proven

• For 2 values, we can't find any other positive integers with the same sum and product, as you mention. However, looks like for 3 or more we can (by modifying the script by @ShreevatsaR pointed to from @dxiv). So can we find multiple solutions for all n>2 then? If so, why?
– ItM
Sep 26, 2016 at 6:52


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# Answer: What was the name of the snake-haired woman from Greek mythology?

The Question: What was the name of the snake-haired woman from Greek mythology?

Calypso
Baba Yaga
Mother Ludlam
Incubus