text:
Math Help - How would I multiply this out?

1. How would I multiply this out?

I have these four terms:

(x+1)(x-1)(x-3)(x-5)

What order would i multiply the terms? Does FOIL apply here? I dont need the answer I just need the steps if anyone can help thanks!

2. Originally Posted by nascar77
I have these four terms:

(x+1)(x-1)(x-3)(x-5)

What order would i multiply the terms? Does FOIL apply here? I dont need the answer I just need the steps if anyone can help thanks!
just go left to right ...

$(x+1)(x-1)(x-3)(x-5)$

$(x^2-1)(x-3)(x-5)$

$(x^3-3x^2-x+3)(x-5)$

keep going ...

3. Originally Posted by nascar77
I have these four terms:

(x+1)(x-1)(x-3)(x-5)

What order would i multiply the terms?
In whatever order you like. Order doesn't matter in multiplication.

Originally Posted by nascar77
Does FOIL apply here?
Only when multiplying two binomials. As soon as you have something with three terms, you're going to have to use regular multiplication.

4. thank you guys

format: plain


text:
# Relationship between vrms and vmax

### Sinewave Voltages - Vpk, Vpk-pk, Vavg, Vrms - RF Cafe

Vmax to Vrms conversion. Vmax of a sine wave voltage waveform is defined as the Positive Amplitudes on the sine wave. The Vrms is defined as square root of. What you refer to as "Vm" I assume is an abbreviation for "Vmax" in a delta connection then, the power is the RMS Line voltage squared and. RMS Voltage or Root Mean Square Voltage of an AC Waveform is the amount of AC power that The RMS or effective value of a sinusoidal waveform gives the same heating effect of an equivalent DC supply . Is there a difference between.

Четыре на шестнадцать, - повторил профессор. - Лично я проходил это в четвертом классе. Сьюзан вспомнила стандартную школьную таблицу.


text:
How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

2. Complete and sign the license agreement.

3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

memo_has_moved_text();Fundamental solutions and local solvability for nonsmooth Hörmander’s operators

About this Title

Marco Bramanti, Luca Brandolini, Maria Manfredini and Marco Pedroni

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 249, Number 1182
ISBNs: 978-1-4704-2559-3 (print); 978-1-4704-4131-9 (online)
DOI: https://doi.org/10.1090/memo/1182
Published electronically: August 8, 2017
Keywords:Nonsmooth Hörmander’s vector fields, fundamental solution, solvability, Hölder estimates.

View full volume PDF

View other years and numbers:

Table of Contents

Chapters

• Chapter 1. Introduction
• Chapter 2. Some known results about nonsmooth Hörmander’s vector fields ec known results
• Chapter 3. Geometric estimatesec geometric
• Chapter 4. The parametrix methodec parametrix
• Chapter 5. Further regularity of the fundamental solution and local solvability of $L$
• Chapter 6. Appendix. Examples of nonsmooth Hörmander’s operators satisfying assumptions A or B

Abstract

We consider operators of the form in a bounded domain of where are nonsmooth Hörmander's vector fields of step such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method we construct a local fundamental solution for and provide growth estimates for and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients we prove that also possesses second derivatives, and we deduce the local solvability of , constructing, by means of , a solution to with Hölder continuous . We also prove estimates on this solution.


text:
# First Order Differential Equation

## Homework Statement

Solve the following first order differential equation for x,

$$\frac{dy}{dx}$$ = 3xy + xy2

## Homework Equations

Methods: Separation of Variables, Define an Integrating Factor

## The Attempt at a Solution

I have been staring at this question for a while now, hoping that somehow the numbers would start looking nicer for me to solve, but alas this has not happened. I am totally stumped here, any help to steer me in the right direction would be greatly appreciated.

Start by seperating the left and right hand side, such that on one side you only have a y-dependence and on the other only an x-dependence. In other words, you want something like this:

$$f(y)dy = g(x) dx$$

Written this way it's possible to integrate both sides (do you see why?). You can post your answer or attempt here.

HallsofIvy
Homework Helper
That's separable: dy/dx= x(3y+ y2).

Thanks for the quick reply, so,

$$\frac{dy}{dx}$$ = x(3y + y2)

$$\frac{1}{3y + y^2}$$dy = x dx

However, integrating the left hand side is beyond the scope of the subject I am taking at the moment. That's why they have asked to solve for x, I think. Is it still possible to solve for x using this method?

Any ideas?

Are you sure about that? I don't really see an easier approach then to solve these integrals.

The integral on the left hand side is probably not 'out of your scope'. The trick is to seperate the fraction. It goes like this:

You have:
$$\frac{1}{3y+y^2} = \frac{1}{y(3+y)}$$
Now make the following 'guess':
$$\frac{1}{y(3+y)} = \frac{a}{y}+\frac{b}{3+y}$$
That is, we have split the fraction into two easier fractions. We just need to determine a and b -- I'll leave this up to you.

$$\frac{1}{3y+y^2} = \frac{1/3}{y}-\frac{1/3}{3+y}$$
You should be able to solve the integrals using this form.

Mark44
Mentor
Thanks for the quick reply, so,

$$\frac{dy}{dx}$$ = x(3y + y2)

$$\frac{1}{3y + y^2}$$dy = x dx

However, integrating the left hand side is beyond the scope of the subject I am taking at the moment. That's why they have asked to solve for x, I think. Is it still possible to solve for x using this method?

If all you need to do is solve for x without solving the differential equation, you can do that in one step, starting with the 2nd equation above.

Ah OK, so you use the partial fraction method then integrate each term. Cheers xepma, I should be able to do it now. Also thanks for the input everyone.


text:
# NDSolve Encountered non-numerical value

Below a sample of some first-order differential equations I try to solve. Although there are no non-numerical values involved in the equations, Mathematica systematically spits out "Encountered non-numerical value at $$\tau$$ == 0".

And I also get the error: "Cannot solve to find an explicit formula for the derivatives. NDSolve
will try solving the system as differential-algebraic equations"... What does that mean exactly and how can I remedy this?

Might the problem lie somewhere else?

eqs = {0 == (1.38 D[r[5/6, 0][\[Tau]], \[Tau]] == -10.18 +
9.61 r[5/6, 0][\[Tau]] - 3.1 r[5/6, \[Pi]/4][\[Tau]]),
0 == (2.95 (D[r[5/6, \[Pi]/4][\[Tau]], \[Tau]]) == -21.7 +
13.8 r[5/6, \[Pi]/4][\[Tau]] +
1.66 (r[5/6, 0][\[Tau]] - 1. r[5/6, \[Pi]/2][\[Tau]])^2 -
1.67 (4.64 .3 r[5/6, \[Pi]/4][\[Tau]]) (r[5/6, 0][\[Tau]] -
1. r[5/6, \[Pi]/2][\[Tau]])^2 -
3.33 (r[5/6, 0][\[Tau]] - 2 r[5/6, \[Pi]/4][\[Tau]] +
r[5/6, \[Pi]/2][\[Tau]])),
0 == (4.53 (D[r[5/6, \[Pi]/2][\[Tau]], \[Tau]]) == -33.2 -
5.10 (2 r[5/6, \[Pi]/4][\[Tau]] - 2 r[5/6, \[Pi]/2][\[Tau]]) +
21.14 r[5/6, \[Pi]/2][\[Tau]])};

Unknowns = {r[5/6, 0], r[5/6, \[Pi]/4], r[5/6, \[Pi]/2]};
Unknowns\[Tau] = Unknowns /. r[a_, b_] :> r[a, b][\[Tau]];

Init = Join[Map[(0 == #) &, Unknowns\[Tau] /. \[Tau] -> 0],
Map[(0 == #) &, D[Unknowns\[Tau], \[Tau]] /. \[Tau] -> 0]];

sols = NDSolve[Flatten[{eqs, Init}], Unknowns, {\[Tau], 1, 3}][[1]];

---------- Edit--------

So the context is that I am trying to solve some differential equation for a function $$r(x,y,t)$$ that I have discretised on a grid along the x-y direction leaving the time dependence continuous. So I end up with a set of differential equations for t with unknowns the point in my grid. The code above is for a sample of points $$r(a,b)$$ in this grid (and the corresponding equations.)

$$\left\{0=\left(1.38 r\left(\frac{5}{6},0\right)'\tau =9.61 r\left(\frac{5}{6},0\right)\tau -3.1 r\left(\frac{5}{6},\frac{\pi }{4}\right)\tau -10.18\right),0=\left(2.95r\left(\frac{5}{6},\frac{\pi }{4}\right)'\tau =-2.32464r\left(\frac{5}{6},\frac{\pi }{4}\right)\tau \left(r\left(\frac{5}{6},0\right)\tau -1.r\left(\frac{5}{6},\frac{\pi }{2}\right)\tau \right)^2+1.66\left(r\left(\frac{5}{6},0\right)\tau -1.r\left(\frac{5}{6},\frac{\pi }{2}\right)\tau \right)^2+13.8r\left(\frac{5}{6},\frac{\pi }{4}\right)\tau -3.33\left(r\left(\frac{5}{6},0\right)\tau -2r\left(\frac{5}{6},\frac{\pi }{4}\right)\tau +r\left(\frac{5}{6},\frac{\pi }{2}\right)\tau \right)-21.7\right),0=\left(4.53r\left(\frac{5}{6},\frac{\pi }{2}\right)'\tau =-5.1\left(2r\left(\frac{5}{6},\frac{\pi }{4}\right)\tau -2r\left(\frac{5}{6},\frac{\pi }{2}\right)\tau \right)+21.14r\left(\frac{5}{6},\frac{\pi }{2}\right)\tau -33.2\right)\right\}$$

• Can you write out the equations you're trying to solve, with their initial conditions? It's not clear from the code what you're actually trying to do, which we'll need to know in order to figure out how to fix things. This Stack accepts MathJaX, so if you know LaTeX, just edit your question to include LaTeX code. Jun 23 at 19:10
• @MichaelSeifert Thanks for taking already a look at my question! I added some context and the equations in tex form Jun 23 at 19:39
• Setting all the differential equations equal to zero makes them into algebraic equations for an equilibrium. Is that what you intend? If so, then use Solve or NSolve or FindRoot. Jun 23 at 19:51
• @ChrisK I don't think I can do that, since I still have a set of genuine differential equations with respect to $\tau$? Jun 23 at 19:59
• @Nomdeplume Not after you set time derivatives (rate of change) equal to zero! Jun 23 at 20:34

Your code is a bit hard to parse and the math seems wrong. You have a system of 3 first order diff-eqs but 6 initial conditions (all variables and their derivatives are 0 at $$t=0$$). That's overconstrained, and if the variables and their derivatives are all 0, then the solution is trivial. They all stay 0 forever.

What's up with the 0==(a==b) formatting of the equations? If that's really what you mean, then you again just get the trivial solution $$r(5/6,0)=r(5/6,\pi/4)=r(5/6,\pi/2)=0$$ for all $$\tau$$.

Let's remove the initial condition on the derivatives and the 0== terms in the equations. Let's also rename $$\tau$$ to $$t$$ since it's ugly to have \[Tau]'s floating around here. And for conciseness let's also call $$r(5/6,0)=r1, \ r(5/6,\pi/4)=r2, \ r(5/6,\pi/2)=r3$$. Then we get

eqs = {1.38 r1'[t] == -10.18 + 9.61 r1[t] - 3.1 r2[t],
2.95 r2'[t] == -21.7 + 13.8 r2[t] + 1.66 (r1[t] - 1. r3[t])^2 - 2.32464 r2[t] (r1[t] - 1. r3[t])^2 - 3.33 (r1[t] - 2 r2[t] + r3[t]),
4.53 r3'[t] == -33.2 - 5.1 (2 r2[t] - 2 r3[t]) + 21.14 r3[t]}
vars[t_] = {r1[t], r2[t], r3[t]};
sols = NDSolve[Flatten[{eqs, init}], vars[t], {t, 1, 3}][[1]]

This only goes to about $$t=2.35$$ where the solution appears to run into divergence issues. You can see this from a plot of the solution

Plot[Evaluate[{r1[t], r2[t], r3[t]} /. sols], {t, 1, 2.35}, PlotRange -> All]

Hope that helps!

Try this:

eqs = {(1.38 D[r[5/6, 0][\[Tau]], \[Tau]] == -10.18 +
9.61 r[5/6, 0][\[Tau]] -
3.1 r[5/6, \[Pi]/4][\[Tau]]), (2.95 (D[
r[5/6, \[Pi]/4][\[Tau]], \[Tau]]) == -21.7 +
13.8 r[5/6, \[Pi]/4][\[Tau]] +
1.66 (r[5/6, 0][\[Tau]] - 1. r[5/6, \[Pi]/2][\[Tau]])^2 -
1.67 (4.64 .3 r[5/6, \[Pi]/4][\[Tau]]) (r[5/6, 0][\[Tau]] -
1. r[5/6, \[Pi]/2][\[Tau]])^2 -
3.33 (r[5/6, 0][\[Tau]] - 2 r[5/6, \[Pi]/4][\[Tau]] +
r[5/6, \[Pi]/2][\[Tau]])), (4.53 (D[
r[5/6, \[Pi]/2][\[Tau]], \[Tau]]) == -33.2 -
5.10 (2 r[5/6, \[Pi]/4][\[Tau]] - 2 r[5/6, \[Pi]/2][\[Tau]]) +
21.14 r[5/6, \[Pi]/2][\[Tau]])};

Unknowns = {r[5/6, 0], r[5/6, \[Pi]/4], r[5/6, \[Pi]/2]};
Unknowns\[Tau] = Unknowns /. r[a_, b_] :> r[a, b][\[Tau]];
Init = Map[(0 == #) &, Unknowns\[Tau] /. \[Tau] -> 0]

sols = NDSolve[Flatten[{eqs, Init}], Unknowns, {\[Tau], 1, 3}][[1]];

Now the integration proceeds until $$\tau\approx 2.28$$


text:
» «

# Easy script switcher [PC v1 only]

169 replies to this topic
rubregg
• ##### rubregg

brainslug

• Members
• Joined: 03 May 2006

### #151 Posted 20 September 2009 - 06:08 PM

 QUOTE (hetment @ Sep 20 2009, 16:00) My C has 31.5 gb now... free space is it alright?

That sounds like loads of free space. At least as long as your C partition is not bigger than 315 GB.

General rule of thumb:
Have at least 10% free disk space on any partition/drive

hetment
• ##### hetment

GTA 4 and gta san andreas. the best gta games ever!

• Members
• Joined: 03 May 2009

### #152 Posted 21 September 2009 - 04:26 AM

My C partison has total 68gb and 31.5 free.. So i guyess its alright.. i have a 250gb HDD

josephajain
• ##### josephajain

Player Hater

• Members
• Joined: 17 Feb 2010

### #153 Posted 13 April 2010 - 09:01 AM

Thanks for the feedback man!!!

Miro
• ##### Miro

Time Killa™

• Zaibatsu
• Joined: 26 Nov 2009
• GTANet Contribution Award [Newbie Guide]
Most Improved 2010

### #154 Posted 08 June 2012 - 10:25 PM

Might I suggest a little update to the links in the first page?

 QUOTE DownloadsScript Switcher v0.7:primary @ GTA Garage  --  mirrorScript files: v1 -- v2 -- Hot Coffee -- Spaceeinstein's All In One Mod

The mirror of the Script Switcher isn't working anymore, and so are the V1 & V2 script files links. Here are some re-uploads - in case some staff member will edit the first post:

lil weasel
• ##### lil weasel

Shoot Looters, Hang Pirates!

• GTA Series Staff
• Joined: 25 Dec 2006
• Contribution Award [San Andreas]

### #155 Posted 08 June 2012 - 10:46 PM

Done!

RusiBoy
• ##### RusiBoy

King of San Andreas

• BUSTED!
• Joined: 11 May 2012

### #156 Posted 14 June 2012 - 04:33 PM

I'm having this problem. No script files were found it says on the screen of the Script Changer.
I created a new folder in my script file at GTA San Andreas folder called V! (I have V1)
I have Widows XP.

lil weasel
• ##### lil weasel

Shoot Looters, Hang Pirates!

• GTA Series Staff
• Joined: 25 Dec 2006
• Contribution Award [San Andreas]

### #157 Posted 14 June 2012 - 04:54 PM

You have V1?
They were discontinued in 2006, where did you get it?

RusiBoy
• ##### RusiBoy

King of San Andreas

• BUSTED!
• Joined: 11 May 2012

### #158 Posted 16 June 2012 - 04:26 PM

 QUOTE (lil weasel @ Thursday, Jun 14 2012, 16:54) You have V1?They were discontinued in 2006, where did you get it?

I have V1 and I want it to convert to V2. So I can participate in the GTA SA chain game

Miro
• ##### Miro

Time Killa™

• Zaibatsu
• Joined: 26 Nov 2009
• GTANet Contribution Award [Newbie Guide]
Most Improved 2010

### #159 Posted 16 June 2012 - 04:29 PM

Make a new folder in you main San Andreas directory called "Script V2" (I think you said that you've done that already) and keep on switching that with your V1. How you do that? Well, create also a folder called "Script V1" and copy paste the main.scm & script files (located in San Andreas directory --> data --> script) and paste them in your V1 folder. Now, you can switch them by either copy pasting each version from your V1 or V2 folders in the data --> script folder, and replace those files, or simply cut and paste.

Tell me if there's something you do not understand.

CCPD
• ##### CCPD

Boss

• Members
• Joined: 01 May 2006
• Contribution Award [Chain Game]

### #160 Posted 21 August 2012 - 10:05 AM

I moved my SA folder from my primary HD C to my files HD D. Since I originally installed it on C I'm guessing the registry says it's where it should be now. As a result I can't use Script Switcher. Any simple way to let it know where the script files are?

OrionSR
• ##### OrionSR

Chain Game Development Team

• Feroci
• Joined: 23 May 2007
• Helpfulness Award [GTA & Modding]

### #161 Posted 21 August 2012 - 10:34 AM

I had problems with this batch file when I installed a new hard disk with Windows 7 but continued to run the game off the old hard drive which is now installed as disk F:. I edited the New Method section of the batch file to avoid querying the registry and assigned the path manually. If you have experience with paths and batch files you might be able to adapt the snippet below to work on your computer (drive F: probably won't work for you).

 CODE ::.................................NEW METHOD (from v0.4)...Hacked.............................::::for /f "skip=2 tokens=3*" %%G in ('reg query "HKLM\SOFTWARE\Rockstar Games\GTA San Andreas\Installation" /v Exepath') do set GAMEPATH=%%G %%H::^ original method (XP) commented out::set GAMEPATH=%GAMEPATH:\gta_sa.exe=%::^ original method commented outset GAMEPATH="F:\Program Files\Rockstar Games\GTA San Andreas"::^ manually assign the game pathset SCRIPTPATH=%GAMEPATH%\data\script::

CCPD
• ##### CCPD

Boss

• Members
• Joined: 01 May 2006
• Contribution Award [Chain Game]

### #162 Posted 26 August 2012 - 06:13 PM

Have no experience so all these look alien to me. Any chance you could edit the game path (D:\Games\Rockstar Games\GTA San Andreas\data\script) for me or it has to be my system file from my computer?

OrionSR
• ##### OrionSR

Chain Game Development Team

• Feroci
• Joined: 23 May 2007
• Helpfulness Award [GTA & Modding]

### #163 Posted 26 August 2012 - 06:39 PM Edited by OrionSR, 30 August 2015 - 05:26 AM.

Use any text editor to find the codes below:

::.................................NEW METHOD (from v0.4)......................................
::
for /f "skip=2 tokens=3*" %%G in ('reg query "HKLM\SOFTWARE\Rockstar Games\GTA San Andreas\Installation" /v Exepath') do set GAMEPATH=%%G %%H
set GAMEPATH=%GAMEPATH:\gta_sa.exe=%
set SCRIPTPATH=%GAMEPATH%\data\script

And change it to look like this:

::.................................NEW METHOD (from v0.4)...Hacked.............................
::
::for /f "skip=2 tokens=3*" %%G in ('reg query "HKLM\SOFTWARE\Rockstar Games\GTA San Andreas\Installation" /v Exepath') do set GAMEPATH=%%G %%H
::set GAMEPATH=%GAMEPATH:\gta_sa.exe=%
set GAMEPATH="D:\Program Files\Rockstar Games\GTA San Andreas"
set SCRIPTPATH=%GAMEPATH%\data\script

My path was "D:\Program Files\Rockstar Games\GTA San Andreas". Your path will probably be different. Drive C: instead of D: is much more common, and SA ends up in Program Files (x86) if you've got a 64bit OS. I can't tell you what your path is. You'll need to figure that on your own.

CCPD
• ##### CCPD

Boss

• Members
• Joined: 01 May 2006
• Contribution Award [Chain Game]

### #164 Posted 26 August 2012 - 06:40 PM Edited by CCPD, 26 August 2012 - 06:50 PM.

After passing my short brainf*ck I successfully tailored switcher to my needs. By the way if you check you'll see that I have a different path than you provided (no Program Files folder) and I had no idea where to actually put it. Just simple steps that could have make it easier for someone who's new to this. Luckily I figured it out, so thanks for code.

Shoumaker
• ##### Shoumaker

y

• The Yardies
• Joined: 09 Jul 2012
• Best Signature 2013
Best Signature 2012

### #165 Posted 11 December 2012 - 04:59 PM Edited by Shoumic, 29 October 2013 - 08:29 PM.

.

rubregg
• ##### rubregg

brainslug

• Members
• Joined: 03 May 2006

### #166 Posted 18 December 2013 - 01:55 PM Edited by rubregg, 18 December 2013 - 02:03 PM.

I had a very very brief look at this a couple of days ago. I found the corresponding registry path on my win7 x64, but when I put it in, I still got an error message.
for /f "skip=2 tokens=3*" %%G in ('reg query "HKLM\SOFTWARE\Microsoft\Windows\CurrentVersion\GameUX\S-1-5-21-2403304607-1631922368-3772349190-1000\{45E676A5-64C7-4279-946D-DC43E901081C}" /v AppExePath') do set GAMEPATH=%%G %%H
Confused, because that command yielded the proper path just fine in a separate shell. I may or may not have the time to eventually fix this.

For now, please go with what OrionSR has suggested 3 posts up, that should do it:
set GAMEPATH="D:\whatever\the\path\to\your\exe\is\typically\Rockstar Games\GTA San Andreas"

HackerDetected
• ##### HackerDetected

I Can See My House From There!!!

• Members
• Joined: 15 Aug 2015

### #167 Posted 30 August 2015 - 03:05 AM

dude it has an error the syestem cannot find the path specified can you help me please!!!

OrionSR
• ##### OrionSR

Chain Game Development Team

• Feroci
• Joined: 23 May 2007
• Helpfulness Award [GTA & Modding]

### #168 Posted 30 August 2015 - 03:31 AM Edited by OrionSR, 30 August 2015 - 05:16 AM.

Yeah, we know. The script was designed in the WinXP era so it can't autodetect the path on newer operating systems. Scroll up a few posts to find my workaround that bypasses the autodetect routine and sets the path explicitly. It's an easy edit if you know the basics of file naming and paths.

DragonpokeZ
• ##### DragonpokeZ

Special Vehicle Hoarder

• Members
• Joined: 24 May 2015

### #169 Posted 13 September 2015 - 02:03 AM

OrionSR
• ##### OrionSR

Chain Game Development Team

• Feroci
• Joined: 23 May 2007
• Helpfulness Award [GTA & Modding]

### #170 Posted 13 September 2015 - 03:15 AM

The primary links to GTAGarage are still valid. Don't forgot to modify you path if you've got an OS newer than XP. See notes just above. You might need to fuss with permissions if you've got SA installed in a Windows program folder or it'll look like you've copied something but the game will be using something in a virtual store, maybe - I forget how this stuff works and used to copy my script manually until I finally had enough HD space to have multiple installs. Having SA installed outside of the program folder certainly makes it a lot easier to work with.

Script links are dead, except All-In-One. Links to V2 scripts can probably be found in the Chain Game topic, maybe v1 too. Don't bother with HC, you're liable to get banned for asking and it's pointless anyway. I used that slot for testing PS2 scripts or custom mains instead.

#### 1 user(s) are reading this topic

0 members, 1 guests, 0 anonymous users


text:
# PRNG determinism?

Looking at getDefaultPRNG() in java.security.SecureRandom makes me very unsure about how to use it. On my desktop, it uses sun.security.provider.NativePRNG, which reads from /dev/random and/or /dev/urandom which should be effectively unpredictable (non-deterministic) on Linux if you don't have direct access to the operating system. But on Windows, who knows? When I run this code on another machine or JVM, it could fall back to SHA1 or another deterministic algorithm.

If the output of SecureRandom is effectively unpredictable (dev/random), then it can be used by a web server to generate secure session tokens which are given to end users more-or-less raw. But if it is deterministic (digest), then exposing enough output to the end user effectively compromises all subsequent output.

For a deterministic PRNG, I should obscure each output with a few random bits (e.g. xor the low bits of the current time onto it). That would make it harder to guess where the deterministic RNG was in it's cycle. OTOH, If the PRNG uses /dev/random and the low bits of the time is already used as part of the seed, then using those same bits again is unnecessary or might even be counter-productive. The use models for the two kinds of "random" numbers are incompatible.

Unless I can be sure what kind of output SecureRandom is producing I can't use it properly. This seems like a design flaw to me. I think that the interface should guarantee either a deterministic, or effectively non-deterministic PRNG. I think this a necessary contract for users of all PRNGs (not just Java's SecureRandom).

Please tell me what I'm missing.

• Do you have access to the SecureRandom object? If so, you could just check the getAlgorithm() method and see if it returns "NativePRNG" or "SHA1PRNG". – cygnusv Dec 30 '14 at 17:38
• @cygnusv Oh - thanks! I should have seen that. Still, I think my question stands because "NativePRNG" behaves differently on Windows. Windows doesn't have a /dev/random and the workings of it's PRNG are proprietary: stackoverflow.com/questions/191335/… It just pushes the question along as to whether "NativePRNG" provides deterministic or effectively non-deterministic "randomness." – GlenPeterson Dec 30 '14 at 17:50
• @cygnusv They should add getAlgorithm() to the Random interface instead of making it specific to SecureRandom. – GlenPeterson Dec 30 '14 at 20:35
• java.util.Random is not an interface, but a class, which has a sole algorithm for generating the random numbers. Quoting the documentation from java.util.Random: "The class uses a 48-bit seed, which is modified using a linear congruential formula. (See Donald Knuth, The Art of Computer Programming, Volume 2, Section 3.2.1.)". – cygnusv Dec 30 '14 at 20:44
• I'm thinking you are underestimating the random number generator within the current Windows operating systems, although the older version certainly had a lot of issues. See this non-authoritative response. I'm not sure how often it is reseeded though. – Maarten Bodewes Dec 31 '14 at 12:31

Assuming that nobody's screwed up the implementation, it should not matter what kind of RNG you get. This is because all java.security.SecureRandom implementations are supposed to be cryptographically strong, as defined in RFC 1750 §6.3 (emphasis mine):

6.3 Cryptographically Strong Sequences
In cases where a series of random quantities must be generated, an adversary may learn some values in the sequence. In general, they should not be able to predict other values from the ones that they know. […]

This includes the SHA1PRNG implementation, which is described as follows:

This algorithm uses SHA-1 as the foundation of the PRNG. It computes the SHA-1 hash over a true-random seed value concatenated with a 64-bit counter which is incremented by 1 for each operation. From the 160-bit SHA-1 output, only 64 bits are used.

Note two details here:

1. The internal state of the PRNG, which an attacker would need to learn in order to predict future output of the generator, consists of the random seed value and the counter. The output never contains these values, but only their SHA-1 hashes. In order to reconstruct the internal state from the output, an attacker would have to carry out a successful preimage attack on SHA-1, which is not currently believed to be feasible.

2. The output of the SHA-1 hash function is truncated to 64 bits, discarding a full 60% of it. Even if the internal state were propagated as in many non-cryptographic PRNGs, by repeatedly feeding the previous state through a mixing function (here, SHA-1), this truncation should be sufficient to prevent any reasonable attacker from reconstructing the full state, since they would effectively have to guess the remaining 96 bits of the state.

Thus, even though the output of SHA1PRNG is deterministic (given the initial seed, which the output never reveals), it should not be possible for any attacker, using currently known techniques and feasible computing resources, to predict its future output based on observed past outputs (or, indeed, even to distinguish its output from a truly random bitstream).

Now, for some caveats:

• The security of SHA1PRNG relies entirely on the seed value being secure (i.e. secretly and randomly chosen from a sufficiently large set of probabilities to withstand brute force guessing attacks). If the generator is seeded improperly, all the security properties may be lost.

Normally, the first call to nextBytes() should automatically seed the generator from a system-provided source of randomness. However, this automatic seeding does not happen if setSeed() is called before the first call to nextBytes(). In this case, the internal PRNG state depends entirely on the user-provided seed, which, if poorly chosen, might not have much entropy. (This is particularly prone to cause confusion because calling setSeed() after the internal seeding just mixes the user-provided seed into the internal state, and so never reduces the randomness of the output.)

Also, while the automatic seeding is supposed to be secure, this might not always be the case in practice. Notably, a flaw in the PRNG seeding on Android was used in 2013 to steal bitcoins from people running bitcoin software on Android devices.

• Even if the SecureRandom instance is used properly, the internal implementation of the PRNG might have bugs that compromise its security. Here's one paper I found that discusses weaknesses in various Java PRNG implementations.

If you don't trust the SHA1PRNG provided by your JRE, one option would be to explicitly request a NativePRNG implementation by using SecureRandom.getInstance("NativePRNG") instead of new SecureRandom(). Of course, this call may fail on systems that do not have such an implementation available, and in any case, still requires you to trust the OS native PRNG, and the Java interface to it provided by your Java crypto implementation.

The other alternative is to implement your own (deterministic) cryptographically secure pseudorandom number generator, such as HMAC_DRBG or CTR_DRBG from NIST SP 800-90A (note: do not use Dual_EC_DRBG!), and use it to generate your random numbers, possibly mixed (i.e. XORed) with the output of the OS native PRNG. The tricky part, here, is seeding the PRNG properly: if you have access to a secure OS RNG, you might as well use it directly; if not, you'd have to implement some ad hoc entropy collection scheme, which is difficult and easy to get wrong.

The key thing here is that even in the case that the final algorithm used is "SHA1PRNG", some entropy will be collected somehow for generating the seed that initializes the PRNG. So, it all depends on the seed. In this case, you can see in the code of sun.security.provider.SecureRandom that the seed is generated by the class sun.security.provider.SeedGenerator. What follows is a description from the documentation of this class:

This class generates seeds for the SHA1PRNG cryptographically strong random number generator. The seed is produced using one of two techniques, via a computation of current system activity or from an entropy gathering device. In the default technique the seed is produced by counting the number of times the VM manages to loop in a given period. This number roughly reflects the machine load at that point in time. The samples are translated using a permutation (s-box) and then XORed together. This process is non linear and should prevent the samples from "averaging out". The s-box was designed to have even statistical distribution; it's specific values are not crucial for the security of the seed. We also create a number of sleeper threads which add entropy to the system by keeping the scheduler busy. Twenty such samples should give us roughly 160 bits of randomness. These values are gathered in the background by a daemon thread thus allowing the system to continue performing it's different activites, which in turn add entropy to the random seed. The class also gathers miscellaneous system information, some machine dependent, some not. This information is then hashed together with the 20 seed bytes. The alternative to the above approach is to acquire seed material from an entropy gathering device, such as /dev/random. This can be accomplished by setting the value of the securerandom.source Security property to a URL specifying the location of the entropy gathering device, or by setting the java.security.egd System property. In the event the specified URL cannot be accessed the default threading mechanism is used.

Note that this procedure only happens if you don't seed the SecureRandom object yourself. If you provide your own seed, then the output of the PRNG will depend on that seed, obviously.

IMHO, I think that this procedure should be completely fine for your application, so you shouldn't be worried about the predictability of the source of randomness.

Other interesting references here:

• "How Random Nextbytes" link is exactly what I was looking for! Second link says that a deterministic PRNG "should not be used indefinitely without reseeding" and specifically suggests periodically reseeding or making a new PRNG. That seems to contradict your closing "shouldn't be worried about the predictability of the source of randomness." I would think it doubly prudent to reseed/recycle a SHA1 PRNG periodically because SHA1 has been deprecated by NIST, Microsoft, Google (and now Mozilla) since that article was written. en.wikipedia.org/wiki/SHA-1 – GlenPeterson Dec 31 '14 at 14:44
• The fact that SHA1 is being deprecated doesn't mean that it is totally broken. In fact, if that were true, you can imagine what consequences that would have. SHA1PRNG would be the least of our worries... – cygnusv Dec 31 '14 at 16:29
• @GlenPeterson: BTW, this link says pretty interesting stuff. Maybe it is not directly related to your question, but I think you will like it :) – cygnusv Dec 31 '14 at 17:18


text:
# kaashif's blog

Programming, software freedom and Unix

### 2014-02-05

Whenever you hear something about Haskell, chances are it sounds arcane and involves lots of complicated and intimidating mathematical language. Well, the truth is that all this talk of 'endofunctors' and 'monoids' is really unnecessary, if the concept of functors is explained using a simple analogy.

If you have a situation where a type contains other types, you are dealing with a functor. For example, when you have a list of integers, the list is the functor. If you have a herd of sheep, the herd is the functor. At its core, this is what a functor is - something which contains types and can be mapped over. That last part is crucial and is central to how functors are defined in Haskell, but are not central to how one should think about functors. 46aa190c513173849b0f6a0dc1eec6d6 At a GHCi prompt, you can import a few modules here and there with the word 'functor' in them, but at this stage, you do not require any of the more complex functions, you only need one - fmap. The functor we'll be using as an example is the 'Maybe' functor, which contains either a single value ('Just' the value) or Nothing. If this is confusing to you, just think of the 'Just' constructor as representing a box you put types into, and the 'Nothing' constructor as representing an empty box. So 'Just 1' can be represented as 'put the value 1 into a box'. This isn't entirely contrived, the 'Maybe' functor does have a use in functions that can fail, so can return an empty box or a box with a result in them. Let's examine the type of 'Just' using ':t'.
Prelude> :t Just
Just :: a -> Maybe a

If you are familiar with Haskell syntax, you should know that this means that 'Just' takes any type and turns it into a 'Maybe' functor containing that type. No surprises here, that is the definition of a functor.

Let's say we have a few functions that return numbers contained in a Maybe functor, and we want to combine them in some way. When mapping over a single functor, the function you use is fmap, like so:

Prelude> fmap (+2) (Just 3)
Just 5

This is fine for function that only take 1 argument, but remember, we want to apply this to multiple arguments. If we define a function like so, which takes three (non-Maybe) numbers and adds them together, it becomes obvious that using fmap on its own won't work:

Prelude> let f x y z = x*y*z

If we just try to fmap our three-operand function over a Maybe functor, and examine its type, we see the following:

Prelude> :t (fmap (f) (Just 3))
(fmap (f) (Just 3)) :: Num a => Maybe (a -> a -> a)

This means that fmap has transformed our function which took things of typeclass Num into a partially applied function, which still takes things of typeclass Num, but the function itself is within the box of the Maybe functor. This means that we want to map a Maybe function over a Maybe Num. If the function takes yet another argument, doing this will return yet another function. If not, it'll give us the answer, wrapped in a functor. If we want to apply a function with multiple arguments to functors, we need to use Applicative functors.

Prelude> import Control.Applicative

And then, we can use the <*> operator, which takes a function in a functor and applies it to a value in a functor. We can use it in more trivial cases like so:

Prelude> Just (*4) <*> Just 3
Just 12

We can see that (*4) takes one value, so we are given the answer we want. Applying this to our situation gives us this:

Prelude> (fmap (f) (Just 3)) <*> Just 4 <*> Just 5
Just 60

We can improve the look of this ugly statement by using fmap as an infix function.

Prelude> f fmap Just 3 <*> Just 4 <*> Just 5

Because this is such a common use case, there is some syntactic sugar provided - fmap can be replaced with <$>. Prelude> f <$> Just 3 <*> Just 4 <*> Just 5

At this point, we just have the function and its arguments as we usually would, just with some extra operators thrown in. The most important thing you should take away from this is that using Applicative is far better than using liftM and its numbered equivalents, as Applicative can be applied to an arbitrary number of arguments. Monads are useful in other cases, and Maybe is actually a monad, but thinking about it like a monad rarely helps.


text:
## Frank Gutch Jr: Rain Perry, Mark Hallman (The Shopkeeper), and Congress House Studio; Spotify Once Again; and Notes Hitting the Spot

Posted in Opinion, Review with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on June 27, 2017 by segarini

You can file this one under “and I thought I knew something.”  I just watched a documentary which starts “When I was a kid, music was everything,” a statement as acute to me as author Scott Turow‘s line “It suddenly hit me how much I missed music for which I once felt a yearning as keen as hunger.”  It struck a note so deep in me that I watched  all one-hour-and-thirty-one minutes feeling a kinship with the narrator (and, as it turns out, producer of the film), almost relieved that I was not alone.

## Frank Gutch Jr: Who’s Making Up the Rules, Marc Cohn at the Aladdin Theater in Portland Oregon, and Whatever the Hell Else I Want To Write About…..

Posted in Opinion with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on January 30, 2013 by segarini

I’m tired.  I mention this at the outset because this inevitably is going to turn in to one of those stream-of-consciousness tirades in which logic does not play a part.  But I don’t care.  I need to get something off my chest and I don’t care if you stop reading right now, but the stench of what is wrong with the music business has me gasping for air and there is little oxygen available.  But before I start…..


text:
# Fast Switching with a Tone Decoder (LM567)

Discussion in 'General Electronics Chat' started by BackyardBrains, May 6, 2011.

1. ### BackyardBrains Thread Starter Member

Aug 26, 2010
33
0
I am making a circuit that will pulse an LED on/off when a tone is present/absent from an iPod/iPhone. The iOS application can switch on and off a pure 10kHz signal with sub-millisecond precision. This will be used for high school kids to stimulate the neurons of genetically-modified fruit flies, you can read more here.

I am using the LM567 Tone Detector to switch the LED. The circuit is attached as an image. It works well, and switches on when the tone is played. Only there is a substantial delay between when the tone comes on to when the LED comes on I measured to be 15ms (or ~150 cycles). For my application, this must be much quicker: 2-3ms.

According to the spec sheet (also attached to this post), the Fastest ON-OFF Cycling Rate is:$
{f_o}/{20}
$

In my design, ${f_o}=10,000Hz$ therefore the switching rate should be $10,000Hz/20 = 500 Hz = 2ms$. The documentation has a graph labeled "Greatest Number of Cycles Before Output". But I am struggling to understand what it means.

Has anyone ever used this chip? I am assuming that C2 may change the number of cycles required to change states, but would like to understand what that capacitor is doing. What does Bandwidth (% of Fo) mean in this situation?

Thanks for all your help. I love this board!

File size:
340.6 KB
Views:
215
File size:
84.5 KB
Views:
231
2. ### Veracohr Well-Known Member

Jan 3, 2011
559
77
I haven't used it, but here's what I see. Bandwidth in the datasheet refers to the input filter. If your center frequency is 10kHz and you set the bandwidth to 10%, it will trigger from any input between 9kHz-11kHz.

It also appears the higher the bandwidth, the lower number of maximum cycles before output. So if you want the lowest number of cycles, and you're using a pure 10kHz sine wave as the input signal, you can set the bandwidth as wide as possible. If your input is only a single frequency then you don't have to worry about it triggering on any other unwanted frequency. The datasheet has a formula for calculating bandwidth.


text:
# Unexpected error related to Unprotecting StringPartition

Posted 5 months ago
449 Views
|
4 Replies
|
8 Total Likes
|
 Unprotect@StringPartition StringPartition[n_Integer, d_Integer] := Function[str, StringPartition[str, n, d]] results in the following error on my machine: SetDelayed::write: Tag StringPartition in StringPartition[n_Integer,d_Integer] is Protected. I have added operator definitions to many other built-in functions using this methodology and StringPartition is the first one that gives me this error.Please let me know if I'm missing something here or if this error message is a bug?Thanks, Rian
4 Replies
Sort By:
Posted 5 months ago
 I have a quite interesting observation of this function as well: In[1]:= Attributes[StringPartition] Out[1]= {Protected, ReadProtected} In[2]:= Unprotect@StringPartition Out[2]= {StringPartition} In[3]:= Attributes[StringPartition] Out[3]= {ReadProtected} In[4]:= StringPartition[n_Integer,d_Integer]:=Function[str,StringPartition[str,n,d]] During evaluation of In[4]:= SetDelayed::write: Tag StringPartition in StringPartition[n_Integer,d_Integer] is Protected. Out[4]= $Failed In[5]:= Attributes[StringPartition] Out[5]= {Protected,ReadProtected} Somehow the function is automatically protected again. It does not happen for Log in this online doc Answer Posted 5 months ago  From V11.3, I feel comfortable to deal with this type of operator definition with Curry In[3]:= op=Curry[StringPartition,{3,1,2}][1][2] Out[3]= Curry[StringPartition,{3,1,2}][1,2] In[4]:= op["abc"] Out[4]= {a,c} Answer  If you look at the definition of StringPartition, you'll see that it is just a stub loading a package, not a real function (in a fresh kernel, of course): ClearAttributes[StringPartition, {Protected, ReadProtected}] Definition@StringPartition StringPartition := SystemDumpAutoLoad[Hold[StringPartition], Hold[StringPart, StringPartition], "StringUtilities"] /; SystemDumpTestLoad The package redefines StringPartition and protects it again: StringPartition; ClearAttributes[StringPartition, {Protected, ReadProtected}] Definition@StringPartition StringPartition[StringUtilitiesDumpargs___]:=With[{StringUtilitiesDumpres=StringUtilitiesDumpoStringPartition[StringUtilitiesDumpargs]},StringUtilitiesDumpres/;StringUtilitiesDump`res=!=$Failed] This explains the behavior you observe. The workaround is to preload the package before making any new definitions as I made above. But remember that modifying built-in functions is dangerous and isn't recommened until you know exactly what you do...
Posted 5 months ago
 Hi Shenghui,Thank you for your response.I am familiar with Curry, however I would like to avoid defining a new symbol (op in your example) and instead would like to add an operator definition to the existing, built-in symbol StringPartition.Yes you are right, the unexpected behavior I was referring to in my question is exactly the fact that StringPartition for some reason automatically re-protects itself.Do you know why this is or how to go about fixing it?Thanks, Rian
Community posts can be styled and formatted using the Markdown syntax.


text:
# zbMATH — the first resource for mathematics

D. c. optimization approach to robust control: Feasibility problems. (English) Zbl 0998.93012
Global optimization algorithms are designed to solve numerically some difficult non-convex problems arising in robust control of linear systems, namely $$H_\infty$$ output feedback control with constant scalings and more general bilinear matrix inequality problems. It is shown how the particular problem structure can be exploited in a branch and bound scheme: branching is performed on a reduced number of variables, whereas bounding is done on d.c. (difference of convex functions) inequalities relaxed to linear or linear matrix constraints. Very preliminary numerical results are described.

##### MSC:
 93B40 Computational methods in systems theory (MSC2010) 90C26 Nonconvex programming, global optimization 93B36 $$H^\infty$$-control 90C22 Semidefinite programming 93D09 Robust stability 93B51 Design techniques (robust design, computer-aided design, etc.)
Full Text:


text:
zbMATH — the first resource for mathematics

Generalized group characters and complex oriented cohomology theories. (English) Zbl 1007.55004
This is a paper of long gestation and wide scope. It examines the functor $$G\mapsto E^*(BG)$$ which takes a finite group $$G$$ into the $$E$$-cohomology of its classifying space where $$E^*(-)$$ is a complex oriented cohomology theory. Complex oriented cohomology theories include ordinary integral cohomology, complex K-theory, complex cobordism, the variety of theories related to Brown-Peterson cohomology, and the Morava K-theories, $$K(n)^*(-)$$.
For a finite group $$G$$, $${\mathcal A}(G)$$ is the category whose objects are the Abelian subgroups of $$G$$. A morphism from $$B$$ to $$A$$ is a $$G$$-map from $$G/B$$ to $$G/A$$. With $$G$$ a finite group of order $$|G|$$ and $$E$$ as above, ${1\over|G|} E^*(BG)\to \lim_{A\in{\mathcal A}(G)} {1\over|G|} E^*(BA)$ is an isomorphism. This result is an analogue of Artin’s theorem; the splitting principle is employed in the proof.
A second result states that for a finite group $$G$$, the Morava K-theory Euler characteristic $$\dim K(n)^{\text{even}}(BG)- \dim K(n)^{\text{odd}}(BG)$$ is the number of $$G$$-orbits in a $$G$$-set $$G_{n,p}$$. The prime $$p$$ is the one associated to the Morava K-theory. $$G_{n,p}$$ is the set of $$n$$-tuples of commuting order-$$p$$ elements of $$G$$. $$G$$ acts on $$G_{n,p}$$ by conjugation.
If we take as our example of complex-oriented $$E^*$$ ordinary cohomology, $${1\over|G|} E^*(BG)$$ is not very interesting. But decades ago, Landweber observed that $$MU^*(BZ/2)$$ is torsion-free – vastly different to its homology analogue. If $$E^*$$ is a $$p$$-complete integral life of $$K(n)^*$$, then $$E^*(BG)$$ will be torsion-free if and only if $$K(n)^*(BG)$$ is concentrated in even degrees. This leads the authors to seek conditions to ensure this even concentration (e.g. if the group $$G$$ is “good”).
The authors interpret their results in terms of generalized characters and invariant rings. The reader is referred to the paper’s introduction for the necessary definitions and precise statements. The authors’ approach can be applied to compute the kernel of $$\pi^0_s(BG)\to MU^0(BG)$$ up to finite index.

MSC:
 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 55N91 Equivariant homology and cohomology in algebraic topology 55N34 Elliptic cohomology
Full Text:
References:
 [1] J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill.-London, 1974. Chicago Lectures in Mathematics. · Zbl 0309.55016 [2] John Frank Adams, Infinite loop spaces, Annals of Mathematics Studies, vol. 90, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. [3] J. F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 483 – 532. · doi:10.1007/BFb0075584 · doi.org [4] Shôrô Araki, Equivariant stable homotopy theory and idempotents of Burnside rings, Publ. Res. Inst. Math. Sci. 18 (1982), no. 3, 1193 – 1212. · Zbl 0527.55020 · doi:10.2977/prims/1195183305 · doi.org [5] M. F. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23 – 64. [6] M. F. Atiyah, \?-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam, 1967. [7] Michael Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671 – 677. · Zbl 0708.19004 · doi:10.1016/0393-0440(89)90032-6 · doi.org [8] Andrew Baker and Urs Würgler, Liftings of formal groups and the Artinian completion of \?_\?$$^{-}$$\textonesuperior \?\?, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 3, 511 – 530. · Zbl 0729.55002 · doi:10.1017/S0305004100068249 · doi.org [9] Andrew Baker, Hecke algebras acting on elliptic cohomology, Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997) Contemp. Math., vol. 220, Amer. Math. Soc., Providence, RI, 1998, pp. 17 – 26. · Zbl 0911.55002 · doi:10.1090/conm/220/03091 · doi.org [10] Raoul Bott and Clifford Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989), no. 1, 137 – 186. · Zbl 0667.57009 [11] Tammo tom Dieck, Kobordismentheorie klassifizierender Räume und Transformationsgruppen, Math. Z. 126 (1972), 31 – 39 (German). · Zbl 0227.57017 · doi:10.1007/BF01580352 · doi.org [12] Tammo tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics, vol. 766, Springer, Berlin, 1979. · Zbl 0445.57023 [13] Tammo tom Dieck, Transformation groups, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. · Zbl 0611.57002 [14] Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0454.14020 [15] Michael J. Hopkins, Characters and elliptic cohomology, Advances in homotopy theory (Cortona, 1988) London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 87 – 104. · Zbl 0729.55003 [16] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel, Morava \?-theories of classifying spaces and generalized characters for finite groups, Algebraic topology (San Feliu de Guíxols, 1990) Lecture Notes in Math., vol. 1509, Springer, Berlin, 1992, pp. 186 – 209. · Zbl 0757.55006 · doi:10.1007/BFb0087510 · doi.org [17] John Hunton, The Morava \?-theories of wreath products, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 309 – 318. · Zbl 0705.55009 · doi:10.1017/S0305004100068572 · doi.org [18] N. M. Osadčiĭ, The algebras \?²_\?(\Gamma ), and the lattice of closed ideals of these algebras, Ukrain. Mat. Ž. 26 (1974), 669 – 670, 717 (Russian). [19] Igor Kriz, Morava \?-theory of classifying spaces: some calculations, Topology 36 (1997), no. 6, 1247 – 1273. · Zbl 0896.55005 · doi:10.1016/S0040-9383(96)00049-3 · doi.org [20] I. Kriz and K. P. Lee. Odd degree elements in the Morava $$K(n)$$ cohomology of finite groups. Preprint, 1998. To appear in Topology and its applications. · Zbl 0963.55004 [21] Nicholas J. Kuhn, The Morava \?-theories of some classifying spaces, Trans. Amer. Math. Soc. 304 (1987), no. 1, 193 – 205. · Zbl 0637.55002 [22] Nicholas J. Kuhn, Character rings in algebraic topology, Advances in homotopy theory (Cortona, 1988) London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 111 – 126. · Zbl 0693.55014 [23] Erkki Laitinen, On the Burnside ring and stable cohomotopy of a finite group, Math. Scand. 44 (1979), no. 1, 37 – 72. · Zbl 0389.16021 · doi:10.7146/math.scand.a-11795 · doi.org [24] Peter S. Landweber, Complex bordism of classifying spaces, Proc. Amer. Math. Soc. 27 (1971), 175 – 179. · Zbl 0207.53601 [25] Peter S. Landweber, Douglas C. Ravenel, and Robert E. Stong, Periodic cohomology theories defined by elliptic curves, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 317 – 337. · Zbl 0920.55005 · doi:10.1090/conm/181/02040 · doi.org [26] Serge Lang, Cyclotomic fields, Springer-Verlag, New York-Heidelberg, 1978. Graduate Texts in Mathematics, Vol. 59. · Zbl 0395.12005 [27] L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. · Zbl 0611.55001 [28] Jonathan Lubin and John Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1965), 380 – 387. · Zbl 0128.26501 · doi:10.2307/1970622 · doi.org [29] Jack Morava, Completions of complex cobordism, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 349 – 361. · Zbl 0394.55007 [30] P. S. Landweber , Elliptic curves and modular forms in algebraic topology, Lecture Notes in Mathematics, vol. 1326, Springer-Verlag, Berlin, 1988. · Zbl 0642.00007 [31] Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549 – 572; ibid. (2) 94 (1971), 573 – 602. · Zbl 0247.57013 · doi:10.2307/1970770 · doi.org [32] Douglas C. Ravenel, Morava \?-theories and finite groups, Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemp. Math., vol. 12, Amer. Math. Soc., Providence, R.I., 1982, pp. 289 – 292. · Zbl 0508.55009 [33] Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. · Zbl 0608.55001 [34] Douglas C. Ravenel and W. Stephen Wilson, The Morava \?-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), no. 4, 691 – 748. · Zbl 0466.55007 · doi:10.2307/2374093 · doi.org [35] Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105 – 112. · Zbl 0199.26404 [36] Graeme Segal, Equivariant \?-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129 – 151. · Zbl 0199.26202 [37] G. B. Segal, Equivariant stable homotopy theory, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 59 – 63. · Zbl 0225.55014 [38] Jean-Pierre Serre, Représentations linéaires des groupes finis, Hermann, Paris, 1967 (French). · Zbl 0205.04001 [39] C. T. Stretch, Stable cohomotopy and cobordism of abelian groups, Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 2, 273 – 278. · Zbl 0493.55002 · doi:10.1017/S0305004100058734 · doi.org [40] N. P. Strickland, Morava \?-theory of symmetric groups, Topology 37 (1998), no. 4, 757 – 779. , https://doi.org/10.1016/S0040-9383(97)00054-2 N. P. Strickland, Correction to: ”Morava \?-theory of symmetric groups” [Topology 37 (1998), no. 4, 759 – 779; MR1607736 (99e:55008)], Topology 38 (1999), no. 4, 931. · Zbl 0912.55012 · doi:10.1016/S0040-9383(98)00043-3 · doi.org [41] Michimasa Tanabe, On Morava \?-theories of Chevalley groups, Amer. J. Math. 117 (1995), no. 1, 263 – 278. · Zbl 0820.55002 · doi:10.2307/2375045 · doi.org [42] Clifford Henry Taubes, \?\textonesuperior actions and elliptic genera, Comm. Math. Phys. 122 (1989), no. 3, 455 – 526. · Zbl 0683.58043 [43] M. Tezuka and N. Yagita, Cohomology of finite groups and Brown-Peterson cohomology, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 396 – 408. · Zbl 0673.55005 · doi:10.1007/BFb0085243 · doi.org [44] Urs Würgler, Commutative ring-spectra of characteristic 2, Comment. Math. Helv. 61 (1986), no. 1, 33 – 45. · Zbl 0622.55003 · doi:10.1007/BF02621900 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
Journal article

### Spectral order for contact manifolds with convex boundary

Abstract:

We extend the Heegaard Floer homological definition of spectral order for closed contact 3–manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact 3–manifolds with convex boundary. We show that the order of a codimension-zero contact submanifold bounds the order of the ambient manifold from above. As the neighborhood of an overtwisted disk has order zero, we obtain that overtwisted contact structures have order zero. We also prove that the order of a small perturbation of a G...

Publication status:
Published
Peer review status:
Peer reviewed
Version:
Publisher's version

### Access Document

Files:
• (pdf, 405.3kb)
Publisher copy:
10.2140/agt.2018.18.3315

### Authors

More by this author
Institution:
University of Oxford
Division:
MPLS Division
Department:
Mathematical Institute
Oxford college:
Keble College
ORCID:
0000-0003-3705-2288
More by this author
Institution:
University of Oxford
Division:
MPLS Division
Department:
Mathematical Institute
More from this funder
Funding agency for:
Juhasz, AI
Publisher:
Mathematical Sciences Publishers Publisher's website
Journal:
Algebraic and Geometric Topology Journal website
Volume:
18
Issue:
6
Pages:
3315–3338
Publication date:
2018-10-18
Acceptance date:
2018-05-25
DOI:
EISSN:
1472-2739
ISSN:
1472-2747
Pubs id:
pubs:853835
URN:
uri:4229c0ab-412b-43ea-9449-65186dc9abba
UUID:
uuid:4229c0ab-412b-43ea-9449-65186dc9abba
Local pid:
pubs:853835
Keywords:


text:
# Representation theory of the Lorentz group

The Lorentz group, a Lie group on which special relativity is based, has a variety of representations. Many of these representations are important in theoretical physics in the description of particles in relativistic quantum mechanics, as well as fields in quantum field theory. This representation theory provides a theoretical ground for the concept of spin (which can be either integer or half-integer in the unit of the reduced Planck constant ℏ); representations with half-integer weights are normally constructed out of spinors.

The group may also be represented[how?] in terms of a set of functions defined on the Riemann sphere. These are the Riemann P-functions, which are expressible as hypergeometric functions. The identity component SO(3;1)+ of the Lorentz group is isomorphic to the Möbius group, and hence any representation of the Lorentz group is necessarily a representation of the Möbius group.

The subgroup SO(3) with its representation theory form a simpler theory, but the two are related (see details below) and both are prominent in theoretical physics as descriptions of spin, angular momentum, and other things related to rotation.

## Finite-dimensional representations

Representation theory of groups in general, and Lie groups is particular, is a very rich subject. The full Lorentz group makes no exception. The Lorentz group has some properties that makes it "agreeable" and other that makes it "not very agreeable" within the context of representation theory. The group is semisimple, but it is not simple. It is not connected, and none of its components are simply connected. Perhaps most importantly, the Lorentz group is not compact.

The presence of semisimplicity means the Lorentz group can be much be dealt with without the presence of simplicity - all representations are known from the fact that the Lorentz group is semisimple. But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, can on the contrary, not be dealt with in all the aspects of the same simple framework that applies to simply connected, compact groups.

### The Lie algebra

According to the general representation theory of Lie groups, one first looks for the representations of the complexification, so(3;1)C of the Lie algebra so(3;1) of the Lorentz group. A convenient basis for so(3;1) is given by the three generators Ji of rotations and the three generators Ki of boosts. These two sets of generators, for the standard representation, are both found in explicit form in the section Conventions below. Now first complexify the Lie algebra, and then change basis to the components of:

$\mathbf{A} = \frac{\mathbf{J} + i \mathbf{K}}{2}\,,\quad \mathbf{B} = \frac{\mathbf{J} - i \mathbf{K}}{2}\,.$

In this new basis, one checks that the components of A = (A1, A2, A3) and B = (B1, B2, B3) satisfy separately the commutation relations of the Lie algebra su(2) and moreover that they commute with each other[1]

$\left[A_i ,A_j\right] = i\varepsilon_{ijk}A_k\,,\quad \left[B_i ,B_j\right] = i\varepsilon_{ijk}B_k\,,\quad \left[A_i ,B_j\right] = 0\,,$

where i, j, k are indices which each take values 1, 2, 3, and εijk is the three-dimensional Levi-Civita symbol. In other words, one has the isomorphism

$\mathfrak{so}(3,1)_C \cong \mathfrak{sl}(2,C) \oplus \mathfrak{sl}(2,C)\,,$[2]

(A1)

where sl(2, C) is the complexification of su(2). The utility of this isomorphism comes from the fact that all irreducible representations of su(2) are, up to isomorphism, known. Every irreducible representation of su(2) is isomorphic to one of the highest weight representations. Moreover, there is a one-to-one correspondence between linear representations of su(2) and complex linear representations of its complexification sl(2, C).[3] Via the displayed isomorphism, all irreducible representations of so(3;1)C, and thus those of so(3;1) are known. Since so(3;1) is semisimple,[2] all its representations, not necessarily irreducible, can be built up as direct sums of the irreducible ones.

Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers m and n, conventionally written as

$(m,n) \equiv D^{(m,n)} \equiv D^{(m)} \otimes D^{(n)}$

where denotes the tensor product of the representations D(m) and D(n), which fix representations of the su(2) subalgebras spanned by the components of A and B respectively. The m and n are the highest weights, respectively, of the complexified su(2) representations, i.e. sl(2,C) representations, of AC and BC. Let π(m,n): so(3;1) → gl(V), where V is a vector space, denote the irreducible representations of so(3;1) according to the this classification. These are, up to a similarity transformation, uniquely given by

\begin{align} \pi_{m,n}(J_i) & = 1_{(2m+1)}\otimes J^{(n)}_i + J^{(m)}_i\otimes 1_{(2n+1)}\\ \pi_{m,n}(K_i) & = i(1_{(2m+1)}\otimes J^{(n)}_i - J^{(m)}_i \otimes 1_{(2n+1)}), \end{align}[4]

(A2)

where the J(n) = (J(n)1,J(n)2,J(n)3) are the (2n + 1)-dimensional irreducible spin n representations of so(3)su(2) and 1n is the n-dimensional unit matrix. Explicit formulas on component form are given at the end of the article.

The highest weight for a sl(2, C) representation coincides with the highest eigenvalue of the representative of the third Pauli matrix σ3sl(2, C) and is in quantum mechanics thus related to the maximum value the spin z-component can take on for a particle whose state vector transform under the representation in question.

### The group

Here V is a finite-dimensional vector space, GL(V) is the set of all invertible linear transformations on V and gl(V) is its Lie algebra. The maps π and Π are Lie algebra and group representations respectively, and exp is the exponential mapping. The diagram commutes only up to a sign if Π is projective.

Given a so(3;1) representation, one may try to construct a representation of SO(3;1)+, the identity component of the Lorentz group, by using the exponential mapping. If X is an element of so(3;1) in the standard representation, then

$\Lambda = e^{iX} \equiv \sum_{n=0}^{\infty} \frac{(iX)^n}{n!}$

(G1)

is a Lorentz transformation by general properties of Lie algebras. Motivated by this, let π: so(3;1) → gl(V) for some vector space V be a representation and attempt to define a representation Π of SO(3)+ by first setting

$\Pi_U(e^{iX}) = e^{i\pi(X)}, \quad X \in \mathfrak{so}(3;1).$

(G2)

There are at least two potential problems with this definition. The first is that it is not obvious that this yields a group homomorphism, or even a well defined map at all. But a theorem[5] based on the inverse function theorem states that the map exp: so(3;1) → SO(3;1)+ is one-to-one for X small enough (A). The qualitative form of the Baker-Campbell-Hausdorff formula then guarantees that it is a group homomorphism, still for X small enough (B).[5] Let U⊂SO(3;1)+ denote image under the exponential mapping of the open set in so(3;1) where conditions (A) and (B) both hold. The second problem is that for a given g ∈ SO(3;1)+ there may not be exactly one Xso(3;1) such that g = eiX. Technically, formula (G2) is used to define Π near the identity. For other elements gU one chooses a path from the identity to g and defines Π along that path by partitioning it finely enough so that formula (G2) can be used again on the resulting factors in the partition. In detail, one sets

$g = g_n = (g_n g_{n-1}^{-1})(g_{n-1}g_{n-2}^{-1})\cdots(g_{2}g_{1}^{-1})(g_{1}g_{0}), \qquad \Pi(g) \equiv \Pi_U(g_{n}g_{n-1}^{-1})\Pi_U(g_{n-1}g_{n-2}^{-1})\cdots\Pi_U(g_{2}g_{1}^{-1})\Pi_U(g_{1}g_{0}), \quad g_0 = 1$[5]

(G3)

where the gi are on the path and the factors on the far right are uniquely defined by (G2) provided that all gigi + 1−1U. For each i take, by the inverse function theorem, the unique Xi such that exp(Xi) = gigi-1-1 and obtain

$\Pi(g) = \Pi_U(e^{iX_n})\Pi_U(e^{iX_{n - 1}})\cdots\Pi_U(e^{iX_2})\Pi_U(e^{iX_1}) = \Pi_U(e^{iX_n}e^{iX_{n - 1}}\cdots e^{iX_2}e^{iX_1}),$

(G4)

where the final step is by the Baker-Campbell-Hausdorff formula. It shows that Π is a group homomorphism for elements close to the identity. By compactness of the path there is an n large enough so that Π(g) is well defined, possibly depending on the partition and/or the path, whether g is close to the identity or not. Likewise, the formula yields a homomorphism for all elements, whether small or not, as seen using expressions similar to (G2). One simply drops the subscript U in formula (G2).

It turns out that the result is always independent of the partitioning of the path.[5] For simply connected groups, the construction will be independent of the path as well, yielding a well defined representation.[5] In that case formula (G2) can unambiguously be used directly. For a group that is connected but not simply connected, such as SO(3;1)+, the result may depend on the homotopy class of the chosen path.[6] The result, when using (G2), will then depend on which X in the Lie algebra is used to obtain the representative matrix for g.

The Lorentz group is doubly connected so that its fundamental group π1(SO(3,1)+), whose elements are the path homotopy classes, has two members. Thus not all representations of the Lie algebra will yield representations of the group, but some will instead yield projective representations.[5] Once these conclusions has been reached, and once one knows whether a representation is projective, there is no need not be concerned about paths and partitions. Formula (G2) applies to all group elements and all representations, including the projective ones.

For a projective representation Π of SO(3;1)+ it holds that

$[\Pi(\Lambda_1)\Pi(\Lambda_2)\Pi^{-1}(\Lambda_1\Lambda_2)]^2 = 1\Rightarrow \Pi(\Lambda_1\Lambda_2) = \pm \Pi(\Lambda_1)\Pi(\Lambda_2), \qquad \Lambda_1,\Lambda_2 \in SO(3;1),$[7]

(G5)

since any loop in SO(3;1)+ traversed twice, due to the double connectedness, is contractible to a point so that its homotopy class is that of a constant map. It follows that Π is a double-valued function. One cannot consistently chose sign to obtain a continuous representation of all of SO(3;1)+, but this is possible to do locally around any point.[8]

### The covering group

Let πg denote the set of path homotopy classes [pg] of paths pg(t), 0 ≤ t ≤ 1, from 1 ∈ SO+(3;1) to g ∈ SO+(3;1) and define the set

$G = \{(g,[p_g]): g\in SO^+(3,1),[p_g]\in \pi_g\}$

and endow it with the multiplication operation

$(g_1,[p_1])(g_2,[p_2]) = (g_1g_2,[p_{12}]),\quad g_1,g_2\in SO(3;1)^+,\quad [p_1]\in\pi_{g_1}, [p_2]\in \pi_{g_2}, [p_{12}]\in \pi_{g_{12}},\quad p_{12}(t) = p_1(t)\cdot p_2(t).$

The dot on the far right denotes path multiplication. With this multiplication, G is a group and G SL(2,C),[9] the universal covering group of SO+(3;1). By the above construction, there is, since each πg has two elements, a 2:1 covering map p:SL(2.C) → SO(3,1)+. According to covering group theory, the Lie algebras so(3;1) and sl(2,C) are isomorphic and the Π(m,n) representations (whether projective or not) correspond to non-projective representations of the simply connected group SL(2,C).

For an algebraic view of the universal covering group, let SL(2,C) act on the set of all Hermitean 2 × 2 matrices h by the operation[10]

$\mathbf{P}(A): \mathbf{h} \rightarrow \mathbf{h}; \quad X \rightarrow A^\dagger XA, \quad X \in \mathbf{h}, A \in SL(2,C).$

Since X ∈ h is Hermitean, AXA is again Hemitean because (AXA) = AXA†† = AXA, and also A(αX + βY)A = αAXA + βAYA, so the action is linear as well. An element of h may generally be written in the form

$X = \bigl(\begin{smallmatrix} \xi_4 + \xi_3&\xi_1 + i\xi_2\\ \xi_1 - i\xi_2&\xi_4 - \xi_3\\ \end{smallmatrix}\bigr)$

for ξi real, showing that h is a 4-dimensional real vector space. Moreover, (AB)X(AB) = BAXAB meaning that P is a group homomorphism into End(h). Thus P:SL(2,C) → End(h) is a 4-dimensional representation of SL(2,C). Now map h to spacetime R4, endowed with the Lorentz metric, via

$X = (\xi_1,\xi_2,\xi_3,\xi_4) \leftrightarrow \overrightarrow{(\xi_1,\xi_2,\xi_3,\xi_4)} = (x,y,z,t) = \overrightarrow{X}.$

The action of P(A) on h preserves determinants since det(AXA) = (det A)(det A)(det X) = det X. The induced representation p of SL(2,C) on R4, via the above isomorphism, given by

$\mathbf{p}(A)\overrightarrow{X} = \overrightarrow{AXA^\dagger}$

will preserve the Lorentz inner product since −det X = ξ12 + ξ22 + ξ32 - ξ42 = x2 + y2 + z2 - t2. This means that p(A) belongs to the full Lorentz group O(3;1). By the main theorem of connectedness, since SL(2,C) is connected, its image under p in O(3;1) is connected as well, and hence is contained in SO(3;1)+. It can be shown that the Lie map of p:SL(2,C) → SO(3;1)+, π:sl(2,C) → so(3;1) is a Lie algebra isomorphism (it's kernel is {∅} and must therefore be an isomorphism for dimensional reasons) and thus SL(2,C), since it is simply connected, is the universal covering group of SO(3;1)+, isomorphic to the group G of above.

#### Representations of SL(2,C) and sl(2,C)

The complex linear representations of sl(2,C) and SL(2,C) are more straightforward to obtain than the SO(3;1)+ representations. If πm is a representation of su(2) with highest weight m, then the complexification of π is a complex linear representation of sl(2,C). All complex linear representation of sl(2,C) are of this form. The group representations are obtained by exponentiation. By simple connectedness of SL(2,C), this always yields a representation of the group as opposed to in the SO(3;1)+ case.

The kernel of the mapping p of above is {I, -I}. By the first isomorphism theorem, a representation Π of SL(2,C) descends to a representation of SO(3;1)+ if and only if ker p ⊂ ker Π. In particular, if Π is faithful, then there is no corresponding proper representation of SO(3;1)+, but there is a projective one as was shown in a previous section.

### Common representations

m=0 1/2 1 scalar Weyl spinor bispinor self-dual 2-form 2-form field Weyl spinor (right-handed) 4-vector Rarita–Schwinger field anti-self-dual 2-form traceless symmetric tensor Purple: (m, n) complex irreps   Black: (m, n) ⊕ (n, m) Bold: (m, m)

Since for any irrep where mn it is essential to operate over the field of complex numbers, the direct sum of representations (m, n) and (n, m) has a particular relevance to physics, since it permits to use linear operators over real numbers.

### Properties of the (m, n) representations

The irreducible (m, n) representations are (2m + 1)(2n + 1)-dimensional, and they are the only irreducible representations.

Since the angular momentum operator is given by J = A + B, the highest weight (or spin modulo ℏ in quantum mechanics) of the rotation subrepresentation will be m + n.

The (m, n) Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary. This is due to the non-compactness of the Lorentz group. It can also be seen directly from the definitions. The representations of A and B used in the construction are Hermitian. This means that J is Hermitian, but K is anti-Hermitian. The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm.[4]

The (m, n) representation is, however, unitary when restricted to the rotation subgroup SO(3), but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an (m, n) representation have SO(3)-invariant subspaces of highest weight (spin) m + n, m + n − 1, …, | mn |,[4] where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) j is (2j + 1)-dimensional. So for example, the (1/2, 1/2) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.

The (m, n) representation is the dual of the (n, m) representation.

### Induced representations

In general representation theory, if (π, V) is a representation of a Lie algebra g, then there is an associated representation of g on End (V), also denoted π, given by

$\pi(X)(A) = [\pi(X),A], \quad A\in \mathrm{End}\,(V),\ X\in\mathfrak{g}.$

Likewise, a representation (Π,V) of a group G yields a representation Π on End (V) of G, still denoted Π, given by

$\Pi(g)(A) = \Pi(g)A\Pi(g)^{-1}, \quad A\in \mathrm{End}\,(V),\ g\in G.$

Applying this to the Lorentz group, if (Π, V) is a projective representation, then direct calculation using (G4) shows that the induced representation on End (V) is, in fact, a proper representation, i.e. a representation without phase factors.

In quantum mechanics this means that if (π, H) or (Π, H) is a representation acting on some Hilbert space H, then the corresponding induced representation acts on the set of linear operators on H. As an example, the induced representation of the projective spin (1/2, 0) ⊕ (0, 1/2) representation on End(H) is the non-projective 4-vector (1/2, 1/2) representation.[4]

For simplicity, consider now only the "discrete part" of End(H), that is, given a basis for H, the set of constant matrices of various dimension, including possibly infinite dimensions. A general element of the full End(H) is the sum of tensor products of a matrix from the simplified End(H) and an operator from the left out part. The left out part consists of functions of spacetime, differential and integral operators and the like. See Dirac operator for an illustrative example. Also left out are operators corresponding to other degrees of freedom not related to spacetime, such as gauge degrees of freedom in gauge theories.

The induced 4-vector representation of above on this simplified End(H) has an invariant 4-dimensional subspace that is spanned by the four gamma matrices.[4] (Note the different metric convention in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, Cℓ3,1(R)M4(C)End(H) spanned by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), a pseudoscalar irrep, a vector irrep, a pseudovector irrep, and a tensor irrep.[4] The dimensions add up to 1 + 1 + 4 + 4 + 6 = 16. This is, in fact, a reasonably convenient way to show that the algebra spanned by the gammas is 16-dimensional.[4] An algebraic proof of this fact is fairly lengthy.[12] For details, see Dirac algebra.

The conclusion is that every linear operator (complex 4×4 matrix) in the simplified End(H) has well defined Lorentz transformation properties. The "left out" part transforms under the infinite-dimensional representations or, if not related to spacetime, not at all under Lorentz transformations, but instead they transform under the corresponding gauge group. The gauge transformations in turn leave the spacetime parts unaffected.

There is also a multitude of other representations that can be said being "induced" by the irreducible ones, such as those obtained in a standard manner by taking direct sums, tensor products, duals, quotients, etc of the irreducible representations. These are not discussed here.

### The full Lorentz group

The (possibly projective) (m, n) representation is irreducible as a representation SO+(3;1), the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If m = n it can be extended to a representation of all of O(3;1), the full Lorentz group, including space parity inversion and time reversal.[4] This follows from considering the adjoint action AdP of P ∈ O(3;1) on so(3;1), where P is the standard representative of space parity inversion, P = diag(1, −1, −1, −1), given by

$\mathrm{Ad}_P(J_i) = PJ_iP^{-1} = J_i, \qquad \mathrm{Ad}_P(K_i) = PK_iP^{-1} = -K_i.$

(F1)

It is these properties of K and J under P that motivate the terms vector for K and pseudovector or axial vector for J. In a similar way, if π is any representation of so(3;1) and Π is its associated group representation, then Π(SO(3;1)+) acts on the representation of π by the adjoint action, π(X) ↦ Π(g) π(X) Π(g)−1 for Xso(3;1), g ∈ SO(3;1)+. If P is to be included in Π, then consistency with (F1) requires that

$\Pi(P)\pi(B_i)\Pi(P)^{-1} = \pi(A_i)$

(F2)

holds, where A and B are defined as in the first section. This can hold only if Ai and Bi have the same dimensions, i.e. only if m = n. When mn then (m, n) ⊕ (n, m) can be extended to an irreducible representation of O+(3;1), the orthocronous Lorentz group. The parity reversal representative Π(P) does not come automatically with the general construction of the (m, n) representations. It must be specified separately. The matrix β = iγ0 may be used in the (1/2, 0) ⊕ (0, 1/2)[4] representation. If parity is included in the (0,0) representation, it is called a pseudoscalar representation.

Time reversal T = diag(−1, 1, 1, 1), acts similarly on so(3;1) by

$\mathrm{Ad}_T(J_i) = TJ_iT^{-1} = -J_i, \qquad \mathrm{Ad}_P(K_i) = TK_iT^{-1} = K_i.$

(F3)

By explicitly including a representative for T, as well as one for P, one obtains a representation of the full Lorentz group O(3;1). The matrix Π(T) is antiunitary and its action on Hilbert space is antilinear.[13]

When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. The Dirac representation, (1/2, 0) ⊕ (0, 1/2), is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation.

The third discrete symmetry entering in the CPT theorem along with P and T, charge conjugation symmetry C, has nothing directly to do with Lorentz invariance.[14]

## Infinite-dimensional unitary representations

### History

The Lorentz group SO(3,1)[clarification needed] and its double cover SL(2, C) also have infinite dimensional unitary representations, first studied independently by Bargmann (1947), Gelfand & Naimark (1947) and Harish-Chandra (1947) (at the instigation of Paul Dirac). The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by Harish-Chandra (1951) and Gelfand & Graev (1953), based on an analogue for SL(2, C) of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in Rühl (1970) and Knapp (2001).

The theory of spherical functions for the Lorentz group, required for harmonic analysis on the 3-dimensional hyperboloid in Minkowski space, or equivalently 3-dimensional hyperbolic space, is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on R. This theory is discussed in Takahashi (1963), Helgason (1968), Helgason (2000) and the posthumous text of Jorgenson & Lang (2008).

### Principal series

The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup B of G = SL(2, C). Since the one-dimensional representations of B correspond to the representations of the diagonal matrices, with non-zero complex entries z and z−1, and thus have the form

$\chi_{\nu,k}(re^{i\theta})=r^{i\nu} e^{ik\theta},$

for k an integer and ν real. The representations are irreducible; the only repetitions occur when k is replaced by k. By definition the representations are realised on L2 sections of line bundles on G / B = S2, the Riemann sphere. When k = 0, these representations constitute the so-called spherical principal series.

The restriction of a principal series to the maximal compact subgroup K = SU(2) of G can also be realised as an induced representation of K using the identification G / B = K / T, where T = BK is the maximal torus in K consisting of diagonal matrices with | z | = 1. It is the representation induced from the 1-dimensional representation zk T, and is independent of ν. By Frobenius reciprocity, on K they decompose as a direct sum of the irreducible representations of K with dimensions | k | + 2m + 1 with m a non-negative integer.

Using the identification between the Riemann sphere minus a point and C, the principal series can be defined directly on L2(C) by the formula

$\pi_{\nu,k}\begin{pmatrix}a& b\\ c& d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-i\nu} \left({cz+d\over |cz+d|}\right)^{-k}f\left({az+b\over cz+d}\right).$

Irreducibility can be checked in a variety of ways:

• The representation is already irreducible on B. This can be seen directly, but is also a special case of general results on ireducibility of induced representations due to François Bruhat and George Mackey, relying on the Bruhat decomposition G = BBsB where s is the Weyl group element $\begin{pmatrix}0& -1\\ 1& 0\end{pmatrix}$.[15]
• The action of the Lie algebra $\mathfrak{g}$ of G can be computed on the algebraic direct sum of the irreducible subspaces of K can be computed explicitly and the it can be verified directly that the lowest dimensional subspace generates this direct sum as a $\mathfrak{g}$-module.[16][17]

### Complementary series

The for 0 < t < 2, the complementary series is defined on L2-functions f on C for the inner product

$(f,g)=\int \int {f(z) \overline{g(w)}\, dz\, dw\over |z-w|^{2-t}}.$

with the action given by

$\pi_{t}\begin{pmatrix}a& b\\ c& d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-t} f\left({az+b\over cz+d}\right).$

The complementary series are irreducible and inequivalent. As a representation of K, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of K = SU(2). Irreducibility can be proved by analysing the action of $\mathfrak{g}$ on the algebraic sum of these subspaces[16][17] or directly without using the Lie algebra.[18][19]

### Plancherel theorem

The only irreducible unitary representations of SL(2, C) are the principal series, the complementary series and the trivial representation. Since I acts (−1)k on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided k is taken to be even.

To decompose the left regular representation of G on L2(G), only the principal series are required. This immediately yields the decomposition on the subrepresentations L2(G/±I), the left regular representation of the Lorentz group, and L2(G/K), the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with k = 0.)

The left and right regular representation λ and ρ are defined on L2(G) by

$\lambda(g)f(x)=f(g^{-1}x),\,\,\rho(g)f(x)=f(xg).$

Now if f is an element of Cc(G), the operator πν,k(f) defined by

$\pi_{\nu,k}(f)=\int_G f(g)\pi(g)\, dg$

is Hilbert–Schmidt. We define a Hilbert space H by

$H=\bigoplus_{k\ge 0} HS(L^2(C)) \otimes L^2(R, c_k(\nu^2 + k^2)^{1/2} d\nu),$

where

$c_0=1/4\pi^{3/2}, \,\, c_k=1/(2\pi)^{3/2}\,\,(k\ne 0)$

and HS(L2(C)) denotes the Hilbert space of Hilbert–Schmidt operators on L2(C).[20] Then the map U defined on Cc(G) by

$U(f)(\nu,k) = \pi_{\nu,k}(f)$

extends to a unitary of L2(G) onto H.

The map U satisfies

$U(\lambda(x)\rho(y)f)(\nu,k) = \pi_{\nu,k}(x)^{-1} \pi_{\nu,k}(f)\pi_{\nu,k}(y).$

If f1, f2 are in Cc(G) then

$(f_1,f_2) = \sum_{k\ge 0} c_k^2 \int_{-\infty}^\infty {\rm Tr}(\pi_{\nu,k}(f_1)\pi_{\nu,k}(f_2)^*) (\nu^2 +k^2) \, d\nu.$

Thus if f = f1f2* denotes the convolution of f1 and f2*, and $f_2^*(g)=\overline{f_2(g^{-1})}$, then

$f(1) = \sum_{k\ge 0} c_k^2 \int_{-\infty}^\infty {\rm Tr}(\pi_{\nu,k}(f)) (\nu^2+k^2)\, d\nu.$

The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively. The Plancherel formula extends to all fi in L2(G). By a theorem of Jacques Dixmier and Paul Malliavin, every function f in $C^\infty_c(G)$ is a finite sum of convolutions of similar functions, the inversion formula holds for such f. It can be extended to much wider classes of functions satisfying mild differentiability conditions.[21]

## Explicit formulas

### Conventions and Lie algebra bases

The metric of choice is given by η = diag(−1, 1, 1, 1), and the physics convention for Lie algebras and the exponential mapping is used in this article. These choices are arbitrary, but once they are made, fixed. The rationale is to allow the use of a single reference[4] for several related formulas. One possible choice of basis for the Lie algebra (which is not fixed by the reference) is, in the 4-vector representation, given by

\begin{align} J_1 &= J^{23} = -J^{32} = i\biggl(\begin{smallmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&-1\\ 0&0&1&0\\ \end{smallmatrix}\biggr),\\ J_2 &= J^{31} = -J^{13} = i\biggl(\begin{smallmatrix} 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&-1&0&0\\ \end{smallmatrix}\biggr),\\ J_3 &= J^{12} = -J^{21} = i\biggl(\begin{smallmatrix} 0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0\\ \end{smallmatrix}\biggr),\\ K_1 &= J^{01} = J^{10} = i\biggl(\begin{smallmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{smallmatrix}\biggr),\\ K_2 &= J^{02} = J^{20} = i\biggl(\begin{smallmatrix} 0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ \end{smallmatrix}\biggr),\\ K_3 &= J^{03} = J^{30} = i\biggl(\begin{smallmatrix} 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ \end{smallmatrix}\biggr). \end{align}

The commutation relations of the Lie algebra so(3;1) are

$[J^{\mu\nu},J^{\rho\sigma}] = i(\eta^{\sigma\mu}J^{\rho\nu} + \eta^{\nu\sigma}J^{\mu\rho} - \eta^{\rho\mu}J^{\sigma\nu} -\eta^{\nu\rho}J^{\mu\sigma}).$[4]

In three-dimensional notation, these are

$[J_i,J_j] = i\epsilon_{ijk}J_k, \quad [J_i,K_j] = i\epsilon_{ijk}K_k, \quad [K_i,K_j] = -i\epsilon_{ijk}J_k.$[4]

The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol J above and in the sequel should be observed.

Let π(m,n): so(3;1) → gl(V), where V is a vector space, denote the irreducible representations of so(3;1) according to the (m,n) classification. In components, with -ma,a′m, -nb,b′n, the representations are given by

\begin{align} (\pi_{m,n}(J_i))_{a'b' , ab} &= \delta_{b'b}(J_i^{(m)})_{a'a} + \delta_{a'a}(J_i^{(n)})_{b'b},\\ (\pi_{m,n}(K_i))_{a'b' , ab} &= i(\delta_{a'a}(J_i^{(n)})_{b'b} - \delta_{b'b}(J_i^{(m)})_{a'a}), \end{align}[4]

where δ is the Kronecker delta and the Ji(n) are the (2n + 1)-dimensional irreducible representations of so(3), also termed spin matrices or angular momentum matrices. These are explicitly given by

\begin{align} (J_3^{(j)})_{a'a} &= a\delta_{a'a},\\ (J_1^{(j)} \pm iJ_2^{(j)})_{a'a} &= \sqrt{(j \mp a)(j \pm a + 1)}\delta_{a',a \pm 1}. \end{align}[4]

### Weyl spinors and bispinors

By taking, in turn, m = 1/2, n = 0 and m = 0, n = 1/2 and by setting

$J_i^{(\frac{1}{2})} = \frac{1}{2}\sigma_i$

in the general expression (G1), and by using the trivial relations 11 = 1 and J(0) = 0, one obtains

\begin{align} \pi_{(\frac{1}{2},0)}(J_i) & = \frac{1}{2}(\sigma_i\otimes 1_{(1)} + 1_{(2)}\otimes J^{(0)}_i) = \frac{1}{2}\sigma_i\quad\pi_{(\frac{1}{2},0)}(K_i) = \frac{i}{2}(1_{(2)}\otimes J^{(0)}_i - \sigma_i \otimes 1_{(1)}) = -\frac{i}{2}\sigma_i,\\ \pi_{(0,\frac{1}{2})}(J_i) & = \frac{1}{2}(J^{(0)}_i\otimes 1_{(2)} + 1_{(1)}\otimes \sigma_i) = \frac{1}{2}\sigma_i\quad\pi_{(0,\frac{1}{2})}(K_i) = \frac{i}{2}(1_{(1)}\otimes\sigma_i - J^{(0)}_i \otimes 1_{(2)}) = +\frac{i}{2}\sigma_i. \end{align}

(W1)

These are the left-handed and right-handed Weyl spinor representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) VL and VR, whose elements ΨL and ΨR are called left- and right-handed Weyl spinors respectively. Given (π(1/2,0),VL) and (π(0,1/2),VR) one may form their direct sum as representations,

\begin{align} \pi_{(\frac{1}{2},0) \oplus (0,\frac{1}{2})}(J_i) &= \frac{1}{2}\biggl(\begin{matrix} \sigma_i&0\\ 0&\sigma_i\\ \end{matrix}\biggr),\\ \pi_{(\frac{1}{2},0) \oplus (0,\frac{1}{2})}(K_i) &= \frac{i}{2} \biggl(\begin{matrix} \sigma_i&0\\ 0&-\sigma_i\\ \end{matrix}\biggr)\\ \end{align}.[22]

(D1)

This is, up to a similarity transformation, the (1/2,0)⊕(0,1/2) Dirac spinor representation of so(3,1). It acts on the 4-component elements L, ΨR) of (VLVR), called bispinors, by matrix multiplication. The representation may be obtained in a more general and basis independent way using Clifford algebras. These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of so(3,1). Expressions for the group representations are obtained by exponentiation.

## Notes

1. ^ Weinberg 2003, Chapter 5
2. ^ a b Hall 2003, Chapter 6
3. ^ Hall 2003, Chapter 4
4. Weinberg 2002, Chapter 5
5. Hall 2003, Chapter 3
6. ^ Weinberg 2002, Appendix B, Chapter 2
7. ^ Weinberg 2002, Section 2.7, Chapter 2
8. ^ Wigner 1937
9. ^ Wigner 1937, p.27
10. ^ Weinberg 2003, Section 2.7
11. ^ The "traceless" property can be expressed as Sαβgαβ = 0, or Sαα = 0, or Sαβgαβ = 0 depending on the presentation of the field: covariant, mixed, and contravariant respectively.
12. ^ Greiner, W (2000), Relativistic Quantum Mechanics, ISBN 3-540-67457-8
13. ^ Weinberg 2002 Chapter 2
14. ^ Weinberg 2002}, Chapter 3
15. ^ Knapp 2001, Chapter II
16. ^ a b Harish-Chandra 1947
17. ^ a b Taylor 1986
18. ^ Gelfand & Naimark 1947
19. ^ Takahashi 1963, p. 343
20. ^ Note that for a Hilbert space H, HS(H) may be identified canonically with the Hilbert space tensor product of H and its conjugate space.
21. ^ Knapp 2001
22. ^ Weinberg 2002, Equations (5.4.19) and (5.4.20)

## References

• Bargmann, V. (1947), "Irreducible unitary representations of the Lorenz group", Ann. Of Math. 48 (3): 568–640, doi:10.2307/1969129, JSTOR 1969129 (the representation theory of SO(2,1) and SL(2, R); the second part on SO(3,1) and SL(2, C), described in the introduction, was never published).
• Dixmier, J.; Malliavin, P. (1978), "Factorisations de fonctions et de vecteurs indéfiniment différentiables", Bull. Sc. Math. 102: 305–330
• Gelfand, I. M.; Naimark, M. A. (1947), "Unitary representations of the Lorentz group", Izvestiya Akad. Nauk SSSR. Ser. Mat. 11: 411–504
• Gelfand, I. M.; M. I. Graev (1953), "[On a general method of decomposition of the regular representation of a Lie group into irreducible representations]", Doklady Akademii Nauk SSSR 92: 221–224
• Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proceedings of the Royal Society A 189 (1018): 372–401, Bibcode:1947RSPSA.189..372H, doi:10.1098/rspa.1947.0047
• Harish-Chandra (1951), "Plancherel formula for complex semi-simple Lie groups", Proc. Nat. Acad. Sci. U. S. A. 37 (12): 813–818, Bibcode:1951PNAS...37..813H, doi:10.1073/pnas.37.12.813
• Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Springer, ISBN 0-387-40122-9
• Helgason, S. (1968), Lie groups and symmetric spaces, Battelle Rencontres, Benjamin, pp. 1–71 (a general introduction for physicists)
• Helgason, S. (2000), Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions (corrected reprint of the 1984 original), Mathematical Surveys and Monographs 83, American Mathematical Society, ISBN 0-8218-2673-5
• Jorgenson, J.; Lang, S. (2008), The heat kernel and theta inversion on SL(2,C), Springer Monographs in Mathematics, Springer, ISBN 978-0-387-38031-5
• Knapp, A. (2001), Representation theory of semisimple groups. An overview based on examples., Princeton Landmarks in Mathematics, Princeton University Press, ISBN 0-691-09089-0 (elementary treatment for SL(2,C))
• Naimark, M.A. (1964), Linear representations of the Lorentz group (translated from the Russian original by Ann Swinfen and O. J. Marstrand), Macmillan
• Paërl, E.R. (1969) Representations of the Lorentz group and projective geometry, Mathematical Centre Tract #25, Amsterdam.
• Rühl, W. (1970), The Lorentz group and harmonic analysis, Benjamin (a detailed account for physicists)
• Takahashi, R. (1963), "Sur les représentations unitaires des groupes de Lorentz généralisés", Bull. Soc. Math. France 91: 289–433
• Taylor, M.E. (1986), Noncommutative harmonic analyis, Mathematical Surveys and Monographs 22, American Mathematical Society, ISBN 0-8218-1523-7, Chapter 9, SL(2, C) and more general Lorentz groups
• Weinberg, S (2002), The Quantum Theory of Fields, vol I, ISBN 0-521-55001-7
• Wigner, E. P. (1939), "On unitary representations of the inhomogeneous Lorentz group", Annals of Mathematics 40 (1): 149–204, doi:10.2307/1968551, MR 1503456.