# Eikonal Blog

## 2010.01.30

### Householder transformation

Filed under: mathematics — Tags: , — sandokan65 @ 18:08

Definition: For an Euclidean vector $v$, the Householder matrix is defined as:

$P :\equiv {\bf 1} - 2 \frac{\underline{v}\otimes\underline{v}}{||\underline{v}||^2}$

It expresses the mirroring transformation:

• $P \underline{v} = -\underline{v}$,
• $P \underline{u} = +\underline{u}$; ($\forall \underline{u} \perp \underline{v}$).
• $P \underline{x} = - \underline{v}\frac{\underline{v}\cdot\underline{x}}{||\underline{v}||^2} + \left(1-\frac{\underline{v}\otimes\underline{v}}{||\underline{v}||^2}\right)\underline{x} = - \underline{x}_{||v} + \underline{x}_{\perp v}$ for an arbitrary vector $\underline{x}$.

Properties: $P^T=P$, $P^2 = {\bf 1}$.

### Historical maps

Filed under: history — Tags: — sandokan65 @ 17:27

## Web Applications and Web Services assessment

Lists of tools:

Application Assessment:

Web services testing:

Fuzzing:

Using browsers as the webapp testing tools:

Misc info:

• Book: “The Web Application Hacker’s Handbook: Discovering and Exploiting Security Flaws” by Dafydd Stuttard and Marcus Pinto – http://www.amazon.com/Web-Application-Hackers-Handbook-Discovering/dp/0470170778/
• The Web Application Security Consortium (WASC): http://www.webappsec.org/ | http://projects.webappsec.org/
• OSSTMM – “Open Source Security Testing Methodology Manual” by Pete Herzog – http://www.isecom.org/osstmm/
• OWASP: http://www.owasp.org/index.php/Main_Page
• OWASP Testing Guide:
• Top 10 Web Vulnerability Scanners – http://sectools.org/web-scanners.html:

• #1 – Nikto : A more comprehensive web scanner:
Nikto is an open source (GPL) web server scanner which performs comprehensive tests against web servers for multiple items, including over 3200 potentially dangerous files/CGIs, versions on over 625 servers, and version specific problems on over 230 servers. Scan items and plugins are frequently updated and can be automatically updated (if desired). It uses Whisker/libwhisker for much of its underlying functionality. It is a great tool, but the value is limited by its infrequent updates. The newest and most critical vulnerabilities are often not detected.
• #2 – Paros proxy : A web application vulnerability assessment proxy
A Java based web proxy for assessing web application vulnerability. It supports editing/viewing HTTP/HTTPS messages on-the-fly to change items such as cookies and form fields. It includes a web traffic recorder, web spider, hash calculator, and a scanner for testing common web application attacks such as SQL injection and cross-site scripting.
• #3 – WebScarab : A framework for analyzing applications that communicate using the HTTP and HTTPS protocols
In its simplest form, WebScarab records the conversations (requests and responses) that it observes, and allows the operator to review them in various ways. WebScarab is designed to be a tool for anyone who needs to expose the workings of an HTTP(S) based application, whether to allow the developer to debug otherwise difficult problems, or to allow a security specialist to identify vulnerabilities in the way that the application has been designed or implemented.
• #4 – WebInspect : A Powerful Web Application Scanner
SPI Dynamics’ WebInspect application security assessment tool helps identify known and unknown vulnerabilities within the Web application layer. WebInspect can also help check that a Web server is configured properly, and attempts common web attacks such as parameter injection, cross-site scripting, directory traversal, and more.
• #5 – Whisker/libwhisker : Rain.Forest.Puppy’s CGI vulnerability scanner and library
Libwhisker is a Perl module geared geared towards HTTP testing. It provides functions for testing HTTP servers for many known security holes, particularly the presence of dangerous CGIs. Whisker is a scanner that used libwhisker but is now deprecated in favor of Nikto which also uses libwhisker.
• #6 – Burpsuite : An integrated platform for attacking web applications
Burp suite allows an attacker to combine manual and automated techniques to enumerate, analyze, attack and exploit web applications. The various burp tools work together effectively to share information and allow findings identified within one tool to form the basis of an attack using another.
• #7 – Wikto : Web Server Assessment Tool
Wikto is a tool that checks for flaws in webservers. It provides much the same functionality as Nikto but adds various interesting pieces of functionality, such as a Back-End miner and close Google integration. Wikto is written for the MS .NET environment and registration is required to download the binary and/or source code.
• #8 – Acunetix WVS : Commercial Web Vulnerability Scanner
Acunetix WVS automatically checks web applications for vulnerabilities such as SQL Injections, cross site scripting, arbitrary file creation/deletion, weak password strength on authentication pages. AcuSensor technology detects vulnerabilities which typical black box scanners miss. Acunetix WVS boasts a comfortable GUI, an ability to create professional security audit and compliance reports, and tools for advanced manual webapp testing.
• #9 – Rational AppScan : Commercial Web Vulnerability Scanner
AppScan provides security testing throughout the application development lifecycle, easing unit testing and security assurance early in the development phase. Appscan scans for many common vulnerabilities, such as cross site scripting, HTTP response splitting, parameter tampering, hidden field manipulation, backdoors/debug options, buffer overflows and more. Appscan was merged into IBM’s Rational division after IBM purchased it’s original developer (Watchfire) in 2007.
• #10 – N-Stealth : Web server scanner
N-Stealth is a commercial web server security scanner. It is generally updated more frequently than free web scanners such as Whisker/libwhisker and Nikto, but do take their web site with a grain of salt. The claims of “30,000 vulnerabilities and exploits” and “Dozens of vulnerability checks are added every day” are highly questionable. Also note that essentially all general VA tools such as Nessus, ISS Internet Scanner, Retina, SAINT, and Sara include web scanning components. They may not all be as up-to-date or flexible though. N-Stealth is Windows only and no source code is provided.
• Jeremiah Grossman blog on web applicaiton’s security: http://jeremiahgrossman.blogspot.com | RSS – http://feeds.feedburner.com/JeremiahGrossman:
• “Web Application Testing” by Russ Klanke (at Aggressive Virus Defense blog) – http://aggressivevirusdefense.wordpress.com/2009/08/02/web-application-testing/ – has numerous good links and tips.
• Browser Security Handbook – http://code.google.com/p/browsersec/wiki/Main
Browser Security Handbook is meant to provide web application developers, browser engineers, and information security researchers with a one-stop reference to key security properties of contemporary web browsers. Insufficient understanding of these often poorly-documented characteristics is a major contributing factor to the prevalence of several classes of security vulnerabilities.

The document currently covers several hundred security-relevant characteristics of Microsoft Internet Explorer (versions 6, 7, and 8), Mozilla Firefox (versions 2 and 3), Apple Safari, Opera, Google Chrome, and Android embedded browser.

Open-source test cases provided alongside with this document permit any other browser implementations to be quickly evaluated in a similar manner.

• “Web Security Testing Cookbook” by Paco Hope – http://websecuritytesting.com/

## IPSec security

• IPSecScan (by NTSecurity.nu) – http://ntsecurity.nu/toolbox/ipsecscan/IPSecScan is a tool that can scan either a single IP address or a range of IP addresses looking for systems that are IPSec enabled.

Tools:

Articles:

## Network testers

See also: list of (other) security tools (in this blog) – https://eikonal.wordpress.com/2010/07/28/security-tools/.

## 2010.01.28

### Cisco “password 7” decryption – Perl code

Filed under: infosec — Tags: , , , — sandokan65 @ 17:19

Source: somewhere from the web.

 #!/usr/bin/perl -w # $Id: ios7decrypt.pl,v 1.1 1998/01/11 21:31:12 mesrik Exp$ # # Credits for orginal code and description hobbit@avian.org, # SPHiXe, .mudge et al. and for John Bashinski # for Cisco IOS password encryption facts. # # Use for any malice or illegal purposes strictly prohibited! # @xlat = ( 0x64, 0x73, 0x66, 0x64, 0x3b, 0x6b, 0x66, 0x6f, 0x41, 0x2c, 0x2e, 0x69, 0x79, 0x65, 0x77, 0x72, 0x6b, 0x6c, 0x64, 0x4a, 0x4b, 0x44, 0x48, 0x53 , 0x55, 0x42 ); while () { if (/(password|md5)\s+7\s+([\da-f]+)/io) { if (!(length($2) & 1)) {$ep = $2;$dp = ""; ($s,$e) = ($2 =~ /^(..)(.+)/o); for ($i = 0; $i < length($e); $i+=2) {$dp .= sprintf "%c",hex(substr($e,$i,2))^$xlat[$s++]; } s/7\s+$ep/$dp/; } } print; } 

### BCS (Baker–Campbell–Hausdorff) formula

Filed under: Uncategorized — Tags: , — sandokan65 @ 17:12

In this posting the matrix $C(t)$ is defined by $e^{C(t)}:\equiv e^{t(A+B)}$ where $A$ and $B$ are constant matrices.

The Baker–Campbell–Hausdorff theorem claims that:

$C(t) = B + \int_0^1 dt g(e^{t a} e^b ) A$,

where $g(z):\equiv \frac{\ln(z)}{z-1} = \sum_{m=0}^\infty \frac{(1-z)^m}{m+1}$, and the lower-case letters $a$ and $b$ represent the adjoint actions of the corresponding matrices $A$ and $B$ (e.g. $a X:\equiv ad(A) X :\equiv [A,X]$).

One is frequently seeing the following series expression:

$C(t) = t (A+B) + \frac{t^2}2 [A,B] + \frac{t^3}{12} ([[A.B],B]-[[A,B],A]) + \cdots = t (A+B) + \frac{t^2}2 a B + \frac{t^3}{12} (b^2 A + a^2 B) + \cdots$

Source: T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).

### The Trotter Formula

Filed under: mathematics — Tags: — sandokan65 @ 17:02

$e^{t(A+B)} = \lim_{m\rightarrow \infty} \left(e^{\frac{t}{m}A}e^{\frac{t}{m}B}\right)^m$.

Source: T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).

### The Zassenhaus formula

Filed under: mathematics — Tags: , — sandokan65 @ 16:49

$e^{A+B} = e^{A}e^{B}e^{C_2}e^{C_3}\cdots$
where

• $C_2 = -\frac12 [A,B] = -\frac12 a B$,
• $C_3 = -\frac13 [[A,B],B] - \frac16 [[A,B],A] = -\frac13 b^2 A + \frac16 a^2 B$, etc.

Here, as elsewhere on this site, I am using notation $a:\equiv ad(A)$ etc.

Sources:

• T1264 = W. Magnus “On the exponential solution of differential equations for a linear operator”; Communications of Pure and Applied mathematcis, Vol VII, 6490673 (1954).
• T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).

### Bookmarklets

Filed under: web tools — Tags: , , , , , — sandokan65 @ 15:24

The most useful bookmarklets:

• Wayback archive bookmarklet (http://www.archive.org/web/web.php):
 javascript:location.href='http://web.archive.org/web/*/'+document.location.href; 
• TinyURL bookmarklet (Source: http://tinyurl.com/#toolbar):
 javascript:void(location.href='http://tinyurl.com/create.php?url='+location.href) 
• All-In-One Video bookmarklet: http://1024k.de/bookmarklets/video-bookmarklets.html
• WordPress.com PressThis bookmarklet:
 javascript: var d=document, w=window, e=w.getSelection, k=d.getSelection, x=d.selection, var s=(e?e():(k)?k():(x?x.createRange().text:0)), var f='https://eikonal.wordpress.com/wp-admin/press-this.php', var l=d.location, e=encodeURIComponent, u=f+'?u='+e(l.href)+'&t='+e(d.title)+'&s='+e(s)+'&v=4'; a=function(){ if (!w.open(u,'t','toolbar=0,resizable=1,scrollbars=1,status=1,width=720,height=570')) l.href=u; }; if (/Firefox/.test(navigator.userAgent)) setTimeout(a,0); else a(); void(0) 

## Library

• Encipher.It – AES Text encryptor for Google Mail or anything else – https://encipher.it/:
• Code:
 javascript:( function(){ document.body.appendChild(document.createElement('script')).src='https://encipher.it/javascripts/inject.v2.js'; } )(); 
• GmailThis! bookmarklet (2010.09.10):
 javascript: popw=''; Q=''; x=document; y=window; if(x.selection)%20{Q=x.selection.createRange().text;} %20else%20if%20(y.getSelection)%20{Q=y.getSelection();} %20else%20if%20(x.getSelection)%20{Q=x.getSelection();} popw%20=%20y.open('https://mail.google.com/mail/?view=cm&fs=1&tf=1&to=& su='%20+%20escape(document.title)%20+%20'&body='%20+%20escape(Q)%20+%20escape('\n')%20+%20escape(location.href)%20+%20'& zx=RANDOMCRAP& shva=1& disablechatbrowsercheck=1& ui=1','gmailForm','scrollbars=yes,width=680,height=510,top=175,left=75,status=no,resizable=yes'); if%20(!document.all)%20T%20=%20setTimeout('popw.focus()',50);void(0); 
• GmailThis! bookmarklet (old):
 javascript: popw=''; Q=''; x=document; y=window; if(x.selection)%20{Q=x.selection.createRange().text;} %20else%20if%20(y.getSelection)%20{Q=y.getSelection();} %20else%20if%20(x.getSelection)%20{Q=x.getSelection();} popw%20=%20y.open('https://mail.google.com/mail?view=cm&tf=0&to=& su='%20+%20escape(document.title)%20+%20'&body='%20+%20escape(Q)%20+%20escape('\n')%20+%20escape(location.href),'gmailForm', 'scrollbars=yes,width=680,height=510,top=175,left=75,status=no,resizable=yes'); if%20(!document.all)%20T%20=%20setTimeout('popw.focus()',50);void(0); 

### Astronomic tables and calculators

Filed under: Uncategorized — Tags: , , , — sandokan65 @ 15:11

Related here: Astronomy & Astrophysics – https://eikonal.wordpress.com/2011/05/09/astronomy-astrophysics/ | Astronomic tables and calculators – https://eikonal.wordpress.com/2010/01/28/astronomic-tables-and-calculators/ | Solar planetary system – https://eikonal.wordpress.com/2011/02/17/solar-planetary-system/ | Inside black holes – https://eikonal.wordpress.com/2011/04/12/inside-black-holes/ | Physics Sites – https://eikonal.wordpress.com/2010/02/12/physics-sites/

## 2010.01.27

### Horizontal gene transfer

Filed under: evolution — Tags: — sandokan65 @ 13:21

There is an interesting article in the New Scientist magazine (“Horizontal and vertical: The evolution of evolution” by Mark Buchanan; 2010.01.26, Issue 2744; http://www.newscientist.com/article/mg20527441.500-horizontal-and-vertical-the-evolution-of-evolution.html) describing new research that claims that the dominant form of evolution on this planet was for long time non-Darwinian. According to the described research by microbiologist Carl Woese and physicist Nigel Goldenfeld, there were three phases of evolution (so far):

• 1) the initial phase when there existed many alternate and compating genetic codes.
• 2) in the second phase the genetic material was exchanged between organisms horizontally, i.e. not via inheritance.
• 3) Only in the most recent, third phase, the major (for transfer of genetic mechanism) mechanism is the Darwinian inheritance.

## 2010.01.26

### Magnus on matrix exponentials

Definition: The Magnus’ bracket is defined as $\{y,x^n\}_M :\equiv (- ad(x))^n y$.

Properties:

• $e^{-ad(x)} y = \sum_{n=0}^\infty \{y, x^n\}_M$
• for a power series $P(x)=\sum_{n=0}^\infty p_l x^l$ one has $\{y, P(x)\}_M = \sum_{n=0}^\infty p_l \{y, x^n\}_M$
• Hausdorf:
• $e^{-x} (y D_x) e^x = \left\{y,\frac{e^x -1}{x}\right\}_M$,
• $(y D_x e^{-x}) e^x = \left\{y,\frac{1-e^{-x}}{x}\right\}_M$,

where the Hausdorff polarization operator $(y D_x)$ is defined as following:

$(y D_x) x^n :\equiv \sum_{k=0}^{n-1} x^k y x^{n-k-1}$.

• Lemma: For two power series $P(x)$ and $Q(x)$ s/t $P(x) Q(x) = 1$ one has the following equivalence:
$\{y,P(x)\}_M=u \Leftrightarrow y = \{u, Q(x)\}_M$.
• $\{\{y,x^n\}_M,x^m\}_M = \{y,x^{n+m}\}_M$.

For $e^{z} :\equiv e^{x}e^{y}$ one has:

• $z=\ln(1+u) = u - \frac12 u^2 + \frac13 u^3 + \cdots$

where
$u:\equiv e^x e^y - 1 = x + y + \frac12 x^2 + xy + \frac12 y^2 + \cdots$
• $z = x + y + \frac12 [x,y] + \frac1{12} \{x,y^2\}_M + \frac1{12} \{y,x^2\}_M -\frac1{24} [\{x,y^2\}_M,x] - \frac1{720} \{x,y^4\}_M - \frac1{720} \{y,x^4\}_M + \frac1{180} [[\Delta_1,x],y] - \frac1{180}\{\Delta_1,y^2\}_M - \frac1{120} [\Delta_1,\Delta] - \frac1{360} [\Delta_2,\Delta] + \cdots$

where

$\Delta=[x,y]$,
$\Delta_1=[\Delta,x]$,
$\Delta_2=[\Delta,y]$.

## The Magnus’ formula for ordered exponentials

And this is really beautiful thing: for the ordered exponential $U(t|A):\equiv \left(e^{\int_0^t dt_1 A(t_1)}\right)_+$ one can observe its logarithm (the Magnus function of A) $\Omega(t|A) :\equiv \ln U(t|A)$ (i.e. $e^{\Omega}=U$) and get following nonlinear ODE for it:

$\frac{d\Omega(t)}{dt} = \{A, \frac{\Omega}{1-e^{-\Omega}}\}_M = \sum_{n=0}^\infty \beta_n \{A,\Omega^n\}_M = A + \frac12 [A,\Omega] + \frac1{12}\{A,\Omega^2\}_M + \cdots$

where $\beta_{2n1+1}=0$ ($\forall n\in{\Bbb N}_0$), $\beta_{2n} = (-)^{n-1} \frac{B_{2n}}{(2n)!}$ and $B_n$‘s are ordinary Bernoulli’s numbers.

Source: T1264 = W. Magnus “On the exponential solution of differential equations for a linear operator”; Communications of Pure and Applied mathematics, Vol VII, 6490673 (1954). Note: This paper is probably mother of the whole this area of matrix exponentials.

Related here: Magnus’ Bracket Operator – https://eikonal.wordpress.com/2010/02/15/magnus-bracket-operator/

### A Theorem of Ellis and Pinsky

Filed under: mathematics — Tags: — sandokan65 @ 13:12

Theorem (Ellis & Pinsky): Let $\latex A$, $B$ be real symmetric $n\times n$ matrices, and assume that $B$ is negative semi-definite. Then as $\epsilon \rightarrow +0$:

• $e^{t(A+i\frac1{\epsilon}B)} e^{-i\frac{t}{\epsilon}B} = e^{t A_0} + {\cal O}(\epsilon)$,

where $A_0 :\equiv \sum_{\lambda\in\sigma(B)} P_\lambda A P_\lambda$, here $P_\lambda$ being the orthogonal projector on the eigenspace of $B$ associated with the eigenvalue $\lambda$.

Source: T1270 = S.L.Campbell “On the limit of a product of matrix exponentials”; Linear and Multilinear Algebra; 1978, Vol 6, p.p. 55-59.

### Expansions of the exponentials of the sums of matrices

Richmond’s formula #1: $e^{t(A+B)} = e^{tA}e^{tB} + \sum_{r=0}^\infty E_r t^r$, where:

• $E_{r+1} = \frac1{r+1}\left((A+B)E_r+[B,F_r]\right) \$ with $E_0=E_1=0$,
• $F_{r+1} = \frac1{r+1}(A F_r + F_r B) \$ with $F_0=1$.

Richmond’s formula #2: $e^{t(A+B)} = \frac12(e^{tA}e^{tB}+e^{tB}e^{tA}) + \sum_{r=0}^\infty E'_r t^r$, where:

• $E'_{f+1} = \frac1{r+1}((A+B)E'_r+\frac12[B,F_r]+\frac12[A,F^{*}_r]) \$ with $E'_0=E'_1=E'_2=0$, and where $F_r$‘s are the same as in the Richmond’s formula #1.
• Following bounds are valid: $||E'_r|| < \frac1{(r-1)!} (||A||+||B||)^r$.

Note that the exponential generating function ${\cal F}(z):\equiv \sum_{r=0}^\infty \frac{z^r}{r!} F_r$ satisfies the second order ODE: $\partial_z (z\partial_z {\cal F}(z)) = A{\cal F}(z)+{\cal F}(z)B$.

—-
Source: T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).

### Exponential splittings

Filed under: mathematics — Tags: — sandokan65 @ 12:06

First order splitting:

• $e^{t\sum_{i=1}^{n} A_i } = \prod_{i=1}^n e^{t A_i} + {\cal O}(t^2).$

Second order splittings:

• The Strang’s Splitting:
$e^{t\sum_{i=1}^{n} A_i } = e^{\frac{t}2 A_1} e^{\frac{t}2 A_2} \cdots e^{\frac{t}2 A_{n-1}} e^{t A_n} e^{\frac{t}2 A_{n-1}} \cdots e^{\frac{t}2 A_2} e^{\frac{t}2 A_1} + {\cal O}(t^3).$
• The parallel Splitting:
$e^{t\sum_{i=1}^{n} A_i } = \frac12(e^{t A_1}\cdots e^{t A_n} + e^{t A_n}\cdots e^{t A_1}) + {\cal O}(t^3).$

Source: T1268 = Qin Sheng “Global Error Estimates for Exponential Splitting”; IMA Journal of Numerical Analysis (1993) 14, 27-56.

### Error estimates for exponential splittings

Filed under: mathematics — Tags: , — sandokan65 @ 11:58

Def: $\varphi(t):\equiv \frac{||e^{t(A+E}-e^{tA}||}{||e^{tA}||}$.

Estimate:
$\varphi(t) \le t ||E|| e^{t(\mu(A)-\eta(A)+||E||)}$
where:

• $\mu(A):\equiv \max\{\Re(\lambda)|\lambda \in \Sigma_{\frac12(A+A^\dagger)}\}$ (i.e. the largest eigenvalue of the matrix $\frac12(A+A^\dagger)$, i.e. “the logarithmic norm” of $A$),
• $\eta(A):\equiv \max\{\Re(\lambda)|\lambda \in \Sigma_A\}$,
• $\mu(A) \ge \eta(A)$.

Source: T1268 = Qin Sheng “Global Error Estimates for Exponential Splitting”; IMA Journal of Numerical Analysis (1993) 14, 27-56.

## 2010.01.24

### Heat conduction equation

Filed under: physics — Tags: — sandokan65 @ 23:38

The problem is defined by following three elements:

• ODE: $\partial_t u = k \partial_x^2 u$,
• Domain: $D=\{(x,t)|x\in{\Bbb R}, t\in{\Bbb R}^{+}\}$,
• Initial condition: $u(x,0)=f(x)$.

Using the Fourier transform one gets to the solution:
$u(x,t) = \frac1{(2\pi)^\mu} \int_{\Bbb R} e^{-ipx-kp^2t}{\bf F}[f](p).$

For example, for $f(x)=A e^{-Bx^2}$ one gets $u(x,t) = \frac{A}{\sqrt{1+4Bkt}} e^{-\frac{Bx^2}{1+4Bkt}}$.

## 2010.01.23

Filed under: tools — Tags: , , , , , , , — sandokan65 @ 00:05

File sharing sites:

Books and other textual material:

## 2010.01.22

### Chuck Norris superman meme

Filed under: martial arts, memetics, mind & brain — Tags: , , — sandokan65 @ 16:50

Various cultural heroes have assigned to them list of “facts” which are extreme hyperboles of the virtues exposed by the specific hero. The extremity of these “facts” makes them infectious and memorizeable, i.e. good material to be memes.

The most familiar example in the web culture is Chuck Norris, martial artist and actor (in quite bad martial arts movies and series). Use Google to find various collections of “Chuck Norris facts”, for example:

Related here: Martial arts sites – https://eikonal.wordpress.com/2010/09/21/martial-arts-sites/ | Martial Arts articles – https://eikonal.wordpress.com/2010/09/08/martial-arts-articles/ | Martial Arts magazines and other sources – https://eikonal.wordpress.com/2010/02/05/martial-arts-magazines-and-other-sources/.

### BCS formula on computer

Filed under: mathematics — Tags: , , — sandokan65 @ 16:06

Def: $e^C :\equiv e^A e^B$

$C = A + \int_0^1 dt \psi(e^{a}e^{t b}) B$,
where $a:\equiv ad(A)$, $b:\equiv ad(B)$ and
$\psi(z):\equiv \frac{z \ln(z)}{z-1} = 1 + \frac{w}{1\cdot 2} - \frac{w^2}{2\cdot 3} + \frac{w^3}{3\cdot 4} - \cdots$ for $w:\equiv z-1$.

Richtmyer and Greenspan obtained the 512 initial terms of that expansion, but here I present only several of them:

$C = A + B + \frac12 a B + \frac1{12} a^2 B - \frac1{12} b a B + \frac1{12,096} a^4 b^2 a B - \frac1{6,048} a^3 b a^3 B - \frac1{3,780} a^3 b a b a B + ...$

Not all terms are independent, due to Jacobi identities.

Sources:

• T1267: R.D. Richtmyer and S. Greenspan: “Expansion of the Campbell-Baker-Hausdorff formula by Computer”; Communications on Pure and Applied Mathematics, Vol XVIII 107-108 (1965).

### Guenin on commutators

Filed under: Uncategorized — Tags: , — sandokan65 @ 15:46

$A B^n = \sum_{j=0}^n \binom{n}{j} (-)^j B^{n-j} \ ad(B)^j A.$

$B^n A = \sum_{j=0}^n \binom{n}{j} (-)^j \ (ad(B)^j A) B^{n-j}.$

$\partial_s f(F(s)) = \sum_{j=0}^\infty \frac{(-)^j}{(j+1)!} f^{(j+1)}(F(s)) \ ad(F(s)^j \partial_s F(s) = \sum_{j=0}^\infty \frac{1}{(j+1)!} \ (ad(F(s))^j \partial_s F(s) ) f^{(j+1)}(F(s)).$

$[H, f(A)] = \sum_{j=0}^\infty \frac{(-)^j}{j!} f^{(j)}(A) \ ad(A)^j H = - \sum_{j=0}^\infty \frac1{j!}\ (ad(A)^j H) f^{(j)}(A).$
—-
Sources:

• T1266: M. Guenin: “On the Derivation and Commutation of Operator Functionals”; Helv. Phys. Act. 41, 75-76, 1968.
Older Posts »