# Eikonal Blog

## 2010.10.29

### Assembly language

Filed under: assembly — Tags: , — sandokan65 @ 15:53

## Tools

• Compendium – http://compendium.open.ac.uk/institute/about.htm [FREE, requires registration]
• perlIBIS – http://eekim.com/software/perlIBIS/ – a suite of Perl modules for processing IBIS dialog maps.
“… Issue-Based Information System (IBIS) is a methodology for discussing and exploring “wicked problems,” problems that are not well understood and that have no straightforward answers. Invented by Horst Rittel and colleagues in the 1970s, IBIS provides a simple grammar for mapping conversations. The grammar consists of questions (issues), possible solutions (ideas), and arguments (pros and cons) for and against an idea.

In the 1980s, Jeff Conklin and others developed software for graphically mapping and representing IBIS conversations. This software eventually evolved into a Windows program called QuestMap….”

## Definitions and other blurbs

• Collective Memory = Also known as group memory. (Source: [1])
• Deep Structure – http://www.cognexus.org/deep_structure.htm
• Dialog mapping
• Knowledge is information that has been internalized. (Source: [2])
• Information is a knowledge artifact (Source: [2])
• IBIS = Issue Based Information System(s)
• Issue mapping: … the process of crafting an issue map, a way of making critical thinking visible. An issue map is a graphical network that integrates many problems, solutions, and points of view and shows the deep structure of an issue… (Source: [4])
• Issue Mapping and Dialogue Mapping: … Issue Mapping is Dialogue Mapping minus group facilitation. (Source: [4])
• Knowledge vs. Information: People often confuse “knowledge” with “information,” but there is an importance distinction between the two. Knowledge is information that has been internalized. If you have a physics textbook on your shelf, then you have information (or a knowledge artifact). If you are capable of doing the problems in the textbook, then you have knowledge. (Source: [2])
• Dynamic Knowledge Repository [DKR] = (Source: [2])
• Shared Understanding = An important pattern where the group has achieved a unity … of goal/mission/vision such that the question “what are we trying to do” doesn’t really come up. (Source: [3])
• Wicked problem – … describe a problem that is difficult or impossible to solve because of incomplete, contradictory, and changing requirements that are often difficult to recognize. Moreover, because of complex interdependencies, the effort to solve one aspect of a wicked problem may reveal or create other problems…. (Source: http://en.wikipedia.org/wiki/Wicked_problem)

## Sources

Related here: Knowledge management – https://eikonal.wordpress.com/2010/05/15/knowledge-management/

## Various

Elsewhere in this blog: Brain games – https://eikonal.wordpress.com/2013/01/03/brain-games/

### Scarcity and Abundance

Filed under: economy — Tags: , , , , — sandokan65 @ 11:44

## 2010.10.20

### Futurism

Filed under: future — Tags: , , , — sandokan65 @ 21:28

### “Quantum” calculus

Currently all this material is retyped from the reference [1].

Definitions:

• The q-analogue of $n$: $[n]:\equiv \frac{q^n-1}{q-1}$;
• The q-analogue of factorial $n!$: $[n]! :\equiv \prod_{k=1}^{n}[k]$ for $k\in{\Bbb N}$ (and $[0]!=1$);
• $(x-a)^{n}_{q} :\equiv \prod_{k=0}^{n-1} (x-q^k a)$ for $n\in{\Bbb N}$.
• $(x-a)^{-n}_{q} :\equiv \frac1{(x- q^{-n}a)^{n}_{q}}$.

Properties:

• $[-n] = - q^{-n} [n]$.
• $(x-a)^{m+n}_{q} = (x-a)^{m}_{q} (x- q^m a)^{n}_{q}$,
• $(a-x)^{n}_{q} = (-)^n q^{n(n-1)/2} (x - q^{-n+1}a)^{n}_{q}$,

## the “quantum” differentials

Definitions: For an arbitrary function $f:{\Bbb R}\rightarrow {\Bbb R}$ define:

• its q-differential: $d_q f(x) :\equiv f(q x) - f(x)$;
• its h-differential: $d_h f(x) :\equiv f(x+h) - f(x)$;
• its q-derivative: $D_q f(x) :\equiv \frac{d_q f(x)}{d_q x} = \frac{f(q x) - f(x)}{(q-1)x}$;
• its h-derivative: $D_h f(x) :\equiv \frac{d_h f(x)}{d_h x} = \frac{f(x+h) - f(x)}{h}$;

Note that $d_q x = (q-1) x$ and $d_h x = h$.

Basic properties:

• all four operators ($d_q$, $d_h$, $D_q$ and $D_h$) are linear, e.g. $d_q (\alpha f(x) + \beta g(x)) = \alpha d_q f(x) + \beta d_q g(x)$.
• $d_q (f(x)g(x)) = (d_q f(x)) g(x) + f(qx) (d_q g(x))$;
• $d_h (f(x)g(x)) = (d_h f(x)) g(x) + f(x+h) (d_h g(x))$;
• $D_q (f(x)g(x)) = (D_q f(x)) g(x) + f(qx) (D_q g(x))$;
• $D_q \left(\frac{f(x)}{g(x)}\right) = \frac{D_q f(x) \ g(x) - f(x) \ D_q g(x)}{g(x)g(qx)} = \frac{D_q f(x) \ g(q x) - f(q x) \ D_q g(x)}{g(x)g(qx)}$;
• there does not exist a general chain rule for q-derivatives
• such rule exists for monomial changes of variables $x\rightarrow x' = \alpha x^{\beta}$, where $D_q f(x'(x)) = (D_{q^\beta}f)(x') \cdot D_q x'(x)$.

Examples and properties:

• $D_q x^n = [n] x^{n-1}$.
• $(D^n_q f)(0) = \frac{f^{(n)}(0)}{n!} [n]!$.
• $D_q f(x) = \sum_{n=0}^{\infty}\frac{(q-1)^n}{(n+1)!} x^n f^{(n+1)}(x)$.
• $P_n(x) :\equiv \frac{x^n}{[n]!}$ satisfies $D_q P_n(x) = P_{n-1}(x)$.
• $D_q (x-a)^{n}_{q} = [n] (x-a)^{n-1}_{q}$,
• $D_q (a-x)^{n}_{q} = - [n] (a- q x)^{n-1}_{q}$,
• $D_q \frac1{(x-a)^{n}_{q}} = [-n] (x-q^n a)^{-n-1}_{q}$,
• $D_q \frac1{(a-x)^{n}_{q}} = \frac{[n]}{(a-x)^{n+1}_{q}}$,

## q-binomial calculus

Definition: The q-binomial coefficient is defined by $\left[{n \atop j}\right] :\equiv \frac{[n]!}{[j]![n-j]!}$.

Properties of q-binomial coefficients:

• $\left[{n \atop n-j}\right] = \left[{n \atop j}\right]$.
• there exist two q-Pascal rules: $\left[{n \atop j}\right] = \left[{n-1 \atop j-1}\right] + q^j \left[{n-1 \atop j}\right]$ and $\left[{n \atop j}\right] = q ^{n-j}\left[{n-1 \atop j-1}\right] + \left[{n-1 \atop j}\right]$.
• $\left[{n \atop 0}\right] = \left[{n \atop n}\right] = 1$.
• $\left[{n \atop j}\right]$ is a polynomial in $q$ of degree $j(n-j)$ with the leading coefficient equal to $1$.
• $\left[{\alpha \atop j}\right] = \frac{[\alpha] [\alpha-1] \cdots [\alpha -j +1]}{[j]!}$ for any number $\alpha$.
• $\left[{m+n \atop k}\right] = \sum_{j=0}^{k} q^{(k-j)(m-j)} \left[{m \atop j}\right] \left[{n \atop k-j}\right]$.
• $x^n = \sum_{j=0}^{n} \left[{n \atop j}\right] (x-1)^j_q$.
• $\sum_{j=0}^{2m} (-)^j \left[{2m \atop j}\right] = (1-q^{2m-1})(1-q^{2m-3})\cdots(1-q)$.
• $\sum_{j=0}^{2m+1} (-)^j \left[{2m+1 \atop j}\right] = 0$.
• The Gauss’s binomial formula: $(x+a)^n_q = \sum_{j=0}^n \left[{n \atop j}\right] q^{j(j-1)/2} a^{j} x^{n-j}$.
• For two non-commutative operators $\hat{A}$ and $\hat{B}$ s/t $\hat{B}\hat{A} = q\hat{A}\hat{B}$ (with $q$ and ordinary number), the non-commutative Gauss’s binomial formula is: $(\hat{A}+\hat{B})^n = \sum_{j=0}^n \left[{n \atop j}\right] \hat{A}^{j} \hat{B}^{n-j}$. Such two operators are $\hat{x}$ and $\hat{M_q}$ defined as $\hat{x} f(x) :\equiv x f(x)$ and $\hat{M_q} f(x) :\equiv f(qx)$.
• The Heine’s binomial formula: $\frac1{(1-x)^n_q} = 1+ \sum_{j=1}^{\infty} \frac{[n][n+1]\cdots[n+j-1]}{[j]!} x^j$.
• $\frac1{(1-x)^{\infty}_q} = \sum_{j=0}^{\infty} \frac{x^j}{(1-q)(1-q^2)\cdots(1-q^j)}$.
• $(1+x)^{\infty}_q = \sum_{j=0}^{\infty} q^{j(j-1)/2} \frac{x^j}{(1-q)(1-q^2)\cdots(1-q^j)}$.

## Generalized Taylor’s formula for polynomials

For given number $a$ and linear operator $D$ on space of polynomials, there exist a unique sequence of polynomials $\{P_0(x), P_1(x), \cdots\}$ such that

• $P_0(a)=1$ and $P_{n>0}(a)=0$;
• $\hbox{deg}P_n = n$;
• $D P_n(x) = P_{n-1}(x)$ ($\forall n\ge 1$) and $D(1)=0$.

Then any polynomial $f(x)$ of degree $n$ has the unique expansion via following generalized Taylor expression: $f(x) = \sum_{j=0}^n (D^j f)(a) P_j(x)$.

Examples:

• If $D$ is $D_q$ we have: $f(x) = \sum_{j=0}^n (D_q^j f)(a) \frac{(x-a)^j_q}{[j]!}$.
• for $f(x)=x^n$ and $a=1$ one gets: $x^n = \sum_{j=0}^n \left[{n \atop j}\right] (x-1)^j_q$.

## Exponentials and trigonometric functions

Definitions: (q-exponentials):

• $e_q^x :\equiv \sum_{k=0}^\infty \frac{x^k}{[k]!} = \frac1{(1-(1-q)x)_q^\infty}$;
• $E_q^x :\equiv \sum_{k=0}^\infty q^{k(k-1)/2} \frac{x^k}{[k]!} = (1+(1-q)x)_q^\infty$.

Properties:

• $e_q^0 = 1$, $E_q^0 =1$.
• $\frac1{(1-x)_0^\infty} = e_q^{x/(1-q)}$.
• $D_q e_q^x = e_q^x$, $D_q E_q^x = E_q^{qx}$.
• $D_q \frac1{(1-(1-q)x)_q^n} = \frac{(1-q)[n]}{(1-(1-q)x)_q^{n+1}}$, $D_q (1+(1-q)x)_q^n = (1-q) [n] (1+q(1-q)x)^{n-1}_q$.
• $e_q^{x}e_q^{y} \ne e_q^{x+y}$; but $e_q^{x}e_q^{y} = e_q^{x+y}$ iff $yx=qxy$.
• $E_q^{-x} = \frac1{e_q^{x}}$, i.e. $E_q^{x} = \frac1{e_q^{-x}}$.
• $e_{\frac1{q}}^{x} = E_q^x$.
• direct consequence of the previous two lines: $e_q^{-x}e_{\frac1{q}}^{x} = 1$.

Definitions: (q-trigonometric functions):

• $sin_q(x) :\equiv \frac{e_q^{ix}-e_q^{-ix}}{2i}$,
• $Sin_q(x) :\equiv \frac{E_q^{ix}-E_q^{-ix}}{2i}$,
• $cos_q(x) :\equiv \frac{e_q^{ix}+e_q^{-ix}}{2i}$,
• $Cos_q(x) :\equiv \frac{E_q^{ix}+E_q^{-ix}}{2i}$.

Properties:

• $cos_q(x) Cos_q(x) + sin_q(x) Sin_q(x) = 1$.
• $D_q sin_q(x) = cos_q(x)$,
• $D_q Sin_q(x) = Cos_q(qx)$,
• $D_q cos_q(x) = - sin_q(x)$,
• $D_q Cos_q(x) = - Sin_q(qx)$.

## Partition functions and product formulas

Definitions:

• The triangular numbers: $\Delta_n :\equiv \frac{n(n+1)}2$,
• the square numbers: $\Box_n :\equiv n^2$,
• the pentagonal numbers: $\Pi_n :\equiv \frac{n(3n-1)}2$,
• the k-gonal numbers: $m^{(k)}_n :\equiv (k-2) \Delta_{n-1} + n = \frac{n(kn -2n -k +4)}2$.

Definition: The classical partition function $p(n):{\Bbb Z}\rightarrow{\Bbb N}$) is defined as

• $p(n) =$ the number of ways to partition an positive integer number $n$ into sum of positive integers (modulo reordering of summands);
• $p(n)=0$ for $n<0$;
• $p(0)=1$.

Properties:

• Examples: $p(1)=1$, $p(2)=2$, $p(3) =3$, $p(4) = 5$, $p(5)=7$.
• Asymptotic behavior: $p(n) \sim \frac1{3\sqrt{3}n} e^{\pi \sqrt{\frac{2n}{3}}}$ \ as \ $n\rightarrow \infty$.
• $\varphi(q)^{-1} = \sum_{n=0}^\infty p(n) q^n$.
• $p(n) = \sum_{n=0}^\infty (-)^{n-1} (p(n-\Pi_n) + p(n-\Pi_{-n}))$.

Definition: the Euler’s product: $\varphi(q) :\equiv \prod_{n=1}^\infty (1-q^{n})$.

Theorem (Jacobi’s triple product identity): For $|q|<1$ following is true:

$\sum_{n\in{\Bbb Z}} q^{n^2}z^n = \prod_{n=1}^\infty (1-q^{2n})(1+q^{2n-1}z)(1+q^{2n-1}z^{-1})$.

Consequences:

• Euler’s product formula: $\sum_{n\in{\Bbb Z}} (-)^n q^{\frac{n(3n-1)}2} = \prod_{n=1}^\infty (1-q^{n})$.
• This can be rephrased as follows: $\varphi(q) = \sum_{n\in{\Bbb Z}} (-)^n q^{\Pi_n}$ where $\Pi_n$ are the pentagonal numbers defined above.
• Following Gauss identities are special cases of the Jacobi’s triple product identity:
• $\sum_{n=0}^{\infty} q^{\Delta_n} = \prod_{n=1}^\infty \frac{1-q^{2n}}{1-q^{2n-1}}$,
• $\sum_{n=0}^{\infty} (-q)^{\Box_n} = \prod_{n=1}^\infty \frac{1-q^{n}}{1+q^{n}}$.

Sources:

• [1] book: “Quantum Calculus” by Victor Kac and Pokman Cheung (Springer) – ISBN 0-387-9534198; QA303.C537 2001

Other references:

## 2010.10.19

### Infosec books

Filed under: books, infosec — Tags: , — sandokan65 @ 12:58
• Book: Enterprise Security For the Executive – https://eikonal.wordpress.com/2010/01/07/book-enterprise-security-for-the-executive/
• “Strategic Cyber Security” by Kenneth Geers (NATO; 2011) – http://www.ccdcoe.org/278.html [FREE: PDF, ePUB]
The book argues that computer security has evolved from a technical discipline to a strategic concept. The world’s growing dependence on a powerful but vulnerable Internet – combined with the disruptive capabilities of cyber attackers – now threatens national and international security.

Strategic challenges require strategic solutions. The author examines four nation-state approaches to cyber attack mitigation:

• Internet Protocol version 6 (IPv6)
• Sun Tzu’s Art of War
• Cyber attack deterrence
• Cyber arms control

The four threat mitigation strategies fall into several categories. IPv6 is a technical solution. Art of War is military. The third and fourth strategies are hybrid: deterrence is a mix of military and political considerations; arms control is a political/technical approach.

The Decision Making Trial and Evaluation Laboratory (DEMATEL) is used to place the key research concepts into an influence matrix. DEMATEL analysis demonstrates that IPv6 is currently the most likely of the four examined strategies to improve a nation’s cyber defence posture.

There are two primary reasons why IPv6 scores well in this research. First, as a technology, IPv6 is more resistant to outside influence than the other proposed strategies, particularly deterrence and arms control, which should make it a more reliable investment. Second, IPv6 addresses the most significant advantage of cyber attackers today – anonymity.

## Misc

This should be titled “some infosec books” – namely the infosec books that I have recently read or used.

There exist many other collections/lists of books in this domain. Some are:

Similar at this blog: Book: Enterprise Security For the Executive – https://eikonal.wordpress.com/2010/01/07/book-enterprise-security-for-the-executive/ | Infosec pages at this blog – https://eikonal.wordpress.com/2011/05/17/information-security-sites/ | Infosec online (= infosec sites) – https://eikonal.wordpress.com/2010/02/01/infosec-online/

### Body language

Filed under: body language, mind & brain, society — Tags: , , , , — sandokan65 @ 10:37

## 2010.10.14

### Sylvester and Lyapunov equations

Definitions: ([1],[2]) For $n\times n$ matrices $A$, $B$, $X$, $Y$:

• the Sylvester equation: $AY+YB=X$;
• the Lyapunov equation: $AYA^{\dagger}-Y=X$;
• the “continuous” Lyapunov equation: $AY+YA^{\dagger}=X$.

More:

## 2010.10.13

### Python

Filed under: programming languages, python — Tags: , , — sandokan65 @ 11:44

### Calabi-Yau

Filed under: mathematics — Tags: — sandokan65 @ 10:46

## 2010.10.12

### Retirement

Filed under: economy — Tags: , , , , , — sandokan65 @ 10:29

## Articles

• Retirement checklist: What to do from 35 to 55+ – http://money.cnn.com/2010/09/21/retirement/retirement_checklist.moneymag/index.htm. Here is the list of bullets this article covers (for details, go read the article itself):

• Mid-30s to early 40s:
• How much you should have in your retirement savings: 1.5 times your annual salary.
• Goal: Develop the habit of saving.
• Savings: 1.5 times your annual salary by age 35.
• Take full advantage of my 401(k) match.
• Boost my 401(k) contribution.
• Find other tax-advantaged ways to save.
• Cover six months of expenses.
• Invest for growth.
• Mid-40s to early 50s:
• How much you should have in your retirement savings: 3 times your annual salary.
• Main goal: Focus on how you invest your money.
• Savings: 3 times annual salary by age 45
• Rebalance my portfolio.
• Go over my investment strategy.
• Make my catch-up contributions.
• Give myself a reality check.
• Consolidate my far-flung retirement accounts.
• Mid-40s to early 50s:
• How much you should have in your retirement savings: 6 times your annual salary.
• Main goal: Decide what type of retirement you want.
• Savings: 6 times your annual salary by age 55.
• Prune my stock portfolio.
• Map out a blueprint for my retirement.
• Run (and rerun) my income plan.
• Look into when to take Social Security.
• Work on my Plan B.

## 2010.10.08

### Running

Filed under: health, running, workout — Tags: , , — sandokan65 @ 12:56
• Instructables web site has a a collection of articles on running:
• Fuel efficiency for marathoners (HarvardScience) – http://news.harvard.edu/gazette/story/2010/10/fuel-efficiency-for-marathoners/ – New research helps runners set the right pace for 26.2 miles
• Running Endurance Calculator – http://endurancecalculator.com/ [Java Applet]
• “Metabolic Factors Limiting Performance in Marathon Runners” by Benjamin I. Rapoport (Harvard Medical School and Department of Electrical Engineering and Computer Science and Division of Health Sciences and Technology) – http://www.ploscompbiol.org/mirror/article/pcbi.1000960.html
Abstract: Each year in the past three decades has seen hundreds of thousands of runners register to run a major marathon. Of those who attempt to race over the marathon distance of 26 miles and 385 yards (42.195 kilometers), more than two-fifths experience severe and performance-limiting depletion of physiologic carbohydrate reserves (a phenomenon known as ‘hitting the wall’), and thousands drop out before reaching the finish lines (approximately 1–2% of those who start). Analyses of endurance physiology have often either used coarse approximations to suggest that human glycogen reserves are insufficient to fuel a marathon (making ‘hitting the wall’ seem inevitable), or implied that maximal glycogen loading is required in order to complete a marathon without ‘hitting the wall.’ The present computational study demonstrates that the energetic constraints on endurance runners are more subtle, and depend on several physiologic variables including the muscle mass distribution, liver and muscle glycogen densities, and running speed (exercise intensity as a fraction of aerobic capacity) of individual runners, in personalized but nevertheless quantifiable and predictable ways. The analytic approach presented here is used to estimate the distance at which runners will exhaust their glycogen stores as a function of running intensity. In so doing it also provides a basis for guidelines ensuring the safety and optimizing the performance of endurance runners, both by setting personally appropriate paces and by prescribing midrace fueling requirements for avoiding ‘the wall.’ The present analysis also sheds physiologically principled light on important standards in marathon running that until now have remained empirically defined: The qualifying times for the Boston Marathon.

Filed under: FaceBook, privacy — Tags: — sandokan65 @ 10:44

2010.10.07:

2010.10.06:

### Lawsuits in mobile space

Filed under: mobile and wireless, patents — Tags: , , , , , — sandokan65 @ 10:21

Related here: “One more example of reactive social role of patent system” – https://eikonal.wordpress.com/2010/07/09/one-more-example-of-reactive-social-role-of-patent-system/ | “Broken patents system” – https://eikonal.wordpress.com/2011/03/29/broken-patents-system/

### Runge-Lenz vector

For a potential $U(\vec{r}) = \frac{k}{r}$, the conserved vectors are (slightly changed definitions from reference [1]):

• the angular momentum verctor: $\vec{L} :\equiv \vec{r}\times\vec{p}$, and
• the (dimensionless) eccentricity vector: $\vec{e} = \frac{\underline{A}}{mk} :\equiv \frac1{mk}\vec{p}\times\vec{L} - \hat{r}$ where $\hat{r} :\equiv \frac1{r} \vec{r}$, and $\underline{A}$ is the LRL vector (the Laplace-Runge-Lenz vector = the Runge–Lenz vector = the Lenz vector).
• As a consequence, the (dimensionless) vector of binormal is also preserved: $\vec{b} :\equiv \vec{l}\times\vec{e} = \frac{L}{mk}\vec{p} +\hat{r}\times\vec{l}$. Here $\vec{l}:\equiv \frac1{L}\vec{L} = \vec{e}\times\vec{b}$.

• [1] Wikipedia: Laplace-Runge-Lenz vector – http://en.wikipedia.org/wiki/Laplace%E2%80%93Runge%E2%80%93Lenz_vector | Eccentricity vector – http://en.wikipedia.org/wiki/Eccentricity_vector
• Laplace-Runge-Lenz Vector – http://scienceworld.wolfram.com/physics/Laplace-Runge-LenzVector.html
• Laplace-Runge-Lenz vector (The Tangent Bundle Physics Wiki) – http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/Laplace-Runge-Lenz_vector
• P.E.S. Wormer (2003). Properties of the quantum mechanical Runge-Lenz vector. – <a href=http://www.theochem.ru.nl/~pwormer/rungelenz.pdf".http://www.theochem.ru.nl/~pwormer/rungelenz.pdf
• “Mysteries of the gravitational 2-body problem” by John Baez (2003.05.03) – http://math.ucr.edu/home/baez/gravitational.html
• “A planar Runge-Lenz vector” by S.G.Kamath (arXiv:hep-th/0112067v1; 2001.12.10; also J.Math.Phys. 43 (2002) 318-324) – http://arxiv.org/abs/hep-th/0112067
Abstract: Following Dahl’s method an exact Runge-Lenz vector M with two components M and M is obtained as a constant of motion for a two particle-system with charges e and e whose electromagnetic interaction is based on Chern-Simons electrodynamics. The Poisson bracket {M, M} = L but is modified by the appearance of the product e e as central charges.
• “Duality of force laws and Conformal transformations” by Dawood Kothawala
(arXiv:1010.2238v3 [physics.class-ph]; 2011.01.07) – http://arxiv.org/abs/1010.2238

Abstract: As was first noted by Isaac Newton, the two most famous ellipses of classical mechanics, arising out of the force laws F~r and F~1/r^2, can be mapped onto each other by changing the location of center-of-force. What is perhaps less well known is that this mapping can also be achieved by the complex transformation, z -> z^2. We give a simple derivation of this result (and its generalization) by writing the Gaussian curvature in its “covariant” form, and then changing the \emph{metric} by a conformal transformation which “mimics” this mapping of the curves. The final result also yields a relationship between Newton’s constant G, mass M of the central attracting body in Newton’s law, the energy E of the Hooke’s law orbit, and the angular momenta of the two orbits. We also indicate how the conserved Laplace-Runge-Lenz vector for the 1/r^2 force law transforms under this transformation, and compare it with the corresponding quantities for the linear force law. Our main aim is to present this duality in a geometric fashion, by introducing elementary notions from differential geometry.
• “Laplace-Runge-Lenz symmetry in general rotationally symmetric systems” by Uri Ben-Ya’acov (arXiv:1005.1817; 2010.05.11) – http://arxiv.org/abs/1005.1817
Abstract: The Laplace-Runge-Lenz symmetry, well known to exist in classical two-body Kepler-Coulomb systems, is also known to be generalizable to all rotationally symmetric systems. It also appears in the computation of the Lorentz boost in relativistic systems. Towards the unification of these properties, the generic (independent of the interaction) properties of the symmetry are verified and extended. The independence of the symmetry on the type of interaction is proven applying only the most minimal properties of the Poisson brackets. Generalized Laplace-Runge-Lenz vectors are definable to be constant (not only piece-wise conserved) for all cases, including open orbits. Also discussed are the transformations generated by the Laplace-Runge-Lenz vectors, the emergence of these vectors in post-Newtonian extensions of general centrally symmetric systems, and the application of these results to relativistic Coulomb systems.
• “Generalized Laplace-Runge-Lenz vector for the three-dimensional classical motions generated by central forces with a monopole” by T. Yoshida (Il Nuovo Cimento B (1971-1996); Volume 104, Number 4, 375-385, DOI: 10.1007/BF02725670) – http://www.springerlink.com/content/k481322u42ln7244/ [REQUIRES PAYMENT FOR ACCESS]
Abstract: This paper is concerned with the general method of constructing conserved vectors, which is applicable to noncentral force problems. The point of this method is to express an orthonormal system of vectors in terms of the position, momentum and angular-momentum vectors. Then conserved vectors, such as the Laplace-Runge-Lenz vector, are obtained as a unit vector of the fixed orthonormal system, if the equations of motion are solvable by the quadrature. To facilitate the procedure for obtaining the conserved vectors a unit dyadic is introduced. A generalized Laplace-Runge-Lenz vector is obtained indeed by this procedure for the three-dimensional classical motions generated by central forces with Dirac’s monopole.
• “Energy spectrum of the two-dimensionalq-hydrogen atom” by Shengli Zhang (International Journal of Theoretical Physics; Volume 34, Number 11, 2217-2221, DOI: 10.1007/BF00673837) – http://www.springerlink.com/content/n251256l88572074/ [REQUIRES PAYMENT FOR ACCESS]
Abstract: The discrete energy spectrum of theq-analog of the two-dimensional hydrogen atom is derived by deforming the Pauli equation. It contracts to that of the ordinary two-dimensional hydrogen atom in the limitq rarr ± 1. The degeneracy is discussed generally and some properties of theq-energy spectrum are studied both for realq and for complexq of magnitude unity.
• “The hidden symmetry of the Coulomb problem in relativistic quantum mechanics: From Pauli to Dirac” by Tamari T. Khachidze and Anzor A. Khelashvili (American Journal of Physics — July 2006 — Volume 74, Issue 7, pp. 628-632) – http://ajp.aapt.org/resource/1/ajpias/v74/i7/p628_s1 [REQUIRES PAYMENT FOR ACCESS]
Abstract: Additional conserved quantities associated with an extra symmetry govern a wide variety of physical systems ranging from planetary motion to atomic spectra. We give a simple derivation of the hidden symmetry operator for the Dirac equation in a Coulomb field and show that this operator may be reduced to the one introduced by Johnson and Lippmann to include the spin degrees of freedom. This operator has been rarely discussed in the literature and has been neglected in recent textbooks on relativistic quantum mechanics and quantum electrodynamics.
• “Rotating Laplace-Runge-Lenz vector leading to two relativistic Kepler’s equations” by Takeshi Yoshida (Phys. Rev. A 38, 19–25 (1988) = Phys. Rev. A » Volume 38 » Issue 1) – http://pra.aps.org/abstract/PRA/v38/i1/p19_1 [just abstract]
Abstract: For the nonrelativistic Kepler problem it is well known that there exists a transcendental equation (the so-called Kepler’s equation) which gives a position in the orbit at a given time. The precessing orbit of the relativistic Kepler problem is reduced to the nonprecessing one by using the transformation related to a rotating Laplace-Runge-Lenz vector. This vector is an extension of the familiar conserved vector and is always oriented to the moving perihelion point of the precessing orbit. From the considerations on Kepler’s equation and the rotating Laplace-Runge-Lenz vector, a relativistic Kepler’s equation is newly defined to find a position in the precessing orbit at a given time. This equation has two expressions corresponding to the relativistic coordinate time and proper time. By using the equation, Lambert’s theorem that gives the required time on the orbit is extended to the relativistic problem.
• “Dynamics of the Laplace-Runge-Lenz vector in the quantum-corrected Newton gravity” by C. Farina, W. J. M. Kort-Kamp, Sebastiao Mauro Filho, Ilya L. Shapiro (arXiv:1101.5611v2 [gr-qc]; 2011.02.02) – http://arxiv.org/abs/1101.5611
Abstract: Recently it was shown that quantum corrections to the Newton potential can explain the rotation curves in spiral galaxies without introducing the Dark Matter halo. The unique phenomenological parameter $\al\nu$ of the theory grows with the mass of the galaxy. In order to better investigate the mass-dependence of $\al\nu$ one needs to check the upper bound for $\al\nu$ at a smaller scale. Here we perform the corresponding calculation by analyzing the dynamics of the Laplace-Runge-Lenz vector. The resulting limitation on quantum corrections is quite severe, suggesting a strong mass-dependence of $\al\nu$.
• “Orbit Determination with the two-body Integrals” by Giovanni Federico Gronchi, Linda Dimare, Andrea Milani (arXiv:0911.3555v2 [math-ph]; 2010.03.31) – http://arxiv.org/abs/0911.3555
Abstract: We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.
• “Orbit Determination with the two-body Integrals. II” by Giovanni F. Gronchi, Davide Farnocchia, Linda Dimare (arXiv:1101.4569v1 [math-ph]; 2011.01.24) – http://arxiv.org/abs/1101.4569
Abstract: The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or radar observations. We write polynomial equations for this problem, that we can solve using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. In this paper we continue the research started in [Gronchi, Dimare, Milani, ‘Orbit determination with the two-body intergrals’, CMDA (2010) 107/3, 299-318], where the angular momentum and the energy integrals were used. A suitable component of the Laplace-Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.
• “The Universal Kepler Problem” by Guowu Meng (arXiv:1011.6609v3 [math-ph]; 2010.12.22) – http://arxiv.org/abs/1011.6609>
Abstract: For each simple euclidean Jordan algebra, the analogues of hamiltonian, angular momentum and Lenz vector in the Kepler problem are introduced. The analogue of hidden symmetry algebra generated by hamiltonian, angular momentum and Lenz vector is also derived. Finally, for the simplest simple euclidean Jordan algebra, i.e., $\bb R$, we demonstrate how to get generalized Kepler problems by combining with the quantizations of the TKK algebra.
• pages 90-95 of book “Hamiltonian dynamics” by Gaetano Vilasi – http://tinyurl.com/648t4zf
• “Determination of the Runge—Lenz Vector” by W. H. Heintz (1974) – http://www.phys.ufl.edu/~maslov/classmech/heinz.pdf [PDF]

## 2010.10.05

### sed tricks

Filed under: scripting, transformers — Tags: , , — sandokan65 @ 15:58

These one-liners are collected from various sites and articles on web – see the list of Sources at the bottom of this posting.

• Deleting all empty lines from the input file:
sed ‘/^$/d’  • In-place replacement: sed –i ‘/^$/d’ INPUTFILE
• In-place replacement with backup of original file:
sed –ibak ‘/^$/d’ INPUTFILE • In-place deletion of all occurences of a string in a file: sed –i ‘/WORDTOBEDELETED/d’ • How to replace the first occurrence only (of a string match) in a file, using sed sed '0,/THISSTRING/s//TOTHATSTRING/' INPUTFILE • Append environment variable PATH with sed: sed -e '/^PATH/s/"$/:\/usr\/lib\/myprog\/bin"/g' -i /etc/environment
• Remove all whitespace from beinning of lines:
sed 's/^[ \t]*//g' foo
• Deleting the / from all html files contained in current folder:
sed -i ‘s/src=”\//src=”/g’ *.html
• Greedy matching:
% echo "foobar" | sed 's///g'
bar

• Non greedy matching:
% echo "foobar" | sed 's/]*>//g'
foobar


Sources:

## References

Related here: Command line based text replace – https://eikonal.wordpress.com/2010/07/13/command-line-based-text-replace/.

Related here: Scripting languages – https://eikonal.wordpress.com/2010/06/15/awk-sed/ | Unix tricks – https://eikonal.wordpress.com/2011/02/15/unix-tricks/ | SED tricks – https://eikonal.wordpress.com/2010/10/05/sed-tricks/ | Memory of things disappearing > nmap stuff > getports.awk – https://eikonal.wordpress.com/2010/06/23/memory-of-things-disappearing-nmap-stuff-getports-awk/ | AWK – https://eikonal.wordpress.com/2011/09/30/awk/

### Transformers

Filed under: transformers — Tags: , , , , , — sandokan65 @ 14:49

## Finding difference of two MS Word documents

### Port knocking

Filed under: firewalls, infosec, security hardening — Tags: — sandokan65 @ 14:20

More on this blog: IpTables – https://eikonal.wordpress.com/2011/01/24/iptables/ | Personal Computer Security > Personal Firewalls – https://eikonal.wordpress.com/2011/02/28/personal-computer-security/ | Firewalls – https://eikonal.wordpress.com/2012/05/04/firewalls/

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