Eikonal Blog

2010.02.12

Some continued fractions

Filed under: mathematics — Tags: — sandokan65 @ 21:01
  • \sqrt{2} = 1+ \frac1{2+}\frac1{2+}\frac1{2+}\cdots = [1;2,2,2,\cdots] = [1;\dot{2}].
  • \sqrt{3} = [1;\dot{1},\dot{2}].
  • \sqrt{5} = [2; \dot{4}].
  • \sqrt{7} = [2; \dot{1},1,1,\dot{4}].
  • e = [2;1,2,1,1,4,1,1,6,1,1,8,\cdots].
  • \frac\pi4 = \frac1{1+} \frac{1^2}{2+} \frac{3^2}{2+} \frac{5^2}{2+} \frac{7^2}{2+} \frac{9^2}{2+} \cdots.
  • [3;\dot{3}] = \frac{3+\sqrt{13}}2 = 3.302,77\cdots.
  • [a;\dot{b}] = a+\frac{\sqrt{b^2+4}-b}2.
  • [4;\dot{6}] = 1+\sqrt{10}.
  • [a;\dot{b},\dot{c}] = ab + \frac{\sqrt{bc(bc+4)}-bc}2.

Theorem: A simple infinite continued fraction [a_0; a_1, a_2, \cdots] converges IFF \sum_n a_n diverges.

Theorem:
Every real number x has the unique simple continued fraction expression. That expression is:

  • finite IFF x \in {\Bbb Q},
  • infinite and repeating IFF x \not\in {\Bbb Q} but algebraic,
  • infinite and non-repeating IFF x is transcendental.

The 50 Most Brilliant Atheists of All Time

Filed under: atheism, critical thinking — sandokan65 @ 20:31

The Brainz.org has an article titled “The 50 Most Brilliant Atheists of All Time” – http://brainz.org/50-most-brilliant-atheists-all-time/.
According to them, following people were some kind of atheists (secularists, atheists, positivists, etc):

  • 1. Democritus – an ancient Greek philosopher
  • 2. Diagoras of Melos – 5th century b.c.e. poet and sophist from Melos known as Diagoras the Atheist.
  • 3. Epicurus – Born in 341 b.c.e. in Athens, Epicurus established the school of philosophy known as Epicureanism
  • 4. Theodorus the Atheist – lived around 300 b.c.e
  • 5. Andrew Carnegie [1835-1919]
  • 6. Ivan Petrovich Pavlov [1849-1936] – a Russian physiologist, psychologist and physician, won the Nobel Prize in medicine in 1904 for research on the digestive system.
  • 7. Sigmund Freud (Sigismund Schlomo Freud) [1856-1939] – an Austrian psychiatrist founded the psychoanalytic school of psychology.
  • 8. Clarence Seward Darrow [1857-1938] – an American lawyer, a leading member of the ACLU and a notable defense attorney.
  • 9. Richard Georg Strauss [1864-1949] – a brilliant German composer who began writing music at the age of six and continued almost until his death.
  • 10. Bertrand Arthur William Russell [1872-1970], 3rd Earl of Russell – a British philosopher, logician, mathematician, historian, pacifist and social activist, awarded the Nobel Prize for literature in 1950.
  • 11. Jawaharlal Nehru [1889-1964] – Prime Minister of an India from 1947 to 1964.
  • 12. Linus Carl Pauling [1901-1994] – one of only 4 individuals ever to have won solo Nobel Prizes in separate and unrelated fields – for chemistry in 1954, and the Nobel Peace Prize for his tireless campaign against atmospheric nuclear bomb testing in 1962.
  • 13. Paul Adrien Maurice Dirac [1902-1984] – a British theoretical physicist who contributed to the early development of quantum mechanics and quantum electrodynamics [QED].
  • 14. Ayn Rand [1905-1982] – Best known for her sweeping intellectual masterpiece Atlas Shrugged, the fiction mystery allowed her to fully develop her philosophy of objectivism.
  • 15. Katherine Houghton Hepburn [1907-2003] – an acclaimed actress in film, television and stage for 73 years of her long life.
  • 16. Jacques Lucien Monod [1910-1976] – awarded the Nobel Prize in Physiology or Medicine in 1965.
  • 17. Padma Vibhushan Subrahmanyan Chandresekhar [1910-1995] – Awarded the Nobel Prize in Physics in 1983 for his important contributions to knowledge about the evolution of stars
  • 18. Alan Mathison Turing [1912-1954] – a mathematician, logician, computer scientist and cryptanalyst from England.
  • 19. Francis Harry Compton Crick [1916-2004] – best known as the co-discoverer of the structure of DNA.
  • 20. Claude Elwood Shannon [1916-2001] – an electronic engineer and mathematician known as “the father of information theory.”
  • 21. Richard Phillips Feynman [1918-1988] – won the Nobel Prize in 1965 for QED
  • 22. Avram Noam Chomsky [b. 1928] – one of the most notable American philosophers of any age. Professor emeritus of linguistics at MIT, and is considered a father of modern linguistics. Also a prolific writer, he has also become famous for being an outspoken political dissident, anarchist, humanist freethinker and libertarian socialist.
  • 23. James Dewey Watson [b. 1928] received the Nobel Prize in physiology or medicine in 1962 as co-discoverer along with Francis Crick and Maurice Wilkins of the molecular structure of DNA.
  • 24. Peter Ware Higgs [b. 1929] – a theoretical physicist and emeritus professor at the University of Edinburgh in Scotland.
  • 25. Warren Edward Buffett [b. 1930] – an American businessman and CEO of Berkshire Hathaway
  • 26. John Rogers Searle [b. 1932] – an American philosopher whose contributions to the philosophy of mind, philosophy of language and social philosophy made him an influential member and spokesperson for the Free Speech Movement in Berkeley during the late 1960s and early ’70s.
  • 27. Steven Weinberg [b. 1933] – an American physicist best known for his work on unification of electromagnetism and the weak force, for which he shared the Nobel Prize in physics in 1979.
  • 28. Carl Edward Sagan [1934-1996] – an American astronomer, astrochemist, and successful popularizer of science.
  • 29. David Takayoshi Suzuki [b. 1936] – a Canadian zoologist, geneticist, science broadcaster and entironmental activist.
  • 30. George Denis Patrick Carlin [1937-2008] – one of the most popular and controversial comedians during his lifetime, having won five Grammy awards for his comedy albums.
  • 31. Bruce Jun Fan Lee [1940- 1973] – an American born Chinese martial artist, philosopher, instructor and actor, the founder of the Jeet Kune Do combat form.
  • 32. Leonard Susskind [b. 1940] – an American physicist specializing in string theory and quantum field theory.
  • 33. Stephen Jay Gould [1941-2002] – a paleontologist, evolutionary biologist and historian of science
  • 34. Clinton Richard Dawkins [b. 1941] – the most prominent scientific atheist in the world today
  • 35. Daniel Clement Dennett [b. 1942] – an American philosopher specializing in the philosophies of mind, science and biology.
  • 36. Stephen William Hawking [b. 1942] – the Lucasian Professor of Mathematics at Cambridge
  • 37. Sir Michael Philip “Mick” Jagger [b. 1943] – singer of Rolling Stones
  • 38. Richard Erskine Frere Leakey [b. 1944] – discovered Australopithecus boisei.
  • 39. David Jon Gilmour [b. 1946] – member of rock group Pink Floyd
  • 40. Brian Eno (Brian Peter George St. John le Baptiste de la Salle Eno) [b. 1948] – an English musician, composer, record producer, music theorist and singer best known as the father of ambient music.
  • 41. David Sloan Wilson [b. 1949] – SUNY Distinguished Professor of Biology and Anthropology at Binghamton University in New York, a prolific popular science writer, and a promoter of evolution by group and multi-level selection.
  • 42. Stephen Gary “Woz” Wozniak [b. 1950] – founder of the Apple computer company with Steve Jobs.
  • 43. Douglas Noel Adams [1952-2001] – an English writer, dramatist and musician, best known for his Hitchhiker’s Guide to the Galaxy series.
  • 44. Steven Arthur Pinker [b. 1954] – an experimental psychologist and cognitive scientist best known for his advocacy of evolutionary psychology and the computational theory of mind.
  • 45. PZ (Paul Zachary) Myers [b. 1957] – an evolutionary developmental biologist and professor of biology at the University of Minnesota, Morris
  • 46. Jodie Foster (Alicia Christian Foster) [b. 1962] – an American film actor
  • 47. Stephen Russell Davies [b. 1963] – a Welsh writer and producer of the modern version of the popular science fiction television series Doctor Who.
  • 48. David John Chalmers [b. 1966] – an Australian philosopher.
  • 49. Sean M. Carroll [b. 1966] – is a theoretical cosmologist at Caltech.
  • 50. Mark Elliot Zuckerberg [b. 1984] – founder of Facebook.

2010.02.11

Hash calculators, online and downloadable

Filed under: infosec — Tags: , , , — sandokan65 @ 12:25

Online calculators:

Downloadable applications:

2010.02.10

Fitness links

Filed under: workout — Tags: , , , , , , , — sandokan65 @ 13:44

Sites:

Pushups

Pull-ups

Various

  • “7 Time-Wasting Mistakes You’re Making at the Gym” by Leta Shy (PopSugar; 2013.06.30) – http://www.fitsugar.com/Gym-Time-Wasters-30637782
  • “8 Essential Strength-Training Exercises to Master” by Leta Shy (PopSugar; 2013.03.09) – http://www.fitsugar.com/Most-Effective-Strength-Training-Exercises-28394198
  • “8 Reasons Why You Should Lift Heavier Weights” by Charlotte Hilton Andersen – http://www.shape.com/fitness/workouts/8-reasons-why-you-should-lift-heavier-weights
    • 1 – You’ll Torch Body Fat
    • 2 – You’ll Look More Defined
    • 3 – You’ll Fight Osteoporosis
    • 4 – You’ll Burn More Calories
    • 5 – You’ll Build Strength Faster
    • 6 – You’ll Lose Belly Fat
    • 7 – You’ll Feel Empowered
    • 8 – You’ll Prevent Injury
  • “People who are more fit during middle age have less chronic illness in later years, study shows” By Michelle Castillo (CBS News) – http://www.cbsnews.com/8301-504763_162-57501737-10391704/people-who-are-more-fit-during-middle-age-have-less-chronic-illness-in-later-years-study-shows/
  • “The Scientific 7-Minute Workout” by Gretchen Reynolds (Wll; NYTimes blog; 2013.05.09) – http://well.blogs.nytimes.com/2013/05/09/the-scientific-7-minute-workout/
    • “In 12 exercises deploying only body weight, a chair and a wall, it fulfills the latest mandates for high-intensity effort, which essentially combines a long run and a visit to the weight room into about seven minutes of steady discomfort — all of it based on science. … “There’s very good evidence” that high-intensity interval training provides “many of the fitness benefits of prolonged endurance training but in much less time,” … The exercises should be performed in rapid succession, allowing 30 seconds for each, while, throughout, the intensity hovers at about an 8 on a discomfort scale of 1 to 10, … Those seven minutes should be, in a word, unpleasant.”
    • These 12 phases are:
      • 1) Jumping jacks
      • 2) Wall sits
      • 3) Push-ups
      • 4) Abdominal crunch
      • 5) Step-up onto chair
      • 6) Squat
      • 7) Triceps dip on chair
      • 8) Plank
      • 9) High knees running in place
      • 10) Lunge
      • 11) Push-ups and rotation
      • 12) Side plank
  • “Strength vs. Endurance: Why You Are Wasting Your Time in the Gym” by Mark Rippetoe (2014.01.29) – http://pjmedia.com/lifestyle/2014/01/29/strength-vs-endurance-why-you-are-wasting-your-time-in-the-gym/
    • Medical professionals still steer older patients towards endurance, to their detriment.
  • “Superset Your Workouts to Save Time, but Add Intensity” by Michele Foley (PopSugar Fitness; June 29, 2011) – http://www.fitsugar.com/Supersets-Add-Intensity-Save-Time-During-Workouts-962125
  • “Warm up, cardio & games” – http://www.7weekstofitness.com/warm-up-cardio-games/
  • “Can Jumping Rope Get You Ripped?” by Nick Bromberg (Yahoo Sports; 2011.05.13) – http://www.thepostgame.com/blog/training-day/201105/double-dutch-treat-can-jumping-rope-get-you-ripped/
  • “What’s the Single Best Exercise?” (New Yourk Times; 2011.04.15) – http://www.nytimes.com/2011/04/17/magazine/mag-17exercise-t.html?_r=1&pagewanted=all
      But when pressed, he suggested one of the foundations of old-fashioned calisthenics: the burpee, in which you drop to the ground, kick your feet out behind you, pull your feet back in and leap up as high as you can. “It builds muscles. It builds endurance.” He paused. “But it’s hard to imagine most people enjoying” an all-burpees program, “or sticking with it for long.”
      And sticking with an exercise is key, even if you don’t spend a lot of time working out. The health benefits of activity follow a breathtakingly steep curve. “The majority of the mortality-related benefits” from exercising are due to the first 30 minutes of exercise, … A recent meta-analysis of studies about exercise and mortality showed that, in general, a sedentary person’s risk of dying prematurely from any cause plummeted by nearly 20 percent if he or she began brisk walking (or the equivalent) for 30 minutes five times a week. If he or she tripled that amount, for instance, to 90 minutes of exercise four or five times a week, his or her risk of premature death dropped by only another 4 percent. So the one indisputable aspect of the single best exercise is that it be sustainable. …
  • “Why Exercise Won’t Make You Thin” by John Cloud (Time online; 2009.08.09) – http://www.time.com/time/health/article/0,8599,1914857,00.html
  • “Are the Religious Prone to Obesity?” by Randy Dotinga (Bloomberg Businessweek; 2011.03.23) – http://www.businessweek.com/lifestyle/content/healthday/651145.html
    • New research finds that people who frequently attend religious services are significantly more likely to become obese by the time they reach middle age. The study doesn’t prove that attending services is fattening, nor does it explain why weight might be related to faith. Even so, the finding is surprising, especially considering that religious people tend to be in better health than others, …
    • Scientists have been studying links between religious behavior and health for years, and have found signs that there’s a positive connection between the two. The studies suggest that religious involvement — whether it’s private or public — is linked to things like better physical health, less depression and more happiness, …
    • After adjusting their statistics to take into account factors such as race, the researchers found that 32 percent of those who attended services the most became obese by middle age … By contrast, only 22 percent of those who attended services the least became obese.
    • What might explain obesity among those who attend services regularly? There are plenty of theories. … one possibility is that those who attend services, along with activities such as Bible study and prayer groups, could be “just sitting around passively instead of being outside engaging in physical activity.” Also, … “a lot of the eating traditions surrounding religion are not particularly healthy; for example, constant feasts or desserts after services or at holidays — fried chicken, traditional kosher foods cooked in schmaltz (chicken fat), and so on.”
    • There’s another question: Why might religious people be obese yet still have good health? The fact that fewer are smokers might help explain that, …

Local: Running – https://eikonal.wordpress.com/2010/10/08/running/ | Yoga – https://eikonal.wordpress.com/2010/09/29/yoga/ | 100 pushups challenge – https://eikonal.wordpress.com/2010/01/04/100-pushups-chalenge/

2010.02.09

A Knichin’s theorem on limit of geometric means for continued fraction form of (the most) real numbers

Filed under: mathematics — Tags: , — sandokan65 @ 15:57

Theorem: (Aleksandr Yakovlevich Khinchin; 1894-1959)
Let \{a_n\in{\Bbb R}|n\in{\Bbb N}\} be a series of numbers. Define the impromptu geometric means GM_n\{a\} :\equiv \left(a_1\cdot a_2 \cdots a_n\right)^{\frac1n}. Then

    \lim_{n\rightarrow \infty} GM_n\{a\} = \prod_{k=1}^\infty \left(1+ \frac1{k(k+2)} \right)^{\frac{\ln(k)}{\ln(2)}} = 2.685,452,001,\cdots

for almost all series \{a\}.

Since \forall x\in{\Bbb R} one can write x = [a_0; a_1, a_2, \cdots] (expansion of x in the continued-fraction form), the above theorem is valid for almost all real x, except for:

  • algebraic numbers with definite patterns for denominators,
  • some transcendental numbers (e.g. e).

Source:

  • Calvin C. Clawson “Mathematical Sorcery” (Revealing the secrets of numbers); ISBN 0-7382-0496-X; Perseus Publishing

Related:

Products

  • \prod_{n=1}^\infty \frac{(2n)^2}{(2n-1)(2n+1)} = \frac\pi2, (John Wallis; 1616-1703)
  • \prod_{n=1}^\infty \frac{(4n)^2}{(4n-1)(4n+1)} = \frac{\pi\sqrt{2}}4,
  • \prod_{n=1}^\infty \frac{(4n+2)^2}{(4n+1)(4n+31)} = \sqrt{2},
  • Question: what is the value of P_m:\equiv \prod_{n=1}^\infty \frac{(mn)^2}{(mn-1)(mn+1)}? Clearly P_2= \frac\pi2 and P_4= \frac{\pi\sqrt{2}}4.
  • \prod_{n=1}^\infty \left(1+\frac{(-)^{n+1}}{2n-1}\right) = \sqrt{2},
  • \prod_{n=1}^\infty \left(1-\frac1{n^2}\right) = \frac12,
  • \prod_{n=1}^\infty \left(1+\frac1{n^2}\right) = \frac{\sinh(\pi)}\pi = 3.0\cdots,
  • x \prod_{n=1}^\infty \left(1- \frac{x^2}{n^2\pi^2}\right) = \sin(x),
  • x \prod_{n=1}^\infty \left(1+ \frac{x^2}{n^2\pi^2}\right) = \sinh(x),
  • \prod_{n=1}^\infty \left(1- \frac{4x^2}{(2n-1)^2\pi^2}\right) = \cos(x),
  • \prod_{n=1}^\infty \left(1+ \frac{4x^2}{(2n-1)^2\pi^2}\right) = \cosh(x),
  • \prod_{n=3}^\infty \left(1-\frac4{n^2}\right) = \frac16,
  • \prod_{n=1}^\infty \left(1+\frac1{n^3}\right) = \frac{\cosh(\\pi\sqrt{3}/2)}\pi = 2.428,\cdots,
  • \prod_{p\in{\Bbb P}} \left(1-\frac1{p^s}\right) = \frac1{\zeta(s)} (Leonard Euler), where {\Bbb P} is the set of prime numbers, and \zeta(s):\equiv\sum_{n=1}^\infty \frac1{n^s} is the Riemann’s zeta function.
  • \prod_{p\in{\Bbb P}} \left(1-\frac1{p^2}\right) = \frac6{\pi^2}.

Strange forms:

  • \frac{\sqrt{2}}2 \frac{\sqrt{2+\sqrt{2}}}2 \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}2 \cdots = \frac{2}{\pi}, (Francois Viete; 1540-1603).
  • \sqrt{x \cdot \sqrt{x\cdot \sqrt{x \cdots}}} = x,
  • \sqrt[n+1]{x \cdot \sqrt[n+1]{x \cdots}} = \sqrt[n]{x},
  • \prod_{n=1}^\infty \begin{pmatrix} \frac1n & 1 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 0 & e \\ 0 & 1\end{pmatrix} ,
  • \prod_{n=1}^\infty \begin{pmatrix} 1 & \frac1{n!}  \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 1 & e \\ 0 & 1\end{pmatrix} , (which is quite trivial).

Sources:

  • Calvin C. Clawson “Mathematical Sorcery” (Revealing the secrets of numbers); ISBN 0-7382-0496-X; Perseus Publishing

Religion is derived from moral instinct

Filed under: religion — Tags: , — sandokan65 @ 11:21

Source: “Morality research sheds light on the origins of religion” (PhysOrg.com; February 8, 2010): http://www.physorg.com/news184857515.html

This article summarizes the paper published in “Cell Press in the journal Trends in Cognitive Sciences” (Feb 2010) that discusses the current studies in relation of religiosity and morality. These paragraphs summarize the paper’s main conclusion:

Citing several studies in moral psychology, the authors highlight the finding that despite differences in, or even an absence of, religious backgrounds, individuals show no difference in moral judgments for unfamiliar moral dilemmas. The research suggests that intuitive judgments of right and wrong seem to operate independently of explicit religious commitments.

“This supports the theory that religion did not originally emerge as a biological adaptation for cooperation, but evolved as a separate by-product of pre-existing cognitive functions that evolved from non-religious functions,” says Dr. Pyysiainen. “However, although it appears as if cooperation is made possible by mental mechanisms that are not specific to religion, religion can play a role in facilitating and stabilizing cooperation between groups.”

Perhaps this may help to explain the complex association between morality and religion. “It seems that in many cultures religious concepts and beliefs have become the standard way of conceptualizing moral intuitions. Although, as we discuss in our paper, this link is not a necessary one, many people have become so accustomed to using it, that criticism targeted at religion is experienced as a fundamental threat to our moral existence,” concludes Dr. Hauser.

2010.02.08

IT tips pages

Filed under: cheatsheets — Tags: , , — sandokan65 @ 14:39

Excel:


Related here: Excel sortIP macro – https://eikonal.wordpress.com/2012/02/07/excel-sortip-macro/ | Excel to text – https://eikonal.wordpress.com/2011/02/14/excel-to-text/ | Excel files processing – https://eikonal.wordpress.com/2011/02/25/excel-files-processing/

2010.02.07

Simple sums

Filed under: mathematics — Tags: — sandokan65 @ 16:43

Definition:

  • S_n^{(m)} :\equiv \sum_{k=1}^n k^m
  • A_n^{(m)} :\equiv \sum_{k=1}^n (-)^k k^m

Several first sums:

  • S_n^{(1)} = \frac12 n (n+1),
  • S_n^{(2)} = \frac16 n (n+1) (2n+1),
  • S_n^{(3)} = \frac14 n^2 (n+1)^2,
  • S_n^{(4)} = \frac1{30} n (n+1) (2n+1) (3n^2+3n-1),
  • S_n^{(5)} = \frac1{12} n^2 (n+1)^2 (2n^2+2n-1),
  • A_n^{(1)} = (-)^n \left[\frac{n+1}2\right],
  • A_n^{(2)} = (-)^n \frac12 n(n+1),
  • A_n^{(3)} = 4\left[\frac{n}2\right]^2 \left(\left[\frac{n}2\right]+1\right)^2 - \frac14 n^2(n+1)^2,
  • A_n^{(4)} = (-)^n \frac12 n(n^3+2n^2-1),
  • A_n^{(5)} = -\frac14 [1+(-)^n(-2n^5 - 5n^4 + 5n^2 -1)],

Related sums:

  • \sum_{k=0}^n (2k+1) = (n+1)^2,
  • \sum_{k=0}^n (2k+1)^2 = \frac13 (n+1)(2n+1)(2n+3),
  • \sum_{k=0}^n (2k+1)^3 = (n+1)^2 (2n^2+4n+1),
  • \sum_{k=0}^n (k+a)(k+b) = \frac16 n(n+1)[2n+1 + 3(a+b)]+nab,
  • \sum_{k=0}^n k(k+1) = \frac13 n(n+1)(n+2),
  • \sum_{k=0}^n k(k+1)(k+2) = \frac14 n(n+1)(n+2)(n+3),
  • \sum_{k=0}^n k(k+3)(k+6) = \frac14 n(n+1)(n+6)(n+7),
  • \sum_{k=0}^n k(k+4)(k+8) = \frac14 n(n+1)(n+8)(n+9),
  • \sum_{k=0}^n (-)^k (2k+1) = (-)^n (n+1),
  • \sum_{k=0}^n (-)^k (2k+1)^2 = (-)^n 2(n+1)^2 - \frac12(1+(-)^n).

General dependency:

  • A_n^{(m)} = 2^{m+1} S_{[n/2]}^{(m)} - S_n^{(m)}.

More:

  • \sum_{k=a}^b k^2 = a b n + \frac{n(n-1)(2n-1)}6 = sum of squares of integer numbers, where n= b-a + 1 the number of terms in the sum.
  • \sum_{k=a/2}^{b/2} (2k)^2 = a b n + \frac{2n(n-1)(2n-1)}3 = sum of squares of even numbers.
  • \sum_{k=(a-1)/2}^{(b-1)/2} (2k+1)^2 = a b n + \frac{2n(n-1)(2n-1)}3 = sum of squares of odd numbers.
  • \sum_{k=a}^b k^3 = \left[a b  + \frac{n(n-1)}2 \right] \frac{n(a+b)}2 = sum of cubes of integer numbers
  • \sum_{k=a/2}^{b/2}  (2k)^3 = \left[a b  + 2 n(n-1) \right] \frac{n(a+b)}2 = sum of cubes of even numbers.
  • \sum_{k=(a-1)/2}^{(b-1)/2} (2k+1)^3 = \left[a b  + 2 n(n-1) \right] \frac{n(a+b)}2 = sum of cubes of odd numbers.
  • \sum_{k=a}^b k^4 = a n \left[n b^2  + ab(n-1) \right] +  \frac{n(n-1)(2n-1)}6 \cdot \frac{3n(n-1) -1}5 = sum of fourth degrees of integer numbers.
  • \sum_{k=a/2}^{b/2}  (2k)^4 = a n \left[2 n b^2 - ab + ab(a-1) \right] +  \frac{8 n(n-1)(2n-1)}6 \cdot \frac{3n(n-1) -1}5 = sum of fourth degrees of even numbers.
  • \sum_{k=(a-1)/2}^{(b-1)/2} (2k+1)^4 = a n \left[2 n b^2 - ab + ab(a-1) \right] +  \frac{8 n(n-1)(2n-1)}6 \cdot \frac{3n(n-1) -1}5 = sum of fourth degrees of odd numbers.
  • \sum_{k=0}^{n-1} (a + d k)^3 = \left[a (a-d)  + \frac{n(n-1) d}2 \right] \frac{n(a+b)}2 = sum of cubes of numbers starting with a, finishing with b with every two neighboring members of sequence spread apart d.

Sources:

  • T1989.02.14
  • “Summing same degrees of subsequent natural numbers” by N.A. Andreyev (Matematical Education #5, Moskva 1960; pp. 201-202) (in Russian)

2010.02.05

Equations of electro+magneto-dynamics in 3+1 dimensions

Filed under: physics — Tags: , — sandokan65 @ 22:49

Maxwell equations:

  • \nabla \vec{D} = \rho_e,
  • \nabla \vec{B} = \rho_m,
  • \partial_t  \vec{B} + \nabla \times \vec{E} = - \vec{j}_m,
  • \partial_t  \vec{D} - \nabla \times \vec{H} = - \vec{j}_e.

Integral form of Maxwell equations:

  • \oint_S \vec{D}\cdot d\vec{S} = Q_e :\equiv \int_V \rho_e dV,
  • \oint_S \vec{B}\cdot d\vec{S} = Q_m :\equiv \int_V \rho_m dV,
  • \oint_l  \vec{E} \cdot d\vec{l} = - \int_S ( \vec{j}_m + \partial_t  \vec{B})
  • \oint_l  \vec{H} \cdot d\vec{l} = + \int_S ( \vec{j}_e + \partial_t  \vec{D})

Moments of the field (aka Energy-Momentum(-Pressure) complex of the electromagnetic field):

  • Flow of energy (Poynting vector): \vec{\Gamma}:\equiv\vec{E}\times\vec{H},
  • Linear momentum of field: \vec{G}:\equiv \vec{B} \times\vec{D},
  • Stress-pressure tensor: {\cal T}:\equiv \frac12 \vec{E}\otimes\vec{D} + \frac12 \vec{H}\otimes\vec{B} - \frac14 {\bf 1} \frac12 (\vec{E}\cdot\vec{D} + \vec{H}\cdot\vec{B}).

Material equations: in the case of the weak fields
the polarizable and magnetizable environment yields the linear responses:

  • \vec{D} = \epsilon \vec{E} + \vec{P} = \hat{\epsilon} \vec{E},
  • \vec{B} = \mu \vec{H} + \mu \vec{M} = \hat{\mu} \vec{H},

where the linear coefficients are the permittivity (\epsilon) and the the permeability (\mu), and the polarization and the magnetization of the substance are measured by vector fields \vec{P} and \vec{M}.

When \rho_m\ne 0 or \vec{\jmath}_m\ne 0 one can not use the signgle complex of potentials (\Phi, \vec{A}) (as used in ordinary electrodynamics), but should add one more complex of potentials (\Psi, \vec{F}). Then:

  • \vec{E} = - \partial_t \vec{A} -\nabla \Phi -c^2 \nabla \times \vec{F},
  • \vec{D} = - \partial_t \vec{L} -\nabla \Psi - \nabla \times \vec{U},
  • \vec{H} = - \partial_t \vec{U} -\nabla \phi + c^2 \nabla \times \vec{L},
  • \vec{B} = - \partial_t \vec{F} -\nabla \psi + \nabla \times \vec{A}.

The source constrainst:

  • \partial_t \rho_e + \nabla \cdot \vec{\jmath}_e = 0,
  • \partial_t \rho_m + \nabla \cdot \vec{\jmath}_m = 0.

Martial Arts magazines and other sources

Filed under: martial arts — Tags: , , , , — sandokan65 @ 16:13

Magazine

Journals

Books


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/ | Chuck Norris superman meme – https://eikonal.wordpress.com/2010/01/22/chuck-norris-superman-meme.

Critical thinking links

Filed under: critical thinking — Tags: , , — sandokan65 @ 14:19

2010.09.09: All content of this posting has been blended into the posting Atheism (https://eikonal.wordpress.com/2010/04/09/atheism/).

2010.02.03

Cheatsheets

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

IT:

Dawkins on Edge

Filed under: evolution — Tags: — sandokan65 @ 14:07

The Edge has biographical excerpt “GROWING UP IN ETHOLOGY” (2009.12.17) by Richard Dawkins: http://www.edge.org/3rd_culture/dawkins09.1/dawkins09.1_index.html.

Threats of cloud computing

Filed under: Uncategorized — Tags: , , , , — sandokan65 @ 12:28

The Edge (edge.org) has an article by Charles Leadbeater (“CLOUD CULTURE: THE PROMISE AND THE THREAT” – http://www.edge.org/3rd_culture/leadbeater10/leadbeater10_index.html) on the dangers of the budding “cloud culture”.

The second replicators (memes) are getting organized better and better.


Related here:

2010.02.01

IT cultural heros

Filed under: it — Tags: — sandokan65 @ 15:03

Computing:

Unix, C:

TCP/IP, Internet, Web:

Perl:

Infosec:

Infosec online (= infosec sites)

Magazines

Knowledge and tools sites

Hacking: a cultural phenomenon

Other sites

Conferences


Related content at this blog:

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