Recent Articles

Dabanese syntax

The kernel of the dabanese syntax uses just four notions: dabagram (dabanese ideogram), unordered phrase, ordered phrase, accent (subject). The language is governed by very simple syntax rules.

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Dabanese — a general introduction

an international language for written communication

Other dabanese knols are listed here: Dabanese–index. Dabanese is an international language for written communication, which I have been developing (not intensively) since the spring of 1985. Dabanese has a lot in common with the written Chinese language. In particular, dabanese like Chinese uses ideograms. The dabanese ideograms are called dabagrams. At this time (october […]

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Fractions and diophantine approximations, I

The original goal of the theory of diophantine approximations is to study how well different real numbers can be approximated by rational numbers in terms of a relation between the precision of the approximation and the size of the denominator of the approximating fraction. The most difficult real numbers x to approximate by rational numbers different from x are rational numbers x. Then the worse x is the so-called golden ratio Φ and irrational numbers related to Φ in a certain way. Almost as bad as Φ are the so-called quadratic irrationalities like √2.

This knol introduces two of the basic notions of the theory of diophantine approximations: 1. pairs of fractions which are neighbors, and 2. the LU encoding of the non-negative real numbers. Only their very first properties are provided in this knol. More advanced properties will be presented in the next knols.

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Mathematics — index

An index of my knols related to mathematics (it starts with “General”, and then the main topics are listed in the alphabetical order); see the details and click on the links just below, in the “Contents”, or on more detailed links in the main body of this knol, below “contents”.

There will be more to come, I hope. Also, an extensive (but not complete) list of outstanding mathematicians (and other scientists, artists, etc.) is included in my knol “WhoWheWha”, which tells you: who? when? what?

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Reflections on mathematics (migma)

My view and sayings about mathematics (migma)

I have posted my first migma in 1994 (years after I made it up), on sci.math, when there was a discussion “*** Definition of Mathematics ***” ;. and my second migma on pl.hum.poezja, in 2001. The remaining ones (and the first two too) I posted on my page:

in 2004, and some of them, in Polish, on pl.hum.poezja, where these migma started a discussion about the material (physical) nature of mathematics, etc., during the week++ from 2004-Oct-23 to Oct-31, with Polish poet Przemysław Łośko being especially involved, and presenting his deep, independent views, which had quite a bit, even a lot in common with my own.

Migma is both singular and plural.

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Special characters & HTML strings

Greek, Polish and Mathematical letters, Russian keyboard, and symbols + html strings

Greek letters, mathematical symbols, and other, are listed here, so that one can relatively easily copy them and paste into their knols or other files (let me know if you want me to add some more, I might).

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Infinitude of primes – 1

Several proofs of the infinitude of the set of all primes, starting with the original proof by Euclid.

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wh reference knols

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Geometric progression

Geometric progression, and Number Theory too.

The geometric progression (g.p.) has the distinction (or stigma?) of belonging to the elementary and high school curriculum. The infinite g. p. plays its role also in Mathematical Analysis and especially in the theory of analytic functions. It is less commonly noted that geometric progressions appear in Number Theory, starting with the Mersenne numbers 2^n – 1 = 1 + 2 + … + 2^(n-1). Mersenne numbers occur in the Euclid-Euler characterization of even perfect numbers, as the sums of divisors of powers 2^(p-1). In general, the sum of divisors σ(p^n), when p is a prime, is the sum of a geometric progression 1 + p + … + p^n, hence the connection between g.p. and the number theory, which motivates a close look at the algebraic properties of a g.p. The standard geometric progressions generalize the Mersenne numbers (here “standard” means that the initial term is 1).

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  1. WhoWheWha (in progress)
  2. The sharpest minds ever
  3. California – State – USA – Knol Authors – Knols – Education
  4. Money — economy (part 1)
  5. “california rains”
  6. Pairs, Kuratowski pairs, and reality
  7. Number Theory — congruences
  8. Matematyka – esej
  9. Congruence x^2 ≡ -1 mod p (Euler), and a super-Wilson theorem.
  10. Dabanese – dictionary (1)
  11. Dabanese syntax
  12. Dabanese — a general introduction
  13. Tentative dabanese dictionary (1)
  14. Fractions and diophantine approximations, I
  15. Mathematics — index
  16. Reflections on mathematics (migma)
  17. Special characters & HTML strings
  18. Infinitude of primes – 1
  19. Reference
  20. Geometric progression
  21. Ergänzungssatz: x^2 ≡ 2 mod p; and more.
  22. Primes in arithmetic progressions (part I)
  23. Euler introduction to Gauss quadratic reciprocity
  24. Trigonometry of a triangle–sinus & cosinus
  25. Wlodzimierz Holsztynski, knolog
  26. Tichonov product
  27. Factorization in semigroups
  28. Nowe knole
  29. Knole w języku polskim
  30. Metric spaces — introduction
  31. Topology — compact spaces II
  32. Topology–singular spaces
  33. the last summer concert in san jose
  34. California in poetry
  35. threeway
  36. san jose blues
  37. san francisco blues
  38. open your arcs
  39. Tom Wachtel, [I was running out of …]
  40. Tom Wachtel, [on the sand at…]
  41. “september”
  42. Topological sequences and convergence
  43. Topology–Arkhangelskii nets
  44. Topology–compact spaces I
  45. Topological product of two spaces. Hausdorff spaces.
  46. Trigonometry
  47. Total logarithmic series
  48. “affinity”
  49. [close your eyes…]
  50. “San Jose”
  51. willow girl
  52. heaven california
  53. “phase transitions”
  54. “california?”
  55. “dimensions”
  56. bachelor life
  57. “spring in california”
  58. [day -]
  59. Singular product of two spaces. Double limit.
  60. Art of Agreeing — painless tax
  61. The Birkhoff lattice of topologies
  62. Left topologies
  63. Topology — Kolmogorov axiom
  64. Topological subbases and bases
  65. Complexity of sorting
  66. ∞-Metrics
  67. Even perfect numbers and Mersenne primes
  68. Fermat sequence base b
  69. Harmonic series & Euler’s gamma
  70. 4-Baroque numbers with utmost 4 different prime divisors
  71. 3-Baroque numbers with utmost 3 different prime divisors
  72. Mathematical notation
  73. Baroque numbers – 2
  74. Number theory — ideals in Z, and the greatest common divisor
  75. Baroque numbers – 1
  76. Infinitude of primes – 2
  77. Number theory — Gothic numbers
  78. log and exp–a constructive and an axiomatic approaches
  79. Aleksandrov 2-point space
  80. Topology — short-short introduction
  81. Art of Agreeing — patent law
  82. Topology — the closure operation
  83. Integration of monotone functions
  84. The ground level properties of integral
  85. Metric spaces universal for 2-point spaces
  86. Products of bounded primes
  87. Sequences of pairwise coprime integers
  88. Mathematics — right triangles
  89. Art of Agreeing — United Nations
  90. Mathematics — two definitions
  91. Government wa(steful wa)ys
  92. Art of Agreeing — index
  93. Metric universality of (R^n d_m) — part 2
  94. Iso-graphs of metric maps into R (part 1)
  95. Metric universality of (R^n d_m) — part 1
  96. Metric spaces universal for 3- and 4-point spaces
  97. Number theory–units, composites, and primes
  98. Social life & energy saving
  99. Topological spaces and continuous mappings. Isolated points.
  100. Topological weight
  101. Mathematics — Euclid-Heron area of a triangle
  102. Connected spaces
  103. Art of Agreeing — marriage versus law & government
  104. Art of Agreeing — introduction
  105. Dabanese — index
  106. Linear orders in topological spaces
  107. Topological cuts and miscuts
  108. Closed sets. T1-spaces.
  109. Topology — the interior operation
  110. Mathematics — triangles
  111. Continuity of the piecewise continuous functions
  112. Topological subspaces
  113. Art of Agreement — medical insurance
  114. Art of Agreeing — business & stock market