Recent Articles

WhoWheWha (in progress)

A who?-when?-what? historical chronological table

This knol should help its readers to realize the density of the interaction or at least coexistence in time of the great contributors to the human knowledge and quality of life, from the same or different fields of activity, how they followed the foot steps of their predecessors. To provide a fuller historical background, some royal figures and political figures are (soon to be) included too, even some pathological murderous monsters.

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The sharpest minds ever

To know the best that the human race has offered means to know about the work of the sharpest human minds ever. Ironically, only a tiny percentage even of the educated(?) people know their names, except for the names of those of these intellectual titans who are known for their contribution to physics: Archimedes, Newton, […]

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Money — economy (part 1)

Real money contra fake money

NOTE (2010-10-03): This knol has been written on September 22 & 24 of 2008 (but for a couple of minor edits). In the past two years my thinking about the presented problems has much advanced, making this knol obsolete. I’ll still keep it since it may help others as a starting point for their own study of the global i local financial questions.


Today (2008-Sept) the meaning of money is somewhat vague, which leads to economic problems. Money used to be equivalent to gold, which had its advantages but also disadvantages, so that the gold based monetary system is abandoned. But a system based on a parallel three kind currency: food+shelter+clothes, would be viable. It would prevent both inflation and deflation., i.e. it would contribute toward a stable and prosperous society.

This text discusses:

1. the reasons behind introducing money;
2. industry
3. the value of money;
4. stable n-s-c money system;
5. secure pension;
6. a smooth transition from the present system to n-s-c system
7. a financially productive life plan

More detailed financial economical moments like for instance a comparison of the cost of living alone and living in a financial community of a group of people (say family or close friends), are not touched upon in this text.

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“california rains”

California in poetry

       you never know california rains they come and go they tease the thirsty Earth   i am waiting for a rain which will drown the hips of details which will amplify the contour                 wh ©1988

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Pairs, Kuratowski pairs, and reality

Every known mode of information assumes order: left to right, top-down, earlier-later, etc. In particular, the basic object of computer science, string, assumes order. Is communication or even a world altogether possible without the notion of ordering built into it, without order as one of its physical, primary features? An answer was de facto given a long time ago, by Kazimierz Kuratowski.

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Number Theory — congruences

Congruences or canonical projections onto quotient rings; Chinese, Fermat and Euler theorems; linear congruences.

Congruences were introduced by Carl Gauss, who presented them in a simple, natural and elegant way in his classic “Disquisitiones Arithmeticae”. In this knol a more modern, algebraic approach is chosen, which helps when one goes beyond the ring of integers Z, while it gives a better understanding of number theory even in the special case of Z.

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Matematyka – esej

Matematyka - esej nigdy nie zakończony

Dziś niniejszy esej zawiera uwagi o definicji matematyki, z jednej strony jako sztuki myślenia, a z drugiej jako teorii formalnej. Ponadto wspomniany jest podział matematyki na działy, i kłopoty z tym związane. (W planie jest dodanie do niniejszego knola także uwag o (nie)skonczoności, oraz z relacji pomiędzy matematyką i zastosowaniami.)

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Congruence x^2 ≡ -1 mod p (Euler), and a super-Wilson theorem.

Wilson Euler Gauss Eisenstein work on related congruences

Euler’s contribution (one of many) Let  p > 2  be a prime (the case of  p=2  is not terribly interesting in the context of the present knol). Thus  p  is odd, and its “smaller half”  h := (p-1)/2 ≥ 1  is an integer. Let  y := z h,  where  z  is an arbitrary integer not […]

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Dabanese – dictionary (1)

Dabanese groups of synonymous dabagrams

An ever growing dictionary of dabagrams (dabanese ideographs), organized into groups of synonyms. Actually, their pigeon version. The number of real dabagrams (which use true graphics) for each group of synonyms most of the time will differ from the respective number of the pigeon dabagrams, they are not in a one to one correspondence.

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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