Friday, November 29, 2013

unprepared for dinner yet (my wife just messaged)

unprepared for dinner yet:
lacking one thing for each menu to cook
no garlic for pasta
no celery for minestrone
no flour for chowder
no meat for stewed meat
and, only cabbage for casserole

Sunday, November 24, 2013

Who's behind this silly site? (egocentric advertisement)

NOTICE
  • Just Kidding
  • Who's behind this silly site? The uneducated blogger answers. Ask me in English (Japanese question might be answered). Anonymous question is allowed, feel free to visit http://ask.fm/TuvianNavy

Wednesday, November 20, 2013

On Personal Matters and Forthcoming Posts (update November 2013)

For my and my family's health concern, I left Tokyo and moved to Kyoto from September 2013. For courtesy of my employer, my labor contract is kept unchanged. I have some heartache, headache, slightly high blood pressure and random pains but feel somewhat eased. My wife's throat turned well. We enjoy bigger rooms and now I have bookshelf!

I am preparing (at least) two essays, one on history of extensive thoughts for recent four centuries and the other on some biographical facts of ergodic theorist Shizuo Kakutani. Other subjects which have appeared in this blog would be rewritten in paper form.

There is no actual plan for forthcoming posts, but dimension theory (for cases of discrete and continuous dimensions) must be treated first.

(added) I also have a long-awaited plan for simple explaination of various versions of fixed point theorems in economical context, but I have no through understanding on this subject yet.

Tuesday, November 19, 2013

Shelah's Galois theory (memorandum)

"As an undergraduate, I found Galois theory beautiful, (more exactly what is in the book of Birkhoff-Maclane), and later I found Morley's theorem (with its proof) beautiful." -- Saharon Shelah, "The Future of Set Theory"
NOTICE
  • DO NOT TRUST THE FOLLOWING
RESEARCH CONTEXT
MOTIVATIONS
  • Abelian variety is product of symmetric groups
  • Galois theory states that solvable algebraic equations have their corresponding abelian groups
  • Lie group is continuous, or infinite group
  • Eigenvalue problem of (infinite dimensional) unitary operator should be "solvable" in certain meaning (gruppenpest)
  • (added) Kakutani-Halmos: unitary oparator can be factorized as product of 4 symmetries, but not always as product of 3 symmetries
  • What is Galois theory involving Lie groups?
  • What is Galois theory involving groups of infinite degree?
HISTORY
  • Sophus Lie and Friedrich Engel's Theorie Der Transformationsgruppen (1888) was motivated from Helmholtz' response to Riemann(1868)
  • Cousin problems
  • Whitehead conjecture
  • Shelah's proof (V=L -> Whitehead group is free abelian group)
SPECULATIONS
  • free group represents classical mechanics (zero curvature)
  • non-free group represents "viscosity" of "elementary" mathematical objects
  • regarding a cardinal as an infinite group which could be factorized into product of several symmetric infinite groups
  • cofinality of singular cardinal might be degree of "normal subgroup" of this cardinal
  • There's some arbitrariness in "factorization", or "cardinal division"
  • (added) Such arbitrariness disappears when GCH holds (so under V=L too)
TODO
  • clarify relations to Silver's works on singular cardinals (Silver was a pupil of Vaught, so his works also might have model theoretic implications)
  • (added) "cardinal division" is inadequate. what is cofinality when GCH holds?
  • (added) and how "ideal thoery of models" is?
  • (added) clarify relations to entscheidungsproblem of transfinite functions (concerning complexity of elementary formula allowing recursively enumerable symbols of transfinite functions)

Saturday, November 9, 2013

John von Neumann and computational fluid dynamics (memorandum)

NOTICE
  • DO NOT TRUST THE FOLLOWING
HISTORY
  • (added) "Recent theories of turbulence" (von Neumann 1949), esp. chap. VI
  • argument on analogy of "ultraviolet catastrophe" appears in chap. II
  • (added) See Ryder and Mattsson's slide: CFD before CFD
POINTS
  • In the wide spectrum of John von Neumann's works, relation between early computational fluid dynamics (including explosives, atomic bomb and weather forecast studies), continuum geometry and other fundamental contributions to logic and quantum mechanics does not seem well explained
  • continuum geometry is unclear to me, so I would consider later
  • any real fluid (except superfluid helium) is somewhat compressible and have certain viscosity
  • regarding fluid as quantum multi-body system, interaction matters
  • Artificial viscosity, as convergence parameter in CFD computations, is one of JvN's inventions and he should have been aware of manipulation of AV is a form of ultraviolet cutoff
  • Mathematical intuitionism (of Brouwer and Weyl) mandates "UV cutoff" to any mathematical objects (no such thing like non-measurable and null set of continuum cardinality might be permitted to appear in intuitionist reasoning): Intuitionist continuum is "viscous"
  • Some efforts on Banach-Tarski paradox and two-dimensional analogues, topological/amenable(at first it was called "measurable") groups and regular rings preceding the huge theory of operator rings seems to be focused on defining "intuitionistically valid" objects in classical mathematics
TODO
  • settle continuum geometry in above account
  • examine Navier and Stokes' reasonings of FD