arXiv · 1112.3861
A very simple but useful almost sure convergence theorem of probability is given.
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Masumi Nakajima. 2011-12-14. A Theorem of Probability. https://arxiv.org/abs/1112.3861
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In this treatise I present the solutions of the third Clay Millennium problem in the computational complexity and the fourth Clay Millennium problem in classical fluid dynamics.
This note takes the probabilistic half of the Riemann Xi story on its own terms. Every object in the subject is a first passage time: Riemann's kernel is the law of the logarithm of a sum of two hitting times of a three dimensional Bessel process, Polya's approximation is the first passage of a Brownian motion with drift, and the reciprocal Xi function, under the Riemann hypothesis, is the Laplace transform of an infinite convolution of exponentials whose rates are the squared zeros. That reciprocal is written as $F_α(s)=ξ(α)/ξ(α+\sqrt s)$, and complete monotonicity, unconditional for $α\ge1$, is conjectured to persist to the critical basepoint $α=1/2$. Kent's eigenvalue expansion says which laws can arise this way, namely those whose Thorin measure is a Dirichlet spectrum with unit atoms, and Krein's inverse spectral theory turns the hypothesis into the existence of a string. The passage from Riemann to Polya is a flow, not a jump: the Cauchy semigroup on Thorin measures, each step an Esscher tilt followed by a Brownian subordination along a curvature family of hyperbolic Bessel processes, with the arithmetic surviving as Fourier modes damped like $e^{-2πk\varepsilon}$. The arithmetic lives in the atoms and nowhere else. Approximations rank by what they keep: Polya keeps neither atoms nor tempering and is off by a factor of three, a fitted Bessel dimension reaches one per cent, and a few atoms with an erfc tempering stay better than one part in a thousand across four decades. And the flow runs backwards: the Thorin measure of the reciprocal is evaluated from a prime sieve with no reference to any zero, and nonnegative deconvolution of it returns the first ten zero ordinates with unit masses, nine of them to four decimals and one to three. All identities are verified with mpmath, and the scripts are included.
We study invariant codimension-one constraints on principal bundles through compatible pairs: a constraint distribution and a nonzero coadjoint field parallel for a principal connection. Pairing the field with the connection gives an invariant one-form whose Levi form separates a horizontal curvature contribution from a vertical coadjoint-orbit contribution. This decomposition yields criteria for integrability, contactness, and characteristic reduction; holonomy and stabilizer reductions describe global existence. For evolving compatible pairs, we identify the mixed-curvature obstruction to compatibility and prove that connection transport preserves the Levi geometry. For initially contact data over a closed base of dimension $2n$, we establish matching bounds for the quadratic $H^{n+1}$ cost of contact degeneration: making the paired curvature vanish on a Darboux ball of radius $r$ in time $T$ costs an amount comparable to $[T\log(R_*/r)]^{-1}$. The constructed paths remain contact before $T$, preserve the curvature class, and force the $L^\infty$ norm of every transporting velocity gradient to grow at least as $1/[2(T-t)]$ on the collapsing region in Darboux coordinates. On a closed three-manifold, a contact form preserved by a locally free circle action admits an invariant adapted metric with any prescribed positive curl eigenvalue and unit circle generator. We parametrize all such metrics and prove that their space is contractible. Along the circle-bundle degeneration paths, every continuous tensor limit of normalized adapted metrics is degenerate above the collapsing region.