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The $\Gamma$-sets and $\zeta$-space.

Assume $X$ is a smooth convex open manifold and $\partial X$ is a convex regular polytope. Def: An *$\mathcal F$-completion,* or *foliational completion* of a *block*, $\mathcal B:=X\cup\partial X$ over its vertex set $V$, is $\mathscr CX_V:=\cup_{v_{ij}\in V} \mathcal F_{v_{ij}}$ where $i,j$ index the vertices.  Example: Set $X=(0,1)^n$. Take a smooth foliation $\mathcal F$ for every pair of points $v_i,v_j\in \partial X=[0,1]^n-(0,1)^n$ satisfying $\mathrm{sup~dist}_n(v_i,v_j)=\sqrt{r}$. Let the leaves in each $\mathcal F$ be mutually diffeomorphic to the class $M=(0,\sqrt{n})\times S^{n-2}$ accumulating to $v_i,v_j$. We see that $r=\lbrace 1,\sqrt{n} \rbrace.$ Allowing only $r=\sqrt{n}$, we can look at sections of $\mathscr CX_V$: Def: A $\Gamma$-set on $\mathscr CX_V$ is the union of all leaf intersections of the foliations i.e.  $$\Gamma_{n}:= \lbrace \mathcal (\mathcal F_1 \cap \mathcal F_2)\cup (\mathcal F_2 \cap \mathcal F_3) \cup (\mathcal F_3 \cap \mathcal F_4) \cup \cdot\cdo
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Gluing leaves in zeta space and automorphisms on the resulting quotient spaces

Define an analytic planar homotopy $h_t: (0,1) \to (0,1)$ for $t\in[1/3,3]$ where $h_t(x)=e^{\frac{t}{\log x}}$ s.t. $h_{1/3}=e^{\frac{1/3}{\log x}}$ and $h_3=e^{\frac{3}{\log x}}$ for $x\in(0,1)$. I want a rotational homotopy that projects onto the planar homotopy. So, define the equivalence relation on the boundary curves of the closed subset of $(0,1)^2$, that is $h_{1/3}\sim h_3$, where points are identified by straight lines with slope $1$, connecting $h_{1/3}$ and $h_3$. This yields a spherical topology $\mathcal S$ embedded in $(0,1)^3$ with the endpoints of the sphere being singularities at $p=(0,1,1)$ and $q=(1,0,0)$.  Conceptually it's clear that a projection $\pi: (0,1)^3 \to (0,1)^2$ takes the leaves on $\mathcal S$ to $h_t(x)$ as a double cover. But my main objective of study is to put a metric on $\mathcal S$. The $1$-parameter metric I derived on the planar space is $g_t(r)=\int_{(0,1)} x^{r-1}h_t(x)~dx = 2\sqrt{\frac{t}{r}}K_1(2\sqrt{tr})$ and it measures the distan
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Extending the base metric to the quotient metric in $\zeta$-space

Define an analytic planar homotopy $h_t: (0,1) \to (0,1)$ for $t\in[1/3,3]$ where $h_t(x)=e^{\frac{t}{\log x}}$ s.t. $h_{1/3}=e^{\frac{1/3}{\log x}}$ and $h_3=e^{\frac{3}{\log x}}.$ I want a rotational homotopy that projects onto the planar homotopy. So, define the equivalence relation on the boundary curves of the closed subset of $(0,1)^2$, that is $h_{1/3}\sim h_3$. This yields a spherical topology $\mathcal S$ embedded in $(0,1)^3$ with the endpoints of the sphere being singularities at $p=(0,1,1)$ and $q=(1,0,0)$.  Conceptually it's clear that a projection $\pi: (0,1)^3 \to (0,1)^2$ can take the leaves on $\mathcal S$ to $h_t(x)$ as a double cover. I would also like to put a metric on $\mathcal S$. The metric I derived on the planar space is $g_t(r)=\int_{(0,1)} x^{r-1}h_t(x)~dx = 2\sqrt{\frac{t}{r}}K_1(2\sqrt{tr})$ and it measures the distance between the $h_t(x)$ leaves. For example the distance between leaf $t=1/3$ and leaf $t=3$ would be a subtraction of integrals $d(1/3,3
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Zeta space as a probabilistic model

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Consider $ S^n,$ the space of Schur-convex, simply connected, closed topological $n-$manifolds as subsets of the unit $(n+1)-$cube, which include $p=(0,0,\cdot\cdot\cdot,0)$ and $q=(1,1,\cdot\cdot\cdot,1).$ Consider the set of all Schur-convexity preserving maps from $S^n$ to itself. Define a class of $S^n,$ denoted $\zeta(\mathbf X),$ in the following way: Let $ L^n_+$ be the set of all $n$-dimensional nonnegative random vectors $\mathbf X = (X_1, X_2,\cdot\cdot\cdot,X_n)^⊤$ with finite and positive marginal expectations, and let $\mathbf Ψ^{(n)}$ be the class of all measurable functions from $\Bbb R^n_+$ to $[0, 1].$ Then $\zeta(\mathbf X)$ of the random vector $\mathbf X$ with joint CDF $F$ is: $$\zeta(\mathbf X)=\bigg\{\bigg(\int \psi(\mathbf x)dF(\mathbf x), \int \frac{x_1\psi(\mathbf x)}{E(X_1)}dF(\mathbf x),\cdot\cdot\cdot,\int \frac{x_n\psi(\mathbf x)}{E(X_n)}dF(\mathbf x):\psi \in \mathbf Ψ^{(n)}\bigg)\bigg\}, $$ $$ =\bigg\{\bigg(E\psi(\mathbf X), \frac{E(X_1\psi(\mathbf X))}{
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Automorphisms and zeta space

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Here I will operate under the assumption that $\phi_s$ is a unique and well-posed solution to the following PDE. This means whatever conditions are needed for that to occur, will be imposed. $$s\frac{\partial^2}{\partial s^2}\phi_s(x)=\mp x\frac{\partial}{\partial x}\phi_s(x) $$ for which we have our pre-assigned unique solution: $$\mathcal \phi_{s}:=\bigg\lbrace e^{\frac{\pm s}{\log x}}:s \in \Bbb R \bigg\rbrace$$ We have singularities at $s=0$ and $x=0,1$ and this will be called a "cross singularity" or  $CS$. A natural idea would be to intialize some distribution on this $CS$ and run time forward giving a wellposed analytic solution for all time. So we choose our pre-assigned solution $\phi_s$ and run the solution back in time where we obtain what we wanted, at least qualitatively. We can throw out our solution for $s<0$ because it doesn't directly apply to the situation, and it does not rapidly decay. Meanwhile, for $s>0$ it is Schwartz i.e. has a rapid decay cr
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Representation of Zeta space from a PDE and combinatorial viewpoint

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Consider the following partial differential equation: $$ s \frac{\partial^2}{\partial s^2} \phi(x,s) = - x \frac{\partial}{\partial x} \phi(x,s) $$  with our classical analytic generator as in the previous post, $\phi_s(x)=e^{\frac{s}{\log x}}$. This local foliation generates zeta space on the base planar space.  Now, each leaf of the foliation directly corresponds to a Schwartz function therefore we can build out corresponding symmetric function equations and gather critical strips for each leaf. The goal going forward will be to establish a very particular generalization of the leaves on the base space and allow them to sit in $\Bbb R^3$. They will form football like structures and the football structures can be seen as sections sitting over top of the base space. In fact, the base space of the foliation will be pasted onto each "face" of the corresponding cube, wherein the football structures will be promoted within the cube. Then we will study the combinatorics of zeta sp
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A description of $\zeta-$space

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I would like to attempt to define a mathematical space I call $\zeta-$space.  $\zeta-$space has a description as the transversal intersection of two isometric pseudo-Riemannian manifolds (Lorentzian submanifolds) both equipped with the Lorentz metric. It is somewhat laborious to describe and derive $\zeta$ because it is not a standard approach and $\zeta$ does not have a clean representation so to speak. However it is a fundamentally important mathematical and physical object in its own right and should be studied. I define $\zeta$ in the following way: $\zeta:=\{\varphi_S \} \cap \{ \varphi_T \}$ Under this first level description, $\zeta$ is the intersection between two class structures denoted as $\varphi_S$ and $\varphi_T.$ These class structures can be thought of as a family of functions and also as lines of constant time and constant space respectively: $$ \varphi_S(x) = e^{\frac{S}{\log(x)}} $$ $$ \varphi_T(x) = e^{\frac{T}{\log(1-x)}}$$ Here I view $S,T$ as fundamen
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