JZdynamics

The following diagram is a high explosive lens mold from the Manhattan project, drawn in secret by David Greenglass, to pass on to the Soviets: Below is a 3D symmetrical extension of this high explosive lens mold: This is precisely $\mathcal I$ which is a special subset of $CX_V$ in dimension $n=3$. You can see the outer colored loops that wrap around $\mathcal I$. This is a subset of $\Gamma$. This also gives a clean example of the coloring function $\phi: E \to \mathcal C$.  The flat local system $\mathcal L$ lives on these colored loops and it is inherited directly from the geometry of $\mathcal I$. Thus, we can form the enrichment $\mathcal X=(\phi,\mathcal L, \rho).$ For more details on this see the previous post:  https://jzdynamics.blogspot.com/2025/04/the-geometry-of-delta-groupoid.html .  Without going into too much detail on the physics - the right thing to do is to take geodesic flows along the white strands and add (i.e. direct sum) the force vectors restricte...

This post serves to marry the last three posts into one cohesive framework. The connection to foliational completions is also made clear. Enriched Graphs and Delta Groupoid Let $\Gamma = (V, E)$ be a finite connected tailless regular planar graph or multigraph, embedded in $\mathbb{R}^3$. Define an enrichment of $\Gamma$ to be a triple: $$\mathcal{X} = (\phi, \mathcal{L}, \rho)$$ where: - $\phi: E \to \mathcal{C}$ is a coloring or stratification into a finite set of colors $\mathcal{C}$, - $\mathcal{L}$ is a local system (functor from the path groupoid of $\Gamma$ to $\mathrm{Vect}_k$), assumed flat, - $\rho: \pi_1(\Gamma) \to \mathrm{Aut}(\mathcal{L})$ is a holonomy representation. Let $\mathcal{E}_\Gamma$ denote the set of all such enrichments. Define the Delta groupoid $\mathcal{D}_\Gamma$ as follows: - Objects: enrichments $\mathcal{X} \in \mathcal{E}_\Gamma$. - Morphisms: invertible maps   $$\delta: (\phi, \mathcal{L}, \rho) \longrightarrow (\phi', \mathcal{L}', \rho')...

 Let $\Gamma = (V, E)$ be a finite connected tailless regular planar graph or multigraph embedded in $\mathbf R^3$. Define an enrichment of $\Gamma$ to be a triple: $$\mathcal{X} = (\phi, \mathcal{L}, \rho)$$ where, $\phi: E \to \mathcal{C}$ is a coloring or stratification of the edge set into a finite set of colors $\mathcal{C}$. $\mathcal{L}$ is a local system (i.e., a functor from the path groupoid of $\Gamma$ to $\mathrm{Vect}_k$). It is assumed a priori that connections are flat. $\rho: \pi_1(\Gamma) \to \mathrm{Aut}(\mathcal{L})$ is a holonomy representation. Let $\mathcal{E}_\Gamma$ denote the set of all such enrichments. The Delta groupoid $\mathcal{D}_\Gamma$ is defined as follows: Objects are enrichments $\mathcal{X} = (\phi, \mathcal{L}, \rho) \in \mathcal{E}_\Gamma$. Morphisms are invertible maps $$\delta: (\phi, \mathcal{L}, \rho) \longrightarrow (\phi', \mathcal{L}', \rho')$$ that satisfy the following properties: The underlying graph $\Gamma$ is fixed, thus t...

Let $\Gamma = (V, E)$ be a finite, planar labeled, undirected, tailless, topological multigraph ($4$-regular) arising from a $z=0$ slice of a stratified foliated manifold. The graph admits a decomposition into two overlapping subgraphs ("ovals") $\mathcal{O}_1, \mathcal{O}_2 \subseteq \Gamma$, such that their union reconstructs $\Gamma$: $$\Gamma = \mathcal{O}_1 \cup \mathcal{O}_2, \quad \text{with } \mathcal{O}_1 \cap \mathcal{O}_2 \neq \varnothing.$$ We define a coloring function $$\phi: E \to \mathcal{C} = \{c_1, c_2\}$$ that assigns each edge a domain label corresponding to one of the two ovals. This coloring encodes the stratified foliation structure by partitioning edge interactions into two local holonomy domains. Each edge $e \in E$ carries a local holonomy map $\tau_e \in \mathrm{Aut}(\mathcal{F})$, where $\mathcal{F}$ is a coefficient system (e.g., a local system or vector bundle). For any based cycle $\gamma = (v_0 \to e_1 \to \cdots \to e_k \to v_0)$, the holonomy...

I'll start off with a definition that might initially sound pretty abstract and maybe confusing. The point in defining things as I do is to make our lives easier later on. The goal is to focus on a symmetrical subspace of what I'll call the $\mathcal F$-completion. This subspace is called $\mathcal I$. So let's begin: Let $X$ be a smooth, open, geodesically convex $n$-dimensional manifold with regular polytope boundary, and let $V = \{v_i\}$ be a finite set of vertices in $\partial X$. A block is defined as $\mathcal{B} := X \cup \partial X$, where $\partial X$ denotes the boundary. A foliational completion (or $\mathcal{F}$-completion) of a block $\mathcal{B}$ is a structure obtained by extending foliations to accumulate at specific vertices in the boundary. Formally, for each pair of distinct vertices $(v_i, v_j) \in V \times V$, there exists a foliation $\mathcal{F}_{v_{ij}}$, whose leaves accumulate at $v_i$ and $v_j$. The $\mathcal{F}$-completion is the union of these ...

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 ...

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...

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...

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))}{...