7
$\begingroup$

(Using the advice from Mathoverflow, I have rephrased and splitted up the question)

  1. (Might be a bit layman because I don't have rigorous math term to describe the concept) Generalize it to mathematical spaces, are there spaces which are sort of like

a. Having multiple $\mathbb{R}^2$ spaces that has a finite size k but acts like as if extending to infinity (see point b) sewed together enter image description here

b. Has a weird property that for each trip wrt a parameter (e.g. time) if each distance of travel $D_i$ exceed a certain value k, you will hop to the corresponding location of the neighboring patch of $\mathbb{R}^2$ plus moving in this new patch by a distance $D \hspace{1mm}mod\hspace{1mm}k$.

If your distance of travel for any of the trips does not exceed k, then no matter how far you travel, you will still be travelling within the same patch and never reach the edge of it enter image description here

Using an example, (letting the parameter be time) property b is basically saying that you only jump to neighbouring patches (and travelling the remainder of it after dividing k) if your velocity exceed the threshold k (the size of the patch), otherwise you will only be moving within a single patch forever, even if the total distance traveled exceed k

An attempt to formulate the above descriptions mathematically:

Question: Is there a class of mathematical spaces that has similar properties as described below or a more generalized version of it?

a. For the case of $\mathbb{R}^2$, each patch is a $\mathbb{R}^2$ space and is described by ordered pair of integers $(p_{1},p_{2})$. Let $(0,0)$ be the central patch.

Therefore every point $P_{x}$ in the space can be described by $(p_{x1},p_{x2},x_1,x_2)$ and the separation $S_{xy}$ between points is described by $(\Delta p_{x1y1},\Delta p_{x2y2},\Delta x_1y_1,\Delta x_2y_2)$ where

$$\Delta x_jy_j=y_j-x_j,j=1,2$$

$$\Delta p_{x_jy_j}=p_{y_j}-p_{x_j},j=1,2$$

Let a path $\mathit{\lambda}$ connecting from $P_x$ to $P_z$ (both points are located on the same patch). $P_x'$ be the actual destination. $D$ is the total distance of the path $\mathit{\lambda}$ which is the sum of the distances of i trips (shorter partitions of the path i.e. $\lambda_i$),

$$D=\sum_{i=0}^nD_i,D \geq D_i$$

where $D_i$ is defined as the usual Euclidean distance between two neighbouring points in each partition

$$D_i=\sqrt{\sum_{j=1}^2\left(x_{(i+1)j}-x_{ij}\right)^2}$$

Let $d$ be the Euclidean separation between $P_x$ and $P_z$ defined as

$$d=\sqrt{\sum_{j=1}^2\left(z_j-x_j\right)^2}$$

b. Each patch has a threshold $k$ where $k \in \mathbb{Z}$ such that

If $\exists D_i>k$ then $P_x'$ has coordinates

$$\left\{\begin{matrix} (p_{x1}+1,p_{x2},x_1+D \hspace{1mm}mod\hspace{1mm}k,x_2),z_1>x_1,z_2=x_2 \\ (p_{x1},p_{x2}+1,x_1,x_2+D \hspace{1mm}mod\hspace{1mm}k),z_1=x_1,z_2>x_2 \\ (p_{x1}+1,p_{x2},x_1-D \hspace{1mm}mod\hspace{1mm}k,x_2),z_1<x_1,z_2=x_2 \\ (p_{x1},p_{x2}+1,x_1,x_2-D \hspace{1mm}mod\hspace{1mm}k),z_1=x_1,z_2<x_2 \\ \\ (p_{x1}+1,p_{x2}+1,x_1\left(1+\frac{D \hspace{1mm}mod\hspace{1mm}k}{d}\right),x_2\left(1+\frac{D \hspace{1mm}mod\hspace{1mm}k}{d}\right)),z_1>x_1,z_2>x_2 \\ (p_{x1}+1,p_{x2}-1,x_1\left(1+\frac{D \hspace{1mm}mod\hspace{1mm}k}{d}\right),x_2\left(1-\frac{D \hspace{1mm}mod\hspace{1mm}k}{d}\right)),z_1>x_1,z_2<x_2 \\ (p_{x1}-1,p_{x2}+1,x_1\left(1-\frac{D \hspace{1mm}mod\hspace{1mm}k}{d}\right),x_2\left(1+\frac{D \hspace{1mm}mod\hspace{1mm}k}{d}\right)),z_1<x_1,z_2>x_2 \\ (p_{x1}-1,p_{x2}-1,x_1\left(1-\frac{D \hspace{1mm}mod\hspace{1mm}k}{d}\right),x_2\left(1-\frac{D \hspace{1mm}mod\hspace{1mm}k}{d}\right)),z_1<x_1,z_2<x_2 \\ \end{matrix}\right.$$

If there is no $D_i>k$ then regardless of whether $D>k$

$$P_x'=P_z$$

.

For the case of $\mathbb{R}^n$

a. Each patch is a $\mathbb{R}^n$ space and is described by a vector $\mathbf{p}$ with integer components. Therefore $\mathbf{p=0}$ is the central patch.

Therefore every point $P_x$ in the space is described by ($\mathbf{p_x}$,$\mathbf{x}$) and the separation $\mathbf{S_{xy}}$ is described in a similar fashion to the $\mathbb{R}^2$ case as ($\mathbf{\Delta p_{xy}}, \mathbf{\Delta xy}$). The path $\lambda$, the points $P_z$, $P_x'$, the distances $D$, $D_i$ and the threshold $k$ are generaliation of the $\mathbb{R}^2$ case. The vector $\mathbf{d}$ which go from $P_x$ to $P_z$ is defined as follows

$$\mathbf{d}=\mathbf{z}-\mathbf{x}$$

such that $d=|\mathbf{d}|$ and $\hat{\mathbf{d}}=\frac{\mathbf{d}}{d}$

b. Each patch has a threshold $k$ where $k \in \mathbb{Z}$ such that

If $\exists D_i>k$ then the coordinates of $P_x'$ is

$$(\mathbf{p_x+}\sum_{j=1}^n \epsilon_j \mathbf{e_j},\mathbf{x+}(D \hspace{1mm}mod\hspace{1mm}k)\hat{\mathbf{d}})$$

where

$$\epsilon_j=\left\{\begin{matrix} 1,\mathbf{d \cdot e_j}>0\\ -1,\mathbf{d \cdot e_j}<0\\ 0,\mathbf{d \cdot e_j}=0 \end{matrix}\right.$$

where $\mathbf{e_j}$ are unit vectors along the $j,m\in [1,n]$ cartesian directions and $\mathbf{e_j}\cdot\mathbf{e_m}=\delta_{jm}$

If there is no $D_i>k$ then regardless of whether $D>k$

$$P_x'=P_z$$

I would like some suggestion or sources for further reading on such objects if they exist in order to learn more about their properties

$\endgroup$
1
  • $\begingroup$ Mathoverflow said this is not research mathematics, thus they suggest me to improve my question and post it here instead $\endgroup$
    – Secret
    Nov 27, 2014 at 4:58

0

You must log in to answer this question.

Browse other questions tagged .