Consider $n$ adversarially chosen points in $\mathbb{R}^{d}$ where $n \gg d$. Let $\mathbf{a}$ be one of the $n$ points. Is there an upper bound on the number of points among the remaining $n-1$ points, that can have $\mathbf{a}$ as the nearest neighbor?

It is easy to see that if $\mathbf{a}$ is the nearest neighbor for both $\mathbf{b}$ and $\mathbf{c}$, then if a triangle is drawn with these three points, the angle subtended at $\mathbf{a}$ is at least $60$ degrees.

I am considering an upper half plane double complex $C^{\bullet}_{\bullet}$ satisfying the following condition: consider the horizontal homology of each position in every row, $H^{i}(C^{\bullet}_{*})=0,\forall i \neq 0,1$(i.e. in each row every position is exact except the 0-th and 1-st position). Then I think there are some relations among $H^{0}(C^{\bullet}_{*}), Tot(C^{\bullet}_{\bullet})$ and $H^{1}(C^{\bullet}_{*})$. And also I am wondering is there any application of this kind of double complex in algebraic topology (like computing the homology of some topological space) or some other fields?

Let $X$ be a locally compact Hausdorff space, $\beta X$ its Stone-Cech compactification and $\Delta: X\to\beta X$ the inclusion map. Given a Borel probability measure $\mu^{\beta}$ over $\beta X$, is it possible to find a Borel probability measure $\mu$ over $X$ so that $L_1(X,\mathscr{B}(X),\mu)$ is isometrically isomorphic to $L_1(\beta X,\mathscr{B}(\beta X),\mu^{\beta})$.

I guess the answer is no (at least in such generality) and perhaps some conditions on the support of $\mu^{\beta}$ are needed in order to get the isomorphism. Probably counter-examples can be found when $\mu^{\beta}(\Delta X)<1$, but I am needing some help to construct them.

I am not that strong in mathematics and trying my best to explain my problem in words. Consider 2 sets of real numbers $X = \{x_1 = 0.1, x_2 = 0.42, x_3 = 0.11\}$ and $Y = \{y_1 = 0.2, x_2 = 0.41, x_3 = 0.12\}$. The size of ach array is $m=4$ (as there are 4 elements). These two sets or array(in programming terms) are the initial conditions for iterating a 1 D chaotic map, $f(.)$ $n$ times, which produces $n$ real valued numbers for each initial condition. Let the iteration time index be $k$.

Using, $x_1 = 0.1$, we can get an orbit $\{x_1[k]\}_{k=1}^n$, similarly for $x_2$ and rest of the numbers in $X$.

For the other array, $Y$ I repeat the same procedure. Then, obtaining the symbolic dynamics for each of the iterates $\{x_i[k]\}_{k=1}^n \in X$ and $\{y_i[k]\}_{k=1}^n \in Y$ for $i = 1,2,...,m$.

The length of each symbolic iterate is $n$. Ignoring the radix or the decimal point, I have for the array $X$, the symbolic dynamics given by $S_X = \{\mathbf{s}_{x_1},\mathbf{s}_{x_2},\mathbf{s}_{x_3},\mathbf{s}_{x_4}\}$. The length or the number of bits in this array $S_X$ is $mn$

where $\mathbf{s}_{x_1} = {s_1[1]s_1[2]...s_1[n]}$ is the symbolic dynamics for the initial condition $x_1 = 0.1$ $\mathbf{s}_{x_2} = {s_2[1]s_2[2]...s_2[n]}$ is the symbolic dynamics for the initial condition $x_2 = 0.42$ $\mathbf{s}_{x_3} = {s_3[1]s_3[2]...s_3[n]}$ is the symbolic dynamics for the initial condition $x_2 = 0.42$ $\mathbf{s}_{x_4} = {s_4[1]s_4[2]...s_4[n]}$ is the symbolic dynamics for the initial condition $x_2 = 0.42$

- Similary, I have another array $S_Y$ denoted as $S_Y = \{\mathbf{s}_{y_1},\mathbf{s}_{y_2},\mathbf{s}_{y_3},\mathbf{s}_{y_4}\}$ containing $mn$ bits representing the symbolic dynamics for $Y$.

My Questions : for the following two scenarios :

case (1) when $X = Y$, the probability that $S_{X} = S_{Y}$ is 1 but in practical sense, the symbolic dynamics are bits and there may be situations when few of the bits in $\mathbf{s}_{x_1} = \mathbf{s}_{y_1}$ or $\mathbf{s}_{x_2} = \mathbf{s}_{y_2}$ etc. How can I calculate the probability that $b$ bits are same i.e, $S_X = S_Y$ in probability sense?

case (2) when $X \neq Y$ and using the fact that due to sensitivity to initial conditions the chaotic orbits would be all different obtained from the $n$ iterates of each initial condition, the probability that $\{x_i[k]\}_{k=1}^n \neq \{y_1[k]\}_{k=1}^n$ is zero and $S_{X} = S_{Y}$ is 0. However, their symbolic dynamics (bits) again few bits can be same so entirely the probability calculated based on symbolic dynamics is not 0.

For instance, let for $X$ and $Y$ different arrays the following symbolic dynamics array is obtained where $mn = 20$

$S_X = [0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,1,0,0]$ $S_Y = [1,1,0,0,0,0,0,1,1,0,0,0,1,1,0,0,1,1,0,0]$

How to calculate the probability that $S_X = S_Y$? Putting it in (more) words, in general how to derive the expression for $N$ arrays $X_1,X_2,...., X_N$, the probability that their symbolic dynamics can be same or dissimilar, $S_{X_p} = S_{X_q}, p \neq q$?

Let $f(x,y) = f_2 x^2 + f_1 xy + f_0 y^2$ be a primitive, positive definite, and reduced binary quadratic form. Put $k_f$ for the fundamental discriminant associated to $f$. That is, $k_f$ is square-free and the splitting field of $f$ is equal to $\mathbb{Q}(\sqrt{k_f})$. Note that $k_f < 0$, by hypothesis.

We say that $f$ is *admissible* if for all primes $p | k_f$, we have $\left(\frac{f_2}{p} \right) = 1$; that is, the leading coefficient $f_2$ is a quadratic residue for all primes $p$ dividing $k_f$. Put

$$\displaystyle N^-(X) = \{f : f \text{ primitive, reduced, and admissible, with} -X < \Delta(f) < 0\}.$$

Is there an asymptotic formula known for $N^-(X)$? If we drop the condition that $f$ is admissible, then the corresponding asymptotic formula was known to Gauss and was given a modern treatment by Siegel.

I am trying to compute the asymptotic variance of OU process $$ d X_t = - H X_t dt + S dW_t $$ where $X_t$ takes value in $R^d$. How to compute its variance at $t$?

Things are classical for one dimensional case as $$ Var (X_t) = \sigma^2 / \mu (1-e^{-\mu t}) $$