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3 votes
2 answers
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If You Go Fishing Everyday - What Is The Probability You Know X% Of The Pond?

This is a problem I have recently come across and have been trying to solve it mathematically. Suppose there is a pond with 100 fish Each day, there is a: 5% chance that population of the pond ...
stats_noob's user avatar
  • 3,206
22 votes
0 answers
570 views
+50

Can difference quotient sets be nowhere dense?

Let $f:\mathbb{R} \to \mathbb{R}$ be a function and consider the difference quotient set $$D_f = \left\{\frac{f(y) - f(x)}{y-x} : (x,y) \in \mathbb{R}^2, y > x\right\}$$ Can $D_f$ be nowhere dense ...
MathematicsStudent1122's user avatar
11 votes
1 answer
297 views
+100

On the map $\operatorname{Top}(X,Y \times Z) \longrightarrow \operatorname{Top}(X,Y) \times \operatorname{Top}(X,Z)$

Disclaimer: We define a topological space $X$ to be compact if every open cover has a finite subcover, but $X$ is otherwise allowed to be arbitrary. I have been faced with the following problem: Let $...
tychonovs-scholar's user avatar
0 votes
1 answer
35 views
+50

Converse of bounds on the spectrum of a Toeplitz matrix

The following is from Robert M. Gray's review (https://ee.stanford.edu/~gray/toeplitz.pdf): Lemma 4.1 Let $\tau_{n,k}$ be the eigenvalues of a Toeplitz matrix $T_n(f)$. If $T_n(f)$ is Hermitian, then ...
gen's user avatar
  • 1,468
0 votes
0 answers
85 views
+50

If integrated tail $F_I$ is subexponential, is $F$ also subexponential?

I’m currently reading Non-Life Insurance Mathematics by Thomas Mikosch and in chapter 4 on Ruin theory there is constant distinction between light and heavy-tailed distributions (as there should be) ...
VlakecTomaz's user avatar
1 vote
0 answers
91 views
+50

Calculation of special subsets in high-dimensional binary matrices

I need to solve a rather specific problem related to binary matrices. The task is to count the number of specific "combinations", where "combination" means the following: this is ...
Disciple's user avatar
  • 349
7 votes
1 answer
226 views
+100

Realized graph of majority of permutations

For any collection of permutations of $\{1,2,\dots,n\}$, we say that it realizes a directed multigraph with $1,2,\dots,n$ as vertices, such that there is an edge from $i$ to $j$ if $i$ appears before $...
user336268's user avatar
  • 2,269
-2 votes
0 answers
510 views
+100

Like Ramanujan : Strange limit and a remarkable behavior .

Problem/Context: Let $f_a\left(x\right)=a^{3-\sqrt{1+2x!\sqrt{1+3x!!\sqrt{1+4x!!!\sqrt{\cdot\cdot\cdot\sqrt{1+(k-1)x!\cdots!\sqrt{1+kx!!\cdots!}}}}}}}, g_a(x)=\left(\frac{f_a\left(x\right)}{f_a\left(0\...
Ranger-of-trente-deux-glands's user avatar
4 votes
0 answers
226 views
+100

What is the Number of Facets of a $d$-Dimensional Cyclic Polytope?

A face of a convex polytope $P$ is defined as: $P$ itself, or a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
D.S.'s user avatar
  • 62
15 votes
0 answers
175 views
+50

8000 congruent convex asteroids can form a "stable cluster". How much better can we do?

The Infinite Case The "Trapped in Thickland" puzzle in Peter Winkler's latest edition of Mathematical Puzzles asks the following (in my own words): The Infinite Asteroid Belt is a region in ...
greenturtle3141's user avatar
5 votes
1 answer
205 views
+50

Long term probability of going bankrupt?

Suppose there is a coin flipping game where you start with 5 dollars. At each turn, there is a $p$ probability of winning 1 dollar and a $1-p$ probability of losing 1 dollar. The game ends at 0 ...
konofoso's user avatar
  • 795
5 votes
3 answers
234 views
+50

Minimizing surface area of revolution for a fixed volume of revolution

Suppose I want an open-topped cup to have the capacity to hold a volume $V$ of liquid. I want to find a shape for my cup that minimises its surface area. My attempt I strongly suspect that the shape ...
Cristof012's user avatar
0 votes
0 answers
96 views
+150

Can $S/\mathfrak{q}^{[p]}$ have embedded primes?

This question stems from another question I asked on MathOverflow. Let $S$ be a regular local ring with prime characteristic $p$. Let $\mathfrak{q}$ be a prime ideal and let $\mathfrak{q}^{[p]}$ ...
Anon's user avatar
  • 448
2 votes
1 answer
109 views
+100

Spectral radius of $I-KH$, where $K$ is a Kalman gain

I have noticed numerically that the spectral radius of $I-KH$, where $K$ is a Kalman gain, is less than or equal to 1. In other words, for some symmetric positive definite matrices $R$ and $C$, and ...
Hypercube's user avatar
  • 476
2 votes
0 answers
72 views
+50

Sobolev functions approximated by ridge functions

Let $f \in W^{k,2}(\mathbb{R}^d)$, a Sobolev space with smoothness $k$ and dimension $d$. We aim to approximate $f$ using ridge functions of the form $g(\mathbf{a}.\mathbf{x})$. Suppose the ...
Maths Freak's user avatar
0 votes
0 answers
42 views
+50

Construction of a graph on even number of vertices with required eccentricities.

I was trying to construct a graph on an even number of vertices $n$ such that center and periphery contain an equal number of vertices, i.e. $|C(G)|=|P(G)| =\frac{n}{2}$. Till now, I was able to draw ...
monalisa's user avatar
  • 4,420
0 votes
0 answers
35 views
+50

bounded density for the determinant of a GOE

Let $M$ a random GOE matrix, i.e. $M=(M_{i,j})$ is a symmetric matrix and the $M_{i,j},i\leq j$ are independent centred Gaussien entries with variance 1, except on the diagonal where the variance is $...
kaleidoscop's user avatar
6 votes
0 answers
175 views
+300

Curvature operator of a Kähler manifold with constant holomorphic sectional curvature

A Riemannian metric $g$ on a manifold $M$ induces a pointwise inner product on $\Lambda^2 (TM)$, given on decomposable elements by $$ \langle X\wedge Y, Z\wedge T \rangle = g(X,Z) g(Y,T) - g(X,T) g(Y,...
Didier's user avatar
  • 19.7k
1 vote
0 answers
24 views
+50

Transition probability density of non-intersecting Brownian bridges are independent of h function from Doob-h transform?

I've been trying to derive the transition probability of a $2$-dimensional Brownian motion $B = (B^{(1)}_{t}, B^{(2)}_{t})_{t \geq 0}$ conditioned to stay in the Weyl chamber and also conditioned to ...
tornt's user avatar
  • 41
2 votes
1 answer
107 views
+50

Derivative of an Infinite Fraction

If $y= \frac{x}{x^2+\frac{x}{x^2+\frac{x}{x^2+....}}}$ and $\int\frac{y-x^2}{(x^2+y)(x+y^2)}dx = f(y)+c$, determine the value of $f'(y)$ for $x=1$ My approach: $y = \frac{x}{x^2+y} \Rightarrow \frac{y}...
Ashwin Krishnan's user avatar
1 vote
0 answers
49 views
+50

Well-posedness result for a linear parabolic equation on torus

Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$ $$ \partial_t u- ...
kumquat's user avatar
  • 129
2 votes
1 answer
97 views
+100

Defining polynomial for a compositum of splitting fields

Let $L_1,...,L_n/K$ be finite separable field extensions. Then the compositum extension $L:=L_1\cdot ...\cdot L_n/K$ is also finite and separable. Thus by the primitive element theorem, there are $\...
Nicolas Banks's user avatar
2 votes
1 answer
110 views
+100

Rigrous proof of the rate and order of convergence of bisection method

For the bisection method, we have the following error bound: $$ |x_n - x| \leq \frac{1}{2^{n+1}} |a_0 - b_0|. $$ This estimate is commonly interpreted in textbooks as indicating that the bisection ...
Veronica's user avatar
  • 474
4 votes
1 answer
83 views
+50

Proof of the Euler characteristic of the real Grassmannian $\mathbf{G}(k, n)$

I'm interested in proving the following statement, Let $\textbf{G}$(k,n) the real grassmannian and $\chi_{n,k} := \chi(\textbf{G}(k,n))$, where $\chi$ is the Euler characteristic, then $$\chi_{k,n} = \...
Fernando Avilés's user avatar
2 votes
0 answers
51 views
+100

Conditional Density Estimation in RKHS

I would like to model the conditional density of two real-valued random variable and estimate it using the empirical conditional mean embedding. I am not sure which of these two are correct way of ...
domath's user avatar
  • 1,215
13 votes
1 answer
150 views
+50

Analogue of roots of unity in n-sphere

The $n$-th roots of unity $z_1,…,z_n$ in $\mathbb{C}\equiv \mathbb{R}^2$ for $n$ prime have an interesting property: for $0\leq p<n$ and $u$ a unit vector, the sum $$\sum_{k=1}^n \langle z_k,u\...
kaleidoscop's user avatar
2 votes
2 answers
68 views
+50

Show convergence of $\sqrt N \int (\hat G_m - G)\,\mathrm d(\hat F_n - F)$ to zero in probability

Let $\hat F_n$ be the empirical CDF of a probability measure with CDF $F$. I want to prove that $$\sqrt N \int (\hat G_m - G)\,\mathrm d(\hat F_n - F)$$ converges to zero in probability as $\min(n,m)\...
Quertiopler's user avatar
3 votes
0 answers
84 views
+50

Probability of Coffee being Hotter than some Temperature?

My cousin and I have been long debating this question: Will a cafe with fresh but inferior quality coffee beans have "better coffee?" compared to a cafe with superior quality coffee beans ...
stats_noob's user avatar
  • 3,206
1 vote
1 answer
41 views
+50

Equivalence of the two functors $Alt^{k}(-*)$ and $(Alt^{k})^{*}$

I am finishing Vector Analysis of Klaus Jänich. I am stuck at chapter $12$ because I am confused about a notation. I hope some of you could untangle it for me. Lemma We can interpret each $\varphi \in ...
Matteo Aldovardi's user avatar
1 vote
1 answer
64 views
+50

Law of Large Numbers for Changing Distributions

I thought of the following problem: Suppose there is a school with 1000 students. A random sample of 50 students is selected and each of these 50 students is asked to run 100 meters and the time is ...
stats_noob's user avatar
  • 3,206
0 votes
0 answers
239 views
+500

How does the quotient affect the complex valued metric?

Take a probability distribution without the normalization factor $f_t(x)=e^{{tL}}$ for suffcient statistic $L(x)=\frac{1}{\log x}.$ The Fisher metric associates a Riemannian metric to a probability ...
zeta space's user avatar
0 votes
0 answers
60 views
+100

Diagonal metric with induced metric being diagonal as well

A nondegenerate cyclic surface in $\Bbb L^3$ (Lorentz-Minkowski space) with constant (Gaussian curvature) $K \ne 0$ is a surface of revolution. Take a surface of revolution $S\subset \Bbb L^3$ with ...
zeta space's user avatar
1 vote
0 answers
37 views
+50

a cool optimization problem involving cubes surfaces and volumes

Consider a codimension one surface of revolution $S$ and an embedding $e:S \hookrightarrow X^n$ for $X^n=[0,1]^n$ with points $p,q$ elements of $\partial X^n$ where $\partial X^n=X^n-(0,1)^n$ for $\...
zeta space's user avatar
0 votes
1 answer
73 views
+50

Degenerate perturbation theory to nonlinear equation

I want to use perturbation theory to find the steady-state solution to the following nonlinear equation: $$ x_i\left(\sum_{j=1}^Nx_j^2\right)-a x_i + \epsilon \sum_{j\neq i}^N J_{ij}x_j=0, $$ where $i=...
Sean's user avatar
  • 51
0 votes
1 answer
88 views
+100

Combinatorial proof for probabilities of random integers in ascending runs

Considering an infinite sequence of real numbers chosed randomly within the interval $ [0,1] $, each value will be higher or smaller then the previous one, and so we will get an infinite sequence of $ ...
user967210's user avatar
0 votes
0 answers
35 views
+50

Convergence rate of Laguerre coefficients for polynomially bounded functions

Suppose $f:[0,\infty)\rightarrow\mathbb{R}$ satisfies: $$f(x)= \sum_{n=0}^\infty \hat{f}_n L_n(x),$$ for some $\hat{f}_0,\hat{f}_1,\dots\in\mathbb{R}$, where $L_n$ is the $n$th Laguerre polynomial for ...
cfp's user avatar
  • 635
0 votes
0 answers
59 views
+150

The condition "mass preserving maps" in the motivation of optimal transportation

I am reading Monge's formulation of the optimal transportation problem. It says that one wishes to find the a transport map $T: (X,\mu) \rightarrow (Y,\nu)$ that minimizes the transport cost $$\int_X |...
Hanh's user avatar
  • 129
4 votes
0 answers
43 views
+50

Characterization of self-conjugate spin$^c$ structures

Let $M$ be an oriented Riemannian $n$-manifold. Then we can choose a trivializing open cover $M=\bigcup_\alpha U_\alpha$ for $TM$ and corresponding transition functions $g_{\alpha \beta}:U_\alpha \...
user302934's user avatar
  • 1,620
0 votes
0 answers
25 views
+50

Experimental Design: Selecting value of $n$ given desired width of credible interval

Suppose I have $n$ IID Bernoulli trials with $k$ successes. Assume that as a prior we are assuming that $P(\theta)$ is uniform on $[0,1]$. We can pretty easily use Bayes theorem to represent the ...
wjmccann's user avatar
  • 3,055
4 votes
1 answer
99 views
+50

If two Stratonovich SDEs are equal in distribution, do they have the same drifts?

General problem: Let $X$ and $Y$ be processes taking values in $\mathbb{R}^n$ which solve the Stratonovich SDEs $$\partial X_t = \sigma(X_t) \partial W_t$$ $$\partial Y_t = \xi(Y_t) \partial B_t,$$ ...
Nap D. Lover's user avatar
  • 1,042
-3 votes
1 answer
69 views
+50

Bounding $S(n) = \sum_{k=1}^n \mu(k) \left( \pi\left(\frac{n}{k}\right) - \pi(\text{gpf}(k)) \right)$

I am trying to bound the sum $$S(n) = \sum_{k=1}^n \mu(k) \left( \pi\left(\frac{n}{k}\right) - \pi(\text{gpf}(k)) \right)$$ In other site, I have been given the following "proof". I would ...
Juan Moreno's user avatar
  • 1,134
3 votes
0 answers
58 views
+50

Weak convergence and integral convergence

Let $f \in L^{p}(\mathbb{R}^{n})$ with $p > \max\{n/2,1\}$. Suppose $u \in L^{2}(\mathbb{R}^{n})$ and let $\{h_{n}\}_{n\in \mathbb{N}}$ be a bounded family of $H^{1}(\mathbb{R}^{n})$ functions ...
Idontgetit's user avatar
  • 1,909