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# Background
I have two parametric curves, and I want to find the parameter values of their intersection point closest to zero.
The first curve is an involute of a circle with parameter theta:
$$
x_1(θ) = \frac{Z}{2} ⋅ \cos(α) ⋅ (\cos(θ − \tan(α) + α) + θ ⋅ \sin(θ − \tan(α) + α)),
$$
$$
y_1(θ) = \frac{Z}... |
Is it possible to find the intersection of this involute and roulette, given their parametric equations? |
Let $(X,d)$ be a metric space and let $(X^*,d^*)$ denote its completion (via equivalence classes of Cauchy sequences in $X$). Let $f:X \to X^*$ be a bijective isometry such that $f(X)$ is dense in $X^*$.
Now define
$$
X'=(X^*\setminus f(X))\cup X
$$
and a map $g\colon X'\to X^*$ by
$$
g(x)=\begin{cases}
f(x) ... |
Can we include a metric space into its completion? |
>Let $P(X=1)=P(X=2)=P(Y=1)=P(Y=2)=\frac{1}{2}$.
Find the smallest and largest possible value of $E[XY]$.
My solution:
If $X$, $Y$ are independent, then $E[XY] = (\frac{3}{2})^2 = \frac{9}{4}$.
Now, let's assume they are dependent in such a way:
1) If $X=1$ then $Y=1$ and if $X=2$ then $Y=2$, then $E[XY] = ... |
The typical definition of expectation requires a **probability space** and a **random variable**
> Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(\mathsf{X}, \mathcal{X})$ be a measurable space and $\mathrm{X}:\Omega\to\mathsf{X}$ be a random variable. The expectation of $\mathrm{X}$ is defined as... |
Let $(X,d)$ be a metric space and let $(X^*,d^*)$ denote its completion (via equivalence classes of Cauchy sequences in $X$). Let $f:X \to X^*$ be an isometry such that $f(X)$ is dense in $X^*$.
Now define
$$
X'=(X^*\setminus f(X))\cup X
$$
and a map $g\colon X'\to X^*$ by
$$
g(x)=\begin{cases}
f(x) & x\in X ... |
There are a _lot_ of ways to understand adjoints, because they show up in many places, each of which gives a different perspective on things. For instance, if you're familiar with some [galois theory](https://en.wikipedia.org/wiki/Galois_theory) or [covering space theory](https://en.wikipedia.org/wiki/Covering_space), ... |
How to derive an inequality for the solution of a differential equation based on a differential inequality? |
Why could I expect $f$ can extend?
I found counterexample in lower dimension.
There is no polynomial function in $k^2$ satisfying
$f(x,y)=y$ on $x=0$, $f(x,y)=0$ on $y=0$, and $f(x,y)=0$ on $y=x$
(because $f$ must divisible by $y(y-x)$)
If rational function is allowed, I have for example $y(y-x)/(y+x)$ (... |
Let $\mathcal F := \{f_\theta\mid\theta\in\mathbb R^p\}$ be a parametric family of real-valued, continuous functions defined on $[0,1]^d $, such that for all $\theta\in\mathbb R^p$,
$$|f_\theta(x)| \le F<\infty \,\,\forall x\in[0,1]^d, $$
for some $F>0$. Now let $\rho$ be a probability distribution on $[0,1]^d\time... |
Let $(X,d)$ be a metric space and let $(X^*,d^*)$ denote its completion (via equivalence classes of Cauchy sequences in $X$). Let $f:X \to X^*$ be an isometry such that $f(X)$ is dense in $X^*$.
Now define
$$
X'=(X^*\setminus f(X))\cup X
$$
and a map $g\colon X'\to X^*$ by
$$
g(x)=\begin{cases}
f(x) & x\in X ... |
How do we find locus of P looking at two foci distance $a$ apart such that
$$ \cos \phi/\cos \theta =- k~? $$
(k positive real). Vaguely remember it comes from Electromagnetic theory. Thanks in advance.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/lgSpZ.png |
The typical definition of expectation requires a **probability space** and a **random variable**
> Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(\mathsf{X}, \mathcal{X})$ be a measurable space and $\mathrm{X}:\Omega\to\mathsf{X}$ be a random variable. The expectation of $\mathrm{X}$ is defined as... |
It can be shown that taking $y = \frac{1}{x}$ and rotating about the origin can produce (part of) a hyperbola (or already is a hyperbola, technically). Is there such a relation in $3$D? Does the equation $xyz = 1$ form a hyperboloid?
I tried figuring this out with a similar approach as one would in $2$D: converting ... |
Does $xyz = 1$ form a hyperboloid? |
Let $F$ be a multiplicative generalized cohomology theory.
Let $\sigma^0 \in F^0(point) \simeq \tilde{F}^0(S^0)$, and let for any $n\in \mathbb{Z}$, $\sigma^n \in \tilde{F}^n(S^n)$ be the element associated to $\sigma^0$ under the suspension isomorphism.
In the famous paper of Atiyah-Bott-Shapiro (Clifford Mod... |
Let $A$ be an abelian group.
The definition of the Pontryagin dual of $A$ is $\text{Hom}_{\text{conti}}(A, \mathbb{Q}/\mathbb{Z})$.
In this context, what are the topologies on $A$ and $\mathbb{Q}/\mathbb{Z}$?
Is it the cofinite topology on $A$ and the discrete topology on $\mathbb{Q}/\mathbb{Z}$, or does that ... |
I try to find a balanced exercise on it with Desmos.
It seems we have:
$$\sum_{n=1}^{\infty}\frac{\gcd\left(1+n!,1+n^{2}\right)}{n!}\overset?=e$$
It seems non trivial as you can remark in replacing the square by a cube.
At first glance we can think it's problem with hidden rearrrangement theorem. Keeping ... |
Let $F$ be a multiplicative generalized cohomology theory.
Let $\sigma^0 \in F^0(point) \simeq \tilde{F}^0(S^0)$, and let for any $n\in \mathbb{Z}$, $\sigma^n \in \tilde{F}^n(S^n)$ be the element associated to $\sigma^0$ under the suspension isomorphism.
In the famous paper of Atiyah-Bott-Shapiro (Clifford Mod... |
I know this question seems too easy to post it here (it is for 9th graders in India) but somehow I had to skip classes at school and couldn't attend geometry and now I am stuck with a question I couldn't solve
> In the given semicircle; ABCDE is a cycle? pentagon in which AB is diameter. Find $\angle C + \angle E$.
[... |
The typical definition of expectation requires a **probability space** and a **random variable**
> Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(\mathsf{X}, \mathcal{X})$ be a measurable space and $\mathrm{X}:\Omega\to\mathsf{X}$ be a random variable. The expectation of $\mathrm{X}$ is defined as... |
My attempt: take the interval $H=(0,1)$. Assume $(0,1)$ is countable, then there exist a bijection from $\mathbb N$ to $H$ where. Define elements of $H$ as a monotonically increasing sequence $\{a_i\}_{i=1}^{\infty}$ , $H=\{a_i | i \in\mathbb N\}$. However $a_1^{2} \in (0,1)$ and is not an element of $H$ since $a_1^{2... |
I am currently learning multivariate function calculus and trying to figure out the relationship between differentiability, continuity, and directional derivatives. My main question is if a directional derivative exists in all directions at a point $(x_0, y_0)$ and satisfies the equation
$$D_{\vec{u}}f(x_0, y_0) = ... |
The equation of a hyperbola is
$ x^2 - y^2 = r $
Assuming $ a = b = 0 $
Suppose now the equation
$ x^2 - y^2 + z^2 = r $
What kind of curve would this equation parametrize ? What is its geometry?
Since $ x^2 + y^2 = r $ is the equation of a circle would this be like a mix between a hyperbola (in... |
There are 9 people such as P1,P2,...,P9 who have to stand in a line but P1 and P2 will each get angry if they are not in the first 3 or last 3 spots in line. How many ways can these people line up such that both P1 and P2 will not get angry?
I am confused as to what formula do I have to use to get the answer. |
Permutation or Combination? |
Given are the points $A, B, C, D$ of an octahedron with equal edge lengths (picture). We call the remaining vertices $S1$ and $S2$ and the center of the octahedron $M$.
Show that $\overline{AM} = \overline{MS_1}$.
There is a solution to this problem in which it is argued that the quadrilateral $AS_2CS_1$ is a squar... |
How do we find locus of P looking at two foci distance $a$ apart such that
$$ \cos \phi/\cos \theta =- k~? $$
(k positive real). Vaguely remember it comes from Electromagnetic theory. Thanks in advance.
[![enter image description here][1]][1]
$$ \frac{r}{\sqrt{r^2+a^2-2 ar \cos \theta}}=\frac{\sin \phi}{\si... |
Set theory - functions: Given that $|A|=|B|$, how to prove that $|A^C|=|B^C|$? |
How do we find locus of P looking at two foci distance $a$ apart such that
$$ \cos \phi/\cos \theta =- k~? $$
(k positive real). Vaguely remember it comes from Electromagnetic theory. Thanks in advance.
[![enter image description here][1]][1]
$$ \frac{r}{\sqrt{r^2+a^2-2 ar \cos \theta}}=\frac{\sin \phi}{\si... |
Lets say I have {a,b,c,d}
I though of (a,b)(b,c) - but this is antisymmetric, right?
Then I though of (a,b)(b,a)(b,c) but this time is transitive
Finally I tried with (a,b)(b,c)(c,d) but again, is this antisymmetric?
|
Can a relation be non reflexive, non symmetric, non antisymmetric and not transitive? |
Let $A$ be an abelian group.
The definition of the Pontryagin dual of $A$ is $\text{Hom}_{\text{conti}}(A, \mathbb{Q}/\mathbb{Z})$.
In this context, what are the topologies on $A$ and $\mathbb{Q}/\mathbb{Z}$?
Is it the cofinite topology on $A$ and the discrete topology on $\mathbb{Q}/\mathbb{Z}$, or does that ... |
Suppose $A$ is a $d\times d$ matrix with IID standard normal entries.
Plots below compare value of $f(AA^TAA^TAA^TAA^T)$ and $f(AAAAA^TA^TA^TA^T)$ using 3 standard Schatten [norms](https://en.wikipedia.org/wiki/Schatten_norm) for $f$ and the former is consistently larger, why? [![enter image description here][1]][1]... |
Why is $AA^TAA^TAA^TAA^T$ larger than $AAAAA^TA^TA^TA^T$ for Gaussian $A$? |
Consider a bipartite quantum system described by the density operator, $\hat{\rho}$, an operator acting on the Hilbert space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. This matrix will be a vector (that in quantum physics is denoted with a double bra or ket $|\rho\rangle\rangle$) in the Liouville space, $\mat... |
Why is this not always true? My rationale is that some vector c (mx1) must either be in the columnspace or nullspace of A. If c is in the nullspace of A, it must be in the nullspace of BA. If c is the in nullspace of BA is must be in the nullspace of A because BAc = zero then B(Ac)=0 and the only vector that can make B... |
Are the columnspace and the nullspace not always the same for some matrix A (mxn) and some product of matrices BA, where B is invertible (mxm)? |
Most often (at least in probability), one defines the $L^p$ space as
> Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a *probability space* and let $p\geq 1$ be a real number. Then
$$
L^p(\Omega, \mathcal{F}, \mathbb{P}) := \left\{f \left| f:\Omega\to\mathbb{R} \text{ is } \mathcal{F}\text{-measurable and } \int |f(\o... |
Is it possible to define $L^p$ spaces using a non-sigma-finite measure space and a Banach space? |
A buzzer sounds every $15$ minutes. If the buzzer sounded at $12:40$PM, which of the following could be a time at which the buzzer sounded?
A. $4:05$
B. $5:30$
C. $6:45$
D. $7:15$
E. $8:10$
I chose E.
This is exactly the same as but I don't know why as if the buzzer sounded at $12:30$PM and the answe... |
In the figure below $DM=DN$ and $\dfrac{S_{\text{ADN}}}{S_{\text{ABC}}}=\dfrac{6}{121}$.
What is $\dfrac{AB}{AD}?$
My attempt:
Knowing that ${S_{\text{ADN}}}={S_{\text{ADM}}}$, and since the distances of $M$ and $N$ to $AB$ are the same, I tried to find a relation between ${S_{\text{ABC}}}$ and ${S_{\text{ABM}... |
What's the ratio? |
Let $\Omega \subset \mathbb{R}^n$ be a (not necessarily bounded) domain and $1 \leq p \leq \infty$. Then define $L^p_{loc}(\Omega)$ to be the set of functions $f: \Omega \rightarrow \mathbb{R}$ such that
$$\Big(\int_K |f|^p\Big)^{1/p} < \infty, \quad \forall K \text{ compact.}$$
I have read that this space is a co... |
How do we find locus of P looking at two foci distance $a$ apart such that
$$ \cos \phi/\cos \theta =- k~? $$
(k positive real). Vaguely remember it comes from Electromagnetic theory. Thanks in advance.
[![enter image description here][1]][1]
$$ \frac{r}{\sqrt{r^2+a^2-2 ar \cos \theta}}=\frac{\sin \phi}{\si... |
Let $A$ be an abelian group.
The definition of the Pontryagin dual of $A$ is $\text{Hom}_{\text{conti}}(A, \mathbb{Q}/\mathbb{Z})$.
In this context, what are the topologies on $A$ and $\mathbb{Q}/\mathbb{Z}$?
Is it the cofinite topology on $A$ and the discrete topology on $\mathbb{Q}/\mathbb{Z}$, or does that ... |
For **any** measure space $(X,\mathfrak{M},\mu)$, and any finite-dimensional normed vector space $V$, one has a very standard theory of integration for maps $f:X\to V$ which are $\mathfrak{M}$-$\mathcal{B}(V)$ measurable ($\mathcal{B}(V)$ being the Borel $\sigma$-algebra on $V$). Literally open up any measure theory bo... |
I can understand that you'd sometimes want to see where the linear subdivisions are on a logarithmic plot, and track e.g. where one y-value is ~3 times as large as another y-value (e.g. 2x10^5 compared to 6x10^5), and so on, but why is this the most common way of drawing such plots?
At least in my view, when drawing... |
Why is it more common to draw in arithmetic subdivisions in logarithmic plots? |
For **any** measure space $(X,\mathfrak{M},\mu)$, and any finite-dimensional normed vector space $V$, one has a very standard theory of integration for maps $f:X\to V$ which are $\mathfrak{M}$-$\mathcal{B}(V)$ measurable ($\mathcal{B}(V)$ being the Borel $\sigma$-algebra on $V$). Literally open up any measure theory bo... |
Let $k$ be a field, $R = k[X_1, \ldots, X_n]$ and $I = \langle X_1, \ldots, X_n\rangle$. I am trying to prove that the quotient ring $R/I^r$, $r\in\mathbb{N}$ is local, i.e., has only one maximal ideal. Here we define $I^r$ as the ideal generated by the products $f_1\ldots f_r$ with each $f_i\in I$.
By the correspon... |
For what values of $p$ does $\int_0^1 \frac{\sin(1/x)}{x^p} \space dx$ converge? |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
Can Markov Chains be used to Disprove the St Petersburg Paradox? |
Part answer to the last side note:
I've managed to plot this [via Wolfram Alpha][1] and the result is quite interesting in that - contrary to my intuition - the probability of having `>= n/2` identical sides of any die face (e.g. half or more than half 1s) decreases rapidly:
[![n=2 to 40][2]][2]
[1]: https... |
No, the intersection does not have to be empty. You have to take disjoint union. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ den... |
In the figure below $DM=DN$ and $\dfrac{Area_{\text{ADN}}}{Area_{\text{ABC}}}=\dfrac{6}{121}$.
What is $\dfrac{AB}{AD}?$
My attempt:
Knowing that ${Area_{\text{ADN}}}={Area_{\text{ADM}}}$, and since the distances of $M$ and $N$ to $AB$ are the same, I tried to find a relation between ${Area_{\text{ABC}}}$ and ... |
In the figure below $DM=DN$ and $\dfrac{Area_{\text{(ADN)}}}{Area_{\text{(ABC)}}}=\dfrac{6}{121}$.
What is $\dfrac{AB}{AD}?$
My attempt:
Knowing that ${Area_{\text{(ADN)}}}={Area_{\text{(ADM)}}}$, and since the distances of $M$ and $N$ to $AB$ are the same, I tried to find a relation between ${Area_{\text{(ABC... |
In a chemical reaction that follows the logistic equation, if the initial velocity is 0, it becomes a steady solution. However, in the equation of motion, the fact that the initial velocity is 0 does not necessarily mean that it is a steady solution.
For example,for the $$\frac{dx}{dt}(t) = 4(1-0.5x(t))x(t)$$,
unde... |
I have always felt moderately "skeptical" about the notion that Euclidean geometry as presented in a mathematics course (whether in a synthetic or an analytic form) actually corresponds to our intuitive, visual notion of flat (2d or 3d) space. By "skeptical" I of course do not mean that I actually *doubt* that it is tr... |
How can I (semi-formally) convince myself that Euclidean geometry comports with visual intuition? |
Given the first fundamental form and the shape operator: $$g = \begin{pmatrix}
1 & 0\\
0 & \cos^2(u)
\end{pmatrix} \text{and} \,S = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix},$$ I want to check, if there exists a surface $f: \Omega \rightarrow \mathbb{R^3}$ satisfying these conditions. We know that $S^T = bg^{... |
I Found the following identity:
$\partial_k g_{ij}=\partial_j g_{ik} $ ?
or at least that is what it looks that it is being used.
g is a riemannian metric by the way.
It's used in eq 3.28 of this book https://radbouduniversitypress.nl/site/books/m/10.54195/EFVF4478/
[![enter image description here][1]][1]
... |
Why is $\partial_k g_{ij}=\partial_j g_{ik} $? |
This is a conclusion I came to through **very** basic knowledge of statistics and error analysis.
Suppose we want to determine the fairness of a coin. To do this, we need to determine the probability of the coin landing on either Heads or Tails and we do this through repeated flipping. We have an amount of uncertain... |
I am studying Lecture 1 of [Stefan Wager's Causal Inference notes][1] and come across a central limit theorem for the difference-in-means estimator, which I am unable to prove.
The mathematical abstraction is as follows. Let $W_1,...,W_n$ be i.i.d. Bernoulli random variables. Let $(Y_1(0),Y_1(1)),...,(Y_n(0),Y_n(1))... |
Let $ABC$ a triangle inscribed în The circle $\Gamma$. The bisector of $\angle BAC$ intersecta $BC$ în $A'$ and $\Gamma$ în $A_1$. Let $AA_2\perp BC, A_2\in BC$ and $A_1A_2\cap \Gamma=\{A_1,A_3\}$, $A_3A'\cap\Gamma=\{A_3,A_4\}$.
IT will be possible that $AA_1$ to be The bisector of $\angle A_2AA_4$?[![enter image de... |
The question is
> Examples of natural proofs by induction exercises which needs 3 or more base cases?
I think an elementary natural proof of the parity of the Somos-4
sequence is a nice exercise. Given the Somos-4 sequence of positive integers that satisfies
$ s_n s_{n-4}=s_{n-1}s_{n-3}+s_{n_2}^2 $
for all inte... |
How do we find locus of P looking at two foci distance $a$ apart such that
$$ \cos \phi/\cos \theta =- k~? $$
(k positive real). Vaguely remember it comes from Electromagnetic theory. Thanks in advance.
[![enter image description here][1]][1]
$$ \frac{r}{\sqrt{r^2+a^2-2 ar \cos \theta}}=\frac{\sin \phi}{\si... |
In the figure below $DM=DN$ and $\dfrac{Area_{\Delta ADN}}{Area_{\Delta ABC}}=\dfrac{6}{121}$.
What is $\dfrac{AB}{AD}?$
My attempt:
Knowing that ${Area_{\Delta ADN}}={Area_{\Delta ADM}}$, and since the distances of $M$ and $N$ to $AB$ are the same, I tried to find a relation between ${Area_{\Delta ABC}}$ and ... |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
Let $Y$ be a $d$-dimensional standard Gaussian vector, i.e. $Y\sim\mathcal{N}(0,I_d)$. For any subset $S\subset[d]=\{1,2,\cdots,d\}$, let $|S|$ be the size of $S$ and $Y_S$ be the subvector $(Y_i:i\in S)$. [This post](https://math.stackexchange.com/questions/3547487/gaussian-width-of-sparse-balls) shows the following: ... |
Matching lower bound on maximum of 2-norm of sub-vectors of standard Gaussian? |
Yes, in fact $X^*\cap X=\emptyset$. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ denotes the power set of $A$. Now what is a sequ... |
Definition of an octahedron according to Wikipedia: "an octahedron is a polyhedron with eight faces".
Given are the points $A, B, C, D$ of an octahedron with equal edge lengths (picture). We call the remaining vertices $S1$ and $S2$ and the center of the octahedron $M$.
Show that $\overline{AM} = \overline{MS_1}$.
... |
In the figure below $DM=DN$ and $\dfrac{Area_{({\Delta ADN})}}{Area_{({\Delta ABC})}}=\dfrac{6}{121}$.
What is $\dfrac{AB}{AD}?$
My attempt:
Knowing that ${Area_{({\Delta ADN})}}={Area_{({\Delta ADM})}}$, and since the distances of $M$ and $N$ to $AB$ are the same, I tried to find a relation between ${Area_{({... |
Yes, in fact $X^*\cap X=\emptyset$. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ denotes the power set of $A$. Now what is a sequ... |
Let $H$ be a topological group acting on a topological space $X$. Is there a general definition of periodicity in this case? Write $X=G/\Gamma$ and consider the orbit $Hg\Gamma$, what does it mean for the orbit $Hg\Gamma$ to be periodic? If there is an abstract definition, I also wish to know the intuition behind it. |
How to define periodicity of orbits for general group actions? |
I am learning about the St Petersburg Paradox https://en.wikipedia.org/wiki/St._Petersburg_paradox - here is my attempt to summarize it:
- A fair coin is tossed at each stage.
- The initial stake begins at 2 dollars and is doubled every time tails appears.
- The first time heads appears, the game ends and the play... |
Given the possibility for automata like [Langton's Ant](https://en.wikipedia.org/wiki/Langton%27s_ant) to lead to complex, intricate structures, I'm curious whether a Penrose tiling can be generated via this sort of local exploration. More formally:
>Suppose we have an "ant" which can take steps of $e^{k\cdot\pi i/5... |
Can Langton's Ant draw a Penrose tiling? |
In the figure below $DM=DN$ and $\dfrac{Area_{({\Delta ADN})}}{Area_{({\Delta ABC})}}=\dfrac{6}{121}$.
What is $\dfrac{AB}{AD}?$
My attempt:
Knowing that $\mathbf{Area_{({\Delta ADN})}}\mathbf{=}\mathbf{Area_{({\Delta ADM})}}$, and since the distances of $\mathbf{M}$ and $\mathbf{N}$ to $\mathbf{AB}$ are the s... |
How do we find locus of P looking at two foci distance $a$ apart such that
$$ \cos \phi/\cos \theta =- k~? $$
(k positive real). Vaguely remember it comes from Electromagnetic theory. Thanks in advance.
[![enter image description here][1]][1]
$$ \frac{r}{\sqrt{r^2+a^2-2 ar \cos \theta}}=\frac{\sin \phi}{\si... |
Definition of an octahedron according to Wikipedia: "an octahedron is a polyhedron with eight faces".
Given are the points $A, B, C, D$ of an octahedron with equal edge lengths (picture). We call the remaining vertices $S_1$ and $S_2$ and the center of the octahedron $M$.
Show that $\overline{AM} = \overline{MS_1}$... |
Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which as ratio of a cubic form and a quadratic form:
$$\frac{1}{\sqrt 2}\frac{x_1^2 x_2 + x_1 x_2^2 + x_1^2 x_3 - 6x_1 x_2 x_3 + x_2^2 x_3 + x_1 x_3^2 + x_2 ... |
Ratio of cubic and quadratic form is approximately normal? |
Yes, in fact $X^*\cap X=\emptyset$. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ denotes the power set of $A$. Now what is a sequ... |
Definition of an octahedron according to Wikipedia: "an octahedron is a polyhedron with eight [triangualar] faces".
Given are the points $A, B, C, D$ of an octahedron with equal edge lengths (picture). We call the remaining vertices $S_1$ and $S_2$ and the center of the octahedron $M$.
Show that $\overline{AM} = \o... |
Definition of an octahedron according to Wikipedia: "an octahedron is a polyhedron with eight [triangualar] faces".
Given are the points $A, B, C, D$ of an octahedron with equal edge lengths (picture). We also know that $ABCD$ is a square with center $M$. We call the remaining vertices $S_1$ and $S_2$.
Show that $\... |
Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which as ratio of a cubic form and a quadratic form:
$$\frac{1}{\sqrt 2}\frac{x_1^2 x_2 + x_1 x_2^2 + x_1^2 x_3 - 6x_1 x_2 x_3 + x_2^2 x_3 + x_1 x_3^2 + x_2 ... |
**Definition -**
A set $S$ is **countable** iff there exists an injective function $f$ from $S$ to the natural numbers $\mathbb{N}$.
**Cantor's diagonal argument -** Briefly, the Cantor's diagonal argument reads as follow:
Take $S = (0, 1) \subset \mathbb{R}$ and suppose that there exists an injective function $... |
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x = \exp(x\ln(x)) \\ a_3 & = x^{x^x} = \exp(x^x \ln(x)) \\
a_4 & = x^{x^{x^x}} = \exp(x^{x^x} \ln(x))\\
& \ldots
\end{align*}
where the definition of general powers of... |
As an undergraduate engaged in applied mathematics research, specifically within mathematical programming, my work has quite some intersection with areas like algebraic geometry for polynomial optimization and the use of manifolds in manifold optimization algorithms.
Producing research is my priority, but I'm afraid... |
How to learn necessary concepts for research and avoiding a fake impression of understanding? |
I want to show that the aplication
$$p: \mathbb{R} \times \cdots \times \mathbb{R} \to \mathbb{R}$$
defined by
$$p(x_{1}, \dots, x_{n})=\prod_{i=1}^{n}x_{i}$$
is a $n$-linear form.
Let's see in particular, for example, if it is bilinear
$$p(\alpha u+\beta v, w)=((\alpha u+\beta v) \cdot w)=\alpha(u \cdo... |
Definition of an octahedron according to Wikipedia: "an octahedron is a polyhedron with eight [triangualar] faces".
Given are the points $A, B, C, D$ of an octahedron with equal edge lengths (picture). We also know that $ABCD$ is a square with center $M$. We call the remaining vertices $S_1$ and $S_2$.
Show that $\... |
This notion of pre-convex depends a bit on our notion of local maximum: is it a point which is not exceeded in a neighborhood, or a point which strictly exceeds every other point in some neighborhood? (E.g., is $(1,0.5)$ a local maximum of $f(x,y)=x$ on the unit square $[0,1]\times [0,1]$?)
I'm going to use the lat... |
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x = \exp(x\ln(x)) \\ a_3 & = x^{x^x} = \exp(x^x \ln(x)) \\
a_4 & = x^{x^{x^x}} = \exp(x^{x^x} \ln(x))\\
& \ldots
\end{align*}
These are powers of general reals. They a... |
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