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Under which set of conditions a function could have just one local minimizer? |
We aim to prove that the set
\{V(n, 2) = \{(a, b) \in R^n \times R^n : \|a\|^2 = \|b\|^2 = 1 \text{ and } <a,b> = 0\}
is a smooth (2n-3)-submanifold of R^n \times R^n. \
This will be demonstrated by considering the function f :R^n \times R^n \rightarrow R^3 given by
f(a, b) = (\|a\|^2, \|b\|^2, <a,b>)
and showin... |
Let $\{X_n\}$ be a sequence of random variables. $X_n\rightarrow X$ a.s. Let $G$ be a $\sigma$-algebra. Suppose each $X_n$ is independent of $G$. Is it true that $X$ is also independent of $G$?
I proved it in this way:
**Fact 1** a random variable $X$ is independent of a $\sigma$-algebra $G$ if and only if $\fora... |
Why is a random walk $(X_n)_{n\in \mathbb{N}_0}$ on $\mathbb{R}^d $(defined below) a time-homogeneous Markov process? In particular, why does $(X_n)$ satisfy
[requirement 3 of Def17.3](https://math.stackexchange.com/questions/183945/why-does-a-time-homogeneous-markov-process-possess-the-markov-property)?
**Definit... |
How can I prove that this stochasitic process is a time-homogeneous Markov process? |
My Question
----
>Define subsets $G$, $G_0$, and $G_1$ of $\mathbb{R}$ by
\begin{align*}
G &= \{x:x=r+n\sqrt{2}\ \text{for some $r$ in $\mathbb{Q}$ and $n$ in $\mathbb{Z}$}\},\\
G_0 &= \{x:x=r+2n\sqrt{2}\ \text{for some $r$ in $\mathbb{Q}$ and $n$ in $\mathbb{Z}$}\},\ \text{and}\\
G_1... |
Probability of not visiting a recurrent state does not go to zero under an infinite measure? |
This is the first exercise in section 6.5 of Robert Ash's abstract algebra, I want to understand the given solution:
Noting that $\psi_n(X^p)= \prod_{w_i} X^p- w_i$, the roots of $X^p −ω_i$ are the $p$-th roots of $ω_i$, which must be primitive $np$-th roots of unity because the map $\theta\to \theta^p$ is a biject... |
The cyclotomic polynomials satisfy $\psi_{pn}(X)= \psi_n(X^p)$ if $p|n$? |
The Constant Elasticity of Variance (CEV) process is a one dimensional diffusion process given by the following stochastic differential equation.
\begin{equation}
d X_t = \mu X_t \cdot dt + \sigma X_t^\beta \cdot d B_t \tag{1}
\end{equation}
where $\mu,\sigma,\beta$ are positive real parameters and $B_t$ is t... |
My Question
----
>Define subsets $G$, $G_0$, and $G_1$ of $\mathbb{R}$ by
\begin{align*}
G &= \{x:x=r+n\sqrt{2}\ \text{for some $r$ in $\mathbb{Q}$ and $n$ in $\mathbb{Z}$}\},\\
G_0 &= \{x:x=r+2n\sqrt{2}\ \text{for some $r$ in $\mathbb{Q}$ and $n$ in $\mathbb{Z}$}\},\ \text{and}\\
G_1... |
> Theorem 4 The discretized signature kernel over $k$,
$$
\mathrm{k}^{+}: X^{+} \times X^{+} \rightarrow \mathbb{R}, \quad \mathrm{k}^{+}(x, y)=\left\langle\mathrm{S}^{+}\left(\mathrm{k}_x\right), \mathrm{S}^{+}\left(\mathrm{k}_x\right)\right\rangle
$$
> 1. is a positive definite kernel,
> 2. $$\mathrm{k}^{+}(x, y... |
> Theorem 4 The discretized signature kernel over $k$,
$$
\mathrm{k}^{+}: X^{+} \times X^{+} \rightarrow \mathbb{R}, \quad \mathrm{k}^{+}(x, y)=\left\langle\mathrm{S}^{+}\left(\mathrm{k}_x\right), \mathrm{S}^{+}\left(\mathrm{k}_x\right)\right\rangle
$$
> 1. is a positive definite kernel,
> 2. $$\mathrm{k}^{+}(x, y... |
Let $V, V'$ be full flags in $\mathbb{C}^n$, let $\lambda, \mu$ be admissible partitions, and let $$\sigma_\lambda(V) = \{\Lambda: \dim(\Lambda \cap V_{n-k+i-\lambda_i}) \geq i\}$$ and $$\sigma_\mu(V) = \{\Lambda: \dim(\Lambda \cap V_{n-k+i-\mu_i}) \geq i\}$$ be general Schubert cycles. Then for each $i$ and any $\Lamb... |
See Theorem 3.3 https://www.sciencedirect.com/science/article/pii/S0022247X12002600?via%3Dihub#s000015 and also a recent preprint https://arxiv.org/abs/2312.07940. |
[Exercise][1]
[1]: https://i.stack.imgur.com/doo2O.png
To prove equation 2 :
I used the following formula :
$$(X \cdot M)_{T^t} - (X \cdot M)_{S^t} = \int_{S^t}^{T^t} X_s \, dM_s$$
$$ \mathbb{E}\left(\left[(X \cdot M)_{T^t} - (X \cdot M)_{S^t}\right]\left[(Y \cdot M)_{T^t} - (Y \cdot M)_{S^t}\right... |
[Exercise][1]
[1]: https://i.stack.imgur.com/doo2O.png
To prove equation 2 :
I used the following formula :
$$(X \cdot M)_{min(T,t)} - (X \cdot M)_{min(S,t)} = \int_{min(S,t)}^{min(T,t)} X_s \, dM_s$$
$$ \mathbb{E}\left(\left[(X \cdot M)_{min(T,t)} - (X \cdot M)_{min(S,t)}\right]\left[(Y \cdot M)_{... |
I have a certain infinite power series: $S(x) = a0+a1*x+a2*x^2+a3*x^3+...$, which is defined for $0 <= x <= 1$ and I have reasons to assume that it is convergent for all such $x$. Is there any way to prove that if one rewrites this series in the form $S(1-y) = b0+b1*y+b2*y^2+b3*y^3+...$ then the infinite expressions (s... |
Given a convergent power series in x, can one prove the equivalent power series in (1-x) is convergent? |
I have a certain infinite power series: $S(x) = a_0+a_1x+a_2x^2+a_3x^3+ \ldots$, which is defined for $0 \leq x \leq 1$ and I have reasons to assume that it is convergent for all such $x$. Is there any way to prove that if one rewrites this series in the form $S(1-y) = b_0+b_1y+b_2y^2+b_3y^3+ \ldots$ then the infinite ... |
Is there a specific terminology for these upper unitriangular matrices? |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/iwaf5.jpg
The slice maps are defined in the above sceenshot.
If we take normal state $\psi$ on $N$, then the slice map $L_\psi$ defined by
$L_\psi(\sum x_i\otimes y_i)=\sum \psi(y_i)x_i$
extends to a normal conditional expectatio... |
I am currently studying martingales with Resnick's book *A Probability Path*. He defines a martingale as closed on the right if there is an $X \in L_1$ such that $X_n = \mathbb{E}[X \mid \mathcal{B}_n]$ for all $n$. One can also find that definition [here](https://math.stackexchange.com/questions/246762), defining it a... |
I'm trying to solve the 2D heat equation in cartesian coordinates with initial condition of the form $1/r$. The problem is as follows:
$$
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right), & x \in \mathbb{R}, ... |
>Finding range of function $\displaystyle f(x)=\bigg|\sin\bigg(\frac{x}{2}\bigg)\bigg|+\bigg|\cos(x)\bigg|$
Using $\displaystyle 0\leq \bigg|\sin\bigg(\frac{x}{2}\bigg)\bigg|\leq 1$ and $\displaystyle 0\leq |\cos(x)|\leq 1$
$\displaystyle 0\leq f(x)\leq 1+1=2$
But minimum value is $\displaystyle \frac{1}{\sqrt... |
How to use variational calculus on a ratio of two functionals, with inequality constraint associated with the denominator? |
I am not understanding how to conclude this problem, I've been tries for days so far.
I have to find the minimum of $f(x, y) = (x-2)^2+y$ subject to $y-x^2 \geq 0, y + x^3 \leq 0, y \geq 0$.
I have tried to give a sketch of the problem, for it's easy to compute: the level curves of $f$ are parabolas (concave), and... |
Is it okay to use an indicator function and have two statistics in the support? And, if this is possible, would the indicator function be a function of both statistics?
We have that $\mathbf{X}$ is a random sample from Uniform$(\theta, \theta+1)$ and we want to re-write the joint pdf from $$f(\mathbf{x}| \theta) =... |
Indicator functions based on one or more statistics (Is it alright?) (Mathematical stats) |
So, lets $A$, be a $C^k-$atlas, and $B$ a set of charts wich each chart is compatible which each chart in A. The thing is that I can take $(V_1,\psi_1),(V_2,\psi_2) \in B$, such that $V_1\cap V_2 \neq \varnothing$. By the definition of $B$, there is $(U,\phi)\in A$ which is compatible with $(V_1,\psi_1),(V_2,\psi_2)$.
... |
>Finding range of function $\displaystyle f(x)=\bigg|\sin\bigg(\frac{x}{2}\bigg)\bigg|+\bigg|\cos(x)\bigg|$
Using $\displaystyle 0\leq \bigg|\sin\bigg(\frac{x}{2}\bigg)\bigg|\leq 1$ and $\displaystyle 0\leq |\cos(x)|\leq 1$
$\displaystyle 0\leq f(x)\leq 1+1=2$
But minimum value is $\displaystyle \frac{1}{\sqrt... |
This is a simple exercise needed to prove the Optional Stopping Theorem that I'm working on.
Suppose $(X_n)$ is a supermartingale and we have stopping times $T, S$. Then we already know in general that $\mathbb{E}(X_T|F_s)\leq X_{\min(S,T)}$ by splitting $X_T=X_{\min(S,T)} +\sum_{k=0}^n (X_{i+1}-X_i)1_{S\leq k\leq T... |
I'm attempting to solve the following ODE:
$$xy''+y'-y=0$$
According to Frobenius' theorem, there exists a solution to this ODE in the form of a series given by the linear combination of two solutions such that: $y=C_1y_1+C_2y_2$, where:
$$y_1=\displaystyle \sum_{n=0}^\infty a_nx^n$$
$$y_2=y_1log|x|+\displays... |
So, lets $A$, be a $C^k-$atlas, and $B$ a set of charts wich each chart is compatible which each chart in A. The thing is that I can take $(V_1,\psi_1),(V_2,\psi_2) \in B$, such that $V_1\cap V_2 \neq \varnothing$. By the definition of $B$, there is $(U,\phi)\in A$ which is compatible with $(V_1,\psi_1),(V_2,\psi_2)$.
... |
I'm looking for a function (preferrably in Python) which takes a complex number $a$ as input, and outputs the zeros of the function $f:x \rightarrow J_x(a)$ where $J_x(a)$ is the Bessel function of the first kind. This problem is relevant to solving a Schoedinger equation in a 1D condensed matter system (N.B. I am **no... |
Joe is playing a game where his given a bag one at a time and has to win the bag by drawing the corresponding color marble from the bag. There are 6 bags. Each bag is assigned a color that contains marbles of that color: The red bag has red marbles, the blue bag has blue marbles and so on. Each bag also has a proporti... |
What's the probability of Joe winning? |
Joe is playing a game where he is given a bag one at a time and has to win the bag by drawing the corresponding color marble from the bag. There are 6 bags. Each bag is assigned a color that contains marbles of that color: The red bag has red marbles, the blue bag has blue marbles and so on. Each bag also has a propor... |
What's the probability of Joe winning a set of outcomes? |
An elliptic curve over a field $K$ is defined as a 'genus $1$ curve with a base point'.
My question is, why don't we adopt the definition 'a smooth curve that is isomorphic to a degree $3$ smooth curve with a base point'?
I think these definitions are equivalent because a genus $1$ curve with a base point is isom... |
How can I prove that this stochastic process is a time-homogeneous Markov process? |
Find the range of function $f(x)=|\sin\bigg(\frac{x}{2}\bigg)|+|\cos(x)|$, using $0\leq \bigg|\sin\bigg(\frac{x}{2}\bigg)\bigg|\leq 1$ and $ 0\leq |\cos(x)|\leq 1, 0\leq f(x)\leq 1+1=2$. But minimum value is $\frac{1}{\sqrt{2}}$.
Other way what I tried:
$ f(x)=\bigg|\sin\bigg(\frac{x}{2}\bigg)\bigg|+\bigg|1-2\sin... |
I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$.
Now, I solved the problem quite easily in a sketching way: the level curves of $f(x, y)$ are concave parabolas ($y = k - (x-2)^2$), and the feasible region is the up... |
Exercise 2.13 on page 46 reads
>(Double Induction) Let $P(x,y)$ be a property. Assume
>
>(**) If $P(k,l)$ holds for all $k,l \in \mathbb{N}$ such that $k < m$ or ($k=m$ and $l<n$), then >$P(m,n)$ holds.
>
>Conclude that $P(m,n)$ holds for all $m,n \in \mathbb{N}$.
Since the authors state on page 44
>(The ... |
I have difficulty understanding the solution below and have already summarized my difficulties as follows,
1. why "the area of the polygon $abcelef$
.... represents the number of ways the three points can be taken, so that the circle circumscribing the triangle will lie wholly within the given polygon." It looks li... |
I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$.
Now, I solved the problem quite easily in a sketching way: the level curves of $f(x, y)$ are concave parabolas ($y = k - (x-2)^2$), and the feasible region is the up... |
[![enter image description here][2]][2]
How do I prove the second part?
___
**My attempt:** I used the following formula :
$$(X \cdot M)_{min(T,t)} - (X \cdot M)_{min(S,t)} = \int_{min(S,t)}^{min(T,t)} X_s \, dM_s$$
$$ \mathbb{E}\left(\left[(X \cdot M)_{min(T,t)} - (X \cdot M)_{min(S,t)}\right]\left[(Y ... |
I managed to solve it using the following function:
given a cartesian point A and point B.
the geodesic path on a sphere is defined as:
r(t) = sin(1-t)*A + sin(t)*B, for t=[0, 1]
Can someone explain me why does it work? |
I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$.
Now, I solved the problem quite easily in a sketching way: the level curves of $f(x, y)$ are concave parabolas ($y = k - (x-2)^2$), and the feasible region is the up... |
What is the counterexample to show that almost sure convergence does not imply mean (L^p space) convergence?
All the examples I've seen were about the fact that convergence in mean does not imply almost sure convergence. I have not found any counterexamples for vice versa, I really want to prove it because if I find... |
>Find the range of function $$f(x)=|\sin\bigg(\frac{x}{2}\bigg)|+|\cos(x)|.$$
Using $0\leq \bigg|\sin\bigg(\frac{x}{2}\bigg)\bigg|\leq 1$ and $ 0\leq |\cos(x)|\leq 1, 0\leq f(x)\leq 1+1=2.$ But minimum value is $\frac{1}{\sqrt{2}}$.
Other way what I tried:
$ f(x)=\bigg|\sin\bigg(\dfrac{x}{2}\bigg)\bigg|+\bigg|... |
Take a Hilbert space $H$, an orthogonal projector $P$, an unbounded self-adjoint operator $A$. Is $A^{-1} P A$ bounded ? |
Why is $\sum_{m=0}^x 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{2(x - sp)^2}{s}\right)}$? |
Below I am trying to make the asked points clear, computations do not stay in focus. First of all a picture:
[![mse 43881764][1]][1]
The problem starts somehow with the middle, making the least number of words in a random language count for exposing the problem. This is not the way we think, and not the way we st... |
I am reading a text that shows this expression
$$\beta_j = (1 - \lambda\sqrt{pj}/||S_j||)_+ S_j$$
From the text, the expression inside the parenesis is a number, but what does the + means on this situation? |
What does a subscript + means? |
How can I obtain a polynomial formula for the product of a triangle's [incircle and excircles](https://en.wikipedia.org/wiki/Incircle_and_excircles), avoiding the use of radicals?
Assume three lines in general position are given by three equations
$$a_ix+b_iy+c_i=0\quad\text{for }i\in\{1,2,3\}$$
A circle with ... |
I encountered a tough integral and I am wondering if anyone has any ideas on how to evaluate it.
$$\displaystyle \int_{0}^{\infty}\frac{x}{x^{2}+a^{2}}\log\left(\left|\frac{x+1}{x-1}\right|\right)dx=\pi\tan^{-1}\frac1a, \;\ a>1$$
I tried breaking up the log and differentiating w.r.t 'a', but did not make consider... |
I am reading this paper about model selection that shows this expression for
$$\beta_j = (1 - \lambda\sqrt{pj}/||S_j||)_+ S_j$$
for the vectors $\beta_j$ of estimations.
From the text, the expression inside the parenesis is a number, but what does the + means on this situation? |
I think this is a good place for thinking about the alternative way of framing induction.
First, note that $\Bbb N \times \Bbb N$ is well ordered (meaning every non-empty subset has a $\lt$-least element) by the lexicographic ordering $(a, b) \lt (c, d) \iff a \lt c \lor (a=c \land b \lt d)$. This is easy to see.
... |
Yes, your first argument works. However, I think this is a good place for thinking about the alternative way of framing induction as using well ordering to show that the set of counterexamples must be empty.
First, note that $\Bbb N \times \Bbb N$ is well ordered (meaning every non-empty subset has a $\lt$-least el... |
I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$.
Now, I solved the problem quite easily in a sketching way: the level curves of $f(x, y)$ are concave parabolas ($y = k - (x-2)^2$), and the feasible region is the up... |
I managed to solve it using the following function:
given a cartesian point A and point B.
the geodesic path on a sphere is defined as:
r(t) = sin(1-t)*A + sin(t)*B, for t=[0, 1]
then normalize r(t)/||r(t)|| to get the points on a sphere
[![enter image description here][1]][1]
Can someone explain me w... |
I derived a formula for SIP calculations myself, but I'm unsure if it's accurate. Could you please help me verify it? |
I'm reviewing some very basic concepts in analysis because I want to get better at it. I am trying to prove that
$$\lim_{x\to 9}(\sqrt{x} + 2) = 5$$
using the rigorous definition of a limit.
Here is my proof:
Let $\epsilon > 0$. And take $\delta = \epsilon$. Then $\lvert x-9\rvert < \delta$ implies that $\lvert ... |
#### Prompt:
> Find the $x$- and $y$-intercepts of the graph of $x = (y+2)\ln(3y+4)$.
#### Given solution:
For $x = 0$, solve $(y+2)\ln(3y+4) = 0$ for $y$:
\begin{align}
y + 2 &= 0 & \ln(3y + 4) &= 0 \\
y &= -2 & 3y + 4 &= e^0 \\
\rlap{\text{not a solution}}\qquad && 3y + 4 &= 1 \\
&& 3y &= -3 \\
&& y ... |
>Can we choose a special $\psi$ that make $M\otimes N$ is $*$-isomorphic to $N$?
No. For starters, most of the time $M\otimes N$ will not be isomorphic to $N$. You could have $M=N=M_2(\mathbb C)$, and it already fails. Or if you want infinite-dimensional examples, take $N$ amenable, for instance the hypefinite II$_... |
Recently I have encountered an paper which impose a subGaussian condition on the covariate vector $\boldsymbol{x}$ as follows:
>There exists some universal constant $\sigma_x \in[1, \infty)$ such that
$$
\forall \boldsymbol{v} \in \mathbb{R}^p, \quad \mathbb{E}\left[\exp \left\{\boldsymbol{v}^{\top} \boldsymbol{\... |
> I have $3$ statements $A, B$, and $C$ and I wish to change them from the form:
>
> $(A \operatorname{xor} B)$
>
> $(A \operatorname{xor} C)$
>
> Into
>
> $A \operatorname{xor} B \operatorname{xor} C$
You may wish to do so, but please note that the conjunction of the first two statement is *not* equi... |
I'm a beginner to number theory, and in a text book, right after proving the fundamental theorem of arithmetic, the following problem is stated:
Let $H_m$ be the subset of real numbers, which can be written in the form of $x + y\sqrt{m}$, where $x$ and $y$ are integers and $m$ isn't a square number.
Show that besid... |
How can I obtain a polynomial formula for the product of a triangle's [incircle and excircles](https://en.wikipedia.org/wiki/Incircle_and_excircles), avoiding the use of radicals?
Assume three lines in general position are given by three equations
$$a_ix+b_iy+c_i=0\quad\text{for }i\in\{1,2,3\}$$
A circle with ... |
We know a hyperbola can be expressed in the form of$$ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$ where $(h,k)$ is it's centre. I've learnt that in the parametric form, we take
$$x= h + a\sec t$$ and $$ y = k + b\tan t $$
These values satisfy the given equation. But so does $$x= h + a\csc t$$ and $$y=k+b\cot t$$
Th... |
I'm a beginner to number theory, and in a text book, right after proving the fundamental theorem of arithmetic, the following problem is stated:
Let $H_m$ be the subset of real numbers, which can be written in the form of $x + y\sqrt{m}$, where $x$ and $y$ are integers and $m$ isn't a square number.
Show that besid... |
I managed to solve it using the following function:
given a cartesian point A and point B.
the geodesic path on a sphere is defined as:
r(t) = sin(1-t)*A + sin(t)*B, for t=[0, 1]
then normalize r(t)/||r(t)|| to get the points on a sphere
[![enter image description here][1]][1]
Can someone explain me w... |
- I am measuring the temperature of a metal plate using 4 thermal sensors.
- The sensor used is specified to have an accuracy of ±0.1°C.
If I have for example those readings:
[![enter image description here][1]][1]
The average of all sensors is: 20.45°C.
However how should I report the accuracy?
My intu... |
How to report the accuracy of the average of multiple sensors? |
Let $x \geq 1$. I wish to show, by interpreting the sum as a Riemann sum, that
$$\log(x) = \int_{1}^{x} \frac{dt}{t} \leq \sum_{n \leq x} \frac{1}{n}.$$.
Certainly, $f(t) = 1/t$ is continuous on $[1, x]$, so is integrable on $[1, x]$ and may be computed as the limit of a sequence of Riemann sums. I have only the id... |
The author of [this](https://math.stackexchange.com/questions/2413529/inverse-of-a-matrix-derivative) question was close to determining the derivative of the function of dual variable, when we consider matrices isomorphic (algebraically and topologically) to dual numbers: $$(a+\epsilon b) \sim \begin{bmatrix}
a ... |
Let $x \geq 1$. I wish to show, by interpreting the sum as a Riemann sum, that
$$\log(x) = \int_{1}^{x} \frac{dt}{t} \leq \sum_{n \leq x} \frac{1}{n}.$$
Certainly, $f(t) = 1/t$ is continuous on $[1, x]$, so is integrable on $[1, x]$ and may be computed as the limit of a sequence of Riemann sums. I have only the idea... |
I managed to solve it using the following function:
given a cartesian point A and point B.
the geodesic path on a sphere is defined as:
r(t) = sin(1-t)*A + sin(t)*B, for t=[0, 1]
then normalize r(t)/||r(t)|| to get the points on a sphere
[![enter image description here][1]][1]
Can someone explain me w... |
Let $x \geq 1$. I wish to show, by interpreting the sum as a Riemann sum, that
$$\log(x) = \int_{1}^{x} \frac{dt}{t} \leq \sum_{n \leq x} \frac{1}{n} \leq 1 + \int_{1}^{x} \frac{dt}{t}.$$
Certainly, $f(t) = 1/t$ is continuous on $[1, x]$, so is integrable on $[1, x]$ and may be computed as the limit of a sequence of... |
How can I obtain a polynomial formula for the product (i.e. union) of a triangle's [incircle and excircles](https://en.wikipedia.org/wiki/Incircle_and_excircles), avoiding the use of radicals?
Assume three lines in general position are given by three equations
$$a_ix+b_iy+c_i=0\quad\text{for }i\in\{1,2,3\}$$
A... |
Let X is Banach space. Let T(t) is a semigroup on X.(t≧0)If there exists M≧1,ω≧0, will T(t) become a C0 semigroup?
I know the converse holds true, but
I'm not sure about this proposition.
I would be glad to have a proof if it is valid, and a counterexample if it is not. |
Suppose we have groups $G,H$, a subsemigroup $S$ of $G$ generating $G$ as a group, and a homomorphism $f\colon S\to H$.
When does $f$ extend to a homomorphism $G\to H$?
If $f$ is injective, when does it extend to an injective homomorphism $G\to H$?
Aside from the trivial case (when $S=G$, which is the case e.g... |
How can I obtain a polynomial formula for the product (i.e. union) of a triangle's [incircle and excircles](https://en.wikipedia.org/wiki/Incircle_and_excircles), avoiding the use of radicals?
Assume three lines in general position are given by three equations
$$a_ix+b_iy+c_i=0\quad\text{for }i\in\{1,2,3\}$$
A... |
I'm working through Problem 4.16 in Armstrong's *Basic Topology*, which has the following questions:
1) Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
2) Are these two isomorphic as topological groups?
Let $\mathbb{M_n}$ denote the set of $n\times n$ matrices with real entries. We identify each matrix ... |
Are $O(n)$ and $SO(n)\times Z_2$ homeomorphic as topological spaces? |
I'm working through Problem 4.16 in Armstrong's *Basic Topology*, which has the following questions:
1) Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
2) Are these two isomorphic as topological groups?
**Some preliminaries:**
Let $\mathbb{M_n}$ denote the set of $n\times n$ matrices with real entrie... |
We know a hyperbola can be expressed in the form of$$ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$ where $(h,k)$ is it's center. I've learned that in the parametric form, we take
$$x= h + a\sec t$$ and $$ y = k + b\tan t $$
These values satisfy the given equation. But so does $$x= h + a\csc t$$ and $$y=k+b\cot t$$
T... |
We know a hyperbola can be expressed in the form of$$ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$ where $(h,k)$ is it's center. I've learnt that in the parametric form, we take
$$x= h + a\sec t$$ and $$ y = k + b\tan t $$
These values satisfy the given equation. But so does $$x= h + a\csc t$$ and $$y=k+b\cot t$$
Th... |
For context:
* $\Omega$ is an open subset of $\mathbb{R}^n$.
* $\mathcal{D}(\Omega)$ is the space of test (smooth, compactly supported) functions $\Omega\to\mathbb{C}$ with a complete, unmetrizable topology $\tau$.
* $\mathcal{D}_K(\Omega)$ is the space of test functions supported on the compact set $K\subseteq \O... |
Why is $\mathcal{D}'$, the space of distributions, closed under addition, scalar multiplication, and multiplication? |
I managed to solve it using the following function:
given a cartesian point A and point B.
the geodesic path on a sphere is defined as:
r(t) = sin(1-t)*A + sin(t)*B, for t=[0, 1]
then normalize r(t)/||r(t)|| to get the points on a sphere
[![enter image description here][1]][1]
Can someone explain me w... |
Everything that follows is from Rudin's _Functional Analysis_: $\def\L{\Lambda} \def\DDD{\mathcal{D}} \def\sbe{\subseteq} \def\W{\Omega} \def\RR{\mathbb{R}} \def\CC{\mathbb{C}} $
---
Below $\DDD$ is the space of test (smooth, compactly supported) functions from an open set $\Omega\sbe\RR^n$ to $\CC$. Such space i... |
Suppose $P < Q$ and $PQ=N$, where both $P$ and $Q$ are prime (including $2$). What's the maximum distance $P$ can ever be from $\sqrt{N}$, if we hold $Q$ constant and let $P$ start at $0$ approach $Q$?
|
So there is a million (essentially equivalent) ways to make it obvious **algebraically** that the axis of symmetry of the parabola $y=x^2+bx+c$, must be at $x=-\frac{b}{2}$, while its real roots, if they exist, are $\pm\sqrt{\left(\frac{b}{2}\right)^2-c}$ away from that axis.
It feels like there must be a way to mak... |
I have to understand a thing about this exercise: find the minimum of $f(x, y) = (x-2)^2 + y$ subject to $y-x^3 \geq 0$, $y+x^3 \leq 0$ and $y \geq 0$.
Now, I solved the problem quite easily in a sketching way: the level curves of $f(x, y)$ are concave parabolas ($y = k - (x-2)^2$), and the feasible region is the up... |
For ellipses having equation in the form of $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ why are the parametric equations are always $$x=a\cos(\theta)$$
$$y=b\sin(\theta)$$
even when $b>a$?
As far as I know, for hyperbola having equation
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=\pm 1$$the parametric equations are $(x(\theta... |
why do we always take the parameter of parametric equation of ellipse as the angle formed with x-axis instead of semi-major axis? |
I'm working through Problem 4.16 in Armstrong's *Basic Topology*, which has the following questions:
1) Prove that $O(n)$ is homeomorphic to $SO(n) \times Z_2$.
2) Are these two isomorphic as topological groups?
**Some preliminaries:**
Let $\mathbb{M_n}$ denote the set of $n\times n$ matrices with real entrie... |
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