instruction stringlengths 12 30k |
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Let $X,Y$ be the topological spaces and $f:\ X\to Y$. Then $f$ is continuous of $X$ if and only if preimage of any open set in $Y$ is also an open set in $X$.
An open map $f:\ X\to Y$ is a map that takes any open set in $X$ to an open set in $Y$. In general topology, we are introduced with some famous results of th... |
Is open map needed to be continuous? |
Let $X$ and $Y$ be two continuous positive random varianbles, that are not independent. In addition, all of the stated expectations are bounded away from zero. We know that $ E(X)^{-1} \neq E(1/X)$ unless we look at trivial cases.
Is there some way to simplify
$$ E\left\{ \frac{1}{E \left(Y|X \right)}
\right\... |
First time posting here so please forgive any lack of adherence to best practices.
sin(x) is a transcendental function. However most common values for the angle x will yield an algebraic number result. In particular, sin(q) (in degrees) or sin(qπ) (in radians) will always be algebraic.
I can reverse-engineer a tr... |
I am trying to find the Fisher information of a binomial distribution where $n=2$ and $n=\theta$. I have the log-likelihood function as $$n\ln2 + \sum^{n}_{x=1}x_i\ln \theta + (2n-\sum^{n}_{x=1}x_i)(ln(1-\theta))-\sum^{n}_{x=1}(\ln x_i!+\ln((2-x_i)!))$$I have the second partial derivative of the log-likelihood function... |
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And conversely, so th... |
You're on the right track, but it's $|\sin x|$ that is scaled by a factor of $3$. The absolute value guarantees that the values are non-negative:
$$
\begin{array}{rcrcl}
-1 & \leq & \sin x \phantom{|} & \leq & 1 \\
\implies \qquad 0 & \leq & |\sin x| & \leq & 1 \\
\implies \qquad 0 & \leq & 3\,|\sin x| & \leq & 3
... |
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And conversely, so th... |
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And conversely, so th... |
I have difficulty understanding the solution below and have already summarized my difficulties at the end of this post
**Problem**:
A circle is circumscribed about a triangle formed by joining three points taken at random in the surface of a circumscriptible polygon of $n$ sides, find the chance that the circle l... |
Let $\mathcal{C}$ be a family of subsets of some finite set $X$, and let $\Omega$ be the $\sigma$-algebra generated from $\mathcal{C}$. Let $\mathcal{A}$ be the set of subsets of $S \subseteq X$ that are "indivisible" in the sense that they always go together as a group, or not at all, in any $C \in \mathcal{C}$, and t... |
> Let $f$ be holomorphic on $D(0,1)$, continuous on $\overline{D}(0,1)$
> s.t $f(z)=0\quad\forall |z|=1$ and $0\leq\arg z\leq\dfrac{2\pi}{n}$
> for some $n\in\mathbb{N}^*.$
>
> a) Let
> $g(z)=\prod\limits_{k=0}^{n-1}f\left(e^{\frac{2k\pi i}{n}}z\right)$. Prove
> $g(z)=0\quad\forall |z|=1$.
>
> b) Prove that $... |
1. Is the set of all open and closed sets in R^2 a σ-algebra for
R^2?
2. Is the set of all closed sets in R^2 a topology for R^2?
I would say no in both cases, since the union of countably many closed sets is not necessarily closed or open again. |
I am an engineer who is currently working with some network optimization problem. In my work, I encounter a strange optimization problem that seems to be convex but it has some cubic and quartic constraint. The problem look like this
$\begin{array}{*{20}{c}}
{\mathop {\min }\limits_{x,y,z,t,y} }&{x + y + z + t + y}... |
Convex optimization with some cubic and quartic constraint? |
How would you solve this?
[![enter image description here][1]][1]
I did the standard way of solving questions like these - I took the Laplace of both sides and used the convolution identity. But the solution I got, $f(t)=3+t^2/2$, does not satisfy the integral equation, nor does it satisfy the initial conditions.... |
__Definitions.__
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And... |
Let $V$ be a vector space with an increasing filtration $V_j, j\in \mathbb Z$, we assume the filtration is Hausdorff $\bigcap_j V_j=0$ and exhaustive $\bigcup_j V_j=V$. Consider the associated graded space $\text{gr}V=\bigoplus_j V_j/{V_{j-1}}$, when is $\text{gr}V\cong V$ as a vector space?
In the case of $V_j$ is ... |
I am attempting to correct for cylinder distortion with a galvo laser. Mathwise, this is a single point where the laser's mirror is located. A certain distance `d` away from the surface with radius `r`. This object is intended to be engraved as if it were flat, while still being rounded. Typically this can be done by h... |
__Definitions.__
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And... |
Why is the Cartesian product of second-order cones self-dual? |
Choose two random numbers from [0,1] and let them be the endpoints of a random interval. Repeat this independently for n times to get n random subintervals (denoted as $U_1,\dots,U_n$).
Let $T_n = \max\{|S|: S\subset\{U_1,\dots,U_n\}, U_i\cap U_j=\emptyset \mbox{ for any }U_i\not =U_j\in S\}$, where $|S|$denote the... |
Let an ideal $I \subset S = k[x_0,\dots,x_n]$ defines a subscheme of $\mathbb{P}_k^n$. There is a one-to-one correspondence between the saturated ideals of $S$ and the closed subschemes of $\mathbb{P}_k^n$ (Hartshorne exercise II.5.10), so $\text{Proj }S/I = \text{Proj }S/\bar{I}$. But in Eisenbud prop 3.13, if $I = \c... |
1. Is the set of all open and closed sets in $\mathbb{R}^2$ a $\sigma$-algebra for $\mathbb{R}^2$?
2. Is the set of all closed sets in $\mathbb{R}^2$ a topology for $\mathbb{R}^2$?
I would say no in both cases, since the union of countably many closed sets is not necessarily closed or open again. |
Let $A$ and $B$ be two rings, and let $L$ be an $A\times B$-module. Prove that $L$ is of the form $M\oplus N$ for some $A$-module $M$ and $B$-module $N$, with $(a,b)\cdot (m,n) = (am,bn)$.
Let $M := \{ (a_1,0)l_1 + \cdots + (a_n,0)l_n \mid n \in \mathbb N, a_i \in A, \text{ and } l_i \in L \text{ for all } 1 \leq i... |
This part is not precise enough:
> there exists some delta > 0 s.t. if abs(x-c) < delta then lim (f(x) - f(c)) / (x - c) > 0
This is how you would phrase this more precisely: For all $\epsilon > 0$, there exists $\delta > 0$ such that for all $x$ such that $|x-c| < \delta$, the following inequality holds:
$$
... |
So I have a question about a sequence.
lets assume $a_n = {sin(\frac 1n)\over 3n}$
does the limit of the series $(-1)^na_n$ converge to zero?
I's pretty sure this is decreasing because of the derivative test which gives me a negative derivative at n > 1/3 and I believe it should be bounded at zero. But if it oscela... |
So I have a question about a sequence.
lets assume $a_n = {sin(\frac 1n)\over 3n}$
does the limit of the sequence $(-1)^na_n$ converge to zero?
I's pretty sure this is decreasing because of the derivative test which gives me a negative derivative and I believe it should be bounded at zero. But if it oscelates betwe... |
Consider the following boundary value problem.
\begin{equation*}
\begin{split}
u=x^2u'+\frac{x^3}{3}u'', &\quad u(0)=A, \quad u(1)=B
\end{split}
\end{equation*}
Is there any analytical method to solve the problem? According to WolframAlpha one can write the equation as a Sturm-Liouville equation a... |
(Following an earlier post [here][1]) I intuit that if we restrict to functions $f(x,y,z)$ that are harmonic (i.e. satisfying $\nabla^2f=0)$ but neither a linear function nor a constant, all the three eigenvalues of the $3\times 3$ Hessian matrix
$$
H(x,y,z)=
\begin{bmatrix}
f_{xx} & f_{xy} & f_{xz}\\
f_{yx} & f... |
Is it possibile to have only zero eigenvalues of the Hessian of a harmonic function that is neither a linear nor a constant? |
Possible to have only zero eigenvalues of the Hessian of a harmonic function that is neither linear nor a constant? |
(Following an earlier post [here][1]) I intuit that if we restrict to functions $f(x,y,z)$ that are harmonic (i.e. satisfying $\nabla^2f=0)$ but neither of the form $ax+by+cz+d$ ($a,b,c,d\in\mathbb{R}$) nor a constant, all the three eigenvalues of the $3\times 3$ Hessian matrix
$$
H(x,y,z)=
\begin{bmatrix}
f_{xx}... |
Possible to have only zero eigenvalues of the Hessian of a harmonic function that is neither of the form $ax+by+cz+d$ nor a constant? |
> Let $X$ be a second countable normal space (where a normal space is not necessarily $T_4$).
Then there is a countable collection of continuous maps $\{f_n:X\to[0,1]\}$ such that for any $x\in X$ and closed subset $A$ with $x\not\in A$, there is $f_k$ satisfying $f_k(x)=0$ and $f_k(A)=\{1\}$.
My attempt:
If we ... |
The round metric on $S^2$ with radius $\rho$ is given by $g_{S^2(r)} = \rho^2 \cdot \left(\begin{matrix} \sin^2(\phi) & 0 \\ 0& 1 \end{matrix} \right)$. Here the top-left entry is given by $g_{\theta\theta}$ from viewing $(X,Y,Z) = (\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi)$ of the inclusion of $S^2... |
$$y^{\prime \prime}+P(x) y^{\prime}+Q(x) y=0$$
Provided we know a solution $y_1$ we assume the second linearly independent solution to be $y_2=v y_1$ substitute this back into the ODE and end up with this relation $y_2=y_1 \int \frac{1}{y_1^2} e^{-\int P(x) d x} d x$
My issue with this is that why is it necessary t... |
Why do second order linear ODE have solutions $y_1$ and $y_2=v y_1$ where $v$ is a function of x? |
The following popped up when I was contemplating the completion of normed linear spaces.
**Question:** Let $E$ be an NLS (normed linear space) and $X$ a topological space. Let $E$ sit inside $X$ as a dense subset. Under what conditions can one ensure that there exists *at most one* NLS structure on $X$ that extends ... |
This summation came up in a problem I was working on. My first attempt in coming up with a formula yielded the result $\binom{n-1}3\binom{n-2}3-\sum\limits_{k=3}^{n-3}\binom{k+1}2\binom k3$ at which point I was unable to simplify any further. I then devised a second method to achieve a formula and came up with the re... |
Does $\sum\limits_{k=5}^n\binom{k-3}2\binom{k-1}3$ have a closed form? |
Let $X$ be an arbitrary set, and let $\mathbb F$ be an arbitrary field. Functions $X \to \mathbb F$ form a vector space over $\mathbb F$, with pointwise addition and scalar multiplication. What is its dimension as a cardinal?
This question has been [answered](https://math.stackexchange.com/a/3817141/718671) for spec... |
Choose two random numbers from $[0,1]$ and let them be the endpoints of a random interval. Repeat this independently for $n$ times to get $n$ random subintervals (denoted as $U_1,\dots,U_n$).
Let $T_n = \max\{|S|: S\subset\{U_1,\dots,U_n\}, U_i\cap U_j=\emptyset \mbox{ for any }U_i\not =U_j\in S\}$, where $|S|$deno... |
The following popped up when I was contemplating the completion of normed linear spaces.
**Question:** Let $E$ be an NLS (normed linear space) and $X$ a topological space. Let $E$ sit inside $X$ as a dense subset. Under what conditions can one ensure that there exists *at most one* NLS structure on $X$ that extends ... |
I was checking the subadditivity of the function $f\colon x\mapsto \min(1, x)$, *i.e.*, whether $f(x + y)\le f(x) + f(y)$ and was expecting that the entire first quadrant would be colored. However, the following was outputted:
[![enter image description here][1]][1]
But clearly, for nonnegative $x, y$ satisfying ... |
Let me answer for my original question. Let $I:=[-\pi,\pi]$. Since $f'$ is (right) continuous at $x_0 \in \operatorname{int}I:=(-\pi, \pi) $, $f'(x_0)=f'(x_0^+)$ ( $\because$ Manfred's book 4.4.3. theorem ) So,
$$ \lim_{t\to 0} \frac{f'(x_0+t)-f'(x_0^+)}{t} = \lim_{t\to 0} \frac{f'(x_0+t)-f'(x_0)}{t} =: f''(x_0)$... |
The tank has the shape of a horizontal cylinder with radius $r$ and length $l$. The water exists through a small opening on the top right side of the cylinder. It seems that there are two ways to solve this problem:
* integrate horizontally
* integrate vertically
My first attempt was done by integrating vertical... |
Note
$$\binom{k-3}{2}\binom{k-1}{3}=\frac{1}{12}(k-1)(k-2)(k-3)^2(k-4)$$
Consider the expression
$$\binom k5(5k-13)-\binom{k-1}{5}(5k-18)$$
Factorise out $(k-1)(k-2)(k-3)(k-4)/120$,
$$\frac{(k-1)(k-2)(k-3)(k-4)}{120}\bigg(k(5k-13)-(k-5)(5k-18)\bigg)$$
Notice that the expression in brackets simply reduces to $30k-... |
__Definitions.__
This is about repeating sequences of natural numbers such as $\overline{6} = 6,6,6, \dots$ or $\overline{2,1,2} = 2,1,2,2,1,2, \dots$. I am aware that these can be faithfully represented as the sequence of coefficients of a power series $f(x)(1 + x^k + x^{2k} + \dots)$ where $k = \deg f + 1$. And... |
Note
$$\binom{k-3}{2}\binom{k-1}{3}=\frac{1}{12}(k-1)(k-2)(k-3)^2(k-4)$$
Consider the expression
$$\binom k5(5k-13)-\binom{k-1}{5}(5k-18)$$
Factorise out $(k-1)(k-2)(k-3)(k-4)/120$,
$$\frac{(k-1)(k-2)(k-3)(k-4)}{120}\bigg(k(5k-13)-(k-5)(5k-18)\bigg)$$
Notice that the expression in brackets simply reduces to $30k-... |
Given a smoothly time-dependent density matrix (positive-semidefinite matrix with trace 1) $\rho(t)$, its von Neumann entropy is defined as
$$
S(t)=\mathrm{Tr}(f(\rho(t))),\ f:[0,\infty)\ni x\mapsto -x\ln x\in\mathbb{R}\ (0\ln0=0).
$$
I want to know when $S(t)$ is differentiable.
If $\rho(t)$ has the same rank, th... |
When is von Neumann entropy differentiable? (Edited) |
*A standard $6$-sided fair die is rolled until the last $3$ rolls are ascending. What is probability that the first such roll is a $1$, $2$, $3$, or $4$?*
**My attempt**
We can investigate the $3-$roll (when there are exactly $3$ rolls), the $4-$roll, the $5-$roll, the $n-$roll.
I did that via SQL, basically n t... |
> Is it the closure $\overline{f(Z)}$?
Yes indeed. Remember that we think of elements $g \in \mathbb C[Y]$ as functions $g: Y \to \mathbb C$. The map $f^*$ is then just composition $f^*g = g \circ f: X \to Y \to \mathbb C$. Now such a function $g$ vanishes on $f(X)$ if and only if $f^*g$ vanishes on $X$, i.e. $f^*g ... |
A standard 6-sided fair die is rolled until the last 3 rolls are strictly ascending. What is probability that the first such roll is a 1,2,3, or 4? |
*A standard $6$-sided fair die is rolled until the last $3$ rolls are strictly ascending. What is probability that the first such roll is a $1$, $2$, $3$, or $4$?*
**My attempt**
We can investigate the $3-$roll (when there are exactly $3$ rolls), the $4-$roll, the $5-$roll, the $n-$roll.
I did that via SQL, basi... |
This question concerns the theory developed in Winning Ways for Your Mathematical Plays. The relevant Volume 1 can be found online [here.](https://annarchive.com/files/Winning%20Ways%20for%20Your%20Mathematical%20Plays%20V1.pdf)
I'm unclear about the intuition behind the authors' "Bypassing Reversible Moves" result,... |
In a round robin tournament with $n$ teams, each match ends in either a win or loss. What is the minimum number of losses to ensure that a team is out of contention of top 2 spots? |
So recently I have encountered the following lemma:
Given $x \in X$ and $x' \in B(x,r)$, and a linear operator from $X$ to $Y$, one has $sup\{ \Vert Tx'\Vert \} \ge \lVert T \rVert r$ which is slightly surprising to me. (the proof is here https://arxiv.org/pdf/1005.1585.pdf ). I wonder (i) Will there be a strict ine... |
Let $A=\begin{pmatrix}1 & 1/2 \\ -1/2 & 0 \end{pmatrix}$. Define $(x_n)$ recursively by
$$
x_{n+1}=A x_n
$$
for some starting vector $x_0\in\mathbb R^2$. Let $\vert\cdot\vert$ be the Euclidean norm on $\mathbb R^2$. I want to show that this iteration converges linearly, which means that
$$
\lim_n \frac{\vert x_{n... |
**Let $G$ be a simple $n$-vertex triangle free graph with $\delta(G) \geq 3$. Prove that if $n \leq 11$, then $G$ is $3$-edge-connected. Show that such a graph could be not $2$-(vertex)-connected.**
Hint: Using Mantel's Theorem can help.
Mantel's Theorem: Let $f(n)$ be the maximum number of edges in a simple $n$-... |
In a round robin tournament with $n$ teams, each match ends in either a win or loss. What is the minimum number of losses to ensure that a team is out of contention of top 2 spots?
**My attempt:** assign them ranks 1 to $n$. There are a total of $\frac{n(n-1)}{2}$ matches and hence as many wins. So, the rank 1 needs... |
Just for a University reference, from the [Cambridge Dictionary Definition of a Prime][1]:
> a number that cannot be divided by any other number except itself and
> the number
Clearly, if negative numbers were allowed as factors, there would be no primes. So the number under consideration to be a prime itself c... |
In the book titled *Classical Theory of Fields*, by famous Physicist Landau, writes,
From $\nabla^2\phi(x,y,z)=0$,
> ...it follows, in particular, that the potential $\phi$ of the electric field can nowhere have a maximum or a minimum. For in order that $\phi$ have an extreme value, it would be necessary that the f... |
Let $y \in X_K$ lying above $x$. In particular, $l:\mathrm{Spec}\,\kappa(y) \rightarrow X_K$ factors through $i: \mathrm{Spec}\kappa(y) \to \mathrm{Spec}\,\kappa(x) \times_k \mathrm{Spec}\,K$, by mobilizing some commutative diagram argument.
Since $\kappa(x)$ is a finite extension of $k$ ( $\because$ $x$ is closed... |
I'm reading the Gortz's Algebraic Geometry, proof of Theorem 6.28 (p.159) and some question arises.
Let $k$ be a field, $X$ a $k$-scheme locally of finite type, and let $x \in X$ be a closed point.
And let $K$ be an algebraically closed extension of $k$. And write $X_K = X \otimes_k K$.
Let $y$ be a poin... |
> Given $10$ random variables $W_1,W_2,\cdots,W_{10}\stackrel{i.i.d}{\sim}N(\mu,\sigma)$, where $\mu,\sigma$ are **unknown**. How can we construct a **known** function $f$ such that $Y=f(W_1,W_2,\cdots,W_{10})\sim N(0,1)$?
> Note, here **known function** means there is no $\mu,\sigma$ in $f$, e.g. $f_1=\frac{X_1-\mu... |
I came across with the following problem: Let $p$ be a prime, prove that $(p-1)^np^{(n^2-n)/2}|\prod_{k=1}^n(p^n-p^{k-1})$ for any $n\in \mathbb{N}$. It is quite hard to prove it using elementary number theory. However, the problem becomes trivial if we notice that $|\operatorname{GL}_n(\mathbb{F}_p)|=\prod_{k=1}^n(p^n... |
Can every divisibility problem solved by Lagrange's theorem? |
I came across with the following problem: Let $p$ be a prime, prove that $$(p-1)^np^{(n^2-n)/2}\mid \prod_{k=1}^n(p^n-p^{k-1})$$ for any $n\in \mathbb{N}$. It is quite hard to prove it using elementary number theory. However, the problem becomes trivial if we notice that $|\operatorname{GL}_n(\mathbb{F}_p)|=\prod_{k=1}... |
>More precisely, if $m\mid n$, can we always find a group $G$ of order $n$ and a subgroup $H$ of $G$ of order $m$?
Yes.
For each $n$, there exists a cyclic group of order $n$. In the cyclic group $G$ of order $n$, for each $d\mid n$, there exists a subgroup of order $d$. |
Let $y \in X_K$ lying above $x$. In particular, the canonical morphism $l:\mathrm{Spec}\,\kappa(y) \rightarrow X_K$ factors through some $i: \mathrm{Spec}\kappa(y) \to \mathrm{Spec}\,\kappa(x) \times_k \mathrm{Spec}\,K$, by mobilizing some commutative diagram argument.
Since $\kappa(x)$ is a finite extension of $k... |
So recently I have encountered the following lemma:
Given $x \in X$ and $x' \in B(x,r)$, and a linear operator from $X$ to $Y$, one has $sup\{ \Vert Tx'\Vert \} \ge \lVert T \rVert r$ which is slightly surprising to me. (the proof is here https://arxiv.org/pdf/1005.1585.pdf ). I wonder (i) Will there be a strict ine... |
For each postive integer $n:$
$a_n=\frac{n^2}{n^2-45n+675}.$
Evaluate $a_1+2a_2+3a_3+\cdots+44a_{44}$
What I have tried:
I have taken $a_1,a_2,\cdots,a_{44}$ and put these values into $a_1+2a_2+3a_3+\cdots+44a_{44}.$
Is there any other way to solve this problem?
Any guidance will be highly appreciated.Thank... |
Let $y \in X_K$ lying above $x$. In particular, the canonical morphism $l:\mathrm{Spec}\,\kappa(y) \rightarrow X_K$ factors through some $i: \mathrm{Spec}\kappa(y) \to \mathrm{Spec}\,\kappa(x) \times_k \mathrm{Spec}\,K$, by mobilizing some commutative diagram argument (maybe involving $\mathrm{Spec}(φ_1)$ and $\mathrm{... |
>More precisely, if $m\mid n$, can we always find a group $G$ of order $n$ and a subgroup $H$ of $G$ of order $m$?
Yes.
For each $n$, there exists a cyclic group of order $n$. In the cyclic group of order $n$, for each $d\mid n$, there exists a subgroup of order $d$.
---
*Note that this renders the questi... |
I need to prove $f(y) \geq f(x) + \langle \nabla f(x), y-x \rangle, \forall x,y \in C \subset \Bbb{R}^n$ where $C$ is a convex set, and $f:C \rightarrow \Bbb{R}$ is a convex function.
I saw someone explain this and he reaches at the following expression using definition of convex function:
$$
\lim_{\lambda \rig... |
>More precisely, if $m\mid n$, can we always find a group $G$ of order $n$ and a subgroup $H$ of $G$ of order $m$?
Yes.
For each $n$, there exists a cyclic group of order $n$. In the cyclic group of order $n$, for each $d\mid n$, there exists a subgroup of order $d$.
---
*Note that this assumes $m\mid n$,... |
What is a Nyström method for an object experiencing friction? |
**Background:**
A thin filament of fluid mixes into a flow field in accord with the so-called "compression-diffusion equation" (CDE), which is essentially an advection-diffusion equation with a particular space-time dependent velocity field $u(x,t) = \dot{s}(t)x/s(t)$. The variable $s(t)$ is the time-dependent thick... |
Here $\mathbb{T}$ denotes the torus $\mathbb{R}\backslash \mathbb{Z}$, and $\mathbb{D}^2$ the closed unit ball in $\mathbb{R}^2$. Since $\mathbb{T}\times [0,1]$ and $\mathbb{D}^2$ are both smooth compact manifolds with boundary (I'm not sure about this statement, if I was wrong, please correct me), can we establish a d... |
Whether $\mathbb{T}\times [0,1]$ is diffeomorphic to $\mathbb{D}^2$? |
>More precisely, if $m\mid n$, can we always find a group $G$ of order $n$ and a subgroup $H$ of $G$ of order $m$?
Yes.
For each $n$, there exists a cyclic group of order $n$. In the cyclic group of order $n$, for each $d\mid n$, there exists a subgroup of order $d$.
---
*Note that this assumes $m\mid n$,... |
What is a converging explicit Nyström method for an object experiencing friction? |
Given a Levi-Civita connection $\nabla$ on the tangent bundle with bundle metric $g$, a spinor representation $(\rho, S)$ and a spin structure $P$, what is the induced connection on a spinor bundle $P_{Spin} \times_\rho S$? I am looking for a coordinate-free definition. I understand that $\nabla$ canonically induces a ... |
**NOTE:** The answer has been improved using Тyma's suggestion in the comment section.
First, a couple of observations. The function $z^z/z$ seems to be continuous. I checked it using the [Desmos][1] graphing calculator. The picture is given below:
[![Desmos][2]][2]
Second, the integral $\int z^z$ dz is called... |
I am new to the study of linear algebra and there may be errors in the description of my question. My question is that:
Are there any of the following methods for generating nilpotent matrices $M^{n\times n}$ from multivariable polynomials.
$M^{n\times n}\leftarrow\begin{pmatrix}f_{1,1}(\overrightarrow{z_1}):=\su... |
How to construct the nilpotent matrix using multi-variable polynomials? |
I have the following coupled non-linear ODE system, which describes a biological system:
$$
\begin{cases}
\dfrac{dp}{dt} = -\gamma p f,\\
\\
\dfrac{df}{dt} = \gamma p f,\\
\\
\dfrac{dT}{dt} = \left( 1 - \dfrac{k}{T} \right) \left\{ b (1 - T) \dfrac{r^n}{f^n + r^n} - m \dfrac{f^n}{f^n + r^n} \right\},
\end{cas... |
Can an ODE system never reach its stable equilibrium in the long run? |
I have the following coupled non-linear ODE system, which describes a biological system:
$$
\begin{cases}
\dfrac{dp}{dt} = -\gamma p f,\\
\\
\dfrac{df}{dt} = \gamma p f,\\
\\
\dfrac{dT}{dt} = \left( 1 - \dfrac{k}{T} \right) \left\{ b (1 - T) \dfrac{r^n}{f^n + r^n} - m \dfrac{f^n}{f^n + r^n} \right\},
\end{cas... |
In the book titled *Classical Theory of Fields*, by famous Physicist Landau, writes,
From $\nabla^2\phi(x,y,z)=0$,
> ...it follows, in particular, that the potential $\phi$ of the electric field can nowhere have a maximum or a minimum. For in order that $\phi$ have an extreme value, it would be necessary that the f... |
I am reading an article in which they state the following fact:
If $a,b,x>0$, then $x\leq a\sqrt{x}+b$ implies that $x\leq a^2+b$.
I think the proof of this should be pretty trivial, but I cannot figure it out. I have tried squaring both sides of the first inequality and then using the quadratic formula, but it d... |
How do you find the intersection of two vector lines in 3D space where the point of intersection is the starting point of one line and is unknown? |
If $(A, \mathfrak{m}) \hookrightarrow (B, \mathfrak{n})$ is a local injection of Noetherian local domains, do we necessarily have $\dim B \geq \dim A$? |
Consider the series $\displaystyle\sum_{k=0}^{\infty}x_k$, where $x_k>0$ for all $k\geq 0$. If $\displaystyle\lim_{k\to\infty}\frac{x_{k+1}}{x_k}=1^+$, i.e. the ratio of consecutive terms converges to 1 from above, can the series converge? Intuitively I think this shouldn't be the case, but judging from the way the pro... |
$\lim_{k\to\infty}\frac{x_{k+1}}{x_k}=1^+ and $\sum_k x_k$ converges? |
Let $(R, m_R, \kappa_R) \subset (B, m_B, \kappa_B)$ be an extension of discrete valuation rings whose corresponding extension of fields of fractions $K :=\text{Frac}(R) \subset \text{Frac}(B) =: L$ is finite of degree $n:=[L:K]$.
It is well known that in general we have only the inequality $[L:K] \ge e_{B/R} \cdot [\k... |
If we have a conic with equation: $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ or S =0, and a point $P(x_1, y_1)$.
Then the equation of the chord with P as its midpoint is given by T = $S_1$.
$S_1$ is obtained by plugging point P in S. ($S_1$ = $ax_1^2 + 2hx_1y_1 + by_1^2 + 2gx_1 + 2fy_1 + c = 0$)
>The 'T' form... |
In the current Wikipedia entry for covers, it says that
>if ${\displaystyle C=\lbrace U_{\alpha }:\alpha \in A\rbrace }$ is an indexed family of subsets ${\displaystyle U_{\alpha }\subset X}$ (indexed by the set ${\displaystyle A}$), then ${\displaystyle C}$ is a cover of ${\displaystyle X}$ if ${\displaystyle \bigcu... |
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