Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
What is the difference between average slope and Instant slope(Instantaneous Rate of Change) I'm starting to learn calculus, and I'm getting confused about what average slope and instant slope(instantaneous rate of change)do and what they're differences are after looking at several sources on the internet. I know that ... | $\Delta y$ and $\Delta x$ represent actual numbers. If you have two points on the graph of a function, then $\Delta y$ is the change in their $y$-coordinates, and $\Delta x$ is the change in their $x$-coordinates. So, when you divide change in $y$ by change in $x$, i.e. $\frac{\Delta y}{\Delta x}$, you get the slope... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3750972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to solve this logarithmic equation with sum of exponential functions? I come across this logarithmic equation recently (solve for $x \in \mathbb{R}$) :
$$ 2x \geq \log_2 \left( \frac{35}{3} \cdot 6^{x-1} - 2 \cdot 9^{x - \frac{1}{2}} \right)$$
With few quick changes, this equation can be rewritten as :
$$ \ln \left... | We need to solve $$2^{2x}\geq\frac{35}{18}\cdot6^x-\frac{2}{3}\cdot9^x,$$ where $$ \frac{35}{18}\cdot6^x-\frac{2}{3}\cdot9^x>0$$ and after substitution $\left(\frac{3}{2}\right)^x=t$ we obtain a quadratic inequality:
$$\frac{2}{3}t^2-\frac{35}{18}t+1\geq0.$$
Can you end it now?
I got the following answer.
$$(-\infty,-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3751082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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values of q for which tangent integral is converges
Finding value of $q$ for which $$\int^{1}_{0}\frac{1}{(\tan (x))^{q}}dx$$ converges
What i try::
Let $\tan x=t.$ Then $\displaystyle dx=\frac{1}{\sec^2 (x)}dx=\frac{1}{1+t^2}dt$
And changing limits
$$I=\int^{\tan (1)}_{0}\frac{1}{(1+t^2)t^{q}}dt<\int^{\tan(1)}_{0}\f... | As $x\to0$, $\tan x\sim x$ and so $I$ converges iff $\int_0^1\frac{dx}{x^q}$ converges,
that is iff $q<1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3751310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Another way to solve $\int \frac{\sin^4(x)}{1+\cos^2(x)}\ dx$ without the substitution $y=\tan\left(\frac{x}{2}\right)$? Is there another way to solve an integral $$\int \frac{\sin^4(x)}{1+\cos^2(x)}\ dx$$ without the substitution $y=\tan\left(\frac{x}{2}\right)$?
$\large \int \frac{\sin^3(x)}{1+\cos^2(x)}\ dx$ is easi... | If you enjoy special functions, using $t=\tan(x)$
$$I_n=\int \frac{\sin^n(x)}{1+\cos^2(x)}\ dx=\int \left(\frac{t}{\sqrt{t^2+1}}\right)^n\frac{dt}{t^2+2}$$
$$I_n=\frac {t^{n+1}}{2(n+1)}\,F_1\left(\frac{n+1}{2};\frac{n}{2},1;\frac{n+3}{2};-t^2,-\frac{t^2}{2}\right)$$ where appears the Appell hypergeometric function of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3751405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
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List of quadratic field with the UFD property Let D be a square free integer let $K=\mathbb{Q}(\sqrt{D})$ and let
$\mathcal{O_{K}}$ be the ring of integer of $K$
My question: where I can find a list of of the value of $D$ which makes the ring of the quadratic field
$K=\mathbb{Q}(\sqrt{D})$ have the UFD property for $D... | The broken one-liner
L = [n for n in [2..300]
if n.is_squarefree() and QuadraticField(n, 'a').class_number() == 1]
computes in sage the list of all squarefree $n\le 300$, so that $\Bbb Q(\sqrt n)$ has class number one, this corresponds to the property of being UFD. Some first few entries in L are:
sage: L[:21]
[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3751518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that a positive operator is invertible Help guys, I need to prove this:
Let $(V,\langle~,~ \rangle)$ a finite n-dimensional euclidean space.Let $T$ be a linear operator defined positive (There exists a non singular operator $S$ such that $T=S^*S$) on $V$, prove that $T$ is invertible
I tried this:
We know by hypo... | $T$ is invertible if $Tx=0$ implies $x=0$. In this case, $T=S^*S$ where $S$ is non-singular. Therefore, if $Tx=0$, it follows that
$$
0 = \langle Tx,x\rangle=\langle S^*Sx,x\rangle=\langle Sx,Sx\rangle=\|Sx\|^2 \implies Sx=0 \implies x=0.
$$
So $T$ is invertible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3751638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
What is Friedberg doing in this proof to show the (ij)th entry of this matrix Here is the Theorem:
Let $V$ and $W$ be finite-dimensional vector spaces over $F$ with ordered bases $\beta = \{x_1, \ldots, x_n\}$ and $\gamma = \{y_1, \ldots, y_m\}$ respectively. For any linear transformation $T : V \to W$, the mapping $T^... | You have not explicitly said so, but I suspect that $(x_1,\dots,x_n),(y_1,\dots,y_m)$ are meant to denote bases for $V$ and $W$, and $(f_1,\dots,f_n),(g_1,\dots,g_m)$ are the corresponding dual bases for $V^*$ and $W^*$. Please correct me if I am wrong.
Recall that for a transformation $\alpha:V \to W$, the entries $a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3751735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Evaluating $\lim _{m\to\infty}\left(\frac1{m^2}+\frac2{m^2}+\frac3{m^2}+\cdots+\frac{m}{m^2}\right)$. Where's my error? Question is following
$$\lim _{m \to \infty}\left(\frac{1}{m^{2}}+\frac{2}{m^{2}}+\frac{3}{m^{2}}+\cdots+\frac{m}{m^{2}}\right)$$
method-1
$$\lim _{m \rightarrow \infty}\left(\frac{m(m+1)}{2 m^{2}}\ri... | First case is correct. Second case wrong: limit of sums equal sum of limits when we have fixed amount of summands.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3751879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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question about similar triangle word problem A person is walking directly away from a light on an $18$-foot tall pole. At this instant, the person casts a shadow $14$ feet long. If they walk 10 feet farther from the pole, they will cast a shadow $20$ feet long. How tall is the person?
I've tried setting up $2$ unknown ... | See the image (not to scale!):
We have from the smaller triangle: $$(x+14):18 = 14:h$$
and from the big triangle: $$(x+30):18 = 20:h$$
which resolves to $$\begin{cases}x=18\times 14:h - 14\\x=18\times 20:h-30\end{cases}$$
Left hand sides are the same, so right hand sides are equal, too:
$$\frac{18\cdot 14}h-14 = \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3752007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I transform this problem into a semidefinite program? $$\begin{array}{ll} \text{minimize} & \dfrac{(c^T x)^2}{(d^Tx)}\\ \text{subject to} & Ax \leq b\\ & d^T x > 0\end{array}$$
I have been stuck on this question for a couple days. I am sharing with you what I tried, although I am pretty sure it's wrong. Please h... | Let $r=d^Tx > 0$ and $s=c^T x$. Your objective $\min t$ subject to $t \geq \frac{s^2}{r}$ is conic representable via the rotated quadratic cone (best for practical optimization):
$(t,r,s) \in \mathcal{Q}_r^3$,
or alternatively via the semidefinite cone:
$\left(\begin{array}{ll}
t & s\\
s & r
\end{array}\right) \in \mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3752182",
"timestamp": "2023-03-29T00:00:00",
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Inverse of $y=1/x$ Can anybody say anything about the inverse function of $y=1/x$ and plot it on a graph and then compare the graphs of the given function and it's inverse? Is $y=1/x$ invertible? If yes then do the graphs of $y=1/x$ and its inverse coincide?
| Solve the equation
$$y=\frac1x$$ for $x$. This immediately gives you
$$x=\frac1y$$ and is valid for $x,y\ne0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3752320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Convergence of double Integral Let's assume that $f \in L^p(\mathbb{R})$ ($1 \leq p < \infty$). Does it then hold that
$$
\lim_{n \rightarrow \infty} \int_0^1 \int_0^1 \left \lvert f\left( \frac{\tilde{r}}{n} \right) -f\left(\frac{r}{n} \right) \right \rvert^p ~\mathrm{d}r \mathrm{d}\tilde{r} = 0 \quad ?
$$
And if yes,... | Well, for fixed $\tilde{r} \in [0,1]$ we have
$$\lim_{n\to\infty} \int_0^1 \left|f\left(\frac{\tilde{r}}n\right) - f\left(\frac{r}n\right)\right|^p\,dr =\int_0^1 \lim_{n\to\infty}\left|f\left(\frac{\tilde{r}}n\right) - f\left(\frac{r}n\right)\right|^p\,dr= 0$$
by the Lebesgue dominated convergence theorem since the int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3752440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Is there a geometric intuition for integration by parts? Is there a geometric intuition for integration by parts?
$$\int f(x)g'(x)\,dx = f(x)g(x) - \int g(x)f'(x)\,dx$$
This can, of course, be shown algebraically by product rule, but still where is geometric intuition? I have seen geometry of IBP using parametric equat... | Note. Edited because Adayah pointed out (correctly, and to my chagrin) that this answer was totally sloppy—sloppier even than I intended it to be. Let's hope it's better now.
When we use integration by parts on an integral
$$
\int u(x) \, \mathrm{d}v(x) = \int u(x) v'(x) \, \mathrm{d}x
$$
we implicitly treat $u$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3752680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 2,
"answer_id": 1
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Show $\log(\det(A))\le \operatorname{tr}(A)-n$ Suppose that $A$ is a real, symmetric, positive definite $n\times n$ matrix. Show that
$$\log(\det(A))\le \operatorname{tr}(A)-n \quad \text{and} \quad \log(\det(I_n+A))\le \operatorname{tr}(A).$$
Since $A=CDC^{-1}$ we can say the following:
$$\det(A)=\det(C)\det(D)\det(C... | We have:
$$\mathrm{Tr}(A) = \sum \lambda_i $$
$$\det(A) = \prod \lambda_i $$
So, if eigenvalues are positive reals, we have to show
$$ \sum \ln \lambda_i \leq \sum \lambda_i -n $$
which is true as $$\ln x \leq x -1 $$
for all $x>0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3752813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If $X$ is $\sigma(\mathcal{G} \cup \mathcal{H})$-measurable can we write $X=f(G,H)$? Let $(\Omega, \mathcal{F}, P)$ be a probability triplet. Let $\mathcal{G}$ and $\mathcal{H}$ be two sub-sigma-algebras of $\mathcal{F}$. Let $X:\Omega\rightarrow\mathbb{R}$ be a random variable such that
$\sigma(X) \subseteq \sigma(\m... | What is true, and this may be sufficient for your purposes, is that under the stated conditions there is a $\mathcal G\otimes\mathcal H$-measurable map $Z:\Omega\times\Omega\to\Bbb R$ such that $X(\omega) = Z(\omega,\omega)$ for all $\omega\in\Omega$. (And conversely, because $\omega\mapsto(\omega,\omega)$ is $\sigma(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3752941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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Can we apply here the Cayley–Hamilton theorem? We have the matrix \begin{equation*}A:=\begin{pmatrix}3 & 1 & 0 & -1& -1 \\ 0 & 2 & 0 & 0 & 0 \\ 1 & 0 & 2 & 0 & -1 \\ 0 & 0 & 0 & 2 & 0 \\ 1 & 0 & 0 & -1 & 1\end{pmatrix}\in M_5(\mathbb{R})\end{equation*}
The characteric polynomial is \begin{equation*}P_A(\lambda)=(2-\lam... | Here, dimension of eigenspace corresponding to the eigenvalue 2 is 2, There are two Jordan canonical forms possible! So, in one Jordan canonical form, there can be two blocks possible which are of order 3 and order 2 and in another of Jordon canonical form, there can also be possiblity for two blocks of order 4 and 1,... | {
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What does `Sitzber. Heidelberg Akad. Wiss., Math.-Naturw. Klasse. Abt. A' stand for? I would like to cite an article from 1914 by Oskar Perron without any abbreviations. I am unable to figure out what `Sitzber. Heidelberg Akad. Wiss., Math.-Naturw. Klasse. Abt. A' is short for. Can anyone here perhaps help me with this... | It is “Sitzungsberichte der Heidelberger Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse: Abteilung A, Mathematisch-physikalische Wissenschaften” – Source
| {
"language": "en",
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Prove that the diagonals of a regular octagon intersect at the angular points of a square. $ABCDEFGH$ is a regular octagon and $AF, BE, CH, DG$ are drawn. Prove that their intersections are the angular points of a square.
This question is pretty trivial to solve with coordinate geometry, but it is time-consuming to fi... | Another way.
Let $BE\cap CH=\{K\},$ $BE\cap DG=\{L\}$, $AF\cap DG=\{M\}$ and $AF\cap CH=\{N\}.$
Thus, since our octagon is cyclic, we obtain: $$\measuredangle EBF=\measuredangle BFA$$ and from here $$BE||AF,$$ which gives $$KL||MN.$$
Similarly $$KN||ML,$$ which gives that $KLMN$ is a parallelogram.
Now, since $$\measu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3753350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the image of a faithful representation of a Lie group a Lie subgroup? Let $\varphi:G\to\operatorname{GL}_n(\mathbb{R})$ be a faithful Lie group representation, i.e. an injective Lie group homomorphism. Then is $\varphi(G)$ a Lie subgroup of $\operatorname{GL}_n(\mathbb{R})$? Somehow I think this shouldn't be to ... | You are perhaps confused about the definitions, specifically the definition of Lie subgroup (which you have not specified in your question; I will do so below following the most common convention).
An immersion is a $C^\infty$ map $\phi:N \to M$ between manifolds such that the differential $d \phi_n$ is injective at ea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3753481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do you find the interval in which a parametric equation will be traced exactly once I have been all over the internet and I can't find an answer to what seems like a simple question. I want to be able to find the interval for a parametric equation so that it is only traced once. My equations are:
\begin{align}
x &=... | You are looking for the smallest value of $p$ such that $x(u)=x(u+p)$ and $y(u)=y(u+p)$ for all $u$. Notice the equations for both contain only trigonometric functions, so this $p$ must be some multiple of $\pi$. Consider $$x(u)=11\cos u-4\cos\left(\frac{11u}{2}\right) $$ The period of the first term is clearly $2\pi$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3753625",
"timestamp": "2023-03-29T00:00:00",
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Simplify $\frac{d}{dt}\int_x^t f(t,y)dy$ I am trying to simplify $\frac{d}{dt}\int_x^t f(t,y)dy$ as a part of a proof.
I am somewhat confused on how I can proceed with this. Do I define a function $g(t,y)$ such that $\frac{\partial g}{\partial y} = f(x,t)$ and then say $\frac{d}{dt}\int_x^t f(t,y)dy = \frac{d}{dt}(g(t,... | You can think of the integral as being a function of three parameters:
$$g(a,b,c) = \int_a^b f(c,y)\:dy$$
Thus the derivative you want can be derived from chain rule
$$\frac{d}{dt}g(a(t),b(t),c(t)) = \frac{\partial g}{\partial a}\frac{da}{dt} + \frac{\partial g}{\partial b}\frac{db}{dt} + \frac{\partial g}{\partial c}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3753735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine the $\lambda \in \mathbb{R}$ for which this integral converges Consider the cuspoidal cubic given by $x^2 - y^3 =0$ in $\mathbb{C}^2$. The log-canonical threshold of the cuspoidal cubic is determined by finding the largest value of $\lambda \in \mathbb{R}$ for which the integral $$\int \frac{1}{| x^2 - y^3|^{... | Here is another approach that I have learned from Donaldson:
For a non-negative integer $r$, consider the annular regions $$\Omega_r = \{ (z,w) \in \mathbb{C}^2 : 2^{-3(r+1)} \leq | x | \leq 2^{-3r}, \ 2^{-2(r+1)} \leq | y | \leq 2^{-2r} \}.$$
Let $I_r = \int_{\Omega_r} | x^2 - y^3 |^{-2 \lambda}$. The substitution $z ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3753806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How can I integrate $\int \frac{u^3}{(u^2+1)^3}du?$ How to integrate following
$$\int\frac{u^3}{(u^2+1)^3}du\,?$$
What I did is here:
Used partial fractions
$$\dfrac{u^3}{(u^2+1)^3}=\dfrac{Au+B}{(u^2+1)}+\dfrac{Cu+D}{(u^2+1)^2}+\dfrac{Au+B}{(u^2+1)^3}$$
After solving I got
$A=0, B=0, C=1, D=0, E=-1, F=0$
$$\dfrac{u^3}{... | Substitute $u=\sinh t$ to integrate
\begin{align}
& \int \dfrac{u^3}{(u^2+1)^3}du= \int \frac{\sinh^3t}{\cosh^5t}dt\\
=&\int\tanh^3td(\tanh t)=\frac14\tanh^4t+C= \frac{u^4}{4(u^2+1)^2}+C
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3753883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 10,
"answer_id": 2
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Proof Check: $x \leq y+ \epsilon$ for all $\epsilon >0$ iff $x \leq y$. Synopsis
I want to be sure I'm utilizing proof by contradiction correctly, so please check my proof of the exercise below. It's relatively simple, so it shouldn't take you too much time.
Exercise
Let $x$ and $y$ be real numbers. Show that $x \leq ... | No, it is not correct. I tried to prove it by contradiction, and you got the conclusion that 0<\varepsilon, claiming that that's a contradiction. Why? There is no contradiction there.
If $x>y$, let $\varepsilon=\frac12(x-y)$. Then $\varepsilon>0$ and therefore $x\leqslant y+\varepsilon$. But this means that$$x\leqslant... | {
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"url": "https://math.stackexchange.com/questions/3754141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove inequality $\tan(x) \arctan(x) \geqslant x^2$ Prove that for $x\in \left( - \frac{\pi} {2},\,\frac{\pi}{2}\right)$ the following inequality holds
$$\tan(x) \arctan(x) \geqslant x^2.$$
I have tried proving that function $f(x) := \tan(x) \arctan(x) - x^2 \geqslant 0$ by using derivatives but it gets really messy an... | It's enough to prove this for $0<x<\pi/2$. Let $f(x)=(\tan x)/x$. Then $f$ is increasing
on $(0,\pi/2)$. To prove this, for instance $f(x)$ has nonnegative Maclaurin coefficients.
Let $x\in(0,\pi/2)$, and let $y=\arctan x$. Then $x=\tan y\ge y$ as $f(y)=(\tan y)/y\ge1$.
Therefore $g(y)\le g(x)$, that is
$$\frac{\tan y}... | {
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"url": "https://math.stackexchange.com/questions/3754270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How do I finish solving $f(x)f(2y)=f(x+4y)$? I'm trying to solve this functional equation:
$$f(x)f(2y)=f(x+4y)$$
The first thing I tried was to set $x=y=0$; then I get:
$$f(0)f(0)=f(0)$$
which means that either $f(0)=0$ or we can divide the equation by $f(0)$ and then $f(0)=1$.
Case 1: If $f(0)=0$ then we can try to se... | Assuming that $ f(0)=1$, you got
for $ y\in \Bbb R $,
$$f(4y)=f(2y)=f(y)=f(\frac y2)$$
$$=...=f(\frac{y}{2^n})$$
for each $ n\ge 0$.
but by continuity of $ f $ at $ 0$,
$$\lim_{n\to +\infty}f(\frac{y}{2^n})=f(0)=1$$
thus
$$(\forall y\in \Bbb R)\;\; f(y)=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3754417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Can we relax hypothesis of Fundamental theorem of calculus? Let $F$ is continuous $[a,b]$ and differentiable on $[a,b]$ and $F'(x)=f(x)$ for $x\in [a,b]$. Assume that $f$ is Riemann integrable. Then Fundamental theorem of calculus say that
$$F(x)-F(a)=\int_{a}^x f(t) dt$$
My Question is: Can we say that $$F(x)-F(a)=\... | Apostol gives the theorem in following manner
FTC: Let $f:[a, b] \to\mathbb {R} $ be Riemann integrable on $[a, b] $ and let $g:(a, b) \to\mathbb {R} $ be such that $g'(x) =f(x) $ for all $x\in(a, b) $. Then the limits $$\lim_{x\to a^{+} } g(x), \lim_{x\to b^{-}} g(x) $$ exist and we have $$\int_{a} ^{b} f(x) \, dx=\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3754502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Solve $x^5\equiv 4\pmod 7$ We know about calculating $x^2\equiv 2\pmod 7$ using quadratic residue properties in order to find out whether a solution exists or not.
I wonder is there any way to determine that $x^n\equiv k\pmod v$, where $v\ge 2$, $k\in\Bbb Z$, and $n\ge 3$? As I asked in title:
Solve $x^5\equiv 4\pmo... | Here is another way to determine this. Over the field $\Bbb F_7$ the Berlekamp algorithm gives the factorisation
$$
x^5-4=(x^4 + 2x^3 + 4x^2 + x + 2)(x - 2).
$$
Hence $x=2$ is the solution of the equation $x^5=4$. We may rewrite this as $x\equiv 2\bmod 7$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3754648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Proving that among any $2n - 1$ integers, there's always a subset of $n$ which sum to a multiple of $n$ How can one prove that among any $2n - 1$ integers, there's always a subset of $n$ which sum to a multiple of $n$?
It is not hard to see this is equivalent to show that among $2n-1$ residue classes modulo $n$ there a... | Posts may potentially alter the general conditions, such as restrict the set of available congruences and use a set of available integers which is considerably larger than necessary, with the idea being that a specific method can be used to solve the problem. The only such post I know of is the following one which deal... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
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Let $A\subset\Omega$ and $\mathcal{B}_{A} = \{B\cap A:B\in\mathcal{B}\}$. Show that $\mathcal{B}_{A}$ is a $\sigma$-algebra on $A$. Let $\Omega$ be a nonempty set and $\mathcal{B}$ be a $\sigma$-algebra on $\Omega$. Let $A\subset\Omega$ and $\mathcal{B}_{A} = \{B\cap A:B\in\mathcal{B}\}$. Show that $\mathcal{B}_{A}$ is... | If $S =A\cap B\in \mathcal B_A$ where $B \in \mathcal B$ then complement of $S$ in $A$ is $A\cap S^{c}$ and $S^{c} \in \mathcal B$ so $S^{c} \in \mathcal B_A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3754871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to integrate $ \int\frac{x-2}{(7x^2-36x+48)\sqrt{x^2-2x-1}}dx$? How to integrate $$ \int\frac{x-2}{(7x^2-36x+48)\sqrt{x^2-2x-1}}dx\,\,?$$
The given answer is $$ \color{brown}I=-\frac{1}{\sqrt{33}}\cdot \tan^{-1}\bigg(\frac{\sqrt{3x^2-6x-3}}{\sqrt{11}\cdot (x-3)}\bigg)+\mathcal{C}.$$
I tried by different substitutio... | $$I=\int \frac{x-2}{(7x^2-36x+48)\sqrt{x^2-2x-1}}\,dx$$ This can be simplifies using
$$\frac{x-2}{7x^2-36x+48}=\frac 1{7(a-b)}\left(\frac{a-2 } {x-a }+\frac{2-b } {x-b } \right)$$ where
$$a=\frac{2}{7} \left(9-i \sqrt{3}\right) \qquad \text{and} \qquad b=\frac{2}{7} \left(9+i \sqrt{3}\right) $$ which makes that we are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3755017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How to calculate $ \lim_{x\to\infty} (\frac{x}{x+1})^x$ using L'Hopitals rule? I am trying to calculate $ \lim_{x\to\infty} (\frac{x}{x+1})^x$ using L'Hopital.
Apparently without L'Hopital the limit is
$$ \lim_{x\to\infty} (\frac{x}{x+1})^x = \lim_{x\to\infty} (1 + \frac{-1}{x+1})^x = \lim_{x\to\infty} (1 - \frac{1}{x... | Caution,
$$\lim fg=\lim f\lim g$$
can only be used when the limits on the right both exist, which is not the case here.
By L'Hospital
$$\lim_{x\to\infty}\log\left(\frac x{x+1}\right)^x=\lim_{x\to\infty}\frac{\log\left(\dfrac x{x+1}\right)}{\dfrac1x}=\lim_{x\to\infty}\frac{\dfrac1x-\dfrac1{x+1}}{-\dfrac1{x^2}}=-\lim_{x\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3755096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Unclear problem with $n$-th power matrix and limit
Find $$\lim\limits_{n \to \infty} \frac{A_n}{D_n}$$ where
$$\begin{pmatrix}
19 & -48 \\
8 & -21 \\
\end{pmatrix} ^{\! n} =
\begin{pmatrix}
A_n & B_n \\
C_n & D_n \\
\end{pmatrix}$$
$n$ - is the power of a matrix, but what is $A_n, B... | Here is an unconventional approach: we have
$$
M = \pmatrix{19 & -48\\ 8 & -21}.
$$
We find that the eigenvalues satisfy
$$
\det(M - xI) = x^2 + 2x - 15 = 0 \implies x = -5,3.
$$
By the Cayley Hamilton theorem, the powers of $M$ satisfy the recurrence
$$
M^n + 2M^{n-1} -15 M^{n-2} = 0
$$
From the theory of constant coe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Players and coins Three players A,B and C flip simultaneously a coin. The coin of A (B,C) gives head with probability $a$ ($b,c$), with $0<a,b,c,<1$. If two of three coins give the same result, the player who flip the third coin is tossed out of play; if the coins are all equal, players flip again the coins.
*
*What ... | Letting $t$ be the probability that a given round ends in a tie, we get
$$
t=abc+(1-a)(1-b)(1-c)=1-a-b-c+ab+bc+ca
$$
Explanation:
*
*$abc$ is the probability that in a given round, all players get heads.$\\[4pt]$
*$(1-a)(1-b)(1-c)$ is the probability that in a given round, all players get tails.
Letting $p$ be the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If a 'distance function' does not possess triangle inequality property, would the limit of a converging sequence still be unique? Let $X$ be a set and $d$ be a function such that $d:X\times X\to \mathbb{R}$ such that it satisfies positivity, that is, $d(x,y)\geq 0$ and $d(x,y)=0 \iff x=y.$ Moreover suppose it satisfies... | Let $d(x,y) = (x-y)^2$ on $\Bbb R$, which satisfies the first two axioms but not the triangle inequality, because:
$$d(0,2)=4\not≤2=d(0,1)+d(1,2)$$
however limits are still unique, in fact you have the same limits as the usual metric $d(x,y)=|x-y|$, since $(x_n-y)^2\to0$ iff $|x_n-y|\to0$ (continuity of the root on pos... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluating $\int_{-\infty}^\infty\frac{\cos(2x)}{x^2+4}\:\mathrm{d}x$ As stated in the title, I want to evaluate the integral
$$I=\int_{-\infty}^\infty\frac{\cos(2x)}{x^2+4}\:\mathrm{d}x$$
I'm pretty sure it evaluates to
$$\frac{\pi}{2e^4}$$
But I'm not sure how to evaluate it.
I have read an Instagram post where 3 dif... | Too long for a comment, just reducing this case, to the case you know from instagram and some notes.
For $t \in \mathbb R,a > 0$ let:$$I(t,a) = \int_{-\infty}^\infty \frac{\cos(tx)}{x^2+a^2}dx $$
Note that it converges for every $t \in \mathbb R,a> 0$. Taking substitution $x=ay$, $dx=ady$ we get:
$$ I(t,a) = \int_{-\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3755533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Projection of space curve shortens Let $C$ be a rectifiable, open curve in $\mathbb{R}^3$,
and let $|C|$ be its length.
Orthogonally project $C$ to a plane $\Pi$ (e.g., the $xy$-plane).
Call the projected curve $C_{\perp}$, and its length $|C_{\perp}|$.
I would like to claim $|C_{\perp}| \le |C|$.
I would appreciate ei... | If you are dealing with rectifible curves, you are taking polygons with vertices
on $C$ and looking at the limit at the lengths of the polygons as the points
become closer. But if you project a line segment to $\{z=0\}$ its length cannot increase,
so the projected polygons are no longer than the original.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability that every player gets 1 queen, jack, and king The four jacks, queens, and kings from a standard deck of cards are shuffled, and three cards are dealt to each of four players. Compute the probability that each player gets one jack, one queen, and one king.
I know that there are $12$ cards total and since ev... | Let's write up what we discussed in chat.
Since we have to distribute three of the twelve cards to the first player, three of the remaining nine cards to the second player, three of the remaining six cards to the third player, and give the fourth player all three of the remaining three cards, there are
$$\binom{12}{3}\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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When does $(xz+1)(yz+1)=az^{3}+1 $ have finitely many solutions in positive integers? Consider the diophantine equation in three variables $x$, $y$ and $z$; ($xz+1$)($yz+1$) $=$ $6z^{3}+1$. The only positive integer solutions I have found are {$x=4,y=10,z=7$} and {$x=10,y=4,z=7$}. From a Maple program, I have iterated ... | Here is a partial answer: If $a=b^3$ is a cube, then there is an infinite family of solutions to $(xz+1)(yz+1)=az^3+1=b^3z^3+1$ given by
$$(x,y,z) = (b, b^2z-b, z),\ b, z\in\mathbb{N}.$$
This arises from the factorization $b^3z^3+1 = (bz+1)(b^2z^2-bz+1) = (bz+1)((b^2z-b)z+1)$.
In addition to the above, for any $a$ ther... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3756018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the angle between vectors $\mathbf x$ and $\mathbf y$ in radians Two unit vectors $\mathbf{x}$ and $\mathbf{y}$ in $\Bbb R^n$ satisfy $\mathbf{x}\cdot\mathbf{y}=\frac{\sqrt{2}}{2}$ in radians. How would I go about finding the angle between $\mathbf{x}$ and $\mathbf{y}$?
As I don't know the $\mathbf{x}$ and $\ma... | No, I believe the unit circle is not really involved here.
It is simple. You already know the $cosinus$ of the angle $\theta$ between the two vectors. It is this expression:
$$cos(\theta) = \dfrac{\mathbf{x}\cdot\mathbf{y}}{\mathbf{||x||\cdot ||y||}}$$
Just plug in the numbers in this formula. Thus you get:
$$cos(\thet... | {
"language": "en",
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Let s be a set of five positive integers at most 9. Prove that the sums of the elements in all the non empty subsets of s cannot be distinct.? Let s be the set of five positive integer the maximum of which is at most 9 prove that the sums of the elements in all the non empty subset of as cannot be distinct?
Note:
I kno... | Hint: Pigeonhole principle. You can form $31$ sums in total from the non-empty subsets of the five numbers. What is the largest such possible sum? What is the smallest?
| {
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Hard Differential Equation Can anyone help me to find the solution of this ODE : $$4(y')^2-y^2+4=0.$$
I've tried to find it's solution by putting $y = e^{at}$ (for null solution) and $y = 2$ (for particular solution). My final solution is $$y = c_1 e^{0.5t} + c_2 e^{-0.5t} + 2,$$
but didn't match with the solution of t... | The general form of equation
$$
A^2-B^2=1
$$
can be parametrized as $A=\pm\cosh(u)$, $B=\sinh(u)$ similar to a circle equation. If $(A,B)$ change smoothly, but remain on this curve, then also $u$ is a smooth function.
Here that gives $$y(x)=\pm2\cosh(u(x)), ~~ y'(x)=\sinh(u(x))$$ from that parametrization. Now take the... | {
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Integration of an improper integral I have an integral that has two individually divergent parts. Wolfram says that the answer is \begin{equation}\frac{\ln(s+1)}{s},\end{equation} but I cannot figure out how it's done. The integral is
\begin{equation}
\int_{1}^{\infty}\left({\frac{1}{x}-\frac{1}{x+s}}\right)dx
\end{equ... | $\int_1^{M} (\frac 1 x -\frac 1 {x+s})dx =\ln M-[\ln (M+s)-\ln (1+s)]$
$=\ln (\frac M {M+s})+\ln(1+s)$.
$Let$ $M \to \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3756451",
"timestamp": "2023-03-29T00:00:00",
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Finding the centre of a circle under a specific condition Question: Consider a circle, say $\mathscr{C}_1$ with the equation $x^2 + (y-L)^2=r^2$. A second circle, say $\mathscr{C}_2,$ with equal radii that has a centre $(x_0,y_0)$ which lies on the line $y=mx$. Find an expression for $x_0$ and $y_0$, in terms of $L$, $... | For the two circles to touch at exactly one point, the distance between the centers of the two circles should be the sum of the radii.
$$4r^2=(0-x_0)^2+(L-mx_0)^2.$$
Now solve for $m$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solving $2\sin\left(2x\right)=3\left(1-\cos x\right)$ Background - this was part of a homework packet for students looking to skip HS pre-calc. There is a text book they use as well, but this particular problem was not in it.
$$2\sin\left(2x\right)=3\left(1-\cos\left(x\right)\right)$$
My first step was to eliminate the... | Rewrite $2\sin\left(2x\right)=3\left(1-\cos x\right)$ as
$$ 4\sin\frac x2\cos\frac x2\cos x= 3\sin^2\frac x2$$
Then, let $t= \tan\frac x2, \> \cos x = \frac{1-t^2}{1+t^2}$ and factorize
$$\sin\frac x2 \frac{4-3t-4t^2-3t^3}{1+t^2}=0
$$
The factor $\sin\frac x2=0 $ yields $\frac x2 =\pi n$ and $4-3t-4t^2-3t^3=0$ has one ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Probability of $\limsup_{n\to \infty} \{X_n X_{n+1}>0\}$ where $\{X_n\}$ are independent Gaussian r.v.'s with mean 0 Let $\{X_n\}$ be a sequence of independent Gaussian random variables with $\mathbb{E}\, X_n = 0$ for all $n \geq 1$. Find the probability of the event
$$ \limsup_{n\to \infty} \big\{ X_n X_{n+1}> 0 \big\... | Note that it is enough to consider only events $\{X_{2k}X_{2k+1} > 0 \}_{k \in \mathbb N}$ and by independence of $\{X_k\}_{k \in \mathbb N}$, those are independent as well. Moreover $\mathbb P(X_{2k}X_{2k+1} > 0 ) = \mathbb P(X_{2k},X_{2k+1} > 0) + \mathbb P(X_{2k},X_{2k+1}<0) = \frac{1}{2}$ by symmetry, so by Borel C... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I show $\lim_{n \to \infty} \int_0^\infty \frac{n}{n^2+x}\sin(\frac{1}{x})\, dx = 0\,$? How do I show $$\lim_{n \to \infty} \int_0^\infty \frac{n}{n^2+x}\sin\left(\frac{1}{x}\right)\, dx = 0\,\,?$$ I've tried splitting into the cases where $x \leq 1$ and $x \geq 1$ but I am having trouble finding bounds so that ... | Edit: The second half of this is nonsense. See the comments below...
Say the integrand is $f$. If $0<x\le1$ then $|f(x)|\le 1$, while if $x\ge1$ then $|f(x)|\le 1/x^2$, since $|\sin(t)|\le|t|$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3757048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Express $\frac{\partial^2F}{\partial x^2} - \frac{\partial^2F}{\partial y^2}$ in terms of the partial derivatives of $F$ with respect to $u$ and $v$. If $F = F(u,v)$ and $u = x - y, v = x + y$, express $\frac{\partial^2F}{\partial x^2} - \frac{\partial^2F}{\partial y^2}$ in terms of the partial derivatives of $F$ with ... | $\frac{\partial u}{\partial x} = 1,\frac{\partial u}{\partial y} = -1, \frac{\partial v}{\partial x} = 1, \frac{\partial v}{\partial y} = 1$
$\frac{\partial F}{\partial x} $
$= \frac{\partial F}{\partial v}\frac{\partial v}{\partial x} + \frac{\partial F}{\partial u}\frac{\partial u}{\partial x}$
$= \frac{\partial F}{\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to find $\int_0 ^ \frac{\pi}{2} \frac{\cot x}{\cot x + \csc x}\,dx \,$?
How to find
$$\int_0 ^ \frac{\pi}{2} \frac{\cot x}{\cot x + \csc x}\,dx \,\,?$$
The integrand $ \frac{\cot x}{\cot x + \csc x} $ is not defined at $x =0$. But the function is bounded on $(0 , \frac{\pi}{2}]$.
$$\lim _{x \to 0} \frac{\cot x}{\... | $$\int_0 ^ \frac{\pi}{2} \frac{\cot x}{\cot x + cosec x} dx=\int_{0}^{\pi/2} \frac{\cos x}{1+\cos x} dx=\pi/2-\int_{0}^{\pi/2} \frac{dx}{1+\cos x} dx$$ $$=\pi/2-\frac{1}{2}\int_{0}^{\pi/2} \sec^2(x/2)~dx=\pi/2-\tan x |_{0}^{\pi/2}=\pi/2-1.$$
| {
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"url": "https://math.stackexchange.com/questions/3757423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\mathbb{Z}(\sqrt{11})$ a UFD?
Is $\mathbb{Z}(\sqrt{11})$ a UFD?
I read about real quadratic filed and about algebraic integers. In all of books that I read tehy show that imaginary filed doesn't have unique factorization such as $$6=2\cdot3=(1+\sqrt{-5})\cdot(1-\sqrt{-5})\,,$$ but not for real.
| The main reason that you won't find anything in the literature on the fact that $\mathbb{Z}[\sqrt{D}]$ is a UFD or not is that we don't know.
For $D>0$ squarefree such that $D\not\equiv 1 \mod 4$, this is equivalent to ask whether $\mathbb{Z}[\sqrt{D}]$ is a PID or not (because this ring is a Dedekind domain in this ca... | {
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Does $\ker T\cap {\rm Im}\,T=\{0\}$ imply $V=\ker T\oplus{\rm Im}\,T$? Let $T: V\rightarrow V$ be a linear operator of the vector space $V$.
We write $V=U\oplus W$, for subspaces $U,W$ of $V$, if $U\cap W=\{0\}$ and $V=U+W$.
If we assume $\dim V<\infty$, then by the rank-nullity theorem, $\ker T\cap {\rm Im}\,T=\{0\}$ ... | Consider the shift operator $s$, defined on $\text{Vect}(e_i, i\in\mathbb{N})$, where $s(e_n)=e_{n+1}$ for $n\in\mathbb{N}$. Note that $\ker(s)=0$ but $s$ is not surjective.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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every trail from start node having the same end node given a directed graph (can have cycles) with:
*
*an arbitrary number of nodes
*an arbitrary number of edges
*that satisfies the condition that there is (at least) one trail (i.e. a walk where no edge is repeated) that visits all nodes.
Would this be a true stat... | Remove the edge from $B$ to $A$ in your second graph. Then from $A$ you have trails
$$A\to B\to C\to B$$
and
$$A\to C\to B\to C\;.$$
I am assuming here that the edge $B\to C$ is considered different from the edge $C\to B$, but if that is not the case, this is still a counterexample.
| {
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Prove that $13\sqrt{2}$ is irrational. I am currently a beginner at proofs and I am having trouble proving this problem...
I know that the square root of $2$ is irrational because the square root of $2$ can be expressed as $\frac{p}{q}$ and once both sides are squared it is true that both $p$ and $q$ are even which is ... | A much less elegant but maybe clearer proof than those involving greatest common denominator (g.c.d.):-
If $\sqrt{2}$ is rational, the we can express it as a ratio of two integers, $A$ and $B$.
So:
$$ \sqrt{2} = \frac {A}{B} $$
where
$$ A, B \in \mathbb{N} $$
$$ \implies A = \sqrt{2}B $$
Since $ A^2 = 2B^2$ we can se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3758045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluate $\lim_{x \to 0} \frac{\sqrt{1 + x\sin x} - \sqrt{\cos x}}{x\tan x}$ What I attempted thus far:
Multiply by conjugate
$$\lim_{x \to 0} \frac{\sqrt{1 + x\sin x} - \sqrt{\cos x}}{x\tan x} \cdot \frac{\sqrt{1 + x\sin x} + \sqrt{\cos x}}{\sqrt{1 + x\sin x} + \sqrt{\cos x}} = \lim_{x \to 0} \frac{1 + x\sin x - \cos ... | We use the elementary limit results $\displaystyle \lim_{x\to 0}\frac{\sin x}{x} =1=\lim_{x\to 0} \frac{\tan x}{x}$
Now coming to the main problem we may write it as $$\begin{aligned}\lim_{x\to 0} \frac{\sqrt{1+x^2\left(\frac{\sin x}{x}\right)}-\sqrt{\cos x}}{x^2\left(\frac{\tan x}{x}\right)}&=\lim_{x\to 0} \frac{\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3758133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 0
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Solution verification: sum at least $4N/7$ times odd
Problem: Let $a_{j}$,$b_{j}$,$c_{j}$ be whole numbers for $ 1 \leq j \leq N$. Suppose that for each $j$ at least one of $a_{j}$,$b_{j}$,$c_{j}$ is odd. Show that there are whole numbers $r$,$s$ and $t$ such that the sum $$r\cdot a_{j} + s\cdot b_{j} + t\cdot c_{j}$$... | Let $$u_1 = (0,0,1)$$ $$u_2 = (0,1,0)$$ $$u_3 = (1,0,0)$$ $$u_4 = (0,1,1)$$ $$u_5 = (1,0,1)$$ $$u_6 = (1,1,0)$$ $$u_7 = (1,1,1)$$
and let for each $i\in \{1,2,...,n\}$ define $v_i = (a_i,b_i,c_i)$. ''Connect'' $u_j$ with $v_i$ iff $u_j\cdot v_i\equiv _2 1$ and we count the number of all connections.
It is easy to see t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What's the gradient of a vector field? Imagine I have the following function
$$ \vec{f}(\vec{x}) = x \vec{x}, x = | \vec{x} |, \vec{x} \in R^3 $$
That is, the function is essentially a quadratic function, but contains a vector direction as well. Intuitively from single variable calculus I would expect the gradient $ \n... | Recall the formula for gradient of scalar times vector:
$$\nabla(a\vec{v}) = \vec{v}\otimes\nabla a + a\nabla\vec{v}.$$
In our case we have $f(\vec{x})= |\vec{x}|\vec{x}$ so
$$\nabla{|\vec{x}|} = \nabla\sqrt{x_1^2+x_2^2+x_3^2}= \frac{\vec{x}}{|\vec{x}|}, \quad \nabla \vec{x} = \nabla(x_1,x_2,x_3) = I$$
where $I$ is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3758352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Can we say that $\text {tr}\ (A) = 0\ $? Let $A$ be an $n \times n$ real matrix with $A^3 + A = 0.$ Can we say that $\text {tr}\ (A) = 0\ $?
I think it's true but can't prove it. Any help will be highly appreciated.
Thanks in advance.
| The most straightforward way to approach the proof is to use the minimal polynomial of $A$. If $K$ is the minimal polynomial of $A$, then any other polynomial $Q$ with $Q(A)=0$ is a multiple of $K$. Hence, the eigenvalues of $A$ are either $0$, $i$ or $-i$. Moreover, as $A$ is a real matrix, the sum of its eigenvalues ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Isometry on inner product space
$V$ is an inner product vector space. If a transformation $T\colon V\to V$ satisfies $\langle T(x), T(y)\rangle = \langle x, y\rangle$ for every vector $x, y \in V$, prove or disprove that $T$ is linear.
Seems true, but can't prove it. Tried plugging $x+y$ into $x,y$ and got
$\langle T... | I found answer.
The point is using that $\langle x,x\rangle=0$ implies $x=0$.
Consider $||T(u+v)-T(u)-T(v)||^2$ then the condition directly gives $||T(u+v)-T(u)-T(v)||^2=||u+v-u-v||^2=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to evaluate $\int \frac{dx}{\sin(\ln(x))}$? I am wondering how to evaluate the indefinite integral
$$\int \frac{dx}{\sin(\ln(x))} \quad (1)$$
Attempt 1
I tried using Weierstrass substitution.
The Weierstrass substitution, (named after K.Weierstrass (1815)), is a substitution used in order to convert trigonometric... | Based on the hypergeometric answers of J.G. and Simply Beautiful Art.
Taking another branch of the solution of the hypergeometric differential equation from those answers, we can get solutions like this:
$$
f(x) = \mathrm{Re}\left[ {\frac { \left( 1+i \right) {x}^{1+i}}{{x}^{2\,i}-1}
\;{\mbox{$_2$F$_1$}\left(1,1;\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3758742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 5
} |
What is the expression for the centroid of an arbitrary parameterized space curve? Let $\gamma:t\in[a,b]\rightarrow (x(t),y(t),z(t))\in \mathbb{R}^3$ be a parametrized curve
I am looking for the expression of the centroid of the curve $\gamma$ in a good reference. (I didn't find a good one.)
| You can use the standard definition of center-of-mass:
$$
r={\int_a^b\gamma(t)\,|\dot\gamma(t)|\,dt\over
\int_a^b|\dot\gamma(t)|\,dt},
$$
where: $\dot\gamma(t)=(\dot x(t), \dot y(t), \dot z(t))$ and $|\dot\gamma(t)|=\sqrt{\dot x^2(t)+\dot y^2(t)+\dot z^2(t)}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the limit of $\frac{(2n)!}{n!}$ So I've tried to use Stirling's approximation and got that $\lim\frac{2n!}{n!}=\frac{1}{1}=1$
Any thoughts?
| $\frac{2n!}{n!} = \frac{2n(2n-1)(2n-2)!}{n(n-1)!} \ge 2.\frac{2(n-1)!}{(n-1)!}$
So, $\frac{2n!}{n!} \ge 2^n$
$\lim\limits_{n \to +\infty} \frac{2n!}{n!} \ge \lim\limits_{n \to +\infty} 2^n = + \infty$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3758931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the absolute extrema of $F(x) = 2x + 5\cos(x)$
Find the absolute extrema of $F(x) = 2x + 5\cos(x)$ on the interval $[0,2\pi]$ using the extreme value theorem.
Answer should be 2 ordered pairs.
I got $\arcsin(2/5)$ for the first value of $x$, but can’t figure out the second.
Thanks in advance.
| If $f(x)=2x+5\cos x , x \in [0,2\pi], f'(x)=2-5 \sin x, f''(x)=-5 \cos x$
$$f'(x)=0 \implies \sin x=\frac{2}{5} \implies x_1= \sin^{-1} (2/5), x_2=\pi-\sin^{-1} (2/5)$$
$$f''(x_1)<0, f''(x_2)>0 \implies f_{max}=f(\sin^{-1}(2/5))=2\sin^{-1}(2/5)+\sqrt{21},$$ $$ f_{min}
=f(x_2)=2[\pi-\sin^{-1}(2/5)]-\sqrt{21}$$
$$f(0)=5,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3759057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Central limit theorem for multi-dimensional martingale difference I know that there is a CLT for $\mathbb R$-valued martingale difference process that goes roughly as follows:
Let $X$ be an $\mathbb F$-martingale difference process, i.e. $\mathbf E [X_t \mid \mathcal F_{t-1}]=0$, and suppose $X$ satisfies some kind of... | Use the Cramer-Wold device. That is, if $t^{\top}S_n\xrightarrow{d}t^{\top}S$ for all $t\in \mathbb{R}^d$, then $S_n\xrightarrow{d}S$, where $S_n:=\sum_{i=1}^n X_i/\sigma_n$ and $\{\sigma_n\}$ is a normalizing sequence. Note that in your case $\{t^{\top} X_n\}$ is a martingale difference sequence w.r.t. $\{\mathcal{F}_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3759152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Functional equation for $\eta(s)$ following Riemann's $2^{nd}$ method. Being
\begin{equation*}
\eta(s)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots
\end{equation*}
and following Riemann's second method, (Edwards p.15), to obtain the functional equat... | Ignoring technicalities of convergence, in Riemann's second proof, you start with the Poisson summation formula $\sum_{n\in\mathbb Z} f(n / x) = x \sum_{n\in\mathbb Z} \hat f (n x)$, take the Mellin transform of both sides, and use the self-dual function $f(x)=e^{-x^2}$.
To get the alternating sum you want, you could e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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small distances between powers of irrationals The value of $$\inf \left\{ |\pi^m-e^n|: m,n\in\mathbb{N} \right\}$$ is a known unsolved problem. But transcendental numbers are known to cause problems of this sort.
Is the value of $$\inf \left\{ |\sqrt{2}^m-\sqrt{3}^n|: m,n\in\mathbb{N} \right\}$$ known? Or at least is ... | Note you have
$$d = \left|\sqrt{2}^m-\sqrt{3}^n\right| = \frac{\left|2^m - 3^n\right|}{\sqrt{2}^m + \sqrt{3}^n} \tag{1}\label{eq1A}$$
As stated near the bottom of Differences Between Powers,
Indeed, Tijdeman proved that there exists a number $c \ge 1$ such that
$$\left|2^m - 3^n\right| \ge \frac{2^m}{m^c}$$
Also, a c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3759471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Help with a differential equation system Given $x' = -x$ and $y' = -4x^3+y$, we want to linearize and show phase portrait at origin.
So I make system $\vec{Y}' = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\vec{Y}$ by just scrapping the $-4x^3$ term. But now we have repeated $0$ eigenvalue, so I try to find an eigenve... | The assertion that the eigenvalues of the matrix $\vec Y'$ are both zero is erroneous. However, we have:
The eigenvectors of the matrix
$\vec Y' = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \tag 1$
are $(1, 0)^T$, with eigenvalue $-1$, and $(0, 1)$, with eigenvalue $1$, as is easily checked, e.g.
$\begin{pmatrix} -... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Linear Algebra books that also covers multilinear algebra I want self-studying linear algebra but I also want to know what tensors are. I don't see any books that cover both linear and multilinear algebra (tensors are part of multilinear algebra right?). So if there are any books that do would be great. I don't mind th... | One book you could use would be Lectures in Geometry, Semester 2: Linear Algebra and Differential Geometry, by Postnikov. It's the second volume in his six-part series Lectures in Geometry.
This might not be a good book to learn linear algebra from by itself, as there are no exercises and some important topics are omit... | {
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"timestamp": "2023-03-29T00:00:00",
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How to prove that if $x>0$ and $y>0$, then $\sqrt{x}+\sqrt{y}>\sqrt{x+y}$, using the relation of arithmetic and geometric means?
How to prove that if $x>0$ and $y>0$, then $$\sqrt{x}+\sqrt{y}>\sqrt{x+y}\,,$$ using the relation of arithmetic and geometric means?
I started by showing that if $x>0$ and $y>0$, based on t... | Is this correct?
Since $x+y≥2\sqrt{xy}$, by AM-GM relationship,
$x+y+2\sqrt{xy}≥x+y$
$(\sqrt{x}+\sqrt{y})^2≥x+y$
Thus, $\sqrt{x}+\sqrt{y}>\sqrt{x+y}$. QED.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3759816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Taking derivative with function of multiple variables? Suppose we are studying the function
$$
f(x,y) = xy + ax^2 + bx^2y^2,
$$
We want to find the maximum $x$ satisfying the equation
$$
f(x,y) = c,
$$
where $a, b, c$ are constants. Somebody suggested to make use of the following auxiliary function
$$
g(x,y) = xy,
$... | I don't want to go into details of different cases, so I will assume that $x$ and $a,b,c$ are all positive, while $y$ is negative.
The canonical procedure would be to solve $f(x,y)=c$ with respect to $x$, to obtain $x=g(y)$, then solve $g'(y)=0$ to obtain $y_0$ (suppose it is unique), finally the solution would be $x_0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to show that $\sum_{n=1}^{N} \cos(2n-1)x = \frac {\sin(2Nx)}{2\sin(x)} $ I am studying Fourier analysis and have been given the following question:
Show that $$\sum_{n=1}^{N} \cos(2n-1)x = \frac {\sin(2Nx)}{2\sin(x)} $$
I used the formula for a finite geometric sum and Euler's formula to get to the following:
$\sum... | Since
$2\sin(a)\cos(b)
= \sin(a+b)+\sin(a-b)
$,
$2\sin(x)\cos((2n-1)x)
= \sin(2nx)+\sin(-(2n-2)x)
= \sin(2nx)-\sin((2n-2)x)
$,
so you get a telescoping sum.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3760074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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Let $ABC$ be a triangle and $M$ be the midpoint of $BC$. Squares $ABQP$ and $ACYX$ are erected. Show that $PX = 2AM$.
$\textbf{Question:}$ Let $ABC$ be a triangle and $M$ be the midpoint of $BC$. Squares $ABQP$ and $ACYX$ are erected. Show that $PX = 2AM$.
I could solve this problem using computational techniques but... | To rotate a vector around some point in the plane by $\alpha$ it's the same to rotate this vector by $\alpha$ around the tail of the vector.
Let $R^{\alpha}(\vec{a})$ be a rotation of $\vec{a}$ by $\alpha$.
Thus, $$R^{90^{\circ}}(\vec{AD})=R^{90^{\circ}}\left(\frac{1}{2}\left(\vec{AB}+\vec{AC}\right)\right)=\frac{1}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3760238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $A$ is a simple finite dimensional $\mathbb{C}$-algebra then $A\cong M_n(\mathbb{C})$ I'm trying to prove the problem 17 of chapter 13 of the book Algebra: a Graduate Course (by Martin Isaacs), which is:
Let $A$ be a simple finite dimensional $\mathbb{C}$-algebra. Show that $A\cong M_n(\mathbb{C})$ for
some integer... | The field $\mathbb{C}$ is not important. The statement is true for any algebraically closed field $F$.
Take a simple right $A$-module $V$. This is a finite dimensional vector space over $F$, because it is a quotient of $A$ modulo some maximal right ideal. Therefore its endomorphism ring is a finite dimensional division... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Let $f$ be an entire function s.t. $F(z) = \lim_{n\to\infty} f^{(n)}(z)$ exists for all $z$ with local uniform convergence. What can we say about $F$? I have stumbled into this problem, without a given answer.
Let $f$ be an entire function such that $F(z) = \lim\limits_{n\to\infty} f^{(n)}(z)$ exists $\forall z \in \ma... | As already worked out in the comments, $F(z) = \lim_{n\to\infty} f^{(n)}(z)$ (locally uniformly) implies that
$$
F'(z) = \lim_{n\to\infty} f^{(n+1)}(z) = F(z)
$$
so that $F(z) = Ce^z$ for some constant $C \in \Bbb C$.
Then $g(z) = f(z) - Ce^z$ satisfies
$$
\lim_{n\to\infty} g^{(n)}(z) = \lim_{n\to\infty} f^{(n)}(z) -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3760485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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How prove that the elementary operations don't change the rank of a matrix One considers certain operations, called elementary row operations, that are applied to a matrix $A$ to obtain a new matrix $B$ of the same size.
These are the following:
*
*exchange rows $i_1$ and $i_2$ of $A$ (where $i_1\neq i_2$);
*replace... | This is a super important linear algebra theorem. The basic idea of the proof is that each of these operations is equivalent to right-multiplication by a matrix of full rank. I'll give an example of each operation in the 2 by 2 case:
*
*Swap the rows by multiplying on the right by
\begin{pmatrix}
0 & 1 \\
1 & 0
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3760618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The integral of a nonnegative measurable function definition: what does $f_n(\omega)\uparrow f(\omega)$ mean? Let $f:\Omega\to\overline{\mathbb{R}}_{+}$ be a nonnegative measurable function on $(\Omega,\mathcal{F},\mu)$. The integral of $f$ with respect to $\mu$ is defined as
\begin{align*}
\int f \, \mathrm{d}\mu = \l... | Yes, it's as you say. This means that the sequence of functions converges "upwards" to $f$, meaning that $f_n\to f$ pointwise and for each $x\in \Omega$, and for each $n$, $f_n(x)\le f_{n+1}(x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3760806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to smooth sine-like data I'm trying to produce a growth graph but I'm getting sine wave artefacts due to the way the data is compared (current 7 days / previous 7 days). I've drawn the red and yellow lines by hand by first connecting the mid point of each sine wave (red), then connecting those mid points (yellow).
... | Probably a logarithmic function ? $log$$y$ $x$ strictly increases for $x, y > 1$ .
Plus, you may even try taking the derivative of the function used and adjust it. This is one of the methods used in machine learning (especially in linear regression) as a learning technique and this is called stochastic gradient descen... | {
"language": "en",
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Alternate way to solve $\lim\limits_{x \to 0} (\sin x) ^x$? My solution:
$$ \lim_{x \to 0} (\sin x)^x = \lim_{x \to 0} e^{(x)(\ln \sin x)} = \exp \left( \lim_{x \to 0} (x) (\ln \sin x) \right)$$
Now we have $\lim_{x \to 0} (x) (\ln\sin x)$.
Now we can say that the limit is $0$ as $\ln$, $\sin x$ decreases more slowly... | Just a small variant on Kavi Rama Murthy's answer, together with a comment on the one-sidedness of the limit:
Note,
$$(\sin x)^x=\left(\sin x\over x\right)^xx^x$$
We have
$$\lim_{x\to0}\left(\sin x\over x\right)^x=1^0=1$$
and
$$\lim_{x\to0^+}x^x=1$$
(from the easy L'Hopital for $x\ln x={\ln x\over1/x}$). Therefore
$$\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3761182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fix point of function over disk Prove that there is a fix point of function $f:B(0,1) \to R^2$, where $B(0,1)$ is circle of radius 1, and $f(x,y)=\frac {1}{4}(ye^{x}-y,cosy)$. I tried to prove that f is contractive mapping (there is $0 < q < 1$ such that $d(f(a),f(b)) \leq qd(a,b)$), and than use Banach theorem.
| Using Brouwer theorem :
You know that $f$ is continous and $f(B(0,1))\subseteq B'\triangleq\overline{B} (0,1)$ where $\overline{B} (0,1)$ denotes the closed disk.
So $f: B' \to B'$ is continous, from a closed disk to itself.
By Brouwer theorem it has a fixed point.
Exercise to end the proof : verify it isn't on the edg... | {
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} |
$G= \langle a, b : a^{7} = b^{3} = 1,\ b^{-1}ab = a^{2} \rangle$ and commutator group Consider the $a$ and $b$ the following permutation in $S_{7}$:
$$a = (1\ 2\ 3\ 4\ 5\ 6\ 7 ),\ b = (2\ 3\ 5)(4\ 7\ 6)$$
Consider the group $G = \langle a, b \rangle$. I know that $a^{7} = b^{3} = 1$ and $b^{-1}ab = a^{2}$. Moreover, wi... | To see the other inclusion, consider the map $G\rightarrow\mathbb{Z}_3$ sending $a$ to $0$ and sending $b$ to $1$. This application preserves relations, and so it induces an homomorphism from $G$ to $\mathbb{Z}_3$. Its kernel coincides with $\langle a\rangle$. Therefore $G/\langle a \rangle\cong\mathbb{Z}_3.$ By defini... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Spectral norm, eigenvalues range I stumbled upon a property in solutions of some exercises which stated that if a hessian of a possibly non-convex function f(x) is bounded in spectral norm then its eigenvalues lie in the interval.
$$ ||\nabla^2f(x)||_2 \leq L $$
$$ eigenvalues \in [-L, L]$$
I fail to understand or more... | The spectral norm of a matrix is, by definition, the largest absolute value of its eigenvalues. If the largest absolute value of the eigenvalues is less than or equal to $L$, then all the eigenvalues have absolute value less than or equal to $L$, so if they are real, they all lie in the interval $[-L,L]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3761544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find the other 2 vectors in a triangle given all the magnitudes and one of the sides. I'm trying to run a simulation in Python of a linkage system. The problem I'm working with returns results that don't make sense (like one of the linkages changes size). I've reduced the issue I'm confused about to the following.
Supp... | SSS determines the shape of the triangle, but the direction can vary.
You may think of $\vec{c}$ as an unoriented segment $AB$ of length $||c||$. Given $||a||$ and $||b||$, the possible locus of $a$ and $b$ are two circles centered at $A$ and $B$ respectively, with radii of $||a||$ and $||b||$. Geometrically, two circl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3761652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Through two given points on a circle, construct two parallel chords with a given sum. The problem is from Kiselev's Geometry exercise 317.
Through two given points on a circle, construct two parallel chords with a given sum.
Here is what I have tried so far:
Mark the two points by $A$ and $C$ respectively. If we have... |
Let Q, R be the given points. QRBA the given circle. Partition the line segment of summed length QP at A.
Draw a parallel through B and parallelly transfer AP to BR. The point R must lie on the circle because $\alpha,\beta$ are opposite supplementary angles in a cyclic quadrilateral.
Likewise transfer AQ to BS. Draw ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3761747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Integral: $\int \dfrac{dx}{(x^2-4x+13)^2}$?
How can I integrate $$\int \dfrac{dx}{(x^2-4x+13)^2}?$$
Here is my attempt:
$$\int \dfrac{dx}{(x^2-4x+13)^2}=\int \dfrac{dx}{((x-2)^2+9)^2}$$
Substitute $x-2=3\tan\theta$, $\ dx=3\sec^2\theta d\theta$
\begin{align*}
&=\int \dfrac{3\sec^2\theta d\theta}{(9\tan^2\theta+9)^2}\... | After square completion and substituting $u=\frac{x-2}{3}$, there is a simple standard trick to evaluate the integral without trigonometric substitutions:
$$\int \dfrac{dx}{(x^2-4x+13)^2} \stackrel{u=\frac{x-2}{3}}{=}\frac 1{27} \underbrace{\int \frac{1}{(u^2+1)^2}du}_{I(u)}$$
Just rewrite the numerator
$$I(u) = \int\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3761986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 9,
"answer_id": 2
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Jordan normal form of powers of Jordan normal form Previous related question: Jordan normal form powers
Let $A$ be a $n\times n$ Matrix such that $A=PBP^{-1}$ where $B$ is in Jordan normal form with $\lambda_i(k)_j$ Where $i$ is the size, $k$ is the eigenvalue and $j$ the order.
From the previous question I know that... | So you want to know the Jordan canonical form of the $i \times i$ matrix
$$ A = \sum_{r=0}^{i-1} \left( n \atop r \right) k^{n-r} t^r .$$
Since $A$ has $k^n$ as an $i$-fold repeated eigenvalue, it is sufficient to find the Jordan form for
$$ A - k^n I = \sum_{r=1}^{i-1} \left( n \atop r \right) k^{n-r} t^r .$$
First co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3762073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\bigcap\mathcal H\subseteq(\bigcap\mathcal F)\cup(\bigcap\mathcal G)$. Not a duplicate of
Prove that $∩\mathcal H ⊆ (∩\mathcal F) ∪ (∩\mathcal G)$.
This is exercise $3.5.17$ from the book How to Prove it by Velleman $($$2^{nd}$ edition$)$:
Suppose $\mathcal F$, $\mathcal G$, and $\mathcal H$ are nonempty fa... | Your proof is okay.
It is more handsome though to prove the contrapositive statement:$$x\notin\left(\bigcap\mathcal{F}\right)\cup\left(\bigcap\mathcal{G}\right)\implies x\notin\bigcap\mathcal{H}$$
Proof:
If $x\notin\left(\bigcap\mathcal{F}\right)\cup\left(\bigcap\mathcal{G}\right)$
then some $A\in\mathcal{F}$ exists $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3762217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I find all functions $F$ with $F(x_1) − F(x_2) \le (x_1 − x_2)^2$ for all $x_1, x_2$? In calculus class we were given this so-called "coffin problem" originally from Moscow State University.
Find all real functions $F(x)$, having the property that for any $x_1$ and $x_2$ the
following inequality holds:
$$F(x_1) ... | Exchanging $x_1$ with $x_2$ in the original inequality shows $F(x_1)-F(x_2)$ is bound by $\pm(x_1-x_2)^2$, i.e. $\left|\frac{F(x_1)-F(x_2)}{x_1-x_2}\right|\le|x_1-x_2|$. This proves the two-sided derivative is $0$.
But you actually don't need derivatives to solve the problem. Since $|F(x)-F(0)|\le x^2$ for all $x$, $|F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3762311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Lagrangian fiber bundles and Poisson commutative subalgebras Let $M$ be a symplectic manifold and $\pi : M \to B$ a fiber bundle (fibers are manifolds). Is it true that $\pi$ is an isotropic fiber bundle i.e. all fibers are isotropic submanifolds if and only if $\pi^*(C^\infty(B))$ is Poisson commutative?
More precisel... | A counterexample is the fiber bundle $\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R}^{3}$ with projection $\pi:(x_1,x_2,x_3,z)\mapsto(x_1,x_2,x_3)$ and symplectic form $\omega\in\Omega^{2}(\mathbb{R}^{3}\times\mathbb{R})$ given by
$$
\omega=dx_1\wedge dx_2 + dx_3\wedge dz.
$$
The fibers are one-dimensional, hence ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3762455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of ternary strings of length n such that number of 0s is greater than or equal to number of occurrences of any other digit I understand how to count this for a binary string of a fixed length using combinations, so I think the way to go with this problem is to use an exponential generating function for each of t... | I don't think generating functions give the best approach. You should exploit the symmetry of the situation instead. The are $3^n$ ternary strings of length $n$. In how many of them is $0$ a winner, that is, in how many of them are there at least as many $0$'s as $1$'s or $2$'s? If we count all the winners, $\frac1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3762543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Optimal Betting Strategy question I am preparing for an exam in probability theory and I bumped against a question I can't solve.
Given are an integer starting capital $k$, an end goal capital $m$ and a period of $n$ days. Each day I can bet some integer amount $X$ of my choosing $(X \leq k)$ on an unfair coin landing ... | It seems to me that this question is much easier than people are making it out to be, although I might be wrong here.
First: The optimal strategy will at each betting day be the one that increases the expected gains the most. Therefore you should bet all your capital if $p>0.5$, since the expected return is the largest... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3762663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 4
} |
Compute the value of $\lim \int_0^1 f_n(x) \sin (nx)dx.$
Let $\{f_n(x)\}$ be a sequence in $L^2[0,1]$ and $\lim f_n=f$ almost everywhere, where $f(x)\in L^2[0,1]$. Then find
$$\lim_{n\to \infty} \int_0^1 f_n(x)\sin (nx)dx.$$
By Riemann-Lebesgue lemma, we can conclude that $$\lim \int_0^1 f(x)\sin(nx)dx=0.$$
But how t... | The limit need not exist. For example if $f_n(x)=n^{2} \chi_{(0,\frac 1n)}$ and $f=0$ then $f_n \to f$ at every point but the given integral tends to $\infty$.
[$\int_0^{1}f_n(x) \sin (nx)dx=n\int_0^{1} \sin y dy=n(1-\cos 1)$ by the substitution $y=nx$].
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3763035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there an ordered field with distinct subfields isomorphic to the reals? Is there an ordered field with distinct subfields isomorphic to the field $\mathbb R$ of real numbers?
| Yes. Let $K$ be any real-closed field that contains $\mathbb{R}$ as a proper subfield. In particular, then, $K$ has a nonzero infinitesimal element $\epsilon$. Let $B$ be a transcendence basis for $\mathbb{R}$ over $\mathbb{Q}$ and let $B'=\{b+\epsilon:b\in B\}$. Then $B'$ is still algebraically independent (a poly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3763151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Determining whether $\iint_{|x|+|y| \leq 1} \ln(x^{2}+y^{2}) \,dx\,dy$ is positive or negative I have an integral:
$$\iint_{|x|+|y| \leq 1} \ln(x^{2}+y^{2}) \,dx\,dy$$
So basically it's:
$$\int_{-1}^{0}\,dx \int_{-x-1}^{x+1} \ln(x^{2}+y^{2})\,dy + \int_{0}^{1}\,dx \int_{x-1}^{-x+1} \ln(x^{2}+y^{2})\,dy$$
But it's two h... | We can compute the value of this integral exactly. By rotational symmetry we have that the integral is equivalent to the integral on the square (and subsequent triangle):
$$I = \iint_{\left[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right]^2} \log\left(x^2+y^2\right)\:dA = 8\int_0^{\frac{1}{\sqrt{2}}} \int_0^x \log\left(x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3763265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Probability selecting three cards out of a deck In Introduction to Probability by Blitzstein & Hwang, Chapter 2 Problem 5:
Three cards are dealt from a standard, well-shuffled deck. The first two cards are flipped
over, revealing the Ace of Spades as the first card and the 8 of Clubs as the second card.
Given this info... | Symmetry: there are 50 cards left. Each has the same probability so the probability to get an ace is $\tfrac{3}{50}$ as there are 3 aces left.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3763502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Convexifying Optimization Problem Let $\mathbf{V} \in \mathbb{R}_{+}^{n \times m}$ and $\mathbf{E} \in \mathbb{R}_{+}^{n \times m}$.
I am trying to convexify the following program which solves for $\mathbf{X} \in \mathbb{R}^{n \times m}$:
\begin{align}
&\max &\sum_{i = 1}^n \log \left(\sum_{j = 1}^n V_{ij}\left( X_{ij... | This can be convexified by Difference of Convex (DC) Programming.
See DC Programming: The Optimization Method You Never Knew You Had To Know.
Variations and extension of the convex–concave procedure, Thomas Lipp1 and Stephen Boyd
A modeling system such as CVXPY which supports Disciplined Convex-Concave Programming (DCC... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3763807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Which irrationals become rational for some positive integer power? Related to Irrationals becoming rationals after being raised to some power. Let $r \in \mathbb{R} \setminus \mathbb{Q}$. True or false: there exists an $n \in \mathbb{N}$ (positive integers) such that $r^n = r \cdot \dots \cdot r \in \mathbb{Q}$. This i... | You have to find the transcendental numbers like exponential number, π, sin(a) , sin h(a) etc. which make your statement wrong! One thing that you have to remind, every transcendental numbers are irrational numbers, but converse is not always true i.e, there can be irrationals which are not transcendental numbers, like... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3763999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to solve polynomial rational relations for $y$ (e.g $\sqrt{4-3y-y^2} = x(y+4)$)? From time to time, I struggle to solve polynomial relations for $y$.
A trivial example is :
$$ \frac{y}{x} = x \iff y = x^2$$
Easy.
But consider this relation:
$$ \sqrt{4-3y-y^2} = x(y+4)$$
No matter how much I mess around it, seems i... | Hint:
$$4-3y-y^2=\dfrac{25-(2y+3)^2}4$$
needs to be perfect square of a rational number
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3764098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Are open sets locally simplicial? So I read parts of Rockafellar's "Convex Analysis".
When introducing "locally simplicial" sets, all the examples stated are convex sets, yet he mentions that they do not need to be convex.
I wonder whether all open sets in $\mathbb{R}^n$ are locally simplicial? I think such a fact woul... | This community wiki solution is intended to clear the question from the unanswered queue.
Yes, your proof is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3764245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find volume of the solid generated by revolving Find volume of the solid generated by revolving the region bounded by the parabola
$ =^2+1 ,=0 $ and the line =3 about the line =3
Using Disk method,I found the answer to be 9.4
| $$V=\pi \int_{y_0}^{y_1} (R(y))^2dy$$ $$=\pi \int_{0}^{\sqrt{2}} (3-(y^2 +1))^2dy$$ $$=\frac{32\sqrt{2}\pi}{15}\approx 9.478$$
You're correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3764404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How does one show that the residue of the first derivative of a holomorphic function is zero in its singularity? I've got a holomorphic function $f:\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C}$ and I want to show that $$res_{0}f'=0$$
| Applying the definition of Residue as the -1'th Laurent series coefficient: $$\operatorname{Res}(f';0)=a_{-1}=\frac{1}{2\pi i} \oint_{\gamma}\,\frac{f'(z)}{(z-c)^{-1+1}}\,dz = \frac{1}{2\pi i} \oint_{\gamma}\,f'(z)\,dz .$$
And $f'$ has a primitive along all of $\gamma$ so this integral is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3764514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to differentiate $g(X)=\operatorname{tr}\left(X^{-1}\right)$? Let $X$ be a square invertible $n \times n$ matrix. Calculate the derivative of the following function with respect to X.
$$
g(X)=\operatorname{tr}\left(X^{-1}\right)
$$
I'm stumped with this. As when I work through it I use these two identities.
*
*$... | The problem is with this equation
$$\frac{\partial}{\partial \boldsymbol{X}} \operatorname{tr}(\boldsymbol{f}(\boldsymbol{X}))=\operatorname{tr}\left(\frac{\partial \boldsymbol{f}(\boldsymbol{X})}{\partial \boldsymbol{X}}\right)$$
Note that on the LHS you are taking the derivative of a function $\mathbb R^{n\times n} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3764596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
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