Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Maclaurin expansion problem I am having problem with a question regarding Maclaurin expansion.
a) Find the term up to $x^4$ in the Maclaurin expansion of $f(x)=\ln(\cos(x))$.
This part I was able to anwser part a. The Maclaurin expansion until $x^4$ is: $f(x)=-x^2/2-x^4/12$
b) Use this series to find an approximation ... | Hint: Use the fact that $\ln(a^b)=b\cdot\ln(a)$. Therefore, $\ln(2)=-\ln\bigl(\frac{1}{2}\bigr)$. Then, when does $\cos(x)=\frac{1}{2}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2540444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Expected number of draws to draw 3 of the same balls out of an urn with replacement
An urn contains twelve balls numbered 1 to 12. We draw a ball, record its
number, and replace it in the urn. We repeat this until we draw any
number three times. What is the expected number of draws?
First post here. Anyhow, I can... | Let's say the first time we have drawn the same number three times is on draw number $X$, so we want to find $E(X)$. Our approach is to find $P(X>n)$ for $n=0,1,2, \dots ,24$, and then apply the theorem $E(X) = \sum_{n>0} P(X>n)$.
$X>n$ if no number has been drawn more then two times by the $n$th draw. There are $12^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2540530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Circular permutations problem, need help
How many ways can $5$ people sit on a circular table of $8$ places?
I know that if both people and places number =$n$ , then the answer $= (n-1)!$
But the situation here is different, since there are more places than people, what should I do?
Thanks for help
| Let $X$ be the set of all possible arrangements of $5$ people in a straight line of $8$ chairs. Now, define a relation on $X$ : two arrangements are similar, if they are related by a cyclic permutation: if there exists a number $0 \leq r\leq 7 $, such that each person in the first arrangement, when shifted by $r$ place... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2540644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
A finite group is nilpotent iff two elements with relatively prime order commute (Question 9 in chapter 6.1 of Dummit and Foote). Prove that a finite group G is nilpotent if and only if whenever $a, b \in G$ with $(|a|, |b|) = 1$ then $ab = ba$. It says to use the following theorem:
Let G be a finite group, $p_1$, $p_2... | Here is the my proof:
I will use theorem which says: If G is finite nilpotent and $P_i$'s are Sylow-$p_i$-subgroups of G then $G= \displaystyle\prod_{p_i} P_i$ (Also this prodocut is direct but we dont need uniqueness).
Let $G$ be a nilpotent group and let $P_1$,$P_2$,..,$P_n$ be Sylow subgroups of G for every prime $p... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Can this definite integral involving series be solved without a calculator? I got this question today but I can't see if there is any good way to solve it by hand.
Evaluate the definite integral
$$\int_2^{12}\frac{\sqrt{x+\sqrt{x+\sqrt{x+...}}}}{\sqrt{x\sqrt{x\sqrt{x}...}}}\,\mathrm{d}x$$
where the series in the numer... | Hint:
1.
For real $y,$ $\sqrt{y^2}=|y|\ge0$
and $\sqrt{1+4x}\ge3$ for $2\le x\le12\implies1-\sqrt{1+4x}<0$
2.
Set $\sqrt{1+4x}=u\implies4x=u^2-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2540947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 0
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Direct Sum Isomorphism: $\mathbf{Z}_4 \oplus \mathbf{Z}_3 = \mathbf{Z}_{12}$ A question I'm working on asks:
If $f$ is an isomorphism from $\mathbf{Z}_4 \oplus \mathbf{Z}_3 = \mathbf{Z}_{12}$, what is $\phi(2,0)$? What are the possibilities for $\phi(1, 0)$? Give reasons for your answer.
I know that isomorphisms pres... | For $Z_n$, any number $m$ where $gcd(m,n)=1$ will generate the entire group. As you noted, $(1,1)$ is a generator of $Z_3\oplus Z_4$. Since both groups are cyclic and of the same order, any map defined by sending a generator to a generator will be an isomorphism.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2541085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Bounded variation of $\frac1f$ when $\inf(|f|)>0$ & $f$ bounded variation I want to show if $\frac{1}{f}\in BV[a,b]$
when $\inf(|f|)>0 \land f\in BV[a,b]$.
I tried to find a partition that $V(\frac{1}{f},P)$ is upper-bounded
using the partition that makes $V(f,P)$ upper-bounded in which I failed.
(in case $f$ is not ... | Hint: Say $f(x)\ge c>0$ for all $x$. Then $$\left|\frac1{f(x)}-\frac1{f(y)}\right|=\left|\frac{f(y)-f(x)}{f(x)f(y)}\right|\le\frac{|f(x)-f(y)|}{c^2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2541155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Diophantine polynomial equations If K is a positive integer, what is the smallest k for which $x^2 + kx = 4y^2 -4y +1$ has a solution (x,y) where x and y are integers
= $(2y-1)^2 = x^2 +kx$
=$(2y-1+x)(2y-1-x) = kx $
I tried equating them to each other
$(2y-1+x) = k; (2y-1-x) = x$
$x= k-2y+1; k= (6y-3)/2$
but then k do... | We need $$k^2+(4y-2)^2$$ to be perfect square
WLOG $k=a(p^2-q^2),2(2y-1)=a(2pq)$
$\implies apq$ is odd
The minimum positive values of $a,p^2,q^2$ will be $1,3^2,1^2$ respectively
$$x=\dfrac{a(q^2-p^2)\pm a(p^2+q^2)}2=?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2541263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Convolution of the PDFs of $2$ Independent Random Variables I'm having trouble getting the correct pdf for $Z$ in the problem $Z = X + Y$ where the pdf of $X$ and $Y$ are
$$
f_x(x) = f_y(x) = \left\{\begin{aligned}
&x/2 &&: 0 < x < 2\\
&0 &&: \text{otherwise}
\end{aligned}
\right.$$
I am solving this using the convolut... | You computed the limits of integration correctly. But you substituted wrong expressions for $f_y(w-x)$ in both integrals. In particular,
$$f_y(w-x) = (w-x)/2$$
when $0<(w-x)<2$. Therefore, your integrals would be
$$f_z(w) = \frac{1}{4}\int_{0}^{w}x(w-x)\,dx$$
for $0 \le w \le 2$ and
$$f_z(w) = \frac{1}{4}\int_{w-2}^{2}... | {
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"timestamp": "2023-03-29T00:00:00",
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Calculation of the volume and surface of a cuboïd knowing the coordinates of two opposite vertices I hope someone will help me with this problem i'm facing right now. So our C language teacher gives us an exercice to solve. The exercice itself looks easy and the programmation part is not the problem. So he did ask us t... | The edges of cuboid are parallel to the coordinate axes.
$$ x_2-x_1= L;\quad y_2-y_1= B; \quad z_2-z_1 = H; $$
$$ V= LBH; \quad A = 2 (LB+BH +HL). $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2541655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Linearize the equation $f^2(x)+1=2f(x+1)$ How can I linearize for the continuous function $f$ the functional equation $f^2(x)+1=2f(x+1)$
I' ve been studying recently the book for functional equations by Christopher G. Small and I' ve encountered the term "Linearization" of functional equations. I cannot figure out how ... | I am not sure that linearization is applicable to this functional equation, but I will depict another way to solve this. First, lets consider trivial solutions.
$$f^2(x)+1=2f(x+1)$$
$$f(x)=c$$
$$c^2+1=2c$$
$$c^2-2c+1=0$$
$$(c-1)^2=0$$
$$c=1$$
$$f(x)=1$$
I assume that the function is meant to be continuous and defined e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2541825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Getting x in terms of y I have this equation:
$$\dfrac{x}{y} = \dfrac{y-x}{x}$$
How would I separate $x$ and $y$ in $x^2+xy-y^2=0$ ?
| Consider your equation, as a quadratic equation with respect to the variable $y$ :
$$-y^2 + xy + x^2$$
handling the variable $x$ as a parameter.
Then, it would be :
$$D=b^2-4ac=x^2+4x^2=5x^2$$
It's easy to check that $\forall x\in \mathbb R, D\geq 0$, so you can safely express $y$ with respect to $x$ via the solution... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2541921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Inversion of Gamma and Zeta Functions through Fourier Transform I have been messing around with the Fourier Transform and wanted to see if I could manipulate this equation: $$\Gamma(s)\zeta(s)=\int_{0}^\infty\frac{x^{s-1}}{e^x-1}\rm dx$$
into an integral over the Gamma and Zeta Functions.
My Work: $$\Gamma(s)\zeta(s)=\... | For $\sigma > 1$, with $x = e^{-u}$
$$\Gamma(\sigma+2i\pi\xi)\zeta(\sigma+2i\pi\xi) = \int_0^\infty \frac{x^{(\sigma+2i\pi\xi)-1}}{e^x-1}dx = \int_{-\infty}^\infty \frac{e^{-(\sigma+2i\pi\xi) u}}{e^{e^{-u}}-1} du= \mathcal{F}[\frac{e^{-\sigma u}}{e^{e^{-u}}-1}](\xi)$$
$$\frac{e^{-\sigma u}}{e^{e^{-u}}-1}= \int_{-\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2542007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Finding the eigenvalues of a 4x4 matrix So I have this matrix $A$, so after I did $\det(xI-A)$, I have this
\begin{pmatrix}
x & 0 & 0 & -x \\
0 & x & 0 & -2x \\
0 & 0 & x & -2x \\
-1 & -1 & -1 & x-1
\end{pmatrix}
At this point I'm contemplating taking out an $x$ out of the ma... | $A = \pmatrix {1&1&1&1\\2&2&2&2\\2&2&2&2\\1&1&1&1}$
Since A is a singular matrix, we know that 0 is an eigenvalue.
So, what is the dimension of the kernel of A?
if we perform row operations on A we get
$A = \pmatrix {1&1&1&1\\0&0&0&0\\0&0&0&0\\0&0&0&0}$
The dimension of the kernel is 3
0 is an eigenvalue of multiplic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2542141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Simple upper bound on the probability that the sum of $n$ dices rolls is equal to the most likely total Suppose $n$ $s$-sided (and fair) dice and are rolled, and consider the most likely value their total will take. Is there a simple / easy to state upper-bound on the probability that this total is rolled?
I know you c... | A good approximation for large $n$ is to note (rescaled) convergence in distribution to a normal distribution where the variance of the sum is $n\frac{s^2-1}{12}$ and so the probability of a value near the expectation of $n\frac{s+1}{2}$ will be about $\frac{1}{2\pi \sigma^2}=\sqrt{\frac{6}{\pi n(s^2-1)}}$.
If you wa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2542263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Convergence of $(x_n)$, $(x_n + y_n)$ and $(y_n)$ Prove that if real sequences $(x_n)$ and $(x_n + y_n)$ converge, then $(y_n)$ converges.
My attempt so far:
Suppose that the limits of $(x_n)$ and $(x_n + y_n)$ are $x$ and $x+y$ respectively. (Intuitively, $y_n \rightarrow y$ as $n \rightarrow \infty$.)
Then given ... | Set $L$ to be the limit of $(x_{n}+y_{n})$ (or we take $y=L-x$ as OP noted), and we claim that $y=L-x$ is the limit of $(y_{n})$: For $\epsilon>0$, choose $N$ such that for all $n\geq N$, $|x_{n}-x|<\epsilon/2$ and $|x_{n}+y_{n}-L|<\epsilon/2$, then for such an $n$, we have
\begin{align*}
|y_{n}-(L-x)|&=|y_{n}+x_{n}-L... | {
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"url": "https://math.stackexchange.com/questions/2542361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Probability of choosing value of variable in equation - 4 tuple ($x_1 +x_2+x_3+x_4=10$) It may be the wording in this problem that is throwing me off but I can't seem to figure out the number of possible successful outcomes to calculate the probabiliy:
Suppose a non-negative integer solution to the equation $w+x+y+z=1... | Put w = 0
Find solution for $x+y+z = 10$ and that is ${12\choose2}=66$
Put w = 1
Find solution for $x+y+z = 9$ and that is ${11\choose2}=55$
Put w = 2
Find solution for $x+y+z = 8$ and that is ${10\choose2}=45$
Add the above cases to a total of $166$
For all non negative w,x,y,z
Find the solution for $w+x+y+z = 10$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2542508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find real solutions in $x$,$y$ for the system $\sqrt{x-y}+\sqrt{x+y}=a$ and $\sqrt{x^2+y^2}-\sqrt{x^2-y^2}=a^2.$
Find all real solutions in $x$ and $y$, given $a$, to the system:
$$\left\{
\begin{array}{l}
\sqrt{x-y}+\sqrt{x+y}=a \\
\sqrt{x^2+y^2}-\sqrt{x^2-y^2}=a^2 \\
\end{array}
\right.
$$
From a math olympiad... | Hint:
Let $u=\sqrt{x-y},v=\sqrt{x+y}$. The system now reads
$$u+v=a,\\\sqrt{\frac{u^4+v^4}2}-uv=a^2$$
Raising the first equation to the fourth power,
$$a^4=u^4+v^4+4uv(u^2+v^2)+6u^2v^2.$$
Then using $u^4+v^4=2(a^2+uv)^2$ and $u^2+v^2=a^2-2uv$, you get an equation in $uv$, which simplifies:
$$a^4=2(a^2+uv)^2+4uv(a^2-2uv... | {
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"timestamp": "2023-03-29T00:00:00",
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Divergence of integral I found in web the following problem:
Let $f,g$ be continuous non-negative decreasing functions on $\mathbb R_{\ge 0}$ such that $$\int_0^\infty f(x)\,dx\to\infty, \int_0^\infty g(x)\,dx\to\infty$$. Define the function $h$ by $h(x)=\min \{f(x),g(x)\},\forall x\in\mathbb R_{\ge 0}$. Prove or disp... | here is my attempt to prove (it's only a rought idea)
let define
$$M(x)=max(f,g)$$
and
$$I_1(x)=\frac{M(x)+h(x)}{2}$$
thus by comparison $I_1(x)$ diverges
then define
$$I_k(x)=\frac{I_{k-1}(x)+h(x)}{2}$$
thus by induction $I_k(x)$ diverges
as $$I_k(x) \rightarrow h(x)$$
$h(x)$ also diverges $\square$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2542777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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If a sequence of bounded operator converges pointwise, then it is bounded in norm $\newcommand{\vertiii}[1]{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1
\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}$
Let $\left(V, \|\cdot\|_V \right)$ be a Banach space and $\left(W, \|\cdot\|_W \right)$... | The last update contains a correct solution to the exercise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2542884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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h-vector of an unmixed ideal We know that:
*
*if a ring is Cohen-Macaulay, then it is unmixed. But the converse is not true.
*if a ring is Cohen-Macaulay, then its $h$-vector is positive. The converse is not true.
Then I expect to find unmixed ideals that have positive $h$-vector, but they are not CM. Now the q... | I solved my question starting from the simplicial complex $$\Delta = \{\{x_1,x_2\},\{x_1,x_3\},\{x_1,x_4\},\{x_2,x_3\},\{x_2,x_4\},x_5\}$$
It is not unmixed but $h=(1,3,1)$, since $f=(1,5,5)$.
So, the ideal that I was looking for is
$$I_{\Delta} = (x_1x_2x_3, x_1x_2x_4,x_3x_4, x_1x_5,x_2x_5,x_3x_5, x_4x_5) \subseteq \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Parametrising the surface enclosed by a parametric curve I have a curve given by $(\cos t, \sin t, \sin 2t)$ with $0 \leq t \leq 2\pi$:
I need to integrate a function over any of the infinitely many surfaces to which this curve is a boundary. How would I find a parametrised form of such a surface?
| You can see that
$$z(t)=\sin 2t = 2\sin (t) \cos (t)=2x(t)y(t)$$
Therefore, a possible surface is
$$
z=2xy, \quad x=x, \quad y=y\quad \mbox{with}\quad x^2+y^2\le 1
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove that $\operatorname{Arsinh}(x) \ge \ln(1+x)$ for $x>-1$ Prove that $\operatorname{Arsinh}(x) \ge \ln(1+x)$ for $x>-1$.
I have solved similar inequalities for other trigonometric functions, but for this one I have no idea where to start, other than the fact that the plot of the functions makes it obvious.
For othe... | Defining $$f(x)=\operatorname{arsinh}(x)-\ln(x+1),$$ then we get by differentiating with respect to $x$: $$f'(x)=\frac{1}{\sqrt{1+x^2}}-\frac{1}{x+1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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What is the difference between $0$ and $\vec{0}$? If we take the approach underlined by Peano axioms, we get:
$$a + (-a) = 0.$$
Likewise, we can get the same thing in Euclidean plane:
$$|AB| + (-|AB|) = 0,$$
where $A$ and $B$ are points, and $|AB|$ is the length of the line segment $AB$ (thus the "operation" $AB - AB$ ... | $\vec{0}$ is a vector while $0$ is a number
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Show that a polynomial function has a minimum $p:\mathbb{R}\rightarrow\mathbb{R}$ is a polynomial function, which only has positive values. Show that p has a minimum.
This is what I tried to do:
I want to proof this statement using the extreme value theorem.
To be able to use it, the function has to be continuous and b... | Since $\lim_{x\to\pm\infty}p(x)=+\infty$ (it follows from the assumption that the polynomial has only positive values), there are $a<0$ and $b>0$ such that $p(x)>p(0)$ on $(-\infty,a)$ and $p(x)>p(0)$ on $(b,+\infty)$.
Consider $p$ on $[a,b]$.
Since all values of $p$ outside this interval are greater than some values ... | {
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"url": "https://math.stackexchange.com/questions/2543520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Limit of multinomial distributions Let $(X_1, X_2, ..., X_n)$ follow a multinomial distribution with parameters $n$ and $\mathbf{p}=(p_1, p_2,..., p_n) = (p+ \frac{1-p}{n}, \frac{1-p}{n},...,\frac{1-p}{n})$.
I am trying to prove that $\mathbb{P}(\max_k X_k \neq X_1) \to 0$ as $n\to \infty$. My idea is to first use the ... | I think Hoeffding's inequality
is enough to show $P(X_k \ge X_1)$ is small.
Let $Z_1,\ldots,Z_n$ be i.i.d., each taking values $1$, $0$, and $-1$ with respective probabilities $\frac{1}{n}(1-p)$, $\frac{n-2}{n}(1-p)$, and $p+\frac{1}{n}(1-p)$.
Then
$$X_k - X_1 \overset{d}{=} Z_1 + \cdots + Z_n.$$
We have $E[Z_i]=-\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Mean value thorem - Showing $\sqrt{1+x} < 1+\frac{x}{2}$ for $x>0$ So as the title states I have to show $\sqrt{1+x} < 1+\frac{x}{2}$ for $x>0$.
This is a example from the book which has to be explained for me, i'm having a hardtime understanding the proof. I do however understand the concept of MVT.
So the rest of the... | $\frac{1}{2}$ is not a random number, it comes from differentiating the square root: $\sqrt{x}\,'=\frac{1}{2\sqrt{x}}$ or $\sqrt{x+1}\,'=\frac{1}{2\sqrt{x+1}}$, the same thing.
Next he noticed that $\frac{1}{\sqrt{1+c}}<\frac{1}{\sqrt{1+0}}=1$ for all $c>0$, which is quite obvious after simple transformation.
| {
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Evaluating the integral $\int_0^{\infty} \frac{\sin(x)}{\sinh(x)}\,dx$ I was trying to evaluate the following integral,
$$I=\int_\limits {-\infty}^{\infty} \dfrac{\sin(x)}{\sinh(x)}\,dx$$ but had no success. I first expanded the the hyperbolic sine:
$$I=2\int_\limits {-\infty}^{\infty} \dfrac{\sin(x)}{e^{x}-e^{-x}}\,... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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Continuity of function defined on $C_c(\mathbb{R})$ Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the sup norm. Define $ f: (X, \vert \vert.\vert \vert_{\infty} )\rightarrow \mathbb{R}$ as $ f= \int_{- \infty}^{\infty} x(t)dt \>\>\> \forall x \in X$
Then $f$ is continuous(T/F).
I th... | Take $x_n \in C_c(\mathbb{R})$ such that
$$x_n(t) = \begin{cases} \frac{1}{n} & x \in [-n,n] \\ -\frac{1}{n}(x-n+1) &x \in (n, n+1] \\ - \frac{1}{n}(x+n-1) & x \in [-n-1, -n) \\ 0 & \text{otherwise}.\end{cases}$$
Then $x_n \to 0$ in $\| \cdot \|_\infty$ (that is $x_n$ tends to $0$ uniformly), but
$$f(x_n) >\frac{1}{n... | {
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Evaluating $\int_0^{\infty} \frac{\tan^{-1} x}{x(1+x^2)} dx$ The question is to evaluate $$\int_0^{\infty} \frac{\tan^{-1} x}{x(1+x^2)} dx$$
I used the substitution $x=\tan a$ then the given integral becomes $\int_0^{\pi/2} \frac{\tan^{-1}(\tan a)}{\tan a} da$
Now $\tan^{-1} (\tan a)=a \forall a \in [0,\pi /2]$ so that... | Proceeding by your method:
$\int_0^{\pi/2}\cfrac{a}{\tan(a)}da=\int_0^{\pi/2}a \tan(a)da$ [Now use Integration by Parts]
$= [a\log(\sin(a))]_0^{\pi/2} - \int_0^{\pi/2}\log\sin(a)da$
$= 0 - \int_0^{\pi/2}\log\sin(a)da$
$= \cfrac{\pi\ln(2)}{2}$
| {
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Convergence of subsequences in a compact set of $\mathbb{R}^n$ For any sequence in a compact set $K$ of $\mathbb{R}^n$, it is well known that there is a subsequence that is convergent to a point in $K$.
Is this true that for any sequence in a compact set $K$ of $\mathbb{R}^n$, there is a partition of this sequence into... | Denote the original sequence by $\{a_n\}$ and call its convergent sub-sequence $\{a_{n_0}\}$
Construct $\{a_{n_1}\}$ by first removing the terms of $\{a_{n_0}\}$ from $\{a_n\}$ and then finding a convergent sub-sequence.
Proceeding in this way, each time you will construct $\{a_{n_k}\}$ by first removing the terms of $... | {
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$E$ is nowhere dense
Let $E$ be the set of all $x\in [0.4,0.777\dots]$ whose decimal expansion contains only digits $4$ and $7$. How can we show that $E$ is nowhere dense in $[0.4,0.777\dots]$? That is, there is no interval $(a,b)$ in $E$.
My Try: $E$ does not contain intervals of the form $(0.4\dots41,0.4\dots3)$ a... | Lets say there is interval (a,b).
Then all numbers between $a$ and $b$ must be in your set.
Lets take decimal expansion of $a$ and $b$ and find first digit where they differ. Put for example $5$ there. You have a number that is between them, but is not in your set.
Therefore there can't be any interval.
| {
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Does pointwise convergence of a sequence of functions in each closed subinterval of an open interval imply pointwise convergence in the open interval? If a sequence {$f_n$} be uniformly convergent in every closed subinterval $[a+ε,b-δ]⊂(a,b)$, then it isn't necessarily uniformly convergent in $(a,b)$. But what about po... | Of course it is. Pick a point $x \in (a,b)$. The interval being open gives you an $ \epsilon>0 $ such that $(x-\epsilon, x+\epsilon) \in (a,b)$, and clearly this open interval contains the closed interval $[x-\frac{\epsilon}{2},x+\frac{\epsilon}{2}]$.
| {
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Probability that everyone shows up for flight?
The probability of a flight reservation being a no-show is unknown but
after observing $10000$ flight reservations we found that $95\%$ of
those people showed up.
If we consider a new sample of $100$ flight reservations, what is the
chance that each of the people sh... | A Bayesian approach could be to take a Beta prior distribution for $p$ the probability of somebody turning up, with a density proportional to $p^{\alpha-1}(1-p)^{\beta-1}$. Common choices are $\alpha=\beta=1$ (a uniform prior), $\alpha=\beta=\frac12$ (a Jeffreys prior) or $\alpha=\beta=0$ (an improper prior), but wit... | {
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Will terms in this sequence always have digital root 1? A recreational math problem, dubbed "Insert and Add", asks: What is the least integer m that requires no less than n insertions of plus signs so that, after performing the addition(s), we arrive at a single digit? (See the last page here: http://orion.math.iast... | I am terrible at proofs, so I won't be surprised when someone points out a glaring hole in this, but what about:
Assume the terms $a(1)$ to $a(n)$ all have digital root $1$, but $a(n + 1) = x$ doesn't. Increment $a(n)$ by one until we reach $x$.
Insert one plus sign into $x$ in the optimal way that guarantees the resu... | {
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$Ass(M) = Ass(N) \cup Ass(M/N)?$ Suppose $M$ is an $R$ module and $N$ is a submodule of $M.$ I am trying to prove the following: $Ass(M) = Ass(N) \cup Ass(M/N)$. This seemed intuitively true but the proof proved to be difficult. If I have a prime ideal $P \in Ass(M)$ then $P = Ann(m)$ for some $m \in M.$ If $m \in N$ t... | Disclaimer: I'm not quite good enough to give a nice hint, or point you in the right direction. Here is the argument I am familiar with, but read on only if you want a spoiler
claim:$ Ass(M) \subset Ass(N) \cup Ass(M/N)$.
Let $P \in Ass(M)$. Note that $P$ is associated to $M$ if and only if there is an injection $f:R/... | {
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The "constants" of a structure (model theory) A structure is composed of the domain, constants, relation symbols, and function symbols. I understand all of the ingredients of a structure immediately except for the constants. The definition in my book (A Shorter Model Theory) is
A set of elements of $A$ called consta... | One good reason to specify constants in a signature for models is that you want to make sure that these are preserved by homomorphisms. Consider, as in your example, the signature of ordered fields $\{ +, -, \cdot, 0,1, \leq \}$. A homomorphism $f:A \to B$ of structures of this signature will always satisfy $f(0_A) = 0... | {
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Working on Homotopy Equivalent Suppose I have spaces $X_{0}, X_{1},Y_{0},Y_{1}$ and $X_{0}$ is homotopy equivalent to $Y_{0}$ and $X_{1}$ is homotopy equivalent to $Y_{1}$.
I would like to show that $X_{0}$ x $X_{1} $ is homotopy equivalent to $Y_{0}$ x $Y_{1} $.
How do I approach this statement? I am fresh new to this... | Since $X_0 \simeq Y_0$ and $X_1 \simeq Y_1$, there are two pairs of functions
$$X_0 \overset{f_0}{\underset{g_0}{\rightleftarrows}} Y_0 \quad \text{and} \quad X_1 \overset{f_1}{\underset{g_1}{\rightleftarrows}} Y_1$$
such that the relevant composites are homotopic to the relevant identities.
From these functions, you c... | {
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Show that $(C_b (0,1],||\cdot||_{\infty})$ is not separable I'm not sure what I'm suppose to do for this question. Any help would be greatly appreciated. I know that there is a similar question like this for $[0,1)$ instead. But is there anyway for me to show this without having to prove the isomorphism between $[0,1)$... | Hint: Think of triangles of height $1$ over the intervals $[1/(n+1),1/n], n = 1,2,\dots.$
| {
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"In a circle of 33, the next 10 people on my right are all liars"
In the land of Truthlandia, each person is either a truth teller who always tells the truth, or a liar who always tells lies. All 33 people who gathered for a meal in Truthlandia at a round table, said: "The next 10 people on my right are all liars." Ho... | A truth-teller, by the given statement, forces the nature of the ten people to the right – a block of eleven. However, there cannot be a block of eleven liars, because the leftmost "liar" would actually be telling the truth. Thus there are three equally-spaced truth tellers and 30 liars.
| {
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The product of three ordered variables is $8!$ (Factorial)? The question asks:
I could get a lowest of 17 with 28, 32, 45, but the answer is 4.
Other than trying again and again, is there a fast and accurate way to solve this?
| Note that $8! = 40320$. Note that if $a < b < c$, then $a,b,c$ must be very close to the cube root of $40320$, which you can easily estimate to be between $30$ and $40$, since $3^3 = 27 < 40 < 4^3 = 64$, so we can predict the first digit easily.
Therefore, the desired numbers have to be somewhere around the $30-40$ ma... | {
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Smallest possible value of expression involving greatest integer function If $a, b, c \gt 0$ then what is the smallest possible value of
$$\left[\frac{a+b}{c}\right]+ \left[\frac{b+c}{a}\right] + \left[\frac{c+a}{b}\right]$$
where $[.]$ denotes greatest integer function.
I tried using the AM GM inequality at first b... | Wolog $a \le b \le c$
$[\frac {b+c}a] \ge [\frac {a+a}a] =2$
And $[\frac {a+c}b] \ge [\frac {a + b}b] \ge [\frac bb] = 1$.
So you can not get less than $3$
If $\frac {a+b}{c} < 1$ then $c > a+b$ and $\frac {c+a}b > \frac {2a + b}b= \frac {2a}b + 1$. So If $\frac {c+a}b < 2$ then $\frac {2a}b < 1$ so $b > 2a$. So $\fr... | {
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What does $\{\infty\}$ mean? I'm being introduced to $\sigma$-algebra's and I came across the following definition:
Let $\mathcal{B}(\overline{\mathbb{R}})$ be the $\sigma$-algebra generated by the sets $\{-\infty\}$,$\{\infty\}$ and $B\in\mathcal{B}(\mathbb{R})$. This $\sigma$-algebra $\mathcal{B}(\overline{\mathbb{R... | The extended real line is $\Bbb R$ adjoined by the two non-number-objects $+\infty$ (or $\infty$) and $-\infty$, along with the declaration that $-\infty<r<\infty$ for any real number $r$.
The text you quote simply describes the Borel sets of the extended real line, and since it turns out that these are exactly the sam... | {
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Problem 9 from Herstein's book Suppose that $H$ is a subgroup of $G$ such that whenever $Ha\neq Hb$ then $aH\neq bH$. Prove that $gHg^{-1}\subset H$ for all $g\in G$.
Remark: Honestly, I had some problems. Firstly, after some thoughts I have realized that condition on $H$ is equiavalent to the following: If $aH=bH$ the... | Here is one part:
$aH=bH \implies a^{-1}b\in H $
Indeed, $b=be \in bH=aH \implies b=ah$ for some $h \in H$ and so $a^{-1}b=h\in H $.
$a^{-1}b\in H \implies ba^{-1}\in H$
Indeed, $a^{-1}b=h \in H \implies b^{-1}a=(a^{-1}b)^{-1}=h^{-1} \in H $.
| {
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Define $F : \mathbb{S}^n \times I \to \overline{\mathbb{B}^{n+1}}$ by $F(x, s) = sx$, show that $F$ is injective
Define $F : \mathbb{S}^n \times I \to \overline{\mathbb{B}^{n+1}}$ by $F(x, s) = sx$, show that $F$ is injective on $\overline{\mathbb{B}^{n+1}} \setminus \{0\}$
I tried to show that by contradiction but i... | Suppose $F(s_1,x_1)=F(s_2,x_2)$, so that $s_1x_1=s_2x_2$. Taking the magnitude of both of these vectors, we get $s_1\|x_1\|=s_2\|x_2\|$, which since $x_1,x_2\in \mathbb S^n$ implies $s_1=s_2$. Therefore, $s_1x_1=s_1x_2$. Finally, since $s_1\neq0$ (which we know since $s_1x_1\in \mathbb B_{n+1}\setminus\{0\}$), we can s... | {
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Find the area of the region bounded by the curves $y =\sqrt x$, $y=x-6$ and the x-axis by integral with respect to x
Find the area (in green) of the region bounded by the curves $f(x) =\sqrt x$, $g(x)=x-6$ and the x-axis by integral with respect to x
Attempt:
since $x-6 =\sqrt x$
$x=9$
$A = \displaystyle\int _a^... | Alternatively, you can use
\begin{align}
f^{-1}(y)&=y^2
,\\
g^{-1}(y)&=y+6
\end{align}
and find the area as
\begin{align}
\int_0^3 \int_{f^{-1}(y)}^{g^{-1}(y)} \,dx \,dy
&=\int_0^3 6+y-y^2\,dy
\\
&= \left. 6y+\tfrac12\,y^2-\tfrac13\,y^3\right|_0^3
\\
&=\frac{27}2
.
\end{align}
| {
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Nonhomeomorphic subsets of the plane I'm trying to find two compact, nonhomeomorphic subsets of the plane, say $X$ and $Y$, such that $X \times [0,1]$ is homeomorphic to $Y \times [0,1]$. I can not think of how a homeomorphism arises when you product with the interval.
| This CW answer is supposed to kick this question from the unanswered queue. I strictly follow the approach mentioned in What to do with questions that are exact duplicates from MathOverflow?
There are indeed counterexamples to which Igor Belegradek gave a reference. Here is another counterexample in the plane, perhap... | {
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Trouble understanding null-homotopic chain maps I'm working through Weibel, and I'm at the part where he defines null-homotopic maps. He says it's essentially topological null-homotopy, but I'm having trouble reconciling that with examples.
Remark: This terminology comes from topology via the following observation. A ... | It doesn't. $f$ induces an isomorphism on $H_0$ because $X$ and $Y$ are both path-connected, and this is true more generally for any map between path-connected spaces. In topology it's actually not possible for a map to induce zero on $H_0$ since the connected components of the source have to map to the connected compo... | {
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Equivalences of formulas under an interpretation Is this proof correct?
Prove $\vDash\exists x\phi\leftrightarrow\lnot\forall x\lnot\phi$
Proof: We want to see the that $\vDash\exists x\phi\leftrightarrow \vDash\lnot\forall x\lnot\phi$, so let I be an interpretation with domain |I|.
Notice that
$\vDash\exists x\phi\iff... | My main criticism with this is that I'd hate to see the use of quantifiers in the semantics. That is, rather than saying
$$\vDash\exists x\phi\iff\vDash\phi[a/x],\exists a\in |I|$$
I would define the semantics as:
$$\vDash\exists x\phi\iff \text{for some } a\in |I|: \ \vDash\phi[a/x]$$
That way, you have a much more ... | {
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Prove $\bigcap_{i=1}^k\ker(f_i)\subset \ker(f)\iff f\in {\rm span}(f_1,...,f_k) $ Let $V$ a $\mathbb{K}$-vector space of finite dimension $n$, with $\{f_1,...,f_n\}$ a set linearly independent of $V^*$ and $f\in V^*$.
Prove $\bigcap_{i=1}^k\ker(f_i)\subset \ker(f)\iff f\in {\rm span}(f_1,...,f_k).$
($\Leftarrow$) Let $... | You start well, but soon make a mistake: the statement $f\in\bigcap_{i=1}^k\ker(f_i)$ is wrong, because the kernels are subspaces of $V$ and $f\in V^*$.
What you have to prove is
If $f\in\operatorname{Span}(f_1,\dots,f_k)$ then $\ker(f)\supset\bigcap_{i=1}^k\ker(f_i)$.
Suppose $f\in\operatorname{Span}(f_1,\dots,f_k)$... | {
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$\tan(x) = 3$. Find the value of $\sin(x)$ I’m trying to figure out the value for $\sin(x)$ when $\tan(x) = 3$. The textbook's answer and my answer differ and I keep getting the same answer. Could someone please tell me what I'm doing wrong?
1.) $\tan(x) = 3$, then $\frac{\sin(x)}{\cos(x)} = 3$.
2.) Then $\cos(x) = ... | If $\tan(x)=3$, then $\tan^2(x)=9$. This means that $\frac{\sin^2x}{1-\sin^2x}=9$. So, $\sin^2(x)=\frac9{10}$; in other words (at least if we're on the first quadrant), $\sin(x)=\frac3{\sqrt{10}}$.
Your error lies in item 4: $\sin^2(x)+\left(\frac1{3\sin(x)}\right)^2=\frac{10}9\sin^2(x)$.
| {
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Group action on a completely regular space I am trying to prove that $X/G$ is completely regular provided that $X$ is a completely regular space and $G$ is a compact hausdorff group acting on $X$.
Let $p: X \rightarrow X/G$ be the canonical quotient map. So I choose a point $\overline{x} \in X/G$ and $C \subseteq X/G$ ... | I recall the following known facts
From [AT]:
I guess [165, Theorem 1.4.13] is the same as [Eng, Theorem 1.4.13].
From [Eng]:
References
[165] Ryszard Engelking General Topology, PWN, Polish Scientific Publ., 1977.
[AT] A.V. Arhangel'skii, M. Tkachenko Topological groups and related structures, Atlantis Press, Par... | {
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There are infinitely many integers $n$ such that $\varphi(n)=n/3$ prove or disprove the following statement:
There are infinitely many integers $n$ such that $φ(n)=n/3$. where $φ(n)$ is Euler Phi-Function.
Could you please help me with the prove of this , I try it many time but I do not how can I start to prove it or ... | Hint: Look at integers in the form $n = 2^a3^b$ where $a$ and $b$ are positive integers. Can you calculate $\varphi(n)$ for integers $n$ in that form?
| {
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} |
Are irrational numbers like π relationships which are established by some rule or criteria? Rational and Irrational of Reals
Irrational numbers appear to fill in the ‘gaps’ between Rational numbers on a Real number line. However they seem to be stipulations or definitions of relationships which are established by some ... | There are uncountably many numbers on the real line. Most of them are just there and serve as a sort of "glue" in order to make the system ${\mathbb R}$ complete. The real numbers that actually do occur in mathematics as individuals of interest are all defined by criteria, formulas, or algorithmic procedures, etc.
Th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Question about Bayesian probabilities. Where did I go wrong? I was solving a question for an acquaintance, about probabilties. The original question goes like this:
Of all threats in an year, $12\%$ are Tier 1 and the remaining $88\%$ are Tier 2. If the probability that a reported Tier 1 threat is actually a Tier 2 is... | Using conditional probability expressions in a Bayesian probability question will lead to much less confusion about the topic.
Use $T_1,T_2$ as the mutually exclusive and exhaustive events that a threat is tier 1 or tier 2 respectively, and $R_1,R_2$ as the m.e.e. events that a threat is reported as such.
Of all threa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Convert a circle to a polygon If I have a circle with a given radius, how calculate I find a regular polygon's vertices, if I know they are all on the circles edge (like in the images below)?
| For a polygon with $n$ vertices (an $n$-gon), each vertex will have an angle of $\frac{2\pi}{n}$ between it and the next one. All we need to do is place a first vertex, then translate it around the origin by this amount $n$ times to get all the vertices.
Place the first vertex at $p_0=(\rho, 0)^T$, where $\rho$ is the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Can the sequence of derivatives $\{f^{(k)}(0)\}_{k\geq 1}$ be any sequence? Let $\{a_{k}\}_{k\geq 1}$ be any sequence of real numbers, must there exist a smooth function $f:]-\epsilon,\epsilon[\rightarrow \mathbb{R}$ (for some positive $\epsilon$) such that for every positive integer $k\geq 1$, we have $f^{(k)}(0)=a_k ... | Yes, this is a special case of a theorem of Borel. Given any sequence $(a_n)$ there is a smooth function on $\Bbb R$ whose Maclaurin series
is $\sum a_nx^n$.
I outline the proof. There is a smooth function $f:\Bbb R\to\Bbb R$
which equals $1$ on $[-1,1]$ and vanishes outside $[-2,2]$. Then consider
$f(x)=\sum_{n=0}^\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Simple inequality over positive reals: $2(x+y+z) \geq 3xyz + xy+yz+zx$ for $xyz=1$ Problem
Let $x,y,z$ be real positive numbers with $xyz=1$. Prove:
$$ 2(x+y+z) \geq 3xyz + xy+yz+zx$$
Note : I don't know whether the inequality is true or not. I couldn't find a prove in the place found it nor a solution to it.
My try
... | If $x=y=4$ and $z=1/16$, then $xyz=1$, and $$2(x+y+z)-(3xyz+xy+yz+zx)=-27/8<0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
} |
Deriving the density of sum of iid Uniform distributions using Laplace Transforms. In Resnick's Adventures in Stochastic Processes, there's this example where the author derives the density of $\sum_i X_i$ where $X_i$ are iid uniform (0,1).
In the picture below, I don't understand 2 things:
*
*What they mean (and w... | 1) They mean that $$\int e^{-\lambda x} (\epsilon_k*g(x)) dx = e^{-\lambda k} \lambda^{-n}$$ This follows from the two equations (one right after "Now," and the other one right after "Furthermore,") ,and the fact that products in Laplace domain are convolutions in "normal" domain.
2) Plug the integral I just typed int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solve $\mathbf{\vec{a}} \, \cdot \mathbf{\vec{x_{i+1}}} \gt \mathbf{\vec{a}} \, \cdot \mathbf{\vec{x_{i}}} $ for $\mathbf{\vec{a}}$ I have a set $X = \{\mathbf{\vec{x_1}},\mathbf{\vec{x_2}},\mathbf{\vec{x_3}},\ldots,\mathbf{\vec{x_k}}\}$ of vectors, all of which are $n$-dimensional.
I want to find an $n$-dimensional ve... | [I write superscripts to index the vectors.]
If the vectors $ \mathbf{\vec{x^{i}}} $ are linearly independent, then for $k\le n$ you can find such a vector $\mathbf{\vec{a}} $. You can do even more: you can fix the overlaps $\mathbf{\vec{a}} \, \cdot \mathbf{\vec{x^{i}}} $. So let
$\mathbf{\vec{a}} \, \cdot \mathbf{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why does $\lim_{n \to \infty} \left(1+\frac{1}{n}\right)^n=e$ but $\lim_{n \to \infty} \left(1-\frac{1}{n}\right)^n=e^{-1}$? Why does $$\lim_{n \to \infty} \left(1+\frac{1}{n}\right)^n=e$$ but $$\lim_{n \to \infty} \left(1-\frac{1}{n}\right)^n=e^{-1}$$
Shouldn't the limits be the same since $\left(1+\frac{1}{n}\right) ... | Expand both using the binomial theorem
$$\left(1+\frac 1n\right)^n=1^n+n\cdot\frac 1n\cdot1^{n-1}+\binom n2\left(\frac 1n\right)^21^{n-2}+\binom n3\left(\frac 1n\right)^31^{n-3}+\dots =$$$$=1+1+\frac {n(n-1)}{2n^2}+\frac {n(n-1)(n-2)}{6n^3}+\dots$$while $$\left(1-\frac 1n\right)^n=1-1+\frac {n(n-1)}{2n^2}-\frac {n(n-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Finding the turning points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$ I've just come across this function when playing with the Desmos graphing calculator and it seems that it has turning points for many values of $a$.
So I pose the following problem:
Given $a \in \mathbb{R}-\{0\}$, fin... | A note about Case $1$ with $\,a:= 2^{-\frac{2}{3}}\,$ and $\,\displaystyle x\to 2^{\frac{1}{3}}\,$ .
Left side $\,=0\,$ for $\,\displaystyle x=2^{\frac{1}{3}}\,$ because of $\, ax^2-1=0\,$ and $\,\displaystyle ax^2-a^2x+1=\frac{3}{2}\ne 0\,$ .
Right side $\,=0\,$ for $\,\displaystyle x\to 2^{\frac{1}{3}}>1\,$:
$\,\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2548041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
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Tangent space as the set of all derivations I am trying to get a grip on the concept of derivations at a point on a manifold by working out some concrete examples.
Let $M$ be a smooth manifold with or without boundary, and let $p \in M$. A linear map $v : C^{\infty}(M) \to \mathbb{R}$ is called a derivation at $p$ if ... | You have effectively done everything correctly.
You recognised that $\frac{\partial}{\partial x}|_{p}$ was a basis for the tangent space at $p$ and $\frac{\partial f}{\partial x}|_{p} = -1$
An arbitrary tangent vector at $p$ is thus $v = c\frac{\partial}{\partial x}|_{p}$ for some $c$ so by linearity $v(f)= c\frac{\par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2548163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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On the asymptotics of $a_n=a_{n-1}^k+a_{n-2}^k$ where $k>1$ and $a_0=0, a_1=1$ Consider the sequence defined by $a_0=0,a_1=1, a_n=a_{n-1}^k+a_{n-2}^k$, where $k$ is a fixed integer larger than $1$.
One finds $a_n\sim a_{n-1}^k$ and thus $a_n\sim \alpha_k^{k^n}$ where $\alpha_k$ is a constant which depends on $k$.
It ... | If $k > 1$, it's obvious that the sequence $a_n$ is increasing and nonnegative. So, because $0 \le a_{n-2} \le a_{n-1}$, we have the inequality $a_{n-1}^k \le a_n \le 2a_{n-1}^k$, and that will be all we need to prove the inequality
$$
2^{k^{n-3}} \le a_n \le 2^{k^{n-1}/(k-1)} \tag{$\star$} \label{eq:star}
$$
for s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2548216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Uniform intgrability and compact convergence Suppose we have a family of uniformly integrable random variables $\{X_n:n\in\mathbb{N}\}$. Suppose also we have a sequence of continuous functions $f_n:\mathbb{R}\to\mathbb{R}$ converging to $f$ and that $\{f(X_n):n\in\mathbb{N}\}$ is uniformly integrable. Also assume that ... | Let $f_n\colon x\mapsto x/n$. Then $f_n$ is continuous for all $n$ and the sequence $\left(f_n\right)_{n\geqslant 1}$ converges to $0$ uniformly on compact sets. It thus suffices to find a sequence of random variables $(X_n)_{n\geqslant 1}$ such that for each $n$, $X_n/n$ is integrable but the family $\left\{X_n/n,n\ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2548325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Numerical range of normal matrices Let $F_1, F_2$ be two normal matrices, we consider
$$W(F_1,F_2)=\{(\langle F_1 y\; ,\;y\rangle,\langle F_2 y ,\;y\rangle):y \in F,\;\;\|y\|=1\}.$$
If $F_1F_2=F_2F_1$. Is $W(F_1,F_2)$ convex?
Thank you!
| It is quite simple: if all $(F_1,\cdots, F_d)$ commute with each other and normal, they all have the same orthonormal eigenbases $(v_1,\cdots,v_n)$ and every state vector $x$ can be decomposed:
$$x=\displaystyle\sum_{i=1}^n x_i v_i$$
Average value of $F_j$ over this state is simply
$$\displaystyle\sum_{i=1}^n |x_i|^2 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2548484",
"timestamp": "2023-03-29T00:00:00",
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The intersection of all subspace of $V$ is $\{0\}$. Let $V$ be a $\mathbb{R}$-subspace with basis $B=\{v_1 ,v_2, \ldots , v_n\}$ and $\overline{v}\in V$, $\overline{v}\neq 0$.
I have shown that if we exchange $\overline{v}$ with a $v_i\in B$ we get again a basis.
I want to show, using this fact, that the intersection... | I am not sure this is the easiest way to answer your original question. The way to do what you’re asking is:
Asume by contradiction that the intersection has a non zero vector $u$. Complete $u$ to a basis $u, v_2, ....v_n$. Now $v_2, ...,v_n$ spans a $n-1$ subspace. $u$ can’t be an element of that sub space because it’... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Integer part of $\sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}}}$ Find the value of the following infinite series:
$$\left\lfloor\sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}}}\right\rfloor$$
Now, my doubt is whether it's $2$ or it's $3$. I'm not sure if it just converges to $3$ but not actually reaches it or if it compl... | Let $$x= \sqrt[3]{24+\sqrt[3]{24+...}}$$ $$x^3 = 24+\sqrt[3]{24+\sqrt[3]{24+...}}$$
Notice $x^3 -24$ will be itself so $$x^3-24=x$$ Solving for $x$ yields $$3, \frac{3+\sqrt{23}i}{2}, \frac{3-\sqrt{23}i}{2}$$ but since we are concerning real solutions, 3 is the answer. It will be infinitely close to 3 amd reaches 3 so ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2548701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 6
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Proving $g(2n)=0$ Given that $f$ is an odd function periodic with period $2$ and continuous for all $x$ and $g(x)=\int_0^x f(t) dt$
then the question is to prove $g(2n)=0$
$g(2n)=n\int_0^2 f(x) dx=n g(2)$ so I have to prove that $g(2)=0$.I could check that $g(x) $ is a even function.Any ideas?
| $$g(2)=\int_0^2f(x)\text{d}x=\int_{-2}^0f(x)\text{d}x$$
Thus,
$$g(2)=\frac{1}{2}\left(\int_0^2f(x)\text{d}x+\int_{-2}^0f(x)\text{d}x\right)=\frac{1}{2}\int_{-2}^2f(x)\text{d}x=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2548813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Showing a set has nonempty interior Let $A$ be a finite set. Let $M : A \times A \to {\bf R}$ be a symmetric function, which is positive-semidefinite when regarded as an $A \times A$ matrix.
Let $P(A)$ be the set of vectors $p = (p_a)_{a \in A} \in {\bf R}^A$ such that for all $a \in A$, $p_a \ge 0$, and $\sum_{a \in ... | This is not true.
If $S^{-1}((-\infty, r])$ is non-empty, then it contains each of the open sets $S^{-1}((-\infty, s])$ for $0\leq s<r$ and is thus of non-empty interior unless all of those sets are empty, in other words when $r$ is the global minimum of $S$ on $P(A)$.
But the global minimum of $S$ might well be $r>0$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Definite Integral $\int_{0}^{1}\frac{\ln(x^2-x+1)}{x-x^2}dx$ for Logarithm and Algebraic.
Evaluate
$$\int_{0}^{1}\frac{\ln(x^2-x+1)}{x-x^2}dx$$
I have been thinking for this question quite some time, I've tried all the methods I have learnt but still getting nowhere. Hope that someone can explain it for me. Thanks ... | You can use, since you're in the range $[0, 1]$, the logarithm expansion
$$\ln(x^2 - x + 1) = \sum_{k = 1}^{+\infty} \frac{(-1)^k}{k} (x^2-x)^k$$
together with the fact that $x - x^2 = -(x^2 - x)$, hence the integration becomes
$$\int_1^0$$
due to the minus sign, incorporated into the extrema.
After that, treat the fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Trying to solve this First Order Differential Equation. I've made several attempts to this question but they've been unsuccessful. The question is shown below:
$$(1+x^2)\frac{\text{d}x}{\text{d}y}+xy=0$$
the answer in the book states when y(0)=2
$$y^2(1+x^2)=4$$ however I gain $c^2=y^2(1+x^2)$ and when the boundary co... | It looks like you're correct - for $x=0$ and $y=2$, we have $c^2 = 2^2 (1 + 0^2)$, so $c^2 = 4$. So, your equation becomes $y^2 (1+x^2) = c^2 = 4$, as required.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determinant of a matrix with odd diagonal and even entries I'm trying the solve the following problem linear algebra problem and I'm not sure where to begin:
Let $B$ be a square matrix with n columns and integer entries. This matrix is constructed so that all diagonal entries are odd and all other entries are even. We ... | $$B = \begin{bmatrix}
\color{red}{odd} & even & \dots & even\\
even & odd & \dots & even \\
\vdots & \ddots & \dots & even\\
even & even &even & odd
\end{bmatrix}$$
$$|B| = \color{red}{odd} + even+even+..........+even=odd\ne 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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How do I show $f_n(x)=n^2 x^n(1-x)$ pointwise converges to $0$ on $[0,1]$?
How do I show $f_n(x)=n^2 x^n(1-x)$ pointwise converges to $0$ on $[0,1]$?
I first started with the case $x=1/2$ to try out. We see that $n^2(1/2)^n(1/2)=n^2/2^{n+1}$ which goes to $0$ as $n$ goes to infinity since $n^2 < 2^{n+1}$.
Now I need... | For $0 < x < 1$ we have $y = 1/x - 1 > 0$ and $ x = 1/ (1 + (1/x - 1))= 1/(1+y).$
Thus,
$$n^2 x^n = \frac{n^2}{(1 + y)^n}.$$
Using the binomial expansion $(1+y)^n = 1 + ny + \frac{1}{2}n(n-1)y^2 + \frac{1}{6}n(n-1)(n-2)y^3 + \ldots,$we have
$$0 \leqslant n^2 x^n = \frac{n^2}{(1 + y)^n} < \frac{n^2}{n(n-1)(n-2)}\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Calculus Derivative—Finding unknown constants
Find all values of $k$ and $l$ such that
$$\lim_{x \to 0} \frac{k+\cos(lx)}{x^2}=-4.$$
Any help on how to do this would be greatly appreciated.
| Note that
$$\lim \frac{1-\cos x}{x^2}=\frac{1}{2}$$
And that $$\frac{k+\cos(lx)}{x^2}=\frac{k+1}{x^2}-\frac{1-\cos(lx)}{x^2}$$
Thus $$k+1=0$$and $$\frac{1}{2}l^2=4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Shifted absorption time If $(X_n)_{n\geq0}$ is a Markov chain, and $T=inf\left\{n\geq0 : X_n=1 \right\}$, why does $$\mathbb{E}_i[T-1|X_1=j]=\mathbb{E}[T|X_0=j] ?$$
This question is partially answered here : Markov chains: Expected time until absorption., but I'd like a rigorous proof of that fact.
| It's enough to prove that
$$
\Pr[T = t \mid X_0 = i, X_1 = j] = \Pr[T = t-1 \mid X_0 = j].
$$
Then $\mathbb E[T]$ will follow by the usual expectation formula.
Both sides of this probability can be decomposed into sums over paths: on the left, paths $(i,j, \dots,1)$ of length $t$, and on the right, paths $(j,\dots,1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof of Generalized Cayley's formula I'd like to prove the equation following:
$$x_1x_2x_3...x_n(x_1 + x_2 + ... x_n)^{n-2} = \sum_Tx_1^{d_{T(1)}}x_2^{d_{T(2)}}...x_n^{d_{T(n)}}\tag 1$$
where the sum is over all spanning trees $T$ in $K_n$ and $d_{T(i)}$ is the degree of $i$ in $T$
I heard this's called Cayley's gen... | As you might have noticed, setting all the $x_i$ to $1$ recovers the fact that the number of trees on $n$ vertices, which is the same as the number of trees spanning $K_n$, is $n^{n-2}$.
To get a bijective proof of (1), let $x_1, \ldots, x_n$ be numbers labeling the vertices of $K_n$. Given a spanning tree $T$, note th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2550064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
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$\det(A^4+I)=29$ is not solvable by any $A\in M_4(\mathbb Z)$ I recently encountered the following problem.
Given any $A \in M_4(\mathbb Z)$, show that $\det(A^4+I)\ne29$, where $I$ denotes the identity matrix.
LHS can be written as the product of $1+{\lambda _i}^4$ where $\lambda _i$ denotes the eigenvalues of A. By... | Consider the matrix $A$ modulo $29$.
The matrix $A^4+I$ has determinant $29$ so has rank $3$ and nullity
zero considered as a matrix over $\Bbb F_{29}$. It has a unique null-vector
$u$ up to scalar multiplication: $(A^4+I)u\equiv 0\pmod{29}$
and $(A^4+I)v\equiv 0\pmod{29}$ implies $\newcommand{\la}{\lambda}v\equiv\la u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2550139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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convergence sequence and continuous functions I have the following question: assume $C$ is a subset (we can assume it is convex and compact) of a Banach space $(X,\|\cdot\|)$, $f:C\longrightarrow C$ continuous and $x_{0}\in C$ a fixed point of $f$. Also, let $(x_{n})_{n\geq 1}\subset C$ be a sequence having a subsequen... | Without additional conditions on the sequence $(c_{n})_{n\geq 1}$ (maybe, that the series $\sum c_n$ converges) the asset may be false even for the segment $[0,1]$ endowed with the standard metric and the identity map $f$. As a counterexample it suffices to put $x_0=0$ and $\{x_n\}$ any sequence of points of $[0,1]$ s... | {
"language": "en",
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Find the formula for the sequence $(a_n)$ that satisfies the recurrence relation $a_n=(n+7)a_{n-1}+n^2$ with $a_0=1$
Find the formula for the sequence $(a_n)$ that satisfies the
recurrence relation $a_n=(n+7)a_{n-1}+n^2$ with the initial condition
$a_0=1$.
This is a non-linear nonhomogeneous recurrence relation. ... | Following @GTonyJacobs remark about the homogeneous sequence,
let us define (this is a discrete method of variation of the parameter)
$${a}_{n} = \left(n+7\right) ! \ {b}_{n}$$
We have
$$\left(n+7\right) ! \ {b}_{n} = \left(n+7\right) \left(n+6\right) ! \ {b}_{n-1}+{n}^{2}$$
hence
$${b}_{n}-{b}_{n-1} = \frac{{n}^{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2550562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
If $f(g(x))=x$ is $f$ an injective function?
Let $ f:\mathbb R\to \mathbb R $ . If $f(g(x))=x$ then is $f$ an injective function?
Well, I proved it to be true. But honestly I have a strong feeling that my proof is wrong.
Here's my proof:
Assume $f(a_1)=f(a_2) = x$ and we want to prove that $a_1 = a_2$ .
if $f(a_1)=f... | Recall the definition of $\arctan$:
$\arctan$ is the only continuous function $\Bbb R\to \left(-\frac\pi2,\frac\pi2\right)$ such that $\tan\arctan x=x$ for all $x\in\Bbb R$.
Now, it is clear that $\tan$ can be extended to a function $\Bbb R\to\Bbb R$ by assigning arbitrary values to $x=k\pi+\frac\pi2$. Would such an ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2550627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Problems with finding limit The function $f(x)$ has a derivative at the point $a$ and $f(a) > 0$. I need to find the limit as n $\to + \infty$ of $$\left(\frac{f(a + \frac1n)}{f(a)} \right)^n$$ Substitution method?
| By definition of derivative we get
$$ \lim_{n\to \infty}\left(\frac{f(a + \frac1n)}{f(a)} \right)^n= \lim_{n\to \infty}
\exp\left[n\log\left(\frac{f(a+1/n)}{f(a)}\right )\right]
=\lim_{h\to 0}\exp\left[\frac{\log\left(f(a+h\right ) -\log f(a)}{h}\right] =\exp\left(\frac{f'(a)}{f(a)}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2550733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Test for divergence of $\int_{0}^{\infty} \frac{\sin^2(x)}{x}dx$ without evaluating the integral I would like to prove that $$\int_{0}^{\infty} \frac{\sin^2(x)}{x}dx$$ diverges without actually evaluating the integral. Is there a convergence test from calculus or real analysis that can show that this integral diverges?... | $$
\begin{align}
\int_0^\infty\frac{\sin^2(x)}{x}\,\mathrm{d}x
&=\sum_{k=1}^\infty\int_{(k-1)\pi}^{k\pi}\frac{\sin^2(x)}{x}\,\mathrm{d}x\\
&\ge\sum_{k=1}^\infty\frac1{k\pi}\int_{(k-1)\pi}^{k\pi}\sin^2(x)\,\mathrm{d}x\\
&=\sum_{k=1}^\infty\frac1{2k}
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2550977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
If $a+b\mid a^4+b^4$ then $a+b\mid a^2+b^2$; $a,b,$ are positive integers. Is it true: $a+b\mid a^4+b^4$ then $a+b\mid a^2+b^2$?
Somehow I can't find counterexample nor to prove it. I try to write it $a=gx$ and $b=gy$ where $g=\gcd(a,b)$ but didn't help. It seems that there is no $a\ne b$ such that $a+b\mid a^4+b^4$. O... | We have that $$a^4+b^4=(a+b)(a^3-a^2b+ab^2-b^3)+2b^4$$ so if $a+b$ is a factor of $a^4+b^4$ it is also a factor of $2b^4$ and (by symmetry) $2a^4$
If the highest common factor of $a$ and $b$ is $y$ so that $a=py$ and $b=qy$ we find that $(p+q)y$ is a factor of $2q^4y^4$. Now $p+q$ can have no factor in common with $q$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Find vectors and a scalar that disprove that $A=\{p(x)\in V:p(0)-p(2)=2\}$ is a subspace of $V=R_3[x]$ So I am supposed to disprove that $A=\{p(x)\in V:p(0)-p(2)=2\}$ is a subspace of $V=R_3[x]$ by finding $v_1$, $v_2$, and $\alpha$ such that $v_1+\alpha v_2\notin A$. I am honestly kind of stumped here. So far, I have ... | $0 \notin A$. So, just take any $v_1 \in A$ and $\alpha=-1$ and $v_2=v_1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Digit sum of a huge number I am helping a high school student to solve some challenging problems. I am stuck on the following problem: let $S(n)$ be a digit sum of the integer $n$, e.g. $S(1234)=10$. Find $S(S(S(S(2018^{2018}))))$.
I spent some time on this problem but was not able to solve it. I found out, that the re... | Remember $n \equiv S(n) \mod 9$. ($a \equiv b \mod n$ means $a$ and $b$ have the same remainder when divided by $n$.)[See postscript]
So $S(S(S(S(2018^{2018})))) \equiv 2018^{2018}\mod 9$. So what is the remainder of $2018^{2018}$ when divided by $9$.
As $2018 \equiv 2 \mod 9$ then $2018^{2018} \equiv 2^{2018} \mod 9$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is $\aleph_0!$? What is $\aleph_0!$ ?
I know that in the original definition the factorial is defined for natural numbers but, what if we extend this concept to cardinal numbers?
This concept has been extended to the real numbers by the $\Gamma$ function but I never see this kind of extension before.
This is a ... | We have $k!=1\cdot2\cdots k$, i.e., it is products of all numbers with size at most $k$.
Therefore
$$\aleph_0! = 1\cdots 2 \cdots \aleph_0 = \prod_{k\le\aleph_0} k$$
seems like a possible generalization. (Although probably taking the number of bijections - as suggested in other answers - is a more natural generalizati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Why is the boundary of an oriented manifold with its (opposite oriented) copy the empty set? Excuse the very basic question:
I'm following Milnor's Lectures on Characteristic Classes. He defines a relation on the collection of compact, smooth, oriented manifolds of dimension $n$ by letting $M_1\sim M_2$ if $M_1\sqcup ... | You're a bit confused about what this definition means. First of all, the $n$-manifolds we're considering here all do not have boundary. So $M$ should be an $n$-manifold without boundary, and you shouldn't be thinking about its boundary at all.
The relation $M_1\sim M_2$ means that $M_1\sqcup -M_2$ is the boundary of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. (which approach to take) Use Green’s Theorem to evaluate the line integral along
the given positively oriented curve.
$$\int_c y^3 \, dx - x^3 \, dy, C \text{ is the circle } x^2+y^2=4$$
Ok, so I'm not sure how to approach thi... | Green's theorem tell us that $$\int_C y^3 \, dx- x^3\, dy = \iint_{x^2+y^2 \leq 4} \left(\frac{\partial (-x^3)}{\partial x}\right)- \left(\frac{\partial y^3}{\partial y}\right) \,dx \, dy$$
While it is not a must to use polar coordinate, I strongly Iencourage you to do so as the form becomes elegant once you use polar ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to determine the range of the following function $\frac{x}{1+ |x|}$? How to determine the range of the following function $\frac{x}{1+ |x|}$?
when I calculated it, it was $\mathbb{R}$, but my professor said that the range is ]-1,1[, could anyone explain for me why?
thanks!
| Let $f(x)=\frac{x}{1+ |x|}$. Then: $|f(x)|=\frac{|x|}{1+ |x|} \le 1$, hence
$f( \mathbb R) \subseteq [-1,1]$.
Furthermore: $\lim_{x \to \infty}f(x)=1$ and $\lim_{x \to -\infty}f(x)=-1$.
Show that $f(x) \ne 1$ and $f(x) \ne -1$ for all $x$.
Are you now in a position to derive $f( \mathbb R) =]-1,1[$ ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Compute the sum fast. How can I compute the following sum in the fastest way possible?
$y = 1 + x + ... + {x}^{{n}^{3}}=\sum_{i = 0}^{n}{x}^{{i}^{3}}$
I wrote that $n^3 - (n-1)^3 = 3n^2-3n+1$, but so far it does not help a lot.
| This is a modified version of my answer here, which was the same question only with square exponents rather than cubed ones.
You have already calculated $x^{(n+1)^3} = x^{n^3}\cdot x^{3(n+1)^2-3(n+1)+1}$ (I shifted the indices by one to conform with my other answer). We also have $x^{3(n+1)^2-3(n+1)+1}=x^{3n^2-3n+1}\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Circle: finding locus of the chord's mid point Find the locus of the middle points of chords of the circle $$x^2+y^2 =a^2$$ which subtends a right angle at the point $(c, 0)$
|
Let $\Gamma$ be a circle centered at $O$ with radius $R$ and $P$ a point inside $\Gamma$. Let $ABCD$ a quadrilateral inscribed in $\Gamma$ with orthogonal diagonals $AC,BD$ meeting at $P$. By Thales' theorem the midpoints of $AB,BC,CD,DA$ are the vertices of a rectangle with sides parallel to $AC,BD$. Since $O$ is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2552047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Show that $\left|\int_{-n}^{n}e^{iy^2}dy\right|\le 2$ for $n\ge 5.$ Question is to show that
$$\left|\int_{-n}^{n}e^{iy^2}dy\right|\le 2$$
when $n\geq5$, $x \in \mathbb R $ and $i$ is an imaginary unit.
My effort:
$$|\int_{-n}^{n}e^{iy^2}dy|\leq \int_{-n}^{n}|e^{iy^2}|dy=\int_{-n}^{n}|\cos(y^2)+i\sin(y^2)|dy$$
$$ \l... | A Contour Integral Estimate
This estimate is valid for all $n\gt0$, and shows that the bound is $2$ for $n\ge4.5$ (since $\sqrt\pi\doteq1.77245385$). The contours in the complex plane are straight lines, and are parametrized linearly.
$$
\begin{align}
\left|\,\int_{-n}^ne^{iy^2}\,\mathrm{d}y\,\right|
&\le\left|\,\int_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2552190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
A matrix raised to a high power ($87$) So, I have this matrix: $$\pmatrix {0&0&0&-1\\1&0&0&0\\0&1&0&0\\0&0&1&0}^{87}$$
My teacher never discussed eigenvalues. So, I do not know what they are and there must be another way to do this (without multiplying the matrix $87$ times).
Thanks for your help.
| Note that your matrix acts like the permutation $\sigma=(2341)$, i.e. $1 \mapsto 2$, $2 \mapsto 3$, $3 \mapsto 4$, $4 \mapsto 1$ such that each time it spits a minus sign to the fourth, third, second and first column respectively.
The reason for this is that you can think of a matrix in the following way:
*
*The fir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2552463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Selecting Reference Orbit for Fractal Rendering with Perturbation Theory I have been trying to implement the perturbation algorithm for rendering fractals, as per this article: http://www.superfractalthing.co.nf/sft_maths.pdf
Let's say we simply focus on the first of the method, part up to equation 1, neglecting the se... | Not only do you need a reference whose iteration count is larger than all the other pixels, sometimes there are "glitches" when pixel dynamics differ significantly from the dynamics of the reference. These glitches can be detected in various ways: the most common heuristic is one developed by Pauldelbrot, though there... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2552605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $\sin x$ lies between $x-x^3/6$ and $x \;$ $\forall x \in R$
Show that $\sin x$ lies between $x-x^3/6$ and $x \;$ $\forall x \in R$
I am getting:
$$\sin(x) = x - \frac{x^3}{3!} + R_4(x)$$
where $R_4(x) = \frac{\cos(c)x^5}{5!}$ for some $c$ between $0$ and $x$
I want to prove $R_4(x)\geq 0$ to arrive at the... | The aim is to show that:
$$ x - \frac{x^3}{3!} \leq \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -\frac{x^7}{7!} + ...\leq x $$
For $sin x\leq x$ it suffices to use MVT:
$$\cos c =\frac{\sin x- \sin 0}{x-0}=\frac{\sin x}{x}\implies -1\leq \frac{\sin x}{x}\leq \implies \frac{\sin x}{x}\leq 1 \implies sin x\leq x$$
For ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2552775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find all complex numbers $z$ such that ($z^6 - i) \in \mathbb R$
Find all complex numbers $z$ such that ($z^6 - i) \in \mathbb R$
My solution:
Let's set $x^6 = (z - i)^6$. Then
$$x^6 = |x| e^{6\theta i} \\
x^6 \in \mathbb R \iff 6\theta = k\pi \land k\in \mathbb Z$$
$$\theta = \frac{k\pi}{6}$$
Therefore $z - i = |z... |
$z^6 = \cot \theta + i\\
z^6 = \csc \theta (\cos \theta + i\sin\theta)\\
z = (\csc \theta)^{\frac 16} e^{i(\frac {\theta}{6}+\frac {k\pi}{3})}\\$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2553022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Measurable set in real numbers with arbitrary lebesgue density at some point I'm not sure if this is easy or not, but i can't see the solution (or that it is wrong!)
Suppose that $\alpha \in (0,1)$ is given, Can you find a Lebesgue measurable set in $\mathbb{R}$, such that at point $0$, it has Lebesgue density $\alpha... | Here’s full justification for the answer provided by Kavi:
Suppose first that $t=\frac 1 n$, then $$|(-t,t)|=2\cdot(\sum \alpha\cdot(\frac 1 n-\frac{1}{n+1}))=2\alpha t,$$ hence $\frac{|E\cap (-t,t)|}{2t}=\alpha.$
Now suppose we’re given arbitrary $t\in (\frac 1 n,\frac{1}{n+1})$. Indeed,
$$\frac{2\cdot (\frac{1}{n+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2553302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove diagonal entries of positive definite matrices cannot be smaller than the eigenvalues The aim is to prove that the diagonal entries of a positive definite matrix cannot be smaller than any of the eigenvalues.
I know a positive definite matrix must have eigenvalues that are > 0, and that just because a matrix has... | Proof by contradiction:
We know that, if $\lambda$ is an eigenvalue of $A$, then $\lambda - p$ is an eigenvalue of $A-pI$.
So if $\lambda$ is an eigenvalue of $A$, then $\lambda - a{_i}{_i}$ is an eigenvalue of $A-a{_i}{_i}I$. Now, if $a{_i}{_i}$ is smaller than all the eigenvalues of $A$, then each $\lambda - a{_i}{_i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2553378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
Hello folks, how should I proceed to solve this differential equation through power series? The problem is:
$$
y'-y = x \\
y(0) = 0
$$
I know I have use the general form and its derivatives
$$
\sum a_n(x-x_0)^n
$$
My problem is with the alone $x$ variable on the right side. Could someone give me any tips?
Thanks in ... | $$y'-y = x $$
Power series may seem cool, but you can easily solve it by another way since it is a standard differential equation.
This is an Euler equation.
$$y'+P(x)y=Q(x)$$
$$\implies ye^{\int Pdx}=\int Qe^{\int Pdx}dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2553568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
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