Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Proof verification (sets & logic) Question:
Let $f$: $\mathbb{N} \rightarrow \mathbb{Z}$ be a function that is eventually zero. i.e: There exists some $N \in \mathbb{N}$ s.t $f(n)=0$ for all $n \geq \mathbb{N}$. Prove that the set of such functions is countable.
Proof:
Define $g$: $\mathbb{N} \rightarrow \mathbb{Z}$ s... | The basic idea is good, but you rather want to take the union of $G_N$'s where
$$G_N:=\{f:\Bbb N\to\Bbb Z:f(n)=0\text{ if } n>N\}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3494041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Is there a formal way for solving these series with |r| < 1 $$\sum _{n=1}^{\infty }\:\left(\frac{5}{6}\right)^{2n-2}=\sum \:_{n=1}^{\infty \:}\:\frac{\left(\frac{5}{6}\right)^{2n}}{\left(\frac{5}{6}\right)^2}=\frac{1}{\left(\frac{5}{6}\right)^2}\sum \:\:_{n=1}^{\infty \:\:}\:\left(\frac{5}{6}\right)^{2n}=\frac{1}{\left... | The formal way of solving this problem is to map it to a geometric progression:
$$ \sum_{n=m}^{\infty} ar^{cn-b}=\sum_{n-m=0}^{\infty} ar^{-b} r^{cm}r^{c(n-m)} = ar^{-b} r^{cm} \sum_{i=0}^{\infty}\left( r^c\right)^{i} = \frac{ar^{cm-b}}{1-r^c} .$$
Here, we first use a change of variable $(n,m)\to(i=n-m,m)$. Then, we re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3494285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$L \vert K$ normal extension with $K \subset M \subset L$. Show that $L \vert M$ is normal Let $L \vert K$ be a field extension and let $K \subset M \subset L$ be a subextension. Show that if $L \vert K$ is normal, so is $L \vert M$.
So at this point I'm asking myself if $L \vert K$ must be a finite extension.
If not, ... | Let $p$ be an irreducible element of $M[X]$ suppose that $P$ has a root $u\in L$, since $L|K$ is algebraic ? $K(u)|K$ is finite and denote by $f$ the minimal polynomial of $u$ over $K$, since $L|K$ is normal, all the roots of $f$ are in $L$ and remark that $p$ divides $f$ since $p$ is irreducible, we deduce that all th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3494382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sequence of measurable $\&$ continuous functions defined on $[0,1]$ Let $\{f_n\}$ be a sequence of measurable $\&$ continuous functions from $[0,1]$ to $[0,1]$. Assume $f_n \rightarrow f$ pointwise. Is it true/false that,
*
*$f$ is Riemann integrable $\& \int _{[0,1]}f_n \rightarrow \int_{[0,1]}f$?
*$f$ is Lebesg... | $$
\text{Let } f_n(x) = \begin{cases} 2n\left(1-|2nx-1| \right) & \text{if } 0\le x\le1/n, \\[6pt] {} 0 & \text{otherwise.} \end{cases}
$$
Then
$$
\lim_{n\to\infty} \int\limits_{[0,1]} f_n(x)\,dx = 1 \ne 0 = \int\limits_{[0,1]} \lim_{n\to\infty} f_n(x)\,dx.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3494545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is $x^2 \geq \alpha(\alpha-1)$?
If $\alpha$ is a nonnegative real and $x$ is a real satisfying $(x+1)^2\geq \alpha(\alpha+1),$ is $x^2 \geq \alpha(\alpha-1)$?
The answer is yes. Consider two cases: $1) \, x < -1$ and $2)\, x\geq -1.$
In case $1,$ taking the square root of both sides of the inequality gives $-(x+1) \... | Here is one way by contraposition that has some advantage in not requiring cases. We can assume $\alpha > 1$ otherwise $\alpha(\alpha-1) \leq 0 \leq x^2$. (This is so we can take the square root.)
Suppose $x^2 < \alpha(\alpha-1)$, so
$$ (x + 1)^2 = x^2 + 2x + 1 < \alpha(\alpha - 1) + 2\sqrt{\alpha(\alpha-1)} + 1. $$
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3494612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Estimation of the $L^2$ norm of a function given the function $$g(x,t)=\frac{e^{\frac{Y}{2}}}{2 \epsilon [1+e^{\frac{Y}{2}}]^3}$$ with $Y=\frac{x-\frac{t}{2}}{\epsilon}$ where $x\in \mathbb{R}$ and $t>0$.
I'm looking for $C_{\epsilon}>0$ where $$||g(.,t)||_2\leq C_{\epsilon}$$
where $C_{\epsilon} \to 0 $ as $\epsilon \... | Integration-by-parts results in
$$\int\frac{z}{(1+z)^6}\text{d}z = - \frac{z}{5(1+z)^5} + \frac{1}{5}\int\frac{1}{(1+z)^5}\text{d}z,$$
where the second integral is straightforward,
$$\int\frac{1}{(1+z)^5}\text{d}z = -\frac{1}{4(1+z)^4}.$$
Taking limits, the improper integral is
$$\int_{0}^{+\infty}\frac{z}{(1+z)^6}\tex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3494886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Election between $2$ candidates ends in a tie: probability one candidate leads until the penultimate vote
Assume there are two candidate $C_1$ and $C_2$. At the end of the election both candidates receive the same amount of votes. What is the probability $P$ that candidate $C_1$ leads during the whole election process... | Let's shift gears and use the ballot theorem instead of reinventing it for the case of ties. Other than the one appeal to the ballot theorem, this will be a conditional probability problem. ^_^
Our sample space will be all cases where each candidate received $p$ votes. Let $A$ be the event that $C_1$ lead all the wa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3495031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to calculate the following limits: $\lim_{x\to0}\frac{\ln\left(\cosh\left(x\right)\right)}{\ln\left(\cos\left(x\right)\right)}$
How to calculate the following limits:
*
*$\lim\limits_{x\to0}\frac{\ln\left(\cosh\left(x\right)\right)}{\ln\left(\cos\left(x\right)\right)}$
*$\lim\limits_{n\to\infty}\sin\le... | Do you like an insane overkill? By the Weierstrass product for the cosine function
$$ \cos(x)=\prod_{n\geq 0}\left(1-\frac{4x^2}{\pi^2(2n+1)^2}\right) $$
we have
$$ \cosh(x)=\prod_{n\geq 0}\left(1+\frac{4x^2}{\pi^2(2n+1)^2}\right) $$
hence
$$ \frac{\log\cosh(x)}{\log\cos(x)}=\frac{\sum_{n\geq 0}\log\left(1-\frac{4x^2}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3495156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
} |
Equation for $x^2+y^2=1$ in exponential space? Consider the unit circle $x^2+y^2=1$ in exponential space. Is there an equation for the set of points under the mapping?
Here's what I tried:
I took each coordinate pair on the unit circle and exponentiated it.
For example, $(x,y)\mapsto(e^x,e^y).$
This is what I have so ... | The equation for the set of points under the prescribed mapping is $\ln(x)^2+\ln(y)^2=1.$
Start with $(x,y) \mapsto (e^x,e^y).$ Define $u=\ln(x)$ and $v=\ln(y).$ In the pre-image space, we simply have $x^2+y^2=1.$ In the image space (after the prescribed mapping) we get $u^2+v^2=1.$ Substituting back to get an explicit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3495371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the group of group-like elements of a quantum group? A quantum group is not a group.
For example, the Drinfeld-Jimbo "quantum doubles" are Hopf algebras obtained by deforming the universal enveloping algebras of Lie algebras.
But in every Hopf algebra, there's a subset of group-like elements that satisfy
$$ \De... | I do not have a complete answer to the question. But the group of group-like elements of a quantum group might be deformed somehow.
I'm refering to Lemma 6.4.1 of V.Chari, A.Pressley A guide to quantum groups.
In the case of $\mathfrak{sl_2}$, if you take the h-adic version of quantum $\mathfrak{sl_2}$, $U_h(\mathfrak{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3495483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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'imperfect' numbers A perfect number (integer) is equal to the sum of its divisors, including 1, and excluding itself. This has been around since Euclid.
Recently, I noticed that at least for the initial integers, it is more common for that sum of divisors to be smaller than the number in question.
However, for example... | The sum of all proper divisors of a number is less than, equal to, or greater than the number,
according as the number is deficient, perfect, or abundant.
The first 28 abundant numbers are $12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60,$
$66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104,$ $ 108, 112, 114, $ and $12... | {
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Linear transformation and dependence of vectors I couldn't find an example or explanation why the following sentence is correct
If a Transformation is linear, and vectors $u_1$,$u_2$,$u_3$ are dependent then
$T(u_1)$,$T(u_2)$,$T(u_3)$
must also be dependent
but
If a Transformation is linear, and $T(u_1)$,$T(u_2)$,$T... | Let $T$ be the null function and take any $3$ linear independent vectors $u_1$, $u_2$, and $u_3$. Then $\bigl\{T(u_1),T(u_2),T(u_3)\bigr\}$ is linearly dependent (since it is equal to $\{0\}$).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$1+\frac{1}{2}+...+\frac{1}{x_n}\geq n$. Determine $\lim\limits_{n\to\infty} \frac{x_{n+1}}{x_n}$ For every $n \in \mathbb{N}$ let $x_n$ be the smallest natural number such that:
$$1+\frac{1}{2}+...+\frac{1}{x_n}\geq n$$
Determine $\lim\limits_{n\to\infty} \frac{x_{n+1}}{x_n}$
It's the harmonic series but I can't figur... | Here's a slightly different approach to the first answer that provides a more precise asymptotic:
I'll write $H_n = 1 + 1/2+ \ldots + 1/n$. It is known that
$$H_n = \log n + \gamma + o(1)$$
where $\gamma$ is the Euler-Mascheroni constant. There therefore exists two sequences $\delta_n, \epsilon_n =o(1)$ such that
$$\lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3495970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the integral by first principle In my math course homework, I encountered this problem:
Find $$\frac{d}{dx}\int^{x^2}_0\frac{dt}{1+e^{t^2}}$$ by first principle
$$\begin{split}\frac{d}{dx}\int^{x^2}_0\frac{dt}{1+e^{t^2}}&=\lim_{h\rightarrow0}\frac{\int^{(x+h)^2}_0\frac{dt}{1+e^{t^2}}-\int^{x^2}_0\frac{dt}{1+e^{t... | It might be easier to look a little more abstractly and then apply to your specific case.
Suppose $f$ is continuous and $g$ is differentiable.
Let $\phi(x) = \int_0^{g(x)} f(t) dt$. The fundamental theorem/Leibniz rule tells us that
$\phi'(x) = f(g(x)) g'(x)$.
So we want to show that $\lim_{h \to 0} |{ \phi(x+h)-\phi(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3496096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Degree $1$ birational map of $\mathbb{P}_k^n$ I wonder why the degree $1$ birational maps of $\mathbb{P}_k^n$ are the automorphisms of $\mathbb{P}_k^n$? In particular, why are they defined everywhere on $\mathbb{P}_k^n$?
I know each degree $1$ birational maps of $\mathbb{P}_k^n$ is of the form $\phi:=[f_0:...:f_n]$, wh... | The key to this is the matrix of linear forms. If the matrix is not full rank, then the image of the map associated to this matrix of linear forms is supported inside some closed linear subvariety, which implies it is not birational. So birationality is equivalent to the matrix being full rank, which is equivalent to i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3496199",
"timestamp": "2023-03-29T00:00:00",
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Solving a system of equations with proof: I can't figure out what exactly the question is asking me.
Question
There is a system such as follows:
$\dot{x}=-2x$, $\dot{y}=y$, given that $0<\eta<1$ and $(x_{\eta}(t),y_{\eta}(t))^T$ is the solution of the system with initial conditions $x(0)=1$ and $y(0)=\eta$.
Prove th... | Yes, the general solution is $x(t)=c_1e^{-2t},y(t)=c_2e^t$ where $c_1,c_2$ are constants.
You now apply the initial conditions to get the particular solution $x_0(t),y_0(t)$.
So we have $x_0(0)=1$ and hence $c_1=1$. Similarly, $y_0(0)=\eta$, so $c_2=\eta$.
We want to find $\tau$ so that $y_0(\tau)=1$. That implies that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3496318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the supremum of $A=\{\cos{n},n\in\Bbb{N}\}$ I would like to find the supremum of the following set:
$$A=\{\cos{n},n\in\Bbb{N}\}$$
Is it $1$ or something smaller?
One way to show that the supremum is $1$ (If it were), is to prove that for any $\epsilon>0$, there are positive integers $p,q$ such as $\vert2\pi q-p... | Well the answer would be $1$ if it were true that for any $\epsilon > 0$ we can find $n_\epsilon$ so that $|\cos n_\epsilon -1| < \epsilon$.
Now each natural $n$ is congruent $\pmod {2\pi}$ to some value $n': -\pi \le n' < \pi$, or in other words $n = n' + 2k\pi$ for some integer $k$. And we want to show that $n'$ can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3496470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Conditional probability: The second ball is black. Find the probability that the first ball was white. Two black balls and three white balls are put in a bag. First, we pull one ball, then the second. The second ball is black. Find the probability that the first ball was white.
I think it could be $\frac{3*2}{3*2+2*1} ... | Your answer is correct
Another way of getting the same answer is to say that, given the second draw without replacement is white, there are three black balls and one white ball for the non-second draws, and these are equally likely to be any position making the conditional probability that the first draw is black $\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3496647",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Evaluating $\int_{-e}^e(x+x^3+x^5+\cdots+x^{999}) dx$
Integrate
$$\int_{-e}^e(x+x^3+x^5+\cdots+x^{999}) dx$$
I converted the integral into geometric sum, but i cannot proceed from there
The geometric sum is
$$\frac{x(x^{1001}-1)}{x^2-1}$$
| Your original functions is and "odd function" (https://en.wikipedia.org/wiki/Even_and_odd_functions#Odd_functions) such that $f(-x) = -f(x)$. When odd functions are integrated over intervals symmetric over the origin, like $[-1, 1]$ or in your case $[-e,e]$, they cancel out and integrate to zero.
Simple example: $\int_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3496795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inequality about the sum of square root of 1 over x This question originates from an adversarial bandit problem. In the proof of self-bounding property of regret, there is a step showing
$$C\sum_{t=1}^T\sum_{i\neq i^*}\sqrt{\frac{x_{i,t}}{t}}\leq 2C\sqrt{KT}$$
where $\sum_{i\neq i^*}$ means summing over all $K$ arms e... | The best way I now know to see the inequality about $t$ is to upbound the sum by integral:
$$\sum_{t=1}^T\sqrt{\frac{1}{t}}\leq \int_0^T\sqrt{\frac{1}{t}} dt= 2\sqrt{T}$$
Of course, the upboundedness is true because $\sqrt{\frac{1}{t}}$ is monotonically decreasing.
A lesson I took from it was while we can approximate ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3496893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determine if the graph is Hamiltonian graph I am supposed to determine if the following graph is a Hamiltonian graph:
$G=(V,E)$ where $V=\{1,2,...,13\}$ and the set of edges is defined as follow: $E=\{(i,j)\in V\times V|11\leq i + j \leq 15\}$.
My Attempt:
I draw it and I found out, that it does not contain bridge o... | The graph is Hamiltonian, here are two Hamilton cycles:
$(7,8,5,10,3,12,1,13,2,11,4,9,6,7)$
$(7,8,5,10,3,12,2,13,1,11,4,9,6,7)$
I constructed this cycle starting from the vertex $13$ whose neighbors are only the vertex $1,2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3497037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that for all acute triangles $\triangle ABC$, $r_a + r_b + r_c \ge m_a + m_b + m_c$.
Let $r_b$ and $m_b$ respectively be the exradius of the excircle opposite $B$ and the median drawn from the midpoint of side $CA$ of acute triangles $\triangle ABC$. Prove that $$\large r_a + r_b + r_c \ge m_a + m_b + m_c$$
We ... | Your inequality is true for any triangle!
Let $a=y+z$, $b=x+z$ and $c=x+y$.
Thus, $x$, $y$ and $z$ are positives and in the standard notation we need to prove that:
$$\sum_{cyc}\frac{2S}{b+c-a}\geq\frac{1}{2}\sum_{cyc}\sqrt{2b^2+2c^2-a^2}$$ or
$$\sum_{cyc}\frac{2\sqrt{xyz(x+y+z)}}{2x}\geq\frac{1}{2}\sum_{cyc}\sqrt{4x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3497183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Example of weak convergence I'd like to understand better the concept of weak convergence.
I know that a sequence of probability measures $\mu_n$ converges weakly to $\mu$ if $\int{f d\mu_n}$ converges to $\int{f d\mu}$ for each $f$ which is continuous and bounded.
Could you please give me an example of a sequence of ... | Let $P(X_n=1/n)=1$, let $P(X=0)=1$. The probability distribution of $X_n$ is $\mu_n=\delta_{1/n}$, the point-mass measure concentrated at $1/n$, that of $X$ is the point mass at $0$, namely $\mu=\delta_0$.
We can check that $\mu_n$ converges to $\mu$ weakly: since $\int f d\mu_n = f(1/n)$ and $\int f d\mu=f(0)$, for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3497341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determine Sum of $\sum_{k=3}^{25}{k^2-5k+9}$ Given $\sum_{k=3}^{28}(k-3)^2 = 14,910$ and $\sum_{k=0}^{25}k = 325$ I am working through some questions from a textbook which states that I should determine the sum without expanding or calculating any sums. I have been given the following information.
$\sum_{k=3}^{28}(k-3)... | We can manually subtract terms from each of the two sums to get the desired range:
$$\sum_{k=3}^{25}(k-3)^2=14910-25^2-24^2-23^2=13180$$
$$\sum_{k=3}^{25}k=325-0-1-2=322$$
The final answer is $13180+322=13502$.
In fact, the given values are completely wrong; the actual correct answer is $4117$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3497491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Union of intersections and intersection of unions Would someone be able to help me understand the (concept of) "union of intersections" and the "intersection of unions".
More specifically, how to approach (prove) the following equality of two sets $X$ and $Y$;
$$
X=\bigcup_{n=1}^{\infty}\bigcap^\infty_{j=n} A_{j} \text... | As pointed out in the comments, the sets are not necessarily equal. Consider the following counterexample :
For even $n$, define $A_n = \{0\}$ and for odd $n$, define $A_n = \{1\}$.
That is, $A_1 = A_3 = A_5 = \cdots = \{1\}$ and $A_2 = A_4 = A_6 = \cdots = \{0\}$.
Let us look at the set $X$ first.
Note that for any gi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3497611",
"timestamp": "2023-03-29T00:00:00",
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Convergence of $\int_{0}^{\pi^{2}}\frac{1}{1-\cos(\sqrt{x})}dx$ I need to compare this integral $\int_{0}^{\pi^{2}}\frac{1}{1-\cos(\sqrt{x})}dx$ to another one in order to ensure its convergence, but I can't find the proper one, because $\frac{1}{1-\sqrt{x}}$ is not continuous.
| @SL_MathGuy already showed that the definite integral diverges.
Using the tangent half-angle substitution and one integration by parts, the antiderivative is given by
$$J(x)=\int\frac{dx}{1-\cos \left(\sqrt{x}\right)}=4 \log \left(\sin \left(\frac{\sqrt{x}}{2}\right)\right)-2 \sqrt{x} \cot \left(\frac{\sqrt{x}}{2}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3497732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
How to solve $|1+1/x| > 2$ I am having a problem with this question:
Find all real $x$ that satisfies $|1 + 1/x| > 2$.
This is clearly not defined in $x = 0$.
By my logic, it should be solved with:
$1+1/x > 2$ or $1+1/x < -2$
But the result I am getting from this is wrong. ($x<1$)
Correct result is
$-1/3 < x <... | Alternatively, $x\ne 0$ and:
$$|x+1|>2|x| \iff (x+1)^2>(2x)^2 \iff (x-1)(3x+1)<0 \iff -1/3<x<1$$
Hence:
$$x\in (-1/3,0)\cup (0,1)$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 7
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How to prove that if $G$ is a group with $|G|= p^k*q$ with $p,q$ distinct primes, that $G$ has a subgroup of order $p^k$ I have to prove with induction on $k$ that if $G$ is a group with $|G|= p^k*q$ with $p,q$ distinct primes, that $G$ has a subgroup of order $p^k$. I already know that if $p$ is not a divisor of the c... | For the induction step: suppose $G$ has order $p^k q$. If $p$ does not divide the order of the center of $G$, then you're done. Else, by your second lemma, $Z(G)$ has an element of order $p$. Call the cyclic subgroup generated by this element $P$. Then $P$ is a normal (in fact, central) subgroup of $G$. The quotient $G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3498078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Pointwise convergence of probability densities imply weak convergence of probability measures Assume $P_n, n\in\mathbb{N}$ and $P$ are absolutely continuous probability measures with respect to a sigma finite measure $\mu$ on $(\mathbb{R},\mathcal{B})$. Let $f_n, n\in \mathbb{N}$ and $f$ be the densities of above measu... | You can apply the lemma of Scheffé.
Observe that: $$\int\left(f-f_{n}\right)d\mu=\int fd\mu-\int f_{n}d\mu=1-1=0$$
implying that: $$\int\left(f-f_{n}\right)^{+}d\mu=\int\left(f-f_{n}\right)^{-}d\mu$$
and consequently: $$\int\left|f-f_{n}\right|d\mu=2\int\left(f-f_{n}\right)^{+}d\mu$$
Then for every measurable set $A$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3498228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$ABCD$ is a trapezium, $MN=\dfrac{AB-CD}{2}$ where $M,N$ are the midpoints of $AB,CD$, respectively; show $\angle BAD+\angle ABC=90^\circ$
$ABCD$ is a trapezium in which $AB$ is parallel to $CD$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $CD$. If $MN=\dfrac{AB-CD}{2}$, I should show that $\angle BAD+\ang... |
Extend $AD$ and $BC$ to meet at $P$. We are given that $m=x-y$. Since $CD\parallel AB$, $\triangle PDN\sim\triangle PAM$ and thus
$$\frac yn=\frac x{n+m}=\frac x{n+x-y}\\nx=y(n+x-y)=yn+xy-y^2\\nx-xy=yn-y^2\\x(n-y)=y(n-y)\\x=y\quad\text{or}\quad n-y=0$$
Since generally $x\neq y$, we conclude that $n-y=0$. Thus $y=n$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Map explicitly a simply connected region with at least two boundary points to unit disk In an old Qualifying exam there is an exercise that says:
"Let $G$ be a simply connected region of $\mathbb{C}$ such that the boundary $\partial G$ has at least two points $a\neq b$. Construct explicitly, in function of $a$ and $b$,... | By the assumption that $G$ is a simply connected region of $\mathbb{C}$ such that the boundary $\partial G$ has at least two points $a\ne b$, we may take a path $\Gamma$: $z=\gamma(t)$ $(0\le t<1)$ with $\gamma(0)=a\, (\text{or }b)$, $\gamma(1/2)=b\, (\text{or }a)$, and $\lim_{t\to 1} \gamma(t)=\infty.$
Let $H=\mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3498543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Local coordinate frame there is this problem that keeps bothering me.
Consider the following local frame of the tangent space of $\mathbb{R}^2\setminus \{0\}$:
$$ E_1 = \frac{x}{\sqrt{x^2+y^2}} \partial_x + \frac{y}{\sqrt{x^2+y^2}}\partial_y, E_2 = \frac{-y}{\sqrt{x^2+y^2}}\partial_x + \frac{x}{\sqrt{x^2+y^2}}\partia... | I'm not quite sure why you have the negative signs, but in both cases, you're off by a factor of $r$. $-df_1 = \dfrac{dr}r$ and $-df_2=d\theta$, so the dual vector fields are $r\dfrac{\partial}{\partial r}$ and $\dfrac{\partial}{\partial\theta}$, but $E_1=\dfrac{\partial}{\partial r}$ and $E_2=\dfrac1r\dfrac{\partial}{... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$X^p + Y^p \equiv Z^p \pmod q$ Is it possible that although fermat's last theorem implies $x^p + y^p = z^p$ has no solution in integers $x,y,z$ and $p\geq3$, for some $q>p, x^p + y^p \equiv z^p \pmod q$. Like maybe $x^p \equiv 2 \pmod q$, $y^p$ is $1 \pmod q$ and $z^p \equiv 3 \pmod q$?
| A. E. Pellet, Mémoire sur la théorie algébrique des équations, Bull. Soc. Math. France 15 (1887) 61-102, showed that for every prime $p$ there is a number $q_0(p)$ such that if $q$ is prime and $q\ge q_0(p)$ then $x^p+y^p+z^p\equiv0\bmod q$ has nontrivial solutions. (Later authors found explicit values for $q_0(p)$, an... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Negligible function I'm studying about complexity and reaching negligible. Can anyone tell me if $f(x)$ is a negligible function and $p(x) \in \mathbb{R}$ then $p(x) . f(x)$ is a negligible function?
| Let $f'(x) = f(x) \cdot p(x)$
*
*f(x) is negligible means that it is smaller than the inverse of any polynomial, for all sufficiently large n. In your case, given any polynomial $q(x)$, it is smaller than $1/p(x)q(x)$
*By using the limit definition of negligibility.
$f(x)$ is negligible than for every polynomial $q... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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When does multiplication by an orthogonal matrix preserves the eigenvalues? Let $A$ be a real $n \times n$ matrix, with rank $\ge n-1$.
Suppose that the eigenvalues (counted with multiplicities) of $A$ are the same as the eigenvalues of $QA$ for some orthogonal matrix $Q$. Must $Q$ be diagonal?
The condition $\text{... | No. Consider
$$
R=\pmatrix{0&1\\ 1&0},\ Q=\pmatrix{R\\ &R},\ A=\pmatrix{R\\ &I},\ QA=\pmatrix{I\\ &R}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3499051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Evaluate $\int_{0}^{1}4x^3\left(\frac{d^2}{dx^2}(1-x^2)x^5\right)dx$ $$\int_{0}^{1}4x^3\left(\dfrac{d^2}{dx^2}(1-x^2)x^5\right)dx$$
$$\int_{0}^{1}4x^3\left(\dfrac{d}{dx}(5x^4-7x^6)\right)dx$$
$$\int_{0}^{1}4x^3\left(20x^3-42x^5\right)dx$$
$$8\int_{0}^{1}10x^6-21x^8 dx$$
$$8\left(\dfrac{10x^7}{7}-\dfrac{21x^9}{9}\right)... | Although you've gotten to the point of knowing that the "actual answer" you've been given is wrong, you might want to consider a different approach to the problem -- integration by parts. You have an integral of the form
$$
\int_0^1 f(x) g'(x) ~ dx
$$
(where $g(x) = ((1-x^2)x^5)'$), so you can convert it, with "parts"... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Application WLLN in a convergence related problem So, I have stumbled upon a question like this:
${X_n}_{\{n\geq1\}}$ be a sequence of iid random variables having common pdf $f_X(x)=xe^{-x}I_{x>0}$. Define $\overline{X_n}=\frac{1}{n}\sum_{i=1}^{n}{X_i},n=1,2,...$. Then $\lim{n\to \infty}P(\overline{X_n}=2)=$
*
*$0... | We say that a sequence $(X_n)_n$ of random variables converges in probability to the random variable $X$ (notation: $X_n \stackrel{\mathbb{P}}{\to} X)$ if
$$\forall \epsilon > 0: \lim_n\mathbb{P}(|X_n-X| \geq \epsilon)=0$$
If $\overline{X_n} = 2$ with probability $1$ then for all $\epsilon > 0$ we have $\mathbb{P}(|\o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $a^{\cos(b)^2}-b^{\cos(a)^2}\leq \frac{4}{\pi}(1-\frac{\pi}{2})b+\frac{\pi}{2}-1$ and the reverse inequality . It'a problem of my own :
Let $a\geq b>0$ such that $a+b=\frac{\pi}{2}$ then we have :
$$a^{\cos(b)^2}-b^{\cos(a)^2}\leq \frac{4}{\pi}(1-\frac{\pi}{2})b+\frac{\pi}{2}-1$$
Let $b\geq a>0$ such th... | put $x=\dfrac{\pi}{4}+a$
$h(a)=(\frac{\pi}{4}-a)^{\frac{1}{4}(\cos{a}-\sin{a})}+(\frac{\pi}{4}+a)^{\frac{1}{4}(\sin{a}+\cos{a})}$
$=i(a)+i(-a)$
$i''(x)$ is increasing in the area $(-\frac{\pi}{4},0)$ and $i'(0)\geq-1$, h(x) is the shape like which add two parabola.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Mathematical equivalent of a one-dimensional array as seen in programming Suppose some kind of structure to hold a (finite) number of elements is needed. The order of the elements matters and duplicate elements are allowed. When programming one would use a one-dimensional array for this.
What mathematical structure com... | A vector is the wrong object to consider. While vectors (in $\mathbb{R}^d$) can be represented as ordered lists, calling something a vector implies that it comes from a space with a great deal more structure. Specifically, vectors can be added together, and can be scaled by field elements (in the case of vectors in $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3499672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sampling from a distribution with given pdf Given a continuous multivariate pdf in analytical form (i.e. in function form), how can one sample from the corresponding distribution? In other words, what are the ways of coming up with random (or psuedo-random) realizations from the distribution with probability matching t... | This is a broad topic. Here are a few examples to initiate some thinking on your part.
(1) To generate a random sample $x$ from a univariate distribution with CDF $F$, first generate a uniformly distributed random number $u \sim U(0,1)$ and take $x = F^{-1}(u)$. Note that $x$ has the desired distribution since
$$\m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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a question about the surface area of two Octagonal pyramids A friend of mine came to me with a problem earlier today and I'm unsure if the answer I arrived at was correct. I simply found a formula on wolfram and used the provided values to work backwards to a solution. I know there has to be an easier way to solve this... | I think there is a problem with this question, as the given area for pyramid A is incorrect (assuming a right pyramid, i.e. the apex above the centre of a regular octagon). Nevertheless, I'll solve this problem using that incorrect figure using as few calculations as possible.
In pyramid A, the octagonal base can be sp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3500018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove by limit definition $\lim _{x\to \infty }\left(\frac{-7x^2+9x}{4x^2+8}\right)=\frac{-7}{4}$ Prove by limit definition
$$\lim _{x\to \infty }\left(\frac{-7x^2+9x}{4x^2+8}\right)=\frac{-7}{4}$$
let $\epsilon > 0$ need to find $M$ such that for every $x>M \implies |f(x) - L|<\epsilon$
$\left|\frac{-7x^2+9x}{4x^2+... | Look for example at pp. 105, Calculus (Third Edition) from Spivak. The definition of $\lim_{x\to\infty} f(x)=L$ is that for every $\varepsilon>0$ there is a number $N$ such that, for all $x$, if $x>N$, then $|f(x)-L|<\varepsilon$. According to this, what you have done proves the limit, but I think that in your attempt ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3500163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving the existence of a line that passes only through two points So I came across this question in a Math Olympiad Book:
Consider a finite set $S$ of points in a plane which are not all collinear. Show that there is a line in the plane which passes only through two points in $S$.
$$S = \{A_1, A_2,...,A_n\}$$
Since... | There is a simple proof without going into any deep knowledge of Mathematics. Just a little common sense and you are done.
If $S$ be such a finite set then area enclosed by all points is finite. If that happens then we can construct an imaginary boundary around the points. Outside this field we draw a straight line , a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the value of $E(\frac{1}{2018})+E(\frac{2}{2018})+\dots+E(\frac{2017}{2018})$ Let $E(n) = \dfrac{4^n}{4^n+2}$.
If the value of $E(\frac{1}{2018})+E(\frac{2}{2018})+\dots+E(\frac{2017}{2018})=\frac{a}{b}$ (in lowest terms), find $b$.
I tried solving this question using telescopic sum but was unable to solve
| Let $\displaystyle f(x)=\frac{4^x}{4^x+2},$ Then $\displaystyle f(1-x)=\frac{4^{1-x}}{4^{1-x}+2}=\frac{2}{4^x+2}$
So $$f(x)+f(1-x)=\frac{4^x+2}{4^x+2}=1.$$
So Put $\displaystyle x=\frac{1}{2018},\frac{2}{2018},\frac{3}{2018},\cdots \cdots ,\frac{1008}{2018}$
So $$f\bigg(\frac{1}{2018}\bigg)+f\bigg(\frac{2}{2018}\bigg)+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3500411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Probability of Independent Normal Random Variables There are two Random Variables X and Y. Both of them are Normally distributed with 0 mean and variance a and b respectively. X and Y are independent of each other.
What is the probability of
What should be the easiest way to do this?
| The joint probability distribution of $X$ and $Y$ is $\frac{1}{2\pi \sqrt{ab}}e^{\frac{-1}{2} \left(\frac{x^2}{a^2} + \frac{y^2}{b^2} \right)}$.
Hence, we are looking for the value of the following integral.
\begin{align*}
\frac{1}{2 \pi \sqrt{ab}} \iint_{x+y>0, y>0}e^{\frac{-1}{2} \left(\frac{x^2}{a^2} + \frac{y^2}{b^... | {
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"timestamp": "2023-03-29T00:00:00",
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Find all squarefree integers whose divisors $d_1 < d_2 < · · · < d_k$ satisfy $d_i − d_{i−1}|n$ for all $2 \leq i \leq k$. Disclaimer: this problem came from USAMTS: https://www.usamts.org/Tests/Problems_31_3.pdf
The contest has ended., in case there is any doubt.
Problem: Find all squarefree integers whose divisors... | We must have $d_1=1.$ So $d_2-1$ must be divisor less than $d_2$ so $d_2-1=1,$ or $d_2=2.$
In general, if $p$ is a prime divisor of $n$ then let $d$ be the previous divisor. Then $p\not\mid d$ and $p-d$ must be a divisor of $n$ and likewise $\frac{n}{d}-\frac{n}{p}=\frac{(p-d)n}{pd}=(p-d)\frac{n}{pd}$ must be a divisor... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Understanding the least squares regression formula?
I've seen the following tutorial on it, but the formula itself had not been explained (https://www.youtube.com/watch?v=Qa2APhWjQPc).
I understanding the intuition behind finding a line that "best fits" the data set where the error is minimised (image below).
However... | very concisely:
*
*if the points were all on a straight line, then you would like that to be the regression line, isn't it ?
*if now you translate rigidly the linear cloud (no rotation), you would like the regression line to translate in the same way;
*the regression line will contain all the cloud point... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3500898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Use Fubini's theorem to reverse the order of the double integral
*
*Use Fubini's theorem to reverse the order of the double integral (the reversed integral may split into a sum of multiple pieces):
a) $\int_0 ^4 \int_0 ^{\sqrt{x}} f(x,y) dy dx$
b) $\int_0 ^2 \int_x ^{3} f(x,y) dy dx$
c) $\int_{-1} ^2 \int_0 ^{1-y^2... | The graph and working for (a) are correct. The graph for (b) is wrong and should look like
The reversed integral is $\int_0^3\int_0^{\min(2,y)}f(x,y)\,dx\,dy$.
For (c) the graph looks like this:
where the negative sign indicates a reversed orientation. This integral needs to be split when reversed, and the result is
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3501001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is the Kronecker symbol $(n/2)$ conventionally defined as it is? Admittedly this may be an extremely naive question, but I am simply puzzled about the motivation behind choosing this function to be one of the Dirichlet characters modulo $8$, rather than the Dirichlet character modulo $2$ (or, indeed, any other char... | The reason is quadratic residues and Legendre's
symbol $\,\big(\frac{a}{p}\big).\,$ In particular the
Legendre symbol $\,\big(\frac2{p}\big).\,$ It can be
proved that $2$ is a quadratic residue of primes
of the form $\,8n+1\,$ or $\,8n+7\,$ and a quadratic
nonresidue of primes of the form $\,8n+3\,$ or
$\,8n+5.\,$ This... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3501439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Verify $\cos(x)=\frac{1-t^2}{1+t^2}$ with $t=\tan(\frac{x}{2})$ I was requested to verify, with $t=\tan(\frac{x}{2})$, the following identity:
$$\cos(x)=\frac{1-t^2}{1+t^2}$$
I'm quite rusty on my trigonometry, and hasn't been able to found the proof of this. I'm sure there may be some trigonometric property I should k... | $$\cos(x)=\cos^2(x/2)-\sin^2(x/2)=(1-t^2)/(\sec^2(x/2))=(1-t^2)/(1+t^2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3501607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Inequality Solution Correctness Let $a,b,c \in \Re $ and $ 0 < a < 1 , 0 < b < 1 , 0 < c < 1 $ & $ \sum_{cyc} a = 2$
Prove that
$$ \prod_{cyc}\frac{a}{1-a} \ge 8$$
My solution
$$ \prod_{cyc}\frac{a}{1-a} \ge 8$$
or
$$ \prod_{cyc}a \ge 8\prod_{cyc}(1-a)$$
From $ \sum_{cyc} a = 2$ we can conclude $\prod_{cyc}a \le ... | As shown by Michael Rozenberg, the proof given is not correct. Here is another AGM approach, using
$$
\sum_{k=1}^n\frac1{p_k}=1\implies\sum_{k=1}^nx_k\ge\prod_{k=1}^n(p_kx_k)^{1/p_k}\quad\text{where}\quad x_k,p_k\gt0\tag1
$$
Suppose $x+y+z=1$, then $(1)$ says that
$$
\begin{align}
&(1-x)(1-y)(1-z)\\
&=1-(x+y+z)+(xy+yz+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3501708",
"timestamp": "2023-03-29T00:00:00",
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Using the equation $\mathbf{A} \cdot \mathbf{B} = AB\cos(\theta)$ to show that the dot product is distributive when the three vectors are coplanar. I am asked to use the equation $\mathbf{A} \cdot \mathbf{B} = AB\cos(\theta)$ to show that the dot product is distributive when the three vectors are coplanar. It seems tha... | Normally we'd prove it by showing $A\cdot B=\sum_iA_iB_i$. You could try an alternative using $(B+C)^2=B^2+C^2-2BC\cos\phi$ with $\phi$ the angle between $B,\,C$, but I doubt that'll help much. Since $AB\cos\theta$ and $\sum_iA_iB_i=(A^TB)_{11}$ are both invariant under rotations of the plane (viz. $A\mapsto RA,\,B\map... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove a finite element space is not conforming Let $\tau=[-1,1]^2$, consider the finite element $\left(\tau, Q_{1}, \Sigma\right)$, $Q=span\{1,x,y,x^2−y^2 \},$ $\Sigma=\{w(−1,0),w(1,0),w(0,−1),w(0,1)\}$.
Show that the unisolvent element leads to a finite element space,
which is not $H^1-$conforming.
I have proved that ... | Denote $V_h$ the finite element space. Recall that if $V_h\subset H^1(\Omega)$ and $Q$ consists of continuous functions, then $V_h\subset C(\Omega)$.
Thus, now we only need to show that $V_h$ is not contained in $C(\Omega)$. It's not difficult to find a function in $V_h$ but not continuous.
For example, let us conside... | {
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Proving that a group of order $p^nq$ for primes $p$ and $q$ is not simple.
Prove that a group of order $p^nq$ for primes $p$ and $q$ is not simple.
I've been able to prove the theorem holds for $p=q$ and $p>q$. If $p<q$ the best I've been able to do is use Sylow to show: $$p^n+p^{n-1}-1\leq q$$
Yet I seem to be stuck... | A consequence of one of Sylow's theorems is that if there is exactly one $p$-Sylow subgroup $H$ of $G$, then is it normal. Can you do the rest?
| {
"language": "en",
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Prove for every positive integer $n$, one of $n, n+1, n+2,...,2n$ is the square of an integer Prove for every positive integer $n$, one of $n, n+1, n+2,...,2n$ is the square of an integer.
This seems like a proof by induction, but I'm more used to using proof by induction with a single equation, where I usually prove i... | Assume $x^2<n$ but $(x+1)^2\ge n$.
Then, if $x\ge3, 2x+1<x^2,$ so $(x+1)^2=x^2+2x+1<2x^2<2n.$
The cases where $x\lt3$ can be easily eliminated.
| {
"language": "en",
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You started with one chip. You need to get 4 chips to win. What is the probability that you will win? This is very similar to the question I've just asked, except now the requirement is to gain $4$ chips to win (instead of $3$)
The game is:
You start with one chip. You flip a fair coin. If it throws heads, you gain
... | This answer only addresses what's wrong with your recursion, since the other answers (both in this question and your earlier question) already gave many different ways to set up the right recursions (or use other methods).
The key mistake is what you highlighted. When you throw $HHT$, you now have $2$ chips. For the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why must absolute value be used in this equation? I have a simple equation:
$(x-2)^2 < 3$
My first solution was:
$x-2 < \sqrt{3}$
But this gives me only:
$x < \sqrt{3} + 2$
Which is only one solution, so it's not enough. I've figured it out that I must use:
$|x-2| < \sqrt{3}$
Then, solutions are:
$-\sqrt{3} + 2 < x < \... | The quantity $(x-2)^2$ is derived by the squaring of the quantity $(x-2)$. But, you do not know whether this expression is positive or not. Thus, by square-rooting (as you did), you must take into consideration both cases, which is only yielded by using the absolute value.
Note, that the absolute value represents the e... | {
"language": "en",
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Show that $\lim_{x \to a} kx^2 + mx + n = ka^2+ ma +n$ Show that $$\lim_{x \to a} kx^2 + mx + n = ka^2+ ma +n$$ (Assume the domain is $\mathbb{R}$)
Let $\epsilon > 0$ be given. Suppose
$$\delta = min\{ \frac{1}{|kx + ka +m|}, 1\}$$
Then we have if $$ |x- a| < \delta$$
$$|(kx^2 + mx + n) - (ka^2 + ma + n)| = |k(x-a)... | By the triangle inequality we have
$$|k(x^2-a^2)+m(x-a)|\leq|k(x^2-a^2)|+|m(x-a)|=|k|\cdot|x^2-a^2|+|m|\cdot|x-a|$$
This equals the follwing, before we estimate with $\delta$
$$|k|\cdot|x-a|\cdot|x+a|+|m|\cdot|x-a|\lt |k|\delta \cdot|x+a|+|m|\delta$$
Now we use a sneaky trick to be able to estimate with with the triang... | {
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One one and continuous function from $\Bbb R$ to $\Bbb R$ Let $f :\mathbb R \to \mathbb R$ be a continuous and one one function. Then which of the following is true?
*
*f is onto.
*f is either strictly increasing or strictly decreasing.
*There exist $x \in \Bbb R$ such that $f (x) = 1$
*f is unbounded
| 1, 3 and 4 are not true.
The function $f(x) = \tan^{-1}x + 10$ is a counterexample for all of them.
2 is true.
To show this, assume the contrary and derive a contradiction using intermediate value theorem.
| {
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Width and thickness of the Samsung Odyssey G9 monitor The width and thickness of the upcoming Samsung monitor have not been released yet.
However, we know it's a part of a circle of 1m radius and we know the length of that part of the circle. I'm guessing we should be able to get the width and thickness of the monitor ... | First, we need to convert $47.17$ inches into metric units. This is $119.8$ centimetres.
The angle subtended by the monitor is
$$360^\circ×\frac{119.8}{2\pi×100}=68.64^\circ$$
The width of the monitor forms the third side of a triangle with the other two sides $1$ metre and included angle $68.64^\circ$. Thus it may be ... | {
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Solving $m\frac{\mathrm d ^2x}{\mathrm d t^2}+b\frac{\mathrm d x}{\mathrm d t}+kx=F_0\cos(\omega_dt)$ for $x(t)$ I am studying damped oscillations and forced oscillations in physics which are represented by the following differential equations in order.
$$\begin{align}m\dfrac{\mathrm d^2x}{\mathrm dt^2}+b\dfrac{\mathr... | You can use Laplace transform if you can. It is the easiest way of solving linear differential equations, I think.
Taking laplace transform of both sides, by noting that $X(s)$ is the Laplace transform of $x(t)$ and Laplace transform of the derivative of $x(t)$ is multiplication by $s$ in Laplace domain (if initial con... | {
"language": "en",
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proof $ \lvert\int_a^bf(t)dt\rvert\le\frac{(b-a)^2M}{4} $ Let $ f \in C^1([a,b],\Bbb R)$ such that $f(a)=f(b)=0$ with $a \lt b $.
If $M=\sup\limits_{x\in[a,b]}\lvert f'(x)\rvert $, show that:
$$ \left|\int_a^bf(t)dt\right|\le\frac{(b-a)^2M}{4} $$
I tried this
for the MVT: $f'(t)=\frac{f(t)-f(a)}{t-a} $
then $f'(t)(t-... | There's nothing wrong with what you've done. $\frac{(b - a)^2M}2$ is the best you can do if $|f'(x)| \le M$ and $f(a) = 0$. Because the MVT says that $|f(x)| \le M(x - a)$ and the integral of that will give you $M(b - a)^2/2$.
To get $\frac14$ you need to use the fact that $f(b) = 0$ as well. Then MVT will give you
$$ ... | {
"language": "en",
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Is every power series infinitely differentiable everywhere? I have found various sources on the internet that say that power series are infinitely differentiable on their interval of convergence:
Wikipedia:
Once a function $f(x)$ is given as a power series as above, it is differentiable on the interior of the domain o... | There is lot of difference between polynomials and power series.
You cannot define $\sum z^{n}$ for $|z| >1$. So there is no question of differentiability of this on $\{z: |z| >1\}$.
| {
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Moderate complexity between polynomial and exponential There are ,,plenty'' of functions growing faster then any polynomial and at the same time growing slower than any exponential function (with base $>1$) e.g. $f(x)=e^{g(x)}$ where $g(x)=\log^{c} x$ where $c>1$ or $g(x)=x^c$ where $c \in (0,1)$. I would like to know ... | The Graph isomorphism problem is an example of a problem unknown to be in $\text{P}$, that was conjectured to not be $\text{NP-Hard}$, and relatively recently (2015) Laszlo Babai published a paper proving it can be done in $\exp(\log(n)^{O(1)})$ which is quasi-polynomial time.
Remark: Someone found a mistake in the pap... | {
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Showing a subgroup of $\mathbb{Z}\times\mathbb{Z}$ is cyclic.
In the group $G = \mathbb{Z}\times\mathbb{Z}$, consider the subgroup $H$ generated by $(-5,1)$ and $(1,-5)$. I want to show that $G/H$ is cyclic and find the standard cyclic group it is isomorphic to.
I haven't much group theory experience, but understand ... | If $H$ is generated by $h_1$ and $h_2$ then $H=\{ah_1+bh_2\}$ where $a$ and $b$ are integers. If you think of $G$ as the set of points in the plane with integer co-ordinates then $H$ is the lattice of points with co-ordinates $(-5a+b, a-5b)$ where $a$ and $b$ are integers.
The elements of $G/H$ correspond to the co-set... | {
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Expectation under conditional distribution I am trying to understand the following equivalence:
\begin{array}{c}{\mathbb{E}_{p(x, y)}\left[\log \frac{q(x | y)}{p(x)} \frac{p(x | y)}{q(x | y)}\right] =\mathbb{E}_{p(x, y)}\left[\log \frac{q(x | y)}{p(x)}\right]+\mathbb{E}_{p(y)}\left[D_{K L}(p(x | y) \| q(x | y))\right]}... | This is just the law of total expectation: Quite generally,
$$
\mathbb E_{p(x,y)}[Z]=\mathbb E_{p(y)}\left[\mathbb E_{p(x\mid y)}[Z]\right]\;.
$$
That is, you can first form the expectation as if you knew $Y$, and then form the expectation of the result using the marginal distribution of $Y$. In the discrete case, this... | {
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Prove that two block matrices over $\mathbb{F}$ are similar
Let $\mathbb{F}$ be a field, $n\in\mathbb{N}_{\geq 1}$ and $A\in M_{2n}(\mathbb{F})$, such that $$A=\begin{pmatrix} 0_n & 0_n \\ B & 0_n \end{pmatrix}$$ with $B\in GL_n(\mathbb{F})$. Show that A is similar to the matrix $$\begin{pmatrix} C & 0_2 & \ldots & 0_... | Denote by $\Gamma$ the matrix $$\begin{pmatrix} C&0_2&0_2&\dots&0_2\\0_2&C&0_2&\dots&0_2\\0_2&0_2&C&\dots&0_2\\\vdots&\vdots&\vdots&\ddots&\vdots\\0_2&0_2&0_2&\dots&C\end{pmatrix}.$$
Let $P$ denote the permutation matrix associated to the permutation $\sigma\in \mathcal{S}_{2n}$ such that
$$\sigma(k)=\left\{\begin{arra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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tail bound of the euclidean norm of multivariate norm distributed random variable Let $x$ has a multivariate norm distribution, i.e, $x\sim \mathcal{N}(\mu,\Sigma)$, what is the following upper bound $\Pr(||x||_2^2\geq M)\leq ?$, where $M$ is a constant.
Thanks a lot!
| $P(\|x\|_2^{2} \geq M) \leq \frac 1 M E\|x\|^{2}=\frac 1 M \sum\limits_{k=1}^{n} Ex_i^{2}=\frac 1 M \sum\limits_{k=1}^{n} (\sigma_i^{2}+\mu_i^{2})$.
| {
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How to apply squeeze theorem to this limit. I'm trying to solve $$\int_0^∞ e^{-x} \cos(x)\,dx$$
It is not hard to find that $$\int e^{-x} \cos(x)=\frac{1}{2}(e^{-x} \sin(x)-e^{-x} \cos(x))+C$$
From all this follows that
$$\lim_{t\to\infty}\int_0^te^{-x} \cos(x) \, dx = \frac{1}{2}\lim_{t\to\infty}(e^{-t} \sin(t) - e^{-... | $$\because\sin x-\cos x=\sqrt 2(\cos\frac{\pi}{4}\sin x-\sin\frac{\pi}{4}\cos x)=\sqrt 2\sin (x-\frac{\pi}{4})$$
$$\therefore\lim_{t\rightarrow +\infty}|e^{-t}\sin t-e^{-t}\cos t|= \lim_{t\rightarrow +\infty}|\sqrt 2e^{-t}\sin (t-\frac{\pi}{4})|\leqslant \lim_{t\rightarrow +\infty}|\sqrt 2e^{-t}|=0$$
$$\therefore\lim_{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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False Σ1-sound theories I was wondering how Σ1-sound theories in the language of first-order arithmetic can go wrong. As far as I can tell, they cannot prove false claims about consistency, since such claims (e.g., that there is a proof of 0=1" from the axioms) are equivalent (in weak theories of arithmetic) to Σ1 se... | Basically, soundness at one level of the arithmetical hierarchy doesn't prevent errors higher up.
For example, let $PA_n$ be PA + all true $\Pi_n$ sentences (incidentally, note that every true $\Sigma_{n+1}$ sentence is a theorem of $PA_n$). Then we can form the Godel-Rosser sentence for $PA_n$:
$(*)_n$: "For every pr... | {
"language": "en",
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Matrix with even integers entries doesn't have odd eigenvalue Let $A \in M_n(\mathbb{Z})$ with even entries. Prove that $A$ doesn't have odd eigenvalue.
| Suppose $A$ has an odd eigenvalue $k$. Then the characteristic polynomial $\det(A-\lambda I)$ can be written as:
$$\lambda^n+a_{n-1}\lambda^{n-1}+\ldots+a_0=(\lambda-k)(b_{n-1}\lambda^{n-1}+b_{n-2}\lambda^{n-2}+\ldots+b_1\lambda+b_0)$$
where each $a_i$ is even $(i=0, \ldots, n-1)$.
The right hand side evaluates to:
$$b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the values for which $A^2 = I_2$, A is a matrix, with $A \neq I_2$ and $A \neq -I_2$ First I tried to find $A^2$ with
$$
A=\begin{bmatrix}
\alpha & \beta\\
\delta & \gamma\\
\end{bmatrix}
$$
I multiplied this by itself and got:
$$
\begin{bmatrix}
\alpha^2+\beta\delta& \beta(\alpha + \gamma)... | You assumed $\delta \neq 0$ when you divided by it (when solving for $\beta$). The first system stills valid whether $\delta = 0$ or not.
In fact, when $\delta = 0, |\alpha| = |\gamma| = 1$. If $\gamma = -\alpha, \beta$ can be anything.
| {
"language": "en",
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"source": "stackexchange",
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Getting probability greater than 1? Consider a random walk starting at a positive integer $k$. Now, trying to calculate the probability of returning to zero (from $k$), I did the following:
\begin{align}
P&(\text{returning to zero}) \\
&=P(\text{returning to zero in } k\text{ steps}) \\
&\quad+ P(\text{returning to z... | Clue :
$P$(Returning to zero in $k+2$ steps) should be $(\frac12)^{k+2}(k+1)$
Returning to zero in $k$ steps can only be done by taking the path $$k, k-1, \dots, 0$$
Returning to zero in $k+2$ steps can only be done by deviating exactly once from the '$k$ steps path'. So it's something like $$k, k-1, \dots, n, n+1, n, ... | {
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Why writing equations in two column explanation is bad math? A question arised while I was reading "A Guide to Write Mathematics" (by Dr. Kevin P. Lee, p. 5) about writing explanation of steps, in equations calculations or listing some equations, without using two columns like this:
I don't think is bad or hard to r... | The author of the guide is trying to teach some elements of compositional style, emphasizing the idea that mathematics should be presented in a way that is readable as sensible English sentences. From this point of view, the two-column display reads as an alternation of declaratives and imperatives (i.e., statements an... | {
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"timestamp": "2023-03-29T00:00:00",
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Relationship between groups and spaces From the definition on Wikipedia: In mathematics, a space is a set (sometimes called a universe) with some added structure.
A group seems to satisfy this definition. For example, we know that Lie groups are differentiable manifolds.
So my question is: is it proper to say that a ... | Sure, it can be useful to attempt to think about mathematical concepts in a way that goes against the grain a little bit. Here are a few ways of thinking of groups as spaces. I suppose it should go without saying that these are very useful ways of thinking about groups.
There are topological groups, which you could def... | {
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How to show the additive property of a sum of a series So I have been working on this proof for a bit now. I found a way to prove it directly, but am struggling to find a way to prove it with limits.
Question:
Let ∑an = A, ∑bn = B, for an, bn, A, B in the Reals,
show that ∑(an+bn) = (A+B)
| If you have two convergent sequences, the limit of the sum is the sum of the limits. So you have
\begin{align}
\sum_{n=1}^\infty(a_n+b_n)&=\lim_{m\to\infty}\sum_{n=1}^m(a_n+b_n)=\lim_{m\to\infty}\left(\sum_{n=1}^ma_n+\sum_{n=1}^mb_n\right)\\ \ \\
&=\lim_{m\to\infty}\sum_{n=1}^ma_n+\lim_{m\to\infty}\sum_{n=1}^mb_n
=\sum... | {
"language": "en",
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Prove that $\det(AB - BA) = \frac{1}{3}\left(\mathrm{Trace}(AB - BA)^3\right)$ I want a correct and feasible answer to this question.
So does anyone have any creative ideas to prove this equation?
$A$ and $B$ are $3\times3$ matrices.
$\det(AB - BA) = \dfrac{1}{3}\operatorname{Trace}\left((AB - BA)^3\right)$
Answer:
... | This follows easily from Cayley-Hamilton theorem. Since $M=AB-BA$ has zero trace, by Cayley-Hamilton theorem, $M^3=cM+dI_3$ where $c$ is some scalar and $d=\det(M)$. Therefore $\operatorname{tr}(M^3)=3d$ and the result follows.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "13",
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A quote by Jacobi I remember reading that Jacobi once said:
If Cauchy says he proved something, you can be 50% sure that he actually did. If Gauss says he proved something, you can be mostly sure that he actually did. But if (insert name) says he proved something, then you can be 100% sure that he actually did.
Does... | The original is in a letter from Jacobi to Alexander von Humboldt dated December 21, 1846:
Dirichlet allein, nicht ich, nicht Cauchy, nicht Gauß, weiß, was ein
vollkommen strenger Beweis ist, sondern wir lernen es erst von ihm.
Wenn Gauß sagt, er habe etwas bewiesen, so ist es mir sehr
wahrscheinlich, wenn Cauch... | {
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Bounding Bernoulli trials by the standard Bernoulli process Suppose we have a Bernoulli-like process $P$. At each step a coin is tossed and the outcome ("success", "failure") is recorded. What differentiate $P$ from the standard Bernoulli process, is that we pick a probability of "success" uniformly at random in range ... | While, kimchi's answer is an answer to the problem as I have stated it in the first place ...
I want to share an approach, that tackles directly the expectation bound.
Suppose, we have a series of independent Bernoulli trials $X_i$ each with a probability of success $p_i \geq \frac{1}{2}$. And a series of standard i.i.... | {
"language": "en",
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What does the !! mean in trigonometric identity? What does the $!!$ mean in:
$$
\int_0^x \sin^n(t) \mathrm dt = \begin{cases}
\frac{(n-1)\color{red}{!!}}{n\color{red}{!!}}\Big[1-\cos(x)\sum_{j=0}^{(n-1)/2}\frac{(2j-1)\color{red}{!!}}{(2j)\color{red}{!!}}\sin^{2j}(x)\Big]&\text{for $n$ odd}\\
\frac{(n-1)\color{red}{!!}}... | In mathematics, the double factorial or semifactorial of a number $n$ (denoted by $n!!$) is the product of all the integers from $1$ up to $n$ that have the same parity (odd or even) as $n$.
Example: $9!! = 9 \cdot 7 \cdot 5 \cdot 3 \cdot 1$
| {
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"url": "https://math.stackexchange.com/questions/3506511",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "11",
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} |
Infinite complex nested radical and it's complex conjugate. Today I want to play with $i$ the imaginary unit I have this :
$$\overline{\sqrt{1+i\sqrt{1+i^2\sqrt{1+i^3\sqrt{1+i^4\sqrt{\cdots}}}}}}=\sqrt{1+\frac{1}{i}\sqrt{1+\frac{1}{i^2}\sqrt{1+\frac{1}{i^3}\sqrt{1+\frac{1}{i^4}\sqrt{\cdots}}}}}$$
Main remarks we have :... | Just a comment on this. Well first and foremost, I see that you're doing a lot of radicals of this form & also much trivially we can get;
$$\sqrt{1+\frac{1}{x}\sqrt{1+\frac{1}{x^2}\sqrt{1+\frac{1}{x^3}\sqrt{1+\frac{1}{x^4}\sqrt{\cdots}}}}}=\frac{1}{x^{2}}\sqrt{x^{4}+\sqrt{x^{6}+\sqrt{x^{8}+\sqrt{x^{10}+\sqrt{x^{12}+..... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3506607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What is the difference between logical statements and statement variables? For the following two question
Let P, Q, and R be logical statements. Use a truth table to prove that_______.
Let P, Q, and R be statement variables, and suppose that the logical expression_______ is false.
The blank is two expressions which I d... | A logical statement like $P$ is meant to be a specific statement. For example, $P$ could mean 'It is raining'.
A statement variable is something we use to indicate that we are dealing with some statement ... but we don't know what it is.
It is like the difference between $2$ and $x$ when doing algebra. The $2$ is a spe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3506852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find $\lim\limits_{n \to \infty} \int\limits_0^n \frac1{1 + n^2 \cos^2 x} dx$. I have to find the limit:
$$\lim\limits_{n \to \infty} \displaystyle\int_0^n \dfrac{1}{1 + n^2 \cos^2 x} dx$$
How should I approach this?
I kept looking for some appropriate bounds (for the Squeeze Theorem) that I could use to determine the ... | Here is a direct approach. Let $f(x)$ be the integrand, and note that $f(x)$ has a period $\pi$. Let $k$ be the largest positive integer such that $(2k+1)\frac{\pi}{2}<n$. Then:
$$\begin{align}
\int_0^nf(x)\,dx&=\int_0^{\pi/2}f(x)\,dx+\int_{\pi/2}^{3\pi/2}f(x)\,dx+\dots+\int_{(2k+1)\pi/2}^nf(x)\,dx \\
&=\int_0^{\pi/2}f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3506964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
What's wrong with my solution to this probability problem Let $S$ be the set of all integers between −10 and 10, inclusive: $\{−10, −9, −8, . . . , 8, 9, 10\}$. If two integers are chosen out of the set at random, then the probability that the product of the two integers is positive is a/b, where a and b share no commo... | The product is $0$ if either sample is $0$. $\frac{1}{21}$ is the chance that the first sample is zero.
Assuming the samples are uniform and independent, the probability of a 0 product is
$$Pr(x_0 = 0 {\rm \ or \ } x_1=0) \\
= Pr(x_0 = 0) + Pr(x_1=0) - Pr(x_0 = 0 {\rm \ and \ } x_1 = 0) \\
= \frac{1}{21} + \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3507127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $g(y) = \int_0^1 f(x,y) dx$ is not continuous at $y =0$.
Exercise 7.5.17 (Introduction to Real Analysis by Jiri Lebl): Define
$$f(x) = \left\{
\begin{array}{lr}
\frac{2xy}{x^4+y^2} & \text{if} & (x,y) \not= (0,0)\\
0 & \text{if} & (x,y) =(0,0)
\end{array}
\right.
$$
Show that $g(y) = \int_0^1... | HINT
Let $x^2 = t$
$$\int_0^1\frac{2xy}{x^4 + y^2}dx = \int_0^1\frac{y}{(t^2 + y^2)}dt = \arctan(\frac{1}{y})$$
Can you complete the rest?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3507247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A man is on block 2, find expected number of steps for him to reach block 1.
There are infinite blocks placed along positive axis at integer points. A man is on position $2$. He moves back one step with probability $p$ and forward with probability $1-p$ where $p > 0.5$.
Find expected number of steps for him to reach... | If you take a step forward, notice how the situation is similar, if you now ask the expected number of steps to return to tile $2$. So it will also take $E_2$ steps on average to go back to tile $2$ when you're on tile $3$, and then again $E_2$ to go from $2$ to $1$. So we can write
$$E_2 = p\times 1 + (1-p)\times(1+2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3507472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Brownian Motion $E[B_t \cdot B_s \cdot B_v]$ with $0 < t < s Can someone derive analytically the following Brownian Motion question? $E[B_t \cdot B_s \cdot B_v]$ with $0 < t < s < v$.
| We can also use the martingale property of the BM. Let us denote $\mathcal{F}_t$ the natural filtration of the Brownian motion $B_t$.
We now that $\{B_t\}_{t\geq0}$ and $\{B_t^2-t\}_{t\geq0}$ are martingales. I recall that $B_t \sim \mathcal{N}(0,t)$.
We have then
\begin{align*}
E\left[B_sB_tB_v\right] &= E\left[\unde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3507560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to compute the limit below by L'Hopital's rule?
Let $\left(X_j\right)_{j\geq1}$ be i.i.d. with $E\{X_j\}=\mu$ and $\text{Var}\{X_j\}=\sigma^2$ (all $j$) with $0<\sigma^2<\infty$. Let $S_n=\sum\limits_{j=1}^{n}X_j$ and $Y_n=\dfrac{S_n-n\mu}{\sigma\sqrt{n}}$. Let $\varphi_j$ be the characteristic function of $X_j-\m... | HINT:
Let $x=1/n$. Then examine the limit
$$\lim_{x\to0}\,\frac{\log\left(1-u^2x/2+(u^2/\sigma^2xh(u\sqrt{x}/\sigma))\right)}{x}$$
Can you proceed now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3507687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Building a Subspace Complementary to Two Given Subspaces Lee's Introduction to Smooth Manifolds claims that given an n-dimensional vector space $V$ and two $k$-dimensional subspaces $P,P'$, we can find a single $n-k$-dimensional subspace $Q$ complementary to both $P$ and $P'$. How do we show this?
| If $P=P'$ then there exists a $0 \neq v \in V \setminus P$, and then span$\{v\}$ is a subspace disjoint from $P$ and $P'$. If they are not equal, there exists a $v_1 \in P \setminus P'$ and a $v_2 \in P' \setminus P$. Then any non-trivial linear combination of these two will suffice to make a new subspace disjoint from... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3507801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\int _0^1\:\frac{2-x^2}{(1+x)\sqrt{1-x^2}} \, dx$
Evaluate $$\int _0^1\:\frac{2-x^2}{(1+x)\sqrt{1-x^2}}dx$$
$$
\int _0^1\:\frac{2-x^2}{(1+x)\sqrt{1-x^2}} \, dx = \int_0^1 \bigg[\sqrt{\frac{1+x}{1-x}}+\frac{1}{(1-x)^{3/2}\sqrt{1+x}}\bigg] \, dx=\int_0^1[f(x)+f'(x)]\,dx
$$
Does making it into the above final ... | Let $x=\sin t$. Then,
$$\int _0^1 \frac{2-x^2}{\left(1+x\right)\sqrt{1-x^2}}dx
=\int _0^1\left(\frac{1}{\left(1+x\right)\sqrt{1-x^2}}
+\frac{\sqrt{1-x^2}}{1+x}\right)dx$$
$$=\int_0^{\pi/2} \left( \frac{dt}{1+\sin t}+1-\sin t\right)dt$$
$$=\left(-\frac{\cos t}{1+\sin t} + t + \cos t\right)_0^{\pi/2}=\frac\pi2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3507993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Linear transformation with respect to basis problem Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ such that $T(\begin{bmatrix} 3 \\ 1 \end{bmatrix}) = \begin{bmatrix} 1 \\2 \end{bmatrix}$ and $T(\begin{bmatrix} -1 \\ 0 \end{bmatrix}) = \begin{bmatrix} 1 \\1 \end{bmatrix}$. Find the matrix $A$ representing $T$.
I understand t... | Relative to the bases $B$ and the standard basis, the matrix is:. $\begin{pmatrix}1&1\\2&1\end{pmatrix}$.
The change of basis matrix is: $\begin{pmatrix}3&-1\\1&0\end{pmatrix}$.
The latter changes basis from $B$ to the standard basis.
Thus you want: $\begin{pmatrix}1&1\\2&1\end{pmatrix}\begin{pmatrix}3&-1\\1&0\end{pma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3508294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Waiting time for a pattern in coin tossing Let $\{X_t\}$ be an iid sequence of fair coin tosses and $\tau_{HTH} = \inf\{t\geq 3: X_{t-2}X_{t-1}X_t=HTH\}$. I want to determine $E\tau_{HTH}$.
I don't understand a part of the explanation which is:
Gamblers place bets on each individual toss. On each bet, gambler pays an e... | We can handle the case of a possibly unfair coin using generating functions.
A pattern for all sequences ending in $HTH$ is
$$
T^\ast\left(H^\ast(HTT)T^\ast\right)^\ast H^\ast HTH\tag1
$$
where $(\dots)^\ast$ represents $0$ or more sequences matching $(\dots)$.
The generating function for $(1)$ is
$$
\overbrace{\ \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3508407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Finding eigenspaces of a matrix with parameter I have the following matrix:
$$
\begin{pmatrix}
3&0&0\\
k+2&1&0\\
5&k-1&1
\end{pmatrix}
$$
The exercise asks to find the eigenvalues of the matrix and, for all $k\in\mathbb{R}$, determine a basis of each eigenspace of the matrix.
Since ths is a lower triangular matrix, the... | Since $-k^2-k-8$ can’t be zero, you can safely divide by it and continue merrily on your way with the row-reduction. However, because you’re working in $\mathbb R^3$ there’s a much simpler way to find a basis for this null space. The row space of $A-3I$ is clearly spanned by its two nonzero rows, and the null space of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3508502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Probabilistic inequality for an antisymmetric function Let $X$ be a random variable with $\mathbb P(-1 \leq X \leq 1) = 1$. Does $\mathbb E(X) \geq 0$ imply
$$\mathbb E[X(1-|X|)] \geq 0?$$
The function is antysymmetric around $0$ and has more probability mass on the positive side. Intuitively, it should be correct.
If ... | The answer is no.
A simple counter-example is letting $X\equiv 1$.
meaning, $X$ is a constant random variable equal to 1.
We get $$\mathbb E[X(1-\vert X\vert)] = \mathbb E[X(1-\vert 1\vert)] = 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3508647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Given positives $a, b, c$, prove that $\frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$.
Given positives $a, b, c$, prove that $$\large \frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$$
Let $x = \dfrac{b + c}{2}, y = \dfrac{c + a}{2... | By Cauchy-Schwarz,
$$\left[\dfrac{a}{(b + c)^2} + \dfrac{b}{(c + a)^2} + \dfrac{c}{(a + b)^2}\right](a+b+c) \ge \left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)^2$$
Then by rearrangement inequality,
$$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\ge\dfrac{b}{b+c}+\dfrac{c}{c+a}+\dfrac{a}{a+b} \\ \dfrac{a}{b+c}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3508789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
} |
Inequality $\binom{n+m}{k}+\binom{n-m}{k}\ge 2\binom{n}{k}$ Is it true that $\binom{n+m}{k}+\binom{n-m}{k}\ge 2\binom{n}{k}$?
I've been checking this for many cases in the Pascal triangle ($n,m,k\in\mathbb{N}^*$ such that $n-m\ge k$) but cannot prove formally.
| Claim. The inequality
$$\binom{n+m}k+\binom{n-m}k \ge 2 \binom nk$$
holds for any integers such that $0\le m,k \le n$.
Proof. It is clear that this is true for $k=0$. So we will assume from now on that $k\ge1$.
Let us denote $$a_j=\binom{n+j}k+\binom{n-j}k$$ for $j=0,1,\dots,n$. We have $a_0=2\binom nk$. It suffices to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3508893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Surjection of map to n-Sphere Consider the map $f\colon S^n \times S^m \times [0,1] \rightarrow S^{n+m+1}$ defined by $f(p,q,t) = p \cos(\frac{\pi}{2}t) +q \sin(\frac{\pi}{2}t)$ where $p \in S^n$, $q \in S^m$ and $t \in[0,1]$. Show this map is surjective, where $n,m \in \mathbb{R}$
I have been able to show it when $n=m... | Consider a point $(x,y) \in S^{n+m+1}$, where $x \in \mathbb R^{n+1}, y \in \mathbb R^{m+1}$. We are looking for a solution $(p,q,t) \in S^n \times S^m \times [0,1]$ of the two equations
$$\cos(\frac{\pi}{2} t) p = x, \sin (\frac{\pi}{2} t) q = y .$$
Note that $\cos(\frac{\pi}{2} t), \sin(\frac{\pi}{2} t) \in [0,1]$ f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3509012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Olympiad problem :Integer roots of $P(P(x))$ in function of the roots of $P(x)$ Here is the original problem :
A polynomial $P(x)$ of degree $n \geq 5$ with integer coefficients and
$n$ distinct integer roots is given. Find all integer roots of $P(P(x))$
given that $0$ is a root of $P(x)$.
Here is my solution, h... | So $P(x)=x\prod_{i=2}^n (x-x_i)$ where $x_2,\ldots, x_n$ are the nonzero integral roots of $P$. Now suppose there is a a nonzero root $y$ of $P(P(x))$ that is distinct from $0,x_2,\ldots, x_n$. Then $y\prod_{i=2}^n(y-x_i)$ must be in $\{x_2,x_3,\ldots, x_n\}$. We show that this is impossible for integral $y \not = 0,x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3509219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $|f(z)| \leq |z|$ on annulus Let $D = \{ z \in \mathbb{C} : 2 < |z| < 3 \}$. Suppose that $f$ is holomorphic on $D$ and $f$ is continuous on $\overline{D}$. Suppose that $\max \{ |f(z)| : |z| = 2\} \leq 2$ and $\max \{ |f(z)| : |z| = 3 \} \leq 3$. Show that $\forall z \in D, |f(z)| \leq |z|$. The solution I a... | Your proof is correct, but you can simplify it. The function $\lvert g(z) \rvert$ assumes its maximum on $\partial D$. For $\lvert z \rvert = 2$ we have $\lvert f(z) \rvert \le 2 = \lvert z \rvert$, thus $\lvert g(z) \rvert \le 1$. Similarly $\lvert g(z) \rvert \le 1$ for $\lvert z \rvert = 3$. Thus necessarily $\lvert... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3509408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
circle envelope tangent in another circle As the picture shows,
One big circle ,$(0,0)$ ,radius=R, there is a small circle in it, $(m,0)$ ,radius=r .
G is on the big circle. From G ,we can do two tangent lines about the small circle. Get the points of intersection E and F. line EF has a envelope about G , which seem... | Here is an analytic geometry solution :
We can assume without loss of generality that $R=1$, i.e., we work inside the unit circle. Therefore, we have the following condition : $0 < m <1$.
Let us introduce some notations.
Let $B$ be the second intersection of line $GC$ with the unit circle.
Let $a$ be the polar angle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3509582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
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