Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
soft question: explaining proportions/percentages in simple terms I know this is a fairly easy question but I haven't been able to word it into Google so as it would give me a substantive list of resources.
Here's my question:
If a process is 25% efficient, I'd multiply (1/0.25) by the output, which would yield what i... | You could say that you "divided out" the $25\%$ to return what $100\%$ would be.
To explain what $25\%$, or any percent, you could think of it as the amount of $\$1.00$ when you cut it up into $100¢$ and take $25$ of those pieces.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Suppose $A$ is a matrix with distinct and positive $n$ eigenvalues. How many real matrices $B$ exist such that $B^k=A$? Suppose $\mathbf{A}$ is an $n\times n$ matrix with distinct and positive $n$ eigenvalues.
How many real matrices $\mathbf{B}$ exist such that $\mathbf{B}^k=\mathbf{A}$?
Maybe diagonalization or Jorda... | By the spectral mapping theorem, the eigenvalues of $B$ must be $k$'th roots of the eigenvalues of $A$, the corresponding eigenvectors also being eigenvectors of $A$. Now if $B$ is real, its complex eigenvalues come in complex-conjugate pairs, but then the $k$'th powers of such a pair would also be complex conjugates,... | {
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If $n > 1$ and all $n$ positive integers $a, a + k, \cdots , a+ (n - 1)k$ are odd primes, show every prime $Background: This is from Rosen 5th edition, $3.2.15$ Number Theory.
This is an important proof because $3$ following problems require it to be correct.
If $i=0$, and $j=p$ then this proof is wrong and $p\mid (i-... | If $i=0$, then $p$ would have to be equal to $-j$. This would be a contradiction since $i, j, p > 0$. WLOG, we can assume $i-j>0$.
Moreover, since $i \le p$ and $j \le p$, then the claim that $i-j < p$ holds, and the proof is true. Does that help?
| {
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Independent with a vector v.s. independent with its components Suppose that $Z$ (a scalar) is independent with $X=(X_1,\ldots,X_n)$, $n>1$. Then, $Z$ is independent with $X_1,\ldots,X_n$ because each of the latter is a function of $X$.
I suspect the reverse direction: $Z$ being independent with $X_1,\ldots,X_n$ implyin... | I is not only about the joint distribution of $(X_1, \dots, X_n)$.
It is possible that all components of $X$, namely $(X_1, \dots, X_n)$ are independent and that all pairs $(Z,X_i)$ are independent, but $Z$ is not independent from $X$.
My favorite example to illustrate this is from Bauer's book Wahrscheinlichkeitsthe... | {
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"timestamp": "2023-03-29T00:00:00",
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Why do we use integration to calculate the average of a function? What I think is that we need to sum up all the values on the curve and we can only do that by integration. Is that correct?
| Suppose we want
to get the average
of $f(x)$
for
$a \le x \le b$.
As a first estimate,
we might use
$A_2
=\dfrac{f(a)+f(b)}{2}
$.
Adding another point,
$A_3
=\dfrac{f(a)+f((a+b)/2)+f(b)}{3}
$.
If we use
$n$ points,
we get
$A_n
=\frac1{n}\sum_{k=0}^{n-1} f(a+k(b-a)/n)
$.
In the limit,
if it exists,
we get
$A
=\lim_{n \t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Isometric Spherical Space Forms I have seen the following "theorem" stated in any places, but haven't been able to find a proof of it:
Theorem: Two spherical space forms $S^{2n-1}/G_1$ and $S^{2n-1}/G_2$ are isometric iff $G_1$ and $G_2$ are orthogonal in O(2n).
Can anyone please share a proof of this?
Thanks.
| I think the word you want is conjugate, not "orthogonal." (I don't know what it means for subgroups of $O(2n)$ to be "orthogonal.")
A more general version of the theorem you're looking for is proved, for example, in Joe Wolf's Spaces of Constant Curvature:
Lemma 2.5.6: Let $P\colon L\to M$ and $Q\colon L\to N$ be univ... | {
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Polynomials and the IMO I have been stuck on this problem, as I am unable to proceed properly. The problem is as follows:
If $f(x) = (x+2x^2+\cdots nx^n)^2 = a_2x^2 + a_3x^3 +\cdots a_{2n}x^{2n},$ prove that $$a_{n+1} + a_{n+2} +\cdots +a_{2n} = \binom{n+1}{2}\frac{5n^2+5n+2}{12}$$
I tried expanding the LHS but o... | Hint
See that:
$$a_j= \sum_{i=1}^{j}i(j-i)=\frac{j^3-j}{6}$$
And then
$$\sum_{j=n+1}^{2n}a_j=\frac{1}{6}\sum_{j=n+1}^{2n}(j^3-j)=\frac{1}{6}\left(\sum_{j=1}^{2n}(j^3-j)-\sum_{j=1}^{n}(j^3-j)\right)$$
| {
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prove by induction; integer division by $5$ I'm having trouble with my math project.
I have to prove that for very natural, $n$, there is an integer $q$ , and an integer $r$ such that:
$n=5q+r$ and $0\le r<5$ using induction
I have tried using the axiom of Archimedes but I can't really get around the problem. Sorr... | Start with the base case: $n=0$. Then, clearly $n=5\cdot 0+0$, so we are okay.
Next, suppose that the result holds for some fixed integer $n\geq 0$. We want to prove that the result holds for $n+1$. I.e., we want to prove that there exist integers $q$ and $0\leq r< 5$ satisfying $n+1=5q+r$.
By the inductive hypothes... | {
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Clarification of a proof as written in Margaris's book In Angelo Margaris's book First Order Mathematical Logic it is written (see below),
Then,
Questions
*
*In the above proof I don't understand what am I supposed to do at the second step. Can anyone explain that to me?
*How from "$P$ admits $t$ for $v$" it... | Answers
*
*In the second step "SC,1" means that the following, $$(\forall v{\sim}P\to{\sim}P(t/v))\to (P(t/v)\to{\sim}\forall v{\sim}P)$$ is a tautology.
*Hint: If ${\sim}P$ doesn't admit $t$ for $v$, then there exists at least one variable $u$ in $t$ such that it occurs in a subformula of the form $\mathtt{Q}uM$ o... | {
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show $\frac{y}{x^2+y^2} $ is harmonic except at $y=0,x=0$ Let $f(z)=u(x,y)+iv(x,y) $
where $$ f(z)=u(x,y)=\frac{y}{x^2+y^2}$$
show $u(x,y)$ is harmonic except at $z=0$
Attempt
$$ u=\frac{y}{x^2+y^2}=y(x^2+y^2)^{-1} $$
Partial derivatives with x
$$\begin{aligned}
u_x&= y *(x^2+y^2)^{-2}*-1*2x
\\ &= -y*2x(x... | While I believe that the "right" answers are those already given, I would like to add yet another one, based on polar coordinates. Introduce
$$
\begin{cases}
x=r\cos \phi\\
y=r\sin \phi
\end{cases}
$$
The given function $f(x, y)=\frac{y}{x^2+y^2}$ is harmonic if and only if
$$
\left(\partial_r^2 +r^{-1}\partial_r +r^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Understanding of the limit of $f\left( x\right) =\sqrt {x-1}$ as $x\rightarrow 1$ What is the limit of $f\left( x\right) =\sqrt {x-1}$ as $x\rightarrow 1$
Wade's intro.to analysis book says that ''A reasonable answer is that the limit is zero. This function, however, does not satisfy Definition3.1 because it is not an ... | What does an open interval around $a=1$ look like? It is symmetric, and this is the problem.
I.e. it looks like
$$
(1-\delta,1+\delta)
$$
for some real $\delta>0$. But as the textbook notes, if $1-\delta<x<1$ then $f(x)$ is not defined.
| {
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Is the limit of $f(n) = n-n$ zero as $n\rightarrow \infty$? I have been working on a proof which involves sums and products going to infinity. I am wondering whether the following proof of a limit is valid, and whether that result would allow me to come to another conclusion.
What is:
$$\lim \limits_{n \to \infty} f(n)... | If $f(n) = n-n$, then $f(n) = 0$ for all $n$. The limit you gave is true, i.e.
$$\lim\limits_{n\to \infty} f(n) = 0$$
is correct.
Furthermore, we have that $n\cdot f(n) = n\cdot 0 = 0$ for all $n$, so the limit
$$\lim\limits_{n\to\infty}n\cdot f(n) = 0$$
is also correct.
The red flag probably stems from the well known... | {
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"timestamp": "2023-03-29T00:00:00",
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Is $f$ continuous on a contour $\Gamma$? I am practicing for my midterm exam. I would appreciate it if someone could point me to any mistakes or flaws in my answer to this question I tried:
Given a contour $\Gamma : |z-\pi|=\frac{\pi}{2}$ traversed once counterclockwise, and define for all $z \in \mathbb{C}$ that are ... | My answer:
We take any point $z_0$ on $\Gamma$ such that $\cos z_0\ne 0$.
Cauchy's integral formula says that
$$f(z)=\cos z$$
for any $z$ in the interior of $\Gamma$.
If $z$ approaches $z_0$ along a path in the interior of $\Gamma$, $$f(z)=\cos z \to \cos z_0 \ne 0$$
since $\cos z$ is continous on $\mathbb{C}$.
Howe... | {
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If one egg is found to be good, then what is the probability that other is also good? A basket contains 10 eggs out of which 3 are rotten. Two eggs are taken out together at random. If one egg is found to be good, then what is the Probability that other is also good?
I applied conditional probability. It says that one... | After picking one good egg.
Rotten eggs = 3
Good ones = 6
Total = 9
Probability (second is also good) = $\frac{6}{9}$ = $\frac{2}{3}$
Edit -
This question has some assumptions also.
How is it found?
Both are picked together not in succession.
When you are picking two eggs together there are two ways to select egg... | {
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"timestamp": "2023-03-29T00:00:00",
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Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists? Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists ? If it does then what is the value ?
I don't think even L'Hospital's rule can be applied . Please help . Thanks in advance
| As en alternative for the powerfull Euler-Maclaurin asymptotics in @robjohn answer one can use simple zero order approximations and the Squeeze theorem.
*
*The estimate for $x+\pi k\le t\le x+\pi(k+1)$
$$
\frac{|\cos t|}{(x+\pi(k+1))^2}\le \frac{|\cos t|}{t^2}\le \frac{|\cos t|}{(x+\pi k)^2}
$$
and integration give... | {
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"timestamp": "2023-03-29T00:00:00",
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Select $k$ items from $n$ such that every item can occur in the combination for at most $k$ times. What is the generic formula for-
Select $k$ items from $n$ items such that every item can occur in the combination for at most $k$ times,
e.g. Let us assume we have $n=3$ items namely $\{A,B,C\}$, of these $3$ items we ha... | The analytical way
Let's try to find a recurrence for said number, call the number $M(n, k)$.
Trivially, if there is only $n=1$ element to chose from, there is only one possible solution:
$$M(1, k) = 1$$
and clearly for $k=0$ there is only one set, $\emptyset$:
$$M(n, 0) = 1$$
Now let's add a new element, $n$ to our se... | {
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About Euclid's proof of infinite primes..... I was checking that if product of first n primes+1 gives a prime again is true to how many n
For example $$2+1=3$$ is a prime$$2\times 3+1=7$$ is a prime$$2\times 3\times 5+1=31$$ is a prime$$2\times 3\times 5\times 7+1=211 $$ is a prime$$2\times 3\times 5\times 7\times 11+1... | The proof relies on the fact that every prime is in that product, and that a prime can't divide both a number and that number plus one.
Assume there are finitely many primes. If $c$ is their product, then $p$ divides $c$ for any prime $p$. Therefore $p$ does not divide $c+1$ for any prime $p$. This is a contradicti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to factorize $a^2-b^2-a+b+(a+b-1)^2$? The answer is $(a+b-1)(2a-1)$ but I have no idea how to get this answer.
| Using $a^2-b^2=(a+b)(a-b)$ we find $a^2-b^2-a+b=(a+b-1)(a-b)$. The first factor also occurs in the remaining summand.
| {
"language": "en",
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Prove that if the absolute value of a sequence converges to $0$, the sequence converges to $0$ as well. Let ($a_n)_{n \in \mathbb{N}}$ be a sequence, prove that if $|a_n|$ converges to $0$ then ($a_n)_{n \in \mathbb{N}}$ converges to 0 as well.
Now let $|a_n|$ converge to $0$ and let $\epsilon > 0$ that means that $||... | You're almost there:
$||a_n|-0| = ||a_n|| = |a_n| = |a_n-0|$
| {
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"timestamp": "2023-03-29T00:00:00",
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let $f : [0,1]^2\rightarrow \mathbb{R}$ be defined by setting $f(x, y) = 0$ if $y \neq x$, and $f(x, y) = 1$ if $y = x$. Show that $f$ is integrable You have tried the following way: Give $\varepsilon>0$, Take any partition P, $0=t_0<t_1<...<t_n=1$ of $[0,1]$ so $U(P,f)-L(P,f)=\sum_{i=1}^{n}(M_i-m_i)\Delta t_i$ And li... | Consider a tagged partition $P \equiv x_{i,j}= (\frac{i}{n},\frac{j}{n})$ with $0 \le i \le n$ and $0 \le j \le n$ where $n \in \mathbb N$ and $t_{i,j} \in (x_{i,j},x_{i+1,j}) \times (x_{i+1,j},x_{i+1,j+1})$ for $0 \le i \le n-1$ and $0 \le j \le n-1$.
Then you can prove that
$$0 \le \sum_{0 \le i \le n-1, 0 \le j \le ... | {
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Notation for length of arbitrarily long sequences in a set Suppose I have the set of arbitrarily long sequences of dice rolls:
\begin{equation}
\Omega = \bigcup_{n = 1}^\infty [6]^n = \{ (1),(2),(3),(4),(5),(6),(1,1),(1,2),\ldots \}
\end{equation}
Given some sequence $\omega \in \Omega$, what's the notation to get the ... | I have seen each of the following notations used to represent the length of a finite sequence $\sigma$:
*
*$\vert\sigma\vert$
*$length(\sigma)$
*$lh(\sigma)$
Personally, I think all three are perfectly fine, although it's worth spending a sentence saying what your notation means.
Note that "$\vert\sigma\vert$" is ... | {
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"url": "https://math.stackexchange.com/questions/2063671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Recovering initial conditions from observation of y(t) Consider the following state-space representation of a plant:
$$
x\dot(t) = Ax(t) + Bu(t)
$$
$$
y(t) = Cx(t)
$$
If the pair (A,C) is observable, is it true that under any causal control law that yields continuous $u(t)$ as a function of current and past values of $... | If $(C,A)$ is observable then the observability gramian $W_O(t):=\int_0^t{e^{A^T\tau}C^TCe^{A\tau}d\tau}$ is positive definite for all $t>0$. The output response is given by
$$y(t)=Cx(t)=Ce^{At}x(0)+\int_0^t{e^{A(t-s)}Bu(s)ds}$$
If we now define the known signal $$z(t):=y(t)-\int_0^t{e^{A(t-s)}Bu(s)ds}$$
then
$$Ce^{At}... | {
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"timestamp": "2023-03-29T00:00:00",
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Function continuous at odd numbers from $0$ to $99$, but discontinuous everywhere else. I have an idea about this but am not entirely sure if it's right.
If I have a function, say $g(x) = (x-1)(x-3)(x-5)...(x-99)$
The roots of g(x) are only odd numbers from 0 to 99.
$$
f(x) = \begin{cases}
g(x) & x \in\mathbb{... | Yes, your approach is fine. We can simplify a bit. Given a finite set of rationals $A$, let $d(x)$ be the distance from $x$ to $A$.
then the function $f(x)= \begin{cases} d(x) & x \in\mathbb Q \\ 0 & x\not\in \mathbb Q \end{cases}$
Is continuous only at $A$.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Using a direct proof to show argumentative validity Can anybody either verify or dispute the my proof for the following argument?
Premise 1: (E • I) v (M •U)
Premise 2: ~E
Conclusion: ~(E v ~M)
Proof:
(1) Applying DeMorgan's Second Law to the Conclusion; The Negation of a Disjunction, it is the case that ~(E v ~M) is... | Although the OP's proof appears correct, it may help to use a proof checker. The drawback of using such a tool is that one is forced to use the inference rules available. The benefit is the added confidence one has that one's proof is correct.
Here is a proof using a Fitch-style proof checker.
Links to the proof chec... | {
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$e + τ$ is irrational proof check I was reading the Tau Manifesto (no offence to pi fans) and realized you could do as follows. Starting with the Euler identity for a full rotation:
$$e^{iτ}=1$$
If $e+τ=\frac{p}{q}$ then:
$$e^{i(p/q-e)}=1$$
$$e^{ip/q}=e^{ie}$$
$$i\frac{p}{q}=ie$$
$$\frac{p}{q}=e$$
Which we know is fals... | As others have already noted, the complex exponential function is not one-to-one; specifically, since $e^{\tau i}=1$, for any $a, b$ with $b = a + n\tau$ for some integer $n$, we would have $e^{ai} = e^{bi}$. Therefore, if $e^{ai}=e^{bi}$ then the most we can conclude is that $ai = bi + n \tau i$ for some $n$.
In your... | {
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Use continuity to show that $f(x)=x^3$ is uniformly continuous on $[0,1]$ but not $[0,\infty]$ I'm trying to use continuity to show that $f(x)=x^3$ is uniformly continuous on $[0,1]$ but not $[0,\infty)$.
I've tried setting up an epsilon-delta proof, but I'm struggling a little:
By definition of uniform continuity, we ... | To show that $x^3$ fails to be uniformly continuous on $[0,\infty)$, we take $\epsilon=\frac{3}{2}$. Then, for all $\delta>0$, and for $x=\frac{1}{\sqrt\delta}$ and $y=\frac{1}{\sqrt \delta}+\frac{\delta}{2}$ we have $|x-y|<\delta$ and
$$\begin{align}|x^3-y^3|&=\left|\left(\frac{1}{\sqrt\delta}+\frac{\delta}{2}\right)... | {
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"timestamp": "2023-03-29T00:00:00",
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What will happen if minimal polynomial co-incides with the characteristics polynomial? If $A$ is a $n \times n$ matrix such that it's minimal polynomial co-incides with the characteristics polynomial then can we claim that any $n-th$ degree polynomial which annihilates $A$ co-incides with the characteristics polynomial... | There are other things that happen when the minimal polynomial and characteristic polynomial coincide; note that we demand both monic...
First, while there may be eigenvalues with multiplicity greater than one, nevertheless each eigenvalue occurs in a single Jordan block.
Second, if we call our matrix $A,$ then any mat... | {
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"source": "stackexchange",
"question_score": "1",
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Is $(\frac{1}{x})\cos(\frac{1}{x})$ continuous? $$f(x) = \begin{cases}\frac{1}{x} \,\cos\frac {1}{x} &\text{if } x>0 \\
0 & \text{if } x = 0 \end{cases}$$
Is this function continuous?
My intuition says no because as $x$ approaches $0$, $f(x)$ approaches $\infty$.
Is that a good enough reason?
What about $f(x) = \left(... | Yes, it is good enough reason. To formalize it just notice that for $f$ to be continuous at $0$ we would need $$0=f(0)=\lim_{k\to+\infty}f\left(\frac{1}{2k\pi}\right)=\lim_{k\to+\infty}2k\pi=+\infty,$$
which is absurd.
For your second question, $f(x)=(1/x)^a=e^{-a\log x}$ is continuous in $(0,+\infty)$ but not at $0$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2064372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing $\lvert \frac{x^2-2x+3}{x^2-4x+3}\rvert\le1\Rightarrow x\le0$
$\lvert \frac{x^2-2x+3}{x^2-4x+3}\rvert\le1\Rightarrow x\le0$
How is the proof. If I separate the denominator with triangle inequality,
$\lvert \frac{x^2-2x+3}{x^2-4x+3}\rvert\le \frac{\lvert x^2-2x+3\rvert}{\lvert x^2-2x+3 \rvert-\lvert 2x\rvert}\... | What you want is
$$\left|\frac{x^2-2x+3}{x^2-4x+3}\right|\le1\iff-1\le\frac{x^2-2x+3}{x^2-4x+3}\le1$$
Beginning with the left inequality:
$$-1\le\frac{x^2-2x+3}{x^2-4x+3}\iff\frac{x^2-2x+3}{x^2-4x+3}+1\ge0\iff\frac{2x^2-6x+6}{x^2-4x+3}\ge0\iff$$
$$\frac{x^2-3x+3}{(x-1)(x-3)}\ge 0\iff (x-1)(x-3)>0\;\text{ (why?)}\implie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2064501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Examples of topology that results in a connected topological space when $X = \{1,2,3\}$ Just starting off a chapter on Connectedness in Adams Introduction to Topology: Pure and Applied and I'm looking to get a concrete example ..
The book states, "Let $X$ be a topological space. We call $X$ connected if there does not ... | The first three examples work fine. Note that your fourth example $T = \{X, \emptyset, \{1\}, \{2\}, \{1, 3\}\}$ is not a topology, since the union of the open sets $\{1\}$ and $\{2\}$ is not open. Your fifth example does work however.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that a local diffemorphism $f$ preserves the Gauss measure if and only if some condition is met I was reading through these lecture notes and found this exercise (page 6)
Let $f:U\rightarrow U$ be a local $C^1$ diffeomorphism, and let $\rho$ be a continuous function. Show that $f$ preserves the measure $\mu=\rho... | I see the confusion: the statement you are trying to prove is general and unrelated to the Gauss measure.
Apply the area formula. For any integrable function $u : U \to \mathbb R$ you have $$\int_{f^{-1}(U)} u(x) |\det Df(x)| \, dx = \int_U \sum_{x \in f^{-1}(y)} u(x) \, dy.$$
If $B \subset U$ is measurable you can tak... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2064819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Exponent of a direct product of groups Prove that if the group $G=\prod_{i=1}^nH_i$, where each $H_i$ is a finite group, then the exponent of $G$ which is
$\exp(G)=\min\{n \in \mathbb{N}:g^n=e, \forall g \in G\}$ is equal with $\operatorname{lcm}(\exp(H_1),\ldots,\exp(H_n))=M$
I proved that $\exp(G) \leqslant M$
Can so... | You show the other direction by identifying an element of the product which has order M. Hint: Find a tuple that maximizes each coordinate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2064948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $a(n)=n^2+1$ then $\gcd(a_n,2^{d(a_n)})=1\text{ or }2$? Let $n\in\mathbf{N}$. I write $a_n=n^2+1$ and let $d(a_n)$ count the number of divisors of $a_n$. Set $$\Phi_n=\gcd\left(a_n,2^{d\left(a_n\right)}\right)$$ I would like to show and I believe it to be true that
$$\Phi_n =
\begin{cases}
1, & \text{if $n$ is eve... | Note that $2^{d(a_n)}$ can only be divisible by $1$ and powers of $2$.
If $n$ is even then $n^2+1$ is odd and in that case $\gcd=1$.
If $n$ is odd, then $n^2+1 \equiv 2 \pmod{4}$. Thus $n$ is not divisible by $4$, hence the $\gcd=2$.
Added explanation:
Using the division algorithm, we can write any integer $n=4k+r$, w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Operations with $\frac{dy}{dx}$. While working on a problem I have found a solution. I am curious about a clean and correct way to write it down. I want to find the derivative of $y(x)=:y$.
$$(5y^4+1)\frac{dy}{dx} + 1 = 0\\ \frac{dy}{dx} = -\frac{1}{5y^4+1}$$
Is it mathematically correct to divide by $(5y^4+1)$? I am a... | Yes, what you're doing is perfectly "legal". Note that $\frac{dy}{dx}$ is the differentiation operation $\frac{d}{dx}$ being applied to $y$, and is thus the result of applying an operator; which is a function. Meanwhile $\frac{d}{dx}$ on its own is the differential operator, which maps functions to functions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2065159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the last Digit of $237^{1002}$? I looked at alot of examples online and alot of videos on how to find the last digit But the thing with their videos/examples was that the base wasn't a huge number. What I mean by that is you can actually do the calculations in your head. But let's say we are dealing with a $3$ di... | You want to know the last digit of $237^{1002}$, which is the same as the remainder of $237^{1002}$ after division by $10$. This calls for modular arithmetic. From $237\equiv7\pmod{10}$ it follows that
$$237^{1002}\equiv7^{1002}\pmod{10}.$$
Now the base number is small; can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2065373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 11,
"answer_id": 0
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Showing that $f \in C^{\infty}(\mathbb{R},\mathbb{R})$ Let $f(x)=\begin{cases}
e^{-(1/x)} & \text{for} \quad x > 0 \\
0 & \text{for} \quad x \leq 0
\end{cases}$
Show that $f \in C^{\infty}(\mathbb{R},\mathbb{R})$
I need to show that $f(x)$ has derivatives of all orders at all points in $\mathbb{R}$. It is trivial for $... | You should be able to show $f$ is smooth for $x > 0$ by hand - just use the chain rule (both $\exp(x)$ and $1/x$ are smooth for $x >0$ is the point).
$x = 0$ is the tricky part. Let's first show it's differentiable at $0$. $$\lim \limits_{h \to 0^+} \frac{f(h)}{h} = \lim\limits_{h \to 0^+} \frac{e^{-1/h}}{h} = \lim_{x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2065439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $\sqrt[n+1]{n+1}-\sqrt[n]{n}\sim-\frac{\ln{n}}{n^2}$
Show that
$\sqrt[n+1]{n+1}-\sqrt[n]{n}\sim-\frac{\ln{n}}{n^2}$,
when $n\to+\infty$
I'm learning Taylor's Formula. The given solution is:
$\sqrt[n+1]{n+1}-\sqrt[n]{n}=e^{\frac{\ln(n+1)}{n+1}}-e^{\frac{\ln{n}}{n}}$
and use Taylor's Formula:
$e^{\frac{\... | You just need to handle the subtraction of series for $\sqrt[n+1]{n+1}$ and $\sqrt[n] {n} $ term by term in a proper manner upto 3 terms. The first term $1$ cancels out in both series. The second terms upon subtraction lead to
\begin{align}
A &= \frac{\log(n+1)}{n+1}-\frac{\log n} {n} \notag\\
&= \frac{\log(n+1)}{n+1}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2065510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to integrate $\int_a^b (x-a)(x-b)\,dx=-\frac{1}{6}(b-a)^3$ in a faster way? $\displaystyle \int_a^b (x-a)(x-b)\,dx=-\frac{1}{6}(b-a)^3$
$\displaystyle \int_a^{(a+b)/2} (x-a)(x-\frac{a+b}{2})(x-b)\, dx=\frac{1}{64}(b-a)^4$
Instead of expanding the integrand, or doing integration by part, is there any faster way to ... | You can first get rid of the integration bounds by the linear transform $a+(b-a)t$:
$$\int_a^b (x-a)(x-b)\,dx=(b-a)^3\int_0^1t(t-1)\,dt.$$
Mentally expanding the polynomial, the integral is $\frac13-\frac12=-\frac16$.
For the other case, $a+(b-a)t/2$:
$$ \int_a^{(a+b)/2} (x-a)(x-\frac{a+b}{2})(x-b)\, dx=\frac{(b-a)^4}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2065639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Alternative to Geogebra In Norwegian schools, Geogebra is widely used for plotting graphcs, calculus, algebra, etc. However, by the looks of it, it is not very commonly used, so the documentation and resources is very limited (especially on the Computer Algebra System). Is there any good alternatives to it? What is mos... | Try Mathematica: https://www.wolfram.com/mathematica/. It might be a little overpowered for what you're using it for, but I can almost guarantee it'll be sufficient.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Proving that $\det(A) \ne 0$ if $a_{i,i} = 0$ and $a_{i,j} = \pm 1$ for $i \neq j$
Let $A$ be an $n\times n$ matrix ($n=2k$, $k \in \Bbb N^*$) such that.
$$a_{ij} =
\begin{cases}
\pm 1, & \text{if $i \ne j$} \\
0, & \text{if $i=j$}
\end{cases}$$
Show that $\det (A) \ne 0$.
P.S. $a_{ij}=\pm 1$ means that it can be $... | Here's an approach which connects to your original idea of writing out the definition of $\det A$ as a sum over permutations:
If you to this, you get $n!$ terms, each of which is either $+1$, $-1$ or $0$. The terms which are zero are the ones where you include at least one matrix entry from the diagonal, so they corres... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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"answer_id": 2
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Condition for $|AB|^2+|BC|^2+|CA|^2$ to be maximum Let $|\vec{OA}|=l,| \vec{OB}|=m,| \vec{OC}|=n$.$O$ be the origin.$A,B,C$ lie on the plane $x+2y-z=0$ and $|AB|^2+|BC|^2+|CA|^2$ is maximum,then the value of $|AB|$ is
I tried to convert the equation of plane in vector.clearly the plane passes through origin.The equatio... | Points $A,B,C$ belong resp. to spheres centered in $O$ with radii $\ell,m,n$. Thus, any plane passing through the origin intersects these spheres along a diameter, i.e., as circles, with the same radii. Let us choose one of them. All the following is a 2D problem with coordinates in this plane.
Let us give names $a,b,c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2066023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$f_x$ is Borel measurable and $f^y$ is continuous then $f$ is Borel measurable I have to prove the following:
Let $f: \mathbb{R^2}\to \mathbb{R}$ such that $f_x:y\to f(x,y)$ is Borel measurable for all $x\in\mathbb{R}$ and that $f^y:x\to f(x,y)$ is continuous for all $y\in\mathbb{R}$. Prove that $f$ is Borel measurable... | By the continuity of $f^y$ we have
$$f(x,y) = \lim_{n \to \infty}f(\lfloor nx \rfloor / n, y).$$
By the measurability of $f^x$, we see that $f$ is the pointwise limit of a sequence of Borel measurable functions, and hence is itself Borel measurable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2066145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Application of Min Cost Flow to hostel bookings
A hostel made a mistake concerning their bookings for 2017 and took many reservations without checking for free rooms in these periods.
Every reservations is made for exactly one room and one period of time.
All rooms are equal but were sold for different prices.
T... | I am unable to comment on Kuifje's answer, but hopefully the following is interesting.
I agree with the modelling approach taken by Kuifje. In addition, to account for the limited room capacity, a further arc that connects the sink, $t$, to the source, $s$, can be added with a capacity equal to the number of available... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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The sum of the fourth powers of the first $n$ positive integers I am studying mathematical induction and most of the times I have to prove something. Like, for example:
$1 + 4 + 9 + ...+ n^2 = \frac{n(n+1)(2n+1)}{6}$
This time I found a question that ask me to find a formula for
$1 + 16 + 81 + .... + n^4$
How can I do... | As $S_0=0$ and $S_n-S_{n-1}=n^4$, $S_n$ must be a polynomial of the fifth degree with no independent term, let
$$S_n=an^5+bn^4+cn^3+dn^2+en.$$
Then
$$S_n-S_{n-1}=\\
a(n^5-n^5+5n^4-10n^3+10n^2-5n+1)+
\\b(n^4-n^4+4n^3-6n^2+4n-1)+\\
c(n^3-n^3+3n^2-3n+1)+\\
d(n^2-n^2+2n-1)+\\
e(n-n+1)=\\
a(5n^4-10n^3+10n^2-5n+1)+
\\b(4n^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2066334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to use the implicit function theorem? Consider a gas field with K > 0 cubic meters of gas at the
start of the planning horizon. The price of gas changes over time: one
cubic meter of gas can be sold for $m \cdot exp\{st\}$ euros at time $t$, where $m$ > 0 and $s \in \mathbb{R}$.
Extracting gas is costly: if the ex... | If we differentiate both sides of $K-S + \frac{m e^{st}}{r} = m e^{st}T^*(S) + \frac{m e^{st}}{r}e^{-rT^*(S)}$ with respect to $S$, we get
$-1 = m e^{st} {d T^*(S) \over dS} - m e^{st} e^{-r T^*(S)} {d T^*(S) \over dS}$.
Factoring out ${d T^*(S) \over dS}$ gives
${d T^*(S) \over dS} = { -1 \over me^{st}(1-e^{-r T^*(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2066474",
"timestamp": "2023-03-29T00:00:00",
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Formula for smallest distance between two parabolas I have been struggling with this problem I came across:
Create a general formula for finding the closest points between two
parabolas. Given that the parabolas have opposing concavity and are not interesecting.
I want to answer this problem in the simplest way pos... | I have an approach,
but there probably is one or more errors,
so I'll enter what I have
and hope that
others can
correct/complete this.
To start,
the two parabolas
have opposite direction,
so we can write them as
$f(x) = x^2+ax+b$
and
$g(x) = -x^2+cx+d$.
The derivatives are
$f'(x) = 2x+a$
and
$g'(x) = -2x+c$.
If the sl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2066803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Problem in solving a question concerning real analysis. The question is :
Does there exist any function $f : \mathbb R \longrightarrow \mathbb R$ such that $f(1) = 1$, $f(-1) = -1$ and $|f(x) - f(y)| \leq |x - y|^{\frac {3} {2}}$?
It is clear that $f$ is continuous over $\mathbb R$ by the given condition and hence it a... | Notice that if $|f(x) - f(y)| \leq {|x - y|}^{\frac {3} {2}}$, then
$$\left|\frac{f(x)-f(y)}{x-y}\right|\leq {|x - y|}^{\frac {1} {2}}$$
and hence $f$ is differentiable... but $f'=0$ everywhere. In other words, $f$ is constant, so one may not have $f(1)\neq f(-1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2066873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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How do I maximize entropy? In the book on probability I am reading, I am asked to prove that the entropy of $X$ is maximized when $X$ is uniformly distributed.
At first I came up empty and decided to check online. Most proofs made use of the AM-GM inequality which the book did not cover, so I was wondering if I could... | One approach:
*
*Say your probability distribution takes values in $\{1,\cdots,n\}$.
*Then $H(X) = \mathbb{E}[\log\frac{1}{p(X)}] \le \log \mathbb{E} \left[ \frac{1}{p(X)} \right] = \log n$, by Jensen's inequality applied to the concave function $f(x)=\log x$.
*Equality holds when $p(X)$ is constant, i.e. when $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2066966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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If $P$ and $Q$ are invertible matrices $PQ=-QP$, then which claim about their traces is true? If $P$, $Q$ are invertible and $PQ=-QP$, then what can we say about traces of $P$ and $Q$.
I faced this question in an exam but according to me this question is wrong as $Q=-P^{-1}QP$, which implies $\det(Q)=0$ and it implies ... | Pre-multiply by $P^{-1}$ to get $Q=P^{-1}(-Q)P$ which implies that matrices $Q$ and $-Q$ are similar, so $tr(Q)=tr(-Q)\implies tr(Q)=0$. Similarly you can get $tr(P)=0$ also.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2067063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Calculate the expected value of the highest floor the elevator may reach. I've been to solve this exercise for a few hours now, and all the methods I use seems wrong, I'll be glad if someone could solve this for me, since I don't know how to approach this correctly.
Given a building with 11 floors while the bottom floo... | The highest floor that the elevator reaches is the maximum $M$ of the floors chosen by the 12 people.
For $m = 1,\dots,10$, the probability that $M \leq m$ is just the probability that all 12 people chose floors less than or equal to $m$, $$\left(\frac {m}{10}\right)^{12}$$
So the probability that $M=m$ is $$\left(\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2067253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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7 distinct trucks are sent to 3 different cities A,B,C.what is the number of possibilities? If exactly 2 trucks were sent to city A ,and exactly 4 trucks for city B and exactly 1 truck for city C
here is my thought process ,i looked at city C first and said there is 7 different possibilities ,then i looked at A and sai... | Your thought process is indeed correct. Expressing your solution in a slightly different fashion we could also say:
There $7$ trucks to choose from and we want to choose $2$ for $A$, out of the remaining $5$ trucks we want to choose $4$ for $B$ and we are left with $1$ truck that must go to $C$.
Then expressing our cho... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Polynomials $\&$ Matrices Assume $A$ is a matrix of order $n$. We know that the characteristic polynomial of matrix $A$ is obtained as follows
$$
P(x)=\det (A-x\,I)\, .
$$
Where $I$ is an identity matrix of order $n$. What about inverse? For a given polynomial
like $P(x)$, Is there an efficient method to find a
mat... | Yes. What you are looking for is the companion matrix.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2067446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $(n+1)^{n-1}(n+2)^n>3^n(n!)^2$
Why is $(n+1)^{n-1}(n+2)^n>3^n(n!)^2$ for $n>1$
I can use $$(n+1)^n>(2n)!!=n!2^n$$ but in the my case, the exponent is always decreased by $1$, for the moment I don't care about it, I apply the same for $n+2$
$(n+2)^{n+1}>(2n+2)!!=(n+1)!2^{n+1}$
gathering everything together,
$... | EDIT: This answer is wrong, because I mixed up my left and right-hand sides right at the end. I think it is salvageable, but it'll be quite a bit of work.
I'll do it without induction.
Rearrange: we want $\left(\frac{(n+1)(n+2)}{3}\right)^{n-1} \frac{n+2}{3} > (n!)^2$
We'll show that this actually holds if we remove th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluation of the integral $\int^{\infty}_{0} \frac{dx}{(x+\sqrt{x^2+1})^n}$ $$
\mbox{If}\ n>1,\ \mbox{then prove that}\quad\int_{0}^{\infty}{\mathrm{d}x \over
\left(x + \,\sqrt{\, x^{2} + 1\,}\,\right)^{n}} = {n \over n^{2} - 1}
$$
Could someone give me little hint so that I could proceed in this question. I tried put... |
Could someone give me little hint so that I could proceed in this question.
Hint. One may perform the change of variable
$$
x \in [0,\infty),\quad x=\sinh u \implies x+\sqrt{x^2+1}=e^u, \quad dx=\cosh u\:du,
$$ giving
$$
\int^{\infty}_{0} \frac{dx}{(x+\sqrt{x^2+1})^n}=\int^{\infty}_{0} e^{-nu}\cosh u\:du.
$$ Can you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2067657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Write a linear optimization problem to find a hyperplane that strictly separates two disjoint polyhedra.
Let $P_{1}=\left\{x\:|\: Ax\leq b\right\}$ and $P_{2}=\left\{x\:|\: Cx\leq d\right\}$ be two disjoint polyhedra. Write a linear optimization problem to find a hyperplane that strictly separates $P_{1}$ of $P_{2}$.
... | We consider the functions
$$p^{*}_{1}(a):=\inf_{x\in P_{1}}a^{T}x \: \:\:\:\: \mbox{ and } \: \:\:\:\: p^{*}_{2}(a):=\sup_{x\in P_{2}}a^{T}x.$$
If hyper-plane $a^{T}x=\alpha $ separates $P_{1}$ of $P_{2}$ then we should have $p^{*}_{1}(a)<\alpha < p^{*}_{2}(a)$, or what is the same, $0<\alpha -p^{*}_{2}(a)<p^{*}_{1}(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2067825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Review of my T-shirt design I'm a graphics guy and a wanna-be mathematician. Is the T-shirt design below okay? Or if there's a bone headed error, I'd appreciate a heads up.
`
| Yet another pleasing way to write the sum:
$$
\left. \pi^2\big /12\right. = \sum_{n=1}^\infty (-1)^{n+1}\big/n^2
$$
say, if you wanted to take more horizontal space.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2067911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "62",
"answer_count": 3,
"answer_id": 1
} |
How do I solve quadratic equations when the coefficients are complex and real? I needed to solve this: $$x^2 + (2i-3)x + 2-4i = 0 $$
I tried the quadratic formula but it didn't work. So how do I solve this without "guessing" roots? If I guess $x=2$ it works; then I can divide the polynomial and find the other root; but... | The square root is not a well defined function on complex numbers. If you want to find out the possible values, the easiest way is probably to go with "Polar form", that is, converting your number into the form
$$r(\cos(\theta) + i \sin(\theta))$$ and then taking root of it,
where $r$ is the modulus of the complex nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2067993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 8,
"answer_id": 2
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What is the intuition behind the formula for the average? Why is the average for $n$ numbers given by $(a+b+c+\cdots)/n$? I deduced the formula for the average of 2 numbers which was easy because its also the mid point, but I couldn't do it for more than 2 numbers.
|
If Bill Gates walked into a crowded bar, on an average, everyone is a millionaire.
Loosely, an average is supposed to be a representative value for a sample. Sort of. But as you can see, it needn't be the case always.
But every time, average is definitely this: if what we collectively have is distributed equally am... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 0
} |
uniqueness of neutral element for matrix addition confusion I'm reading Basic Linear Algebra 2e (T.S.Blyth and E.F.Robertson) and have come across the following theorem:
Theorem
There is a unique $m \times n$ matrix $M$ such that, for every $m \times n$ matrix $A$ one has $A + M = A$.
Proof
Consider the matrix $M = [m... | This uses a technique very common in uniqueness proofs. It goes like this:
*
*Assume $X$ and $Y$ both have the properties that we want
*Show that actually, $X = Y$, so $X$ was unique.
In this case, they are using the fact that that $A + B = A$ for any matrix $A$ to say that $M + B = M$. This is allowed because $M$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $x, y \leq 500$ then find the number of nonnegative integer solutions to $4 x - 17y = 1$
If $x, y \leq 500$ then find the number of nonnegative integer solutions to $4 x - 17y = 1$.
I don't know how to proceed. Please help me out. Thank you.
| We are asked to find integer solutions for $x,y$.
We can get particular solutions simply by plugging in values. Our first solution is obtained as: $y’ = 3$ and $x’ = 13$.
Thus, the general solution for x will be:
$$x = x’ + bn = 13 + 17n$$
Hence lower limit for $n$ is $n \geq 0$.
Now we’ll find upper limit using given ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2068233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$3$-Sylows of a simple group of $168$ elements I'm trying to find how many $3$-Sylows there are in a simple group of $168$ elements.
Let $n_3$ denote the number of $3$-Sylows.
By the Sylow theorems I know that the possibilities for $n_3$ are $1,4,7,28$. $1$ is not possible since the group is simple. $4$ is also not pos... | The group is isomorphic to ${\rm PSL}(2,7)$, which is a doubly-transitive group of degree $8$. Since $168 = 8 \times 7 \times 3$, a $2$-point stabilizer has order $3$.
By considering a diagonal $2 \times 2$ matrix of determinant one over ${\mathbb F}_7$, whose entries are $w$ and $w^{-1}$ with $w$ of order $3$, you can... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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continuous surjective functions from $(a,b$] to $(a,b)$ The problem is does there exist a continuous surjective function from
$(a,b]$ to $(a,b)$
I am really not sure how to prove it but I do not think that it is possible. As
$f(b)$ has to equal something but the function has to get close to $a$ and also $b$.
Many t... | Yes, there is a continuous, surjective function $f:(a, b]\to (a, b)$. Such an $f$ is given by, for instance,
$$
f(x) = \frac{b-a}2e^{-x+a}\sin\left(\frac{1}{x-a}\right) + \frac{b+a}{2}
$$
As $x$ increases (toward $b$), the $e^{-x+a}$ factor will flatten the first term out so that $f(x)$ comes close to $\frac{b+a}2$. As... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2068544",
"timestamp": "2023-03-29T00:00:00",
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Prove that for any $k \mathbb \in N^*$ Prove that for any $k \mathbb \in N^*$: $$\frac{1}{2\sqrt{(k+1)^3}} \leq \frac{1}{\sqrt{k}}-\frac{1}{\sqrt {k+1}} \leq \frac{1}{2\sqrt{k^3}}$$
I have tried to use simple induction but i didn't get a good result
| Notice that
$$
\frac1{\sqrt{k}}-\frac1{\sqrt{k+1}}=\frac1{\sqrt{k}\sqrt{k+1}(\sqrt{k}+\sqrt{k+1})}
$$
and we have
$$
\sqrt{k}\sqrt{k+1}(\sqrt{k}+\sqrt{k+1})\le \sqrt{k+1}\sqrt{k+1}(\sqrt{k+1}+\sqrt{k+1}) = 2\sqrt{(k+1)^3}$$
and similarly
$$
\sqrt{k}\sqrt{k+1}(\sqrt{k}+\sqrt{k+1})\ge \sqrt{k}\sqrt{k}(\sqrt{k}+\sqrt{k}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2068618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it possible to stack perfect spheres? Is it possible to stack a perfect sphere on top of another? It is easy to stack a cube on top another, but as the faces of the shapes increase, it seems more and more difficult to stack. So, is a sphere with (infinite sides or no sides?) "stackable?" This scenario does not have ... | A sphere technically has infinite sides if it is "perfect", because it is entirely smooth. This doesn't ever occur in reality, as once we magnify any given edge of matter that is smooth we will find imperfections in the material. Even if we didn't, then at the atomic level we would find the nuclear structure that mak... | {
"language": "en",
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Find a sequence such that this tower of of exponent is convergent Context
*
*We already know that if we take a sequence $(x_n)\in{\mathbb R_+^*}^{\mathbb N}$ such that
$$x_n=O\left(\frac 1{n^2}\right)$$
then
$$\sum_{n=0}^\infty x_n <+\infty.$$
*
*We also now that if we take for instance for all $n\in \mathbb N^*$
... | The lame answer is to have $x_0=1$ or $x_0=0,x_1>0$. Then the result is trivial.
If $x_n=x_0$ for all $n$ and $x_0>0$, then it converges iff $e^{-e}\le x_0\le e^{e^{-1}}$, which actually does not require $\lim_{n\to\infty}x_n=1$. These are found in the Wikipedia for tetration. More information on the exact nature of... | {
"language": "en",
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"question_score": "6",
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Given a positive number n, how many tuples $(a_1,...,a_k)$ are there such that $a_1+..+a_k=n$ with two extra constraints The problem was: Given a positive integer $n$, how many tuples $(a_1,...,a_k)$ of positive integers are there such that $a_1+a_2+...+a_k=n$. And $0< a_1 \le a_2 \le a_3 \le...\le a_k$. Also, $a_k-a_1... | The answer is $n$.
Given $n$, you need to proof that for each $k$, where$k\leq n$, there exists exactly one tuple.
First, you can proof that for each $k$, there exists at least one tuple.
$n=kt+r$
where $r<k$.
Make the tuple $(a_1,...,a_k)=(t,...,t)$. Then add to the last $r$ components $1$ unit to get a valid tuple.
S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2068875",
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"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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What is an example of infinite dimensional subspace that is not closed? In a theorem I am reading about closed subspace the author states that an infinite dimensional subspace need not be closed.
What is an example of infinite dimensional subspace that is not closed?
| Let $\ell^2$ be the space of all square-summable real (or complex) sequences $x = (x_1,x_2, \ldots)$ with norm $\|x\| = \displaystyle ( \sum |x_i|^2)^{1/2}$. Let $V \subset \ell ^2$ be the subspace of all sequences with all but finitely many entries equal to zero. Then $V$ is infinite-dimensional but not closed. It is ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Getting certain derivative value from a system of differential equations I have the following differential equations system:
$\frac{dS}{dt} = -0.001SI$
and
$\frac{dI}{dt} = 0.001SI - 0.3I$
How do I retrieve the value of $\frac{dI}{dS}$ ?
I know its supposed to be $\frac{dI}{dS} = -1 + \frac{300}{S}$
| $$\dfrac{dI}{dS} = \dfrac{dI/dt}{dS/dt} = \dfrac{0.001 SI -0.3I}{-0.001SI}= -1 + \dfrac{300}{S}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2069183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving momentum and mass equation
A rocket has velocity $v$. Burnt fuel of mass $\Delta m$ leaves at
velocity $v-7$. Total momentum is constant:
$$mv=(m-\Delta m)(v+\Delta v) + \Delta m(v-7).$$
What differential equation connects $m$ to $v$? Solve for $v(m)$ not $v(t)$, starting from $v_0 = 20$ and $m_0 = 4$.
... | To formulate the equation correctly, you should be considering the mass of the rocket at time $ t+\delta t$ as $ m\color{red}{+}\delta m$ (and velocity as $v+\delta v$). By conservation of mass, the particle of ejected mass is$\color{red}{-}\delta m$.
All such variable mass equations should be set up this way. This is ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A natural number is a perfect square as well as a perfect cube. Show that it is $0$ or $1$ $ ($mod $7$ $)$. A natural number is a perfect square as well as a perfect cube. Show that it is $0$ or $1$ $($mod $7$$)$.
I tried the following.
There are integers numbers $x,y$ such that $n=x^{2}=y^{3}.$ By using Euclidean div... | You know that all the squares are equals to $0,1,2$ or $4$ mod $7$ because:
$$0^2=0\pmod 7$$
$$1^2=1\pmod 7$$
$$2^2=4\pmod 7$$
$$3^2=2\pmod 7$$
$$4^2=2\pmod 7$$
$$5^2=4\pmod 7$$
$$6^2=1\pmod 7.$$
And all the cubes are equals to $0,1$ or $6$ mod $7$ because:
$$0^3=0\pmod 7$$
$$1^3=1\pmod 7$$
$$2^3=1\pmod 7$$
$$3^3=6\pmo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2069374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $2^b-1=2^{b-1}+2^{b-2}+...+1$? Can I get an intuitive explanation why the formula in the title holds?
I know that it works but I am not sure why
$2^b-1=2^{b-1}+2^{b-2}+...+1$
| You can visualize it by looking this identity as identity of polynomials:
$$
(1-x)(1+x)=1-x^2
$$
$$
(1-x)(1+x+x^2)=1-x^3
$$
and in general
$$
(1-x)(1+x+x^2+x^3+\ldots+x^{n-1})=1-x^n
$$ Now on substituting $x=2$, we get the desired result. Hope it helps.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 5
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How can we determine the point at which the distance between vectors is equal to a certain constant? Consider the following points:
$$A(-3,0)\hspace{1cm}
B(3,0)\hspace{1cm}
C(x,y)$$
Now consider the following vectors:
$$CA\hspace{1cm}
CB\hspace{1cm}
CO$$
where $O$ is the origin $O(0,0)$.
Consider the vector $HF$, of ma... | I won't do all the calculations since I didn't find a nice method. However, the reasoning I follow is simple to understand, despite involving horrible expressions.
In the sequel, I denote by $d_{XY}$ a line passing thgrouh distincts points $X$ and $Y$. I also take $C(\alpha,\beta)$ to avoid confusion and since we don't... | {
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Power tower question $$x^{x^{x^{.^{.^{.}}}}} = 8$$
Then how to solve for x?
I first tried like this
$x^8=8$ but I don't get any way to solve.
| There is no way to obtain an analytical solution in terms of elementary functions.
However, one can find an expression in terms of the Lambert W function. This expression evaluates to $8=-\frac{W(-ln(x))}{ln(x)}$. This expression can be solved using numerical methods.
However, as noted by others you may notice that inf... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find what level of the Calkin–Wilf tree a number is on The Calkin–Wilf tree is a tree of fractions where to get the two child nodes, the first child is (the parent's numerator / x) and the second child is (x / the parent's denominator), where x is the sum of the parent's numerator and denominator. (This part was added ... | Let $h(x,y)$ be the number of steps required to get to the pair $(x,y)$, assuming this is possible, or $h(x,y) = -1$ if $(x,y)$ is not reachable.
Clearly $h(x,y) = h(y,x)$, since you can always add in either direction, so we may as well assume $x \leq y$ when evaluating this function.
Also, if either $x$ or $y$ is less... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Summation of a Sequence using previous term You have a data table going from 1st term to 1000th term.
The 1st term is -6 and in the second third of the data table you see a term that gives 1.
From 2nd term to 999th term the value equals the sum of the value of the term right before and right after it.
Which of the foll... | Starting off with our formula $a_n = a_{n-1} + a_{n+1}$, I expanded out the $a_{n-1}$ to get
$$a_n = a_{n-2} + a_n + a_{n+1} \implies a_{n+1} = -a_{n-2}$$
so now our sequence looks like $-6, a_2, a_3, 6, a_5, a_6, -6, \dots$. Since $991 \equiv 1 \mod 6$, $a_{991} = -6$, $a_{994} = 6$, and $a_{997} = -6$. None of those ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the value of $f'(0)$ If $f$ is a quadratic function such that $f(0)=1$ and
$$\int\frac{f(x)}{x^2(x+1)^3}dx$$
is a rational function, how can we find the value of $f'(0)$? I am totally clueless to this. Any tip on how to start? If you wish to give details, then many thanks to you.
| Note that
$${f(x)\over x^2(x+1)^3}={xf(x)\over(x^2+x)^3}$$
Let $u=x^2+x$. Suppose
$$xf(x)=u{du\over dx}=u(2x+1)=(x^2+x)(2x+1)=2x^3+3x^2+x=x(2x^2+3x+1)$$
Then
$$\int{f(x)\over x^2(x+1)^3}dx=\int{xf(x)\over(x^2+x)^3}dx=\int{udu\over u^3}=\int{du\over u^2}=-{1\over u}+C=-{1\over(x^2+x)}+C$$
is a rational function. So $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2070180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Difficulty understanding a financial question
To have $\$50,000$ for college tuition in $20$ years, what gift $y_o$ should a grandparent make now? Assume $c = 10\%$. What continuous deposit should a parent make during $20$ years? If the parent saves $s = \$1000$ per year, when does he or she reach $\$50,000$ and retir... | *
*$y_0$ is the present value of $y=\$\, 50,000$, that is
$$
y_0=y\,\mathrm e^{-ct}=50,000 \times \mathrm e^{-0.1 \times 20}\approx \$\, 6,766.76
$$
*starting with $y_0=0$ the continuous deposit $s$ to obtain $y$ in $t=20$ years is found by
$$
y=\frac{s}{c}\left(e^{ct}-1\right)
$$
that is
$$
s=\frac{yc}{e^{ct}-1}=\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2070262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Green's theorem for piecewise smooth curves Green's theorem is usually stated as follows:
Let $U \subseteq \mathbb{R}^2$ be an open bounded set. Suppose its boundary $\partial U$ is the range of a closed, simple, piecewise $C^1$, positively oriented curve $\phi: [0,1] \to \mathbb{R}^2$ with $\phi(t) = (x(t),y(t))$. Le... | In "A First Course in Real Analysis" by Murray H. Protter and Charles B. Jr. Morrey Green's theorem is proved in paragraph 16.4 prior to proving Stokes' theorem. They prove Green's Theorem for so-called regular regions, which are regions whose boundary is given by piecewise-differentiable curves (along with some other ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determinant of a non-square block matrix $M_{n\times k}$ is defined as a matrix whose all elements are '-1'.
The following block matrix is as such:
$A =\begin{bmatrix}m\cdot I_{n-1} & M_{n-1\times m}\\M_{m\times n-1} & n\cdot I_{m}\end{bmatrix}$
prove the following: $det A = n^{m-1}\cdot m^{n-1}$
| Let more generally let $C(\gamma) = \gamma 1_{n-1}1_{m}^T $ denote the ${n-1}\times m$ matrix with each element equal to $\gamma$ (here $1_k$ denotes the k-dimensional column vector of all ones) and let $A(\gamma) = \begin{pmatrix} mI_{n-1} & C(\gamma) \\ C(\gamma)^T & nI_m\end{pmatrix}$. Your problem is to compute th... | {
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How can I prove that $\mathbb Z[x]/(1+x^2)\mathbb Z[x]$ is a free module with basis $\{1,\bar x\}$? 1) How can I prove that $\mathbb Z[x]/(1+x^2)\mathbb Z[x]$ is a free $\mathbb{Z}$-module with basis $\{1,\bar x\}$?
I wanted to prove that $$\mathbb Z[x]/(1+x^2)\mathbb Z[x]\cong \mathbb Z^2,$$
but it looks complicate.
... | 1) Over any commutative ring $R$, the quotient ring $R[X]/(f(X))$ of $R[X]$ by a monic polynomial is a finitely generated free $R$-module, with rank equal to the degree of the polynomial.
Denoting by $x$ the class of $X$ in the quotient, you just have to prove that any $x^n$, with $n\ge \deg f$ lies in the submodule g... | {
"language": "en",
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"answer_count": 2,
"answer_id": 0
} |
What does this formula/notation mean in boolean algebra/bayesian probability? I am reading Jaynes' "probability theory: the logic of science"
He uses a notation that I do not understand.
He says that if $A_i$ and $A_j$ are two mutually exclusive events, then:
$p(A_i A_j |B) = p(A_i |B)δ_{ij} $
How am I to understand th... | $$P(A_iA_j)=\left.\begin{cases}P(A_iA_i)=P(A_i), &\text{if } i=j\\ P(A_iA_j)=P(\emptyset)=0, &\text{if } i\neq j\end{cases}\right\}=P(A_iA_j)δ_{ij}$$ where $δ_{ij}$ is called Kronecker delta and has the purpose to indicate the event $i=j$. That is $$δ_{ij}=\begin{cases}1, &i=j\\0, &i\neq j\end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2070641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Choose $a, b$ so that $\cos(x) - \frac{1+ax^2}{1+bx^2}$ would be as infinitely small as possible on ${x \to 0}$ using Taylor polynomial $$\cos(x) - \frac{1+ax^2}{1+bx^2} \text{ on } x \to 0$$
If $\displaystyle \cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{4!} - \cdots $
Then we should choose $a, b$ in a such way that it's T... | Hint: Notice that, by its Taylor expansion, $\big(\cos(x)-1\big)\to0$ as $x\to0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2070748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
Is $\sum\limits_{n=1}^{\infty}\frac{1}{n^k+1}=\frac{1}{2} $ for $k \to \infty$? This series :$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^k+1}$ is convergent for every $k>1$ , it's seems that it has a closed form for every $k >1$, some calculations here in wolfram alpha show to me that the sum approach to $\frac{1}{2}$... | Yes, indeed
$$
0\le \lim_{k\to \infty}\sum_{n\ge 2}\frac{1}{n^k+1}\le \lim_{k\to \infty} \int_1^\infty \frac{1}{x^k+1}\mathrm{d}x=\int_1^\infty \lim_{k\to \infty}\frac{1}{x^k+1}\mathrm{d}x=0
$$
by the dominated convergence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2070991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Closed compact unit ball I am reading this proof about compact closed unit ball and finite dimentional space
I am confused about that last paragraph because I am not sure what would change in the proof if $\dim X<\infty$. Would we still have $||x_m-x_n||\geq \frac{1}{2}$?
| The infinite dimension assumption says that whenever we have finitely many $v_1,\ldots,v_n$, the subspace they generate, $X_n$, cannot contain all of $X$. There must be points outside of it, which allows us to apply Riesz, as the subspace is proper. So the recursive construction of the sequence can never halt, we keep ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How many rounds are required in a "Swiss tournament sorting algorithm"? You're organizing a Swiss-style tournament with N players of a game.
The game is a two-player game, and it results in one winner and one loser. The players are totally ordered by skill, and whenever two players play against each other, the more ski... | Just like you seem to have already realized, asking for the number of tournaments $Swiss(n)$ is the same as asking for the span of an optimal parallel sorting network.
I'll just point you to a simple sorting network, the Bitonic Sorter, which gives an $O(\log^2n)$ span.
There is a famous result by Ajtai, Kolmos and Sze... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
approximation for $\Gamma (\alpha) / \Gamma (\beta) $ where $\alpha$ and $\beta$ are arbitrary numbers in $R^{+}$ I am working on implementation of a machine learning method that in part of the algorithm I need to calculate the value of $\Gamma (\alpha) / \Gamma (\beta) $. $\alpha$ and $\beta$ are quite large numbers (... | I think that a good solution would be Stirling approximation that is to say $$\log(\Gamma(x))=x (\log (x)-1)+\frac{1}{2} \left(-\log \left({x}\right)+\log (2 \pi
)\right)+\frac{1}{12 x}+O\left(\frac{1}{x^3}\right)$$ Now, consider $$y=\frac{\Gamma(\alpha)}{\Gamma(\beta)}\implies \log(y)=\log(\Gamma(\alpha))-\log(\Gam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find The eigen value of P Let P,M,N be n$\times$n matrices such that M and N are non singular.If
x is an eigenvector of P corresponding to eigen value $\lambda$,then
an eigenvector of N$^{-1}M$PM$^{-1}N$ corresponding to eigenvalue
$\lambda$is
(a) MN$^{-1}$x (b) M$^{-1}Nx$ (c) NM$^{-1}x$ (d) N$^{-1}Mx$
One more thing t... | As already said $Px=\lambda x$.
$$N^{-1}MPM^{-1}N=K \rightarrow N^{-1}MP=KN^{-1}M \rightarrow N^{-1}MPx=K(N^{-1}Mx) \rightarrow \lambda (N^{-1}Mx)=K(N^{-1}Mx) $$ and so, $K$ has eigenvalue $\lambda$ and eigenvector $N^{-1}Mx$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If you didn't already know that $e^x$ is a fixed point of the derivative operator, could you still show that some fixed point would have to exist? Let's suppose you independently discovered the operator $\frac{d}{dx}$ and know only its basic properties (say, the fact it's a linear operator, how it works on polynomials,... | You want a function $f$ such that $f'=f$. Let's hope that such function exists and has an inverse, $g$ (this is just for motivation: we will prove it in the long run).
We then have that, since $f \circ g =Id$, by the chain rule, $f'(g(x)) g'(x)=1$. Therefore,
$$g'(x)=\frac{1}{f'(g(x))}=\frac{1}{f(g(x))}=\frac{1}{x}.$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
Continuity and derivability of a piece wise f unction Let $f(x)=x^3-9x^2+15x+6$ and
$$g(x)=
\begin{cases}
\min f(t) &\mbox{if } 0\leq t \leq x, 0 \leq x \leq6 \\
x-18 &\mbox{if }x\geq 6. \\
\end{cases}
$$
Then discuss the continuity and derivability of $g(x)$.
Could someone explain be how to deal with $... | To figure out $\min f(t)$ for $0 \leq t \leq x$, it would help to know when $f(t)$ is decreasing and when it is increasing, so we need to know the derivative. We have:
$$f'(t)=3x^2-18x+15=3(x-5)(x-1)$$
This means $f(t)$ is increasing for $0 < x < 1$, decreasing from $1 < x < 5$ and increasing from $5 < x < 6$.
This mea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
4-manifold with $w_1\neq 0$, $w_1^2=0$, $w_2\neq 0$ I wonder if there is a 4-manifold whose Stiefel-Whitney
classes satisfy $w_1\neq 0$, $w_2\neq 0$, and $w_1^2=0$?
There is no 3-manifold whose Stiefel-Whitney
classes are given by the above. For $\mathbb{R}P^4$, $w_1^2\neq 0$.
| Let $M$ be the non-orientable $S^3$ bundle over $S^1$. Its $\Bbb Z/2$ Betti numbers are $b_1 = 1, b_2 = 0, b_3 = 1$. It's non-orientable, so $w_1 \neq 0$, but clearly $w_1^2 = 0$. Now take $M \# \Bbb{CP}^2$. $\Bbb{CP}^2$ has $w_1 = 0$ but $w_2 \neq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
What are the conditions for a second order nonhomogenous ODE with $q(x) = A_1\sin(Bx)+A_2\cos(Bx)$? In my notes it states that it
$y_1$ $\neq$ $\mathbf e$sin(Bx)
as one of the conditions and this is the one I am confused about. There is no $\mathbf e$ in the original equation so is this a typo in my lecture?
| I will interpret the question as follows:
Given that $y=c_1y_1+c_2y_2+y_p$ is a solution of
\begin{equation}
y^{\prime\prime}+ay^\prime+by=A_1\sin(Bx)+A_2\cos(Bx)
\end{equation}
and that there is no solution containing a term of the form $e^{\alpha x}\cos(Bx)$, find a general solution.
Since if $a^2<4b$ the general so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is this an allowed step for working with infinite sequences? I just started learning about infinite sequences, which are of course very interesting. Out of curiosity, I tried doing a proof that:
$${1,-1,1,-1 ...} = 0$$
This was pretty easy to do, if I could make a certain step. Now, this step makes a lot of intuitive s... | What you're trying to prove is not true.
If you have learned the formal definition of limit, it ought to be easily for you to prove directly from the definition that $0$ is not the limit of $(-1)^n$.
(Set $\varepsilon=\frac12$ and see that no possible $N$ can even begin to work).
In fact, your proposed rule
$$ \lim_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Left/Right Eigenvectors Let $M$ be a nonsymmetric matrix; suppose the columns of matrix $A$ are the right eigenvectors of $M$ and the rows of matrix $B$ are the left eigenvectors of $M$.
In one of the answers to a question on left and right eigenvectors it was claimed that $AB=I$. Is that true, and how would you prove... | Try e.g. $$M = \pmatrix{3 & 2\cr -1 & 0\cr}$$
Eigenvalues are $1$ and $2$. Normalized right eigenvectors form the matrix
$$A = \pmatrix{-1/\sqrt{2} & -2/\sqrt{5} \cr 1/\sqrt{2} & 1/\sqrt{5}\cr}$$
Normalized left eigenvectors form
$$ B = \pmatrix{1/\sqrt{5} & 2/\sqrt{5}\cr 1/\sqrt{2} & 1/\sqrt{2}\cr}$$
These are not in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2071964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 1
} |
Average angle of vectors Given the following set ($n$ dimensional vectors with the length $1$ where each component is positive):
$
S=\{x\in\mathbb{R}_{\geq0}^n: \|x\|=1\}
$
What is the average / expected angle between two of these vectors?
For the $1$-dimensional case it is trivial, but for the $2$-dimensional case it ... | In two dimensional case, the problem becomes computing the mean of $|x-y|$ where $x$ and $y$ are independently drawn from a uniform random distribution in $[0, \frac{\pi}{2}]$, which is $$\left(\frac{2}{\pi}\right)^2\int_0^{\pi/2}\int_0^{\pi/2}|x-y|\,dx\,dy = \frac{\pi}{6}$$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2072027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
completing a primitive integer vector into an integer matrix of determinant 1 This is probably well known in algebraic number theory, in particular Minkowski lattice theory, but I am given an integer vector of dimension $n$ whose components are relatively prime, meaning there is an integer linear combination of them th... | HINT: This is a particular case of Smith normal form theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2072119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Combinatorics - $5$ cards, $4$ different suits I have the following question:
In a deck of $52$ cards with $4$ suits ($13$ of each), how many different ways are there to choose $5$ different cards such that every suit appears at least once.
the correct answer is:
$4×13^3×{13\choose 2}=685464$
My question is, why is... | Here is an alternative solution.
Use inclusion/exclusion principle:
*
*Include the number of combinations with at most $\color\red4$ suits: $\binom{4}{\color\red4}\cdot\binom{13\cdot\color\red4}{5}$
*Exclude the number of combinations with at most $\color\red3$ suits: $\binom{4}{\color\red3}\cdot\binom{13\cdot\col... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2072183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Analysis of cubic /complex equation If $z^3+(3+2i)z+(-1+iy)=0$ ( where $i^2=-1$) has one real root, then the value of $y$ does not belongs to :
*
*$(2,3)$
*$(-5,-1)$
*$(0,1)$
*$(-2,-1)$
My try: I would like you guys to tell me how to analyze a cubic equation; not just for this particular question but how to... | An idea: suppose $\;x\;$ is the real root, then:
$$x^3+3x+2xi-1+iy=0\implies\begin{cases}x^3+3x-1=0\\{}\\2x+y=0\end{cases}\implies y=-2x$$
Now, the function $\;x^3+3x-1\;$ is strictly monotone increasing (why?) , and by the MVT it has a root in $\;\left(0,\frac12\right)\;$ which is then its unique real root (again, why... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2072251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Check whether the given series is conditionally convergent or absolutely convergent or divergent? Check whether the given series is conditionally convergent or absolutely convergent or divergent?
(i)$\displaystyle\sum_{n=1}^\infty (-1)^n \frac 1 {2n+3}$
(ii)$\displaystyle\sum_{n=1}^\infty (-1)^n \frac n {n+2}$
(iii)$\... | Hints:
(i) $\;\frac1{2n+3}\;$ is monotone descending, so this is a Leibniz series. Without the absolute value though compare to the harmonic series
(ii) What is the limit of the series' sequence?
(iii) Use the ratio test without the $\;(-1)^n\;$ . What can you deduce from this?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2072424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Calculate the limit $\lim_{x\to 0}\frac{\arcsin(2x)}{\ln(e-2x)-1}$ without using L'Hôpital's rule I need to calculate the limit without using L'Hôpital's rule:
$$\lim_{x\to 0}\frac{\arcsin(2x)}{\ln(e-2x)-1}$$
I know that: $$\lim_{a\to 0}\frac{\arcsin a}{a}=1$$
But, how to apply this formula?
| We have, $$\lim_{x \to 0} \frac{\arcsin 2x}{\ln(e-2x)-1} = \lim_{x \to 0}\frac{\arcsin 2x}{2x} \frac{2x}{\ln(e-2x)-1} = \lim_{x \to 0}\frac{\arcsin 2x}{2x}\frac{-\frac{2x}{e}}{\ln(1+(-\frac{2x}{e}))}\times (-e)$$ This can be easily simplified to get the answer as $-e$. Hope it helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2072505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Calculating $\int_{-\infty}^x\frac{1}{\sqrt{2\pi}} e^{-(\frac{x^2}{2}+2x+2)}\,dx$ I want to solve this integral from $-\infty$ to $x$ .
$$f_X(x)=\frac{1}{\sqrt{2\pi}} e^{-(\frac{x^2}{2}+2x+2)}, -\infty<x<\infty$$
I have searched as much as I could and I found a solution in wikipedia
$$\int_{-\infty}^{\infty} x e^{-a(x-... | Complete the square in the exponential function, then let $y=\frac{x+2}{\sqrt{2}}$
\begin{align}
\frac{1}{\sqrt{2 \pi}} \int\limits_{-\infty}^{z} \mathrm{e}^{-(\frac{1}{2}x^{2}+2x+2)} dx &=
\frac{1}{\sqrt{2 \pi}} \int\limits_{-\infty}^{z} \mathrm{e}^{-\frac{1}{2}(x+2)^{2}} dx \\
&= \frac{1}{\sqrt{\pi}} \int\limits_{-\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2072617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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