Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Completing a proof Say we are given this:
Impossibility of ordering the complex numbers. As yet we have not defined a relation of the form $x < y$ if $x$ and $y$ are arbitrary complex numbers, for the reason that it is impossible to give a definition of $<$ for complex numbers which will have all the properties in Axio... | The passage proves only for the case when $i>0$ was considered, and the $i<0$ doesn't readily follow from this, though it is not hard neither:
By axiom 7., and if $i<0$, we have $0=i+(-i)<0+(-i)=-i$, that is, $-i>0$. But then the passage can be applied again, as $(-i)^2=i^2=-1$.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to LU facctorisation of a 4 by 4 matrice using gaussian eilimination! I have a 4 by 4 matrice,
A = [2 -2 0 0]
[2 -4 2 0]
[0 -2 4 -2]
[0 0 2 -4]
How would I use Gaussian Elimination to find the LU factorisation of the matrix
Please could someone explain how to do this!? I have an exam where a similar que... | Here's your matrix $A$, and you multiply it on the left with a 4 by 4 Identity matrix (it's always going to be the same dimensions as your $A$ matrix). So it'll look like $[I]*[A]$, and then you do Gaussian Elimination (GE) to your $A$ matrix, and make sure you keep track of your row operations that you do. Ex that's n... | {
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Asymptotic bound $T(n)=T(n/3+\lg n)+1$ How would I go about finding the upper and lower bounds of $T(n)=T(n/3+\lg(n))+1$?
| Not sure how tight a bound you need, but here is an idea.
Compare your recurrence to the one that satisfies $X_n = X_{n/3} + 1$ (without the log) - what is the relationship between $T_n$ and $X_n$? Note that when you solve, $X_n = \Theta(\log n)$.
Next item is in the other direction. Compare $T_n$ to $Y_n = Y_{n-1}+1$,... | {
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Power series infinity at every point of boundary Is there an example of a power series $f(z)=\sum_{k=0}^\infty a_kz^k$ with radius of convergence $0<R<\infty$ so that $\sum_{k=0}^\infty a_kw^k=\infty$ for all $w$ with $|w|=R$
Thank you kindly.
| No there is not. In fact, there is no example of such power series $\sum_n a_n z^n$ such that $\sum_n a_n w^n = \infty$ for all $w$ in a set of positive measure in $\partial D$, where $D=\{|z|<R\}$. Indeed, suppose there exists such a power series $f$. By Abel's Theorem, we deduce that $f(z)$ has non-tangential boundar... | {
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Evaluate the limit $\lim\limits_{n\to\infty}{\frac{n!}{n^n}\bigg(\sum_{k=0}^n{\frac{n^{k}}{k!}}-\sum_{k=n+1}^{\infty}{\frac{n^{k}}{k!}}\bigg)}$
Evaluate the limit
$$ \lim_{n\rightarrow\infty}{\frac{n!}{n^{n}}\left(\sum_{k=0}^{n}{\frac{n^{k}}{k!}}-\sum_{k=n+1}^{\infty}{\frac{n^{k}}{k!}} \right)} $$
I use $$e^{n}=1+... | In this answer, it is shown, using integration by parts, that
$$
\sum_{k=0}^n\frac{n^k}{k!}=\frac{e^n}{n!}\int_n^\infty e^{-t}\,t^n\,\mathrm{d}t\tag{1}
$$
Subtracting both sides from $e^n$ gives
$$
\sum_{k=n+1}^\infty\frac{n^k}{k!}=\frac{e^n}{n!}\int_0^n e^{-t}\,t^n\,\mathrm{d}t\tag{2}
$$
Substtuting $t=n(s+1)$ and $u^... | {
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How to prove that $\lim\limits_{n\to\infty} \frac{n!}{n^2}$ diverges to infinity? $\lim\limits_{n\to\infty} \dfrac{n!}{n^2} \rightarrow \lim\limits_{n\to\infty}\dfrac{\left(n-1\right)!}{n}$
I can understand that this will go to infinity because the numerator grows faster.
I am trying to apply L'Hôpital's rule to this... | Dominic Michaelis is the 'right' answer for such a simple problem. This is just to demonstrate a trick that is often helpful in showing limits going off to $\infty$. Consider $$\sum_{n=1}^{\infty} \frac{n^2}{n!}$$ By the ratio test this converges. So the terms $\frac{n^2}{n!} \to 0$.
| {
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How can i prove this identity (by mathematical induction) (rational product of sines) I would appreciate if somebody could help me with the following problem:
Q: proof? (by mathematical induction)
$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n}=\frac{n}{2^{n-1}}~(n\geq 2)$$
| Let
$$S_n=\prod_{k=1}^{n-1}\sin \frac{k\pi}{n}.$$
We solve the equation $(z+1)^n=1$ for $z\in\mathbb{C}$ we find
$$z=e^{i2k\pi/n}-1=2ie^{ik\pi/n}\sin\frac{k\pi}{n}=z_k,\quad 0,\ldots,n-1.$$
Moreover $(x+1)^n-1=x\left((x+1)^{n-1}+(x+1)^{n-2}+\cdots+(x+1)+1\right)=xP(x).$
The roots of $P$ are $z_k, k=1,\ldots ,n-1$. By t... | {
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Given the product of a unitary matrix and an orthogonal matrix, can it be easily inverted _without_ knowing these factors? Given the product $M$ of a unitary matrix $U$ (i.e. $U^\dagger U=1$) and an orthogonal matrix $O$ (i.e. $O^TO=1$), can it be easily inverted without knowing $U$ and $O$?
Sure enough, if $M=UO$, the... | Note that
$$M^\dagger M = O^\dagger\underbrace{U^\dagger U}_{=1} O = O^\dagger O$$
Therefore,
$$(M^\dagger M)(M^\dagger M)^T = O^\dagger \underbrace{O O^T}_{=1} O^* = (OO^T)^* = 1$$
So that
$$(M^\dagger)^{-1} = M(M^\dagger M)^T$$
And thus
$$M^{-1} = M^T M^* M^\dagger$$
| {
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How to choose the starting row when computing the reduced row echelon form? I'm having hell of a time going around solving matrices to reduced row echelon form. My main issue is which row to start simplifying values and based on what? I have this example
so again, the questions are:
1.Which row to start simplifying ... | Where you start is not really a problem.
My tip:
*
*Always first make sure you make the first column: 1,0,0
*Then proceed making the second one: 0,1,0
*And lastly, 0,0,1
Step one:
$$\begin{pmatrix} 1&2&3&9 \\ 2&-1&1&8 \\ 3&0&-1&3\end{pmatrix}$$
row 3 - 3 times row 1
$$\begin{pmatrix} 1&2&3&9 \\ 2&-1&1&8 \\ 0&-6&-1... | {
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Fibonacci identity proof I've been struggled for this identity for a while, how can I use combinatorial proof to prove the Fibonacci identity $$F_2+F_5+\dots+F_{3n-1}=\frac{F_{3n+1}-1}{2}$$
I know that $F_n$ is number of tilings for the board of length $n-1$, so if I rewrite the identity and let $f_n$ be the number of ... | This may not be the quickest approach, but it seems fairly simple, using only the recursion equation $F_i+F_{i+1}=F_{i+2}$ and initial conditions, which I take to be $F_0=0$ and $F_1=1$. Notice first that you can apply the recursion equation to replace, on the left side of your formula, each term by the sum of the two... | {
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How to minimize the amount of material used to make a shape of a given volume? A metal can company will make cylindrical shape cans of capacity 300 cubic centimeters. What is the bottom radius of the cans in order to use the least amount of the sheet metal in the production? Accurate to 2 decimal places.
| Hint:
*
*Write out the expressions for surface area and volume of cylinders. Here they are for reference:
$ A = 2 \pi r h + 2 \pi r^2 $
$ V = \pi r^2 h $
*
*We already know what the required volume is so we can set $ V = 300 $.
*Can we combine our expressions for $ A $ and $V $ and make progress that way?
ETA... | {
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Linear Transformations - Direct Sum Let $U, V$, and $W$ be finite dimensional vectors spaces over a field.
Suppose that $V\subset U$ is a subspace. Show that there is a subspace $W\subset U$ such that $U=V\oplus W$.
only thing i know about this problem is that you have to use the null space. I'm pretty much lost! any ... | Hint: Start with a basis for $V$, $\{v_1,\ldots,v_k\}$, and extend it to a basis of $U$, $\{v_1,\ldots,v_k,u_{k+1},\ldots,u_n\}$. Now find a subspace of $U$ which has the properties that you want by using that extended basis.
| {
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Common basis for subspace intersection
Let $ W_1 = \textrm{span}\left\{\begin{pmatrix}1\\2\\3\end{pmatrix}, \begin{pmatrix}2\\1\\1\end{pmatrix}\right\}$, and $ W_2 = \textrm{span}\left\{\begin{pmatrix}1\\0\\1\end{pmatrix}, \begin{pmatrix}3\\0\\-1\end{pmatrix}\right\}$. Find a basis for $W_1 \cap W_2$
I first though... | The two sub spaces are not the same, because $W_2$ has no extent in the second axis while $W_1$ does. The intersection is then a line in the $xz$ plane, which $W_2$ spans. If you can find a vector in $W_1$ in that plane that is your basis
| {
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Is there a bijection between $\mathbb N$ and $\mathbb N^2$?
Is there a bijection between $\mathbb N$ and $\mathbb N^2$?
If I can show $\mathbb N^2$ is equipotent to $\mathbb N$, I can show that $\mathbb Q$ is countable. Please help. Thanks,
| Yes. Imagine starting at $(1,1)$ and then zig-zagging diagonally across the quadrant. I'll leave you to formulate it.
Hint: for every natural number $>1$ there's a set of elements of $\mathbb{N}^2$ that add up to that number. For $2$, there's $(1,1)$. For $3$, there's $(2,1)$ and $(1,2)$...
| {
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Prove that if matrix $A$ is nilpotent, then $I+A$ is invertible. So my friend and I are working on this and here is what we have so far.
We want to show that $\exists \, B$ s.t. $(I+A)B = I$. We considered the fact that $I - A^k = I$ for some positive $k$. Now, if $B = (I-A+A^2-A^3+ \cdots -A^{k-1})$, then $(I+A)B = I-... | It's the usual polynomial identity
$$
1 - x^{k} = (1 - x)(1 + x + x^{2} + \dots + x^{k-1}),
$$
where you are substituting $x = -A$.
| {
"language": "en",
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"source": "stackexchange",
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Intersection points of a Triangle and a Circle How can I find all intersection points of the following circle and triangle?
Triangle
$$A:=\begin{pmatrix}22\\-1.5\\1 \end{pmatrix} B:=\begin{pmatrix}27\\-2.25\\4 \end{pmatrix} C:=\begin{pmatrix}25.2\\-2\\4.7 \end{pmatrix}$$
Circle
$$\frac{9}{16}=(x-25)^2 + (y+2)^2 + (z-3)... | The side $AB$ of the triangle has equation $P(t) = (1-t)A + tB$ for $0 \le t \le 1$. The $0 \le t \le 1$ part is important. If $t$ lies outside $[0,1]$, the point $P(t)$ will lie on the infinite line through $A$ and $B$, but not on the edge $AB$ of the triangle. Substitute $(1-t)A + tB$ into the circle equation, as ot... | {
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Prove $f(x)=ax+b$ Let $f(x)$ be a continuous function in $\mathbb R$ that for all $x\in(-\infty,+\infty)$, satisfies
$$ \lim_{h\rightarrow+\infty}{[f(x+h)-2f(x)+f(x-h)]}=0. $$
Prove that $f(x)=ax+b$ for some $a,b\in\mathbb R$.
This is a problem from my exercise book, but I can't figure out the solution of it, I think ... | Given $x\in\mathbb{R}$, the limit can be rewritten as
$$f(x)=\frac{1}{2}\lim_{h\to\infty}[f(x+h)+f(x-h)].\tag{1}$$
Given $y\in\mathbb{R}$, replacing $h$ with $h+y$ or $h-y$ in $(1)$, we have
$$f(x)=\frac{1}{2}\lim_{h\to\infty}[f(x+y+h)+f(x-y-h)],\quad \forall x\in\mathbb{R}.\tag{2}$$
and
$$f(x)=\frac{1}{2}\lim_{h\to\i... | {
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"timestamp": "2023-03-29T00:00:00",
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Find $x$ such that $\sum_{k=1}^{2014} k^k \equiv x \pmod {10}$ Find $x$ such that $$\sum_{k=1}^{2014} k^k \equiv x \pmod {10}$$
I knew the answer was $3$.
| We are going to compute the sum mod $2$ and mod $5$. The Chinese Remainder Theorem then gives us the result mod $10$.
Mod $2$, obviously $$k^k \equiv \begin{cases}0 & \text{if }k\text{ even,}\\1 & \text{if }k\text{ odd,}\end{cases}$$
so
$$\sum_{k = 0}^{2014} \equiv \frac{2014}{2} = 1007 \equiv 1 \mod 2.$$
By Fermat, $k... | {
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Is there an analytic function applying formula? Is there an analytic function $f$ in $\mathbb{C}\backslash \{0\}$ s.t. for every $z\ne0$: $$|f(z)|\ge\frac{1}{\sqrt{|z|}}\, ?$$
| How about this:
Since $f(z)$ is analytic on $\mathbb{C}-\{0\}$, $g(z) = \frac{1}{(f(z))^2}$ is analytic on
$\mathbb{C}-\{0\}$. Also $\bigg|\frac{g(z)}{z}\bigg| \leq 1$.
I am sure you will finish the rest (think about the order of the pole at $0$ and use Liouville's Theorem).
| {
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$f(z)= az$ if $f$ is analytic and $f(z_{1}+z_{2})=f(z_{1})+f(z_{2})$ If $f$ is an analytic function with $f(z_{1}+z_{2})=f(z_{1})+f(z_{2})$, how can we show that $f(z)= az$ where $a$ is a complex constant?
| It's true under weaker assumptions, but let's do it by assuming that $f$ is analytic.
Fix $w \in \mathbb{C}$. Since $f(z+w) = f(z)+f(w)$, it follows that $f'(z+w) = f'(z)$ for all $z$. Hence $f'$ is constant, say $f'(z) = a$ which implies that $f(z) = az+c$.
Plug in $z_1 = z_2 = 0$ in the defining equation to conclude ... | {
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A probability question that involves $5$ dice For five dice that are thrown, I am struggling to find the probability of one number showing exactly three times and a second number showing twice.
For the one number showing exactly three times, the probability is:
$$
{5 \choose 3} \times \left ( \frac{1}{6} \right )^{3} \... | First, I assume they wil all come out in neat order, first three in a row, then two in a row of a different number. The probability of that happening is
$$
\frac{1}{6^2}\cdot \frac{5}{6}\cdot\frac{1}{6} = \frac{5}{6^4}
$$
The first die can be anything, but the next two have to be equal to that, so the $\frac{1}{6^2}$ ... | {
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Eigenvalues of a rotation How do I show that the rotation by a non zero angle $\theta$ in $\mathbb{R}^2 $ does not have any real eigenvalues. I know the matrix of a rotation but I don't how to show the above proposition.
Thank you
| The characteristical polynomial is
$$x^2-2\cos(\theta) x+1$$
and $x^2\geq 0$ and $1>0$ and $|\cos(\theta)|\leq 1$ the polynomial can only have a zero
when $|\cos(\theta)|=1$.
As $x^2-2x+1=(x-1)^2$ and $x^2+2x+1=(x+1)^2$
| {
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Property of $10\times 10 $ matrix Let $A$ be a $10 \times 10$ matrix such that each entry is either $1$ or $-1$. Is it true that $\det(A)$ is divisible by $2^9$?
| Answer based on the comments by Ludolila and Erick Wong as an answer:
The answer follows from three easily proven rules:
*
*Adding or subtracting a row of a matrix from another does not change its determinant.
*Multiplying a line of the matrix by a constant $c$ multiplies the determinant by that constant.
*The det... | {
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Separation of function When can a function of 2 variables say $h(x,y)$ can be written as $$\sum_i f_i(x)g_i(y)$$ I want to know what conditions on $h$ would ensure this kind of separation.
| If you don't require the sum to be finite then essentially anything expandable in a two dimensional Fourier Series will satisfy what you want. For example if $f(x,y)$ is defined on the unit square, then
$$f(x,y)=\sum_{m,n\in\mathbb{Z}}a_{m,n}e^{2\pi i m x}e^{2\pi i n y}$$
for appropriate coefficients $a_{m,n}$. For ex... | {
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Is there any specific formula for $\log{f(z)}$? Let $f(z)$ be a nonvanishing analytic function on a simply connected region $\Omega$. Then there is an analytic function $g(z)$ such that $e^{g(z)}=f(z)$. Is there any specific formula for $g(z)$?
(By specific formula I mean, for example, on the region $\mathbb{C}-\{x\le ... | Hint: Try defining your function of $z$ as an integral from a certain function to a fixed point $z_0$ (the well-definedness [i.e. path independence] of which comes from simple connectedness).
| {
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Probability of rolling three dice without getting a 6 I am having trouble understanding how you get $91/216$ as the answer to this question.
say a die is rolled three times
what is the probability that at least one roll is 6?
| There are two answers already that express the probability as $$1-\left(\frac56\right)^3 = \frac{91}{216},$$
I'd like to point out that a more complicated, but more direct calculation gets to the same place. Let's let 6 represent a die that comes up a 6, and X a die that comes up with something else. Then we might di... | {
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What is a Ramsey Graph? What is a ramsey graph and What is its relation to RamseyTheorem?
In Ramsey Theorem:
for a pairs of parameters (r,b) there exists an n such that for every (edge-)coloring of the complete graph on n vertices with colors r(ed) and b(lue) there will exist a complete subgraph on r vertices colored r... | Instead of considering a complete graph $K_n$ whose edges are red and blue, just consider some graph $G$ with $n$ vertices. Let $\bar G$ be the complement of $G$. $\bar G$ contains a clique of size $s$ if and only if $G$ contains an independent set of size $s$.
Now consider $G$ as a subgraph of $K_n$. Color the edg... | {
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$\lim_{n\to\infty}(\sqrt{n^2+n}-\sqrt{n^2+1})$ How to evaluate $$\lim_{n\to\infty}(\sqrt{n^2+n}-\sqrt{n^2+1})$$
I'm completely stuck into it.
| A useful general approach to limits is, in your scratch work, to take every complicated term and replace it with a similar approximate term.
As $n$ grows large, $\sqrt{n^2 + n}$ looks like $\sqrt{n^2} = n$. More precisely,
$$ \sqrt{n^2 + n} = n + o(n) $$
where I've used little-o notation. In terms of limits, this means... | {
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} |
Solution of $19 x \equiv 1 \pmod{35}$ $19 x \equiv 1 \pmod{35}$
For this, may I know how to get the smallest value of $x$. I know that there is a theorem like $19^{34} = 1 \pmod {35}$. But I don't think it is the smallest.
| Hint $\rm\,\ mod\ 35\!:\,\ 19x\equiv 1\iff x\equiv \dfrac{1}{19}\equiv \dfrac{2}{38}\equiv \dfrac{2}3\equiv \dfrac{24}{36}\equiv\dfrac{24}1$
Remark $\ $ We used Gauss's algorithm for computing inverses $\rm\:mod\ p\:$ prime.
Beware $\ $ One can employ fractions $\rm\ x\equiv b/a\ $ in modular arithmetic (as above) only... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
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How to write the following expression in index notation? I would like to know how can I write $ ||\vec{a} \times(\nabla \times \vec{a})||^2 $ and $(\vec{a} \cdot (\nabla \times \vec{a}))^2$ in index notation if $\vec{a}=(a_1,a_2,a_3)$
Thank you for reading/replying
EDIT: found the second one: $(\vec{a} \cdot (\nabla \t... | To do this I would use the Levi-Civita symbol and its properties in 3 dimensions.
(from Wikipedia:)
Definition:
\begin{equation}
\varepsilon_{ijk}=
\left\{
\begin{array}{l}
+1 \quad \text{if} \quad (i,j,k)\ \text{is}\ (1,2,3),(3,1,2)\ \text{or}\ (2,3,1)\\
-1 \quad \text{if} \quad (i,j,k)\ \text{is}\ (1,3,2),(3,2,1)\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Radicals and direct sums. Let A be a K-algebra and M, N be right A-submodules of a right A module L, $M \cap N =0$. How to show that $(M\oplus N) \text{rad} A = M \text{rad} A \oplus N \text{rad} A$? Let $m \in M, n\in N, x\in \text{rad} A$. Since $(m+n)x=mx+nx$, $(m+n)x \in M \text{rad} A \oplus N \text{rad} A$. Since... | Sure, it's obvious that for any $S \subseteq A$, we have $MS\subseteq M$ and $NS\subseteq N$. That means $MS\cap NS\subseteq M\cap N =\{0\}.$
But this has little to do with the sum being direct: the real question here is how to get the equality
$$(M\oplus N) \text{rad} A = M \text{rad} A \oplus N \text{rad} A.$$
The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove $a\sqrt[3]{a+b}+b\sqrt[3]{b+c}+c\sqrt[3]{c+a} \ge 3 \sqrt[3]2$ Prove $a\sqrt[3]{a+b}+b\sqrt[3]{b+c}+c\sqrt[3]{c+a} \ge 3 \sqrt[3]2$ with $a + b+c=3 \land a,b,c\in \mathbb{R^+}$
I tried power mean inequalities but I still can't prove it.
| Here is my proof
by AM-GM inequality,we have
$$ a\sqrt[3]{a+b}=\frac{3\sqrt[3]{2}a(a+b)}{3\sqrt[3]{2(a+b)(a+b)}}\geq 3\sqrt[3]{2}\cdot \frac{a(a+b)}{2+2a+2b} $$
Thus,it's suffice to prove that
$$ \frac{a(a+b)}{a+b+1}+\frac{b(b+c)}{b+c+1}+\frac{c(c+a)}{c+a+1}\geq 2 $$
Or
$$ \frac{a}{a+b+1}+\frac{b}{b+c+1}+\frac{c}{c+a+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Limit of definite sum equals $\ln(2)$ I have to show the following equality:
$$\lim_{n\to\infty}\sum_{i=\frac{n}{2}}^{n}\frac{1}{i}=\log(2)$$
I've been playing with it for almost an hour, mainly with the taylor expansion of $\ln(2)$. It looks very similar to what I need, but it has an alternating sign which sits in ... | Truncate the Maclaurin series for $\log(1+x)$ at the $2m$-th term, and evaluate at $x=1$. Take for example $m=10$. We get
$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots+\frac{1}{19}-\frac{1}{20}.$$
Add $2\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\cdots +\frac{1}{20}\right)$, and subtract the sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 2
} |
How to calculate the asymptotic expansion of $\sum \sqrt{k}$? Denote $u_n:=\sum_{k=1}^n \sqrt{k}$. We can easily see that
$$ k^{1/2} = \frac{2}{3} (k^{3/2} - (k-1)^{3/2}) + O(k^{-1/2}),$$
hence $\sum_1^n \sqrt{k} = \frac{2}{3}n^{3/2} + O(n^{1/2})$, because $\sum_1^n O(k^{-1/2}) =O(n^{1/2})$.
With some more calculations... | Let us substitute into the sum
$$\sqrt k=\frac{1}{\sqrt \pi }\int_0^{\infty}\frac{k e^{-kx}dx}{\sqrt x}. $$
Exchanging the order of summation and integration and summing the derivative of geometric series, we get
\begin{align*}
\mathcal S_N:=
\sum_{k=1}^{N}\sqrt k&=\frac{1}{\sqrt \pi }\int_0^{\infty}\frac{\left(e^x-e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
How to prove **using SVD** that $\mathbb{C}^{n \times n}$ is dense for nonsingular matrices? How to show using SVD that the set of nonsingular matrices is dense in $\mathbb{C}^{n \times n}$? That is, for any $A \in \mathbb{C}^{n \times n}$, and any given $\varepsilon > 0$, there exists a nonsingular matrix $A_\varepsi... | In finite dimensional space all norms are equivalent. We choose an algebra norm.
Let $A=U\Sigma V^*$ the singular value decomposition where $U$ and $V$ are unitary matrices that's $||U||=||V||=1$ and $\Sigma=diag(\sigma_n,\ldots,\sigma_1)$ with
$$\sigma_n\geq\cdots\geq\sigma_1\geq0.$$
Let $\Sigma_p=diag(\sigma_n+1/p,\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Need help proving this integration If $a>b>0$, prove that :
$$\int_0^{2\pi} \frac{\sin^2\theta}{a+b\cos\theta}\ d\theta = \frac{2\pi}{b^2} \left(a-\sqrt{a^2-b^2} \right) $$
| I'll do this one $$\int_{0}^{2\pi}\frac{cos(2\theta)}{a+bcos(\theta)}d\theta$$if we know how to do tis one you can replace $sin^{2}(\theta)$ by $\frac{1}{2}(1-cos(2\theta))$ and do the same thing. if we ae on the unit circle we know that $cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2}$ so by letting $z=e^{i\theta}$ we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Laplace transform of the Bessel function of the first kind I want to show that $$ \int_{0}^{\infty} J_{n}(bx) e^{-ax} \, dx = \frac{(\sqrt{a^{2}+b^{2}}-a)^{n}}{b^{n}\sqrt{a^{2}+b^{2}}}\ , \quad \ (n \in \mathbb{Z}_{\ge 0} \, , \text{Re}(a) >0 , \, b >0 ),$$ where $J_{n}(x)$ is the Bessel function of the first kind of ... | Everything is correct up until your computation of the residue. Write
$$bz^2+2az-b=b(z-z_+)(z-z_-)$$
where
$$z_\pm=-\frac{a}{b}\pm\frac{\sqrt{a^2+b^2}}{b}$$
as you have determined. Now,
$${\rm Res}\Bigg(\frac{z^n}{b(z-z_+)(z-z_-)};\quad z=z_+\Bigg)=\lim_{z\to z_+} (z-z_+)\frac{z^n}{b(z-z_+)(z-z_-)}$$
and here you get t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
On the Definition of Posets... In my book, the author defines posets formally in the following way:
Let $P$ be a set, and let $\le$ be a relationship on $P$ so that,
$a$. $\le$ is reflective.
$b$. $\le$ is transitive.
$c$. $\le$ is antisymmetric.
Say for $a$, does this merely mean that if some element $x\in P$, $x$ sho... | Your statements are correct.
On your last question: A poset is a pair ($P$, $R$), where $P$ is a set and $R$ is a relation on $P$ (which must have properties a, b and c). So the relation is a part of the poset, there is no freedom to choose it.
Hence to determine if something is a poset, you don't have to determine if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Discrete math: Euler cycle or Euler tour/path? Could someone help explain to me how I can figure out if the graphs given are Euler cycle or Euler path? Is it through trial and error?
Here are some examples:
Would appreciate any help.
| a graph is Eulerian if its contains an Eulerian circuit, where Eulerian circuit is an Eulerian trail. By eulerian trail we mean a trail that visits every edge of a graph once and only once. now use the result that "A connectded graph is Eulerian if and only if every vertex of G has even degree." now you may distinguish... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Continuous real-valued function and open subset Let $f$ be a continuous real-valued function defined on an open subset $U$ of $\mathbb{R}^n$.
Show that $\{(x,y):x\in{U},y>f(x)\}$ is an open subset of $\mathbb{R}^{n+1}$
Let $\forall{x}\in{X}, X\subset{U}$
Using the theorem, for a function $f$ mapping $S\subset{\mathbb... | Hint: the function
$$
g:(x,y)\longmapsto y-f(x)
$$
is defined and continuous on $U\times \mathbb{R}$, which is open in $\mathbb{R}^{n+1}$. Now try to express your set with this function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/326978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Dilogarithm Identities Is there a cleaner way to write:
$$
f(x) = \operatorname{Li}_2(i x) - \operatorname{Li}_2(-i x)
$$
in terms of simpler functions? I don't know enough about dilogarithms, and the basic identities I see on wikipedia are not helping me.
| I can show you some approximations which might help:
For the case of $|x|<1$ we have that
$ i(\operatorname{Li}_2(-i x) - \operatorname{Li}_2(i x)) ≃ 2x $
For the case of $|x|\ge1$ we have that
$ i(\operatorname{Li}_2(-i x) - \operatorname{Li}_2(i x)) ≃ π\cdot log(x) $
Another good approximation for any x is
$ i(\op... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Understanding branch cuts for functions with multiple branch points My question was poorly worded and thus confusing. I'm going to edit it to make it clearer, and then I'm going to give a brief answer.
Take, for example, the function $$f(z) = \sqrt{1-z^{2}}= \sqrt{(1+z)(1-z)} = \sqrt{|1+z|e^{i \arg(1+z)} |1-z|e^{i \ar... | I would recommend chapter 2.3 in Ablowitz, but I can try to explain in short.
Let
$$w : = (z^2-1)^{1/2} = [(z+1)(z-1)]^{1/2}.$$
Now, we can write
$$z-1 = r_1\,\exp(i\theta_1)$$ and similarly for
$$z+1 = r_2\,\exp(i\theta_2)$$ so that
$$w = \sqrt{r_1\,r_2}\,\exp(i(\theta_1+\theta_2)/2). $$
Notice that since $r_1$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
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Are there infinite sets of axioms? I'm reading Behnke's Fundamentals of mathematics:
If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.
I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible existence of unknown a... | Perhaps surprisingly even the classical (Łukasiewicz's) axiomatization of propositional logic has an infinite number of axioms. The axioms are all substitution instances of
*
*$(p \to (q \to p))$
*$((p \to (q \to r)) \to ((p \to q) \to (p \to r)))$
*$((\neg p \to \neg q) \to (q \to p))$
so we have an infinite nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 7,
"answer_id": 0
} |
Find $E(\max(X,Y))$ for $X$, $Y$ independent standard normal Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$.
I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$.
Now I need to find $EZ$.
Should I start like this ?
$$EZ=\int_{-\infty}^{\infty}\int_{-\infty}^\infty \max(... | To go back to $Z$ as a function of $(X,Y)$ once one has determined $F_Z$ is counterproductive. Rather, one could compute the density $f_Z$ as the derivative of $F_Z=\Phi^2$, that is, $f_Z=2\varphi\Phi$ where $\Phi$ is the standard normal CDF and its derivative $\varphi$ is the standard normal PDF, and use
$$
\mathbb E(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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Correct way of saying that some value depends on another value x only by a function of x I would like to know what good and valid ways there are to say (in words) that some value f(x), which depends on a variable x, in fact only depends on x "through" some function of x.
Example: For $x\in\mathbb{R}$ let $\hat{x}:=\min... | Translating the formulas to words is one way: f is a function of x such that if we denote g(x) by y, f can be written as a function of y only. Perhaps not exactly what you are looking for.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/327337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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What is the order of $2$ in $(\mathbb{Z}/n\mathbb{Z})^\times$? Is it there some theorem that makes a statement about the order of $2$ in the multiplicative group of integers modulo $n$ for general $n>2$?
| Let me quote from this presentation of Carl Pomerance:
[...] the multiplicative order of $2 \pmod n$
appears to be very erratic and difficult to get hold of.
The presentation describes, however, some properties of this order. The basic facts have already been elucidated by @HagenvonEitzen in his comment.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/327412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 0
} |
Convergence of $\sum_{n=1}^\infty (-1)^n(\sqrt{n+1}-\sqrt n)$ Please suggest some hint to test the convergence of the following series
$$\sum_{n=1}^\infty (-1)^n(\sqrt{n+1}-\sqrt n)$$
| We have
$$u_n=(-1)^n(\sqrt{n+1}-\sqrt{n})=\frac{(-1)^n}{\sqrt{n+1}+\sqrt{n}}$$
So the sequence $(|u_n|)_n$ converges to $0$ and is monotone decreasing then by Alternating series test the series $\sum_n u_n$ is convergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/327474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Number of Different Equivalence Classes? In the question given below, determine the number of different equivalence classes. I think the answer is infinite as $\ b_1$ and $b_2\;$ can have either one 1's or two 1's or three 1's etc. I just want to clarify if this is right as some says the number of different equivalence... | The proof that the correct answer is infinite is quite easy.
First of all, let's note that the set of equivalence classes is not empty, since, trivially, "1" generates a class.
Suppose that the number of classes is finite. Then we can build over the set of the equivalence classes an order relation, defined as follows:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculus, integration, Riemann sum help? Express as a definite integral and then evaluate the limit of the Riemann sum lim
$$
\lim_{n\to \infty}\sum_{i=0}^{n-1} (3x_i^2 + 1)\Delta x,
$$
where $P$ is the partition with
$$
x_i = -1 + \frac{3i}{n}
$$
for $i = 0, 1, \dots, n$ and $\Delta x \equiv x_i - x_{i-1}$.
I am comple... | Let $f$ be a function, and let $[a,b]$ be an interval. Let $n$ be a positive integer, and let $\Delta x=\frac{b-a}{n}$. Let $x_0=a$, $x_1=a+\Delta x$, $x_2=a+2\Delta x$, and so on up to $x_n=a+n\Delta x$. So $x_i=a+i\Delta x$.
So far, a jumble of symbols. You are likely not to ever understand what's going on unless yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Monomial ordering problem I've got the following problem:
Let $\gamma$, $\delta$ $\in$ $\mathbb R_{> 0}$. The binary relation $\preceq$ on monomials in $X,Y$ is defined: $X^{m}Y^{n} \preceq X^{p}Y^{q}$ if and only if $\gamma m + \delta n \leq \gamma p + \delta q .$ Show that this is a monomial ordering if and only if... | Every monomial order is a total order; in particular when $\gamma/\delta$ is a rational $a/b$, we have $X^b\preceq Y^a\preceq X^b$, so $\preceq$ is not a total order, so $\preceq$ is not a monomial order.
But not every total order is a monomial order. To verify that $\preceq$ is a monomial order you have to show:
*
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Functions $f$ satisfying $ f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R}$. How to prove that the continuous functions $f$ on $\mathbb{R}$ satisfying
$$f\circ f(x)=2f(x)-x,\forall x\in\mathbb{R},$$
are given by
$$f(x)=x+a,a\in\mathbb{R}.$$
Any hints are welcome. Thanks.
| If $f$ were bounded from below, so were $x=2f(x)-f(f(x))$. Therefore $f$ is unbounded, hence by IVT surjective.
Also, $f(x)=f(y)$ implies $x=2f(x)-f(f(x))=2f(y)-f(f(y))=y$, hence $f$ is also injective and has a twosided inverse. With this inverse we find $$\tag1f(x)+f^{-1}(x)=2x.$$
We conclude that $f$ is either strict... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
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Taylor polynomials expansion with substitution I am working on some practice exercises on Taylor Polynomial and came across this problem:
Find the third order Taylor polynomial of $f(x,y)=x + \cos(\pi y) + x\log(y)$ based at $a=(3,1).$
In the solution provided, the author makes a substitution such that $x=3+h$ and $y=1... | It's generally a good policy to "always expand around zero". This means that you want to have variables that go to zero at your point of interest.
In your case, you want to have $h$ go to zero for $x$ and $k$ go to zero for $y$. For this to happen at $x=3$ and $y=1$, you want to use $x=3+h$ and $y = 1+k$.
One reason th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Manipulating the equation!
The question asks to manipulate $f(x,y)=e^{-x^2-y^2}$ this equation to make its graph look like the three shapes in the images I attached.
I got the first one: $e^{(-x^2+y^2)}\cos(x^2+y^2)$.
But I have no idea what to do for second and the third one. Please help me.
| Is the cosine in the exponent? In case not, Alpha gives something that looks like the original function
If so, Alpha gives something closer to your second target
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/327879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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r.v. Law of the min On a probability space $(\Omega, A, P)$, and given a r.v. $(X,Y)$ with values in $R^2$. If the law of $(X,Y)$ is $\lambda \mu e^{-\lambda x - \mu y } 1_{R^2_+} (x,y) dx dy$, what is the law of the min $(X,Y)$?
| HINT
Let $Z=\min(X,Y)$. Start by computing, for some $z>0$
$$
1-F_Z(z) = \mathbb{P}\left(Z>z\right) = \mathbb{P}\left(X>z, Y>z\right)
$$
Notice that the probability of $(X,Y)$ factors, i.e. $X$ and $Y$ are independent, hence
$$
\mathbb{P}\left(X>z, Y>z\right) = \mathbb{P}\left(X>z\right) \mathbb{P}\left(Y>z\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327956",
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Question on measurable set I am having a hard time solving the following problem. Any help would be wonderful.
If $f: [0, 1] \rightarrow [0, \infty)$ is measurable, and we have that $\int_{[0, 1]}f \mathrm{d} m = 1$, must there exist a continuous function $g: [0, 1] \rightarrow [0, \infty)$ and a measurable set $E$ wi... | $m([f > 10]) \le 1/10$ (Chebyshev)
Uniformly approximate $f$ on $[0 < f \le 10]$ by a simple function, within say 1/200.
By linearity, reduce to characteristic function of a measurable set. But a measurable set is close to a finite union of intervals (the measure of their symmetric difference can be made small).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/328039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Problem involving combinations. In how many ways can $42$ candies (all the same) be distributed among 6 different infants such that each infant gets an odd number of candies?
I seem to think that we have 42 different objects, and 6 choices. So it should be 42C6. However, I'm not factoring in the "odd number of candies"... | You are looking for compositions of $42$ into six odd parts. If you give each child one candy and put the rest in pairs, this will be the same as the weak compositions of 18 into six parts, which is given by ${23 \choose 5}=33649$. To prove the formula, put $24$ (pairs) in a row, then you select five places to split ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Algebra Question from Mathematics GRE I just started learning algebra, and I came across a question from a practice GRE which I couldn't solve. http://www.wmich.edu/mathclub/files/GR8767.pdf #49
The finite group $G$ has a subgroup $H$ of order 7 and no element of $G$ other than the identity is its own inverse. What c... | Since $\rm a=a^{-1}\iff a^2=1$, you can use Cauchy's theorem to eliminate even orders.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/328134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Solving equation $A^{(B^{(C^x)})} = C^{(B^{(A^x)})}$ for $x$ Recently I came across the equation
$$A^{(B^{(C^x)})} = C^{(B^{(A^x)})}$$
where $A \neq B \neq C$, and if $A, B, C > 1$ or if $0 < A,B,C < 1$, there exists
a unique solution for $x$.
Here is my attempt:
$$A^{(B^{(C^x)})} = C^{(B^{(A^x)})}$$
$$B^{(C^x)} = \log... | By symmetry in $A\leftrightarrow C$, we may assume that either $0<C<A<1$ or $1<A<C$.
So far, by taking logarithms and reordering
$$\tag0A^{(B^{(C^x)})} = C^{(B^{(A^x)})}$$
$$B^{(C^x)}\ln A = B^{(A^x)}\ln C$$
$$C^x\ln B + \ln\ln A = A^x\ln B +\ln\ln C$$
$$\tag1C^x-A^x = \frac{\ln\ln C-\ln\ln A}{\ln B}$$
where the right ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/328189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Differentials and implicit differentiation Consider this example. Suppose $x$ is a function of two variables $s$ and $t$,
$$x = \sin(s+t)$$
Taking the differential as in doing implicit differentiation [1],
$$dx = \cos(s+t)(ds+dt) = \cos(s+t)dt + \cos(s+t)ds$$
I know the right way of taking the differential of $x$ is by... | This is a special case: you may consider a simple substitution $y = s + t$
$x = \sin(y)$
$dx = \cos(y)dy$
$dx = \cos(s+y)(ds + dt)$
if you do this with $x=s^t$
$y = s^t$
$x = y$
$dx = dy$
$dx = ts^{t-1}ds + s^t\ln(s)dt$
which isn't useful at all...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/328245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
find maximum for a function on a ball I have a question thats giving me a hard time, can someone please help me out with it ?
I need to find maximun to the func:
$$
f(x) = 2-x+x^3+2y^3+3z^3
$$
on the ball
$$
\{ (x,y,z) | x^2+y^2+z^2\leq1 \}
$$
what i have tried to do is this $$ (f_x,f_y,f_z)=0 $$ and then to check th... | At first search for extremas in the interior. (With the normal way)
Than observe that for a maximum on the bound $x^2+y^2+z^2=1$ the derivative doesn't need to be zero, use lagrange multipliers here.
Look at the function
$$g(x,y,z,\lambda)= 2-x+x^3+2y^3+3z^3+ \lambda (x^2+y^2+z^2-1)$$
and search find for a maximum for... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Linearly independent Let $S$ be a linearly independent subset of a vector space $V$, and let $v$ be a vector in $V$ that is not in $S$. Then $S\cup \{v\}$ is linearly dependent if and only if $v\in span\{S\}$.
proof)
If $S\cup \{v\}$ is linearly dependent then there are vectors $u_1,u_2,\dots,u_n$ in $S\cup \{v\}$ su... | Your intuition is good but there is a problem with stating that $v$ is in the span of $S$ directly. Mainly if we have some linear combination:
$$(\ast) \qquad a_1s_1 + \cdots a_ns_n + a_{n+1}v = 0$$
where not all the $a_i$ are zero, we want to show that $a_{n+1}$ is not zero so that we can subtract that term over and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Math question functions help me? I have to find find $f(x,y)$ that satisfies
\begin{align}
f(x+y,x-y) &= xy + y^2 \\
f(x+y, \frac{y}x ) &= x^2 - y^2
\end{align}
So I first though about replacing $x+y=X$ and $x-y=Y$ in the first one but then what?
| Hint:
If $x+y=X$ and $x-y=Y$ then
\begin{align*}
x&=X-y \implies Y= X-y-y \implies 2y=X-Y \dots
\end{align*}
And be careful with the second for $x=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Groupoids isomorphism Let $G, G'$ be two groups and $X=\{x,y\}$ be a set of two elements. Consider a groupoid $\mathcal{G}$ with objects from $X$ such that Hom$(x,x)=G$ and Hom$(y,y)=G'$.
Suppose Hom$(x,y) \neq \emptyset$, i.e. there is a morphism between $x$ and $y$. I think this is possible if and only if this morphi... | You should just construct the functor. The objects map to themselves as do the endomorphisms. All you really have to decide is how the map $F\colon\hom_{\mathcal G_1}(x, y) \to \hom_{\mathcal G_2}(x, y)$ is going to work.
Pick $\phi_1 \in \hom_{\mathcal G_1}(x, y)$ and $\phi_2 \in \hom_{\mathcal G_2}(x, y)$. What yo... | {
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Does it make sense to talk about $L^2$ inner product of two functions not necessarily in $L^2$? The $L^2$-inner product of two real functions $f$ and $g$ on a measure space $X$ with respect to the measure $\mu$ is given by
$$
\langle f,g\rangle_{L^2} := \int_X fg d\mu, $$
When $f$ and $g$ are both in $L^2(X)$, $|\langl... | Yes it does make sense. For example, if you take $f\in L^p(X)$ and $g\in L^{p'}(X)$ with $\frac{1}{p}+\frac{1}{p'}=1$, then $\langle f,g\rangle_{L^2} $ is well defined and $$\langle f,g\rangle_{L^2} <\|f\|_p\|g\|_{p'}$$
With this notation it is said the the inner product on $L^2$ induces the duality $p$ and $p'$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/328592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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} |
Ideal of ideal needs not to be an ideal Suppose I is an ideal of a ring R and J is an ideal of I, is there any counter example showing J need not to be an ideal of R? The hint given in the book is to consider polynomial ring with coefficient from a field, thanks
| Consider $R=\mathbb Q[x]$, and $I=xR$ be the most obvious ideal of $R$.
Note that we can define $J$ as a subset of $I$ to be an ideal of $I$ if $J$ is a subgroup of $(I,+)$ and $IJ\subseteq J$. Find a $J$ that is a super-set of $x^2R$ but does not contain all of $I=xR$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/328670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 0
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Simple probability problems which hide important concepts Together with a group of students we need to compose a course on probability theory having the form of a debate. In order to do that we need to decide on a probability concept simple enough so that it could be explained in 10-15 minutes to an audience with basic... | There are three prisoners in Cook Maximum security prison. Jack, Will and Mitchel. The prison guard knows the one who is to be executed. Jack has finished writing a letter to his mother and asks the guard whether he should give it to Will or Mitchel. The prison guard is in a dilemma thinking that telling Jack the name ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Existence of a sequence that has every element of $\mathbb N$ infinite number of times I was wondering if a sequence that has every element of $\mathbb N$ infinite number of times exists ($\mathbb N$ includes $0$). It feels like it should, but I just have a few doubts.
Like, assume that $(a_n)$ is such a sequence. Find... | Let $a_n$ be the largest natural number, $k$ such that $2^k$ divides $n+1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/328821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
Einstein Summation Notation Interpretation A vector field is called irrotational if its curl is zero. A vector field is called solenoidal if its divergence is zero. If A and B are irrotational, prove that A $ \times $ B is solenoidal.
I'm having a hard time the proof equation that is required, and the steps that would ... | This doesn't work; $(\nabla\times A)+(\nabla\times B)$ doesn't correspond to anything, and $\nabla\cdot(A\times B)$ doesn't expand to it (as noted by Henning Makholm in a comment, one of these is effectively a scalar and one effectively a vector). Instead, you want a form of the triple product identity $A\cdot(B\times... | {
"language": "en",
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Rotating Matrix by $180$ degrees through another matrix To rotate a $2\times2$ matrix by $180$ degrees around the center point, I have the following formula:
$PAP$ = Rotated Matrix, where
$$P =\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}$$
$$A= \begin{bmatrix}
a & b\\
c & d
\end{bmatrix}$$
And the resulting matrix wil... | There is no such $2 \times 2$ matrix $P$ which will do what you want for a general $A$. This is because you need to rearrange both the rows and columns and so need a matrix action on the left and on the right. To see this explicitly, define $$P = \left[\begin{array}{cc} p_1 & p_2 \\ p_3 & p_4 \end{array} \right] $$
C... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How do I prove: If $A$ is an infinite set and $x$ is some element such that $x$ is not in $A$, then$ A\sim A\cup \left\{x\right\}$. How do I prove: If $A$ is an infinite set and $x$ is some element such that $x$ is not in $A$, then$ A\sim A\cup \left\{x\right\}$.
| Hint: Since $A$ is infinite there is some $A_0\subseteq A$ such that $|A_0|=|\Bbb N|$. Show that $\Bbb N$ has the wanted property, conclude that $A_0$ has it, and then conclude that $A$ has it as well.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Representing an element mod $n$ as a product of two primes Given a positive integer $n$ and $x \in (\mathbb{Z}/n\mathbb{Z})^*$ what is the most efficient way to find primes $q_1,q_2$ st
$$q_1q_2 \equiv x \bmod n$$
when $n$ is large?
One option is just to take $q_1=2$ and then find the least prime $q_2 \equiv x/2 \bmod... | If $n$ is not unreasonably large, I'd take advantage of the birthday paradox. You should only need to collect about $O(\sqrt{n})$ distinct residues of primes before you find two that multiply to $x$. This does require $O(\sqrt{n})$ memory unlike your Linnik's solution which is essentially constant memory.
Since you w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Normalized cross correlation via FHT - how can I get correlation score? I'm using the 2D Fast Hartley Transform to do fast correlation of two images in the frequency domain, which is the equivalent of NCC (normalized cross correlation) in the spatial domain.
However, with NCC, I can get a confidence metric that gives m... | Without knowing what you actually computed I can only assume that the following is probably what you want. I interpret "correlation image" as the cross correlation $I_1 \star I_2$ of the two images $I_1$ and $I_2$. Depending on normalizations in your FHT and IFHT there might be an additional scale factor. In what fol... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $\frac{1}{x-1}+ \frac{2}{x-2}+ \frac{3}{x-3}+\cdots+\frac{10}{x-10}\geq\frac{1}{2} $ I would appreciate if somebody could help me with the following problem:
Q: find $x$
$$\frac{1}{x-1}+ \frac{2}{x-2}+ \frac{3}{x-3}+\cdots+\frac{10}{x-10}\geq\frac{1}{2} $$
| If the left side is $$f(x)=\sum_{k=1}^{10} \frac{k}{x-k},$$
then the graph of $f$ shows that $f(x)<0$ on $(-\infty,1)$, so no solutions there. For each $k=1..9$ there is a vertical asymptote at $x=k$ with the value of $f(x)$ coming down from $+\infty$ immediately to the right of $x=k$ and crossing the line $y=1/2$ betw... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I show that $T$ is invertible? I'm really stuck on these linear transformations, so I have $T(x_1,x_2)=(-5x_1+9x_2,4x_1-7x_2)$, and I need to show that $T$ is invertible. So would I pretty much just say that this is the matrix: $$\left[\begin{matrix}-5&9\\4&-7\end{matrix}\right]$$ Then it's inverse must be $\fra... | I think it would be more in the spirit of the question (it sounds like it is an exercise in a course or book) to write down a linear map $S$ such that $S\circ T$ and $T\circ S$ are both the identity - the matrix you have written down tells you how to do this. Then such an $S$ is $T^{-1}$.
You should also note that ther... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
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How to create a generating function / closed form from this recurrence? Let $f_n$ = $f_{n-1} + n + 6$ where $f_0 = 0$.
I know $f_n = \frac{n^2+13n}{2}$ but I want to pretend I don't know this. How do I correctly turn this into a generating function / derive the closed form?
| In two answers, it is derived that the generating function is
$$
\begin{align}
\frac{7x-6x^2}{(1-x)^3}
&=x\left(\frac1{(1-x)^3}+\frac6{(1-x)^2}\right)\\
&=\sum_{k=0}^\infty(-1)^k\binom{-3}{k}x^{k+1}+6\sum_{k=0}^\infty(-1)^k\binom{-2}{k}x^{k+1}\\
&=\sum_{k=1}^\infty(-1)^{k-1}\left(\binom{-3}{k-1}+6\binom{-2}{k-1}\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Integral of irrational function $$
\int \frac{\sqrt{\frac{x+1}{x-2}}}{x-2}dx
$$
I tried:
$$
t =x-2
$$
$$
dt = dx
$$
but it didn't work.
Do you have any other ideas?
| Let $y=x-2$ and integrate by parts to get
$$\int dx \: \frac{\sqrt{\frac{x+1}{x-2}}}{x-2} = -2 (x-2)^{-1/2} (x+1)^{1/2} + \int \frac{dy}{\sqrt{y (y+3)}}$$
In the second integral, complete the square in the denominator to get
$$\int \frac{dy}{\sqrt{y (y+3)}} = \int \frac{dy}{\sqrt{(y+3/2)^2-9/4}}$$
This integral may be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Summations involving $\sum_k{x^{e^k}}$ I'm interested in the series
$$\sum_{k=0}^\infty{x^{e^k}}$$
I started "decomposing" the function as so:
$$x^{e^k}=e^{(e^k \log{x})}$$
So I believe that as long as $|(e^k \log{x})|<\infty$, we can compose a power series for the exponential. For example,
$$e^{(e^k \log{x})}=\frac{... | If we consider values $\alpha,\beta$ and $t$ greater then zero with $\alpha\beta=2\pi$ then your series can be expressed in relation to several other sums under the following double exponential series identity:
$$\alpha \sum_{k=0}^\infty e^{te^{k\alpha}}=\alpha\left(\frac{1}{2}-\sum_{k=1}^\infty\frac{(-1)^{k}t^k}{k!(e^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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On the meaning of the second derivative When we want to find the velocity of an object we use the derivative to find this. However, I just learned that when you find the acceleration of the object you find the second derivative.
I'm confused on what is being defined as the parameters of acceleration. I always thought ... | The position function is typically denoted $r=x(t)$. Velocity is the derivative of the position function with respect to time: $v(t)=\dfrac{dx(t)}{dt}$. Acceleration is the derivative of the velocity function with respect to time: $a(t)=\dfrac{dv(t)}{dt}$. This is equivalent to the second derivative of the position fun... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 4
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$\forall m \exists n$, $mn = n$ True or False
Identify if the statement is true or false. If false, give a counterexample.
$\forall m \exists n$, $mn = n$, where $m$ and $n$ are integers.
I said that this statement was false; specifically, that it is false when $m$ is any integer other than $1$
Apparently this is inc... | As others have noted, the statement is true:
$$\forall m \exists n, \; mn = n, \;\;\; m,\,n \in \mathbb Z \tag {1}$$
For all $m$, there exists an $n$ such that $mn = n$. To show this is true we need only to find the existence of such an $n$: and $n = 0:\;\; m\cdot 0 = 0 \forall m$.
Since there exists an $n$ ($n = 0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 2
} |
Equilateral triangle geometric problem I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle $P$. The distance $|AP|=3$ cm, $|BP|=4$ cm, $|CP|=5$ cm.
It is the red triangle in the picture. The exercise is to calculate the area of the Equilateral triangl... | Well, since the distances form a Pythagorean triple the choice was not that random. You are on the right track and reflection is a great idea, but you need to take it a step further.
Check that in the (imperfect) drawing below $\triangle RBM$, $\triangle AMQ$, $\triangle MPC$ are equilateral, since they each have two e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
How to simplify a square root How can the following:
$$
\sqrt{27-10\sqrt{2}}
$$
Be simplified to:
$$
5 - \sqrt{2}
$$
Thanks
| Set the nested radical as the difference of two square roots so that $$\sqrt{27-10\sqrt{2}}=(\sqrt{a}-\sqrt{b})$$ Then square both sides so that $$27-10\sqrt{2}=a-2\sqrt{a}\sqrt{b}+b$$ Set (1) $$a+b=27$$ and set (2) $$-2\sqrt{a}\sqrt{b}=-10\sqrt{2}$$ Square both sides of (2) to get $$4ab= 200$$ and solve for $b$ to g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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How to find $f:\mathbb{Q}^+\to \mathbb{Q}^+$ if $f(x)+f\left(\frac1x\right)=1$ and $f(2x)=2f\bigl(f(x)\bigr)$
Let $f:\mathbb{Q}^+\to \mathbb{Q}^+$ be a function such that
$$f(x)+f\left(\frac1x\right)=1$$
and
$$f(2x)=f\bigl(f(x)\bigr)$$
for all $x\in \mathbb{Q}^+$. Prove that
$$f(x)=\frac{x}{x+1}$$
for all $x\in \mathb... | Some ideas:
$$\text{I}\;\;\;\;x=1\Longrightarrow f(1)+f\left(\frac{1}{1}\right)=2f(1)=1\Longrightarrow \color{red}{f(1)=\frac{1}{2}}$$
$$\text{II}\;\;\;\;\;\;\;\;f(2)=2f(f(1))=2f\left(\frac{1}{2}\right)$$
But we also know that
$$f(2)+f\left(\frac{1}{2}\right) =1$$
so from II we get
$$3f\left(\frac{1}{2}\right)=1\Longri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
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Rotating a system of points to obtain a point in a given place Given an arbitrary number of points which lie on the surface of a unit sphere, one of which is arbitrarily <0, 0, 1> (which I will call K) in a rotated system (i.e. the rotation matrix is unknown), I'm trying to figure out how to (un-)rotate it so that the ... | Having the vector $K$ and knowing that it corresponds to $(0,0,1)$ the rotation matrix can be obtained by taking the cross product $M = K\times(0,0,1)$, this yields a vector perpendicular to both $K$ and $(0,0,1)$, so that you can rotate by the angle between $K$ and $(0,0,1)$ around vector $M$. For that, expression in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
(Probability Space) Shouldn't $\mathcal{F}$ always equal the power set of $\Omega$? This is from the wikipedia article about Probability Space:
A probability space consists of three parts:
1- A sample space, $\Omega$, which is the set of all possible outcomes.
2- A set of events $\mathcal{F}$, where each event is a s... | If $\Omega$ is an infinite set, you can run into problems if you try to define a measure for every set. This is a common issue in measure-theory, and the reason why the notion of a $\sigma-$algebra exists. See, for instance:
The Vitali Set: http://en.wikipedia.org/wiki/Vitali_set
Banach Tarski Paradox: http://en.wikipe... | {
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How to evaluate $\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$ Find the value of
$$I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$$
We have the information that
$$J=\displaystyle\int_0^{\pi/2}x\ln(\sin x)\ln(\cos x)\ \mathrm dx=\dfrac{\pi^2}{8}\ln^2(2)-\dfrac{\pi^4}{192}$$
| This is not quite a complete answer but goes a good way towards showing that the idea of @kalpeshmpopat is not so far off the mark - if we want to answer the question that was orginally asked.
First, numerical investigation indicates that the correct integral is
$$I=\displaystyle\int_0^{\pi/2}x\ln(\sin x)\ln(\cos x)dx=... | {
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Relating volume elements and metrics. Does a volume element + uniform structure induce a metric? AFAIK a metric uniquely determines the volume element up to to sign since the volume element since a metric will determine the length of supplied vectors and angle between them, but I do not see a way to derive a metric fro... | The volume form tells you very little about the metric. Let $V$ be an $n$-dimensional vector space, with volume form $v_1\wedge \ldots \wedge v_n$, where $\{v_1,\ldots,v_n\}$ are linearly independent elements of $V$ (any volume form can be written this way). Now define a metric by the condition that this be an orthon... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Example of a (dis)continuous function The following thought came to my mind: Given we have a function $f$, and for arbitrary $\varepsilon>0$, $f(a+\varepsilon)= 100\,000$ while $f(a) = 1$. Why is or isn't this function continuous?
I thought that with the epsilon-delta definition, where we would chose delta just bigger ... | Hint:
Choose $0<r<10000-1$
Then $\forall~\epsilon>0,~|f(a+\epsilon)-f(a)|=10000-1>r.$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exercise in propositional logic. Which of the following arguments is valid?
A. If it rains, then the grass grows. The worms are not happy unless it rains. Therefore, If the worms are happy , then the grass grows.
B. If the wind howls, then the wolf howls. If the wind howls, then the birds sing. Therefore, if the birds... | A. $(U \wedge (\neg V \Rightarrow \neg U) \wedge (V \Rightarrow W)) \Rightarrow W$
True. If the worms are happy, then it rains, then the grass grows.
B. $((U \Rightarrow V) \wedge (U \Rightarrow W)) \Rightarrow (V \Rightarrow W)$
Wrong. If the birds sing, you don't know anything else. Especially, you don't know if the ... | {
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"url": "https://math.stackexchange.com/questions/330278",
"timestamp": "2023-03-29T00:00:00",
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Find at least two ways to find $a, b$ and $c$ in the parabola equation I've been fighting with this problem for some hours now, and i decided to ask the clever people on this website.
The parabola with the equation $y=ax^2+bx+c$ goes through the points $P, Q$ and $R$. How can I find $a, b$ and $c$ in at least two diff... | You can find normal equation using least square method
First you have to obtain sum of the square of deviation say S and then put the values zero of partial differentiation of S with respect to a,b and c to get three normal equation then solve three normal equation to get a,b and c
| {
"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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Exact differential equations. Test to tel if its exact not valid, am I doing something wrong? I got this differential equation:
$$(y^{3} + \cos t)'y = 2 + y \sin t,\text{ where }y(0) = -1$$
Tried to check for $dM/dY = dH/dY$ but I cant seem to get them alike. So what would the next step be to solve this problem?
| A related problem. Here is how to start. Write the ode as
$$ (y^3 + \cos t)\frac{dy}{dx} = 2 + y \sin t \implies (y^3 + \cos t){dy} - (2 + y \sin t)dx=0 .$$
Now, you should be able to proceed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/330401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Complement is connected iff Connected components are Simply Connected Let $G$ be an open subset of $\mathbb{C}$. Prove that $(\mathbb{C}\cup \{ \infty\})-G$ is connected if and only if every connected component of $G$ is simply connected.
| If the connected open set $H \subset \mathbb C$ is not simply connected, there is a simple closed curve $C$ in $H$ that is not homotopic to a point in $H$. Therefore there must be points inside $C$ that is not in $H$. Such a point and $\infty$ are in different connected components of $({\mathbb C} \cup \{\infty\} - G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/330479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Diophantine equation on an example I do have one task here, that could be solved my guessing the numbers. But the seminars leader said, also Diophantine equation would lead to solution. Has anyone an idea how it works? And could you please show it to me on that example here?
The example:
The number $400$ shall be divid... | You are looking for solutions to the diophantine equation:
$$7x+13y = 400.$$ Clearly, if you can find integer values (x,y) that satisfies this equation, then 7x is your first summand and 13y your second summand.
To solve such a linear diophantine equation, you should use the Euclidean algorithm. Have you done this or ... | {
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"timestamp": "2023-03-29T00:00:00",
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I need to show that the intersection of $T$ and $ \bar{I}$ is equal to empty as well. Lets assume that $T⊂R^{n}$ is open and $I⊂R^n$ & $T\cap I=\varnothing$.
I need to show that the intersection of $T$ and $\bar{I}$ (the closure of $I$) is empty as well.
How to show this? I have seen this question in a book.but I have... | Hint: Since $T$ is open, the complement of $T$ is closed. So now what happens when you take the closure of $I$? How could it possibly have non-empty intersection with $T$?
(Note that $I$ is contained in the complement of $T$, since $I\cap T = \emptyset$.)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
How many strings are there that use each character in the set $\{a, b, c, d, e\}$? How many strings are there that use each character in the set $\{a, b, c, d, e\}$ exactly once and that contain the sequence $ab$ somewhere in the string?
My intuition is to do the following:
$a \; b \; \_ \; \_ \; \_ + \_ \; a \; b \; \... | As Darren aptly pointed out:
Note that in treating $\{a, b, c, d, e\}\;$ like a set of four objects where $a$ and $b$ are "glued together" to count as one object: $\{ab\}$, then we have a set of four elements $\{\{ab\}, c, d, e\}$, and the number of possible permutations (arrangements) of a set of $n = 4$ objects equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/330755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
probability density function In a book the following sentence is told about probability density function at point $a$: "it is a measure of how likely it is that the random variable will be near $a$." What is the meaning of this ?
| The meaning is that if the probability density function is continuous at $a$,
then the probability that the random variable $X$ takes on values in a short interval of length $\Delta$ that contains $a$ is approximately $f_X(a)\Delta$. Thus, for example,
$$P\left\{a - \frac{\Delta}{2} < X < a + \frac{\Delta}{2}\right\} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/330826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
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How do I transform the left side into the right side of this equation? How does one transform the left side into the right side?
$$
(a^2+b^2)(c^2+d^2) = (ac-bd)^2 + (ad+bc)^2
$$
| Expand the left hand side, you get
$$(a^2+b^2)(c^2+d^2)=a^2c^2+a^2d^2+b^2c^2+b^2d^2$$
Add and substract $2abcd$
$$a^2c^2+a^2d^2+b^2c^2+b^2d^2=(a^2c^2-2abcd+b^2d^2)+(a^2d^2+2abcd+b^2c^2)$$
Complete the square, you can get
$$(a^2c^2-2abcd+b^2d^2)+(a^2d^2+2abcd+b^2c^2)=(ac-bd)^2+(ad+bc)^2$$
Therefore,
$$(a^2+b^2)(c^2+d^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 5
} |
Geometry of a subset of $\mathbb{R}^3$ Let $(x_1,x_2,x_3)\in\mathbb{R}^3$. How can I describe the geometry of vectors of the form
$$
\left( \frac{x_1}{\sqrt{x_2^2+x_3^2}}, \frac{x_2}{\sqrt{x_2^2+x_3^2}}, \frac{x_3}{\sqrt{x_2^2+x_3^2}} \right) \, ?
$$
Thank you all!
| To elaborate on user1551's answer a bit, notice that this locus you describe is a fibration of unit circles in the $yz$-plane over the $x$-axis. More precisely, you are interseted in the image of the map $$(x_1,x_2,x_3)\mapsto \left(\frac{x_1}{\sqrt{x_2^2+x_3^2}},\frac{x_2}{\sqrt{x_2^2+x_3^2}},\frac{x_3}{\sqrt{x_2^2+x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/330933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Euclidean Algorithm - GCD Word Problem
An Oil company has a contract to deliver 100000 litres of gasoline. Their tankers can carry 2400 litres and they can attach on trailer carrying 2200 litres to each tanker. All the tankers and trailers must be completely full on this contract, otherwise the gas would slosh around ... | Step 1. Divide everything through by $\gcd(2200,4600)$. (If that greatest common divisor doesn't divide 100000, then there's no solution - do you see why?) In this case, we get $500 = 11x + 23y$.
Step 2. With the remaining coefficients, use the extended Euclidean algorithm to find integers $m$ and $n$ such that $11m + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/330987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Is this function quasi convex I have a function $f(x,y) = y(k_1x^2 + k_2x + k_3)$ which describes chemical potential of a species ($y$ is mole fraction and $x$ is temperature)
I only want to check quasi convexity over a limited range.
k1,k2 and k3 are coefficients of a polynomial function to calculate Gibbs energy o... | Let $k_{1}=1, k_{2}=k_{3}=0$ and consider the level set
$$
\{(x,y)\in R^{2}: f(x,y)\leq 1\}
$$
This condition is equivalent to $y\leq\tfrac{1}{x^{2}}$ and the set
$$\{(x,y)\in R^{2}: y\leq\tfrac{1}{x^{2}}\}$$ is convex in $R^{2}$.
This can be generealized. If the function $k_{1}x^{2}+k_{2}x+k_{3}$ has two different re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Inverse Laplace Transform of $(s+1)/z^s$ I'm trying to compute this ILT
$$\mathcal{L}^{-1}\left\{\frac{s+1}{z^s}\right\},$$
where $|z|>1$. However, I'm not sure this is possile? Any help would be appreciated.
| This is an odd one. I worked the actual Bromwich integral directly because the residue theorem is no help here. You get something in terms of $c$, the offset from the imaginary axis of the integration path. So I appealed to something more basic. Consider
$$\hat{f}(s) = \int_0^{\infty} dt \: f(t) e^{-s t}$$
Just the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
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