Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Given that $\gcd(a,b) =1$, show that $\gcd(a+2b,b)=1$ without using prime factorization theorem If $\gcd(a,b) =1$, show $\gcd(a+2b,b)=1$. I need help figuring how to showing from just that $\gcd(a,b) =1$. Does it have to do with Euclidean formula and that $\gcd(a,b) = am + bn$ for some $m,n$? Thanks.
| If $d|a+2b$ and $d|b$ then $d|a+2b-2(b)=a$ so $\gcd(a+2b,b)|a$ and $\gcd(a+2b,b)|b$. So that gcd divides $\gcd(a,b)=1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Questions about the quotient ring $(\mathbb{Z}/2\mathbb{Z})[x]/\langle x^2+x+1\rangle$ I'm just starting to learn about quotient rings. I was able to think about what type of of elements are generated by some $a$ when $\langle a \rangle$ is simply an integer, and also with simple quotient rings, but I just don't get th... | *
*The set you're written is correct. In any commutative ring ideal generated by one element $a$ is just $<a>=\{ra|r\in R\}$
*Ideal and quotient ring is similar to normal subgroup and quotient group , the element in $R/I$ (here your $R$ is $Z/2Z[x]$ and $I$ is $<x^2+x+1>$) is of the form $I+r$ with $r\in R$ and $I+r... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to see this matrix equality When doing some formulas for regression I encountered this which I think is true by trying some examples:
$$X^T (X X^T + \lambda I)^{- 1} = (X^T X + \lambda I)^{- 1} X^T$$
Here $\lambda > 0$. I'm stuck on how to prove this. Help/counterexamples appreciated.
| By multiplying from the left with $(X^T X + \lambda I)$ and from the right with $(X X^T + \lambda I)$, we obtain
$$
(X^TX + \lambda I) X^T = X^T (X X^T + \lambda I),
$$
and from here it's obvious.
| {
"language": "en",
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Does anyone have any advice as to what measure-theoretic Probability Theory books there are with lots of worked examples? I know that measure-theoretic probability book reference requests have been mentioned quite a few times on this site. However, I was wondering if anyone knew of any good books out there with lots of... | The closest books I know with lots of worked exercises on measure-theoretic probability theory are:
"Problems and Solutions in Mathematical Finance: Volume 1 - Stochastic Calculus" by Chin, Nel and Olafsson. As you can see from the title, it contains more than just probability theory, but I think chapter I covers exact... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Arranging cards so that no two consecutive values remain consecutive Let us say we have 52 cards with values ranging from 1-13 (4 sets of cards from 1-13). Assume that you wanted no two consecutive values to be next to each other in the pile of cards. For example, a 3 cannot be next to a 2 or a 4.
How many ways can I ... | total ways are 52!. now let us assume we have two cards consecutive . consider them as a single group we can have 104 ways . they can be arranged in 2! ways between themselves.We have 1 set of 13 so number of ways of 1 pair of two consecutive cards is 2!.11! such arrangements in 104 ways can be done and of four differe... | {
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A collection of subsets of $\mathbb{R}$ that is not a topology I want to show that the collection of sets of the form: $$\{(-\infty, x] : x \in \mathbb{R}\}$$ together with the empty set and $\mathbb{R}$, is not a topology for $\mathbb{R}$. But if I take the infinite union of sets of the form $\{(-\infty, x] : x \in \m... | Take the union $$\bigcup_{x<y} (-\infty,x],$$ this is $(-\infty,y)$ which is not in your collection of open sets.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1492502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Use fourier transform to solve second-order differential equation -- an "easy" integral? I have scoured the internet for a fully-explained solution to this problem but have found none:
The problem asks to solve this differential equation for $y(t)$ using Fourier Transforms, and then consider cases where $b > w_0$, $b <... | Such kind of integral usually evaluate via residue.
$$
y(t) = -\frac{1}{2\pi}\int_{-\infty}^{\infty}{\frac{e^{-iwt} }{(w-ib)^2 - w_0^2-b^2}dw}
$$
The function $\frac{e^{-iwt} }{(w-ib)^2 - w_0^2-b^2}$ is a holomorphic function. Tis function has two residue $w_{\pm}=i b \pm\sqrt{w_0^2+b^2}$
You should consider two differ... | {
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Square root of complex number. The complex number $z$ is defined by $z=\frac{9\sqrt3+9i}{\sqrt{3}-i}$. Find the two square roots of $z$, giving your answers in the form $re^{i\theta}$, where $r>0$ and $-\pi <\theta\leq\pi$
I got the $z=9e^{\frac{\pi}{3}i}$. So I square root it, it becomes $3e^{\frac{\pi}{6}i}$. But the... | Notice, $$z=\frac{9\sqrt 3+9i}{\sqrt 3-i}$$ $$=\frac{9(\sqrt 3+i)(\sqrt 3+i)}{(\sqrt 3-i)(\sqrt 3+i)}$$
$$=\frac{9(\sqrt 3+i)^2}{3-i^2}=\frac{9(2+2i\sqrt 3)}{3+1}$$$$=9\left(\frac{1}{2}+i\frac{\sqrt 3}{2}\right)=9\left(\cos\frac{\pi}{3}+i\sin \frac{\pi}{3}\right)=9e^{i\pi/3}$$
hence, the square roots of $z$ are found a... | {
"language": "en",
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Prove that if ${s_n}$ is bounded and monotonic, then $t_n =(s_1 + \cdots+ s_n)/n$ converges to the same limit as ${s_n}$ I have already shown that $t_n$ is convergent using the monotonic convergence theorem. Let's say ${s_n}$ converges to $L_1$ and ${t_n}$ converges to $L_2$. How can I show that $L_1$=$L_2$?
| $$s_n-t_n=s_n-\frac1n\sum_{i=1}^n s_i=s_n-\frac{ns_n-\sum_{i=1}^n s_i}n
=\frac1n\sum_{i=1}^n (s_n - s_i)
$$
$$|s_n-t_n|\le \frac1n\sum_{i=1}^n |s_n - s_i| = \frac1n\sum_{i=1}^N |s_n - s_i| + \frac1n\sum_{i=N+1}^n |s_n - s_i|
$$
For any $\epsilon>0$ there is $N>0$, such that for every $n>N$ we have $|s_n - s_i|<\epsilon... | {
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heading angle calculation using atan2 I am trying to find heading angle for my three wheeled robot my robot setup is as below
I know all co-ordinate values.
(x1 y1) (x2 y2) two back wheels and (x3 y3) is front wheel co-ordinate
(xm ym) is the midpoint of (x1 y1) and (x2 y2)
(xt yt) is the target point
I am trying to f... | You have a line from (xm,ym) to (x3,y3) and another line between (xm,ym) and (xy,yt).
Now all you interesting in is the angle between the two lines.
The answer is here.
This is officially unanswered question, but the correct answer is there.
Pick up the one with highest votes (Rory Daulton) and you good. Just use $\m... | {
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Possibility that two students answer the same on a 25 question multiple choice test I am trying to get some approximation of the possibility that two students would answer $25$ questions in a row exactly the same. There are 4 possible responses for each question. I calculate the random possibility as $4^{25}=1.13\tim... | While you can come up with a mathematical answer. It would only show how unlikely the possibility of the two students getting the same answers assuming they were to randomly guess. In reality, it is not as unlikely as you would think. For example some questions might have a really deceptive choice and both students pic... | {
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Let $f(x)= \sin^{-1}(2e^x)$. What is $f '(\ln(3/10)$? So I tried switching $y= \sin^{-1}(2e^x)$ with $x= \sin(2e^y)$ and then finding the derivative. The answer does not seem to match up. Thanks for the help
| It should be $x=\log\left(\frac{1}{2}\sin y\right)$.
$$\sin y = 2e^x\\
\frac{1}{2}\sin y = e^x\\
\log\left(\frac{1}{2}\sin y\right) = x$$
| {
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Find a particular solution of a nonhomogenous equation Find a particular solution to $y''-2y' + y= {e^{px}}$
where p is any real constant.
My attempt/idea is as follows:
Since $Y_p$ is ${e^{px}}$ and it appears in the complementary solution, $Y_c$ is $c_1{e^x} + c_2x{e^x}$ , we will have to multiply $Y_p$ by $x^2$ so ... | Working the differential equation in the most general manner, the particular solution is $$y_p=\frac{e^{p x}}{(p-1)^2}$$ which shows that the case of $p=1$ is totally different from the other possible cases just as Dylan pointed it out.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What exactly did Hermann Weyl mean? "The introduction of numbers as coordinates is an act of violence." - Hermann Weyl.
A lot of people like this quote, apparently. They also seem to associate it to the manifold context in the obvious way: they interpret the quote as saying that focusing on coordinate charts is insight... | From a Google search, it appears the quote is from Hermann Weyl's Philosophy of Mathematics and Natural Science. I found a copy online here; the relevant passage is on page 90 (search "act of violence"):
The introduction of numbers as coordinates by reference to the particular division scheme of the open one dimension... | {
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Find the nullity and rank of a 3x5 matrix C where three columns = 0? Tricky problem. So I have a 3x5 matrix C, where {s,t,u,v,w} (I'm assuming those are its columns) is a linearly independent set of vectors in R^5, and that Cu=0, Cv=0, Cw=0.
What is the rank and nullity of C?
I'm guessing with the latter part of the qu... | *
*The elements $\{s,t,u,v,w\}$ are not the columns of $C$, they are just linearly independent vectors in $\mathbb{R}^5$.
*The nullity of $C$ is the dimension of its nullspace, which is the subspace of $\mathbb{R}^5$ consisting of vectors $x$ satisfying $Cx=0$. You already have three linearly independent vectors in t... | {
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How to evaluate $\int_0^A \frac{\tanh x}{x}dx$? How to evaluate $$\int_0^A \frac{\tanh x}{x}dx$$
Where $A$ is a large positive number.
The answer is: $$\ln (4e^\gamma A/\pi)$$, where $\gamma$ is Euler constant.
I have no idea how to get this result. Here is a numerical result, blue is the original integral.
| That formula is indeed an asymptotic expansion. Indeed, integration by parts yields
$$ \int_{0}^{A} \frac{\tanh x}{x} \, dx = \tanh A \log A - \int_{0}^{A} \frac{\log x}{\cosh^2 x} \, dx $$
for $A > 0$, and we easily see
$$ \int_{0}^{A} \frac{\log x}{\cosh^2 x} \, dx = \int_{0}^{\infty} \frac{\log x}{\cosh^2 x} \, dx +... | {
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integrate over a cube given some differential form What is process of integrating a differential form given some cube (hyperdimensional obejcts)? I read a lot qualitative problems on this, but seem to find rare examples on how to compute such integrations step by step.
For example, if I have a 2-cube$$[0,1]^2->R^3$$ wi... | Pull back the form to $[0,1]^2$ and integrate it there. The computation looks like this: we have $(x_1,x_2,x_3) = (t_1^2, t_1 t_2, t_2^2)$ and thus $(dx_1,dx_2,dx_3) = (2t_1 dt_1, t_1 dt_2 + t_2 dt_1, 2t_2 dt_2)$. Differentiating $\alpha$ we have $$d\alpha = dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_2 \wedge dx_3.$$ Sub... | {
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Question about the derivative definition The derivative at a point $x$ is defined as:
$\lim\limits_{h\to0} \frac{f(x+h) - f(x)}h$
But if $h\to0$, wouldn't that mean:
$\frac{f(x+0) - f(x)}0 = \frac0{0}$ which is undefined?
| Actually, the $ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $ is the value which $\frac{f(x+h) - f(x)}{h}$ approaches when you keep reducing $h$ as much as you can.
Here is how my teacher explained me the idea:
Think of drawing a tangent. It's a line touching a curve at only one point.
But how could that be possible? Two p... | {
"language": "en",
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Finding the reflection that reflects in an arbitrary line y=mx+b How can I find the reflection that reflects in an arbitrary line, $y=mx+b$
I've examples where it's $y=mx$ without taking in the factor of $b$
But I want to know how you can take in the factor of $b$
And after searching through for some results, I came to... | One way to do this is as a composition of three transformations:
*
*Translate by $(0,-b)$ so that the line $y=mx+b$ maps to $y=mx$.
*Reflect through the line $y=mx$ using the known formula.
*Translate by $(0,b)$ to undo the earlier translation.
The translation matrices are, respectively,
$$
\begin{pmatrix}
1 & 0... | {
"language": "en",
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Finding the coordinate C. A triangle $A$, $B$, $C$ has the coordinates:
$A = (-1, 3)$
$B = (3, 1)$
$C = (x, y)$
$BC$ is perpendicular to $AB$. Find the coordinates of $C$
My attempt:
Grad of $AB$ =
$$\frac{3-1}{-1-3} = -0.5$$
Grad of $BC = 2$ ($-0.5 \times 2 = -1$ because AB and BC are perpendicular).
Equation of $BC$... | The equation of $AC$ is $x-3y=-10$ as slope of any line is $-\frac{a}{b}$ where $a$ is $x$-coordinate and $b$ is $y$-coordinate so slope is $-\left(\frac{-1}{3}\right)$. $C$ is the point where $AC$ and $BC$ so meet we have two simultaneous equations $2x-y=5$ and $x-3y=-10$ solving them you get $x=5$ and $y=5$.
| {
"language": "en",
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"source": "stackexchange",
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Radius of convergence of this power series We're given the power series $$ \sum_1^\inf \frac{j!}{j^j}z^j$$
and are asked to find radius of convergence R. I know the formula $R=1/\limsup(a_n ^{1/n})$, which leads me to compute $\lim \frac{j!^{1/j}}{j}$, and then I'm stuck.
The solution manual calculates R by $1/\lim|\fr... | Use Stirling's formula
$$n! \sim {n^n e^{-n}\over\sqrt{2\pi n}}.$$
| {
"language": "en",
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A and B play a series of games. Find the probability that a total of 4 games are played. A and B play a series of games. Each game is independently
won by A with probability $p$ and by B with probability $1 - p$. They stop when the
total number of wins of one of the players is two greater than that of the other player.... | I would consider two games at a time. Two games can result in AA (team A wins, probability $p^2$) or BB (team B wins, probability $(1-p)^2$) or AB, BA considered together (match is back to starting state, with probability $2p(1-p)$).
Then the probability that the match goes for four games (two pairs) is $(2p(1-p))(p^2... | {
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Is it possible to find the absolute value of an integer using only elementary arithmetic? Using only addition, subtraction, multiplication, division, and "remainder" (modulo), can the absolute value of any integer be calculated?
To be explicit, I am hoping to find a method that does not involve a piecewise function (i.... | I think you need this: https://stackoverflow.com/questions/9772348/get-absolute-value-without-using-abs-function-nor-if-statement.
Please note that remainder is modulo, absolute is modulus, so please correct the question, because I can't suggest edit.
| {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate $\int\frac {\sin 4x }{\sin x}\ dx$
Evaluate $$\int \frac{\sin 4x}{\sin x} dx$$
Attempt:
I've tried to use the double angle formulas and get it all into one identity, which came out as:
$$
\int \left( 8\cos^3x - 4\cos x \right) dx
$$
But I'm not sure if this is the right way to go about it. Any help would be ... | We can put the complex identity
$$\sin \alpha := \frac{\exp(i \alpha) - \exp(-i \alpha)}{2i}$$ to efficient use here.
Taking $\alpha = 4 x$ gives
$$\sin 4x = \frac{\exp(4 i x) - \exp(-4 i x)}{2i},$$
We can use a difference-of-squares factorization to write the numerator as
\begin{align*}
\exp(4 i x) - \exp(-4 i x)
&= [... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that a matrix $A=\pmatrix{a&b\\c&d}$ satisfies $A^2-(a+d)A+(ad-bc)I=O$
Let $A=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
,a,b,c,d\in\mathbb{R}$ . Prove that every matrix $A$ satisfies the condition $$A^2-(a+d)A+(ad-bc)I=O .$$ Find $$ \begin{bmatrix}
a & b \\
... | One can of course prove this directly by substituting and computing the entries of the $2 \times 2$ matrix on the left-hand side, but here's an outline for a solution that reduces the amount of actual computation one needs to do.
Hint
*
*Prove the claim for diagonal matrices.
*Prove the claim for matrices similar t... | {
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Regular categories and epis stable under pullbacks The definition of a regular epi is an arrow which coequalizes some parallel pair. Isn't this just another name for a coequalizer?
One of the (usual) axioms for a regular category says each arrow has a kernel pair. Another says regular epis are stable under pullbacks. B... | I think your confusion is ultimately a confusion about what it means for the various notions to be "stable under pullbacks". First, "regular epis are stable under pullbacks" means that if
$$\require{AMScd}
\begin{CD}
A @>{}>> B\\
@V{f'}VV @V{f}VV \\
C @>{}>> D
\end{CD}$$
is a pullback square and $f$ is regular epi, t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Find $\lim_{x\to0}{\frac{\arctan (2x)}{3x}}$ without using '0/0=1' Find limit as
$$
\lim_{x \to 0} \frac{\arctan (2x)}{3x}
$$
without using $\frac{0}{0} = 1$.
I wanted to use
$$
\frac{2}{3} \cdot \frac{0}{0} = \frac{2}{3} \cdot 1,
$$
but our teacher considers $\frac{0}{0}$ a "dangerous case" and we are not allowed t... | You may use L'Hospital's rule:
$$
\lim_{x\to0}{\frac{\arctan (2x)}{3x}}=\lim_{x\to0}{\frac{\frac{2}{1+4x^2}}{3}}=\frac23.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1494521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Other ways to compute this integral? The following (improper) integral comes up in exercise 2.27 in Folland (see this other question): $$I = \int_0^\infty \frac{a}{e^{ax}-1} - \frac{b}{e^{bx}-1}\,dx.$$
I computed it as follows.
An antiderivative for $a(e^{ax}-1)^{-1}$ is $\log(1-e^{-ax})$, found by substituting $u = e... | Here's a nice solution.
If we let $$f(x) = \frac{x}{e^x-1} = \frac{1}{1+\frac{x}{2}+\frac{x^2}{6}+\dotsb}$$
then $f(0) = 1$ and $f(\infty) = 0$ and $$\frac{f(ax) - f(bx)}{x} = \frac{a}{e^{ax}-1} - \frac{b}{e^{bx}-1},$$
and now apply Frullani's theorem.
| {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 1
} |
What is $x^2-1$ applied n times For the function $F(x)=x^2-1$. How do I write $F^n(x)$ ($F$ applied $n$ times) in terms of $x$?
| A somewhat trivial way would be $F^n(x) = x^2-1$ if $n=1$ and $F^n(x)=(F^{n-1}(x))^2-1$ otherwise.
Presumably you want a closed form, though, and it is probably pretty obvious that in the closed form the leading coefficient is 1, the degree is $2n$, and the constant term is $-1$ if $n$ is odd and 0 if $n$ is even. Per... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1494683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Isn't every set a G set? It is clear to me that every G-set (X) can be described as an action on X(a subgroup of $S_X$). The trouble I am having is why specify a set X as a G set without specifying that their must exist an embedding $F:G\to S_X$. It seems not very useful to just say X is a G-set.
| A $G$-set isn't just a set $X$ such that there exists a homomorphism $G\to S_X$; it is a set $X$ equipped with a specific homomorphism $G\to S_X$. That is, a $G$-set is more properly speaking a pair $(X,\rho)$ where $X$ is a set and $\rho:G\to S_X$ is a homomorphism, and only by abuse of terminology do we say "$X$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1494833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How to prove $\frac xy + \frac yx \ge 2$ I am practicing some homework and I'm stumped.
The question asks you to prove that
$x \in Z^+, y \in Z^+$
$\frac xy + \frac yx \ge 2$
So I started by proving that this is true when x and y have the same parity, but I'm not sure how to proceed when x and y have opposite partiy
Th... | Here is "another" simple method(this is essentially same as completing the square.) of proving $\frac{x}{y}+\frac{y}{x}\geq 2$ by the help of AM-GM Inequality :
Consider the set $\{\frac{x}{y},\frac{y}{x}\}.$
Applying AM-GM Inequality on these two values , we have :
$$\frac{\frac{x}{y}+\frac{y}{x}}{2} \geq \sqrt {\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1494887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 10,
"answer_id": 7
} |
Envelope of Projectile Trajectories
For a given launch velocity $v$ and launch angle $\theta$, the trajectory of a projectile may be described by the standard formula
$$y=x\tan\theta-\frac {gx^2}{2v^2}\sec^2\theta$$
For different values of $\theta$ what is the envelope of the different trajectories? Is it a parabol... | Although there are several very nice solutions provided above, here is another approach. No calculus is needed.
Consider a projectile launched with angle $\theta$ with relation to an inclined
plane with tilt angle $\varphi$. The initial speed is $v_0$.
We want to know at what distance from the launch point the projecti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 7,
"answer_id": 3
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Using CRT to reason about powers of $2$ where the power is two less than an odd prime Let $p$ be any odd prime.
Using the Chinese Remainder Theorem and Fermat's Little Theorem, we know that:
$$\frac{2p}{2} + \left(\frac{2p}{p}\right)\left(\frac{2p}{p}\right)^{p-2} = p + 2^{p-1} \equiv 1 \pmod {2p}$$
There exists $c$ wh... | In the first line, you have a term $\frac{2p}{2}$ mod $2p$. That is a dangerous thing to write, because we cannot divide by $2$ mod $2p$. This is because both $2\cdot 0 = 2p$ mod $2p$ and $2\cdot p = 2p$ mod $2p$, so both 0 and $p$ can be said to be equal to $2p/2$ mod $2p$. In general, it is a bad idea to divide by ze... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is a harmonic conjugate unique up to adding a constant? If $v$ and $v_0$ are harmonic conjugates of $u$, then $u + iv$ and $u + iv_0$ are analytic functions. Then $i(v - v_0)$ is analytic, but how does this imply $v - v_0$ is a constant function?
| It does not follow from $v-v_0$ being (real) analytic, if that is what you are asking for. $u+iv$ and $u+iv_0$ are complex analytic, i.e. holomorphic, so they satisfy the Cauchy Riemann differential equations, $u_x = v_y$ an $u_y = -v_x$, the same holds for $v_0$ and $u$. So, consequently, $(v-v_0)_x = 0 = (v-v_0)_y$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Laplace's equation after change of variables
Show that if $u(r, \theta)$ is dependent on $r$ alone, Laplace's
equation becomes $$u_{rr} + \frac{1}{r}u_r=0.$$
My first reaction is to replace $r=x$ and $\theta=y$, but obviously it does not work. Then I recall $x=r\cos \theta$ and $y=r\sin \theta$. Then I obtain the f... | You have
$$ \frac{\partial f}{\partial r} = \frac{\partial x}{\partial r} \frac{\partial f}{\partial x} + \frac{\partial y}{\partial r} \frac{\partial f}{\partial y} = f_x \cos{\theta} + f_y \sin{\theta}. $$
Then differentiating again (and using that $\partial \theta/\partial r=0$),
$$ f_{rr} = \cos{\theta} (\cos{\thet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras. Is it true that $A^+ \subset B^+$ if $A \hookrightarrow B$ is inclusion of $C^*$ algebras, where $A^+$ denotes the positive elements in $A$.
I read in Murphy 2.1.11 that this is true if $B$ is unital and $A$ contains the unit of $B$.
Does it ... | You don't need to look at the spectra. You can characterize postive elements as those of the form $z^*z$. So $a\in A^+$. then $a=z^*z$ for some $z\in A\subset B$. So $a$ is positive in $B$ too.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding real numbers such that $(a-ib)^2 = 4i$ Prove that $(a^2 - b^2) = 0$ I sometimes find myself overcomplicating my life... overthinking simple concepts.
Here I don't use what's given, i.e., $$(a − ib)^2 = 4i$$
So I might say let $a = 1$ and $b = 1$
then $a = b$ and $a^2 = b^2$ thus $a^2 - b^2 = 0$
Now that see... | You are on the right track, but you are overthinking it. The basic idea behind these sorts of questions is the fact that if $z = x + iy$ and $w = q + ip$ where $z,w \in \mathbb{C}$ and $x,y,q,p \in \mathbb{R}$ then $z = w$ iff $x = q$ and $y = p$.
Equipped with this we may deal with your problem. Indeed first we comput... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a way to parametrise general quadrics? A general quadric is a surface of the form:
$$ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Eyz + 2Fxz + 2Gx + 2Hy + 2Iz + J = 0$$
It can be written as a matrix expression
$$ [x, y, z, 1]\begin{bmatrix}
A && D && F && G \\
D && B && E && H \\
F && E && C && I \\
G && H && I && J
\end{bma... | Given a point $p$ on the quadratic surface $Q$, every line $L$ through $p$ is either tangent to $Q$, or it intersects $Q$ in another point $p_L$. In this way the lines not tangent to $Q$ parametrize $Q-\{p\}$. These lines are in turn parametrized by $\Bbb{R}^2$, so this is possible if you omit one point from the parame... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving a quadratic equation in $\Bbb{Z}/19\Bbb{Z}$. I am working on a problem and I have written down my thoughts but I am having trouble convincing myself that it is right. It very may well be it is also completely wrong, which Is why I thought id post it and my attempt and try to get some feedback.
I want to find al... | It seems to me that you are doing too much work here. You have $y = 1-x$, so $x(1-x) = c$. So now you just have to list all values of $x(1-x)$ mod $19$ for $0 \le x < 19$, and you are done.
This is hardly more difficult than listing all the squares mod $19$, which you did as the first step of your solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Can this function be integrated? Can't seem to figure out this integral! I'm trying to integrate this but I think the function can't be integrated? Just wanted to check, and see if anyone is able to find the answer (I used integration by parts but it doesn't work). Thanks in advance; the function I need to integrate is... | Probably you'd want to use the calculus of residues to do this.
But below I do it using first-year calculus methods.
The cumbersome part may be the algebra, and that's what I concentrate on here.
\begin{align}
x^5 + 2 & = \left( x+\sqrt[5]{2} \right) \underbrace{\left( x - \sqrt[5]{2} e^{i\pi/5}\right)\left( x - \sqrt[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can one do anything useful with a functional equation like $g(x^2) = \frac{4x^2-1}{2x^2+1}g(x)$? I got
$$g(x^2) = \frac{4x^2-1}{2x^2+1}g(x)$$
as a functional equation for a generating function. Is there a way to get a closed form or some asymptotic information about the Taylor coefficients from such an equation?
Here... | I get that
the only solution is
$g(x) = 0$
if we can write
$g(x)
=\sum_{n=0}^{\infty} a_n x^n
$.
Here is my proof:
We have
$g(x^2)
= \frac{4x^2-1}{2x^2+1}g(x)
$
or
$(2x^2+1)g(x^2)
= (4x^2-1)g(x)
$.
From this,
as copper.hat pointed out,
$g(0) = 0$.
If
$g(x)
=\sum_{n=1}^{\infty} a_n x^n
$,
(since
$g(0) = 0$)
the left s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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How to determine image of the fundamental group of a covering space of $S^1 \vee S^1$
Consider the covering space of $S^1 \vee S^1$ in $(1)$. Then distinct loops in $(1)$ are represented by $\langle a, b^2, bab^{-1} \rangle$.
Thus elements of the fundamental group are words generated by these distinct loops.
This fu... | 1)is $S^1\vee S^1\vee S^1$ its fundamental group is $Z*Z*Z$ and it is the subgroup of $\pi_1(S^1\vee S^1)$ generated by $a,b^2,bab^{-1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1496065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to estimate of coefficients of logistic model Consider model $logit(p)=a+bx$. I would like to get a analytic formula of $a$ and $b$ like in linear regression. In linear regression, we can get a formula of estimates of $a$ and $b$.
I tried using MLE to estimate it. But it is too complicated for me.
| For estimate the values of $a$ and $b$ in your model:
$$logit(p)=a+bx^{(i)}$$
For simplify you can consider that the $a$ is multiplying by $x^{(0)}$ with value $1$, and use the matrix notation.
$$ Z = \theta\cdot x$$
In logistic regression you can use the sigmoid function as below.
$$
h_\theta (x) = \dfrac{1}{1 + e^{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1496177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Extensions of $\mathbb{Q}((T))$ and $\mathbb{F}_p((T))$ Okay, I'm having some trouble finding good references for this, so here goes:
Is every finite extension of $\mathbb{Q}((T))$ isomorphic to $K((T^{1/e}))$ where $K$ is finite over $\mathbb{Q}$, and $e$ is an integer?
Is every finite extension of $\mathbb{F}_p((T))$... | Answer to edited question: Yes and it works for arbitrary field $k((T))$ if either $k$ is of characteristic 0 or when the ramification index is not divisible by characteristic of $k$, assuming that your notation $k((T))$ refers to power series ring with coefficients in $k$. This is theorem 6 of chapter 4 section 1 (pag... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1496236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If a function of two variables has a unique critical point, which is a local maximum, is it a global maximum?
$f(x,y)$ has partial derivatives in all $\mathbb R^2$ and a unique critical point at $(x_0,y_0)$ (local maximum). Is it a global maximum?
I know that in compact sets, it isn't enough to say that if a point is... | Loose description of the geometry: imagine a flat plane and then you put a lone hill with a peak on it. Now tilt the plane a little. Now you still have one peak, but hopefully you can also see that you have introduced a saddle point. Imagine sliding the saddle point location off to infinity to make it effectively no lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1496321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Proving an intuitively true statement. Let $X \subseteq Y$ and $X\neq Y$ Also let $f: Y → X$ define a bijection. Prove that $Y$ is infinite.
Here's what I have as a proof, but I'm not really sure if it's enough.
Let $X \subseteq Y$ and $X\neq Y$, this must mean that $|X| < |Y|$ which also implies there's an injectio... | Using fundamental theorems of cardinality, if $Y$ happens to be finite then so is $X$. Then, from the inclusion and using the fact that $Z_1\subset Z_2$, $|Z_1|=|Z_2|$ implies $Z_1=Z_2$ if $Z_1$ and $Z_2$ are finite, we see that $|X|<|Y|$ on the other hand, using the bijection $f$ we get $|X|=|Y|$ which is a contradict... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1496400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $n_1 \sqrt{2} +n_2 \sqrt{3} + n_3 \sqrt{5} + n_4 \sqrt{7} $ never zero? Here $n_i$ are integral numbers, and not all of them are zero.
It is natural to conjecture that similar statement holds for even more prime numbers. Namely,
$$ n_1 \sqrt{2} +n_2 \sqrt{3} + n_3 \sqrt{5} + n_4 \sqrt{7} + n_5 \sqrt{11} +n_6 \s... | If $p_1,\dots,p_n$ are prime numbers with $p_i\ne p_j$ for $i\ne j$, then the field extension $\mathbb Q\subset\mathbb Q(\sqrt{p_1},\dots,\sqrt{p_n})$ has degree $2^n$ and a basis is given by the set $$\{1,\sqrt{p_{i_1}\cdots p_{i_k}}:1\le i_1<\cdots<i_k\le n, 1\le k\le n\}.$$ See Proving that $\left(\mathbb Q[\sqrt p_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1496567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Three lines are concurrent (or parallel) $\iff$ the determinant of its coordinates vanishes. I'm trying to prove the concurrency condition for three lines lying on a plane. This condition says that:
Let
\begin{cases} ax + by + cz=0 \\ a'x – b'y + c'z=0 \\ a''x + b''y + c''z=0 \end{cases}
be three lines (barycentr... | I have also tackled this problem recently. It looks very easy and a basic problem, and yet I haven't been able to convince myself of this proposition.
What I did:
If the determinant |A| is 0, then the rank of the coefficient matrix
is $\leq 2$.
Thus, the zero space of the given system of equations A x = 0 has dimensio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1496688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Show that maximal abelian normal subgroup of $p$-group contains the commutator subgroup
Let $P$ be a $p$-group and $A$ a maximal abelian normal subgroup of $P$. If $|A : C_A(x)| \le p$ for all $x \in P$, then $P' \le A$.
As far as I know I have no idea how to bring the condition about the index of $C_A(x)$ in $A$ int... | This seems quite challenging! For $x,y \in P$, it is enough to prove that $[x,y]$ centralizes $A$, or equivalently that the automorphisms of $A$ induced by conjugation by $xy$ and $yx$ are the same, because then the fact that $A$ is maximal abelian implies $C_G(A)=A$, so $[x,y] \in A$.
This is easy if $C_A(x)=C_A(y)$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving using mean value theorem Let f be a function continuous on [0, 1] and twice differentiable on (0, 1).
a) Suppose that f(0) = f(1) =0 and f(c) > 0 for some c ∈ (0,1).
Prove that there exists $x_0$ ∈ (0,1) such that f′′($x_0$) < 0.)
b) Suppose that $$\int_{0}^{1}f(x)\,\mathrm dx=f(0) = f(1) = 0.$$
Prove that the... | We solve the first problem only. The function $f$ reaches a (positive) maximum at some point $p$ strictly between $0$ and $1$. At any such $p$, we have $f'(p)=0$.
Since $f(1)=0$, by the Mean Value Theorem there is a $q$ strictly between $p$ and $1$ such that $f'(q)\lt 0$.
Thus by the Mean Value Theorem there is a point... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1497086",
"timestamp": "2023-03-29T00:00:00",
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Mutually exclusive countable subsets of a countable set This is part of a bigger problem I'm trying to prove, but my argument relies of the validity of the following idea. Note that when I say countable, I don't mean finite -- I mean countable infinity.
Consider the set of natural numbers, $\mathbb{N}$. Now take a cou... | Edit: @ErickWong is right (see his comment below). This doesn't answer the question. I did cover myself with the caveat "If I understand ...". I may leave this nonanswer up for a while since others may learn from it.
If I understand your question correctly here's another way to show you can have an infinite set left ov... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove $x^{4}-x+1=0$ has no solution
I would like to prove that the following equation has no solution in $\mathbb{R}$
$$x^{4}-x+1=0$$
my question : could we use Intermediate Value Theorem to prove it otherways I'm interested in more ways of prove that has no solution in $\mathbb{R}$.
without :
i know that we can p... | If $x = x^{4}+1$, then certainly $x \geq 1$ as $x^{4} \geq 0$ for all real $x$. But then $x^{4} = x^{3}x \geq x$ and $x^{4}+1 \geq x+1 > x$, a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1497270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Determining if $973$ is prime
Without a calculator, determine if $973$ is prime or not
I was given this question to solve. I know $973$ is not prime. I was told a strategy to solve whether a number is prime or not is to test all the numbers less than the square root of $973$
So I would have to test till $32$
and i fi... | Another way, $973 = 910+63$
This gives $973 = 7.(130+9) = 7 . 139$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1497404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Proving convergence by the comparison test I need to prove the convergence of the following series :
$$\sum_{n=1}^{\infty} \frac{1}{(3n-2)(3n+1)}$$
I suppose I need to find two series like
$$\sum_{n=1}^{\infty}(k_1a_n + k_2b_n) = \sum_{n=1}^{\infty} = k_1\sum_{n=1}^{\infty}a_n + k_2\sum_{n=1}^{\infty}b_n$$
I thought t... | One may just observe a telescoping sum here
$$
\sum_{n=1}^N \frac{1}{(3n-2)(3n+1)}=\frac13\sum_{n=1}^N \left(\frac{1}{3(n-1)+1}-\frac{1}{3n+1}\right)=\frac13-\frac{1}{3(3N+1)}
$$
giving the convergence and the sum of your initial series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1497528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $v$ is an eigenvalue of $e^{t A}$ for all $t \geq 0$, is it also an eigenvalue of $A$? I'm currently writtng a proof and I wont use that, if $v$ is an eigenvector of $e^{tA}$ for all $t \geq 0$ where $A$ is some generating matrix, then $v$ is an eigenvector of $A$ itself. However, I found neither the proof nor a con... | If $Av = \lambda v$, it's easy to see $\exp(tA)v = \exp(t\lambda) v$.
Conversely, if $v$ is an eigenvector of $\exp(tA)$ for all non-zero $t$, then $v$ is an eigenvector of $\frac{1}{t}(\exp(tA) - I) = A + O(t)$ for all $t \neq 0$, and by continuity $v$ is an eigenvector of $A$.
Erratum: Though the question stipulates... | {
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"timestamp": "2023-03-29T00:00:00",
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Understanding modulos and polynomials when dividing two polynomials I understand the idea of modulo with integers but I am having an incredibly hard time wrapping my head around the idea of modulus with polynomials. I am taking a course in which I need to divide polynomials in terms of a specific modulus. I am able to ... | All you need is computing the inverse of the leading coefficient modulo $7$. The operations are the same as in $\mathbf R$, but you compute modulo $7$. Here is an example:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1497715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If $p=4k+1$ then $p$ divides $n^2+1$ I am stuck in one step in the proof that if $p$ is congruent to $1 \bmod 4$, then $p\mid (n^2+1)$ for some $n$.
The proof uses Wilson's theorem, $(4k)!\equiv -1 \pmod p$.
The part I am stuck is where it is claimed that $(4k)!\equiv (2k)!^2 \bmod p$. Why is this so?
| You group $\{1,p-1\},\{2,p-2\},...,\{2k,2k+1\}$ together, and the product of the first terms are equal to to product of second terms as there are even number of terms.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help me find my mistake when finding the exact value of the infinite sum $\sum_{n=0}^{\infty}\frac{e^{n-2}}{5^{n-1}}$
Finding the exact value of the infinite sum:
$$\sum_{n=0}^{\infty}\frac{e^{n-2}}{5^{n-1}}$$
My Approach:
First term (a);
$$\frac{e^{-2}}{5^{-1}}=\frac{5}{e^2}$$
Second term:
$$\frac{e^{-1}}{5^{0}}=\... | Your sum is correct.
One may recall a standard result concerning geometric series
$$
\sum_{n=0}^\infty r^n=\frac{r}{1-r},\qquad |r|<1.
$$ Applying it with $r=\dfrac{e}5\,\,\left(\left|\dfrac{e}5\right|<1\right)$, gives
$$
\sum_{n=0}^{\infty}\frac{e^{n-2}}{5^{n-1}}=\frac{e^{-2}}{5^{-1}}\sum_{n=0}^{\infty}\frac{e^{n}}{5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1497953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Guy with $7$ friends Some guy has $7$ friends$(A,B,...,G)$. He's making dinner for $3$ of them every day for one week. For how many ways can he invite $3$ of them with condition that no couple won't be more then once on dinner. (When we took $A,B,C$ we cannot take $A$ with $B$ or $C$ and $B$ with $C$). My idea: We have... | Either I am missing something or the book answer is wrong. I take it that when you write "no couple won't be more than once on dinner", you actually mean "no couple will ....." as has been amply clarified in your examples.
Now there can only be $\dbinom72 = 21$ distinct couples, and each day $3$ such couples get elimin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1498062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Bounded linear functional is necessarily continuous proof verification I want to prove that a bounded linear functional $f$, must be continuous.
I have defined: $f$ is bounded means that $\exists c> 0, |f(x)|\leq c\|x\|, \quad \forall x\in X$ and continuous means that $x_n\to x \implies f(x_n)\to f(x)$.
Proof: Let $f$ ... | (1) Yes, that's all.
(2) Without linearity, the statement becomes wrong, even in the $X = \mathbf R$ case. Consider for example, $f \colon \def\R{\mathbf R}\R\to \R$ given by
$$ f(x) = \begin{cases} x & x \in [-1,1] \\ 0 & x \not\in [-1,1] \end{cases} $$
Then $f$ is bounded, as $|f(x)| \le |x|$ for all $x \in \R$, bu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Formula for Nested Radicals I know that: $$\sqrt{2+\sqrt{2+\sqrt{2+...\sqrt{2}\; (upto\; n\; times)}}}=2\cos(2^{-n-1}\:\pi)$$
I was wondering whether such a formula exists for $$\sqrt{3+\sqrt{3+\sqrt{3+...\sqrt{3}\; (upto\; n\; times)}}}$$
or in general for,
$$\sqrt{k+\sqrt{k+\sqrt{k+...\sqrt{k}\; (upto\; n\; times)}}}... | Let $$f(k,n)=\underbrace{\sqrt{k+\sqrt{k+\sqrt{\ldots+\sqrt k}}}}_n $$
We know that $\cos\frac x2=\pm\frac12\sqrt{1+\cos x}$. Therefore if $f(n,k)=a\cos b$ then $$f(n+1,k)=\sqrt{k+f(n,k)}=2\sqrt k\cdot\frac12\sqrt{1+\frac{f(n,k)}{k}}=2\sqrt k\cdot\frac12\sqrt{1+\frac ak\cos b}. $$
This works out nicely with our haf-ang... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Coin change problem Given a set of coins $S = \{w^0, w^1, w^2, ....., w^n\}$, for a given $w$, how to test whether an amount $X$ can be changed for i.e. find such subsets $S1, S2$ such that $$\sum_{c \in S1} c + X = \sum_{c \in S2} c$$ $S1, S2 \subseteq S$ and $S1 \cap S2 = \emptyset$ i.e. each coin can be used at most... | The amounts of change that can be made are the numbers of the form
$$\sum_{k=0}^n\epsilon_kw^k\;,\tag{1}$$
where each $\epsilon_k\in\{-1,0,1\}$, and if $\ell=\max\{k:\epsilon_k\ne 0\}$, then $\epsilon_\ell=1$. Equivalently they are the non-negative integers that can be expressed with at most $n+1$ digits, each of which... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving that if $ab=e$ then $ba=e$ Suppose that instead of the property $ab=ba=e$ a group G has the condition that for every element $a$ there exists an element $b$, such that $ab=e$. Prove that $ba=e$. Is the following a valid proof?
Since $ab=e$ then under the condition of the group there exists an element $k$ such t... | I assume that you mean if for SOME $a,b$ (not every) $ab = e$ then $ba = e$.
Your proof is valid, but you could write it much easier without playing with $k$. $$ab = e \Rightarrow bab = b.$$ If you already know the cancellation law, then we are done. Otherwise you may continue by writing $baba = ba$, so that $(ba)^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that if $AB = 0$, then rank(A) + rank(B) ≤ p Let $A$ be an $m \times$ n matrix and $B$ be an $n \times p$ matrix.
I understand that since $AB=0$, the column space of $B$ is contained within the nullspace of $A$. Does this mean that $\operatorname{rank}(B) \leq \operatorname{nullity}(A)$?
How do I proceed to show... | Yes, you may indeed deduce that the rank of $B$ is less than or equal to the nullity of $A$.
From there, simply apply the rank-nullity theorem (AKA dimension theorem).
Counterexample to question as stated:
$$
A = \pmatrix{0&1&0\\0&0&1\\0&0&0}
,\quad
B = \pmatrix{1\\0\\0}
$$
$B$ is $3 \times 1$ and $AB = 0$, but $\oper... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Prove that $a \equiv b \mod{p^j} \iff |a-b|_{p}\leq p^{-j}$ Prove that $a \equiv b \mod{p^j} \iff |a-b|_{p}\leq p^{-j}$
Typically, when dealing with a congruence I go to the division statement. i.e
$$a\equiv b\mod{p^j}\Rightarrow p^j|a-b \;\;\;(\star)$$
Moreover, I know that
\begin{align}
|a-b|_{p}&=p^{-vp(a-b)}\\
&\... | Instead of thinking of $a-b$ as a difference of two numbers, think of it as a single $p$-adic number. Then $a-b \equiv 0 \pmod {p^j}$ is exactly the statement that $p^j \mid (a-b)$, which is exactly the statement that the $p$-adic norm of $a-b$ is at least $j$ (and is more than $j$ if and only if additional factors of ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $f(0)=0$ and $\left|f^\prime(x)\right|\leq\left|f(x)\right|$ imply $f(x)=0$? Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(0)=0$ for all real numbers $x$, $\left|f^\prime(x)\right|\leq\left|f(x)\right|$. Can $f$ be a function other than the constant zero function?
I coudn't find any other function sa... | Continuing the idea that I mentioned after proposing the question:
Let's define $y_1:=y$. By the mean value theorem there's a real number $y_2$ between $0$ and $y_1$ such that $\left|f(y_1)\right|\leq\left|y_1f^\prime(y_2)\right|\leq\left|y_1f(y_2)\right|$ so $\left|f(x)\right|\leq\left|xy_1f(y_2)\right|\leq\left|x^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Finding region for Change of Variables and Double integral problem I'm running into some trouble on a problem in Vector Calc by Marsden and Tromba. I don't think I am correctly finding the region for my change of variables and the book doesn't have a similar example.
Question:
Let $D$ be the region $0 \leq y \leq x$ an... | If $x=u+v$ and $y=u-v$, you get the Jacobian
$$J=\begin{vmatrix}\dfrac{\partial x}{\partial u}&\dfrac{\partial y}{\partial u}\\[1ex]\dfrac{\partial x}{\partial v}&\dfrac{\partial y}{\partial v}\end{vmatrix}=\begin{vmatrix}1&1\\1&-1\end{vmatrix}=-2~~\implies~~|J|=2$$
So,
$$\begin{align*}\iint_D(x+y)\,\mathrm{d}x\,\mathr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How should I read and interpret $A = \{\,n^2 + 2 \mid n \in \mathbb{Z} \text{ is an odd integer}\,\}$ As my question states, I need to interpret: $A = \{\,n^2 + 2 \mid n \in \mathbb{Z} \text{ is an odd integer}\,\}$
Does this mean that any odd integer $n$ will work, or does this mean that the output of $n^2 + 2$ must ... | The condition on the right of the "$|$" is not about what is on the left of it. So for each $n\in \Bbb Z$ that is an odd integer (that is, for $n=\ldots,-5,-3,-1,1,3,5,\ldots$) we form the expression $n^2+2$ (that is, $\ldots, 27,11,3,3,11,27,\ldots$) and collect the results in the set $A$. In orther words, $$A=\{3,11,... | {
"language": "en",
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Union of set and interval I'm working on finding the boundaries of sets, I feel like I understand this. However, one problem asks for the boundary of $\{1,2,3\}\cup(2,4)$ and I'm unsure as to how to take the union of an interval and a set.
Here's my thoughts:
The union will include points $1$ and $2$ and then the inte... | Let's think formally about what a boundary is. If you have a set $A$, with closure $\bar{A}$ and interior $\mathring{A}$, then the boundary of $A$ is $\partial{A} = \bar{A} \setminus \mathring{A}$.
Let $A = \{1,2,3\} \cup (2,4)$.
What is the closure of this set? The easy way is to find the points whose neighborhoods al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1499244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Linear Algebra - Reflection in a hyperplane
We have a matrix $A$: $$ A = \dfrac{1}{7} \cdot
\begin{pmatrix}
5 & -4 & -2 & 2 \\
4 & -1 & -4 & 4 \\
-2 & -4 & 5 & -2 \\
2 & 4 & 2 & 5
\end{pmatrix} $$
The map $f_a : \mathbb{R}^4 \to \mathbb{R}^4 $ is a reflection in the hyperplane $H \subset \mathbb{R}^4 $.... | A reflection about the hyperplane $H$ will fix vectors in $H$ and reflect other vectors of $\mathbb{R}^4$ across $H$. Since vectors in $H$ are fixed by $f_A$, they are eigenvectors of $f_A$ with eigenvalue 1. To find $H$, you must find the eigenspace of $f_A$ associated to the eigenvalue 1.
An example in a smaller di... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Some questions in measure theory. I got two questions.
Suppose we have the statement
If a property Q holds a.e, then P holds a.e.
Wouldd the contrapositive be,
If ~P a.e, then ~Q a.e.
Or would we remove a.e and say it holds everywhere?
Also can someone check if this argument is valid (I've seen the proof elsewhere,... | You should insert quantifiers: "$P$ holds a.e." can be written as
$$(\exists N \in \mathcal{A}) \: \mu(N)=0 \wedge \left [ (\forall x \in X \setminus N) \: P(x) \right ].$$
So the negation of that can be written as
$$(\forall N \in \mathcal{A}) \: \mu(N) \neq 0 \vee \left [ (\exists x \in X \setminus N) \: \neg P(x) \r... | {
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Solution of $\tan(nx)=k\tan(x)$ Could you kindly suggest me ways to find out closed form solution of the following equation type:
$$\tan(nx)=k\tan(x),$$
where $n$ and $k$ are some positive real numbers (that excludes zero). I can solve it using series expansion, but that gives me only an approximate solution for $x$.
T... | I think i have a trivial solution . If $n=k=1$ and $x=45$ then we have $\tan(1\cdot45)=1\tan(45)$ so $1=1$.
| {
"language": "en",
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Second order differential equation Is this correct
Is this solution in this pic correct or not?
I want to solve this equation and that is my understanding.
| Continuing from my comment, if there is no typo and the equation is really as you have written it, then the right approach would be to assume $z=y'$.Then we can write $y''-6y'+13=0$ as $z'-6z+13=0$.
Therefore, we solve the above equation as follows:
$$z'= 6z-13$$
$$6z'= 6(6z-13)$$
$$\frac{d(6z-13)}{(6z-13)}=6 \,\ dt$$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Encrypt/Compress a 17 digit number to a smalller 9(or less) digit number. I have a unsigned long integer(8 bytes) which is guaranteed to be of 17 digits and i want it to store in int(4 bytes) which is of 9 digits at max. Basically i want to encrypt or compress the number so that i could retrieve the number without any ... | There are $9 \times 10^{16}$ different decimal integers with $17$ decimal digits.
There are $2^{4\times 8} \lt 4.3 \times 10^9$ possible values of four bytes, a much smaller cardinality.
So you cannot find a $1-1$ injection from the former set to the latter.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Write on my own my first mathematical induction proof I am trying to understand how to write mathematical induction proofs. This is my first attempt.
Prove that the sum of cubic positive integers is equal to the formula
$$\frac{n^2 (n+1)^2}{4}.$$ I think this means that the sum of cubic positive integers is equal to a... | Your inductive assumption is such that the formula marked $\color{red}{\mathrm{red}}$ (several lines below) holds for $i=k$: $$\sum^{i=k}_{i=1} i^3=\frac{k^2 (k+1)^2}{4}$$
You need to prove that for $i=k+1$: $$\sum^{i=k+1}_{i=1} i^3=\color{blue}{\frac{(k+1)^2 (k+2)^2}{4}}$$
To do this you cannot use: $$\sum^{i=n}_{i=1}... | {
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"timestamp": "2023-03-29T00:00:00",
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Behaviour of ($\overline{a}_n)_{n=1}^{\infty}$ and ($\underline{a}_n)_{n=1}^{\infty}$ I have been trying to understand the following definition and just needed some clarification.
For each bounded sequence $(a_n)_{n=1}^{\infty}$ we define the sequences ($\overline{a}_n)_{n=1}^{\infty}$ and ($\underline{a}_n)_{n=1}^{\... | Note that
$$\bar a_2 = \sup\{ a_2, a_3, \cdots \} \le \sup\{ a_1, a_2, a_3, \cdots \} = \bar a_1$$
as the set $\{ a_2, a_3, \cdots \}$ is contained in $\{ a_1, a_2, a_3, \cdots \}$. Similarly we have $\bar a_{n+1} \le \bar a_n$ for all $n$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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Three point numerical differentiation Is there any generalized way to calculate numerical differentiation using a certain number of points? I have found 2-point and 5-point methods, but could not find information about using any other number of points. I am interested in doing 3-point, but am not sure if this would be ... | The general method is as follows.
*
*Decide which points you want to use: maybe $x-2h$, $x+h$ and $x+3h$ for some reason. Here $x$ refers to the point at which I want to compute the derivative.
*Write down Taylor expansions for those points, centered at $x$. Use as many terms as you have points:
$$f(x-2h) = f(... | {
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Find an Ideal of $\mathbb{Z}+x \mathbb{Q}[ x ]$ that is NOT principal
The ring $\mathbb{Z}+x \mathbb{Q}[ x ]$ cannot be a principal ideal domain since it is not a unique factorization domain. Find an ideal of $\mathbb{Z}+x \mathbb{Q}[ x ]$ that is not principal.
My book gives no examples of how to show an ideal is n... | Call $R= \Bbb{Z}+ x \Bbb{Q}[x]$. I highly suspect that $R$ is a Bezout domain, (i.e. every finitely generated ideal is principal), so I give you a non finitely generated ideal. Consider the ideal
$$I=(x, x/2 , x/4 , x/8 , \dots) = \bigcup_{k \ge 1} \left( \frac{1}{2^k}x \right)$$
Clearly, for all $k \ge 1$ we have $$\l... | {
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"timestamp": "2023-03-29T00:00:00",
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Study the monotonicity of this function The function is $$y=x^2-5x+6$$
I have made
$$[f(x_2)-f(x_1)]/(x_2-x_1)$$
It results in
$$x_1+x_2-5.$$
What should I do next?
| Now suppose that $x_2+h =x_1$ with $ h \to 0$. For a positive monotonicity it has to be
$$\lim_{h \to 0 } x_1+x_1+h -5\geq 0$$
Solving for $x_1$
$x_1 \geq \frac{5}{2}$
Therefore for $x \geq \frac{5}{2}$ the function is positive monotonic. And for $x < \frac{5}{2}$ the function is strictly negative monotonic.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $\binom{n}{k} \frac{1}{n^k}\leqslant \frac{1}{k!}$ holds true for $n\in \mathbb{N}$ and $k=0,1,2, \ldots, n$ $$\binom{n}{k} \frac{1}{n^k}\leqslant \frac{1}{k!}$$
How would I prove this? I tried with induction, with $n$ as a variable and $k$ changing, but then I can't prove for $k+1$, can I?
Is there a better... | The simplest way I see is the following:
$$\binom{n}{k}\frac{1}{n^k} = \frac{\overbrace{n(n-1)\dots(n-k+1)}^{\text{$k$ positive terms, each no larger than $n$} }}{k!} \frac{1}{n^k} \le \frac{1^k}{k!}$$
| {
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Proof that given equation(quartic) doesn't have real roots $$
(x^2-9)(x-2)(x+4)+(x^2-36)(x-4)(x+8)+153=0
$$
I need to prove that the above equation doesn't have a real solution. I tried breaking it up into an $(\alpha)(\beta)\cdots=0$ expression, but no luck. Wolfram alpha tells me that the equation doesn't have real r... | Your polynomial is
$$P(x) = ({x^2} - 9)(x - 2)(x + 4) + ({x^2} - 36)(x - 4)(x + 8) + 153\tag{1}$$
Now consider theses
$$\eqalign{
& f(x) = ({x^2} - 9)(x - 2)(x + 4) \cr
& f({x \over 2}) = \left( {{{\left( {{x \over 2}} \right)}^2} - 9)} \right)\left( {\left( {{x \over 2}} \right) - 2} \right)\left( {\left( {{x \o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1500705",
"timestamp": "2023-03-29T00:00:00",
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Definition of the ordered triple (a, b, c) according to Kuratowski's Set Theory. Can someone give Kuratowski's definition of the ordered triple $(a,b,c)$ assuming $A \times B \times C$ is rewritten as $(A \times B) \times C$, please? I noticed there is already an answered question for the ordered $n$-tuple, but (as I'm... | Ordered triples are defined recursivley, so that
$(x,y)=\{\{x\},\{x,y\}\}$ and $(x,y,z)=((x,y),z)$. Observe that $((x,y),z)$ only has two elements, $(x,y)$ and $z$, so we can just apply the definition. To make our lives easier, let $q=(x,y)=\{\{x\},\{x,y\}\}$. Then the substitution is simple:
$$\begin{align}\label{e... | {
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Questions about infinite intersections of sets I am making some examples to make sure I understand these ideas correctly, but some of them im unsure of, I posted what I think the intersection are. Are they correct?
*
*$\displaystyle\bigcap^{\infty}_{k=1}\left[1,1+\frac{1}{k}\right]=\{1\}.$
*$\displaystyle\bigcap^{\... | All four are correct. They are not justified, but they are correct
An example of a justification (for 3.):
Let $\displaystyle A = \bigcap_{k=1}^\infty(1, 1+\frac1k)$, and let $x\in\mathbb R$.
If $x\leq 1$, then obviously, $x\notin A$.
If $x>1$, then $x=1+\epsilon$ for some $\epsilon>0$. There then exists some $k$ for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1500868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Real Analysis, Folland Proposition 2.7 If $\{f_j\}$ is a sequence of $\overline{\mathbb{R}}$-valued measurable functions on $(X,M)$, then the functions
$$\begin{aligned}
g_1(x) = \sup_{j}f_j(x), \ \ \ \ g_3(x) = \lim_{j\rightarrow \infty}\sup f_j(x)
\end{aligned}$$
$$\begin{aligned}
g_2(x) = \inf_{j}f_j(x), \ \ \ \ ... | One has
\begin{align*}
\limsup f_j(x) \geq a
&\iff \inf \sup_{j \geq k} f_j(x) \geq a\\
&\iff \forall k: \quad \sup_{j \geq k} f_j(x) \geq a \\
&\iff \forall k, \forall \epsilon > 0, \exists j \geq k: \quad f_j(x) \geq a - \epsilon\\
&\iff \forall k, \forall n > 0, \exists j : \quad f_j(x) \geq a - \frac{1}{n}\\
&\iff... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1500998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Given the function $ f(x) = \sin^2(\pi x) $, show that $ f \in BV[0, 1] $ I'm learning about functions of bounded variation and need to verify my work to this problem since my textbook does not provide any solution :
Given the function $ f(x) = \sin^2(\pi x) $, show that $ f $ is of bounded variation on $ [0, 1] $.
... | For a proof from scratch, suppose $f:[0,1]\rightarrow \mathbb R$ has a Lipschitz constant $M$.
Let $\mathcal P=\left \{ 0,x_{1},\cdots ,x_{n-2},1 \right \}$ be a partition of $[0,1]$.
Then using MVT, and the Lipschitz assumption, we have with $x_i<x_i^{*}<x_{i+1}$,
$\sum_{i=0}^{n-1}\left | f(x_{i+1})-f(x_i) \right |... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1501208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Orders of Elements in GL(2,R) Let A =
$$\begin{pmatrix}
0&1\\
-1&0\\
\end{pmatrix}$$
and B =
$$\begin{pmatrix}
0&-1\\
1&-1\\
\end{pmatrix}$$
be elements in $GL(2, R)$. Show that $A$ and $B$ have finite orders but AB does not.
I know that $AB$ =
$$\begin{pmatrix}
1&-1\\
0&1\\
\end{pmatrix}$$
and that $GL(2,R)$ is a grou... | To have an order n means that n is the smallest positive number such that $A^n = I$. In this case, $I$ is of course the identity matrix.
$A^2$ = $\begin{pmatrix}
0&1\\
-1&0\\
\end{pmatrix}\begin{pmatrix}
0&1\\
-1&0\\
\end{pmatrix} = \begin{pmatrix}
-1&0\\
0&-1\\
\end{pmatrix} = -I$
and so $A^4 = I$, order of A is 4
$B^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1501288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
For what values of $p$, the series $\sum_{n=1}^{\infty}|\frac{\sin(n)}{n}|^p$ is convergent? Let $p>1$ , $p\in\mathbb{R}$. For what values of $p$, the series $\sum_{n=1}^{\infty}|\frac{\sin(n)}{n}|^p$ is convergent?
When $p=1$, I know the series is divergent but how about other cases? Thanks!
| If $p > 1$ we can write:
$0 < \sum\limits_{n=1}^{\infty}|\frac{\sin(n)}{n}|^p \le \sum\limits_{n=1}^{\infty}|\frac{1}{n}|^p$
and the right part converges.So the middle part also converges as $\sum\limits_{n=1}^{k}|\frac{\sin(n)}{n}|^p$ is an increasing function of $k$ and it's limited by $\sum\limits_{n=1}^{\infty}|\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1501389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Expanding $(x-2)^3$ I was trying to expand $(x-2)^3$. This is what I did
*
*Expanded the term so $(x-2)(x-2)(x-2)$
*Multiplied the first term and second term systematically through each case
The answer I got did not match the one at the back of the book, can someone show me how to do this please?
| Make use of the identity: $$(x-y)^3=x^3-3x^2y+3xy^2-y^3.$$
Letting $y=2$, we obtain: $$(x-2)^3=x^3-6x^2+12x-8.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1501496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to prove that the set $A = \{\ q \in \mathbb{Q}\ |\ q = n + \frac{1}{2n} \mathrm{\ for\ }n\in\mathbb{N}\ \}$ is closed in $\mathbb{R}$? I am working in the metric space $\mathbb{R}$ equipped with the distance function $d(x,y)=|x-y|$.
Let $A = \{\ q \in \mathbb{Q}\ |\ q = n + \frac{1}{2n} \ \}$. How do I formally pr... | $A$ is set of the solutions of the equation $$\sin\big(\pi \dfrac{x\pm\sqrt{x^2-2}}{2}\big)=0$$and so is the union of the zero sets of two continuous functions, hence closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1501727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Can polar coordinates always be used to calculate a limit in a multivariable function? Are polar coordinates always a viable way to calculate the limit of a multivariable function?
In lecture, it appeared as if converting a function into polar coordinates and then checking the limit as r approaches 0 would be a foolpr... | If your function is $\frac{x}{x^2+y^2}$, then the polar substitution makes the denominator $r^2(\cos^2\theta+\sin^2\theta)=r^2$.
If your function is $\frac{x}{x^2+2y^2}$, then the polar substitution makes the denominator $r^2(\cos^2\theta+2\sin^2\theta)$, from which you can't just eliminate $\theta$ the same way.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1501823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
show that for any prime p: if $p|x^4 - x^2 + 1$, with $x \in \mathbb{Z}$ satisfies $p \equiv 1 \pmod{12}$? show that for any prime p: if $p|x^4 - x^2 + 1$ satisfies $p \equiv 1 \pmod{12}$
I suppose that if $p$ divides this polynomial we can see that:
$x^4 - x^2 + 1 = kp$ for some $k \in \mathbb{N}$. But then $x^4 - x^2... | If $p\mid x$, then $p\mid x^4-x^2+1\implies p\mid 1$, contradiction. Therefore $p\nmid x$.
$(2x^2-1)^2\equiv -3\pmod{p}$ and $\left(\left(x^2-1\right)x^{-1}\right)^2\equiv -1\pmod{p}$. We can't have $p=3$, because $2x^2-1\equiv 0\pmod{3}$ has no solutions. We also can't have $p=2$, because $x^4-x^2+1$ is always odd, so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to find the determinant of this $5 \times 5$ matrix? How can I find the determinant of this matrix?
I know in matrix $3 \times 3$
$$A= 1(5\cdot 9-8\cdot 6)-2 (4\cdot 9-7\cdot 6)+3(4\cdot 8-7\cdot 5) $$
but how to work with a $5\times 5$ matrix?
| Multiplying the 1st row by $3$ and then adding it to the 4th row, and then multiplying the 3rd row of the resulting matrix by $-\frac 1 4$ and adding it to the 5th row, we obtain
$$\det \begin{bmatrix} 1 & 2 & 3 & 4 & 1\\ 0 & -1 & 2 & 4 & 2\\ 0 & 0 & 4 & 0 & 0\\ -3 & -6 & -9 & -12 & 4\\ 0 & 0 & 1 & 1 & 1\end{bmatrix} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Calculating two specific limits with Euler's number I got stuck, when I were proving that
$$\lim_{n \to \infty} \frac {\sqrt[2]{(n^2+5)}-n}{\sqrt[2]{(n^2+2)}-n} = \frac {5}{2}$$
$$\lim_{n \to \infty}n(\sqrt[3]{(n^3+n)}-n) = \frac {1}{3}$$
First one I tried to solve like
$$\lim_{n \to \infty} \frac {\sqrt[2]{(n^2+5)}-n... | $$(a)\;\;\lim_{n\rightarrow \infty}\frac{\sqrt{n^2+5}-n}{\sqrt{n^2+2}-n} =\lim_{n\rightarrow \infty}\frac{\sqrt{n^2+5}-n}{\sqrt{n^2+2}-n}\times \frac{\sqrt{n^2+5}+n}{\sqrt{n^2+5}+n}\times \frac{\sqrt{n^2+2}+n}{\sqrt{n^2+2}+n} $$
So we get $$=\lim_{n\rightarrow \infty}\frac{5}{2}\times \frac{\sqrt{n^2+2}+n}{\sqrt{n^2+5}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Understanding telescoping series? The initial notation is:
$$\sum_{n=5}^\infty \frac{8}{n^2 -1}$$
I get to about here then I get confused.
$$\left(1-\frac{3}{2}\right)+\left(\frac{4}{5}-\frac{4}{7}\right)+...+\left(\frac{4}{n-3}-\frac{4}{n-1}\right)+...$$
How do you figure out how to get the $\frac{1}{n-3}-\frac{1}{n-1... | Looking a little closer at the question, he is asking about partial fraction decomposition, as opposed to the value of the sum itself. For this particular example, it's fairly straight forward.
When given a fraction which contains a polynomial denominator, you can factor this fraction and break it into a sum of other f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
What's the Summation formulae of the series $2*2^0 + 3*2^1 + 4*2^2 + 5*2^3.......$? I faced this question where I was asked to find a summation formulae for $n$ terms of $2*2^0 + 3*2^1 + 4*2^2 + 5*2^3.......$ I did try generalizing it with $$a_n = (n + 1)2^{n - 1}; n2^{n - 2}$$ but to no avail then I tried subtracting ... | I can provide another idea:
$$
\begin{array}{cccccccc}
a= & 2\times2^{0}+ & 3\times2^{1}+ & 4\times2^{2}+ & 5\times2^{3}+ & \cdots & \left(M+1\right)\times2^{M-1}+ & \cdots\\
2a= & & 2\times2^{1}+ & 3\times2^{2}+ & 4\times2^{3}+ & \cdots & \left(M\right)\times2^{M-1}+ & \cdots
\end{array}
$$
Then, by subtracting the f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Where, if ever, does the decimal representation of $\pi$ repeat its initial segment? I was wondering at which decimal place $\pi$ first repeats itself exactly once.
So if $\pi$ went $3.143141592...$, it would be the thousandth place, where the second $3$ is.
To clarify, this notion of repetition means a pattern like ... | This is unknown, but conjectured to be false; see e.g. Brian Tung's answer to PI as an infinite set of integers.
An interesting point here is the different kinds of "patternless-ness" numbers can exhibit.
On the one hand, there is randomness: where the idea is that the digits of a number are distributed stochastically... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
$\bigcup\limits_{i=1}^n A_i$ has finite diameter for each finite $A_i$ Let $(X,d)$ be a metric space. The diameter of a set $A\subset X$ is defined to be
$\operatorname{diam}(A)= \sup\{d(x,y):x,y\in A\}$.
Suppose $A_1, \dots, A_n$ is a finite collection of subsets of $X$ each with finite diameter.
Prove that $\bigcup\l... | It is not true that $\operatorname{diam}\left(\bigcup\limits_{i=1}^n A_i\right)\leq \operatorname{diam}(A_1)+\operatorname{diam}(A_2)+\cdots+\operatorname{diam}(A_n)$. For example, suppose the diameter of each of ten sets is two inches. But one of those ten sets is in Constantinople and another is in Adelaide.
Pick a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Problem related to $L^p$ space Problem Let $k: \mathbb R^{d\times d} \to \mathbb R^d$ be a measurable function such that there is $c>0$ with $$\sup_{x \in \mathbb R^d}\int |k(x,y)|dy \leq c, \space \sup_{y \in \mathbb R^d}\int |k(x,y)|dx \leq c$$
Show that for $1<p<\infty$, the function $K:L^p(\mathbb R^d) \to L^p(\mat... | Use (Riesz-Thorin) interpolation. Show that $\|K\|_{L^1(\Bbb R^d) \to L^1(\Bbb R^d)} \le c$ using $\sup_y \int |k(x,y)|\, dx \le c$, and show that $\|K\|_{L^\infty(\Bbb R^d)\to L^\infty(\Bbb R^d)} \le c$ using $\sup_x \int |k(x,y)|\, dy \le c$. These imply $\|K\|_{L^p(\Bbb R^d) \to L^p(\Bbb R^d)} \le c$ for all $1 < p ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Elementary Infinite Limit Question Suppose $f:\mathbb{R}\to\mathbb{R}$ is a continuous odd function. Does the following statement hold (assuming the limits exist):
$$ \lim_{x\to-\infty}f(x)=-\lim_{x\to\infty}f(x)$$
This certainly seems to be true to me, because we can make this statement for any arbitrarily large, fin... | Clearly $$\lim_{x \to -\infty}f(x) = \lim_{y \to \infty}f(-y) = \lim_{y \to \infty}(-f(y)) = -\lim_{x \to \infty}f(x)$$ The above holds without any regard to continuity of $f$ and also without any regard to the existence of the limit.
However, when the limit does not exist we need to interpret the equality in a differe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What is the limit of $x^{e^{x}}$ if $x$ tends to $-\infty$? I was looking at the function $f(x)=x^{e^{x}}$ and was curious as to why its domain is defined for $x\geq0$, when it looks like there are no problems with negative values of $x$. Also, when plugging in negative values for $x$, I noticed that it looks like $f(x... | First of all:
The function at stake is not defined on the negative reals.
However, if one considers complex numbers then a negative real $(x<0)$ as a complex number can be written as $$z=-|x|+i0,\,\,\text{ or } \,\, z=|x|e^{i\pi}.$$
With this in mind
$$z^{e^z}=e^{\ln\left(z^{e^z}\right)}=e^{e^z\ln(z)}=e^{e^z\ln\left(|x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1503101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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