Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Integrate $\frac{\sin^3 \frac{x}{2}}{\cos \frac{x}{2} \sqrt{\cos x+\cos^2 x+\cos^3 x}}$ Evaluate $$\int \frac{\sin^3 \frac{x}{2}}{\cos \frac{x}{2} \sqrt{\cos x+\cos^2 x+\cos^3 x}}dx$$ I saw terms like $1+t+t^2$ in the denominator , so I thought of $t^3-1$ and then converting back into half angle but it doesn't help me.... | Let $$\displaystyle I = \int \frac{\sin ^3(x/2)}{\cos(x/2)\sqrt{\cos^3 x+\cos^2x+\cos x}}dx = \frac{1}{2}\int\frac{2\sin^2 \frac{x}{2}\cdot 2\sin \frac{x}{2}\cdot \cos \frac{x}{2}}{2\cos^2 \frac{x}{2}\sqrt{\cos^3 x+\cos^2 x+\cos x}}dx$$
So we get $$\displaystyle I = \frac{1}{2}\int\frac{(1-\cos x)\cdot \sin x}{(1+\cos ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1781313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is a pointwise limit of $L^p$ functions in $L^p$ under certain conditions? Let $I=[0,1]$ with Lebesgue measure and $1\leq p<\infty$. If $f_k$ is a sequence in $L^p(I)$ with $\|f_k\|_p\leq 1$ for all $k$, and $f(x)=\lim_{k\rightarrow\infty} f_k(x)$ exists almost everywhere, must $f$ belong to $L^p(I)$?
| By Fatou's lemma,
$$\int_0^1|f(x)|^p\;dx\leq \liminf_{n\to\infty}\int_0^1|f_n(x)|^p\;dx\leq 1$$
so $f$ is in $L^p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1781453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Geometric Probability Problem, Random Numbers $0$-$1+$ Triangles. Randy presses RANDOM on his calculator twice to obtain two random numbers between $0$ and $1$. Let $p$ be the probability that these two numbers and $1$ form the sides of an obtuse triangle. Find $p$.
At first, I thought that the answer would be $.25$, b... | I bet they are looking for you to use the Pythagorean Theorem. Since the random number will be less than 1, you can assume 1 is the largest side length. For obtuse triangles, $a^2+b^2<c^2$. In this case, because c is 1, you can simplify this to $a^2+b^2<1.$ This is a two-variable probability problem, so the easiest way... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1781582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can any presentation of a finitely presented group be reduced to a finite one? Suppose $G = \langle x_1, \ldots, x_n \mid p_1, \ldots, p_m \rangle$ is a finitely presented group, and let $\langle A \mid R \rangle$ be another presentation of $G$, with $A$ and $R$ possibly infinite. Do there always exist finite subsets ... | Not every presentation of a finitely presented group can be reduced to a finite one.
In his highly advisable notes on Geometric Group theory, Charles F. Miller III provides the presentation:
\begin{equation}
\langle a,b,c_o,c_1,c_2,\ldots \mid a^4=1, b^3 =1, c_0^{-1} b c_0 = a^2, c_1^{-1} c_0 c_1 =b, c_2^{-1} c_1 c_2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1781695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
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Give a graph model for a permutation problem
Describe a graph model for solving the following problem: Can the permutations of $\{1,2,\ldots,n\}$ be arranged in a sequence so that the adjacent permutations $$p:p_1,\ldots,p_n \text{ and } q:q_1,\ldots,q_n$$ satisfy $p_i\neq q_i$ for all $i$?
I have problem understand... | I think they mean permutations that are adjacent in the sequence. If the sequence is $\pi_1,\pi_2,\ldots,\pi_{n!}$, then e.g. $\pi_5$ and $\pi_6$ would be adjacent. And where it says "the adjacent permutations" I think they mean "any two adjacent permutations".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1781911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
Covariance of stochastic integral I have a big problem with such a task:
Calculate $\text{Cov} \, (X_t,X_r)$ where $X_t=\int_0^ts^3W_s \, dW_s$, $t \ge 0$.
I've tried to do this in this way: setting up $t \le r$
$$\text{Cov} \, (X_t,X_r)=\text{Cov} \, \left(\int_0^ts^3W_s \, dW_s,\int_0^rs^3W_s \, dW_s \right)=\int_... |
$\text{Cov} \, (X_t,X_r)=\text{Cov} \, (\int_0^ts^3W_sdW_s,\int_0^rs^3W_sdW_s)=\int_0^t s^6 W^2_s ds$
This identity does not hold true. Note that the left-hand side is a fixed real number whereas the right-hand side is a random variable.
If you apply Itô's isometry correctly, you find
$$\text{cov} \, (X_t,X_r) = \col... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1782068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Unramified primes of splitting field I would like to show the following:
Theorem: Let $K$ be a number field and and $L$ be the splitting field of a polynomial $f$ over $K$. If $f$ is separable modulo a prime $\lambda$ of $K$, then $L$ is unramified above $\lambda$.
This should follow from the following theorem:
Theo... | I'll start by reformulating the first theorem:
Theorem: Let $F = K(\alpha)$, where $f$ is the minimal polynomial of $\alpha$ over $K$. Suppose that $\mathfrak p$ is a prime of $K$, and the $f(X)$ splits as a product of distinct irreducibles modulo $\mathfrak p$ (i.e. $f$ is separable mod $\mathfrak p$). Then $\mathfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1782313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Is it possible that Gödel's completeness theorem could fail constructively? Gödel's completeness theorem says that for any first order theory $F$, the statements derivable from $F$ are precisely those that hold in all models of $F$. Thus, it is not possible to have a theorem that is "true" (in the sense that it holds i... | Given a sentence $\phi$ that holds in all models of a first-order theory $F$, you can effectively find a proof of $\phi$: just enumerate all proofs in $F$ until you find the one that proves $\phi$ (there is such a proof, since you are given that $\phi$ holds in all models of $F$ and hence, by the completeness theorem, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1782423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Largest root as exponent goes to $+\infty$ Let $a\geq 1$ and consider
$$
x^{a+2}-x^{a+1}-1.
$$
I am interested to see what is the largest root of this polynomial as $a\to +\infty$.
In order to find a root, we surely have to have
$$
x^{a+2}-x^{a+1}=x^{a+1}(x-1)=1.
$$
Hence, I guess we have to look for which $x$ we have... | If we set
$$ p_a(x) = x^{a+2}-x^{a+1}-1 $$
we may easily see that $p_a(x)$ is negative on $[0,1]$, increasing and convex on $[1,+\infty)$, so the largest real root is in a right neighbourhood of $x=1$. We may also notice that:
$$ p_a\left(1+\frac{\log(a+1)}{a+1}\right) = \frac{\log(a+1)}{a+1}\left(1+\frac{\log(a+1)}{a+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1782497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Extremum of $f (x,y) := x^2+y^2$ subject to the equality constraint $x+y=3$ I had to find the extremum of $z=x^2+y^2$ subject to the constraint $x+y=3$. I used Lagrange multipliers to reach the conclusion that $(1.5,1.5)$ is an extremum point, but had no way of determining whether it's a maximum or a minimum (we did no... | By CS inequality:
$$
x+y=(x,y)\cdot (1,1)\le \sqrt{x^2+y^2}\sqrt{2}
$$
Since $x+y=3$:
$$
3\le \sqrt{2}\sqrt{x^2+y^2}
$$
Now, squaring both sided yields
$$
9\le 2(x^2+y^2)
$$
In other words
$$
x^2+y^2\ge \frac{9}{2}
$$
This lower bound is attained when $x=y=3/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1782615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluation of $\sum_{n=1}^\infty \frac{(-1)^{n-1}\eta(n)}{n} $ without using the Wallis Product In THIS ANSWER, I showed that
$$2\sum_{s=1}^{\infty}\frac{1-\beta(2s+1)}{2s+1}=\ln\left(\frac{\pi}{2}\right)-2+\frac{\pi}{2}$$
where $\beta(s)=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^s}$ is the Dirichlet Beta Function.
In th... | Observation $1$:
A suggestion made in a comment from @nospoon was to expand one of the integrals in a series and exploit Frullani's Integral. Proceeding accordingly, we find that
$$\begin{align}
\int_0^\infty \frac{1-e^{-x}}{x(1+e^x)}\,dx&=\int_0^\infty \left(\frac{(e^{-x}-e^{-2x})}{x}\right)\left(\sum_{n=0}^\infty (-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1782731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
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Can fractions be written as over 1? I know that all whole numbers can be written as the whole number divided by one. I was wondering if fractions could be written the same way, for example..
Can $1\over2$ be written as
$1/2\over1$
Or $2\over3$ as
$2/3\over1$
If we wanted to solve this problem,
$3 \times 1/2$
Could i... | Of course! All real numbers, when divided by one, equal the number itself. This is true regardless of how the real number is expressed, whether it be a fraction, a mixed number, or any other representation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1782858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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$A\times B$ connected component implies $A,B$ connect components $X,Y$ topological spaces. I want to show that if $C\subseteq X\times Y$ is a connected component, then $C=A\times B$ where $A,B$ are connected components of $X,Y$.
What I have so far, is that any continuous $f:\;C\to\{0,1\}$ must be constant - so pick $f_... | One can indeed show that $A, B$ connected implies $A \times B$ connected. There are several answers on this site that address this.
So if $C$ is a connected component of $X \times Y$, then consider the continuous projections $p_X,p_Y$ onto the component spaces. Continuous images preserve connectedness so $A = p_X[C], ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Inverse of $\tan^{2}\theta$? I re-arranged: $$3\tan^{2} x -1=0$$ to get $\tan^{2}\theta = \frac{1}3$. I noticed the inverse of the $cos, sin$ and $tan$ functions are written as $\cos^{-1}\theta, \sin^{-1}\theta$ and $\tan^{-1}\theta$ respectively, does this mean the inverse of $\tan^{2}\theta$ would equal $\tan^{(2-1=... | REMEMBER: $tan^2\;x$ is a simplification of $(tan(x))^2$.
It's easier than it seems, root both sides so $tan(x) = \frac{\pm 1}{\sqrt3}$
Now inverse tan $\frac{1}{\sqrt{3}} $ ... $tan^{-1}(\frac{1}{\sqrt{3}})$
and you get: $\theta = 30$ this is the principal value (closest to the origin); you can find the limitless oth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1783130",
"timestamp": "2023-03-29T00:00:00",
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Use direct proof to prove: If $A \cap B = A \cap C$ and $A \cup B = A \cup C$, then $B = C$ I'm interested in knowing if the method I used is correct. I've been teaching myself proofs lately and I am having difficulties with how to approach a problem so any general tips would be awesome as well! Here is what I have:
A... | I believe the proof should go like this:
Assume $x\in B$. Then $x\in A \cup B=A\cup C$. If $x\in C$ then good. If $x\in A$ then it is in $A\cap B=A\cap C$ so it is in $C$.
Then assume $x\in C$. By a symmetric argument to the one just given, it is in $B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1783226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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How do I get the goal $(A \land B) \lor (A \land C)$ from the premises $A \land (B \lor C)$? (Using Fitch) I have
$$\begin{array} {r|c:l}
1. & A \land (B \lor C)
\\ 2. & A \land B
\\ 3. & A & \land \textsf{ Elim } 2
\\ 4. & B & \land \textsf{ Elim } 2
... | Hint 1: If the only assumption you are given is $A \land B$, can you prove the conclusion? What about only being given $A \land C$?
Hint 2 : Combine the 2 above proofs with 1 more step from the fitch rules.
Long version: Looking at the last step,
$$(A \land B) \lor (A \land C)$$
it seems like the final step might b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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All subgroups normal $\implies$ abelian group This is , I think an easy problem just that I am not getting the catch of it. How to show whether or not the statement is true?
All subgroups of a group are normal$\implies$ the group is an abelian group?
I have been able to show the other way round.
| This is actually not true. A group for which all subgroups are normal is called a Dedekind group, and non-abelian ones are called "Hamiltonian". The smallest example is the quaternion group $Q_8$. See this MO discussion for more info.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1783418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
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Resolving $\sec{x}\left(\sin^3x + \sin x \cos^2x\right)=\tan{x}$ Going steadily through my book, I found this exercise to resolve
$$ \sec{x}\left(\sin^3x + \sin x \cos^2x\right)=\tan{x}$$
Here's how I resolve it ($LHS$) and again bear with me as I truly reverting to a feeling of vulnerability, like a child actually
A... | You missed nothing. The book is doing the same you did, only in a more cumbersome way.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1783555",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Evaluate $\int\frac{\sqrt{x^2+2x-3}}{x+1}d\,x$ by trig substitution I am preparing for an exam and found this integral in a previous test. Did I do it correctly?
My attempt.
$$
\int\frac{\sqrt{x^2+2x-3}}{x+1}\,dx
$$
Complete the square of $x^2+2x-3$; I changed the integral to
$$
\int\frac{\sqrt{(x-1)^2-4}}{x+1}\,dx
$$
... |
You've made a mistake. I hope you can find it using my answer
$$\int\frac{\sqrt{x^2+2x-3}}{x+1}\space\text{d}x=\int\frac{\sqrt{(x+1)^2-4}}{x+1}\space\text{d}x=$$
Substitute $u=x+1$ and $\text{d}u=\text{d}x$:
$$\int\frac{\sqrt{u^2-4}}{u}\space\text{d}u=$$
Substitute $u=2\sec(s)$ and $\text{d}u=2\tan(s)\sec(s)\space... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1783684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Alternative Derivation of Recurrence Relation for Bessel Functions of the First Kind How can the recurrence relation
$J^{'}_n(x) = \frac{1}{2} [J_{n-1}(x) - J_{n+1}(x)]$
be derived directly from the following?
$J_n(x) = \frac{1}{\pi} \int_0^\pi \cos(n\theta - x \sin\theta) \text{d} \theta$
| Hint: $J'_n(x)={1\over\pi}\int_0^{\pi}\sin(\theta)\sin(n\theta-x\sin\theta)d\theta$
Hint: use $\sin x\sin y ={1\over 2}(\cos(x-y)-\cos(x+y))$
$={1\over \pi}\int_0^{\pi}{1\over 2}(\cos(-\theta+n\theta-x\sin\theta)-\cos(\theta+n\theta-x\sin\theta))d\theta$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1783824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Finding the fixed subfield (Galois theory) Let's say we are working with the field extension $\mathbb{Q}(\gamma)$, where $\gamma$ is the seventh root of unity. I know my basis for this extension will thus be:
$\{1, \gamma, \gamma^2, \gamma^3, \gamma^4, \gamma^5, \gamma^6 \}$
And the Galois group will consist of auto... | Your above writings imply that $\mathbb{Q}(\zeta)$ has degree $7$ over $\mathbb{Q}$. What is wrong?
Try writing $\zeta^{6}$ in terms of $1,\dots,\zeta^{5}$. Remember the minimal polynomial for $\zeta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1783926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Given $x$ and $y$, let $m = ax + by$, $n = cx + dy$, where $ad-bc = \pm 1$. Prove that $(m, n) = (x, y)$. This is exercise 10 Chapter 1 the book Introduction
to Analytic
Number Theory by Tom M. Apostol. All alphabets represent integers and by $(w,z)=g$ we mean the greatest common divisor of w and z is g. What I tried... | Following my suggestion in the comments, suppose without loss that $(x,y)=1$ (divide both equations by this if necessary). We have $$
\begin{bmatrix} m \\ n \end{bmatrix} = \begin{bmatrix}a & b \\c & d \end{bmatrix}
\begin{bmatrix}x \\ y\end{bmatrix}
$$
By hypothesis the matrix $\begin{bmatrix}a & b \\c & d \end{bmat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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In mathematics, what is an $N \times N \times N$ matrix? In mathematics, what is an $N \times N \times N$ matrix? I think this is a tensor but definitions of tensors that I have read are so overly complicated and verbose that I have trouble understanding them.
| I'ts a tensor. But so are normal numbers, vectors, and matrices which can be considered 1X1 tensor, 1Xn tensor, mXn Tensor. All data structures of this form (mXnX...)are what are called Tensors.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Find the rank and nullity of the following matrix Find the rank and the nullity of the following linear map from $\mathbb{R}^{4}$ to $\mathbb{R}^{3}$
$\left(x,y,z,t\right)\rightarrow(x-t,z-y,x-2y+2z-t)$
I understand how to find the rank and nullity, since the nullity is the dimension of the null space and the rank is ... | You can write the transformation as
$$T(\mathbf{x})=\mathbf{A}\mathbf{x}=\begin{bmatrix}1&0&0&-1\\0&-1&1&0\\1&-2&2&-1\end{bmatrix}\begin{bmatrix}x\\y\\z\\t\end{bmatrix}$$
where $\mathbf{A}$ is the transformation matrix. This matrix acts on a given vector $\mathbf{x}$ to give a transformed vector. If you carry out the m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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How to find the probability of elementary events of a random experiment when they are not equally likely? Consider this question
Consider the experiment of tossing a coin. If the coin shows head, toss it
again but if it shows tail, then throw a die. Find the conditional probability of the event that ‘the die shows a... | Let's look at just one elementary event, to get the idea: What is the probability of $(T,1)$.
Well, first yo have to flip a tails -- that is probability $\frac12$ to happen. Then if you do, you have to roll a 1; that is probability $\frac16$ if you rolled a tails in the first place. So the probability of actually ro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Integral of Wiener Squared process I don't have a background of stochastic calculus.
It is known fact that definite integral of standard Wiener process from $0$ to $t$ results in another Gaussian process with slice distribution that is normal distributed with mean equal to $0$ and variance $\frac{T^3}{3}$ i-e
$$ \int_... | That should not be true.
The problem is that you're not considering the square of the stochastic integral,
i.e. the random variable $J= (\int_0^1 {W_s} ds)^2$, which would indeed be the square of a normally distributed random variable, but the r.v. $S= (\int_0^1 {W_s}^2 ds)$.
A note in the direction of finding a soluti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
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Show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous. I tried to use the definition and arrived this far:
$|f(x)-f(y)|=\left|\frac{x}{1+|x|}-\frac{y}{1+|y|}\right|=\frac{|x-y+x|y|-y|x||}{(1+|x|)(1+|y|)}\leq|x-y+x|y|-y|x||$.
Any suggestion for ending the proof?
I also tried to prove that $\frac{x}{1+x}$ is uniformly... | Here is another approach: Your function $f$ is differeentiable on all of ${\mathbb R}$, whereby $$f'(x)={1\over\bigl(1+|x|\bigr)^2}\qquad(-\infty<x<\infty)\ .$$
As $|f'(x)|\leq1$ for all $x$ the MVT implies that $|f(x)-f(y)|\leq|x-y|$; hence $f$ is even Lipschitz continuous on ${\mathbb R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Whats the difference between a series and sequence? I was looking at a question earlier that involved sequences and found out that the sequence converged to 0 but the series diverged to infinity. How is that possible? for example the sequence was $a_n$ $=$ $\frac{1}{x_n}$
The question was here: Convergence of $x_n=f\l... | A series is in some sense a type of sequence. However, it is a sequence of partial sums.
For example, if we take some sequence $\{a_n\}_{n\geq 1}$, then we can in turn retrieve a series from this sequence by considering the following partial sums:
$S_N=a_1+...+a_n=\sum_{n=1}^{N}a_n$
Then if we consider the following se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Evaluation of $\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}$
Evaluation of $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} = $$
$\bf{My\; Try::}$ I have solved Using Direct formula::
$$\sin \frac{\pi}{n}\cdot \sin \frac{2\pi}{n}\cdot......\sin \frac{(n-1)\pi}{n} = \frac{... | Using $2\sin a\sin b=\cos(a-b)-\cos(a+b)$ and $2\sin a\cos b=\sin(a+b)+\sin(a-b)$, write
$$\sin \frac{\pi}7\cdot\sin \frac{2\pi}7\cdot \sin \frac{3\pi}7 = \frac12\left(\cos\frac{\pi}7-\cos\frac{3\pi}7\right)\sin\frac{3\pi}7=\frac14\left(\sin\frac{4\pi}7+\sin\frac{2\pi}7-\sin\frac{\pi}7\right)\\=\frac14\left(\sin\frac{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
What is a particular use of Gram-Schmidt orthogonalization? We have a linear space V of $m \times n$ matrices. I know that we can use Gram-Schmidt to construct an orthonormal basis, but the natural basis for this space (where every ij-th element is $1$ and the rest $0$) is just that - every matrix there is orthogonal t... | You can apply Gram Schmidt in order to obtain decomposition of a matrix $A \in \Re^{n\times m}, n>m$ as:
\begin{align}
QR = A \quad Q \in \Re^{n \times n},R \in \Re^{n \times m}
\end{align}
where $Q$ is orthogonal matrix obtained by Gram Schmidt orthogonalisation and $R$ is right upper matrix with zero raws $r_i$ for $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
why the quotient space is finite $X/\ker T$ Let $T:X\rightarrow Y$ be a linear operator from Banach space to Banach space, if $Y$ is finite dimensional, show $X/\ker T$ is finite dimensional, moreover has same dimension with $Y$.
Any help is appreciated.
| What you seem to be looking for is a proof of the first isomorphism theorem.
Let $\{k_i\}_{i \in I}$ be a basis of $\ker T$. This extends to a basis $\{k_i\}_{i \in I} \cup \{x_j\}_{j \in J}$ of $X$ (where $\operatorname{span} \{k_i\}_{i \in I} \cap \operatorname{span} \{x_j\}_{j \in J} = \{0\}$). Hence $\{ x_j + \ker ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1784940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Let$ (X; T_{\text{cocountable}})$ be an infinite set, show that it is closed under countable intersections. Also give an example to show that $\mathcal{T}_{\text{cocountable}}$ need not be closed under arbitrary intersections.
I was looking for some feedback on my proof:
$X\setminus\bigcap_{\alpha \in I} A_{\alpha}$, w... | I concur with the countable intersections, the equivalent formulation that the closed sets are closed under countable unions is even easier.
As to arbitrary intersections, your examples make no sense (the results are both in the topology?).
Instead consider $X_x = X\setminus \{x\}$ for all $x \in X$, all of which are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1785019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Basis maps to Linearly Independent Set How do I prove that a linear map f will map a basis to a linearly independent set?
Supposedly this is examinable but damned if it's not on either set of notes.
Thanks!
| Suppose that $T:V\rightarrow W$ is a injective linear map. Let $\left\{v_1,\dots ,v_n\right\}$ be a basis of $V$ (You can do a similar argument if $V$ is infinite-dimensional). Suppose that $\sum_{i=1}^n\lambda_iT(v_i)=0$, then $T(\sum_{i=1}^n\lambda_iv_i)=0$. Hence $\sum_{i=1}^n\lambda_iv_i\in \text{Ker}(T)$, but sinc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1785088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Why do equatons of two variables specify curves in $\mathbb{R}^2$? I suppose to more formally characterize the question more formally, why are all points of the set $\{ (x,y) \mid F(x,y) = 0 \}$ always boundary points (and I believe also never isolated points) in the standard topology of $\mathbb{R}^2$ -- for simplicit... | Suppose $p(x,y)$ is a polynomial that is not identically $0.$ Let $Z$ be the zero set of $p.$ Then $Z$ is closed, hence $Z = \text { int } Z \cup \partial Z.$ Suppose $\text { int } Z $ is nonempty. Then there is an open disc $D(a,r) \subset Z.$ Then for any nonzero $v\in \mathbb R^2,$ the function $p_v(t) = p(a+tv)$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1785324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Express last equation of system as sum of multiples of first two equations The question says to 'Express the last equation of each system as a sum of multiples of the first two equations."
System in question being:
$ x_1+x_2+x_3=1 $
$ 2x_1-x_2+3x_3=3 $
$ x_1-2x_2+2x_3=2 $
The question gives a hint saying "Label the e... | NOTE: $r_i$ is the original $i^{th}$ equation as stated in your question above.
Well, let's go through the process of finding the extended echelon form using Gauss-Jordan elimination. Here's the matrix:
$$\left[\begin{matrix}1 \ 1 \ 1 \ 1 \\ 2 \ -1 \ 3 \ 3\\ 1 \ -2 \ 2 \ 2\end{matrix}\right]\left[\begin{matrix}1 \ 0 \ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1785444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Bott&Tu Definition: "Types of Forms: In Bott&Tu's well-known book "Differential forms in Algebraic topology", they note
-(p34): every form on $\mathbb{R}^n \times \mathbb{R}$ can be decomposed uniquely as a linear combination of two types of forms:
Type 1: $\pi^*(\phi) f(x,t)$
Type 2: $\pi^*(\phi) f(x,t) dt.$
Here $\... | Hint: Take a partition of the unity $(f_{\alpha})$ subordinate to $(U_{\alpha})$, and for each form $v$, let $v_{\alpha}$ be the restriction of $v$ to $U_{\alpha}$, write $v_{\alpha}=v^1_{\alpha}+v^2_{\alpha}$ where $v^1_{\alpha}$ is of type 1 and $v^2_{\alpha}$ is of type 2. Then write $v_1=\sum_{\alpha}f_{\alpha}v^1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1785568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What is the smallest $d$ such that $\int\cdots\int \frac{1}{(1+x_1^2 + \cdots + x_n^2)^d} \, dx_1\cdots dx_d$ converges?
What power $d>0$ is the smallest integer such that in $\mathbb{R}^n$,
$$I(n) =\int_{-\infty}^\infty\int_{-\infty}^\infty \cdots\int_{-\infty}^\infty \frac{1}{(1+x_1^2 + \cdots + x_n^2)^d} \,dx_1\... | Think of $dx_1 \cdots dx_n$ as a volume. Given the the integrand has rotational symmetry, partition the space into spherical shells. The integrand is constant on the surface of this shell.
$$
I(n) = \int_0^\infty \frac{1}{(1+ r^2)^d} dV_r
$$
where $dV_r$ denotes the volume of the thin shell spanning radius range $(r, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1785665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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If $I$ is finitely generated nilpotent and $R/I^{n-1}$ is noetherian then $R$ is noetherian
If $I$ is a finitely generated ideal of a commutative ring $R$ with $1$ such that $I^n = \{0\}$ and $R/I^{n-1}$ is noetherian, then $R$ is also noetherian.
I don't know what I should do. If I can prove for example that $I^{n-1... | Since $I^n=0$ the ideal $I^{n-1}$ is a finitely generated $R/I^{n-1}$-module: it is finitely generated (any power of a finitely generated ideal is finitely generated) and $I^{n-1}\cdot I^{n-1}=0$; see also here. Now use the exact sequence of $R$-modules $$0\to I^{n-1}\to R\to R/I^{n-1}\to 0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1785889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Interpretation of definitions and logical implication in Calculus - e.g. monotonic strictly increasing function I read definitions in Calculus books that often confuse me from a logical perspective. For example, the definition of a monotonic function, e.g. a strictly increasing function, is defined as follows.
$$
\fora... | To be a strictly increasing function, every pair $(a,~b)$ must pass the test. The test is that if $a < b$ then $f(a) < f(b)$.
In your plot (d), you've shown a pair that passes the test: the pair $(x_1,~x_2)$. But the plot is clearly not strictly increasing. This suggests that there must be a pair that doesn't pass t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Alternative formula for sample covariance Is this an equivalent formula for the sample covariance?
$$\frac{1}{n-1}\left(\sum_{i=1}^nx_iy_i -n\overline{x}\overline{y}\right)$$
Thanks
| It is an equivalent formula to $\frac{1}{n-1}\sum_{i=1}^{n} (x_i-\overline x)(y_i-\overline y)$
Firstly you can multiply out the brackets.
$\sum_{i=1}^{n} (x_i-\overline x)(y_i-\overline y)=\sum_{i=1}^{n}x_iy_i-\sum_{i=1}^{n}x_i\overline y-\sum_{i=1}^{n}\overline x y_i+ \overline x \ \overline y\sum_{i=1}^{n} 1$
$\over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to expand $x^n$ as $n \to 0$? I am trying to expand $x^n$ in small $n$ using Taylor series.
Using wolfram alpha, I found that it is $1+ n\log(x) + \cdots$
I tried to Taylor expand $x^n$ around $n=0$ but I cannot get this result.
| Hint. One may recall that
$$
e^z=1+z+\cdots+\frac{z^k}{k!}+\cdots,\quad z \in \mathbb{C},
$$ then, for $x>0$ and for $0<n<1$,
$$
\begin{align}
x^n&=e^{n\ln x}=1+n\ln x+\frac{n^2(\ln x)^2}{2!}+\cdots+\frac{n^k(\ln x)^k}{k!}+\cdots,
\\\\x^n&=1+n\ln x+\cdots+\mathcal{O}\left(n^k\right),
\end{align}
$$ as announced.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Intuition why Eigenvector of Covariance matrix points into direction of maximum variance In context of principal component analysis, I am using the Eigenvectors of the Covariance matrix to project the data.
I am able to prove that the Eigenvector of the Covariance Matrix is the direction of the greatest variance in the... | Yes, the way I captured this is:
Nature has no metric system in itself, so when you measure something, you're doing it through a super-imposed metric that does not, in principle, have any meaning
However, one could measure things in a "more natural way" taking the distance from the mean divided by the standard deviatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Probability that $A \cup B$ = S and $A \cap B = \phi $ Let $S$ be a set containing $n$ elements and we select two subsets: $A$ and $B$ at random then the probability that $A \cup B$ = S and $A \cap B = \varnothing $ is?
My attempt
Total number of cases= $3^n$ as each element in set $S$ has three option: Go to $A$ or $B... | Pick any subset $A$, and there is only one subset $B$, namely $S \setminus A$ which satisfies $A \cup B = S$ and $A \cap B = \emptyset $.
There are $2^n$ subsets to choose from so the probability of selecting such a pair is $1/2^n$.
(Or, $1/(2^n - 1)$ if one constrains that $A \ne B$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 0
} |
Estimation the integration of $\frac{x}{\sin (x)} $ How to prove that integral of $\frac{x}{\sin (x)} $ between $0$ to $\frac{\pi}{2}$ lies within the interval $\frac{\pi^2}{4}$ and $\frac{\pi}{2}$ ?
| The function is:
$$f(x)=x\csc x$$
Using the Taylor series of $\csc x$ with radius $\pi$,meaning it converges for $x \in (-\pi,0) \cup (0,\pi) \supseteq (0,\frac{\pi}{2}]$:
$$f(x)=x(\frac{1}{x}+\frac{1}{6}x+....)$$
$$=1+\frac{1}{6}x^2+...$$
In the specified interval,
$$1 < \frac{x}{\sin x}$$
Also the Taylor series expan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Solution to a simple system of quadratic equations I am hoping to find a closed-form solution to the following system of $n$ quadratic equations:
$$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$
for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial solution at $x=0$ but I am looking for others. Any help would be much a... | Let $B\in\mathbb{R}^{n\times n}$ a matrix with entries $B_{ij}$. Let $x\in\mathbb{R}^n$ such that $x=(x_1,\ldots,x_n)$, define $y=(x_1^2,\ldots,x_n^2)$ and note that your system is equivalent to the system $B^Tx = y$. This is a linear system and there is a lot about this in the literature.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Negative sign that appears in integration I have to solve the following definite integral $$\int_{0}^{4}r^3 \sqrt{25-r^2}dr=3604/15$$
I have tried a change of variables given by $u=\sqrt{25-r^2}$ where then I find that $dr=-u du/r$ and $r^2=25-u^2$. That change of coordinates then give the following integral $$-\int_{3... | You've already been answered about the confusion with the limits. Now you can try the following and not make a substitution and thus not change the limits. Integrate by parts:
$$\begin{cases}u=r^2&u'=2r\\{}\\v'=r\sqrt{25-r^2}&v=-\frac13(25-r^2)^{3/2}\end{cases}\;\;\implies$$$${}$$
$$\int_0^4r^3\sqrt{25-r^2}\,dr=\left.-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
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On the Y-Combinator in Lambda Calculus I am trying to follow this explanation on the Y-combinator
Fairly at the beginning the author shows this function definition and claims that stepper(stepper(stepper(stepper(stepper()))) (5) were equal to factorial(5).
stepper = function(next_step)
return function(n)
... | Well, assuming that $n$ will reach $0$ while we get to the innermost stepper function, that would allow us to input anything as next_step is not used then. (Probably nil is passed this way to the function.)
Now stepper() is a function that basically expects the input 0 and returns 1.
The next step is stepper(stepper())... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1786910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Prove ths sum of $\small\sqrt{x^2-2x+16}+\sqrt{y^2-14y+64}+\sqrt{x^2-16x+y^2-14y+\frac{7}{4}xy+64}\ge 11$ Let $x,y\in R$.show that
$$\color{crimson}{f(x,y)=\sqrt{x^2-2x+16}+\sqrt{y^2-14y+64} + \sqrt{x^2-16x+y^2-14y+\frac{7}{4}xy+64} \ge 11}$$
Everything I tried has failed so far.use Computer found this inequality $\c... | For convenience, we make the translation $x=2+a$ and $y=6+b$, so that the equality case is $a=b=0$. Then the expression to bound is:
$$\sqrt{(a+1)^2+15}+\sqrt{(b-1)^2+15}+\sqrt{\frac{7}{8}(a+b)^2+\frac{1}{8}(a-6)^2+\frac{1}{8}(b+6)^2} $$
Now recall the following form of Cauchy-Schwarz for $n$ nonnegative variables $x_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1787007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Set Interview Question, Any Creative Way to solve? I ran into a simple question, but I need an expert help me more to understand more:
The following is True:
$ A - (C \cup B)= (A-B)-C$
$ C - (B \cup A)= (C-B)-A$
$ B - (A \cup C)= (B-C)-A$
and the following is False:
$ A - (B \cup C)= (B-C)-A$
this is an interview ques... | To check quickly, you can use the boolean logic, i.e. replacing the set $A$ with statement $a = (x \in A)$. Thus you get (using $A - B = A \cap B^C \Leftrightarrow a*\neg b$ ):
$$a * \neg (b+c) = (b*\neg c)*\neg a$$
$$a * \neg b * \neg c = (b*\neg c)*\neg a$$
this should be equal for all $a,b,c$, but for $(1,0,0)$ we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1787353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 9,
"answer_id": 3
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Diagonalisation proof
Suppose the nth pass through a manufacturing process is modelled by the linear equations $x_n=A^nx_0$, where $x_0$ is the initial state of the system and
$$A=\frac{1}{5} \begin{bmatrix} 3 & 2 \\ 2 & 3 \end{bmatrix}$$
Show that
$$A^n= \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \fra... | Hint :
$A^{n}=(P^{-1} D P)^{n}=P^{-1} D^{n} P$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1787460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Choice of a horizontal tangent space of a principal bundle Let $\pi:P\to M$ be a principal bundle with group $G=\pi^{-1}(p)$, and let $u\in P$ and $p=\pi(u)$.
As I understand it, the choice of the vertical tangent space $V_uP=\mathrm{ker}(\pi_*)$ is natural, while there's no natural choice of a horizontal subspace $H_u... | The point is that the local trivialization is defined on an open subset of $M$ not on $M$ unless the principal bundle is trivial. Yes, you can use a trivialization $(U_i)_{i\in I}$ where $\pi^{-1}(U_i)\simeq U_i\times G$ of the principal bundle, define the connection like that on $U_i\times G$ and use a partition of th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is this Inequality True for all Positive Real Numbers? I was reading Folland's Real Analysis and came across the following inequality in chapter 6 (it is from lemma 6.1)
Let $0<\lambda<1$. Then $t^\lambda\leq \lambda t +(1-\lambda)$, $t$ a
real positive number.
Is this inequality always true? I'm not sure how t... | If you want to avoid concavity arguments just note that the function $$\phi(t) = t^\lambda - \lambda t - (1-\lambda)$$
has derivative $$\phi'(t) = \lambda t ^{\lambda - 1} - \lambda$$ which is positive if $0 < t < 1$ and negative if $t > 1$. Thus $\phi$ increases to its maximum at $t = 1$ and then decreases so that $$t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $a$ and $b$ are roots of $x^4+x^3-1=0$, $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$. I have to prove that:
If $a$ and $b$ are two roots of $x^4+x^3-1=0$, then $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$.
I tried this :
$a$ and $b$ are root of $x^4+x^3-1=0$ means :
$\begin{cases}
a^4+a^3-1=0\\
b^4+b^3-1=0
\end{cases}$
whic... | Let $a,b,c,d$ be the roots of $x^4+x^3-1=0$.
By Vieta's formula,
$$a+b+c+d=-1\quad\Rightarrow\quad c+d=-1-(a+b)\tag1$$
$$abcd=-1\quad\Rightarrow \quad cd=-\frac{1}{ab}\tag2$$
Since we have
$$a^4+a^3=1\quad\text{and}\quad b^4+b^3=1$$
we can have
$$1=(a^4+a^3)(b^4+b^3)$$
$$(ab)^4+(ab)^3(a+b+1)=1,$$
i.e.
$$a+b=\frac{1-(ab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1787850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Using Logic Laws to prove $p \leftrightarrow q \equiv (p\lor q)\to(p \land q)$ I am trying to prove that
$p \leftrightarrow q \equiv (p\lor q)\to(p \land q)$
and am really lost in the steps to solve this.
So far I have:
$p \leftrightarrow q \equiv (p\to q)\land(q\to p) \qquad$|equivalence
$p \leftrightarrow q\equ... | \begin{align}
(p\lor q)\to(p \land q) & \equiv (p\lor q)'\lor(p \land q)\\
& \equiv (p'\land q')\lor(p \land q)\\
& \equiv (p'\lor(p \land q))\land (q'\lor(p \land q))\\
& \equiv ((p'\lor p) \land (p'\lor q))\land ((q'\lor p) \land (q'\lor q))\\
& \equiv (T \land (p'\lor q))\land ((q'\lor p) \land T)\\
& \equiv (p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1787962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $f(x)=\frac{1}{x}$ is not uniformly continuous on (0,1) So I'm having difficulties understand and utilizing the definition of uniform continuity:
$\forall \epsilon \gt 0,$ $\exists \delta>0 $ such that
$$ |x_1-x_2|\lt \delta \Rightarrow |f(x_1)-f(x_2)|\lt \epsilon $$
Asides from plugging in the function I... | Consider the elements $\{1/n^2 : n \geq 1\}$ of $(0,1)$. As $n$ goes to $\infty$ these elements converge to $0$ and get as close as you want. Therefore you'll satisfy the $|1/n^2-1/(n+1)^2|\leq \delta$ part. On the other hand if you take $f(1/(n+1)^2)-f(1/n^2) = 2n+1 \to \infty$. Thus, for any $\delta$ you find a count... | {
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Counting Turns in a Rectilinear Spiral Graph So consider a rectangular spiral graph which starts at the origin, goes right 1, up 1, left 2, down 2, right 3, ... (in units). How can we tell how many turns there have been given a point? For example, if we give the point $(1,1)$, then there have been a total of $1$ turn t... | Most of the corners lie on circles of radius $k\sqrt 2$, as shown in the diagram below:
There is a set of "rogue corners" (coloured red) that do not lie on these circles.
The value of $(k-1)$ gives you the number of complete turns made around the origin, but this is not the same as the number of "turns" as you describ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proportion in sets We have $3$ sets of positive integers.
$$A = \{x_1,y_2,z_3\},\quad{} B = \{x_2,y_2,z_2\}, \quad{} C= \{x,y,z\}$$
Which proportion do we use for adding $A$ and $B$ ($x_1+x_2$ and so on), so the proportion in their numbers gets as close as possible to the proportion of the numbers in $C$ ($x,y,z$ make ... | Let's make some assumptions about things to approach this problem. We need a notion of comparing proportions, so let's say, given two triples $(a,b,c)$ and $(d,e,f)$, they are in-proportion equivalent to $(1, b/a, c/a)$ and $(1, e/d. f/d)$, so I will use the 2-norm to measure the difference and say that the distance $D... | {
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Is there any Algorithm to Write a Number $N$ as a Sum of $M$ Natural Numbers? I have a number $N$ (for example $N=64$).
Is there any algorithm to find all the feasible ways for writing the number $N$ as a sum of $M$ positive numbers?
(For example $M=16$) --- Repetition is allowed.
$n_1 + n_2 + n_3 + ... + n_{16} = 6... | This is the stars and bars problem. Imagine a line of $64$ stars and $63$ candidate bars between them. Pick $15$ of those bars to be real and read off the groupings. There are ${64 \choose 15}=122131734269895$ To make the list, there are many algorithms on the web to generate the combinations of $15$ choices out of ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to find normal subgroups from a character table? I know that normal subgroups are the union of some conjugacy classes
Conjugacy classes are represented by the the columns in a matrix
How could we use character values in the table to determine normal subgroups?
| This is quite well-known and can be found in books on representation theory. Here is an explanation, which is far from being original.
First fact : $N$ is a normal subgroup of a finite group $G$ if and only if there exists a character $\chi$ of $G$ such that $N = \ker \chi := \{g \in G | \chi(g)=\chi(1)\}$. Indeed, if ... | {
"language": "en",
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"source": "stackexchange",
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Check equivalence of quadratic forms over finite fields How to check whether the two quadratic forms \begin{equation} x_1^2 + x_2^2 \quad \text{(I)}\end{equation} and
\begin{equation} 2x_1x_2 \quad \text{(II)} \end{equation}
are equivalent on each of the spaces $\mathbb{F}_3^2\, \text{and}\,\mathbb{F}_5^2$?
I know t... | You've wrtten the other way round. For the quadratic form $ x_1^2 + x_2^2 $, the corresponding matrix is the identity while that for $ 2x_1x_2 $ is $ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $. The forms are equivalent iff the matrices are congruent. (over $ \mathbb{F}_3 $ and $ \mathbb{F}_5 $)
So, suppose $ A... | {
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Limit of $\sqrt{\frac{\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$ when $x\to 1^-$? I am trying to understand if
$$\sqrt{\frac{2\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$$ is convergent for $x\to 1^-$. Any help?
Update: Given the insightful comments below, it is clear it is not converging, he... | Using the binomial series you know that $\sqrt{\frac{2\pi}{1-x}}=\sqrt{2\pi}+\sum_{k=1}^\infty\begin{pmatrix}-\frac{1}{2}\\k\end{pmatrix}(-x)^k=\sqrt{2\pi}+\sqrt{2\pi}\sum_{k=1}^\infty \frac{(2k)!}{4^k(k!)^2}x^k$. Using Stirling's formula you know that $\sqrt{2\pi}\frac{(2k)!}{4^kk!}=\frac{\sqrt{2\pi}}{4^k}(1+\epsilon_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Dimension of irreducible affine variety is same as any open subset Let $X$ be an irreducible affine variety. Let $U \subset X$ be a nonempty open subset. Show that dim $U=$ dim $X$.
Since $U \subset X$, dim $U \leq$ dim $X$ is immediate. I also know that the result is not true if $X$ is any irreducible topological spac... | You should note that any nonempty open subset of an irreducible variety (or topological space) is dense : https://en.wikipedia.org/wiki/Hyperconnected_space.
Then you can use the definition of the dimension to conclude.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The fact that surface of spherical segment only depends on its height follows from symplectic geometry It is quite quite well known that the surface of the piece of a sphere with $z_0<z<z_1$ for some values of $z_0,z_1$ is given by
$ S = 2\pi R (z_1-z_0) $. So this surface area only depends on the height of the spheri... | Consider a hamiltonian $S^1$-action on a symplectic manifold $(M, \omega)$. Denote by $\mu : M \to \mathbb{R}$ the associated moment map. Since $Lie(S^1) \cong T_0S^1 \cong \mathbb{R}$ is generated by $\frac{\partial}{\partial \theta}$, the moment map is determined by the hamiltonian function $H : M \to \mathbb{R} : m ... | {
"language": "en",
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Quantum group notation I was jumping into the deep end and reading a few papers and lectures on quantum groups. My knowledge on Lie algebras is a bit thin but I was just wondering the notation used in the starting of this document:
One should have $U_\hbar(g) = U(g)[[\hbar]]$ as a vector space [...]
My question is, i... | In this case $\hbar$ is just a parameter. Frequently, one also denotes $U_{\hbar}(g)$ as $U_q(g)$.
$U(g)[[\hbar]]$ denotes the space of formal power series over $U(g)$, i.e. if $f\in U(g)[[\hbar]]$, then $$f=\sum_{n\in\mathbb{Z}}a_n \hbar^n$$ with $a_n\in U(g)$ for every $n$. (Sometimes people reindex so that the sum i... | {
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Rational Distance Problem triple -- irrational point Many points with rational coordinates are known with rational distances to three vertices of a unit square. For example, the following points are rational distances from $a=(0,0)$, $b=(1,0)$, and $c=(0,1)$.
$(945/3364, 225/841)$, $(99/175, 297/700)$, $(8288/12675, 16... | I would say no consider only two points which form an ellipse then by the erdos anning theorem all points of mutual rational distance (ellipse) which includes the other point. all points then have rational coordinates.
this is poorly worded take the first two points as foci. take the radius such that the ellipse hits t... | {
"language": "en",
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Find the sum of the infinite series $\sum n(n+1)/n!$ How do find the sum of the series till infinity?
$$ \frac{2}{1!}+\frac{2+4}{2!}+\frac{2+4+6}{3!}+\frac{2+4+6+8}{4!}+\cdots$$
I know that it gets reduced to $$\sum\limits_{n=1}^∞ \frac{n(n+1)}{n!}$$
But I don't know how to proceed further.
| You could also consider that $$A(x)=\sum\limits_{n=1}^∞ \frac{n(n+1)}{n!}x^n=\sum\limits_{n=0}^∞ \frac{n(n+1)}{n!}x^n=\sum\limits_{n=0}^∞ \frac{n(n-1)+2n}{n!}x^n$$ $$A(x)=\sum\limits_{n=0}^∞ \frac{n(n-1)}{n!}x^n+2\sum\limits_{n=0}^∞ \frac{n}{n!}x^n=x^2\sum\limits_{n=0}^∞ \frac{n(n-1)}{n!}x^{n-2}+2x\sum\limits_{n=0}^∞ \... | {
"language": "en",
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"source": "stackexchange",
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Lim Sup and Measurability of one Random Variable with respect to Another Here, there is a common proposition in probability theory :
Let $X,Y: (\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{R})$ where $\mathcal{R}$ are the Borel Sets for the Reals. Show that $Y$ is measurable with respect to $\sigma(X) = \{ ... | We have, by construction, $f_n(X)=Y_n$ and therefore $\lim f_n(X)=Y$, but $\lim f_n(x)$ may not exist for all $x\in\mathbb R$ if the range of $X$ doesn't cover the entire real line. (This is related to the fact that the choice of $B_{i,n}$ may not be unique.) So you take limsup outside the range of $X$ (on the range of... | {
"language": "en",
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How do you pack circles into a circle? I want to know how many small circles can be packed into a large circle.
Looking at Erich's Packing Center it seems that packing is a non-trivial problem to solve.
My interest is practical rather than theoretical, so I don't really care about the absolute upper bound. I just want ... | Have you tried just packing them on the boundary? (I'll explain what I mean below).
Let $C$ be the circle of radius $R$ and order your $N$ circles as $C_1,\cdots,C_N$ such that $r_1 \ge r_2 \ge \cdots \ge r_N$. Place $C_1$ inside $C$ tangent to $C$. Then place $C_2$ tangent to $C_1$ and $C$, $C_3$ tangent to $C_2$ and... | {
"language": "en",
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Differentiation under the integral sign for $\int_{0}^{1}\frac{\arctan x}{x\sqrt{1-x^2}}\,dx$ Hello I have a problem that is:
$$\int_0^1\frac{\arctan(x)}{x\sqrt{1-x^2}}dx$$
I try use the following integral
$$ \int_0^1\frac{dy}{1+x^2y^2}= \frac{\arctan(x)}{x}$$
My question: if I can do $$\frac{\arctan(x)}{x\sqrt{1-x^2}... | $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
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\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
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Another of $\frac{1^2}{1^2}+\frac{1-2^2+3^2-4^2}{1+2^2+3^2+4^2}+\cdots=\frac{\pi}{4}$ type expressable in cube? Gregory and Leibniz formula
(1)
$$-\sum_{m=1}^{\infty}\frac{(-1)^m}{2m-1}=\frac{\pi}{4}$$
We found another series equivalent to (1)
This is expressed in term of square numbers
$$-\sum_{m}^{\infty}\frac{\sum_{... | A proof of mahdi's result:
$$ \sum_{k=1}^{n}k^3 = \frac{n^2(n+1)^2}{4}\tag{1}$$
$$ \sum_{k=1}^{2n}(-1)^{k+1} k^3 = -n^2(4n+3),\qquad \sum_{k=1}^{2n-1}(-1)^{k+1} k^3 = n^2(4n-3)\tag{2}$$
lead to:
$$\begin{eqnarray*}\sum_{n\geq 1}\frac{1^3-2^3+\ldots}{1^3+2^3+\ldots+n^3} &=& \sum_{m\geq 1}\frac{-(4m+3)}{(2m+1)^2}+\sum_{m... | {
"language": "en",
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Non-trivial explicit example of a partition of unity Does exist a non-discrete paracompact example where is possible to give a partition of unity with the functions defined explicitly for a specific non trivial cover of the space?
| Here is another simpler example that may help you visualize what partitions of unity are. Consider the open cover of $\mathbb{R}$ given by $U = (-\infty,2)$ and $V = (-2,\infty)$. A partition of unity associated to $\{U,V\}$ could be given by $\{ f_U, f_V \}$, where
$$ f_U, f_V \colon \mathbb{R} \to [0,1]$$
$$ f_U (x) ... | {
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s-arc transitive graph is also (s-1)-arc transitive I read on a paper that an s-arc transitive graph is also (s-1)-arc transitive and thus (s-2)-transitive, which was stated as obvious. However, I was thinking that a path of 2 edges, $P_2$, is 2-arc transitive but not vertex-transitive, right?
Could anyone advise me wh... | The right answer is that the s-arc transitive graph must also be regular (all vertices have the same degree). with this extra condition it is easy to prove the claim. Note in some trivial cases the result can be concluded by resorting to trivial LOGICAL rules.
| {
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Parametrizing the surface $z=\log(x^2+y^2)$
Let $S$ be the surface given by $$z = \log(x^2+y^2),$$ with $1\leq x^2+y^2\leq5$. Find the surface area of $S$.
I'm thinking the approach should be $$A(s) = \iint_D \ |\textbf{T}_u\times \textbf{T}_v| \,du\,dv ,$$ where $T_u, T_v$ are tangent vectors to the surface (and ... | If a geometric object has a particular symmetry, generally parameterizations that reflect that symmetry result in easier computations. In our case, $S$ is the surface of a graph whose domain $A := \{1 \leq x^2 + y^2 \leq 5\}$ is an annulus centered at the origin. This suggests letting one of our parameterization variab... | {
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Unit quaternion multiplied by -1 If all components of a unit quaternion (also known as versor) are multiplied by -1, so it still remains a versor, does the resulting versor is considered equivalent to the original versor?
| If we think about rotation quaternion which are also unit quaternion then multiplying it with $-1$ will result it in additional $2\pi$ rotation and in consequence this will not affect original rotation hence it is equivalent to original unit quaternion.
$$q = \cos{\frac{\theta}{2}}+\sin{\frac{\theta}{2}}\frac{\vec{u}}... | {
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How to show that a limit exisits, and evaluate the limit of trig functions as $x$ tends to point For each of the following functions $f$, determine whether $\lim_{x\to a}f(x)$
exists, and compute the limit if it exists. In each case, justify your answers.
$$f(x)= x^2\cos\frac{1}{x} (\sin x)^4, \text{ where } a=0$$
$$f(... | Part (a):
$$\lim_{x\to 0}x^2\cos(1/x)(\sin x)^4$$ as $\cos(\mathrm{anything})=$ is defined, but here it is oscillating between $-1$ and $1$ so we can apply limit
$$\lim_{x\to 0}x^2\cos(1/x)(\sin x)^4=(0)*n*(0)$$ where $n=$ some oscillating(but finite) number
hence answer is $0$.
Part (b):
this one is quite straight for... | {
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Prove that an $n \times n$ matrix $A$ over $\mathbb{Z}_2$ is diagonalisable and invertible if and only if $A=I_n$ Through some facts, when $A$ is invertible, I found out that the eigenvalue can't be $0$, since if the eigenvalue is $0$, then $\det(A)=0$, which means that is is not invertible. Since it is over $\mathbb{Z... | Since $0$ and $1$ are the only scalars available, and a diagonal entry$~0$ in a diagonal matrix makes it singular (for instance because there is then a zero column), the only invertible diagonal matrix is the identity matrices$~I_n$. A diagonalisable matrix is by definition similar to some diagonal matrix, and invertib... | {
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Number of rectangles to cover a circle After searching around I found this is similiar to the Gauss Circle but different enough (for me anyway) that it doesn't translate well.
I have a circle, radius of 9 that I need to completely cover with rectangles 4 x 8. What is the minimum number of whole rectangles required.
My ... | It is possible to cover the circle by $11$ rectangles.
We can construct the $11$ rectangles by following procedure.
*
*Center the circle of radius $9$ at origin.
*Start covering the circle with rectangle $C_0 = [5,9] \times [-4,4]$ (the red one).
*Rotate $C_0$ with respect to origin for angles $\frac{2k}{7} k \pi ... | {
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Prove that $f$ is periodic if and only if $f(a) =f(0)$.
Suppose that $f'$ is periodic with period $a$. Prove that $f$ is periodic if and only if $f(a) =f(0)$.
I understand the direction assuming that $f(a) = f(0)$, but the other direction I don't get in the solution below. How do they get $f(na) = nf(a) - (n-1)f(0) =... | Slightly differently worded:
As $g$ is constant, we have $f(x+a)-f(x)=g(x)=g(0)$ for all $x$, hence $f(na)=f(0)+ng(0)$ by induction on $n$. If $g(0)\ne 0$, this is unbounded, whereas a continuous periodic function must be bounded. Hence we conclude $g\equiv 0$, i.e., $f(x+a)=f(x)$ for all $x$.
| {
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How to prove that $\int_0^{\infty}\cos(\alpha t)e^{-\lambda t}\,\mathrm{d}t=\frac{\lambda}{\alpha^2+\lambda^2}$ using real methods? Could you possibly help me prove $$\int_0^{\infty}\cos(\alpha t)e^{-\lambda t}\,\mathrm{d}t=\frac{\lambda}{\alpha^2+\lambda^2}$$
I'm an early calculus student, so I would appreciate a thor... | Explicitly, with the choice $$u = \cos \alpha t, \quad du = -\alpha \sin \alpha t \, dt, \\ dv = e^{-\lambda t}, \quad v = -\lambda^{-1} e^{-\lambda t},$$ the first integration by parts gives
$$I = \int e^{-\lambda t} \cos \alpha t \, dt = -\frac{1}{\lambda} e^{-\lambda t} \cos \alpha t - \frac{\alpha}{\lambda} \int e^... | {
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Example of a projective variety that is not projectively normal but normal I want to prove the following statement:
Let $Y$ be the quartic curve in $\mathbb{P}^3$ given parametrically by $(x,y,z,w)=(t^4,t^3u,tu^3,u^4)$. Then $Y$ is normal but not projectively normal.
To show it is normal, we can consider the affine p... | A good way to visualize this is to look the set $Q$ of all possible exponents. In this case, it's the set of all $\Bbb N$-linear combinations of $(4,0), (3,1),(1,3),(0,4)$. For our ring to have even a chance to be normal, $Q$ certainly (prove it!) must satisfy the following criterion (in which case we say $Q$ is satura... | {
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Double Integral $\int\limits_0^\pi \int\limits_0^\pi|\cos(x+y)|\,dx\,dy$ First off, I apologize for any English mistakes.I've come across a double integral problem that I haven't been able to solve: Find
$$\int_0^\pi \int_0^\pi|\cos(x+y)|\,dx\,dy$$
Intuitively, I thought it could be solved in a similar manner to what ... | A handful of hints to get you started, each for different directions.
Most require you to just use some alternative definition of $\cos(x)$, as mentioned in the comments...
*
*Use the piecewise definition of $\left|\cos(x+y)\right|$, and add together the separate integrands
*Note that if $x,y\in\Bbb R$, then $\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1791264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Recurrent sequence limit Let $a_n$ be a sequence defined: $a_1=3; a_{n+1}=a_n^2-2$
We must find the limit: $$\lim_{n\to\infty}\frac{a_n}{a_1a_2...a_{n-1}}$$
My attempt
The sequence is increasing and does not have an upper bound. Let $b_n=\frac{a_n}{a_1a_2...a_{n-1}},n\geq2$. This sequence is decreasing(we have $b_{n+1}... | since take $a_{1}=x+\dfrac{1}{x},x>1$,then
$$a_{2}=x^2+\dfrac{1}{x^2},a_{3}=x^4+\dfrac{1}{x^4}\cdots,a_{n}=x^{2^{n-1}}+\dfrac{1}{x^{2^{n-1}}}$$
so we have
$$a_{1}a_{2}\cdots a_{n}=\left(x+\dfrac{1}{x}\right)\left(x^2+\dfrac{1}{x^2}\right)\cdots\left(x^{2^{n-1}}+\dfrac{1}{x^{2^{n-1}}}\right)=\dfrac{x^{2^n}-\dfrac{1}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1791396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
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In how many ways can $2t+1$ identical balls be placed in $3$ boxes so that any two boxes together will contain more balls than the third?
In how many ways can $2t+1$ identical balls be placed in $3$ boxes so that any two boxes together will contain more balls than the third?
I think we have to use multinomial theore... | Notation: A box $X$ has $x$ balls inside.
First of all, note that no box can contain less than $1$ and more than $t$ balls. Also, the sum of two boxes mustn't be lower than $t + 1$.
Let's say box $A$ has $1 \leq a \leq t$ balls. Box $B$ must have then at least $t - a + 1 \leq b \leq t$ balls. This gives you $a$ ways of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1791653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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What's the chance that a number randomly drawn from a set of numbers is bigger than one drawn before it? It has been a long time since last I opened a book on probability/statistics, so this might actually be a very basic question.
Let's say I have a set $S = \{1, 2, \cdots, 5000\}$. Clearly, $S$ has 5000 elements.
Now... | The probability is $\frac12$ by symmetry. Since all numbers are by assumption equiprobable, for each pair you have the same probability of drawing first the one number and then the other or vice versa.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1791734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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One partial derivative is continuous (at a single point) implies differentiable? Let $f:\mathbb{R}^2\to\mathbb{R}$ and $(p,q)\in\mathbb{R}^2$ such that both $f_x$ and $f_y$ exists at $(p,q)$.
Assume that $f_x$ is continuous at $(p,q)$.
How do we prove/disprove that $f$ is differentiable at $(p,q)$?
I do note that it i... | A simpler example: f(x,y)= 1 if xy= 0, otherwise, f(x,y)= 0. Both partial derivatives at (0, 0) are 0 but f is not differentiable at (0, 0).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1791813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Norm equivalence on $l^1$. Suppose that $\|\cdot\|$ is a norm on $l^1$ such that:
a) $(l^1, \|\cdot\|)$ is a Banach space,
b) for all $x \in l^1$ $\|x\|_{\infty} \leq \|x\|$.
Prove that the norms $\|.\|$ and $\|.\|_1$ are equivalent.
($\|\cdot\|_{\infty}$ - supremum norm and $\|\cdot\|_1$ - standard $l^1$)
Becouse... | At first I thought one or the other inequality must be obvious, but I don't see it after a little thought. Big Hint: It's trivial from the Closed Graph Theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1791925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove that $x = a_1+\dfrac{a_2}{2!}+\dfrac{a_3}{3!}+\cdots$
If $x$ is a positive rational number, show that $x$ can be uniquely expressed in the form $$x = a_1+\dfrac{a_2}{2!}+\dfrac{a_3}{3!}+\cdots\text{,}$$ where $a_1,a_2,\ldots$ are integers, $0 \leq a_n \leq n-1$ for $n > 1$, and the series terminates.
I don't se... | It's the archmedian principal. For any two natural number $m$ and $n$ there exist a unique natural number (including 0) $k$ such that $k*n \le m < (k+1)n$.
Or in other words for any two natural numbers $m$ and $n$ there are unique natural $k$ and $a$ such that $m = k*n + a; 0\le a < n$.
Or in other words for any two n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1792136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimize $a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2$ with distinct positive integers
Find the minimum value of the following:
$$a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2$$
where all numbers are different/distinct positive integers.
I know the answer (see below), but wan... | the smallest is
76913 squares of: 263 88
76913 cubes of: 40 17 20
76913 fourth powers of: 16 3 6 10
76913 fifth powers of: 9 1 2 4 7
76913
the second smallest is
1560402 squares of: 1239 159
1560402 cubes of: 101 45 76
1560402 fourth powers of: 35 5 12 14
1560402 fifth powers of: 17 1 6 8 10
156040... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1792231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Cardano's method returns incorrect answer for $x = u + v$ I'm trying to use Cardano's method to solve this equation:
$$x^3+6x=20 \tag{1}$$
As described on Wikipedia, I let $x = u + v$ and expand in $(1)$:
$$(u+v)^3+6(u+v)=20$$
$$u^3 + v^3 + (3uv+6)(u+v)-20=0 \tag{2}$$
I then let $3uv + 6 = 0$ and substitute in $(2)$:
$... | You forgot the condition $u^3v^3=-8$. The correct solution is the first you enunciated.
That said, your way of solving the system of equations $\; \begin{cases}u^3+v^3=20\\u^3v^3=-8\end{cases}\;$ is over complicated.
Just use what any high-school student knows to solve the problem of finding two numbers the sum $s$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1792298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why am I getting two different answers (and the textbook a third) on this 3D trig problem?
Simone is facing north and facing the entrance to a tunnel through a mountain. She notices that a $1515$ m high mountain is at a bearing of $270^\circ$ from where she is standing and its peak has an angle of elevation of $35^\c... | From the right angled triangle directly, connect two triangles in horizontal plane:
$$ 1515 \sqrt{ \cot ^2 31^0 - \cot^2 35^0} \approx 1294 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1792388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
} |
Factor out (m+1) in the following so that the final answer is $\frac{(2m+1) (m+2) (m+1)} {6}$ Question: $\frac{m (m+1) (2m+1) + 6(m+1)^2}{6}$=$\frac{(2m+3)(m+2)(m+1)}{6}$
I must multiply by 6 on both sides and expand the brackets and collect like terms. I'm I correct?
Edit notes: The original problem was posed as:
$$ \... | The description as given is correct. Multiplying both sides by $6$ yields
$$(2m+1) (m+1) + 6(m+1)^2= (2m+3)+(m+2)(m+1)$$
Now what remains is to determine the value of both sides. Let $L_{m}$ be the left and $R_{m}$ be the right.
\begin{align}
L_{m} &= (2m+1)(m+1)m + 6(m+1)^2 \\
&= (2m^2 + 3m + 1)m + 6(m^2 + 2m +1) \\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1792482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Module of finite length $\implies$ finite direct sum of indecomposable modules I'm trying to solve the following question.
Let $M$ be an $R$-module of finite length (i.e, both Artinian and Noetherian). Prove that it is isomorphic to a finite direct sum of indecomposable submodules.
I was thinking I could induct on th... | If $M$ is indecomposable, you're done. Otherwise, $M=X\oplus Y$, with $X$ and $Y$ both nonzero.
Now prove that the lengths of $X$ and $Y$ are less than the length of $M$.
Hint: build composition series for $X$ and $Y$ and paste them up.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1792619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Tokens in boxes problem Tokens numbered $1,2,3...$ are placed in turn in a number of boxes. A token cannot be placed in a box if it is the sum of two other tokens already placed inside that box. How far can you reach for a given number of boxes?
For two boxes it's quite straightforward to see that it's impossible to pl... | These numbers are tabulated at https://oeis.org/A072842 – but not very far. All that's given there is 2, 8, 23, 66, 196, and that last number is a little shaky. Several references are given for further reading. Some bounds are given for large numbers of boxes, but the upper and lower bounds are very far apart.
So, if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1792725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Student test statistic and self normalizing sum. I study the asymptotic distribution of self normalizing sums which are defined as
$S_n/V_n$ where
$S_n=\sum_{i=1}^n X_i$ and $V_n^2 = \sum_{i=1}^n X_i^2$
for some i.i.d RV's $X_i$.
Motivation to study such sums comes from the fact that the classical Student $T_n$ sta... | The authors reference a 1969 paper by Efron. The relevant section in Efron appears to be a reference to what at that time was an unpublished paper by Logan, Mallows, Rice and Shepp (see p.16, last paragraph of Efron).
However, the article you are reading give the actual paper, published in 1973 :Limit distributions of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1792795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Definition of $\frac{dy}{dx}$ where we cannot write $y=f(x)$ Informally, I can say that "for a small unit change in $x$, $\frac{dy}{dx}$ is the corresponding small change in $y$". This is however a bit vague and imprecise, which is why we have a formal definition, such as:
Where $y=f(x)$, we define $$\frac{dy}{dx}=\li... | Apparently you are mixing two things: "definition of derivative" and "a practical method of calculate derivative in a certain context". The limit definition is not supposed to be used for calculating derivatives of complicated functions, rather it is supposed to be used to prove theorems and these theorems turn out to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1792878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Power Diophantine equation: $(a+1)^n=a^{n+2}+(2a+1)^{n-1}$ How to solve following power Diophantine equation in positive integers with $n>1$:$$(a+1)^n=a^{n+2}+(2a+1)^{n-1}$$
| If the equation holds, then $(a+1)^n>(2a+1)^{n-1}$ and $(a+1)^n>a^{n+2}$.
Multiplying gives $(a+1)^{2n} > (2a+1)^{n-1}a^{n+2}$. This can be rewritten as $(a^2+2a+1)^n > (2a^2+a)^{n-1} a^3$. However, if $a\geq 3$, then $a^2+2a+1 < 2a^2+a$ and $a^2+2a+1 <a^3$, which gives a contradiction.
Hence $a=1$ or $a=2$. The first... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1793142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
recurrence relation number of bacteria Assume that growth in a bacterial population has the following properties:
*
*At the beginning of every hour, two new bacteria are formed for each bacteria that lived in the previous hour.
*During the hour, all bacteria that have lived for two hours die.
*At the beginning of... | We know $A_1 = 100$, and in the second hour there will be $200$ newborn and $100$ that will die the third hour. So, $A_2 = 300$. Those $300$ alive in the second hour mean that in the third hour there will be $600$ newborn and $200$ still alive; $A_3 = 800$.
Following this logic, we can define:
$$A_n = 2 \, A_{n-1} + 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1793349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Prove the Radical of an Ideal is an Ideal I am given that $R$ is a commutative ring, $A$ is an ideal of $R$, and $N(A)=\{x\in R\,|\,x^n\in A$ for some $n\}$.
I am studying with a group for our comprehensive exam and this problem has us stuck for two reasons.
FIRST - We decided to assume $n\in\mathbb{Z}^+$ even though t... | You seem to have everything else, so here’s one approach to the question of why $x\in N(A)$ implies that $-x\in N(A)$.
Since $(-a)b +ab =0b=0$, we see that $(-a)b=-(ab)$, and so $(-x)^n=x^n$ if $n>0$ is even, and it’s $-x^n$ if $n$ is odd. In either case, if $x^n\in A$, then $(-x)^n\in A$, so that $x\in N(A)\Longrighta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1793633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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What is the probability that all $n$ colors are selected in $m$ trials? I have a concrete problem, say, there are $n$ different balls ($n$ different colors to distinguish them), each ball will be selected uniformly at random. The way I choose a ball is that I randomly get a ball, and write down its color．Then I put th... | There are $n^m$ possible colour strings, all equally likely. Now we count the favourables. The favourables are the functions from a set of $m$ elements to a set of $n$ elements that are onto.
The number of such functions is given by $n!$ times the Stirling Number of the Second Kind often denoted by $S(m,n)$. There is n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1793701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Trying to Understand Van Kampen Theorem
Theorem. Let $X$ be the union of two path-connected open sets $A$ and $B$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x_0$ be a point in $A\cap B$ and all fundamental groups will be written with respect to this base point. Let $\Phi:\pi_1(A)\sqcup \pi_1(B)\... | If I am understanding you correctly, you are asking whether the canonical inclusion map $\pi_1(A)\to \pi_1(A)\sqcup\pi_1(B)$ is injective. This is true, and follows from the concrete description of elements of the free product as reduced words. Explicitly, an element of $\pi_1(A)\sqcup\pi_1(B)$ is a finite sequence $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1793853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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