Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to interpret this Mathematica command input as given in book? I am unable to understand what the book stands for by the given command shown as input in Mathematica, as given at its Googlebooks limk here.
I mean the line given by:
$lim_{x\rightarrow \infty} cos2(2x)2x -3.$
I tried to look into Mathematica syntax fo... | I searched in the book, Exploring Calculus: Labs and Projects with Mathematica Crista Arangala Karen A. Yokley, for Lab-2, and here is what those functions were intending.
In Mathematica syntax, the last item $i$ is defined as (of course they want you to use the items listed and not this approach), so you can see how... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3267863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integration using substitution - applying integral of symmetric functions properties or second substitution? I need help with this integral: $\int_0^2 (x-1)e^{(x-1)^2}\;\mathrm{d}x$
Okey. I'm choosing $u=x-1$, so $du=dx$.
$a=0$ and $b=2$
$u=g(x)=x-1$
$g(a)=g(0)=0-1=-1$
$g(b)=g(2)=2-1=1$
$$\int_0^2 (x-1)e^{(x-1)^2}\;\ma... | note:
$$I=\int_0^2(x-1)e^{(x-1)^2}dx$$
by letting $u=x-1$ like you suggested this can be turned into:
$$I=\int_{-1}^1ue^{u^2}du$$
now notice that splitting this up gives:
$$\int_{-1}^0f(u)du+\int_0^1f(u)du=\int_0^1f(u)du+\int_0^1f(-u)du=\int_0^1f(u)du-\int_0^1f(u)du=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3268272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to calculate Net Present Value? So I have this question :
Your boss asked you to evaluate a project with an infinite life.
Sales and costs project to$ \$1,000 $ and $\$500$ per year, respectively. (Assume sales and costs occur at the end of the year, i.e., profit of $\$500$ at the end of year one.)
There is no ... | The present value of the series of cash flows is as follows;
$$-3000+\frac{(1000-500)\cdot (1-0.3)}{1.1^1}+\frac{(1000-500)\cdot (1-0.3)}{1.1^2}+\frac{(1000-500)\cdot (1-0.3)}{1.1^3}+\frac{(1000-500)\cdot (1-0.3)}{1.1^4}+\ldots$$
$$=-3000+\sum_{k=1}^{\infty}\frac{(1000-500)\cdot (1-0.3)}{1.1^k}$$
For simplicity let $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3268435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$dxdy=rdrd\theta$ I'm trying to show that $dx\,dy=r\,dr\,d\theta$ using differentials.
$x=r\cos(\theta)$ and $y=r\sin(\theta)$
thus $dx=\cos(\theta)dr-r\sin(\theta)d\theta$ and $dy=\sin(\theta)dr+r\cos(\theta)d\theta$
$\begin{align}dx\,dy&=(\cos(\theta)dr-r\sin(\theta)d\theta)(\sin(\theta)dr+r\cos(\theta)d\theta)\\&
=\... | You are missing the important point on the difference between the partition elements in Cartesian and Polar systems.
While $dxdy$ is the area of a rectangle $rdrd\theta $ is the area of the curved section between circles of radii $r$ and $r+dr$ and the central angle of $d\theta$
The so called Jacobian gives you the m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3268573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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proof of binomial identity involving double sum It's asked to simplify the sum
$$\displaystyle\sum_{0\le i <j\le n+1} \binom{n}{i} \times \binom{n+1}{j} $$
inspecting first values of $n$ shows the sum if apparently equal to $4^n$
I tried re-writing the sum as
$$\displaystyle\sum_{j=1}^{n+1} \binom{n+1}{j} \displaystyl... | Starting from $$\sum_{0\le i <j\le n+1} \binom{n}{i} \binom{n+1}{j}$$
we split the second binomial to
$$\sum_{0\le i <j\le n+1} \binom{n}{i} \binom{n}{j} + \sum_{0\le i <j\le n+1} \binom{n}{i} \binom{n}{j-1}$$ and reindex to
$$\sum_{0\le i <j\le n} \binom{n}{i} \binom{n}{j} + \sum_{0\le i \le k \le n} \binom{n}{i} \bin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3268712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine if there exist rational number a and irrational number A such that $A^3+aA^2+aA+a=0$. Determine if there exist a rational number a and irrational number A such that $A^3+aA^2+aA+a=0$. If so, can we say something about them? Are there infinitely many of them?
| For any integer $a$ except $0$ or $1$, the polynomial $x^3 + a x^2 + a x + a$ has no rational roots. Any rational root $A$ would have to be an integer (by Gauss's lemma, or the Rational Root Theorem). Now $A^3 + a A^2 + a A + a = 0 $ means
$$a = - \frac{A^3}{A^2 + A + 1} = -A + 1 - \frac{1}{A^2 + A + 1}$$
which, if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3268806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Does $\forall i\in \mathbb Z^+:\left \lfloor log_{2}(i) \right \rfloor = \left \lfloor log_{2}(i+0.999999) \right \rfloor$ Is it true that
$\forall i\in \mathbb Z^+:\left \lfloor log_{2}(i) \right \rfloor = \left \lfloor log_{2}(i+0.999999) \right \rfloor$ ?
The following is false:
$\forall i\in \mathbb Z^+:\left \lfl... | Let $$\lfloor\lg(i)\rfloor\le n\land n<\lfloor\lg(i+e)\rfloor$$ for some integer $n$.
This is equivalent to
$$\lg(i)<n+1\land n+1\le\lg(i+e)$$
or, when $i$ is an integer
$$ i\le2^{n+1}-1\land 2^{n+1}\le i+e.$$
By subtraction of the inequalities,
$$1\le e.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3268998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A strange result if $|G/\mathrm{Z}(G)|=p$ I came across something strange, which I would like to share.
Let's take a group $G$ such that $|G/\mathrm{Z}(G)|=p$, where $p$ is a prime number.
Then, we can show that $G$ is abelian $\iff \mathrm{Z}(G)=G$.
But then $|G/\mathrm{Z}(G)|=|G/G|=1$ and we have a contradiction.
W... | We have that if $G/Z(G)$ is cyclic then $G$ is abelian. But since $G$ abelian means $G=Z(G)$, this forces $|G/Z(G)|=1$. Now, if $|G/Z(G)|$ were a prime number then $G/Z(G)$ would be cyclic and then $|G/Z(G)|$ would be 1, which is impossible. Hence we can never have $|G/Z(G)|$ prime. We have proved that either $G$ is ab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3269102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Continuous mapping of Cantor Set to $[0,1]$ I read that $[0,1]$ is a continuous image of the Cantor set using the dyadic expansion $f$ of real numbers. $f$ is an onto function, since all $x \in [0,1]$ can be represented by an element in the Cantor set.
But $f$ should not be injective right? since otherwise we will have... | The Cantor set $K$ and $[0,1]$ are both compact Hausdorff spaces. A continuous bijection from one compact Hausdorff space to another must be a homeomorphism. So if any $f:K\to [0,1]$ was continuous, surjective, and injective then $f$ would be a homeomorphism, implying that $f^{-1}:[0,1]\to K$ is continuous and surjec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3269258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove or disprove $\sum_{i=0}^\infty \frac{1}{i+j+1}\frac{1}{\sqrt{i+\frac{1}{2}}}<\frac{\pi}{\sqrt{j+1}}$ In Example 2.3.5; Functional Analysis book by S. Kesavan it was shown that for $j\gt -\frac{1}{2}$
$$s(j) := \sum_{i=0}^\infty \frac{1}{i+j+1}\frac{1}{\sqrt{i+\frac{1}{2}}}<\frac{\pi}{\sqrt{j+\frac{1}{2}}}\tag{1}$... | In S. Kesavan's example, we do no need the summation to begin from $i=0$. As $a_{ij}$ is not even defined there. Therefore, I was able to prove a relaxed version of the inequality.
$$\sum_{i=1}^{\infty}\frac{1}{(i+j+1)\sqrt{i+1/2}} = \frac{1}{(j+2)\sqrt{3/2}}+\sum_{i=2}^{\infty}\frac{1}{(i+j+1)\sqrt{i+1/2}} \tag{1}$$
F... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to check if a set is compact? I am struggling in finding a way to check if a set is compact. I know by definition that the set is compact if it is closed and bounded, but what about practice?
Especially if I have something like that:
$$\{(x,y)\mid x^2+y^2 < 2\}$$
Do you know how to approach the problem?
| Sure. That specific set is not compact since it is not a closed set: $\lim_{n\to\infty}\left(\sqrt{2}-\frac1n,0\right)=(\sqrt{2},0)$, which does not belong to your set, whereas each $\left(\sqrt{2}-\frac1n,0\right)$ does.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3269512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there algebraic functions with infinitely many roots? For example, a rational function is zero if and only if its numerator (which is a polynomial) is zero. Thus, a rational function which is not identically zero have only a finite number of roots.
Is the same conclusion valid for smooth algebraic functions? If so, ... | There are even non-zero polynomials $f(x)$ having infinitely many roots. This can happen when we do not consider polynmials over fields, but, say, over the real algebra of quaternions $\mathbb{H}$. The polynomial
$$
f(x) = x^2+1
$$
has infinitely many roots in $\mathbb{H}[x]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3269621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Finding shaded triangle areas in a parallelogram There is the following parallelogram involving two shaded triangles.
If I found rightly, angles of $AMD$, $BMN$ and $CDM$ are $45$. But I can’t go further.
| $\angle AMD$, $\angle BMN$ and $\angle CDM$ are not necessarily $45^\circ$.
$[\triangle AMD]=\dfrac12\times\dfrac23\times[ABCD]$
$[\triangle BMN]=\dfrac12\times\dfrac13\times\dfrac13\times[ABCD]$
$[\triangle CND]=\dfrac12\times\dfrac23\times[ABCD]$
$[\triangle DMN]=\left(1-\dfrac13-\dfrac1{18}-\dfrac13\right)\times[AB... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3269897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Arrangements of the word $ABCDEFGGGG$ If we consider the word $ABCDEFGGGG$. To find the number of arrangments for that word, we just calculate: $\frac{10!}{4!}$.
But if now we want to find the total number of arrangements for that word such that $2$ $G$'s must come together and the two other $G$'s be separated. One of ... | the answer is not $75600$. The reason is some of the combinations are repeated.
For example $G G GG$ are separated but when we permute them the first and the second $G$ will make the same combination. So, the solution is $2! * 7C3 * 6!$ .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3270001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Let $(\textbf{a}_{n})_{n = 1}^{\infty}$ be a sequence in $\mathbb{R}^k$. Show $\{\textbf{a}_{n} : n \geq 1 \} \cup \{\textbf{a}\}$ is closed. Let $(\textbf{a}_{n})_{n = 1}^{\infty}$ be a sequence in $\mathbb{R}^k$ and $$\lim_{n \rightarrow \infty}\textbf{a}_{n} = \textbf{a}.$$ Show $B = \{\textbf{a}_{n} : n \geq 1 \} \... | Suppose there is a different limit point $\mathbf b$ of $\mathbf B$. Set $\varepsilon=\frac12|\mathbf b-\mathbf a|$. Then by convergence of the sequence there are only finitely many points outside $B_ε(\mathbf a)$.
However at the same time, as $\mathbf b$ is a limit point, there need to be infinitely many points of $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3270118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Using Weierstrass theorem to prove having a finite optimal solution I was wondering if anyone has used Weierstrass theorem to prove that we have a finite optimal solution or has any reference for this claim.
I read in a paper that if an objective function is convex a, using Weierstrass theorem, we can conclude that the... | The Weierstrass extreme value theorem asserts that if you minimize a continuous function over a closed and bounded set in $\mathbb R^{n}$, then the minimum will be achieved at some point in the set.
I read in a paper that if an objective function is convex a, using
Weierstrass theorem, we can conclude that the opt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3270330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Cycle type of a permutation in $S_n$ and its relation to partition of $n$ and its Young diagram I know that it's possible to assign to each permutation its cycle type. I found two definitions of the cycle type and its relation to a partition of $n$:
First definition
Given $\sigma \in S_n$ written as product of $l$ cycl... | I have never seen the second definition to be honest. For me the cycle type is given by the first one and at least for me that is the classical one. Most often one defines the cycle type to see that the cycle type determines the conjugacy class, i.e. two permutations have the same cycle type iff they are conjugate and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3270465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is it obvious that the plane $z=0$ is tangent to the surface $z=x^{2}+y^{2}$ Why is it obvious that the plane $z=0$ is tangent to the surface $z=x^{2}+y^{2}$
I don't quite understand, is this obvious? I have a problem with the background knowledge, I don't even know how to deal with the surface $z=x^{2}+y^{2}$.
| In fact at $z=0$ the equation $x^2+y^2=0$ defines only $1$ point, which is $(0,0)$.
So the plane $z=0$ and the surface intersect in a single point.
But since the surface is smooth (the equation is polynomial), it cannot have singular points, and the touching point is automatically a tangent point.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3270540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Generating function of a parametrized binomial coefficient Let be $m$ an integer and $A_p(m) = \binom{mp}{p}$.
I'd like to know more about $B_m(z) = \sum_{p \geq 0} A_p(m) z^p$.
At least, I'd love to be able to compute $B_m\left(\dfrac{1}{q}\right)$ for some $q$ integers.
What I tried:
*
*Look at Fuss-Catalan number... | This is not an answer.
For $m=1,2,3$ there are closed forms for $B_m(z)$.
For $m\geq 4$ come again hypergeometric functions with interesting patterns
$$B_m(z)=\,
_{m-1}F_{m-2}\left(\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}m; \frac{1}{m-1},\frac{2}{m-1},\cdots,\frac{m-2}{m-1};\frac{m^m}{(m-1)^{m-1}}z\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3270761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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If $T$ is terminal object, is also $\text{hom}(X,T)$ a terminal object? In a category with internal Hom and terminal object $T$. Is it true that also $\text{hom}(X,T)$ is a terminal object for any object $X$?
It is definitely true for $\mathbf{Set}$ or $\mathbf{Vec}$, but I'm not sure if it is true in general.
I'm pro... | By definition of the terminal object, $\hom(X,T)$ is a one element set for every $X$.
And one element sets are exactly the terminal objects in $Set$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3270858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Does $\mathbf{N}$ with the reverse divisibility order form a Heyting algebra?
Consider the nonnegative integers $\mathbf{N}$ with the reverse divisibility order (i.e. $\mathrm{a} \leq \mathrm{b}$ $\iff$ $\mathrm{b} \mid \mathrm{a}$). Is this a Heyting algebra?
One advantage of the reverse ordering is that the elemen... | Yes, the lattice of subgroups of the infinite cyclic group is relatively pseudocomplemented.
It is easy to check the pseudocomplement $(p^m)\to(p^n)$ is $(p^n)$ if $m<n$ and $\mathbb{Z}$ if $m\geq n$. So
$$
((p_1^{m_1}\dots p_k^{m_k})\to(p_1^{n_1}\dots p_k^{n_k})) = \biggl(\prod_{\substack{j\\ m_j<n_j}} p_j^{n_j}\bigg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3270998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Clarification of L'Hopital Proof Pugh I am self-studying Real Analysis right now via Pugh's Real Mathematical Analysis but am having trouble understanding a step of the author's proof of L'Hopital's rule.
The theorem is stated as:
If $f$ and $g$ are differentiable functions defined on an intveral $(a,b)$, both of whic... | This is a good question. I actually don't think it follows from what he has written. Take, for example, $g(t) = 1/2, f(t) = -1/2, g(x) = 1, f(x) = 1$. Then $|f(t)+g(t)| < \frac{g(x)^2\epsilon}{4(|f(x)|+|g(x)|)}$ and $|g(t)| < \frac{|g(x)|}{2}$, but $|\frac{g(x)f(t)-f(x)g(t)}{g(x)(g(x)-g(t))}| = |\frac{-1/2-1/2}{1/2}| =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3271142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 2
} |
Proving that an integral is a holomorphic function Let $U\in \mathbb C$. Let $f:U\to\mathbb C$ be analytic on $U$ and continuous on the boundary of $U$. I want to prove that, for each $a\in U$, and sufficiently small $r>0$,
$$
g(w)=\frac{1}{2\pi i}\int_{|z-a|=r}\frac{zf'(z)}{f(z)-w}dz
$$
defines a holomorphic function... | Let $n$ be the order of the zero of $f(z)-f(a)$ at $z=a$.
$$f(z)- f(a)= f^{(n)}(a) (z-a)^n+O((z-a)^{n+1})$$
For $r$ small enough then $f(z)-f(a)-w, |z-a|=r$ doesn't vanish on $|w| < R= \frac12 |f^{(n)}(a)| r^n$ so that $$g(f(a)+w)=\frac{1}{2\pi i}\int_{|z-a|=r}\frac{zf'(z)}{f(z)-f(a)-w}dz$$ is analytic on $|w| < R$.
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3271272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\sum_{n=1}^\infty \int_{-\infty}^\infty\cos(n^2x)I(x)dx$ converges absolutely.
Let $I$ be a measurable subset of $\mathbb R$. We define
$$ I(x)=\int_I\frac{\chi_{(-1\le x-y\le 1)}}{1+y^2}dy. $$
For $n\ge 1$ we define
$$a_n=\int_{-\infty}^\infty\cos(n^2x)I(x)dx.$$
Prove that $\sum_{n=1}^\infty a_n$ ... | We need resort to oscillatory nature of the integrand. By Fubini's theorem1),
$$ a_n
= \int_{I}\int_{\mathbb{R}} \frac{\cos(n^2 x)\mathbf{1}_{\{\left|x-y\right|\leq 1\}}}{1+y^2}\,\mathrm{d}x\mathrm{d}y
= \int_{I} \frac{\sin(n^2(y+1)) - \sin(n^2(y-1))}{n^2(1+y^2)}\,\mathrm{d}y, $$
and so, $\left|a_n\right| \leq c/n^2$ f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3271390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$ 5r + 4s + 3t + 6u = 100, \:\: r \ge s \ge t \ge u \ge 0 $ maximum and minimum possible of $r + s + t + u$? We have
$$ 5r + 4s + 3t + 6u = 100, \:\: r \ge s \ge t \ge u \ge 0 $$
What is the sum of the maximum and minimum possible of $r + s + t + u$?
Attempt:
Assume that $r'+s'+t'+u'$ is the maximum. Now if $u > 0$, ... | Hint: We get $$20\le r+s+t+u\le 25$$, where the minum will be attained for $$r=20,s=t=u=0$$ and the maximum by $$r=s=t=\frac{25}{3},u=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3271522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Two sided normal p-value question From Statistical Inference by Casella and Berger:
Let $X_1 , \dots, X_n$ be a random sample from a $n(\mu, \sigma^2)$ population. Consider testing $:H_0: \mu = \mu_0$ verses $H_1 : \mu \neq \mu_0$. $W(X) = |\bar X - \mu_0| / (S / \sqrt n)$ is a test statistic that rejects $H_0$ for ... | The distribution of $T_{n-1}$ is independent of $\sigma$, hence for every value of $\sigma$ the p.value equals the probability of $T_{n-1}$ being larger than $|\bar{x} - \mu_0|/(s/\sqrt{n})$. Due to symmetry of $T_{n-1}$ around $0$, it is suffice to calculate only one sided probaility and multiplying it by $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3271603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
if the sum of two numbers $\alpha$ and $\beta$ is algebraic, and their product is transcendental, what do we know about these numbers? These are elements of a field. My intuition says that $\alpha=a+b$, $\beta=a-b$, where, $a$ is algebraic and $b$ is transcendental, but I can't prove it. I don't even know where to star... | If $\alpha + \beta$ is algebraic and $\alpha \beta$ is transcendental, then
$(\alpha - \beta)^2 = (\alpha+\beta)^2 - 4 \alpha \beta$ is transcendental, so $\alpha - \beta$ is transcendental.
Thus $\alpha = ((\alpha + \beta) + (\alpha - \beta))/2$ is transcendental, and so is
$\beta = ((\alpha + \beta) - (\alpha - \beta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3271729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does the congruence hold? Let $p$ be a prime number and $\mathbb{Q}(\zeta)$ be the pth cyclotomic number field where $\zeta$ is any primitive pth root of unity.
Writing $$t=b_0+b_1\zeta+...+b_{p-2}\zeta^{p-2} $$ with $b_j \in \mathbb{Z}$ , we get $$t^p \equiv b_0^p+b_1^p+...+b_{p-2}^p \pmod{p\mathbb{Z}[\zeta]}$$
I ... | First prove that $t^p = b_0^p + (b_1\zeta)^p + \cdots + (b_{p-2}\zeta^{p-2})^p$.
Hint: write it out and note what terms get a coefficient divisible by $p$, then note that all those coefficients are in your ideal $(p)$.
Another hint: Maybe start small and show $(a + b)^p = a^p + p(\cdots) + b^p$.
Next, use $\zeta^p = 1$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$F$ is algebraically closed $\iff$ $\nexists$ $K$ s.t. $F \leq K$, $K \neq F$ and $[K:F] < \infty$
$F$ is algebraically closed $\iff$ $\nexists$ $K$ s.t. $F \leq K$, $K \neq F$ and $[K:F] < \infty$.
Proof:
($\implies$) assume by way of contradiction that $F$ is an algebraically closed field and there does exist such ... | The correct statement is :
$F$ is algebraically closed if and only if there does not exist a finite field extension $K$ of $F$, i.e. there does not exist $K$ such that $F\leq K$ and $[K:F]<\infty$.
Proof : $(\Rightarrow)$ Say $K$ is a finite field extension of $F$ of degree $n$. Let $\alpha\in K-F$. Then $\{1,\alph... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate the sum of series with square roots Calculate the sum of the following series using partial sums:
$$\sum_{n=1}^\infty \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n} \sqrt{n+1}} $$
I rationalized the upper part of the fraction but I got lost. Could you please help me showing the steps of the how to transform the fracti... | HINT:$$\frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n} \sqrt{n+1}} = \frac{1}{\sqrt{n}}-\frac{1}{\sqrt {n+1}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3272127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is $\sqrt [3]{-1}$ and how does one obtain its value? I know that $i=\sqrt{-1}$. I was wondering what the $\sqrt [3] {-1}$ is.
I went on wolfram alpha, and it gave me values for $a$ and $b$ such that $\sqrt [3] {-1}=a+bi$. After some experimenting, I am almost absolutely certain we have:
$$\sqrt [3] {-1}=\frac12+... | Assuming you're not familiar with polar coordinates as suggested by @J.W.Tanner, this algebraic way might be simpler to understand.
Define a complex number to represent the cube root:
$\sqrt [3] {-1} = a + b i$
so
$-1 = (a + b i)^3$
which simplifies to
$-1 = a^3 - 3 a b^2 + (3a^2 b - b^3 )i$
Therefore:
$-1 = a^3 - 3 a ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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$A=\{\frac ab | a,b \in Z^+ , \frac{a^2}{b^2}<2 \}$
Show that the set $$A=\left\{\frac ab | a,b \in Z^+ , \frac{a^2}{b^2}<2 \right\}$$ has a least upper bound $L$
My try:
$$\frac{a^2}{b^2}<2$$
$$\frac{a^2}{b^2}<(\sqrt2)^2$$
$$-\sqrt2<\frac{a}{b}<\sqrt2$$
But $a,b >0$ so $\frac ab >0$
Thus, we get
$$0<\frac ab<\sqrt2$... | The least upper bound $L$ exists by definition of the real numbers, which have as their defining axiom that every set which has some upper bound actually has a least upper bound.
This is, in fact, how one defines $\sqrt{2}$ - so strictly speaking your proof is incorrect, since you're invoking the existence of some real... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Are countable topological spaces second-countable? Are countable spaces (i.e. $\mathbb{N}$ with any topology) second-countable? A countable space can have at most $2^\omega$ open subsets which suggests that a counterexample may exist. On the other hand both discrete and anti-discrete (or more generally with countable t... | Consider $ω$ many convergent sequences, and glue their limits. The resulting space won't have countable base at the common limit point.
Also note that a countable space is second-countable if and only if it is first-countable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3272545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Extraction of coefficient from Generating Function Determine the coefficient of $~x ^ {12}~$ in:
$(1+^2+^4+^6+^8+^{10}+^{12})(1+^4+^8+^{12})(1+^6+^{12})(1+^8)(1+^{10})(1+^{12})$
How to proceed with the resolution of this type of question when there is the product of more than two functions?
| The coefficient of $x^{12}$ is equal to the number of partitions of $12$ in which all summands are even.
Given a partition of $12$ in which all summands are even we can divide each summand by $2$ to get a partition of $6$. And given a partition of $6$ we can find a partition of $12$ with even summands by doubling each... | {
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Implicit Function Theorem Intersection of Hyperbolas Let $$M:=\{(x,y,z)^T\in\mathbb{R}^3:x^2+2yz=3, x^2+y^2+yz=z^2+5\}$$ and $(x_0,y_0,z_0)\in M,\ y_0z_0 \neq 0$
Show that there is an open neighborhood $U \subseteq \mathbb{R}$ around $x_0$ and continuously differentiable functions $g,h:U\rightarrow \mathbb{R}$ with $g(... | Set $F:(x,y,z)\mapsto (u(x,y,z),v(x,y,z))=(x^2+2yz,x^2+y^2+yz-z^2)\ $ so $F(x_0,y_0,z_0)=(3,5).$ To apply the implicit function theorem, we use the Jacobian (in $y$ and $z$) and check that $(x_0,y_0,z_0)$ is a regular point. This follows by hypothesis and the fact that
$\begin{bmatrix}
2z & 2y\\
2y+z&y-2z
\end{bmatr... | {
"language": "en",
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"source": "stackexchange",
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The image of a functor need not be a subcategory Warning 1.2.19 gives an example when the image of a functor is not a subcategory:
But I'm confused: the author defines a functor $F$ right away without saying what the codomain category is. This causes the question: the image of that functor is not subcategory of which ... | The codomain is the category depicted on the right, and the image of $F$ is not a subcategory because it contains the morphisms $p$ and $q$ but not their composition $qp$. This is explained under the diagram.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3273147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is this matrix invertible? I'm following Intro to stochastic processes by Lawler, page 27.
It says if we have a matrix Q such that $Q^n \rightarrow0$, then the eigenvalues of Q have absolute value less than $1$. That part I understand.
Then it says: "Hence, $I-Q$ is invertible." How does that follow?
P.S. I unders... | Hint: If $Q$ has no eigenvalue of $1$ then $I - Q$ has no eigenvalue of $0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A definite integral: $\int_{0}^{\pi/2} \frac{\sin x~\mathrm dx}{\sin x+\cos x+ e^x}$ Mathematica can do this integral,
$$\int_{0}^{\pi/2} \frac{\sin x~ \mathrm dx}{\sin x+\cos x+ e^x}\,,$$
the question is: how to do it by hand?
| $$\int_{0}^{\pi/2} \frac{\sin x~ dx}{\sin x+\cos x+ e^x}dx$$
$$=\int_{0}^{\pi/2} \frac{e^{-x}\sin x~ dx}{e^{-x}(\sin x+\cos x)+ 1}dx$$
Put $1+e^{-x}(\sin x+\cos x)=t$. Then, $-2e^{-x}\sin x dx=dt$.
The integral changes to
$$=\int_{2}^{1+e^{-\pi/2}} \frac{-1}{2t}dt$$
$$=\frac{1}{2}\ln\left(\frac{2}{1+e^{-\pi/2}}\right)... | {
"language": "en",
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Dimension of a subspace of $2\times2$ Matrices The question is asking to find the dimension of the subspace $W$, where, $V = M_{2,2}$,
$$
W = \{A \in V: AB= BA\}
$$
where
$$B=\begin{bmatrix}1&2\\3&4\\\end{bmatrix}$$
I defined an arbitrary matrix $A$ which contains the entries $a,b,c,d$. Then I considered the equailty a... | Your equations give
\begin{align*}
3b&=2c\\
a+c&=d\\
\end{align*}
Meaning that once $a$ and $c$ are known you can deduce $b$ an $d$.
Consequently the matrix $A$ has the following form
\begin{pmatrix}
a & 2c/3 \\
c & a+c
\end{pmatrix}
Can you conclude from there ?
| {
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If $\tan x=3$, then what is the value of ${3\cos{2x}-2\sin{2x}\over4\sin{2x}+5\cos{2x}}$?
If $\tan x=3$, then what is the value of
$${3\cos{2x}-2\sin{2x}\over4\sin{2x}+5\cos{2x}}$$
So what I did is change all the $\sin{2x}$ and $cos{2x}$ with double angle formulas, getting
$${3\cos^2{x}-3\sin^2{x}-4\sin{x}\cos{x}\o... | The answer is $\displaystyle \frac 94$.
Alternative method.
I like this half angle identity: $\displaystyle \tan \frac 12 y = \frac{\sin y}{1 + \cos y}$
So $\displaystyle 3 = \tan x = \frac{\sin 2x}{1 + \cos 2x}$, giving $\displaystyle \sin 2x = 3 + 3\cos 2x$.
Substituting that into the original expression transforms i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Analytical expression for the shape of the rounded pyramid. I'm searching for an analytical equation approximating the pyramid with rounded tip.
In particular, I have a pyramid whose base is an equilateral triangle with side "a", and height "h". The tip of the pyramid is rounded by some radius "r". (The point is to mo... | Since I was not satisfied with existing formulations, I decided to approximate the pyramid with the rounded cone:
where a describes the slope of the cone's side and delta is the rounding parameter. Inside the square root it is reponsible for roundness, outside for having a tip of the approximated indenter in the origi... | {
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Radical equation solve $\sqrt{3x+7}-\sqrt{x+2}=1$. Cannot arrive at solution $x=-2$ I am to solve $\sqrt{3x+7}-\sqrt{x+2}=1$ and the solution is provided as -2.
Since this is a radical equation with 2 radicals, I followed suggested textbook steps of isolating each radical and squaring:
$\sqrt{3x+7}-\sqrt{x+2}=1$
$(3x+... | The big error is that $4x^2+16x+16=4(x+2)$ is the same as $4x^2+12x+8=0.$ You somehow got $4x^2+12x+14=0.$ Did you treat $4(x+2)$ as the same as $4x+2?$ The equation $4x^2+12x+8=0$ has $x=-1$ and $x=-2$ as roots.
There's an earlier error where you write: $3x+7=(1-\sqrt{x+2})^2.$ The right side should be $(1+\sqrt{x+2})... | {
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Integral from infinity to infinity My physics professor today wrote on the blackboard:
$$ \int_{\infty}^{\infty} f(x) dx = 0 $$
for every function $f$.
And the proof he gave was:
$$ \int_{\infty}^{\infty} f(x) dx = \int_{\infty}^{a} f(x) dx + \int_{a}^{\infty} f(x)dx = - \int_{a}^{\infty} f(x) dx + \int_{a}^{\infty}f(... | This is not necessarily true. Take the following example;
$$\int_a^{2a}\frac1x\mathrm{d}x=[\ln{|x|}]_a^{2a}=\ln{(2)}$$
If we take $a\to\infty$ then the integral becomes
$$\int_\infty^\infty\frac1x\mathrm{d}x=\ln{(2)}$$
as the integral is constant for all $a\in\mathbb{R}$. What I guess your professor meant was that
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3274013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove this integration is not zero Let {$f_{n}$}$_{n=1}^{\infty}$ be a sequence of non-zero elements of $L^{2}[0,1]$. Prove that there is a function $g\in L^{2}[0,1]$ such that for all $n\ge1$ we have $\int_{0}^{1}g(x)f_{n}(x)dx\neq0$.
I try to assume there isn't such function $g$ and get contradiction. But I co... | Note that
$$U_n = \left\{ f\in L^2([0,1]) : \int_0^1 f f_n dx\neq 0 \right\}$$
is an nonempty (since $f_n$ is nonzero) open sets which is dense in $L^2 ([0,1])$. The Baire Category theorem says that
$$ \bigcap U_n$$
is nonempty. Thus there is $g\in L^2([0,1])$ so that
$$ \int_0^1 g f_n dx\neq 0$$
for all $n$.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to integrate $\int_0^{\infty} \frac{\sin(x^{-p})}{x^2}dx$ where $p>1$? How to integrate the following integral:
$$\int_0^{\infty} \dfrac{\sin(x^{-p})}{x^2} dx, p>1 ?$$
Thank you for any help.
Attempt: I have tried simple sub: $x^{-p} =u \implies du=dx (-p)x^{-p-1}.$
$$\int_0^{\infty} \dfrac{\sin(x^{-p})}{x^2} dx =... | From the change of variable $u=x^{-p}$, the integral becomes
$$
-\frac{1}{p}\int_{0}^\infty \sin(u)u^{1/p-1}du=-\frac{1}{p}\mathcal{M}\{\sin(u)\}(1/p),
$$
where $\mathcal{M}$ denotes the Mellin transform.
Since $0<1/p<1$, one can easily infer from http://mathworld.wolfram.com/MellinTransform.html
that
$$
-\frac{1}{p}\G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3274219",
"timestamp": "2023-03-29T00:00:00",
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Calculating inverse trigonometric values without a calculator (AEA 2016)
Find the value of
$$\arccos(1/\sqrt2) + \arcsin (1/3) + 2 \arctan(1/\sqrt2).$$
Give your answer as a multiple of $\pi$.
This was the least well answered question on Edexcel's Advanced Extension Award annual paper in 2016. The next paper is t... | $$\sin x=\frac13\implies \tan x=\frac{\frac13}{\sqrt{1-(\frac13)^2}}=\frac1{2\sqrt2}\implies
x=\arctan\frac1{2\sqrt2}\tag1$$
$$\tan(y/2)=\frac1{\sqrt2}\implies\tan y=\frac{2\frac1{\sqrt2}}{1-(\frac1{\sqrt2})^2}=2\sqrt2\implies y=\arctan(2\sqrt2)\tag2$$
$$(1)\& (2) \implies x+y=\frac\pi2.$$
Somewhat shorter aproach:
$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$p^2 - 2 q^2 = 5039$ for primes $p, q$ Are there primes $p$ and $q$ for which $p^2 - 2 q^2 = 5039$?
This is the least prime $r$ for which I don't know whether $p^2 - 2 q^2 = r$ has a solution in primes.
The solutions of the Pell-type equation $x^2 - 2 y^2 = 5039$ are $x_n, y_n$ given by the recurrences
$x_{n+4} = 6 ... | COMMENT.- I do not handle powerful calculators but I want to suggest what seems to me a way to calculate solutions or impossibility of such with more comfort maybe (I am not sure of this!).
The complementary formulas of the law of quadratic reciprocity allow to say that $2$ is a square module the prime $5039$. In effe... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Find $ \frac{1}{2^2 –1} + \frac{1}{4^2 –1} + \frac{1}{6^2 –1} + \ldots + \frac{1}{20^2 –1} $ Find the following sum
$$
\frac{1}{2^2 –1} + \frac{1}{4^2 –1} + \frac{1}{6^2 –1} + \ldots + \frac{1}{20^2 –1}
$$
I am not able to find any short trick for it.
Is there any short trick or do we have to simplify and add it?
| Alternatively to the telescoping sum decomposition, there is an easy pattern
$$\frac13$$
$$\frac13+\frac1{15}=\frac25$$
$$\frac13+\frac1{15}+\frac1{35}=\frac37$$
$$\frac13+\frac1{15}+\frac1{35}+\frac1{63}=\frac49$$
$$\cdots$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Suppose $B_j = \sum_{i=1}^{r} a_{ij} A_i, j= 1,2,....,t$. How does showing that $B_i's$ are dependent prove that $r \geq n$? I am reading 'Galois Theory by Emil Artin', and while reading the proof of Theorem $2$ on Pg. No. $5$, I couldn't grasp the following step :
Now, let $B_1,. . ., B_t$ be any system of vectors in... | Suppose we can show that $B_1,\ldots,B_t$ are dependent for all choises of $t>r$ and $B_1,\ldots, B_t$. The claim is that we can conclude $r\ge n$.
Indeed, assume $r<n$. Then by definition 1, we can find $t:=n$ independent vectors $B_1,\ldots, B_t$ - and can show that they are dependent!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Writing a matrix as a product of two matrices Consider the matrix
$$ A = \begin{pmatrix}
0 & y & -x\\
y & y^2 & -xy\\
-x & -xy & x^2
\end{pmatrix}. $$
Is it possible to find matrices $X = X(x)$ and $Y=Y(y)$ such that $A = XY$ (or $A = YX$)?
A possibly unrelated observation of mine is that if we consider the vector $v ... | Becasue of the symmetry, if it is possible with $A=YX$ then it is possible with $A=XY$ too. So without loss of generality let us assume that we can write $A(x,y)=X(x)Y(y)$ for some matrix-valued functions $X$ and $Y$.
Now setting $x=-1$ we get
$$ X(-1)Y(y)\begin{pmatrix}1-w\\0\\w\end{pmatrix} = \begin{pmatrix} 0 & y & ... | {
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Expanding random N(0,1) variable If I have an expression
$$\frac{1}{1+\sigma m(z/l)}$$
where $m(z/l)$ is a random $N(0,1)$ variable, $\sigma$ is dimensionless, can I rewrite this via an expansion to bring up the random variable on the numerator?
| If you want to use $$\frac1{1+x}=1-x+x^2-x^3+\cdots$$
then you need to remember that this only works for $|x|\lt 1$
and that a random variable with a normal distribution has a positive probability of being outside this
| {
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What values of $\alpha$ make this improper integral convergent? I'm having trouble discussing what values of $\alpha$ make
$$\int_{0}^{+\infty}\frac{1-\cos{x}}{x^{\alpha}}dx$$
convergent. The problem explicits that $\alpha \gt 1$.
I've seen that the integral can be written like
$$\lim_{a \to 0^+}\int_{a}^{b}\frac{1-\c... | Regarding your first integral, for $x$ near zero, we know $1-\cos x \approx x^2/2$ so your integrand is close to $x^2/x^\alpha$. This puts a condition on $\alpha$ that I will let you figure out.
In your second integral, the first term of the integrand is $1/x^\alpha$. I think you can see that this is divergent for t... | {
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How to find the week day of (any) given date? How to find the week day of any given date?
Say we need to know in which week-day was June $25,2019$?
| To determine the week-day of a given date, we need to:
*
*find out whether the given year is "common" or "leap".
*know $\mod(a,b)$.
*know $\left \lfloor a \right \rfloor$.
To find out whether the given year is "common" or "leap", we can use the following chart:
$\mod(a,b)$ means the remainder when dividing $a$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3275126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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absolutely continuous and increasing function Im having a really hard time trying to solve this problem that appears in the book of Royden guy. Any help would be extremely appreciated.
Let $f:[a,b]\rightarrow \mathbb{R}$ an absolutely continuous and increasing function. Show that $\lambda(f(A)) = \int_{A}f'd\lambda$ ... | It suffices to prove this for Borel sets. Without loss of generality, $a=0,\ b=1.$ Set $\mathscr S = \{A\subset\mathscr B([0,1]) : \lambda(f(A)) = \int_A f' \}.$ Absolute continuity of $f$ implies that $f(b)-f(a)=\int^b_af'$ for all $0 \le a\le b\le 1$ and this in turn implies that $\mathscr S$ contains the intervals,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3275270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Proof of Bound for Growth of Divergent Trajectory in $3x+1$ Problem In this paper, Lagarias makes the following claim in section 2.7 (Do divergent trajectories exist?).
Context
$$T(x) = \left\{ \begin{array}{rl} \dfrac{3x + 1}{2}, & 2 \nmid x \\ \dfrac{x}{2}, & 2 \mid x \end{array} \right.$$
$$\begin{align*} \tag{2.30}... | I contacted the author, and he was kind enough to write up a proof for me. I have attempted to simplify his proof for presentation here. I also use some notation without explanation to reduce clutter; the meanings should be clear. The trick is to use an apparently well known result from lattice theory.
Proposition 1 (L... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3275361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Solving a Fractional Equation Involving a Logarithm I may be being stupid right now, so I've come to Stack to see if this elementary algebra holds up.
Suppose I have the equation $$\frac{\ln x}{(1+ \ln x)^2} = \frac{1}{4}$$
My chosen way to solve this would be to cross multiply and expand brackets, solve the quadratic... | $$\frac{\ln x}{(1+\ln x)^2}=\frac{1}{4}$$
with $u=\ln(x)$ we get:
$$\frac{u}{(1+u)^2}=\frac{1}{4}$$
$$4u=1+2u+u^2$$
$$u^2-2u+1=0\Rightarrow (u-1)^2=0$$
$$\therefore u=1$$
$$x=e^u\Rightarrow x=e,$$
This appear to be the only solution
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3275438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Calculus, water poured into a cone: Why is the derivative non-linear? If water is poured into a cone at a constant rate and if $\frac {dh}{dt}$ is the rate of change of the depth of the water, I understand that $\frac {dh}{dt}$ is decreasing. However, I don't understand why $\frac {dh}{dt}$ is non-linear. Why can't it ... | The notion by Mike is nearly correct. Christian points out the correct result without being overly specific.
If you consider the volume of a cone of maximum height $h$ and maximum radius $R$ but only calculate it to the height $h'<h$, you can shuffle the equation to give that height depending on the volume of that frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3275538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 8,
"answer_id": 5
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Is there a positive integer $n \ge 2$ for which $\frac{k}{\pi(k)} = n$ has no solution? For a given positive integer $n \ge 2$ let $a_n$ be the number of integers $k$ such that $\dfrac{k}{\pi(k)} = n$ where $\pi(x)$ is the prime counting function. The first few values of $(n,a_n)$ are
$$(2, 4), (3, 3), (4, 3), (5, 6), ... | Not an answer but too long for comment: here is a Mathematica script to look for solutions systematically:
solve[] := Module[
{i, n},
i = 2;
n = 2;
While[True,
While[i/PrimePi[i] != n,
i++
];
Print["solve(", n, ")=", i];
n++;
];
];
The first few solutions for $n=2,3,4,...$
solv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Null of Quotient Map is the Subspace Consider a subspace $U$ of $V$, where $V$ is a finite-dimensional vector space over a generic field, $F$.
Let the quotient map $\pi$ be the linear map $\pi:V\rightarrow V/U$ such that $\pi(v)=v+U$ for $v\in V$. Note $V/U$ is the quotient space such that $V/U=\{v+U:v\in V\}$.
$\textb... | Remember that addition in the quotient space is defined as
$$(a+U) + (b+U) := (a+b) + U$$
This also corresponds to addition of sets $a+U = \{a+u : u \in U\}$ and $b+U = \{b+u : u \in U\}$ obtaining $(a+b)+U = \{a+b+u : u \in U\}$.
Hence the zero element in $V/U$ is $0+U = \{0+u : u \in U\} =U$.
We have
$$v \in \ker \pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3275794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove $H\circ N = H\cap N$. Where $H$ and $N$ are two subgroups of a group $G$. I was trying to prove another theorem where I thought the above result could be helpful and started trying to prove it.
I am not sure whether the above statement is true or not, but I am unable to prove it. I'll be very thankful if somebody... | Thanks to Mindlack, the doubt is now clear, and it was very silly.
$H\circ N$ as defined above should contain all elements of $H\cup N$, and only in the case $H=N$, $H\circ N = H\cap N$ is true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3275945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Derivative when $(\sqrt{x})^2$ is involved? Problem: If $f(x)=\frac{1}{x^2+1}$ and $g(x)=\sqrt{x}$, then what is the derivative of $f(g(x))$?
My book says the answer is $-(x+1)^{-2}$. This answer seems flawed because $(\sqrt{x})^2$ is being simplified to $x$ when it should really be simplified to $|x|$. If $(\sqrt{x})^... | The domain of $f(g(x))$ is $[0,\infty)$ so the absolute value does not do anything here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3276055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma =1$
Let be $\alpha, \beta, \gamma$ the angles between a generic direction in 3D and the axes $x,y,z$, respectively.
Prove that
$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma =1$.
PS: the 2D case is trivial. But I can't prove the 3D case.
| Let $$\vec{v}=[v_1,v_2,v_3]$$ then we get
$$\cos(\alpha)=\frac{\vec{v}\cdot\vec{e_1}}{|\vec{v}|\cdot|\vec{e_1}|}=\frac{{v_1}}{|\vec{v}|}=\frac{v_1}{\sqrt{v_1^2+v_2^2+v_3^2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3276199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Reference for basic result on algebraic dimension of a complex manifold In his book Complex Geometry, Huybrechts states and proves a classical theorem of Siegel that a compact complex manifold of complex dimension n has algebraic dimension at most n. Unfortunately, I don't understand his proof, so I'd like another refe... | Try Shafarevich's Basic Algebraic Geometry II: Schemes and Complex Manifolds. The result you're looking for is Theorem 3 on page 175.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3276327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Prime Numbers. Show that if $a \mid 42n + 37$ and $a \mid 7n +4$, for some integer $n$, then $a = 1$ or $a = 13$ Show that if $a \mid 42n + 37$ and $a \mid 7n +4$, for some integer $n$, then $a = 1$ or $a = 13$
I know most of the rules of divisibility and that any integer number can be expressed as the product of prime... | $$a|7n+4 \implies a|6(7n+4)=42n+24$$
$$ a|42n+37\text { and, } a|42n+24 \implies a|(42n+37)-(42n +24) = 13 $$
$$ a|13 \implies a=\pm 1, \text {or, } a=\pm 13 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3276410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Burnside's Lemma on octagon, using one of two colors on each side I am trying to find why the number of colorings of a regular octagon there are such that each side is colored either red or blue is 10. As well as this condition, each color must be used once. Essentially, out of the 8 edges, 4 are red and 4 are blue. If... | We may apply PET here since we require the cycle index $Z(C_8)$ of the
cyclic group $C_8$ anyway in order to apply Burnside. We have
$$Z(C_n) = \frac{1}{n} \sum_{d|n} \varphi(d) a_d^{n/d}$$
With $n=8$ this works out to
$$Z(C_8) = \frac{1}{8} a_1^8 + \frac{1}{8} a_2^4
+ \frac{1}{4} a_4^2 + \frac{1}{2} a_8.$$
We get
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3276521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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I have a question about partial differential equations How to find $\frac{\partial^2 e^u}{\partial y^2} =$? When $u = u(x,y)$
| $$\frac{\partial}{\partial y}e^u=e^u\frac{\partial u}{\partial y},$$
$$\frac{\partial^2}{\partial y^2}e^u
=\frac{\partial}{\partial y}\left(e^u\frac{\partial u}{\partial y}\right)
=\left(\frac{\partial}{\partial y}e^u\right)\frac{\partial u}{\partial y}+e^u\frac{\partial}{\partial y}\frac{\partial u}{\partial y}
=e^u\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3276798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Normal distribution: Weight of a package of cookies Suppose the mass of a cookie is a normal random variable X. Let's say a cookie weighs $20g$ on
average with a standard deviation of $2g$. A packet contains exactly $25$ cookies, with the weight
of the packaging also being a normal random variable Y with mean $100g$ an... | $(i) \ \ X \ $~$ \ N(20,4), \ \ Y \ $~$ \ N(100,36)$. A packet contains $25$ cookies, so the weight of the packet $W \ $ ~ $\ N(100+25\times20,36+25\times 4)=N(600,136).$
So the variance of the packet is $136$.
$(ii)$ If you buy $3$ packets, their combined weight follows $K \ $~ $\ N(1800,408)$ and operating wit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3276914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find the shortest path between several nodes in a particular order without ever using any edge twice? Given are a number of ordered nodes in a bidirectional graph with known only positive cost for each edge.
I need to find the shortest path through the given nodes in the particular order that are given while nev... | Even finding out whether there is a solution at all is NP-complete. I will show this by reducing from 3SAT.
Given a 3SAT instance, create a copy of the following 18-node network for each 3-literal clause:
The edges $Y_1\leftrightarrow Z_1$, $Y_2\leftrightarrow Z_2$, and $Y_3\leftrightarrow Z_3$ represent the literal; ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3277016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simplifying $\sqrt\frac{\left(a^2\cos^2t+b^2\sin^2t\right)^3}{\left(b^2\cos^2t+a^2\sin^2t\right)^3}$ I am looking to simplify these term [ I forgot the 3 :( ]
$$\sqrt\frac{\left(a^2\cos^2t+b^2\sin^2t\right)^3}{\left(b^2\cos^2t+a^2\sin^2t\right)^3}$$
where $a$ and $b$ are two non-negative reals.
(This is not homewor... | The form is
$$\sqrt{\frac{f^3}{g^3}}$$
which we can write as
$$\left(\frac{f}{g}\right)^{3/2}$$
so let's just worry about that inner quotient, $f/g$.
The quotient is definitely not constant; different $t$ values give different results:
$$t = 0 \;\to\; \frac{a^2}{b^2} \qquad\qquad t= \frac{\pi}{2}\;\to\;\frac{b^2}{a^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3277124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Poincaré inequality for Lipschitz functions with bounded domain Let $u\in W^{1,\infty}(B_h(0),\mathbb R^n)$, where $B_h(0)=\{x\in\mathbb R^n:|x|<h\}$.
From the Poincaré inequality we know that
$$
\|u-\mathrm{Id}-\frac{1}{\mathrm{Vol}(B_h(0))}\int_{B_h(0)}(u-\mathrm{Id})\|_{L^2}
\leq C\|du-\mathrm{Id}\|_{L^2}
$$
for som... | Is is not true. Consider a constant function u(x)=N. The estimate would lead to a contradiction for a sufficiently large N.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3277238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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What is the midsection of a frustum and how do you calculate its perimeter? A doubt while reading "How to solve it" by George Polya.
Given the figure below:
What is the midsection of this figure and how would you calculate its perimeter? (Would be great if you could tell me how to find it on the diagram).
Quoting from... | Just $$2\pi\cdot\frac{R+r}{2}=\pi(R+r).$$
Because a perimeter of the circle with radius $x$ it's $2\pi x$.
The needed midsection it's a circle with diameter, which is a midline of the trapezoid with bases $2R$ and $2r$ and this midline is equal to $\frac{2R+2r}{2}=R+r.$
Id est, the radius of the circle is equal to $\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3277311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Compute the following sum in closed form : $\sum_{n=1}^{\infty}\frac{n\binom{2n}{n}}{4^{n}(2n+1)(2n-1)(4n+1)}$ $$\text{Find : }\sum_{n=1}^{\infty}\frac{n\binom{2n}{n}}{4^{n}(2n+1)(2n-1)(4n+1)}$$
I know that $\displaystyle\sum_{n=0}^{\infty}\binom{2n}{n}x^{2n}=\frac{1}{\sqrt{1-4x^{2}}}$ so $\displaystyle\sum_{n=1}^{\inf... | As remarked in the OP,
\begin{equation}
\sum_{n=0}^{\infty}\binom{2n}{n}x^{2n}=\frac{1}{\sqrt{1-4x^{2}}}
\end{equation}
or, by changing $x\to x/2$,
\begin{equation}
\sum_{n=0}^{\infty}\frac{1}{4^n}\binom{2n}{n}x^{2n}=\frac{1}{\sqrt{1-x^{2}}}
\end{equation}
We use the decomposition
\begin{equation}
\frac{n}{\left( 2n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3277455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Having trouble finding the range of this function. $$f(x)=\frac{e^{2x}-e^x+1}{e^{2x}+e^x+1}$$
Let, $e^x=t$ . Then,
$$f(x)=\frac{t^{2}-t+1}{t^{2}+t+1}=y\quad where,\ t>0$$
$$(y-1)t^2+(y+1)t+(y-1)=0$$
so from the discriminent of the quadratic equation of $t$ I get,
$$(y+1)^2-4(y-1)^2\ge0$$
$$(3y-1)(y-3)\le0$$
$$\frac{1}{... | Note that
$$h(t)=\frac{t^{2}-t+1}{t^{2}+t+1}=1-\frac{2t}{t^{2}+t+1}=1-\frac{2}{t+\frac{1}{t}+1}$$
Now by AGM inequality, for $t=e^x>0$, $t+\frac{1}{t}\in [2,+\infty)$ and therefore
$$f(\mathbb{R})=h((0,+\infty))=[1/3,1).$$
P.S. By solving the quadratic equation
$$(1-y)t^2-(1+y)t+(1-y)=0$$
(for $y=1$, we have that $t=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3277602",
"timestamp": "2023-03-29T00:00:00",
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Commutativity of $\bigcup$ and $\wp$ In his Naive Set Theory, Halmos in Section 5, Complements and Powers, asks the following.
Show that $E$ is always equal to $\bigcup_{X\in\wp (E)} X$ (that is $E=\bigcup\wp (E)$), but that the result of applying $\wp$ and $\bigcup$ to $E$ in the other order is a set that includes $E... | A simple illustration of the second fact: let $E=\{\{\emptyset\}\}$, then $\bigcup E = \{\emptyset\}$ (all elements of elements of $E$ together) and $\mathscr{P}\left(\bigcup E)\right) = \{\emptyset, \{\emptyset\}\}$ which indeed properly contains $E$ as a subset.
That $E \subseteq \mathscr{P}(\bigcup E)$ is clear:
su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3277728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An exercise on Möbius transformations I had an exam on Complex Analysis and I could not solve the following exercise on Möbius transformations:
Let $f(z)=\frac{z+1}{z-1}$ and $A=\{z : \operatorname{Im}(z) >0\}\setminus \{|z|<1\}$. Find $f(A)$
I know this probably is not that difficult but I don't know how to solve thes... | Moebius transformations map circles* (meaning circles or lines) to circles*. The problem tells about two circles* in the $z$-plane. Compute $f(-1)$, $f(0)$, $f(1)$, and $f(i)$ in order to obtain three points of each of the two image circles*. It turns out that $f(A)$ is one of the quadrants in the $w$-plane. In order t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3277894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Determine the character of the singularity at $z=-2$ for function$\frac{1}{(z+2)^{2} \sin z}.$ Determine the character of the singularity at $z=-2$ for function
$$\frac{1}{(z+2)^{2} \sin z }.$$
Since function $ z \mapsto \frac{1}{\sin z} $ is holomorphic in some neighbourhood around $z=-2$, its development to Laurent's... | What you say is correct.
The function $1/\sin z$ is analytic in a suitable neighborhood of $z = -2$. Therefore, in such a neighborhood we have:
\begin{align}
f(z) & = \frac{1}{(z+2)^{2}}\frac{1}{\sin z} \\ & = \frac{1}{(z+2)^{2}} \left\{ a_0 +a_1 (z+2)+a_2 (z+2)^{2}+ a_3 (z+2)^{3} + \cdots \right\} \\ & = \frac{a_0}{(z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3278012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$G\leqslant GL(n,\mathbb R)\cap\operatorname{Sym}(n,\mathbb R)$ and $|G|=m<\infty$. Prove that $G\cong (\mathbb Z/2\mathbb Z)^k$ for some $k\ge 0$.
Let $G$ be a finite subgroup of the group of real $n\times n$ matrices with nonzero determinant such that all elements of $G$ are symmetric matrices. Prove that $G$ is iso... | It is an exercise in first courses in group theory to prove $g^2=e$ for all $g\in G$ implies $gh=hg$ for all pairs of elements $g,h\in G$. The way to do it is expand out $(gh)^2=g^2h^2$ and cancel $g,h$ on the sides. (So, actually, in general it's sufficient for $x\mapsto x^2$ to be a homomorphism for $G$ to be commuta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3278288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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On closed forms for the binomial sum $\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}$ for general $p$? Define the function,
$$A_p(z)=\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}$$
I've asked about the special case $z=1$ of this function before. At the end of this post, we find for $p\geq 2$ a closed-form in terms o... | Hoping that you enjoy hypergeometric functions,
$$ A_p(z)=\sum_{n=1}^\infty \frac{z^n}{n^p\,\binom {2n}n}=\frac{z}{2} \, \, _{p+1}F_p\left(1,\cdots,1;\frac{3}{2},2,\cdots,2;\frac{z}{4}\right)$$ and what you wrote in comments is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3278448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Evaluate $\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x^2}dx$
How to prove $$\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x^2}\ dx=\text{Im}\left(\operatorname{Li}_3(1+i)\right)-\frac{\pi^3}{32}-G\ln2 \ ?$$
where $\operatorname{Li}_3(x)=\sum\limits_{n=1}^\infty\frac{x^n}{n^3}$ is the trilogarithm and $G=\sum_{n=0}^\infty\frac{(-1)^n}{(2... | Different approach:
Start with subbing $x\mapsto \frac{1-x}{1+x}$
$$\small{\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x^2}dx=\ln2\underbrace{\int_0^1\frac{\ln\left(\frac{1-x}{1+x}\right)}{1+x^2}dx}_{-G}-\int_0^1\frac{\ln x\ln(1+x)}{1+x^2}dx+\int_0^1\frac{\ln^2(1+x)}{1+x^2}dx}\tag1$$
where
$$\int_0^1\frac{\ln^2(1+x)}{1+x^2}dx=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3278573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
} |
Existence of infinite set of positive integers s.t sum of reciprocals is rational and set of primes dividing an element is infinite Does there exist a sequence $(a_i)_{i \geq 0}$ of distinct positive integers such that
$\sum_{i\geq 0}\frac{1}{a_i} \in \mathbb{Q}$ and
$$\{ p \in \mathbb{P} \text{ }|\text{ } \exists\te... | Let $f=(f_1,f_2): \mathbb{N} \rightarrow \mathbb{N}^2$ be a bijection.
Define $a_n=(2^{3+f_1(n)}+1)^{f_2(n)}$.
By Bang’s theorem (https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem), if $n > m > 3$, then there exists some prime $p$ dividing $2^{2n}-1$ but neither $2^{2m}-1$ nor $2^n-1$, thus $p|2^n+1$ but not $p|2^m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3278677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 3,
"answer_id": 2
} |
For $0\lt\theta\lt1$, $\frac 1\theta\notin\mathbb Z$, there exists $f\in C[0, 1]$ such that $f(0)=f(1)$ and $f(x+\theta)-f(x)\ne0$ Prove that for each $0\lt\theta\lt1, \dfrac{1}{\theta} $ isn't an integer, there exists $f \in C[0, 1]$ such that $f(0)=f(1)$, and $ \forall x\in[0,1-\theta] , f(x+\theta)-f(x)\ne0 $
(If $... | Parting from predicates, as $0 < \theta < 1$ in strict order, we can establish a function $\phi : ]0,1[ \rightarrow \mathbb{R}$ such that $\phi(\theta) = \frac{1}{\theta}$. As we know that $\phi(\theta)$ tends to $1$ as $\theta$ tends to $1$, and $+ \infty$ as $\theta$ tends to $0$, we can restrict the function as foll... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3278772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Develop into Laurent series around $0$: $\frac{e^{1/z}}{z - 3i} .$ Develop into Laurent series around $0$:
$$\frac{e^{\frac{1}{z}}}{z - 3i} .$$
I was thinking of developing $e^{\frac{1}{z}}$ first and then $\frac{1}{z - 3i}$, but I got stuck while writing its multiple as one sum. Is that a good way of solving this?
I'm... | If you have two functions $f$ and $g$ holomorphic on an annulus $$\mathcal{A}=\{z\in\mathbb{C}: r_0<|z-z_0|<r_1\},$$ then given Laurent series' $$f(z)=\sum\limits_{n=-\infty}^\infty a_n (z-z_0)^n$$ and $$g(z)=\sum\limits_{n=-\infty}^\infty b_n (z-z_0)^n$$ valid on $\mathcal{A}$, we will have $fg$ hololomorphic on $\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3279050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Writing $a^2b^0c^0+a^0b^2c^0+a^0b^0c^2+a^1b^1c^0+a^1b^0c^1+a^0b^1c^1$ using $\sum$? $$a^2+b^2+c^2+ab+ac+bc$$
$$=a^2b^0c^0+a^0b^2c^0+a^0b^0c^2+a^1b^1c^0+a^1b^0c^1+a^0b^1c^1$$
Been messing around with some probability stuff and that popped up. I couldn't figure out how to write it in summation form so I can generalize it... | $$\sum_{\substack{i,j,k\ge0\\ i+j+k=2}}\mkern-9mua^i b^j c^k$$
seems to be what you're after.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3279194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Find all primes $(p,q)$ such that $p|q+6$ and $q|p+7$ Find all primes $(p,q)$ such that $p|q+6$ and $q|p+7$
I haven't found any. I initially started from $p,q\gt 3$ since, by a simple substitution you get, if of $p=2$ then $q|8$ and $q=2$, but then $2|9$ and it's a contradiction. Similarly happens with 3. Then, $p,q$ a... | From $q\mid p+7$ we have $p+7=qk$ for some positive integer $k$.
*
*If $k=1$ then $p+7=q$ so $p,q$ one is even, so $p=2$ and $q=9$. Not good.
*If $k=2$ then $p+7=2q$ and since $p\mid 2q+12$, then $p\mid 19$, so $p=19$ and
$q=13$.
*If $k=3$ then $p+7=3q$ so $p,q$ one is even, so $p=2$ and $q=3$ which doesn't work... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3279284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Prove by epsilon-delta that $\lim \limits_{x \to 1} x^3-4=-3$ I need to prove that $\lim \limits_{x \to 1} x^3-4=-3$ with epsilon-delta.
My work
$\forall \varepsilon > 0 ,\exists \space \delta > 0: 0<|x-1|< \delta \implies |x^3-4+3| < \varepsilon$
Working with the consequent:
$|x^3-4+3| < \varepsilon \iff |x^3-1| < \v... | Assume that $\delta < 1/2$, then $1/2<x<3/2$.
$$|x^2+x+1|=x^2+x+1 < 5$$
$$ |x-1||x^2+x+1|< \delta |x^2+x+1|<5\delta =\epsilon$$
You can take it from here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3279354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How can a transformation be linear transformation without linearity? My teacher at the University gave me a question I could not understand completely. Here is the question:
Let $T: \mathbb R^3 \to P[x]$ be a linear transformation with $$T([1, 0,
0])=x+1, \quad T([0, 1, 0])=x^2-x, \quad T([0, 0, 1])=x^2,$$ find
$T([... | Linearity is nothing more or less than requiring $T(u+v)=T(u) +(v), T(au)=aT(u)$. This does not rule out
$T(u)=x+1$.
In fact consider the function $M\colon P[x]\to P[x]$ which has the effect of multiplying anything by $x^2+3x+1$ (or any random, fixed polynomial).
$M(f(x)+g(x) )= M(f(x) ) + M(g(x))$. ANd M(af(x) ) = a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3279529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Minimize the Sum of Reciprocal of Probabilities I need a probability distribution over $ n $ events such that the sum of expected values is minimized. That is, minimize
The problem is given by:
$$\begin{aligned}
\arg \min_{ {p}_{i} } & \; && \sum_{i = 1}^{n} \frac{1}{ {p}_{i} } \\
\text{subject to} & \; && \sum_{i = ... | Here is an argument that tells you that the minimum value can only be attained the $p_i$'s are equal. Suppose the minimum value is attained when $p_i=q_i, 1 \leq i \leq n$. If possible let $q_i \neq q_j$. Note that $\frac 1 {q_i} +\frac 1 {q_j} >\frac 1 {\frac {q_i+q_j} 2}+\frac 1 {\frac {q_i+q_j} 2}$. [This is simply... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3279641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Solve the equation $|2x^2+x-1|=|x^2+4x+1|$ Find the sum of all the solutions of the equation $|2x^2+x-1|=|x^2+4x+1|$
Though I tried to solve it in desmos.com and getting the requisite answer but while solving it manually it is getting very lengthy.
I tried to construct the two parabola and mirror image the region bel... | The expressions between the absolute value bars have the same or opposite signs. Hence there are two independent cases (by addition and subtraction):
$$3x^2+5x=0$$ and $$x^2-3x-2=0.$$
Then by Vieta,
$$-\frac53+3.$$
For complete rigor, one should show that no root is repeated. This is true, because the polynomials have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3279752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Asymptotic formula for $\sum_{k=N}^\infty \frac{x^k}{k!}$ as $N \to \infty$ This seems like a weird question, because the series has good convergence and there's no need to use other methods to estimate it for $N \to \infty$.
However, after seeing this question, I tried to come up with some approximation which could al... | Note that
$$
\frac{n!}{x^n}\sum_{k=n}^\infty\frac{x^k}{k!}
=1+\sum_{k=n+1}^\infty\frac{x^{k-n}n!}{k!}\\
$$
and for $n\ge|x|$,
$$
\begin{align}
\left|\sum_{k=n+1}^\infty\frac{x^{k-n}n!}{k!}\right|
&\le\sum_{k=1}^\infty\frac{|x|^k}{(n+1)^k}\\
&=\frac{|x|}{n+1-|x|}
\end{align}
$$
Thus,
$$
\sum_{k=n}^\infty\frac{x^k}{k!}
=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3279850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Interesting topological spaces to calculate the homology groups. Interesting topological spaces to calculate the homology groups.
I am calculating homology groups of several topological spaces to learn and I have already calculated the homology groups of $\mathbb{S}^m$, $\mathbb{R}P^2$, the Klein bottle, $\mathbb{R}^2... | Not sure whether you consider them to be interesting, but these are some spaces whose homology groups I once computed in the past when I was studying algebraic topology:
1) The torus $T^2 = S^1 \times S^1$ or more generally $T^n = S^1 \times \dots \times S^1$
2) The space you get when you take $S^2$ and identify the no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3280042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
How to write recursive functions in mathematics Okay this is a really stupid question, but anyways.
Action isn't just the effect of motivation; it is also the cause of
it.
~ "The subtle art of not giving a f*ck", Mark Manson
If I take variable action as a and variable motivation as m, I want to write a simple equa... | $$a_t=m(a_{t-1})$$
This recursive formula treats motivation as the function and actions as both the input and output to that function. In other words, motivation has a predefined relationship with action, whereby actions cause motivation which produces the next action.
With inspiration:
$$a_t = m(i(a_{t-1}))$$
or in En... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3280166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that MLE of $\theta$ is consistent for $N(\theta, \theta)$ distribution I want to show that the MLE of $N(\theta, \theta) $, namely :
$$\theta_1 = \frac { \sqrt{1+\frac 4 n \sum^n x_i^2} } 2 $$ converges in probability towards the true parameter $ \theta$. I thought about showing that the mean square error converg... | By the WLLN
$$
1/n \sum X_i^2 \xrightarrow{p} \mathbb{E}X^2=Var(X)+\mathbb{E}^2X=\theta+\theta^2.
$$
as $n \to \infty$,
and
$$
g(x) = \frac{\sqrt{1 + 4 x}}{2}
$$
it a continuous transformation. Hence by the continuous mapping theorem,
$$
g\left( \sum X_i^2/n \right) \xrightarrow{p}g(\theta + \theta^2)=\frac{ \sqrt{(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3280268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proof verification: the angle subtended by a chord can never be 90 degrees I couldn't find any sources of this online, so I would like to ask if what I'm proposing below is correct, or if a similar theorem has been proven before.
We know that the angle subtended by the diameter of a circle is always $90^\circ$ (Thales... | not the prettiest solution but perhaps this helps:
basically I assumed that we have input of 90 degree angle without assuming we are at the center, and we got that $y=x$ thus it indeed must be the center - proving back Thales' theorem
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3280387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $S=\sum_{n=2}^\infty\frac{_nC_2}{(n+1)!}.$
Prove that $$S=\sum_{n=2}^\infty\frac{_nC_2}{(n+1)!}=\frac{e}{2}-1.$$
$$
S=\sum_{n=2}^\infty\frac{_nC_2}{(n+1)!}=\sum_{n=2}^\infty\frac{n!}{2(n-2)!(n+1)!}=\sum_{n=2}^\infty\frac{1}{2(n+1)(n-2)!}\\
=\frac{1}{2}\bigg[\frac{1}{3.0!}+\frac{1}{4.1!}+\frac{1}{5.2!}+\frac{... | First, it must be:
$$S=\sum_{n=2}^\infty\frac{^nC_2}{(n+1)!}=\sum_{n=2}^\infty\frac{n!}{2(n-2)!(n+1)!}=\sum_{n=2}^\infty\frac{1}{2(n+1)(n-2)!}=\\
\color{blue}{=\sum_{n=2}^\infty\frac{(n+1)-n}{2(n+1)(n-2)!}=\frac12\left[\sum_{n=2}^\infty\frac{1}{(n-2)!}-\sum_{n=2}^\infty\frac{n}{(n+1)(n-2)!}\right]=}\\
=\frac{1}{2}\bigg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3280661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
What does P@1 mean in this scientific article? In this scientific article: Label Filters for Large Scale Multilabel Classification by Alexandru Niculescu-Mizil and Ehsan Abbasnejad, they use the notations P@1, P@5 and P@10 as the following table shows
I was thinking that they were $p$-values at the beginning, but i... | On the same page of the figure (Section 4) in the article, the notation is defined as follows:
"Following previous work on large scale multilabel classification (Weston et al., 2013; Prabhu and Varma, 2014; Bhatia et al., 2015) we use precision at k (P@k) as the
evaluation metric. Precision at k is defined as the f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3280781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$f$ is holomorphic in $B(z_0,r)\setminus\{z_0\}$ and does not except real values. Then $z_0$ is a removable singularity $f$ is holomorphic at $B(z_0,r)\setminus\{z_0\}$ and $f$ doesn't except real values - i.e $f(z)\notin\mathbb{R}$ for all $z\in \mathbb{R}$. Then $z_0$ is a removable singularity point ($f$ can be exte... | EDITED: If $f$ has a pole at $z_0$, $1/f$ has a zero there, and by the Open Mapping Theorem $1/f$ would take all values in some interval near $0$.
If $f$ has an essential singularity at $z_0$, Picard says it can omit at most one value near $z_0$.
Removable is all that's left.
EDIT If you don't want to use the heavy art... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3280930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Average relative to quantity Not really sure how to ask my question. I have a list of average transaction wait times which are averaged over the count of the transactions for each zone, as shown in the list below which is sorted by the avg wait highest to lowest. When I look at the top entry I think "yuck, they waited ... | The question has many answers. It depends on what is important. I can give a score as number of transactions plus number of seconds wait time, but that is probably not what you want. I would suggest giving separate scores for wait time and number of transactions (how you score these is also subjective), then take an av... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3281013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Expression for ( or approximation of ) series of odd terms in series representation of Bessel function of first kind? We know:
$$J_v(z) = \sum_{k=0}^{\infty}\frac{(-1)^k}{\Gamma(k+v+1)k!}\bigr(\frac{z}{2}\bigl)^{2k+v} \ \ (Eq. 1)$$
$$s.t.\ (v,k)\in \mathbb N,z\in \mathbb R$$
courtesy of Introduction to Bessel Fu... | This is a generalized hypergeometric function, with
\begin{align}\sum_{k=0}^\infty\frac{(-1)^{2k+1}}{\Gamma(2k+v+2)(2k+1)!}\left(\frac z2\right)^{4k+v+2}&=-\left(\frac z2\right)^{v+2}\sum_{k=0}^\infty\frac1{\Gamma(2k+v+2)(2k+1)!}\left(\frac z2\right)^{4k}\\&=-\left(\frac z2\right)^{v+2}{}_0F_2\left(;-\frac32,\frac{v+3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3281135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve this multivariable exponential equation? I searched if this was asked before but couldn't find a solution. I have this equation
$y^{70} = x + 500 $
$y^{50} = x + 1 $
Is it possible to solve this equation? The only thing I could do is to bring it into this form and then cross-multiply which didn't yield ... | The best I can think of at the moment is
$$
\eqalign{
& \left\{ \matrix{
50\ln y = \ln \left( {1 + x} \right) \hfill \cr
20\ln y = \ln \left( {{{x + 500} \over {x + 1}}} \right) = \ln \left( {1 + {{499} \over {x + 1}}} \right) \hfill \cr} \right. \cr
& \ln y = {1 \over {50}}\ln \left( {1 + x} \right) = {1 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3281237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How do I find x as a function of x? Sorry for the title, I don't know how else to put this into words.
Basically I wanted to know how to get the result below:
I have no idea about why the graph is showing X as a diagonal line. How can X by itself be a line which is not constant?
| $$x=462+0.085x$$
$$(1-0.085)x=462$$
$$0.915x=462$$
$$x=\frac{462}{0.915}=\frac{30800}{61}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3281372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
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