Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find the solution of the $x^2+ 2x +3=0$ mod 198 Find the solution of the $x^2+2x+3 \equiv0\mod{198}$
i have no idea for this problem i have small hint to we going consider $x^2+2x+3 \equiv0\mod{12}$
| Here's a more-or-less generalizable, manual way of finding all of the solutions:
First, as P. Vanchinathan does, change variable to $a := x + 1$, which transforms the equation into one with zero linear term:
$$a^2 + 2 \equiv 0 \pmod {198} .$$
(This step is option, but reduces the amount of later work.)
Now, we exploit ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2227709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Jacobson radical of group rings
$4 b)$
(i) Any hints?
(ii) Well $R$ is not semi-simple since $|\mathbb{Z}/3|=3=0 \in F_3$ by the converse of Maschke's theorem.
(iii) The surjective $\mathbb{C}$-algebra map $\phi:R \to M_2(\mathbb{C})\times\mathbb{C}\times\mathbb{C}: (a_{i,j}) \mapsto \begin{bmatrix}
a_{11} ... | (i) If you know Maschke's theorem (as you hint in (ii)) then you already know the answer to this since the order of $D_8$ is $8$.
(ii) Yes. And it would not even be too hard to exhibit some nonzero nilpotent element to prove that the Jacobson radical is nonzero.
(iii) You can use exactly the same logic here at this sim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2227921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Counting integers in a sequence with a least prime factor = $p$ Let $p > 2$ be a prime. It is very easy to count the integers in a sequence that are divisible by $p$.
Let $m \ge 0, n > 0$ be integers. The count of $x$ where $m < x \le (m+n)$ and $p | x$ is at most $1 + \left\lfloor\frac{n}{p}\right\rfloor$.
For examp... | By sieving I find $$\displaystyle w =\sum_{l\in A_{p}} \mu(l)\left(\left\lfloor \frac{n}{pl} \right\rfloor-\left\lfloor \frac{m-1}{pl} \right\rfloor\right)$$ where $\ \mu\ $ is the Möbius function and $\ A_{p}\ $ are the ( squarefree ) integers whose largest prime factor is $< p$.
So if $2^p$ is much smaller than $n-m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2228044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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An infinite summation involving binomial coefficient The question is to find out the value of $$\sum_{n=k}^{\infty} P^n \binom nk (1/2)^n$$
I tried to break down the binomial coefficient and bring it in form of some known sequence but could not get anything out of it.Please help me in this regard.Thanks.
| $$\begin{align}
\sum_{n=k}^{\infty}P^n{n\choose k}\left({1\over 2}\right)^n &= {P^k\over 2^kk!}\sum_{n=k}^\infty n(n-1)\dots(n-k+1)\left({P\over 2}\right)^{n-k}
\\&= {P^k\over 2^kk!}f^{(k)}({P\over 2})
\end{align}$$
Where $f : x\mapsto \sum_{n=0}^\infty x^n = {1\over 1-x}$
Thus $$\sum_{n=k}^{\infty}P^n{n\choose k}\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2228171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What topics does Elliptic curve cryptography lie under? I know this is a weird question to ask. Basically for my Math Internal Assessment, I want to explore Elliptic curve cryptography. (
Due to the lack of time, I'm unable to properly study it and I'm suppose to hand in a proposal very soon.
Therefore I was wondering ... | Put it under Number Theory. I would recommend Ketheth Rosen's Book as it is a pleasure to read, even as an undergrad, and its section on ECC is written by Larry Washington, an expert on the subject matter.
To see this stuff is real and in use, from the linux command line if you type
openssl s_client -host website -port... | {
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"timestamp": "2023-03-29T00:00:00",
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If $x$ be the $A.M$ between $y$ and$z$... If $x$ be the AM between $y$ and $z$, $y$ be the GM between $z$ and $x$, then $x$, $y$, $z$ are in :
$1$). A.P
$2$). G.P
$3$). H.P
$4$). None.
My Attempt:
$x$ is the AM between $y$ and $z$
$$x=\dfrac {y+z}{2}$$
$$2x=y+z$$
$y$ is the G.M between $z$ and $x$.
$$y=\sqrt {zx}$$
$$y... | HINT:
We have $$y+z=2x=2\cdot\dfrac{y^2}z\implies0=z^2+yz-2y^2=(z-y)(z+2y)$$
| {
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"url": "https://math.stackexchange.com/questions/2228579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Coordinate free proof for $a\times (b\times c) = b(a\cdot c) - c(a\cdot b)$ The vector triple product is defined as $\mathbf{a}\times (\mathbf{b}\times \mathbf{c})$. This is often re-written in the following way:
\begin{align*}\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mat... | Adapted from my previous proof of $\nabla \times (\vec{A} \times \vec{B})$:
\begin{align}
\vec a \times (\vec b \times \vec c)
& = a_l \hat{e}_l \times (b_i c_j \hat{e}_k \epsilon_{ijk}) \\
& = a_l b_i c_j \epsilon_{ijk} \underbrace{ (\hat{e}_l \times \hat{e}_k)}_{(\hat{e}_l \times \hat{e}_k) = \hat{e}_m \epsilo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 4
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Definition of SQUARE ML (㎖) This is a very simple thing, I suppose, I'm having hard time to find out, what is the meaning of Square ML (㎖) symbol? Square MiLe, Square MilliLiter, Square Maximum Likelihood, or totally something else?
I can find these symbols belonging to the physical symbol set in unicode set starting f... | Disclaimer: This post will use a lot of unicode characters that may not display properly in some environments.
The characters in the CJK Compatability block are mostly symbols for units used in Japanese, with some crossover into other languages like Chinese and Korean. Most of them have a name in unicode with "SQUARE"... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2228789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\alpha$ algebraic over field $F$ implies $[F(\alpha):F]=\text{degree of minimal polynomial over $\alpha$}$ Let $\alpha$ be some element algebraic over a field $F$. Then $F(\alpha)$ is isomorphic to $F[x]/\langle m_{\alpha,F}\rangle$, where $m_{\alpha,F}$ is the minimal polynomial with root $\alpha$ over $F$. Moreover,... | It seems the other answer and comment are about $F[\alpha]$ being a $F$-vector space of dimension $deg(p)$ ($F[\alpha]$ is the smallest ring containing $F$ and $\alpha$, i.e. $F[\alpha] = \{ \sum_{n =0}^d c_n \alpha^n, c_n \in F\}$).
For showing that $F[\alpha] = F(\alpha)$ you need to prove that $\varphi : F[x]/(p(x))... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2228910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof the Levi-Civita symbol is a tensor A tensor of rank $n$ has components $T_{ij\cdots k}$ (with $n$ indices) with respect to each basis $\{\mathbf{e}_i\}$ or coordinate system $\{x_i\}$, and satisfies the following rule of change of basis:
$$
T_{ij\cdots k}' = R_{ip}R_{jq}\cdots R_{kr}T_{pq\cdots r}.
$$... | I met this problem today and this is what I am trying:
$$
\epsilon_{ijk}=det(e_i\ e_j \ e_k)
$$
Let A be an orthogonal transformation, then:
$$
\begin{aligned}
\epsilon'_{ijk}&\equiv\epsilon_{lmn}=det( e_l \ e_m \ e_n)\\
&=det(Ae_i\ Ae_j \ Ae_k)\\
&=(det(A))^3 det(e_i e_j e_k)\\
&=det(A)\cdot \epsilon_{ijk}\\
&=\pm \ep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2228996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How Hahn-Banach theorem implies that the dual space is non-trivial? Why does the theorem of Hahn-Banach implies that the dual space is not empty ($X^*\neq\emptyset)$ ?
Is there an important corollary which I've missed ?
| The dual space is always non-empty, as it contains the zero functional. The Hahn-Banach theorem implies that if $X \neq \{0\}$, then also $X^* \neq \{0\}$.
Choose a non-zero vector $a \in X$. Denote the subspace $Y := \mathrm{span} (a) \subseteq X$ and the bounded functional $\varphi \in Y^*$ defined by $\varphi(a) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2229175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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What is meant by "The Levi-Civita Connection is an $\mathfrak{so}(n)$-Valued 1-form"? this is a statement that I've seen around, and I thought it's time that I understand it. I know that the LCC is locally given by a matrix $ \omega = (\omega_i^j)$ of 1-forms in a preferred frame $e_i$, so that $$ \nabla f_ie_i = df... | The Levi Civita connection is a particular case of an Ehresmann connection defined on the bundle of frames $F$, such a connection is defined by a $1$-form $\omega$ defined on the tanent bundle of $FM$ andwhich is $gl(n,\mathbb{R}$ valued. A Levi Civita connection means that $\omega$ takes its values in $so(n,\mathbb{R}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The Bent Washer Problem -- divide a shape into 2 pieces of the same volume. Fans of the Ham sandwich theorem know that any set of points can be divided by a plane into two equal halves.
Consider instead a 3-D shape that must be divided into 2 equal pieces by a single cut. A sufficiently bent spring washer or keyring... | The Ham Sandwich Theorem says that given three measurable subsets of $\mathbb{R}^3$ can be cut into two equal (with respect to measure) pieces by a single plane. In particular, we can choose two of our sets to be empty. So any measurable subset of $\mathbb{R}^3$ can be cut in half by a plane.
EDIT: I originally misunde... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating $\lim_{r \to \infty} \int_0^\frac{\pi}{2} e^{-r\sin\theta} d\theta$ Evaluating $$\lim_{r \to \infty} \int_0^\frac{\pi}{2} e^{-r\sin\theta} d\theta$$
First for all, I need to show that $e^{-r\sin\theta}$ converges uniformly to a function $F(r)$ Then I can easily take the limit inside, since $e^{-r\sin\theta}$... | Note that for $0 \le \theta \le \frac{\pi}{2}$, we have $\sin \theta \ge \frac{2}{\pi}\theta$, so:
$$0<\int_0^\frac{\pi}{2} e^{-r\sin\theta} d\theta \le \int_0^\frac{\pi}{2} e^{-(2r/\pi) \theta} d\theta \le \int_0^\infty e^{-(2r/\pi) \theta} d\theta=\frac{\pi}{2r}\to0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2229462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Whether the stochastic process has continuous sample paths I am considering the following question and want to convince myself that the stochastic process $X$ has continuous sample paths. I hope someone could give me some hints or references, many thanks!
Suppose that $\{B_t\}_{t\ge 0}$ is a standard Brownian motion an... | Let $F(T)$ be defined in the way you've defined it. Consider $\epsilon > 0$, we have that $$\begin{align}
F(T+\epsilon) &= \int_0^{T+\epsilon}\boldsymbol{1}_{\{X_t(\omega)\le a\}}dt\\
&= \int_0^{T}\boldsymbol{1}_{\{X_t(\omega)\le a\}}dt + \int_{T}^{T+\epsilon}\boldsymbol{1}_{\{X_t(\omega)\le a\}}dt \\
&\le F(T) + \int_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove that in the natural numbers "if $a = b$ then $a + c = b + d$ if and only if $c = d$" using Peano Axioms? I want to prove using the Peano axioms that in the natural numbers if $a = b$ then $a + c = b + d$ if and only if $c = d$ preferably by induction.
| Once you assume $a = b$, then showing $c = d \rightarrow a + c = b + d$ is just a matter of using the $=$-rules.
The more interesting part is showing $a + c = b + d \rightarrow c = d$. Since $a = b$, that reduces to showing that $b + c = b + d \rightarrow c = d$, and to show that, you need to use Induction over $b$.
Fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2229674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Gauss prime divides exactly one integer prime in $\mathbb{Z}[i]$ I am asked to show that a Gauss prime $\pi$ divides exactly one integer prime in $\mathbb{Z}[i]$.
To show existence, I have tried to use the fact that the product $\pi \overline{\pi}$ is equal to either an integer prime $p$ or the square of integer prime... | This works in any ring of integers $\mathcal{O}_K$ of a number field $K$ :
Take a proper ideal $I$ of $\mathcal{O}_K$ (here $I = \pi \mathcal{O}_K$).
Note that $J =I \cap \mathbb{Z}$ is a proper ideal of $\mathbb{Z}$, thus $J = n\mathbb{Z}$ for some $n \in \mathbb{N}_{> 1}$.
If $p \in I$ then $p \in I \cap \mathbb{Z}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2229786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Trying to prove the real-valued function has a limit. I am trying to show that the function $f:\mathbb{R}^2\to\mathbb{R}$ by the formula
$$f(\mathbf{x}) = \dfrac{x_1x_2^2}{x_1^4+x_2^2} \textrm{ if }\mathbf{x} \neq \mathbf{0}$$
$$f(\mathbf{0}) = 0$$
has a limit of $0$ as $\mathbf{x}\to\mathbf{0}$.
My book says I have ... | Mario's answer is good, but it is also good to practice $\varepsilon-\delta$ proofs. Let $x:=x_1$ and $y:=x_2$ for easy typing. Fix an arbitrary $\varepsilon>0$. First notice that for any $(x,y)\neq (0,0)$
\begin{align*}
|f(x,y)-0|& = \left| \frac{xy^2}{x^4+y^2}\right | \\
& \leq \frac{|x|y^2}{y^2}~\text{noting that}~ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Size of a convergent series restricted to primes Let $(a_n)_{n=1}^\infty$ be a decreasing sequence of positive real numbers such that $\sum_{n=1}^\infty a_n$ converges. I am interested in the sum $\sum_{p}a_p$, where $p$ ranges over the primes. This subsum obviously converges, but I am interested in how quickly it conv... | What you need is that $\pi(x) \sim \sum_{n < x} \frac{1}{\ln n}$ for saying that since $a_n > 0$ and is non-increasing, then $\sum_{p > x} a_p \sim \sum_{n > x} \frac{a_n}{\ln n}$.
Summing by parts : $$\sum_{p > x} a_p = \sum_{n > x} a_n 1_{n \in P} = \sum_{n > x} \pi(n) (a_n-a_{n+1}) =\sum_{n > x} ((1+o(1))\sum_{k < n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2230000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Remarkable/unexpected rational numbers Consider the Riemann $\zeta$ function. We know that $\zeta(2n)$ is a rational multiple of $\pi^{2n}$ (in particular is transcendental). We also know that $\zeta(3)$ is irrational, and we expect $\zeta(n)$ to be irrational (if not even transcendental) for every $n\in\mathbb{N}$, or... | The average distance between two randomly chosen points in the Sierpinski triangle (of side $1$) is
$$\frac{466}{885}$$
(where "distance" means the length of the shortest path between the points that lies within the Sierpinski triangle).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2230120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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"answer_id": 1
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Solving $ax+b$ (mod n) = $cx+d$ (mod n) How does one solve questions of $ax+b$ (mod n) = $cx+d$ (mod n) form?
I know that if $b,d$ are $0$ then, I can take multiples of $n$ as solutions for x.
but what if $b,n$ are not zero?
Under what conditions do solutions exist?
| $$ ax + b \pmod n \equiv cx + d \quad \pmod n $$
$$ (a - c)x \equiv d-b \pmod n$$
$$ Ax \equiv B \quad \pmod n$$
where $A = a-c, \; B = d - b$. Thus,
1) If $ a \equiv c \pmod n$:
*
*If $b \equiv d \pmod n$: $x$ is any integer.
*Otherwise, no solution.
2) If $a \not\equiv c \pmod n$:
*
*If $A$ has a multiplica... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2230297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $f(x)=x^2$ is a contraction on each interval in $[0,0.5]$ I need to formally prove that $f(x)=x^2$ is a contraction on each interval on $[0,a],0<a<0.5$. From intuition, we know that its derivative is in the range $(-1,1)$ implies that the distance between $f(x)$ and $f(y)$ is less then the distance between $... | Remark that
$$
|f(x)-f(y)| = |(x+y)(x-y)| \leq (|x|+|y|)|x-y| < 2a |x-y|
$$
if $x$, $y \in [0,a]$. Now $2a<1$ by assumption.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2230417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Matlab code to compute the smallest nonzero singular value of the matrix without using SVD I want to compute the smallest nonzero singular value of the matrix A, which is defined as follows.
Let $B = rand(500, 250)$, $A = B*B^t$, where $t$ denotes the transpose of the matrix.
I found the following matlab code to co... | Summary
The answer is...use svds.
What are the singular values?
There may be some confusion over how you get the singular values. The command svd computes the singular values and the components that you don't want. The command svds only computes the singular values.
As explained here Computing pinv, the first step in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2230507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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Which infinity do integrals diverge to? When we say $\int{f}dx = \infty$, what is the cardinality of that $\infty$?
| You are mixing up two notions of "infinity". One concerns the sizes of sets. A set is infinite (in that sense) when there's a one to one correspondence with a proper subset. That's the infinity that you mean when you talk about cardinality.
The other "infinity", written $\infty$, is sometimes confusing shorthand used w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2230630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Trigonometric Inequality I have a problem about this trigonometric inequality, which I cannot completely solve. In particular, I cannot get the whole solution the book provides and what a bad luck: I don't have the book with me, because this problem arose from one of my student's problem during a private lesson.
$$\si... | Hint: Going with the comment regarding use of the 'unit circle method', it may be easier to think about this geometrically rather than algebraically. For instance, if we pick a point on the upper half of the unit circle then the angle $x/2$ corresponds to a point in the first quadrant with positive sine. On the other h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2230848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Evaluate $\int_0^{2\pi}\frac{\cos(\theta)}{1 + A\sin(\theta) + B\cos(\theta)}\, \mathrm{d}\theta \ \text{ for }\ (A^2+B^2) <<1$ I need to evaluate the definite integral $\int_0^{2\pi}\frac{\cos(\theta)}{1 + A\sin(\theta) + B\cos(\theta)}\, \mathrm{d}\theta \ \text{ for }\ A<<1, B<<1, (A^2+B^2) <<1,$
For unresricted (... | HINT: set $$t=\tan(x/2)$$, $$\sin(x)=\frac{2t}{1+t^2}$$, $$\cos(t)=\frac{1-t^2}{1+t^2}$$ and $$dx=\frac{2dt}{1+t^2}$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Reference related to Hirzebruch surfaces I was reading about the Hirzebruch surfaces construction to understand a few examples of Delzant construction, and need some help. I am using this reference Hirzebruck surfaces and can't understand the followinga few things about the final step: $P(L_{-n} \oplus \bar{\mathbb{C}}... | $\overline{\mathbb C}$ is the trivial line bundle over $X$. It's a line bundle over $X$ (in your case, $X = \mathbb P^1$). So the total space of $\overline{\mathbb C}$ is $X\times \mathbb C$, with the projection given by $\pi(x, z) = x$.
In your case, $X = \mathbb P^1$ and the Hirzebruch surfaces is by definition
$$P... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Algebra rules when finding inverse modulo The idea is to find an inverse modulo for two numbers, $660$ and $43$ in this case. I find that the GCD is easy to calculate, but the step after that, calculating the inverse modulo when calculating back trough the GCD calculation.
The thing I do not get, is that 'by algebra' t... | The way I like to describe the process is this:
When finding the GCD of two numbers, begin writing a table where the first row is the first number we are interested in, followed by a 1 followed by a 0. The second row will be the second number we are interested in, followed by a 0 followed by a 1.
$$\begin{array}{c|c|c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2231155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Noetherian hypothesis when permuting elements of a regular sequence? One of the first results showed when studying regular sequences is that we are allowed to shuffle the elements of the sequence if the ring is noetherian, local, and the module is finite (see Proposition 2 in here for a proof).
I know the ring being l... | The assumption that the ring be noetherian is used when Krull's intersection theorem is applied. And the assumption is necessary, as Stacks Project's tag 00LH shows, for example: consider $k[x,y,w_1,w_2,\ldots]/(yw_1,yw_2,\ldots,w_1-xw_2,w_2-xw_3,\ldots)$ and localise in the maximal ideal generated by $x,y$ and all the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2231274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to explain for my daughter that $\frac {2}{3}$ is greater than $\frac {3}{5}$? I was really upset while I was trying to explain for my daughter that $\frac 23$ is greater than $\frac 35$ and she always claimed that $(3$ is greater than $2$ and $5$ is greater than $3)$ then $\frac 35$ must be greater than $\frac 23$... | It's quite impressive that an 8-year old can state a clear reason (even if wrong) for a mathematical conclusion. I suggest you should be pleased rather than upset, and should begin with the respectful approach of addressing her reasoning.
So you might try asking her which is the greater of $1/3$ and $2/5$. If she choo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2231366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "67",
"answer_count": 29,
"answer_id": 4
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How can one generate an open-ended sequence of low-discrepancy points in 3D? I'd like a low-discrepancy sequence of points over a 3D-hypercube $[-1,1]^3$, but don't want to have to commit to a fixed number $n$ of points beforehand, that is just see how the numerical integration estimates develop with increasing number... | another good solution to get an open-ended sequence is using the Halton method. It is also very easy to implement, even for any dimension! For d<8 it has usually good properties, beyond this more difficult will typically outperform Halton.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2231391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Motivation Of Correlation Coefficient Formula Definitions
correlation coefficient $= r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2\sum_{i=1}^{n}(y_i - \bar{y})^2}}$
My Question
What is the motivation of this formula? It's supposed to measure linear relationships on bivari... | Suppose we have a scatterplot of heights X and weights Y of n subjects.
The 'center of the data cloud' is at the point $(\bar X,\,\bar Y)$.
One might expect a positive association between heights and weights.
Points above and to the right of the center make a positive
contribution to the sum $\sum (X_i -\bar X)(Y_i - \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2231517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving inequalities on $L^P$ spaces Suppose $p,q,r \in[1,\infty)$ and $ 1/r = 1/p +1/q$. Prove that
$$\|fg\|_r \leq \|f\|_p*\|g\|_q.$$
I am assuming that this proves involves using Hölder inequality , but so far I am unable to proceed in the proof. Maybe so because this is my first problem about using the Hölder/Mink... | Clearly $p,q>r$, so one may apply Holder's inequality to $|f|^r$, $|g|^r$ with $p'=\frac{p}r$, $q'=\frac{q}r$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2231648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Methods for finding $\cos(4x)$ given that $\sin(2x) = \frac{3}{5}$ If $\sin(2x) = \frac{3}{5}$
Find $\cos(4x)$..
I tried by : $\cos(4x)= \cos(2\cdot2x)$..
And $\cos (2\cdot2x) = 1-2\sin^2(2x)$ ..
From it ---- $\cos(4x)=0.28$.
Is there any other ways ?
| *
*Form the 3-4-5 triangle.
*Reflect it about the "4" side. to get a 5-5-6 triangle.
*apply the cosine law to it.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Rearranging $\left| \sqrt{x} -\sqrt{y} \right| $ I'm just going through an example of a Holder function ($f(x) = \sqrt{x}$), and a step in the example goes as follows,
$$\left| \sqrt{x} -\sqrt{y} \right| = \frac{\left|x-y\right|}{\sqrt{x}+\sqrt{y}}$$
I've been fiddling around with this for half an hour and cannot see h... | $(\sqrt x-\sqrt y)(\sqrt x+\sqrt y) = (\sqrt x)^2 - (\sqrt y)^2 = x - y$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2231958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Universal Property of Quotients I couldn't recall the UPQ from my memory exactly, so I wrote the following:
Let $R, S$ be rings, $I$ be an ideal of $R$, $\pi:R\to R/I$ be the canonical quotient ring homomorphism and $\varphi:R\to S$ be a surjective ring homomorphism. Then there exists isomorphism $\overline{\varphi}:R/... | The general universal property of quotients states:
For a ring homomorphism $\varphi:R\to S$, and and ideal $I\subseteq\ker\varphi$, there exists a unique homomorphism $\overline{\varphi}:R/I\to S$ such that $\bar{\varphi}\circ\pi=\varphi$, where $\pi:R\to R/I$ is the usual projection.
In the case where $\varphi$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2232070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Since the $\omega$-limit is invariant, then it must contain only points with $\dot{V}(x)=0$ I am trying to understand the following proof:
I do not fully get why he says: since $\Gamma^+$ (which I think is usually referred to as the $\omega$-limit) is an invariant set, then $\dot{V}(x)=0$ in $\Gamma^+$.
Intuitively, i... | The preceding sentence says that $V(p)=c$ for each $p \in \Gamma^+$. And since $\Gamma^+$ is invariant, any orbit $y(t)$ which starts in $\Gamma^+$ stays in $\Gamma^+$. Thus $V(y(t))=c$ (identically) for such an orbit, which implies $\frac{d}{dt} \Bigl( V(y(t)) \Bigr)=0$. And this is exactly what the phrase “$\dot V=0$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding value of limit involving cosine I know we can use Maclaurin expansion and l'hopital's rule for solving it but I want another way . Find value of $\lim_{x \to 0} \frac{\cos^2x - \sqrt{\cos x}}{x^2}$.
My try : I multiplied and then divided by conjugate of numerator but it didn't help .
| Two standard limits $$\lim_{x\to a} \frac{x^{n} - a^{n}} {x-a} =na^{n-1},\,\lim_{x\to 0}\frac{1-\cos x} {x^{2}}=\frac{1}{2}$$ come to our rescue here. We have
\begin{align}
L&=\lim_{x\to 0}\frac{\cos^{2}x-\sqrt{\cos x}} {x^{2}}\notag\\
&=\lim_{x\to 0}\frac{\cos x - 1}{x^{2}}\cdot\left(\frac{\cos^{2}x-1}{\cos x - 1}-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2232291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Filtered colimit and directed colimit Is there an example that in some abelian category $A$, direct limit always exist, but filtered colimit does not always exist?
| No. Every filtered category admits a cofinal functor from a directed category, so the existence of directed colimits implies that of filtered colimits.
The construction of this directed category is slightly technical. It can be read in the first part of Adamek and Rosicky's monograph on locally presentable and accessi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"Compositional roots" of functions, how to define them and how many are there? Assume I am interested in solving $$(\underset{k \text{ times}}{\underbrace{g\circ \cdots \circ g)}}(x) = g^{\circ k}(x) = f(x)$$
That is, $g$ is in some sense a function which is a $k$:th root to applying the function $f$. Applying $g$ $k$ ... | Let $f(x)=4x$. That's a useful function, though hardly more so than 0; more importantly, it is nontrivial.
Define $g(x)$ as some freehand monotone curve running from (1,2) to (2,4), and then use the functional equation to expand its domain to (2,4), then to (4,8) and so on. You see that it may be made infinitely differ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that this fractal contains every odd, positive integer Let $f(x)=2x+\frac{1}{3}$
Let $g(x)=\frac{2x-1}{3}$
$f^n$ represents composition of $f$
Let $G_0=\{1\}$
Let $F_{m}=\{f^n(x):x\in G_{m}, n\in\mathbb{N_{\geq0}}\}$
Let $G_{m+1}=\{g(x):x\in F_m\}$
Show that for any given odd, positive integer $p$ there is some s... | Since $n=0$ is allowed, $\bigcup F_n$ is the set of numbers that ca be obtained by an arbitrary finite sequence of $f$ and $g$ from $1$.
Thus the claim is that for every $p\in\Bbb N$, there exists a sequence $x_0,x_1,\ldots, x_N$ with $x_0=1$, $x_N=p$ and $x_{k+1}\in\{2x_k+\frac13, \frac{2x-1}3\}$.
Let $y_k=6x_k+2$. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2232774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show $(1+x_1)(1+x_2)...(1+x_n)\geq2^n$ given $x_1x_2...x_n=1$
Show $(1+x_1)(1+x_2)...(1+x_n)\geq2^n$ given $x_1x_2...x_n=1$ and that all $x_i $ are positive reals.
I think simple AM-GM-HM must work, but I am missing on something trivial.
| The brute force approach is:
$$(1+x_1)(1+x_2)\cdots(1+x_n)=\sum_{S\subseteq \{1,\dots,n\}}\prod_{i\in S}x_i$$
But by AM/GM, since there are $2^n$ subsets of $\{1,\dots,n\}$, you get:
$$\frac{1}{2^n}\sum_{S\subseteq \{1,\dots,n\}}\prod_{i\in S}x_i \geq \sqrt[2^n]{x_1^{2^{n-1}}\cdots x_n^{2^{n-1}}}=1$$
This is because:
$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
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Are the Elementary Functions Dense in $C^n((0,1))$ with the Compact-Open Topology? My question is really a basic (and, I dare say, naive) question about the theory of ordinary differential equations and is likely answered in a book on functional analysis or the like, so any pointers to such a book with the answer would... | The polynomials are dense by the Stone-Weierstrass theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2232973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $\sin(x) = x + O(x^3)$? I was doing an exercise in which I have to find the rate of convergence of
$$\lim\limits_{h\to 0}\dfrac{\sin h}{h} = 1$$
and the answer is $O(h^2)$. I don't understand why. The only thing I have found is that $$\sin(x) = x + O(x^3)$$ when $x$ tends to zero, and with that the exerci... | Start with
$\sin' = \cos
$,
$\cos' = -\sin
$,
$\sin(0) = 0$,
$\cos(0) = 1$,
and
$\sin^2+\cos^2 = 1$.
For small $t$,
$1 \ge \cos(t)
\ge 0
$
so
$\sin(x)
=\int_0^x \cos(t)dt
\le x
$.
Therefore
$1-\cos(x)
=\int_0^x \sin(t) dt
\le \int_0^x t dt
= \frac{x^2}{2}
$
so
$\cos(x)
\ge 1-\frac{x^2}{2}
$.
Therefore
$\sin(x)
=\int_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2233089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that the series is less than the square of a series. Prove that $(\sum_{i=1}^n a_i)^2 \leq n \sum_{i=1}^na_i^2$ for $a_1, ..., a_n$. Hint: You may want to use the triangle inequality or Cauchy-Shwartz inequality.
I'm trying to prove this preposition. here's what I've done so far...
$(\sum_{i=1}^n a_i)^2$ = $\sum_... | Hint:
Consider the vectors $\underbrace{(1,1,\dots,1)}_{n\;1\text{s}}$ and $\;(a_1, a_2,\dots,a_n)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2233131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding a non-zero vector in Col A Definiton: The column space of an $m \times n$ matrix $A$, written as $\operatorname{Col} A$, is the set of all linear combinations of the columns of $A$. If $A = [ a_1 \ldots a_n ]$, then $\operatorname{Col} A = \operatorname{Span}\{a_1,\ldots,a_n\}$.
$$A= \begin{bmatrix} 2 & 4 & -... | As you stated the column space is the set of all linear combinations of the columns of $A$. So if the matrix is not the zero matrix, you will be able to find some non zero vector in $Col A$. Colum 3 is non zero and in fact your column space is the set of all vectors in 3D because $x\times col1 + y \times col2 + z\times... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2233230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\int ^{\infty}_0 \frac{\ln x}{x^2 + 2x+ 4}dx$ We have to find the integration of
$$\int ^{\infty}_0 \frac{\ln x}{x^2 + 2x+ 4}dx$$
In this I tried to do substitution of $x=e^t$
After that got stuck .
| Let $I$ be defined by the integral
$$I=\int_0^\infty \frac{\log(x)}{x^2+2x+4}\,dx \tag1$$
and let $J$ be the contour integral
$$J=\oint_{C}\frac{\log^2(z)}{z^2+2z+4}\,dz \tag2$$
where the contour $C$ is the classical "key-hole" contour for which the keyhole coincides with the branch cut along the positive real axis. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Prove：any smooth map $f:M \rightarrow \mathbb{R}$ can't be one to one. Let $M$ be a connected smooth manifold, $\dim M \ge 2$. Prove：any smooth map $f:M \rightarrow \mathbb{R}$ can't be one-to-one.
| No continuous map from $M$ to $\mathbb{R}$ can be injective.
As $M$ has dimension at least $2$ it contains a subspace $C$ homeomorphic to a circle. Now there is no continuous injective map $g$ from $C$ to $\mathbb{R}$. If there were, then there are $a$ and $b$ on $C$ with $g(a)<g(b)$. Then on each arc of $C$ with endpo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2233453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Class number of the compositum of a quadratic extension of a cyclotomic field Let $d\in\mathbb{Z}$ be a square-free integer and let $p$ be an odd prime. Let $K = \mathbb{Q}(\sqrt{d})$ and let $\zeta_p$ be a primitive $p$th root of unity. I am interested in knowing the class number of $L = K(\zeta_p)$.
1) I tried to use... | All your fields are abelian CM fields. There is a complete determination of all CM fields of class number one. (The case of imaginary quadratic fields is well known, but the other cases are somewhat easier, because they avoid issues concerning Siegel zeros.) For example, $\mathbf{Q}(\zeta_p)$ has class number bigger th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A continuous a.e orientation-preserving isometry is locally injective? Let $M,N$ be $d$-dimensional Riemannian manifolds. Let $f:M \to N$, and suppose $f$ is continuous, differentiable almost everywhere (a.e
) and that $df$ is an orientation-preserving isometry a.e.
Question: Is it true that there exist a ball $B_{\ep... | I want to reply to your comment in
https://mathoverflow.net/questions/264873/do-curvature-differences-obstruct-a-e-orientation-preserving-isometries/267163#267163
Question : Assume that $$f :B\rightarrow
\mathbb{E}^2$$ is a map s.t.
(1) $B$ is a geodesic ball in $S^2(1)$ of radius $\varepsilon$
(2) $df$ is isometric a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2233696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$\text{Ext}(H,G)$ in the universal coefficient theorem for cohomology In the universal coefficient theorem for cohomology with a chain complex $C$ of free abelian groups having homology groups $H_n(C)$,
\begin{eqnarray}
0\rightarrow\text{Ext}(H_{n-1}(C),G)\rightarrow H^n(C,G)\xrightarrow{h}\text{Hom}(H_n(C),G)\rightar... | We typically get examples with non-trivial cokernels where there is torsion around. For a simple example, let $C_0$ and $C_1$ both equal $\mathbb{Z}$ with all other groups $C_n$ being zero. Let $d:C_1\to C_0$ be the map taking $m$ to $2m$. Take $G$ to be $\mathbb{Z}$. In this case $Z_0=\mathbb{Z}$ and $B_0=2\mathbb{Z}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2233810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Trigonometric identity $2\cos(\alpha)\cos(\beta) = \cos(\alpha + \beta) + \cos(\beta - \alpha)$ in a book. Is it correct? The identity should be $2\cos(\alpha)\cos(\beta) = \cos(\alpha + \beta) + \cos(\alpha - \beta)$, but in the book (see attached picture) states it as follows
$2\cos(\alpha)\cos(\beta) = \cos(\alpha +... | It's correct. $\cos x = \cos(-x) \forall x \in \mathbb R$ which means $\cos(A-B) = \cos (B-A) \forall A,B \in \mathbb R$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2233919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Rate of convergence of fixed-point iteration in higher dimensions Consider the fixed-point iteration process in $\mathbb{R}^n$.
Given a sufficiently smooth function $f:\mathbb{R}^n\to\mathbb{R}^n$ and an initial value $x_0\in\mathbb{R}^n$, define the iteration sequence $x_{k+1}=f(x_k)$. Suppose that
$$\lim_{k\to\infty}... | As in many similar situations in higher dimensional spaces, it helps to look at the simplest case where the function can be decoupled. That is, the function requires no interaction between the variables. For two dimensions this is $f(x,y) =(f_1(x),f_2(y))$. For even more simplicity, assume $f(x,y) = (ax,by)$. Now the r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2234006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Sum of correlated normal random variables Suppose I have two correlated random variables, that were generated in the following way:
\begin{align*}
X_1 &\sim \mathcal{N}(0,1)\\
X_1' &\sim \mathcal{N}(0,1)\\
X_2 &= \rho X_1+\sqrt{1-\rho^2}\cdot X_1'\\
Y_1 &= \mu_1+\sigma_1 X_1\\
Y_2 &= \mu_2+\sigma_2 X_2.
\end{align*}
No... | $\alpha_1 Y_1 + \alpha_2 Y_2$ is a linear combination of $X_1$ and $X_1^\prime$ - that is $\alpha_1 Y_1 + \alpha_2 Y_2 = \beta X_1 + \beta^\prime X_1^\prime$ for some $\beta, \beta^\prime$ that are a bit of a pain to calculate. Linear combinations of independent normal random variables are normal; there are several pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2234078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Proving the inequality $0\leq \frac{\sqrt{xy}}{1-p}\frac{x^{\frac{1}{p}-1}-y^{\frac{1}{p}-1}}{x^{\frac{1}{p}}-y^{\frac{1}{p}}} \leq 1$ Suppose $p\in(0,1)$. How might one show that
\begin{equation}\tag{1}
0\leq \frac{\sqrt{xy}}{1-p}\frac{x^{\frac{1}{p}-1}-y^{\frac{1}{p}-1}}{x^{\frac{1}{p}}-y^{\frac{1}{p}}} \leq 1
\end{... | I've figured out the correct integral representation to use here. For $a\in(-1,1)$, consider the following integral representations:
\begin{align*}
\frac{x^a-y^a}{x-y} &= \frac{\sin(a\pi)}{\pi}\int_{0}^{\infty}\frac{t^a}{(x+t)(y+t)}dt\\
\text{and}\qquad ax^{a-1} &= \frac{\sin(a\pi)}{\pi}\int_{0}^{\infty}\frac{t^a}{(x+t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that the range of $f$ is all of $\mathbb{R}$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continous function such that $|f (x)−f (y)| \geqslant |x −y|$, for all real $x$ and $y$. I need to prove that the range of $f$ is all of $\mathbb{R}$. I tried to solve the inequality to get a general form for the functio... | Well, let's think.
Is it possible that that $f(x)$ is bounded above or below? Probably not as $|f(x)- f(y)| \ge |x-y|$ and $|x-y|$ seems likely to be arbitrarily large.
So Lemma 1: $f(x)$ is unbounded. (We'll prove that.)
If $f$ is unbounded does it have to take on every value. This seems reasonable as a variation o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2234286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Are there any continuous “totally rational” functions besides piecewise first-order polynomial or rational functions? Let $I \subseteq \mathbb{R}$ be an interval and $f: I \to \mathbb{R}$ a continuous function. We’ll say that $f$ is totally rational if the following propositions are true for any $x\in I$:
*
*If $x \... | Here's a different type of example with the property:
$f(n + 0.a_1 a_2 a_3 \dots) = 0.0 a_1 0 a_2 0 a_3 \dots$
or
$f(n + \sum_{i \ge 1}{a_i \cdot 10^{-i}}) = \sum_{i \ge 1}{a_i \cdot 100^{-i}}$
In other words, the function takes the decimal expansion of the fractional part and inserts a $0$ between every digit. Or it ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2234425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Relation Between Eigenvalues of a Matrix and its Real Part Let $A$ be an $n \times n$ complex matrix. The real part of $A$ is $\frac{A^H + A}{2}$. What is the relation between the eigenvalues of $A$ and $\frac{A^H + A}{2}$? I know that the eigenvalues of $A^H$ are the complex conjugates of $A$'s eigenvalues.
In fact th... | If $A$ is normal, the eigenvalues of $\frac{A + A^{*}}{2}$ are the real parts of the eigenvalues of $A$. If $A$ is not normal, there isn't any nice relation. For example, if
$$ A = \begin{pmatrix} 0 & 2\varepsilon \\ 0 & 0 \end{pmatrix} $$
then the eigenvalues of $A$ are zero but the eigenvalues of
$$ \frac{A + A^{*}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2234535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Help on Algebra Problem I need help on this problem that 8/9 Grade Math Teacher Assigned me (I'm in 7th Grade). Here's the problem -
Today is Latisha's birthday. Fifty friends have thrown her a surprise party. Before the party, her friends got together and decided to secretly hide several presents in 50 separate boxes... | A box is flipped once for every divisor its number has and it starts from closed, so it is closed if it has an even number of divisors and open if it has an odd number of divisors.
So this is progress, but it would be good to simplify the answer further. So we need to figure out how to tell whether a number has an eve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2234683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Student's Age in Fictional Stats Class Problem (Probability) Problem:
In a fictional stats class, 40% of students are female, and the rest are male. Of the female students, 30% are less than 20 years old and 90% are less than 30 years old. Of the male students, half are less than 20 years old and 70% are less than 30 y... | Let's follow @BGM's suggestion in the comments.
Let $F$ denote female; let $M$ denote male; let $A$ denote age.
Since $40\%$ of the students are female and $30\%$ of them are less than $20$ years old, the probability that a student is female and less than $20$ years old is
$$P(F~\cap A < 20) = P(F)P(A < 20 \mid F) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2234809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding E((X-2)²) This is a practice quiz as I have a test in a few days.
I've already completed the first and second questions and my answers are:
(1) h = 0.4
(2) k = 30
I'm struggling on question 3, I know how to find E(X) using the x•p(x) method but how do I find E((X-2)²). I tried (X-2)²•p(x) but I'm not sure tha... | $\begin{align}E[(X-2)^2] &= E[X^2-4X+4]\\
&= E[X^2]-4E[X]+E[4]\\
&= \sum_xx^2p(x)-4\sum_xxp(x)+4\\
&= 93.8 - 4\times4+4\\
&= 81.8
\end{align}$
On a side note: $E[g(X)]=\displaystyle{\sum_x}g(x)p(x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2234908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
union of proper subgroup is proper?
Question:
Prove that a finite group is the union of proper subgroups IFF the group is not cyclic.
Let G be a finite group.
Suppose G is the union of proper subgroups $b_{i}$.
This means that there is an element in G that is not in $b_{1}$.
Iterating this reasoning, we see that ther... | Let $G$ be a group, finite or infinite. Observe that the following statements are equivalent.
*
*$G$ is the union of some proper subgroups.
*$G$ is the union of all of its proper subgroups.
*Each element of $G$ belongs to some proper subgroup of $G.$
*For each element $g\in G,$ $\langle g\rangle$ is a proper subg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2235062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $X$ is uniform on $(0,1)$ how can I prove $2X$ is uniform on $(0,2)$? If $X$ is uniform on $(0,1)$ how can I prove $2X$ is uniform on $(0,2)$?
I am struggling with this question and similar but harder variants.
It makes a lot of sense to me intuitively but I am unfamiliar with rigorous arguments that are needed to p... | What you say is a way, and here I give another point, but both are similar.
The distribution function $F_{2X}(x)=P(2X\le x)=P(X\le \frac{1}{2}x)=F_X(\frac{1}{2}x)$ and we can start from here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2235180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solve$\int\frac{x^4}{1-x^4}dx$ Question: Solve $\int\frac{x^4}{1-x^4}dx.$
My attempt:
$$\int\frac{x^4}{1-x^4}dx = \int\frac{-(1-x^4)+1}{1-x^4}dx = \int 1 + \frac{1}{1-x^4}dx$$
To integrate $\int\frac{1}{1-x^4}dx,$ I apply substitution $x^2=\sin\theta.$ Then we have $2x \frac{dx}{d\theta} = \cos \theta.$ which implies... | By doing long division, you'll get $$-\int \left(-\frac{1}{2(x^2+1)}-\frac{1}{4(x+1)}+\frac{1}{4(x-1)}+1\right) dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2235305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Find the constant term in the expansion of $(x^2+1)(x+\frac{1}{x})^{10}$ I can't solve this problem. How to solve it?
The Problem is
"Find the constant term in the expansion of $
\left({{x}^{2}\mathrm{{+}}{1}}\right){\left({{x}\mathrm{{+}}\frac{1}{x}}\right)}^{\mathrm{10}}
$"
| $f(x)=(x^2+1)(x+\dfrac{1}{x})^{10}$
We can rewrite $f(x)$ like below:
$f(x) = x^2(x+\dfrac{1}{x})^{10} + (x+\dfrac{1}{x})^{10}$
In the first term, the power of $x$, must be $-2$ in the parenthesis, so the when it's multiplyed by $x^2$ the power of $x$ becomes $0$. And we know that:
$(x+\dfrac{1}{x})^{10}=\sum_{k=0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2235394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Series convergence test $$\sum_{n=1}^{\infty} \frac{1}{(\log{n})^{(\log n)}}$$
I tried using Cauchy's condensation test:
$$\sum_{n=1}^{\infty} 2^n\frac{1}{(\log{2^n})^{(\log 2^n)}}$$
Assume that the log is of base 2:
$$\sum_{n=1}^{\infty} 2^n\frac{1}{n^{n}}$$
$$\sum_{n=1}^{\infty} \left(\frac{2}{n}\right)^n$$
And now I... | The series $\sum_{n=1}^\infty(\frac{2}{n})^n$ converges, since $(\frac{2}{n})^n\leq(\frac{1}{2})^n$ for $n\geq 4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2235477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Jacobi fields are linearly independent if and only if...
If the dimension of the Riemannian manifold $M$ is $n$, there exist exactly $n$ linearly independent Jacobi fields along the geodesic $\gamma : [0,a] \to M$, which are zero at $\gamma(0)$. This follows from the fact, easily checked, that the Jacobi fields $J_1,\... | Here is an alternative proof. We know the explicit form of a Jacobi field $J$ with $J(0)=0$. See Corollary 2.5 of Chapter 5 of Do carmo. Now use this explicit formula with the fact that derivative is a linear map. At $t=0$ they are linearly independent by definition. At other points $t\neq 0$ therefore we can use formu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2235597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to prove that $\sin\left(\frac{\pi}{n}\right)$ is decreasing, $\forall n\in\mathbb{N}$ (series test) How to prove that $\sin\left(\dfrac{\pi}{n}\right)$ is decreasing, $\forall n\in\mathbb{N}$.
My original question was to determine the convergence of $\sum (-1)^{n+1}\sin\left(\dfrac{\pi}{n}\right).$
I showed that t... | One may observe that
$$
x \mapsto \sin x \quad \text{is increasing over} \quad \left[0,\frac \pi2\right]
$$ and that
$$
x \mapsto \frac \pi x \quad \text{is decreasing over} \quad \left[1,\infty\right)
$$ giving that $ \sin \circ \:\frac \pi x$ is decreasing over $\left(2,\infty\right)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2235869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
simultaneous equation, solve for x & y I'm stuck solving what appears to be a simple simultaneous equation. A point in the right direction would be appreciated.
Solve the simultaneous equations for x and y:
$y=x^{2}+7x-11$,
$y=x-1$
my workings:
$0=x^{2}+6x-10$
$10=x^{2}+6x$
$10/x=x+6$
From here i go around in circles ... | Once you have
$x^2+6x-10=0$,
you have a standard quadratic equation.
Solve it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A coin is tossed 3 times. What is the probability of getting 3 heads or at least 1 head?
*
*A coin is tossed $3$ times. Let $A=[3 \text{ heads occur}]$ and $B = [\text{at least 1 head occurs}]$. What is $P(A \cup B)$?
This is a SAT MATH 2 questions from Barron's book. The answer is $\frac{7}{8}$ but I do not... | This is a common mistake. You likely memorized
$$P(A\cup B) = P(A)+P(B)$$
but this is only true if $A$ and $B$ are disjoint: $A\cap B = \varnothing$.
Instead, we have by inclusion-exclusion
$$P(A\cup B) = P(A)+P(B)-P(A\cap B).$$
Also recall that $$P(AB) = P(A)P(B)$$
if $A$ and $B$ are independent.
*
*Notice that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Why the singular values have to appear in descending order across the diagonal matrix? This question is about Singular Value Decomposition.
Given an arbitrary matrix $A$
$$A \in \mathbb{R}^{m\times n} $$
Its reduced SVD form is:-
$$A = U\Sigma V^T$$
Where as $\Sigma$ is the diagonal matrix containing scaling factors ac... | It is just convention. If you change the order, and permute the columns of $U$ and $V$ correspondingly, you do get the same matrix $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is this definition of a complete bipartite graph correct? The book I am using defined a complete graph $K_{m,n} = \overline{K}_m + \overline{K}_n$. Is this correct? I am confused since the complement of a complete graph is an empty graph. How is the addition of two empty graph a complete bipartite graph if we follow th... | Some authors use $G+H$ to indicate the graph join, which is a copy of $G$ and a copy of $H$ together with every edge between $G$ and $H$. This is IMO unfortunate, since $+$ makes more sense as disjoint union. (Authors who use $+$ for join probably use either $G\cup H$ or $G\sqcup H$ for the disjoint union.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Triangle - length of the sides - proof a, b and c are the lengths of the sides of a triangle. Prove that
$$a^2+b^2 \ge \frac{1}{2}c^2$$
Let $\gamma$ be the angle between sides a and b. then:
$$a^2 + b^2 - 2ab\cos(\gamma) = c^2$$
Hence we need to prove that
$$a^2+b^2 \ge \frac{1}{2}c^2$$
$$2a^2+2b^2 \ge a^2+b^2 -... | You're already there. Your last line is
$$a^2+b^2+2ab\cos(\gamma) \geq (a-b)^2$$
And so you only have to note $(a-b)^2\geq 0$ to see that
$$a^2+b^2+2ab\cos(\gamma) \geq 0$$
Note that we can make this inequality strict ($>$) since $\cos(\gamma)=-1$ can't happen, for one angle is then stretched, so we could've taken $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Question on infinite geometric series. Problem- The sum of the first two terms of an infinite geometric series is 18. Also, each term of the series is seven times the sum of all the terms that follow. Find the first term and the common ratio of the series respectively.
My approach- Let $a+ar+ar^2+\dots$ be the serie... |
The sum of the first two terms of an infinite geometric series is 18.
$$
S(a, r) = \sum_{k=0}^\infty a r^k = \frac{a}{1-r} \\
a + a r = 18 \quad (*)
$$
Also, each term of the series is seven times the sum of all the terms
that follow.
$$
a r^n = 7 \sum_{k=n+1}^\infty a r^k \quad (**)
$$
Find the first term and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to determine the optimal control law?
Given the differential equation
$$\dot x = -2x + u$$
determine the optimal control law $u = - kx$ that minimizes the performance index
$$J = \int_0^{\infty} x^2 \, \mathrm d t$$
My approach was to find the state feedback $k$. But since the value of $R$ (positive semidefinite ... | Using the state feedback law $u = - \kappa \, x$, where $\kappa$ is to be determined, we obtain
$$\dot x = -(\kappa + 2) \, x$$
Integrating,
$$x (t) = \exp \left( -(\kappa + 2) t \right) \, x_0$$
where $x_0$ is the initial condition. Hence,
$$\int_0^{\infty} \left( x (t) \right)^2 \, \mathrm d t = \cdots = \dfrac{x_0^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Proof of Lyapunov Matrix Equation I read the book A Linear Systems Primer by Antsaklis, Panos J., and Anthony N. Michel (Vol. 1. Boston: Birkhäuser, 2007) and I am confused by a part of the proof of a theorem about the Lyapunov Matrix Equation.
\begin{equation}
\dot{x}=Ax\tag{4.22}
\end{equation}
\begin{equation} ... | This is a variant of the fundamental theorem of calculus i.e. for an absolutely continuous function $f$ on $[0,T]$ we have for $t \in [0,T]$
$$f(t)=f(0)+\int_0^t \frac{d}{dy} f(y) ~dy$$
So in your case for $f(t)=\phi(t)^T P\phi(t)$ we get
\begin{align} \phi(t)^T P\phi(t)&=\phi(0)^T P\phi(0)+\int_0^t \frac{d}{dy} \phi(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How many binary words of length n are there in which 0 and 1 occur the same number of times and in which no two 0's are adjacent? I understand that, in order to satisfy the first two conditions (length n, same number of 0's and 1's) all that needs to be done is
$$ \frac {n!} { \frac {n} {2}! \times \frac {n} {2}!} $$
b... | Not many. If you have too many consecutive 1s, there will be more 1s than 0s in the word. So we can count them:
*
*If the word starts with a 1, the only option is 1010...1010.
*If the word starts with a 0 and ends with a 1, the only option is 0101...0101.
*If the word starts and ends with a 0, we need to have one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2237025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Finding out the numbers under given conditions Let $M$ be a $2$ digit number $ab$, $N$ is a $3$ digit number $cde$ , and $X=M\times N$ is such that $9(X)=abcde$.The question is to find out the ratio $\frac NM$
I tried to solve it using trial and error and examined a number of cases but couldn't reach the answer so far.... | Rewrite $9(X) = abcde$ as $9MN = abcde = 1000ab + cde = 1000M + N$, then divide by $M$ to get $9N = 1000 + \frac{N}{M}$ or $\frac{N}{M} = 9N - 1000$. Notice then that $\frac{N}{M}$ must be a whole number, call it $k$.
Replace $N$ with $kM$ to get $9kM = 1000 + k$. Since the left hand side is divisible by $k$, the right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2237149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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For which continuous functions $f:\mathbb R\to\mathbb R$ does there exist a discontinuous function $g$ such that $f=g\circ g$? Inspired by a bad approach to a homework problem, I'm wondering for which which continuous functions $f:\mathbb R\to\mathbb R$ does there exist a discontinuous function $g$ such that $f=g\circ ... | For examples of a function $f$ that has no $g$ with $f = g \circ g$ (continuous or not), consider a case where $f$ has exactly one fixed point $p$ and exactly one point $q \ne p$ such that $f(q) = p$. Since $g(p)$ would have to be a fixed point of $f$, we need $g(p) = p$, and since $f(g(q)) = g(f(q)) = p$ we need $g(q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2237321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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homomorphism keeps the unit and commutativity?
Let $R,S$ be a rings and let $\varphi:R\to S$ be a rings homomorphism on $S$. Prove or disprove with counter example:
A. if $R$ is a commutative ring then $S$ is commutative
B. if $R$ has a unit then $S$ has a unit.
Attempt:
A. Take $r_1,r_2\in R$ then $\varphi(r_1\cdot ... | Hints:
*
*For $A$, this only works for elements in the image of $\varphi$. What if $S$ has more elements than are images of $\varphi$? As an example, consider the map
$$
\mathbb{Z}\rightarrow M_{2,2}
$$
the map from the integers to $2\times 2$ matrices where $a\mapsto\begin{bmatrix}a&0\\0&a\end{bmatrix}$.
*For $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2237430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Is $\exp(-1/z^2)$ differentiable at $0$? Let $f: \mathbb{C} \rightarrow \mathbb{C}: \begin{cases} \exp(-1/z^2) & z \neq 0 \\ 0 & z=0 \end{cases}$ be a function. Is $f$ differentiable in $0$?
Suppose $f$ is differentiable in $a$, then $\lim_{z \rightarrow 0} \frac{f(z)-f(0)}{z-0}=\lim_{z \rightarrow 0} \frac{\exp(-1/z^2... | You can try and evaluate the limit where $z \to 0$ along the real line and along the imaginary line. We have
$$ \lim_{x \to 0} \frac{e^{-1/x^2}}{x} = 0 $$
while
$$ \lim_{x \to 0} \frac{e^{-1/(ix)^2}}{ix} = (-i) \cdot \lim_{x \to 0} \frac{e^{1/x^2}}{x} = \infty. $$
The first limit can be calculated using L'Hopital as a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2237535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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The differential equation with a matrix We have the differential equation $ \dot x = Ax$,
$$
A =\begin{pmatrix}
8 & 12 & -2 \\
-3 & -4 & 1 \\
-1 & -2 & 2 \\
\end{pmatrix}
$$
*
*I calculated the determinant $det(A-\lambda E) = - (\lambda - 2)^3$, to find the eigenvalues $\lambd... | As Moo has explained in his comment to your question, the matrix is deficient and can’t be diagonalized, so you have to go to the Jordan decomposition instead. If your textbook is giving you such a problem to solve, I’m sure that this has been covered somewhere in the preceding material. The link Moo provided has a goo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2237644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A Borel-Cantelli lemmas problem
Let $X_1$, $X_2$, ... be independent random variables. Show that sup$X_n$ < ∞ almost surely if and only if $\sum_{n=1}^∞$Pr($X_n$ > A) < ∞ for some A > 0.
Here is my idea:
(Forward direction): Since sup$X_n$ is bounded almost surely, $X_n$ is bounded almost surely for each n. So there ... | ($X_n : \Omega \to \bar{\mathbb{R}}$).
For the other direction, note that
$$
\sum_{n\geq 1}\mathbb{P}(X_n > A)<\infty \implies \mathbb{P}\left(\{w : X_n(w) > A, i.o.\}\right) = 0.
$$
Now, read this as, for almost every $w$ (i.e. except on a set of measure $0$); $X_n(w) > A$ happens finitely many times. For instance, su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2237754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof for any two ordered fields with the least upper bound property are order isomorphic I will write down the definitions first and then what I have done.
Order isomorphic:
$A$ and $B$ are ordered integral domains. They are order isomorphic if $\exists$ a bijection $f: A \to B$ such that
$$f(x+y) = f(x) + f(y)$$
$$f(... | There is another name for an ordered field that satisfies the least upper bound property -- the system of real numbers. So we are trying to construct an order isomorphism between any two systems of real numbers, say $\mathbb{R}_A$ and $\mathbb{R}_B$, in other words, the system of real numbers is unique.
To extend the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2237833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Counterexample of pullback lemma. I'm looking for a counterexample of the pullback lemma, i.e., a diagram
$$
\require{AMScd}
\begin{CD}
A @>{}>> B @>>> C\\
@VVV @VVV @VVV\\
D @>>> E @>>> F
\end{CD}
$$
such that left square and outer square are pullbacks but right square is not a pullback. I have tried hard but I can't ... | Here's a counterexample that works in (almost) any category : let $p:X\to Y$ be a split epimorphism that is not an isomorphism, with section $s:Y\to X$. Then in the diagram
$$\require{AMScd}
\begin{CD}
Y @>{id_Y}>> Y @>{id_Y}>> Y\\
@V{id_Y}VV @VV{s}V @VV{id_Y}V\\
Y @>>{s}> X @>>_p> Y,
\end{CD}$$
the outer rectangle is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2237915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How do you find the sum of this alternating series? $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)(n+1)!}.$$
I found out from my fellow peers at stack exchange see here, that this series converges from the alternating series test. But how do you find the sum? I know if you use wolfram alpha you get: 0.861528, but my question... | Here's a familiar trick for summing such series. This is $f(1)$ where
$$f(x)=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)(n+1)!}.$$
Then
$$f'(x)=\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(n+1)!}$$
which you can write in closed form. Integrate to get $f(x)$ etc.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2238011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Help calculating a math limit Can anyone help with this limit?
\begin{equation*}
\lim_{x \rightarrow 4}
\frac{16\sqrt{x-\sqrt{x}}-3\sqrt{2}x-4\sqrt{2}}{16(x-4)^2}
\end{equation*}
I've tried a variable change of \begin{equation*} y=\sqrt{x} \end{equation*} but this didn't help.
| Hint:
Set $x=4+h\;(h\to 0)$ and apply repeatedly Taylor's formula at order $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2238165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Finding the number of edges $k$-connected graph with certain number of vertices I'm trying to figure out giving a $k$-connected graph how to find the number of edges needed for a certain number of vertices. For example, the number of edges in $2$-connected graph giving that it has $8$ vertices. I guess this is relevant... | You can do this directly if you take a look at a picture. It's easy to see the minimal number of vertices to gain a connected graph is $k-1$ just by chaining them in a straight line. Now each edge you add adds connectivity to both vertices to which it is connected, but it also clearly increases the connectivity to exac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2238309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How good is Polya's "How to Solve it"? I will be going for math major this year, and I am hoping to start this book. But after reading some reviews, they say its mostly for teachers. Can it be used by undergrads? If possible include your brief review of it.
Some other questions in my mind regarding the same book:
*
... | I'm surprised to hear that people say its mostly for teachers. It's one of my go-to recommended books for someone who wants to start learning proofs (example). However, reading a book about thinking can only do so much for you. As I stress in that answer, the only way to become a proficient proof writer is to read and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2238392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Let $G$ be a group. Show that $\forall a, b, c \in G$, the elements $abc, bca, cab$ have the same order.
Let $G$ be a group. Show that $\forall a, b, c \in G$, the elements
$abc, bca, cab$ have the same order.
I thought that my solution ($?$) was enough to show that $abc, bca, cab$ have the same order, but my teach... | In general, conjugate elements have the same order:
$o(ghg^{-1}) = o(h)$
because $ x \mapsto gxg^{-1}$ is an automorphism.
Now note that
$bca = a^{-1} (abc) a$
and
$cab = c (abc) c^{-1}$
are conjugates of $abc$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2238562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Let $A\subset \mathbb{R}^n$ be a convex set. Show that $f:\mathbb{R}^n\to\mathbb{R}$ defined by $f(x) = d(x,A)$ is convex
Let $A\subset \mathbb{R}^n$ be a convex set. Show that
$f:\mathbb{R}^n\to\mathbb{R}$ defined by $f(x) = d(x,A)$ is convex
All I need to prove is that:
$$f((1-t)x+ty) = d((1-t)x+ty, A) \le (1-t)f... | Let $E = \{(x,t) | \exists a \in A \text{ such that } t \ge \|x-a\| \}$. Since $\|\cdot\|$ is convex, it is straightforward to check that $E$ is
convex.
Let $l(x) = \inf_{(x,t) \in E} t $ and notice that $d(x,A) = l(x)$.
Suppose $\lambda \in [0,1]$ and
$(x_k,t_k) \in E$ for $k=1,2$, then we have
$\lambda (x_1,t_1)+(1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2238882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Difficulty in finding interval of convergence with power series I have the following power series:
$$\sum_{n=1}^\infty \frac{(4x+1)^{n}}{n} $$
When finding the interval of convergence, I am left with the following inequality:
$$ |4x+1|\lt1 $$
How do I go about finding the values of $x$ for which this series converges ... | Another approach, differentiating our series term-by-term we find:
$$\frac{\partial }{\partial x}\sum_{n=1}^\infty \frac{(4x+1)^{n}}{n} = \sum_{n=1}^\infty 4\frac{n(4x+1)^{n-1}}{n} = 4\sum_{n=0}^\infty (4x+1)^{n}$$
This is the geometric series for $\cases{\frac{1}{1-r}\\ r = 4x+1}$
which converges on $|r|<1 \Leftrighta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2239007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Geodesics on Cylinders I have a question about Geodesics on Cylinders and think I have the right answer but am unsure. The question reads:
Let $C_r:=[(x,y,z)\in\mathbb{R}^3: x^2+y^2=r]$ be the infinite cylinder of radius $r$. Show that $C_{r_1}$ is isometric to $C_{r_2}$ iff $r_1=r_2$.
Now I understand the logic behi... | Let $S_r$ be the circle of radius $r$ in $\mathbb R^2$. Let $C$ be a curve inside this $S_r$ and have length equals that of $S_r$. Then $S_r\times \mathbb R$ is isometric to $C\times \mathbb R$: Let $i: S_r \to C$ be the unit length parametrization of $C$, then
$$ \phi : S_r\times \mathbb R \to C\times \mathbb R, \ \ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2239167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Why does parenthesis before exponents not apply to squaring a binomial? This must be a stupid question with an obvious answer that is hidden from me. No one I have found even mentions a conflict.
lets say I want to find the square $(2 + 3)^2 = 2^2 + 2\times2\times3 + 3^2$
Why do I not simplify first? Parenthesis first?... | By the rules of precedence, we compute
$$ (2+3)^2 = 5^2 = 25.$$
By the same rules, we compute
$$2^2+2\cdot 2\cdot 3+3^2=4+12+9=25. $$
The very fact that both computations produce the same result justifies us to write down the interesting fact
$$ (2+3)^2=2^2+2\cdot 2\cdot 3+3^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2239238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Parameterizing a circle Suppose I wanted to parameterize $S = \{x^2 + y^2 \leq 1, 0 \leq z \leq 1\}$.
Would this parameterization be given by $G(u,v) = (u \cos(v), u \sin(v), u)$ for $0 \leq u \leq 1$ and $0 \leq v \leq 2\pi$?
The confusion I am having is with regards to $x^2 + y^2 \leq 1$. If this were simply one, it... | In order to parameterize the solid $S$ in $\mathbb{R}^3$, you have to have 3 parameters. In your current parameterization, your $z$ coordinate depends also on the radius of the circle. This would cause it have a cone-like shape. If you wanted to parameterize the solid it would be something like:
$g(u,v,z) = (u\cos{v},u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2239341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
proving unity is unique First The textbook I am using (Contemporary Abstract Algebra) already gave a proof for this, but I am trying a different way. I am unsure how to go about one of the cases or if the proof is correct. I would appreciate your help.
Claim: If $R$ is a ring with unity, it is unique. If $\alpha \in R$... | The equation $\alpha(\beta_1-\beta_2)=0$ does not imply $\alpha=0$ or $\beta_1-\beta_2=0$ in a general ring $R$. (For example, in $Z_6$ we have $2\cdot 3=0$. We say that $2$ and $3$ are zero divisors because they are nonzero elements which divide $0$. Rings with identity and no zero divisors are called integral domains... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2239445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the probability of a functioning circuit
I keep getting that
$$P(\text{Circuit works}) = P(\geq\text{1 of subcircuit 1}) +P(\geq\text{1 of subcircuit 2}) +P(\geq\text{1 of subcircuit 3}) $$
$$= (0.9*0.1^2 + 0.9^2*0.1 + 0.9^3)*(0.95^2 + 0.95*0.05)*(0.99) = 0.770$$
But this is the wrong answer
| I think you have to use the binomial distribution.
$P(\text{A=at least one device functions in the first subcircuit})$
$=1-P(\text{no device function in the first subcircuit})$
So $P(A)=1-\binom 30 0.9^0 0.1^3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2239536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Sum of $X\sim\chi^2_ \nu$ and $Y\sim\chi^2_ k$ and gamma function Suppose $X\sim\chi^2_ \nu$ and $Y\sim\chi^2_ k$ are independent.
(a) Show that $X+Y\sim\chi^2_{\nu+k}.$
(b) Additionally, find the value of
$$\int_0^1u^{\frac{\nu}{2}-1}(1-u)^{\frac{k}{2}-1}du$$
as a ratio of Gamma functions. This formula was discovere... | Suppose the joint distribution of $R,S$ is
$$
\underbrace{\frac 1 {\Gamma(\nu/2)} \left( \frac r 2 \right)^{\nu/2-1} e^{-r/2} \, \frac{dr} 2} \times \underbrace{ \frac 1 {\Gamma(\kappa/2)} \left( \frac s 2 \right)^{\kappa/2} e^{-s/2} \, \frac{ds} 2} \tag 1
$$
so $R,S$ are independent and each has a chi-square distribut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2239670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A question about Taylor expansion Is this statement true? Statement: Let $n$ be a positive integer. Consider the Taylor expansion of $\sqrt[n]{1+x}$ to the $k$th order, that is,
\begin{gather*}
\sqrt[n]{1+x}=\sum_{j=0}^{k}\binom{\frac{1}{n}}{j}x^j+o(x^k), \qquad \text{as $x\to 0$.}
\end{gather*}
Then the Taylor polyno... | This is actually a very good exercise in combinatorics and in multiplying polynomials so I will post an answer.
Firstly note that the following identity holds true:
\begin{equation}
1= \sum\limits_{J=0}^{n k} \delta_{j_1+\cdots+j_n,J}
\end{equation}
Now we insert the unity into the left hand side of (1) and we expand t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2239782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
M/M/1 Queue with new arrivals Let us have a simple FIFO M/M/1 queue. There is an initial poisson arrival stream with arrival rate $\lambda_1$.
Now after some time $t$ there is an additional stream with arrival rate $\lambda_2$, independent of the original stream. How do i analyze the waiting times after this new arriva... | Maybe the multiclass M/G/1 queue model is helpful.
See example slides 50-62.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2239909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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