Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Does the recurring succession $X_{n+1} = 1+\frac{1}{X_n}$ converge? I'm working on some calculus exercises and came across this one. It is familiar yet not the same as the $\mathcal e$ one. The question states:
Given $X_0 = 1$ and $X_{n+1} = 1+\frac{1}{X_n}$, for $n \geq 0$.
Show that it converges and find its limit.
... | It does converge. Here is an sketch of how one can proceed. From definition, it is easy to check that
$$1\leq X_n\leq2$$
The right-hand side for example follows by induction from $1\leq X_n$ for then $X_{n+1}=1+\frac{1}{X_n}\geq2$.
It is easy to check that $X_{2n}\leq X_{2(n+1)}$ and $X_{2(n+1)-1}\leq X_{2n-1}$.
From t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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non-trivial solution of an integral equation The values of $\lambda$ for which the following equation has a non-trivial solution $\phi(x)=\lambda\int_0^{\pi} K(x,t)\phi(t)dt$ where $0\leq x\leq \pi$ and $K(x,t)= \begin{cases}
\sin x\cos t & 0\leq x\leq t \\
\cos x\sin t & t\leq x\leq \pi
\end{cases}
$ a... | Unfortunately there seems to be a mistake in the above. Substituting the kernel in and given it's form, it is easy to prove that:
$$\phi(x)=\lambda\sin x\int_{x}^\pi dt \cos t~\phi (t)+\lambda\cos x\int_{0}^x dt \sin t~\phi (t)$$
Then applying derivatives successively we can also prove that:
$$\phi'(x)=\lambda\cos x\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3252724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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how to justify the logistic function is the inverse of the natural logit function? per wiki
The logistic function is the inverse of the natural logit function
The standard logistic function looks like (equation_1)
$$
{\displaystyle
{\begin{aligned}
f(x)&={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{e^{x}+1}}={\frac {1}{2}}+... | just calculate $f\big(\operatorname{logit}(p)\big)=p$ and $\operatorname{logit}\big(f(x)\big)=x$;
$$f\big(\operatorname{logit}(p)\big) = \frac{1}{1+e^{-\log\left(\frac{p}{1-p}\right)}}=\frac{1}{1+\frac{1-p}{p}}=\frac{1}{\frac{1}{p}}=p$$and$$\operatorname{logit}\big(f(x)\big)=\log\left(\frac{\frac{1}{1+e^{-x}}}{1-\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3252945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Can a conditional statement have multiple premises? Is it correct to say that a conditional statement of the form p ^ q -> r has multiple premises (where the premises are the individual conjuncts p and q) or is that an incorrect use of the term "premise" when referring to implications (as the conjunction itself could b... | To prove a mathematical theorem you assume your premise and do your best to get to the conclusion. Very often, the premise you are assuming that's true contains a lot of claims. If you let each of those claims be a logical letter such as p,q, and so on, you'll get the structure that you're asking, a "multiple premises"... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that for every $v\in\mathbb R^6$ there is a $0\ne w\in\mathbb R^6$ such that $B(v,w)=B(w,w)=0$.
Let $B$ be a nondegenerate symmetric bilinear form on $\mathbb R^6$ with signature $0$. Show that for every $v\in\mathbb R^6$ there is a $0\ne w\in\mathbb R^6$ such that $B(v,w)=B(w,w)=0$.
My attempt:
Since $B$ has ... | Hint Here's a geometric approach that requires proving a short lemma, namely that the punctured null cone $N := \{u \in \Bbb R^6 : B(u, u) = 0\}$ is (path-)connected.
Now, pick any $X \in N$: If $B(v, x) = 0$, we are done; otherwise $B(v, x)$ and $B(v, -x)$ have opposite signs. Since $N$ is (path-)connected, there is s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3253192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is it possible to have 2 different but equal size real number sets that have the same mean and standard deviation? By inspection I notice that
*
*Shifting does not change the standard deviation but change mean.
{1,3,4} has the same standard deviation as {11,13,14} for example.
*Sets with the same (or reversed) sequ... | A simple way to find a counterexample to the conjecture is to focus on sets whose values are symmetrical about zero. This ensures that the two sets have the same mean, and also simplifies calculation of their standard deviations.
Let $a,b$ be any real numbers. Then the set $\{a,b,-a,-b\}$ has mean zero and SD $\sqrt{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
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Counterexample to a Set that cannot be Written as a Cross Product Let $f:\mathbb{R} \to \mathbb{R}$ be a function, and let $S = \{(x,f(x)):x \in \mathbb{R}\}$. Then for any $A \subseteq \mathbb{R}$ and $B \subseteq \mathbb{R}$, $S\neq A \times B$.
My first thought was that this statement was false and I was searching ... | $S$ is of the form $A \times B$ iff $f$ is a constant: suppose $B$ has only one point $b_0$. Then, for any $x$, $(x,f(x)) \in A \times B$ so $f(x)=b_0$ and $f$ is a constant. If $B$ has two distinct points $b_1$ and $b_2$ pick any $a \in A$. Then $(a,b_1)\in A\times B=S$ so $b_1=f(a)$. Similarly, $b_2=f(a)$ so we get a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why $A-I $ invertible implies $ I + A + A^2 + \cdots + A^{k-1} = 0 $ Let $A$ be a square complex matrix such that $1$ is not an eigenvalue of $A$ and $A^k=I_n$ for some positive integer $k$. I want to show that $$
I + A + A^2 + \cdots + A^{k-1} = 0
$$.
I was reading an answer but I do not understand the last step. I un... | Since $(A - I)(I + A + A^2 + \cdots + A^{k-1})=0$ and $A-I$ is invertible, by composing by the inverse $(A-I)^{-1}$ of $A-I$, we have:
$(A-I)^{-1}(A - I)(I + A + A^2 + \cdots + A^{k-1}) = I + A + A^2 + \cdots + A^{k-1}=0$
| {
"language": "en",
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A Poisson process with a fixed maximum number of counts? For a Poisson process, we have the pmf of arrival in (0,t] as:
$$p(n)=\frac{(\lambda t)^n \exp(-\lambda t)}{n!}$$
where $\lambda$ is the arrival rate and the PDF of the inter-arrival times is given as:
$$f(t)=\lambda \exp(-\lambda t)$$
What if I assume that the m... | If I understand your question correctly, then I believe such a distribution is possible. Your original probability mass function was defined by
$$P(n\text{ arrivals in }(0,t])=p(n)=\frac{(\lambda t)^n e^{-\lambda t}}{n!}$$
If you wish to limit the “number of arrivals” to $N$, then the new probability mass function will... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3254135",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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A non-exponential proof of $\lim_{x \to \infty} c^{1/x} = 1$ when $c>1$ My textbook asks me to prove this when we are just beginning to learn limits.
$\lim \limits_{x \to \infty} c^{1/x} = 1, \ \ \ \ \ c>1$
Suppose otherwise,
$$\forall N > 0, \exists x, \epsilon \ | \ \ \ \ \ \ \ x> N \ \ \ \ \
\Rightarrow \ \ \ \ ... | I shall assume that we are taking the limit only over natrual $x$, i.e., we can write it more suggestively as
$$ \lim_{n\to\infty}c^{1/n}.$$
I shall also assume that you define $c^{1/n}$ as the unique positive real number $y$ such that $c=y^n$ (where the latter is just the product of $n$ identical factors).
As $0<y\le1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to use L Hospital's rule for $\lim_{x \to \infty} \sqrt{x} \sin( \frac{1}{x}) $ I want to find the following limit using L Hospital's rule:
$$ \lim_{x \to \infty} \sqrt{x} \sin( \frac{1}{x}) $$
I know that this can be solved using squeezed theorem from Cal 1:
$$ 0 < \sqrt{x}\sin( \frac{1}{x} ) < \frac{1}{x} $$
s... | Consider the Taylor series expansion for $\sin(x)$.
$$\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$$
For $x^{-1}$, this series is
$$\frac{1}{x} - \frac{1}{3! *x^3} + \frac{1}{5! * x^5} - \ldots$$
This is equivalent to $O\Big(\frac{1}{x}\Big)$ (meaning $\sin(\frac{1}{x}) \to \frac{1}{x}$ as $x \to \infty$). So r... | {
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Proving $\int_0^{\pi/4}\sin^n(x)\,dx>\frac{1}{2^{n/2}(n+2)}$ I want to prove:$$\int_0^{\pi/4}\sin^n(x)\,dx>\frac{1}{2^{n/2}(n+2)}$$
This came up when I was working on this
question that only asked for elementary calculus solution. Some trivial lower bounds such as:
$$ \sin(x)\geq\frac{2\sqrt{2}}{\pi}x$$
or even a slig... | I would (optionally) substitute $\sin x=t$ and integrate by parts twice:
\begin{align}\int_0^{1/\sqrt{2}}\frac{t^n\,dt}{(1-t^2)^{1/2}}&=\frac{1}{n+1}\left(\left.\frac{t^{n+1}}{(1-t^2)^{1/2}}\right|_0^{1/\sqrt{2}}-\int_0^{1/\sqrt{2}}\frac{t^{n+2}\,dt}{(1-t^2)^{3/2}}\right)\\&=\frac{1}{n+1}\left(2^{-n/2}-\frac{1}{n+3}\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3254540",
"timestamp": "2023-03-29T00:00:00",
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Integral $\int_{0} ^{3} \dfrac {1} {(x-1)^{2/3}}dx$
Evaluate the Integral $$\int_{0} ^{3} \dfrac {1} {(x-1)^{2/3}}dx$$
I got the answer to be $3(x-1)^{1/3}+C$ by integral techniques, but my friend says that it's false because the integral is improper. I couldn't understand what he meant so can you explain?
| One should be wary that the integrand has a singularity at $x=1$ and thus it is wise to split the integral (call it $I$) between the subdomains $(0,1)$ and $(1,3)$, so we have
$$I = \int_0^1 \frac{1}{(x-1)^{2/3}}dx + \int_1^3 \frac{1}{(x-1)^{2/3}}dx.$$
Since we have just removed a single point, nothing really changes. ... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $x \in \mathbb{R}$ is a limit point of a set $A \subset \mathbb{R}$ if and only if $d(x, A \setminus {x})=0$.
Prove that $x \in \mathbb{R}$ is a limit point of a set $A \subset \mathbb{R}$ if and only if $d(x, A \setminus {x})=0$.
I think I have it right but I would like to have it checked.
We assume that ... | Improvement suggestion, staying close to the definition of infimum (no "letting $r$ get smaller and smaller" vagueness, but definitions):
If $x \in A'$ then let $s=d(x,A\setminus\{x\}) \ge 0$.
Suppose for a contradiction that $s >0$, and consider $B(x,s)$, which is an open ball around $x$, and as $x$ is a limit point ... | {
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Is this language recognizable (Turing machines) $L = \{ \langle M \rangle \mid \text{ M is a TM, M accepts some string of length 3 \}}$ Is this language recognizable? A string is $\Sigma = \{0, 1\}$.
my attempt to prove its recognizable:
let $w_1, w_2, w_3, \dots$ be an effective enumeration of $\Sigma^*$ where $\Sigm... | 1) R = "On input <M>
2) for s = 1 to infinity
3) for each string w of length 3
4) run M on w for s steps
5) if M accepts w within s steps then
6) accept
You are using a "dove tailing" technique.
Note that there are only 8 possible strings of length 3.
So, the loop in line 3 is guaranteed to... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Vector fields on the $5$-sphere What is the maximum number of smooth, linearly-independent vector fields that can exist on the $5$-sphere?
| 1) There certainly exists at least $1$ smooth linearly independent vector field on $S^5$ (i.e. nonzero at each $x\in S^5$), namely $x=(x_1,x_2,x_3,x_4,x_5,x_6)\mapsto (x_2,-x_1,x_4,-x_3,x_6,-x_5)$ .
2) There do not even exist $2$ linearly independent continuous vector fields on $S^5$.
This is proved in Steenrod's The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3255095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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variance of number of isolated vertices in random graph $G(n,p)$ Suppose we have random graph $G(n,p)$ from a uniform distribution with $n$ vertices and independently, each edge present with probability $p$. Calculating it's expected number of isolated vertices proves quite easy, chance of single vertex to be isolated ... | Let $P_{n,k}$ be the probability of exactly $k$ isolated vertices in $G(n,p)$. Look at what happens when we add a new vertex gives:
$$
P_{n+1,k}=q^n P_{n,k-1} + (1-q^{n-k})q^k P_{n,k} + \sum_{i=1}^{n-k}\binom{k+i}{i}p^iq^kP_{n,k+i}
$$
where
*
*$q=1-p$ as usual
*the first term is the new vertex being isolated
*the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3255436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Defining a measure on the space of infinite words over a finite alphabet All words in this question are over the alphabet $\{0,1\}$.
Let $X$ be the set of words of infinite length.
For a finite word $w$, write $w0$ for the word obtained by appending $0$ to the end of $w$. Define $w1$ similary.
For each finite word $w$,... | Yes, there are probability measures on sequence space satisfying your conditions. For instance, if $m_w=2^{-|w|}$, where $|w|$ is the length of $w$. And many many others. Part of the key is the Kolmogorov extension theorem, which in your case results in the $m_w$ for all finite length words inducing a measure on sequ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If compact embedding $H^1_0(\Omega) \hookrightarrow H^{-1}(\Omega)$ is not surjective, how can $H^{-1}(\Omega)$ be the dual of $H^1_0(\Omega)$? Let $\Omega$ be bounded with Lipschitz boundary.
*
*By Rellich compactness, $\iota: H^1_0(\Omega) \hookrightarrow L^2(\Omega)$ is compact embedding. It is also dense.
*By R... | There exists a compact embedding $H^1_0(\Omega)\to H^{-1}(\Omega)$. There also exists an isometric isomorphism $H^1_0(\Omega)\to H^{-1}(\Omega)$. There is nothing contradictory about this, because these are two different maps. (If you view a map from a normed space to its dual as a bilinear form on the space, the fi... | {
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"timestamp": "2023-03-29T00:00:00",
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Solve the following Algebraic Logarithmic inequalities Solve the following logarithmic inequality with all log base $10$. $$\left(\frac12\right)^{(\log x^2)} + 2 > 3\times2^{(-\log(-x))}$$
I have done many logarithmic inequalities but in this i am not able to crack the problem.please give the hint or th... | Hint:
We need $-x>0$.
Let $\displaystyle y=2^{-\log(-x)}$.
Then $\displaystyle \left(\frac12\right)^{\log x^2}=\left(2^{-\log(-x)}\right)^2=y^2$.
The inequality can be written as $\displaystyle y^2 -3y+ 2 > 0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Definitions of exists unique
$\exists!x_0 \in S,P(x_0)$
Definitions:
1.$\exists x_0 \in S, P(x_0)\wedge(\forall x_1,x_2 \in S, P(x_1)\wedge P(x_2)\rightarrow x_1=x_2)$
2.$\exists x_0 \in S, P(x_0)\wedge (\forall x_1 \in S,P(x_1)\rightarrow x_0=x_1)$
$3. \exists x_0 \in S, \ \forall x_1 \in S, (P(x_1) \leftrightarrow x... | Passing from (1) to (2) is trivial, it suffices to let $x_2=x_0$. Now we will start with assuming (2) as true. We have
$$\forall x_1,x_2 \in S \big( P(x_1) \land P(x_2) \Rightarrow x_0=x_1 \land x_0=x_2\big)$$which leads to$$\forall x_1,x_2 \in S \big( P(x_1) \land P(x_2) \Rightarrow x_1=x_2\big)$$and we have just demo... | {
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Calculate $\sum_{n=-\infty}^{\infty}\frac{1-\cos(an)}{(an)^2}$ After playing with some series in a numerical math website, it seems to me like the following identity holds:
$$\sum_{n=-\infty}^{\infty}\frac{1-\cos(an)}{(an)^2}=\frac{\pi}{a}$$
It seems a little bit surprising to me, and I was wondering if there is an ele... | Define
$$f(x)=\sum_{n=-\infty}^{\infty}\frac{1-\cos(nx)}{(nx)^2}$$
for $x\in (0,2\pi]$. We can differentiate this function to get
$$f'(x)=\frac{d}{dx}\left(\sum_{n=-\infty}^{\infty}\frac{1-\cos(nx)}{(nx)^2}\right)=\sum_{n=-\infty}^{\infty}\frac{d}{dx}\left(\frac{1-\cos(nx)}{(nx)^2}\right)$$
$$=\sum_{n=-\infty}^{\infty}... | {
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"timestamp": "2023-03-29T00:00:00",
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Polish spaces are continuous images of the Baire space I'm having some troubles understanding the proof of Theorem 7.9 (pag. 39) in Kechris' "Classical Descriptive Set Theory":
There are two points of the proof proposed that I don't quite understand.
First of all, when we define the Lusin scheme, why do we need to spe... | (ii) is somewhat superfluous. (iii) is already the intended construction: each $F_s$ is partitioned by its successors $F_{s\smallfrown i}$ and these sets are all $F_\sigma$ too. (ii) is just to reinforce and anticipate (iii), I think. (iii) is a statement of intent, not the definition of $F_s$.
The final point is more ... | {
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Variational principal question regarding functions that have a minimum at the origin under a restriction. I'm going over some old assignments from a couple terms ago and have come across a problem from my variational principals module.
I looked at the function in the hint and noticed that for some points $f(x,y)<0$ b... | Recall the definition of a restriction of a function, for instance, from Wikipedia. Let $f$ be a function from a set $E$ to a set $F$. If a set $A$ is a subset of $E$, then the restriction of $f$ to $A$ is the function $f|A:A\to F$ given by $f|_A(x) = f(x)$ for (each) $x$ in $A$.
That is, we have to consider restricti... | {
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Recursive formula for points of algebraic curves over finite fields Given $f$ an algebraic curve, let $N_f(p,k)$ the points of $f$ over $\mathbb{F}_{p^k}$ I need to prove that there exists a recursive formula of order $2g+2$ where $g$ is the genus of $f$. I know that according to Shparlinkski (page 160 from https://www... | Surely you mean
$$N_f(p,k)=p^k+1-\sum_{i=1}^{2g}w_i^k?$$
If $b_1,\ldots,b_m$ and $c_1,\ldots,c_m$ are any numbers and we define
$$a_n=\sum_{k=1}^m b_kc_k^n$$
then the sequence $(a_n)$ satisfies the recurrence
$$a_n=-\sum_{k=1}^m u_ka_{m-k}$$
where
$$\prod_{j=1}^m(X-c_j)=X^m+\sum_{j=1}^n u_jX^{m-j}.$$
In your example, t... | {
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Card picking dependant probability without replacement - P(6,6,Red) I'm trying to teach myself statistics online @ khan academy and although its a wonderful resource I have no one to turn to for help when I don't understand. I hope someone here might be able to give me some advice.
I thought I understood it all and th... | You got it already. Since you are aware that first two picks being red or not affects the probability of third pick being red, you should consider it case by case:
Case 1: First two 6's are black. First, we can find the probability of having two black 6's in our two picks. This is nothing but $\frac{2}{52}\cdot\frac{1}... | {
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Find a matrix of (any) linear map $ \varphi : \mathbb{R}^{4} \rightarrow \mathbb{R}^{3} $, so that conditions apply I'm trying to find a matrix of (any) linear map $ \varphi : \mathbb{R}^{4} \rightarrow \mathbb{R}^{3} $, for which the following conditions apply:
*
*the dimension of the image $ \varphi = 2$
*$ \var... | Let's pretend for a moment that points two and three require that:
*
*$\phi(e_1) = w_1$
*$\phi(e_2) = w_2$
Where $e_i$ are the elements of the canonical basis for $\mathbb R^4$, $w_1 = (1,2,1)$ and $w_2 = (2,1,0)$. An immediate solution for this problem would be the linear function defined by the associations above... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Intersection and distance between two affine subspaces Given are two $m$-dimensional affine subspaces embedded in $\mathbb{R}^n$
$$a_{1i}x_{1i} + a_{2i}x_{2i} + \cdots + a_{ni}x_{ni} = a_{0i}$$
$$b_{1i}x_{1i} + b_{2i}x_{2i} + \cdots + b_{ni}x_{ni} = b_{0i}$$
where $i = 1 ... m$ define the $m$ hyperplanes defining eac... | Let's start with 3D and 2 lines. If they are parallel, normalizing their coefficients (the normal to both) , and dividing by the same factor the known term to the right, then the difference of the normalized known terms (taken in case with the absolute value) is the distance between the lines.
Can you then proceed a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3256872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is the series $\sum_{n=1}^\infty \frac{(i)^n}{n}$ convergent or divergent Does the series
$$\sum_{n=1}^\infty\frac{(i)^n}{n}$$
converge or diverge? The only theorems I've covered thus far are:
Theorem: A series $\sum_{n=1}^\infty (x_n+i y_n)$ converges if and only if $\sum_{n=1}^\infty x_n$ and $\sum_{n=1}^\infty y_n$ ... | We can write
$$\begin{align}
\sum_{n=1}^{2N} \frac{i^n}{n}&=\sum_{n=1}^{2N} \frac{\left(e^{i\pi/2}\right)^n}{n}\\\\
&=\underbrace{\sum_{n=1}^{2N} \frac{\cos(n\pi/2)}{n}}_{\text{All of the odd terms are zero}}+i \underbrace{\sum_{n=1}^{2N} \frac{\sin(n\pi/2)}{n}}_{\text{All of the even terms are zero}}\\\\
&=\sum_{n=1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3256964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Suppose $f(x)\geq 0$, and $\int_0^{+\infty} f^2(x)dx$ is convergent. Prove that $\lim\limits_{x \to \infty}\frac{\int_0^x e^t f(t) dt}{e^x}=0.$ Suppose $f(x)\geq 0$, and $\int_0^{+\infty} f^2(x)dx$ is convergent. Prove that $\lim\limits_{x \to \infty}\dfrac{\int_0^x e^t f(t) dt}{e^x}=0.$
Notice that we are not given th... | A proof from Kavi Rama Murthyk
Since $\displaystyle\int_0^{+\infty}f^2(x)dx$ is convergent，by Cauchy's convergence test, we have
$$\forall \varepsilon>0,\exists \xi>0,\forall x>\xi ~~~s.t.~~~ \int_{\xi}^{x} f^2(t)dt< 2\varepsilon^2.$$
Thus，as per Cauchy-Schwarz's inequality，we obtain
$$\int_{\xi}^x e^t f(t)dt \leq \lef... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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What is the theme of analysis? It is safe to say that every mathematician, at some point in their career, has had some form of exposure to analysis. Quite often, it appears first in the form of an undergraduate course in real analysis. It is there that one is often exposed to a rigorous viewpoint to the techniques of c... | Mathematical analysis is a mental edifice built up to describe and understand phenomena of geometry, physics, and technics in terms of formulas involving finite mathematical expressions. The core of this all is the study of functions $f:\>{\mathbb R}\to{\mathbb R}$ and their properties.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3257398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "56",
"answer_count": 7,
"answer_id": 3
} |
Compute $\lim_{n \to \infty}n\int_{0}^{1}\frac{\cos x}{1+e^{nx}},\ n\in\mathbb{N}$ Let
$$
I_n = n\int_{0}^{1}\frac{\cos x}{1+e^{nx}}\,dx\,,\quad n\epsilon \mathbb N^*.
$$ Calculate $\lim_{n \to \infty}I_n$
| Integrating by parts,
$$
I_n=n\int_0^1\frac{\cos x}{1+e^{nx}}\,dx=-\int_0^1 \cos x\,\frac{d}{dx}\log(1+e^{-nx})\,dx
\\=\log2-\cos(1)\log(1+e^{-n})-\int_0^1\sin x\,\log(1+e^{-nx})\,dx\,.
$$
Now,
$
\cos(1)\log(1+e^{-n})
$
tends to zero as $n\to\infty$ by continuity, while
$$
\int_0^1\left|\sin x\,\log(1+e^{-nx})\right|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3257544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Solve $\frac{f'(x)f'''(x)}{(f''(x))^2}=C$ Let $f$ be twice differentiable almost everywhere and continuous. $C$ is a constant.
Solve for the following differential equation:
$\frac{f'(x)f'''(x)}{(f''(x))^2}=C$ almost everywhere.
If we remove the "almost everywhere", the solution is pretty standard:
$f$ can be a log fun... | Updated answer after the OP changed the wording of his question.
$$\frac{f'(x)f'''(x)}{(f''(x))^2}=C$$
$y(x)=f'(x)$
$$\frac{y''}{y'}=C\frac{y'}{y}$$
$$\ln|y'|=C\ln|y[+\text{constant}$$
$$y'=c_1y^C$$
$$\frac{y'}{y^C}=c_1$$
$$\frac{y^{1-C}}{1-C}=c_1x+c_2\quad\text{in case of}\quad C\neq 1$$
$$y=(1-C)(c_1x+c_2)^{1/(1-C)}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3257782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Uniqueness in the Perron–Frobenius Theorem I'm working through proving the basics of Perron's theorem, but I'm stuck on uniqueness for the positive eigenvector.
Given a positive square matrix $A$, I see how to use Brouwer's fixed point theorem to show the existence of a positive eigenvector $\mathbf{v}$. Call its eigen... | I found a nice solution here.
Let $\mathbf{v}$ and $\mathbf{w}$ be positive eigenvectors of a positive matrix $A$, both associated with eigenvalue $\lambda$. And suppose for reductio that they are linearly independent.
Because they're independent, we can find a constant $c$ so that $\mathbf{v}-c\mathbf{w}$ is nonnegati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3257863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Matrix function from canonical base to base $B$ Let's say I have a function $F=\begin{bmatrix}\frac{1}{2} & 1 & \frac{1}{2}\\ 0 & 1 & 0\\ -\frac{1}{2} & 1 & \frac{3}{2}\end{bmatrix}$ defined with respect to the canonical basis.
I found out that the respective function defined with three variables is: $$f(x,y,z)=(\frac... | Think of $f$ as a function that eats a vector $\mathbf v$ and spits out another vector $f(\mathbf v)$. If we represent $\mathbf v$ by its coordinates relative to the standard basis, which I’ll denote by $[\mathbf v]_{\mathcal E}$, then you have the formula $[f(\mathbf v)]_{\mathcal E} = F[\mathbf v]_{\mathcal E}$: the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Could someone verify my proof that if $f:(a,b) \rightarrow \mathbb{R}$ is uniformly continuous, then $f$ is bounded? For my proof, I had to inverse the definitions for uniform continuity and sequence convergence, so it feels a bit sketchy to me. I'd also appreciate suggestions if you have any.
Let $f:(a,b) \rightarrow ... | Suppose $f$ were unbounded on $(a,b).$ Let $x_1$ be any member of $(a,b).$ For $n\in \Bbb N$ let $x_{n+1}\in (a,b)$ with $|f(x_{n+1})|\ge 1+|f(x_n)|.$
Observe that this implies that $|f(x_m)-f(x_n)|\ge 1$ when $m\ne n.$
Let $(j(n))_{n\in \Bbb N}$ be some sub-sequence of $\Bbb N$ such that $(x_{j(n)})_{n\in \Bbb N}$ con... | {
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"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Why isn't this function affine? I'm studying convex optimization using Convex Optimization (Boyd & Vandenberghe) and had a question from an example used in Chapter 4.2: Convex Optimization.
The specific example is as follows:
$$\begin{array}{ll} \text{minimize} & f_0(x) = x_1^2 + x_2^2\\ \text{subject to} & f_1(x) = \... | The function $h_{1}(x)$ is not an affine function because it can't be written in the form $h_{1}(x)=a^{T}x+b$. Or, if you let $x=(1, 1)$ and $y=(2,2)$, you'll see that $(h_{1}(x)+h_{1}(y))/2=10$, while $h_{1}((x+y)/2)=9$. It's true that these points don't satisfy $h_{1}(x)=0$ and $h_{1}(y)=0$, but that doesn't make an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3258172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Numbers which cannot be formed We are given two numbers $a,b$ such that $a<b$. Now we have a set $\{a,a+1,a+2,\ldots, b\}$ (all number between a and b including them). Then, we have to find how many numbers cannot be formed from the above set. The only operation allowed on the set elements is addition.
Note : We can ... | Let us consider the numbers that can be formed by $\alpha-1$ additions:
\begin{align}
\alpha =1 \text{ gives }& \mathcal{A}_1 =\{a,a+1,\ldots, a+(b-a)\}\\
\alpha =2 \text{ gives } & \mathcal{A}_2 =\{2a,2a+1,\ldots, 2a+2(b-a)\}\\
\alpha =3 \text{ gives } & \mathcal{A}_3 =\{3a,3a+1,\ldots, 3a+3(b-a)\}\\
&\vdots\\
\text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3258301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Existence of an homeomorphic between [0,1] to X × Y I'm doing a practice exam questions and am stuck at this question:
Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?
I'm not even sure how to start tackle it, any help and clues... | You had the right idea. $X$ and $Y$ are the image under the projections of $X\times Y$, so they must be path connected.
Now on $[0,1]$ there are many points that after removing them make this space disconnected. Can this happen with $X\times Y$, assuming both have more than one point?
(Hint: make a picture and try to c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Uniform convergence of: $f_n(x) = \frac{1}{(1+x^2)^n},\text{ with } x\in \mathbb{R}$ Does the function defined by:
$f_n(x) = \frac{1}{(1+x^2)^n},\text{ with } x\in\mathbb{R}$ converge uniformly in $\mathbb{R}$ ?
| Every nonzero point must converge to $0$, but for any $n$, a sufficiently small positive $x$ satisfies $f_n(x) > \frac{1}{2}$. So the function does not uniformly converge.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3258495",
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"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Write the linear application Is given the basis of $\mathbb{R^2}$ $v_1=(1,0) v_2=(1,1)$. Write the only linear application $f: \mathbb{R^2} \to \mathbb{R^2}$ with $f(v_1)=(0,1) f(v_2)=(1,0)$.
I tried to solve it like $(x,y)= x(1,0) + y(1,1)$ and I found $f(x,y) = (y,x)$ that is wrong. Where am I wrong?
| The linear function ("application") will be of the form $$f(v)=Mv$$ for some matrix $M$. You want this to satisfy $f(v_1)=(0,1)^T$ and $f(v_2)=(1,0)^T$. In other words, you want $$M\left(\begin{smallmatrix} 0&1\\1&1\end{smallmatrix}\right)=\left(\begin{smallmatrix} 0&1\\1&0\end{smallmatrix}\right)$$
To find a matrix ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How to approach the proof of this formula for triangulations. I am currently working on an assignment for my discrete mathematics lecture.
It is specifically about graphs which are triangulations and have a minimum degree of 3. Triangulation specifically means here, that every area in the graph is bordered by exactly 3... | If you take all the terms to one side, you get
$$-3v_3-2v_4-v_5+v_7+\cdots+(X-6)v_X+12.$$
This can be expessed as
$$\sum_{i=3}^{X}(i-6)v_i+12,$$
i.e. every vertex $v$ counts for $d(v)-6$ in the sum. Now use the fact that the sum of the degrees is twice the number of edges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3258798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is the use or importance of continuity and differentiability? In a lot of mathematical proofs I often see things like "Assume $f$ is continuous" or "Assume $f$ is differentiable" and sometimes I've even seen both "Assume $f$ is continuous and differentiable", though I believe (differentiability implies/requires co... | A function being differentiable is important in a lot of analysis as a lot of theorems (such as Rolle's theorem, for instance) simply don't hold when a function is not differentiable. It's basically so that we can assume the function is "well-behaved" in a way so that we can apply standard theorems to the function; oth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3258917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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challenging sum $\sum_{k=1}^\infty\frac{H_k^{(2)}}{(2k+1)^2}$ How to prove that
\begin{align}
\sum_{k=1}^\infty\frac{H_k^{(2)}}{(2k+1)^2}=\frac13\ln^42-2\ln^22\zeta(2)+7\ln2\zeta(3)-\frac{121}{16}\zeta(4)+8\operatorname{Li}_4\left(\frac12\right)
\end{align}
where $H_n^{(m)}=1+\frac1{2^m}+\frac1{3^m}+...+\frac1{n^m}$... | Different approach:
\begin{align}
S&=\sum_{n=1}^\infty\frac{H_n^{(2)}}{(2n+1)^2}\\
&=\sum_{n=1}^\infty H_n^{(2)}\int_0^1-x^{2n}\ln x\ dx\\
&=-\int_0^1\ln x\sum_{n=1}^\infty(x^2)^nH_n^{(2)}\\
&=-\int_0^1\frac{\ln x\operatorname{Li}_2(x^2)}{1-x^2}\ dx,\quad \operatorname{Li}_2(x^2)=2\operatorname{Li}_2(x)+2\operatorname{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3259011",
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"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing? Using the Cauchy-Schwarz Inequality, we have
$$
\begin{align}
1
&=\left(\int_n^{n+1}1\,\mathrm{d}x\right)^2\\
&\le\left(\int_n^{n+1}x\,\mathrm{d}x\right)\left(\int_n^{n+1}\frac1x\,\mathrm{d}x\right)\\
&=\left(n+\frac12\right)\log\left(1+\frac1n\right)
\end{align}
$$
which... | Preliminaries: A couple of extensions to Bernoulli's Inequality.
Bernoulli's Inequality says that $(1+x)^n$ is at least as big as the first two terms of its binomial expansion. It turns out, at least for $n\in\mathbb{Z}$, that a sharper inequality can be obtained using any partial sum with an even number of terma.
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3259175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
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An urn contains $12$ balls, of which $7$ are black and $5$ are white. How many ways can we take $6$ balls out of the urn, $2$ of which are white? Uma urna contém $12$ bolas, das quais $7$ são
pretas e $5$, brancas. De quantos modos podemos tirar $6$ bolas
da urna, das quais duas sejam brancas?
The original text is in P... | If the balls are indistinguishable, then there are $1$ ways, because you could take $2$ white balls and $4$ black balls.
If the balls are distinguishable, then you would have $\binom{5}{2}$ ways to choose $2$ white balls and $\binom{7}{4}$ ways to choose the $4$ black balls, leading to $350$ ways total.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3259275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding distribution of $y$ intercept given by line generated by two uniformly chosen points in $[0,1]^2$ The method I'm using seems super inefficient. What I did was define $4$ RVs namely $X_1,X_2,Y_1,Y_2$ and thus my two uniformly random points are $(X_1,Y_1),(X_2,Y_2)$ and hence the $y$ intercept is $-X_1(Y_2-Y_1)/(... | Partial answer: Proofs that (A) $y$-intercept $\in [0,1]$ with probability $\frac12$, and (B) it is uniform within $[0,1]$.
For shorthand, write $E=$ the event that $y$-intercept $\in [0,1]$.
Easy proof of (A): Given any two points $P_1, P_2 \in [0,1]^2$, the line $L$ through them hits a corner of the square with proba... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3259536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Is this equation correct? And if so, is this famous?
$$\int_1^{\infty}x^n e^{-x}dx=\frac{1}{e} \sum_{k=0}^n\frac{n!}{(n-k)!}$$
My proof.1
\begin{align*}
&\int_{1}^{\infty} e^{-\alpha x} \, \mathrm{d}x = \frac{e^{-\alpha}}{\alpha}, \\
&\Rightarrow \qquad \int_{1}^{\infty} (-x)^n e^{-\alpha x} \, \mathrm{d}x = \sum_{k=... | Is it famous? probalby not. Is it known? Yes.
In:
Gradshteyn and Rhyzik, Table of Integrals, Series, and Products
formula 3.351.2 is
$$
\int_u^\infty x^n e^{-\mu x} dx = e^{-u \mu} \sum_{k=0}^n \frac{n!}{k!}\;
\frac{u^k}{\mu^{n-k+1}}
$$
Now if you take $u=\mu=1$ you get your formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3259696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Why is there no solution set for $|x-7|<-4$? I am asked to find the solution set fot $|x-7|<-4$.
I arrived at $(-\infty, 3)\cup(11, \infty)$
For $x - 7 > 0$:
$x-7<-4$
=> $x<3$
For $x-7 < 0$:
$-(x-7)<-4$
=> $-x+7<-4$
=> $-x<-11$
=> $x>11$
So, I arrive at a solution set of:
$(-\infty, 3)\cup(11, \infty)$
However, my... | When the absolute value is on the "less than" side, the conjunction is "and", not "or." You've discovered that $x<3$ AND $x>11$, which is impossible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3259852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Show $\frac{\sin x_1\sin x_2\cdots\sin x_n}{\sin(x_1+x_2)\sin(x_2+x_3)\cdots\sin(x_n+x_1)}\le\frac{\sin^n(\pi/n)}{\sin^n(2\pi/n)}$, for $\sum x_i=\pi$ Let $x_{i}>0$, ($i=1,2,\cdots,n$) and such that
$$x_{1}+x_{2}+\cdots+x_{n}=\pi.$$
Show that
$$
\dfrac{\sin{x_{1}}\sin{x_{2}}\cdots\sin{x_{n}}}{\sin{(x_{1}+x_{2})}\sin{(x... | Just to make it clear, this is not really an answer, more of a failed approach using an obvious idea to try
We will assume $n \geq 3$, since otherwise the statement does not make a lot of sense (for $n = 2$ we would get $x_1 + x_2 = \pi$ so $\sin(x_1+x_2) = 0$ in the denominator, and similarly for $n = 1$).
First note ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3259944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 7,
"answer_id": 1
} |
Order of the eigenvalues of Sturm-Liouville operator Suppose we are working on a bounded interval $[a,b]$ with Sturm-Liouville operator $L$ given by
$$
Lf = \frac{1}{w(x)}\left[-\frac{d}{dx}\left(p(x)\frac{df}{dx}\right)+q(x)f\right].
$$
How can I prove that the eigenvalues of the operator can be ordered as an inc... | The problem needs to be a regular problem on $[a,b]$; otherwise what you say may not be true at all. If the problem is regular, and if you have endpoint conditions
$$
\cos\alpha f(a)+\sin\alpha f'(a) = 0 \\
\cos\beta f(b)+\sin\beta f'(b) = 0,
$$
then you basically get what you want. On... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3260065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find $\int_0^1 \frac{\ln^2x\arctan x}{x(1+x^2)}\ dx$ I came across this integral while i was working on a tough series.
a friend was able to evaluate it giving:
$$\int_0^1 \frac{\ln^2x\arctan x}{x(1+x^2)}\ dx=\frac{\pi^3}{16}\ln2-\frac{7\pi}{64}\zeta(3)-\frac{\pi^4}{96}+\frac1{768}\psi^{(3)}\left(\frac14\right)$$ usi... | Start with breaking the denominator
$$I=\int_0^1 \frac{\ln^2x\arctan x}{x}\ dx-\int_0^1 \frac{x\ln^2x\arctan x}{1+x^2}\ dx$$
For the first integral, use $\arctan x=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{2n+1}$ and for the second integral, use the identity $\frac{\arctan x}{1+x^2}=\frac12\sum_{n=0}^\infty(-1)^n\left(H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3260176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Solving DLP by Baby Step, Giant Step I want to solve the DLP $6\equiv 2^x\pmod {101}$ using Baby Step, Giant Step.
$$$$
I have done the following:
We have that $n=\phi (101)=100$, since $101$ is prime.
$m=\lceil \sqrt{100}\rceil=10$.
For each $j\in \{0,1,\ldots , 9\}$ we calculate $(j,2^j)$:
\begin{align*}j \ \ \... | Using An Introduction to Mathematical Cryptography, J. Hoffstein, J. Pipher, J. H. Silverman, let's use Shank's Babystep, Giantstep Algorithm:
*
*Let $G$ be a group and let $g \in G$ be an element of order $N \ge 2$. The following algorithm solves the discrete logarithm problem in $\mathscr{O}(\sqrt{N} . \log N)$ st... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3260370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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maximum and minimum value of $\frac{x^2+y^2}{x^2+xy+4y^2}$
If $x,y\in\mathbb{R}$ and $x^2+y^2>0.$ Then maximum and minimum value of $\displaystyle \frac{x^2+y^2}{x^2+xy+4y^2}$
Plan
Let $$K=\frac{x^2+y^2}{x^2+xy+4y^2}$$
$$Kx^2+Kxy+4Ky^2=x^2+y^2\Rightarrow (4K-1)y^2+Kxy+(K-1)x^2=0$$
put $y/x=t$ and equation is $(4K-1)t... | From the restriction $x^2+y^2>0$, we get that $x,y$ are not both zero.
If $x=0$, then $K={\large{\frac{1}{4}}}$.
Suppose $x\ne 0$.
Letting $t={\large{\frac{y}{x}}}$, and following your approach, we get
$$(4K-1)t^2+Kt+(K-1)=0$$
which has a real solution for $t$ if and only $K={\large{\frac{1}{4}}}$ or the discriminan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3260474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
I have a doubt regarding the group {4,8,12,16}under multiplication modulo 20. I faced a question show that {4,8,12,16} is a group under multiplication mod 20.It is fine.I have solved the problem also.But I am feeling something strange in it.The idenity element of multiplication modulo 20 should be 1 which is not in the... | Set $S=\{4,8,12,16\}$
The identity element of algebraic structure $(S,\times_{20})$ is $16$.
*
*Is this group closed under modulo multiplication: YES
*Is the operation associative: $(4\times_{20}8)\times_{20}12=12\times_{20}12=\textbf{4}=4\times_{20}(8\times_{20}12)=4\times_{20}16=\bf{4}$
*Is there exists identit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3260615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Why is the deleted comb space still connected? I read from wikipedia that the deleted comb space:
$$
(\{0\} \times \{0,1\}) \bigcup (K \times [0,1]) \bigcup ([0,1] \times \{ 0 \})
$$
has a well known result of being connected.. But how is it connected if there's a separate point $p=<0,1>$? Or is it because since $p$ is... | Clearly, if $Y=\bigl(K\times[0,1]\bigr)\cup\bigl([0,1]\times\{0\}\bigr)$, then $Y$ is path connected, and therefore connected. But your set lies between $Y$ is its closure. Therefore, it is connected too.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3260755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I solve $x^2 \equiv 19 \pmod {59}$. How can I solve $x^2 \equiv 19 \pmod {59}$?
I know that we can just try squaring numbers from 1 to 58 , but this is a very slow method, is not their a quicker one?
| Hint: You Gauss reciprocity theorem:
$$\Big({19\over 59}\Big)\Big({59\over 19}\Big) = (-1)^{{59-1\over 2}\cdot {19-1\over 2}}= -1$$
Since $$59 \equiv 2\pmod{19}$$ and if $$x\equiv 0,\pm1,\pm2,\pm3,\pm4,\pm5,\pm6, \pm7,\pm8, \pm9 \pmod{19}$$ we have $$x^2\equiv 0,1,4,9,-3,6,-2, -8,6, 5 \pmod{19}$$
we see that $$\Big({59... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3260996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
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When is $y=|P(x)|$ is differentiable?
Let $P:\mathbb{R}\to\mathbb{R}$ denote the polynomial function. When is $y=|P(x)|$ differentiable?
I found out that $y=|P(x)|$ may be differentiable through-out $\mathbb{R}$ or it may not be. When it is not differentiable thorough out $\mathbb{R}$, the points of non- differentiab... | I've upvoted Arthur's answer, but I'd like to add a perspective on why Arthur's answer is correct, since none of the answers seem to address the why of it.
Let $$\newcommand\sgn{\operatorname{sgn}}\sgn(x) = \begin{cases} -1 &x<0\\1&x\ge0\end{cases}$$
be the sign function that returns the sign of its argument. Observe t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3261140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Inverse Laplace transformation of Bessel and exponential I'm trying to find the inverse Laplace transform of
$$\mathrm{e}^{-\beta\alpha}J_0(\frac{\beta\alpha}{2})^2$$ where $\alpha=const$. Would an idea be because since I'm looking for the $\alpha\rightarrow\infty$, to calculate the asymptotic form of $J_0(x)$ i.e., $\... | I think your idea works, so asymptotic form of it is
$$\dfrac{2e^{-\alpha\beta}}{\pi\alpha\beta}\left(1+\sin \alpha\beta\right)\tag{1}$$
also $\dfrac{1+\sin \alpha\beta}{\alpha\beta}$ is bounded and
$$\dfrac{e^{-\alpha\beta}}{\alpha\beta}\left(1+\sin \alpha\beta\right)\to0$$
as $\alpha\to\infty$ where $\beta$ is a con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3261250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is this a proper rendering for "$b$ is a power of 2" using TNT from Gödel, Escher, Bach In the book "Gödel, Escher, Bach" by Hofstadter introduces the system of "Typographical Number Theory". One of the exercises is to write 'b is a power of 2', I ended up coming up with the following $<(a \cdot c)=b \supset \exists a'... | Your formula doesn't quite work as written, because every number has $1$ as a factor. For example, $8$ is a power of $2$, but we have $1\cdot 8 = 8$, and $1$ is not even.
But the basic idea is correct. You can express "$b$ is a power of $2$" by "every factor of $b$ except for $1$ is even". I would write this as follow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3261398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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what is the symbol $⊭$ in logic? I know that $⊨$ symbol is entails symbol and $A⊨B$ means that if A is True then B must be True.
But I'm confused about the $⊭$ symbol. which one is it?
*
*$A⊭B$ means if A is true then B is False? = $A⊨¬B$
*$A⊭B$ means the Trueness of A is not any guarantee for B?
*$A⊭B$... | $A \vDash B$ means
For all assignments $v$, if $A$ is true under $v$, then $B$ is true under $v$.
$A \not \vDash B$ simply is the (meta-logical) negation of this statement, that is
Not for all assignments $v$ it is the case that if $A$ is true under $v$, then $B$ is true under $v$
which is equivalent to
There is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3261512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Zeroes of $\sin(z)-z^2$ I am studying for my prelims exam. I stumbled upon the following question.
Show that there are infinitely many zeroes of $\sin(z)-z^2$ in the complex plane.
Had it just been $f(z)=\sin(z)-z$, one can observe that $f(z+2\pi)=f(z)-2\pi$. One can, therefore, see that if $f$ takes zeros only finit... | Questions like this can typically be answered using the Hadamard factorization theorem. The function $f(z)=\sin z-z^2$ has finite order, so if it has finitely many zeroes then its Hadamard factorization has the form $P(z)e^{Q(z)}$ for some polynomials $P$ and $Q$. So, we would have the equation $$P(z)e^{Q(z)}=\sin z-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3261812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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What is the general formula for a sequence that is neither arithmetic nor geometric? If a sequence is such that the next term is given by $a_n*a +b$, what is the general formula of the given sequence?
e.g.: $-$ for $a=4,\;b=6$, the sequence is:
$1,\quad1*4+6=10,\quad10*4+6=46,\quad46*4 +6=190,\;\dots$
What would the fo... | You have recursion $$x_{n+1} = a x_n + b \qquad a \neq 1, \quad b\neq 0$$
Note that $$ x_{n+1} - \frac{b}{1-a} = a x_n + b - \frac{b}{1-a} = a x_n - \frac{ab}{1-a} = a\big(x_{n+1} - \frac{b}{1-a}\big)$$
which means that $y_n = x_n - \frac{b}{1-a} $ is a geometric series:
$$ y_{n+1} = a y_n$$
It can be solved $$ y_n = a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3261910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 5
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Solve $\arctan(x)-\ln(x)=0$ $f:(0, \infty)\rightarrow \mathbb{R}; f(x)=\arctan(x)-\ln(x).$
An interval $I$ in which the equation $f(x)=0$ has a real solution is: a. $(0,1)$; b.$(1,e)$; c.$(e,e^{2})$; d.$(e^{2}, \infty)$;
I tried to solve $f(x)=0$ with Rolle's Theorem but the derrivative has no real solutions.
Also, I t... | For $x>0$, the function is indeed growing. Then
$$f(0)>0,$$
$$f(1)>0,$$
$$f(e)>0,$$
$$f(e^2)<0,$$
$$f(\infty)<0$$ gives you the answer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3262095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Convergence of weak solution for second order parabolic equation (Evans) I am trying to understand how weak convergence works in the theorem of existence of weak solution from Evans (Theorem 3, chapter 7.1), in particular how pass from (30) to (31). Here the theorem with proof
Does (31) follows from (30)? Is it necess... | (31) does follow from (30). If the two sides of (31) are different on a positive measure set of times, then possibly restricting to a smaller set you can assume without loss of generality that $$ \langle \mathbf{u}',v\rangle + B[\mathbf{u},v;t] > \langle\mathbf{f},v\rangle$$ on a positive measure set of times. Then you... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f$ is invertible is $f\left(f^{-1}(x)\right)=x$ and $f^{-1}(f(x))=x$ always True? If $f$ is invertible, is $f\left(f^{-1}(x)\right)=x$ and $f^{-1}(f(x))=x$ always True?
I have doubt in the second one since $$\sin^{-1}(\sin x)=x$$ is not always true.
Any comments on this?
| Yes, it is always true. It turns out that $\sin$ is not invertible. The function which is denoted by $\sin^{-1}$ (or, more generally, by $\arcsin$) is the inverse of the restriction of $\sin$ to $\left[-\frac\pi2,\frac\pi2\right]$. And so$$\left(\forall x\in\left[-\frac\pi2,\frac\pi2\right]\right):\arcsin\bigl(\sin(x)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3262408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why should the equality of mixed partials be "intuitively obvious"? I am reading Ted Shifrin's excellent book Multivariable Mathematics. It claims that the equality of mixed partials is "an intuitively obvious result, but the proof is quite subtle". However, I guess I must be thinking in the wrong way, because I do not... | If you write the difference quotient for a small change $\Delta x$ in $x$ and then the difference quotient for that when you change $y$ by $\Delta y$ the result is the symmetric expression
$$\frac{
f(x + \Delta x, y + \Delta y)
-f(x + \Delta x, y )
-f( x, y + \Delta y)
+f(x,y)
}
{\Delta x \Delta y} .
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3262481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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A question on "unlabeled Cayley graphs" Suppose $G$ is a finite group $S \subset G$. Let's define $UCG(G, S)$ (unlabeled Cayley graph) as a finite unordered simple graph $\Gamma(V, E)$, where $V = G$ and $E = \{(x, y) \in G \times G| x \neq y \text{ and } (y^{-1}x \in S \text{ or } x^{-1}y \in S)\}$.
Unlabelled Cayley... | According to Lauri, Josef; Scapellato, Raffaele (2003), Topics in graph automorphisms and reconstruction, the Petersen graph is not a Cayley graph of any group $G$ and generating set $S$. There are only two groups of order $10$, $\mathbb Z/10\mathbb Z$ and $D_5$. For each, you can look at all generating sets for which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3262634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving $\arctan (-\sqrt3) = \frac{2\pi}{3}$ using basic methods Background
A recent exam I'm looking at, has the following trigonometric equation:
$$\sin(\pi x) + \sqrt3 \cos(\pi x) = 0, \quad x \in [0, 2]$$
After a simple re-write, we get $$\tan(\pi x) = -\sqrt 3$$
Note, on this exam, the student doesn't have any t... | I agree with one of the comments to your question that it’s worth memorizing some of the basic triangles and the associated sines and cosines. However, a 30-60-90 triangle can be quickly reverse-engineered from $\tan\theta = -\sqrt3$: we have $$\frac yx = -\sqrt3 \\ x^2+y^2=1$$ from which $4x^2=1$, so $x=\pm\frac12$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3262813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Linear program with $\pm 1$ constraints I am trying to formulate a constraint as follows ($X, Y, Z$ are either $-1$ or $1$):
If $Z$ and $Y$ both equal $-1$, then $X$ must be $1$. But, if either $Z$ or $Y$ are not $-1$, then $X$ can be $-1$ or $1$.
I came up with $Z\cdot Y \le X$, but that limits $X$ to $1$ if both $Z$... | Note: If you're trying to keep a linear program, do not add constraints of the form ZY
My solution: $-3(Y+Z)\leq 5+X$
I am not sure if a better solution exists, however!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3262918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability Dilemma The teacher gave us a question:
We flip a coin. If it was Heads, we roll a die and if it was Tails, we flip three other coins. What's the probability of exactly having one coin as Heads?
I first calculated $n(S)= 1
\cdot 6 + 1 \cdot 2 \cdot 2 \cdot 2$, then $n(a) = 1 \cdot 6 + 1 \cdot 1 \cdot 1 ... | If the first flip is head, then the die won't produce coins showing head, hence we will have exactly one head. (You could just leave the die alone).
If the first flip is tails, then there are exactly three out of eight outcomes from the three extra coins that result in exactly one head.
Hence
$$p=\frac12\cdot 1+\frac1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3263051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Cartesian product of $S^1$ and $S^2$ Can anyone help me imagine $S^1 \times S^2$? I understand that $T^2=S^1 \times S^1$, but I don't know what to do with the spheres. I am not even sure if it's in 3D. Thank you!
| This is a good example of a 3-dimensional manifold that cannot be embedded into the Euclidean space $\mathbb R^3$.
The best way you can think of it is as a circle such that it has a sphere associated to each and every point of it.
I bet you already know how to parameterise both the circle $S^1$ (call the parameter $\rh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3263299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the number of routes from leftmost vertex to rightmost vertex
Suppose that one can move from one vertex to another if, and only if,
the two vertices are connected by a unique common edge. The number of
routes that one can take from the leftmost vertex L through 6 edges
and 5 intermediate vertices to the ri... | Try labeling the vertices and writing down the resulting adjacency matrix. Then raise it to the power of 6, and read the entry in the (i, j) position, where i and j are the labels of the left- and right- most vertices, respectively. Since the adjacency matrix is 14 x 14, you should probably use a computer. I recommend ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3263441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Range of a function $|\sin x|+|\cos x|$ What is the range of function $Y=[|\sin x|+|\cos x|]$ where $[ ]$ denotes the greatest integer function.
And what is range of function $Y=|\sin x|+|\cos x|$
For the second one,
I have tried squaring :
I got $Y^{2}=1+|\sin 2x|$
and range will be between $1$ and $\sqrt{2}$
Please... | The most intuitive way to proceed might be to sketch the shape of the level sets
$$ |x|+|y| = 0 \\
|x|+|y| = 1 \\
|x|+|y| = 1.5\\
|x|+|y| = 2 $$
and so forth. Which of them have points in common with the unit circle?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3263568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How to compute $\sum_{k=1}^{\infty}{\frac{1}{k^2+2k}}$? To whom this may concern,
i am struggling with partial sum formulas. I don't really get why you would need to perform a partial fraction decomposition or how you know that you have to.
I started by getting trying to get a grasp of the series:$$\sum_{k=1}^{n}{\frac... | Note that $\frac 1 {k^{2}+2k} =\frac 1 2(\frac 1 k -\frac 1 {k+2})$. In $(1+\frac 1 2+...+\frac 1 n) -(\frac 1 3+...+\frac 1 {n+2})$ all terms except the first two in the first term and the last two terms in the second term cancel. Can you now compute the partial sum. The answer given in WolframAlfa is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3263815",
"timestamp": "2023-03-29T00:00:00",
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Proof about monotonicity of functions
Let $f:[a,b]\rightarrow\mathbf{R}$
Suppose $f'(x)>0$ for all x $\in (a,b)$
$\ \ \ \ \ \ \ $and $f$ is differentiable on $(a,b)$
$\ \ \ \ \ \ \ $and $f$ is continuous on $[a,b]$
Show $f$ is strictly increasing on $[a,b]$.
What i tried so far:
Let $x_1,x_2 \in (a,b)$,
Assume $x_1<... | Your proof is finished with the the line that begins "With $\;f'(c)>0\;$ ..."! Nevertheless, I would write it down as follows:
Take any two different points $\;x_1,\,x_2\in [a,b]\,,\,\,x_1<x_2\;$ ( of course, it can be $\;x_1=a\;$ or $\;x_2=b\;$). Since the conditions of the MVT are fulfilled in $\;[x_1,x_2]\;$ , there... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3263956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculate $\lim\limits_{x\rightarrow 0^+} \int\limits_0^1 \ln(1+\sin(tx))dt$
Calculate $$\lim_{x\rightarrow 0^+}\int_0^1\ln(1+\sin(tx))dt$$
My try:
$$\lim_{x\rightarrow 0^+} \int_0^1 \ln (1+ \sin (tx)) dt=\lim_{x\rightarrow 0^+} ([t \ln (1+\sin (tx))]^1_0 - \int_0^1 \frac{t \cos (tx) x}{1+ \sin (tx)} dt)$$
Then I wan... | It is not necessary to compute the integral explicitly. The following estimate is sufficient to determine the limit:
If $0 \le x \le \pi$ then $\sin(tx) \ge 0$ for all $t \in [0, 1]$, so that
$$
0 \le \ln (1+ \sin (tx)) \le \sin(tx) \le tx
$$
and therefore
$$
0 \le \int_0^1 \ln (1+ \sin (tx)) dt \le \frac x2 \, .
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3264081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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The min of $x + k/x$ $f(x) = x + \dfrac{k}{x} $ ($k > 0$ and $x > 0$)
given that the k is a constant, how do I solve the x that makes f(x) minimum?
| To avoid differentiating, notice that
$$x+\frac{k}{x}=\bigg(\sqrt{x}-\sqrt{\frac{k}{x}}\bigg)^2+2\sqrt{k}$$
and that $\sqrt{x}=\sqrt{k/x}$ when $x=\sqrt{k}$.
(Note that we can use $\sqrt{x}$, since it is given that $x\gt 0$)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3264200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Proving binomial sum equals $0$ My hypothesis is that, when $n \equiv 0 \mod 6$, then
$$\sum_{k=0}^{n/3-1} \Bigg( \binom{n-2}{3k+1} 2^{3k+1}-\binom{n-2}{3k-1}2^{3k-1} \Bigg) = 0, \quad \binom{n-2}{3k-1} = 0 \text{ when } k=0$$
But I get stuck at finding a proof. I have tried induction, but that does not seem to work be... | Hint: Let $$f_n(x)=\sum_{j=0}^n\binom{n}{j}x^j=(1+x)^n$$ and consider $f_n(2e^{2\pi i/3})-f_n(2e^{-2\pi i/3})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3264293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Mathematics in Other Languages The intention of this community wiki is to aid in learning another language in the restricted topic of mathematics. It does matter whether one wants to learn another language for mathematics or use mathematics as a stepping stone into another language. This post focuses only on the writte... | French
This is one of the easier languages to learn as a native English speaker. Many of its words are recognizable by most English speakers, and its grammar is similar to the grammar of English. It also has one of the highest payoffs. Currently, more mathematical works are being published in French than any other lang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3264421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 6,
"answer_id": 3
} |
How to calculate the number of elements in all subsets of a set There are $2^N$ subsets of a set.
For instance the set $\{ 1, 2 \}$ has the following subsets:
$\{ \}$
$\{ 2 \}$
$\{ 1 \}$
$\{ 1, 2 \}$
I'm trying to calculate the total number of elements in all of the subsets. In the above example, it would be $4$:
$\{ 2... | Let $A=\{a_1,a_2,\dots,a_N\}$. For $i\in\{1,2,\dots,N\}$ and subset $B$ of $A$. We either have $a_i\in B$ or $a_i\notin B$. There are two possibilities for each $i$. So there are totally $2^N$ possibilities. $A$ has $2^N$ subsets.
For each $a_i$, there are $2^N\div2=2^{N-1}$ subsets of $A$ containing it. As there are $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3264585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Explain why a line can never intersect a plane in exactly two points. Why can a line never intersect a plane in exactly two points?
I know this seems like a really simple question, but I'm having a hard time figuring out how to answer it. I also tried googling the question but I couldn't find an answer for exactly what... | A plane is a convex set. A convex (i.e. linear) combination of any two points in a convex set will be inside the set. So if you have two distinct points in the plane, you automatically have infinite points inside the plane.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3264677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 10,
"answer_id": 3
} |
Evaluate a power series by relating it to a geometric series (where coefficients depend on the index of summation). I am trying to evaluate the power series $f\left(z\right)=\sum_{n\geq0}n^{2}z^{n}$ by relating it to a geometric series. By the ratio test, the series converges (absolutely) for $\left|z\right|<1$.
I assu... | Start with the fact that
$$\frac{1}{1-x} = \sum_{n\geq 0} x^n$$
(This is the simplest formula for an infinite geometric series)
Differentiate to obtain
$$\frac{1}{(1-x)^2} = \sum_{n\geq 0} nx^{n-1}$$
and multiply by $x$:
$$\frac{x}{(1-x)^2} = \sum_{n\geq 0} nx^{n}$$
Repeat the top two steps and you get a closed formula... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3264870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A question regarding the group $G = GL_2(\mathbb Z/ p\mathbb Z)$. For any prime $p$ , consider the group $G =\mathrm{ GL}_2(\mathbb Z/ p\mathbb Z)$. Then which of the following are true?
1) $G$ has an element of order $p$.
2) $G$ has exactly one element of order $p$.
3) $G$ has no p-Sylow subgroups.
4) Every element of... | The following is a description of the group $G =\mathrm{ GL}_2(\mathbb Z/ p\mathbb Z)$ that answers all your questions:
1) The order $n$ of $G$ is $n=p(p+1)(p-1)^2$. Indeed , for the first row we have the choice between $p^2-1$ possibilities. For the second row we have the choice of $p^2$ vectors minus the $p$ vectors ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3264983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculate local maximum and minimum of $f(x)=(2+\sin x)(5-\sin x)$ I am working on my scholarship exam practice which assumes high school or pre-university math knowledge. Could you please have a look on my approach?
The minimum of the function $f(x)=(2+\sin x)(5-\sin x)$ is ......
First, I began with some basic appr... | $$f(x)=(2+\sin x)(5-\sin x) = -\sin^2 x + 3\sin x + 10$$
this function has $D<0$. And obviously minimum when $\sin x$ is $-1$ and max when $\sin x$ is $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3265097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Convert system of second order PDEs to a system of first order PDEs I am attempting so convert
$$H_t = D_H H_{xx} - vH_x - \alpha H Z$$
$$Z_t = D_Z Z_{xx} + \beta H Z $$
into a system of first order equations. $D_H$, $D_Z$, $\alpha$ and $v$ are constants and $H$ and $Z$ describe the population of humans and zombies whi... | The basic idea is to introduce new variables $X = H_x$, $Y = Z_x$, that gives directly:
$$
H_t = D_H X_x - v H_x - \alpha H Z \\
X = H_x \\
Z_t = D_Z Y_x + \beta H Z \\
Y = Z_x
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3265231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $p$, $q$, $r$ and $s$ are four sides of a quadrilateral, then find the minimum value of $\frac{p^2+ q^2 + r^2}{s^2}$ with logic If $p$, $q$, $r$ and $s$ are four sides of a quadrilateral then find the minimum value of $\frac{p^2+ q^2 + r^2}{s^2}$ with logic.
Please help me with this.
| By the triangle inequality and by C-S we obtain:
$$\frac{p^2+q^2+r^2}{s^2}>\frac{p^2+q^2+r^2}{(p+q+r)^2}=\frac{(1+1+1)(p^2+q^2+r^2)}{3(p+q+r)^2}\geq\frac{(p+q+r)^2}{3(p+q+r)^2}=\frac{1}{3}.$$
The equality does not occur, but easy to see that $\frac{1}{3}$ is an infimum.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3265362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Does $\frac{z^2}{(z^2-1)^{\frac{1}{2}}}$ have a simple pole at $\infty$? If we define $w=\frac{1}{z}$ and Laurent expand the function about $w=0$ we have:
$$\frac{1}{w(1-w^2)^{\frac{1}{2}}}=\frac{1}{w}+\frac{1}{2}w+\frac{3}{8}w^3+...$$
This implies that there is a simple pole about $w=0$, because the inverse powers of ... | I think your error here is where you expanded the function. Notice that
$$g(z)=\frac{z^2}{(z^2-1)^{\frac{1}{2}}}$$
has no singularity at zero, but it does have a singularity at $z=-1$ and $z=1$. You cannot use the Laurent series about $z=0$ of
$$f(z)=\frac{1}{({z^2-1})^{\frac{1}{2}}}$$
to evaluate something as $z\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3265470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Can you help with this integration problem? Integration of $\displaystyle \int \dfrac{x}{x^2-\sqrt{x}} dx.$
I attempted substituting $\sqrt{x}$ with $u$ and then solving it using partial fractions but I'm not getting the right answer. I know I could used conjugates but I want to understand why the way I solved it at fi... | You were by the right way, let me continue your work, if
$$I=\displaystyle \int \dfrac{x}{x^2-\sqrt{x}} dx.$$
First you make $u=\sqrt{x}$, then $du=\dfrac{1}{2\sqrt{x}}dx$, so
$$I=2\displaystyle \int \dfrac{u^2}{u^3-1} du.$$
Then, you make the substitution $s=u^3-1$, so $ds=3u^2du$ and
$$I=\frac{2}{3}\displaystyle\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3265713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Quadratic inequality with a variable consisting of more than one unknown The initial inequality was
$$2^{2ax+1}+2^a ≤ 2^{ax}+2^{ax+a+1},$$
after substituting $2^{ax}$ with $y$ (so that $y>0$), and transferring all the terms from the RHS to the left I got
$$2\cdot y^2 − y\cdot({2^{a+1} +1)} + 2^a ≤ 0.$$
I am stuck the... | Setting $y=2^{ax}$ and a rearranging a bit yields
$$2y^2-y\leq 2^{a+1}y-2^a,$$
and both sides factor nicely, giving
$$y(2y-1)\leq2^a(2y-1).$$
Can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3265819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Improper Lebesgue integral $\int_{\mathbb R}\frac{\sin(x)}x~\mathrm dx$ While thinking over the idea of an improper Lebesgue integral, I came up with the following:
$$\int_0^\infty\mu\{x~|~f(x)>t\}-\mu\{x~|~f(x)<-t\}~\mathrm dt$$
with $\mu$ being the Lebesgue measure.
And I wanted to tackle this integral:
$$\int_{\math... | Partial answer:
We can attempt to compare the difference of the improper Lebesgue integral and the improper Riemann integral as follows:
\begin{align}R(n)&=\int_{-2N\pi}^{2N\pi}\frac{\sin(x)}x~\mathrm dx-\int_{\mathbb R}f_{1/2N\pi}(x)~\mathrm dx\\&=2\int_0^{2N\pi}\min\left\{\frac1{2N\pi},\left|\frac{\sin(x)}x\right|\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3265954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Limit of sequence using taylor's formula on trigonometric functions I had the following question in an exam. I wasn't able to solve the problem and I don't understand the solution.
I am supposed to find the limit of the following sequence:
$$ \lim_{x\to 1} \frac{\ln(x) - \sin(\pi x)}{\sqrt{x -1}} $$
where x > 1.
Now... | Since $\ln(x+1)=x+O(x^2)$, $\ln(x)=\ln\bigl((x-1)+1\bigr)=x-1+O\bigl((x-1)^2\bigr)$. And since $\sin(x)=x+O(x^2)$,\begin{align}\sin(\pi x)&=-\sin(\pi x-\pi)\\&=-\sin\bigl(\pi(x-1)\bigr)\\&=-\pi(x-1)+O\bigl((x-1)^2\bigr).\end{align}So\begin{align}\frac{\ln(x)-\sin(\pi x)}{\sqrt{x-1}}&=\sqrt{x-1}\frac{\ln(x)-\sin(\pi x)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3266133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Hahn-Banach extension for positive functionals Let $H$ be a Banach space and $C\subset H$ be a convex cone (I'm thinking of $H=L^2(U)$ or $C(U)$ and $C=\{\text{positive functions on }U\}$ for some open $U\subset\mathbb R^d$).
Moreover, let $H'\subset H$ a subspace and $\phi:H'\to\mathbb R$ be a functional which is
*
... | This is not true in every infinite-dimensional Banach space.
Let $x^*$ be a linear functional on $H$ which is not continuous.
We choose $C=\{x\in H : x^*(x)\ge 0\}$.
Let $x\in C\subset H$ be such that $x^*(x)=1$.
For $H'$ we choose a one-dimensional subspace generated by $x$.
We then choose $\phi$ as the restriction of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3266259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How is this induction argument here valid I have this proof of 'existence of splitting fields' from Gallian's Contemporary Abstract Algebra (I have seen it elsewhere as well, and it goes in similar lines):
Let $F$ be a field and let $f(x)$ be a non-constant element of $F[x]$. Then there exists a splitting field $E$ fo... | Elaborating on Lee Mosher's comment, consider the following assertion, $P(n)$, about a natural number $n$:
$P(n)$: Let $F$ be a field and let $f(x)$ be a non-constant element of $F[x]$ of degree at most $n$. Then there exists a splitting field $E$ for $f(x)$ over $F$.
The statement you want to prove can be written as $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3266530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
What is the solution to this equation: $e^x+2x=0$ I cant find any program that actually solves this type of equations and I cant find anything helpful about this type. What is the name of these equations and how do I solve this one? Thanks.
| Such an equation does not have an easy formula or a "closed form" like quadratic formula for example. If you only care about the answer, you can try to plot it as in Plot using Desmos .It turns out that it has a real root at approximately: $x=-0.352$.
Alternatively you could try a "numerical method" such as Newtons's i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3266785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Calibrations vs. Riemannian holonomy I've began to study the relationship between calibrations and holonomy, mainly through D.D. Joyce's Riemannian Holonomy Groups and Calibrated Geometry and partly through internet material.
Pretty much everyone explains this relationship by the holonomy principle: if $H=\text{Hol}_x... | (1) A generic metric has (restricted) holonomy group $SO(n)$ (More proper statement: the set of holonomy $SO(n)$ metrics is comeagre in the space of all Riemannian metrics). Hence the adjective special is coined (as in the opposite of "generic") when we can reduce it to smaller subgroups. It definitely predates Joyce... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3266923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Linear maps take 0 to 0 Suppose $T$ is a linear map from $V$ to $W$ where $V$ and $W$ are subspaces of a finite-dimensional vector space over some generic field $\mathcal{F}$.
Show $T(0)=0$.
Proof:
By additivity of a linear map,
(1) $T(0)=T(0+0)=T(0)+T(0)$.
Since $T(0)\in W$ and $W$ is a subspace, there exists an addit... | Well, given two groups $(G,\cdot,e)$ and $(G',\circ,e')$ (think of the additive groups of vector spaces) and a homomorphism $\phi:G\rightarrow G'$, i.e., $\phi$ is a mapping with
$\phi(g\cdot h) =\phi(g)\circ \phi(h)$.
Then $e'\circ \phi(e) = \phi(e) = \phi(e\cdot e) = \phi(e)\circ \phi(e)$. By multiplying with $\phi(e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3267233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conformal circles (A property of inversions) I came across the following proposition in this book:
If C is a circle in the Euclidean plane, iC is conformal, that is it preserves angles. Also, iC takes circles not containing the center of C to circles, circles containing the center to lines, lines not containing the ce... | A holomorphic mappingg $\mathbb{\hat{C}} \rightarrow \mathbb{\hat{C}}$ that has a holomorphic inverse is called a conformal mapping of $\mathbb{\hat{C}}$. It turns out that every conformal mapping of $\mathbb{\hat{C}}$ is a Möbius transformation.
The proposition you ask about asserts that a conformal map will send the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3267384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Why can an algebraic extension of a local field be written as a tower of finite extensions? Let $K$ be a local field and $L$ an algebraic extension of $K$.
I found a statement that $L$ can be written as
$$
L = \bigcup_{n=0}^\infty L_n,
$$
where $L_n$ are finite extensions of $K$ with $L_n \subseteq L_{n+1}$.
I could ... | $K$ is a completion of a global field $K'$. Every finite extension $E$ of $K$ can be obtained as a completion of a finite extension $E'$ of $K'$. This implies that there are only countably many finite extensions of $K$, as $K'$ and hence $K'[x]$ are countable. Every algebraic extension $L$ of $K$ is the directed union ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3267563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that there is no isometry from $\mathbb R^3$ to $\mathbb R^2$ How should I start of? I know if $T$ is an isometry then $T=A \vec x+\vec b$ where A is orthogonal matrix and $\vec b$ is a column vector. If I can prove that there is no orthogonal matrix from $\mathbb{R^3}$ to $\mathbb{R^2}$ then will it suffice?(whi... | Consider a tetrahedron in $\mathbb R^3$. Its vertices are four points equidistant from each other.
An isometry with $\mathbb R^2$ would create such a set in $\mathbb R^2$, but this is clearly impossible.
One way to see is that since distinct circles in $\mathbb R^2$ only intersect in at most two locations. If you pic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3267713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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