Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Why is the graph of $f(x) = \frac{x^3}{x^2 - 1}$ approaching negative infinity below $x = -1$? I'm studying calculus at elementary level .
While drawing a graph, I had little confusing thing.
The function is $f(x) = \dfrac{x^3}{x^2-1}$ ($x$ is real number)
and I did this in wolfram alpha ,
and I wonder why the graph... | I see you already got it, but for the benefit of others who may come here with the same question, and just to be thorough, the reason is that since $f(x) = \dfrac{x^3}{x^2-1}$, then we can do polynomial long division (divide $x^3$ by $x^2-1$) to see that $y=x$ is the oblique asymptote of $f(x)$. Since asymptotes descr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Dual spaces and weak solutions. I have two questions:
(1) $H^{-1}$ space is defined as the dual space of $H_{0}^{1}$, so is the dual space of $H^{1}$ also $H^{-1}$? Or is it correct to act an $T\in H^{-1}$ on a function $u\in H^{1}$?
(2) For the Poisson problem $-\Delta u=f$ with boundary condition $u=g$ for proper con... | *
*$H^1_0$ is contained in $H^1$ and has the same norm, so $H^{-1}$ must be contained in the dual of $H^1$. It turns out that it is a proper subset of the actual dual of $H^1$.
*Sure, this is usually the definition of the weak solution (though we have to interpret "$w=g$ on the boundary" using the trace operator). In... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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M. Ross problem 12 chapter 5 - Exponential distribution I have a question regarding problem 12(b) and (c) of chapter 5 of M.Ross "Introduction to probability models".
The question is as follows:
If $X_1, X_2, X_3$ are independent exponential random variables with rates $\lambda_i$, $i = 1,2,3$, find
(b) $P(X_1 < X_2 \... | (b) The following might help to understand your mistake.
Let $U_{1},U_{2},U_{3}$ be independent random variables where $P\left(U_{1}=1\right)=P\left(U_{1}=3\right)=\frac{1}{2}$
and $P\left(U_{2}=2\right)=1=P\left(U_{3}=2\right)$.
Then it is evident that: $$P\left(U_{1}<U_{2}\mid\max\left(U_{1},U_{2},U_{3}\right)=U_{3}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the Matrix $A^{482}$ in terms of $A$ Given $$A=\begin{bmatrix}
-4 & 3\\
-7 & 5
\end{bmatrix}$$ Find $A^{482}$ in terms of $A$
I tried using Characteristic equation of $A$ which is $$|\lambda I-A|=0$$ which gives
$$A^2=A-I$$ so $$A^4=A^2A^2=(A-I)^2=A^2-2A+I=-A$$ so
$$A^4=-A$$ but $482$ is neither multiple of $4$ n... | If $A^4=-A$, then $$A^{482}=A^2A^{480}=A^2(A^4)^{120}=A^2(-A)^{120}=A^2(A^4)^{30}=A^2(-A)^{30}=A^{32}=(A^4)^8=(-A)^8=(A^4)^2=(-A)^2=A^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Maximum and minimum distance of two points Consider six distinct points in a plane. Let $m$ and $M$ denote respectively the minimum and the maximum distance between any pair of points.
Show that $M/m \geqslant \sqrt{3}$.
| Assume otherwise.
Pick two points $A,B$ at distance $M$ (The small dots in the following image)
Then all other points are in the closed lens-shaped area bounded by the circles of radius $M$ around $A$ and $B$ (big circles in above image). In particular, they must be in the ten blue triangles shown.
As the blue trian... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1744983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Intersection of a line and line at infinity in projective space I understand parallel lines in Euclidean space intersect at the line at infinity in terms of projective space.
My question is for a single line. A single line if extended to infinity must intersect the line at infinity at some point (correct me if this wr... | Each line has just one point at infinity, which is approached by going in either direction along the line. Two lines share the same point at infinity if and only if they are parallel to each other. Two lines not parallel to each other have different points at infinity.
When one adds to the affine line a point at infi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1745225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Why is the Laplacian of $1/r$ a Dirac delta? How does one show that $\nabla^2 1/r$ (in spherical coords) is the Dirac delta function ? Intuitively, it would seem that the function undefined at the origin and I'm not able to construct a limiting argument that avoids this problem.
| Delta function is zero everywhere except at origin, and integration over space is zero. The first property is easy to prove with vector identities. For the second property:
$$
I = \iiint_V \nabla^2 \frac{1}{r} d V = \iiint_V \nabla \cdot \nabla \frac{1}{r} dV
$$
With divergence theorem:
$$
I = \oint_S \nabla \frac{1}{r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$
Show that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ and find all $w\in \mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ such that $\mathbb{Q}(w)=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$.
It is clear that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5... | IMO, Galois theory is the way to go here, but I want to give you an idea of how to do this in an elementary way. The key idea below is that squaring should "eliminate" some of the $\sqrt{2}$, but won't really get rid of any of the $\sqrt[3]{5}$ (I will use $\alpha=\sqrt[3]{5}$ for the rest of this answer).
If we squar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1745628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 4
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Prove the intersection of events to be 1 $B_n$, $n$ from $1$ to infinity is countably infinite sequence of events and each event has probability $1$. How do I formally prove that the probability of intersection of $B_n$ from $n = 1$ to infinity is also $1$. Intuitively I know because $P[B_i] = 1$, so all events are equ... | Hint: The probability of at least one of the events not happening is at most the sum of the probabilities of each event not happening, which is zero because there are only countably many of them each with probability zero of not happening.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1745841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Limit of $f(x)=|\log x|$ My textbook solved this problem:
Find $f'(1^{-})$ if
$$f(x)=|\log x|$$
for the interval $x>0$
The textbook solved it by using the method described below:
$$f'(1^{-})=\lim\limits_{x\to 1^{-}} \frac{f(x)-f(1)}{x-1}$$
Which becomes:
$$f'(1^{-})=\lim\limits_{x\to 1^{-}} \frac{|\log x|-|\log 1|}{x-1... | An alternative answer which does not require Taylor series or the knowledge of the derivative of $\text{log}$ is to use the limit definition for $\frac{1}{e}$:
$$\frac{1}{e} = \lim_{n\rightarrow \infty} \left( 1- \frac{1}{n} \right)^n = \lim_{m\rightarrow 0} \left( 1- m \right)^{1/m} $$
So that:
$$ -\lim_{h\rightarrow ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Difference between Increasing and Monotone increasing function I have some confusion in difference between monotone increasing function and Increasing function. For example
$$f(x)=x^3$$ is Monotone increasing i.e, if $$x_2 \gt x_1$$ then $$f(x_2) \gt f(x_1)$$ and some books give such functions as Strictly Increasing fu... | As I have always understood it (and various online references seem to go with this tradition) is that when one says a function is increasing or strictly increasing, they mean it is doing so over some proper subset of the domain of the function. To say a function is monotonic, means it is exhibiting one behavior over t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1746077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Prove that $\oint_{|z|=r} {dz \over P(z)} = 0$ I got stuck on this problem, hope anyone can give me some hints to go on solving this:
P is a polynomial with degree greater than 1 and all the roots of $P$ in complex plane are in the disk B: $|z| = r$. Prove that: $$\oint_{|z| = r} {{dz}\over{P(z)}} = 0$$
Here, the di... | Using the ML inequality:
$$\left|\oint_{|z|=R}\frac{dz}{p(z)}\right|\le2\pi R\cdot\max_{|z|=R}\frac1{|p(z)|}\le2\pi R\frac1{R^n}\xrightarrow[R\to\infty]{}0$$
since $\;n\ge 2\; $ .
Why? Because of the maximum modulus principle:
$$p(z)=\sum_{k=0}^na_kz^k=z^n\sum_{k=0}^na_kz^{k-n}\stackrel{\forall\,|z|=R}\implies\left|p(z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1746172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Elliptic curves with trivial Mordell–Weil group over certain fields. I am looking for elliptic curves $E,E'$ defined over $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$ respectively and given by a Weierstrass equation such that their Mordell-Weil group is trivial, i.e. such that
$$
E(\mathbb{F}_{3})=\{\mathcal{O}\}\quad\text{a... | By Hasse's bound we know that $1\le |E(\mathbb{F}_3)|\le 7$; and indeed there is an elliptic curve with $E(\mathbb{F}_{3})=\{\mathcal{O}\}$, given by
$$
y^2=x^3-x-1.
$$
Actually, since we know that all such curves are given by the long Weierstrass equation $y^2=x^3+ax^2+bx+c$ with nonzero discriminant, we can just try ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1746278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the power series of a complex function So I have the function
$$\frac{z^2}{(z+i)(z-i)^2}.$$
I want to determine the power series around $z=0$ of this function.
I know that the power series is $\sum_{n=0}^\infty a_n(z-a)^n$, where $a_n=\frac{f^{(n)}(a)}{n!}$. But this gives me coefficients, how can I find a se... | It's also convenient to perform a partial fraction decomposition followed by a binomial series expansion.
We obtain by partial fraction decomposition
\begin{align*}
\frac{z^2}{(z+i)(z-i)^2}&=\frac{1}{4(z+i)}+\frac{3}{4(z-i)}+\frac{i}{2(z-i)^2}\\
&=-\frac{i}{4}\cdot\frac{1}{1-iz}+\frac{3i}{4}\cdot\frac{1}{1+iz}-\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1746395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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How to test the convergence of $\sum^{\infty}_{n=2}\frac{1}{\ln{n}}$ and $\sum^{\infty}_{n=2}\frac{\ln{n}}{n^{1.1}}$? How to test the convergence of $\sum^{\infty}_{n=2}\frac{1}{\ln{n}}$ and $\sum^{\infty}_{n=2}\frac{\ln{n}}{n^{1.1}}$?
For the first one, I use basic comparison and compare it to $\frac{1}{n}$, since $\... | Use the $n^{\alpha}-$test with $\alpha = 1.05$.
The $n^{\alpha}-$test says that if $\alpha > 1$, and $a_n \ge0$ eventually, and if:
$$\lim n^{\alpha}a_n = 0$$
Then $\sum a_n$ converges.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability and expectation of three ordered random variables I am really stuck on the following question during my exam preparation.
Let $X_1$, $X_2$ and $X_3$ be independent exponential random variables with respective rates $\mu_1$, $\mu_2$ and $\mu_3$. Find:
a) $P(X_1 < X_2 < X_3)$
b) $E(X_2 | X_1 < X_2 < X_3) $
Qu... | Your first approach has a flaw, because:
$$\mathsf P(X_2 <X_3 ,X_1 <X_2 ,X_1 <X_3 )\neq \mathsf P(X_2 <X_3 )~\mathsf P(X_1 <X_2 )~\mathsf P(X_1 <X_3 )$$
The random variables are independent, but the events of their pairwise ordering are not. When given that $X_3$ is somewhat larger than $X_2$ (whatever that is) then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1746575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Improper convergence of $ \cos(x)/{x^{1/2}} $ I have to evaluate the convergence of the improper integral $ \int_1^\infty \frac {\cos(x)}{x^{1/2}}dx $.
As the function is continuous on every $ [1, M] $, I can tell that this function is Riemann integrable on every interval $ [1,M] $, M > 1. So all I have to do is to ev... | We prove convergence. Integrate by parts, letting $u=x^{-1/2}$ and $dv=\cos x\,dx$. Then $du=(-1/2)x^{-3/2}$ and we can take $v=\sin x$. So our integral from $1$ to $M$ is
$$\left. x^{-1/2}\sin x\Large\right|_1^M-\int_1^M (-1/2)x^{-3/2}\sin x\,dx.$$
Both parts behave nicely as $M\to\infty$, because $|\sin x|\le 1$.
Rem... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Continuous strictly increasing function with derivative infinity at a measure 0 set Let $E\subset [0,1]$ with $\mu(E)=0$. Does there exist a continuous, strictly increasing function $f$ on $[0,1]$ so that $f'(x)=\infty$ for all $x\in E$ (in Lebesgue sense)?
I think there exist such a function, but I don't know how to c... | I couldn't figure how the general case works but if $E$ is countable consider $E=\{q_n\;:\;n\in\mathbb{N}\}$ and take $f$ a function which is $0$ on $(-\infty,-1/2]$, $1$ on $[1/2,\infty)$, monotonic and has a derivative $f':\mathbb{R} \to [0,\infty]$ with $f'(0)=\infty$, additionally take a sequence $(a_n)_{n\in\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1746772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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is this question Using chinese remainder theorem?
I think that it will use Chinese remainder theorem
but I don't know how to put...
well by CRT, there exist $x=k\mod (m_1)(m_2)(m_3)$
which is $x=a_1^3 \mod m_1$ and $x=a_2^3 \mod m_2$ and $x=a_3^3 \mod m_3$
but $k$ must be $a^3$ ?
| Yes, one uses the Chinese Remainder Theorem, but not quite in the way partly described in the post.
Consider the system of congruences $y\equiv a_i\pmod{m_i}$ ($i=1,2,3$). By the Chinese Remainder Theorem, this system of congruences has a solution. Call it $a$.
Then $a^3\equiv a_i^3\equiv x\pmod{m_i}$ for all $i$, and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1746898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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why is the geometric mean less than the logarithmic mean? Can someone explain why the geometric mean is less than the logarithmic mean?
$$\sqrt{ab} \leq \frac{b-a}{\log b-\log a}
$$
| There might be a geometric interpretation that you are looking for, but I still prefer an algebraic approach. So let's suppose $0 < a < b$, and put $b = ta, t > 1$. Thus: $LHS = \dfrac{1}{\sqrt{ab}} = \dfrac{1}{\sqrt{ta^2}} = \dfrac{1}{a\sqrt{t}}$,and $RHS = \dfrac{\log(at) - \log a}{at- a}= \dfrac{\log t}{a(t-1)}$. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1746976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Numerical solver for maxwell equations? Just curious if someone has come across a package where I can simply solve the basic maxwell equations(just the curl equations). I'm just interested in solving it on a 2-d plate out of interest. Anyone come across such a package in their travels?
Thanks
| You mean "numerically solve", right? In this case I would suggest you to give FiPy an earnest shot. It's reasonably simple and fairly well documented. It may be the answer to your problem.
PS: FiPy is based on the Finite Volume Method (FVM).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1747226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Frobenius Norm and Relation to Eigenvalues I've been working on this problem, and I think that I almost have the solution, but I'm not quite there.
Suppose that $A \in M_n(\mathbb C)$ has $n$ distinct eigenvalues $\lambda_1... \lambda_n$. Show that $$\sqrt{\sum _{j=1}^{n} \left | {\lambda_j} \right |^2 } \leq \left \|... | You are in the right way. The corresponding Schur decomposition is $A = Q U Q^*$, where $Q$ is unitary and $U$ is an upper triangular matrix, whose diagonal corresponds to the set of eigenvalues of $A$ (because $A$ and $U$ are similar). Now because Frobenius norm is invariant under unitary matrix multiplication:
$$||QA... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Question about $\aleph$-fixed point I am working through a proof on cardinals I found and can't reason some of the steps.
The proposition is that there is an $\aleph$-fixed point, i.e. there is an ordinal $\alpha$ (which is necessarily a cardinal), so that $\aleph_{\alpha} = \alpha$. The proof goes as follows:
Let $\a... | I think the key piece is that the $\alpha_n$s are increasing: $\alpha_n\le\alpha_{n+1}$.
More broadly, the following is true:
$\aleph_\beta\ge\beta$, for all $\beta$.
So we know $\alpha_n\le\alpha_{n+1}\le\alpha$ (the latter inequality since $\alpha$ is the sup of the $\alpha_n$s), so if $\alpha_n=\alpha$ then $\alp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1747465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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What is the correct way of adjoining an "infinite" point to a totally-ordered field, such that the end result is still totally-ordered? Let $Q$ denote a totally-ordered field. Now adjoin an infinite element $\infty$ to $Q$, by equipping the field of rational functions $Q(\infty)$ in the indeterminate $\infty$ with the ... | For a more general construction : let $\Gamma$ be a totally ordered abelian group, and $\chi: \Gamma\to \{ \pm 1\}$ be a group morphism.
Then we can form the Hahn series field $Q((\Gamma))$ as the set of formal series $\sum_{\gamma\in \Gamma} a_\gamma X^\gamma$ with $a_\gamma\in Q$ such that the support (ie $\{\gamma\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1747603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Consecutive integers sum with different steps First of all: beginner here, sorry if this is trivial.
We know that $ 1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2 $ .
My question is: what if instead of moving by 1, we moved by an arbitrary number, say 3 or 11? $ 11+22+33+44+\ldots+11n = $ ?
The way I've understood the usual ... | Assume you have a sequence $$x, x+a, x+2a, x+3a, ..., x+na.$$
Let us note $$S = \sum_{i=0}^n (x+ia).$$ Then by the trick you mentioned, we see that $$S+S = (2x+na)+(2x+na)+...+(2x+na) = (n+1)(2x+na).$$ Hence $$S = \frac{(n+1)(2x+na)}{2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1747696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 3
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Question regarding projective coordinate transformation While reading Kunz's commutative algebra book, I came across a statement I can't understand. First, let me define the notations.
Let $L/K$ be extension of fields, and let $\mathbb{P}^n (L)$ denote the projective n-space over $L$.
*
*A Projective coordinate tra... | $V$ is certainly changing in the sense that after applying a coordinate transformation $A$ to $\mathbb{P}^n$ you will find that $A(V)$ is not (in general) the zero locus of the same homogeneous polynomials $F_1,\ldots,F_m$ that $V$ was, but the author is saying that there still exist other homogeneous polynomials $G_1,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Localization Preserves Euclidean Domains I'm wanting to prove that given a ring $A$ (by "ring" I mean a commutative ring with identity) and a multiplicative subset $S \subset A$:
if $A$ is an Euclidean Domain, and $0 \notin S$ then $S^{-1}A$ (localization of A at S) is also an Euclidean Domain.
I'm trying to produce... | In wikipedia's language, we may assume that $N$ satisfies $N(a)\le N(ab)$ for $a,b\in A$. Let us denote the candidate function for the localization by $N_S\colon (S^{-1}A)\setminus\{0\}\to\mathbb N$.
We will also replace $S$ by its saturation, i.e. by $S_{\mathrm{sat}}:=\{ a\in A \mid \exists b\in A: ab\in S\}$. Notice... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1748012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Why does $\tan\left(\frac{\pi}{2} - \beta\right) = \frac{t}{s}$ imply that $\tan\beta = \frac{s}{t}$? I have been preparing for the SAT on KhanAcademy. For one of the trigonometry problems, the following conversion is made:
$$\tan\left(\frac{\pi}{2} − \beta\right)= \frac{t}{s}$$
$$\tan(\beta) = \frac{s}{t}$$
There is n... | Observe that $$\tan(\pi/2-\beta)=\frac{\sin(\pi/2-\beta)}{\cos(\pi/2-\beta)}=\frac{\cos\beta}{\sin\beta}=\frac{1}{\tan\beta}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1748117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can every partially ordered set (POSET) take the form of a directed acyclic graph (DAG)? A POSET (Partially ordered set) is a set on the elements of which we have established a partial order relation ($\leq$), i.e. a relation which is:
*
*reflexive: $x\leq x,$ for every x in S
*anti-symmetric: $x \leq y \wedge y \l... | YES. Every POSET can take the form of a DAG.
However, in order to obtain a less cluttered graph, you’d better avoid drawing every single edge. (see figure)
You can omit the edges that can be inferred from the reflexive and transitive properties.
Moreover you can arrange nodes in order to orient every edge upward and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1748204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Actuarial Mathematics proof error? Is there something wrong with this proof or is it just me? Did they forget the -ve sign or does it cancel somehow? (From the second to last to last line)
| Remember that the survival function (and consequently the deterministic number of lives) is a nonincreasing function with $s(\infty) = l_\infty = 0$. Consequently, by the fundamental theorem, $$\int_{y=x}^\infty l_y \mu(y) \, dy = g(\infty) - g(x),$$ where $g$ is an antiderivative of the integrand, i.e., $g$ satisfies... | {
"language": "en",
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Find all the parameters and such that the line $y = ax + \frac{1}{2}a - 2$ intersects the hyperbola $xy = 1$ at right angles in at least one point . Problem:
Find all the parameters and such that the line
$y = ax + \frac{1}{2}a - 2$ intersects the hyperbola $xy = 1$ at right angles
in at least one point.
My work:
We tr... | It would be better to use the functional equation
$$y=\frac 1x$$
The derivative is
$$y'=-\frac 1{x^2}$$
So at $x_0=r$ we get $y_0=\dfrac 1r$, $f'(x_0)=-\dfrac 1{r^2}$. For the perpendicular line, we want the slope to be $m=-\dfrac 1{f'(x_0)}=r^2$. Putting that into the point-slope equation of a line, the perpendicular ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1748452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Rolling a die until two rolls sum to seven Here's the question:
You have a standard six-sided die and you roll it repeatedly, writing
down the numbers that come up, and you win when two of your rolled
numbers add up to $7$. (You will almost surely win.) Necessarily, one of the
winning summands is the number roll... | While not exactly stated, it seems that you win only if the sum of consecutive numbers sums to 7.
Obviously you can't win on the first roll. For every roll thereafter, you have a $1/6$ chance of winning on the next roll, (and a $5$ in $6$ chance of the game continuing).
Now, $P(N=2) = \frac{1}{6}$, $P(N=3) = \frac{5}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1748549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 2
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Cover $(0, +\infty )$ by open sets
Cover $(0, +\infty)$ by open sets $U_\alpha$ such that for any $\epsilon > 0$ there are points $x, y \in (0, +\infty)$ with $|x-y|<\epsilon$, not both belonging to the same $U_\alpha$
The distance beteen $x$ and $y$ is $\epsilon$ so would it work if we took our open setss $U_\alpha=... | Your definition of $U_{\alpha}$ is not clear, is $\alpha$ a point or is it an index?
Besides that, you want to be able to find these two points for every $\epsilon$.
If I understand correctly, what you need is a family of sets such that you can always find two points very close to each other such that they correspond ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
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Where is a good source for serious math (wall-size) posters? Where is a good source for math wall posters that give glimpses of serious and beautiful mathematics?
I'm a faculty member looking to find some wall posters (e.g. 2 ft x 3 ft) to hang in a handful of display cases around our department. I'd like the posters t... | One source for posters that may fit your description (though I don't know about the size) could be http://www.ams.org/samplings/mathmoments/mathmoments
| {
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"source": "stackexchange",
"question_score": "6",
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For every nonzero vector $v$ there exists a linear functional $f$, sucht that $f(v) \neq 0$.
I want to prove that for all $v \in V$ with $v \neq 0 \implies \exists f \in V^{*} : f(v) \neq 0$.
I know that if $V$ is finite-dimensional we can choose a basis $\{e_i\}$ of $V$ and construct the corresponding dual basis $\{... | $f$ is actually nonzero on $\operatorname{span}(v)$ (except at $0$, of course). What they did not say explicitly is that you extend $f$ to $\operatorname{span}(v)$ by linearity, i.e. $f(\alpha v) \equiv \alpha f(v)$ for all $\alpha\in\mathbb{F}$. So the proof is more or less a definition, in effect.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1748904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Convergence of $\int_0^{\infty} \frac{x^n}{(1+x^2)^m}dx$ Suppose we have an integral of the form $$I(n,m):= \int_0^{\infty} \frac{x^n}{(1+x^2)^m}dx$$
Most of my test cases computed seem to indicate if $n\geq 2m-1$, then this integral diverges.
I have a non-rigorous justification for this: just expand the bottom and loo... | If $n<2m-1$, we can just directly bound the integrand:
$$\frac{x^n}{(1+x^2)^m}\leq \frac{x^n}{x^{2m}+1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1748991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show that a periodic, completely multiplicative arithmetic function is a Dirichlet character to some module $q$
Show that if $f$ is a periodic, completely multiplicative arithmetic function, then $f$ is a Dirichlet character to some modulus $q$.
A Dirichlet character modulo $q$ is an arithmetic function $\chi$ which ... | Note that if $f\equiv1$ we have obviously that $f$ is the principal character mod $1$. So assume that $f\not\equiv1$, assume that $f\not\equiv0$ and also assume that $\left(n,q\right)=1$. Since $f$ is completely multiplicative and $q$ periodic (and we consider $q$ as the minimum period), we have (remember we can work i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving $n - \frac{_{2}^{n}\textrm{C}}{2} + \frac{_{3}^{n}\textrm{C}}{3} - ...= 1 + \frac{1}{2} +...+ \frac{1}{n}$ Prove that $n - \frac{_{2}^{n}\textrm{C}}{2} + \frac{_{3}^{n}\textrm{C}}{3} - ... (-1)^{n+1}\frac{_{n}^{n}\textrm{C}}{n} = 1 + \frac{1}{2} + \frac{1}{3} +...+ \frac{1}{n}$
I am not able to prove this. Plea... | Hint. One may observe that
$$
\frac1k=\int_0^1t^{k-1}dt,\quad k\geq1,
$$ giving
$$
\sum_{k=1}^n\frac1k=\int_0^1\sum_{k=1}^nt^{k-1}dt=\int_0^1\frac{1-t^{n}}{1-t}dt=\int_0^1\frac{1-(1-u)^{n}}udu
$$ then use the binomial theorem in the latter integrand to conclude.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1749182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Roots of a Quartic (Vieta's Formulas)
Question: The quartic polynomial $x^4 −8x^3 + 19x^2 +kx+ 2$ has four distinct real roots denoted
$a, b, c,d$ in order from smallest to largest. If $a + d = b + c$ then
(a) Show that $a + d = b + c = 4$.
(b) Show that $abcd = 2$ and $ad + bc = 3$.
(c) Find $ad$ and $bc.$
(d) F... | Hint If you know that $ad = 2$ and $a + d = 4$, then $$(x - a)(x - d) = x^2 - (a + d) x + ad = x^2 - 4 x + 2 ,$$ so finding $a, d$ is just finding the roots of that quadratic. Of course, finding $b, c$ is analogous.
| {
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"timestamp": "2023-03-29T00:00:00",
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Integrability of a function in $L^2$ we consider $V $ a polynomial in $R[x_1,x_2,..,x_n]$ such that $e^{-V(X)}\in L^2(R^n)$
I want to prove that this implies that $e^{-V(X)} $ vanish as $|X|$ goes to $+\infty$
For that we suppose that the function didn't vanish,and we prove that this implies that
it's value would be... | If you are taking the Lebesgue integralof $f$ then what $f$ does on a set of measure $0$ is irrelevant. So if $S$ is an unbounded null set, let $f(x)=0$ for $x\not \in S,$ and let $f(x)$ be anything at all for $x\in S.$ Then $\int f=0.$ Example: Let $S=Q^n$ and let $f(x_1,...,x_n)=\sum_{j=1}^n|x_j|$ for $(x_1,...,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$(\mathbb{Z}/77 \mathbb{Z})^{\times} \cong \mathbb{Z}/10 \mathbb{Z} \times \mathbb{Z}/6 \mathbb{Z}$ - Group Let $A$ a ring with unit element $1 \ne 0$ let $A^{\times}=\{a \in A: a$ invertible$\}$. Show that $(\mathbb{Z}/77 \mathbb{Z})^{\times} \cong \mathbb{Z}/10 \mathbb{Z} \times \mathbb{Z}/6 \mathbb{Z}$ as a group.
... | Hint: When $p$ is a prime, we have that $(\mathbb{Z}/p\mathbb{Z})^\times \cong (\mathbb{Z}/(p-1)\mathbb{Z})$. Notice that $(\mathbb{Z}/77\mathbb{Z}) \cong (\mathbb{Z}/7\mathbb{Z})\times (\mathbb{Z}/11\mathbb{Z})$.
Can you show that $$(\mathbb{Z}/7\mathbb{Z}\times \mathbb{Z}/11\mathbb{Z})^\times \cong (\mathbb{Z}/7\ma... | {
"language": "en",
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Single variable $C^1$ locally invertible function is globally invertibe? I am wondering if and why a single variable $C^1$ locally invertible function on the entire real line is globally invertible.
I've been told that for single variable function, if the derivative is always non zero and continuous, then the inverse c... | That's correct for $f:\mathbb{R}\rightarrow \mathbb{R}$. If $f^\prime$ is everywhere nonzero $f$ is strictly increasing or decreasing. It's easy to see that this implies that $f$ is one to one (injective), hence a unique inverse $f(\mathbb{R}) \rightarrow \mathbb{R}$ exists (this is just set theory). That this is cont... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Difficult Inverse Laplace Transform I've had this question in my exam, which most of my batch mates couldn't solve it.The question by the way is the Laplace Transform inverse of
$$\frac{\ln s}{(s+1)^2}$$
A Hint was also given, which includes the Laplace Transform of ln t.
| $f(s,a) = L(t^a) = \frac{\Gamma(a+1)}{s^{a+1}} $
Differentiating with respect to a, we get
$L(t^a\cdot lnt) = \frac{\Gamma'(a+1) - \Gamma(a+1)\cdot lns}{s^{a+1}} $
set a = 1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1750104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Can't seem to solve a radical equation? Question is : $\sqrt{x+19} + \sqrt{x-2} = 7$ So there is this equation that I've been trying to solve but keep having trouble with.
The unit is about solving Radical equations and the question says
Solve:
$$\sqrt{x+19} + \sqrt{x-2} = 7$$
I don't want the answer blurted, I want... | Multiply both sides by $\sqrt{x+19} -\sqrt{x-2} $ to get
$$x+19 -(x-2) = 7 (\sqrt{x+19} - \sqrt{x-2}) \\
21 = 7 (\sqrt{x+19} - \sqrt{x-2}) \\
3 =\sqrt{x+19} - \sqrt{x-2}$$
Adding this to the original equation you get
$$2\sqrt{x+19}=10 \Rightarrow x+19=25 \Rightarrow x=6$$
P.S. You can find the same method employed in m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1750192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What equation produces this curve? I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline.
The spline is pretty simple - a gentle curve which begins and ends horizontal.
Is there a simple equation for this curve?
Or perhaps two equations, o... | Assuming you mean
$$\begin{align}
y(0) &= 40 \\
y(120) &= 0 \\
\dot{y}(0) &= 0 \\
\dot{y}(120) &= 0
\end{align}$$
then a simple cubic will do:
$$y(x) = \frac{x^3}{21600} - \frac{x^2}{120} + 40$$
At range $x=0\dots120$, it looks like this:
You can find these very easily. In general, a cubic curve is
$$y(x) = C_3 x^3 +... | {
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"timestamp": "2023-03-29T00:00:00",
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Determine the largest open set to which $f(z)=\sum_{n=1}^{\infty}(-1)^n(2n+1)z^{n}$ can be analytically continued Let $U=B_1(0)$ and $$f:U \rightarrow \mathbb{C},\qquad f(z)=\sum_{n=1}^{\infty}(-1)^n(2n+1)z^{n}.$$
Determine the largest open set to which $f$ can be analytically continued
Remark: I was given following su... | The series converges for $\;|z|<1\;$ , for example using the $\;n\,-$ th root test. Now, for any $\;|z|=1\implies z=e^{it}\;,\;\;t\in\Bbb R\;$ , we have that
$$\lim_{n\to\infty}|(-1)^n(2n+1)e^{nit}|=\lim_{n\to\infty}(2n+1)=\infty\neq0$$
and thus the series doesn't converge on the unit circle, so the maximal open set wh... | {
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Prove that $c^{1/n} \rightarrow 1$ for $c > 0$. I can show a straightforward $\epsilon-\delta$ proof that when $c > 1$, $c^{1/n} \rightarrow 1$. But not when $c < 1$?
WTS
For all $\epsilon$ there exists $N$ st. for $n \geq N$, $|c^{1/n} - 1| < \epsilon$
When $c < 1$, the absolute value can be removed and the inner expr... | (very unoriginal)
By Bernoulli's inequality,
$(1+a/n)^n
\ge 1+a
$
so
$(1+a)^{1/n}
\le 1+a/n
$.
If $c > 1$,
let
$c = 1+a$.
Then
$c^{1/n}
= (1+a)^{1/n}
\le 1+a/n
\to 1
$
as
$n \to \infty$.
If
$0 < c < 1$,
let
$c^{1/n} = \dfrac1{1+b}
$.
Then
$c
= \dfrac1{(1+b)^n}
\le \dfrac1{1+nb}
\lt \dfrac1{nb}
$
so
$b < \dfrac1{nc}
$
a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\mathbb R \times \mathbb R_{sorg}$ normal? I know that $\mathbb R \times \mathbb R$ is normal and $\mathbb R_{sorg} \times \mathbb R_{sorg}$ is not. But what about $\mathbb R \times \mathbb R_{sorg}$ ?
$\mathbb R_{sorg}$ is the Sorgenfrey line.
| Yes, it is. By Dowker's theorem we know that $\mathbb{R_l} \times [0,1]$is normal ( $\mathbb{R_l}$ is the left limit topology, another name for the Sorgenfrey line), because $\mathbb{R_l}$ is generalised ordered, so normal and countably paracompact.
Then a theorem by Morita (paper) shows that $\mathbb{R_l} \times \... | {
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Prove that: $\vdash \forall x(\forall y\alpha)\to \forall y(\forall x \alpha)$ How does one prove that
$$\vdash \forall x(\forall y\alpha)\to \forall y(\forall x \alpha)$$
in first order logic?
I have tried using the specialization and generalization rules on various wffs but they don't lead me to anything concrete.
| If we have a formula like $\forall x \forall y\ \alpha$, where $\alpha$ is another arbitrary formula, then we can exchange the order of the quantifiers without changing its semantics. Note, however, that this would NOT be possible if the quantifiers were different, e.g. $\forall x \exists y$.
Now why can we do this? We... | {
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"timestamp": "2023-03-29T00:00:00",
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Question about Vitali Covering (from a Lemma in Royden and Fitzpatrick's book)
Definition. For a real valued function $f$ and an interior point $x$ of its domain, the upper derivative of $f$ at $x$ denoted by $\overline{D}f(x)$ is defined as follows: $$\overline{D}f(x)=\lim_{h\rightarrow0}\left[ \sup \left \{\frac{f(x... | Take any $x \in E_\alpha$. Now, since $\overline{D}f(x)\geq\alpha$, it follows that for some small $\delta$, $t<\delta \implies\frac{f(x+t)-f(x)}{t}\geq\alpha'$.
(The definition for the upper derivative above is slightly wrong, I will edit it)
Putting $t=d-x$, this means that $t<\delta \implies f(d)-f(x) \geq \alpha'(d... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving a differential equation through substitution In a book an example is given:
Solve $\frac{dy}{dx} = (x+y-4)^2$ by first making an appropriate substitution.
In the solution a step is given which I don't understand:
We let $u = x+y-4$ and thus $\frac{dy}{dx} = u^2$. We need to calculate $\frac{du}{dx}$. For th... | Given that $u=x+y-4$, clearly, differentiating with respect to $x$ gives $$\frac{du}{dx}=1+\frac{dy}{dx}.$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Convergence and limit of $\sum_{j=1}^n\sin\left(\frac{j-1}n\right)\left\{\cos\left(\frac jn\right)-\cos\left(\frac{j-1}n\right)\right\}$ The title says it all - I'm trying to find a way of proving the convergence and evaluating the limit of $a_n=\sum_{j=1}^n\sin\left(\frac{j-1}n\right)\left\{\cos\left(\frac jn\right)-\... | We have:
$$ \lim_{n\to +\infty} \sum_{j=1}^{n}\sin\left(\frac{j-1}{n}\right)\left[\cos\left(\frac{j}{n}\right)-\cos\left(\frac{j-1}{n}\right)\right]=\color{red}{\frac{\sin(2)-2}{4}}.$$
Sketch of proof: we have:
$$ \cos\left(\frac{j}{n}\right)-\cos\left(\frac{j-1}{n}\right) = \int_{\frac{j-1}{n}}^{\frac{j}{n}}(-\sin x)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1751207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
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How to square both the sides of an equation? Question: $x^2 \sqrt{(x + 3)} = (x + 3)^{3/2}$
My solution: $x^4 (x + 3) = (x + 3)^3$
$=> (x + 3)^2 = x^4$
$=> (x + 3) = x^2$
$=> x^2 -x - 3 = 0$
$=> x = (1 \pm \sqrt{1 + 12})/2$
I understand that you can't really square on both the sides like I did in the first step, howe... | You can square it like that, and the equality will still hold - remember these expressions are equal, so squaring them mean they are still equal. This can, however, produce spurious solutions - if you do this you should check that the values you get do indeed solve the given equation.
Note however, that $\sqrt{x+3} = (... | {
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"timestamp": "2023-03-29T00:00:00",
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Three nested summations I'm not sure of how to solve three nested summations and I came up with the following. Is it wrong?
$$\sum\limits_{i=1}^n\sum\limits_{j=1}^n\sum\limits_{k=1}^{2i+j} {1}=\sum\limits_{i=1}^n\sum\limits_{j=1}^n(2i+j)=\sum\limits_{i=1}^n(2in+\frac{n(n+1)}{2})=3\frac{n^2(n+1)}{2}$$
| Since
$$\sum_{i=1}^n\sum_{j=1}^n i=\sum_{i=1}^n\sum_{j=1}^n j=\frac{n^2(n+1)}2$$
we can do this:
$$\sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^{2i+j}1=\sum_{i=1}^n\sum_{j=1}^n(2i+j)=3\sum_{i=1}^n\sum_{j=1}^ni=\frac{3n^2(n+1)}2\quad\blacksquare$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof of $[0,1]~\text{disconnected}\implies(0,1)~\text{disconnected}$ I want to prove the following implication
$$[0,1]~\text{disconnected}\implies(0,1)~\text{disconnected}.$$
My try:
Suppose $[0,1]=U\cup V$ with $U,V$ open, disjoint and nonempty.
Using the subspace topology of $\mathbb{R}$ we also have $U=U'\cap[0,1]$... | You just take off the points $0,1$.
$(0,1)=(U\setminus\{0,1\})\cup(V\setminus\{0,1\})$. Prove $U\setminus\{0,1\}$ and $V\setminus\{0,1\}$ are open in $(0,1)$ (follows almost trivially), non-empty (trivial) and disjoint (more trivial).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1751678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$E_{k} \subset [0,1]$ such that $\lim_{k \to \infty} m(E_{k}) = 1$ but $\bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}E_{k} = \phi $ I am trying to find an example of collection $\left \{ E_{k} \right \}_{k=1}^{\infty}$ such that each $E_{k} \subset [0,1]$ satisfying $\lim_{k \to \infty} m(E_{k}) = 1$ but $\bigcup_{n=1}^... | There's a classic example. Let $E_k=[0,1]\setminus F_k$, where $F_k=\left[\sum_{n=1}^k\frac{1}{n},\sum_{n=1}^{k+1}\frac{1}{n}\right] \operatorname{mod}1$. Mod 1 means to move points over by whole numbers until you're in $[0,1]$.
Clearly $m(E_k)=1-\frac{1}{k+1}\to1$. Meanwhile, if there's any point, $x$, in $\cup\cap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1751806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using the Intermediate Value Theorem and derivatives to check for intersections. I have the following question:
Prove that the line $y_1=9x+17$ is tangent to the graph of the function $y_2=x^3-3x+1$. Find the point of tangency.
So, what I did was:
Let's construct a function $h$, such that $$h(x)=x^3-3x+1-(9x+17)=x^3-1... | I have a comment, and I know this has been looked at:
The statement involving Bolzano's Theorem, where you find that there is $c\in (0,5)$ where $h(c)=0$, implies that the graphs of the two functions intersect. It doesn't imply tangency, and it's actually a distraction, in that it gives you a point in $(0,5)$, which is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1752012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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De Morgan's Laws proof
My only problem with this proof is that they seem to be assuming that $(A \cup B)'$ is nonempty (similar idea for $A' \cap B')$. Is that because this holds trivially if $(A \cup B)'$ is empty? In other words, $A \cup B = U$ where $U$ is the universal set.
| You are correct that this proof is assuming that $(A\cap B)'$ is nonempty. Luckily, we can quite easily check that it holds in that case, as it means that there are no elements not in both $A$ and $B$, and so $A'\cap B'=\emptyset$. Likewise for the other direction, if $A'\cap B'=\emptyset$ then there are no elements th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1752111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Express $1/(x-1)$ in the form $ax^2+bx+c$
Let $x$ be a root of $f=t^3-t^2+t+2 \in \mathbb{Q}[t]$ and $K=\mathbb{Q}(x)$. Express $\frac{1}{x-1}$ in the form $ax^2+bx+c$, where $a,b,c\in \mathbb{Q}$.
I have proved that $f$ is the minimal polynomial of $x$ over $\mathbb{Q}$ but I am stuck showing the above claim. I tri... | Using the Extended Euclidean Algorithm as implemented in this answer, we get
$$
\begin{array}{r}
&&x^2&1&-(x+2)/3\\\hline
1&0&1&-1&(1-x)/3\\
0&1&-x^2&x^2+1&(x^3-x^2+x+2)/3\\
x^3-x^2+x+2&x-1&x+2&-3&0\\
\end{array}
$$
which means that
$$
\left(\vphantom{x^2}x-1\right)\left(x^2+1\right)+\left(x^3-x^2+x+2\right)\cdot\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1752195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Is the absolute Galois Group of $\Bbb Q$ countable? Is $\text{Gal} (\overline{\Bbb Q}/\Bbb Q)$ countable or uncountable? It seems like it should be countable (because the algebraic closure of $\Bbb Q$ is countable and there are countably many permutations of the irrational algebraic numbers, and a countable union of co... | Let $I\subseteq \Bbb N$ be any subset and let $K_I=\Bbb Q(\{\sqrt{p_i}\}_{i\in I})$ where $p_i$ is the $i^{th}$ prime. Then there are precisely $2^{\Bbb N}$ in fact the Galois group of the compositum of this extension is exactly isomorphic to
$$\prod_{i\in\Bbb N}\Bbb Z/2\Bbb Z$$
and this of course indicates there are u... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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The Area of Trapezium is given by $A=\frac{1}{2}(4-x^2)(2x+4)$ The Area of Trapezium is given by: $$A=\frac{1}{2}(4-x^2)(2x+4)$$ Find the Maximum area of Trapezium.
Hi, can anyone help me with this question. I know we differentiate the equation, but i don't know what to do next. Can anyone help me.
Thanks.
| Hint. One may observe that
$$A(x)=\frac{1}{2}(4-x^2)(2x+4)$$
gives
$$A'(x)=-(3x-2)(x+2)$$ If there exists a maximum of $A$, it has to be found among the values $A(x_0)$ for which $A'(x_0)=0$.
Can you take it from here?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $(a_n)_{n \geq 0}$ be a strictly decreasing sequence of positive real numbers , and let $z \in \mathbb C$ , $|z| < 1$. Let $(a_n)_{n \geq 0}$ be a strictly decreasing sequence of positive real numbers , and let $z \in \mathbb C$ , $|z| < 1$. Prove that the sum $a_0 + a_1z + a_2z^2 + \cdots + a_nz^n +\cdots $ is nev... | Simply note that $(1-z)\sum_{k=0}^\infty a_kz^k=a_0+\sum_{k=1}^\infty (a_k-a_{k-1})z^k$.
If $\sum_{k=0}^\infty a_kz^k=0$ for some $|z|<1$, $$a_0=\sum_{k=1}^\infty (a_{k-1}-a_k)z^k\implies$$ $$ \begin{align} a_0\leq \left| \sum_{k=1}^\infty (a_{k-1}-a_k)z^k\right| &\leq \sum_{k=1}^\infty (a_{k-1}-a_k)|z|^k \\ &< \sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1752531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Hermitian Matrix and nondecreasing eigenvalues I am studying for finals and looking at old exams. I found this question and am not sure how to proceed.
Let $A$ be an $n\times n$ Hermitian matrix with eigenvalues $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. Prove that if $a_{ii}=\lambda_1$ for some $i$, th... | Let $ \langle \cdot, \cdot \rangle $ be the standard inner product on $ \mathbb{C}^n $. For the Hermitian matrix $ A $, we will show that $ \inf_{||v||=1} \langle Av,v \rangle = \lambda_1 $ for $ v \in \mathbb{C}^n $ having unit norm.
First note that $ \langle Av,v \rangle $ is real, since $ \langle Av,v \rangle = \lan... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding a term in a sequence A strictly increasing sequence of positive integers $a_1, a_2, a_3,...$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ is geometric and the subsequence $a_{2k}, a_{2k+1}, a_{2k+2}$ arithmetic. If $a_{13}=539$. Then how can I find $a_5.$
An... | Hint: Let $a_1=a,a_2=ka$. Then it is not hard to show that $a_{2n+1}=a(nk-n+1)^2$. We have $539=11\cdot7^2$, so evidently we take $k=2,a=11$. That gives $a_5=11(2\cdot 2-1)^2=99$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Limit ordinal that cannot be written as $\alpha \cdot \omega$? I learned from this thread that every limit ordinal can be written as $\omega\cdot\alpha$ for some $\alpha$ but is this also true for $\alpha\cdot\omega$ even though ordinal multiplication is not commutative?
If it does not hold there must be a limit ordina... | Think about the following, if $n<\omega$, what is $n\cdot\omega$? On the other hand, if $\alpha\geq\omega$, it is certainly the case that $\alpha\cdot\omega\geq\omega\cdot\omega$.
So where does $\omega+\omega$ fit in? (Hint: It doesn't.)
| {
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compute probability density function of a bivariate function without sampling Suppose $X_1 \sim f_{X_1}(x_1)$, $X_2 \sim f_{X_2}(x_2)$ are random variables with known probability density function.
Is there any way to compute the probability density function of a bivariate function $g(x_1,x_2)$, assuming on specific fin... | HINT
Let's rename your variables $X,Y$ for easier notation. So you are defining
$$
Z = g(X,Y)
$$
and asking what is the probability density function of $Z$. For simplicity, let's assume a particular (very simple) $g$, let's say $Z = X+Y$.
$Z=z$ means $X+Y=z$, so $X=z-Y$; let's condition on $Y=y$, then you end up with
$... | {
"language": "en",
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Can they both be perfect squares? Let $X$ and $Y$ be positive integers. Prove that at least one of $X^2+Y$ and $Y^2+X$ is not a perfect square.
| If $X^2+Y=A^2$ then $Y\ge{2X+1}$ and thus $Y>{X}$.
If $Y^2+X=B^2$ then $X\ge{2Y+1}$ and thus $X>{Y}$.
And this is a contradiction.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Discuss about compactness of these sets My question is: How can I see if (in $\mathcal H=\mathcal l (\mathbb{N} )$
$B_1=\left\{ u | \frac{|u_k|}{k^2}\leq1 \right \}$
,$B_2=\left\{ u | \frac{|u_k|}{log(1+k)}\leq1 \right \}$
are compacts or not.
I know that I should show that they admit a convergent subsequence. Any help... | A metric space with an infinite closed discrete subspace is not compact. Let $v_k=(x_{k,n})_{n\in N}$ where $x_{k,n}$ is $1$ when $k=n$, and is $0$ when $k\ne n.$ Then $\|v_k-v_j\|=1$ when $k\ne j.$ So $\{v_k: 2\leq\ k \in N\}$ is an infinite closed discrete space of $B_1$ and of $B_2.$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What if we define function of moderate decrease as function satisfying $|f|\le\frac{A}{x^2}$ In Stein's Fourier Analysis, he defines the a continuous function $f$ as of moderate decrease if there exists $A>0$ such that
$$|f(x)| \le \frac{A}{1+x^2} \forall x\in\mathbb{R}$$
I am wondering what if we just define it as $|f... |
I am wondering what if we just define it as $|f(x)|\le \frac{A}{x^2}$?
The function $ x \mapsto \frac1{x^2}$ is not in $L^1(\mathbb{R})$ whereas $ x \mapsto \frac1{1+x^2}$ is in $L^1(\mathbb{R})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1753430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Numerical method with convergence greater than 2 It is a well-known fact that, for solving algebraic equations, the bisection method has a linear rate of convergence, the secant method has a rate of convergence equal to 1.62 (approx.) and the Newton-Raphson method has a rate of convergence equal to 2.
Is there any nume... | In fact, one can go further. Joseph Traub has an entire book devoted to the construction of a family of methods that are effectively high-order generalizations of the Newton-Raphson and secant methods. On a more modern front, Bahman Kalantari constructed a "basic iteration" family that is a generalization of the Newton... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How can one show that a möbius transformation maps the unit circle to some axis? Suppose I have the möbius transformation f(z) = $\frac{1+z}{1-z}$ and I want to show that it will map the unit circle (excluding the point 1) to the imaginary axis. Would it help to express my unit circle in polar form?
Here's what I d... | Point $z=1$ corresponds to $\theta=0$. $e^{i\theta}$ is non zero, and so is $e^{i\theta/2}$. Dividing both numerator and denominator yield:
$$
f(e^{i\theta})=\frac{e^{-i\theta/2}+e^{i\theta/2}}{e^{-i\theta/2}-e^{i\theta/2}}=\frac{\cos(-\theta/2)+i\sin(-\theta/2)+cos(\theta/2)+i\sin(\theta/2)}{\cos(-\theta/2)+i\sin(-\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1753627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is it that the interval of convergence is half open? I am given the following power series and asked to find the radius of convergence and determine the exact interval of convergence
$$\sum\biggr(\frac{3^n}{n\cdot 4^{n}}\bigg)x^n \Leftrightarrow \sum\bigg(\frac{3}{n^{1/n}\cdot 4}\bigg)^{n}x^n$$
If $a_n=(\frac{3}{n^... | Series convergence tests, e.g., Hadamard's formula for checking radius of convergence, are often inconclusive along the boundary; you must check the boundary separately, by plugging in boundary values, thus obtaining a new series (with no x's involved anymore) and again using some series convergence test, to check for ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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f(x) is a function such that $\lim_{x\to0} f(x)/x=1$ $f(x)$ is a function such that $$\lim_{x\to0} \frac{f(x)}{x}=1$$ if
$$\lim_{x \to 0} \frac{x(1+a\cos(x))-b\sin(x)}{f(x)^3}=1$$
Find $a$ and $b$
Can I assume $f(x)$ to be $\sin(x)$ since $\sin$ satisfies the given condition?
| Hint. You may use the standard Taylor series expansions, as $x \to 0$,
$$
\begin{align}
\cos x&=1-\frac{x^2}2+O(x^4)\\
\sin x&=x-\frac{x^3}6+O(x^4)
\end{align}
$$ giving
$$
\begin{align}
\frac{x(1+a\cos x)-b\sin x}{(f(x))^3}&=\frac{(1+a-b) x+\frac16 (-3 a+b) x^3+O(x^5)}{(f(x))^3}
\\\\&=\frac{(1+a-b) x+\frac169 (-3 a+b)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h).
Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also determine the structure of the group formed by these point... | Your curve $y^2 + 7xy = x^3 + 16x$ is isomorphic to $y^2 = x^3 - 44091x + 3304854$. Maybe you can find the transformation between these two models yourself?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A team squad combination and probability problem A team of 11 is chosen randomly from a squad of 18.
Two of the squad are goal keepers and one of them must be chosen. If neither of the goalkeepers is captain or vice captain, what now is the probability that both the captains and vice captains are selected?
The working ... | Apparently we’re supposed to understand that exactly one of the goalkeepers is chosen. Once that choice has been made, we must choose the other $10$ members of the team from the $16$ members of the squad who are not goalkeepers; this can be done in $\binom{16}{10}$ different ways. How many of these $\binom{16}{10}$ pos... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding the monthly payment for fixed-rate mortgage, but with first month interest free. I'm trying to calculate the monthly payment of a fixed-rate (annuity) loan, but with the twist that the first month is interest free.
I.e., I have a principal $P_0$ - the total sum that I've loaned - and I want to pay it off comple... | Let's write the Loan $P_0$ as the present value of the $N$ payments
\begin{align}
P_0&=c+cv^2+cv^3+\cdots+cv^N=c+c\left(\sum_{k=2}^N v^k\right)=c+c\left(a_{\overline{N}|r}-v\right)=c\left(1-v+a_{\overline{N}|r}\right)\\
&=c+cv\left(v+v^2+\cdots+v^{N-1}\right)=c+cv\left(\sum_{k=1}^{N-1} v^k\right)=c+cva_{\overline{N-1}|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1754190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Are we allowed to compare infinities? I'm in middle school and had a question (my dad is helping me with formatting).
We're learning about infinity in math class and there are a lot of problems like how it's not a number and how if you add one to infinity it doesn't change value.
But can you have one infinity be more t... | There have been many answers already, but I'd thought maybe I could try to provide an easier explanation of how cardinality works.
Consider the three infinite sets $A = \{1,2,3,...\}$, and $B = \{1,2,3,...\}$ and $C = \{1,2,3,...\}$, that is to say, each of them are just all natural numbers. Surely you agree that they ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1754313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "148",
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Solve for $\alpha$: $P = \frac{1}{\sigma}\int_{0}^{\alpha} \exp (\frac{ -2 x^{\beta}}{\sigma} ) dx$ I need to solve:
$$P = \frac{1}{\sigma}\int_{0}^{\alpha} \exp ( \frac{ -2 x^{\beta}}{\sigma} ) \;dx $$
This simplifies to:
$$P = \frac{1}{\sigma} \int_{0}^{\alpha} \exp (- B x^{\beta}) \;dx $$
But if we let:
$$t^{2} = Bx... | As Robert Israel answered, the result is given in terms of the incomplete gamma function. This can can also simplify to $$P = \frac{1}{\sigma}\displaystyle\int_{0}^{\alpha} \exp \Big(\frac{ -2 x^{\beta}}{\sigma}\Big ) dx=-\frac{\alpha }{\beta \sigma }\,E_{\frac{\beta -1}{\beta }}\left(\frac{2 \alpha ^{\beta }}{\sigma ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1754373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Is there any well-ordered uncountable set of real numbers under the original ordering? I know that the usual ordering of $\mathbb R$ is not a well-ordering but is there an uncountable $S\subset \mathbb R$ such that S is well-ordered by $<_\mathbb R$?
Intuitively I'd say there is no such set but intuitively I'd also sa... | There can't be. If $S\subseteq \Bbb R$ is well-ordered by the usual ordering, for every element $s_{\alpha}\in S$ that has an immediate successor $s_{\alpha+1}\in S$ (every element of $S$ except the greatest element if there is one), the set of rationals $Q_{\alpha}$ between the element and its successor is nonempty: $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1754463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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On solution to simple ODE Consider the ODE $$\frac{dx}{dt} = ax + b$$ where $a$ and $b$ are two parameters. The way to solve this is to divide both sides by $ax+b$ and integrate:
$$\int \frac{\dot x}{ax+b}dt = t+C \\ \frac{\log|ax+b|}{a} = t+C \\ x(t) = Ke^{at}-\frac ba$$
Easy enough. But I'm not sure why we're not e... | You're right, and this is a good issue to point out. In this case, it's straightforward to show uniqueness, though: Suppose that $x(t)$ is a solution and notice that if $y(t) = e^{-at} x(t)$, we have
\begin{align*}
y'(t) &= -ae^{-at} x(t) + e^{-at} x'(t) \\
&= -ae^{-at} x(t) + e^{-at} \big(ax(t) + b\big) \\
&= be^{-at}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1754572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Orthogonal matrix norm If $H$ is an orthogonal matrix, then $||H||=1$ and $||HA||=||A||, \forall A$-matrix (such that we can writ $H \cdot A$).
What norm is this about?
| The operator norm
$$
\|A\|=\max\{\|Ax\|_2:\ \|x\|=1\},
$$
where $\|\cdot\|_2$ is the Euclidean norm, also satisfies those two equalities. They follow easily from the fact that $\|y\|_2^2=y^Ty$, so $$\|Hx\|_2^2=(Hx)^THx=x^TH^THx=x^Tx=\|x\|_2^2.$$
| {
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"url": "https://math.stackexchange.com/questions/1754712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Open cover; why is the following open? I am asked to look at the following
For $n=0,1,2...$ define $U_n \subseteq [0,1]$ by $U_0=[0,\frac{1}{2})$ and $U_n=(2^{-n},1]$ for $n \geq 1$.
So, along the unit interval, we have a cover from zero to half and other sets $U_n$ covering $1/2,1/4...$ to $1$. Here's a question I ... | They are open in the subspace topology of $[0,1]$
For example, in your example $U_1 = (\dfrac{1}{2}, 1]$ is open, because it is an intersection of an open set in $\mathbb{R}$ say $(\dfrac{1}{2}, \dfrac{3}{2})$ with $[0,1]$
| {
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"url": "https://math.stackexchange.com/questions/1754812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to verify this relationship between area under the graph and the preimage? Define $h : \mathbb{R} \to [0, \infty)$,
Let $H = \{(x,y)| 0 \leq y \leq h(x)\}$ be the area under the graph (including the boundary)
I wish to show the following is true:
$$H = \bigcup_{c>0} h^{-1}[c,\infty) \times [0, c]$$
Attempt:
($\s... | If $h = 0$ (so the constant function), then $h^{-1}[c,\infty)] = \emptyset$ for all $c > 0$, as no $x$ has $h(x) \ge c > 0$. And then the product set is empty too and we'd get $H = \emptyset$ which is false, as $H = X \times \{0\}$. So we should have a union for all $c \ge 0$ not just $c > 0$. The rest of the post assu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1754898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What are some applications of large cardinals? Most mathematicians don't often encounter cardinalities larger than that of the continuum, it seems? What are some results outside of pure set theory/logic that rely on the properties of larger cardinals? In a similar vein, can anyone give examples of types of mathematical... | Some objects whose cardinal is at least $2^c,$ where $c$ is the cardinal of the reals : (1). The set of all real functions. (2). The set of all filters on an infinite set. (3). The dual space $l_{\infty}^*$ of the Banach space $l_{\infty}.$ (4). The maximal compactifications of $N$,of $Q$,and of $R$. (5). The free gro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1754985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Proof related to pigeon hole principle to be done with induction
since the question is about a positive integer m, it's obvious that the use of mathematical induction needed, but to prove the fact for n = k+1 we have to use the pigeon hole principle, i am so confused in using these both at once, some one help me in so... | Hint:
The base case is easy, so let's look at the inductive step. Assuming $P(m)$ is true, let's consider any set $S$ of $2m+3$ distinct integers from among $[-2m-1, 2m+1]$. If less than $3$ of those are from among $R = \{\pm(2m+1), \pm2m\}$, the induction hypothesis takes care of it. So that leaves cases where $|S \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1755084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Another characterization of the cofinality? Is it true that $cf(\kappa)=\min \{\lambda:\ \kappa^{\lambda}>\kappa\}$?
$cf(\kappa)$ is certainly $\geq$ than that minimum since $\kappa^{cf(\kappa)}>\kappa$, but I don't know how to tackle the inverse inequality. What's worse, I'm not even sure if my hypothesis is true :) ... | For infinite cardinal $k,$ it's also consistent with ZFC that it is true because of Konig's Theorem (aka Konig's Lemma): $k^{cf(k)}>k,$ and because GCH implies that $k^l=k$ for $0<l<cf(k).$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1755339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How many 2-dimensional subspaces is a 1-dimensional subspace contained in? V is a 3-dimensional vector space over some field K of order 2.
There are seven 2-dimensional subspaces, and seven 1-dimensional subspaces, using
${n\choose k}_q = \frac{(q^n-1)(q^n-q)...(q^n-q^{k-1})}{(q^k-1)(q^k-q)...(q^k-q^{k-1})}$.
I can sh... | Hint 1: You know there are $7 \times 3$ pairs $(X,Y)$ of two dimensional subspaces $X$ including one dimensional subspaces $Y$. You know how many one dimensional spaces there are. Count those pairs another way.
Possible hint 2: There's a duality that switches one and two dimensional subspaces. You might be able to use ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1755455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For what maximum positive $k$ is $2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$ true? I am trying to find the maximum value of $k$ such that the inequality $$2n \sin^{2} \frac{\pi}{n} > \tan \frac{k\pi}{n}$$ is satisfied. I impose restrictions that $n \in \mathbb{Z}$ with $n \geq 5$, and $k \in \mathbb{Z}$ with $k ... | Let
$$
f(x)=\frac{x}{\pi}\,\arctan\Bigl(2\,x\sin^2\frac{\pi}{x}\Bigr),\quad x>0.
$$
Then $k=\lfloor\, f(n)\rfloor$.
Using that $\arctan x<x$ and $\sin x<x$ if $x>0$, we see that
$$
f(x)<\frac{x}{\pi}\,\arctan\Bigl(\frac{2\,\pi^2}{x}\Bigr)<2\,\pi<7.
$$
Moreover, $\lim_{x\to\infty}f(x)=2\,\pi$. Thus $k\le6$ for all $n$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1755581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Why is there a factor of 1.7159 with the tanh function used in neural network activation? I was reading about neural networks when I came across the line :
Recommended f (x) = 1.7519 tanh (2/3 * x). How do we arrive at these values (we can fix the other once the other is obtained using the condition f(1) = 1) ?
Pg 10 ... | If you read further, at the top of page 14 it states that the required conditions for the sigmoid are:
*
*$f(\pm1)=\pm1$
*The second derivative is a maximum at $x=1$
*The effective gain is close to 1
Once you've decided that a $\tanh$ curve is a useful curve to try to fit to your sigmoid, then it is a case of choo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1755711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Eigenvalues of a tridiagonal block matrix When a tridiagonal matrix is also Toeplitz, there is a simple closed-form solution for its eigenvalues, being $$\lambda_k= a + 2 \sqrt{bc} \, \cos(k \pi / {(n+1)}) , fork=1,...,n$$. Now my question, is there formula for eigenvalues of a tridiagonal block matrix as well? for ex... | Write $$C:=\left[
\begin{matrix}
0 & 1 & 0 \\
1 &0 & 1 \\
0 & 1 & 0\\
\end{matrix}
\right],$$
so that
$$A=B\otimes I + I\otimes C.$$
Now $B$ and $C$ are symmetric and diagonalisable by orthogonal $\Omega$, $\Delta$ and you tell us you have a formula for the eigenvalues $\lambd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1755832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$xy^{-1}$ In order to prove the following:
$$x<y \iff x^{-1}>y^{-1}$$
*for $x>0$ and $y>0$
I tried this:
$$x<y\implies y-x>0$$
I have to prove that this, implies that $\frac{1}{x}>\frac{1}{y}$, that is, $\frac{1}{x}-\frac{1}{y}>0$.
Well, we know that $$\frac{1}{x}-\frac{1}{y} = \frac{y-x}{xy}$$
The numerator $y-x>0$ by... | Some of the answers here threw me off. I think it's a lot easier than some of the answers here. Simply use the properties of multiplication. Another answer here showed that the statement is true with one step, but you can step through the intermediary "calculations" to see what is going in.
If $x < y$ then $1 < x^{-1}y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1755945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Show a given analytic function is constant Suppose that $f$ is analytic on some region $R\in\mathbb{C}$. If Im$(f)$ = $k\cdot$Re$(f)$ for some nonzero constant $k\in\mathbb{C}$, then show that $f$ is constant on $R$.
I know that if $f'(z)=0$ for all $z$ in some region $R$, then $f(z)$ is constant. However, I'm not sure... | If $f$ is analytic, so is the real-valued function
$$g(z) = \frac{f(z)}{1 + ik}$$
Now there's a straightforward application of the Cauchy-Riemann equations to conclude that a real-valued analytic function is constant.
Notice that there is one value of $k$ for which this doesn't work.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is $f:SO(n)\rightarrow S^{n-1}$, $f(A)=(A^n_i)_i$ a submersion? Let $f:SO(n)\rightarrow S^{n-1}$, $f(A)=(A^n_i)_i$, that is $f(A)$ is the last row of $A$. Show that $f$ is a submersion.
I'm not sure how to calculate $df$, because I only know how to calculate the differential using local charts, but I don't know how to ... | In fact, if $e_n$ is the $n-$th base vector $f(A)= Ae_n$. Let us extend this formula to the set of all matrices, $F(A)=A.e_n$. So $F$ is linear and is its own differential $F'(A)B= Be_n$. The derivative of $f$ at the point $A$ is the restriction of this map to the tangent space $ \cal {A}(n). A$ of $O(n)$ at $A$, wher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find all possible real solutions. Find all possible real solutions of $a, b, c, d$ and $e$ if:
$3a= (b+c+d)^3$
$3b= (c+d+e)^3$
$3c= (d+e+a)^3$
$3d= (e+a+b)^3$
$3e= (a+b+c)^3$
Well I believe the solutions are possible only if $a=b=c=d=e$. In that case the solutions possible are $0, \frac{1}3$ and $-\frac{1}3$, but I am ... | HINT
If $x, y$ real then $x^3 < y^3 \Leftrightarrow x<y$.
I consider the equalities to be numbered.
Now suppose $a>b$.
Then from equality (1) and (2) we get $b>e$.
Because $a>e$ from (1) and (5) we get $d>a$.
So $d > a > b > e$.
Now, because $d>e$ from (4) and (5) we get $e > c$, so $d > a > b > e > c$.
Because $b > c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ is continuous and can be differentiated ad infinitum We have $f:(0,\infty) \rightarrow \mathbb{R}$ defined by infinite series $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$
Prove that $f$ is continuous and can be differentiated infinit... | By "$\log$" you mean the "natural logarithm".
From Wiki of "Natural logarithm", we have the following inequality for all $x\in(0,\infty)$,
$$\frac{x}{x+1}\le\log(1+x)\le x.$$
Then we see that each term $u_n(x):=\frac{x}{n}-\log(1+\frac{x}{n})$ satisfies
$$0\le u_n(x)\le\frac{x}{n}-\frac{\frac{x}{n}}{\frac{x}{n}+1}=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
ODE $y(1+2xy)dx+x(1-xy)dy=0$
$$y(1+2xy)dx+x(1-xy)dy=0$$
I have tried to isolate $\frac{dy}{dx}$ and got the following:
$$\frac{dy}{dx}=-\frac{y(1+2xy)}{x(1-xy)}$$
but I understand that the terms have to be in the same order and that is not the case.
What should I do?
| Set $u=xy$. Hence $\frac{du}{dx}=x\frac{dy}{dx}+y.$ Hence your equation became $$\frac{x}{u}\frac{du}{dx}-1 = \frac{1+2u}{u-1}.$$ This can be rewritten $$x\frac{du}{dx} = u\frac{3u}{u-1}$$ or else $$\frac{u-1}{3u^2}du=\frac{1}{x}dx.$$ Now you can integrate both sides.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Evaluating $\lim\limits_{x \to -\infty} \sqrt{x^2 + 3x} - \sqrt{x^2 + x}$. Is Wolfram wrong or is it me? What am I doing wrong?
My attempt
$$\begin{align}
\lim_{x \to -\infty} \sqrt{x^2 + 3x} - \sqrt{x^2 + x} &= \lim_{x \to -\infty} \sqrt{x^2 + 3x} - \sqrt{x^2 + x} \cdot \frac{\sqrt{x^2 + 3x} + \sqrt{x^2 + x}}{\sqrt{x^... | On your way to the last line, you're tacitly assuming that, for example, $\frac1x\sqrt{x^2+x} = \sqrt{\frac{x^2}{x^2}+\frac{x}{x^2}}$. But that is only true when $x$ is positive!
When $x$ is negative, $\frac1x\sqrt{\cdots\vphantom{x}}$ will be negative, and thus it can never be written as a sum of square roots.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
How to check if $n!$ is $ O(2^n)$ How can I check if $n! \in O(2^n)$?
The definition of $f$ being $O(g)$ is $f(n) \le c g (n)$, where $c>0$.
So it would mean $n! \le c 2^n$.
What is the clearest way to solve this? (As I am solving the past papers for my next examen.)
Thank you!
| As you said you need to check if there is a constant $C>0$ such that $n! \le C 2^n$ for all $n$ (or at least all sufficiently large $n$, but it does not matter much).
Note it is crucial that $C$ does not depend on $n$.
In this case it turns out there is no such $C$. There are various ways to see this. One of them coul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
How to rewrite $\frac{\partial \rho u_i u_j}{\partial x_j}$ in vector notation I want to rewrite this index notation expression to a vector notation /symbolic notation.
$$\frac{\partial \rho u_i u_j}{\partial x_j}=\frac{\partial \rho }{\partial x_j}u_i u_j+\rho\frac{\partial u_i}{\partial x_j} u_j+\rho u_i\frac{\parti... | Given that I prefer index notation, that presents less ambiguities, I would write expression 1 as:
$$
\mathbf{u}(\mathbf{u}\cdot\nabla)\rho+\rho(\mathbf{u}\cdot\nabla)\mathbf{u}+\rho\mathbf{u}(\nabla\cdot\mathbf{u})
$$
and expression 2 as
$$
\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u})
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
linear ODE problem
A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius $\frac{1}{2}$ cm has radius $0.4$ cm after $6$ months, how long will it take:
*
*For the radius to be $\frac{1}{4}$ cm?
*For the volume of the mothball to be half of what it was originally?
... | $$\frac{d}{dt}(\frac43\pi r^3) = -4k\pi r^2, $$
$$3r^2\frac43\pi \frac{dr}{dt} = -4k\pi r^2,$$
$$\frac{dr}{dt} = -k $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1756932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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