Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How do I find the slope of a curve of intersection at a given point? I am asked to find the slope of the curve of intersection between the upper half of the unit sphere and the plane $y=\frac12$ at the point $P(\frac12, \frac12,\frac {1} {\sqrt2})$
I made a drawing to illustrate how I visualize the problem.
I'm confus... | Assuming that you mean the unit sphere given by $x^2+y^2+z^2=1$, you have for $y=\frac{1}{2}$ the equation $x^2+z^2=\frac{3}{4}$. Solving this wrt $z$ we have $z=\pm\sqrt{\frac{3}{4}-x^2}$, and since you are looking for the upper half dome of your sphere we use the $+$-solution, that is, $z(x)=\sqrt{\frac{3}{4}-x^2}$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1044602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I differentiate this equation? $y = \sqrt[4]{\frac{(x^3+2\sqrt{x})^2(x-sinx)^5}{(e^{-2x}+3x)^3}}$ $y = \sqrt[4]{\frac{(x^3+2\sqrt{x})^2(x-sinx)^5}{(e^{-2x}+3x)^3}}$
I tried removing the root but that got me no where
| the solution of the given problem should be this here
$$\frac{\left(x^3+2 \sqrt{x}\right) (x-\sin (x))^4 \left(-3 \left(3-2 e^{-2
x}\right) \left(x^3+2 \sqrt{x}\right) (x-\sin (x))+5 \left(3 x+e^{-2 x}\right)
\left(x^3+2 \sqrt{x}\right) (1-\cos (x))+2 \left(3 x+e^{-2 x}\right) \left(3
x^2+\frac{1}{\sqrt{x}}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1044728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integral of $\frac{1}{\sqrt{x^2-x}}dx$ For a differential equation I have to solve the integral $\frac{dx}{\sqrt{x^2-x}}$. I eventually have to write the solution in the form $ x = ...$ It doesn't matter if I solve the integral myself or if I use a table to find the integral. However, the only helpful integral in an in... | $$\int \frac{dx}{\sqrt{x^2-x}} = \int \frac{dx}{\sqrt{\left(x-\frac{1}{2}\right)^2 - \frac{1}{4}}}$$
Setting $ x - \frac{1}{2} = t $
$$ \int \frac{dx}{\sqrt{\left(x-\frac{1}{2}\right)^2 - \frac{1}{4}}} = \int \frac{dt}{\sqrt{t^2 - \frac{1}{4}}} $$
It's possible to do this using a trig substitution, but if you want the ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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"answer_id": 5
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Let $n\geq4$ how many permutations $\pi$ of $S_n$ has the property that $1,2,3$ appear in the same cycle of $\pi$.... Let $n\geq4$ how many permutations $\pi$ of $S_n$ has the property that $1,2,3$ appear in the same cycle of $\pi$,while $4$ appears in a different cycle of $\pi$ from $1,2,3$?
My attempt:For $n=4$ I hav... | If a permutation $\sigma \in S_n$ consists of a $j$-cycle with $1, 2, 3$, call that cycle $\gamma_j$ and the set of numbers it is cycling $\Gamma_j = \{ 1, 2, 3, ... \} \not\ni 4$. The number of such $j$-cycles is $\color{blue}{(j-1)!}$
We can write $\sigma = \gamma_j\sigma'$ where $\sigma'$ is a member of all the perm... | {
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For a given finite group $Г$ , determine an infinite number of mutally nonisomorphic graphs whose groups are isomorphic to $Г$. For a given finite group $Г$ , determine an infinite number of mutally nonisomorphic graphs whose groups are isomorphic to $Г$.
I know that $Г$ is generated by $\Delta$ and for any finite $Г$,... | Given a group $\Gamma = \{g_1,\ldots,g_n\}$, construct an edge-colored digraph on vertex set $\Gamma$, and with arcs $(g_i,g_j)$ having color $g_k$ iff $g_k=g_i g_j^{-1}$. Then the edge-colored digraph has automorphism group $R(\Gamma)$, the right regular representation of $\Gamma$, which is isomorphic to $\Gamma$. Th... | {
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the greatest value of an implicit function $$\frac{\exp(x)\sin(x) - \exp(y)\sin(y)}{\exp(y)-\exp(x)} \le P$$ where $x \le y$
how would you find the smallest value of $P$ - the greatest value of the LHS implicit function without using a graphical calculator
In addition $x$ is less than or equal to $y$ that means the gra... | As $y\to x+$, the function approaches the derivative of $-u\sin(\log u)$.
Otherwise, the two partial derivatives $\partial /\partial x$ and $\partial /\partial y$ of $$\exp x\sin x-\exp y\sin y=P(\exp y-\exp x)$$ both hold.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Noetherian property for exact sequence Let $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ be an exact sequence of $A$-modules. Then $M$ is Noetherian is equivalent to $M'$ and $M''$ are Noetherian.
For the ''$\Leftarrow$'' case: I guess if we let $(L_n)_{n\geq 1}$ be an ascending chain of submodules o... | This follows immediately from the Five Lemma applied to
$$\begin{array}{c} 0 & \longrightarrow & \alpha^{-1}(L_n) & \longrightarrow & L_n & \longrightarrow & \beta(L_n) & \longrightarrow & 0 \\ & & \downarrow && \downarrow && \downarrow & \\ 0 & \longrightarrow & \alpha^{-1}(L_{n+1}) & \longrightarrow & L_{n+1} & \long... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1045202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Could someone point me in the right direction for this proof? I need to prove that Q (rational numbers) is countable by applying the function $f(m/n) = 2^m3^n$ with $m,n$ being relatively prime numbers.
I honestly have no idea where to start. Any pointers would be great!
| Injectivity
$f\left(\dfrac{m_1}{n_1}\right)=f\left(\dfrac{m_2}{n_2}\right) \implies 2^{m_1}3^{n_1}=2^{m_2}3^{n_2}$
Case 1 $m_1\geq m_2, n_1\leq n_2$
$\therefore 2^{m_1}3^{n_1}=2^{m_2}3^{n_2} \implies 2^{m_1-m_2}=3^{n_2-n_1} \implies m_1=m_2, \ n_1=n_2$
Case 2 $m_1\geq m_2, n_1\geq n_2$
$\therefore 2^{m_1}3^{n_1}=2^{m_2... | {
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measurability with zero measure
Let $f : [0,1] \rightarrow \Bbb R$ is arbitrary function , and $E \subset \{ x \in [0,1] | f'(x)$ exists$\}$.
How to prove this statement:
If $E$ is measurable with zero measure then $f(E)$ is measurable with zero measure.
If $E$ is measurable with zero measure then $m(E)=0$ that $m... | hot_queen's answer looks fine to me. This is not an independent answer, it's a minor variation and a comment on hot_queen's answer; I'm posting it as an answer for better readability (and who knows, someone might be fooled into upvoting it).
I'm reorganizing the covering lemma part of the argument. As observed earlier,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1045382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
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"Honest" introductory real analysis book I was asked if I could suggest an "honest" introductory real analysis book, where "honest" means:
*
*with every single theorem proved (that is, no "left to the reader" or "you can easily see");
*with every single problem properly solved (that is, solved in a formal (exam-lik... | Possibly Abbott, Understanding Analysis
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 10,
"answer_id": 4
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Finding the vertical asymptote of $y = \pm2\sqrt\frac{x}{x-2}$ Find the value of vertical asymptotes.
$$ y = \pm2\sqrt\frac{x}{x-2}$$
There are two fuctions right? so,
$$ y = f_1(x)= +2\sqrt\frac{x}{x-2}$$
$$ y = f_2(x)= -2\sqrt\frac{x}{x-2}$$
I'm good until here.
Now, the book says
$$\lim_{x\to2^+}f_1(x) = \lim_{x\to2... | Because if $x\to 2^-$ then the radicand is negative.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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connection between discrete valuation rings and points of a curve. Let $C$ be a projective irreducible non-singular curve over a field $k$ and let K be its function field. It applies that $(k[X,Y]/I(C))_{(X-a,Y-b)}$ (i.e. the localization of $k[X,Y]/I(C)$ at $(X-a,Y-b)$) is a discrete valuation ring w.r.t. $K$ and $k$,... | Suppose that $C/k$ is an integral projective curve. Then, there is a bijection
$$\left\{\text{points of }C\right\}\longleftrightarrow\left\{\begin{matrix}\text{discrete valuations}\\\text{of }K(c)\\ \text{which are trivial}\\ \text{on }k\end{matrix}\right\}$$
The mapping being, as you indicating, sending $p$ to the val... | {
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Dense subset of continuous functions Let $C([0,1], \mathbb R)$ denote the space of real continuous functions at $[0,1]$, with the uniform norm. Is the set $H=\{ h:[0,1] \rightarrow \mathbb R : h(x)= \sum_{j=1}^n a_j e^{b_jx} , a_j,b_j \in \mathbb R, n \in \mathbb N \}$ dense in $C([0,1], \mathbb R)$?
| Yes, by the general Stone-Weierstrass Approximation Theorem. Indeed, $H$ is an algebra of continuous real-valued functions on the compact Hausdorff space $[0,1]$, and $H$ separates points and contains a non-zero constant function. That is all we need to conclude that $H$ is dense in $C([0,1],\mathbb{R})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1045778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove $g(a) = b, g(b) = c, g(c) = a$
Let
$$f(x) = x^3 - 3x^2 + 1$$
$$g(x) = 1 - \frac{1}{x}$$
Suppose $a>b>c$ are the roots of $f(x) = 0$. Show that $g(a) = b, g(b) = c, g(c) = a$.
(Singapore-Cambridge GCSE A Level 2014, 9824/01/Q2)
I was able to prove that
$$fg(x) = -f\left(\frac{1}{x}\right)$$
after which I have co... |
Note: this is only a Hint
Using vieta's formulas
$$a+b+c=3$$
$$ab+bc+ca=0$$
$$g(a) = 1 - \frac{1}{a}=\frac{a-1}{a}=b\implies a-1=ab\tag{1}$$
$$g(b) = 1 - \frac{1}{b}=\frac{b-1}{b}=c\implies b-1=bc\tag{2}$$
$$g(c) = 1 - \frac{1}{c}=\frac{c-1}{c}=a\implies c-1=ca\tag{3}$$
Adding $(1),(2),(3)$
$$a+b+c-3=ab+bc+ca$$
Which... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1045961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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Find area under the curve of a standard normal distribution Given a standard normal distribution, how can I find the
area under the curve that lies:
1. to the left of z = −1.39;
2. to the right of z = 1.96;
3. between z = −2.16 and z = −0.65;
| The areas for one, two, and three standard deviations about the mean are well known, as .68, .95, .997 roughly. But in general, there's no way to find the area under the standard normal distribution without using a computer, calculator, or a chart.
edit1: most charts will give the area to the left of a given $z$-score... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1046046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a shape with infinite area but finite perimeter? Is this really possible? Is there any other example of this other than the Koch Snowflake? If so can you prove that example to be true?
| One can have a bounded region in the plane with finite area and infinite perimeter, and this (and not the reverse) is true for (the inside of) the Koch Snowflake.
On the other hand, the Isoperimetric Inequality says that if a bounded region has area $A$ and perimeter $L$, then
$$4 \pi A \leq L^2,$$
and in particular, f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1046108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "79",
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"answer_id": 5
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Bochner Integral: Approximability Problem
Given a measure space $\Omega$ and a Banach space $E$.
Consider a Bochner measurable function $S_n\to F$.
Then it admits an approximation from nearly below:
$$\|S_n(\omega)\|\leq \vartheta\|F(\omega)\|:\quad S_n\to F\quad(\vartheta>1)$$
(This is sufficient for most cases regard... | This is a modified version from: Cohn: Measure Theory
Enumerate a countable dense set:
$$\#S\leq\mathfrak{n}:\quad S=\{s_1,\ldots\}\quad(\overline{S}=F\Omega)$$
Regard the finite subsets:
$$S_K:=\{s_1,\ldots,s_K\}$$
Construct the domains by:
$$A_k:=A_n(s_k):=\{\omega:\|s_k\|\leq\|F(\omega)\|\}\cap\{\omega:\|F(\omega)-s... | {
"language": "en",
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Which Sobolev-Space to use to formulate weak biharmonic equation, $H^2_0$ or $H_0^1\cap H^2$? For the weak formulation of the biharmonic equation on a smooth domain $\Omega$
$$
\Delta^2u=0\;\text{in}\;\Omega\\
u=0, \nabla u\cdot \nu=0\; \text{on}\; \partial\Omega
$$
why does one take $H^2_0(\Omega)=\overline{C_c^\infty... | Here is what I thought. I am not very sure but I am happy to discuss with you.
The way we cast $\Delta^2u=0$ into a weak formulation, so that we could use Lax-Milgram, tells us that $H_0^2$ is a suitable space.
Suppose we have a nice solution already, then we test $\Delta^2u=0$ with a $C^\infty$ function $v$ and see w... | {
"language": "en",
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Linear independence of a set of mappings $Map(\mathbb{R},\mathbb{R}):=$ The set of all mappings from $\mathbb{R} \rightarrow \mathbb{R}$
For every $a \in \mathbb{R}$ there is a Funktion $f_{a}:\mathbb{R} \rightarrow \mathbb{R}$ with:
$$
f_{a}(x) = \left\{
\begin{array}{l l}
(a-x)^{3} & \quad \text{if $x \le a$}\... | You have to show that if you have a finite linear combination of functions of the form $f_a$, which is identically zero, then all coefficients have to be zero. So let $\lambda_1f_{a_1}+\ldots+\lambda_nf_{a_n}=0$, that is, $\lambda_1f_{a_1}(x)+\ldots+\lambda_nf_{a_n}(x)=0$ for each $x\in\mathbb R$. We want to show that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1046396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Determining the value of ECDF at a point using Matlab I have a data $X=[x_1,\dots,x_n].$
In Matlab, I know by using
[f,x]=ecdf(X)
plot(x,f)
we will have the empirical distribution function based on $X$.
Now, if $x$ is given, how will I know the value of my ECDF at this point?
| Using interp1 is a nice idea. But we should not use 'nearest' option. Instead, to get the right result we must use 'previous' option because ecdf functions are flat except their jumping points. Also don't forget dealing with. I recommend, if [f, x] is given from ecdf command, to use
y = interp1(x, f, vec_eval, 'previou... | {
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Let f(x) be a non negative continuous function on R such that $f(x) +f(x+\frac{1}{3})=5$ then calculate ........ Problem :
Let f(x) be a non negative continuous function on R such that $f(x) +f(x+\frac{1}{3})=5$
then calculate the value of the integral $\int^{1200}_0 f(x) dx$
My approach :
Given that : $f(x) +f(x+\f... | Hint: Note
$$\int_{1/3}^{2/3}f(x)dx= \int_0^{1/3}f(x+\frac{1}{3})dx$$
and hence
$$ \int_0^{1/3}f(x)dx+\int_{1/3}^{2/3}f(x)dx=\int_0^{1/3}(f(x)+f(x+\frac{1}{3}))dx=\frac{5}{3}.$$
Then write
$$\int_0^{1200}f(x)dx=\int_0^{1/3}f(x)dx+\int_{1/3}^{2/3}f(x)dx+\cdots$$
and you will get the answer.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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3 balls in a box We have a box with $3$ balls, that can be black or white.
We extract a ball, and it's white. Then we put the ball in the box, we extract again a ball and it's white.
What is the probability that in the box there are $3$ white balls ?
I obtain $\frac{9}{14}$ but I'm not sure about this result.
| Assume each ball has the same probability of being black or white. The prior probability of having $n$ white balls is $\binom 3n\frac18$, $n=0,1,2,3$.
Given there are $n$ white ball(s) in the box, then the probability of drawing two white balls with replacement is $\left(\frac n3\right)^2$.
The posterior probability of... | {
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Proving that the order of a group is greater than or equal to the product of orders of 2 subgroups. Let $H$ and $K$ be subgroups of a finite group $G$ such that $H \cap K = \{e\}$
Prove that $|G|\ge |H||K|$
What I think is the correct step is to consider the cosets $hK, h \in H$, and then using properties of cosets to ... | Hint: Consider $\phi: H \times K \to G$ given by $\phi(h,k)=hk$. Prove that $\phi$ is injective.
| {
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"timestamp": "2023-03-29T00:00:00",
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System of congruence equations I have a system of congruence eqs
$$
\begin{cases}
x \equiv 14 \pmod{98} \\
x \equiv 1 \pmod{28}
\end{cases}
$$
I have calculated $\text{gcd}(98,28) = 14$.
I can from the congruence eqs get $x = 14+98k$ and $x = 1+28m$.
I equate these
$$
14+98k = 1+28m \Leftrightarrow 28m - 98k = 13
$$
I ... | You are right, but this is faster:
$x\equiv 14\pmod{98}$ implies that $x$ is a multiple of $7$, but $x\equiv 1\pmod{28}$ implies that it isn't.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1046815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Finding Continuous Functions Find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x \in \mathbb{R}$, $f(x) + f(2x) = 0$
I'm thinking;
Let $f(x)=-f(2x)$
Use a substitution $x=y/2$ for $y \in \mathbb{R}$.
That way $f(y)=-f(y/2)=-f(y/4)=-f(y/8)=....$
Im just not sure if this is a good approa... | By induction we prove that
$$f(x)=(-1)^nf\left(\frac x{2^n}\right)$$
but by the sequential characterization of the continuity we have
$$f\left(\frac x{2^n}\right)\xrightarrow{n\to\infty}f(0)$$
so since for $x=0$ we get $f(0)=0$ so $f(x)$ is constant
$$f(x)=f(0)=0\quad\forall x\in\Bbb R$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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Show that $x^a+x-b=0$ must have only one positive real root and not exceed the $\sqrt[a]{b-1}$ If we take the equation $$x^3+x-3=0$$ and solve it to find the real roots, we will get only one positive real roots which is $(x=1.213411662)$. If we comparison this with $\sqrt[3]{3-1}=1.259921$, we will find that $x$ is les... | If $f(x)=x^a+x-b$ then $f'(x)=ax^{a-1}+1>0,$ since $a\in\mathbb{R}_+.$ Thus $f$ is strictly increasing and so it must take the value $0$ at most one time. Now, since $f(0)=-b<0$ and $f(b)>0$ it follows from the Intermediate Value Theorem that there exists a root in the interval $(0,b).$ So, $f$ has exactly one root.
Fi... | {
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Prove there exists a real number $x$ such that $xy=y$ for all real $y$
Prove: There exists a real number $x$ such that for every real number $y$, we have $xy=y.$
In class I learned that I can prove a statement by:
*
*proving the contrapositive,
*proof by contradiction,
*or proof by cases.
Can I do something al... | Questions like this are difficult to give good answers to, not because the proofs themselves are difficult or deep, but because it's unclear what you're allowed to assume as already known, hence what constitutes an "acceptable" proof. It's tempting to say the proofs here boil down to saying "Let $x=1$" (for the first ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1047154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 1
} |
Show that $F$ and $G$ differ by a constant Suppose $F$ and $G$ are differentiable functions defined on $[a,b]$ such that $F'(x)=G'(x)$ for all $x\in[a,b]$. Using the fundamental theorem of calculus, show that $F$ and $G$ differ by a constant. That is, show that there exists a $C\in\mathbb R$ such that $F(x)-G(x)=C$.
I'... | As you said, based on the fundamental theorem of calculus: $\int_a^xF'(x) = F(x) + C_1$ and $\int_a^xG'(x) = G(x) + C_2$, where $C_1$ and $C_2$ are constants from $\mathbb R$. As $F'(x) = G'(x)$, the above expressions are equal too, hence: $F(x) + C_1 = G(x) + C_2 \Leftrightarrow F(x) - G(x) = C_2 - C_1 = C_3$, where C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1047255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding Exact Value $7\csc(x)\cot(x)-9\cot(x)=0$ The values for $x$ on $[0,2\pi)$ solving $7\csc(x)\cot(x)-9\cot(x)=0$ are?
I think that $\dfrac{\pi}2$ is one but I can't find the others. what are the others?
| $7\csc x\cot x-9\cot x=0\implies7\cos x-9\cos x\sin x=0\implies (\cos x)(7-9\sin x)=0$,
so $\cos x=0\implies x=\frac{\pi}{2}$ or $x=\frac{3\pi}{2}$
and $\sin x=\frac{7}{9}\implies x=\sin^{-1}\frac{7}{9}$ or $x=\pi-\sin^{-1}\frac{7}{9}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1047419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Car's airbag deployment system. Let's discuss the math! I'm trying to explore the concepts and/or Calculus/Physics behind auto airbag deployment... I am assuming the computer feeds real time velocity points and the ECU is monitoring for a sudden drop in rate of change / sudden spike in negative acceleration?
Does thi... | It is more complicated than that. They probably have 3 axis measurements of acceleration and angular acceleration, so can trigger on a side impact (where the forward velocity doesn't change much) and a rear impact (where the acceleration is unrealistically positive). They also read very frequently so they can filter ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1047509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum of series involving 1+cos(n)... $$\sum_{n=2}^\infty{1+\cos(n)\over n^2}$$
I justified that it converges absolutely by putting it less than to $\frac{1}{n^2}$ where $p=2>1$ meaning that it converges absolutely. Would this be a correct way to solve this problem?
| Yes, as you say, $$\sum_{k=2}^\infty\left \lvert \frac{1+\cos(n)}{n^2}\right \lvert\leq \sum_{k=2}^\infty \frac{2}{n^2}=\frac{\pi^2}{3}-2,$$
which shows absolute convergence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1047587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Does a diagonal matrix commute with every other matrix of the same size? Does a diagonal matrix commute with every other matrix of the same size?
I'm stuck on one line of a proof that I am writing, and I would like to switch order between a non-diagonal and a diagonal matrix.
Thanks,
| In general, a diagonal matrix does not commute with another matrix. You can find simple counterexamples in the comments. For a matrix to commute with all the others you need the matrix to be scalar, i.e. diagonal with entries on the diagonal which are all the same.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1047681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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} |
Can derivative of Hurwitz Zeta be expressed in Hurwitz Zeta? Can the derivative of Hurwitz Zeta function by the first argument be expressed in terms of Hurwitz Zeta and elementary fuctions?
There is a formula which expresses Hurwitz Zeta through its derivative:
$$\zeta '\left(z,\frac{q}{2}\right)-2^z \zeta '(z,q)+\zeta... | Note that the derivative of the Hurwitz Zeta Function can be calculated as following (similiar to calculating the derivative of the Riemann Zeta):
\begin{align} \frac{d}{ds} \zeta(s,q) &= \frac{d}{ds} \sum_{n=0}^{\infty} (n+q)^{-s} \\ &= \sum_{n=0}^{\infty} \frac{d}{ds} (n+q)^{-s} \\ & = \sum_{n=0}^{\infty} - \ln(n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1047797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
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A basic measure theory question on lebesgue integral Let $\mu$ and $\nu$ are probability measures on a complete separable space $S$. Suppose, for every real-valued continuous function on $S$ we have that
$$\int fd\mu = \int fd\nu$$
does it imply $\mu = \nu$. looks like approximating characteristic functions by continuo... | Let $U$ be open and consider
$$
f_n(x)=(n\cdot d(x,U^c))\wedge1,\quad n\geqslant 1.
$$
Then $f_n$ is non-negative and continuous for all $n\geqslant 1$ and $f_n\uparrow \mathbf{1}_U$. Thus by the monotone convergence theorem we have $\mu(U)=\nu(U)$. Since the Borel sigma-algebra is generated by the open sets (which are... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it possible to simulate a floor() function with elementary arithmetic? I'm using a "programming language" that only allows basic operations: addition, subtraction, multiplication, and division.
Is it possible to emulate a floor function (i.e. drop the decimals a number) by using only those operators? I'm only intere... | Programming languages typically use floating-point arithmetic defined by the IEEE 754 standard. Roughly speaking, if the exact result of an operation calculated using mathematical real arithmetic has an absolute value $2^k ≤ x < 2^{k+1}$, then x will be rounded to the nearest integer multiple of $2^{k-52}$. The excepti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1048054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "51",
"answer_count": 10,
"answer_id": 5
} |
How to find quotient groups? I am struggling to understand how do quotient groups work. For example - for the quotient of real numbers (without zero) under multiplication and positive real numbers, the correct quotient should be isomorphic to Z2. Why is that? How can I find what is it isomorphic to? How does in this ca... | Define
$$\phi:\Bbb R^*\to\{-1\,,\,1\}\le\Bbb C^*\;,\;\;\phi(r):=\begin{cases}\!\!-1&,\;\;r<0\\{}\\\;\,1&,\;\;r>0\end{cases}$$
Check the above is a group homomorphism, that $\;\ker\phi=\Bbb R^*_+\;$ and apply the first isomorphism theorem.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Easiest proof for $\sum_{d|n}\phi(d)=n$ To prove $\sum_{d|n}\phi(d)=n$. What is the easiest proof for this to tell my first year undergraduate junior. I do not want any Mobius inversion etc only elementry proof. Tthanks!
| Write $n = \prod p^{a_p}$
The divisors of $p$ are
$$
\prod p^{b_p}, b_p \le a_p
$$
so the sum on the left hand side is
$$
\sum_{b_2=0}^{a_2}\dots \sum_{b_P=0}^{a_P}
\phi(\prod_{p \text{ prime}, p|n, =2}^P p^{b_p})
= \sum_{b_2=0}^{a_2}\dots \sum_{b_P=0}^{a_P}
\prod_{p \text{ prime}, p|n, b_p>0} \left(1-\frac 1p\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1048209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Compute the following sum $ \sum_{i=0}^{n} \binom{n}{i}(i+1)^{i-1}(n - i + 1) ^ {n - i - 1}$? I have the sum
$$ \sum_{i=0}^{n} \binom{n}{i}\cdot (i+1)^{i-1}\cdot(n - i + 1) ^ {n - i - 1},$$
but I don't know how to compute it. It's not for a homework, it's for a graph theory problem that I try to solve.
| Look up Abel's binomial theorem,
for example, here:
http://en.wikipedia.org/wiki/Abel's_binomial_theorem
I find this funny because,
immediately preceding this,
I answered a question
by giving a reference to
Abel's summation formula.
This means,
of course,
that somebody will ask
about solving the quintic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1048289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Identifying $\mathbb{C}^*/\mathbb{R}^{+}$ I define a Surjective Homomorphism $\varphi:\mathbb{C}^*\to S^1$ by $z\mapsto {z\over |z|}$
Ker$\varphi=\{z\in\mathbb{C}^*:\varphi(z)=1\}\Rightarrow\{z: z=|z|\}=\mathbb{R}^{+}$
So, $S^1=\mathbb{C}^*/\mathbb{R}^{+}$
am I right?
| Yes, you are right. You can visualize this result by identifying each element of $\mathbb{C}^*/\mathbb{R}^+$ with a ray out from $(0, 0)$ in $\mathbb{C}^*$. Then to multiply two rays, multiply representatives from them and take the ray containing the result. Clearly the product of two rays in this sense only depends on... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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orthogonal subspaces in $\mathbb R^2$ Here is the question:
Let W be a subspace of $R^n$ with an orthogonal basis {$w_1$,...,$w_p$}, and let {$v_1$,...,$v_q$} be and orthogonal basis for W$\perp$. Assume that both p >= 1 and q>=1.
(1) For the case when n=2, describe geometrically W and W$\perp$. In particular, what are... | If W is a space of two dimensions then it is not a subspace of R^2 it is the whole space.
The space perpendicular to W is null.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1048553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Continuity and adherence For two topological spaces $E$ and $F$
Please how to prove that $$f(\overline{A})\subset \overline{f(A)},\forall A\Rightarrow f:E\rightarrow F ~\text{is continuous}$$
Thank you
| Suppose that $f$ is not continuous. Then there is some closed set $C\subset F$ such that $f^{-1}(C)$ is not closed. Call $A=f^{-1}(C)$. Then there is some $x\in\overline A$ such that $f(x)\notin C$. That is, $f(\overline A)\not\subset C=\overline C=\overline{f(A)}$.
Arternatively, suppose that $f$ is not continuous at ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1048767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Rank of a linear transformation $T(x_1,x_2)=(x_1-x_2,5x_1)$
Find the associated matrix and compute the rank and nullity of the linear transformation $T:\mathbb{R}^2\to\mathbb{R}^2$ given by $T(x_1,x_2)=(x_1-x_2,5x_1)$.
The associated matrix $A$ is
$$
\left[\begin{matrix} T(\mathbf{e_1}) & T(\mathbf{e_2}) \end{matrix}... | To find the the rank, you can use the rank-nullity theorem:
$$rank (T)+nullity(T)=2.$$
Since $nullity(T)=0$ as you have found, we have $rank (T)=2$.
Or you can argue by saying that the columan space of $T$, $C(T)$, is spanned by $T(e_1)=\left[\begin{matrix} 1 \\ 5 \end{matrix}\right]$ and $T(e_2)=\left[\begin{matri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1048884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compute $\bar z - iz^2 = 0$ $\bar z - iz^2 = 0, i = $ complex unit.
I've found 2 solutions to this, like this:
$x - iy - i(x+iy)^2 = 0$
$i(-x^2+y^2-y)+2xy+x=0$
$2xy + x = 0$ --> $y=-\frac{1}{2}$
$x^2 - y^2 + y = 0$ --> $x_1=\sqrt\frac34$ $x_2=-\sqrt\frac{3}{4}$
Solution 1: $z = \sqrt\frac34 - \frac12i $
Solution 2: $z ... | $$\bar z=iz^2$$
Equate absolute values:
$$|z|=|\bar z|=|iz^2|=|z|^2\Longrightarrow|z|=0,1.$$
One solution is $z=0$; other solutions satisfy $|z|=1$, so $\bar z=\frac{|z|^2}z=\frac1z$ and the equation simplifies to
$$\frac1z=iz^2,$$
that is,
$$z^3=\frac1i=i^3.$$
One solution obviously is$$z=i.$$ The other cube roots of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1048955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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$(X,d)$ is a complete metric space iff $(X,d')$ is a complete metric space. I recently got asked this question on an exam and I wasn't able to give a solution.
If $(X,d)$ is a metric space and we define
$$d'(x,y)= \frac {d(x,y)}{1+d(x,y)}$$
(I've already proved that $(X,d')$ is a metric space.)
Prove that $(X,d)$ is ... | You can see that $d' = {d \over 1+ d} \le {d \over 1} = d$, hence a $d$-Cauchy sequence is a $d'$-Cauchy sequence.
Note that if $d'(x,y) \neq 1$, then since $d'(x,y) = { d(x,y) \over 1 + d(x,y) }$, then $d(x,y) = { d'(x,y) \over 1 - d'(x,y) }$.
Now suppose $x_n$ is a $d'$-Cauchy sequence, and choose $N$ such that if $m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A proof pertaining to the projector operator Let $H_{1}$ be any subspace of a Hilbert space $H$, and let $H_{2} = H_{1}^{\bot}$ be the orthogonal complement of $H_{1}$, so that an arbitrary element $h \in H$ has a unique representation of the form $h = h_{1} + h_{2}$ where $h_{1} \in H_{1}$ and $h_{2} \in H_{2}$. Let $... | Let's agree that a completely continuous operator $T:X\rightarrow Y$ between Banach spaces is one that sends weakly convergent sequences to norm convergent ones.
For the first part, suppose that $H_1$ is finite dimensional. Consider a weakly convergent sequence $(x_n)_n$ in $H$. The projection $P:H \rightarrow H$ can b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find the equation for a parabola for which you are given two points and the vertex? I was originally given the value $(4,-2)$ as the vertex of a parabola and told that it also includes the value $(3,-5)$. From this point, I deduced that the next point would have the same y-value as the point whose x-value is equ... | Vertex form: $g(x) = a(x-h)^2+k$ where $(h,k)$ is your vertex.
Here $$g(x) = a(x-4)^2-2 $$ and $$g(3) = -5 \Rightarrow -5 = a(3-4)^2-2 \Rightarrow -3 = a \Rightarrow g(x) = -3(x-4)^2-2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Trouble understanding how $\int_c^d f \leq 0$ implies $f \leq 0$. We are asked to suppose that we have a function $f:[a,b] \rightarrow \Bbb R$ which is continuous and has the property that
$$\int_{c}^{d} f \leq 0 \quad \text{ whenever } a \leq c < d \leq b.$$
We want to prove that $f(x) \leq 0$ for all $x \in [a,b]$.
... | For $\forall x\in(a,b)$, there is $\varepsilon >0$ such that $(x-\varepsilon,x+\varepsilon)\subset(a,b)$. Thus by the Mean Value Theorem for Integrals, we have
$$ \int_{x-\varepsilon}^{x+\varepsilon}f(x)dx=2\varepsilon f(\xi)\le0$$
for some $\xi\in(x-\varepsilon,x+\varepsilon)$. Thus $f(\xi)\le 0$. Letting $\varepsilon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Matlab wrong cube root How can I get MATLAB to calculate $(-1)^{1/3}$ as $-1$? Why is it giving me $0.5000 + 0.8660i$ as solution? I have same problem with $({-1\over0.1690})^{1/3}$ which should be negative.
| In the following I'll assume that $n$ is odd, $n>2$, and $x<0$.
When asked for $\sqrt[n]{x}$, MATLAB will prefer to give you the principal root in the complex plane, which in this context will be a complex number in the first quadrant. MATLAB does this basically because the principal root is the most convenient one for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 1
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What is $\langle 3u+2v , -u+4v\rangle$ Given $\lVert u\rVert$ , $\lVert v\rVert$ , and $\langle u,v\rangle$? What is $\langle 3u+2v , -u+4v\rangle$ Given $\lVert u\rVert$ , $\lVert v\rVert$ , and $\langle u,v\rangle$?
\begin{align}
\lVert u\rVert &= 3\\
\lVert v\rVert &= 5\\
\langle u,v\rangle &= -4
\end{al... | $\langle 3u+2v,-u+4v\rangle =\langle 3u,-u\rangle +\langle 3u,4v\rangle +\langle 2v,-u\rangle +\langle 2v,4v\rangle$
\begin{align}
&=-3\langle u,u\rangle +12\langle u,v\rangle -2\langle v,u\rangle +8\langle v,v\rangle\\
&=-3\cdot 9 + 12\cdot -4 - 2\cdot 4 + 8\cdot 25
\end{align}
since $\lVert... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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proving Riemann-Lebesgue lemma I have looked at proofs of the Riemann-Lebesgue lemma on the internet; all of these proofs use the technique of Riemann integration and making step functions.
E.g.(https://proofwiki.org/wiki/Riemann-Lebesgue_Lemma)
But if we know Parseval's identity (or Plancherel's identity for Fourier t... | There's a really nice proof that "avoids step functions".
With a $u-$substitution you can conclude that $$ \hat{f}(\xi) = \int_{\mathbb{R}} f(x)e^{ix\xi} dx
\\
= \int_{\mathbb{R}} f(x + \frac{\pi}{\xi})e^{ix\xi}e^{i \pi} dx \\
= - \int_{\mathbb{R}} f(x + \frac{\pi}{\xi})e^{ix\xi} \ dx $$
By averaging these tw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Examples of categories which naturally include End(O) as object I want examples of categories $\textbf C$ which naturally include $End_{\textbf C}(O)$ as object for objects $O$ in the category. The set of all endomorphims is always a monoid under the composition of morphisms, but beside from that.
Counterexamples is al... | $End(O)$ is a set (or a monoid), it belongs to the category of sets and nothing else.
Perhaps you are interested in the notion of a cartesian closed category. There one has an internal hom object $\underline{\hom}(x,y)$ for all objects $x,y$, in particular $\underline{\mathrm{End}}(x):=\underline{\hom}(x,x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How many ways can 25 red balls be put into 3 distinguishble boxes if no box is to contain more than 15 balls? I'm reading Martin's: Counting: The Art of Enumerative Combinatorics.
How many ways can 25 red balls be put into 3 distinguishble boxes if no box is to contain more than 15 balls?
I understand that it's the s... | The number of solutions to $x + y + z = 25 - 15$ is also the number of ways
to put the balls in the boxes if there must be at least $15$ balls in the first box.
But some of those solutions correspond to arrangements with only $15$ balls
in the red box, which we do not want to exclude from our final answer.
We want to s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do you integrate the function $\frac 1{x^2 - a^2}$, where a is a constant? My problem with integrating by parts is that it always ends up being recursive, as I feel like I'm going in loops. Would appreciate it if someone can help me understand the process.
$$\int \dfrac 1{x^2-a^2}dx$$
I know the answer is supposed... | The integrand factors as
$$\frac{1}{(x - a) (x + a)},$$
so we can decompose it via the Method of Partial Fractions---
$$\frac{1}{(x - a) (x + a)} = \frac{K}{x + a} + \frac{L}{x - a}$$
---and cross-multiply and collect like terms to solve for $K$ and $L$. Then, we can integrate the terms separately using the elementary ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1049944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Is this c the same as that c? Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.
| I am not very good at analysis, but the answer to your question is clearly "the $c$'s are the same", by plain elementary logic. I'll try to explain it to you:
call $\varphi(\alpha)$ the "sentence"
$$\forall x \ \left[ \overline{\lim}_{r \to 0} \mu(B(x,r)/r^s < \alpha \right]$$
and call $\psi(\beta)$ the "sentence"
$$\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1050033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to efficiently determine if two propositions are equivalent? Given any two arbitrary propositional formulas only using $\land, \lor, \lnot$, e.g., $\lnot(A \land (B \lor \lnot B) \land C)$ and $\lnot C \lor \lnot A$, how can I (or a computer) efficiently determine if they are equivalent?
The formulas may contain ma... | Efficient : NO.
See Boolean satisfiability problem, abbreviated as SATISFIABILITY or SAT :
There is no known algorithm that efficiently solves SAT, and it is generally believed that no such algorithm exists.
To ask for two propositional formulae $\mathcal A, \mathcal B$ are equivalent can be reduced to asking if :
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1050127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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In a binary vector $ x \in \{0,1\}^{k}$ what does the $^{k}$ mean? In a binary vector $ x \in \{0,1\}^{k}$ what does the $^{k}$ mean? I understand that $\in$ means 'is a possible outcome' or 'in' so x can be 0 or 1, but I'm not sure what the $^{k}$ means.
| Just as points in $\mathbb R^k$ consist of ordered tuples of $k$ real numbers, so points in $\left\{0,1\right\}^k$ consist of ordered tuples of $k$ "bits".
The typical point $x = (x_1,x_2,\ldots,x_k)$, where each $x_i\in\left\{0,1\right\}$ (that is, each $x_i$ is $0$ or $1$).
| {
"language": "en",
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For which $n\in\mathbb{Z}^+$ is there a $\sigma\in S_{14}$ such that $|\sigma|=n$? For which $n\in\mathbb{Z}^+$ is there a $\sigma\in S_{14}$ (the group of permutations on $\{1,2,\dots,14\}$) such that $|\sigma|=n$ (where $|\sigma|$ is the order of $\sigma$)?
I know you could just enumerate all elements of $S_{14}$, bu... | The order of a cycle is the length of the cycle, and every permutation is a product of commuting cycles. Thus the possible orders are all least common multiples of sets of positive integers whose sums are less than or equal to 14.
By hand I found 29 distinct combinations yielding 13, 11, 22, 33, 9, 18, 36, 45, 8, 24, 4... | {
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what can be said about the solution of the following differential equation $y(x)=f(x)+\log(1+y'(x))$ with $\lim_{x \to 0}f(x)=\infty,f(\infty)=0$ and $f(x)$ is contiuoues and decreasing. In particular, can we prove that the solution $y(x)$ is convex in general?
from $e^{y(x)-f(x)}=1+y'(x)$ we get $1+y'(x) > 0$ and $y'(... | If you put $z(x)=x+y(x)$ and $g(x)=x+f(x)$, the equation becomes $$z'(x)=e^{z(x)-g(x)},$$
which is separable, with general solution
$$z=-\ln\bigl(C-h(x)\bigr),\qquad h(x)=\int_0^x e^{-g(t)}\,dt.$$
Then
$$z'(x)=\frac{e^{-g(x)}}{C-h(x)},\quad
z''(x)=\frac{e^{-2g(x)}-(C-h(x))e^{-g(x)}g'(x)}{(C-h(x))^2},$$
so the sign of... | {
"language": "en",
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Find a point on the line: $y=2x-5$ that is the closest to $P(1,2)$ This is our line: $f(x)=2x-5$
I have to find a point on this line that is the closest to the point $P(1,2)$. How do I go on about solving this? Should I use derivative and distance from the point to the line? I've done this so far:
$d\to$ distance betwe... | Actually, there is a simpler way. Since $f$ is a line, the point $P$ and the point on $f$ closest to $P$ will define a line perpendicular to $f$. That means that you can get the equation for the line with slope $m=-\frac{1}{2}$ (the inverse of the reciprocal of the slope of $f$) through the point $(1,2)$, and then find... | {
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Prove that $\frac{\binom{p}{k}}{p}$ is integral for $k\in \{1,..,p-1\}$ with $p$ a prime number with a particular method I started by induction on $k$
For $k=1$ then : $1\in \mathbb{N}$
For $k=2$ then : $\frac{(p-1)}{2!} \in \mathbb{N}$ , indeed for all $p>{2}$, $p-1$ is even. (We still have $k<p$ it's important).
Sup... | By definition,
$$\binom{p}{k} = \frac{p!}{k!(p-k)!}$$
Since $p$ is prime, it's relative prime to all smaller numbers, and therefore for $k<p$, it cannot be a factor of $k!$. Now if further $k>0$, then $p-k<p$, and thus $p$ is not a factor of $k!(n-k)!$. Since by definition, $p$ is a factor of $p!$, and $p$ is prime, it... | {
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Proof involving chords of a circle
In a circumference with center $O$, three chords $\overline{AB},\overline{AD}$ and $\overline{CB}$ such that the last two intersect in $E$. Show that $AE·AD+BE·BC=AB^2 $.
Added: $O\in\overline{AB}$.
Hi, I have been trying to solve this problem with the power of a point with respect... | You were on a good track. You found that
$$2 \cdot AB^2 = AD^2 + BD^2 + BC^2 + AC^2.$$
But you want something with $AD \cdot AE,$ not $AD^2,$ you don't want to see $BD^2,$
and so forth.
But notice that $AC^2 = AE^2 - CE^2$ and $BD^2 = BE^2 - DE^2,$
and also $AD = AE + DE$ and $BC = BE + CE.$
So this suggests you could ... | {
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Prove that $1 + 4 + 7 + · · · + 3n − 2 = \frac {n(3n − 1)}{2}$ Prove that
$$1 + 4 + 7 + · · · + 3n − 2 =
\frac{n(3n − 1)}
2$$
for all positive integers $n$.
Proof: $$1+4+7+\ldots +3(k+1)-2= \frac{(k + 1)[3(k+1)+1]}2$$
$$\frac{(k + 1)[3(k+1)+1]}2 + 3(k+1)-2$$
Along my proof I am stuck at the above section where it w... | Here is @Shooter's answer shown a different way. Let's take the example of n = 8:
1 + 4 + 7 + 10 + 13 + 16 + 19 + 22 = 92
Let's rearrange this and group:
(1 + 22) + (4 + 19) + (7 + 16) + (10 + 13) = 92
Now let's add the groups and look for a pattern:
23 + 23 + 23 + 23 = 92
That's it. The n = 8 example is just what... | {
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Check:etermine the number of Sylow $2$-subgroups and Sylow $3$-subgroups that $G$ can have. Let $G$ be a group of order $48$. By the $1$st Sylow theorem $G$ has a Sylow $2$-subgroup and a Sylow $3$-subgroup. Suppose none of these are normal. Determine the number of Sylow $2$-subgroups and Sylow $3$-subgroups that $G$ c... | Assume that $n_3 = 16$. If $P_i, P_j \in Syl_3(G)$ and $P_i \neq P_j$, then $P_i \cap P_j = 1$. Each Sylow $3$-subgroup has $3$ elements, the identity and two others. The intersections only contain the identity, and hence from each Sylow $3$-subgroup you can count $2$ new elements.
Now $n_2 = 3$. If $Q_i, Q_j \in Syl_2... | {
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Finding the probability of a selecting at least 1 of an element. If there are 18 red and 2 blue marbles what is the probability of selecting 10 marbles where there is either 1 or 2 blue marbles in selected set.
Also, it seems intuitive that the probability should be twice that of selecting a single blue marble in a set... | Another approachment is through hypergeometric distribution.
*
*The probability that there is exactly one blue ball in the selected set is:
$$P(A_1)=\dfrac{\color{blue}{\dbinom 2 1}\cdot \color{red}{\dbinom {18} {10-1}}}{\dbinom {20}{ 10} }$$
*The probability that there are exactly 2 blue balls in the selected set... | {
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Is the symmetric definition of the derivative equivalent? Is the symmetric definition of the derivative (below) equivalent to the usual one?
\begin{equation}
\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}
\end{equation}
I've seen it used before in my computational physics class. I assumed it was equivalent but it seems like it w... | As I noted in a comment to the other answer, Milly's computation is incorrect. I am posting this answer to rectify the situation. The symmetric derivative is defined to be
\begin{align}
\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h}.
\end{align}
If $f$ happens to be differentiable, then the symmetric derivative reduces to th... | {
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Parametrizing intersection I'm working on a question that require parametrizing some curves:
C is the curve of intersection of the hyperbolic paraboloid $z = y^2 − x^2$ and the cylinder
$x^2 + y^2 = 1$ oriented counterclockwise as viewed from above. Let $x$ and $y$ be in terms of $t$ where $0 ≤ t ≤ 2π$.
I thought ... | Your answer is correct.
Try entering the answer $\bigl(\sin t, \cos t, -\cos 2t \bigr)$. This answer is equivalent to yours, but maybe the computer system is too dumb to realize this.
| {
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Is the exponential map for $\text{Sp}(2n,{\mathbb R})$ surjective? For $\mathfrak{g} := {\mathfrak s}{\mathfrak p}(2n,\mathbb{R})$ and $G = \text{Sp}(2n,{\mathbb R})$, is the exponential map
\begin{equation}
\text{exp} : \mathfrak{g} \to G
\end{equation}
surjective?
If not then is it possible to access all of the sympl... | In his comment, user148177 already explained that the exponential function of $\text{Sp}_{2n}({\mathbb R})$ is not surjective.
Two factors, however, suffice:
Theorem (Polar decomposition) Any $g\in\text{Sp}_{2n}({\mathbb R})$ can uniquely be written as a product $g = h\cdot\text{exp}(X)$ with $h\in SO_{2n}({\mathbb ... | {
"language": "en",
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How many ambiguous dates exist? How many ambiguous dates are there in a year? An ambiguous date is one like 8/3/2007 which could either mean the 8th of March or the 3rd of August.
Is it right to say that 1/1/2007 must mean the first of January so there are 11 ambiguous dates for each month (1/2/2007, 1/3/2007, ..., 1/1... | The important question here, is when is a date ambiguous?
A date is ambiguous if the day (of the month) and the month (of the year) can be confused for one another; when the day can be a month as well.
Since there are 12 months in a year, the day can only be confused with the month if the day is in $[1, 12]$, so there... | {
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How to get ellipse cross-section of an ellipsoid I'm trying to get the major and minor radius of an ellipse which represents the cross-section of a given ellipsoid. This is particularly of interest in the field of RF propagation in terms of Fresnel zones and how they interact with a ground surface.
In my particular use... | You have sections with planes perpendicular to a principal axis of the ellipsoid. These ellipses are all similar. If $r$ is the radius of the Fresnel ellipsoid and $h$ is the Earth's height ( or bulge) at the midpoint between the transmitter and the receiver then the cross section perpedicular to the radius $r$ at dist... | {
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Let $G$ be a group. If $H\leq G$ is a subgroup and $N\vartriangleleft G$, then $HN$ is a subgroup of $G$. Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $G$. Prove that $HN=\{hn\:|\:h \in H, n \in N\}$ is a subgroup.
Here what I have so far. It is not really much I understand what I need to show but I am stu... | It is true that $h^{-1}nh=n'\in N\;\implies nh=hn'$ , because $\;N\lhd G\;$ , then
$$h_1n_1h_2n_2=h_1(n_1h_2)n_2=h_1(h_2n')n_2=(h_1h_2)(n'n_2)\in HN$$
| {
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Proving $\frac{200}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)\cosh\left(\frac{\pi}{2}(2n+1)\right)}=25$ I solve a partial differential equation (Laplace equation) with specific boundary conditions and I finally found the answer:
$$U(x,y)=\frac{400}{\pi}\sum_{n=0}^{\infty}\frac{\sin\left((2n+1)\pi x\right)\sinh\left... | We will evaluate
$$S=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)\cosh\left(\frac{\pi}{2}(2n+1)\right)}$$
Since the series is alternating, use $f(z)=\pi\csc(\pi z)$ and we have $$\oint\frac{\pi \csc(\pi z)}{(2z+1)\cosh\left(\frac{\pi}{2}(2z+1)\right)}\,dz$$
The poles are at $z=-\frac{1}{2}, \;\ z=n, \;\ z=\frac{(2n+1)\,i}... | {
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"timestamp": "2023-03-29T00:00:00",
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Problem involving system of differential equations Solve following system of diferential equations$$\begin{cases}
\frac{ds}{dt}=y+z\\ \frac{dy}{dt}=s+z\\ \frac{dz}{dt}=z-s.
\end{cases}.$$
I tried many tehniques without any success. I would appreciate some help with this problem. One of my tries
$$\frac{dz}{dt}=z-s\Rig... | Consider the vector $\boldsymbol V=\left[\begin{array}{r}s\\y\\z\end{array}\right]$. Then your system of equations is equal to:
$$\boldsymbol V'(t)=\left[\begin{array}{r}0&1&1\\1&0&1\\-1&0&1\end{array}\right]\boldsymbol V.$$
Let's call that matrix $\boldsymbol A$. Suppose that there is a solution $\boldsymbol V=\boldsy... | {
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"timestamp": "2023-03-29T00:00:00",
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Geometry Problem involving Midpoints and Analytic Geoometry In triangle $ABC, AB = AC, D$ is the midpoint of $\overline{BC}$, E is the foot of the perpendicular from D to $\overline{AC}$, and F is the midpoint of $\overline{DE}$. Prove that $\overline{AF}$ is perpendicular to $\overline{BE}$.
I am required to solve thi... | Here's a start.
Set point $C$ at $(a,b)$ (with $b \neq 0$). You can also specify $a \gt 0$ without loss of generality.
Now, you're in a position to define $D = ((a+x)/2, b/2).$
The slope of $\overline{AC}$ is $m_{AC} = b/a$, so the slope of a perpendicular to that line is $-a/b$.
Now, try to finish.
From there, you ca... | {
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Question regarding basis in vector spaces How can one prove the following proposition ?
$
B = (e_{1,...,} e_n )\, $ forms a basis for a space $V$ if and only if each vector of $V$ can only be written as an unique linear combination of elements from $B$ .
I'm really confused here, any ideas?
| For the first part, suppose $B$ is a basis. By definition a basis is a set of linearly independent vectors that spans $V$. Take then $v\in V$: $v$ can be written as a linear combination of elements of $B$ in this way $v=\sum_{i=1}^n\alpha_ie_i$. The problem here is to prove that this way is unique. Suppose then that it... | {
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Does the following alternating series converge or diverge? I have the following series that I have to check for convergence or divergence:
$$
\sum\limits_{n=1}^{\infty} \frac{\sin (n+1/2)\pi}{1 + \sqrt{n}}
$$
I know that it is an alternating series therefore I have to check for two conditions to be satisfied in order f... | $$
\sum\limits_{n=1}^{\infty} \frac{\sin (n+1/2)\pi}{1 + \sqrt{n}}=\sum\limits_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt n}
$$
so this is an alternating series. Generic term converges to $0$ $$\underset{n\to \infty }{\text{lim}}\frac{1}{\sqrt{n}+1}=0$$
Therefore the series converges and
$$\sum _{n=0}^{\infty } \frac{(-1)^n... | {
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Evaluation of $ \lim_{x\to 0}\left\lfloor \frac{x^2}{\sin x\cdot \tan x}\right\rfloor$ Evaluation of $\displaystyle \lim_{x\to 0}\left\lfloor \frac{x^2}{\sin x\cdot \tan x}\right\rfloor$ where $\lfloor x \rfloor $ represent floor function of $x$.
My Try:: Here $\displaystyle f(x) = \frac{x^2}{\sin x\cdot \tan x}$ is an... | Let me continue where you stopped in your post.
Expand the denominator for a few terms and perform the long division. You should arrive to $$\frac{x^2}{\sin x\cdot \tan x} =1-\frac{x^2}{6}-\frac{7 x^4}{120}+O\left(x^5\right)$$ which is definitely smaller than $1$.
I am sure that you can take from here.
| {
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what is the order and the degree of the given diff. equation? the given diff. equation is $$y=1+ \frac{dy}{dx}+\frac{1}{2!}\left(\frac{dy}{dx}\right)^2+\cdots+\frac{1}{n!}\left(\frac{dy}{dx}\right)^n$$
it is given that the order is $1$ and the degree is also $1$. But how it happens because the the highest order $1$ has... | If you really do mean that those (first) derivatives are raised to powers from $1$ to $n$, then it is a first order differential equation. However, it would be $n$th degree.
Here are the definitions of order and degree for an ODE.
| {
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Given radius, and many vertices on it, how can I find center of a sphere? I have a sphere, I know its radius. I also have the coordinates of 500 vertices which are on the sphere. How can I find the center coordinates of a sphere? Is there an easy way to do that? Thanks.
| You need just four vertices $A,B,C,D$. Let $O_A$ be the circumcenter of $BCD$ and $l_A$ the line through $O_A$ that is orthogonal to the $BCD$-plane. Let $O_D$ be the circumcenter of $ABC$ and $l_D$ the line through $O_A$ that is orthogonal to the $ABC$-plane. Then $l_A$ and $l_D$ meet in the centre of the sphere.
| {
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How to compute $\lim_{x \to 1}x^{\frac{1}{1-x}}$? I need to compute $$\lim_{x \to 1}x^{\frac{1}{1-x}}$$
The problem for me is negative $x$.
| Set $x-1=h$
So, we have $$\lim_{h\to0}(1+h)^{-\frac1h}=\left([1+h]^{\frac1h}\right)^{-1}=e^{-1}$$
Alternatively, if $A=\lim_{x\to1}x^{\dfrac1{1-x}},$
$\ln A=\lim_{x\to1}\dfrac{\ln\{1-(1-x)\}}{1-x}$
Setting $x-1=h,$
$\ln(A)=-\lim_{h\to0}\dfrac{\ln(1+h)}h=-1$
| {
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Maximum ratio between diameter of shape and diameter of enclosing circle For a 2-dimensional closed convex shape $C$, define:
*
*$d(C)$ = the diameter of $C$ (the largest distance between two points in $C$).
*$D(C)$ = the diameter of the smallest circle containing $C$.
Often $d(C)=D(C)$, for example in a unit squ... | I think $2/\sqrt 3 $ is the largest possible value: For simplicity only consider $C$ to be a convex polygon. If $C$ is a line, then the ratio is $1$.
Now think about triangles: If we fix the diameter of the enclosing circle, then the triangle that produces the largest ratio (i.e. the smallest $d(C)$) is equilateral.
... | {
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Candidates in an exam 443 candidates enter the exam hall. There are 20 rows of seats I'm the hall. Each row has 25 seats. At least how many rows have an equal number of candidates.
My attempt
Seat 25 in the first row 24 in the second and so on...but I cannot find the worst case scenario.
| Let $a_1,a_2\dots a_{20}$ be non-negative integers such that $a_1+a_2+\dots a_{20}=443$. Denote by $s$ the size of the largest subset of the $a$'s which all have the same value. I think we want to minimize this.
We shall prove it is three. To show it is more than two we notice that with $s=2$ the maximum sum would be ... | {
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Finding matrix for given recurrence For the recurrence relation:
$f(0)=1$
$f(1)=1$
$f(2)=2$
$f(2n)=f(n)+f(n+1)+n$
$f(2n+1)=f(n)+f(n−1)+1$
How to find square matrices $M_0, M_1$ and vectors $u, v$ such that if the base-2 expansion of $n$ is given by $e_1 e_2 \cdots e_j$, then $$f(n) = u M_{e_1} \cdots M_{e_j} v.$$ ??
| This is how I would approach this question, and I guess my answer would not be unique.
For each $n = e_1e_2\cdots e_j$, there are $7$ numbers I should store (as a row vector) to calculate $f(2n)$ and $f(2n+1)$:
$$uM_{e_1}M_{e_2}\cdots M_{e_j} = w(n) = \pmatrix {1&n&f(n-2)&f(n-1)&f(n)&f(n+1)&f(n+2)}.$$
To calculate $f(2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1053112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $f$ is a loop in $\mathbb{S}^{n}$, then is $f^{-1}(\{x\})$ a compact set in $[0, 1]$? If $f$ is a loop in $\mathbb{S}^{n}$, then is $f^{-1}(\{x\})$, $x \in \mathbb{S}^{n}$, a compact set in $[0, 1]$?
| This community wiki solution is intended to clear the question from the unanswered queue.
Since $\mathbb S^n$ is Hausdorff, singletons are closed.
The loop $f : [0, 1] \to \mathbb S^n$ is a continuous map, so the inverse image of closed sets is closed, in particular $f^{-1}(\{x\})$ is closed.
But $[0, 1]$ is compact ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How to isolate $n$ in the inequality $ 3n + 7n^3\gt c(17 + 34n^2) $? I have an equation
$$ 3n + 7n^3\gt c\left(17 + 34n^2\right)
$$
and I want to turn this inequality into something like
$$ n \gt c(\mbox{something that does not have}\ n)
$$
I don't know why but always get stumped at these types of questions. Any help... | Hint: This is equivalent to finding the minimum of
$$\frac{n(7n^2+3)}{34n^2+17}=\frac{n}{34}\left(7-\frac1{2n^2+1}\right)$$
Which for positive numbers, is clearly increasing. So you need to find only the least $n$ satisfying the cubic.
Checking for $n$ near $\frac{34}7c$ may quickly solve it, depending on the value of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1053355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Average number of trials until drawing $k$ red balls out of a box with $m$ blue and $n$ red balls
A box has $m$ blue balls and $n$ red balls. You are randomly drawing a ball from the box one by one until drawing $k$ red balls ($k < n$)? What would be the average number of trials needed?
To me, the solution seems to b... | For your sum, we have
$$\sum_{i=k}^{m-1} i \binom{i-1}{k-1}
= \sum_{i=k}^{m-1} \frac{i!}{(k-1)!(i-k)!} = k\sum_{i=k}^{m-1} \frac{i!}{k!(i-k)!}
= k \sum_{i=k}^{m-1} \binom{i}{k}.$$
By the hockey stick formula, this simplifies further:
$$k \sum_{i=k}^{m-1} \binom{i}{k}
= k\sum_{i=k}^{m-1}\binom{i}{i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1053443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Z transform piecewise function I have this piecewise function:
$$x(n)= \left\{ \begin{array}{lcc}
1 & 0 \leq n \leq m \\
\\ 0, &\mbox{ for the rest} \\
\\
\end{array}
\right.$$
How do I calculate the $z$-transform?
| $X(z) = \sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{m}z^{-n} = \frac{1-z^{-(m+1)}}{1-z^{-1}} = \frac{1}{z^{m}}\frac{z^{m+1}-1}{z-1}$, with $z \neq 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1053564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
"Integer average" of two integer numbers Suppose two arbitrary integer numbers $a$ and $b$. I'm looking for some function $f(a,b)$ with the following properties:
*
*$f(a,b)\in\mathbb{Z}$.
*$f(a,a)=a$.
*$f(a,b)=f(b,a)$.
*$\min\{a,b\}< f(a,b)< \max\{a,b\}$, accepting "$\leq$" if $a$ and $b$ are consecutive ($a=b\pm... | I would use "scientific rounding" (round to even) on the arithmetic mean. So if $(a+b)/2$ is an integer, use that, otherwise use the even integer out of $(a+b+1)/2$ and $(a+b-1)/2$. So that would likely be something like
$$\left\lfloor a+b+1\over4\right\rfloor + \left\lceil a+b-1\over4\right\rceil$$
The advantage of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1053695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 8,
"answer_id": 0
} |
Find limit of $\frac {1}{x^2}- \frac {1}{\sin^2(x)}$ as x goes to 0 I need to use a taylor expansion to find the limit.
I combine the two terms into one, but I get limit of $\dfrac{\sin^2(x)-x^2}{x^2\sin^2(x)}$ as $x$ goes to $0$. I know what the taylor polynomial of $\sin(x)$ centered around $0$ is… but now what do I... | Since you already received good answers and being myself in love with Taylor series for more than 55 years, let me add a small trick.
Considering the expression $$\dfrac{\sin^2(x)-x^2}{x^2\sin^2(x)}$$ you have, as already given in answers, for the numerator $$-\frac{x^4}{3}+\frac{2 x^6}{45}-\frac{x^8}{315}+O\left(x^9\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1053754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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prove $f(x)=x$ has a unique solution Question: Let $f$ be a continuous function from $\mathbb{R^2} \rightarrow \mathbb{R^2}$ such that $| f (x)− f (y)| ≤ \frac {1}{3} |x−y|$.
Prove $f(x)=x$ has a unique solution.
My sketch: There exists $\epsilon, \delta$ such that the following holds because $f$ is continuous
$|f(... | Assume there are two such points $x_1,x_2$ such that $f(x_1)=x_1$ and $f(x_2)=x_2$. Then $|f(x_1)-f(x_2)|=|x_1-x_2|\leq \frac{1}{3}|x_1-x_2|$. This is true if and only if $|x_1-x_2|=0$. Therefore $x_1=x_2$ and the fixed point, if it exists, is unique.
To show that a solution does exist consider the sequence $\{x_n\}$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1053841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How to prove that there are infinitely many primes without using contradiction
How can I prove that there are infinitely many primes without using contradiction?
I know the proof that is (not) by Euclid saying there are infinitely many primes. It assumes that there is a finite set of primes and then obtains one that ... | The simplest direct proof I know is one which involves infinite series. One can show that $\sum\limits_{p\text{ prime}}\frac{1}{p}$ diverges. Particularly, it means that there cannot be finitely many primes since the sum of finitely many nonzero numbers is finite. (There is a very slight contradiction - or contrapositi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1053902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Disjoint sets of vertices of a polygon This is the question:
Suppose that the vertices of a regular polygon of 20 sides are coloured with three colours – red, blue and green – such that there are exactly three red vertices. Prove that there are three vertices A,B,C of the polygon having the same colour such that triang... | For your second question; answer is yes. Pigeonhole principle repeatedly used to reach the answer.
By "four disjoint sets", answerer means that "four sets such that no two of them contains same element". Call the vertices $1$, $2$, $3$, $4$, etc. such that adjacent vertices differ by one, except vertices $20$ and $1$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\lim_{n \to +\infty}\frac{\left(1 + \sin\alpha\right)^n}{\left(1+\frac{\sin\alpha}n\right)^n}$
Let, for $n \ge 2$,
$$a_n = \frac{\left(1 + \sin\alpha\right)^n}{\left(1+\frac{\sin\alpha}n\right)^n}$$
Evaluate $\lim_\limits{n \to +\infty}a_n$, as $\alpha$ varies in $[0, 2\pi)$.
I proceeded as follows.
T... | Your first step is correct, with the simpler denominator.
It was a good idea to convert to exp and log.
Also to break into quadrants with particular interest in multiples of $\pi/2$.
But then, when $x=0,\pi$, you have exp(n(0)-0) = exp(0)=1.
When $\sin x>0$, then $1+\sin x>1$, so $\ln(1+\sin x)>0$ and $n\ln(1+\sin x)\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$a_{n+1}=1+\frac{1}{a_n}.$ Find the limit. Let $a_1=1$ and $a_{n+1}=1+\frac{1}{a_n}.$
Is $a_n$ convergent?
How could i find its limit?
I found even terms of the sequence decrease and odd terms are increase. But i cant find upper and lower bounds to use monotone convergence theorem.
| You can prove by induction that
$$a_n = \frac{F_{n+1}}{F_n}$$
where $\{F_n\}_{n\geq 1}=\{1,1,2,3,5,8,\ldots\}$ is the Fibonacci sequence.
Binet formula hence gives:
$$\lim_{n\to +\infty} a_n = \phi = \frac{1+\sqrt{5}}{2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Number of $n-1$-dimensional subspaces of $n$-dimensional space over finite field I got a question with two parts.
Let $V$ be a $n$-dimensional vector space over $\mathbb{F}_{p}$ - finite field with $p$ elements.
a) How many $1$-dimensional subspaces $V$ has.
b) How many $n-1$-dimensional subspaces $V$ has.
I solved (... | Hint:
There is a bijection between subspaces of dimension $k$ and $k\times n$ matrices of rank $k$ in reduced row-echelon form.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
how to point out errors in proof by induction I have searched for an answer to my question but no one seems to be talking about this particular matter..
I will use the all horses are the same color paradox as an example.
Everyone points out that the statement is false for n=2 and that if we want to prove the propositio... |
"But, (as I see it..), you have to use reason to figure that out. My question is, is there anything wrong with induction itself? except for the fact that we can use reason to understand why the proof is faulty."
I'm not entirely sure of what you mean with using 'reason', but I'm interpreting it as "is there way to sp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
the purpose of induction After getting an answer (in a comment) from peter for this question I have a follow up question.
If, in all horses are the same color problem for example, we need to use reason, reason which is specific to the case, in order to find that "hole" between the correct base case and the correct ind... | You can prove statements inductively from which you did not know before being true or false.
You just need to make sure that there is no "hole" in your proof.
In the case of your example the problem occured because the proof of the inductive step was simply wrong.
But if you check that both p(0) and the inductive step ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
Closed form of $\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$ How would you recommend me to tackle the series
$$\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$$?
Can we possibly express it in terms of known constants? What do you think about it?
| You may recall the Lerch transcendent function
$$
\Psi(z,s,a)=\sum_{k=0}^\infty\frac{z^k}{(a+k)^s}
$$
and use
$$
\frac{1}{n^2+1}=\frac{i}{2}\left(\frac{1}{n+i}-\frac{1}{n-i}\right)
$$
to get
$$
\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}=-1-\Im \: \Psi\left(\frac12,1,i\right)
$$
which gives your series in terms of a known... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Inverse laplace transform excercise I want to find the inverse transform of $$\frac{1}{(2s-1)^3}$$
I first applied a shifting theorem to get $$(e^t)\mathcal{L}^{-1}\left( \frac{1}{(2s)^3} \right)$$
I am just wondering is it possible to take the 2 out from here so it becomes
$$\frac{1}{8}(e^t)\mathcal{L}^{-1}\left( \fr... | We could use the inverse Laplace transform integral/Bromwich Integral/Mellin-Fourier integral/Mellin's inverse formula (many names) and then use Residue theory. The pole is at $s = 1/2$ of order three.
The Bromwich contour is a line from $\gamma - i\infty$ to $\gamma + i\infty$ and then a partial circle connecting the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Can we prove that axioms do not contradict? We construct many structures by chosing a set of axioms and deriving everything else from them. As far as I remember we never proved in our lectures that those axioms do not contradict. So:
Is it always possible to prove that a set of axioms does not contain a contradiction? ... | For a first order theory, basically a set of "axioms" formulated in a first order language, you have Gödel's completeness theorem. The theorem establishes that a first order theory is consistent(i.e. non-contradictory) is and only if you have a model for that theory(i.e. a "real" mathematical object satisfying the axio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1054857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$? If $a$ is a complex number s.t. $a\notin \mathbb R$, then $\mathbb R(a)=\mathbb C$?
I'm asked to give a proof or a counterexample. I'm a bit confused on the notation of $\mathbb R(a)$, what does this mean exactly? I'm also unclear on wh... | For $a \in \Bbb C \setminus \Bbb R$, we indeed have $\Bbb R(a) = \Bbb C$; indeed, we have $i \in \Bbb R(a)$; this may be seen as follows:
With such $a$, we have
$0 < \bar a a \in \Bbb R \subset \Bbb R(a); \tag{1}$
since $\Bbb R(a)$ is a field and $a \ne 0$, we further have
$\bar a = a^{-1} (\bar a a) \in \Bbb R(a); \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1055033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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