Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Darboux Theorem Proof http://en.wikipedia.org/wiki/Darboux%27s_theorem_%28analysis%29
I'm having a bit of trouble understanding the proof of Darboux's Theorem on the IVP of derivatives.
Why should there be an extremum such that $g'(x) = 0$ from the fact that $g'(a)>0$ and $g'(b)<0$ ?
| Suppose that $g$ attains a local maximum at $a$. Then $$\lim_{x \to a+} \frac{g(x)-g(a)}{x-a} \leq 0.$$ Analogously, if $g$ attains a local maximum at $b$, then $$\lim_{x \to b-} \frac{g(x)-g(b)}{x-b}\geq 0.$$ But both contradict $g'(a)>0$ and $g'(b)<0$. Hence the maximum, which exists since $g$ is continuous on $[a,b]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Sum of the sequence What is the sum of the following sequence
$$\begin{align*}
(2^1 - 1) &+ \Big((2^1 - 1) + (2^2 - 1)\Big)\\
&+ \Big((2^1 - 1) + (2^2 - 1) + (2^3 - 1) \Big)+\ldots\\
&+\Big( (2^1 - 1)+(2^2 - 1)+(2^3 - 1)+\ldots+(2^n - 1)\Big)
\end{align*}$$
I tried to solve this. I reduced the equation into the followi... | Let's note that $$(2^1 - 1) + (2^2 - 1) + \cdots + (2^k - 1) = 2^{k+1} - 2-k$$ where we have used the geometric series. Thus, the desired sum is actually $$\sum_{k=1}^n{2^{k+1}-2-k}$$. As this is a finite sum, we can evaluate each of the terms separately. We get the sum is $$2\left(\frac{2^{n+1}-1}{2-1}-1\right) - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Can you help me understand this definition for the limit of a sequence? I'm reading the textbook "Calculus - Early Transcendentals" by Jon Rogawski for my Calculus III university course.
I'm trying for the life of me to understand the wording of this definition, and I wonder if it can be said in simpler terms to get th... | Let’s call $\{a_n,a_{n+1},a_{n+2},\dots\}$ the $n$-tail of the sequence.
Now suppose that I give you a target around the number $L$: I pick some positive leeway $\epsilon$ want you to hit the interval $(L-\epsilon,L+\epsilon)$. We’ll say that the sequence hits that target if some tail of the sequence lies entirely in... | {
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"url": "https://math.stackexchange.com/questions/192586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Prove with MATLAB whether a set of n points is coplanar I need to find a way to prove if a set of n points are coplanar. I found this elegant way on one of the MATLAB forums but I don't understand the proof. Can someone help me understand the proof please?
" The most insightful method of solving your problem is to find... | If you put all the points as columns in a matrix, the resulting matrix will have rank equal to 2 if the points are coplanar. If such a matrix is denoted as $\mathbf A$ then $\mathbf{AA}^T$ will have one eigenvalue equal to or close to 0.
Consider that V*U = 0 yields the equation of the plane. Then consider that V'*V*... | {
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"url": "https://math.stackexchange.com/questions/192641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$p$ is polynomial, set bounded open set with at most $n$ components Assume $p$ is a non constant polynomial of degree $n$. Prove that the set $\{z:|(p(z))| \lt 1\}$ is a bounded open set with at-most $n$ connected components. Give example to show number of components can be less than $n$.
thanks.
EDIT:Thanks,I meant co... | Hints:
*
*For boundedness: Show that $|p(z)| \to \infty$ as $|z|\to \infty$
*For openness: The preimage $p^{-1}(A)$ of an open set $A$ under a continuous function $p$ is again open. Polynomials are continuous, so just write your set as a preimage.
*For the connected components: Recall fundamental theorem of algebr... | {
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"url": "https://math.stackexchange.com/questions/192709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to solve $x^3=-1$? How to solve $x^3=-1$? I got following:
$x^3=-1$
$x=(-1)^{\frac{1}{3}}$
$x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
| Observe that $(e^{a i})^3 = e^{3 a i}$ and $-1=e^{\pi i}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 8,
"answer_id": 2
} |
How to prove that this function is continuous at zero? Assume that $g : [0, \infty) \rightarrow \mathbb R$ is continuous and $\phi :\mathbb R \rightarrow \mathbb R$ is continuous with compact support with $0\leq \phi(x) \leq 1$, $\phi(x)=1$ for $ x \in [0,1]$ and $\phi(x)=0$ for $x\geq 2$.
I wish to prove that
$$
\l... | Let $h(x)=\phi(x)g(x)$. Then $h\colon[0,\infty)\to\mathbb R$ is continuous and bounded by some $M$ and $h(x)=0$ for $x\ge2$.
Given $\epsilon>0$, find $\delta$ such that $x<\delta$ implies $|h(x)-h(0)|<\frac\epsilon3$.
Then for $m\in \mathbb N$
$$\sum_{n=1}^\infty \frac1{2^n} h(nx)-h(0)=\sum_{n=1}^{m} \frac1{2^n} (h(nx)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I need some help solving a Dirichlet problem using a conformal map I'm struggling here, trying to understand how to do this, and after 4 hours of reading, i still can't get around the concept and how to use it.
Basically, i have this problem:
A={(x,y) / x≥0, 0≤y≤pi
So U(x,0) = B; U(x,pi) = C; U'x(0,y) = 0;
I know that ... | The domain is simple enough already. Observe that there is a function of the form $U=\alpha y+\beta$ which satisfies the given conditions.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compute: $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)\cdot (2n+1)}$ Compute the sum:
$$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)\cdot (2n+1)}$$
At the moment, I only know that it's convergent and this is not hard to see if you look at the answers here I rece... | Starting with the power series derived using the binomial theorem,
$$
(1-x)^{-1/2}=1+\tfrac12x+\tfrac12\tfrac32x^2/2!+\tfrac12\tfrac32\tfrac52x^3/3!+\dots+\tfrac{1\cdot2\cdot3\cdots(2n-1)}{2\cdot4\cdot6\cdots2n}x^n+\cdots
$$
and integrating, we get the series for
$$
\sin^{-1}(x)=\int_0^x(1-t^2)^{-1/2}\mathrm{d}t=\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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"answer_id": 2
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about Jacobian and eigenvalues I am studying the dynamical system on a discrete standard map
$$x_{n+1} = f(x_n, y_n)$$
$$y_{n+1} = g(x_n, y_n)$$
First of all, could anyone explain the difference between the stationary point and the fixed point? As stated in some book, for the points which satisfying $f(x_0, y_0)=0$ an... | I will concatenate $x$ and $y$ and work with a single state-transition equation
$$x_{k+1} = f (x_k)$$
where $f : \mathbb{R}^n \to \mathbb{R}^n$. Given a state $x$, function $f$ gives you the next state $f (x)$. It's an infinite state machine! Suppose that $f (\bar{x}) = \bar{x}$ for some isolated point $\bar{x} \in \ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Probability of throwing multiple dice of at least a given face with a set of dice I know how to calculate the probability of throwing at least one die of a given face with a set of dice, but can someone tell me how to calculate more than one (e.g., at least two)?
For example, I know that the probability of throwing at ... | You are asking for the distribution of the number $X_n$ of successes in $n$ independent trials, where each trial is a success with probability $p$. Almost by definition, this distribution is binomial with parameters $(n,p)$, that is, for every $0\leqslant k\leqslant n$,
$$
\mathrm P(X_n=k)={n\choose k}\cdot p^k\cdot(1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193050",
"timestamp": "2023-03-29T00:00:00",
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Picking out columns from a matrix using MAGMA How do I form a new matrix from a given one by picking out some of its columns, using MAGMA?
| You can actually do this really easily in Magma using the ColumnSubmatrix command, no looping necessary. You can use this in a few ways.
For example if you have a matrix $A$ and you want $B$ to be made up a selection of columns:
1st, 2nd, $\ldots$, 5th columns:
B := ColumnSubmatrix(A, 5);
3rd, 4th, $\ldots$, 7th col... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Completeness and Topological Equivalence How can I show that if a metric is complete in every other metric topologically
equivalent to it , then the given metric is compact ?
Any help will be appreciated .
| I encountered this result in Queffélec's book's Topologie. The proof is due to A.Ancona.
It's known as Bing's theorem. We can assume WLOG that $d\leq 1$, otherwise, replace $d$ by $\frac d{1+d}$. We assume that $(X,d)$ is not compact; then we can find a sequence $\{x_n\}$ without accumulation points. We define
$$d'(x,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Prove that $x^2 + 5xy+7y^2 \ge 0$ for all $x,y \in\mathbb{R}$ This is probably really easy for all of you, but my question is how do I prove that $x^2 + 5xy+7y^2 \ge 0$ for all $x,y\in\mathbb{R}$
Thanks for the help!
| $$x^2+5xy+7y^2=\left(x+\frac{5y}2\right)^2 + \frac{3y^2}4\ge 0$$
(not $>0$ as $x=y=0$ leads to 0).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/193275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
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Bijection between the set of classes of positive definite quadratic forms and the set of classes of quadratic numbers in the upper half plane Let $\Gamma = SL_2(\mathbb{Z})$.
Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$.
Let $f(x, y) = ax^2 + bxy + cy^2 \in \mathfrak{F}$.
Let $\alpha = \lef... | Proof of (1)
We define a map $\psi\colon \mathcal{H}(D) \rightarrow \mathfrak{F}^+_0(D)$ as follows.
Let $\theta \in \mathcal{H}(D)$.
$\theta$ is a root of the unique polynomial $ax^2 + bx + c \in \mathbb{Z}[x]$ such that $a > 0$ and gcd$(a, b, c) = 1$.
$D = b^2 - 4ac$.
We define $\psi(\theta) = ax^2 + bxy + cy^2$.
Cle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193329",
"timestamp": "2023-03-29T00:00:00",
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When is the set statement: (A⊕B) = (A ∪ B) true? "When is the set statement:
(A⊕B) = (A ∪ B)
a true statement? Is it true sometimes, never, or always? If it is sometimes, state the cases where it is."
How would you go about finding the answer to the question or ones like this one?
Thanks for your time!
| If I've made the right assumptions in my comment above, a good way to approach this problem is by drawing a Venn diagram.
Here's $A\oplus B$:
Here's $A\cup B$:
So, the area that's filled in in $A\cup B$ but not $A\oplus B$ is $A\cap B$. What do I need to be true about $A\cap B$ to make the two Venn diagrams have the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
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A matrix is diagonalizable, so what? I mean, you can say it's similar to a diagonal matrix, it has $n$ independent eigenvectors, etc., but what's the big deal of having diagonalizability? Can I solidly perceive the differences between two linear transformation one of which is diagonalizable and the other is not, either... | Up to change in basis, there are only 2 things a matrix can do.
*
*It can act like a scaling operator where it takes certain key vectors (eigenvectors) and scales them, or
*it can act as a shift operator where it takes a first vector, sends it to a second vector, the second vector to a third vector, and so forth,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Isomorphic subgroups, finite index, infinite index Is it possible to have a group $G,$ which has two different, but isomorphic subgroups $H$ and $H',$ such that one is of finite index, and the other one is of infinite index?
If not, why is that not possible. If there is a counterexample please give one.
| Yes. For $n\in\Bbb N$ let $G_n$ be a copy of $\Bbb Z/2\Bbb Z$, and let $G=\prod_{n\in\Bbb N}G_n$ be the direct product. Then $H_0=\{0\}\times\prod_{n>0}G_n$ is isomorphic to $H_1=\prod_{n\in\Bbb N}A_n$, where $A_n=\{0\}$ if $n$ is odd and $A_n=G_n$ if $n$ is even. Clearly $[G:H_0]=2$ and $[G:H_1]$ is infinite.
Of cours... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Detect when a point belongs to a bounding box with distances I have a box with known bounding coordinates (latitudes and longitudes): latN, latS, lonW, lonE.
I have a mystery point P with unknown coordinates. The only data available is the distance from P to any point p. dist(p,P).`
I need a function that tells me whet... | The distance measurement from any point gives you a circle around that point as a locus of possible positions of $P$.
Make any such measurement from a point $A$.
If the question is not settled after this (i.e. if the circle crosses the boundary of the rectangle), make a measurement from any other point $B$. The two int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Proving that $\sum_{j=1}^{n}|z_{j}|^{2}\le 1$ when $|\sum_{j=1}^{n}z_{j}w_{j}| \le 1$ for all $\sum_{j=1}^{n}|w_{j}|^{2}\le 1$ The problem is like this:
Fix $n$ a positive integer.
Suppose that $z_{1},\cdots,z_{n} \in \mathbb C$ are complex numbers satisfying $|\sum_{j=1}^{n}z_{j}w_{j}| \le 1$ for all $w_{1},\cdots... | To have a chance of success, one must choose a family $(w_j)_j$ adapted to the input $(z_j)_j$. If $z_j=0$ for every $j$, the result holds. Otherwise, try $w_j=c^{-1}\bar z_j$ with $c^2=\sum\limits_{k=1}^n|z_k|^2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c} : (a, b, c) > 0$ Please help me for prove this inequality:
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c} : (a, b, c) > 0$$
| Inequality can be written as:
$$\left(a+b+c\right) \cdot \left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geq 9 .$$
And now we apply the $AM-GM$ inequality for both of parenthesis. So:
$\displaystyle \frac{a+b+c}{3} \geq \sqrt[3]{abc} \tag{1}$ and $\displaystyle \frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{3} \geq \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 3
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What's the probability that the other side of the coin is gold? 4 coins are in a bucket: 1 is gold on both sides, 1 is silver on both sides, and 2 are gold on one side and silver on the other side.
I randomly grab a coin from the bucket and see that the side facing me is gold. What is the probability that the other sid... | 50%. GIVEN that the first side you see is gold, what is the chance that you have the double-gold coin?
Assume you do this experiment a hundred times. In 50% of the cases you pull out a coin and see a gold side; the other 50% you see a silver side. In the latter case we have to discard the experiment and only count the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/193851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I get sequence $4,4,2,4,4,2,4,4,2\ldots$ into equation? How can I write an equation that expresses the nth term of the sequence:
$$4, 4, 2, 4, 4, 2, 4, 4, 2, 4, 4, 2,\ldots$$
| $$
f(n) =
\begin{cases}
4 \text{ if } n \equiv 0 \text{ or } 1 \text{ (mod 3)}\\
2 \text{ if } n \equiv 2 \text{ (mod 3)}
\end{cases}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/193897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 17,
"answer_id": 9
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Example of a bijection between two sets I am trying to come up with a bijective function $f$ between the set : $\left \{ 2\alpha -1:\alpha \in \mathbb{N} \right \}$ and the set $\left \{ \beta\in \mathbb{N} :\beta\geq 9 \right \}$, but I couldn't figure out how to do it. Can anyone come up with such a bijective functio... | Given some element $a$ of $\{ 2\alpha -1 \colon \alpha \in \mathbb{N} \}$, try the function $f(a)=\frac{a+1}{2}+9$.
| {
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"url": "https://math.stackexchange.com/questions/193950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
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(Help with) A simple yet specific argument to prove Q is countable I was asked to prove that $\mathbb{Q}$ is countable. Though there are several proofs to this I want to prove it through a specific argument.
Let $\mathbb{Q} = \{x|n.x+m=0; n,m\in\mathbb{Z}\}$
I would like to go with the following argument: given that we... | Your argument can work, but as presented here there are several gaps in it to be closed:
*
*Your definition of $\mathbb Q$ does not work unless you exclude the case $m=n=0$ -- otherwise everything is a solution. (Thanks, Brian).
*You need to point out explicitly that each choice of $n$ and $m$ has at most one solut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/194018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Any concrete example of ''right identity and left inverse do not imply a group''? In the abstract algebra class, we have proved the fact that right identity and right inverse imply a group, while right identity and left inverse do not.
My question:
Are there any good examples of sets (with operations on) with right ide... | $$\matrix{a&a&a\cr b&b&b\cr c&c&c\cr}$$ That is, $xy=x$ for all $x,y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/194083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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If $f(n) = \sum_{i = 0}^n X_{i}$, then show by induction that $f(n) = f(n - 1) + X_{n-1}$ I am trying to solve this problem by induction. The sad part is that I don't have a very strong grasp on solving by inductive proving methods. I understand that there is a base case and that I need an inductive step that will set ... | To prove by induction, you need to prove two things. First, you need to prove that your statement is valid for $n=1$. Second, you have to show that the validity of the statement for $n=k$ implies the validity of the statement for $n=k+1$. Putting these two bits of information together, you effectively show that your st... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/194128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Show that in a discrete metric space, every subset is both open and closed. I need to prove that in a discrete metric space, every subset is both open and closed. Now, I find it difficult to imagine what this space looks like. I think it
consists of all sequences containing ones and zeros.
Now in order to prove that ... | Let $(X,d)$ be a metric space. Suppose $A \subset X$. Let $ x\in A$ be arbitrary. Setting $r = \frac{1}{2}$ then if $a \in B(x,r)$ we have $d(a,x) < \frac{1}{2}$ which implies that $a=x$ and so a is in A. (1)
To show that A is closed. It suffices to note that the complement of A is a subset of X and by (1), it is open ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/194201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_count": 4,
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Hamiltonian Cycle Problem At the moment I'm trying to prove the statement:
$K_n$ is an edge disjoint union of Hamiltonian cycles when $n$ is odd.
($K_n$ is the complete graph with $n$ vertices)
So far, I think I've come up with a proof. We know the total number of edges in $K_n$ is $n(n-1)/2$ (or $n \choose 2$) and w... | What you are looking for is a Hamilton cycle decomposition of the complete graph $K_n$, for odd $n$.
An example of how this can be done (among many other results in the area) is given in: D. Bryant, Cycle decompositions of complete graphs, in Surveys in Combinatorics, vol. 346, Cambridge University Press, 2007, pp. 67... | {
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Show me some pigeonhole problems I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve.
I would be thankful if you can send me links\books\or just a lone problem.
| This turned up in a routine google search of the phrase "pidgeonhole principle exercise" and appears to be training problems for the New Zealand olympiad team. It contains numerous problems and has some solutions in the back.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Character of $S_3$ I am trying to learn about the characters of a group but I think I am missing something.
Consider $S_3$. This has three elements which fix one thing, two elements which fix nothing and one element which fixes everything. So its character should be $\chi=(1,1,1,0,0,3)$ since the trace is just equal to... | The permutation representation is reducible. It has a subrepresentation spanned by the vector $(1,1,1)$. Hence, the permutation representation is the direct sum of the trivial representation and a $(n-1=2)$-dimensional irreducible representation.
| {
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"source": "stackexchange",
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Finding $\pi$ through sine and cosine series expansions I am working on a problem in Partha Mitra's book Observed Brain Dynamics (the problem was originally from Rudin's textbook Real and Complex Analysis, and appears on page 54 of Mitra's book). Unfortunately, the book I have does not contain any solutions... Here is ... | How to show that $\sin (x_0)=1$ if $\cos (x_0)=0$? Quite simply:
$$\sin^2 x+\cos^2 x=1$$
(you may also want to specify that $\sin x$ is positive in the given range)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Exercise on finite intermediate extensions Let $E/K$ be a field extension, and let $L_1$ and $L_2$ be intermediate fields of finite degree over $K$.
Prove that $[L_1L_2:K] = [L_1 : K][L_2 : K]$ implies $L_1\cap L_2 = K$.
My thinking process so far:
I've gotten that $K \subseteq L_1 \cap L_2$ because trivially both ar... | The assumption implies $[L_1L_2:L_1]=[L_2:K]$. Hence $K$-linearly independent elements $b_1,\ldots ,b_m\in L_2$ are $L_1$-linearly independent, considered as elements of $L_1L_2$. In particular this holds for the powers $1,x,x^2,\ldots ,x^{m-1}$ of an element $x\in L_2$, where $m$ is the degree of the minimal polynomia... | {
"language": "en",
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The integral relation between Perimeter of ellipse and Quarter of Perimeter Ellipse Equation
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
$x=a\cos t$ ,$y=b\sin t$
$$L(\alpha)=\int_0^{\alpha}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$$
$$L(\alpha)=\int_0^\alpha\sqrt{a^2\sin^2 t+b^2 \cos^2 t}\,dt $$
... | I proved the relation via using analytic way. I would like to share the solution with you.
$$\int_0^{\pi/2}\sqrt{b^2+(a^2-b^2)\sin^2 4u}\,du=K$$
$u=\pi/4-z$
$$K=\int_{-\pi/4}^{\pi/4}\sqrt{b^2+(a^2-b^2)\sin^2 (\pi-4z)}\,dz$$
$\sin (\pi-4z)=\sin \pi \cos 4z-\cos \pi \sin 4z= \sin 4z$
$$\int_{-\pi/4}^{\pi/4}\sqrt{b^2+(a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/194596",
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Show that the group $G$ is of order $12$ I am studying some exercises about semi-direct product and facing this solved one:
Show that the order of group $G=\langle a,b| a^6=1,a^3=b^2,aba=b\rangle$ is $12$.
Our aim is to show that $|G|\leq 12$ and then $G=\mathbb Z_3 \rtimes\mathbb Z_4=\langle x\rangle\rtimes\langle y... | The subgroup $A$ generated by $a^2$ is normal and order 3. The subgroup $B$ generated by $b$ is of order 4. The intersection of these is trivial so the product $AB$ has order 12. So $G$ has order at least 12. To show it has order 12, we need to see that $a\in AB$; but $b^2=a^3=a^2a$ so $$a=a^{-2}b^2\in AB.$$ Thus the g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$? If $k$ is a positive natural number then $\phi(k)$ denotes the number of natural numbers less than $k$ which are prime to $k$. I have seen proofs that $n = \sum_{k|n} \phi(k)$ which basically partitions $\mathbb{Z}/n\mathbb{Z}$ into subsets of elements of... | Claim:Number of positive integers pair $(a, b) $ satisfying :
$n=a+b$ (for given $n$)
$\gcd(a, b) =d$ and $d|n$
is $\phi(n/d) $.
Proof:
Let $a=xd$ and $b=yd$
We want number of solution for
$x+y=\frac{n}{d}$ such that $\gcd(x, y) =1$.
$\gcd(x,y)=\gcd(x,x+y)=\gcd(x,n/d)=1$
Solution for $x+y=n/d$, $\gcd(x,y)=1$ is $\p... | {
"language": "en",
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The limit of the sum is the sum of the limits I was wondering why the statement in the title is true only if the functions we are dealing with are continuous.
Here's the context (perhaps not required):
(The upper equation there is just a limit of two sums, and the lower expression is two limits of those two sums.), a... | The limit of the sum is not always equal to the sum of the limits, even when the individual limits exist.
For example:
Define $h(i)=1\sqrt{(n^2)+i}$.
For each $i=1,\cdots,n$, the limit of $h(i)$ is zero as n goes to infinity.
But the limit of the sum $[h(1)+h(2)+\cdots+h(n)]$ as n goes infinity is not zero.
The limit ... | {
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Calculate $\lim\limits_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a}$ Please help me solving $\displaystyle\lim_{x\to a}\frac{a^{a^{x}}-{a^{x^{a}}}}{a^x-x^a}$
| The ratio is $R(x)=\dfrac{u(t)-u(s)}{t-s}$ with $u(z)=a^z$, $t=a^x$ and $s=x^a$. When $x\to a$, $t\to a^a$ and $s\to a^a$ hence $R(x)\to u'(a^a)$. Since $u(z)=\exp(z\log a)$, $u'(z)=u(z)\log a$. In particular, $u'(a^a)=u(a^a)\log a$. Since $u(a^a)=a^{a^a}$, $\lim\limits_{x\to a}R(x)=a^{a^a}\log a$.
| {
"language": "en",
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Linearly disjoint vs. free field extensions Consider two field extensions $K$ and $L$ of a common subfield $k$ and suppose $K$ and $L$ are both subfields of a field $\Omega$, algebraically closed.
Lang defines $K$ and $L$ to be 'linearly disjoint over $k$' if any finite set of elements of $K$ that are linearly independ... | The condition of being linearly disjoint or free depends much on the "positions" of $K, L$ inside $\Omega$, while the isomorphism class of the $k$-algebra $K\otimes_k L$ doesn't. For instance, consider $\Omega=\mathbb C(X,Y)$, $K=\mathbb C(X)$, $L_1=\mathbb C(Y)$ and $L_2=K$. Then
$$K\otimes_\mathbb C L_1\simeq K\otim... | {
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Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m\,$ [LCM = product for coprimes] Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$
I thought that $m=ab$ but I was given a counterexample in a comment below.
So all I really know is $m=ax$ and $m=by$ for some $x,y \in \mathbb Z$. Also, ... | Write $ax+by=1$, $m=aa'$, $m=bb'$. Let $t=b'x+a'y$.
Then $abt=abb'x+baa'y=m(ax+by)=m$ and so $ab \mid m$.
Edit: Perhaps this order is more natural and less magical:
$m = m(ax+by) = max+mby = bb'ax+aa'by = ab(b'x+a'y)$.
| {
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Question about Minimization Let be $J$ a convex functional defined in Hilbert space H and with real values. What hypothesis I should assume to exist solution for the problem?:
$J(u) = \inf \left\{{J(v); v \in K}\right\} , u \in K$
For all closed convex $K \subset H.$
I begin using the theorem
A functional $J:E\righta... | You get equality by taking $u_n$ such that $J(u_n)\to \inf_K J$. Indeed, the weak limit is also an element of $K$ and therefore cannot have a smaller value of the functional than the infimum.
The term is "lower semicontinuous", by the way. What you need from $J$ is being bounded from below, and lower semicontinuous wi... | {
"language": "en",
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evaluation of the integral $\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $ Can the integral $$\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $$ be expressed in terms of elemental functions or in terms of the sine and cosine integrals ? if possible i would need a hint thanks.
From the fractional calculus i guess this integral... | Let $t=x-y^2$. We then have $dt = -2ydy$. Hence, we get that
\begin{align}
I = \int_0^x \dfrac{\cos(ut)}{\sqrt{x-t}} dt & = \int_0^{\sqrt{x}} \dfrac{\cos(u(x-y^2))}{y} \cdot 2y dy\\
& = 2 \cos(ux) \int_0^{\sqrt{x}}\cos(uy^2)dy + 2 \sin(ux) \int_0^{\sqrt{x}}\sin(uy^2)dy\\
& = \dfrac{\sqrt{2 \pi}}{\sqrt{u}} \left(\cos(ux... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Inequality. $\sum{(a+b)(b+c)\sqrt{a-b+c}} \geq 4(a+b+c)\sqrt{(-a+b+c)(a-b+c)(a+b-c)}.$ Let $a,b,c$ be the side-lengths of a triangle. Prove that:
I.
$$\sum_{cyc}{(a+b)(b+c)\sqrt{a-b+c}} \geq 4(a+b+c)\sqrt{(-a+b+c)(a-b+c)(a+b-c)}.$$
What I have tried:
\begin{eqnarray}
a-b+c&=&x\\
b-c+a&=&y\\
c-a+b&=&z.
\end{eqnarray}
... | Notice that the inequality proposed is proved once we estabilish
$$\frac{(a+b)(b+c)}{\sqrt{a+b-c}\sqrt{-a+b+c}}+\frac{(b+c)(c+a)}{\sqrt{a-b+c}\sqrt{-a+b+c}}+\frac{(c+a)(a+b)}{\sqrt{a-b+c}\sqrt{a+b-c}}\geq 4(a+b+c).$$
Using AM-GM on the denominators in the LHS, we estabilish that
*
*$\sqrt{a+b-c}\sqrt{-a+b+c}\leq ... | {
"language": "en",
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Average Distance Between Random Points on a Line Segment Suppose I have a line segment of length $L$. I now select two points at random along the segment. What is the expected value of the distance between the two points, and why?
| You can picture this problem from a discrete approach, then extend it to the real number line.
For the first L natural numbers, the difference between any two of them may range from 1 through L-1. Exactly N pairs of numbers from our set will be L-N units apart. Taking that into account we sum:
Sum of distances =
sum (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/195245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Open cover rationals proper subset of R? If I were to cover each rational number by a non-empty open interval, would their union always be R? It seems correct to me intuitively, but I am quite certain it is wrong. Thanks
| If you enumerate the rationals as a sequence $x_1, x_2, \dots$, you can then take a sequence of open intervals $(x_1-\delta, x_1+\delta), (x_2-\delta/2, x_2+\delta/2), (x_3-\delta/4, x_3+\delta/4), \dots$ which gives an open cover for $\mathbb{Q}$ of total length $4\delta$, which can be made as small as you wish, by ch... | {
"language": "en",
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Is there a geometric proof to answers about the 3 classical problems? I know that there is a solution to this topic using algebra (for example, this post).
But I would like to know if there is a geometric proof to show this impossibility.
Thanks.
| No such proof is known. Note that this would in fact be meta-geometric: You do not give a construction of an object from givens, but you make a statement about all possible sequences of operations with your drawing instruments. Therefore it is a good idea to classify all points constructable from standard given points.... | {
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Trouble with formulation of an analytic geometry question I'm having trouble understanding a certain question, so I am asking for an explanation of it. The question is asked in a different language, so my translation will probably be mistake-ridden, I hope you guys can overlook the mistakes (and of course correct them)... | A slightly different take on your question would be to realize that if your circle touches the $Y$ axis, it must do so at a point $(0, y)$. Substitute $x=0$ in the equation of your circle; can you find a value for $y$? The answer for touching the $X$ axis can be found in a similar way.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/195439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Kernel of Linear Functionals Problem: Prove that for all non zero linear functionials $f:M\to\mathbb{K}$ where $M$ is a vector space over field $\mathbb{K}$, subspace $(f^{-1}(0))$ is of co-dimension one.
Could someone solve this for me?
| The following is a proof in the finite dimensional case: The dimension of the image of $f$ is 1 because $\textrm{im} f$ is a subspace of $\Bbb{K}$ that has dimension 1 over itself. Since $\textrm{im} f \neq 0$ it must be the whole of $\Bbb{K}$. By rank nullity,
$$\begin{eqnarray*} 1 &=& \dim \textrm{im} f \\
&=& \di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/195504",
"timestamp": "2023-03-29T00:00:00",
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Bypassing a series of stochastic stoplights In order for me to drive home, I need to sequentially bypass $(S_1, S_2, ..., S_N)$ stoplights that behave stochastically. Each stoplight, $S_i$ has some individual probability $r_i$ of being red, and an associated probability, $g_i$, per minute of time of turning from red t... | Let $T_i$ denote the time you wait on each stop-light. Because $\mathbb{P}(T_i = 0) = 1-r_i > 0$, $T_i$ is not a continuous random variable, and thus does not have a notion of density.
Likewise, the total wait-time $T = T_1+\cdots+T_N$ also has a non-zero probability of being equal to zero, and hence has no density.
In... | {
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Publishing an article after a book? If I first publish an article, afterward I may publish a book containing materials from the article.
What's about the reverse: If I first publish a book does it make sense to publish its fragment as an article AFTERWARD?
| "If I first publish a book does it make sense to publish its fragment as an article AFTERWARD?"
Sure, why not? You might write a book for one audience, and very usefully re-publish a fragment in the form of a journal article for another audience. I have done this with some stuff buried near the end of a long textbook b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/195721",
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Compute expectation of certain $N$-th largest element of uniform sample A premier B-school has 2009 students.The dean,a math enthusiast,asks each student to submit a randomly chosen number between 0 and 1.She then ranks these numbers in a list of decreasing order and decides to use the 456th largest number as a fracti... | This is a question on order statistics. Let $U_i$ denote independent random variables, uniformly distributed on unit interval. The teacher picks $m$-th largest, or $n+1-m$-th smallest number in the sample $\{U_1, U_2, \ldots, U_n\}$, which is denoted as $U_{n-m+1:n}$. It is easy to evaluate the cumulative distribution ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In Need of Ideas for a Small Fractal Program I am a freshman in high school who needs a math related project, so I decided on the topic of fractals. Being an avid developer, I thought it would be awesome to write a Ruby program that can calculate a fractal. The only problem is that I am not some programming god, and ... | Maybe you want to consider iterated function systems
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/195830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Integer solutions to $ x^2-y^2=33$ I'm currently trying to solve a programming question that requires me to calculate all the integer solutions of the following equation:
$x^2-y^2 = 33$
I've been looking for a solution on the internet already but I couldn't find anything for this kind of equation. Is there any way to c... | Hint $\ $ Like sums of squares, there is also a composition law for differences of squares, so
$\rm\quad \begin{eqnarray} 3\, &=&\, \color{#0A0}2^2-\color{#C00}1^2\\ 11\, &=&\, \color{blue}6^2-5^2\end{eqnarray}$
$\,\ \Rightarrow\,\ $
$\begin{eqnarray} 3\cdot 11\, &=&\, (\color{#0A0}2\cdot\color{blue}6+\color{#C00}1\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/195904",
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What is the point of the Thinning Rule? I am studying predicate calculus on some lecture notes on my own. I have a question concerning a strange rule of inference called the Thinning Rule which is stated from the writer as the third rule of inference for the the formal system K$(L)$ (after Modus Ponens and the Generali... | After a lot of research here and there I think I have found the correct answer thanks to Propositional and Predicate Calculus by Derek Goldrei. So I will try to answer my own question.
The fact is that when we are dealing with Predicate Calculus we have the following Generalization Rule:
GR) If $x_i$ is not free in any... | {
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Analysis problem with volume I'm looking for a complete answer to this problem.
Let $U,V\subset\mathbb{R}^d$ be open sets and $\Phi:U\rightarrow V$ be a homeomorphism. Suppose $\Phi$ is differentiable in $x_0$ and that $\det D\Phi(x_0)=0$. Let $\{C_n\}$ be a sequence of open(or closed) cubes in $U$ such that $x_0$ is i... | Assume $x_0=\Phi(x_0)=0$, and put $d\Phi(0)=:A$. By assumption the matrix $A$ (or $A'$) has rank $\leq d-1$; therefore we can choose an orthonormal basis of ${\mathbb R}^d$ such that the first row of $A$ is $=0$.
With respect to this basis $\Phi$ assumes the form
$$\Phi:\quad x=(x_1,\ldots, x_d)\mapsto(y_1,\ldots, y_d)... | {
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how to calculate the exact value of $\tan \frac{\pi}{10}$ I have an extra homework: to calculate the exact value of $ \tan \frac{\pi}{10}$.
From WolframAlpha calculator I know that it's $\sqrt{1-\frac{2}{\sqrt{5}}} $, but i have no idea how to calculate that.
Thank you in advance,
Greg
| Your textbook probably has an example, where $\cos(\pi/5)$ (or $\sin(\pi/5)$) has been worked out. I betcha it also has formulas for $\sin(\alpha/2)$ and $\cos(\alpha/2)$ expressed in terms of $\cos\alpha$. Take it from there.
| {
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How to prove if function is increasing Need to prove that function $$P(n,k)=\binom{n}{k}= \frac{n!}{k!(n-k)!}$$ is increasing when $\displaystyle k\leq\frac{n}{2}$.
Is this inductive maths topic?
| As $n$ increases, $n!$ increases. As $n$ increases $(n-k)!$ increases. $(n+1)!$ is $(n+1)$ times larger than n!. $(n+1-k)!$ is however only $(n+1-k)$ times greater than $(n-k)!$. Therefore the numerator increases more than the denominator as n increases. Therefore P increases as n increases.
As $k$ increases, $k!$... | {
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Bi-Lipschitzity of maximum function Assume that $f(re^{it})$ is a bi-Lipschitz of the closed unit disk onto itself with $f(0)=0$. Is the function $h(r)=\max_{0\le t \le 2\pi}|f(re^{it})|$ bi-Lipschitz on $[0,1]$?
| It is easy to prove that $h$ is Lipschitz whenever $f$ is. Indeed, we simply take the supremum of the uniformly Lipschitz family of functions $\{f_t\}$, where $f_t(r)=|f(re^{it})|$.
Also, $h$ is bi-Lipschitz whenever $f$ is. Let $D_r$ be the closed disk of radius $r$. Let $L$ be the Lipschitz constant of the inverse... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Using the integral definition
Possible Duplicate:
Natural Logarithm and Integral Properties
I was asked to prove that ln(xy) = ln x + ln y using the integral definition.
While I'm not asking for any answers on the proof, I was wondering how to interpret and set-up this proof using the "integral definition" (As I am... | By definition,
$$\ln w=\int_1^w \frac{dt}{t}.$$
Thus
$$\ln(xy)=\int_1^{xy} \frac{dt}{t}=\int_1^x \frac{dt}{t}+\int_x^{xy}\frac{dt}{t}.$$
Now make an appropriate change of variable to conclude that the last integral on the right is equal to $\ln y$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$? Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?
| Consider, $z_1= \frac{1+2i}{\sqrt{5}}$, $z_2= \frac{1+3i}{\sqrt{10} }$, and $z_3= \frac{1+i}{\sqrt{2} }$, then:
$$ z_1 z_2 z_3 = \frac{1}{10} (1+2i)(1+3i)(1+i)=-1 $$
Take arg of both sides and use property that $\arg(z_1 z_2 z_3) = \arg(z_1) + \arg(z_2) + \arg(z_3)$:
$$ \arg(z_1) + \arg(z_2) + \arg(z_3) = -1$$
The L... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $\int_0^\infty f\text{d}x$ exists, does $\lim_{x\to\infty}f(x)=0$? Are there examples of functions $f$ such that $\int_0^\infty f\text{d}x$ exists, but $\lim_{x\to\infty}f(x)\neq 0$?
I curious because I know for infinite series, if $a_n\not\to 0$, then $\sum a_n$ diverges. I'm wondering if there is something similar... | If $\lim_{x\to+\infty}f(x)=l>0$, then $\exists M>0:l-\varepsilon<f(x)<l+\varepsilon\quad \forall x>M$, and so
$$
\int_M^{+\infty}f(x)dx>\int_M^{+\infty}(l-\varepsilon)dx=+\infty
$$
if $\varepsilon$ is sufficiently small.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/197450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Balanced but not convex? In a topological vector space $X$, a subset $S$ is convex if \begin{equation}tS+(1-t)S\subset S\end{equation} for all $t\in (0,1)$.
$S$ is balanced if \begin{equation}\alpha S\subset S\end{equation} for all $|\alpha|\le 1$.
So if $S$ is balanced then $0\in S$, $S$ is uniform in all directions ... | The interior of a regular pentagram centered at the origin is balanced but not convex.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/197521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Entire functions representable in power series How to prove that an entire function f, which is representable in power series with at least one coefficient is 0, is a polynomial?
| Define $F_n:=\{z\in \Bbb C, f^{(n)}(z)=0\}$. Since for each $n$, $f^{(n)}$ is holomorphic it's in particular continuous, hence $F_n$ is closed. Since we can write at each $z_0$, $f(z)=\sum_{k=0}^{+\infty}\frac{f^{(k)}(z_0)}{k!}(z-z_0)^k$, the hypothesis implies that $\bigcup_{n\geq 0}F_n=\Bbb C$. As $\Bbb C$ is complet... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Semi-direct product of different groups make a same group? We can prove that both of:
$S_3=\mathbb Z_3\rtimes\mathbb Z_2$ and $\mathbb Z_6=\mathbb Z_3\rtimes\mathbb Z_2$
So two different groups (and not isomorphic in examples above) can be described as semi-direct products of a pair of groups ($\mathbb Z_3$ and $\mat... | Such a group of smallest order is $D_8$, the Dihedral group of order 8.
Write $D_8=\langle x,y\colon x^4=y^2=1, y^{-1}xy=x^{-1}\rangle=\{1,x,x^2,x^3, y,xy,x^2y,x^3y \}$.
*
*$H=\langle x\rangle$,$K=\langle y\rangle$, then $D_8=H\rtimes K\cong C_4\rtimes C_2$.
*$H=\langle x^2,y\rangle$, $K=\langle xy\rangle$, then $D... | {
"language": "en",
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$\int\frac{x^3}{\sqrt{4+x^2}}$ I was trying to calculate
$$\int\frac{x^3}{\sqrt{4+x^2}}$$
Doing $x = 2\tan(\theta)$, $dx = 2\sec^2(\theta)~d\theta$, $-\pi/2 < 0 < \pi/2$ I have:
$$\int\frac{\left(2\tan(\theta)\right)^3\cdot2\cdot\sec^2(\theta)~d\theta}{2\sec(\theta)}$$
which is
$$8\int\tan(\theta)\cdot\tan^2(\theta)\... | You have not chosen an efficient way to proceed. However, let us continue along that path.
Note that $\tan^2\theta=\sec^2\theta-1$. So you want
$$\int 8(\sec^2\theta -1)\sec\theta\tan\theta\,d\theta.$$
Let $u=\sec\theta$.
Remark: My favourite substitution for this problem and close relatives is a variant of the one use... | {
"language": "en",
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What is the value of $w+z$ if $1I am having solving the following problem:
If the product of the integer $w,x,y,z$ is 770. and if $1<w<x<y<z$ what is the value of $w+z$ ? (ans=$13$)
Any suggestions on how I could solve this problem ?
| Find the prime factorization of the number. That is always a great place to start when you have a problem involving a product of integer. Now here you are lucky, you find $4$ prime numbers to the power of one, so you know your answer is unique.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Maclaurin expansion of $\arcsin x$ I'm trying to find the first five terms of the Maclaurin expansion of $\arcsin x$, possibly using the fact that
$$\arcsin x = \int_0^x \frac{dt}{(1-t^2)^{1/2}}.$$
I can only see that I can interchange differentiation and integration but not sure how to go about this. Thanks!
| As has been mentioned in other answers, the series for $\frac1{\sqrt{1-x^2}}$ is most easily found by substituting $x^2$ into the series for $\frac1{\sqrt{1-x}}$. But for fun we can also derive it directly by differentiation.
To find $\frac{\mathrm d^n}{\mathrm dx^n}\frac1{\sqrt{1-x^2}}$ at $x=0$, note that any factor... | {
"language": "en",
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3-dimensional array I apologize if my question is ill posed as I am trying to grasp this material and poor choice of tagging such question. At the moment, I am taking an independent studies math class at my school. This is not a homework question, but to help further my understanding in this area. I've been looking aro... | There is no single transformation corresponding to taking the transpose. The reason is that while there is only one non-identity permutation of a pair of indices, there are five non-identity permutations of three indices. There are two that leave none of the fixed: one takes $a_{ijk}$ to $a_{jki}$, the other to $a_{kij... | {
"language": "en",
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"source": "stackexchange",
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A dubious proof using Weierstrass-M test for $\sum^n_{k=1}\frac{x^k}{k}$ I have been trying to prove the uniform convergence of the series
$$f_{n}(x)=\sum^n_{k=1}\frac{x^k}{k}$$
Obviously, the series converges only for $x\in(-1,1)$. Consequently, I decided to split this into two intervals: $(-1,0]$ and $[0,1)$ and see... | Weierstrass M-test only gives you uniform convergence on intervals of the form $[-q,q]$, where $0<q<1$. Your proof shows this.
You also get uniform convergence on the interval $[-1,0]$, but to see this you need other methods. For example the standard estimate for the cut-off error of a monotonically decreasing alternat... | {
"language": "en",
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Which axiom shows that a class of all countable ordinals is a set? As stated in the title, which axiom in ZF shows that a class of all countable (or any cardinal number) ordinals is a set?
Not sure which axiom, that's all.
| A combination of them, actually. The proof I've seen requires power set and replacement, and I think union or pairing, too, but I can't recall off the top of my head. I can post an outline of that proof, if you like.
| {
"language": "en",
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From set of differential equations to set of transfer functions (MIMO system) I want to know how I can get from a set of differential equations to a set of transfer functions for a multi-input multi-output system. I can do this easily with Matlab or by computing $G(s) = C[sI - A]^{-1}B + D$. I have the following two eq... | I am guessing that you are looking for the transfer function from $u$ to $y$, this would be consistent with current nomenclature.
Taking Laplace transforms gives
$$ (s^2+2s) \hat{y_1} + s\hat{y_2} + \hat{u_1} = 0\\
(s-1)\hat{y_2} + \hat{u_2}-s \hat{u_1} = 0 $$
Solving algebraically gives
$$\hat{y_1} = \frac{1-s-s^2}{s... | {
"language": "en",
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How do I simplify this limit with function equations? $$\lim_{x \to 5} \frac{f(x^2)-f(25)}{x-5}$$
Assuming that $f$ is differentiable for all $x$, simplify.
(It does not say what $f(x)$ is at all)
My teacher has not taught us any of this, and I am unclear about how to proceed.
| $f$ is differentiable, so $g(x) = f(x^2)$ is also differentiable. Let's find the derivative of $g$ at $x = 5$ using the definition.
$$
g'(5) = \lim_{x \to 5} \frac{g(x) - g(5)}{x - 5} = \lim_{x \to 5} \frac{f(x^2) - f(25)}{x - 5}
$$
Now write $g'(5)$ in terms of $f$ to get the desired result.
| {
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Basic set questions I would really appreciate it if you could explain the set notation here
$$\{n ∈ {\bf N} \mid (n > 1) ∧ (∀x,y ∈ {\bf N})[(xy = n) ⇒ (x = 1 ∨ y = 1)]\}$$
1) What does $∀x$ mean?
2) I understand that $n ∈ {\bf N} \mid (n > 1) ∧ (∀x,y ∈ {\bf N})$ means $n$ is part of set $\bf N$ such that $(n > 1) ∧ (∀... | 1) $(\forall x)$ is the universal quantifier. It means "for all $x$".
2) $[ ]$ is the same as a parenthesis. Probably, the author did not want to use too many round parenthesis because it would get too confusing.
$\Rightarrow$ is implies.
3) Suppose $x \in A$. Since $A \subset B$, by definition $x \in B$. Since $B \sub... | {
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Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective?
I tried the following: $$f:\mathbb{R}\rightarrow \mathbb{R^{+}}
$$ $$f(x)=x^{2}$$
and $$g:\mathb... | If we define $f:\mathbb{R}^2 \to \mathbb{R}$ by $f(x,y) = x$ and $g:\mathbb{R} \to \mathbb{R}^2$ by $g(x) = (x,0)$ then $f \circ g : \mathbb{R} \to \mathbb{R}$ is bijective (it is the identity) but $f$ is not injective and $g$ is not surjective.
| {
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Graphing Cubic Functions I'm having a Little bit of trouble in Cubic Functions, especially when i need to graph the Turning Point, Y-intercepts, X-intercepts etc. My class teacher had told us to use Gradient Method:
lets say: $$f(x)=x^3+x^2+x+2$$
We can turn this equation around by using the Gradient Method:
$$f'(x)=3x... | I used to think that the Gradient Method is for plotty functions in 2 variables. However, this answer may give you some pointers.
You could start by examining the function domain. In your case, all $x$ values are valid candidates. next, set $x=0$ then $y=0$ to get the intercepts. Setting $x=0$, yields $y=2$, so the po... | {
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Is Fractal perimeter always infinite? Looking for information on fractals through google I have read several time that one characteristic of fractals is :
*
*finite area
*infinite perimeter
Although I can feel the area is finite (at least on the picture of fractal I used to see, but maybe it is not necessarly tr... | If they have infinite sides, than they must have an infinite perimeter, especially if they are perfectly straight because the formula of perimeter of most shapes is adding up the amount of sides, and the fractal has infinite sides, then it should have an infinite perimeter.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/198591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exactly one nontrivial proper subgroup Question: Determine all the finite groups that have exactly one nontrivial proper subgroup.
MY attempt is that the order of group G has to be a positive nonprime integer n which has only one divisor since any divisor a of n will form a proper subgroup of order a. Since 4 is the on... | Let $H$ be the only non-trivial proper subgroup of the finite group $G$. Since $H$ is proper, there must exist an $x \notin H$. Now consider the subgroup $\langle x\rangle$ of $G$. This subgroup cannot be equal to $H$, nor is it trivial, hence $\langle x\rangle = G$, that is $G$ is cyclic, say of order $n$. The number ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Multigrid tutorial/book I was reading Press et. al., "Numerical Recipes" book, which contain section about multigrid method for numerically solving boundary value problems.
However, the chapter is quite brief and I would like to understand multigrids to a point where I will be able to implement more advanced and faster... | First please don't be bluffed by those fancy terms coined by computational scientists, and don't worry about preconditioning or conjugate gradient. The multigrid method for numerical PDE can be viewed as a standalone subject, basically what it does is: make use of the "information" on both finer and coarser mesh, in or... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Multiplicative but non-additive function $f : \mathbb{C} \to \mathbb{R}$ I'm trying to find a function $f : \mathbb{C} \to \mathbb{R}$ such that
*
*$f(az)=af(z)$ for any $a\in\mathbb{R}$, $z\in\mathbb{C}$, but
*$f(z_1+z_2) \ne f(z_1)+f(z_2)$ for some $z_1,z_2\in\mathbb{C}$.
Any hints or heuristics for finding su... | HINT: Look at $z$ in polar form.
| {
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Check that a curve is a geodesic. Suppose $M$ is a two-dimensional manifold. Let $\sigma:M \rightarrow M$ be an isometry such that $\sigma^2=1$. Suppose that the fixed point set $\gamma=\{x \in M| \sigma(x)=x\}$ is a connected one-dimensional submanifold of $M$. The question asks to show that $\gamma$ is the image of a... | Let $N=\{x\in M:\sigma(x)=x\}$ and fix $p\in N$.
Exercise 1: Prove that either $1$ or $-1$ is an eigenvalue of $d\sigma_p:T_p(M)\to T_p(M)$.
Exercise 2: Prove that if $v\in T_p(M)$ is an eigenvector of $d\sigma_p:T_p(M)\to T_p(M)$ of sufficiently small norm, then the unique geodesic $\gamma:I\to M$ for some open inte... | {
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"timestamp": "2023-03-29T00:00:00",
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Find delta with a given epsilon for $\lim_{x\to-2}x^3 = - 8$ Here is the problem. If
$$\lim_{x\to-2}x^3 = - 8$$
then find $\delta$ to go with $\varepsilon = 1/5 = 0.2$.
Is $\delta = -2$?
| Sometimes Calculus students are under the impression that in situations like this there is a unique $\delta$ that works for the given $\epsilon$ and that there is some miracle formula or computation for finding it.
This is not the case. In certain situations there are obvious choices for $\delta$, in certain situation... | {
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Diameter of Nested Sequence of Compact Set
Possible Duplicate:
the diameter of nested compact sequence
Let $(E_j)$ be a nested sequence of compact subsets of some metric space; $E_{j+1} \subseteq E_j$ for each $j$.
Let $p > 0$, and suppose that each $E_j$ has diameter $\ge p$ . Prove that
$$E = \bigcap_{j=1}^{\inft... | For each $j$ pick two points $x_j, y_j \in E_j$ such that $d(x_j,y_j) \ge p$. Since $x_j \in E_1$ for all $j$, and $E_1$ is compact, the sequence $(x_j)$ has a convergent subsequence $(x_{\sigma(j)})$ say, and likewise $(y_{\sigma(j)})$ has a convergent subsequence $(y_{\tau \sigma(j)})$.
What can you say about the lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Breakdown of solution to inviscid Burgers equation
Let $u = f(x-ut)$ where $f$ is differentiable. Show that $u$ (amost always) satisfies $u_t + uu_x = 0$. What circumstances is it not necessarily satisfied?
This is a question in a tutorial sheet I have been given and I am slightly stuck with the second part. To show ... | We have
\[
u_t = f'(x-ut)(x-ut)_t = -f'(x-ut)(u_t t + u)
\iff \bigl(1 + tf'(x-ut)\bigr)u_t = -uf'(x-ut)
\]
and
\[
u_x = f'(x-ut)(x-ut)_x = f'(x-ut)(1 - u_xt)
\iff \bigl(1 + tf'(x-ut)\bigr)u_x = f'(x-ut)
\]
Which gives that
\[
\bigl(1 + tf'(x-ut)\bigr)(u_t +uu_x) = 0
\]
so at each point either $1 + tf... | {
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How do we know how many branches the inverse function of an elementary function has? How do we know how many branches the inverse function of an elementary function has ?
For instance Lambert W function. How do we know how many branches it has at e.g. $z=-0.5$ , $z=0$ , $z=0.5$ or $z=2i$ ?
| Suppose your elementary function $f$ is entire and has an essential singularity at $\infty$ (as in the case you mention, with $f(z) = z e^z$). Then Picard's "great" theorem says that $f(z)$ takes on every complex value infinitely often, with at most one exception. Thus for every $w$ with at most one exception, the inv... | {
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How to find the least path consisting of the segments AP, PQ and QB Let $A = (0, 1)$ and $B = (2, 0)$ in the plane.
Let $O$ be the origin and $C = (2, 1)$ .
Consider $P$ moves on the segment $OB$ and
$Q$ move on the segment $AC$.
Find the coordinates of $P$ and $Q$ for which the length of the path consisting of the se... | Hint: Let $A'$ be the point one unit above $A$.
Let $B'$ be the point one unit below $B$.
Join $A'$ and $B'$ by a straight line. Show that gives the length of the minimal path.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/199230",
"timestamp": "2023-03-29T00:00:00",
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Child lamp problem A street lamp is 12 feet above the ground. A child 3 feet in height amuses itself by walking in such a way that the shadow of its head moves along lines chalked on the ground. (1) How would the child walk if the chalked line is (a) straight, (b) a circle, (c) a square? (2) What difference would it ma... | Similar triangles show that from each point on the line you draw a line to the base of the lamp. When the child's head makes a shadow at a given point it is $\frac 14$ of the way along the line from the point to the lamp. So the child walks in the same shape: line, circle, or square, with size $\frac 34$ of the figur... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the inverse function of $\ x^2+x$? I think the title says it all; I'm looking for the inverse function of $\ x^2+x$, and I have no idea how to do it. I thought maybe you could use the quadratic equation or something. I would be interesting to know.
| If you want to invert $x^2 + x$ on the interval $x \ge -1/2$, write $y = x^2 + x$, so $x^2 + x -y = 0$.
Use the quadratic equation with $a=1$, $b=1$, and $c=-y$ to find $$ x= \frac{-1 + \sqrt{1+4y}}{2}.$$
(The choice of whether to use $+\sqrt{4ac}$ rather than $-\sqrt{4ac}$ is because we are finding the inverse of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the necessary and sufficient conditions on $a$, $b$ so that $ax^2 + b = 0$ has a real solution. This question is really confusing me, and I'd love some help but not the answer. :D
Is it asking: What values of $a$ and $b$ result in a real solution for the equation $ax^2 + b = 0$? $a = b = 0$ would obviously work, b... | I assume the question is "find conditions that are necessary and sufficient to guarantee solutions" rather than "find necessary conditions and also find sufficient sufficient conditions for a solution." If the former is the case, then you're asked for constraints on $a$ and $b$ such that (1) if the conditions are met t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Homomorphism of free modules $A^m\to A^n$ Let's $\varphi:A^m\to A^n$ is a homomorphism of free modules over commutative (associative and without zerodivisors) unital ring $A$. Is it true that $\ker\varphi\subset A^m$ is a free module?
Thanks a lot!
| Here is a counterexample which is in some sense universal for the case $m = 2, n = 1$. Let $R = k[x, y, z, w]/(xy - zw)$ ($k$ a field). This is an integral domain because $xy - zw$ is irreducible. The homomorphism $R^2 \to R$ given by
$$(m, n) \mapsto (xm - zn)$$
has a kernel which contains both $(y, w)$ and $(z, x)$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this AM/GM refinement correct or not? In Chap 1.22 of their book Mathematical Inequalities, Cerone and Dragomir prove the following interesting inequality. Let $A_n(p,x)$ and $G_n(p,x)$ denote resp. the weighted arithmetic and the weighted geometric means, where $x_i\in[a,b]$ and $p_i\ge0$. $P_n$ is the sum of all $... | Your modesty in suspecting that the error is yours is commendable, but in fact you found an error in the book. The "simple calculation" on p. $49$ is off by a factor of $2$, as you can easily check using $n=2$ and $p_1=p_2=1$. Including a factor $\frac12$ in the inequality makes it come out right.
You can also check th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof: Symmetric and Positive Definite If $A$ is a symmetric and positive definite matrix and matrix $B$ has linearly independent columns , is it true that $B^T A B$ is symmetric and positive definite?
| If the matrices are real yes: take $x\in\Bbb C^d$. Then $Bx\in \Bbb C^d$ hence $x^tB^tABx=(Bx)^tA(Bx)\geq 0$ and if $x\neq 0$, as $B$ is invertible $Bx\neq 0$. $A$ being positive definite we have $x^tB^tABx>0$.
But if the matrices are real it's not true: take $A=I_2$, $B:=\pmatrix{1&0\\0&i}$, then $B^tAB=\pmatrix{1&0\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Cardinality of $R[x]/\langle f\rangle$ via canonical remainder reps. Suppose $R$ is a field and $f$ is a polynomial of degree $d$ in $R[x]$. How do you show that each coset in $R[x]/\langle f\rangle$ may be represented by a unique polynomial of degree less than $d$? Secondly, if $R$ is finite with $n$ elements, how d... | Hint $ $ Recall $\rm\ R[x]/(f)\:$ has a complete system of reps being the least degree elements in the cosets, i.e. the remainders mod $\rm\:f,\:$ which uniquely exist by the Polynomial Division Algorithm.
Therefore the cardinality of the quotient ring equals the number of such reps, i.e. the number of polynomials $\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Solving $(x+y) \exp(x+y) = x \exp(x)$ for $y$. While thinking about the Lambert $W$ function I had to consider
Solving $(x+y) \exp(x+y) = x \exp(x)$ for $y$.
This is what I arrived at:
(for $x$ and $y$ not zero)
$(x+y) \exp(x+y) = x \exp(x)$
$x\exp(x+y) + y \exp(x+y) = x \exp(x)$
$\exp(y) + y/x \exp(y) = 1$
$y/x \exp(y... | The solution of $ (x+y) \exp(x+y) = x \exp(x) $ is given in terms of the Lambert W function
Let $z=x+y$, then we have
$$ z {\rm e}^{z} = x {\rm e}^{x} \Rightarrow z = { W} \left( x{{\rm e}^{x}} \right) \Rightarrow y = -x + { W} \left( x{{\rm e}^{x}} \right) \,. $$
Added: Based on the comment by Robert, here are the g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculus and Physics Help! If a particle's position is given by $x = 4-12t+3t^2$ (where $t$ is in seconds and $x$ is in meters):
a) What is the velocity at $t = 1$ s?
Ok, so I have an answer:
$v = \frac{dx}{dt} = -12 + 6t$
At $t = 1$, $v = -12 + 6(1) = -6$ m/s
But my problem is that I want to see the steps of using th... | You see the problem here is that the question is asking for a velocity at $t=1$. This means that they require and instantaneous velocity which is by definition the derivative of the position function at $t=1$. If you don't want to use derivative rules for some reason and you don't mind a little extra work then you can ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Replacing one of the conditions of a norm Consider the definition of a norm on a real vector space X.
I want to show that replacing the condition
$\|x\| = 0 \Leftrightarrow x = 0\quad$
with
$\quad\|x\| = 0 \Rightarrow x = 0$
does not alter the the concept of a norm (a norm under the "new axioms" will
fulfill the "old a... | All you need to show is that $\|0\|=0$. Let $x$ be any element of the normed space. What is $\|0\cdot x\|$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/199956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
Show that the discrete metric can not be obtained from $X\neq\{0\}$ If $X \neq \{ 0\}$ is a vector space. How does one go about showing that the discrete metric on $X$ cannot be obtained from any norm on $X$?
I know this is because $0$ does not lie in $X$, but I am having problems. Formalizing a proof for this.
This is... | You know that the discrete metric only takes values of $1$ and $0$. Now suppose it comes from some norm $||.||$. Then for any $\alpha$ in the underlying field of your vector space and $x,y \in X$, you must have that
$$\lVert\alpha(x-y)\rVert = \lvert\alpha\rvert\,\lVert x-y\rVert.$$
But now $||x-y||$ is a fixed number... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Confusion related to the concatenation of two grammars I have this confusion. Lets say I have two languages produced by type 3 grammar such that
L(G1) = <Vn1,Vt,P1,S1>
L(G2) = <Vn2,Vt,P2,S2>
I need to find a type3 grammar G3 such that
L(G3) = L(G1)L(G2)
I can't use $S3 \rightarrow S1S2$ to get the concatenaion, becau... | First change one of the grammars, if necessary, to make sure that they have disjoint sets of non-terminal symbols.
If you’re allowing only productions of the forms $X\to a$ and $X\to Ya$, make the new grammar $G$ generate a word of $L(G_2)$ first and then a word of $L(G_1)$: replace every production of the form $X\to a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A problem involving Laplace operator $\Omega$ is a bounded open set in $\mathbb R^n$, consider the number
$ r = \inf \{ \left\| {du} \right\|_{{L^2}(\Omega )}^2:u \in H_0^1(\Omega ),{\left\| u \right\|_{{L^2}(\Omega )}} = 1\}$
If for some $v\in H_0^1(\Omega )$ the infimum is achieved, then is $\Delta v\in L^2(\Omega)$?... | Let
$$
f, g: H_0^1(\Omega) \to \mathbb{R}, f(u)=\|\nabla u\|_{L^2(\Omega)}^2,\ g(u)=\|u\|_{L^2(\Omega)}^2.
$$
Then
$$
r=\inf\{f(u):\ u \in H_0^1(\Omega),\ g(u)=1\}.
$$
If
$$
r=f(v),
$$
where $v$ belongs to $H_0^1(\Omega)$ and satisfies $g(v)=1$, then, there is a $\lambda \in \mathbb{R}$ such that
$$
Df(v)\cdot h=\lamb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is an abstract simplicial complex a quiver? Let $\Delta$ be an abstract simplicial complex. Then for $B\in \Delta$ and $A\subseteq B$ we have that $A\in\Delta$. If we define $V$ to be the set of faces of $\Delta$, construct a directed edge from $B$ to $A$ if $A$ is a face of $B$ (i.e. $A\subseteq B$) and define $E$ to ... | Yes, and it's the poset of faces ordered by inclusion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/200200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Problem with Ring $\mathbb{Z}_p[i]$ and integral domains Let $$\Bbb Z_p[i]:=\{a+bi\;:\; a,b \in \Bbb Z_p\,\,,\,\, i^2 = -1\}$$
-(a)Show that if $p$ is not prime, then $\mathbb{Z}_p[i]$ is not an integral domain.
-(b)Assume $p$ is prime. Show that every nonzero element in $\mathbb{Z}_p[i]$ is a unit if and only if $x^2+... | Note that $(a+bi)(a-bi)=a^2+b^2$. If $a^2+b^2\equiv0\pmod p$, then $a+bi$ is not a unit. And if $a^2+b^2$ is not zero modulo $p$, then it's invertible modulo $p$, so $a+bi$ is a unit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/200259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
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