Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$ I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$
Which function/ contour should I consider ?
| Let us integrate the function $e^{-Az}$, where $A=\sqrt{a^2+b^2}$ on a circular sector in the first quadrant, centered at the origin and of radius $\mathcal{R}$, with angle $\omega$ which satisfies $\cos \omega = a/A$, and therefore $\sin \omega = b/A$.
Let this sector be called $\gamma$.
Since our integrand is obvious... | {
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"timestamp": "2023-03-29T00:00:00",
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$T: \mathbb R^n \to \mathbb R^n$, $\langle Tu,v\rangle=\langle u,T^*v\rangle$, is $T^*=T^t$ regardless of inner product? Basic question in linear algebra here. $T$ is a linear transform from $\mathbb R^n$ to $\mathbb R^n$ defined by $T(v)=Av$, $A\in \mathrm{Mat}_n(\mathbb R)$. We are given some inner product $\langle ,... | No it does not hold for any inner product.
It is hard hard to show that any other inner product $\langle\cdot,\cdot\rangle_*$ can be represented as
$$
\langle x,y\rangle_\star=\langle x,Sy\rangle,
$$
where $S$ is a positive definite matrix. So
$$
\langle Tx,y\rangle_\star=\langle x,T^*Sy\rangle \ne \langle x,ST^*y\rang... | {
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"url": "https://math.stackexchange.com/questions/663218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Linear Algebra - check my solution, product of symmetric positive-definite matrices We are given $A,B \in Mat_n(\mathbb R)$ are symmetric positive-definite matrices such that $AB=BA$
Show that $AB$ is positive-definite
What I did:
First I showed that $AB$ is symmetric, this is easily shown from $(AB)^t=B^tA^t=BA=AB$
No... | $AB=BA$ is a interesting condition. Let $v$ be an eigenvector of $A$ such that $Av=\lambda v$. Then $A(Bv)=B(Av)=B(\lambda v)=\lambda(Bv)$. This implies that the eigenvector space $V_{\lambda}$ of $A$ is invariant space of $B$. Now we can find all eigenvalues of $B$ on $V_{\lambda}$. Take the basis of $V_{\lambda}'s$ f... | {
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"timestamp": "2023-03-29T00:00:00",
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Sufficient Statistic Basics If I know the value of a sufficient statistic, but not the sample that generated it, am I right to suspect that the conditional distribution of any other statistic given the sufficient statistic will not depend on the parameter of interest? Formally speaking:
Let $\theta$ be the parameter of... | That is correct, PROVIDED that the statistical model is right. But the sufficient statistic is not where you will find evidence that the model doesn't fit. For example, in estimating the mean and variance of a normally distributed population, the sufficient statistic is the pair whose components are the sum of the ob... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that a group where $a^2=e$ for all $a$ is commutative Defining a group $(G,*)$ where $a^2=e$ with $e$ denoting the identity class....
I am to prove that this group is commutative. To begin doing that, I want to understand what exactly the power of 2 means in this context. Is the function in the group a power or s... | The trick with these types of problems is to evaluate the 'product' of group elements in two different ways.
So for this problem, we interpret $(ab)^2$ two different ways, where $a,b \in G$.
First, we have this rule in $G$ that an element 'squared' is the identity. So we know that
$$
(ab)^2=e
$$
But
$$
(ab)^2=abab
$$... | {
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"timestamp": "2023-03-29T00:00:00",
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Combinatorics - possibly pigeon hole, 100 by 100 matrix with numbers from 1 to 100 We are given a $100$ by $100$ matrix. Each number from $\{1,2,...,100\}$ appears in the matrix exactly a $100$ times.
Show there is a column or a row with at least $10$ different numbers.
I'd like a small tip how to tackle this problem. ... | Managed to solve it.
first, understand that each number appears in at least 20 different rows and columns (maximum is 101, minimum is 20). The most extreme and compact case, is that a number fills a $10$ by $10$ matrix, so that's 20 holes
But we have 100 different numbers, 100*20=2000. $\frac{2000}{200}=10$ and so ther... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/663677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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find $P\{P\{0\}\}$. $P$ represents the power set. I'm assuming that I'm trying to find the power set of a power set?
I start from the inner power set, $P\{0\}$. $P\{0\}= \{ 0, \{0\} \}$.
Now I do $P\{ 0, \{0\} \}$ which is $\{ 0, \{0\}, \{\{0\}\} \}$.
0 is the empty set.
Is this correct?
So I'm taking it that P{0}={0, ... | Almost. You forgot one subset of $\{0,\{0\}\}.$ (Hint: It isn't proper.)
| {
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Continuity of $f(x)=x^p$ when $p$ is a real number and $x\in (0,\infty)$ Here is my final answer.
Definition Let $x>0$ be a real, and $\alpha$ be a real number. We define the quantity $x^{\alpha}$, by the formula $\text{lim}_{n\rightarrow\infty} x^{q_n}$ where $(q_n)$ is a sequence of rationals which converges to $\al... | The book's hint is spot on: suppose you have proven that $x^p$ is continuous at $1$; and choose some $\alpha>0$ distinct from $1$. Prove that $x^p$ is continuous at $\alpha$ iff $\alpha^{-p}x^p$ is continuous at $\alpha$, and note that this last is equivalent to $$\left(\frac x\alpha\right)^p\to 1$$ as $x\to \alpha$. B... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Give me a example of a function Lebesgue Integrable over [a,b] that is not bounded in any subinterval of [a,b] Give me a example of a function Lebesgue Integrable over [a,b] that is not bounded in any subinterval of [a,b].
*I'm thinking about this but without progress...
| $f(x)=0,\ x\in{\Bbb R}\backslash{\Bbb Q}$
$f(r/s)=s$, $r/s$ irreducible fraction.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove an equality using combinatorial arguments $$n \cdot {2^{n - 1}} = \sum\limits_{k = 1}^n {k\left( {\begin{array}{*{20}{c}}
n \\
k \\
\end{array}} \right)} $$
The left-hand side can describe the number of possibilities choosing a committee with one chairman.
How can the right-hand side feet to this story?
| It is equivalent because for each group with the size of $k$ we can choose $k$ different chairmans.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Basic analysis - sequence convergence I'm taking a course entitled "Concepts in Real Analysis," and I'm feeling pretty dumb at the moment, because this is obviously quite elementary...
The example in question shows $\lim_{n\to\infty} \frac{3n+1}{2n+5}=\frac{3}{2}$, and, setting $\left|\frac{3n+1}{2n+5}-\frac32\right|= ... | It may be a little subtle, but it doesn't actually work unless you pick the limit.
To see this, lets write our limiting value as
$$ \left|\frac{3n+1}{2n+5} - \left(\frac{3}{2} + \delta\right)\right| < \epsilon $$
for some $\delta\neq 0$. Simpilfying, we have
$$ \left| \frac{-13 - (2n+5)\cdot 2\delta}{4n+10} \right| < \... | {
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Is $\mathbb Q \times \mathbb Q $ a denumerable set? How can one show that there is a bijection from $\mathbb N$ to $\mathbb Q \times \mathbb Q $?
| Yes, $\mathbb{Q} \times \mathbb{Q}$ is countable (denumerable). Since $\mathbb{Q}$ is countable (this follows from the fact that $\mathbb{N} \times \mathbb{N}$ is countable), taking the cartesian product of two countable sets gives you back a countable set. This link: http://www.physicsforums.com/showthread.php?t=487... | {
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Exponents and fractions pre-calculus How would I go on about solving this:
${x^4y^7\over x^5y^5}$
When $x = {1\over3}$ and y = ${2\over 9}$
My working out:
Firstly I simplify.-
${xy^2\over x}$
Then substitute
${{{1\over3} * {2\over9}}^2\over{1\over3}}$
Further,
${{{1\over3} * {4\over81}}\over{1\over3}}$
and
${{4\over24... | The first step is
$$\frac{y^2}{x}$$
So the answer is
$$
\frac{4/81}{1/3}= \frac{4 \cdot 3}{81} = \frac{4}{27}$$
You just had the extra $x$. Must be an oversight
| {
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Partial derivative of a function with another function inside it? What is
$\cfrac {\partial f(x, y, g(x))} {\partial x}$ expanded out?
I want to say $\cfrac {\partial f(x, y, g(x))} {\partial g(x)} \times \cfrac {\partial g(x)} {\partial x}$ but I don't think that's quite right.
| $d_{x}f(x, y, g(x)) + \frac{dg}{dx} d_{z}f(x, y, g(x))$
f = f(x, y, z) and g(x) = f(x, y, g(x)). When x moves by dx, you are evaluating f to a new point where x AND z have changed.
(x, y, g(x)) -> (x+dx, y, g(x+dx)) ~ (x+dx, y, g(x) + g'(x)dx)
So when x moves a little bit, we will see a change in x and z.
Generally spe... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove if $A \in Mat_{n,n}(\mathbb F)$ is both symmetric and skew-symmetric then $A=0$ Prove if $A \in Mat_{n,n}(\mathbb F)$ is both symmetric and skew-symmetric then $A=0$
I know $A^T = A = -A \Rightarrow A = -A \Rightarrow A_{i,j} = -A_{i,j}$.
Since $\mathbb F$ is a field we have $2A_{i,j} = 0 \Rightarrow 2 = 0 \lor ... | [from my comment]
You're exactly right. This holds in every case but characteristic $2$. To see this, $A=−A\implies2A=0$, and that is true for every matrix in characteristic $2$. All you need is an explicit example to prove that a nonzero $A$ exists, e.g. the identity.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Inconsistent inequalities I want to prove that the following two inequalities cannot hold simultaneously
$\beta_0^2(\beta_1-1)\geq \beta_1$
and
$\beta_1^2(\beta_0-1)\leq \beta_0$,
where $1<\beta_0<\beta_1<2$
| That is a statement rather than a question...
Suppose that they do hold 'simultaneously' and I will use $a:=\beta_0$ and $b:=\beta_1$.
Add them together to get
$$\begin{align}
b+ab^2-b^2& \leq a^2b-a^2+a
\\\Rightarrow a^2b-ab^2-a^2+b^2-b+a&\geq0
\\ \Rightarrow ab(a-b)-(a-b)(a+b)+1(a-b)&\geq 0
\\ \Rightarrow (a-b)(ab-a-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $y = x$ is continuous For every $\epsilon > 0$ there exists a $\delta > 0$ such that $|x - c| < \delta$ implies $|f(x) - f(c)| < \epsilon$.
Start with $|f(x) - f(c)| < \epsilon$ which gives $|x - c| < \epsilon$. We also know $|x - c| < \delta$ but how can we connect $\epsilon$ and $\delta$?
| (For future visitors) You can prove it as follows.
For any $x \in \mathbb R$ we have:
$x_n \to x_0 \implies f\left(x_n\right) = x_n \to x_0 = f\left(x_0\right)$
Meaning, if sequence $x_n$ somehow approaches $x_0$, then corresponding sequence $f\left(x_n\right)$ always approaches $f\left(x_0\right)$, which is the defini... | {
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How many strings contain every letter of the alphabet? Given an alphabet of size $n$, how many strings of length $c$ contain every single letter of the alphabet at least once?
I first attempted to use a recurrence relation to work it out:
$$
T(c) = \left\{ \begin{array}{cr}
0 &\mbox{ if $c<n$} \\
n! &\mbox{ if $c = n$}... | Let $W(c,n)$ denote the number of words of length $c$ from an alphabet of $n$ letters. Then $W(c,n)=n^c$.
Out of these, the number of words of the same size that do not contain one of the letters is $W(c,n-1)=(n-1)^c$. The number of ways of choosing which letter is missing is $\binom{n}{1}$.
The number of words of the ... | {
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How many arrangements of the digits 1,2,3, ... ,9 have this property? How many arrangements of the digits 1,2,3, ... ,9 have the property that every digit (except the first) is no more than 3 greater than the previous digit?
(For example, the arrangement 214369578 has this property. However, 312548697 does not have th... | Let $a_n$ denote the number of valid configurations.
Note that if $n\ge 4$, $a_{n+1}=4a_n$. $a_4=4!$, so $a_9=4! \cdot 4^5=\boxed{24576.}$
| {
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How to calculate the hight by number of nodes Imagine that I have something like following structure and I keep adding more to it, so the level 1 has only one node and level 2 has 2 and level n had n node, how can i calculate the n from the total number of tokens (size).
for example if I want to add j, knowing that th... | If you want to add the $n^{th}$, node, it has to go on the $\left\lceil\frac{-1+\sqrt{1+8n}}{2}\right\rceil^{th}$ floor.
Proof : The last element of the $n^{th}$ row is the $n^{th}$ triangular number, so $\frac{n(n+1)}{2}$.
Now let's consider the $k^{th}$ node. What should be $k$ to be the last element of the $n^{th}$ ... | {
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What is the square root of complex number i? Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I wrong?
EDIT: my question is rather related to nomenclature of definition, while ... | I think this would be easier to see by writing $i$ in its polar form,
$$i=e^{i\pi/2}$$
This shows us that one square root of $i$ is given by
$$i^{1/2}=e^{i\pi/4} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/664962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is there any function that has the series expansion $x+x^{\frac{1}{2}}+x^{\frac{1}{3}}+\cdots$? $$\frac{1}{1-x} = 1+x+x^2+x^3+ \cdots$$
Is there a $f(x)$ that has the series of $n$th roots?
$$f(x)= x+x^{\frac{1}{2}}+x^{\frac{1}{3}}+ \cdots$$
Wolfram Alpha seemed to not understand my input.
| For $x>0,~x+x^{\frac{1}{2}}+x^{\frac{1}{3}}+...$ can't converges since $x^{\frac{1}{n}}\to1.$ Neither does it for $x<0$ since $x^{\frac{1}{2}}$ is undefined then. So the only possibility is the trivial one: $$f:\{0\}\to\mathbb R:0\mapsto 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/665108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 3
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Finding the $\limsup$ of a sequence of sets Is my proof if this equivalence correct?
$$\limsup_{n\to\infty} A_n=\bigcap_{i=1}^\infty \bigcup_{j=i}^\infty A_i=\{\text{ elements that belong to infinitely mane } A_i \text{'s }\}$$
Pf.
Let $B_i=\bigcup_{j\ge i} A_j$. Let $x \in \limsup$, then:
($\rightarrow$) $x\in \lims... | I don't think your proof is correct. Just note that
$$
x \in \limsup_{n \to \infty} A_n \quad \iff \quad \forall i \geq 1, \exists j \geq i : x \in A_j \quad \iff \quad x \text{ belongs to infinitely many $A_i$'s}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/665267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ways to induce a topology on power set? In this question, two potential topologies were proposed for the power set of a set $X$ with a topology $\mathcal T$: one comprised of all sets of subsets of $X$ whose union was $\mathcal T$-open, one comprised of all sets of subsets of $X$ whose intersection was $\mathcal T$-ope... | Let a set be open iff it is empty or of the form $\mathcal{P}(X)\backslash\big\{\{x\}:x\in C\big\}$ for some closed set C.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "16",
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Does every prime $p \neq 2, 5$ divide at least one of $\{9, 99, 999, 9999, \dots\}$? I was thinking of decimal expressions for fractions, and figured that a fraction of the form $\frac{1}{p}$ must be expressed as a repeating decimal if $p$ doesn't divide $100$. Thus, $\frac{p}{p}$ in decimal would equal $0.\overline{99... | Essentially, you wish to find a $k$ such that $10^k - 1 \equiv 0 \pmod p$. This is equivalent to $10^k \equiv 1 \pmod p$. There are many reasons that such a $k$ exists: (for $p \ne 2,5$), but I'd argue the cleanest one is this:
*
*$\mathbb{Z}_p$ is a field, and so $(\mathbb{Z}_p)^\times$ is a group under multiplicat... | {
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How is $n^{1.001} + n\log n = \Theta (n^{1.001})$? I am studying for an exam and stumbled across this here:
https://cs.stackexchange.com/questions/13451/few-big-o-example
(I cant comment there since commenting needs 50 reps and I am a new user. Thought math exchange would help)
The chosen answer says for large $n$, $n^... | For intuition, consider something like $n = 2^{10 ^ 6}$. We have $log(n) = 10^6$ and $n^{0.001} = 2^{10^3} \approx 10^{301}$
In general
$\displaystyle \lim_{n\to \infty} \frac{(\log n)^a}{n^b} = 0$ if $b > 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/665551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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probability density function for a random variable For a given probability density function $f(x)$, how do I find out the probability density function for say,
$Y = x^2$?
$$f(x)=\begin{cases}cx&,0<x<2\\2c&,2<x<5\\0&,\text{otherwise}\end{cases}$$
| HINT 1: Note that $P(Y < 0) = 0$ (since $Y = X^2 \geq 0$) and for $y >0$,
\begin{gather*}
P(Y \leq y) = P(X^2 \leq y) = P(-\sqrt{y} \leq X \leq \sqrt{y}) = \dots
\end{gather*}
HINT 2: For any $z \in \mathbb{R}$, $P(X \leq z) = \int_0^z f(x) dx$. Now evaluate the integral as a piecewise function, from $0$ to $2$ and the... | {
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A formula for a sequence which has three odds and then three evens, alternately We know that triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36... where we have alternate two odd and two even numbers. This sequence has a simple formula $a_n=n(n+1)/2$.
What would be an example of a sequence, described by a similar alge... | The sequence $$n \mapsto 4n^6+n^5+6n^3+4n \pmod 7$$ for $n \geq 1$ gives $$1,1,1,0,0,0,1,1,1,0,0,0,\ldots.$$
| {
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"url": "https://math.stackexchange.com/questions/665722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that the foot of the perpendicular from the focus to any tangent of a parabola lies on the tangent to the vertex
Prove that the foot of the perpendicular from the focus to any tangent of a parabola lies on the tangent to the vertex
I've been trying to prove this by plugging in the negative reciprocal of the slo... | Let $F$ be the focus of the parabola, $HG$ its directrix, with vertex $V$ the midpoint of $FH$. From the definition of parabola it follows that $PF=PG$, where $P$ is any point on the parabola and $G$ its projection on the directrix.
The tangent at $P$ is the angle bisector of $\angle FPG$, hence it is perpendicular to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/665837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
modular exponentiation where exponent is 1 (mod m) Suppose I know that $ax + by \equiv 1 \pmod{m}$,
why would then, for any $0<s<m$ it would hold that $s^{ax} s^{by} \equiv s^{ax+by} \equiv s \pmod{m}$?
I do not understand the last step here. Is it some obvious exponentiation rule I'm overlooking here?
Thanks,
John.
| It's false. $2*5 + 1*7 = 17 \equiv 1 \pmod{4}$, $0<2<4$ and $2^{17} \equiv 0 \pmod{4}$.
However, what is true is that for $s$ and $m$ coprime, $s^{\phi(m)} \equiv 1 \pmod{m}$, where $\phi$ is Euler's totent function (http://en.wikipedia.org/wiki/Euler%27s_totient_function).
$s^{\phi(m)+1} \equiv s \pmod{m}$ holds even ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find a basis for this matrix I have a matrix that only contain variables and zeros, like this:
$$
\begin{bmatrix}
0 & -a & -b \\
a & 0 & -c \\
b & c & 0 \\
\end{bmatrix} $$
I usually would find the basis for this by row reduction and then take the columns with leading ones as basis, but how do I do when there is just... | The variables are also just numbers, so this isn't much different from doing what you described. (And the process you described finds a basis for the rowspace, which is the interpretation I'll use for my solution. I'm also assuming the matrix has enties in a field.)
The only complication is that the size of the basis m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Rolle theorem on infinite interval We have :
$f(x)$ is continuous on $[1,\infty]$ and differentiable on $(1,\infty)$
$\lim\limits_{x \to \infty}f(x) = f(1)$
we have to prove that : there is $b\in(1,\infty)$ such that $f'(b) = 0$
I'm sure we have to use Rolle's theorem so, I tried using Mean Value theorem and using the... | Let $g(x)=f\bigl(\frac 1x\bigr)$ for $x\in (0,1]$ and $g(0)=f(1)$. Then $g$ is continuous in $[0,1]$ and derivable in $(0,1)$. By Rolle's Theorem there exists $c\in (0,1)$ such that $g'(c)=0$, hence
$$0=g'(c)=-f'\Bigl(\frac 1c\Bigr)\frac 1{c^2}$$
thus if $b=\frac 1c$, then $f'(b)=0$ and $b\in (1,+\infty)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
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How to prove the modulus of $\frac{z-w}{z-\bar{w}} < 1$? Given that $\Im{(z)} > 0$ and $\Im{(w)} > 0$, prove that $|\displaystyle\frac{z-w}{z-\bar{w}}|<1$ .
Please help me check my answer:
$z - w = a + ib$
$z - \bar{w}$ = $a + i(b+2\Im(w)) $
$|\displaystyle\frac{z-w}{z-\bar{w}}$| = $\displaystyle\frac{|a+ib|}{|a+i(b+... | An example of almost pretty proof (In my eyes of course) using all the givens implicitly:
$$x \equiv z-w \implies y \equiv z - \overline{w}=x+2i\Im{(w)} \implies 0 <|x|^2 < |y|^2 \implies |\frac{x}{y}|^2=\frac{|x|^2}{|y|^2}<1 \implies |\frac{z-w}{z - \overline{w}}|=|\frac{x}{y}|<1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Measure Theory - condition for Integrability A question from my homework:
Let $\,f:X\to [0,\infty)\,$ be a Lebesgue measurable function.
Show that $\,\int_X f\,d\mu < \infty\,\,$ iff $\,\,\sum\limits_{n=1}^\infty \mu\big(\{x\in X:n\leq f(x)\}\big)<\infty.$
I've managed to solve this, but with a relatively complex and... | Set
$$
E_n=\{x:f(x)\ge n\}, \,\,\,F_n=\{x\in X: n-1\le f(x)< n\}
$$
Then the $F_n$'s are disjoint and
$$
E_n=\bigcup_{j\ge n+1}F_j\quad\text{and}\quad \mu(E_n)=\sum_{j=n+1}^\infty\mu(F_j),
$$
and also $\bigcup_{n\in\mathbb N}F_n=X$ and hence $\sum_{n=1}^\infty \mu(F_j)=\mu(X)$. Thus
$$
\sum_{n=0}^\infty \mu(E_n)=\sum... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that for all natural $a$, $2008\mid a^{251}-a$. How to show, that for all natural $a$ coprime to 2008 the following occurs: $2008\mid a^{251}-a$?
This means, that $a_{251} \equiv_{{}\bmod 2008} a$, right?
It's obvious if $a\mid 2008$.
In the other case I'm totally at a loss.
I thought about using Euler totient fu... | Note that $2008 = 8\cdot 251$. By the Chinese Remainder Theorem it is necessary and sufficient to show $a^{251}\equiv a\pmod 8$ and $a^{251}\equiv a\pmod {251}$. The second is true for all $a$ by Fermat's little theorem.
The first is true for odd $a$ or if $8\mid a$, but otherwise it is not!
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Definition of Independence in probability and how its affected if one of the events has zero probability Usually its stated that two events are independent if and only if their joint probability equals the product of their probabilities. i.e:
$$P(A \cap B) = P(A)P(B)$$
However, I was not sure if that was just a definit... | As you noticed, if we use the definition "$A$ and $B$ are independent iff $P\left(A|B\right)=P\left(A\right)$", we might encounter some difficulties in case $P(B)=0$.(here is a discussion about $P(A|B)$ in case $P(B)=0$)
On the other hand, the definition "$A$ and $B$ are independent iff $P(A\cap B)=P(A)P(B)$" doesn't i... | {
"language": "en",
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"source": "stackexchange",
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Divisibility proof problem I need assistance with the following proof.
Let a,b,c,m be integers, with m $\geq$ 1. Let d = (a,m). Prove that m divides ab-ac if and only if $\frac md $ divides b-c.
Alright, I know that since d = (a,m) there exists an r and t such that $ar + mt = d$
I figure since we're trying to prove m ... | Let $k=b-c$. We want to show that $m$ divides $ak$ if and only if $\frac{m}{d}$ divides $k$.
Let $a=a_1d$ and let $m=m_1 d$. Note that $\gcd(a_1,m_1)=1$.
One direction: We show that if $\frac{m}{d}$ divides $k$, then $m$ divides $ak$. By assumption we have $k=\frac{m}{d}q=m_1q$ for some $q$. Thus $ak=(a_1 d)(m_1q)=(a_1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Relationships between mean and standard deviation when one variable is linear function of another
Let $a$ and $b$ be constants and let $y_j = ax_j + b$ for $j=1,2,\ldots,n$.
What are the relationships between the means of $ya$ and $x$, and the standard deviations of $y$ and $x$?
I'm slightly confused with how to appr... |
The mean of $x$ = mean of $y$
This is not true.
The way you should approach this problem is to use the formulas for mean and standard deviation directly:
\begin{align*}
\text{Mean}(y_1, y_2, \ldots, y_n)
&=
\frac{y_1 + y_2 + \cdots + y_n}{n} \\
&=
\frac{(ax_1 + b) + (ax_2 + b) + \cdots + (ax_n + b)}{n} \\
&=
\frac{a(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/666731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Why does $\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$? Here is a standard identity:
$$\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$$
Why does it hold true?
| A brief answer: Let's consider the exponential function $e^x$. The definition of $e$ is that $\frac{d}{dx}e^x = e^x$. Now let's assume that $e^x$ can be written as an infinite sum of the form $\sum_{i=0}^{\infty}a_ix^i$. Using the sum rule for derivatives, we have $\sum_{i=0}^{\infty}a_ix^i = \sum_{i=0}^{\infty}\frac{d... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Subgroup of an abelian Group I think I have the proof correct, but my group theory is not that strong yet. If there is anything I am missing I would appreciate you pointing it out.
Let $G$ be an abelian group (s.t. $gh = hg$ $\forall g,h\in G$). Show that $H = \{g\in G:g^2=e_G\}$ is a subgroup of $G$ where $e_G$ is th... | Looks very good, but it would be best to explictly establish, in proving closure, that $$(gh)^2 = \underbrace{(gh)(gh) = g^2h^2}_{\large G \;\text{is abelian}}$$
I'm not sure if that's what you meant but left that detail out of your proof(?), or if you erroneously made an immediate move by distributing the exponent: $... | {
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Which of the folowing is true (NBHM-2014)
*
*If $f$ is twice continously differentiable in (a,b) and if for all x $\in (a,b)$ $f''(x) + 2f'(x) + 3f(x) = 0$, the $f$ is infinitely differentiable on (a,b)
*Let $f \in C[a,b]$ be a differentiable in (a,b). If $f(a) = f(b) = 0$, then for any real number $\alpha$, there e... | For (3) $f'(0) = lim_{h\rightarrow 0} \frac{f(h) - f(0)}{h} =\lim_{h\rightarrow 0} \frac{h^2|cos\frac{\pi}{h}|}{h}$ = $0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
} |
The characteristic of a subdomain is the same as the integral domain in which it is contained? Perhaps I have a misunderstanding of what a subdomain and an integral domain are, but I'm having a hard time figuring this out.
I'm asked to show that the characteristic of a subdomain is the same as the characteristic of the... | Notice $1+1+1=0$ in $Z_3$ but $1+1+1\color{Red}{\ne}0$ in $Z_7$. Just because the symbols used to represent the elements of $Z_3$ are a subset of the symbols used to represent the elements of $Z_7$ doesn't mean that one structure sits inside the other structure. Not even remotely close. For a subset to be a subring its... | {
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Which primepowers can divide $3^k-2$? I tried to get a survey which primepowers $p^n$ divide $3^k-2$
for some natural k.
PARI has a function znlog, but there are some issues :
Instead of returning 0, if the discrete logarithm does not exist,
an error occurs. So I cannot filter out, for which primes $3^n\equiv2\ (mod\... | To answer your last question:
It well known that if $a$ is a primitive root mod $p^2$ then $a$ is a primitive root mod $p^n$ for all $n\ge 2$.
$3$ is a primitive root mod $5^2$ and mod $7^2$. This explains why $3^k\equiv 2 \bmod p^n$ can be solved for $p=5$ and $p=7$ for all $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/667199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Can somebody check my proof of this theorem about the derivative? I proved the following theorem and would greatly appreciate it if someone could check my proof:
Theorem: Let $f:[a,b]\to \mathbb R$ be differentiable and $\alpha$ such that $f'(a) < \alpha < f'(b)$ (or $f'(a) > \alpha > f'(b)$ ) then there exists $c\in (... | This is the Darboux's theorem
One point that is false : you divide by $a-x$ but this is negative so you must change the inequality.
This leads to $\frac{f(x) - f(a)}{x-a} <= \alpha$. Take a look at the min of $f(x)-x\alpha$ similarly to complete the proof
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\sqrt{x}$ isn't Lipschitz function A function f such that
$$
|f(x)-f(y)| \leq C|x-y|
$$
for all $x$ and $y$, where $C$ is a constant independent of $x$ and $y$, is called a Lipschitz function
show that $f(x)=\sqrt{x}\hspace{3mm} \forall x \in \mathbb{R_{+}}$ isn't Lipschitz function
Indeed, there is no such constant... | $\sqrt{}$ is monotonous, so just assume $x \geq y$, then you can drop the absolute values and it simplifies to $1 \leq C(\sqrt{x} + \sqrt{y}$. Since you can make the sum of square roots arbitrarily small (by suitably decreasing $x$ and $y$), as soon as it's smaller than $1/C$ the inequality no longer holds.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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"answer_id": 3
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Evaluate using cauchy's integral formula How can we evaluate this expression using cauchy's integral formula $\int_C \frac{e^{\pi Z}}{ ( {Z^2 + 1}) ^2} dZ$ where $C$ is $|Z-i|=1$
| Clearly
$$
\int_{|z-i|=1} \frac{e^{\pi z}}{ ( {z^2 + 1}) ^2} dz=\int_{|z-i|=1} \frac{\frac{\mathrm{e}^{\pi z}}{(z+i)^2}}{ ( {z-i}) ^2} dz.
$$
According to the Cauchy Integral formula $f'(a)=\frac{1}{2\pi i}\int_{|z-a|=r}\frac{f(z)}{(z-a)^2}dz$ we have for $a=i$:
\begin{align}
\frac{1}{2\pi i}\int_{|z-i|=1} \frac{\fra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Translations in two dimensions - Group theory I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as :
$$ \left( \begin{array}{ccc}
x+a \\
y+a \end{array} \right) =
\left( \begin{array}{ccc}
x \\
y \en... | As suresh mentioned if the vector is just a two component object then you can't translate it without expanding the vector. However, if you consider the vector to be variable (which are essentially infinite vectors) then it can be translated.
To find the differential form of a translation, start with the translation of... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
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Does negative zero exist? In the set of real numbers, there is no negative zero. However, can you please verify if and why this is so? Is zero inherently "neutral"?
| My thought on the problem is that all numbers can be substituted for variables. -1 = -x. "-x" is negative one times "x". My thinking is that negative 1 is negative 1 times 1. So in conclusion, I pulled that negative zero (can be expressed by "-a") is negative 1 times 0, or just 0 (-a = -1 * a).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/667577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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"answer_id": 1
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Mean value theorem for vectors I would like some help with the following proof(this is not homework, it is just something that my professor said was true but I would like to see a proof):
If $f:[a,b]\to\mathbb{R}^k$ is continuous and differentiable on $(a,b)$, then there is a $a< d < b$ such that $\|f(b)-f(a)\|\le\|f '... | Consider
$$
f(b) - f(a) = \int_a^b f'(t) dt.
$$
Take a dot product with $u$ on both sides, to get
$$
\| f(b) - f(a) \| = \left|\int_a^b u \cdot f'(t) \, dt \right|.
$$
Now suppose that $\|f'(d)\| < \|f(b) - f(a) \| / |b - a|$:
$$
\| f(b) - f(a) \| = \left|\int_a^b u \cdot f'(t) \, dt \right| \\
\le \int_a^b \|u\| \|f'... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Inner product question We are given an inner product of $\mathbb R^3$:
$f\left(\begin{pmatrix} x_1\\x_2\\x_3\end{pmatrix},\begin{pmatrix} y_1\\y_2\\y_3\end{pmatrix}\right) = 2x_1y_1+x_1y_2+y_2x_1+2x_2y_2+x_2y_3+y_3x_2+x_3y_3$
We are given a linear transformation $T$ such that:
$$T\begin{pmatrix} \;\;1\\\;\;0\\-1\end{pm... | Note that the vectors you're given the effect of T on are a basis for $ \mathbb{R}^3$ of eigenvectors of T. Since everything is linear, you need only prove that $f(Tv,w) = f(v,Tw)$ iff $a = b$ for $v, w$ a pair (that could be equal) of these basis vectors. Note that you do need the equation to hold with the vectors use... | {
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Analytical solution for nonlinear equation Simple question:
Does $\alpha = \frac{x}{\beta} - \left(\frac{x}{\gamma}\right) ^{1/\delta}$ have an analytical solution? ($\alpha,\beta,\gamma,\delta$ are constant)
I'm working on big data arrays and either need to solve this equation analytically or spend resources crunchin... | For certain values of the constants, yes. In general, no.
| {
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"timestamp": "2023-03-29T00:00:00",
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How to tackle a recurrence that contains the sum of all previous elements? Say I have the following recurrence:
$$T(n) = n + T\left(\frac{n}{2}\right) + n + T\left(\frac{n}{4}\right) + n + T\left(\frac{n}{8}\right) + \cdots +n + T\left(\frac{n}{n}\right) $$
where $n = 2^k$, $k \in \mathbb{N} $ and $T(1) = 1$.
simplifie... | I'd just start with $T(1)$ and look for a pattern:
$$T(2^1) = 1 \cdot 2^1 + 2^{1-1}T(1)$$
$$T(2^2) = 2\cdot 2^2 + 1\cdot 2^1 + (2^{2-1}) T(1)$$
$$T(2^3) = 3\cdot 2^3 + 2\cdot 2^2 + 2 \cdot 1 \cdot 2^1 + 2^{3-1} T(1)$$
$$T(2^4) = 4\cdot 2^4 + 3\cdot 2^3 + 2 \cdot 2 \cdot 2^2 + 4 \cdot 1 \cdot 2^1 + 2^{4-1} T(1)$$
so tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/667929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Scaling of Fractional ideals For fractional ideals of a Dedekind Domain, are each of the elements that generate the ideal (ie. form the basis of the lattice associated with the ideal) always scaled by the same amount? That is to say, scaled by the same element from the field of fractions?
| I'm not sure if I understand the question, but a fractional ideal is a fraction times an actual ideal, by the definition, i.e. for a fractional ideal $I$ over a ring $R$ we have some $r\in R$ such that $rI\unlhd R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/668036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How prove this $|ON|\le \sqrt{a^2+b^2}$ let ellipse $M:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$,and there two point $A,B$ on $\partial M$,and the point $C\in AB$ ,such $AC=BC$,and the Circle $C$ is directly for the AB circle,for any point $N$ on $\partial C$,
show that
$$|ON|\le\sqrt{a^2+b^2}$$
my try: let
$$A(x_{1},y_{1... | Jack has already written quite an answer to this problem, but I couldn't believe that it didn't have more elegant solution. As my sense of beauty didn't leave me at peace, I couldn't help but find some geometrical ideas behind this problem. I must also add that I really enjoyed solving it.
As Tian has already noted, it... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using Induction, prove that $107^n-97^n$ is divisible by $10$ Using Induction, prove that $107^n-97^n$ is divisible by $10$
We need to prove the basis first, so let $ n = 1 $
$107^1-97^1$
$107-97 = 10$
This statement is clearly true when $ n = 1 $
Now let's use $P(k)$
$107^k-97^k$
So far so good... next I have to use... | Note that
$$107^{k+1}-97^{k+1}\\=107^k\cdot 107-97^k\cdot 97\\
=107^k\cdot (10+97)-97^k\cdot 97\\
=107^k\cdot 10+107^k\cdot 97-97^k\cdot 97\\
=107^k\cdot 10+(107^k-97^k)\cdot 97$$
where the first term is divisible by $10$, and the second term is also divisible by $10$ by induction.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are functions with the property $f(f(x)) = x$ called? Do functions which, when composed with themselves, are equivalent to the identity function (i.e. functions for which $f(f(x)) = x$ in general) have a name and if so, what is it?
Additionally, am I correct in saying that a such function has a splinter of two, or... | These are involutions. The orbits of an involution all have size $1$ or $2$.
What is a splinter?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/668307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Equation with the big O notation How I can prove equality below?
$$
\frac{1}{1 + O(n^{-1})} = 1 + O({n^{-1}}),
$$
where $n \in \mathbb{N}$ and we are considering situation when $n \to \infty$.
It is clearly that it is true. But I don't know which property I have to use to prove it formally. I will appreciate if someone... | If $n \rightarrow \infty$ then $O(\frac{1}{n}) \rightarrow 0$ and using $$\frac{1}{1-z} = \sum_{n=0}^{\infty}z^{n} \ \ \ \ , \ \ |z|<1$$ we have $$\frac{1}{1 + O(n^{-1})} = \sum_{k=0}^{\infty}(-1)^{k}O(\frac{1}{n})^{k} = 1 +O(\frac{1}{n})$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Openness w.r.t. these two metrics are equivalent. Suppose $(X,d)$ is a metric space. Define $\delta:X\times X\rightarrow[0,\infty)$, as $$\delta(x,y)=\frac{d(x,y)}{1+d(x,y)}.$$ It is easy to show that $\delta$ is a metric as well, but I am having difficulty in showing that if a subset of $X$ is $d$-open , then it is $\... | Let us take the $d$-open ball $B=\{x\,:\,d(x,x_0)<R\}$ and show that it is $\delta$-open.
Note that the function $f(x)=\frac{x}{1+x}=\frac{1}{1+\frac{1}{x}}$ is strictly increasing. Therefore $d(x,x_0)<R$ is equivalent to $\delta(x,x_0)=f(d(x,x_0))<f(R)$, i.e.
$$B=\{x\,:\,\delta(x,x_0)<f(R)\}$$
That is, every $d$-ball... | {
"language": "en",
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"source": "stackexchange",
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Center of Mass via integration for ellipsoid I need some help with the following calculation:
I have to calculate the coordinates of the center of mass for the ellipsoid
$$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 + \left( \frac{z}{c} \right)^2 \le 1, \quad z \ge 0$$
with mass-density $\mu(x,y,z)=z^... | The mass density is invariant under $x\rightarrow -x$ and $y\rightarrow -y$, so the center of mass must have $x=y=0$. You do still need to find its $z$-coordinate, but since the mass density is only a function of $z$, you can reduce this to a one-dimensional integral. At a given value of $z$, the cross-section is an ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using "we have" in maths papers I commonly want to use the phrase "we have" when writing mathematics, to mean something like "most readers will know this thing and I am about to use it". My primary question is whether this is too colloquial. My secondary question is what the alternatives are if it is too colloquial.
Fo... | I would replace " we have" by ", then" or just ", "
| {
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"timestamp": "2023-03-29T00:00:00",
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Equivalence of Two Lorentz Groups $O(3,1)$ and $O(1,3)$ How can I prove that $O(3,1)$ and $O(1,3)$ are the same group?
| The matrices $M$ in $O(3,1)$ and $O(1,3)$ are defined by the condition
$$ M G M^T = G $$
for
$$ G=G_{1,3} ={\rm diag} (1,1,1,-1)\text{ and } G=G_{3,1} = {\rm diag} (1,-1,-1,-1)$$
respectively. I use the convention where the first argument counts the number of $+1$'s in the metric tensor and the second one counts the ne... | {
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Multiply (as a Babylonian): 141 times 17 1/5 How do we multiply 141 times 17 1/5 as a Babylonian?
I wasn't sure the space between 17 and 1/5, now I see that 17 1/5 is 17.2 in our notation.
Is there a formula that I can solve this?
Any hint, comment would be very appreciated it!
| Hint: Note that ${1\over5} = {12\over 60}$, and try to write both numbers in sexagesimal notation. Then multiply in the same way as we multiply numbers in decimal notation. (The Babylonians would have multiplication tables to help them with this.)
| {
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The elements of finite order in an abelian group form a subgroup: proof check If G is an abelian group, show that the set of elements of finite order is a subgroup of G.
Proof:
Let G be an abelian group and H be the set of elements of finite order.
(1) nonempty
Now e ∈ H, since $a^n$ = e, by definition of order, and |e... | Inverse: It is sufficient to show $ka=0$ iff $k(-a)=0$ (and thus $|a|=|-a|$ ) .
But inversion is an invertible function so it is sufficient to show $ka=0$ implies $k(-a)=0$ . Assume $ka=0$. $ka=2ka+(k)(-a)$ by repeated application of the identity axiom. But this is $0=2*0+k(-a), k(-a)=0 $.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the determinant of a solving matrix I have such ODE:
$$\frac{dy}{dt}=\begin{pmatrix}
\sin^2t & e^{-t} \\
e^t & \cos^2t
\end{pmatrix} y=A(t)y(t)$$
and let $M(t,1)$ be the solving matrix (a matrix whose columns generate a fundamental system of solutions), where $M(1,1)=E$. Find $\det M(-1,1)$.
I d... | @Max, read your book. The wronskian $W(t)=\det(M(t,1))=\exp(\int_{1}^t trace(A(u))du)$.
Here $trace(A(t))=1$ and $W(t)=\exp(t-1)$.
| {
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Solving $P(x,y)dx + Q(x,y)dy =0$: interpretation in terms of forms I asked a similar question here which I will formulate more sharply:
When we write a differential equation as $P(x,y)dx + Q(x,y)dy = 0$, what is the interpretation in terms of differential forms?
(I suppose the language of differential forms is the pr... | The interpretation is the following: given a differential 1-form $\omega=Pdx+Qdy$ in the plane, you are asked to find its integral curves, i.e. 1-dim submanifolds of $\mathbb R^2$ whose tangent line at each point is annihilated by the 1-form. For example, the integral curves of $xdx+ydy$ are the concentric circles a... | {
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Suppose $f$ is a thrice differentiable function on $\mathbb {R}$ . Showing an identity using taylor's theorem Suppose $f$ is a thrice differentiable function on $\mathbb {R}$ such that $f'''(x) \gt 0$ for all $x \in \mathbb {R}$. Using Taylor's theorem show that
$f(x_2)-f(x_1) \gt (x_2-x_1)f'(\frac{x_1+x_2}{2})$ for a... | Using the Taylor expansion to third order, for all $y$ there exists $\zeta$ between $(x_1+x_2)/2$ and $y$ such that
$$
f(y) = f \left( \frac{x_1+x_2}2 \right) + f'\left( \frac{x_1+x_2}2 \right)\left(y - \frac{x_1 + x_2}2 \right) \\
+ \frac{f''(\frac{x_1+x_2}2)}2 \left( y - \frac{x_1 + x_2}2 \right)^2 + \frac{f'''(\zeta... | {
"language": "en",
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Probability of the union of $3$ events? I need some clarification for why the probability of the union of three events is equal to the right side in the following:
$$P(E\cup F\cup G)=P(E)+P(F)+P(G)-P(E\cap F)-P(E\cap G)-P(F\cap G)+P(E\cap F\cap G)$$
What I don't understand is, why is the last term(intersection of all) ... | One of the axioms of probability is that if $A_1, A_2, \dots$ are disjoint, then
$$\begin{align}
\mathbb{P}\left(\bigcup_{i=1}^{\infty}A_i\right) = \sum\limits_{i=1}^{\infty}\mathbb{P}\left(A_i\right)\text{.}\tag{*}
\end{align}$$
It so happens that this is also true if you have a finite number of disjoint events. If yo... | {
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Prove a number is even using the Pigeonhole Principle Let n be an odd integer and let f be an [n]-permutation of length n, where [n] is the set of integers 1, 2, 3,...n
Show that the number
x = (1-f(1))*(2-f(2))*...*(n-f(n))
is even using the pigeonhole principle
In this case, I don't understand what this function f ... | There are $(n+1)/2$ odd numbers $i\in[n]$, and equally many numbers $i$ such that $f(i)$ is odd. Since that makes $n+1$ in all, the pigeonhole principle says that at least one $i$ is counted twice: both $i$ and $f(i)$ are odd. But then $i-f(i)$ is even, and so is the entire product.
Here is a proof without the pigeonho... | {
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Prove that if $C$ is a convex set containing $B(r)$, then $\sup\{d(y,0)\mid y\in C\}=\infty$ Let $0<p<1$. Define a metric on $l^p$ by $d((a_k)_{k=1}^\infty,(b_k)_{k=1}^{\infty})=\sum_{k=1}^\infty |a_k-b_k|^p$. For any $r>0$, let $B(r)=\{x\in l^p\mid d(x,0)<r\}$. Prove that if $C$ is a convex set containing $B(r)$, then... | Hint: denoting by $e_n$ the $n$-th vector of "canonical basis" of $\ell^p$, compute $d(x_N, 0)$ for each $N\in\mathbb N$, where $x_N=\frac 1N \sum_{n=1}^N e_n$.
| {
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Prove that the product of some numbers between perfect squares is $2k^2$ Here's a question I've recently come up with:
Prove that for every natural $x$, we can find arbitrary number of integers in the interval
$[x^2,(x+1)^2]$ so that their product is in the form of $2k^2$.
I've tried several methods on proving thi... | This is a community wiki answer to point out that the question was answered in comments by benh: This question is a duplicate of this one; the latter question was answered by Gerry Meyerson who found a proof in this paper of Granville and Selfridge.
| {
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"timestamp": "2023-03-29T00:00:00",
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Calculate sum $\sum\limits_{k=0}k^2{{n}\choose{k}}3^{2k}$. I need to find calculate the sum Calculate sum $\sum\limits_{k=0}k^2{{n}\choose{k}}3^{2k}$.
Simple algebra lead to this $\sum\limits_{k=0}k^2{{n}\choose{k}}3^{2k}=n\sum\limits_{k=0}k{{n-1}\choose{k-1}}3^{2k}$. But that's still not very helpful. This binomial sc... | We have $\displaystyle k^2=k(k-1)+k$
So, $\displaystyle k^2 \binom nk=k(k-1)\binom nk+k\binom nk$
Now $\displaystyle k\cdot\binom nk=k\frac{n!}{(n-k)!k!}=kn\frac{(n-1)!}{[n-1-(k-1)]!(k-1)!\cdot k}=n\binom{n-1}{k-1}$
and $\displaystyle k(k-1)\cdot\binom nk=k(k-1)\frac{n!}{(n-k)!k!}=k(k-1)n(n-1)\frac{(n-2)!}{[n-2-(k-2)]... | {
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Is the function continuous at x=0? Check if the function $f$ is continuous.
$f(x)=$\begin{matrix}
0 & ,x=0\\
\frac{1}{[\frac{1}{x}]} & ,0<x\leq 1
\end{matrix}.
For $0<x\leq 1$,,f is continuous because it is fraction of continuous functions.
How can I check if it is continuous at $x=0$?
| $$|f(x) - f(0)| = |\frac 1 {[\frac 1 x]} - 0| = |\frac 1 {[\frac 1 x]}| = \frac 1 {[\frac 1 x]} \le \frac 1 {\frac 1 x} = x = x - 0\;\; (\text {since $x > 0$ and $\frac 1 {[\frac 1 x]} \le \frac 1 {\frac 1 x}$})$$
Given any $\epsilon \gt 0 $ however small, $\exists \ \delta( = \epsilon) \gt 0 $ such that $|f(x) - f... | {
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What does "\" mean in math In a Linear Algebra textbook I am reading, the following is stated: $b\notin \operatorname{span}(A \cup \{a\})\setminus \operatorname{span}(A)$. It does so without explaining what "$\setminus$" means. I apologize if this question does not belong here but I just want to understand what it mean... | $\setminus$ (\setminus) as its name implies is the set-theoretic difference: $A\setminus B$ is the set of all elements which are in $A$ but not in $B$. ($A-B$ is also used for this.) Be careful to not confuse $\setminus$ with $/$ (quotient).
| {
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"timestamp": "2023-03-29T00:00:00",
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Proof that if $a^n|b^n$ then $a|b$ I can't get to get a good proof of this, any help?
What I thought was:
$$b^n = a^nk$$
then, by the Fundamental theorem of arithmetic, decompose $b$ such:
$$b=p_1^{q_1}p_2^{q_2}...p_m^{q_m}$$
with $p_1...p_m$ primes and $q_1...q_n$ integers.
then
$$b^n=(p_1^{q_1}p_2^{q_2}...p_m^{q_m})... | Hint $\ p^{n\alpha}\!\mid p^{n\beta}\! \iff n \alpha \le n\beta \iff \alpha \le \beta \iff p^\alpha\mid p^\beta.\,$ Apply it to prime factorizations of a,b.
Simpler: $\, (b/a)^n = k\in\Bbb Z\Rightarrow b/a\in \Bbb Z\,$ by the Rational Root Test applied to $\,x^n - k.$
| {
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Summation proof (with binomial coefficents) I am trying to prove that $\sum_{k=2}^n$ $k(k-1) {n \choose k}$=$n(n-1)2^{n-2}$.
I was initially trying to use induction, but I think a more simple proof can be done using the fact that $\sum_{k=0}^n {n \choose k}$=$2^n$.
This is how I begin to proceed:
$\sum_{k=2}^n$ $k(k-1)... | By the binomial theorem
$$\sum_{k=0}^n {n\choose k} x^k=(1+x)^n$$
Take two derivatives in $x$ and plug in $x=1$.
| {
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"timestamp": "2023-03-29T00:00:00",
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How can the point of inflection change before the vertical asymptote? I have to draw a graph of a function which seems to have an inflection point AFTER the vertical asymptote.
i.e. f(x) = $\tan^{-1}\left({\frac{x-1}{x+1}}\right)$
Using the quotient rule, I get...
$$f'(x) = \frac{1}{1+\left(\frac{x-1}{x+1}\right)^2}.\f... | Note that your function is equivalent to arctanx - (pi/4)
| {
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Proof that rational sequence converges to irrational number Let $a>0$ be a real number and consider the sequence $x_{n+1}=(x_n^2+a)/2x_n$.
I have already shown that this sequence is monotonic decreasing and thus convergent, now I have to show that $(\lim x_n)^2 = a$ and thus exhibit the existence of a positive square r... | The recursive definition can be solved exactly. Write a^2 instead of a and consider the sequence $(x(n)+a))/(x(n)-a)$.
Then $(x(n)+a)/(x(n)-a) =
= (.5*x(n-1)+.5*a^2/x(n-1)+a)/(.5*x(n-1)+.5*a^2/x(n-1)-a) =
= (x(n-1)^2+2*a*x(n-1)+a^2)/(x(n-1)^2-2*a*x(n-1)+a^2) =
= ((x(n-1)+a)/(x(n-1)-a))^2 =$
Thus $(x(n)+a)/(x(n)-a) = ((... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/670083",
"timestamp": "2023-03-29T00:00:00",
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Question about finding the volume of a Sphere to a certain point
I've done a few things but I cant seem to figure out how to solve this. Any help please?
| Area=$pi(r)^2$
Area=$pi(2Ry-y^2)$
integral of Area from $y=0$ to $y=R/3$ =Volume.
$R=sqrt((R-y)^2+r^2)$
$R^2=(R-y)^2+r^2$
$R^2=R^2-2Ry+y^2+r^2$
$2Ry-y^2+r^2$
$r=sqrt(2Ry-y^2)$
integral of $pi(2Ry-y^2)dy$ from $0$ to $R/3 = pi(Ry)^2-y^3/3$ from $0$ to $R/3$
after plugging in, your answer should be $pi((R(R/3))^2-(R/3)^3... | {
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Would it be any concern if we find correlation between intercept and other regression coefficients? During a multiple linear regression analysis, I found correlation between intercept (beta-0) and two of the other regression coefficients. Is there any problem or concern in this case? If no, please explain me why.
| Such correlations are guaranteed if you have not standardized your predictors to take the value 0 at their means. However, this correlation is not really a mathematical/statistical problem, per se, but it may be easier to interpret the coefficients if you first standardize the variables.
Therefore, the short answer is... | {
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Condition under which the subring of multiples of an element equal the ring itself? Let $R$ be a commutative ring with identity with $b\in R$. Let $T$ be the subring of all
multiples of $b$, $T=\{r\cdot b : r \in R\}$. If $u$ is a unit in $R$ with $u \in T$,
prove that $T=R$.
Could you help me some suggestions?
I rea... | The critical thing to realize here is that if $u$ is a unit, and $u = ab$, then $a$ and $b$ are both units. For if $u$ is a unit, then $uv = 1$ for some $v \in R$, so that $1 = (ab)v = a(bv) = b(av)$, where we have used the commutativity of $R$. So $u \in T$ a unit implies $b \in T$ is a unit is well, since $u = ab$ ... | {
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Showing that the square root is monotone I've shown the existence of unique square roots of all positive rational numbers, so now I want to prove that the square root is monotone:
$0<a<b$ if and only if $\sqrt{a} < \sqrt{b}$
| We know that if $p, q, r, s$ are positive and $p < q, r < s$ then $pr < qs$. Let $\sqrt{a} < \sqrt{b}$ using $p = r = \sqrt{a}, q = s = \sqrt{b}$ we get $a < b$.
Let $a < b$. Clearly we can't have $\sqrt{a} = \sqrt{b}$ as this will mean $a = b$ (by squaring). Also we can't have $\sqrt{a} > \sqrt{b}$ as by previous par... | {
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A somewhat general question about homeomorphisms. I have been asked to prove that $(0,1)$ is homeomorphic to $(0,1)$. Seems easy enough. If we assume the order topology an both, along with an identity mapping $f:x\to x$, we can show that both $f$ and $f^{-1}$ are continuous.
Similarly, using the identity mapping $f$ ... | Well $(0,1)$ and $[0,1]$ have the same cardinality, so there exists a bijection $\Phi:(0,1)\rightarrow[0,1]$. Now, equip $(0,1)$ with any topology and define $U\subset [0,1]$ to be open if and only if $\Phi^{-1}(U)$ is open. Then it can easily be checked that $\Phi$ is a homeomorphism.
| {
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Efficient diagonal update of matrix inverse I am computing $(kI + A)^{-1}$ in an iterative algorithm where $k$ changes in each iteration. $I$ is an $n$-by-$n$ identity matrix, $A$ is an $n$-by-$n$ precomputed symmetric positive-definite matrix. Since $A$ is precomputed I may invert, factor, decompose, or do anything to... | EDIT. 1. The Tommy L 's method is not better than the naive method.
Indeed, the complexity of the calculation of $(kI+A)^{-1}$ is $\approx n^3$ blocks (addition-multiplication).
About the complexity of $P(D+kI)^{-1}P^{-1}=QP^{-1}$ (when we know $P,D,P^{-1}$); the complexity of the calculation of $Q$ is $O(n^2)$ and th... | {
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"answer_id": 1
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A tricky Definite Integral What is the value of $$\int_{\pi/4}^{3\pi/4}\frac{1}{1+\sin x}\operatorname{d}x\quad ?$$
The book from which I have seen this has treated it as a problem of indefinite integral and then directly put the values of the limits. I am not sure that this is the correct way.
Kindly help.
| $$\frac1{1+\sin x}=\frac1{1+\sin x}\cdot\overbrace{\frac{1-\sin x}{1-\sin x}}^1=\frac{1-\sin x}{1-\sin^2x}=\frac{1-\sin x}{\cos^2x}=\frac1{\cos^2x}-\frac{\sin x}{\cos^2x}=$$
$$=\frac{\sin^2x+\cos^2x}{\cos^2x}+\frac{\cos'x}{\cos^2x}=(1+\tan^2x)-\left(\frac1{\cos x}\right)'=\tan'x-\left(\frac1{\cos x}\right)'\iff$$
$$\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/670709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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$K$-theory exact sequence. Let $Y$ be a closed subspace of a compact space $X$. Let $i:Y \to X$ the inclusion and $r:X \to Y$ a retraction ($r \circ i = Id_Y$). I have to prove that exists this short exact sequence
$$ 0 \to K(X,Y) \to K(X) \to K(Y) \to 0.$$
Then I have to verify that $K(X) \simeq K(X,Y) \oplus K(Y)$.... | This is purely formal and relies on the fact that $K$ is a contravariant functor from topological spaces to abelian groups (actually to commutative rings but this is not needed here).
Since $r\circ i=Id_Y$ we get $i^*\circ r^*=Id_{K(Y)}$ so that $r^*:K(Y)\to K(X)$ is a section of $i^*:K(X)\to K(Y)$ and your exact seque... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/670813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Given a matrix $A$, show that it is positive.
Show that
$$A := \begin{bmatrix}7 & 2 & -4\\2 & 4 & -2\\-4 & -2 & 7 \end{bmatrix}$$
is positive definite.
Could this be proven by showing that each of the vectors of the standard basis gives a positive result, e.g.:
$$\begin{bmatrix}1 & 0 & 0\end{bmatrix} \begin{bmatr... | No, checking the standard basis does not guarantee positive definiteness of your scalar product. It would work if the standard basis was orthogonal with respect to your bilinear form (this is Sylvester's theorem): but, in our case, this is equivalent to have the matrix being diagonal.
By definition, a $n\times n$ matr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/670872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
} |
Translating and scaling a line based on two grabbed points Say there is a line segment going from 0 to 10, now imagine that point 7 and 8 are 'grabbed' and translated to respectively 6 and 11. The effect would this would be that the line segment get's scaled and translated. How can I determine the new defining points o... | Let
$$x_{new} = h_1 + (x - g_1) \frac{h_2 - h_1}{g_2 - g_1}.$$
where I've used $g_i$ for your starting points (7, 8) and $h_i$ for your ending points (6, 11). The formula shows how to take a point $x$ in the pre-stretch coordinates and tell its post-stretch coordinates. So
$$x_1^{end} = h_1 + (x_1^{start} - g_1) \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/670967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Unbounded linear functional maps every open ball to $\mathbb{R}$? I can't get my head wrapped around this:
Let $X$ be a normed linear space. Let $f:X\rightarrow\mathbb{R}$ be a linear functional on $X$. Prove that $f$ is unbounded if and only if $\forall y\in X$ and $\forall \delta>0$ we have $\{f(x)\,:\,|x-y|<\delta\}... | Jonathan's suggestion is spot on, but let me give you a more explicit argument. Consider first the unit ball $X_1$ of $X$. As $f$ is unbounded, there exists a sequence $\{x_n\}\subset X_1$ with $|f(x_n)|>n$. By replacing $x_n$ with $-x_n$ if necessary, we can get $f(x_n)>n$.
Given $t\in[0,1]$, $tx_n\in X_1$, and $f(tx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/671043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
The graph of the function is $g(x)=x^3-2x^2+x+1$ and the tangent to the curve is at $x=2$? a) Find the equation of the tangent to the curve at $x=2$
HELP
and then
b) Determine the angle that this tangent makes with the positive direction of the $x$-axis
Please help I really need to know how to do this
Please inclu... | (a)
Ok so we have the equation $$f(x) = x^3 - 2x^2 + x + 1$$
Taking the derivative we get $$f'(x) = 3x^2 - 4x + 1$$ At $x = 2$, $f'(2) = 3*4 - 4*2 + 1 = 5$, meaning that our equation of the tangent line is $$y = 5x + c$$
$f(2) = 8 - 2*4 + 2 + 1 = 3$, so the graph passes through the point $(2, 3)$. Our tangent line equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/671116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Proving $n^2$ is even whenever $n$ is even via contradiction? I'm trying to understand the basis of contradiction and I feel like I have understood the ground rules of it.
For example: Show that the square of an even number is an even number using a contradiction proof.
What I have is: Let n represent the number.
n is... | We prove the contrapositive. In this case, we want to prove
$n^2$ even implies $n$ even
which is equivalent to the contrapositive
$n$ not even implies $n^2$ not even
or in other words
$n$ odd implies $n^2$ odd.
If $n$ is odd, then $n=2k+1$ then
\begin{align*}
n^2 &= (2k+1)^2 & \text{substituting in } n=2k+1 \\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/671243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What are all possible values of $ x \equiv a^\frac{p-1}{2} \pmod p$? Suppose p is an odd prime and a $\in$ $\mathbb{Z}$ such that $ a \not\equiv 0 \pmod p$. What are all the values of $ x \equiv a^\frac{p-1}{2} \pmod p$ ?
This is what I got so far:
$ x^2 \equiv a^{p-1} \pmod p$
By Fermat's Little Theorem,
$ x^2 \equiv... | Do you know about quadratic residues ?
The values of $x$ are $1$ and $-1$.
$\frac{p-1}{2}$ values of $1$ and also $\frac{p-1}{2}$ values of $-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/671285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Finding Limit Points, Interior Points, Isolated Points, the Closure of Finding Limit Points, Interior Points, Isolated Points, the Closure of $ A \subset \mathbb{R}^2$, where $A$ is the graph of the function $f: \mathbb{R} \rightarrow \mathbb{R}$, $f(x)= \sin(1/x)$ if $x$ doesn't equal $0$ and $0$ if $x=0$. (The distan... | I suggest you graph the function, and check what happens as $x \rightarrow \pm \infty$ (can you explain this behavior?). To find the closure of $A$, find all the points that the graph approaches arbitrarily closely. These are your boundary points, and the closure is the interior of $A$ along with the boundary of $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/671457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Each vertex of this tree is either red or blue. How many possible trees are there? The question: Let $X$ denote the set of 'coloured trees' which result when each vertex of the tree is assigned one of the colours red or blue. How many different coloured trees of this kind are there?
I am not quite sure where to begin ... | It sounds like a tree with $1,2,4$ blue is the same as a tree with $1,2,5$ blue. I would count as follows:
First consider the subtree of $2,4,5$. Vertex $2$ has two choices and vertices $4,5$ have three choices, so there are six ways to color the subtree. Now for the whole tree, you have two choices for $1$. You ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/671555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
embedding of a finite algebraic extension In one of my courses we are proving something (so far, not surprising) and using the fact:
if $F$ is a finite algebraic field extension of $K$, there is an embedding of $F$ into $K$. Well, doesn't seems to me that we can really embed $F$ into $K$, since $F$ is bigger, but can w... | Any homomorphism of fields must be zero or an embedding as there are no nontrivial ideals of any field. There is always the natural inclusion $i: K\rightarrow F$ if $K\subseteq F$, but rarely do we have an embedding $F \rightarrow K$.
For a simple example, there is no embedding $\Bbb C\rightarrow \Bbb R$, as only one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/671640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Given an odd integer $a$ , establish that $a^2+(a+2)^2+(a+4)^2+1$ is divisible by $12$? Given an odd integer $a$ , establish that $a^2+(a+2)^2+(a+4)^2+1$ is divisible by $12$?
So far I have:
$a^2+(a+2)^2+(a+4)^2+1$
$=a^2+a^2+4a+4+a^2+8a+16+1 $
$=3a^2+12a+21$
$=3(a^2+4a+7) $
where do I go from here.. the solution I ha... | If $a$ is odd, then $a = 2b+1$ for some integer $b$.
Then $a^2 + 4a + 7 = 4b^2 + 4b + 1 + 8b + 4 + 7 = 4(b^2 + 3b + 3)$, which is evenly divisible by $4$.
Combine this with the divisibility by $3$ that you already have, and you're done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/671733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Find the point of inflection Will there be an inflection point if there is no solution for $x$ when $f ''(x) = 0$? For example,
$$
f(x)=\frac{x^2-x+1}{x-1}
$$
with domain $\mathbb{R}-\{1\}$
Also, is that when $x$ is smaller than $1$, $f(x)$ is concave down?
| There is no inflection point if there is no solution for $x$ when $f''(x) = 0$. For your case, if $x > 1$, then $f''(x) > 0$. If $x < 1$, then $f''(x) < 0$.
Here is the double-derivative of $\dfrac{x^2 - x + 1}{x - 1}$ and its graph
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/671815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determining if $W$ is a subspace of $\mathbb R^3$ I'm trying to determine whether $W = {\{(x, y, z) | x = 3y, z = -y}\}$ a subspace of $\mathbb R^3$.
If someone can help me understand how to go about doing this that would be great!
| As an alternative to performing a subspace test, we can just note that it's a span: we see that
\begin{align*}
W &= \{(3y,y,-y):y \in \mathbb{R}\} \\
&=\mathrm{span}\{(3,1,-1)\}
\end{align*}
and spans are subspaces.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/671879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
What is the probability to win? Die game You have a die. If you get one pip at any point in the game you lose. If you get two,..., six pips you start adding the number of pips to a sum. To win the sum must get greater or equal to 100. What is the probability to win the game?
| An alternative approach, which still requires numerical calculation is to define a system state as the cumulative score. The system starts in state $0$ and there are a couple of absorbing states $L$ (lost) and $W$ (won). This yields 102 possible system states ${0,1,2,...,99,L,W}$. Each roll of the dice transforms the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/672063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Value of $u(0)$ of the Dirichlet problem for the Poisson equation Pick an integer $n\geq 3$, a constant $r>0$ and write
$B_r = \{x \in \mathbb{R}^n : |x| <r\}$. Suppose that
$u \in C^2(\overline{B}_r)$ satises
\begin{align}
-\Delta u(x)=f(x), & \qquad x\in B_r, \\
u(x) = g(x), & \qquad x\in \partial B_r,
\end{a... | I would suggest an alternative approach. Do you know the Green's function for the domain $B_r$, specifically, the value at the center $G(0; y)$? It is
$$G(0;y)=\frac{1}{n(n-2)\alpha_n} \left(\frac{1}{r^{n-2}} -\frac{1}{|y|^{n-2}} \right).$$
Now the solution to the Poisson equation $u$ is the sum of the solution of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/672160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Why does maximum likelihood estimation for uniform distribution give maximum of data? I am looking at parameters estimation for the uniform distribution in the context of MLEs. Now, I know the likelihood function of the Uniform distribution $U(0,\theta)$ which is $1/\theta^n$ cannot be differentiated at $\theta$. The ... | Your likelihood function is correct, but strictly speaking only for values of $\theta \ge x_\max$.
As you know the likelihood function is the product of the conditional probabilities $P(X_i=x_i|\theta)$.
$$L = P(X_1=x_1|\theta) \times P(X_2=x_2|\theta) \times \ldots \times P(X_2=x_2|\theta) $$
We can look at three diff... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/672266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
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