Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Are $|X|$ and $\operatorname{sgn}(X)$ independent? Let $X$ be a real valued random variable.
Let $\operatorname{sgn}(x)$ be $1$ when $x>0$, $-1$ when $x<0$ and $0$ when $x=0$.
Why are $|X|$ and $\operatorname{sgn}(X)$ independent, when the density function of $X$ is symmetric with respect to $0$?
Are $|X|$ and $\opera... | If $X$ is a continuous random variables (absolutely continuous with respect to Lebesgue),
$$P(|X| \leq x|sgn(X)=1) = P(|X| \leq x|X > 0) = \frac{P(0 < X \leq x)}{P(X > 0)} =
\frac{\int_{0}^{x}{f(x)dx}}{\int_{0}^{\infty}f(x)dx}$$
$$P(|X| \leq x|sgn(X)=-1) = P(|X| \leq x|X < 0) = \frac{P(-x \leq X < 0)}{P(X < 0)} =
\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Existence of Consecutive Quadratic residues For any prime $p\gt 5$,prove that there are consecutive quadratic residues of $p$ and consecutive non-residues as well(excluding $0$).I know that there are equal number of quadratic residues and non-residues(if we exclude $0$), so if there are two consecutive quadratic residu... | The number of $k\in[0,p-1]$ such that $k$ and $k+1$ are both quadratic residues is equal to:
$$ \frac{1}{4}\sum_{k=0}^{p-1}\left(1+\left(\frac{k}{p}\right)\right)\left(1+\left(\frac{k+1}{p}\right)\right)+\frac{3+\left(\frac{-1}{p}\right)}{4}, $$
where the extra term is relative to the only $k=-1$ and $k=0$, in order to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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A book useful to learn lattices (discrete groups) Does anyone know a good book about lattices (as subgroups of a vector space $V$)?
| These notes of mine on geometry of numbers begin with a section on lattices in Euclidean space. However they are a work in progress and certainly not yet fully satisfactory. Of the references I myself have been consulting for this material, the one I have found most helpful with regard to basic material on lattices i... | {
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"question_score": "1",
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Tensors: Acting on Vectors vs Multilinear Maps I have the feeling like there are two very different definitions for what a tensor product is. I was reading Spivak and some other calculus-like texts, where the tensor product is defined as
$(S \otimes T)(v_1,...v_n,v_{n+1},...,v_{n+m})= S(v_1,...v_n) * T(v_{n+1},...,v_{... | I can't comment yet (or I don't know how to if I can), but echo Thomas' response and want to add one thing.
The tensor product of two vector spaces (or more generally, modules over a ring) is an abstract construction that allows you to "multiply" two vectors in that space. A very readable and motivated introduction ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
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${10 \choose 4}+{11 \choose 4}+{12 \choose 4}+\cdots+{20 \choose 4}$ can be simplified as which of the following? ${10 \choose 4}+{11 \choose 4}+{12 \choose 4}+\cdots+{20 \choose 4}$ can be simplified as ?
A. ${21 \choose 5}$
B. ${20 \choose 5}-{11 \choose 4}$
C. ${21 \choose 5}-{10 \choose 5}$
D. ${20 \choose 4}$
Plea... | What is the problem in this solution?
If $S=\binom{10}{4} + \binom{11}{4} + \cdots + \binom{20}{4}$ we have
\begin{eqnarray}
S&=& \left \{ \binom{4}{4} + \binom{5}{4} \cdots + \binom{20}{4}\right \} - \left \{\binom{4}{4} + \binom{5}{4} \cdots + \binom{9}{4}\right \} &=& \binom{21}{5} - \binom{10}{5}
\end{eqnarray}
| {
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How can a Bézier curve be periodic? As I know it, a periodic function is a function that repeats its values in regular intervals or period. However Bézier curves can also be periodic which means closed as opposed to non-periodic which means open. How is this related or possible?
| A curve $C$ parameterised over the interval $[a,b]$ is closed if $C(a) = C(b)$. Or, in simpler terms, a curve is closed if its start point coincides with its end-point. A Bézier curve will be closed if its initial and final control points are the same.
A curve $C$ is periodic if $C(t+p) = C(t)$ for all $t$ ($p \ne 0$ i... | {
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Burnside's Lemma I've been trying to understand what Burnside's Lemma is, and how to apply it, but the wiki page is confusing me. The problem I am trying to solve is:
You have 4 red, 4 white, and 4 blue identical dinner plates. In how
many different ways can you set a square table with one plate on each
side if tw... | There are four possible rotations of (clockwise) 0, 90, 180 and 270 degrees respectively. Let us denot the 90 degree rotation by $A$, so the other rotations are then its powers $A^i,i=0,1,2,3$. The exponent is only relevant modulo 4, IOW we have a cyclic group of 4 elements. These rotations act on the set of plate arra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/165184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Separatedness of a composition, where one morphism is surjective and universally closed. I'm stuck with the following problem:
Let $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ be scheme morphisms such that f is surjective and universally closed and such that $g \circ f$ is separated. The claim is then that g is also sep... | You know that the image $\Delta_X (X)$ of $X$ under $\Delta_X$is closed in $X\times_Z X$, because $g\circ f$ is separated .
Take the image $Im=(f\times f) (\Delta_X (X)) \subset Y\times_Z Y$ of this closed set $\Delta_X (X)$ under $f\times f$ .
This image $Im$ is then closed in $Y\times_Z Y$ (because $f$ is universal... | {
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"timestamp": "2023-03-29T00:00:00",
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Does a closed form formula for the series ${n \choose n-1} + {n+1 \choose n-2} + {n+2 \choose n-3} + \cdots + {2n - 1 \choose 0}$ exist. $${n \choose n-1} + {n+1 \choose n-2} + {n+2 \choose n-3} + \cdots + {2n - 1 \choose 0}$$
For the above series, does a closed form exist?
| Your series is
$$\sum_{k=0}^{n-1}\binom{n+k}{n-k-1}=\sum_{k=0}^{n-1}\binom{2n-1-k}k\;,$$
which is the special case of the series
$$\sum_{k\ge 0}\binom{m-k}k$$
with $m=2n-1$. It’s well-known (and easy to prove by induction) that
$$\sum_{k\ge 0}\binom{m-k}k=f_{m+1}\;,$$
where $f_m$ is the $m$-th Fibonacci number: $f_0=... | {
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How to prove if a function is bijective? I am having problems being able to formally demonstrate when a function is bijective (and therefore, surjective and injective). Here's an example:
How do I prove that $g(x)$ is bijective?
\begin{align}
f &: \mathbb R \to\mathbb R \\
g &: \mathbb R \to\mathbb R \\
g(x) &= 2f(x) +... | The way to verify something like that is to check the definitions one by one and see if $g(x)$ satisfies the needed properties.
Recall that $F\colon A\to B$ is a bijection if and only if $F$ is:
*
*injective: $F(x)=F(y)\implies x=y$, and
*surjective: for all $b\in B$ there is some $a\in A$ such that $F(a)=b$.
Ass... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/165434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
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When will these two trains meet each other I cant seem to solve this problem.
A train leaves point A at 5 am and reaches point B at 9 am. Another train leaves point B at 7 am and reaches point A at 10:30 am.When will the two trains meet ? Ans 56 min
Here is where i get stuck.
I know that when the two trains meets... | We do not need $S$.
The speed of the train starting from $A$ is $S/4$ while the speed of the train starting from $B$ is $S/(7/2) = 2S/7$.
Let the trains meet at time $t$ where $t$ is measured in measured in hours and is the time taken by the train from $B$ when the two trains meet. Note that when train $B$ is about to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/165513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Square units of area in a circle I'm studying for the GRE and came across the practice question quoted below. I'm having a hard time understanding the meaning of the words they're using. Could someone help me parse their language?
"The number of square units in the area of a circle '$X$' is equal to $16$ times the num... | Let the diameter be $d$. Then the number of square units in the area of the circle is $(\pi/4)d^2$. This is $16\pi d$. That forces $d=64$.
Remark: Silly problem: it is unreasonable to have a numerical equality between area and circumference. Units don't match, the result has no geometric significance.
"The number of sq... | {
"language": "en",
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Is this vector derivative correct? I want to comprehend the derivative of the cost function in linear regression involving Ridge regularization, the equation is:
$$L^{\text{Ridge}}(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T\beta)^2 + \lambda \sum_{j=1}^k \beta_j^2$$
Where the sum of squares can be rewritten as:
$$L^{}(\b... | You have differentiated $L$ incorrectly, specifically the $\lambda ||\beta||^2$ term. The correct expression is: $\frac{\partial L(\beta)}{\partial \beta} = 2(( X \beta - y)^T X + \lambda \beta^T)$, from which the desired result follows by equating to zero and taking transposes.
| {
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Your favourite application of the Baire Category Theorem I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications that confirm its neatness and/or power.
Here's the theorem (wi... | One of my favorite (albeit elementary) applications is showing that $\mathbb{Q}$ is not a $G_{\delta}$ set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/165696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "248",
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"answer_id": 3
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Combinatorics thinking I saw this question in a book I've been reading: in a group of four mathematicians and five physicians, how many groups of four people can be created if at least two people are mathematicians?
The solution is obtained by ${4 \choose 2}{5 \choose 2} + {4 \choose 3}{5 \choose 1} + {4 \choose 4}{5 \... | You have counted the number of ways to chose two mathematicians leading the group, plus two regular members which can be mathematicians or physicians.
| {
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"source": "stackexchange",
"question_score": "1",
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Trying to find the name of this Nim variant Consider this basic example of subtraction-based Nim before I get to my full question:
Let $V$ represent all valid states of a Nim pile (the number of stones remaining):
$V = 0,1,2,3,4,5,6,7,8,9,10$
Let $B$ be the bound on the maximum number of stones I can remove from the Ni... | These are known as subtraction games; in general, for some set $S=\{s_1, s_2, \ldots s_n\}$ the game $\mathcal{S}(S)$ is the game where each player can subtract any element of $S$ from a pile. (So your simplified case is the game $\mathcal{S}(\{1\ldots B\})$) The nim-values of single-pile positions in these games are k... | {
"language": "en",
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Question About Concave Functions It easy to prove that no non-constant positive concave function exists (for example by integrating:
$ u'' \leq 0 \to u' \leq c \to u \leq cx+c_2 $ and since $u>0$ , we obviously get a contradiction.
Can this result be generalized to $ \Bbb R^2 $ and the Laplacian? Is there an easy way ... | Let $u$ strictly concave and twice diferentiable in $\mathbb{R^{2}}$ então $v(x) = u(x,0)$ is strictly concave and twice diferentiable in $\mathbb{R}$. Hence assume negative value.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Images of sections of a sheaf I'm currently reading a paper by X. Caicedo containing an introduction to sheaves.
On page 8 he claims, that for every sheaf of sets $p:E\to X$ and every section $\sigma:U\to E$ (U being open in X) the image $\sigma(U)$ is open.
This statement is proved by picking a point $e\in\sigma(U)$, ... | In order to prove $\sigma(U)$ is open, since $p(S)\cap U$ is open in $X$ and $p|S$ is a homeomorphism, it suffices to show $p|S(S\cap \sigma(U))=p(S)\cap U$. Clearly, this is true($p\sigma(u)=u$, for $u\in U$).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Average number of times it takes for something to happen given a chance Given a chance between 0% and 100% of getting something to happen, how would you determine the average amount of tries it will take for that something to happen?
I was thinking that $\int_0^\infty \! (1-p)^x \, \mathrm{d} x$ where $p$ is the chance... | Here's an easy way to see this, on the assumption that the average actually exists (it might otherwise be a divergent sum, for instance). Let $m$ be the average number of trials before the event occurs.
There is a $p$ chance that it occurs on the first try. On the other hand, there is a $1-p$ chance that it doesn't h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/165993",
"timestamp": "2023-03-29T00:00:00",
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Linear system with positive semidefinite matrix I have a linear system $Ax=b$, where
*
*$A$ is symmetric, positive semidefinite, and positive. $A$ is a variance-covariance matrix.
*vector $b$ has elements $b_1>0$ and the rest $b_i<0$, for all $i \in \{2, \dots, N\}$.
Prove that the first component of the soluti... | I don't think $x_1$ must be positive.
A counter example might be a positive definite matrix $A = [1 \space -0.2 ; \space -0.2 \space 1]$ with its inverse matrix $A^{-1}$ having $A_{11}, A_{12} > 0$.
-
Edit:
Sorry. A counter example might be a normalized covariance matrix
$ A= \left( \begin{array}{ccc}
1 & 0.6292 & 0.67... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Determine whether $\sum\limits_{i=1}^\infty \frac{(-3)^{n-1}}{4^n}$ is convergent or divergent. If convergent, find the sum. $$\sum\limits_{i=1}^\infty \frac{(-3)^{n-1}}{4^n}$$
It's geometric, since the common ratio $r$ appears to be $\frac{-3}{4}$, but this is where I get stuck. I think I need to do this: let $f(x) = ... | If $\,a, ar, ar^2,...\,$ is a geometric series with $\,|r|<1\,$ ,then
$$\sum_{n=0}^\infty ar^n=\lim_{n\to\infty} ar^n=\lim_{n=0}\frac{a(1-r^n)}{1-r}=\frac{a}{1-r}$$since $\,r^n\xrightarrow [n\to\infty]{} 0\Longleftrightarrow |r|<1\,$ , and thus
$$\sum_{n=1}^\infty\frac{(-3)^{n-1}}{4^n}=\frac{1}{4}\sum_{n=0}^\infty \lef... | {
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$10+10\times 0$ equals $0$ or $10$ I thought $10+10\times 0$ equals $0$ because:
$$10+10 = 20$$
And
$$20\times 0 = 0$$
I know about BEDMAS and came up with conclusion it should be $$0$$ not $$10$$
But as per this, answer is $10$, are they right?
| To elucidate what you said above in the original post, consider that $20\times0=0$, and consider also that $10+10=20$. If we have two equations like that, with one number $n$ on one side of an equation by itself with $m$ on the other side of the equation, and $n$ also appearing in the middle of a formula elsewhere, we... | {
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Proving $\mathrm e <3$ Well I am just asking myself if there's a more elegant way of proving $$2<\exp(1)=\mathrm e<3$$ than doing it by induction and using the fact of $\lim\limits_{n\rightarrow\infty}\left(1+\frac1n\right)^n=\mathrm e$, is there one (or some) alternative way(s)?
| It's equivalent to show that the natural logarithm of 3 is bigger than 1, but this is
$$
\int_1^3 \frac{dx}{x}.
$$
A right hand sum is guaranteed to underestimate this integral, so you just need to take a right hand sum with enough rectangles to get a value larger than 1.
| {
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"source": "stackexchange",
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De-arrangement in permutation and combination This article talks about de-arrangement in permutation combination.
Funda 1: De-arrangement
If $n$ distinct items are arranged in a row, then the number of ways they can be rearranged such that none of them occupies its original position is,
$$n! \left(\frac{1}{0!} – \fr... | A while back, I posted three ways to derive the formula for derangements. Perhaps reading those might provide some insight into the equation above.
| {
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How to solve infinite repeating exponents How do you approach a problem like (solve for $x$):
$$x^{x^{x^{x^{...}}}}=2$$
Also, I have no idea what to tag this as.
Thanks for any help.
| I'm just going to give you a HUGE hint. and you'll get it right way. Let $f(x)$ be the left hand expression. Clearly, we have that the left hand side is equal to $x^{f(x)}$. Now, see what you can do with it.
| {
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FFT with a real matrix - why storing just half the coefficients? I know that when I perform a real to complex FFT half the frequency domain data is redundant due to symmetry. This is only the case in one axis of a 2D FFT though. I can think of a 2D FFT as two 1D FFT operations, the first operates on all the rows, and f... | Real input compared to complex input contains half that information(since the zero padded part contains no information). The output is in the complex form, that means a double size container for the real input. So from the complex output we can naturally eliminate the duplicate part without any loss. I tried to be as s... | {
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Are there problems that are optimally solved by guess and check? For example, let's say the problem is: What is the square root of 3 (to x bits of precision)?
One way to solve this is to choose a random real number less than 3 and square it.
1.40245^2 = 1.9668660025
2.69362^2 = 7.2555887044
...
Of course, this is a ve... | There are certainly problems where a brute force search is quicker than trying to remember (or figure out) a smarter approach. Example: Does 5 have a cube root modulo 11?
An example of a slightly different nature is this recent question where an exhaustive search of the (very small) solution space saves a lot of grief ... | {
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Existence of such points in compact and connected topological space $X$ Let $X$ be a topological space which is compact and connected.
$f$ is a continuous function such that;
$f : X \to \mathbb{C}-\{0\}$.
Explain why there exists two points $x_0$ and $x_1$ in $X$ such that $|f(x_0)| \le |f(x)| \le |f(x_1)|$ for all $x... | Let $g(x)=|f(x)|$, observe that the complex norm is a continuous function from $\mathbb C$ into $\mathbb R$, therefore $g\colon X\to\mathbb R$ is continuous.
Since $X$ is compact and connected the image of $g$ is compact and connected. All connected subsets of $\mathbb R$ are intervals (open, closed, or half-open, half... | {
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How to get the characteristic equation? In my book, this succession defined by recurrence is presented:
$$U_n=3U_{n-1}-U_{n-3}$$
And it says that the characteristic equation of such is:
$$x^3=3x^2-1$$
Honestly, I don't understand how. How do I get the characteristic equation given a succession?
| "Guess" that $U(n) = x^n$ is a solution and plug into the recurrence relation:
$$
x^n = 3x^{n-1} - x^{n-3}
$$
Divide both sides by $x^{n-3}$, assuming $x \ne 0$:
$$
x^3 = 3x^2 - 1
$$
Which is the characteristic equation you have.
| {
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Unitisation of $C^{*}$-algebras via double centralizers In most of the books I read about $C^{*}$-algebras, the author usually embeds the algebra, say, $A$, as an ideal of $B(A)$, the algebra of bounded linear operators on $A$, by identifying $a$ and $M_a$, the left multiplication of $a$.
However, in Murphy's $C^{*}$-a... | The set of double centralizers of a $C^*$-algebra $A$ is usually also called the multiplier algebra $\mathcal{M}(A)$. It is in some sense the largest $C^*$-algebra containing $A$ as an essential ideal and unital. If $A$ is already unital it is equal to $A$. (Whereas in your construction of a unitalisation we have that ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Detailed diagram with mathematical fields of study Some time ago, I was searching for a detailed diagram with mathematical fields of study the nearest one I could find is in this file, second page.
I want something that shows information like: "Geometry leads to I topic, Geometry and Algebra leads do J topic and so on.... | Saunders Mac Lane's book Mathematics, Form and Function (Springer, 1986) has a number of illuminating diagrams showing linkages between various fields of mathematics (and also to some related areas.)
For example,
p149: Functions & related ideas of image and composition;
p184: Concepts of calculus;
p306: Interconnecti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In set theory, what does the symbol $\mathfrak d$ mean? What's meaning of this symbol in set theory as following, which seems like $b$?
I know the symbol such as $\omega$, $\omega_1$, and so on, however, what does it denote in the lemma?
Thanks for any help:)
| It is the German script $\mathfrak{d}$ given by the LaTeX \mathfrak{d}. It probably represents a cardinal number (sometimes $\mathfrak{c}$ is used to represent the cardinality of the real numbers), but it would definitely depend on the context of what you are reading.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that there exists $n$ such that $i^n = 0$ for all $i$ in the ideal $I$ I'm new to this medium, but I'm quite stuck with an exercise so hopefully someone here can help me.
This is the exercise:
Let $I$ be an ideal in a Noetherian ring $A$, and assume that for every $i\in I$ there exists an $n_i$ such that $i^{n_i... | Pritam's binomial approach is the easiest way to solve this problem in the commutative setting. In case you're interested though, it's also true in the noncommutative setting.
Levitzky's theorem says that any nil right ideal in a right Noetherian ring is nilpotent. This implies your conclusion (in fact, even more.) How... | {
"language": "en",
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Extension of morphisms on surfaces Consider two regular integral proper algebraic surfaces $X$ and $Y$ over a DVR $\mathcal O_K$ with residue field $k$. Let $U \subset X$ be an open subset, s.t. $X\setminus U$ consists of finitely many closed points lying in the closed fiber $X_k$. Assume that all points in $X\setminus... | Suppose $f : U\to Y$ is dominante. Then $f$ extends to $X$ if and only if $Y\setminus f(U)$ is finite. In particular, in your situation, $f$ extends to $X$ only when $g$ is an isomorphism (no component is contracted).
One direction (the one that matters for you) is rather easy: suppose $f$ extends to $f' : X\to Y$. As... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/167151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can we possibly combine $\int_a^b{g(x)dx}$ plus $\int_c^d{h(x)dx}$ into $\int_e^f{j(x)dx}$? I'm wondering if this is possible for the general case. In other words, I'd like to take $$\int_a^b{g(x)dx} + \int_c^d{h(x)dx} = \int_e^f{j(x)dx}$$ and determine $e$, $f$, and $j(x)$ from the other (known) formulas and integral... | Here's a method that should allow one a large degree of freedom as well as allowing Rieman Integration (instead of Lesbegue Integration or some other method):
Let $\tilde{g}$ be such that: $$\int_a^b{g(x)dx} = \int_e^f{\tilde{g}(x)dx}$$
...and $\tilde{h}$ follows similarly. Then they can be both added inside a single ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Limit of a sequence of real numbers If $(a_n), (b_n)$ are two sequences of real numbers so that $(a_n)\rightarrow a,\,\,(b_n)\rightarrow b$ with $a, b\in \mathbb{R}^+$. How to prove that $a_n^{b_n}\rightarrow a^b$ ?
| Note: The statement doesn't require $b > 0$. We don't assume it here.
If we take the continuity of $\ln$ and $\exp$ for granted, the problem essentially boils down to showing $b_n x_n \to bx$ where $x_n = \ln a_n, x = \ln a$ since $a_n^{b_n} = e^{b_n x_n}$. This is what the work in the first proof below goes toward (... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})[X]$ Is it possible to determine how many irreducible factors has $X^p-1$ in the polynomial ring $(\mathbb{Z}/q \mathbb{Z})[X]$ has and maybe even the degrees of the irreducible factors? (Here $p,q$ are primes with $\gcd(p,q)=1$.)
| It has one factor of degree $1$, namely $x-1$. All the remaining factors have the same degree, namely the order of $q$ in the multiplicative group $(\mathbb{Z}/p \mathbb{Z})^*$. To see it: this is the length of every orbit of the action of the Frobenius $a\mapsto a^q$ on the set of the roots of $(x^p-1)/(x-1)$ in the a... | {
"language": "en",
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What does "increases in proportion to" mean? I came across a multiple choice problem where a function $f(x) = \frac{x^2 - 1}{x+1} - x$ is given. One has to choose the statement that is correct about the function. The different statements about the function included:
*
*(1) the function increases in proportion to $x^... | To answer your question about examples where $f(x)$ would be proportional to $x$, and $x^2$, we only need to slightly modify the original function.
$f(x) = \frac {x^2-1}{x+1} $ is interesting between $x=-2$ and $x=2$, but as $x$ becomes really large or really small, the "-1" term and the "+1" term are insignificant and... | {
"language": "en",
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Normal Field Extension $X^4 -4$ has a root in $\Bbb Q(2^{1/2})$ but does not split in $\Bbb Q(2^{1/2})$ implying that $\Bbb Q(2^{1/2})$ is not a normal extension of $\Bbb Q$ according to most definitions.
But $\Bbb Q(2^{1/2})$ is considered a normal extension of $\Bbb Q$ by everybody. What am I missing here?
| You are missing the fact that $x^4-4$ is not irreducible over $\mathbb{Q}$: $x^4-4 = (x^2-2)(x^2+2)$.
The definition you have in mind says that if $K/F$ is algebraic, then $K$ is normal if and only if every irreducible $f(x)\in F[x]$ that has at least one root in $K$ actually splits in $K$. Your test polynomial is not ... | {
"language": "en",
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Exercise on compact $G_\delta$ sets I'm having trouble proving an exercise in Folland's book on real analysis.
Problem: Consider a locally compact Hausdorff space $X$. If $K\subset X$ is a compact $G_\delta$ set, then show there exists a $f\in C_c(X, [0,1])$ with $K=f^{-1}(\{1\})$.
We can write $K=\cap_1^\infty U_i$, ... | As you have said, we can use Urysohn's lemma for compact sets to construct a sequence of functions $f_i$ such that $f_i$ equals $1$ in $K$ and $0$ outside $U_i$.
Furthermore, $X$ is locally compact, so there is an open neighbourhood $U$ of $K$ whose closure is compact. We can then assume without loss of generality that... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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proof for $ (\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{B}\cdot\vec{C})\vec{A}$ this formula just pop up in textbook I'm reading without any explanation
$ (\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{B}\cdot\vec{C})\vec{A}$
I did some "vector arithmetic" ... | vector $ \vec A\times \vec B$ is perpendicular to the plane containing $\vec A $ and $\vec B$.Now, $(\vec A\times \vec B)\times \vec C$ is perpendicular to plane containing vectors $\vec C$ and $\vec A\times \vec B$, thus $(\vec A\times \vec B)\times \vec C$ is in the plane containing $\vec A$ and $ \vec B$ and hence $... | {
"language": "en",
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Idea behind factoring an operator? Suppose if we have an operator $\partial_t^2-\partial_x^2 $ what does it mean to factorise this operator to write it as $(\partial_t-\partial_x) (\partial_t+\partial x)$ When does it actually make sense and why ?
| In the abstract sense, the decomposition $x^2-y^2=(x+y)(x-y)$ is true in any ring where $x$ and $y$ commute (in fact, if and only if they commute).
For sufficiently nice (smooth) functions, differentiation is commutative, that is, the result of derivation depends on the degrees of derivation and not the order in which ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/167814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$. Let $a,b,c>0$ how to prove that :
$$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$$
I find that
$$\ \frac{ab}{a^{2}+3b^{2}}=\frac{1}{\frac{a^{2}+3b^{2}}{ab}}=\frac{1}{\frac{a}{b}+\frac{3b}{a}} $$
By AM-GM
$... | I have a Cauchy-Schwarz proof of it,hope you enjoy.:D
first,mutiply $2$ to each side,your inequality can be rewrite into
$$ \sum_{cyc}{\frac{2ab}{a^2+3b^2}}\leq \frac{3}{2}$$
Or
$$ \sum_{cyc}{\frac{(a-b)^2+2b^2}{a^2+3b^2}}\geq \frac{3}{2}$$
Now,Using Cauchy-Schwarz inequality,we have
$$ \sum_{cyc}{\frac{(a-b)^2+2b^2}{a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Quick probability question If there is an $80\%$ chance of rain in the next hour, what is the percentage chance of rain in the next half hour?
Thanks.
| You could assume that occurrence of rain is a point process with constant rate lambda. Then the process is Poisson and the probability of rain in an interval of time [0,t] is given by the exponential distribution and if T = time until the event rain then
P[T<=t] = 1-exp(-λt) assuming λ is the rate per hour then P(ra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Splitting field and subextension Definition: Let $K/F$ be a field extension and let $p(x)\in F[x]$,
we say that $K$ is splitting field of $p$ over $F$ if $p$ splits
in $K$ and $K$ is generated by $p$'s roots; i.e. if $a_{0},...,a_{n}\in K$
are the roots of $p$ then $K=F(a_{0},...a_{n})$.
What I am trying to understand ... | This is false. Let $F=\mathbb{Q}$, let $E=\mathbb{Q}(\sqrt{2})$, let $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$, and let $p=x^2-3\in F[x]$. Then the splitting field for $p$ over $E$ is $K$, but the splitting field for $p$ over $F$ is $\mathbb{Q}(\sqrt{3})\subsetneq K$.
Let's say that all fields under discussion live in an algeb... | {
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Big List of Fun Math Books To be on this list the book must satisfy the following conditions:
*
*It doesn't require an enormous amount of background material to understand.
*It must be a fun book, either in recreational math (or something close to) or in philosophy of math.
Here are my two contributions to the li... | Both of the following books are really interesting and very accessible.
Stewart, Ian - Math Hysteria
This book covers the math behind many famous games and puzzles and requires very little math to be able to grasp. Very light hearted and fun.
Bellos, Ales - Alex's Adventures in Numberland
This book is about maths in so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "58",
"answer_count": 27,
"answer_id": 16
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Ratio of sides in a triangle vs ratio of angles? Given a triangle with the ratio of sides being $X: Y : Z$, is it true that the ratio of angles is also $X: Y: Z$? Could I see a proof of this? Thanks
| No, it is not true. Consider a 45-45-90 right triangle:
(image from Wikipedia)
The sides are in the ratio $1:1:\sqrt{2}$, while the angles are in the ratio $1:1:2$.
| {
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In a field $F=\{0,1,x\}$, $x + x = 1$ and $x\cdot x = 1$ Looking for some pointers on how to approach this problem:
Let $F$ be a field consisting of exactly three elements $0$, $1$, $x$. Prove that
$x + x = 1$ and that $x x = 1$.
| Write down the addition and multiplication tables. Much of them are known immediately from the properties of 0 and 1, and there's only one way to fill in the rest so that addition and multiplication by nonzero are both invertible.
| {
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"timestamp": "2023-03-29T00:00:00",
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Given that $x=\dfrac 1y$, show that $∫\frac {dx}{x \sqrt{(x^2-1)}} = -∫\frac {dy}{\sqrt{1-y^2}}$ Given that $x=\dfrac 1y$, show that $\displaystyle \int \frac 1{x\sqrt{x^2-1}}\,dx = -\int \frac 1{\sqrt{1-y^2}}\,dy$
Have no idea how to prove it.
here is a link to wolframalpha showing how to integrate the left side.
| Substitute $x=1/y$ and $dx/dy=-1/y^2$ to get $\int y^2/(\sqrt{1-y^2}) (-1/y^2)dy$
| {
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Finding the critical points of $\sin(x)/x$ and $\cosh(x^2)$ Could someone help me solve this:
What are all critical points of $f(x)=\sin(x)/x$ and $f(x)=\cosh(x^2)$?
Mathematica solutions are also accepted.
| Taken from here :
$x=c$ is a critical point of the function $f(x)$ if $f(c)$ exists and if one of the following are true:
*
*$f'(c) = 0$
*$f'(c)$ does not exist
The general strategy for finding critical points is to compute the first derivative of $f(x)$ with respect to $x$ and set that equal to zero.
$$f(x) = \f... | {
"language": "en",
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Finding a function that fits the "lowest points" of another one I came up with this problem, which I cannot solve myself.
Consider the function:
$\displaystyle f(x) = x^{\ln(|\pi \cos x ^ 2| + |\pi \tan x ^ 2|)}$, which has singularities at $\sqrt{\pi}\sqrt{n + \dfrac{1}{2}}$, with $n \in \mathbb{Z}$.
Looking at its gr... | As $a^{\ln b}=\exp(\ln a\cdot\ln b)=b^{\ln a}$ the function $f$ can be written in the following way:
$$f(x)=\bigl(\pi|\cos(x^2)|+\pi|\tan(x^2)|\bigr)^{\ln x}\ .$$
Now the auxiliary function
$$\phi:\quad{\mathbb R}\to[0,\infty],\qquad t\mapsto \pi(|\cos(t)|+|\tan(t)|)$$
is periodic with period $\pi$ and assumes its mini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168321",
"timestamp": "2023-03-29T00:00:00",
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A limit question related to the nth derivative of a function This evening I thought of the following question that isn't related to homework, but it's a question that seems very challenging to me, and I take some interest in it.
Let's consider the following function:
$$ f(x)= \left(\frac{\sin x}{x}\right)^\frac{x}{\s... | The Taylor expansion is
$$f(x) = 1 - \frac{x^2}{6} + O(x^4),$$
so
\begin{eqnarray*}
f(0) &=& 1 \\
f'(0) &=& 0 \\
f''(0) &=& -\frac{1}{3}.
\end{eqnarray*}
$\def\e{\epsilon}$
Addendum:
We use big O notation.
Let
$$\e = \frac{x}{\sin x} - 1 = \frac{x^2}{6} + O(x^4).$$
Then
\begin{eqnarray*}
\frac{1}{f(x)} &=& (1+\e)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168369",
"timestamp": "2023-03-29T00:00:00",
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Show inequality generalization $\sum (x_i-1)(x_i-3)/(x_i^2+3)\ge 0$ Let $f(x)=\dfrac{(x-1)(x-3)}{x^2+3}$.
It seems to be that:
If $x_1,x_2,\ldots,x_n$ are positive real numbers with $\prod_{i=1}^n x_i=1$ then $\sum_{i=1}^n f(x_i)\ge 0$.
For $n>2$ a simple algebraic approach gets messy. This would lead to a generaliza... | Unfortunately, this is not true.
Simple counterexample:
My original counterexample had some ugly numbers in it, but fortunately, there is a counterexample with nicer numbers. However, the explanation below might still prove informative.
Note that for $x>0$,
$$
f(x)=\frac{(x-1)(x-3)}{x^2+3}=1-\frac{4x}{x^2+3}\lt1\tag{1}... | {
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"timestamp": "2023-03-29T00:00:00",
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A problem dealing with even perfect numbers. Question: Show that all even perfect numbers end in 6 or 8.
This is what I have. All even perfect numbers are of the form $n=2^{p-1}(2^p -1)$ where $p$ is prime and so is $(2^p -1)$.
What I did was set $2^{p-1}(2^p -1)\equiv x\pmod {10}$ and proceeded to show that $x=6$ or $... | $p$ is prime so it is 1 or 3$\mod 4$.
So, the ending digit of $2^p$ is (respectively) 2 or 8
(The ending digits of powers of 2 are $2,4,8,6,2,4,8,6,2,4,8,6...$
So, the ending digit of $2^{p-1}$ is (respectively) 6 or 4; and
the ending digit of $2^p-1$ is (respectively) 1 or 7.
Hence the ending digit of $2^{p-1}(2^p-1)$... | {
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Orthogonal projection to closed, convex subset in a Hilbert space I don't understand one step in the proof of the following lemma (Projektionssatz):
Let $X$ a Hilbert space with scalar product $(\cdot)_X$ and let $A\subset X$ be convex and closed. Then there is a unique map $P:X\rightarrow A$ that satisfies: $\|x-P(x... | It is non-negative! Hence, just drop it.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Weak*-convergence of regular measures Let $K$ be a compact Hausdorff space. Denote by $ca_r(K)$ the set of all countably additive, signed Borel measures which are regular and of bounded variation. Let $(\mu_n)_{n\in\mathbb{N}}\subset ca_r(K)$ be a bounded sequence satisfying $\mu_n\geq 0$ for all $n\in\mathbb{N}$. Can... | We cannot.
Let $K = \beta \mathbb{N}$ be the Stone-Cech compactification of $\mathbb{N}$, and let $\mu_n$ be a point mass at $n \in \mathbb{N} \subset K$. Suppose to the contrary $(\mu_n)$ has a weak-* convergent subsequence $\mu_{n_k}$. Define $f : \mathbb{N} \to \mathbb{R}$ by $f(n_k) = (-1)^k$, $f(n) = 0$ otherwis... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate $\int_{-a}^{a} \sqrt{a^2-x^2}\ln(\sqrt{a^2-x^2})\mathrm{dx}$ Well,this is a homework problem.
I need to calculate the differential entropy of random variable
$X\sim f(x)=\sqrt{a^2-x^2},\quad -a<x<a$ and $0$ otherwise. Just how to calculate
$$
\int_{-a}^a \sqrt{a^2-x^2}\ln(\sqrt{a^2-x^2})\,\mathrm{d}x
... | [Some ideas]
You can rewrite it as follows:
$$\int_{-a}^a \sqrt{a^2-x^2} f(x) dx$$
where $f(x)$ is the logarithm.
Note that the integral, sans $f(x)$, is simply a semicircle of radius $a$. In other words, we can write,
$$\int_{-a}^a \int_0^{\sqrt{a^2-x^2}} f(x) dy dx=\int_{-a}^a \int_0^{\sqrt{a^2-x^2}} \ln{\sqrt{a^2-x^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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} |
How to approach integrals as a function? I'm trying to solve the following question involving integrals, and can't quite get what am I supposed to do:
$$f(x) = \int_{2x}^{x^2}\root 3\of{\cos z}~dz$$
$$f'(x) =\ ?$$
How should I approach such integral functions? Am I just over-complicating a simple thing?
| Using the Leibnitz rule of differentiation of integrals,
which states that if
\begin{align}
f(x) = \int_{a(x)}^{b(x)} g(y) \ dy,
\end{align}
then
\begin{align}
f^{\prime}(x) = g(b(x)) b^{\prime}(x) - g(a(x)) a^{\prime}(x).
\end{align}
Thus, for your problem $a^{\prime}(x) = 2$ and $b^{\prime}(x) = 2x$ and, therefore, ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 4
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How many elements in a ring can be invertible?
If $R$ is a finite ring (with identity) but not a field, let $U(R)$ be its group of units. Is $\frac{|U(R)|}{|R|}$ bounded away from $1$ over all such rings?
It's been a while since I cracked an algebra book (well, other than trying to solve this recently), so if someone... | $\mathbb{F}_p \times\mathbb{F}_q$ has $(p-1)(q-1)$ invertible elements, so no.
Since $\mathbb{F}_2^n$ has $1$ invertible element, the proportion is also not bounded away from $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/168875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$. Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
I need this theorem to prove something.
| We know that
$$HK=\bigcup_{h\in H} hK$$
and each $hK$ has the same cardinality $|hK|=|K|$. (See ProofWiki.)
We also know that for any $h,h'\in G$ either $hK\cap h'K=\emptyset$ or $hK=h'K$.
So the only problem is to find out how many of the cosets $hK$, $h\in H$, are distinct.
Since
$$hK=h'K \Leftrightarrow h^{-1}h'\in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "48",
"answer_count": 6,
"answer_id": 4
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Uniqueness of morphism in definition of category theory product (etc) I'm trying to understand the categorical definition of a product, which describes them in terms of existence of a unique morphism that makes such-and-such a diagram commute. I don't really feel I've totally understood the motivation for this definiti... | This is a question which you will be able to answer yourself after some experience ... anyway:
The cartesian product $X \times Y$ of two sets $X,Y$ has the property: Every element of $X \times Y$ has a representation $(x,y)$ with unique elements $x \in X$ and $y \in Y$. This is the important and characteristic property... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Can mathematical definitions of the form "P if Q" be interpreted as "P if and only if Q"?
Possible Duplicate:
Alternative ways to say “if and only if”?
So when I come across mathematical definitions like "A function is continuous if...."A space is compact if....","Two continuous functions are homotopic if.....", etc... | Absolutely. The definition will state that we say [something] is $P$ if $Q$. Thus, every time that $Q$ holds, $P$ also holds. The definition would be useless if the other direction didn't hold, though. We want our terms to be consistent, so it is tacitly assumed that we will also say $P$ only if $Q$. Many texts prefer ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Differential Inequality Help I have the inequality
$f''(x)x + f'(x) \leq 0$
Also, $f''(x)<0$ and $f'(x)>0$ and $x \in R^+$. And I need to figure out when it is true. I know it is a fairly general question, but I couldn't find any information in several textbooks I have skimmed. Also, I am not sure if integrating would ... | $$0\geq f''(x)x+f'(x)=(f'(x)x)'$$
that is why the function $$f'(x)x$$ decreases for positive reals. Then the maximum should be at $0$, so $$f'(x)x\leq 0,\quad x \in R^+$$ which is a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/169219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Convergence of $\sum_{n=0}^\infty(-1)^n\frac{4^{n-2}(x-2)}{(n-2)!}$ What theorem should I use to show that $$\sum_{n=0}^\infty(-1)^n\frac{4^{n-2}(x-2)}{(n-2)!}$$ is convergent no matter what value $x$ takes?
| Note that $(-1)^n = (-1)^{n-2}$. Hence, $$\sum_{n=0}^\infty(-1)^n\frac{4^{n-2}(x-2)}{(n-2)!} = (x-2) \sum_{n=0}^\infty\frac{(-4)^{n-2}}{(n-2)!} = (x-2) \left(\sum_{n=0}^\infty\frac{(-4)^{n}}{n!} \right)$$
where we have interpreted $\dfrac1{(-1)!} = 0 = \dfrac1{(-2)!}$.
This is a reasonable interpretation since $\dfrac1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Combinations - at least and at most There are 20 balls - 6 red, 6 green, 8 purple
We draw five balls and at least one is red, then replace them. We then draw five balls and at most one is green. In how many ways can this be done if the balls are considered distinct?
My guess:
$${4+3-1 \choose 3-1} \cdot {? \choose ?}... | Event A
Number of ways to choose 5 balls = $ _{}^{20}\textrm{C}_5 $
Number of ways to choose 5 balls with no red balls = $ _{}^{14}\textrm{C}_5 $
Hence the number of ways to choose 5 balls including at least one red ball =$ _{}^{20}\textrm{C}_5 - _{}^{14}\textrm{C}_5 $
Event B
Number of ways to choose 5 balls with no g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Fermat's theorem on sums of two squares composite number Suppose that there is some natural number $a$ and $b$. Now we perform $c = a^2 + b^2$. This time, c is even.
Will this $c$ only have one possible pair of $a$ and $b$?
edit: what happens if c is odd number?
| Not necessarily. For example, note that $50=1^2+7^2=5^2+5^2$, and $130=3^2+11^2=7^2+9^2$. For an even number with more than two representations, try $650$.
We can produce odd numbers with several representations as a sum of two squares by taking a product of several primes of the form $4k+1$. To get even numbers with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Please explain this notation equation I am confused by this equation as I rarely use math in my job but need this for a program that I am working on. What exactly does the full expression mean? Note that $m^*_i{_j}$ refers to a matrix whose values have already been obtained.
Define the transition matrix $M = ${$m_i{_j}... | To obtain the transition matrix $M$ from the matrix $M^*=(m^*_{ij})$, the rule gives us two steps. First, for all off-diagonal terms $m^*_{ij}$ where $i\neq j$ we simply divide the existing entry by $\lvert U\rvert$ (in this case $\lvert U\rvert =24$), and we temporarily replace the diagonal entries $m^*_{ii}$ by $0.$ ... | {
"language": "en",
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When is $(6a + b)(a + 6b)$ a power of two?
Find all positive integers $a$ and $b$ for which the product $(6a + b)(a + 6b)$ is a power of $2$.
I havnt been able to get this one yet, found it online, not homework!
any help is appreciated thanks!
| Assume, (6a + b)(a + 6b) = 2^c
where, c is an integer ge 0
Assume, (6a + b) = 2^r
where, r is an integer ge 0
Assume, (a + 6b) = 2^s
where, s is an integer ge 0
Now, (2^r)(2^s) = 2^c
i.e. r + s = c
Now, (6a + b) + (a + 6b) = 2^r + 2^s
i.e. 7(a + b) = 2^r + 2^s
i.e. a + b = (2^r)/7 + (2^s)/7
Now, (6a + b) - ( a + 6b) = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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limit question on Lebesgue functions Let $f\in L^1(\mathbb{R})$. Compute $\lim_{|h|\rightarrow\infty}\int_{-\infty}^\infty |f(x+h)+f(x)|dx$
If $f\in C_c(\mathbb{R})$ I got the limit to be $\int_{-\infty}^\infty |f(x)|dx$. I am not sure if this is right.
| *
*Let $f$ a continuous function with compact support, say contained in $-[R,R]$. For $h\geq 2R$, the supports of $\tau_hf$ and $f$ are disjoint (they are respectively $[-R-h,R-h]$ and $[-R,R]$ hence
\begin{align*}
\int_{\Bbb R}|f(x+h)+f(x)|dx&=\int_{[-R,R]}|f(x+h)+f(x)|+\int_{[-R-h,R-h]}|f(x+h)+f(x)|\\
&=\int_{[-R,R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$ I posted this incorrectly several hours ago and now I'm back! So this time it's correct. Im trying to show that for $n\geq 1$:
$$\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt... | Potato's answer is what's going on geometrically. If you want it analytically:$$\sqrt{1+\left(\sin\left(\frac{\pi}{t}\right) -\frac{\pi}{t}\cos\left(\frac{\pi}{t}\right)\right)^2} \geq \sqrt{\left(\sin\left(\frac{\pi}{t}\right) -\frac{\pi}{t}\cos\left(\frac{\pi}{t}\right)\right)^2}$$
$$ = \bigg|\sin\left(\frac{\pi}{t}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Find Elementary Matrics E1 and E2 such that $E_2E_1$A = I I am studying Linear Algebra part-time and would like to know if anyone has advice on solving the following type of questions:
Considering the matrix:
$$A = \begin{bmatrix}1 & 0 & \\-5 & 2\end{bmatrix}$$
Find elementary Matrices $E_1$ and $E_2$ such that $E_2E_1... | Just look at what needs to be done.
First, eliminate the $-5$ using $E_1 = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}$. This gives
$$ E_1 A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}.$$
Can you figure out what $E_2$ must be?
| {
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"question_score": "2",
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When does a morphism preserve the degree of curves? Suppose $X \subset \mathbb{P}_k^n$ is a smooth, projective curve over an algebraically closed field $k$ of degree $d$ . In this case, degree of $X$ is defined as the leading coefficient of $P_X$, where $P_X$ is the Hilbert polynomial of $X$.
I guess Hartshorne using ... | Chris Dionne already explained the condition for $\phi$ to be an isomorphism from $X$ to $\phi(X)$. Suppose $\phi$ is a projection to $Q=\mathbb P^{n-1}$. Let $H'$ be a hyperplane in $Q$, then the schematic pre-image $\phi^{-1}Q=H\setminus \{ O\}$ where $H$ is a hyperplane of $P:=\mathbb P^n$ passing through $O$. Now
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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how to solve system of linear equations of XOR operation? how can i solve this set of equations ? to get values of $x,y,z,w$ ?
$$\begin{aligned} 1=x \oplus y \oplus z \end{aligned}$$
$$\begin{aligned}1=x \oplus y \oplus w \end{aligned}$$
$$\begin{aligned}0=x \oplus w \oplus z \end{aligned}$$
$$\begin{aligned}1=w \oplus... | The other answers are fine, but you can use even more elementary (if somewhat ad hoc) methods, just as you might with an ordinary system of linear equations over $\Bbb R$. You have this system:
$$\left\{\begin{align*}
1&=x\oplus y\oplus z\\
1&=x\oplus y\oplus w\\
0&=x\oplus w\oplus z\\
1&=w\oplus y\oplus z
\end{align*}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 1
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Proving that for every real $x$ there exists $y$ with $x+y^2\in\mathbb{Q}$ I'm having trouble with proving this question on my assignment:
For all real numbers $x$, there exists a real number $y$ so that $x + y^2$ is rational.
I'm not sure exactly how to prove or disprove this. I proved earlier that for all real numb... | Hint $\rm\ x\! +\! y^2 = q\in\mathbb Q\iff y^2 = q\!-\!x\:$ so choose a rational $\rm\:q\ge x\:$ then let $\rm\:y = \sqrt{q-x}.$
Remark $\ $ Note how writing it out equationally makes it clearer what we need to do to solve it. Just as for "word problems", the key is learning how to translate the problem into the correc... | {
"language": "en",
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Exponentiation of Center of Lie Algebra Let $G$ be a Lie Group, and $g$ its Lie Algebra. Show that the subgroup generated by exponentiating the center of $g$ generates the connected component of $Z(G)$, the center of $G$.
Source: Fulton-Harris, Exercise 9.1
The difficulty lies in showing that exponentiating the center... | $G$ is connected and so is generated by elements of the form $\exp Y$ for $Y \in g$. Therefore it is sufficient to show that for $X \in Z(g)$ $\exp X$ and $\exp Y$ commute. Now define $\gamma: \mathbb R \to G$ by $\gamma(t) = \exp(X)\exp(tY)\exp(-X)$. Then
$$
\gamma'(0) = Ad_{\exp(X)} Y = e^{ad_X} Y = (1 + ad_X + \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170035",
"timestamp": "2023-03-29T00:00:00",
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Kernel of $T$ is closed iff $T$ is continuous I know that for a Banach space $X$ and a linear functional $T:X\rightarrow\mathbb{R}$ in its dual $X'$ the following holds:
\begin{align}T \text{ is continuous } \iff \text{Ker }T \text{ is closed}\end{align}
which probably holds for general operators $T:X\rightarrow Y$ wit... | The result is false if $Y$ is infinite dimensional. Consider $X=\ell^2$ and $Y=\ell^1$ they are not isomorphic as Banach spaces (the dual of $\ell^1$ is not separable). However they both have a Hamel basis of size continuum therefore they are isomorphic as vector spaces. The kernel of the vector space isomorphism is cl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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$\ell_0$ Minimization (Minimizing the support of a vector) I have been looking into the problem $\min:\|x\|_0$ subject to:$Ax=b$. $\|x\|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time has been spent looking for a representation different from the on... | Consider the following two problems
$$ \min:\|x\|_0 \text{ subject to } Ax = b \tag{P0} $$
$$ \min:\|x\|_1 \text{ subject to } Ax = b \tag{P1} $$
The theory of compressed assert that the optimal solution to the linear program $(P1)$ is an optimal solution to $(P0)$ i.e., the sparsest vector, given the following conditi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Nth derivative of $\tan^m x$ $m$ is positive integer,
$n$ is non-negative integer.
$$f_n(x)=\frac {d^n}{dx^n} (\tan ^m(x))$$
$P_n(x)=f_n(\arctan(x))$
I would like to find the polynomials that are defined as above
$P_0(x)=x^m$
$P_1(x)=mx^{m+1}+mx^{m-1}$
$P_2(x)=m(m+1)x^{m+2}+2m^2x^{m}+m(m-1)x^{m-2}$
$P_3(x)=(m^3+3m^2+2... | The formula used to obtain the exponential generating function in Robert's answer is most easily seen with a little operator calculus. Let $\rm\:D = \frac{d}{dx}.\,$ Then the operator $\rm\,{\it e}^{\ zD} = \sum\, (zD)^k/k!\:$ acts as a linear shift operator $\rm\:x\to x+z\,\:$ on polynomials $\rm\:f(x)\:$ since
$$\rm ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Distance in vector space Suppose $k≧3$, $x,y \in \mathbb{R}^k$, $|x-y|=d>0$, and $r>0$. Then prove
(i)If $2r > d$, there are infinitely many $z\in \mathbb{R}^k$ such that $|z-x| = |z-y| = r$
(ii)If $2r=d$, there is exactly one such $z$.
(iii)If $2r < d$, there is no such $z$
I have proved the existence of such $z$ for ... | If there's a $z$ satisfying $|z-x|=r=|z-y|$ then by the triangle inequality, $d=|x-y|\le|z-x|+|z-y|=2r$
So if $d>2r$ there would've been no such $z$!
| {
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"url": "https://math.stackexchange.com/questions/170243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How many triangles with integral side lengths are possible, provided their perimeter is $36$ units? How many triangles with integral side lengths are possible, provided their perimeter is $36$ units?
My approach:
Let the side lengths be $a, b, c$; now,
$$a + b + c = 36$$
Now, $1 \leq a, b, c \leq 18$.
Applying multinom... | The number of triangles with perimeter $n$ and integer side lengths is given by Alcuin's sequence $T(n)$. The generating function for $T(n)$ is $\dfrac{x^3}{(1-x^2)(1-x^3)(1-x^4)}$. Alcuin's sequence can be expressed as
$$T(n)=\begin{cases}\left[\frac{n^2}{48}\right]&n\text{ even}\\\left[\frac{(n+3)^2}{48}\right]&n\tex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Question related to regular pentagon My question is-
ABCDE is e regular pentagon.If AB = 10 then find AC.
Any solution for this question would be greatly appreciated.
Thank you,
Hey all thanks for the solutions using trigonometry....can we also get the solution without using trigonometry? –
| Here's a solution without trig:
[Edit: here is a diagram which should make this more intuitive hopefully.]
First, $\angle ABC=108^{\circ}$; I won't prove this here, but you can do it rather trivially by dividing the pentagon into 3 triangles, e.g. $\triangle{ABC}$, $\triangle{ACD}$, and $\triangle{ADE}$, and summing a... | {
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How to compute the min-cost joint assignment to a variable set when checking the cost of a single joint assignment is high? I want to compute the min-cost joint assignment to a set of variables. I have 50 variables, and each can take on 5 different values. So, there are 550 (a huge number) possible joint assignments. F... | In general, if the costs of different assignments are completely arbitrary, there may be no better solution than a brute force search through the assignment space. Any improvements on that will have to come from exploiting some statistical structure in the costs that gives us a better-than-random chance of picking low... | {
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Ergodicity of the First Return Map I was looking for some results on Infinite Ergodic Theory and I found this proposition. Do you guys know how to prove the last item (iii)?
I managed to prove (i) and (ii) but I can't do (iii).
Let $(X,\Sigma,\mu,T)$ be a $\sigma$-finite space with $T$ presearving the measure $\mu$, $Y... | Let be $B \subset Y$. To prove the invariance of $\mu_A$ it is sucient to prove that,
$$\mu(T_Y^{-1}B)=\mu(B)$$
First,
$$\mu(T_Y^{-1}B)=\sum_{n=1}^{\infty}\mu(Y\cap\{\varphi_Y =n \}\cap T^{-n}B)$$
Now,
$$\{ \varphi_Y\leq n\}\cap T^{-n-1}B=T^{-1}( \{\varphi_Y\leq n-1\}\cap T^{-n}B)\cup T^{-1}(Y \cap\{\varphi_Y =n... | {
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Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$
For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of $p$
So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ then I am done .But I haven't been able to ... | Note that an integer $r$ with $\gcd(r,p)=1$ is a primitive root modulo $p^k$ when the smallest $b$ such that $r^b\equiv1\bmod p^k$ is $b=p^{k-1}(p-1)$.
Suppose that $r$ is not a primitive root modulo $p$, so there is some $b<p-1$ such that $r^b\equiv 1\bmod p$.
In other words, there is some integer $t$ such that $r^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 6,
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Find a function that satisfies the following five conditions. My task is to find a function $h:[-1,1] \to \mathbb{R}$ so that
(i) $h(-1) = h(1) = 0$
(ii) $h$ is continuously differentiable on $[-1,1]$
(iii) $h$ is twice differentiable on $(-1,0) \cup (0,1)$
(iv) $|h^{\prime\prime}(x)| < 1$ for all $x \in (-1,0)\cup(0,1... | Let $h$ satisfy (i)-(iv) and $x_0$ be a point where the maximum of $|h|$ is attained on $[-1,1]\;$. WLOG we can assume that $x_0\ge0$. Then
$$
|h(x_0)|=|h(x_0)-h(1)|=\left|\int_{x_0}^1h'(y)\,dy\right|=
$$
$$
\left|\int_{x_0}^1\int_{x_0}^yh''(z)\,dz\;dy\right|\le
\sup_{(0,1)}|h''|\int_{x_0}^1\int_{x_0}^y\,dz=\frac{(1-x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Counting zero-digits between 1 and 1 million I just remembered a problem I read years ago but never found an answer:
Find how many 0-digits exist in natural numbers between 1 and 1 million.
I am a programmer, so a quick brute-force would easily give me the answer, but I am more interested in a pen-and-paper solution.... | Just to show there is more than one way to do it:
How many zero digits are there in all six-digit numbers? The first digit is never zero, but if we pool all of the non-first digits together, no value occurs more often than the others, so exactly one-tenth of them will be zeroes. There are $9\cdot 5\cdot 10^5$ such digi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Application of Banach Separation theorem
Let $(\mathcal{H},\langle\cdot,\cdot\rangle)$ be a Hilbert Space, $U\subset \mathcal{H},U\not=\mathcal{H}$ be a closed subspace and $x\in\mathcal{H}\setminus U$. Prove that there exists $\phi\in\mathcal{H}^*$, such that\begin{align}\text{Re } \phi(x)<\inf_{u\in U}\text{Re }\phi... | There is more general result which proof you can find in Rudin's Functional Analysis
Let $A$ and $B$ be disjoint convex subsets of topological vector space $X$. If $A$ is compact and $B$ is closed then there exist $\varphi\in X^*$ such that
$$
\sup\limits_{x\in A}\mathrm{Re}(\varphi(x))<\inf\limits_{x\in B}\mathrm{R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Looking for a 'second course' in logic and set theory (forcing, large cardinals...) I'm a recent graduate and will likely be out of the maths business for now - but there are a few things that I'd still really like to learn about - forcing and large cardinals being two of them.
My background is what one would probably ... | I would recommend the following as excellent graduate level introductions to set theory, including forcing and large cardinals.
*
*Thomas Jech, Set Theory.
*Aki Kanamori, The higher infinite. See the review I wrote of it for Studia Logica.
I typically recommend to my graduate students, who often focus on both f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/170946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Factorize $f$ as product of irreducible factors in $\mathbb Z_5$ Let $f = 3x^3+2x^2+2x+3$, factorize $f$ as product of irreducible factors in $\mathbb Z_5$.
First thing I've used the polynomial reminder theorem so to make the first factorization:
$$\begin{aligned} f = 3x^3+2x^2+2x+3 = (3x^2-x+3)(x+1)\end{aligned}$$
Ob... | If $f(X) = aX^2 + bX + c$ is a quadratic polynomial with roots $x_1$ and $x_2$ then $f(X) = a(X-x_1)(X-x_2)$ (the factor $a$ is necessary to get the right leading coefficient). You found that $3x^2-x+3$ has a double root at $x_1 = x_2 = 1$, so $3x^2-x+3 = 3(x-1)^2$. Your mistakes were
*
*You forgot to multiply by th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Functional equations of (S-shape?) curves I am looking for the way to "quite easily" express particular curves using functional equations.
What's important (supposing the chart's size is 1x1 - actually it doesn't matter in the final result):
*
*obviously the shape - as shown in the picture;
*there should be exactl... | Have you tried polynomial interpolation? It seems that for 'well-behaved' graphs like the ones you are looking for (curves for image processing?), it could work just fine.
At the bottom of this page you can find an applet demonstrating it. There is a potential problem with the interpolated degree 4 curve possibly becom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Throwing balls into $b$ buckets: when does some bucket overflow size $s$? Suppose you throw balls one-by-one into $b$ buckets, uniformly at random. At what time does the size of some (any) bucket exceed size $s$?
That is, consider the following random process. At each of times $t=1, 2, 3, \dots$,
*
*Pick up a ball... | I just wrote some code to find the rough answer (for my particular numbers) by simulation.
$ gcc -lm balls-bins.c -o balls-bins && ./balls-bins 10000 64
...
Mean: 384815.56 Standard deviation: 16893.75 (after 25000 trials)
This (384xxx) is within 2% of the number ~377xxx, specifically
$$ T \approx b \left( (s + \log b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
If $f:D\to \mathbb{R}$ is continuous and exists $(x_n)\in D$ such as that $x_n\to a\notin D$ and $f(x_n)\to \ell$ then $\lim_{x\to a}f(x)=\ell$? Assertion: If $f:X\setminus\left\{a\right\}\to \mathbb{R}$ is continuous and there exists a sequence $(x_n):\mathbb{N}\to X\setminus\left\{a\right\}$ such as that $x_n\to a$ ... | You need to have $f(x_n) \to l$ for all sequences $x_n \to a$, not just one sequence.
For example, let $a=(0,0)$ with $f(x,y) = \frac{x y}{x^2+y^2}$. This is continuous on $\mathbb{R}^2 \setminus \{a\}$, and the sequence $x_n=(\frac{1}{n},0) \to a$, with $f(x_n) \to 0$ (excuse abuse of notation), but $f$ is not continu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Preorders, chains, cartesian products, and lexicographical order Definitions:
A preorder on a set $X$ is a binary relation $\leq$ on $X$ which is reflexive and transitive.
A preordered set $(X, \leq)$ is a set equipped with a preorder.... Where confusion cannot result, we refer to the preordered set $X$ or sometimes ju... | Suppose $(c_1, c_1')$ and $(c_2, c_2')$ are pairs with $c_1, c_2 \in C$ and $c_1', c_2' \in C'$. Then $c_1 \le c_2$ or $c_2 \le c_1$ and $c_1' \le c_2'$ or $c_2' \le c_1'$. If $c_1 \le c_2$ but $c_2 \not\le c_1$ then $c_1 < c_2$ and $(c_1, c_1') \le_{lex} (c_2, c_2')$. Similarly, if $c_1 \not\le c_2$ and $c_2 \le c_1$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding two numbers given their sum and their product
Which two numbers when added together yield $16$, and when multiplied together yield $55$.
I know the $x$ and $y$ are $5$ and $11$ but I wanted to see if I could algebraically solve it, and found I couldn't.
In $x+y=16$, I know $x=16/y$ but when I plug it back in... | Given $$ x + y = 16 \tag{1} $$
and $$ x \cdot y = 55. \tag{2} $$
We can use the identity: $$ ( x-y)^2 = ( x+y)^2 - 4 \cdot x \cdot y \tag{3} $$
to get
$$ x - y = \sqrt{256 - 4 \cdot 55} = \sqrt{36} = \pm \, 6. \tag{4} $$
Finding half sum and half difference of equations (1) and (4) gives us
$$ (x,y)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 9,
"answer_id": 8
} |
Is there any way to find a angle of a complex number without a calculator? Transforming the complex number $z=-\sqrt{3}+3i$ into polar form will bring me to the problem to solve this two equations to find the angle $\phi$: $\cos{\phi}=\frac{\Re z}{|z|}$ and $\sin{\phi}=\frac{\Im z}{|z|}$.
For $z$ the solutions are $\c... | The direct calculation is
$$\arg(-\sqrt 3+ 3i)=\arctan\frac{3}{-\sqrt{3}}=\arctan (-\sqrt 3)=\arctan \frac{\sqrt 3/2}{-1/2}$$
As the other two answers remark, you must learn by heart the values of at least the sine and cosine at the main angle values between zero and $\,\pi/2\,$ and then, understanding the trigonometr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Where are good resources to study combinatorics? I am an undergraduate wiht basic knowledge of combinatorics, but I want to obtain sound knowledge of this topic. Where can I find good resources/questions to practice on this topic?
I need more than basic things like the direct question 'choosing r balls among n' etc.; I... | Lots of good suggestions here. Another freely available source is Combinatorics Through Guided Discovery. It starts out very elementary, but also contains some interesting problems. And the book is laid out as almost entirely problem-based, so it useful for self study.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/171543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 8,
"answer_id": 2
} |
Solving Recurrence $T_n = T_{n-1}*T_{n-2} + T_{n-3}$ I have a series of numbers called the Foo numbers, where $F_0 = 1, F_1=1, F_2 = 1 $
then the general equation looks like the following:
$$
F_n = F_{n-1}(F_{n-2}) + F_{n-3}
$$
So far I have got the equation to look like this:
$$T_n = T_{n-1}*T_{n-2} + T_{n-3}$$
I ju... | Numerical data looks very good for
$$F_n \approx e^{\alpha \tau^n}$$
where $\tau = (1+\sqrt{5})/2 \approx 1.618$ and $\alpha \approx 0.175$. Notice that this makes sense: When $n$ is large, the $F_{n-1} F_{n-2}$ term is much larger than $F_{n-3}$, so
$$\log F_n \approx \log F_{n-1} + \log F_{n-2}.$$
Recursions of the f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Existence of valuation rings in a finite extension of the field of fractions of a weakly Artinian domain without Axiom of Choice Can we prove the following theorem without Axiom of Choice?
This is a generalization of this problem.
Theorem
Let $A$ be a weakly Artinian domain.
Let $K$ be the field of fractions of $A$.
L... | Lemma 1
Let $A$ be an integrally closed weakly Artinian domain.
Let $S$ be a multiplicative subset of $A$.
Let $A_S$ be the localization with respect to $S$.
Then $A_S$ is an integrally closed weakly Artinian domain.
Proof:
Let $K$ be the field of fractions of $A$.
Suppose that $x \in K$ is integral over $A_S$.
$x^n + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Do equal distance distributions imply equivalence? Let $A$ and $B$ be two binary $(n,M,d)$ codes. We define $a_i = \#\{(w_1,w_2) \in A^2:\:d(w_1,w_2) = i\}$, and same for $b_i$. If $a_i = b_i$ for all $i$, can one deduct that $A$ and $B$ are equivalent, i.e. equal up to permutation of positions and permutation of lette... | See Example 3.5.4 and the "other examples ... given in Section IV-4.9."
EDIT: Here are two inequivalent binary codes with the same distance enumerator. Code A consists of the codewords $a=00000000000$, $b=11110000000$, $c=11111111100$, $d=11000110011$; code R consists of the codewords $r=0000000000$, $s=1111000000$, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A strangely connected subset of $\Bbb R^2$ Let $S\subset{\Bbb R}^2$ (or any metric space, but we'll stick with $\Bbb R^2$) and let $x\in S$. Suppose that all sufficiently small circles centered at $x$ intersect $S$ at exactly $n$ points; if this is the case then say that the valence of $x$ is $n$. For example, if $S=... | I claim there is a set $S \subseteq {\mathbb R}^2$ that contains exactly three points in every circle.
Well-order all circles by the first ordinal of cardinality $\mathfrak c$ as $C_\alpha, \alpha < \mathfrak c$. By transfinite induction I'll construct sets $S_\alpha$ with
$S_\alpha \subseteq S_\beta$ for $\alpha < \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 2,
"answer_id": 0
} |
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