Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is $\int_E \frac{1}{(x^2+y^2)^2}dxdy$ convergent? I have to tell whether this integral is convergent:
$$\int_E \frac{1}{(x^2+y^2)^2}dxdy$$
where $E=\{0\leq y \leq x^a\} \cap \{x^2+y^2\leq 1\} $.
I'm asked for which $a \geq 0$ the integral converges. How am I supposed to act when I find this kind of integrals? I mean, t... | If $x<0$ we have that $x^a$ is not necessarily defined, so I am going to assume that the actual problem is to discuss the convergence of
$$I(a)=\iint_E \frac{dx\,dy}{(x^2+y^2)^2},\qquad E=\{(x,y):x^2+y^2\leq 1, x> 0, 0<y<x^a\}.$$
With these assumptions we have
$$ I(a) = \int_{0}^{1}\frac{L(\rho)}{\rho^4}\,d\rho $$
he... | {
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"url": "https://math.stackexchange.com/questions/3702259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Guidance requested for vector dot product question. Helping my child out with their year 11 exam preparation, specifically vectors and dot products, I think I may have figured out the answer but I'd like to get some confirmation or, more likely, a short sharp shock of education :-)
Keep in mind it's some thirty-plus ye... | I don't know what your son is supposed to know about the dot product... which is critical for the answer.
However if he knows that dot product is distributive vs. addition then $$a \cdot (b-a) = a \cdot b - a \cdot a=0.$$
Therefore 2. is clear and 4. also as $a \cdot a = \vert a \vert^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3702395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Affine function is a diffeomorphism? Given an affine function $f:\mathbb{R}^n \to \mathbb{R}^n$ defined for all $x\in \mathbb{R}^n$ by $$f(x)=T(x)+a$$ such that $T$ is an invertible Linear map and $a\in \mathbb{R}^n$, is $f$ a diffeomorphism?
| It is a diffeomorphism iff $T$ is invertible.
It is easy to see that $f$ is invertible iff $T$ is invertible with inverse
$$f^{-1}(x) = T^{-1}(x - a).$$
As $T^{-1}$ is a linear transformation, it is differentiable everywhere.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3702579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Do Tarski's (geometry) axioms imply that all zero segments are congruent? Tarski's axioms are an alternate formalization of geometry (similar to axiom sets of Euclid and later Hilbert). Do these axioms imply:
$$\forall\; x,y\in \text{points},\; x x\equiv y y?$$
If yes, what is the proof? My feeling is that the proof m... | This seems surprisingly tricky. Identity of Congruence isn't enough by itself since it goes in the wrong direction.
I want to use the Five Segment Axiom. Let me call your two points $p,q$ instead, to avoid a conflict with Wikipedia's notation. If $p=q$ then we are done by reflexivity, so assume $p \ne q$. Set $u=z=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3702720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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SL-Eigenvalue/function problem with arbitrary boundary values The problem is to find all eigenvalues and eigenfunctions for the following SL system.
$u'' + \lambda u = 0, x \in [a,b]$
$u'(a) = u'(b) = 0$
I know the general idea of how to do these problems and can do them for more simple boundary conditions say $u(0) ... | If you want a solution with $u'(a)=0$, then you may safely assume $u(a)=1$ because $u'(a)=u(a)=0$ will force $u\equiv 0$. Any non-trivial solution will be a multiple of the solution where $u'(a)=0,u(a)=1$, and that solution is
$$
u(x) = \cos(\sqrt{\lambda}(x-a)).
$$
To complete the solution, set $u'(b)=0$, wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3702924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of $\frac{e^{xy}}{x+1}$ as it goes to $0$ I have to do the following limit:
$$\lim_{(x,y)\rightarrow (0,0)}\frac{e^{xy}}{x+1}$$
I did that if $x = 0 \Rightarrow \lim \rightarrow 1$ but if $y = 0 \Rightarrow \lim \rightarrow$ undefined. By taking $x=y$ it gets me to the same result. Is the limit then $1$ or in wh... | Actually, if $y=0$, it's $\lim\limits_{x\to0}\dfrac1{x+1}=\color{red}1$ too.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3703042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Shilov Chapter 4 Problem 16 I am working on the captioned problem which is reproduced below.
And the hint for this problem is the following:
But I have no idea of how to use Chapter 3 Prob 12 for this problem. That problem is reproduced here:
I have no problem in finding out that three equations for unknown elements... | The equation
$$
\pmatrix{a&b\\ c&d}\pmatrix{x&y\\ z&w}=\pmatrix{P&-Q\\ R&-P}
$$
can be rewritten as
\begin{aligned}
bz-cy &= P,\\
(a-d)y-b(x-w) &= -Q,\\
c(x-w)-(a-d)z &= R.
\end{aligned}
That is, if
$$
A=\pmatrix{b&c&a-d\\ y&z&x-w},\tag{$\ast$}
$$
then the minors obtained by deleting respectively the third column, the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3703181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An identity of Arithmetic Functions Problem: Show that for all positive integers $n$,
$$ \sum_{a=1, (a,n)=1}^{n} (a-1, n) = d(n)\phi(n)$$
where $(a, b)$ stands for $\text{gcd}(a, b)$ and $d, \phi$ are the divisor and Euler's totient function, i.e., number of numbers co-prime to n and less than n = $\phi(n)$.
I find th... | For a divisor $d|n$, the number of $a>1$ such that $gcd(a-1,n)=d$ is equal to $|\{1 \leq q \leq \dfrac{(n-1)}{d}|(qd,n)=d\}|=|\{1 \leq q \leq \dfrac{(n-1)}{d}|(q,n/d)=1\}|=\varphi(n/d)$.
Now there's a multiplicative function.
Also, it is known that if $f$ is multiplicative, then $\sum_{d|n}f(d)$ is also muliplicative, ... | {
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"timestamp": "2023-03-29T00:00:00",
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Calculate $\sum_{n=1}^\infty\frac{n^x}{n!}$ I want to evaluate function defined by following sum: $$\sum_{n=1}^\infty\frac{n^x}{n!}$$ I was thinking about writing Taylor series expansion for it. However my try resulted in sum that looks even harder to calculate:
$$\sum_{n=1}^{\infty}\frac{\ln^k(n)}{n!}$$Thanks for all ... | Proof of the Dobiniski's formula:
See, $$\sum_{n=0}^{\infty} \frac{n^k}{n!}=\underbrace{\frac{d}{dx}(x\frac{d}{dx}(x.....(\frac{d}{dx}e^x)))),}_\text{$k$ times}$$ at $x=1$.
$$=\underbrace{\frac{d}{dx}(x\frac{d}{dx}(......\frac{d}{dx}(x^2e^x+xe^x))),}_\text{$k-2$ times}$$ at $x=1$.....(1)
Also , $eB_k=\frac{d^k}{dt^k}(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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"answer_id": 2
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Bound on a complex integral using polar form I'm a little bit confused and also a bit rusty on complex analysis... Here is my problem;
Consider a function $f:\mathbb{R}\to\mathbb{C}$. We can express this function in polar coordinates as: $f(x) = A(x)e^{ia(x)}$, with $A(x)\in\mathbb{R}_+$ and $a(x)\in \mathbb{R}$. My qu... | I found a counter example that shows that this does not holds true...
Choose $A(x)=e^{-x^2}$, $a(x)=x$ and $p=-q=10$. Then the inequality would state that $1.38...\le 1.08...$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is considering only quadratic in one of the variables of a two variable quadratic sufficient for calculating roots Find the $positive$ integral solutions to $7x^2-2xy+3y^2-27=0$
My solution:
Assuming the quadratic in $x$ , if we assume one root to be integral , the other has to be rational (as y must be an integer ... | Assuming $a$ and $b$ are nonzero, we can solve the equation$$ax^2+bxy+cy^2+d=0 \tag{1}\label{1}$$for $x$ by considering the equation as a quadratic in $x$, namely$$x=\frac{-by \pm \sqrt{(by)^2-4a(cy^2+d)}}{2a}, \tag{2}\label{2}$$and for $y$ by considering the equation as a quadratic in $y$, namely$$y=\frac{-bx \pm \sqr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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find the dim S. Problem taken from Apostol calculas Volume $2$ page No: $13$ books
Let $P$ denote the linear space of all real polynomials of degree $\le n$, where $n$ is fixed. let $ S$ denote the set of all polynomials $f$ in $P$, satisfying the condition given below . find the dim S.
$1.$$f$ is even.
$2.$ $f$ is od... | Case 1: $n$ is even, say $n=2k$
*Even $f$ will be of the type $ c+c_1x^2+c_2x^4+...c_kx^{2k}$ hence dim(S) =$k+1=n/2+1$
**Odd $f$ will be of the type $d_1x+d_3x^3+...+d_{2k-1}x^{2k-1}$, hence dim(S) =$k=n/2$
Similarly consider, case 2: $n$ is odd.
PS: * because even $ f(x) =\frac{ f(x) +f(-x)} {2}$
** odd $f(x) =\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3704221",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Poisson Arrival Process and Uniform Distribution I'm brushing up on some basic probability and have this question:
If we have a Poisson arrival process with arrivals $A_{1}, A_{2}, \dots$, and we know that there is one and only one arrival in a time period, say $[t_{1}, t_{2}]$. Does this mean that the one arrival is ... | Let's call your process $N(t)$. i.e. $N(t)$ is the number of arrivals that have happened up to and including time $t$. And given an interval $(a,b]$, let $N((a,b])$ denote the number of arrivals in the interval $(a,b]$.
The way you would go about proving it is to fix some number $s \in (t_1,t_2]$ and try to calculate
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3704384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Resolving indeterminacy of map on projective spaces induced by a linear map Let $V$ and $W$ be two $k$-vector spaces, where $k$ can be assumed as $\mathbb{C}$, and
$f:V\longrightarrow W$ a non null $k$-linear map with kernel $K$. The map $f$ naturally induces a rational morphism $F:\mathbb{P}(V)\dashrightarrow\mathbb{... | Let me assume $f$ is surjective (otherwise, replace $W$ by $Im(f)$). Then
$$
Bl_{\mathbb{P}(K)}(\mathbb{P}(V)) =
\mathbb{P}_{\mathbb{P}(W)}(K \otimes \mathcal{O} \oplus \mathcal{O}(-1)).
$$
Next, if $H_V$ and $H_W$ are the pullbacks of the hyperplane classes of $\mathbb{P}(V)$ and $\mathbb{P}(W)$, respectively, and $E... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3704531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove$:$ $\sum\limits_{cyc} (\frac{a}{b+c}-\frac{1}{2}) \geqq (\sum\limits_{cyc} ab)\Big[\sum\limits_{cyc} \frac{1}{(a+b)^2}\Big]-\frac{9}{4}$ For $a,b,c$ are reals and $a+b+c>0, ab+bc+ca>0, (a+b)(b+c)(c+a)>0.$ Prove$:$
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} -\frac{3}{2} \geqq (\sum\limits_{cyc} ab)\Big[\sum\limit... | From
$$ \sum \frac{a}{b+c} -\frac{3}{2} - \left(\sum \frac{ab+bc+ca}{(a+b)^2} -\frac{9}{4}\right) = \frac14 \sum \frac{(a-b)^2}{(a+b)^2} \geqslant 0.$$
We can see, the inequality also true for all $a,\,b,\,c$ are real numbers.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Conditions that a function is analytic in the complex plane of its independent variable? I have no a mathematic undergraduate background, so I am very sorry if this question is too naive.
Consider a simple example: $f(x)=\vert x \vert^3$ and $g(x)=x^3$ where $x\in \mathbb{C}$. Why $f(x)$ is not analytic in the complex ... | Start with the definition of analytic. The function of a complex variable $z$ is analytic at $z \in \mathbb C$ if it is differentiable at $z$, which means
$$\begin{align} \frac{f(z+h) - f(z)}{h} \tag 1 \end{align}$$
has a unique limit as $\lvert h \rvert \to 0$, denoted $f'(z)$. The limit has to exist regardless of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3704966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Cardinality of $\operatorname{Hom}_{\mathbb{C}}(A,\mathbb{C})$ Question
Prove there is no finite generated algebra $A$ over $\mathbb{C}$ such that the cardinality of $\operatorname{Hom}_{\mathbb{C}}(A,\mathbb{C})$ is exactly $\aleph_0$.
I need to prove it using commutative algebra tools such as Hilbert basis & Nullste... | Hint: Noether normalization says there's an injective map $R=\Bbb C[x_1,\cdots,x_n]\to A$ which makes $A$ into a finite $R$-algebra (where $n=\dim A$). This induces a map on the hom-sets $\operatorname{Hom}_{\Bbb C}(A,\Bbb C)\to\operatorname{Hom}_{\Bbb C}(R,\Bbb C)$. What can you say about this map? What can you say ab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3705123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$f(x)=(\sin(\tan^{-1}x)+\sin(\cot^{-1}x))^2-1, |x|>1$
Let $f(x)=(\sin(\tan^{-1}x)+\sin(\cot^{-1}x))^2-1, |x|>1$. If $\frac{dy}{dx}=\frac12\frac d{dx}(\sin^{-1}(f(x)))$ and $y(\sqrt3)=\frac{\pi}{6}$, then $y(-\sqrt3)=?$
$$f(x)=(\frac{x}{\sqrt{x^2+1}}+\frac{1}{\sqrt{x^2+1}})^2-1=\frac{2x}{1+x^2}$$
$$\frac{dy}{dx}=\frac... | $\sin(\cot^{-1}x)=\sin\left(\dfrac\pi2-\tan^{-1}x\right)=\cos(\tan^{-1}x)$
$$\implies f(x)=\left(\sin(\tan^{-1}x)+\sin(\cot^{-1}x)\right)^2-1=\sin2\left(\tan^{-1}x\right)$$
Now $\sin^{-1}\left(\sin(2\tan^{-1}x )\right)=\begin{cases} \pi-2\tan^{-1}x &\mbox{if } 2\tan^{-1}x>\dfrac\pi2\iff x>\tan\dfrac\pi4 \\
-\pi-2\tan^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3705302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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1st order linear differential equation $y'+\frac{xy}{1+x^2} =x$ Can anyone help me with this one task. I need to resolve 1st order linear equation of this equation.
$$y'+\frac{xy}{1+x^2} = x.$$
I stopped when this result came out
$$e^{\ln|y|}=e^{-\frac{1}{2}\ln|1+x^2|}\cdot e^C.$$
I try solve this by wolfram
$$y=\frac... | This is a linear ODE then
$$
y=y_h+y_p\\
y'_h + \frac{x}{1+x^2}y_h = 0\\
y'_p + \frac{x}{1+x^2}y_p = x
$$
the homogeneous is separable giving
$$
y_h = \frac{c_0}{\sqrt{1+x^2}}
$$
now using the method of constants variation due to Lagrange we make $y_p = \frac{c_0(x)}{\sqrt{1+x^2}}$ and substituting we obtain
$$
\frac{c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A problem regarding the ratios $\frac{f(x)}{g(x)}$ and $\frac{g(x)}{h(x)}$, assuming $f(x)g(y) = h\big(\sqrt{x^2+y^2}\big)$ $\mathbf {The \ Problem \ is}:$ Let, $f,g,h$ be three functions defined from $(0,\infty)$ to $(0,\infty)$ satisfying the given relation $f(x)g(y) = h\big(\sqrt{x^2+y^2}\big)$ for all $x,y \in (0,\... | To make the formulas look simpler, define the functions $ \tilde f $, $ \tilde g $ and $ \tilde h $ from $ ( 0 , + \infty ) $ to $ ( - \infty , + \infty ) $ by:
$$ \tilde f ( x ) = \log \frac { f \left( \sqrt x \right) } { f ( 1 ) } \qquad \tilde g ( x ) = \log \frac { g \left( \sqrt x \right) } { g ( 1 ) } \qquad \til... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 1
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Name of the set that forms a topological space with a topology Let's assume I have a topological space $(X, \tau)$, where $X$ is a set and $\tau$ is a topology. Now I have $Y\subset X$, but $Y$ is not necessarily element of $\tau$. What do I call $X$ in my publication?
It is not the "topological space", because that w... | You can say that $Y$ is a subset of the underlying set of the topological space $(X, \tau)$. Usually, when a set $X$ is endowed with some "structure", you can address the set itself by calling it the underlying set. What you are doing is "forgetting" the structure and considering just the set of elements. This works fo... | {
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"timestamp": "2023-03-29T00:00:00",
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Pushout of categories in Set I found this answer Understanding an Example of a Pushout in very helpful regarding disjoint unions, but wanted to know how to understand a pushout of categories in Set, say from : A→B and : A→C, where f and g are functors with shared domain A. There is an equivalence relation on objects, ... | As you say, the pushout $P$ has objects $\mathrm{ob}(P)=\mathrm{ob} B\sqcup \mathrm{ob} C/\sim$, where $\sim$ is the equivalence relation generated by the relation containing $(b,c)$ whenever $b$ and $c$ are the image of the same object in $A$. Now to construct morphisms, if $p_1,p_2\in \mathrm{ob}(P)$, then we start w... | {
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Prove that $A=\{(x,y)\in\mathbb{R}:(y\neq 0)\vee (x>0)\}$ is connected. I'm working on a multivariable integration problem and, at some point, I need to use Poincaré Lemma. To justify that I verify the Lemma's hypothesis conditions, I need to prove that the set
$$A=\{(x,y)\in\mathbb{R}:(y\neq 0)\vee (x>0)\}$$
is connec... | The space you presented is just $\mathbb{R}^2\setminus \{ (x,0) : x\leq0 \}$. That is you are removing from the real plane the half line starting at the origin and going through the negative part of the $x$ axis.
You can easily show it is path connected, thus it is connected.
It is also simply connected but from the p... | {
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"timestamp": "2023-03-29T00:00:00",
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An Interesting Property Concerning a Sequence of Integers A non-decreasing sequence of positive integers $a_1,a_2,\dots a_n\ (n\geq 3)$ is good if for each $3\leq k\leq n$ there are $1\leq i\leq j<k$ such that $a_i+a_j=a_k$.
Let $\ell,m$ be positive integers, and consider the set $[\ell]=\{1,2,\dots,\ell\}$. We say tha... | Partial solution: $f(3) \neq 13$ because $\{1,4,7,10,13\}, \{2,3,11,12\}, \{5,6,8,9\}$
Note the first part is $1 \pmod 3$, and we certainly cannot get a good sequence out of that. I wonder if that's a generally good approach...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3706029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to calculate average growth when it's negative? We have annual reports for company's revenue and can calculate annual growth as
$yg = {y_{i+1} \over y_i}$.
And then we can calculate the average monthly growth as $mg = ({y_{i+1} \over y_i})^{1 \over 12}$.
So for reports 2000-12 $1m and 2001-12 $2m the average month... | An example for negative growth rate: $y_0=100, y_1=80$
The growth rate from $t=0$ to $t=1$ is $g_{01}=\frac{80}{100}-1=0.8-1=-0.2$
So you can use the formula for growth rate no matter whether the growth rate is positive or negative:
$$g_{t,t+1}=\frac{y_{t+1}}{y_t}-1$$
Btw, the growth factor $1+g_{01}$ is still positiv... | {
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"timestamp": "2023-03-29T00:00:00",
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Compute an $n\times n$ determinant with factorial and powers of $x$ Compute
$$
D_{n}=
\begin{vmatrix}
1 & 0 & 0 & 0 & 0 & \ldots & 1\\
1 & 1! & 0 & 0 & 0 & \ldots & x\\
1 & 2 & 2! & 0 & 0 & \ldots & x^{2}\\
1 & 3 & 3\cdot2 & 3! & 0 & \ldots & x^{3}\\
\ldots & \ldots & \ldots & \ldots & \ldots & \ddots & \ldots\\
1 & n ... | Let $M_n$ be the matrix of interest (the argument of the determinant in your question). We claim that
$$M_n = A_n B_n, ~ A_n = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ 1 & 1! & 0 & \cdots
& 0 \\ 1 & 2 & 2! & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & n & n(n - 1) & \cdots & n! \end{bmatrix}, ~ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3706311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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If $A = [-5, 3)$ and $B = (1, \infty)$, is $3$ a member of $A \cap B$?
If $A = [-5, 3)$ and $B = (1, \infty)$, what is $A \cap B$?
Since $3$ is not a member of $A$, do we include it in the intersection?
| We do not include it in the intersection because it is not a member of $A$, so it is not a member of both sets, so it is not in the intersection.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For $G \sim G(n, 0.5)$, $I$ a $k$-set of vertices, and event $E_I = \{G[I] \cong K_k \text{ or } K_k^c \}$, how many events does $E_I$ depend on? The note in my lecture says that $E_I$ is independent of all events with disjoint edge-sets, and so it depends on at most ${k \choose 2}\left({{n-2}\choose {k-2}}-1\right)$ o... | The formula is an overestimate.
There are $\binom k2$ ways to pick two vertices in $I$. For each one of them, there are $\binom{n-2}{k-2}$ sets of size $k$ containing those two vertices and possibly other vertices in $I$. One of those sets of size $k$ is $I$ itself; if we don't count it, there are $\binom{n-2}{k-2} -1$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Countability of the set $(0,1)$ I am trying to prove that the set $(0,1)$ is uncountable from "A First Course in Analysis by Yau". I have a question about a particular step.
In the text, the result is proved by contradiction. It is supposed that the set $(0,1)$ is countable, which it is then written that there must exi... | There is a 1-1 correspondence between $A$ and $B$, if and only if a) there is an injection from $A$ to $B$ and b) at the same time there is an injection from $B$ to $A$.
So, if you can demonstrate that injection from $(0,1)$ to $\mathbb N$, then yes, you have demonstrated that $(0,1)$ is countable -- but more, you have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3706918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate $\mathbb E(Y^2\mid X)$
Let $X,Y$ be random variables with the total normal distribution such that $$\mathbb EX=\mathbb EY=0, \operatorname{Var} X=1, \operatorname{Var} Y=5, \operatorname{Cov}(X,Y)=-2$$Calculate $\mathbb E(Y^2\mid X)$
From this task I can also calculate
*
*$\mathbb E(XY)=\operatorname{Co... | $Y|X\sim N(\mu_Y+\rho\frac{\sigma_Y}{\sigma_X}(x-\mu_X);\sigma_Y^2(1-\rho^2))$
And
$\mathbb{E}[Y^2|X]=\mathbb{V}[Y|X]+\mathbb{E}^2[Y|X]$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How do you find the coefficient of $x$ in $(x + 1)^2$? I want to learn how can I find out the coefficient of the variable $x$ in the expression $(x + 1)^2$. It is a case of a perfect square expansion.
| This is the distributive property of the multiplication: $(a+b)c=ac+bc$.
What is $(x+a)^2$? Well,
$$(x+a)^2=(x+a)(x+a)$$
Let's write one of the factors $(x+a)=c$, and distribute:
$$(x+a)^2=(x+a)c=xc+ac$$
Replacing $c$ back to $x+a$ we get
$$(x+a)^2=(x+a)(x+a)=x(x+a)+a(x+a)$$
And we distribute again
$$(x+a)^2=x(x+a)+a(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Using three points to create a quadratic equation produces an equation, which doesn't seem to pass through the original points I've got three (x,y) points from which I am trying to create a quadratic equation, which are:
(2325, 5500)
(1880, 3700)
(1400, 2360)
Using those three points, I create three simultaneous equat... | You're simply using too few significant digits in the most important part, the coefficient $a$:
I have it as $0.0013548942$ to 8sf on an online solver. Note that the error to $0.0014$ is a full $3.3\%$, which is enough to see the deviations in your crosscheck.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3707511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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equivalent condition for a quadratic number field to have a solution of the equation $x^2+y^2=-1$ My question is related to the realization problem of the quaternion group $Q_{8}$.
Let $d$ be a positive square-free integer. It is well known that the unique irreducible 2-dimensional representation of $Q_{8}$ realizes ov... | When $d\equiv-1\pmod 8$, $K$ embeds into the $2$-adic numbers $\Bbb Q_2$, and in
$\Bbb Q_2$ the equation $x^2+y^2+1=0$ is insoluble (this boils down to congruences
modulo $8$).
Suppose $d\not\equiv-1\pmod 8$. Then the equation $x^2+y^2=-1$
is soluble in $K$ iff the quadratic form $X^2+Y^2+Z^2$ is isotropic over $K$.
By... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the minimum of $\sum_{i=1}^{n} d_{i}{x_{i}}^2$ where $d_{i}>0$ $\forall i=1, ... , n $, when $x$ is solution of $Ax=b$ For $A\in\mathcal M_{m\times n}(\Bbb R)$, let $x$ be solution of $Ax=b$ when $A$ has full row rank. Find the minimum of $\sum\limits_{i=1}^n d_{i}x_i^2$ where $d_i>0$ $\forall i\in\{1,\ldots,n\}$... | Consider the Lagrangian $\min_x x^TDx - \lambda^T (Ax-b)$, the problem is convex.
Differentiate with respect to $x$, we have
$$2Dx-A^T\lambda =0$$
$$x=\frac12D^{-1}A^T\lambda$$
Along with $Ax=b$, we have
$$\frac12 AD^{-1}A^T\lambda = b$$
Now, you can solve for $\lambda$ and then solve for $x$.
Alternatively,
Express... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How can we express that a partial order is more complete than another one? Suppose we have two partial orders $R$ and $I$ on $\mathbb{C}$ (conplex numbers) such that:
*
*$R$ is a total order on $\mathbb{R}$ (real numbers).
*$I$ is also a total order on $\mathbb{R}$ and, additionally, on $\mathbb{I}$ (imaginary numb... | From Partially ordered set - Wikipedia:
A partial order $\leq^*$ on a set $X$ is an extension of another partial order $\leq$ on $X$ provided that for all elements $x$ and $y$ of $X,$ whenever $x \leq y,$ it is also the case that $x \leq^* y.$ A linear extension is an extension that is also a linear (i.e., total) orde... | {
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How do you compute the value of the right derivative of $f(x)= \sin (x)^{\cos (x)} +\cos (x)^{\sin (x)}$ when $x=0$. How do you compute the value of the right derivative of $f(x)= \sin (x)^{\cos (x)} +\cos (x)^{\sin (x)}$ when $x=0$. I'm trying to learn calculus so some explanations wouldn't be so bad. I got stuck comp... | You should learn logarithmic differentiation:
$$h(x)={(\sin{x})}^{\cos{x}}$$
$$\ln{(h(x))}=\cos{x} \cdot \ln{\sin{x}}$$
$$\frac{h'(x)}{h(x)}=-\sin{x} \cdot \ln{\sin{x}}+\frac{\cos^2{x}}{\sin{x}}$$
$$h'(x)={(\sin{x})}^{\cos{x}} \left(-\sin{x} \cdot \ln{\sin{x}}+\frac{\cos^2{x}}{\sin{x}}\right)$$
Do this for $g(x)={(\cos... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Use the Chinese Remainder Theorem to determine the value of $x$. I'm trying to solve the following modular arithmetic question using the Chinese Remainder Theorem, using this link. (We learned a different method in our class, but I found this easier to grasp).
$$x \equiv 1 (\text{mod} \ 5)$$
$$x \equiv 2 (\text{mod} \ ... | There should be $x = (7 \times 9 \times 11)\cdot(7 \times 9 \times 11)^{-1}_5\cdot 1 $ ${}+ (5 \times 9 \times 11)\cdot(5 \times 9 \times 11)^{-1}_7\cdot 2 $ ${}+ (5 \times 7 \times 11)\cdot(5 \times 7 \times 11)^{-1}_9\cdot 3 $ ${}+ (5 \times 7 \times 9)\cdot (5 \times 7 \times 9)^{-1}_{11}\cdot 4$ for this approach.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Deriving the equation of a plane $Ax+By+Cz=D$. Defining a plane as the span of two linearly independent vectors, I've been trying to derive the equation $$Ax+By+Cz=D$$ without much success. The equation seems to indictate that a vector
$$\vec{v}=\begin{bmatrix} x \\ y \\ z\end{bmatrix}$$
is in the plane if and only if ... | The span of vectors contains the origin, but in general
$$A\,0+B\,0+C\,0\ne D.$$
A plane can be defined as the affine set
$$\vec p=\lambda\vec a+\mu\vec b+\vec c.$$
We can eliminate $\lambda,\mu$ by forming the dot product with $\vec a\times\vec b$,
$$\vec a\times\vec b\cdot\vec p=\vec a\times\vec b\cdot\vec c,$$
whic... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does $\left|A\right|=\left|\mathbb{N}\right|$ and $\left|\mathbb{N}\right|=\left|\mathbb{Z}^+\right|$ imply $\left|\mathbb{Z}^+\right|=\left|A\right|$ Suppose $A$ is a countably infinite set. Is it true that if $\left|A\right|=\left|\mathbb{N}\right|$ and $\left|\mathbb{N}\right|=\left|\mathbb{Z}^+\right|$, then $\left... | Yes: generally, if you know $\lvert A\rvert=\lvert B\rvert$ and $\lvert B\rvert=\lvert C\rvert$, you can conclude $\lvert A\rvert=\lvert C\rvert$ because if $f:A\to B$ is a bijection and $g:B\to C$ is a bijection, then $g\circ f:A\to C$ is a bijection.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3708973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Cauchy problem for PDE Cauchy problem $$y z_x-xz_y=0$$
and $ x_0(s)=cos (s),y_0(s)=sin(s),z_0(s)=1,s>0$
I use Lagrange's method
$$\frac{dx}{y}=\frac{dy}{-x}=\frac{dz}{0}$$
From 1st and 2nd
$x^2+y^2=c_1$ and from last relation $z=c_2$
So solution is of the type $z = f(x^2+y^2)$
By initial condition
$$1=f(cos^2s+sin^... | Thankyou very much I got your point
One more question : is there are condition which quickly tell that Cauchy problem has unique solution, no solution, or infinitely many solution without solving Cauchy problem ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3709106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Weak-* convergence in $L^{\infty}$ implies weak convergence in $L^p$ on bounded set In a lecture I found the following result: "We remark that when $\Omega$ is bounded the weak-* convergence of $u_{n}$ in $L^{\infty}(\Omega)$ to some $u \in L^{\infty}(\Omega)$ implies weak convergence of $u_{n}$ to $u$ in any $L^p$, $1... | I don't know of a reference but that might be because the result isn't so hard to see. Since $\Omega$ is bounded, we have that $L^q(\Omega) \subseteq L^{1}(\Omega)$ for every $1 \leq q \leq \infty$.
Now weak-$*$ convergence in $L^\infty(\Omega)$ means that for every $v \in L^1(\Omega)$,
$$\int u_n v dx \to \int u v d... | {
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Cone over $X$, equivalence relation I have a question about the definition of the cone over $X$.
If $X$ is a space, define an equivalence relation $X\times [0,1]$ by $(x,t)\sim (x',t')$ if $t=t'=1$. Denote the equivalence class of $(x,t)$ by $[x,t]$. The cone over $X$, denoted by $CX$, is the quotient space $X\times [... | The short answer to your question "which points are equivalent to $(x,1/2)$?": Only $(x,1/2)$.
The cone should really look like the geometric cone you know. For a generic example, start with a cylinder and identify the points at the top face. This is the motivation for the name cone for $CX$. Up to pathological example... | {
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Separated morphisms are stable under base change Suppose that the map $f$ in the following diagram is a separated morphism (i.e. $\Delta_{X/S}:X\rightarrow X\times_{S}X$ is a closed immersion). I want to prove that $p_{2}$ is also a separated morphism.
$$\require{AMScd}$$
\begin{CD}
X\times_{S}Y @>{p_{1}}>> X\\
@VV{p_... | I guess that your right horizontal $g$ in your first diagram is your $f: X \to S$ and your $p_2: X \times_S Y \to Y$ is the pullback of $f$ along horizontal $Y \to S$ (you called it also $f$ but in your first sentence you reserved $f$ for $X \to S$. Let call the horizontal arrow $Y \to S$ $h$. Clearly the diagram below... | {
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Don't understand how to solve line integral dot product I am trying to solve the integral:
$$\int_{C} \bf F \cdot dr$$
between $x=0,y=0,z=0$ and $x=0,y=1,z=1$ where $\textbf {F} = (0,y,1-y^2-z)$
and $C$ is $z=2y-y^2$
I have the solution however I do not understand why the "$-z$" is replaced by $C$. I know that it is ... | When we're solving line integrals, it is often useful to use parametrizations even if the parametrizations are obvious, as it is in your case. Your curve $C$ is given by $z = 2y-y^{2}$, so if $\gamma= \gamma(t)$ denotes your parametrized curve, you should have:
$$\gamma(t) = \begin{cases} \displaystyle x = 0 \\ \displa... | {
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"timestamp": "2023-03-29T00:00:00",
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Explanation on the limitation of the proof of the Law of Cosines Regarding this method of proving the Law of Cosine:
It is noted on that page that
This proof isn't perfect. We should have been worried about angles. This can be avoided by using directed angles.
I don't understand why we need to worry about the angle... | When $\pi/2 < \alpha < \pi$, we have an obtuse angle at $A$ and we must consider the geometry of the figure accordingly.
The altitude $h$ from $B$ to $AC$ is no longer "inside" the triangle. It extends to some point, say $B'$, on the line containing $AC$ such that $|B'C| > |AC|$; in other words, the signed distance $r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Triangles and inequalities Let ABC be a triangle and let O be any point in space. How can I show that $AB ^ 2 + BC ^ 2 + CA ^ 2 \leq 3 (OA ^ 2 + OB ^ 2 + OC ^ 2)$?
I know this prove by inner products, but is possible to show this by euclidean geometry?
| I'll use $P$ instead of $O$ through the course of this answer; $O$ is a little confusing because it usually denotes a centre.
Let $D$, $E$ and $F$ be midpoints of $BC$, $CA$, and $AB$ respectively; let $G$ be the intersection of $AD$, $BE$, and $CF$, i.e. the triangle's centroid. It is well known that $AG$ = $2GD$ (and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Describing a kind of probability scenario: chance of two heads over three coin flips A fair coin is tossed three times. What is the probability it will land on heads exactly twice?
We know that flipping a single coin has two mutually exclusive
outcomes, and that multiple flips constitute independent events.
This langua... | The language is the language of 'random variables' and 'probability distributions'. These are fundamental ideas in probability theory.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a "non-curved" manifold? I've seen some visualizations of manifolds. It seems that they are all "curved" shapes. Is there a "non-curved" manifold?
| Yes, you can get a flat, compact manifold embedded in $\Bbb R^3$. This is a consequence of the Nash embedding theorem. In particular, a couple of years ago, this was done in practice with a torus, and the results are quite visually interesting. Here are the first three steps in the construction:
It may look curved, b... | {
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Find an angle in the given quadrilateral In the following problem, I want to find the angle marked as $x$. It seems so simple and yet I am out of ideas. It is very easy to get all angles except two of them: angle ADB and angle CBD.
Is there a calculation for the angle $x$ that only uses parallel lines and no circles?... |
Given:
1) $\angle ABC=30^\circ$
2) $\angle BAD=80^\circ$
3) $\angle DAC=20^\circ$
4) $\angle ACB=50^\circ$
5) $\angle BCD=50^\circ$
1) (the key point of this solution) Let $F\in AB$ such, that $\angle BCF$ $=\angle FBC=30^\circ$, $\angle ACF$ $=\angle CAD=20^\circ$
2) $\angle ADC $ $= 180^\circ-\angle DAC... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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An increasing sequence whose terms contain only odd digits Consider the increasing sequence: $13579, 13597, \dots,199153773,\dots$, where every term contains all (and only) the digits $1,3,5,7,9$ (every digit must appear at least once in every term, so repetition is allowed).
What is the $1992^\text{nd}$ term in the se... | $1992^{nd}$ term is 1137597.
Index of term 199153773 is 306430.
def isValidNumList(numList):
for x in [1, 3, 5, 7, 9]:
if not x in numList:
return False
return True
def nextNumList(numList):
numList.reverse()
valid = False
while not valid:
overflow = 1
for i in r... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof to extending a dynamical system with discrete time to one with continuous time Let $S$ be a dynamical system on a metric space $X$ with discrete time $\mathbb{N}_0$.
In our script we have a theorem that says one can extend such a system to one, here called $\tilde{S}$, with continuous time $[0,\infty)$ on a larg... | You construct an equivalence relation on $X\times [0,1]$ so that $(x,1)$ is equivalent to $(f(x),0)$ where $f=S(1,\cdot)$ is the step-1 map of the dynamical system $S$. Then define $$\tilde S(t,(x,s))=(f^n(x),\alpha)$$ where $s+t=n+\alpha$, $n\in\Bbb N_0$ and $\alpha\in[0,1)$.
Adapt to your notation convention if I gue... | {
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Index of commutator subgroup in the commutator group I need to prove (or find a counter example) that if $G$ is solvable and $H \leq G$ is a subgroup of finite index, then the commutator subgroup $D(H)$ is also a subgroup of finite index in $D(G)$. This would allow me to say :
If $G_1$ and $G_2$ are commensurable (i.e... | This MathOverflow question answers your question about nilpotent groups in the positive, by induction on the Hirsch length.
https://mathoverflow.net/questions/107679/index-of-derived-subgroup-in-derived-group
| {
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$ f(x)=\begin{cases}\sin x& \text{if $x$ is rational}\\{\cos x} &\text{if $x$ is irrational}\end{cases} $, finding continuous point Problem: Where the following function continuous?$$f(x)=\begin{cases}\sin {x}& \text{$x\in\mathbb Q$}\\ \cos{x} &\text{$x\notin\mathbb Q$}\end{cases} $$
We can predict that $f$ might be co... | $f$ is continuous at a point $p$ if and only if for every convergent sequence $p_n$ such that $\lim p_n = p$, $\lim f(p_n) = f(p)$. Now we can approach a point $p$ by both rational sequences and irrational sequences. So given $p$, there is a rational sequence $p'_n$ and an irrational sequence $p_n''$ both of which conv... | {
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$f(x)= \sqrt{\frac{(x+1)^3}{x}}$ find constant values $a,b,c \in\mathbb{R}$ such that $f(x)=ax+b+\frac{c}{x}+o(\frac{1}{x})$ when $x \to +\infty$ Would anyone be kind to explain to me how to do this types of problems?
My proffessor gave us couple of this problems for homework and we dont have solutions. This is my firs... |
I thought that it might be instructive to present a way forward that forgoes use of calculus and relies instead on elementary, pre-calculus tools only. To that end, we proceed.
For $x>-1$, we can write
$$\sqrt{\frac{(x+1)^3}{x}}=(x+1)\sqrt{1+\frac1x}\tag1$$
As $x\to \infty$, $t=1/x \to 0$. So, let us examine the b... | {
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"source": "stackexchange",
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What's $-2^{-2^{-2^{-2...}}}$? Pretty simple question that I know someone has probably asked before:
What's $-2^{-2^{-2^{-2...}}}$? Or, more specifically, what is the number that this repeated sequence approaches? I have calculated it out to about $-0.641185$, but I have no idea what that number is or where you could... | Recursively, we might define this as the limit as $n \to \infty$ of
$$a_n = -(2)^{a_{n-1}}$$
with $a_1 = -2$. Letting $n \to \infty$ on the assumptions of continuity and convergence of $a_n$ to some value $L$ gives us
$$L = -2^L$$
This value $L$ is fundamentally what you seek. This equation will have to be solved by me... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3711763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find solution set of $\dfrac{8^x+27^x}{12^x+18^x}=\dfrac{14}{12}$ What I've done is factoring it.
$$\dfrac{2^{3x}+3^{3x}}{2^{2x}\cdot 3^{x}+3^{2x}\cdot{2^{x}}}=\dfrac{7}{2\cdot 3}$$
This looks like it can be factored more but it doesn't work from my attempts.
| $$\dfrac{2^{3x}+3^{3x}}{2^{2x}\cdot 3^{x}+3^{2x}\cdot{2^{x}}}=\dfrac{7}{2\cdot 3}$$
$$3\cdot 2^{3x+1}+2\cdot 3^{3x+1}=7\cdot 2^{2x}\cdot 3^{x}+7\cdot 3^{2x}\cdot 2^{x}$$
Divide both the sides by $2^{3x+1}$, we get
$$3+3 \left(\frac{3}{2}\right)^{3x}=\frac72 \left(\frac{3}{2}\right)^{x}+\frac72 \left(\frac{3}{2}\right)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3711923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Simplifying: $\sum_{n=0}^\infty\sum _{i=0}^n \left(\frac{a}2\right)^{2n}\frac1{i!n!(n-i)!}\cdots$
I'm trying to simplify the following infinite sum:
$$\sum_{n=0}^\infty\sum_{i=0}^n \left(\frac{a}2\right)^{2n}\frac1{i!n! (n-i)!}\frac{\partial^{2n}f(x,y)}{\partial x^{2i}\, \partial y^{2(n-i)}},\ \ a,x,y\in\Bbb R$$
Where... | From
$$\sum_{n=0}^\infty\frac{\left(\frac{a}2\right)^{2n}}{(n!)^2}\sum_{i=0}^n \binom{n}i\frac{\partial^{2n}f}{\partial x^{2i}\, \partial y^{2(n-i)}}$$
We can identify
$$\Delta^n(f)=\sum_{i=0}^n \binom{n}i\frac{\partial^{2n}f}{\partial x^{2i}\, \partial y^{2(n-i)}}$$
where $\Delta^n$ is the Laplacian composed $n$ time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3712064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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For $A, B \subset \mathbb{R}^+$, $\sup(A \cdot B) = \sup A \sup B$. I am trying to prove that:
For nonempty subsets of the positive reals $A,B$, both of which are bounded above, define
$$A \cdot B = \{ab \mid a \in A, \; b \in B\}.$$
Prove that $\sup(A \cdot B) = \sup A \cdot \sup B$.
Here is what I have so far.... | Hint:
Rather than $\sup A - \varepsilon$ and $\sup B - \varepsilon,$ subtract appropriate multiples of $\varepsilon$ from $\sup A, \sup B$ respectively. You'll need to assume that $\varepsilon$ isn't too big.
Full proof:
[I'm sorry, I can't get the wretched spoiler mechanism to work, so I'm afraid you'll have to avert ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3712256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Combinatorics question where someone has to be at least one seat away from anyone else? In a doctor’s waiting room, there are 14 seats in a row. Eight people are waiting to be seen.
There is someone with a very bad cough who must sit at least one seat away from anyone else. If all arrangements are equally likely, what ... | There are $8 \cdot \frac{14!}{8! \cdot 6!}$ configurations possible in total, where the added $8$ arises from the fact that person 8 is 'distinguishable'.
If person 8 sits on the edges, this leaves $\frac{12!}{7! \cdot 5!}$ configurations open for the other people to sit ($12 = 14 - 1 - 1$, where $1$ is the edge seat a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3712414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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what is $\dim \{f\in P_n|T(f)=f\}$ Let $P_n$ be polynomials of degree $\leq2n$ on $\mathbb{R}$.Clearly $P_n$ is a finite dimensional vector space.Let
$$
T(f)=\int_{\mathbb{R}}e^{(x^2-y^2)\pi}\cos(2\pi yx)f(y)dy
$$
It's easy to show that $T$ maps $P_n$ to itself.
The problem is what is $\dim \{f\in P_n|T(f)=f\}$.
My i... | Write $ \mathcal{F}f(x) = \int_{\mathbb{R}} e^{-2\pi i x y} f(y) \, \mathrm{d}y $ for the Fourier transform, and let $\mathcal{M}f(x) = e^{-\pi x^2}f(x)$. Also let
$$ E_n = \{ f \in P_n : \text{$f$ is even} \}. $$
Then it is easy to check that:
*
*If $f \in P_n$ solves $Tf = f$, then $f \in E_n$.
*For $f \in E_n$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3712528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Group element that normalizes finite subgroup that is generated by a subset of $G$ Let $N$ be a finte subgroup of a group $G$, and assume $N = \langle S \rangle$ for some subset $S$ of $G$. Prove that an element $g \in G$ normalizes $N$ if and only if $gSg^{-1} \subset N$.
My question is about the forward direction. Th... | If $S$ generates $N$, then each element of $N$is of the form $s_1\ldots s_n$, where each $s_i$ is either an element of $S$ or the inverse of an element of $S$ or the identity element of $G$. In particular, $S$ is contained in $N$. Hence if an element $g$ of $G$ normalizes $N$ then in particular it normalizes every ele... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3712700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find the number of solutions of $6|\cos x|=x$? Now, I think the only way to solve this problem for a high school student was graphically.
However using pen and paper to draw the graph, it was virtually impossible to justify or refute the existence of the "Fourth" solution.
Using desmos, I realised that we must m... | [Using some calculus]
It's enough to consider $3\pi/2 \lt x \lt 2\pi$. In that range the cosine is positive, so we can dispense with the absolute value and consider the function $f(x) = x - 6\cos(x)$, show that its minimum is positive and that its second derivative is positive.
To get the minimum, we set the derivativ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3712823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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A group of order PQ, both primes with Q>P (help solution) I want to prove that if G has order pq, primes, with q>p and p DOES NOT divides q-1, then G is cyclic.
My attempt:
By Sylow's theorem,$n_p \equiv 1 (\textrm{mod}\ p)$ and $n_q \equiv 1 (\textrm{mod}\ q)$, and more, $n_p \equiv 0 (\textrm{mod}\ pq)$ and $n_q \equ... | This is correct, and the normality of the Sylow subgroups is where the condition that $p$ not divide $q-1$ comes in. If $p$ does divide $q-1$, then we can have $q$ subgroups of order $p$ and a unique nonabelian group structure arises out of this.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3713009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finitely generated projective resolution Let $K$ be a field, $A$ be a finite dimensional $K$-algebra and $M$ be a finitely generated $A$-module. Is it true that $M$ admits a projective resolution by finitely generated projective $A$-modules?
| As $A$ is a finite algebra over $K$ it is noetherian. As $M$ is finitely generated there is a surjection $A^{\oplus n} \longrightarrow M$. $A^{\oplus n}$ is noetherian as $A$ is. Let $N$ be the kernel of this map. By noetherianness, it is finitely generated, so there is a surjection $A^{\oplus m} \longrightarrow N$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3713163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Proving that $2+\sqrt{2}$ is irreducible in $\mathbb{Z}[\sqrt{2}]$. I'm asked to show that $x=2+\sqrt{2}$ is irreducible in $\mathbb{Z}[\sqrt{2}]$ by using the norm map $$N:\mathbb{Z}[\sqrt{2}]\rightarrow \mathbb{Z}^+:a+\sqrt{2}b\mapsto |a^2-2b^2|$$
Now, if $x=yz$, then $2=N(x)=N(y)N(z)$ forcing wlog $N(y)=1$. I'm now ... | Use the definition of the norm. If $y=c+d\sqrt{2}$ then $N(y)=(c+d\sqrt{2})(c-d\sqrt{2})=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3713331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Find three sets $A$, $B$, $C$, each of them non-empty, such that $(A\cap B)\cup C=A\cap(B\cup C)$ and $(A\cap B)\cup C\neq A\cap(B\cup C)$ I need to find three sets for both statements I have above. I have tried drawing Venn diagrams and shading appropriately then adding numbers in each shaded region to try and guess a... | $$B \subset C \subset A$$
$$A \subset B \subset C$$
For general view is good to write right hand side as:
$$(A\cap B)\cup C=(A \cap B) \cup(A \cap C)$$
now you see, that difference between sides is difference between sets from right hand of union: on left you have $C$ on right $(A \cap C)$ and on this way you can creat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3713443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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To find $A$ given that $2I + A +A^2 = B$ where $B$ is given. How to find a matrix $A$ such that the following holds:
$$2I + A +A^2 = B,$$ where the matrix $B$ is given. I tried with char poly of $B$ but not getting any idea.
Note that it is also given that $B$ is invertible.
P.S. $B = \begin{pmatrix}-2&-7&-4\\ \:12&22&... | Here is an ad hoc method:
It is straightforward (if tedious) to find null spaces of
$B-2I, B-4I, (B-4I)^2$ and determine the Jordan form.
With $V=\begin{bmatrix} 3 & -1 & 1 \\ -6 & 1 & 0 \\ 6 & -1 & -1 \end{bmatrix}$ we see that
$V^{-1}BV = \begin{bmatrix} 4 & 1 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 2 \end{bmatrix}$, and since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3713545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Interval of convergence and integration of a power series The $\arctan(x)$ can be expanded as a MacLaurin series starting from the integral
$$\arctan(x) = \int \frac{1}{1 + x^2} \mathrm{d}x$$
and using
$$\frac{1}{1 + x^2} = \sum_{n = 0}^{\infty} (-1)^n x^{2n}$$
as suggested in this answer. This series converges for $x ... | In general, you won't be able to say things about endpoints of an interval after integrating.
In this case, however, we can just check that the sum converges at the endpoints directly; if $x = \pm 1$, then $x^{2n+1} = x$, so the sum becomes
$$x \sum_{n=0}^\infty \frac{(-1)^n}{2n+1},$$
which converges because $\frac{(-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3713705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Floor function of a product I'm reading a book about proofs and I'm currently stuck in this problem.
Prove that for all real numbers $x$ and $y$ we have that:
$$\lfloor x\rfloor \lfloor y\rfloor \leq \lfloor xy\rfloor \leq \lfloor x\rfloor \lfloor y \rfloor + \lfloor x \rfloor + \lfloor y \rfloor$$
I though I could do ... | It can help to write $x=a+r$ where $a$ is an integer and $0\leq r <1$, so that $\lfloor x \rfloor = a.$
Similarly, let $y=b+s$ where $b$ is an integer and $0\leq s <1.$
Then $$\lfloor x \rfloor \lfloor y \rfloor = ab$$ and
$$\lfloor x y \rfloor = \lfloor ab+as + br +sr \rfloor = ab +\lfloor as + br +sr \rfloor.$$
(As... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3714040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Asymptotic behavior of recursive sequence Suppose, the real sequence
$x_{n+1}=\frac{1}{2}(x_n+\sqrt{x_n^2+c})$ with c>0 is given.
Find the asymptotic behavior of this sequence.
I have shown, that this sequence goes to infinity as $n\to\infty$ per contradiction. I guess, that it holds
$x_n \approx \frac{1}{2}\sqrt{cn}$
... | If we set $y_n=x_n^2$ then
$$y_{n+1}=\frac{y_n}2+\frac c4+\frac{y_n}2\sqrt{1+\frac{c}{y_n}}=y_n+\frac{c}2+O(y_n^{-1}).$$
This proves that $(y_n)$ grows at least linearly, so that $y_n^{-1}=O(n^{-1})$.
Therefore
$$y_n=\frac{cn}2+O(\ln n)$$
and
$$x_n=\sqrt{\frac{cn}2}+O\left(\frac{\ln n}{\sqrt n}\right).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3714229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving $A - B \subset A - (B - A)$ I believe I have been able to prove that for sets $A$ and $B$, $A - (B - A) \supset A - B$, but my proof is not particularly elegant. I was hoping someone knew of a more clever or straightforward way to show this. My proof is:
Let $x \in A - B$. Then $x \in A$ and $x \not \in B$. S... | If $C \subset D$, then we have $A - D \subset A-C$.
We have $B - A \subset B$, hence $A- B \subset A - (B-A)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3714375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Generalisation of a former question concering holomorphic continuation of $z\mapsto \dfrac{1}{z^k}$ Let $\mathbb{D} = \{ z\in \mathbb{C}: |z| < 1\}$ be the unit disk.
I want to show that for any $k\in\mathbb{N}$, there is no holomorphic function $f$ which extends continuously to $\partial \mathbb{D}$, such that
$$f(z) ... | First proof:
Let $h(z)=z^kf(z)-1$. Then $h$ is holomorphic in $\mathbb{D}$, continuous on $\mathbb{D}\cup\partial\mathbb{D}$ and $h(\partial\Bbb{D})=0$. Applying the maximum modulus principle, we obtain that $h\equiv 0$, which implies that $\lim_{z\to 0}f(z)=\infty$, contradicting the fact that $f$ is holomorphic near ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3714530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the trace of this object non-negative? $\phi$ is an arbitrary superposition of powers of a single hermitian matrix, $M^k$:
$$\phi =\sum_{k} \alpha_{k} M^{k}$$
Why is the following statement true?
$$\langle \mathrm{tr}\left(\phi^\dagger \phi \right)\rangle \geq 0$$
I have recently asked a related question, Why is... | Actually if $M$ is any matrix we have $\text{tr(M*M)} \geq 0$. This is clear if $M$ is diagonalizable and diagonalizable matrices are dense in all matrices.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3714708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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A divergent Definite Integral I am trying the study a definite integral
$$\int_{0}^{\pi}\frac{1}{(x-\frac{\pi}{2})^3+\cos{x}} dx$$
It is a divergent integral, but I am struggling to show that fact. Since the discontinuous point is at $\frac{\pi}{2}$, so
$$\int_{0}^{\pi}\frac{1}{(x-\frac{\pi}{2})^3+\cos{x}} dx= \lim_{a ... | One approach may be to change variables to $u = x-\pi/2$
$$
\begin{split}
\int_0^\pi \frac{dx}{(x-\frac{\pi}{2})^3+\cos{x}}
&= \int_{-\pi/2}^{\pi/2} \frac{du}{u^3+\cos(u + \pi/2)} \\
&= \int_{-\pi/2}^{\pi/2} \frac{du}{u^3-\sin u} \\
&= 2 \int_0^{\pi/2} \frac{du}{u^3-\sin u},
\end{split}
$$
where the last step is bec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3714849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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volume of solid generated by the regin bounded by curve $y=\sqrt{x},y=\frac{x-3}{2},y=0$ about $x$ axis
Using sell method to find the volume of solid generated by revolving the region bounded by $$y=\sqrt{x},y=\frac{x-3}{2},y=0$$ about $x$ axis, is (using shell method)
What I try:
Solving two given curves $$\sqrt{x}... | So I would instead split this up into two integrals:
$$\pi\int_0^3{(\sqrt{x})^2}dx + \pi\int_3^9{(\sqrt{x})^2-\left(\frac{x-3}{2}\right)^2}dx$$.
Using the shell method:
$$\int_0^3{2\pi y(2y+3-y^2)}dy$$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3714995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Hypothesis testing with an exponential distribution I have the following problem:
Given the data $X_1, X_2, \ldots, X_{15}$ which we consider as a sample from a distribution with a probability density of $\exp(-(x-\theta))$ for $x\ge\theta$.
We test the $H_0: \theta=0$ against the $H_1: \theta>0$. As test statistic $... | You know that the test statistic under the null hypothesis has distribution
$$T\sim \operatorname{Exp}(n)=\operatorname{Exp}(15)$$
The weight of the tail of this distribution is
$$\operatorname{P}(T>t)=\exp(-15t)$$
We reject null at significance level
$$\alpha \leq \operatorname{P}(T>t)=\exp(-15t)$$
p-value is maximum ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3715139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Expanding linear factors of a polynomial Expanding
$$a(x-r_1)(x-r_2)\cdots (x-r_n)$$
should give
$$ax^n-a(r_1+r_2+\cdots r_n)x^{n-1}+a(r_1 r_2+r_1 r_3+\cdots r_{n-1}r_n)x^{n-2}+\cdots (-1)^{n}ar_1 r_2\cdots r_n$$
but I fail to prove it. I was only able to do cases $n=1$, $n=2$ and $n=3$ (and even the result for $n=3$ s... | Let us first examine some examples to guess the general case:$$(x-r_1)(x-r_2)=x^2-(r_1+r_2)x+r_1r_2$$ $$(x-r_1)(x-r_2)(x-r_3)=x^3-(r_1+r_2+r_3)x^2+(r_1r_2 + r_1r_3+r_2r_3)x-r_1r_2r_3.$$So, we can guess the following identity:$$\prod_{i=1}^n(x-r_i)=\sum_{k=0}^n\sum_{1 \le j_1 \lt ... \lt j_k \le n}(-1)^kr_{j_1} ... r_{j... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3715254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all the endomorphisms of the multiplicative group $\mathbb R^+$. I am looking for the set $End(\mathbb R^+)$i.e. the set of all endomorphisms from $\mathbb R^{+}$ to itself where the operation is multiplication.Can someone help me to find them explicitly.I doubt whether they can be found explicitly or are existent... | $\mathbb R^+$ is isomorphic to the additive group $\mathbb R$ via the logarithm. This is a $\mathbb Q$-vector space. Any endomorphism of this as a group will be $\mathbb Q$-linear, since it is $\mathbb Z$-linear. Thus the set of endomorphisms is all $\mathbb Q$-linear maps. You won't find an explicit description of all... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3715438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is a group of prime-power order always abelian? Let $G$ be a group of order $p^n$, with $p$ prime. By Sylow's first theorem, there exists at least one subgroup of order $p^n$ (the number of subgroups of order $p^i$ is $1$ mod $p$ per $i$). The subgroups with order $p^n$ are all Sylow-$p$ groups.
Now, by Sylow's third ... | You can construct a nonabelian group of order $p^n$, for $p$ an odd prime, $n\gt2$, by selecting a nontrivial homomorphism $\varphi:\Bbb Z_p\to\rm{Aut}(\Bbb Z_{p^{n-1}})\cong\Bbb Z_{p^{n-1}-p^{n-2}}$. Let $G=\Bbb Z_{p^{n-1}}\rtimes_\varphi\Bbb Z_p$.
If, on the other hand, $p=2$, consider dihedral groups.
| {
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"url": "https://math.stackexchange.com/questions/3715585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Showing that disjoint intervals implies independent number of arrivals for random point set If we have a number of i.i.d. random variables, $X_1, X_2, ..., X_z$ (where $z$ is the realisation of a random variable $Z\sim Po(\lambda)$ independent of each $X_i$) with pdf $f$ that form a random point set, then I want to sho... | The point process you are considering is not a Poisson point process (PPP) but rather a binomial point process (BPP). Indeed, for a PPP, $N(A)$ is Poisson distributed while in your case $N(A)$ is a binomial random variable. In particular, even if $A_1$ and $A_2$ are disjoint, $N(A_1)$ and $N(A_2)$ are dependent since
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3715756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
} |
Let $x$ be an eigenvector of $A.$ Is it true that if $x^{\perp}$ is invariant under $A,$ then $A$ is normal? Let $A$ be an $n \times n$ matrix with entries in $\mathbb C,$ and let $x$ be an eigenvector of $A.$ If $x^{\perp}$ is invariant under A, is it true that $A$ is normal?
Here is my idea.
Let $\lambda$ be an eige... | This is false. $x=(1,0,0)^T$ is an eigenvector of $A=\pmatrix{0&0&0\\ 0&0&1\\ 0&0&0}$ and $x^\perp$ is $A$-invariant, but $A$ isn't normal.
The statement can be corrected by requiring that $x^\perp$ is $A$-invariant for every eigenvector $x$ of $A$. The corrected statement can be proved by mathematical induction on the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3715855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the domain and range of $f(x) = \frac{x+2}{x^2+2x+1}$: The domain is: $\forall x \in \mathbb{R}\smallsetminus\{-1\}$
The range is: first we find the inverse of $f$:
$$x=\frac{y+2}{y^2+2y+1} $$
$$x\cdot(y+1)^2-1=y+2$$
$$x\cdot(y+1)^2-y=3 $$
$$y\left(\frac{(y+1)^2}{y}-\frac{1}{x}\right)=\frac{3}{x} $$
I can't find t... | Why? Just write the function as follows:
$$f(x) = \frac{(x+1)+1}{(x+1)^2} = \frac{x+1}{(x+1)^2}+ \frac{1}{(x+1)^2}
$$
later,
$$f(x)= \frac{1}{(x+1)}+\frac{1}{(x+1)^2}=u+u^2 = (u+0.5)^2-0.25 = \left(\frac{1}{x+1}+0.5\right)^2-0.25$$
Since here you should be able to continue, just see the variation of $x$ and transform... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3715987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
} |
Eigenvalues of offset multiplication tables Consider the $n$ x $n$ 'multiplication table' constructed as (using Mathematica language)
$$
M_n^s = \text{M[n_,s_]:=Table[ k*m , {k,1+s, n+s}, {m, 1+s, n+s} ] }
$$
For example,
$$M_4^0 = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\3 & 6 & 9 & 12 \\4 & 7 & 12 & 16 \end... | All lines are linearly dependent (construction with entries $k*m$), so there is only one non zero eigenvalue.
This eigenvalue is the trace of the matrix (theorem that sum of eigenvalues is trace) so $\lambda=\sum_{j=1}^n (s+j)^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3716140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maximizing the probability of Bernoulli Sums Suppose I have $X_1,\cdots X_n$ with $X_i \sim \text{Bernoulli}(p_i)$, here we only assume $X_1,\cdots X_n$ are independent but not neccessarily identical so there are n degree of freedom.
Now, I want to show that the probability that
$$\mathbb{P}(\sum_{i=1}^n X_i = m)$$ i... | Your question does not make sense. Consider these changes to you exercise and tell me if it can be what you are looking for:
*
*the $n$ Bernulli's iid rv $X_i$
*$$Y= \sum_i X_i \sim Bin(n;p)$$
So Y is again a rv taking values in $y=\{0;1;2;...;n\}$. Let's suppose that the result of the sum is a fixed $0 \leq m \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3716308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why does $x=e^t+2e^{-t},y=e^t-2e^{-t}$ plot to a straight line? This parametrization satisfies $x^2-y^2=8$, so I was expecting a hyperbola. But what I got was a straight line. Why though?
https://www.wolframalpha.com/input/?i=parametric+plot+%28e%5Et%2B2e%5E%28-t%29%2Ce%5Et-2e%5E%28-t%29%29
EDIT- I tried a different ra... | Try to make the plot for $t$ between for example -3 and 3, the you should see it. Also, the point $(\sqrt 8,0)$ should be on there, not $(8,0)$.
Regarding the comments, it does not give a straight line, since the constant term depends on $t$, which varies. However, the reason why it looks like a straight line is that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3716516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving: $\lim_{x\to 0}\left(\frac{\pi ^2}{\sin ^2\pi x}-\frac1{x^2}\right)=\frac{\pi ^2}3$ without L'Hospital Evaluating
$$\lim_{x\to 0}\left(\frac{\pi ^2}{\sin ^2\pi x}-\frac{1}{x^2}\right)$$
with L'Hospital is so tedious. Does anyone know a way to evaluate the limit without using L'Hospital? I have no idea where to ... | Well, you've got your answer, and it's a good one, I'd use series expansions always in such case, but then, the answerer couldn't know you've ever heard of those expansions, and some of your comments show you aren't too familiar with them. That's why SE encourages sharing information about your mathematical background,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3716619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Factorize $(a+b+c)^5-(a+b-c)^5-(a+c-b)^5-(b+c-a)^5$ We can see that $a=0$ makes expression $0$, similarly others make expression $0$.This implies that its factorization has $abc$ in it.I substituted $a+b+c=s$ tried to find remaining factors.From here I did not progress any further
Later I went on finding this on wolfra... | Let $a+b=x,\,a-b=y$ \begin{align*}
(a + b + c)^5 - (a + b - c)^5 - (a + c - b)^5 - (b + c - a)^5=\\
(x+c)^5+(c-x)^5-(y+c)^5+(y-c)^5=\\
c^5 + 5 c^4 x + 10 c^3 x^2 + 10 c^2 x^3 + 5 c x^4 + x^5+\\
c^5 - 5 c^4 x + 10 c^3 x^2 - 10 c^2 x^3 + 5 c x^4 - x^5+\\
-c^5 - 5 c^4 y - 10 c^3 y^2 - 10 c^2 y^3 - 5 c y^4 - y^5+\\
-c^5 + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3716780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Let $G$ be a group. Let $x,y,z \in G$ such that $[x,y]=y$, $[y,z]=z$, $[z,x]=x$. Prove that $x=y=z=e$. Let $G$ be a group. Let $x,y,z \in G$ such that $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ (the commutators; $[x,y]=xyx^{-1}y^{-1}$). Prove that $x=y=z=e$.
I tried to show it by proving that $zx^mz^{-1}=x^{2m}$ with induction. T... | We have
$$xyx^{-1}y^{-1}=y$$
so
$$xyx^{-1}=y^2$$
Similarly
$$yzy^{-1}=z^2$$
and
$$zxz^{-1}=x^2$$
Note also that since $xyx^{-1}y^{-1}=y$, we have
$$yx^{-1}y^{-1}=x^{-1}y$$
so
$$yxy^{-1}=y^{-1}x$$
Thus
$$yzxz^{-1}y^{-1}=z^2y^{-1}xz^{-2}=y^{-1}xy^{-1}x$$
hence
$$yzxz^{-1}y^{-1}x^{-1}=y^{-1}xy^{-1}$$
so that
$$yzxz^{-1}x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3716955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
solving this probability paragraph Let $B_n$ denotes the event that n fair dice are
rolled once with $P(B_n)=1/2^n$ where n is a natural number.
Hence $B_1,B_2,B_3,..B_n$are pairwise mutually exclusive events as n approaches infinity.
The event A occurs with atleast one of the event
$B_1,B_2,B_3,..B_n$ and denotes tha... |
probability that greatest number on the dice is 4 if three dice are known to have been rolled is $37/216$
This is about a conditional probability $|B_3$. We have $P=P’-P’’$, where $P’=\left(\frac 46\right)=\frac {64}{216}$ is a probability that the greatest number on a dice is at most $4$ and $P’’=\left(\frac 36\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3717071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Suppose M is a finitely generated non-zero R-module, where R is a commutative unital ring. Show that the tensor product of M with itself is non-zero. Suppose M is a finitely generated non-zero R-module, where R is a commutative unital ring. Show that the tensor product of M with itself is non-zero.
I know one way to sh... | Not sure this is the best proof, but here goes. Note that $M\otimes_R M = 0$ if and only if $(M\otimes_R M)_{\mathfrak p} = 0$ for all primes $\mathfrak p\subset R$ (Atiyah-MacDonald 3.8). But
$$
(M\otimes_R M)_{\mathfrak p}\cong M_\mathfrak p\otimes_{R_{\mathfrak p}}M_{\mathfrak p}
$$
by Atiyah-MacDonald 3.7.
Hence we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3717144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Rouché's theorem in annulus $1<|z|<2$ I have to find number of roots of the polynomial $p(z)=z^4-8z+10$ in the annulus $1<|z|<2$ .
I'm trying to do this using Rouché's theorem.And, by this theorem , I have that $p$ does not have zeros in $|z|<1$ , which means that number of zeros in the given annulus is the same as the... | We plan to use the version of Rouché's theorem which says that if $|g(z)| < |f(z)|$ on the boundary of our region then $f$ and $f+g$ have the same number of zeros with multiplicty in the region.
Let's first set a goal for ourselves. If we can show that $|p(z)| > 3$ for $|z| = 2$, then Rouché's theorem tells the number ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3717307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Signature and Environment in Type Theory Signature and Environment are both related to the description of constants. I feel confused about the two notions in type theory. Could anyone explain their main difference? Thanks!
| Functionally, environment (or contexts) and signatures behave quite similarly, and in some settings you can emulate signatures using environments to place the axioms as variables in the context. However, they should be thought of differently.
An environment or context is typically a list of typed variables, e.g. $x_1 :... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3717470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Name of the rule allowing the exchanging $\sin$ and $\cos$ in integrals with limits $0$ and $\pi/2$? As in $0$ to $\frac{\pi}{2}$ limits the area under curve of $\sin \theta$ and $\cos \theta$ are same, so in integration if the limits are from $0$ to $\frac{\pi}{2}$ we can replace $\sin \theta$ with $\cos \theta$ and v... | This is just plain wrong. Indeed, if you evaluate your original integral numerically, you get a negative answer.
What is correct is this: For any continuous function $f(x,y)$, it is the case that
$$\int_0^{\pi/2} f(\sin\theta,\cos\theta)\,d\theta = \int_0^{\pi/2} f(\cos\theta,\sin\theta)\,d\theta.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3717583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 4
} |
Find the condition where the product of two factorials is minimized I have to find the maximum value of a combination ${N \choose H} = \dfrac{N!}{H!(N-H)!}$ where $1 \leq H \leq N \leq 16$. My reasoning was to find the maximum value you have to maximize the numerator $N!$ and minimize the denominator $H!(N-H)!$. The ma... | From the complementary combination $\displaystyle\binom{N}{H} = \binom{N}{N-H}$ we conclude that
$\displaystyle\binom{N}{H}$ is symmetric with respect to $\displaystyle H=\bigg\lceil{\frac{N}{2}}\bigg\rceil$
Now we show that $\displaystyle\binom{N}{H}$ is monotonically increasing for $\displaystyle 0\le H \le \bigg\lce... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3717719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show that $2^{\aleph_0}\neq \aleph_{\alpha+\omega}$ for any ordinal $\alpha$. Show that $2^{\aleph_0}$ $\neq$ $\aleph_{\alpha+\omega}$ for any ordinal $\alpha$.
What I did was the following:
I first used ordinal addition where $\alpha+\omega$ = $\sup\{\alpha+n:n \in \omega\}=\sup \: \omega$ = $\omega$. Thus, $\aleph_{\... | You got zero points, as you should have.
It is true that $\alpha+\omega=\sup\{\alpha+n\mid n<\omega\}$ the rest is absolutely false. Note, for example, that $\omega_1+n$ is uncountable, for any $n<\omega$, but you are claiming that $\sup\{\omega_1+n\mid n<\omega\}$ is a countable ordinal. How is that even possible?
Wha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3717862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maximal ideal in valuation ring In the lectures I've attended on algebraic number theory, there is a standard definition of a valuation ring,
$$\mathcal{O} = \{x \in K: v(x) \geq 0\}$$
$$\mathcal{P} = \{x \in K: v(x) > 0\},$$
where $K$ is a field and $v$ an exponential valuation. $\mathcal{P}$ is then easily proven to ... | For 1., note that every $x\in R$ not in $\mathcal{P}$ is invertible in $R$, so once one has that $\mathcal{P}$ is an ideal it is automatically the unique maximal ideal. Now if $x,y\in\mathcal{P}$ then since either $\frac{x}{y}\in\mathcal{O}$ or $\frac{y}{x}\in\mathcal{O}$, you get $\frac{x+y}{x}\in\mathcal{O}$ or $\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3718197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How many solutions to the $a+b+c+d=100$ exist? Given a,b,c,d belonging to the set of whole numbers and given the equation
$a+b+c+d=100$
How many solutions like
$(a,b,c,d)=(80,10,5,5)\; ; \; (a,b,c,d)=(0,1,2,97)$
exist? We can repeat elements and the order does not matter.
| We're looking for the non-negative integer solutions for the equation $x_1 + x_2 + x_3 + x_4 = 100$.
Instead of using numbers for writing the solutions, we will use strokes, so for instance we represent the solution $ x_1 = 1, x_2 = 1, x_3 = 1, x_4 =97 $, or 1 + 1 + 1 + 97, like this: | + | + | + ||| $ \cdots$ [97 stro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3718369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Modulus operation to find unknown If the $5$ digit number $538xy$ is divisible by $3,7$ and $11,$ find $x$ and $y$ .
How to solve this problem with the help of modulus operator ?
I was checking the divisibility for 11, 3:
$5-3+8-x+y = a ⋅ 11$ and $5+3+8+x+y = b⋅3$ and I am getting more unknowns ..
| From modulus 11,
$$
53800 + 10x + y \equiv 5 - 3 + 8 - x + y \equiv -1-x+y \equiv 0 \pmod {11}\\
\implies y\equiv 1+x \pmod {11}
$$
but $y$ and $x$ are digits, so $0\le y\le 9$ and $1\le 1+x \le 10$, so it must hold that $y=1+x$.
From modulus 3,
$$
53800 + 10x + y \equiv 5 + 3 + 8 + x + (x+1) \equiv 2+2x \equiv 0 \pmod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3718474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Sum of first K primes is triangle number I was reading something this morning and came across the fact that 28 is both the sum of the first five prime numbers and of the first seven natural numbers. Naturally, I then tried to find other numbers U such that for some integers n and k
$$U=\sum_{a=1}^{n}a=\sum_{a=1}^{k}p_... | You noted that $10$, $28$, and $133386$ were the first three numbers that were initial sums both of primes and of naturals. We can then search Sloane's (the On-line Encyclopedia of Integer Sequences) for those terms and get A066527.
That page reveals that the next terms are $4218060$, $54047322253$, $14756071005948636$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3718591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\{f_n\} _{n=1}^{\infty}$ uniformly converges to $ f(x)=\int_{0}^{1}g(x,t)\mathrm{dt}$ Let $g:(0,\infty)\times [0,1]\to {\mathbb{R}}$ be continuous with respect to each variable separately and $$f_n=\frac{1}{n}\sum_{i=1}^{n}g\left(x,\frac{i}{n}\right)$$
How can show that $\{f_n\} _{n=1}^{\infty}$ uniformly... | I will show you an example that $f$ is not continuous on $[0, 1]$. It is easy to adapter it to $(0,\infty)$. In particular, your statement is not valid in general.
For $x > 0$ let $g(x, \cdot)$ be the hat function with support on $[0, 1/x]$ and maximum value $x$. Further let $g(0, \cdot) = 0$. Then, $g$ is separately c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3718715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
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