Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Question about Isomorphism I was reading this example on Wikipedia.
It says the following:
Another example is the quotient of $\mathbb{R}^n$ by the subspace spanned by the first $m$ standard basis vectors. The space $\mathbb{R}^n$ consists of all $n$-tuples of real numbers $(x_1, ..., x_n)$. The subspace, identified wi... | You can write the quotient space as $\mathbb{R}^n/\mathbb{R}^m$={v + $\mathbb{R}^m$} with v $\in$ $\mathbb{R}^n$. When we consider the projection p from $\mathbb{R}^n$ to $\mathbb{R}^n/\mathbb{R}^m$, we can see that its kernel is: ker(p)={v $\in$ $\mathbb{R}^m$}, since p(v)=0 for all v $\in$ $\mathbb{R}^m$. Therefore... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1948311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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How to find the transition matrix for ordered basis of 2x2 diagonal matrices The problem:
For the vector space of lower triangular matrices with zero trace,
given ordered basis:
$B=${$$
\begin{bmatrix}
-5 & 0 \\
4 & 5 \\
\end{bmatrix},
$$
\begin{bmatrix}
-1 & 0 \\
1 & 1 \\
\e... | Hint:
If you know how to solve the problem for $n\times 1$ vectors than consider that your matrices can be considered as vectors of a vector space with standard basis the $2\times 2$ matrices that have only one element $=1$ and the other elements $=0$. In this basis, the matrix:
$$
\begin{bmatrix}
-5 & 0 \\
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the empty set homeomorphic to itself? Consider the empty set $\emptyset$ as a topological space. Since the power set of it is just $\wp(\emptyset)=\{\emptyset\}$, this means that the only topology on $\emptyset$ is $\tau=\wp(\emptyset)$.
Anyway, we can make $\emptyset$ into a topological space and therefore talk abo... | Your map $h$ does exist, and is a homeomorphism. In fact, it's the identity map: for every element $x\in\emptyset$, $h(x)=x$. So since $h\circ h=h=\operatorname{id}$, $h$ is its own inverse. Since both $h$ and $h^{-1}=h$ are continuous, $h$ is a homeomorphism.
(Incidentally, checking that $h$ is continuous isn't ent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1948742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
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Showing Directly that Dyck Paths Satisfy the Catalan Recurrence How would one show, without appealing to a bijection with a well known problem, that Dyck Paths satisfy the Catalan recurrence?
| A slightly novel approach to this: Considering your paths as mountain ranges (cf. this), let me first show:
Lemma. The number of mountain ranges (i.e. lattice paths over $\mathbb Z^2$ between $(0,0)$ and $(m,0)$ with only steps $(1,-1)$ and $(1,1)$) that is always contained in the region $\mathbb Z\times\mathbb Z_{\le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1948805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Integration of Partial Fraction Expansion Hi This is my first time posting a question on this website. Thank you advance for helping me out here.
My question is
Suppose the density of $X$ is $$f(x) = \frac{Kx^2}{(1 + x)^5}$$ when $x > 0$. Find the
constant $K$ and the density of $Y = \frac{1}{(1 + X)}$.
++one more thin... | Hint: Use $u$-sub first to get
\begin{align}
\int \frac{Kx^2}{(1+x)^5}\ dx = \int \frac{K(u-1)^2}{u^5}\ du.
\end{align}
| {
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Plotting sets in $\Bbb R^2$ or $\Bbb C$ in some CAS Im curious about how to plot sets of $\Bbb R^2$ or $\Bbb C$ in some computer algebra systems, mainly sage, axiom, mathematica and maple, by order of preference.
To make the question not too broad it is enough for me just one example. How you can plot a set like
$$\{z\... | In SageMath, we can use region_plot as follows:
sage: f = lambda x, y: (x-3)^2 + y^2 < x^2 + (y+2)^2
sage: f(3, 2)
True
sage: region_plot(f, (-5, 5), (-5, 5))
Launched png viewer for Graphics object consisting of 1 graphics primitive
Note that we would get the same plot for the "complex" version:
sage: g = lambda x, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1949105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving the correspondence theorem for groups The correspondence theorem in my notes is:
Let $N < G$, $\pi = \text{can}:G \rightarrow G/H$. The map $H \rightarrow \pi(H)$ is a bijection between subgroups of $G$ containing $N$ and subgroups of $G/N$. Under this bijection, normal subgroups match with normal subgroups.
... | I think I found the answer, so I'll post it here.
Firstly I'd like to prove that the map $\phi :H \rightarrow\pi(H)$ is bijective, where $\pi$ is the map $G \rightarrow G/N$. This $\phi$ bijectively maps subgroups of $G$ containing $N$ to subgroups of $G/N$.
Injective: $\phi(H_1) = \phi(H_2) \implies \pi(H_1) = \pi(H_2... | {
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Splitting Field Problem I have a problem on splitting field as follows:
Determine the degree of the splitting field of the polynomial $x^6-7$ over:
(a) $\mathbb{Q},$
(b) $\mathbb{Q(\alpha)},$ where $\alpha$ is primitive 3rd root of unity,
(c) $\mathbb{F_3},$ (field with 3 elements).
I try to prove (a) in the following ... | As $\sqrt[6]{7}$ is a zero of the polynomial and $\sqrt[6]{7} \not \in \mathbb{Q}$ we adjoin it to $\mathbb{Q}$ to obtain $\mathbb{Q}(\sqrt[6]{7})$. Now in $\mathbb{Q}(\sqrt[6]{7})$ the polynomial factors into $(x - \sqrt[6]{7})(x + \sqrt[6]{7})(x^2 + \sqrt[6]{7}x + \sqrt[3]{7})(x^2 - \sqrt[6]{7}x + \sqrt[3]{7})$. Now ... | {
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"timestamp": "2023-03-29T00:00:00",
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Simplify this expression fully How would i simplify fully the following expression?
$\dfrac{{\sqrt 2}({x^3})}{\sqrt{\frac {32}{x^2}}}$
So far i have got this
$\dfrac{{\sqrt 2}{x^3}}{{\frac{\sqrt 32}{\sqrt x^2}}}$ = $\dfrac{{\sqrt 2}{x^3}}{{\frac{4\sqrt 2}{x}}}$
Am not quite sure if this is correct however, could someon... | There is a mistake in the OP. Recall that $\sqrt{x^2}=|x|\ne x$ when $x<0$. To simplify, we can write
$$\frac{\sqrt 2 x^3}{\sqrt{\frac{32}x}}=\frac{\sqrt 2 x^3}{\frac{4\sqrt 2}{|x|}}=\frac{x^3|x|}{4}$$
| {
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What's funny about $\forall \forall \exists \exists$?
So, what's the joke in number $9$?
$9$. You understand the following joke: $\forall \forall \exists \exists$
| What struck me (personally) as droll was:
*
*It can be construed as redundant, in that two "for every" clauses in succession can be replaced by a single "for every", and similarly for two successive "there exists" clauses.
*It looks like the cry of a cartoon character falling upside down from a great height.
After ... | {
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Is there a method for solving equations like $x^{2}-\sqrt{x}-2=0$? Is there a method for solving equations like $x^{2}-\sqrt{x}-2=0$?
As far as I can remember, I don't know any method for equations like this.
| If you use the substitution $u=\sqrt x$ you see that your equation is basically a quartic. Since we can solve all polynomials of degree less than or equal to $4$, there is a "method" to solve it, but it's not necessarily easy.
As it turns out, there are two real solutions to $u^4-u-2=0$. The first is $-1$, but that tur... | {
"language": "en",
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Calculus - Increasing function Problem Find all values for $a\in \mathbb R$ so the function $\, f(x) = x^3 + ax^2 + 3x - 1\,$ is always increasing in $\mathbb R$:
$\ f'(x) = 3x^2 + 2ax + 3 $ , So for the function to be increasing, $\,f'(x) $ must be greater than $ 0.$
Therefore $ a\gt -3(x^2+1)/2x $
Is this right for ... | You should think about whether dividing by x was actually allowed. Remember that x could be 0, since you need that inequality to hold for all real x. For $3x^2+2ax+3$ to be above the x axis, it cannot have any real roots. So it must have this quantity $b^2-4ac$ (the discriminant) be negative. You know what b and c are ... | {
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A formula for the $n$th term of a sequence
Find a formula for the $n$th term of the sequence
$$1,2,2,3,3,3,4,4,4,4,\ldots.$$
Let $x_n$ denote the $n$th term of the sequence. If
$$1+2+\cdots+m < n \leq 1+2+\cdots+m+(m+1)$$
then $x_n = m+1$. Is this a formula or not?
| This is not considered a formula, because it does not give $x_n$ as a function of $n$ but instead gives (sharp) bounds on what $x_n$ is. Instead, note that
*
*$x_1=1=T_1$ and it is the last $x_n$ equal to 1
*$x_2=3=T_2$ and it is the last $x_n$ equal to 2
*$x_3=6=T_3$ and it is the last $x_n$ equal to 3, etc.
He... | {
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Multiplication by One Throughout school we are taught that when something is multiplied by 1, it equals itself.
But as I am learning about higher level mathematics, I am understanding that not everything is as black and white as that (like how infinity multiplied by zero isn't as simple as it seems).
Is there anything ... | It may not be what you're looking for, but I believe this is related. In the case of limits, repeated multiplication by a limiting value of $1$ can have surprising effects.
Consider this:
$$\left(1+\frac{1}{n}\right)^{n}=\left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)\ldots$$
For large va... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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"answer_id": 4
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Combined events and Venn diagrams: If $A$ and $B$ satisfy these conditions what is $P(A \cap B')$? Events $A$ and $B$ satisfy $P((A \cup B)') = 0.2$ and $P(A) = P(B) = 0.5$. Find $P(A \cap B')$.
| Here's some potentially useful formulae
$P(A'\cap B')=P((A\cup B)')$
$P(A')=1-P(A)$
$P(A|B)=\frac{P(A\cap B)}{P(B)}$
| {
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How to show that if $f$ is rapidly decreasing, we have $|f(x-y)|\le C_N/(1+|x|)^N$ when $|y|\le |x|/2$. Let $u(x,t)=(f*\mathcal{H}_t)(x)$ for $t>0$ where $f$ is a function in the Schwartz space and $\mathcal{H}_t$ is the heat kernel. Then we have the following estimate (from Stein and Shakarchi's Fourier Analysis):
$$|... | Use the binomial theorem and the rapidly decreasing property of $f$ to prove that for all $u\in \Bbb R^N$, $(1/2 + \lvert u\rvert)^N\lvert f(u)\rvert \le c_N$ for some constant $c_N$ depending only on $N$. Then $\lvert f(x - y)\rvert \le c_N (1/2 + \lvert x - y\rvert)^{-N}$. Since $\lvert y\rvert \le \lvert x\rvert/2$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1950238",
"timestamp": "2023-03-29T00:00:00",
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Fibonacci numbers: proof4 I need to proove the following
$F_{n+k}=F_{k-1}F_{n}+F_{k}F_{n+1}$
Firstly, I wanted to use mathematic induction, but I do not know, to which letter ($n$ or $k$) should be $1$ added, or it does not matter?
I also tried to find out the solution on the Internet, but unsuccessfully.
Thanks
| There are a couple of things you need to note:
*
*The proposition you're assuming is $P(k)$ where $n, k \in \mathbb Z^+$
*You need to set the "domain" as $k\geq2$ (i.e. your base case will be $P(2)$)
*You need to use strong mathematical induction for this. You should consider two consecutive generic cases such as ... | {
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Travelling through all edges in a complete graph I have a complete graph with 14 vertices (all edges have equal weight). What is the shortest way to go through all edges?
Edit: I've got a lower bound: as I go through every edge, I visit every vertex at least $\left\lceil\frac{13}2\right\rceil$ times so the path contai... | A complete graph with 14 vertices has $\frac{14(13)}{2}$ edges. This is 91 edges.
However, for every traversal through a vertex on a path requires an in-going and an out-going edge. Thus, with an odd degree for a vertex, the number of times you must visit a vertex is the degree of the vertex divided by 2 using ceiling ... | {
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Find the generating function for $c_r = \sum^r_{i=1}i^2$ Find the generating function of
where $c_0 = 0, c_r = \sum^r_{i=1}i^2$.
Hence show that
$\sum^r_{i=1}i^2 = C^{r+1}_3 + C^{r+2}_3$
Attempt:
$c_r = \sum^r_{i=1}i^2$
= $x + 4x^2 + 9x^3 + ... + r^2x^r$
= $x(1 + 4x + 9x^2 + ... + r^2x^{r-1})$
= $x(\frac{1}{1-2x})$
How... | Let $C(t) := \sum_{n\ge0}c_nt^n$ be the generating series. Then, sung the fact that
$$c_n = c_{n-1} + n^2$$
for $n>0$, we have
\begin{align}
C(t) = & \sum_{n\ge0}c_nt^n\\
= & c_0 + \sum_{n\ge1}(c_{n-1}+n^2)t^n\\
= & tC(t) + \sum_{n\ge0}n^2t^n\\
= & tC(t) + t\frac{d}{dt}\left(t\frac{d}{dt}\sum_{n\ge0}t^n\right)\\
= & tC... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the explicit solution for the following differential equation. I have the following differential equation
$$\frac{dx}{dt} = x^2-4$$
Separating the variables, I get
$$\frac{dx}{x^2-4} = dt$$
Let us write it in partial form
$$\frac{dx}{(x-2)(x+2)} = dt$$
$$\frac{dx}{4(x-2)} - \frac{dx}{4(x+2)} = dt $$
$$ \frac{dx}{(... | If $$ \dfrac{a}{b}=\dfrac{c}{d}, $$
then
$$ \dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}. $$
This is Componendo& Dividendo Rule of elementary algebra which if applied to your last but one equation gives the last equation of the manual.
| {
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On $\sum_{n_1,\dots,n_k = 1}^\infty \frac{1}{(n_1+...+n_k)^p}$, $p \in \mathbb{R}^+$. Consider the "multiple harmonic series" $$\sum_{n_1,\dots,n_k = 1}^\infty \frac{1}{(n_1+...+n_k)^p}.$$ How can one study the behavior of this series for various values of $p \in \mathbb{R}^+$?
| The series can be written
$$
\sum_{n=1}^\infty \frac{a_{n,k}}{n^p},
$$
where $a_{n,k}$ is the number of ways $n$ can be written as an ordered sum of $k$ positive integers. We have $a_{n,k}={n-1\choose k-1}\asymp n^{k-1}$, so the series converges if and only if $p>k$, and by expressing ${n-1\choose k-1}$ as a polynomial... | {
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Introduction to viscosity solutions theory Can you recommend an introduction to viscosity solutions theory? More specifically, I'm looking for a modern treatment similar to Chapter 10 of Evans's Partial Differential Equations, but somewhat more detailed and comprehensive.
(Of course, I'm aware of the User's Guide, but... | I would recommend the book by Bardi and Capuzzo-Dolcetta.
"Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations" Bardi, Martino, Capuzzo-Dolcetta, Italo
For something shorter and more introductory you can check out my notes:
http://math.umn.edu/~jwcalder/222BS16/viscosity_solutions.pdf
| {
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Finding $\sum\limits_{n=1}^{k}\frac{1}{n(n+1)}$ in terms of $k$ So I found a question which asked me to find the sum
$$\sum_{n=1}^{k}\frac{1}{n(n+1)}$$
The only hint given was to rewrite the summation (the fraction after the sigma, does anyone know what it's called?) using fraction decomposition, so I did:
$$\frac{1}{n... | You have completed the problem. The sum becomes
$$S=1-\frac12+\frac12-\frac13+\frac13-\frac14+\frac14+-\cdots$$
The $N^{th}$ partial sum is
$$S_N=1-\frac{1}{N+1}$$
Thus,
$$S=\lim_{n\to\infty}S_N=1$$
| {
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Why are homeomorphisms important? I attended a guest lecture (I'm in high school) hosted by an algebraic topologist. Of course, the talk was non-rigorous, and gave a brief introduction to the subject. I learned that the goal of algebraic topology is to classify surfaces in a way that it is easy to tell whether or not s... | The notion of homeomorphism is of fundamental importance in topology because it is the correct way to think of equality of topological spaces. That is, if two spaces are homeomorphic, then they are indistinguishable in the sense that they have exactly the same topological properties.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1951074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 5,
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How can one use the chain rule to integrate? I am trying to calculate the anti-derivative of $y=\sqrt{25-x^2}$, for which I believe I may need the chain rule $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. How I would use it, however, is a different matter entirely.
I used this website for a tutorial, however m... | I lied in my comment - the substitution rule is what you need but it's not used in the conventional way. See, we usually sub $u = g(x)$ so we make a hard problem easier. However, we need something different here.
We start with $\int \sqrt{25 - x^2} \ dx$. Well, let's remember a trig identity: $1 - \sin^2 x = \cos^2 x.$... | {
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Set of closed intervals $[x, y] \subset [0, 1]$ $\pi$-system which generates Borel $\sigma$-algebra on $[0, 1]$? How do I see that the set of closed intervals $[x, y] \subset [0, 1]$ is a $\pi$-system which generates the Borel $\sigma$-algebra on $[0, 1]$?
| To see this, first verify that the set of closed intervals is a $\pi-$system which is the result of having finite intersection of closed set as closed.
To see that it generates Borel $\sigma-$algebra, it is enough to show that every open set $(x,y)$ (and $(x,1]$ and $[0,x)$ belongs to $\sigma-$algebra of closed set. $(... | {
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"url": "https://math.stackexchange.com/questions/1951273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Will the "closed" unit ball $\left\| x \right\| \le 1$ in $\Bbb R^n$ be a compact set for any norm?
Will the "closed" unit ball $\left\| x \right\| \le 1$ in $\Bbb R^n$ always be a compact set for any norm?
I am asking this question because the induced matrix norm is originally defined as a supremum,
Given a vector... | Yes, any two norms are equivalent on a finite dimensional Banach space, so they generate the same topology.
Perhaps for more detail, it is fairly easy to show that the unit ball under the $l_1$ norm, $B(l_1^n)$, is compact. If a sequence is bounded in the $l_1$-norm then it bounded coordinate-wise, and now apply Bolz... | {
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Prove that $F_n$ satisfies the recurrence relation $F_n = aF_{n-1} - \frac{1}{n}$ Suppose that $F_n = \int_0^1 \frac{x^n}{a-x}dx$, where $a>2$ and $n=0, 1, 2, 3,...$
Prove that $F_n$ satisfies the recurrence relation $F_n = aF_{n-1} - \frac{1}{n}$
My thought is that this can be proved by using an induction proof. We ca... | Integration by parts ($(x^{n+1})'=(n+1)x^{n}$)
| {
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"answer_id": 2
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Why does $\sum_{n=0}^k \cos^{2k}\left(x + \frac{n \pi}{k+1}\right) = \frac{(k+1)\cdot(2k)!}{2^{2k} \cdot k!^2}$? In the paper "A Parametric Texture Model based on Joint Statistics of Complex Wavelet Coefficients", the authors use this equation for the angular part of the filter in polar coordinates:
$$\sum_{n=0}^k \cos... | Suppose we seek to verify that
$$\sum_{k=0}^n \cos^{2n}\left(x+\frac{k\pi}{n+1}\right)
= \frac{n+1}{2^{2n}} {2n\choose n}.$$
The LHS is
$$\sum_{k=0}^n \cos^{2n}\left(x+\frac{k\times 2\pi}{2n+2}\right).$$
Observe also that
$$\sum_{k=0}^n
\cos^{2n}\left(x+\frac{(k+n+1)\times 2\pi}{2n+2}\right)
\\ = \sum_{k=0}^n
\cos^{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1951708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Show that the random variables $Y_1$ and $Y_2$ are independent
Let $X_1,X_2$ be i.i.d with pdf
$$f_X(x)=\begin{cases}
e^{-x} & \text{for } 0< x<\infty{}
\\0 & \text{elsewhere }
\end{cases}$$
Show that the random variables $Y_1$ and $Y_2$ with $Y_1=X_1+X_2$ and $Y_2=\frac{X_1}{X_1+X_2}$ are independent.
I know tha... | Here is a simulation of 100,000 $(Y_1, Y_2)$-pairs from R statistical software.
The $X_i$ are iid $Exp(rate=1),$ $Y_1 \sim Gamma(shape=2, rate=1)$
and $Y_2 \sim Unif(0, 1).$ Also, $Y_1$ and $Y_2$ are uncorrelated. (If these
distributions are not covered in your text, you can see Wikipedia articles
on 'exponential distr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1951806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Show that two sets are disjoint I'm trying to solve this problem about two different sets. I have to show that they are disjoint. I understand that they need to be disjoint, because they are two different real ranks, but I dont know how to prove it.
Thank you so much for the help. Any hint is welcome!
Show that if two... | Without loss of generality we can assume $n>m$ (the case $n<m$ follows by symmetry). Then observe that $m-1<m\leq n-1<n$.
We proceed using proof by contradiction. Suppose that there is some $x\in(m-1,m]\cap(n-1,n]$. Then $m-1<x\leq m$ and $n-1<x\leq n$. We have $x>n-1\geq m$ and $x\leq m$, absurd!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1951925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prime modulo maximum Prove that the remainder of division of a positive number $n$ by a prime $p \le n$ is maximized when $p$ is the smallest prime larger than $\frac{n}{2}.$
It is easy to see that for any number of the form $\frac{n}{2}+k$ where $k \gt 0$, if $k$ is increased remainder will decrease. How to prove th... | This does not hold true. Consider $n=14$, the smallest prime larger than $\frac{14}{2}=7$ is $11$, but the maximum remainder is attained for prime $5 \lt 7$:
$$14 \bmod 5 = 4 \;\;\gt\;\; 14 \bmod 11 = 3$$
[ EDIT ] The following shetches the proof to the related question asked in a comment below.
The remainder of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1952193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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finding whether this sequence converges Let
$$a_n=\left(1+\frac{1}{n}\right)^n$$
Does this converge?
When $n$ goes to $\infty$ we end up with $1$ to the power of $\infty$. What does this mean to us? Is it $1$ or undefined. I remember from calculus course that this is not defined, so we take it as divergent right? But w... | We first prove that the sequence $a_n=\left(1+\frac1n\right)^n$ converges. To do so, we will show that it is monotonically increasing and bounded above. The Monotone Convergence Theorem guarantees that such a sequence converges.
SHOWING THAT $a_n$ MONOTONICALLY INCREASES
To see that $a_n$ is monotonically increasi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1952290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Why one of NP-Complete problem had a polynomial solution then everyone of them has? Each NP problem is different, if one (even the hardest one) NP problem could be solved in polynomial time, I guess some related NP problems that could reduced to this one could also be solved in polynomial time. But why all? Does that m... | NP completeness means exactly that "all other NP problems could be reduced [in polynomial time] to the one", so yes, if a single NP-complete problem has a polynomial-time solution, then all NP problems do. See the formal definition.
Note that it is not obvious that NP-complete problems exist in the first place! E.g. ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1952517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Show that $\| f(x) - f(y) \| \leq 2 \| x - y \| $ Let $V$ be a vector space with a given norm $\| \cdot \|$. Define a function $f:V \to V$ in the following way:
$$
f(x) =
\begin{cases}
x, & \|x\| \leq 1 \\
x/ \|x\|, & \|x\| > 1 \end{cases}
$$
Prove, that $\| f(x) - f(y) \| \leq 2 \|x-y\|$ for all $x,y \i... | We will consider two cases.
First case: WLOG, assume $\| x\| \geq 1$ and $ \|y \| \geq 1$, then we see that
\begin{align}
\| f(x) -f(y)\| \leq&\ \Bigg\| \frac{x}{\|x\|}-\frac{y}{\|y\|}\Bigg\| \leq \frac{1}{\|x\| \|y\|}\Big\| x\|y\|-y\|x\|\Big\|\\
\leq&\ \frac{1}{\|y\|} \Big\| x\|y\|-y\|y\|+y\|y\|-y\|x\|\Big\|\\
\leq&\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1952642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Why is $O(\log(\log(n)))$ upper bound $\Theta(\log(n))$? This question more has to do with math than computer science, but there is a computer science component to it.
If I have an algorithm with a time complexity function $$O(\log(\log(n)))$$Why would its upper bound be$$\Theta (\log(n))$$I do not follow the rules of ... | Suppose that $f$ is $O(\log(\log n))$, and $g$ is $\Theta(\log n)$. Then there are positive constants $c_0$ and $c_1$ and an $m\in\Bbb Z^+$ such that
$$f(n)\le c_0\log(\log n)$$
whenever $n\ge m$ and
$$g(n)\ge c_1\log n$$
whenever $n\ge m$. Moreover,
$$\lim_{n\to\infty}\frac{\log(\log n)}{\log n}=0\;,$$
so we may furth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1952748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to formally prove that we cannot find a polynomial in $\textbf Z[x]$ with degree $2$ with such a root? I am trying to find the kernel of the map from $\textbf Z[x]$ to $\textbf C$. The map is evaluating at $\sqrt 2 + \sqrt 3$.
A solution says that we cannot find polynomials of degree $2$ or $3$ that has such a root... | Let's work, for the moment, in $\mathbb{Q}[x]$.
The number $b=\sqrt{2}+\sqrt{3}$ is certainly a root of $f(x)=x^4-10x^2+1$, because $(b-\sqrt{2})^2=3$, so $b^2-1=2b\sqrt{2}$ and, squaring again, $b^4-2b^2+1=8b^2$.
Therefore there is a monic polynomial $p(x)$ of minimal degree (with coefficients in $\mathbb{Q}$) which $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1952831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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Uniqueness of unitarily upper triangular matrix Suppose A is a matrix in $\mathbb{C}^{n\times n}$ with n distinct eigenvalues $\lambda_1,\dots,\lambda_n$. Then by Schur's theorem, for any fixed order of $\lambda_1,\dots,\lambda_n$, we know there exists an unitary matrix $U$ s.t. $U^*AU$ is an upper triangular matrix wi... | Key fact: if $BT=TB$ and $T$ is upper triangular with distinct diagonal entries, then $B$ is upper triangular (proof below). Now, if $$UTU^*=VTV^*,$$ then $V^*UT=TV^*U$. So $V^*U$ is an upper triangular unitary. As such, it is diagonal. Thus, $V$ is of the form $DU$ with $D$ diagonal and $|D_{kk}|=1$. In other words, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1952939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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show that $X/(X+Y) $ has a cauchy distribution if $X $ and $X+Y $ are standard normal A random variable $X$ has a cauchy distribution with parameters $a$ and $b$ is the density of $X$ is $f(x\mid a,b)=\dfrac{1}{\pi b}\dfrac{1}{1+(\frac{x-a}{b})^2}$ where $-\infty <x< \infty $, $-\infty <a <\infty$, $b>0$
Suppose $X$ a... | You had almost reached the end. It suffices to transform:
$$\left(\dfrac{1}{2\pi}\right)\dfrac{1}{v^2-v+\frac{1}{2}}=\left(\dfrac{1}{2\pi}\right)\dfrac{1}{(v-\frac{1}{2})^2+(\frac{1}{2})^2}=\left(\dfrac{1}{2\pi}\right)\dfrac{4}{1+\left(\frac{v-\frac{1}{2}}{\frac{1}{2}}\right)^2}$$
giving
$$\left(\dfrac{1}{\pi \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Parametric to non parametric conversion of a line in 3d I can't for the life of me figure out how to convert this parametric equation to a non parametric equation for a line in 3D. Our lecture notes didn't cover it and I feel like it should be simple but whenever I try to figure it out I end up with nothing. Can someon... | A non-parametric representation of a line in 3D won't be a single equation but rather a system of (two) equations in the Cartesian coordinates $x$, $y$ and $z$.
As the name suggests, you should try to eliminate the parameter by first solving for $t$:
$$\left\{ \begin{array}{rcl}
x &=& 2 + 3t \\
y &=& -1 + t \\
z &=& -2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Introducing probability vectors When I studied probability I did it in a classical manner. My course used Loève's book on probability theory and I used Allan Gut's book for a modern version of it. Now I'm following a course in information theory and the notation, terms and concepts change so that it is difficult for me... | Let $X$ be a random variable with finite support, i.e. the possible values of $X$ are $x_1,\dots,x_n$. Let $p_i=P(X=x_i)$. Strictly speaking, the distribution of $X$ is the measure $\sum_{i=1}^np_i \delta_{x_i}$.
Once you know the support of $X$, in order to describe its distribution, you can just give the vector $(p_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Lusin Theorem Conclusion (Confusion on different versions)
According to Royden, the conclusion of Lusin Theorem is that:
(1) $f=g$ on $F$, where $g$ is a continuous function on $\mathbb{R}$.
However, according to Wikipedia, the conclusion of Lusin Theorem is:
(2) $f$ restricted to $F$ is continuous.
I have seen somewh... | Lusin's theorem contains the condition that $F$ be a closed set. That makes a difference.
In sufficiently nice spaces $X$ ($T_4$ or normal spaces, whichever is the weaker in the nomenclature in use), every continuous function $f \colon F \to \mathbb{R}$, where $F\subset X$ is closed, has a continuous extension $F \colo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Unit square inside triangle. Some time ago I saw this beautiful problem and think it is worth to post it:
Let $S$ be the area of triangle that covers a square of side $1$. Prove that $S \ge 2$.
| Intuitively when vertices of square
$S:=v_1v_2v_3v_4$ of side length $1$ are in sides of triangle
$T=\Delta\ ABC$, then ${\rm area}\ T$ is smallest
Notation : Edge $xy$ is $[xy]$ And length of edge $xy$ is $|xy|$
Here we have two cases :
(1) $ v_1\in [AB],\ v_2\in [AC]$ and edge $[v_3v_4]$ is in edge
$[BC]$
(2) $v_1\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 6,
"answer_id": 5
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how many ways are there to arrange $8$ pennies and $5$ nickels in a How can you arrange a line of $8$ pennies and $5$ nickels, so that no $2$ nickels are next to each other (pennies are indistinguishable and nickels are too)?
Answer: I set nickels and pennies next to each other so I get ${9 \choose 5}*8!*5!=609638400$.... | If you first lay out the $8$ pennies, there are $9$ spaces where the nickels can go ($7$ spaces between adjacent pennies and $2$ spaces at the two ends). So the answer is simply $9\choose5$. You would multiply this by $8!5!$ only if the pennies and nickels were all distinguishable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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determinant of a $2\times2$ matrix, sufficiency of inverse What is the simplest example of a nontrivial ring in which these two conditions are not equivalent for any $2\times 2$ matrix $A$:
(1) there is a $2\times2$ matrix $B$ with $A\cdot B=1$
(2) $a_{11}a_{22}\neq a_{21}a_{21}$
To make the question more interesting: ... | A matrix $A\in M_2(R)$, where $R$ is a ring, is invertible iff $\det A\in U(R)$, where $U(R)$ is the set of the units of the ring $R$.
So a simple example is given by the ring $\Bbb Z$, any matrices whose determinant is different to $\pm 1$ is not invertible.
Example
$$ A=
\begin{pmatrix}
2 & 1\\
3 & 4
\end{pmatrix... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Solving equation in three variables please help me understand how the following equation with 3 variables and power of 2 is solved and what solution approach is the quickest.
$$3y^2 - 3 = 0$$
$$4x - 3z^2 = 0$$
$$-6xz+ 6z = 0 $$
| A quick solution:
By simplification, rewrite
$$\begin{cases}y^2=1\\(1-x)z=0\\3z^2=4x.\end{cases}$$
Then mentally,
$$y=\pm1,\\x=z=0\lor x=1,z=\pm\frac2{\sqrt3}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Show that the following sum of legendre symbols is -1. Let. $p$ be an odd prime. Consider the following sum of Legendre Symbols:
$(\frac{1}{p})(\frac{2}{p}) + (\frac{2}{p})(\frac{3}{p}) + \cdots + (\frac{p-2}{p})(\frac{p-1}{p})$.
Show that this sum is equal to $-1$.
Using the algebra of the Legendre symbol i can show ... | With the assumptions $\left(\frac{0}{p}\right)=0$, $p\equiv 1\pmod{2}$, by exploiting the multiplicativity of the Legendre symbol we have
$$ \sum_{k=1}^{p-2}\left(\frac{k}{p}\right)\left(\frac{k+1}{p}\right)=\sum_{k=1}^{p-2}\left(\frac{k^2+k}{p}\right)=\sum_{k=1}^{p-2}\left(\frac{1+k^{-1}}{p}\right)$$
where $k^{-1}$ st... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum of a squared dot product I have a constant $d \times n$ matrix $\textbf{A}$, and a variable $d \times 1$ vector $\textbf{v}$
$$\sum_{i=0}^n (\mathbf{A}_i^\top \mathbf{v})^2$$
Is there a way to simplify this? Can I pull out any A's?
I know that I could use this property:
$$\sum_{i=0}^n\left( \mathbf{A}_i^\top\mathb... | It depends upon what you mean by the 'square'. If it is scalar product then it
equals:
$$ \sum_i v^T A_i A_i^T v = v^T \left( \sum_i A_i A_i^T\right) v = v^T B v $$
where $B$ is a semi-positive definite matrix. It will be definite positive under reasonable conditions on the $A_i$'s.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1953969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Pascal's Triangle and Binary Representations In the article that I am currently reading, it is stated as a well-known fact that positions $2^i$ or equivalently $(n-2^i)$ in the $n^{th}$ row in Pascal's Triangle modulo $2$ spell out the binary representation of $n$:
$$
\newcommand{\red}{\color{red}}
\newcommand{\blue}{\... | Another approach uses generating functions (for a similar example, see the proof of Lucas's Theorem). Let $p(x) = \sum_{k=0}^n\binom{n}{k}x^k$.
It is easy to check that for primes $p$ and nonnegative integers $k$, we have $(1+x)^{p^k}\equiv 1 + x^{p^k}\pmod p$.
Then
$$p(x) = (1+x)^n = \prod_{i=0}^t \left((1+x)^{2^i}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1954098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
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Two matrices with special eigenvalues We say $\lambda$ is an eigenvalue of a square matrix $A$ if
$Ax = \lambda x$.
Now, i want two examples of a matrix like $A$.
The first one, $A$ should have just one eigenvalue which should be $0$.
The second one, $A$ should be a matrix in which $a_1,\dots,a_n$ are
eigenvalues. (... | So, with the hint andrew gave me, i solved the problem myself...
If a matrix is diagonal and $a_1,\dots,a_n$ are on the diagonal, then we have equations like :
($a_i-\lambda )(x_i)=0$
And one matrix which has zero as an eigenvalue and zero is its only eigenvalue is the zero matrix itself !!! ( For example, zero matrix ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1954192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Can all complex expressions be simplified to the form $a+jb$? Are there any complex expressions that cannot be simplified to the form $a+jb$, where a and b are real numbers?
For example, $$\frac{1}{j}=0+j(-1),\hspace{0.5cm}e^j=\cos(1)+j\sin(1),\hspace{0.5cm}\sin(j)=0+j\frac{e^2-1}{2e}$$
From what I understand, all comp... | To get
$\ln(z)$
for any complex $z$
(where $z \ne 0$),
write
$z = |z|y$.
Then $|y| = 1$,
so there is a real $t$
such that
$y = e^{it}
=\cos(t)+i\sin(t)
$,
so
$z
= |z|y
= |z|(\cos(t)+i\sin(t))
= |z|\cos(t)+i|z|\sin(t)
$.
Since
$|1+j| = \sqrt{2}$,
$(1+j)
=\sqrt{2}\frac{1+j}{2}
=\sqrt{2}e^{i\pi/4}
$,
so
$\ln(1+j)
=\ln(\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1954349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A lower bound in variational inequality Let $f\in C^2([a.b])$, assume that $$f(a)=0=f(b),~f'(a) = 1,f'(b) =0.$$ Prove that $$\int_{a}^{b}\Big|f''(x)\Big|^2\mathrm{d}x\geq \frac{4}{b-a}.$$
Is there a way to convert it into an eigenvalue problem and then one may look for the first eigenvalue as the lower bound?
| Change the variables: $x=(b-a)t+a$ and $g(t)=f((b-a)t+a)$
$$\int_a^b|f''(x)|^2dx = \int_0^1 |f''((b-a)t+a)|^2 (b-a) dt =\frac{1}{(b-a)^3} \int_0^1 |g''(t)|^2 dt$$
Therefore the problem is equivalent to proving
$$\int_0^1|g''(t)|^2dt \geq4 (b-a)^2$$
for $g\in C^2([0,1])$ with $g(0)=g(1)=0$, $g'(0)=(b-a)=:c$ and $g'(1)=0... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that for any set with $m$ elements, if the average of any $nProve that for any set with $m$ elements, if the average of any $n < m$ elements is equal to the constant $k$, then each of the $m$ elements are equal to $k$.
Attempt: Considering some specific examples, say $m = 4$, $n = 3$, and $k=4$. Let the set be ... | Call the elements $a_1,a_2,\dots,a_m.$ Note that
$$a_1+a_2+\cdots+a_n=a_2+a_3+\cdots+a_{n+1}$$
since both sums are equal to $nk.$ It follows that $a_1=a_{n+1}.$ Since the indexing is arbitrary, it follows that any two elements are equal. And of course the common value must be $k.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1954628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Question about homomorphisms between symmetric groups Let $f: S_A \to S_B$ be a homomorphism from the symmetric group on $A$ to the symmetric group on $B$, where $A$ and $B$ may be infinite. For $X\subseteq A$ and $b_1,b_2\in B$, say that $b_1\sim_X b_2$ if and only if $f(g)(b_1)=b_2$ for some $g$ s.t. $g(x) = x$ for a... | I think the answer to Question 1 (and hence Question 2) is "yes", unless I've made a mistake.
For $X,Y\in\mathcal{P}(\mathbb{N})$ and $g\in\mathbb{N}^\mathbb{N}$, let $g(X) = \{g(x): x\in X\}$, $E = \{\langle X,Y\rangle: |(X\cup Y)\backslash(X\cap Y)|<\omega\}$, and $[X] = \{Y: \langle X,Y\rangle\in E\}$. Now let $A =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1954726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
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How can the surd $\sqrt{2-\sqrt{3}}$ be expressed? I was wondering how $\sqrt{2-\sqrt{3}}$ could be expressed in terms of $\frac{\sqrt{3}-1}{\sqrt{2}}$.
I did try to solve both the expressions separately but none of them seemed to match.
I would appreciate it if someone could also mention the procedure
| Theorem: Given a nested radical of the form $\sqrt{X\pm Y}$, it can be rewritten into the form $$\sqrt{\frac {X+\sqrt{X^2-Y^2}}{2}}\pm\sqrt{\frac {X-\sqrt{X^2-Y^2}}{2}}\tag{1}$$
Where $X>Y$.
Therefore, we have $X=2,Y=\sqrt{3}$ because $2>\sqrt{3}$. So plugging that into $(1)$ gives us $$\sqrt{\frac {2+\sqrt{4-3}}{2}}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1954822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Correctly integrating the product of two functions I'm trying to determine the probability of finding a function inside another function.
While plotting the results do give me a sign of there being an actual non-zero number, integrating to determine the number gives me zero.
Plotting these two functions gives me
$$ f(x... | Your orange plot is incomplete. If you were to plot your $f_1$ over the interval $[-15,15]$ you would see that it has a negative lobe that is symmetric with the positive lobe you plotted. There are two ways to fix this:
1) Do what you suggest and only integrate over the intersection of the supports of the two functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1954941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that if $(a+b)$ divides $a^2$ then $(a+b)$ divides $b^2$ I'm trying to solve the following exercise: "Prove that if $(a+b)$ divides $a^2$ then $(a+b)$ divides $b^2$".
It's quite obvious how to prove divisibility for a product, but how to do it for a sum?
| If $a+b$ divides $a^2$, then you can write $a^2=(a+b)k$.
But then $$b^2= a^2-a^2+b^2 = a^2 -(a^2-b^2)$$
$$=\underbrace{(a+b)k}_{a^2} - (a+b)(a-b)$$
$$=(a+b)(k-(a-b))$$
$$=(a+b)(k-a+b)$$
so $(a+b)$ divides $b^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1955034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Complex Analysis Branches I have a single valued function is defined as a branch of the multivalued function
$$f=\ln[z(z+1)] \tag{*}$$ on the complex plane with the segment $[-1,0]$ of the real axis removed, and require $$f(1)=\ln[2] +2\pi i \tag{**}$$
Show that * and ** and the proposed branch cut do not describe a si... | I suppose you mean $f(1)=\ln(2) + 2\pi i$ which is a consistent choice
for $f$ at $1$. As is mentioned the function is not single valued. When you go once around the cut the arg increases with $4\pi$. One way to see this is to take the derivative $f'(z)=\frac{1}{z} + \frac{1}{z+1}$ and note that if you choose a contour... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Strictly increasing function and its derivative We have "$f'(x)>0$ in interval $I\implies f(x)$ strictly increasing in interval $I$".
However, the converse is not true.
A frequently cited example is the function $f(x)=x^3$, which is strictly increasing but $f'(0)=0$.
Here comes my question:
What is the necessary and... | If $f(x)$ satisfies $f'(x)\ge 0$ for all $x\in \mathbb R$ and if for every $x_0\in \mathbb R$ with $f'(x_0)=0$, there is a neighborhood around $x_0$ , such that $f'(x)\ne 0$ within this neighborhood (except $x_0$) (in other words : isolated roots of the derivate ) , then $f(x)$ is strictly increasing on $\mathbb R$.
No... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Plotting complex inequalities in maple I'm having trouble plotting a set of complex numbers in maple. I'm trying to plot the set
$$S = \lbrace z \in \Bbb C : 1 \leq \lvert z\rvert \leq 2, \frac{\pi}{4} \leq \lvert \arg(z)\rvert \leq \frac{\pi}{2}\rbrace.$$
I know what it should look like from a drawing I produced but ... | restart;
z := x + I*y:
plots:-implicitplot( piecewise( (abs(z) <= 2) and (abs(z) >= 1)
and (abs(argument(z)) >= Pi/4)
and (abs(argument(z)) <= Pi/2),
false, true),
x=-3...3, y=-3...3,
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$f:[0,2]\to\mathbb{R}$ continuous, then if $\int_0^x |f(t)| \ dt = \int_x^2 |f(t)| \ dt$ then $f(x) = 0$ I have that
$f:[0,2]\to\mathbb{R}$ continuous, then if $\int_0^x |f(t)| \ dt = \int_x^2 |f(t)| \ dt$, $f(x) = 0$ for $x\in [0,2]$
I tried to substitute $x=0$ in the equality to get:
$$0 = \int_0^0 |f(t)| \ dt = \int... | For an alternative approach, using the Fundamental theorem of calculus and the given identity:
$$|f(x)| = \left(\int_0^x |f(t)|\;dt\right)' = \left(\int_x^2 |f(t)|\;dt\right)' = -|f(x)|$$
Therefore $\;\;|f(x)| = -|f(x)|$ $\;\;\implies\;\; |f(x)| = 0$ $\;\;\implies\;\; f(x) = 0\;\;$ for $\;\; x \in [0,2]$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Comparison principle for $-\varepsilon \, \Delta u + u_t + H(x,Du) = 0$ . Given $u$ is a classic solution for
$$-\varepsilon \, \Delta u + u_t + H(x,Du) = 0$$
and $\varphi\in C^2$ (called a super solution?) satisfies
$$-\varepsilon \, \Delta \varphi + \phi_t + H(x,D\phi) \geq 0$$
with the same initial condition $u(x,... | The answer is no, without some additional growth constraints on $u$ and $\phi$. You can just take $H\equiv 0$, and then you have the heat equation, which has infinitely many smooth solutions for the same initial data if you do not impose growth constraints.
If $u$ and $\phi$ are bounded, then the answer will be yes. Th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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acceleration formula of parametric curve? I have a parametric curve as follows:
x(t)= 0.236t³-0.645t²+0.909t+0
y(t)= 0.189t³-0.792t²+0.603t+0
which looks like:
Now, I want to find the acceleration of this curve, from when it starts at 0,0 and ends at .5,0. HOW would I do this? It has been years since i took a math co... | Hint: The acceleration is the second derivative of position. As such, the acceleration in the x direction would be x''(t), and the acceleration in the y direction would be y''(t).
| {
"language": "en",
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How To Set Up A Card Probability Problem Rosa draws a five-card hand from a $52-$card deck. For each scenario, calculate the total possible outcomes:
Rosa’s hand has three red cards and three face cards.
The only way I could think about doing this was to add together the $26$ red cards, along with the face cards, si... | Since Rosa's hand has $5$ cards and the conditions are $3$ red and $3$ face cards, at least one card has to be a "red, face card". Following hands satisfy the given conditions (please check that I did not forget something):
*
*$1$ red face card, $2$ red not-face cards, $2$ black face cards.
*$2$ red face cards, $1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1956114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of integer triangles with sides 4 or less
Consider triangles having integer sides such that no side is greater than 4 units. How many such triangles are possible?
I suspect a relation to the following question:
How many ways can $r$ things be taken from $n$ with repetition and without regard to order?
| Without loss of generality, assume that $a \leq b \leq c < a+b$. This leaves us with very few choices.
1) All three could be equal. That gives us four choices.
2) $a=1$. Then, $b+1 > c$, so $b=c$ must happen, this gives three choices.
3) $a=2$. Then, $b \leq c < b+2$, so $b=2$,$c=3$ and $b=3, c=3,4$ , and $b=c=4$ are t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1956257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability : Venn diagrams; independent Are both the Venn diagram's (i and ii) showing dependent properties (showing that A and B are dependent on one another), I am told that the quartered one is independent (Venn diagram i), however I do not see any difference between the 2 diagram's. Surely P(A|B) (probability of B... | Edit: as @Henry commented, in general Venn diagrams do not rely on surface in any way, the only thing that matters are the intersections. If you consider your diagrams are pure Venn diagrams, then it is impossible to tell apart your two diagrams, and so it is to deduce independence.
However, if you impose your diagram ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Monotone convergence when $f_n$ is decreasing I know that if $f_n\geq 0$, that $(f_n)$ is increasing and that if $\lim_{n\to \infty }f_n=f\in L^1$, then,
$$\lim_{n\to \infty }\int f_n=\int f.$$
Does this result also work when $(f_n)$ is decreasing ?
| Let $f_{n}=\frac{1}{n}χ[0,n]$.
For every $\epsilon > 0$ and $x \in \mathbb{R}$ there exists $N > 1$
such that
$|f_n(x)| < \epsilon $ for every $n>N$. Hence, $f_n$ converges to $f = 0$ ,uniformly. However,
$$0 = \int
f d\lambda \not= \lim \int
f_n d\lambda = 1$$
The MCT does not apply because the sequence is not mono... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1956484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Generating function for the number of ordered triples Let an be the number of ordered triples $(i, j, k)$ of integer numbers
such that i ≥ 0, j? ≥ 1, k ≥ 1, and $i+3j +3k = n$. Find the generating
function of the sequence (a0, a1, a2, . . .) and calculate a formula for $a_{n}$.
First I used polynomials to express the p... | Yes, your generating function is correct. To extract its coefficients, there are many things you could do. The factor of $\frac1{1-x}$ always leads to sums of coefficients of other terms. For the term $\left(\frac{x^3}{1-x^3}\right)^2$, you can first remove the $x^6$ in the numerator and then deal with the denominator.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Increasing sequence of events and the probability of their limit
If $A_1\subset A_2\subset A_3\subset\cdots$ is an increasing sequence of events with limit $A=\bigcup_{i=1}^\infty A_i$. Prove that $$\lim_{n\rightarrow\infty} P(A_n)=P(A)$$
My attempt so far:
Since $A_1\subset A_2\subset A_3\subset\cdots$ is increasing... | Let $B_1 = A_1$ and $B_{n+1} = A_{n+1}\setminus A_n$ for $n\ge 1$. Then
$$
A_N = \bigcup_{n=1}^N B_n
$$
and
$$
\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty B_n,
$$
and
$$
B_n \cap B_m = \varnothing \text{ for } n\ne m.
$$
So
\begin{align}
P(A) & = P\left( \bigcup_{n=1}^\infty A_n \right) = P\left( \bigcup_{n=1}^\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$f\in L^2(\mathbb{R})\Rightarrow f\to 0, x\to\pm\infty$? As the title already suggests: Let $f\in L^2(\mathbb{R})$. Does this imply that
$$
f\to 0\text{ as }x\to\pm\infty?
$$
| The answer is no.
Consider $$f(x)=\sum_{n=1}^\infty \chi_{[n-\frac 1 {n^2},n+\frac 1 {n^2} ]}(x)$$
With $\chi$ the charasteristic function. $f$ is in $L^2$ but $f$ does not have a limit at $\infty$. You can also find similar examples which create a continuous function.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $S = \{a^2 + b^2: a,b \in \Bbb N\}$ is closed under multiplication. Is it true?
Can you prove or disprove this?
$S = \{a^2 + b^2: a,b \in \Bbb N\}$ is closed under multiplication.
| Suppose that $x=a^2+b^2$ and $y=c^2+d^2$. Then
$$xy=\det
\begin{pmatrix}
a&b\\-b&a
\end{pmatrix}
\det\begin{pmatrix}
c&d\\-d&c
\end{pmatrix}
=\det
\begin{pmatrix}
a&b\\-b&a
\end{pmatrix}
\begin{pmatrix}
c&d\\-d&c
\end{pmatrix}
=\det\left(
\begin{array}{cc}
a c-b d & b c+a d \\
-b c-a d & a c-b d \\
\end{array}
\right... | {
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How does an exponent work when it's less than one? I'm rather familiar with exponents, I know that $y^x = y_1 \cdot y_2 \cdot y_3 .... y_x$, but what if the exponent is less than one, how would that work?
I put in my computer $25^{1/2}$ anyway, expecting it to give me an error, and I got an answer!! And even more surp... | $$y^x = y\cdots y$$
only if $x$ is a positive integer.
In general:
$$y^x = \exp(x\log y)$$
provided that $y >0$ (the functions $\exp$ and $\log$ can be defined via power series).
Also one can prove:
$$\sqrt[x]{y} = y^{1/x}$$
as you noted (this can also be taken as the definition of roots).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
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"answer_id": 2
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Does this make mathematical sense? For a given set $A$,
An element such that $a \in A $ exists.
If $A$ is a set of all natural numbers, then:
$$ a \in A \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}. $$
Would maths normally be written like this, if it is correct?
| This question is a bit confusing and no it doesn't make a lot of "sense" overall. Especially given that $A$ being defined as the set of all natural numbers means $A\not\in\mathbb{N}$ but that $A=\mathbb{N}$.
As for your question, would maths normally be written like this... yes, those are all valid mathematical symbols... | {
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How to know when a quintic is solvable So according to Abel-Ruffini Theorem, it states that there is no algebraic solution, in the form of radicals, to general polynomials of degree $5$ or higher.
But I'm wondering if there is a way to decide whether a polynomial, such as $$x^5+14x^4+12x^3+9x+2=0$$
has roots that can b... | As the others have commented, to know when a quintic (or higher) is solvable in radicals requires Galois theory. However, there is a rather simple aspect when it is not solvable that is easily understood and can be used as a litmus test.
Theorem: An irreducible equation of prime degree $p>2$ that is solvable in radical... | {
"language": "en",
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how is it possible for $a^n\ne a$? If an ideal $a$ is an additive subgroup of a ring $A$ satisfying $Aa\subset a$, then in particular, for any $a_1, a_2, \dots, a_n \in A$ and any $x\in a$
$$y_1=a_nx\in a$$
so
$$y_2=a_{n-1}y_1=a_{n-1}a_nx\in a$$
$$y_3=a_{n-2}y_2=a_{n-2}a_{n-1}a_nx\in a$$
$$\vdots$$
$$y_n=a_1y_{n-1}... | Your argument shows that $a^n\subseteq a$, but it does not show that $a\subseteq a^n$, which may in fact be false. For instance take $A=\mathbb{Z}$ and $a=2\mathbb{Z}$. Then an element of $a^2$ is a sum of products of two even integers. Any such product is divisible by $4$, and so is any sum of such products, so eve... | {
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Evaluate $\lim_{x\to0}\frac{1-\cos3x+\sin 3x}x$ without L'Hôpital's rule I've been trying to solve this question for hours. It asks to find the limit without L'Hôpital's rule.
$$\lim_{x\to0}\frac{1-\cos3x+\sin3x}x$$
Any tips or help would be much appreciated.
| Taylor expansion is always a good solution since the method will provide the limit and more.
Remembering that $$\cos(t)=1-\frac{t^2}{2}+O\left(t^4\right)$$ $$\sin(t)=t-\frac{t^3}{6}+O\left(t^4\right)$$ replace $t$ by $3x$ to get $$1-\cos (3 x)+\sin (3 x)=1-\left(1-\frac{9 x^2}{2}+O\left(x^4\right) \right) +\left( 3 x-\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 4
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Minimum value of $2^{\sin^2x}+2^{\cos^2x}$ The question is what is the minimum value of
$$2^{\sin^2x}+2^{\cos^2x}$$
I think if I put $x=\frac\pi4$ then I get a minimum of $2\sqrt2$. But how do I prove this?
| Let $y=2^{\sin^2x}+2^{\cos^2x}=2^{\sin^2x}+2^{1-\sin^2x}$
$$(2^{\sin^2x})^2-y\cdot2^{\sin^2x}+2=0$$ which is a Quadratic Equation in $2^{\sin^2x}$
So, the discriminant must be $\ge0$
$$(y)^2\ge4\cdot2\implies y^2\ge8$$
As $y>0,y\ge2\sqrt2$
The equality occurs if $$2^{\sin^2x}=\dfrac{2\sqrt2}2=\sqrt2=2^{1/2}$$
i.e., if ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Correct use of the implication symbol A lecturer mentioned that a common mistake people make in assignments is the incorrect use of the implication notation, $\Rightarrow $. I would like to clarify the correct use of the symbol as I am responsible for marking some first year assignments this term, and have been advis... | I would not deduct any marks for the first answer.
What is an implication? It simply says "If A is true, then B is true". This is symbolically written as $A \implies B$.
When the implication is false, there is some object having property $B$ that does not have property $A$.
In the implication in question, it is clear... | {
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"timestamp": "2023-03-29T00:00:00",
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Question in complex numbers from GRE This is a question motivated from GRE subgect test exam.
if f(x) over the real number has the complex numbers $2+i$ and $1-i$ as roots,then f(x) could be:
a) $x^4+6x^3+10$
b) $x^4+7x^2+10$
c) $x^3-x^2+4x+1$
d) $x^3+5x^2+4x+1$
e) $x^4-6x^3+15x^2-18x+10$
What I thought at first was ... | 1.
$(x-(2-i))(x-(2+i))$
$x^2-x(2+i)-x(2-i)+(2-i)(2+i)$
$x^2-2x-xi-2x+xi+(4-2i+2i+1)$
$x^2-4x+5$
2.
$(x-(1-i))(x-(1+i))$
$x^2-x(1+i)-x(1-i)+(1-i)(1+i)$
$x^2-x-xi-x+xi+(1+i-i+1)$
$x^2-2x+2$
3.
$(x^2-4x+5)(x^2-2x+2)$
$x^4-2x^3+2x^2-4x^3+8x^2-8x+5x^2-10x+10$
$x^4-6x^3+15x^2-18x+10$
| {
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Why $\frac{\log4}{\log b}$ can't be simplified to $\frac4b$?
I want to know why $\frac{\log4}{\log b}$ can't be simplified to $\frac4b$.
I am a high school student. Please do not quote some theories that are too advanced for me. Thank you!
| Well, suppose you could do such simplification:
$$
\frac{\log 4}{\log b}=\frac{4}{b}\tag{1}
$$
You would end up with (do you know why?)
$$
b\cdot \log 4=4\cdot\log b,
$$
which implies (do you know why?) that
$$
\log 4^b=\log b^4\tag{2}.
$$
If (1) were true for every $b>0$, then (2) must also be true for every $b>0$ and... | {
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"answer_count": 7,
"answer_id": 6
} |
Simple line integral Let $f(x,y)=x$ and $C=[0,1]\times\{0\}$ (the line segment joining the point $(0,0)$ and $(1,0)$).
I want to calculate $\boxed{\displaystyle\int_C f(x,y)\,ds}$.
I calculate the following:
$$\displaystyle\int_C f(x,y)\,ds=\int_0^1 x\,dx=\frac{1}{2}$$ its ok?
| It is correct if the orientation of the contour is left-to-right (starting at $(0,0)$ and ending at $(0,1)$).
If the orientation is reversed, the value will be $-1/2$.
For line integrals, you should imagine a point in motion along the contour, so there is a "time orientation" of the path. It's not enough to specify the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1958347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is it possible to elementarily parametrize a circle without using trigonometric functions? Just out of curiosity: Is it possible to parametrize a full circle or part of one with elementary functions but without using trigonometric functions? If so, what are advantages/disadvantages compared to the standard parametrizat... | What about $f(x,\pm)=\pm\sqrt{1-x^2}$, where $f(\cdot,\cdot)$ has a discrete and continous parameter defined in $[-1,1]$...
You may also use $e^{it}=\cos(t)+i\sin(t)$ to represent a circle in the complex plane. With this calculating Fourier transforms becomes handy...
Just a comment to H.H. Rugh answer that needs grap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1958573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 3,
"answer_id": 2
} |
Model theory: What is the signature of `Category theory` I'm studying model theory nowadays, and I understand how one-sorted (classical) signatures and structures work. However I am also interested in groupoids, which can not be described as a structure for a one-sorted signature.
Looking up online, I came to the notio... | Just to give you a name to search for: Categories are models for an essentially algebraic theory. Because they require partially defined functions, essentially algebraic theories don't fit into the standard formalism of model theory.
But, as described in Eric Wofsey's answer, they can be simulated in many-sorted logic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1958695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Why is $\int_{-\infty}^\infty |g(x)|\,dx = \int_0^\infty \mu(\{x : |g(x)| \ge t\})\,dt$ true? Why do we have the following equality$$\int_{-\infty}^\infty |g(x)|\,dx = \int_0^\infty \mu(\{x : |g(x)| \ge t\})\,dt,$$where $\mu$ is Lebesgue measure?
| We claim that for a nonnegative measurable function $g:\Bbb{R}\to[0,\infty)$,
$$
\int_\Bbb{R}g(x)\ d\mu(x)=\int_{[0,\infty)}\mu(\{x\in\Bbb{R}\mid g(x)\geq s\}) \ d\mu(s).
$$
This is a good example of applications of the Fubini-Tonelli's Theorem.
Let $\nu:=g_*\mu$ be the pushforward of $\mu$, i.e., $\nu=\mu\circ g^{-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1958794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proving a graph has no Hamiltonian cycle Show that $ G = (V, E)$ has no Hamiltonian cycle, where the vertices are $ V = \{ a, b, c, d, e, f, g \} $ and
the edges are $E = \{ ab, ac, ad, bc, cd, de, dg, df, ef, fg \}$.
From my working out, the vertices $ a, b, c, d, e, f$ are odd degrees of 3 and 1. Moreover $g $ being ... | As discussed in the comments, the three points are not definitions. They are just handy facts you can use to show that a graph is not Hamiltonian. If the facts don't apply to a given graph, it doesn't imply that it is Hamiltonian either - the test is just inconclusive.
Fortunately enough, we can use facts 2 and 3 to pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1958887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Show that the equation $|z-z_0|=R$ of a circle centered at $z_0$ of radius $R$ can be written as $|z|^2-2\text{Re}(z\bar{z_0})+|z_0|^2=R^2$. Show that the equation $|z-z_0|=R$ of a circle centered at $z_0$ of radius $R$ can be written as $|z|^2-2\text{Re}(z\bar{z_0})+|z_0|^2=R^2$.
I tried squaring both sides, but then ... | We have that
$$R^2=|z-z_0|^2=(z-z_0)\cdot \overline{(z-z_0)}=(z-z_0)\cdot (\overline{z}-\overline{z_0})\\
=z\cdot \overline{z}-z\cdot \overline{z_0}
-\overline{z}\cdot z_0 +z_0\cdot\overline{z_0}\\
=|z|^2-\left(z\cdot \overline{z_0}+\overline{z\cdot \overline{z_0}}\right)+|z_0|^2\\
=|z|^2-2\mbox{Re}\left(z\cdot \overl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1959025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$
Let $a,b,c,d>0$. Prove that $$(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$$
I don't know how to begin to solve this problem
| We can assume that $a+b+c+d=2$. Then the inequality becomes
$$
(1-d)(1-c)(1-b)(1-a)\le\left(1-\tfrac{c+d}2\right)\left(1-\tfrac{a+d}2\right)\left(1-\tfrac{a+b}2\right)\left(1-\tfrac{c+b}2\right)\tag{1}
$$
If any of $a$, $b$, $c$, or $d$ is greater than $1$, then the left side is negative and the inequality is trivial. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1959149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Given $\frac1{x+y+z}=\frac1x+\frac1y+\frac1z$ what can be said about $(x+y)(y+z)(x+z)$?
If $\frac1{x+y+z}=\frac1x+\frac1y+\frac1z$ where $xyz(x+y+z)\ne0$, then the value of $(x+y)(y+z)(z+x)$ is
(A) zero
(B) positive
(C) negative
(D) non-negative
I substituted $x=-y$ and the equality was established. In the given expr... | $$\frac1{x+y+z}=\frac1x+\frac1y+\frac1z$$
$$\to \frac1{x+y+z}=\frac{yz+xz+xy}{xyz}$$
$$\to xyz=(yz+xz+xy)(x+y+z)$$
$$\to xyz=(yz+xz+xy)(x+y+z)$$
$$\to xyz=(x+y)(y+z)(x+z)-xyz$$
$$\to 0=(x+y)(y+z)(x+z)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1959364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Dimensions of submanifolds of SO(n) I would like to calculate the dimension of
\begin{align*}
\mathcal{M}_k=\{R\in\mathsf{SO}(n,\mathbb{R})\,|\,\sigma(R)=\{-1,1\},\,m(-1)=k\},
\end{align*}
where $\sigma$ is the spectrum and $m$ is the algebraic multiplicity for all $k=0,\ldots,\lfloor\frac{n}{2}\rfloor$. Clearly $\dim\... | $\newcommand{\Reals}{\mathbf{R}}\newcommand{\calm}[1][k]{\mathcal{M}_{#1}}$Hints: Since $\det R = (-1)^{k}$ for all $R$ in $\calm$, the index $k$ is even.
Each element of $\calm$ determines a splitting $\Reals^{n} = E_{-1} \oplus E_{1}$ into eigenspaces, of respective dimension $k$ and $n - k$. Conversely, each splitti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1959466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is it true that one should not write the universal quantifier behind a statement? We all know that there are different ways to say that e.g. an element $x$ belongs to each member of a family of sets $(A_j)_{j \in J}$ for some index set $J$. The most common ways I know are the following:
*
*$\forall j \in J \colon x ... | It depends whether you are writing on a blackboard or in a formal article. In a formal article, I would write
$x$ belongs to $A_j$ for all $j \in J$
to have a fluent sentence. See Halmos' recommendations on How to write mathematics.
On a blackboard, I may simply write
$x \in \bigcap_{j\in J} A_j$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1959606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proof of limits as one function goes to zero and one is bounded Suppose that $$ \lim_{x\to +\infty} f(x) = 0$$ and $$g(x)$$ is a bounded .
Show that $$\lim_{x\to +\infty}f(x)g(x)=0$$
Thanks in advance
| $$\lim _{ x\to +\infty } f(x)=0\Rightarrow \quad \forall x\in R,\exists M>0\quad \left| f\left( x \right) \right| \le M\\ \forall x\in R,\exists \frac { \epsilon }{ M } >0\quad \left| g\left( x \right) \right| \le \frac { \epsilon }{ M } \\ \left| f\left( x \right) g\left( x \right) \right| \le \left| f\left( x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1959736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Expressing a recurrence relation as a polynomial
Let us define $u_0 = 0, u_1 = 1$ and for $n \geq 0$, $u_{n+2} = au_{n+1}+bu_n$, $a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b$. Prove the result: Given that $b$ is prime, prove that $b$ divides $a(u_b-1)$.
How do we deal with the case... | Let's see if we find a pattern.
\begin{align}
u_0&=0\\
u_1&=1\\
u_2&=au_1+bu_0=a\\
u_3&=au_2+bu_1=a^2+b\\
u_4&=au_3+bu_2=a(a^2+b)+ab=a^3+2ab\\
u_5&=au_4+bu_3=a(a^3+2ab)+b(a^2+b)=a^4+3a^2b+b^2
\end{align}
It seems reasonable to assert that, for $n\ge2$,
$$
u_n=a^{n-1}+bf_n(a,b)
$$
where $f_n(a,b)$ is a polynomial in $a$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1959837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Queen’s random walk (Queen’s random walk). A queen can move any number of squares horizontally,
vertically, or diagonally. Let Xn be the sequence of squares that results if we
pick one of queen’s legal moves at random.
$(a)$ Find the stationary distribution
$(b)$ Find the expected number of moves to return to corner $... | The stationary distribution is defined as the normalized number of moves from a given position. In symbols, for a given position $x $ it is $\frac {deg (x)}{\sum_{x∈S} deg(x)}$, where $deg $ indicates the number of possible moves from $x $.
For a queen on a chessboard, if it is on any of the $28$ squares adjacent to t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1959948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why does solving this equation a certain way yield only complex roots instead of real ones? For the system of equations
$$
4x^2 + y = 4,\quad
x^4 - y = 1
$$
if I attempt to solve by solving each equation for $y$ and setting them equal to each other, I obtain
$$
4 - 4x^2 = x^4 - 1
$$
$$
-4(x^2 - 1) = (x^2 - 1)(x^2 + 1)
... | You divided by $x^2-1$ in the third step. That expression is not necessarily non-zero..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Roots of Unity Filters Suppose we want to evaluate
$$\sum_{k\geq 0} \binom{n}{3k}$$
This can be done using roots of unity filters, i.e. showing the sum is equivalent to:
$$\frac{(1+1)^n+(1+\omega)^n+(1+\omega^2)^n}{3}$$
where $\omega$ is a primitive 3rd root of unity.
Using the fact that $1+\omega+\omega^2=0$, we can s... | $$\sum \binom{n}{3k+1} = \frac{1^2 (1+1)^n + \omega^2(1+\omega)^n + \omega(1+\omega^2)^n}{3}$$
Basically, apply the same approach to $f(x)=x^2(1+x)^n$.
Similarly, taking $g(x)=x(1+x)^n$ we get:
$$\sum\binom{n}{3k+2}=\frac{1(1+1)^n + \omega(1+\omega)^n + \omega^2(1+\omega^2)^n}{3}$$
You should be able to get nice formul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Calculating $\sum_{n=1}^∞ \frac{1}{(2 n-1)^2+(2 n+1)^2}$ using fourier series of $\sin x$ I have to calculate $\frac{1}{1^2+3^2}+\frac{1}{3^2+5^2}+\frac{1}{5^2+7^2}+...$ using half range Fourier series $f(x)=\sin x$ which is:
$f(x)=\frac{2}{\pi}-\frac{2}{\pi}\sum_{n=2}^\infty{\frac{1+\cos
n\pi}{n^2-1}\cos nx}$
I hav... | A different approach. Since $\frac{1}{(2n-1)^2+(2n+1)^2}=\frac{1}{8n^2+2}$ we have:
$$\begin{eqnarray*} \sum_{n\geq 1}\frac{1}{(2n-1)^2+(2n+1)^2}&=&\frac{1}{8}\sum_{n\geq 1}\frac{1}{n^2+\frac{1}{4}}\\&=&\frac{1}{8}\int_{0}^{+\infty}\sum_{n\geq 1}\frac{\sin(nx)}{n}e^{-x/2}\,dx\\&=&\frac{1}{8(1-e^{-\pi})}\int_{0}^{2\pi}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Galois group of a splitting field of a polynomial over $\mathbb{F}_7$ Question: Find the Galois group of the splitting field $(x^2-3)(x^3+x+1)$ over $\mathbb{F}_7$.
I know the splitting field is $K:=\mathbb{F}_7(\sqrt{3},\alpha_1)$, where $\alpha_1$ is one of the roots of the polynomial $x^3+x+1$. I know that the poss... | Note that the Galois group is some subgroup of the direct product of the Galois groups of each factor considered individually. Since the splitting field of $x^2 - 3$ over $\Bbb{F}_7$ has degree two, the splitting field of $x^3 +x+1$ has degree three, and the degrees are coprime the splitting field of their product has ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
The number of solutions to a system of linear equations Can anyone suggest a formal proof that a system of linear equations can have no solution, one solution or infinitely many solutions?
| Consider a linear system in $\mathrm x \in \mathbb R^n$
$$\mathrm A \mathrm x = \mathrm b$$
where $\mathrm A \in \mathbb R^{m \times n}$ and $\mathrm b \in \mathbb R^m$ are given. Suppose that the system is feasible and that $\mathrm x^{(1)}$ and $\mathrm x^{(2)}$ are two solutions. Hence, $\mathrm A \mathrm x^{(1)} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
About the parity of the product $(a_1-1)(a_2-2)\cdots(a_n-n)$ An exercise from Chapter 20 of "How to Think Like a Mathematician" by Kevin Houston:
Let $n$ be an odd positive integer. Let $(a_1,a_2,\dots,a_n)$ be an arbitrary arrangement (i.e., permutation) of $(1,2,\dots,n)$. Prove that the product $(a_1-1)(a_2-2)\cdo... | It suffices to show that one of the terms $a_k-k$ is even.
For that to fail, we would need each $a_k$ be of a parity distinct to that of $k$.
Since $n$ is odd, in $\{1,2,\dots, n\}$ there are $\frac{n+1}2$ odd numbers and $\frac{n-1}2$ even numbers.
But in order for all the $a_k$ to be a distinct parity that of $k$, we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 5
} |
Is an expanding map on a compact metric space continuous? I got inspired by this question Existence of convergent subsequence to think about the following problem: Suppose you have a compact metric space $(X,d)$ and an expanding map $T:X\rightarrow X$, i.e. $d(Tx,Ty)\geq d(x,y)$ for every $x,y\in X$. Is the map $T$ the... | Question: Let $(X,d)$ be a compact metric space and let $f:X\to X$ be a map such that $d(f(x),f(y))\geq d(x,y)$ for all $x,y\in X$. Show that $f$ is an isometry onto $X$.
Solution: First we will see that $f$ is an isometry, then that $f$ is onto $X$.
1. Given a (small) $r>0$, let $K_n(r)$ be the set of $n$-tuples $(x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
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