Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Power series, how to shift the index of a summation with a powers 2n+1 I am taking an ordinary differential equation class, and we are currently learning about power series. One thing that comes up is index shifting, for the most part, I can shift the index quite easily, but in the following case, I end up with a fract... | Hint
you can downshift or upshift.
$$\sum_{n=k}^N f(n)=$$
$$\sum_{n=k-1}^{N-1}f(n+1)=\sum_{n=k+1}^{N+1}f(n-1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
line integral hard function to differentiate $$\int_\gamma \frac{(x^2+y^2-2)\,dx+(4y-x^2-y^2-2) \, dy}{x^2+y^2-2x-2y+2}$$
where $\gamma = 2\sin\left(\frac{\pi x}{2}\right)$ from $(2,0)$ to $(0,0)$.
I think it should be a shortcut to this problem that I cannot see , if that is not the case I will keep trying to simplify... | Complete the squares in the denominator $(x-1)^2+(y-1)^2$ and change the variables $x-1\mapsto x$ and $y-1\mapsto y$. You get the vector field
\begin{align}
&\left[\frac{x^2+2x+y^2+2y}{x^2+y^2},\frac{4y-x^2-2x-y^2-2y}{x^2+y^2}\right]=\\
&=\left[1+\frac{2x}{x^2+y^2},-1+\frac{2y}{x^2+y^2}\right]-2\left[\frac{-y}{x^2+y^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Probability of a random graph being triangle-free
Let $S$ be the set of all graphs with vertex set $\{1,2,\ldots,n\}$. A random graph $G\in S$ has probability $2^{-{n \choose 2}}$. Show that a random graph almost surely contains a triangle.
My attempt so far: we want to show that as $n$ goes to infinity, the probabil... | Let me preface this answer with the following: The overestimation of triangles is a much better argument for your particular problem as the probability of an edge appearing is fixed for all $n$. If you allow $p=p(n)$ then you have to be more careful. The technique that follows is technical, but it can be adapted very e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
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Expression for every combination by multiple sets I'd like to express every combination represented using union or intersection between multiple sets.
For example, we have 3 sets $A_1,A_2,A_3$, the following combinations can be generated by unions and/or intersections between $A_1, A_2, A_3$.
$$A_1 \setminus (A_2 \cup ... | Your expression is correct. Given any point $x\in\bigcup_{i=1}^n A_i$, let $N$ be the (necessarily nonempty) set of all $1\le i\le n$ for which $x\in A_i$; then $x$ is in the set $\bigcap_{r\in N} A_r \setminus \bigcup_{s\in N^c} A_s$ for this $N$ and for none of the others. That proves your disjoint union claim, and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Finding lengths in a diagram. Find $x$ and $y$.
Given that:
$$a_1+a_2+a_3-a_4+a_5=180^\circ$$
$$\text{cos}(180^\circ-a_5)=0.4$$
I managed to solved this problem using Mathematica based on an equation with whole bunch of ArcTan. However, I am looking for an easier way to solve it based on its geometrical properties.
Th... | $$x=t\cos a_5,\\y=t\sin a_5 $$ where $a_5$ is given by the second equation.
Then there remains a sum of four angles which can indeed be expressed as arc tangents of terms $t_k:=y/(x-x_k)$.
As
$$\tan(a_1+a_2+a_3-a_4)=\frac{\tan(a_1+a_2)+\tan(a_3-a_4)}{1-\tan(a_1+a_2)\tan(a_3-a_4)}
=\frac{\dfrac{t_1+t_2}{1-t_1t_2}+\dfrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can we show $\cos^6x+\sin^6x=1-3\sin^2x \cos^2x$? How can we simplify $\cos^6x+\sin^6x$ to $1−3\sin^2x\cos^2x$?
One reasonable approach seems to be using $\left(\cos^2x+\sin^2x\right)^3=1$, since it contains the terms $\cos^6x$ and $\sin^6x$. Another possibility would be replacing all occurrences of $\sin^2x$ by $1... | Any symmetric polynomial in $X$ and $Y$ can be expressed as a polynomial in $S=X+Y$ and $P=XY$. If $X=\cos^2x$ and $Y=\sin^2x$, then $S=\cos^2x+\sin^2x=1$, so a symmetric polynomial expression in $\cos^2x$ and $\sin^2x$ can be written as a polynomial in $P=\cos^2x\sin^2x$.
If the symmetric polynomial is also homogeneou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2023931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 11,
"answer_id": 8
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Show that $1^3 + 2^3 + ... + n^3 = (n(n+1)/2)^2$ by induction I have some with proving by induction. I cannot find a solution for the inductive step: $1^3 + 2^3 + ... + n^3 = (n(n+1)/2)^2$
I already did the induction steps:
Basis: P(1) = $1^3 = (1(1+1)/2)^2$ (This is true)
Inductive step: Assume $P(k) = ((k)(k+1)/2)^2$... | \begin{align}
\underbrace{1^3+2^3+\ldots+n^3}_{\left[\frac{n(n+1)}{2} \right]^2}+(n+1)^3
=&
\left[\frac{n(n+1)}{2} \right]^2+(n+1)^3
\\
=&
\frac{n^2\color{red}{(n+1)^2}}{4}
+
(n+1)\color{red}{(n+1)^2}
\\
=&
\left\lgroup
\frac{n^2}{4}
+
(n+1)
\right\rgroup
\color{red}{(n+1)^2}
\\
=& \left\lgroup \frac{n^2 +4(n+1)}{4} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2024028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Probability of drawing exactly 13 black & 13 red cards from deck of 52 We have a normal deck of $52$ cards and we draw $26$. What's the probability of drawing exactly $13$ black and $13$ red cards?
Here's what I have so far. Consider a simplified deck of $8$ (with $4$ $B$'s and $4$ $R$'s), we have 6 permutations of $BB... | The OP should be commended for approaching the problem by first thinking about a smaller analog that's easy to solve explicitly -- and then for rejecting an idea because it gives the wrong answer for the smaller analog. That is exactly the right thing to do when faced with a problem that seems complicated by its very ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2024175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Infimum of probability and probability of infimum I am studying the Borel Cantelli proof and there is the following step:
$$\Pr\left( \bigcap \limits_{N=1}^{\infty} \bigcup\limits_{n=N}^{\infty}E_n\right) \le \inf_{N\ge1} \Pr\left( \bigcup\limits_{n=N}^{\infty} E_n\right)$$
What happened here? I guess that:
$$\Pr\left(... | Note that for any positive integer $K$ you have the inclusion $\bigcap \limits_{N = 1}^\infty \bigcup \limits_{n = N} E_n \subset \bigcup \limits_{n = K}E_n$. Using the monotonicity of measures we can deduce $P\left(\bigcap \limits_{N = 1}^\infty \bigcup \limits_{n = N} E_n \right) \le P\left(\bigcup \limits_{n = K}E_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2024268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Pointwise convergence doesn't imply $L^p$ convergence if $p=\infty$ under some hypothesis I just proved that if $1\leq p<\infty$ and $f,f_n$ is a sequence of measurable functions such that $f_n(x)\rightarrow f(x)$ a.e $x\in X$ and $\exists g\in L^p(\mu)$ such that $|f_n(x)|\leq g(x)$ a.e $x\in X$, then $f_n\rightarrow ... | When $p=+\infty$, the statement reads as follows:
If $\left(f_n\right)_{n\geqslant 1}$ is a sequence of measurable functions such that $f_n(x) \to f(x)$ almost everywhere and there exists a constant $M$ such that $\left|f_n(x)\right|\leqslant M$ for every $n$ and almost every $x$, then $\lVert f_n-f\rVert_{\mathbb L^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2024411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove: If $p$ is prime and $ (a,p)=1 $, then $\ 1 + a + a^2 + ... + a^{p-2}=0$(mod $p$) Prove: If $p$ is prime and $ (a,p)=1 $, then $\ 1 + a + a^2 + ... + a^{p-2}≡0 \pmod p$
As an example, for $a=2$ and $p=5$, then:
$$1+2+2^2+2^3=1+2+4+8=15≡0 \pmod 5$$
This can also be written as:
$$1 \pmod 5 + 2 \pmod 5 + 4 \pmod 5 +... | If $p=2$, then $a^{p-2}=a^0=1$, so in this case is false, because $1\not\equiv 0(mod \ 2)$.
If $p\neq 2$, then $1+a+\dots+ a^{p-2}=\frac{a^{p-1}-1}{a-1}$. From the fermat theorem, if $(a,p)=1$, then $a^{p-1}\equiv 1 (mod\ p)$ (if this doesn't happen, maybe the $p$ factor of $a^{p-1}-1$ will dissapear), so you need now ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2024524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
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A bijective arcwise isometry is an isometry? Let $X,Y$ be length spaces. Suppose $\, f:X \to Y$ is an arcwise isometry, i.e $L_X(\gamma) = L_Y(f \circ \gamma)$ for every continuous path $\gamma:I \to X$. (In particular, $f$ takes non-recitifiable paths to non-recitifiable paths).
In addition, assume $f$ is a bijection.... | Hint: consider the age old counterexample $f : [0,2\pi) \to S^1$ given by $f(t) = (\cos(t),\sin(t))$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2024623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Convexity Geometric Brownian Motion Let $B(t)$ be a standard Brownian motion with filtration $\{ \mathcal{F}_t: t \geq 0\}$ and $\mu, \sigma >0$ are real parameters. $S(t)$ is modeled by
\begin{align}
S(t) = e^{\mu t + \sigma B(t)- \frac{1}{2}\sigma^2 t}.
\end{align}
In order to find an expression for $\mathbb{P}\big(\... | By taking the logarithm on the left and right side we have
$P(S(2t)>2S(t))=P(B_{2t}-B_t>\frac{log(2)-\mu t+0.5\sigma^2t}{\sigma})$
We know that $B_{2t}-B_t$ is normally distibuted with mean 0 and variance t, therefore we can replace $B_{2t}-B_t$ by $\sqrt{t}Y$ where Y is centered and normally distributed.
Finally ,
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2024737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Continuous function and Borel sets If $f:\mathbb{R}^p \rightarrow \mathbb{R}$ is continuous and $B\subset \mathbb{R}$ is Borel set, how to show that $f^{-1}(B)$ is also Borel set?
I was trying to construct a $\sigma$-ring of sets whose preimage is Borel, but they're not necessarily all borel in that sigma ring...
What ... | A Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement.
As the inverse images of open sets under a continuous function are open sets and inverse images of a countable union is the countable union of the i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2024867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Finding $\lim_\limits{x \rightarrow +\infty}\frac{\log_{1.1}x}{x}$ analytically
$$\lim_{x \rightarrow +\infty}\frac{\log_{1.1}x}{x}$$
I can solve this easily by generating the graph with my calculator, but is there is a way to do this analytically?
| Use the definition: $\;\log_{1.1}x=\dfrac{\ln x}{\ln 1.1}$, so
$$\frac{\log_{1.1}x}{x}=\frac 1{\ln 1.1}\dfrac{\ln x}{x}\xrightarrow[x\to+\infty]{}\frac 1{\ln 1.1}\cdot 0=0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2024997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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Find the ordinary generating funtion for thenumber of at most binary trees on n unlabelled nodes. An at most binary tree is an unordered rooted tree in whic each node has at most two children.
I want to find the ordinary generating function for the number of at most binary trees on n unlabelled nodes.
I am trying to s... | Your $t_n$ are the Wedderburn–Etherington numbers, OEIS A001190. Specifically, $t_n=a_{n+1}$, where the $a_n$ are the Wedderburn-Etherington numbers. The OEIS entry has copious references but neither a closed form nor an explicit generating function. It does note that the generating function $A(x)$ for the $a_n$ satisf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Throwing darts probability Assume a sequence from $1$ to $n$, with $e_i$ denoting the $i$-th element. We throw a dart
at random into the array: if we hit $e_i$ or $e_j$ then $X_{i,j}$ becomes $1$, if we hit between $e_i$ and $e_j$ then
$X_{i,j}$ becomes 0, and otherwise we throw another dart. Once $X_{i,j}$ is assigned... | The game is a sequence of trials, each one of three states, $A_t,B_t,C_t$. State $A_t$ is that $\{X_{i,j}=1\}$ , state $B_t$ that $\{X_{i,j}=0\}$ and state $C_t$ that the value is undefined, for a given trial of the game, $t$.
Assuming each section of the array is equally likely to be hit, and the space "between" $e_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Should I study projective geometry or commutative algebra as prerequisite to start algebraic geometry? I am looking to study Algebraic Geometry but some books list projective geometry as a prerequisite and some list commutative algebra.
I have taken one semester of abstract algebra, real analysis, complex analysis, top... | Marco Flores has given a very good answer.
I would like to echo one aspect of his answer to suggest that if one wants to learn algebraic geometry, one can just start learning it. There are several good books available beginning at an undergraduate level, such as Reid's book that Marco Flores mentions. Later on, one... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Finding $n$ for the given probability. A certain explosive device will detonate if anyone of $n$ short-lived fuses lasts longer than $0.8$ seconds. Let $X_i$ represent the life of the $ith$ fuse. It can be assumed that each $X_i$ is uniformly distributed over the interval $(0,1)$ , also $X_i $'s are independent.
We nee... | Probability of a fuse lasts less than $0.8$ seconds is $0.8$. The probability of none of $n$ independent fuses last longer than $0.8$ seconds is therefore $0.8^n$. So the probability of at least one of them last longer than $0.8$ becomes $1-0.8^n$ which should be greater than or equal to $0.95$.
That is $$1-0.8^n\ge0.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving a Lebesgue integral exists: $\int_0^{\pi/2}\left(\frac{1}{2(e^x -1)}-\frac{1}{\tan(x)\sin(x)}\right)d\lambda(x)$ Can you tell me how I would go about proving that the Lebesgue integral $$\int_{0}^{\pi/2}\left(\frac{1}{2(e^x -1)} - \frac{1}{\tan(x)\sin(x)}\right)d\lambda(x)$$ exists? I've already shown that $\df... | As written, the given integral is divergent since the integrand, as $x \to 0^+$, admits the Laurent series expansion
$$
\frac{1}{2(e^x -1)} - \frac{1}{\tan(x)\sin(x)}=-\frac1{x^2}+\frac1{2x}-\frac1{12}+O\left(x\right)
$$ which is not Lebesgue integrable over $(0,\pi/2)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Applications of words? What are some real-life applications of (Sturmian) words? I'm doing an undergraduate thesis on the Fibonacci infinite word $f$, and although what I'm doing is purely theoretical (by counting maximal occurrences of factors of $f$), I want to put in the introduction a one- or two-sentence applicati... | Let me quote the introduction of J. Berstel, Sturmian and Episturmian Words. A Survey of Some Recent Results,
Algebraic Informatics LNCS 4728, pp 23-47 (2007)
Sturmian words have a geometric description as digitized straight lines.
Computer representation of lines has been an active subject
of research, although e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Path on the torus
Prove that the following map is smooth and analyse its image as the real number $\alpha$ varies (how does it look like? Is it a submanifold?)
\begin{align*}
f_{\alpha}:\mathbb{R}&\to\mathbb{S}^1\times\mathbb{S}^1\\
t&\mapsto (e^{2\pi it}, e^{2\pi \alpha it})
\end{align*}
I've managed to prove that ... | Should it be $f_{\alpha}(t)=(e^{2\pi i t}, e^{2\pi \alpha i t})$ ?
If so, sure it is a manifold when $\alpha\in\mathbb Q$ (with the proper identification it is simply a curve on $\mathbb R^2$. In this case, it is a curve that spirals around the torus until it closes, much like in the figure:
When $\alpha\not\in\mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Arranging colored balls in a line. 2 balls from 5 distinct colors are collected.
*
*In how many ways can the balls be arranged?
*In how many ways can the balls be arranged so that no two balls of the same color are next to one another?
The first is easy. If I have 10 spots and 10 balls, then I have 10 choices in... | For the first case, we could arrange the balls in $10!$ different ways if they were unique. However, they aren't unique. For each color, there are two balls. Therefore, we can swap any of the balls that are the same color and have the same arrangement, so there are $2!$ different arrangements that are the same for ea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2025990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Polynomial division in $\Bbb Z/n\Bbb Z$ So, I know how polynomial division works in principle, but I have currently no Idea what is asked here:
We have to divide two polynomials: f = $4t^4-2t^3-3$ and g = $2t^2-3$ but in the polynomial ring $F_{p}[t]$ with p prime. (F = $\mathbb{Z/pZ}$).
So how does the algorithm for ... | The high-school polynomial division algorithm works over any commutative coefficient ring, as long as the divisor $g$ has invertible leading coefficient. Then the leading term of $g$ always divides any monomial of higher degree, thus the high-school division algorithm works to kill all higher degree terms in the divide... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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projection matrix using A = QR in a projection matrix where P = $A(A^TA)^{-1}A^T$, if you allow A=QR where Q is orthogonal and R is reversible, P can be then expressed as P = $QQ^T$. I can follow all those steps, but I seem to also be able to come up with an incorrect solution, and can't figure out what rule I'm violat... | Hint: you are inverting $Q$. Is this well-defined for nonsquare matrices?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2026348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Change of mean for a standard normal variable Beginning to learn a bit of probability theory and I'm a little confused with this one. I know that given a Standard Normal variable X
$X \sim \mathcal{N}(0, 1)$, if $Y = X + c$ then $Y \sim \mathcal{N}(c, \sigma^2)$
Now if,
$Y = aX+b $, is $Y \sim \mathcal{N}($b/a$, \sigm... | Let $X \sim \mathcal{N}(0, 1)$. Then, $E[X+c] = E[X] + E[c] = E[X] + c = c$. Also, $\text{Var}[X+c] = \text{Var}[X] = 1. $ So $Y \sim \mathcal{N}(c,1)$.
If instead we have $Y=aX+b$, then $E[Y] = E[aX+b] = E[aX] + E[b] = a E[X] + b = a \cdot 0 + b$. Also, $\text{Var}[aX+b] = \text{Var}[aX] = a^2 \text{Var}[X] = a^2 $. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2026474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Deformation retraction restricted to a path Let $X$ be a topological space with subspace $Y$. Suppose that we have a (strong) deformation retraction onto $Y$. That is, a continuous map $F:X\times [0,1]\to X$ such that for all $x\in X$, $F(x,0)=x$, $F(x,1)\in Y$, and for all $y\in Y$ and all $t\in [0,1]$, $F(y,t)=y$.
Is... | Thank you for your answers. Indeed I now realize that the conjecture is false. In fact, I came up with a quite a silly (but easy) counterexample myself:
Let $X=\{a,b,c\}$ with trivial topology, $Y=\{a,b\}$. Define $F:X\times [0,1]\to X$ (continuous because $X$ has the trivial topology) by
$$
F(a,t)=a,\quad F(b,t)=b \qu... | {
"language": "en",
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"question_score": "1",
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"answer_id": 2
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Homeomorphism of the cone on a space The unreduced cone $\text{Cone}(X)$ on a space $X$ is given by $$ \text{Cone}(X)=X\times I/ X\times \{1\}$$ where $I$ is a unit interval.
Show that Cone($S^{n-1}$) is homeomorphic to $D^{n}$ where $S^{n-1}=\left\{x\in R^{n}| \left\| x \right\|=1 \right\}$ and $D^{n}=\left\{x\in R^{n... | Let $p$ denote the apex of the cone $\text{Cone}(S^{n-1})$. For each $x\in S^{n-1}$ and each $t\in[0,1]$ let $c(x,t)\in \text{Cone}(S^{n-1})$ satisfying
$$ c(x,t)=(1-t)p+tx$$
Then let $f: \text{Cone}(S^{n-1})\to D^n$ be defined by
$$ f(c(x,t))=tx $$
This continuously maps $\text{Cone}(S^{n-1})$ onto $D^n$ and its inve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2026728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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$\mathbb R^3$ minus a line is connected. Let $S\subseteq \mathbb R^3$ be homeomorphic to $\mathbb R$. Prove that $\mathbb R^3 \setminus S$ is connected.
I haven't been able to solve this, although my topology skills are pretty weak. My friend told me he managed to prove this using results from his "dimension topology" ... | Here's a nice hint: The zero-th homology $H_0(\mathbb{R}^3 - S)$ must be a direct sum of $n$ copies of $\mathbb{Z}$, where $n$ denotes the number of connected components of $\mathbb{R}^3 - S$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2026842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
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"answer_id": 2
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Using ZFC axioms, prove that the set $\{ \emptyset \}$ exists. As the title describes, I want to prove that the set $\{ \emptyset \}$ exists, using ZFC axioms. I have an answer that I wish to check if I understood ZFC correctly.
Is it that simple as:
1) The empty set axiom - There is a set having no elements. we get $... | This really depends on what axioms you have at your disposal.
Your suggestion works, if you have the empty set axiom and power set. You can also use the empty set and pairing. You can also use just the axiom of infinity (well, an extensionality, you can't do anything without extensionality!):
Let $A$ be an inductive se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2026971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Real-valued function with strictly positive integral on every subinterval of [0, 1] Assume we have a Riemann-integrable function $h: [0, 1] \to \mathbb R$ such that
\begin{equation}
\int_a^b h(x)dx > 0\quad \forall a, b: 0 \leq a < b \leq 1.
\end{equation}
Further assume that $f, g: [0, 1] \to \mathbb R$ are continuous... | I have found a proof that works even under weaker conditions. $f$ and $g$ need to be merely Riemann-integrable and $\int_a^b h(x) \, dx \ge 0$ needs to hold (i.e. the integral does not have to be strictly positive).
We can assume wlog $g \equiv 0$.
Let $Z_n$ be a sequence of meshes of $[0, 1]$ with $|Z_n| \to 0$. Since... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2027073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Finitely generated as a field vs. as an algebra? Let $K$ be a field, $L$ a $K$-algebra. For $x_1, \ldots, x_n \in L$, denote $K[x_1, \ldots, x_n]$ for the smallest subalgebra of $L$ containing $K$ and $x_1, \ldots, x_n$. If $L$ is a field, denote $K(x_1, \ldots, x_n)$ for the smallest subfield of $L$ containing $K$ and... | No. For example, $\;L=\Bbb Q(\pi)\;$ is a field, but it can't be that $\;L=\Bbb Q[\alpha]\;$ , as:
(1) If $\;\alpha\;$ is algebraic over $\;\Bbb Q\;$ then $\;\Bbb Q[\alpha]=\Bbb Q(\alpha)\neq\Bbb Q(\pi)\;$ , as the last one is infinite dimensional over $\;\Bbb Q\;$ whereas the first one is finite dimensional, and
(2) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof why $\frac{1}{n^x}+\frac{1}{(n+1)^x}=\frac{H_{{n+1,x}}}{n^xH_{n,x}}+\frac{H_{{n-1,x}}}{(n+1)^xH_{n,x}}$ I found this in a not very straightforward way, and it seems like a rather strange-looking identity, but there is probably a simple proof.
$$\frac{1}{n^x}+\frac{1}{(n+1)^x}=\frac{H_{{n+1,x}}}{n^xH_{n,x}}+\frac{... | Using the simple computation
$$ \frac{H_{n,x}}{n^x}+\frac{H_{n,x}}{(n+1)^x}=\frac{H_{n+1,x}-(n+1)^{-x}}{n^x}+\frac{H_{n-1,x}+n^{-x}}{(n+1)^x}=\frac{H_{n+1,x}}{n^x}+\frac{H_{n-1,x}}{(n+1)^x}, $$
the result follows immediately.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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System of four equations of four variables including second powers. I've been tasked with solving the following system of equations and it seems like I am stuck:
$$a-x^2=y$$$$a-y^2=z$$$$a-z^2=t$$$$a-t^2=x,$$where $a$ is a real number, for which $0\leq a\leq 1$. I thought the best way would be to subtract some equations... | Because of the symmetry, it is natural to assume $x=y=z=t$, which gives you an easily solvable quadratic. The solutions are $\frac 12(-1 \pm \sqrt{4a+1})$, which are real when $a \ge -\frac 14$, covering your region of interest. Another approach is to assume $x=z, y=t$, which gives $a-(a-x^2)^2=x$ and the additional ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2027492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Equivalence of convergence of a series and convergence of an infinite product
Let $(a_n)_n$ be a sequence of non-negative real numbers.
Prove that $\sum\limits_{n=1}^{\infty}a_n$ converges if and only if the infinite product $\prod\limits_{n=1}^{\infty}(1+a_n)$ converges.
I don't necessarily need a full solution, jus... | A way is to use logarithm and the equivalence
$$\ln(1+x) \sim x$$ around the origin.
Interesting to notice is that for complex series (or series not having a constant sign) $\sum z_n$, the absolute convergence of the series implies the convergence of the product $\displaystyle \prod (1+z_n)$, but the converse is not al... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Bode plot of the open loop given the state space - SIMO I have a SIMO system. $\xi$ is the input and $Y$ is the output. The state space model is given by
\begin{align}
\dot{X} &= AX + B\xi\\
y&=r-Cx
\end{align}
$A$ is $5 \times 5$ matrix. $B$ is $5 \times 1$. The controller $K$ is a state feedback controller such that ... | Given a state space model of the following form,
$$
\dot{x} = A\,x + B\,u, \tag{1}
$$
$$
y = C\,x + D\,u. \tag{2}
$$
The openloop transfer function of this system can be found by taking the Laplace transform and assuming all initial conditions to be zero (such that $\mathcal{L}\{\dot{x}(t)\}$ can just be written as $s\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2027809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\forall n\in\mathbb N$ prove that at least one of the number $3^{3n}-2^{3n}$,$3^{3n}+2^{3n}$ is divisible by $35$. $3^{3n}-2^{3n}=27^n-8^n=(27-8)(27^{n-1}+27^{n-2}\cdot 8+...+27^1\cdot8^{n-2}+8^{n-1})$
If $n$ is even,
$3^{3n}+2^{3n}=27^n+8^n=(27+8)(27^{n-1}-27^{n-2}\cdot 8+...-27^1\cdot8^{n-2}+8^{n-1})$
If $n$ is eve... | Hint $\ {\rm mod}\ 35\!:\,\ 27\equiv -8\,\ $ so $\,\ \overbrace{27^{\large n}}^{\Large 3^{\Large 3n}}\equiv (-8)^{\large n}\equiv \pm\! \overbrace{8^{\large n}}^{\Large 2^{\Large 3n}}\,$ depending on parity of $n.\ $ QED.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2027993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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heterogeneous recurrence with f(n) as constant How to solve this $s_{n+1}=4s_{n-1}-3s_n+5$ where f(n)=5 conditions $s_0=-3$ $s_1=3$
I calculated the general solution $s_n=c_1*(-4)^n+c_2*1^n$ of this recurrence. The roots are $q_1=-4$ and $q_2=1$ but I have problem with particular solution with method of prediction .
I ... | $s_{n+1}=4s_{n-1}-3s_n+5$
$q^2-4+3q=0$
$\Delta=3^2-4*1*(-4)=25$
$\sqrt\Delta=5$
$q_1=\frac{-3-5}2=-4$
$q_2=\frac{-3+5}2=1$
homo general $s_n=c_1*1^n+c_2*(-4)^n$
$k=1$ where $k$ is multipiclity of root
hetero particular $s_n=Q(n)*q^n*n^k$
$s_n=A*1^n*n^1=An$; $1^n=1$ because $1^0=1$, $1^1=1$ and so on
$A(n+1)=4(A(n-1))... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Resultant contains all common roots as linear factors? Let $f,g \in \mathbb{C}[x,y,z]$ be homogeneous polynomials, so they define projective plane curves $C$ and $D$ in $\mathbb{C}P^2$. We are interested in Bezout's theorem applied to $C \cap D$. Write $f$ and $g$ as polynomials in $z$:
$$
f(x,y,z) = \sum_{i = 0}^ma_i(... | The multiplicity of that factor would be the sum of the intersection multiplicities at the $P_i$. To get individual intersection multiplicities, you want that number $k$ to be 1. In $\mathbb{C}[x,y,z]$ you can accomplish that by first applying a generic linear transformation, $(x,y,z) \mapsto$ linear combinations of ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Why don't quaternions contradict the Fundamental Theorem of Algebra? I don't pretend to know anything much about the Fundamental Theorem of Algebra (FTA), but I do know what it states: for any polynomial with degree $n$, there are exactly $n$ solutions (roots).
Well, when it comes to quaternions, apparently $i^2=j^2=k... | Let me put my comments in an answer. As SpamIAm said, the FTA has a generalization in any integral domain :
If $R$ is an integral domain, then any polynomial $P \in R[x]$ of degree $n$ has at most $n$ roots (counted with multiplicity)
The proof is that : $\quad $(a field is commutative, otherwise we say a non-commut... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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Inequality with $x,y,z\geq 0$, $x+y+z=1.$ With $x,y,z\geq 0$, $x+y+z=1$.Prove that $$\sqrt{x+y^2}+\sqrt{y+z^2}+\sqrt{z+x^2}\geq 2 \tag{i}$$
The hint is using a lemma:
If $a,b,c,d\geq 0 $satisfying $a+b=c+d$ and$|a-b|\leqslant|c-d|$ then we have $\sqrt{a}+\sqrt{b}\geq \sqrt{c}+\sqrt{d}$
How to prove this lemma? And is ... | There is the following Vo Quoc Ba Can's solution.
We need to prove that
$$\sum\limits_{cyc}\left(\sqrt{x+y^2}-y\right)\geq1$$ or
$$\sum\limits_{cyc}\frac{x}{\sqrt{x+y^2}+y}\geq1.$$
Now, by AM-GM
$$\sum\limits_{cyc}\frac{x}{\sqrt{x+y^2}+y}=\sum\limits_{cyc}\frac{x(x+y)}{(x+y)\sqrt{x+y^2}+y(x+y)}\geq\sum\limits_{cyc}\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2028360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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ending zeros in 100! I'm working through Hammack's Book of Proof. Section 3.2 has an weird question, and unfortunately it's even-numbered, so there is no answer key.
"There are two 0's at the ned of 10! = 3,628,800. Using only pencil and paper, determine how many 0's are at the end of the number 100!."
I used the spe... | I believe the best way is what you did! Indeed the simple formula applied to that gives you
$$n = \text{int}\left(\frac{100}{5^1}\right) + \text{int}\left(\frac{100}{5^2}\right) = 20 + 4 = 24$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2028464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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What is exactly the relation between vectors, matrices, and tensors? I am trying to understand what Tensors are (I have no physics or pure math background, and am starting with machine learning).
In an introduction to Tensors it is said that tensors are a generalization of scalars, vectors and matrices:
Scalars are 0-... | Vectors and Tensors are mathematical objects invariant under coordinate system chosen.
Matrices and Arrays for that matter are representations of rank-2 tensor or vectors in a given coordinate system.
As for the intuitive understanding part, take the analogy of a vector. By definition a vector is a magnitude and a dir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2028572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Fourier transform of $F(x)$ $$F(x)=\int^{\infty}_{-\infty} e^{-i \xi x}f(x)\ dx$$
Fourier transform of $F$:
$$\int^{\infty}_{-\infty}\left[\int^{\infty}_{-\infty} f(\xi)e^{-x^2\xi^2}\ d\xi\right]\ dx$$
Is that the way to proceed? If yes, how do I continue and if No, show me how I should proceed.
| $=2 \pi [\int^{\infty}_{-\infty} F(x) e^{ix(-\xi)}$ $dx]=2\pi$ $f(-\xi)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2028825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find $\lim \limits_{n \to \infty}{1*4*7*\dots(3n+1) \over 2*5*8* \dots (3n+2)}$ I am trying to find
$$\lim \limits_{n \to \infty}{1*4*7*\dots(3n+1) \over 2*5*8* \dots (3n+2)}$$
My first guess is to look at the reciprocal and isolate factors:
$${2 \over 1}{5 \over 4}{8 \over 7} \dots {3n+2 \over 3n+1}= {\left(1+1\right)... | You are on the right track. Use the inequality $\ln(1+x)\ge x-x^2/2$ for $x\ge0$:
$$
\sum_{k=0}^n\ln\Bigl(1+\frac1{3\,k+1}\Bigr)\ge\sum_{k=1}^n\Bigl(\frac{1}{3\,k+1}-\frac12\,\frac{1}{(3\,k+1)^2}\Bigr)=\sum_{k=1}^n\frac{1}{3\,k+1}-\frac12\sum_{k=1}^n\frac{1}{(3\,k+1)^2}.
$$
The first sum goes to $+\infty$ because the s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Is there a bijection from $[0,1]$ to $\mathbb{R}$? I'm looking for a bijection from the closed interval $[0,1]$ to the real line. I have already thought of $\tan(x-\frac{\pi}{2})$ and $-\cot\pi x$, but these functions aren't defined on 0 and 1.
Does anyone know how to find such a function and/or if it even exists?
Than... | A continuous bijection can't exists because $[0,1]$ is a compact set and continuous functions send compacts in compacts. You can look for a non-continuous bijection, that exists because $[0,1]$ and $\mathbb R$ have the same cardinality. It follows from Cantor-Bernstein theorem, that states "given two sets $A, B$ if exi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2029021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Advice on proof in linear algebra. I just wrote my first proof in linear algebra so I'd love some advice on the things that go well and what could be improved upon. It's a proof by induction.
Theorem:
Let $A_n$ be a $n\times n$ matrix of the form:
$\begin{pmatrix}
2 & 1 & 0 & 0 && & \cdots & 0\\
1 & 2 & 1 & 0 && & \cdo... | You could remark some of the easy steps you did, making the proof more understandable yet (even though they are very simple). For example, you could write $3=3(2-1)$ (same with the other 2) to show that the formula holds for the smaller values of $n$.
Moreover, and more important, you should explain how you used Lapla... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Examples of pairewise independent but not independent continuous random variables By considering the set $\{1,2,3,4\}$, one can easily come up with an example (attributed to S. Bernstein) of pairwise independent but not independent random variables.
Counld anybody give an example with continuous random variables?
| An answer of mine on stats.SE gives essentially the same answer as the one given by vadim123.
Consider three standard
normal random variables $X,Y,Z$ whose joint probability
density function
$f_{X,Y,Z}(x,y,z)$ is not $\phi(x)\phi(y)\phi(z)$ where
$\phi(\cdot)$ is the standard normal density, but rather
$$f_{X,Y,Z}(x,y,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2029257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 1
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Do the even cycles of a graph form a subspace of the cycle space? From Wikipedia:
The cylce space of an undirected graph is the set of its Eulerian subgraphs. This set of subgraphs can be described algebraically as a vector space over the two-element finite field.
The vector addition operation is the symmetric differ... | Let $\mathscr{C}_0(G)=\{E\in\mathscr{C}(G):|E|\text{ is even}\}$. Let $E_0,E_1\in\mathscr{C}_0(G)$, with $|E_0|=2m$ and $|E_1|=2n$. If $k=|E_0\cap E_1|$, then
$$|E_0\mathbin{\triangle}E_1|=|E_0\setminus E_1|+|E_1\setminus E_0|=(2m-k)+(2n-k)=2(m+n-k)\;,$$
so $E_0\mathbin{\triangle}E_1\in\mathscr{C}_0(G)$, and $\mathscr{... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How thick should a cylindrical coin be for it to act as a fair three-sided die? When flipping a coin of radius $r>0$ and thickness $t>0$ in the real world, there is some non-zero probability of getting neither heads nor tails, but instead landing on the thin lateral side. My question is, how thick does this lateral fac... | This is an old and charming question. The thick coin is a cylinder. Cut it vertically by its diameter. You get a rectangle by section with sides $e\wedge D\ $. The thickness is $e$ and $D$ its diameter.
When you flip the coin the decision variable are the angles that form the rectangle diagonals. These angles determine... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2029476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 3
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Lagrange maximization with inequalities I need to prove the maxima of the following summation, using Lagrange.
$$\max_{x_m} \left( \sum_m a_m log(x_m)\right) $$
s.t.
$$0 \le x_m \le 1$$
$$ \sum_m x_m = 1$$
The solution is a closed form $ x_m = \frac{a_m}{\sum_m a_m} $ .
I formulated the Lagrange equation but I am co... | I think that the Lagrangian is:
\begin{equation}
L(x,\lambda,\mu) = \sum_m a_m log(x_m) + \mu \left(\sum_mx_m-1\right)
\end{equation}
Now we have:
\begin{equation}
\begin{cases}
\frac{a_1}{x_1}+\mu=0\\\frac{a_2}{x_2}+\mu=0\\\\\frac{a_m}{x_m}+\mu=0\\\sum_mx_m=1
\end{cases}
\end{equation}
so
\begin{equation... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2029621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the Lie algebra of the Euclidean group? I am trying to find the Lie algebra for $E(n) = \left\{\begin{bmatrix}1 & 0^t \\
\mathbf{x} & A
\end{bmatrix}: A \in SO(n), \mathbf{x} \in \mathbb{E}^n \right\}$. In particular, I would like to show that $\mathfrak{e}(n) = \left\{\begin{bmatrix}0 & 0^t \\ \mathbf{b} & B \... | Proof: suppose that $\gamma(t)$ is a path in the Lie Group with $\gamma(0) = I$. $\gamma$ must have the form
$$
\gamma(t) = \pmatrix{1&0\\ \mathbf x(t) & A(t)}
$$
It follows that $\gamma'(0)$ has the form
$$
\gamma'(0) = \pmatrix{0&0\\ \mathbf x'(t) & A'(0)}
$$
which is of the desired form.
On the other hand, take an... | {
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"timestamp": "2023-03-29T00:00:00",
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Radius of convergence of the power series $\sum_n a_n x^n$ where $a_n={{\sin (n!)}\over {n!}}$
Find the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n x^n$ where $a_n={{\sin (n!)}\over {n!}}.$
Now using the ratio test $$R=\lim_{n\rightarrow \infty}\left|{{a_n}\over {a_{n+1}}}\right|\\=\lim_{n\righ... | If you use the root test;
since
$|\sin(n!)| < 1$
and
$n!^{1/n} \to n/e$,
$|{{\sin (n!)}\over {n!}}|^{1/n}
<\frac{1}{n/e}
\to 0
$
so the series converges
everywhere.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2029805",
"timestamp": "2023-03-29T00:00:00",
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"answer_count": 2,
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$R^{3}$ is open and closed I have thought about 2 ways to prove this, but there are not complete. What should I add to finish the proof?
Proof 1: For all $x\in \mathbb{R^{3}}$ there is $r>0$ such that $B(x,r)\subset \mathbb{R^{3}}$
any $r>0$ will satisfy the requirement because $\mathbb{R^{3}}$ is dense? which argument... | Take $x\in\mathbb{R}^{3}$. Then, for all $\varepsilon >0$, $B_{\varepsilon}(x)\subset\mathbb{R}^{3}$ By contradiction. Suposse that exist $x\in B_{\varepsilon}(x)$ such that $x\notin\mathbb{R}^{3}$, then, $x\in{\mathbb{R}^{3}\setminus\mathbb{R}^{3}}=\emptyset$. Clearly this is a contradiction.
Now, since $\mathbb{R}^{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2029916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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How to prove that $\sin(x)≤\frac{4}{\pi^2}x(\pi-x) $ for all $x\in[0,\pi]$? How to prove the inequality
$$
\sin(x)≤\frac{4}{\pi^2}x(\pi-x)
$$
for all $x\in[0,\pi]$?
As both functions are symmetric to $\frac{\pi}{2}$ it suffices to prove it for $x\in\left[0,\frac{\pi}{2}\right]$. Furthermore one can see that $\frac{4}{\... | Let $f(x)=\frac{4x(\pi-x)}{4\pi^2}-\sin{x}$.
Since $f(\pi-x)=f(x)$, it's enough to prove our inequality for $x\in\left[0,\frac{\pi}{2}\right]$
$f'(x)=\frac{4(\pi-2x)}{\pi^2}-\cos{x}$ and $f''(x)=\sin{x}-\frac{8}{\pi^2}$,
which says that $f$ is a concave function on $\left[0,\arcsin\frac{8}{\pi^2}\right]$
and $f$ is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2030069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
How to relate areas of circle, square, rectangle and triangle if they have same perimeter?? I was given a question which was like:
Suppose that a circle, square, rectangle and triangle have same perimeter. How area their areas related??
My work:
I broke the question in parts and tried to prove it seperately:
STEP $1$... | To generalize: It is easily shown that any regular $n$-gon with side length $a$ the radius $r_n$ of its inscribed circle is
$$r_n=\frac{a/2}{\tan(\pi/n)},$$
hence
the ratio of its perimeter $P$ and the diameter of its inscribed circle is
$$\pi_n:=n\cdot\tan(\pi/n).$$
Let $r_n$ be the radius of its inscribed circle. No... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2030161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Holomorphic function $f$ on the unit disc such that $|f(z)|\leq 1.$ Let $f$ be a holomorphic function on $D=\{z\in\mathbb{C}:|z|\leq 1\}$ such that $|f(z)|\leq 1.$
$$g(z)=\begin{cases} \dfrac{f(z)}{z} &\text{ if } z\neq 0\\[8pt] f'(0) &\text{ if } z=0.\end{cases}$$
Which of the following statements are true
$1.$ $g$ is... | Hint: Maximum modulus principle for your function on $\mathbb{D}$ . The answer is $g$ is holomorphic and $|f'(0)|\le 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2030316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proof that sum of squares of error for simple linear regression follows chi-square distribution I can understand that if Y1~Yn are random samples from N(μ,σ), then the sum of squares of difference between Yi and bar(Y) divided by sigma^2 follows chi-square distribution with n-1 degress of freedom.
But I can't easily pr... | Without involving asymptotic results, you have to assume that the error terms follow Gaussian (uncorrelated) distribution with a constant variance, i.e., $\epsilon \sim N(0, \sigma^2)$. Next, you can show that each fitted value $\hat{Y}_i$ is normally distributed as well, which is follows from
$$
\hat{Y}_i = Y_i + \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2030401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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A very short proof of $e$ is irrational I was talking with my friend and he came up with this very short proof
Given $x\in \mathbb{R}$, if $xy \notin \mathbb{Z}$ for any $y\in \mathbb{Z}$, then $x$ is irrational. Since $e = \sum \frac{1}{n!}$, we see that $ey \notin \mathbb{Z}$ for any $y \in \mathbb{Z}$. So $e$ is ir... | Perhaps you can take the fractional part of $n! \times e$ that number is
$$ 0< n! e - \text{whole number } = \frac{1}{n+1}
+ \frac{1}{(n+1)(n+2)}+\dots <1$$
if $e$ is rational this number has to be integer eventually. However, this error term is strictly between $0$ and $1$ always. So $e \notin \mathbb{Q}$.
https://... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2030524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Does $\int_\limits{0}^{\infty} \frac{\sin(2x)}{x}dx $ absolutely converge? My teacher gave me this task to prove it, but I have no idea how to begin.
Can I have any clue?
| Note that we can write
$$\begin{align}
\int_0^{N\pi/2}\left|\frac{\sin(2x)}{x}\right|\,dx&=\sum_{n=1}^N \int_{(n-1)\pi}^{n\pi}\frac{|\sin(x)|}{x}\,dx\\\\
&\ge \frac1\pi \sum_{n=1}^N\frac1n \int_{(n-1)\pi}^{n\pi}|\sin(x)|\,dx\\\\
&=\frac2\pi \sum_{n=1}^N \frac1n
\end{align}$$
Inasmuch as the harmonic series diverges, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2030658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Implicit Differentiation: Find $\frac{d^2y}{dx^2}$ Find $\frac{d^2y}{dx^2}$ of:
$$4y^2+2=3x^2$$
My Attempt
I attempted the probelm my first solving for the first derivative:
$8y*y'=6x$
$y'=\frac{3x}{4y}$
Then I tried it again; however I was a bit confused, and ended up getting
$y''=\frac{6y-3x(2*y')}{16y^2}$
Woul... | For future reference,
$F(x,y)=0$
$\frac{d^2[F(x,y)]}{dx^2}=-\frac
{\frac{\partial^2 F}{\partial x^2}\left(\frac{\partial F}{\partial y}\right)^2
-2·\frac{\partial^2 F}{\partial x\partial y}·\frac{\partial F}{\partial y}·\frac{\partial F}{\partial x}
+\frac{\partial^2 F}{\partial y^2}\left(\frac{\partial F}{\partial x}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2030739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $X$ contains a strictly increasing sequence which converges to $\sup X$
Suppose that $X ⊂ \mathbb{R}$ is bounded above and that $\sup X \notin X$. Prove that $X$ contains a strictly increasing sequence which converges to $\sup X$.
I've started by assuming there to be an increasing sequence and using its de... | Assuming you have a sequence to begin with doesn't seem like a good way to go -- you need to CONSTRUCT a sequence that satisfies the given properties.
Focus on the definition of supremum as least upper bound. Show that for each $n\in\mathbb{N}$, there must exist $a_n\in X$ so that $\lvert a_n-\sup X\rvert<\frac{1}{n}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2030851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Distributing 12 distinct balls to 5 different persons. Actually , my doubt is about number of ways of distributing 12 distinct balls to 5 different persons such that 3 persons get 2 balls each and 2 persons get 3 balls each.
There is another question in this site about "How many ways to divide group of 12 people into 2... | I believe that the correct answer is the product of:
*
*The number of ways to choose the persons: $\frac{5!}{3!2!}=10$
*The number of ways to choose the balls: $\frac{12!}{2!2!2!3!3!}=1663200$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2031029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
find the maximum $\frac{\frac{x^2_{1}}{x_{2}}+\frac{x^2_{2}}{x_{3}}+\cdots+\frac{x^2_{n-1}}{x_{n}}+\frac{x^2_{n}}{x_{1}}}{x_{1}+x_{2}+\cdots+x_{n}}$ give the postive intger $n\ge 2$,and postive real numbers $a<b$ if the real numbers such $x_{1},x_{2},\cdots,x_{n}\in[a,b]$ find the maximum of the value
$$\dfrac{\frac{x^... | Let $M$ is a maximum value (it exists because continuous function on compact gets there a maximum value) and
$$f(x_1,x_2,...x_n)=\frac{x_1^2}{x_2}+\frac{x_2^2}{x_3}+...+\frac{x_n^2}{x_1}-M(x_1+x_2+...+x_n).$$
Since $f$ is a convex function for all $x_i$, we obtain
$$0=\max_{\{x_1,x_2,...,x_n\}\subset[a,b]}f=\max_{\{x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2031113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
} |
Proof: If a function is in the Schwartz Space, then this function is uniformly continuous I don't have this really clear, I want to justify if $f\in\mathcal{S}(\mathbb{R})$ then $f$ is uniformly continuous. So far, I know how can I bound $|x|$ for $f$ is in the Schwartz space, but I can't proceed with the uniformly con... | I'll prove a result slightly stronger. Consider the space $C_0(\mathbb R)$ of the continuous functions from $\mathbb R$ to $\mathbb C$ such that $\lim_{|x|\to \infty} f(x) = 0$. The functions from this space are uniformly continuous.
Consider $\epsilon >0$ and $f\in C_0(\mathbb R)$. There exist, from the limit above, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2031235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Quotient of finitely presented group Suppose $G, H$ be two groups such that $G$ is finitely presented(with number of defining relators must be at least 1) and let $\phi$ : $G \rightarrow$ $H$ be an epimorphism. Does it imply that $H$ a finitely presented group?
| A method for constructing an example which shows that the answer is negative (see for instance this reference) is to go for a finitely presented group $G$ whose centre $Z(G)$ is not finitely generated. Then, by a standard result (which I think has been recorded by Bernhard Neumann), $G/Z(G)$ is not finitely presented. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2031371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the remainder when a large number is divided by 13 Let a number $x = 135792468135792468$. Find the remainder when $x$ is divided by $13$.
Is it possible to use Fermat's little theorem on this? I notice that the number is also repeating after $8$.
Would really appreciate any help, thanks!
| You noticed how the number repeats, so you can see that it equals $135792468\times1000000001$. Now test $1000000001$ for divisibility by $13$ (repeatedly add $4$ times the rightmost digit to the rest of the number, and if you reach a multiple of $13$ (you reach 26 in this case), the original number is divisible by $13$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2031461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 0
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LPP auxiliary problem optimal solution Let's look at a linear programming problem
$$\max\{\langle c,x\rangle \ \colon Ax=b, \ x\geq 0\}$$
and its auxiliary problem
$$\max\{\langle \overline{c},\overline{x}\rangle\ \colon \overline{A}\overline{x}=b, \ \overline{x}\geq 0\}.$$
I want to prove that if the LPP feasible regi... | Let $x$ be an optimal problem to the original problem with objective value $p := c^Tx$, and let $\tilde{x}$ be the vector $(x,0,0,\ldots,0)$ (the corresponding solution in the auxiliary problem), which is feasible and also yiels an objective value of $p$. Using revised simplex, the reduced cost of $x_{n+j}$ for $\tilde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2031559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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How to show that $\iint_{S}\vec{F}\cdot d\vec{S}=0$ with the vector field $\vec{F}=\big\langle0,0,z\big\rangle$? Problem: If $S$ is the cylindrical surface parametrized by $\phi(\theta,u)=(\cos{(\theta)},\sin{(\theta)},u)$, $\theta\in[0,2\pi]$ and $u\in[0,1]$, then $\iint_{S}\vec{F}\cdot d\vec{S}=0$ for the vector fiel... | $d\vec{S} = \frac{\partial \phi}{\partial u} \times \frac{\partial \phi}{\partial v} \ du \ dv$ this will be in the direction perpendicular to the $\vec{F}$ which you are given.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2031643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
The "lower part" of $BV$ function is always lower semi-continuous. Let $u\in BV(\Omega)$ be a function of bounded variation, where $\Omega\subset\mathbb R^N$ is open bounded smooth boundary. Define
$$
u^-(x):=\sup\left\{t\in\mathbb R:\,\lim_{r\to 0}\frac{\mathcal L^N(B(x,r)\cap\{u<t\})}{r^N}=0\right\}.
$$
Then $u^-$ i... | Not necessarily. Consider $u(x)=\chi_E(x)$ where $E$ is a set which has an inner cusp, for instance the subset of $\mathbb R^2$ given by
$$E=\left\{(x_1,x_2):\,x_2< \sqrt{|x_1|}\right\}. $$
Then $u$, restricted to a bounded open set, is BV, and $u^-(x)=u(x)$ for every $x$ except for the origin, in which $u^-(0)=1$, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2031732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Winding number: times f(w) goes around unit disc I have a problem which is like this:
Let $S^1=\partial D(0,1)$ and consider a differentiable function $f:S^1\rightarrow S^1$. Can you compute the number of times that $f(w)$ goes around $S^1$ per each time that $w$ goes around $S^1$?
I have tried to reparameterize the ... | In fact, holomorphic functions/residue theory isn't required. See Baby Rudin, exercises 23-26 of ch. 8. A brief summary:
(1) Given a closed differentiable curve $\gamma:[a,b]\longrightarrow\Bbb C\setminus\{0\}$, let be
$$
\text{Ind}(\gamma) =
\frac1{2\pi i}\int_a^b\frac{\gamma'(t)}{\gamma(t)}\,dt.
$$
(yes, is a disguis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2031884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Group (co)homology and classyfing spaces I would like to ask where I can find in the literature the proof of the following fact: the group cohomology of the group $G$ is naturally isomorphic with the ordinary (say singular) cohomology of the classyfing space $BG$ of $G$.
| This is explained in Chater 8 of Weibel's "Introduction to Homological Algebra", specifically Example 8.2.3.
The idea is to revise the construction of $BG$ as the geometric realization of a certain simplicial set — the nerve $N G$ of $G$ viewed as a category:
$$BG = |NG|.$$
For any simplicial set $X$ its simplicial hom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2032002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Help with $\int \cos^6{(x)} \,dx$ Problem:
\begin{eqnarray*}
\int \cos^6{(x)} dx \\
\end{eqnarray*}
Answer:
\begin{eqnarray*}
\int \cos^4{(x)} \,\, dx &=& \int { \cos^2{(x)}(\cos^2{(x)}) } \,\, dx \\
\int \cos^4{(x)} \,\, dx &=& \int { \frac{(1+\cos(2x))^2}{4} } \,\, dx \\
\int \cos^4{(x)} \,\, dx &=&
\int { \fra... | In your solution, when substituting the already known expression for $I_4$ (in the third line from the bottom), you forgot to multiply it by $5$. That's the only error there. Put it back in there, and you'll have a correct answer. Your answer would still look different from the output of that online integrator, but the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2032178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Solving inequality involving floor function We have this inequality (over real numbers) :
$$x^2-2x\le \frac{\sqrt{1-\lfloor x\rfloor^2}}{\lfloor x \rfloor + \lfloor -x \rfloor}$$
How we can solve it using both of algebraic and geometric methods ?
| Hint: $x$ cannnot be integer, otherwise the denominator of the RHS would be $0$.
So $x$ is not integer, which gives $\lfloor x \rfloor+\lfloor -x \rfloor=-1$
Also $\lfloor x \rfloor\in\{-1,0,1\}$ since it must be integer and $1-\lfloor x\rfloor^2>0$ which is equivalent to $\lfloor x\rfloor^2<1$.
Then you solve on each ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2032279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
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How can the Airy's equation $y'' - xy = 0$ be used to model diffraction of light? Differential Equations with Boundary-Value Problems by Dennis G. Zill, Michael R. Cullen states that the equation is used to model diffraction of light. It doesn't explain how, it just goes on to solve it using a series solution.
Has any... | Here is a reference to a 1977 article by H. M. Nussenzveig in Scientific American about the Theory of the Rainbow. The captions of the figures in the article are informative.
The differential equation also appears in Professor Nussenzveig's 1992 book Diffraction Effects in Semiclassical Scattering.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2032406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimum value of angles of a triangle In a triangle $ABC$, if $\sin A+\sin B+\sin C\leq1$,then prove that $$\min(A+B,B+C,C+A)<\pi/6$$
where $A,B,C$ are angles of the triangle in radians.
if we assume $A>B>C$,then $\sum \sin A\leq 3 \sin A$,and $ A\geq \frac{A+B+C}{3}=\pi/3$.also $\sum \sin A\geq 3\sin C$ and $ C\leq \... | Since you assumed $A\geq B\geq C$, it must be that $\dfrac{A}{2}+C\leq\dfrac{\pi}{2}.$ Hence, $\sin\tfrac{A}{2}<\sin(\tfrac{A}{2}+C) = \cos(\tfrac{B-C}{2})$.
Finally, $$1\geq \sin A+\sin B+\sin C = \sin A+2\sin\tfrac{B+C}{2}\cos\tfrac{B-C}{2} = 2\cos\tfrac{A}{2}\big(\sin\tfrac{A}{2}+\cos\tfrac{B-C}{2}\big)>4\cos\tfrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2032711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Rearranging angular velocity equation to make $T$ the subject I want to rearrange the formula for angular velocity $\omega = \dfrac{2\pi}{T}$, to make $T$ the subject as I wish to find the period.
Would the correct answer be $T = \frac{\omega}{2\pi}$ or would it be $T = \frac{2\pi}{\omega}$?
And is there a certain rul... | It is quite simple. You have $\omega = \dfrac{2\pi}{T}$ since you want to make T the subject, multiply the whole equation by T and you will get $$\omega{\cdot T} = \dfrac{2\pi}{T}{\cdot T} = 2\pi$$
On bringing ${\omega}$ on right you will have $T = \frac{2\pi}{\omega}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2032923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Is $(x,y) \mapsto 0$ on $\mathbb{Q}\backslash\{0\}$ associative and commutative? I have the following definition of operations on the following sets:
*
*$(x,y) \mapsto 9xy$ on $\mathbb{Z}$
*$(x,y) \mapsto 0$ on $\mathbb{Q}\backslash\{0\}$
I have to determine whether the operations on the given sets are associativ... | For the first one, it is indeed associative and commutative because the usual multiplication of integers is som. It does not have a neutral element though, for the following reason: if $u \in \Bbb Z$ is this neutral element, then $9xu = x$ for all $x \in \Bbb Z$. For $x=1$ this would imply $9u = 1$, whence $u = \frac 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2033051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to show that $\left|\sum_{k=0}^\infty\frac{(ix)^k}{(k+1)!}\right|\le \left|\sum_{k=0}^\infty\frac{(ix)^k}{k!}\right|=|e^{ix}|=1$ with restrictions
How to show that $\left|\sum_{k=0}^\infty\frac{(ix)^k}{(k+1)!}\right|\le \left|\sum_{k=0}^\infty\frac{(ix)^k}{k!}\right|=|e^{ix}|=1$ with restrictions, for $x\in\Bbb R$... | Just for the record I will add a second proof. We knows that
$$e^{ix}=\cos(x)+i\sin(x)$$
and $|e^{ix}|=1$ for all $x\in\Bbb R$. And we want to prove
$$\frac{|e^{ix}-1|}{|x|}\le 1$$
From the last inequality we have the bound
$$|e^{ix}-1|\le |e^{ix}|+1=2\le|x|$$
then for $|x|\ge 2$ the inequality is clear. Now, we will s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2033168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find a inverse of a multivariable function? I have a function $f:\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined as:
$$f(x,y) = (3x-y, x-5y)$$
I proved that it's a bijection, now I have to find the inverse function $f^{-1}$.
Because $f$ is a bijection, it has a inverse and this is true:
$$(f^{-1}\circ f)(x,y)... | You can split this into two separate functions $u, v:\Bbb R^2\to \Bbb R$ the following way:
$$
u(x, y) = 3x-y\\
v(x, y) = x-5y
$$
and we have $f(x, y) = (u(x, y), v(x, y))$. What we want is $x$ and $y$ expressed in terms of $u$ and $v$, i.e. solve the above set of equations for $x$ and $y$, so that you get two function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2033272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
} |
How to determine the number of coin tosses to identify one biased coin from another? If coin $X$ and coin $Y$ are biased, and have the probability of turning up heads at $p$ and $q$ respectively, then given one of these coins at random, how many times must coin A be flipped in order to identify whether we're dealing wi... | Some factors to think about:
*
*How different are the probabilities?
*How sure do we want to be?
If the probabilities of heads are close together, like $p=.501$ and $q=.500$, it will take many trials to really see any difference, but if the probability of heads are drastically different, like $p=.9$ and $q=.4$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2033370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Uniquness of multiplicative inverses in $\Bbb Z_n$ (or any abeliean monoid)
Assume that an integer $a$ has a multiplicative inverse modulo an integer $n$. Prove that this inverse is unique modulo $n$.
I was given a hint that proving this Lemma:
\begin{align}
n \mid ab \ \wedge \ \operatorname{gcd}\left(n,a\right) =... | What you have shown already is a technique that, using the Extended Euclidean Algorithm will give you the inverse (if it exists). Note also that you can determine if an element $a \mod n$ has an inverse by checking that they have no common divisors (that is, $\gcd(a,n)=1$).
Now to answer your question. Suppose that you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2033470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
For any prime number $p$ and natural number $i < p$, prove that $p$ divides ${p \choose i}$. For any prime number $p$ and natural number $i < p$, prove that $p$ divides ${p \choose i}$.
Also, what happens when $p$ is not a prime. Is this still true?
I tried writing out the formula for combination but couldn't get furth... | By a combinatorial argument, or by manipulating factorials, we have
$$i\binom{p}{i}=p\binom{p-1}{i-1}\ .$$
Since $\binom{p-1}{i-1}$ is an integer,
$$p\mid i\binom{p}{i}\ .$$
But $p$ is prime and $1\le i<p$, so $p$ and $i$ have no common factor, so
$$p\mid \binom{p}{i}\ .$$
For the case when $p$ is not prime, just take ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2033748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How does Laplace transform ℒ{ sin(t)/t } solves definite integral 0 to ∞ ∫ (sin(t)/t) dt How does the answer of the Laplace transform $$\mathcal L \left\{ \frac{\sin t}{t} \right\}= \frac{\pi}{2}-\tan^{-1}(s)$$ solve the definite integral
$$\int_0^{\infty} \frac{\sin t}{t} dt = \frac{\pi}{2} $$
How are they related? w... | Your statement is
$$\int_0^{\infty} dt \frac{\sin{t}}{t} e^{-s t} = \frac{\pi}{2} - \arctan{s} $$
Plug in $s=0$ to both sides.
There are lots of ways to prove the LT. One way to do it is to use the FT relation for the sinc term.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2033868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Length of Hilbert Curve in 3 Dimensions The Hilbert Curve is a continuous space filling curve. The length of the $n^{th}$ iteration in two dimensions can be calculated by $2^n-\frac{1}{2^n}$. The curve can be generalized to fill volumes; what is the length of the $n^{th}$ iteration of the Hilbert Curve in three dimensi... | In $2$ dimensions the square is divided into $2^n$ by $2^n$ smaller squares ($n$ is the iteration). The majority of squares contain $2$ lines of length $1/2^n$ except for two, giving us $2(2^n * 2^n -1)/2^n$.
In 3 dimensions there is still $2$ lines in the majority of cubes so wee need only adjust things to account fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2034027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Determine the original group according to its quotient Suppose that $G$ is a group and the group of integers $\mathbb Z$ is its normal subgroup with $G/\mathbb Z\cong\mathbb Z$. Then can I say that $G$ is isomorphic to $\mathbb Z\times\mathbb Z$?
| The answer is no. ${\mathbb Z}^2$ is the only abelian group with that property, but there is a nonabelian group defined by the presentation $\langle x,y \mid y^{-1}xy=x^{-1} \rangle$.
Let $x$ generate the normal infinite cyclic subgroup $N$ and choose $y \in G$ such that $yN$ is a generator of $G/N$. Then $G=\langle x,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2034150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $A^2=0$, then $\mathrm{rank}(A) \le \frac{n}{2}$ For my matrix algebra class I need to prove the following:
If $A^2=0$, prove $\mathrm{rank}(A) \le \frac{n}{2}$. So if A is nilpotent prove $\mathrm{rank}(A) \le \frac{n}{2}$.
I know already how to solve this, but my initial way of solving is false. I am looking for... | Pretty late to the party but adding another way of solving since it's different and uses a nice idea, in case anyone stumbles upon this post for help.
We know from the Sylvester inequality $$\mathrm{rank}(XY)\geq\mathrm{rank}(X)+\mathrm{rank}(Y)-n$$
Setting $X=Y=A$ we get $$\mathrm{rank}(AA)\geq\mathrm{rank}(A)+\mathrm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2034296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Gradient of quadratic forms involving matrix powers Let $f:\mathbb{R}^{n \times n} \to \mathbb{R}$ be defined as:
$$
f(A)= x^T (A^2)^i y + v^T A^i w,
$$
where $i \in \mathbb{N}$ and $x,y,v,w$ are some fixed column vectors. One can assume that $A$ is a symmetric matrix. I am interested in computing the gradient of $f$ w... | Warning: Lots of tedious algebra below, so there's definitely room for errors. I'll see if I can validate this result by other means.
To make the notation less confusing, I'll write the exponent as $N$ rather than $i$ so that I can use lower-case Latin letters such as $i$ for indices. I will also adopt the Einstein co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2034415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Find symmetric matrix of polynomial equation Let f be a polynomial such that $x, y, z, t$ belong to $\mathbb{R}^4$.
$$f(x) = 4x^2+4y^2+4z^2+4t^2+8xy+6xt+6yz+8zt$$
Find the symmetric matrix and determine whether $A$ is a positive definite or not.
I understand how to find a symmetric matrix and check whether or not it is... | $\begin{eqnarray*}f(x) &=& 4x^2+4y^2+4z^2+4t^2+8xy+6xt+6yz+8zt\\
&=&\begin{bmatrix} x&y&z&t \end{bmatrix}
\begin{bmatrix} 4&4&0&3\\4&4&3&0\\0&3&4&4\\3&0&4&4 \end{bmatrix}
\begin{bmatrix}x\\y\\z\\t\end{bmatrix}
\end{eqnarray*}
$
It's eigenvalues are -3, 3, 5, 11,so it's not positive definite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2034569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Prove that a linear subspace of $C([0,1])$ is closed
Prove that the set
$$ W= \{f \in C([0, 1]):f(0)=0\}$$
is a closed linear subspace of $C([0,1]).$
Here $C([0, 1])$ is the space of continuous functions equipped with the uniform norm $||f||_\infty = \sup_{x\in[0,1]}|f(x)|$.
I need some help in how to proceed ... | The linear functional $L(x)=x(0)$ is a continuous linear map from $C[0,1]$ to $\mathbb{C}$ because it is bounded, i.e., $|T(x)|=|x(0)| \le \|x\|$. Therefore the inverse image of $\{0\}$ under $T$ is closed, which is the subspace $W$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2034747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
A condition for irreducibility Let $X$ be a closed projective set. Prove that $X$ is an irreducible set if and only if $X \cap U_i$ is irreducible for every i=0,...,n; where $\cup U_i$ is an open cover of $\mathbb{P^n}$. For "$\Rightarrow $" I succeded.
| It is a topological exercise!
Let $X$ be a topological space, let $\{U_i\}_{i\in I}$ be an open covering of $X$; a subset $Y$ of $X$ is irreducible only if $\forall i\in I,\,Y\cap U_i=Y_i$ is irreducible. Vice versa, if $Y_i$ is irreducible and $\forall i,j\in I,\,Y\cap U_i\cap U_j=Y_{ij}\neq\emptyset$ then $Y$ is irr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2034844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Functional in Hilbert space Let $H$ be a Hilbert space and $0\neq x\in H$. I want to prove that there is an unique $f\in H^*$, such that $\|f\|=1$ and $f(x)=\|x\|$.
Any ideas on how to approach this problem
| Let $K=\Bbb R\text{ or } \Bbb C$ the field of scalars. Let $V$ be the subspace of $H$ defined by $$V=\{\lambda x\}_{\lambda\in K}$$ Consider the continuous form $\phi\in V^*$ defined by $$\phi(\lambda x)=\lambda\space ||x||$$
It is clear that $\phi( x)= ||x||$.
Besides, for all $\lambda x\in V$ one has
$$|\phi(\lambda ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2035080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula As far as every question I've seen concerning "what is $\sum_{k=1}^nk^p$" is always answered with "Faulhaber's formula" and that is just about the only answer. In an attempt to make more interesting answers, I ask that this question concern the problem o... | By the binomial theorem, $$(x+1)^{n+1}=\sum_{h=0}^{n+1} {n+1 \choose h}x^h$$
$$(x+1)^{n+1}-x^{n+1}=\sum_{h=0}^n {n+1 \choose h}x^h$$
Sum this equality for $x=0,1\dotsb k$
$$\sum_{x=1}^k((x+1)^{n+1}-x^{n+1})=(k+1)^{n+1}-1=\sum_{x=1}^k\sum_{h=0}^n {n+1 \choose h}x^h=\sum_{h=0}^n{n+1 \choose h}\sum_{x=1}^kx^h=(n+1)\sum_{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2035188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 14,
"answer_id": 1
} |
Interchanging the sides of a Heegaard splitting Suppose that $M$ is a closed orientable connected $3$-manifold and that $M = U \cup V$ is a Heegaard splitting of $M$ (i.e. $U$ and $V$ are both handlebodies with a common boundary). What are some necessary/sufficient conditions for there to exist a homeomomrphism $h : M... | For what it's worth, your question can be translated into a group theoretic criterion, expressed in the mapping class group $\text{MCG}(S)$ of the surface $S = U \cap V$; this is the group of homeomorphisms of $S$ modulo the normal subgroup of homeomorphisms isotopic to the identity.
A homeomorphism $h : M \to M$ whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2035252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integrate $\int_0^{\pi/2}\frac{\cos^2x}{a\cos^2x + b\sin^2x}\,dx$ I don't know how to deal with this integral
$$I=\displaystyle\int_0^{\pi/2}\frac{\cos^2x}{a\cos^2x + b\sin^2x}\,dx$$
I reached the step
$$I
=\displaystyle\ \int_0^{\pi/2}\frac{1}{a + b\tan^2x}dx$$
Now what should I do? Please help.
| Now substitute
$\displaystyle\ u=\tan x \implies x= tan^{-1} u \implies dx=\frac{1}{1+u^2} du$
$\displaystyle\ =\int_0^{\infty}\frac{1}{(1+u^2)(a + bu^2)}du$
Partial Fractions
$\displaystyle\ =\int_0^{\infty}\frac{1}{(1+u^2)(a + bu^2)}du= \frac{1}{a-b} \left(\int_{0}^{\infty} \frac{du}{1+u^{2}} - b \int_{0}^{\infty} \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2035381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
I'm looking for some mathematics that will challenge me as a year $12$ student. I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.
I want a challenge, some mathematics that I can attempt to lear... | You could look at some of the Questions tagged recreational-mathematics, soft-question and big-list (sort them by "votes").
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2035454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 16,
"answer_id": 15
} |
Calculate the expected value of X I have no idea how to solve this problem, any help would be greatly appreciated:
During the course of $9$ lessons the teacher randomly selects one student (from a class of $30$), asks him several questions and either grades him (with the probability of $1/3$) or not (with the probabil... | Let $X_i$ be the indicator random variable that the $i$-th student is selected and graded at least once in the nine lessons. (Having a value of $1$ if the event happens, or $0$ otherwise.)
Then the count of students graded is: $\sum\limits_{i=1}^{30} X_i$
Now find the expectation of this count. (Hint: use the Linea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2035596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question on proof that $|G| = pqr$ is not simple
Assume $|G| = pqr$ where $p,q,r$ are primes with $p < q < r$. Then $G$
is not simple.
I have a problem understanding the proof (see for example here). In the proof one assumes that $n_p,n_q,n_r > 1$ (number of each $p,q,r$-Sylow subgroups respectively) and then by Sy... | Since $n_r$ is not $1$, we have $k>0$ which implies $n_r=1+kr > r > p$ and $q$, so the only possible divisor of $pq$ is $pq$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2035688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Determinant of a block anti-diagonal matrix Let $a = \begin{pmatrix}
O & \cdots & O & A \\
O & \cdots& B & O\\
\vdots & \ddots & O & O\\
C & \dots & O & O
\end{pmatrix}$,
where $A, B, C$ are $2n \times 2n $ matrices over ring of integers modulo $m$ that is, $\mathbb{Z}_m$. Is $det(a) = det(AB\dots C)?$
| Think of rearranging the matrix, by swapping columns, say, so that it becomes block diagonal. You will have to keep track of the signs.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2035936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
A doubt about ramification in cyclotomic fields I'm studying the Algebraic Number Theory Notes of Robert B. Ash. I really like his notes, but I don't understand a suggestion he gives at page 7 of chapter 8 (Factoring of prime ideals in Galois Extensions).
He's using all the theory of that chapter to discover new proper... | All finite extensions of $\Bbb F_p$ are separable (in fact Galois), so all polynomials are separable--note some authors demand separable to mean "distinct roots" I use the "irreducible factors have distinct roots" definition, if you are using the former, then clearly this is not always true as $x^{pk}-1$ has repeated r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2036084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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