Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
The uniqueness of homogeneous matrices of some geometric transformations Given the description of a specific geometric transformation, the homogeneous square matrix of it can be obtained per the methods described by many textbooks.
For example, many text books suggest the homogeneous matrix of a reflection be constru... | As long as the origin is fixed, these are all ordinary linear transformations, so they are uniquely determined by their matrix, as any linear map $\varphi:\Bbb R^n\to\Bbb R^m$ is just a (left) multiplication by the matrix $M=[\varphi(e_1)\,|\dots|\,\varphi(e_n)]$ where $e_1,\dots,e_n$ is the standard basis of $\Bbb R^n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3402003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show every proper subgroup is cyclic? Let $|G| = p^2$ where $p$ is prime. Show that every proper subgroup of $G$ is cyclic.
I don't know how to approach this problem.
Here is what I have:
From Lagrange Theorem, for any subgroup $H\subset G$, $\frac{|G|}{|H|}$. Now, the order of $G$ is $p^2$. We need to show that... | If $H$ is a subgroup, by Lagrange's theorem $|H|$ is either $1$ or $p$ ($p^2$ is ruled out because $H$ is proper). Now, every group of prime order is cyclic (why?).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3402181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Books about synthetic projective geometry Are there books in English about synthetic projective geometry? More specifically, results of Karl von Staudt (imaginary elements theory through elliptical involutions, imaginary circle, infinity's imaginary circle)
| There's lots to learn from "modern" texts such as Richter-Gebert, Coxeter, et al. But the heyday of synthetic projective geometry appears to have been in the 19th and early 20th centuries. After that, both research and pedagogy moved to other topics in math.
For the older texts archive.org is your friend, going back ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3402323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does there exist a field $K$ and $x \in K$ with $K(x^{1/4})=K(x^{1/2}) \neq K$ I want to know if there exists a field $K$ and $x \in K$ with $K(x^{1/4})=K(x^{1/2}) \neq K$.
My attempt thus far: if $K(x^{1/2}) \neq K$ then $x^{1/2} \notin K$ so the minimal polynomial of $x^{1/2}$ over $K$ has degree $>1$ and is therefor... | $s=x^{1/4}$ , $d= x^{1/2}$
s£ k(d)==> s=a+bd ===> s^2 =a^2 +b^2d^2 +2abd ==> d=a^2+b^2x+2adb
===> d( 1-2a) = a^2 +x
d= ( 1-2ab)^(-) ( a^2+xb^2) , if 1-2ab#0 ==> d£K but this absurde since k( d) #k sorry i have note Latex for better redaction
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3402460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Prove that if $x$ is the greatest lower bound of $U$, then $x$ is the least upper bound of $B$
Suppose $R$ is a partial order on $A$, $B \subseteq A$. Let $U$ be the set for all upper bounds for $B$.
Prove that if $x$ is the greatest lower bound (or g.l.b) of $U$, then $x$ is the least upper bound (or l.u.b) of $B$.
... | My attempt:
Consider L to be the set defined as L = {a $\in$ A | $\forall$ u $\in$ U (aRu)}. Or, L is the set of all lower bounds on U. If x is the greatest lower bound of U, x $\in$ L and thus $\forall$ u $\in$ U (xRu). Suppose u $\in$ U. Then, in particular, xRu. Since u was arbitrary, $\forall$ u $\in$ U (xRu). Or, ... | {
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Heat Equation with Neumann condition Suppose u solves $u_t-u_{xx} = 16u$ on the interval $(0, \pi),$ with the homogenous Neumann condition $u_x=0$ at $x=0,\pi$. Characterize the initial data $u_0=u(x,0)$ for which $u(x,t)$ stays bounded as $t \rightarrow\infty$.
I am not sure how to approach this PDE, thus any help/gu... | $$\begin{cases}
u_t -u_{xx} = 16u\\
u_x(0,t) = u_x(\pi,t)=0\\
u(x,0) = u_0
\end{cases}$$
Let $u(x,t) = X(x)T(t)$ and $\frac{dT(t)}{dt} = \mathring{T}(t)$
Therefore,
$u_t -u_{xx} = 16u$ becomes $\frac{\mathring{T(t)}}{T(t)}-16 = \frac{X''(x)}{X(x)} = \lambda$.
Here, $\lambda$ is a constant because "[t]he left hand sid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3402798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Polynomial solution $xy''+(1-x)y'+ \lambda y=0$
For which values of the constant $\lambda$ does the differential
equation $$xy''+(1-x)y'+ \lambda y=0$$ have a polynomial solution?
I was thinking about solving this problem with the Theorem of Frobenius (even thought I am not sure if I should use it in this case sin... | Suppose that $\lambda$ gives a polynomial solution $y(x)=\sum_{k=0}^na_kx^k$ with $a_n\ne 0$. Then
$$y'(x)=\sum_{k=0}^nka_kx^{k-1}=\sum_{k=0}^n(k+1)a_{k+1}x^k$$
and
$$y''(x)=\sum_{k=0}^nk(k-1)a_kx^{k-2}=\sum_{k=0}^n(k+1)ka_{k+1}x^{k-1},$$
where we set $a_{n+1}=0$. Hence
$$xy''(x)=\sum_{k=0}^n(k+1)ka_{k+1}x^k$$
and
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3402921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $z(t)=\frac{1+it}{1-it}$ describes a circumference on the complex plane. I am asked to prove that $z(t)=\frac{1+it}{1-it}$ describes a circumference on the complex plane when $t$ takes every value in the extended real number line. That is, $\mathbb{R} \cup\{\pm\infty\}$. I don't have any idea how to proceed w... | Method 1
Observe that $$|z|=\frac{|1+it|}{|1-it|}$$
$$=1$$
because denominator and numerator are conjugates. This equation of a circle centred at origin and of radius $1$.
Method 2
Put $$t=\tan \frac{\theta}{2}, 0\leq \theta<2π, \theta\neq π$$
Now,
$$z=\frac{1-t^2}{1+t^2}+i\frac{2t}{1+t^2}$$
$$=\cos\theta+i\sin\theta$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3403016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Existence of limit of a function from existence of another limit of a function
Let $a \in \mathbb{R}$ and suppose that $f:\mathbb{R} \to \mathbb{R}$
is a function such that $f(x) \ge 1,\ \forall x \in \mathbb{R}$. If
$\lim_{x \to a}\bigg[\frac{1}{\sqrt{f(x)}} + \sqrt{f(x)}\bigg]$
exists, then prove that $\lim_{x... | Let's write $u=f(x), v=\sqrt{f(x)} $. Then it is known that $u\geq 1,v\geq 1$ and $v+(1/v)=w$ tends to a limit $l$. Next note that $w=v+(1/v)\geq 2$ via AM-GM inequality and hence $l\geq 2$.
Solving for $v$ we get $$v=\frac{w\pm\sqrt{w^2-4}}{2}$$ Since $v\geq 1$ the $+$ sign must be choosen above. Hence we have $$v=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3403140",
"timestamp": "2023-03-29T00:00:00",
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How to prove $xP(|X_1|>x) \leq E(|X_1|1(|x_1|>x))$? It is from one proof of theorem in Probability: Theory and Examples by Durret.
The theorem is stated as below.
Let $X_1, X_2,\dots$ be i.i.d with $E|X_i|<\infty$. Let $S_n=X_1+\dots+X_n$ and let $u=E(X_1)$. Then $S_n/n \rightarrow u$ in probability.
The proof in the... | The first inequality $x\mathbb P(|X_1|>x) \le \mathbb E\left[|X_1|1_{(|X_1|>x)}\right]$ is just a use of Markov inequality:
$$\mathbb E\left[|X_1|1_{(|X_1|>x)}\right] \ge \mathbb E\left[x\cdot 1_{(|X_1|>x)}\right] \ge x\mathbb P(|X_1|>x)\qquad (\because |X_1| >x\text{ on the support}).$$
The dominated convergence theor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3403254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find real parameter $a$, such that the solution of the linear system lies in the second quadrant
For which real parameter $a$ lies the solution of the system of equations
$$\begin{aligned} \frac{x}{a+1} + \frac{y}{a-1} &= \frac{1}{a-1}\\ \frac{x}{a+1} - \frac{y}{a-1} &= \frac{1}{a+1} \end{aligned}$$
in the second quad... | Solve your system : you should find
$$x=\frac{a}{a-1},\quad y=\frac{1}{a+1}$$
You want $x<0$, so $a$ and $a-1$ must be of opposite signs. So you should have $a>0>a-1$, hence $0<a<1$, and this leads to $y>0$.
The solution is : $(x,y)$ is in the second quadrant if and only if $0<a<1$, or $0\le a<1$ if you look for the c... | {
"language": "en",
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"source": "stackexchange",
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How to compute the normalizer $N(H)$?
Let $H$ be a subgroup generated by $(12)$ in $S_3$. Compute the normalizer $N(H)$ of $H$.
My attempt:
$\sigma$ $\in N(H) $ then $ \sigma H \sigma^{-1}=H=(12)$
implies $\sigma (12)= (12)\sigma$
Now applying both sides to $3$
$\sigma (3) (12) =(12) \sigma(3)$
$(12)$ c... | Hint: For any $\sigma\in S_n$, we have $$\sigma(12)\sigma^{-1}=(\sigma(1)\sigma(2)),$$ where $\sigma(i)$ is $\sigma$ applied to $i$, with $i$ from the set $S_n$ acts on.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3403491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Proving that a function is injective and strictly increasing I have recently started doing proofs, and I find it quite hard to construct one. I am not really clear on how proofs should generally look like. What information should it include? Are we always need to suppose something is true and then arrive at the conclus... | You have the right ideas, but you need to be more rigorous in how you make the arguments.
Let's start with the definition of being injective. A function is injective when for all $a,b$ in the domain, $a\neq b\to f(a)\neq f(b)$. So, let $a\neq b$. By the law of trichotomy, $a<b$ or $a>b$. Without loss of generality let ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3403611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Without calculus, show that the sign of $(1+a)^{1+x}-a^{1+x}-1$ matches the sign of $x$, for any positive $a$ Let $a$ be a positive real constant. Consider the function
$$
f(x) = (1+a)^{1+x}-a^{1+x}-1
$$
We have for any choice of $a$, that $f(x) >0$ for $x>0$ and $f(x) <0$ for $x<0$, which can be proved using calculus.... | For positive $x$ we have
$$
\begin{align}
\frac{1+a^{1+x}}{(1+a)^{1+x}} &= \left( \frac{1}{1+a}\right)^{1+x} + \left( \frac{a}{1+a}\right)^{1+x}\\
&=
\frac{1}{1+a} \left( \frac{1}{1+a}\right)^{x} + \frac{a}{1+a}\left( \frac{a}{1+a}\right)^{x} \\
&< \frac{1}{1+a} + \frac{a}{1+a} = 1 \, .
\end{align}
$$
For negative ... | {
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Does every vector space with a weak topology contain a dense subspace which is a direct sum of real lines? Is it true that given an index set $A$ and a topological vector subspace $E$ of $\mathbb{R}^A$ (that is, $E$ carries weak topology), there is a dense topological vector subspace of $E$ which is isomorphic as a to... | If $E$ is a normed space, then in its weak topology there is no subspace isomorphic as a topological vector space to $\mathbb R^{\mathbb N}$.
We can prove this using: in a normed space, if a sequence converges weakly then it is bounded (in norm).
So let $E$ be a normed space. Let $e_1,e_2,\cdots$ be any sequence of no... | {
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Which mappings preserve convex bodies? Let $$f:\mathbb{R}^n\to\mathbb{R}^n,$$ $n\geq 2$, be a mapping which maps every convex body (compact convex set with nonempty interior) to a convex body.
If we assume $f$ to be a homeomorphism, it needs to be affine. Is there something we can say without this assumption?
| Let $h: {\mathbb R}^2\to [0,1]\subset {\mathbb R}$ be a continuous surjective map which is nonconstant on all nondegenerate intervals, for instance, we can take $h(x,y)=|\sin(x^2+y^2)|$. Let $g: [0,1]\to Q=[0,1]\times [0,1]\subset {\mathbb R}^2$ be a discontinuous function which is maps each nondegenerate interval onto... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove $\forall a,b$ where $a < b$, $\exists x\in \mathbb{Q}$ such that $a
$\forall a,b$ with $a < b$, $\exists x\in \mathbb{Q}$ such that $a<x<b$ and $x$ has a finite decimal expansion with some number of $7$'s in it.
I have shown that there are infinite number of rational numbers in between an interval of two arbitrar... | In order to find such rational number $x$, we modify suitably the classical proof of the density of $\mathbb{Q}$ in $\mathbb{R}$.
Since $b>a$ then there is an positive integer $n$ such that $10^n(b-a)>10$. It follows that the interval $(10^na,10^nb)$, whose size is greater than $10$, contains an integer $m$ with $7$ a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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I have to find maximum and minimum of a certain sequence. I was given a sequnce and have to identify maximum and minimum of this sequence:
And I think that maximum is 1 and minimum -1/2
*Speaking about maximum -> every other number of this sequence is equal or smaller and 1 is a part of sequence = OK
*Speaking about m... | There is a flaw in your reasoning, because you handle the two subsequences completely independently. Indeed $1$ is the largest of the decreasing sequence of terms of odd rank.
But it could be that some term of the subsequence of terms of even ranks exceeds that. This is actually not the case as this sequence is always ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Growth rate of primes of the form $a n+1$, with $n$ fixed If we fix a natural $n$, and let $a$ vary, how big can $a n + 1$ be before the first prime is encountered, in terms of $n$? In other words, let $n = 32$. Then, we have:
$$1(32) + 1 = 33 = 3 * 11 \text{ is composite}$$
$$2(32) + 1 = 65 = 5 * 13 \text{ is compos... | Let $ak+b$ be an arithmetic progression, $k = 1,2,3,\ldots$. Dirichlet theorem of primes in arithmetic progression says that number of prime $\le x$ which are of the form $ak+b$ is
$$
\pi_a(x) \approx \frac{\pi(x)}{\varphi(a)} \approx \frac{x}{\varphi(a)\log x}
$$
where $\varphi(n)$ is the Euler totient function. Hence... | {
"language": "en",
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Showing $5a + 2b \equiv 0 \pmod 7$ is symmetric, pictorially I have a problem regarding the understanding of modulus.
So someone proved $5a + 2b \equiv 0\pmod 7$ is symmetric using the modulo circle,
and my brain cant comprehend how he meant that. I'm talking about circles like this:
I know there is a other way to pro... | The relation is defined on $\Bbb Z$ like this: $a \sim b$ if and only if $5a + 2b \equiv 0 \pmod 7$. On the assumption this holds, we wish to show $b \sim a$, i.e. $5b+2a \equiv 0 \pmod 7$.
Using the definition of modular congruence, then, $\exists k \in \Bbb Z$ such that
$$\frac{5a+2b-0}{7} = k \iff 5a + 2b = 7k$$
Sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3404734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A is an open set and B is a closed set. Prove that A\B is open and B\A is closed. The problem consists of using the epsilon neighborhood definition to prove that A\B is open and B\A is closed.
I'm thinking of using a proof by contradiction and assuming that A\B is not open and vice versa for B\A. Any help would be gre... | $$A\setminus B=A\cap B^c$$
Note that $B^c$ is open and $A^c$ is closed. The result naturally follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3404853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Show linearized Riemann tensor is gauge invariant I'm working in general relativity and figured this problem would be more suited to mathematics than physics.
The linearized Riemann tensor is given by:
$$R_{\alpha \beta \mu \nu}=-\frac{1}{2}\left[h_{\alpha \mu, \beta \nu}+h_{\beta \nu, \alpha \mu}-h_{\alpha \nu, \beta... | The first term contributes a difference of $$ -(\xi_{\alpha,\mu\beta\nu}+\xi_{\mu,\alpha\beta\nu})$$ Similarly, the second term $$ -(\xi_{\beta,\nu\alpha\mu}+\xi_{\nu,\beta\alpha\mu})$$ and the third $$+(\xi_{\alpha,\nu\beta\mu}+\xi_{\nu,\alpha\beta\mu})$$ Hey, stop right there. The second term of the second one $-\xi_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3404980",
"timestamp": "2023-03-29T00:00:00",
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If $E_1 \cup E_2$ is measurable with $λ^\ast(E_1 \cup E_2) = λ^\ast(E_1)+ λ^\ast(E_2)$, then $E_1$ and $E_2$ are measurable. Denote by $\lambda^\ast$the Lebesgue outer measure on $\mathbb{R}^m$. Assume
that $E_1, E_2 \subset \mathbb{R}^m$ satisfy $E_1 \cap E_2 = \phi$, $E_1 \cup E_2$ is Lebesgue measurable with $λ^\ast... | Choose a $G_\delta$ set $G$ such that $E_{1}\subseteq G$ and $\lambda^{\ast}(E_{1})=\lambda(G)$. In particular, $G$ is measurable.
Now let $H=(E_{1}\cup E_{2})\cap G$ so $H$ is measurable. We have $E_{1}\subseteq H\subseteq G$ and hence $\lambda^{\ast}(E_{1})\leq\lambda(H)\leq\lambda(G)=\lambda^{\ast}(E_{1})$, so $\lam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3405152",
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$10000=abc$ with $a,b,c$ natural numbers don't have digit $0$. Smallest possible value of $a+b+c$ is? Let $10000=abc$ with $a,b,c$ natural numbers don't have digit $0$. If $a,b,c$ does not have to be all distinct, then the smallest possible value of $a+b+c$ is?
Attempt:
First write as prime factors: $10000 = 2^{4} 5^{... | Well. "no zero" means none of the $a,b,c$ have both factors of $2$ and $5$ so you either have two numbers that are powers of $2$ and one that is a power of $5$ or you have $2$ that are powers of $5$ and one that is a power of $5$.
CASE 1: $\{a,b,c\} = \{2^k, 2^{4-k}, 5^4\}$.
So $a + b + c = 2^k + 2^{4-k} + 625$.
We ... | {
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Can we find an example where $U$ is not connected and the partition of $p^{-1}(U)$ into slices is not unique? This is an extension of the following question from Munkres. Section 53-problem 2.
Let $p: E \to B$ be continuous and surjective. Suppose that $U$ is an open set of $B$
that is evenly covered by $p$. Show that ... | Let $p : E \to B$ be a continuous surjection and let $U \subset B$ be a nonempty open set which is evenly covered, with slices $V_\alpha, \alpha \in A$. Let $U_1, U_2$ be two disjoint nonempty open subsets of $U$. As an example take $p : \mathbb R \to S^1$ , $U = S^1 \setminus \{1\}$ and $U_i$ any two disjoint nonempty... | {
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How to find x from the equation $\sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} = m \sqrt{\frac{x}{x+\sqrt{x}}}$ For m are real number,
Find x from the equation
$$\sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} = m \sqrt{\frac{x}{x+\sqrt{x}}}$$
I tried to multiply $\sqrt{x + \sqrt{x}}$ to the both sides and I get
$$x + \sqrt{x} -... | As you got (for $x\neq0$):
$$ x+\sqrt{x} + \sqrt{x^2-x} = m\sqrt{x} $$
$$ x+\sqrt{x} + \sqrt{x}\sqrt{x-1} = m\sqrt{x} $$
Setting $ u = \sqrt{x} $ ($ u\neq0 $) we get:
$$ u^2+u+u\sqrt{u^2-1} = mu $$
$$ u+1+\sqrt{u^2-1} = m $$
$$ \sqrt{u^2-1} = (m-1) - u$$
Squaring both sides:
$$ u^2-1 = (m-1)^2-2u(m-1)+u^2 $$
We get:
$$... | {
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"timestamp": "2023-03-29T00:00:00",
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How do I solve this integral via integration by parts when it shows a recursion-like behavior? I've come across an integral while solving a question for ODE's. The integral is as follows:
$$\int e^{-2x}x\,dx$$
From here I set:
$$
\begin{align}
f(x) = e^{-2x}\quad & f'(x) = -2e^{-2x} \\
g(x) = \frac{1}{2}x^2\quad & g'(x... | Let $u=x$ and $dv=e^{-2x}dx$. Then, $du=dx$ and $v=\dfrac{e^{-2x}}{-2}$
$$\int e^{-2x}xdx=-\frac{xe^{-2x}}{2}+\frac{1}{2}\int e^{-2x}dx=-\frac{xe^{-2x}}{2}-\frac{e^{-2x}}{4}+C=-\frac{e^{-2x}}{4}\left(2x+1 \right)+C$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A problem about connected components‘ image under onto and continous map Here is a problem on Topoloy written by Marco Manetti, page89, question 5.4.
Let $f:X \rightarrow Y$ be an identification. Show that if the connected components of $X$ are open, so are the connected components of $Y$.
I think that if we can prov... | The answer is no, the claim is not true if you replace $f$ by continuous and onto.
The problem is the fact that you still need to show that the image of open is open, continuity does not satisfies.
Consider the following example: pick $Y$ to be any topological space whose components are NOT open. Pick $X$ to be the sa... | {
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Two statements are equivalent if... Two statements "If A, then B." and "If P, then R." are equivalent if the validity of one statement implies the validity of the other?
That is, I assume that "If A, then B." is valid, then proceed to show the second statement is valid, and vice versa?
Or this done differently?
To show... |
Two statements "If A, then B." and "If P, then R." are equivalent if the validity of one statement implies the validity of the other?
Not quite right. It's not the the validity of the one statement implies the other (and vice versa), but rather that the assumed truth of the one implies the other (and vice versa)
Th... | {
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How to find the difference between the son's and mother's age if it(ages of son and mother) is reversible for a total of 8 times? There is a quiz question for which I need to write a python code. I don't need help for the code but this is what I need help with.
A son had recently visited his mom and found out that the ... | He is his mother's age and presently he is $66$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\{n!\alpha\},n \in\mathbb{N}$ dense in $R$ ? where $\alpha$ is irrational Is $\{n!\alpha\},n \in\mathbb{N}$ dense in $[0,1]$ ? where $\alpha$ is irrational.
I know that $\{n\alpha\}$ is dense in $[0,1]$? I wanted to generalise it.So, first I thought about $\{n!\alpha\},n \in\mathbb{N}$, but couldn't make any reason... | You can work out, through a somewhat careful argument, that if a sequence $s_n$ increases without bound, then $\{s_n\alpha\}$ is dense in $[0,1]$ for almost every real $\alpha$. However, in general, this sequence is not dense for every irrational $\alpha$.
We can actually find a particular notable counterexample for $n... | {
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Quadratic Modular Arithmetic I want to prove that
$$ w^2 \equiv 2 \quad (\bmod{5})$$
has no solutions in integers.
What I tried:
$$ w^2 \equiv 2 \quad (\bmod{5})$$
$$ \Rightarrow w^2 = 2 + 5k, \quad k \in \mathbb{Z} $$
Now, I don't know what to do, so I considered $x^2 = 2+ 5y$ over $\mathbb{R}$.
We can solve this t... | Without knowing how you finished the proof it can't be verified.
This should prove it though:
All multiples of $5$ end with the digit $5$ or $0$ in decimal base.
$$5+2=7$$
$$w^2\ne7+10n$$
$$w^2\ne2+10n$$
because no perfect squares end in $7$ or $2$ in decimal base.
| {
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Solving indefinite integral $\int \frac{1+\sqrt{1+x^2}}{x^2+\sqrt{1+x^2}}dx$ How can we solve this integration?
$$\int \frac{1+\sqrt{1+x^2}}{x^2+\sqrt{1+x^2}}dx$$
I tried to make the following substitution
$$1+x^2=w^2$$
but this substitution complicated the integral.
| Like the answer of @AmerYR, taking $1+x^2=u^2$
$$I=\int \frac{1+\sqrt{1+x^2}}{x^2+\sqrt{1+x^2}} dx= \int \sec u ~du + \int \frac{sec u}{\tan^2 u+ sec u} du$$ $$ \implies I =\log (\tan u+ sec~ u)+J(u)$$
Next, using $\sin u=\frac{2 \tan(u/2)}{1+\tan^(u/2)},~~ \cos u= \frac{1-\tan^(u/2)}{1+\tan^2(u/2)}$ and $t=\tan(u/2), ... | {
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Probability of more than k balls in any m buckets given n total balls Say there are 100 balls that are randomly distributed across 64 buckets, with the same probability of ending up in any bucket and trials are independent. What is the probability that at least one of the 64 buckets will have more than 20 balls?
A simi... | For $i ∈ \{1,…,64\}$ Let $E_i$ be the event that the $i$-th bucket contains more than $20$ balls. You want to compute: $P[\bigcup_{i = 1}^{64} E_i]$. By the union bound we have:
$P[\bigcup_{i = 1}^{64} E_i] ≤ \sum_{i = 1}^{64} P[E_i] = 64 · 7.33 · 10^{-18}.$
However, any outcome where several buckets overflow at the sa... | {
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Advanced : Compute $\sum_{n=1}^\infty\frac{H_n^4-6H_n^2H_n^{(2)}+8H_nH_n^{(3)}+3\left(H_n^{(2)}\right)^2-6H_n^{(4)}}{n^5}$ How to prove the following equality
$$\mathcal S=\sum_{n=1}^\infty\frac{H_n^4-6H_n^2H_n^{(2)}+8H_nH_n^{(3)}+3\left(H_n^{(2)}\right)^2-6H_n^{(4)}}{n^5}\\=672\zeta(9)-240\zeta(2)\zeta(7)-105\zeta(3... | Different approach by Cornel:
By master theorem we have
$$\sum_{k=1}^\infty\frac{H_k^4-6H_k^2H_k^{(2)}+8H_kH_k^{(3)}+3\left(H_k^{(2)}\right)^2-6H_k^{(4)}}{(k+1)(k+n+1)}\\=\frac{H_n^5+10H_n^3H_n^{(2)}+15H_n\left(H_n^{(2)}\right)^2+20H_n^2H_n^{(3)}+20H_n^{(2)}H_n^{(3)}+30H_nH_n^{(4)}+24H_n^{(5)}}{5n}$$
Multiply both side... | {
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Minimal no of people to guarantee 4 consecutive chairs occupied at round table There are 2019 chairs at a round table. What is the minimum number of chairs that must be occupied such that there are some consecutive set of 4 chairs (or more) occupied.
The chairs can be divided in sets $S_{k} ={4k+1,4k+2,4k+3,4k+4}$ for ... | I think the only gap in your solution is the claim "If we have 3 seats occupied in the last set they will always be adjacent to an occupied seat in either $S_0$ or $S_{503}$." That need not be true; the first seat of $S_0$ and the last seat of $S_{503}$ could both be empty.
You then need to prove that in this case ther... | {
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Motivation of the definition of Euclidean Domain We know the definition of Euclidean Domain is
An Euclidean Domain is an Integral domain $(E,+,\ast )$ together with a function $v: E\setminus \{0\} \to \mathbb{N} \cup \{0\} $ such that
(i) for all $a,b \in E$ with $b \neq 0$, there exist $q,r \in E$ such that $a = qb ... | The motivation comes from a strong property of natural (and integer) numbers: the Euclidean algorithm that allows to find a gcd between two numbers.
The function $v$ is then just a degree function that gives you a weight comparison between these "abstract" elements of the ring. For the integers $v=|\cdot|$, for the rin... | {
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Proving a much stronger version of AM-GM for three variables Here, @MichaelRozenberg stated the following inequality without proof:
Theorem. For all non-negative $a,b,c\in\mathbb R$,
$$a^6+b^6+c^6-3a^2b^2c^2\geq16(a-b)^2(a-c)^2(b-c)^2.$$
In my answer below I give a complete brute-force proof. However, more elegant ... | I will use the Buffalo method: By symmetry of the inequality, we can assume without loss of generality that $a\le b\le c$. So we can substitute $a=x,b=x+y,c=x+y+z$ for some non-negative reals $x,y,z$.
Putting this in our inequality and expanding results in
$$\color{green}{12 x^4 y^2 + 12 x^4 y z + 12 x^4 z^2 + 28 x^3 y... | {
"language": "en",
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If two sides of a triangle are given, how many triangles are possible? I thought myself that you could rotate one arbitrary side, which is connected to the other side, over 90 degrees to the left and 90 degrees to the right, so basically there are infinitely many triangles possible.
Am I right?
| I agree with the other answers but would like to present a geometric argument which is very similar to the analysis already indicated by the OP.
Assume that two of the sides have a length of $x$ and $r$, where $x$ and $r$ are both positive. Assume that one of the vertices of the triangle will have Cartesian coordinate... | {
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Demonstrate that there are no perfect squares ending with $8$ A number n will always end in some digit of the set {$0,1,2,3,4,5,6,7,8,9$}. The last digit of $n^2$ is the last digit of its last squared digit. Like this:
$$\ldots 0^2 = \ldots 0$$
$$\ldots 1^2 = \ldots 1$$
$$\ldots 2^2 = \ldots 4$$
$$\ldots 3^2 = \ldots 9... | Your proof is not bad. You could make a more formal argument with modular arithmetic.
For example, $ n\equiv 0, \pm1, $ or $\pm2 \pmod 5$,
so $n^2\equiv 0, 1, $ or $-1\pmod 5$,
so $5\nmid n^2-3,$ so $10\nmid n^2-8$.
| {
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Formulate two equations as a system of first-order ODEs I have two equations that I want to formulate into a system of first-order ODEs.
\begin{cases}
\displaystyle
x''(t) = \frac{x(t)}{\Big( \sqrt{x(t)^2 + y(t)^2} \Big)^{3}} \\
\displaystyle
y''(t) = \frac{y(t)}{\Big( \sqrt{x(t)^2 + y(t)^2} \Big)^{3}}
\end{ca... | Yes, your work is correct.
Usually the system is written as
$$\begin{cases}
x'(t)=z(t) \\
y'(t)=w(t) \\
\displaystyle
z'(t) = \frac{x(t)}{(\sqrt{x(t)^{2} + y(t)^{2}})^{3}} \\
\displaystyle
w'(t) = \frac{y(t)}{(\sqrt{x(t)^{2} + y(t)^{2}})^{3}} \\
\end{cases}$$
Or they use $u_1, u_2, u_3, u_4$ for $x,y,z,w$
$$\begin{... | {
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Relating orthogonal column distances to basis distances I have two $d\times r$ matrices $X$ and $Y$, with $X^TX=Y^TY=I_r.$ It is also known that the columns $X_i$ and $Y_i$ for $i=1,\dots,r$ satisfy $$\underset{i}{\max}\Vert X_i-Y_i\Vert_2\le \varepsilon.$$ Can I then say that $\Vert XX^T-YY^T\Vert_2\le c\varepsilon,$ ... | No, there is no such $c$ in general. Consider the case $d = r+1$ (though the following construction works in any $d > r$). Let $e_1, \dots, e_{r+1}$ be the the standard basis vectors of $\mathbb{R}^{r+1}$ (considered as column vectors), so they are orthonormal. Also define $u = \frac{1}{\sqrt{r}}e_{r+1} - \frac{1}{r}(e... | {
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If $f(f(x))$ is linear, is $f(x)$ linear? If f(x) is closed on the set of natural numbers? If I have a function defined on the natural numbers $\mathbb Z_{>0}$, does $f(f(x))=ax+b$ imply that $f(x)=rx+q$, where $a,b,r,q$ are naturals? This came to mind in the context of a different problem, but I don't know where to st... | No: consider the function $f$ that swaps every odd natural number with its successor. Then $f\circ f=id$ but $f$ is not monotone.
| {
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Are Cosets Isomorphic to One Another A basic question, I seem to be struggling with the concept of quotient groups and cosets.
Suppose $G$ is a group, and $N$ is a normal subgroup.
I know that is $x$ and $y$ are in the same coset of $G/N$, then $xN = yN$.
I also know that cosets are either disjoint or equivalent, $G/... | A coset $xN$ of $N\unlhd G$ for nontrivial $x$ cannot be isomorphic to $N$ if $xN\neq N$, since $xN$ has no identity element (why?) and is thus not a group.
There is, however, a bijection $\varphi:N\to xN$ given by $n\mapsto xn$. So the sets are of the same cardinality.
| {
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How to write $x^m+x^{-m}$ as a polynomial in $x+x^{-1}$. I did find this question Proving that $x^m+x^{-m}$ is a polynomial in $x+x^{-1}$ of degree $m$. but it only shows by induction that this is possible, not the actual form of the solution. How, for example, would you write $x^2+x^{-2}$ in a polynomial of $x+x^{-1}$... | The inductive argument does tell you how to do this - and, it's worthwhile to see that induction very often gives you the answers you are after, especially since the fact that inductive proofs can be unrolled is often overlooked.
In the linked answer, it is noted that
$$x^{k+1}+x^{-k-1} = (x^k +x^{-k})(x+x^{-1}) - (x^{... | {
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Prove that in the ring $Z[\sqrt2]$: $\langle3+8\sqrt2, 7\rangle$ = $\langle3+\sqrt2\rangle$ So,
I know that $\langle a\rangle$ = $aR$ and $\langle a,b\rangle$ = $aR + bR$ for any ring R.
I then multiplied each side by $\sqrt2$ and computed them.
These are the steps that I took:
$\langle3+8\sqrt2, 7\rangle$ = $\langle3... | Hint: To show $\ \langle 3+8\sqrt{2}, 7\rangle=\langle 3+\sqrt{2}\rangle\ $, you need to show:
*
*$\ 3+8\sqrt{2} = r_1\left(3+\sqrt{2}\right)\ $ and $\ 7 = r_2\left(3+\sqrt{2}\right)\ $ for some $\ r_1\ $ and $\ r_2\ $ in $\ \mathbb{Z}\left[\sqrt{2}\right]\ $; and
*$\ 3+\sqrt{2}=s_1\left(3+8\sqrt{2}\right) + s_2 7\... | {
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Calculate $[K(\alpha^2):K]$ Let $K\subseteq F$ be a field extension. $K(\alpha)/\ K$ an extension of degree $4$ where $\alpha^2$ is not a root of $m_{(\alpha,K)}(x)$. I am asked to calculate $[K(\alpha^2):K]$
Using the towers formula $$[K(\alpha^2):K]=[K(\alpha^2):K(\alpha)][K(\alpha):K]$$
And we already know that $[K... | The correct tower formula is
$$4=[K(\alpha):K]=[K(\alpha):K(\alpha^2)][K(\alpha^2):K]$$
Since $[K(\alpha):K(\alpha^2)]\le2$, we have $[K(\alpha^2):K]\ge 2$ and so there are two possibilities:
*
*$[K(\alpha^2):K]=2$: this happens iff $K(\alpha^2)\ne K(\alpha)$. For instance, for $K=\mathbb Q$ and $\alpha$ a root of $... | {
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Understanding how $() = C^{\log(x)}$ a polynomial function Is $f(x) = C^{\log(x)}$ a polynomial function?
I assume it must be because I learnt that you can rewrite this as $f(x) = x^{\log(C)}$ and given that $\log(C)$ is a constant, this looks like a polynomial function. However, I'm struggling to understand intuitive... | That function is not a polynomial, but its asymptotic growth is polynomial, since (as you show) it can be written as a constant power of $x$.
The fact that it can also be written some other way that does not look like polynomial growth is what makes the question interesting.
The function
$$
f(x) = x^{2+ \sin(x)}
$$
... | {
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How can the integral $\int_0^\frac{\pi}{2} x \cos ^n{x} dx$ be reduced? I tried to write $cos^nx$ as $\cos {x} \cos ^{n-1}{x}$ and then integrate by parts with the first function being $\cos ^{n-1}{x}$ and the second function being $x \cos {x}$. I was able to solve upto a point but then I got stuck since I was not abl... | To get a recurrence for the given integral, call it $I_n$, observe that $$I_{n-2}-I_n=\int_0^{\pi/2}x\cos^{n-2}x\sin^2 x\,dx=\int_0^{\pi/2}(x\sin x)(\cos^{n-2}x\sin x)\,dx,$$ which is ready to be integrated by parts: $$I_{n-2}-I_n=\frac{1}{n-1}\int_0^{\pi/2}\cos^{n-1}x(\sin x+x\cos x)\,dx=\frac{1}{n-1}\left(\frac{1}{n}... | {
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What is the basic lemma on composition of probability generating functions? What is the basic lemma on composition of probability generating functions and how is it most clearly proved?
(I'm posting this mainly to see if I can write an answer as clearly and simply as possible, but maybe other people know things about t... | Do you mean this? Let $S = \sum_{i\le N} Y_i$ where $N, Y_1, Y_2, \ldots$ are independent integer-valued random variables where $N\ge 0$ a.s., and $\mathbb E[z^N] = f(z)$ while $\mathbb E[z^{Y_i}] = g(z)$, then
$$ \mathbb E[z^S] = f(g(z))$$
| {
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Burgers' equation / Finite difference method Let's consider the Burgers' equation:
$$u_t(x,t)+u(x,t).u_x(x,t)=\epsilon.u_{xx}(x,t) \; , \; x\in \mathbb{R} \; ,\;t>0$$
And with the following initial condition:
$$u(x,0)=1_{\{x<0\}}(x)$$
I want to solve it using (Matlab, C, ... or any software) numerically and have a grap... | Why would a finite difference method not work? It can even be applied to the non-linear Navier-Stokes equations in multiple dimensions. Non-linearity is not an impediment even with time-implicit schemes as a system of non-linear difference equations can be solved iteratively.
As a starting point, a simple explicit di... | {
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A proof that $f(K)\le H$ if $K\le G$ for an isomorphism $f: G\to H$.
Let $f:G \rightarrow H$ be a group isomorphism, and $K \subset G$ be a subgroup. Show that $f(K) \subset H$ is a subgroup.
My attempt:
Let $a \in K$. Then since $K$ is a subgroup, $a^{-1} \in K \rightarrow f(a^{-1}) \in f(K)$. Therefore $f(K)$ is cl... | A critique of your proof is already given in the comments.
Here's another way to prove the theorem. It uses the one-step subgroup lemma.
We have $e\in K$ since $K\le G$. Thus $f(e)\in f(K)$, so $f(K)$ is nonempty. As the image of an isomorphism, $f(K)$ is a subset of $H$.
Now let $a, b\in f(K)$. Then $a=f(h), b=f(k)$ ... | {
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Find the infimum and supremum of $\{x\in Q \mid x² < 9\}$ I have to find (and prove) the infimum and supremum of the following set:
$M_1:=\{x\in\mathbb{Q} \mid x^2 < 9\}$
On first glance, I would say:
$\inf M_1=-3 $
$\sup M_1=3$
Now I have to prove that these really are the infimum and supremum of the set, and that's t... | Let $I= \{ x\in\mathbb{Q} : x^2 <9\}.$ $x^2<9\Rightarrow -3 < x < 3.$ Assume that $3-p,-3+p\in\mathbb{Q}$ are $\sup I$ and $\inf I$ respectively, where $p>0.$ By definition, the greatest lower bound $l$ of $I$ is the value satisfying $\forall \epsilon >0,\exists x\in I\;(x<l+\epsilon).$ Similarly, the least upper bound... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3409068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Definite integration evaluation of $\int_0^{\pi/2} \frac{\sin^2(x)}{(b^2\cos^2(x)+a^2 \sin^2(x))^2}~dx$. $$\int_0^{\pi/2} \frac{\sin^2(x)}{(b^2\cos^2(x)+a^2 \sin^2(x))^2}~dx$$
how to proceed please help
The answer given is $\dfrac{\pi}{4a^3b}$.
| Here is my solution:
$$\int_0^\frac{\pi}{2}\frac{\sin^2 x}{(b^2\cos^2 x+a^2\sin^2 x)^2}dx=\int_0^\frac{\pi}{2}\frac{\tan^2 x\sec^2 x}{(b^2 +a^2\tan^2 x)^2}dx\overset{\tan x=t}=\int_0^\infty \frac{t^2}{(b^2+a^2 t^2)^2}dt$$
$$\overset{at=bp}=\frac{1}{a^3b}\int_0^\infty\frac{p^2}{(p^2+1)^2} dp=\frac{1}{a^3b} \left(\int_0^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3409447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Curious about definition of well-ordered set We know that the definition of a well-ordered set guarantees the existence of least element in every non-empty subset of a set.
This isn't a creative or good question, but I am just wondering why it using the existence of least element but not the greatest element or even b... | About greatest element, well ordering comes to generalize in a sense the natural number, which has no maximum, so it makes sense to take least element.
In addition, it makes sense when looking on well founded sets, a well founded set is a set where when order by $∈$ it has a minimal element(not minimum, it need not be ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does $R\subseteq A\times A$ being antisymmetric imply the same for $S$? Given two sets $A, B$ let $f:A\to B$ be surjective and suppose $R\subseteq A\times A$ is an antisymmetric relation. Does it follow that $S=\{(b,b')\in B\times B\ \vert \exists a,a'\in A: a R a', f(a) = b, f(a') = b' \}$ is also antisymmetric?
Her... | You've done an appropriate amount of work to be done, but you should express your work as a counterexample rather than as a condition that any counterexample would have. For instance, your $A=\{a_1, a_2,a_3,a_4\}$, $B=\{b,b'\}$, and so on. Then if you explicitly show that your order on $A$ is antisymmetric but your o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Definition of Tychonoff space or $T_{3\frac{1}{2}}$ space Definition: A topological space $X$ which is $T_1$ space is called Tychonoff space or completely regular space ($T_{3\frac{1}{2}}$ space) if for any point $x\in X$ and $A$ closed in $X$ with $x\notin A$ exists continuous function $f:X\to [0,1]$ such that $f(x)=1... | Right.
Though you could circumvent this problem if you require $f(A)\subseteq\{0\}$, instead of $f(A)=\{0\}$. If $A$ is empty then $f(\varnothing)=\varnothing\subseteq\{0\}$.
The definition requires that your could find a suitable $f$ (in general depending on $A$ and $x$) for every $x$ and every non-empty closed $A\s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Extinction probability in a population with competition Suppose we have a colony of bacteria. At the end of each day, each bacterium produces an exact copy of itself with probability $p$ and then dies with probability $q$. However, $q$ is not constant, but a function of $N$, the total number of bacteria:
$$q=p\bigg(1-\... | (query on the exact terms of the problem, too long for a comment)
One bacterium, at time $t$ , with an extant population $n(t)$, has
- $p$ , constant, probability to generate an additional bacterium, so that it contributes $+1$ to $n(t+1)$;
- $q=p(1-1/n(t))$, depending on $n(t)$, probability to die and contributing ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3410112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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How to check if the number after decimal point of the result of a division goes to infinity? Particularly, is this number any special? I divide a number by a prime number and I observed the division in the below picture. I suspect that the result, quotient, goes to infinity. But how can I check if so? The big calculato... | It's not infinity, and it's not a question at all, it's just a rational number which is either integer or number with a repetitive decimal number.
like $\frac{1}{3}=0.33333333333\cdots$, do you think it's infinity??
But if you mean number of decimal points, it's obviously infinite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3410237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The kernel of this evaluation map. Consider $$f: \mathbb{Z}[x] \longrightarrow \mathbb{Z}[2^{1/3}]$$ which is a evaluation map for $x = 2^{1/3}$.
Question: What is the kernel of $f$?
My work so far: I take the ideal $(x^3-2)$ as my answer, and I want to show that $\ker(f) = (x^3-2)$. One side is trivial, i.e., $(x^3-2)... | Hint:
The kernel has to be a non-maximal prime ideal in $\mathbf Z[x]$, and $x^3-2$ is irreducible in $\mathbf Z[x]$, hence, as it is a U.F.D., the ideal $x^3-2$ is prime, and not maximal. Furthermore, $\mathbf Z[x]$ has Krull dimension $2$, so non-zero, non-maximal prime ideals all have height $1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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uncountable ring with countable quotient ring? Insipired by this post: Countable number of subgroups $\implies $ countable group
I'm wondering if it is possible to have an uncountable commutative ring with identity $R$, such that $R/I$ is countable for some two-sided proper ideal $I$?
It seems to be a really basic prob... | Yes. For instance, for any rings $A$ and $B$, the projection $A\times B\to B$ gives an isomorphism between $B$ and a quotient ring of $A\times B$. In particular, if $A$ is uncountable then $A\times B$ will be uncountable, but $B$ could be any ring at all. Or, for any ring $B$, you can form a polynomial ring $R$ over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3410525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving $A \subset B \implies A \cup B = B$ I have to prove this:
$A \subset B \implies A \cup B = B$
One direction:
$x\in A \subset B$
$\implies x \in A \land x \in B$
$\implies x \in A \cup B$
$\implies x \in (A \cup B = B)$
Other direction:
$x \in (A \cup B = B)$
$\implies x \in A \land x \in B$ (I used $\land$ he... | Below a proof using the natural deduction method.
The goal is read as a simple implication : if A is included in B , then ... ( not as a double implication).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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If $k$ is the smallest integer such that $[a^k]>[a]^k$, which of the following is true? This is a question from the KVPY(SX)-2014 (an examination to get into various research institutes in India) paper.
Q. For a real number $r$ let $[r]$ denote the largest integer less than or equal to $r$. Let $a > 1$ be a real numbe... | You can argue for (D) by ruling out (A), (B), and (C), on the basis that they depend only on $[a]$, and not $[a]-a$. Any bound on $k$ depending only on $[a]$ can be defeated by choosing $a-[a]$ small enough. This kind of reasoning is quick and may be valuable on a timed test.
To see why (D) works (this part is unnecess... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3410890",
"timestamp": "2023-03-29T00:00:00",
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Calculate $\lim_{x\to\infty}\frac{x^2}{x+1}-\sqrt{x^2+1}$ I am stuck on a limit of the indeterminate form $\infty-\infty$. I have tried many approaches, such as multiplying with conjugates etc. and I am unable to find a solution. I suspect that there is an elementary trick that I am plainly missing right here. Can anyb... | 1) $x^2=(x+1)-1)^2=$
$(x+1)^2-2(x+1)+1$;
$\dfrac{x^2}{x+1}=(x+1)-2 +\dfrac{1}{x+1};$
2) $(x^2+1)^{1/2}=x(1+1/(x^2))^{1/2}=$
$ x +O(1/x)$;
3) $\dfrac{x^2}{x+1} -(x^2+1)^{1/2}= $
$ -1+O(1/x)$;
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3411076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The principle of inclusion and exclusion: How many permutations of the set $\{1, 2,. . . , 8\}$ do not leave any even number in its place? I am supposed to solve the following problem:
How many permutations of the set $\{1, 2,. . . , 8\}$ do not leave any even number in its place?
What I tried:
$$8!-\left ( \binom{8}{1... | Let $A_k$ be the number of permutations that fix the $2k$-th element. By Inclusion-Exclusion, $$|A_1\cup A_2\cup A_3\cup A_4|=\sum_{1\leq w\leq4}|A_w|- \sum_{1\leq w<x\leq4}|A_w\cap A_x| +\sum_{1\leq w<x<y\leq4}|A_w\cap A_x\cap A_y|- \sum_{1\leq w<x<y<z\leq4}|A_w\cap A_x\cap A_y\cap A_z|$$ $$=\binom{4}{1}\cdot 7!-\bino... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3411188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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why here $x^2$ is used ? why not $x ?$
Does there exists a function $f : \mathbb{R }\rightarrow \mathbb{R}$ which is differentiable only at the point $0.$?
My attempt : I found the answer here Is there a function $f: \mathbb R \to \mathbb R$ that has only one point differentiable?
But i didn't understands the a... | Because while $x p(x)$ is continuous at $0$, it is not differentiable.
In particular, the fraction
$$
\frac{(0+h)p(0+h)-0p(0)}{h}
$$
has value $0$ or $1$ depending on whether $h$ is rational or not. So it has no limit as $h\to 0$, which by definition of derivative means that $xp(x)$ had no derivative at $0$.
On the oth... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Reduction of a Power Series Solution Separated by an Integer? I am given the following equation: $$x(x-1)\ddot y + 6x^2\dot y + 3y = 0 $$
and have solved/found the regular singular points, the exponents at singularity, the first series solution and the form of the second series solution, which are all as follows:
$x = ... | Just take derivative and separate:
Terms that $\ln$ has no derivative in them, they produce first term.
Terms that $\ln$ has derivative in them, make middle terms.
Terms that summation has derivative make last term.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3411435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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prove this series is convergent prove that the series $(a_1-a_2)+(a_2-a_3)+(a_3-a_4)+...$ converges if and only if the sequence $\{a_n\}$ converges.
I know a series is convergent if the sequence of its partial sums is bounded. I do not know how to apply that to this question.
| Hints:
Note that the series you have is
$$\sum_{k=1}^\infty (a_k - a_{k+1})$$
It's partial sum, in turn, is
$$\sum_{k=1}^n (a_k - a_{k+1})$$
Consider trying a few small values for $n$, and consider the pattern that emerges. Use this to make a formula that is equivalent to the partial sum. In the context of that formula... | {
"language": "en",
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Has this basic number theory conjecture been discovered or proved before? So I was sitting in a car bored for two hours playing around with the calculator on my phone, and I discovered that by picking two numbers (at least one being odd and both being not divisible by each other) I could always add them together to mak... | This is false: Try $15$ and $21$. Note that $3\mid15x-21y$ for all $x,y$.
However, the rule you are thinking of is basically Bezout's Identity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3411896",
"timestamp": "2023-03-29T00:00:00",
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Proving $\sum_{k = 1}^{\infty} k^{-2} = \pi^2 / 6$ using Fourier analysis I'm doing a homework for my analysis class, and a problem says the following:
Let $f : \mathbb{T}^1 \to \mathbb{C}$ be the function
$$f(\theta) = \begin{cases} 1 & 0 < \theta < \pi \\ 0 & - \pi < \theta < 0 \end{cases}$$
Compute the Fourier ... | Thank you to @GReyes for his help. He pointed out that the correct formula for $\| \cdot \|_{L^2}$ should've been
$$\| f \|_{L^2} = (2 \pi)^{-1} \int_{- \pi}^{\pi} |f(\theta)|^2 \mathrm{d} \theta .$$
With this fix, the solution to the problem is as follows:
\begin{align*}
\hat{f}(k) & = (2 \pi)^{-1} \int_{- \pi}^{ \pi}... | {
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Proving implicit function theorem with the fixed point theorem I am trying to solve an exercise in Kolmogorov's analysis text asking for a proof of the implicit theorem using the fixed point theorem. I am struggling with how to get started on this, though. It's clear that I need to define a mapping, $T$, demonstrate th... | Consider a continuously differentiable map
$$\begin{array}{l|rcl}
f : & \mathbb R^n \times \mathbb R^m & \longrightarrow & \mathbb R^m \\
& (x,y) & \longmapsto & f(x,y) \end{array}$$
a point $(a,b) \in \mathbb R^n \times \mathbb R^m$ such that $f(a,b) = 0$ and suppose that the partial Jacobian $J_{f,y}(a,b)$ is inv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Betting game: $3$ $20$-sided dice versus $2$ $30$-sided dice
Alice and Bob play a dice game. Alice rolls three $20$-sided dice and
Bob rolls two $30$-sided dice. Whoever has the highest sum wins.
Who has a better chance of winning?
This was a question asked in an interview. So I don't have much time to answer and I... | The flipping dice argument is straight forward. When flipping a dice you have that $n$ turns into $21-n$ (or $31-n$ depending on which type of die).
What you do is note that if Alice win for some outcome Bob will win in the equally probable(*) outcome when all dice are flipped. This is because if Alice rolled $a$ and B... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Dot product cancellation Given vectors $a,b,c$, I know that
$$
c\cdot a=c\cdot b
$$
does not imply $a=b$ (take three orthogonal vectors, for example).
However, if I say that $c\cdot a=c\cdot b$ holds for any vector $c$, is it then true that $a=b$? How should I argue?
| We have that
$$c\cdot a=c\cdot b \iff c\cdot (a-b)=0 \iff c=0 \quad \lor \quad a-b=0 \quad \lor \quad c\perp (a-b)$$
and since $c$ can assume any value only the case $a-b=0$ remains therefore
$$\forall c\,,\quad c\cdot a=c\cdot b \iff a=b $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Can the sum of two irrational roots of two coprime natural numbers greater than 1 be an irrational root of a rational number? After thinking more about my last question and reading the answers I reformulated it. I'm positive I wanted to ask the following:
Let $x,y\in \mathbb{N};\: x,y>1;\: gcd(x,y)=1; \: \sqrt{x},\sq... | In addition to the elementary solution mentioned in the comments (square both sides and find an irrational number on one side and a rational on the other), you might be interested in more of the theory behind this sort of question. The short version is that any polynomial equation $P(x)=0$ with integer coefficients sa... | {
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"timestamp": "2023-03-29T00:00:00",
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Let $S=\{\frac{\sqrt{x}}{x+1}:x>0\}$. Show that $\inf{S}=0$. Let $S=\{\frac{\sqrt{x}}{x+1}:x>0\}$. Show that $\inf{S}=0$.
My solution so far:
Since $\frac{\sqrt{x}}{x+1}>0$ for all $x>0$, $0$ is a lower bound of $S$. Next I know I need to show that $0$ is the largest lower bound of the set but I don't know how.
| We have that $\forall x>0$
$$\frac{\sqrt{x}}{x+1}>0$$
and
$$\lim_{x\to \infty}\frac{\sqrt{x}}{x+1}=0$$
then $\inf{S}=0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that $N$ is independent of $\{N_1 < N_2\}$ A random experiment has exactly three possible
outcomes, referred to as outcomes $1, 2,$ and $3,$ with probabilities $p_1 > 0, p_2 > 0,$ and $p_3 > 0,$ where $p_1 +p_2 +p_3 = 1.$ We consider a sequence of independent
trials, at each of which the specified random experimen... | HINT
$N_i$ is the trial at which $i$ first appears. So $N = \min(N_1, N_2)$ is the trial at which $1$ or $2$ first appears. In other words, before the $N$th trial, only $3$ has been appearing. In other words, $N>n$ is the event that the first $n$ trials are all $3$s.
(a) I would directly calculate $P(N>n \mid N_1 < ... | {
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If A commutes with both of these matrices, then A must be a scalar multiple of the identity matrix I am working on the following problem:
Let $A$ be a $4 \times 4$ matrix with entries in a field of characteristic zero. Suppose that $A$ commutes with both $\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 3 & 0\\... | Hint:
Left multiplication of a square matrix by $D=$ first (diagonal) matrix amounts to multiply its rows by the diagonal elements (for this matrix, the first row is multiplied by $1$, the second row by $2$, &c.). Right multiplication amounts to multiplying its columns by the diagonal elements. If both results are e... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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What are some interesting topological properties about the set of all convergent real sequences? Let C be the set of all convergent sequences in $\mathbb{R}$ under the product topology or uniform topology. So $C \subset \prod_{n=1}^\infty \mathbb{R}$. For one, Hilbert's Cube $\prod_{n=1}^\infty \left[0,\frac{1}{n}\righ... | Let $\ \lim:C\to\mathbb R\ $ be the limit function (functional). Then
$$ \lim(G)\ =\ \mathbb R $$
for every non-empty open subset of the product space. It follows that the limit function is not continuous under the product topology.
| {
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Express the polynomial in the form p(x) = (x+1) Q(x) +R where (x+1) is the divisor, Q(x) is the quotient and R is the remainder Express the polynomial in the form p(x) = (x+1) Q(x) +R where (x+1) is the divisor, Q(x) is the quotient and R is the remainder,
Hey I would just like to know how to solve this as the questio... | Put $\,a = -1\,$ in $\ p(x)\, =\, (x-a)\,\dfrac{p(x)-p(a)}{\ x\,-\,a} + p(a),\,$ the Polynomial Remainder Theorem.
The quotient is exact by the Polynomial Factor Theorem, and is computable by the Polynomial Division Algorithm (which also computes the remainder).
| {
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Find a vector NOT perpendicular to a given set of vectors
Let us suppose to have a finite set of vectors $S=\{v_1,\ldots,v_m\}$ in $\mathbb{R}^n$ (with $m \gg n$ in general). I need to find a vector $x \in \mathbb{R}^n$ that is NOT perpendicular to any vector in $S$. The existence of such a vector $x$ is guaranteed, m... | If you can implement on a computer, perhaps the following would work.
Create an $m \times n$ matrix $A$, the rows of which contain your vectors.
Now pick a random uniform vector $b > \vec{0} \in \mathbb{R}^m$, for example by generating the entrees of $\vec{b}$ uniformly, and solve $A \vec{x} = \vec{b}$.
If any solution... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof clarification: For any r.v.s $X$ and $Y$, $-1 \le \text{Corr}(X, Y) \le 1$. My textbook, Introduction to Probability by Blitzstein and Hwang, gives the following theorem in a section on covariance and correlation:
Theorem 7.3.5 (Correlation bounds). For any r.v.s $X$ and $Y$,
$$
-1 \le\operatorname{Corr}(X, Y) \... | If you write
$$0 \leq \text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2 \text{Cov}(X, Y) = 2 + 2 \rho,$$
and
$$0 \leq \text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y) - 2 \text{Cov}(X, Y) = 2 - 2 \rho,$$
then it might be clearer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3413712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Explanation for $\mathcal{L} (\theta(e^t))$ is holomorphic on $Re(s) > 1$ Let, $\theta(x)=\sum_{p\leq x} \log p$, it is known, $\theta(x) \sim x$, therefore, $\theta(e^t) = O(e^t)$. Recall the definition of the Laplace transform. I found in a lecture note (click and go to Page 6) that the Laplace transform$\mathcal{L... | The Laplace transform is $$\mathcal L(\vartheta(e^t))=\int_0^\infty e^{-st}\mathcal O(e^t)\,dt=\int_0^\infty\mathcal O(e^{-t(s-1)})\,dt$$ which converges whenever $\Re(s)>1$.
The last equality is justified by defining $$g(t)=\mathcal O(e^t)\iff|g(t)|\le Me^t\quad\forall t\ge t_0$$ and $M,t_0\in\Bbb R$, so $$e^{-st}|g(t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3413847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Probability generating functions for a linear change of variable? I am struggling to understand, why, in textbook exercise on probability generating functions it says that if $X$ is a random variable (rv) with probability generating function (pgf) $G_{X}(s)$, then for rv $Z = X + k$ for positive integer $k$ the pgf is... | Since $\mathbb P(X\geqslant0)=1$ we have
\begin{align}
G_Z(s) &= \sum_{n=0}^\infty \mathbb P(Z=n)s^n\\
&=\sum_{n=0}^\infty \mathbb P(X+k=n)s^n\\
&=\sum_{n=0}^\infty \mathbb P(X=n-k)s^n\\
\end{align}
and with the change of variables $m=n-k$, this is equal to
$$
\sum_{m=0}^\infty \mathbb P(X=m)s^{m+k} = s^k\sum_{m=0}^\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3413942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Which is bigger: a googolplex or $10^{100!}$ A googol is defined as $ 10^{100}$
Let x = $10^{100}$
A googolplex is defined as $10^{x}$
Which is bigger: a googolplex or $10^{100!}$
I only know that:
$100! = 1×2×3×...×98×99×100$
$10^{100} = 10×10×10×...×10×10×10$
I think its easier to approach if I only compare the expon... | The question boils down to whether $10^{100}$ is greater than $100!$ (same bases). But by Stirling's approximation,
$$100!\approx\sqrt{200\pi}(100/e)^{100}$$
and $100/e>33>10$ and $\sqrt{200\pi}<30$ so $100!>10^{100}$. Hence $10^{100!}$ is the bigger number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3414013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
} |
If $G$ is an infinite simple group then any proper subgroup of $G$ has infinite index.
If $G$ is an infinite simple group then any proper subgroup of $G$ has infinite index.
This question's hint is use the $n!$-theorem but i dont understand how i use it for answer.
$n!$-theorem: Let $G$ be a group and $H$ be a subgroup ... | From a stylistic perspective, quoting a "theorem" here this is unnecessarily complicated and makes things harder to understand. The proof of this statement relies on two basic facts; and these are facts that everyone who studies groups should know anyways. 1. There are only finitely many subgroups of a given finite i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3414099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Confusion about Phantom Maps A phantom cohomology operation (originally read phantom map) $f: X \rightarrow Y$ is a non-nullhomotopic map such that the induced cohomology operation on the cohomology theories for spaces is trivial. It is known that phantom maps exist.
Could someone tell me where my argument for the oppo... | I'll translate my argument to the language of suspension spectra which makes it more clear where the error lies:
Let's suppose I decompose $X$ as a colimit of suspension spectra. Then this phantom operation $f$ has the property that the restriction of $f$ to any of the objects in the colimit is trivial. This is precise... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3414520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Compute the derivative from the definition of $f(x)=e^{-1/x^2}$ for $x\neq 0.$ Is $f'(x)$ continuous?
Compute the derivative from the definition of $f(x)=e^{-1/x^2}$ for $x\neq 0.$ Is $f'(x)$ continuous?
I tried using the definition of continuity and I wasn't sure how to show formally that $\lim\limits_{h\to 0}\dfrac... | Note that $$ \frac 1 {x^2} - \frac 1 {(x+h)^2} = \frac {h (2x + h)}{x^2 (x+h)^2} \xrightarrow {h \to 0} 0 \cdot \frac {2x}{x^4} =0,$$ so the difference of the two fractions is an infinitesimal. Now apply the limit you have learned: $$\frac {\mathrm e^h - 1}h \to 1 [h \to 0],$$
by substituting the infinitesimals to a s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3414630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Suppose $f(x) > 0, f'(x) > 0,$ and $f''(x) >0.$ Then $\lim_{x \to \infty} f(x) = \infty.$ Intuitively I see why this is true, in terms of a graph, but I can't seem to get a handle on a proof. I know that I need to show for every $M > 0$ there's a $d > 0$ such that whenever $x > d$, $f(x) > M.$ I attempted proceeding by... | Hint:
Prove that $f(n+1)-f(n)\ge f'(0)$.
Remark: The condition that $f(x)>0$ is not needed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3414789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Any hint for this product of complex numbers I have to calculate the modulus of $z$. I've already tried to find a general formula for $\left(\frac{1+i}{2}\right)^{2^n}$, which seems to be $\frac{1}{2^{2^{n-1}}}, \forall \geq3$, using the trigonometric form and de Moivre's formula.
$z=\left[1+\frac{1+i}{2}\right] \left[... | Note that
$$z=
\frac {[1-(\frac {1+i}{2})]\left[1+\frac{1+i}{2}\right] \left[1+\left(\frac{1+i}{2}\right)^2\right] \left[1+\left(\frac{1+i}{2}\right)^4\right] \cdots \left[1+\left(\frac{1+i}{2}\right)^{2^n}\right] }{[1-(\frac {1+i}{2})]}=$$
$$\frac { \left[1-\left(\frac{1+i}{2}\right)^{2^{n+1}}\right] }{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3414924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Infinite sum with binomial coeffcient - Hypergeometric function ${}_2F_1$ and gauss theorem From an urn and balls problem, I end up with the need to compute the following sum
$$S = \sum_{n\geq 1} \frac{1}{n}\binom{2n}{n+1}2^{-2n}$$
Using Maple I discovered $S=1$. Starting with some basics transformations, I get
$$S= \s... | The given series is telescopic. If we set $a_n=\frac{1}{4^n}\binom{2n}{n}$ we have
$$ \frac{a_{n+1}}{a_n} = \frac{(2n+2)(2n+1)}{4(n+1)^2} = \frac{2n+1}{2n+2}=1-\frac{1}{2n+2}$$
hence
$$ a_n-a_{n+1} = \frac{1}{2n+2}\binom{2n}{n}2^{-2n}= \frac{1}{2}\cdot\frac{1}{n}\binom{2n}{n+1}2^{-2n}$$
and since $a_n\to 0$ we have
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3415081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Polynomial bijections between $\mathbb{N}^{m}$ and $\mathbb{N}$
*
*Is it known if a polynomial bijection from $\mathbb{N}^{m}$ to $\mathbb{N}$ must necessarily be a polynomial of degree $m$?
*Are there two polynomial bijections from $\mathbb{N}^{m}$ to $\mathbb{N}$ that are not obtainable from each other by a polyn... | Though I don't know of any particular reference, your questions can be answered for $m\geq 3$ by using the existence of a such a polynomial on $m=2$.
First, let $f(x,y)={x+y+1 \choose 2}+x$ be the usual polynomial bijection $\mathbb N^2\rightarrow \mathbb N$. We can make two observations: first, this generalizes to all... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3415207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Recursive compactification: Is the direct limit still compact? Thinking about the question whether there is something like a “maximal compactification” (the one-point compactification could be seen as “minimal” compactification), I've come across the possibility to recursively compactify a set (so that you get multiple... | Sequential direct limits are almost never compact. In particular, a direct limit of a sequence of inclusions of $T_1$ spaces is never compact (unless the sequence eventually stabilizes); see Compact subset in colimit of spaces. So, if the initial compactification you use is $T_1$, then your direct limit will not be c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3415520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Matroids-Discrete Optimization I happen to stumble across this question where they ask to give an example of an independent system that has two maximal independent sets $X$ and $Y$ such that $|X|\neq |Y|$. How exactly would you formulate such an example?
| An independence system is a collection $C$ of subsets of a finite set $E$ such that if $c \in C$ and $b \subset c$ then $b \in C$. Independence systems are also known as finite (abstract) simplicial complexes. A simplicial complex whose inclusion-maximal subsets all have the same cardinality is called pure. The questio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3415784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find $n$ points on a circle with integer distances.
Let $n$ be a positive integer, prove that it is possible to put $n$ points on a circle so that the distances among them are all integers.
For $n \leq 3$ this is trivial. I have shown it for $n=4$ by considering a rectangle.
I don't know how to do it for larger numbe... | This is with radius 5 and edge length 6. It appears that the final edge (not drawn in) also has rational length since it is vertical and both endpoints have rational coordinates
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3415894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Saturation of the reals as a dense linear order I've been given the following problem (note $T_{ord}$ is the theory of dense linear orders without endpoints):
Show that $\langle \mathbb{R}, < \rangle$ (the reals with their usual ordering) is not a saturated model of $T_{ord}$ of cardinality $\aleph_1$.
I don't think ... | Yes, the "of cardinality $\aleph_1$" here is not really relevant (it simply gives a much shorter proof in the event that $\mathbb{R}$ does not have cardinality $\aleph_1$ at all, i.e. if the continuum hypothesis is false). It is possible that the intended statement was instead that $\langle \mathbb{R}, < \rangle$ is n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3416006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Coordinatable continuous functions in terms of orthogonal systems $\{e^{in\theta}\}$ Question. Let $f$ be an arbitrary $2\pi$-periodic integrable function on reals. Does there exists any sequence $\{t_n\}$ with $f(\theta)=\sum t_ne^{in\theta}$?
If NOT, what about if we replace integrable function with a continuous o... | There is the following theorem of de la Vallée-Poussin, which I quote from page 880 of J. Marshall Ash's Uniqueness of Representation by Trigonometric Series:
If $S=\sum_n t_n e^{inx}$ converges to $f$ at each $x$, and if $f$ is finite at each $x$ and if $\int_{\mathbb T} |f(x)| dx< \infty$, then $S$ is the Fourier s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3416153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
When two curves touch each other at a single point, are they called intersecting? When any two curves touch each other at a single point, are they called intersecting, or just called they are touching each other? Are these terms - intersecting curves and touching curves used interchangeably? Is there any difference bet... | Let me clarify my above comment.
The discussion might be endless because there is no consensus on the definitions of "intersecting" and "touching".
One cannot give a definitive answer to the question raised until the definitions of these two words be clear and accepted by everybody.
Presently I doubt that the respectiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3416237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
What condition for $n$ such that $G_n$ is graph planar? Let $G_n$ be the simple graph with the set of vertices $V_n=\{1, 2, . . . , n\}$. Two different vertices $x$ and $y$ are adjacent if $x$ is a multiple of $y$ or $y$ is a multiple of $x$. For what $n$ is $G_n$ planar?
I tried to draw graph starting from small $n$ u... | If $n\geq18$ the graph $G_n$ is not planar since the vertices $\{1,2,3,6,12,18\}$ are the vertices of a $K_{3,3}$ inside $G_n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3416408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A conceptual reason for why the Jacobian of a rotation by a changing angle is $1$? Consider
$$f(x,y)=(x\cos r^2+y\sin r^2, y\cos r^2-x\sin r^2)\qquad\text{with }r=\sqrt{x^2+y^2},$$
as a map $\mathbb{R}^2 \to \mathbb{R}^2$. Geometrically, $f(x,y)$ is obtained from the vector $(x,y)$ by rotating it by angle $r^2$. (we ro... | Partition the plane into very thin annular sets $A_k: \>r_{k-1}\leq\sqrt{x^2+y^2}< r_k$ $(k\in {\mathbb N}_{\geq1})$. The map $f$ (approximately) rotates each $A_k$ by an angle $\phi_k$ around the origin, hence is (approximately) an isometry on $A_k$. This implies $\mu\bigl(f(X)\bigr)=\mu(X)$ for any $X\subset A_k$. F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3416520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Evaluation of Integral sums Hey I am supposed to evaluate: $$\lim_{n \to \propto }\sum_{i=1}^{n}\frac{i}{n^{2}+i^{2}}$$
What I did:
$$\lim_{n \to \propto }\sum_{i=1}^{n}\frac{i}{n^{2}+i^{2}}=\lim_{n \to \propto }\frac{1}{n^{2}}\sum_{i=1}^{n}\frac{i}{1+\left ( \frac{i}{n} \right )^{2}}$$
But I do not know, what to do ne... | $$\dfrac r{n^2+r^2}=\dfrac1n\cdot\dfrac{r/n}{1+(r/n)^2}$$
Now use The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3416612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
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