Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How many subgroups of $\Bbb Z_5\times \Bbb Z_6$ are isomorphic to $\Bbb Z_5\times \Bbb Z_6$ I am trying to find the answer to the question in the title. The textbook's answer is only $\Bbb Z_5\times \Bbb Z_6$ itself. But i think like the following: Since 5 and 6 are relatively prime, $\Bbb Z_5\times \Bbb Z_6$ is isomor... | The underlying sets of these groups are by definition $$\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5=\{(a,b,c):a \in \mathbb{Z}_2 \text{ and } b \in \mathbb{Z}_3 \text{ and } c \in \mathbb{Z}_5\}$$ and $$\mathbb{Z}_5 \times \mathbb{Z}_6=\{(a,b):a \in \mathbb{Z}_5 \text{ and } b \in \mathbb{Z}_6\}.$$ So while $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Maximal compact subgroups of $GL_n(\mathbb{R})$. The subgroup $O_n=\{M\in GL_n(\mathbb{R}) | ^tM M = I_n\}$ is closed in $GL_n(\mathbb{R})$ because it's the inverse image of the closed set $\{I_n\}$ by the continuous map $X\mapsto ^tX X$. $O_n$ is also bounded in $GL_n(\mathbb{R})$, for example this is clear by conside... | Let $G \subset GL_n(\mathbb{R})$ be a compact group containing $O(n)$ and let $M \in G$. Using polar decomposition, $$M=OS \ \text{for some} \ O \in O(n), \ S \in S_n^{++}(\mathbb{R}).$$
Since $O(n) \subset G$, we deduce that $S \in G$.
Because $G$ is compact, $(S^n)$ has a convergent subsequence; $S$ being diagonaliz... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Help with proof that $I = \langle 2 + 2i \rangle$ is not a prime ideal of $Z[i]$ (Note: $Z[i] = \{a + bi\ |\ a,b\in Z \}$)
This is what I have so far.
Proof:
If $I$ is a prime ideal of $Z[i]$ then $Z[i]/I$ must also be an integral domain.
Now (I think this next step is right, I'm not sure though),
$$
Z[i]/I = \{a+bi ... | A much simpler argument:
Note that $2,1+i\notin \langle 2+2i\rangle$, yet $2\cdot (1+i)=2+2i\in \langle 2+2i\rangle$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/361875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
A question concerning fundamental groups and whether a map is null-homotopic. Is it true that if $X$ and $Y$ are topological spaces, and $f:X \rightarrow Y$ is a continuous map and the induced group homomorphism $\pi_1(f):\pi_1(X) \rightarrow \pi_1(Y)$ is the trivial homomorphism, then we have that $f$ is null-homotopi... | Take $X=S^{2}$, $Y=S^{2}$, and the map $f(x)=-x$. This map has degree $-1 \neq 0$, therefore it is not nullhomotopic. However, $\pi_{1} (S^{2})$ is trivial, so the induced map will be between trivial groups, and is thus trivial.
The claim you're making is too strong because it asserts that whenever $Y$ is simply conne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/361949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Find the last two digits of $ 7^{81} ?$ I came across the following problem and do not know how to tackle it.
Find the last two digits of $ 7^{81} ?$
Can someone point me in the right direction? Thanks in advance for your time.
| $\rm{\bf Hint}\ \ mod\,\ \color{#c00}2n\!: \ a\equiv b\, \Rightarrow\, mod\,\ \color{#c00}4n\!:\ \, a^2 \equiv b^2\ \ by \ \ a^2\! = (b\!+\!\color{#c00}2nk)^2\!=b^2\!+\!\color{#c00}4nk(b\!+\!nk)\equiv b^2$
$\rm So,\, \ mod\,\ \color{}50\!:\, 7^{\large 2}\!\equiv -1\Rightarrow mod\ \color{}100\!:\,\color{#0a0}{7... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
} |
Power Series Solution for $e^xy''+xy=0$ $$e^xy''+xy=0$$
How do I find the power series solution to this equation, or rather, how should I go about dealing with the $e^x$? Thanks!
| When trying to find a series to represent something, it's important to decide what kind of a series you want. Even if you're dealing with power series, a small change in notation between $\displaystyle \sum a_n x^n$ and $\displaystyle\sum \frac{a_nx^n}{n!}$ can lead to substantial changes. In particular, we have that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Proving two graphs are isomorphic I need to prove that the following two countable undirected graphs $G_1$ and $G_2$ are isomorphic:
Set of vertices of $G_1$ is $\mathbb{N}$ and there is an edge between $i$ and $j$ if and only if the $j$ th bit of the binary representation of $i$ is $1$ or the $i$ th bit of the binary ... | HINT: These are both the Rado graph, which is the unique countable graph with the following extension property: if $U$ and $V$ are disjoint finite sets of vertices of the graph, there is a vertex $x$ connected to each vertex in $U$ and to no vertex in $V$. The link actually demonstrates this for $G_1$, and the same art... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Coefficients of series given by generating function How to find the coefficients of this infinite series given by generating function.$$g(x)=\sum_{n=0}^{\infty}a_nx^n=\frac{1-11x}{1-(3x^2+10x)}$$
I try to expand like Fibonacci sequences using geometric series and binomial theorem but without any success.
| Use (1) the fact that $1-3x^2-10x=(1-ax)(1-bx)$ for some $a$ and $b$, (2) the fact that
$$
\frac{1-11x}{(1-ax)(1-bx)}=\frac{c}{1-ax}+\frac{d}{1-bx},
$$
for some $c$ and $d$, and (3) the fact that, for every $e$,
$$
\frac1{1-ex}=\sum_{n\geqslant0}e^nx^n.
$$
Then put all these pieces together to deduce that, for every $n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Number of bases of an n-dimensional vector space over q-element field. If I have an n-dimensional vector space over a field with q elements, how can I find the number of bases of this vector space?
| There are $q^n-1$ ways of choosing the first element, since we can't choose zero. The subspace generated by this element has $q$ elements, so there are $q^n-q$ ways of choosing the second element. Repeating this process, we have $$(q^n-1)(q^n-q)\cdots(q^n-q^{n-1})$$ for the number of ordered bases. If you want unordere... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
How to calculate max iterations needed to equally increase row of numbers per some value each iteration? I don't know whether title describes the main idea of my question, so apologize me for it.
I have 6 numbers whose values can vary from 0 to 100, but initial value cannot be more than 35. As example, here is my numbe... | You start with $a_1, \ldots, a_n$, have to distribute $d\ge n$ in each round and stop at $m>\max a_i$.
Then the maximal number of rounds is bounded by two effects:
*
*The maximal value grows by at least one per round, so it reaches $m$ after at most $m-\max a_i$ rounds.
*The total grows by $d$ each round, so you ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Nice examples of groups which are not obviously groups
I am searching for some groups, where it is not so obvious that they are groups.
In the lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't think that these examples are helpful to understand the real prop... | I always found the fact that braid groups are groups at all quite interesting. The elements of the group are all the different braids you can make with, say, $n$ strings. The group operation is concatenation. The identity is the untangled braid. But the fact that inverses exist is not obvious.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/362446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "325",
"answer_count": 31,
"answer_id": 25
} |
Counter-examples of homeomorphism Briefly speaking, we know that a map $f$ between $2$ topological spaces is homeomorphic if $f$ is a bijection and the inverse of $f$ and itself are both continuous.
So, can anyone give me $2$ counter examples(preferably simple ones) of non-homeomorphic maps $f$ between 2 topological sp... | $1.$ Let $X$ be the set of real numbers, with the discrete topology, and let $Y$ be the reals, with the ordinary topology. Let $f(x)=x$. Then $f$ is continuous, since every subset of $X$ is open. But $f^{-1}$ is not continuous.
$2.$ In doing $1$, we have basically done $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/362592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
Nicolas Boubarki, Algebra I, Chapter 1, § 2, Ex. 12
Nicolas Boubarki, Algebra I, Chapter 1, § 2, Ex. 12:
($E$ is a Semigroup with associative law (represented multiplicatively), $\gamma_a(x)=ax$.)
Under a multiplicative law on $E$, let $ a \in E $ be such that $\gamma_a $ is surjective.
(a) Show that, if there exist... | I have a Russian translation of Bourbaki. In it Ex.12 looks as follows:
"For $\gamma_{ba}$ to be an one-one mapping of $E$
into $E$, it is necessary and sufficient that $\gamma_{a}$ be an one-one mapping of
$E$ onto $E$ and $\gamma_{b}$ be an one-one mapping of
$E$ into $E$."
So I guess that there is a misprint in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Differential equation must satisfy its edge conditions. I have this variation problem
$$\text{Minimize} \; \int_0^1 \left( 12xt- \dot{x}^2-2 \dot{x} \right) \; dt$$
With the edge conditions $x(0)=0$ and $x(1)$ is "free".
And from here solve it:
$$x(t)\to -t^3 +c_1t+c_2$$
From here it should've been correctly. Now I mus... | Denote the functional as $J(x)$:
$$
J(x) = \int_0^1 \left( 12xt- \dot{x}^2-2 \dot{x} \right)
$$
Then the minimizer $x$ satisfies the following (perturbing the minimum with $\epsilon y$):
$$ \frac{d}{d\epsilon} J(x + \epsilon y)\Big\vert_{\epsilon = 0} =0$$
Simplifying above gives us:
$$
\int^1_0 (12 y t - 2\dot{x}\dot{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Numbers that are divisible So I am given the following question: For natural numbers less than or equal to 120, how many are divisible by 2, 3, or 5? I solved it by inclusion-exclusion principle and by using the least common multiple by having it as (2, 3, 5)=120 which is equal to 30. Are these the right way to solve t... | It is a right way. Inevitably there are others. For example, there are $\varphi(120)$ numbers in the interval $[1,120]$ which are relatively prime to $120$. Here $\varphi$ is the Euler $\varphi$-function.
The numbers in our interval which are divisible by $2$, $3$, or $5$ are precisely the numbers in our interval whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Convergence of $\prod_{n=1}^\infty(1+a_n)$ The question is motivated by the following exercise in complex analysis:
Let $\{a_n\}\subset{\Bbb C}$ such that $a_n\neq-1$ for all $n$. Show that if $\sum_{n=1}^\infty |a_n|^2$ converges, then the product $\prod_{n=1}^\infty(1+a_n)$ converges to a non-zero limit if and only ... | I shall try to give examples where $\sum|a_n|^2$ is divergent and all possible combinations of convergence/divergence for $\prod(1+a_n)$ and $\sum a_n$.
Let $a_{2n}=\frac1{\sqrt n}$ and $a_{2n+1}=\frac1{1+a_{2n}}-1=-\frac{1}{1+\sqrt n}$. Then $(1+a_{2n})(1+a_{2n+1})= 1$, hence the product converges. But $a_{2n}+a_{2n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Concise proof that every common divisor divides GCD without Bezout's identity? In the integers, it follows almost immediately from the division theorem and the fact that $a \mid x,y \implies a \mid ux + vy$ for any $u, v \in \mathbb{Z}$ that the least common multiple of $a$ and $b$ divides any other common multiple.
In... | We can gain some insight by seeing what happens for other rings. A GCD domain is an integral domain $D$ such that $\gcd$s exist in the sense that for any $a, b \in D$ there exists an element $\gcd(a, b) \in D$ such that $e | a, e | b \Rightarrow e | \gcd(a, b)$. A Bézout domain is an integral domain satisfying Bézout'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 6,
"answer_id": 1
} |
Finding the median value on a probability density function Quick question here that I cannot find in my textbook or online.
I have a probability density function as follows:
$\begin{cases}
0.04x & 0 \le x < 5 \\
0.4 - 0.04x & 5 \le x < 10 \\
0 & \text{otherwise}
\end{cases}$
Now I understand that for the median, the v... | I don't know, let's find out. Maybe the median is in the $[0,5]$ part. Maybe it is in the other part. To get some insight, let's find the probability that our random variable lands between $0$ and $5$. This is
$$\int_0^5(0.04)x\,dx.$$
Integrate. We get $0.5$. What a lucky break! There is nothing more to do. The median... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Generating $n\times n$ random matrix with rank $n/2$ using matlab Can we generate $n \times n$ random matrix having any desired rank?
I have to generate a $n\times n$ random matrix having rank $n/2$.
Thanks for your time and help.
| Generate $U,V$ random matrices of size $n \times n/2$, then almost surely $A = U \cdot V^T$ is of rank $n/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/363103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
$f$ is a bijective function with differentiable inverse at a single point
Let $\Omega \subseteq \mathbb{R}^n$ and $p \in \Omega$. Let $f:U \to V$ be a bijection of open sets $p \in U \subseteq \Omega$ amd $f(p) \in V \subseteq \mathbb{R}^n$. If $f^{-1}: V \to U$ is differentiable at $p$, then $df_p: \mathbb{R}^n \to... | The problem statement is either incorrect, incomplete, or both. Certainly, in order to say anything about $df_p$, the assumption on $f^{-1}$ should be made at $f(p)$. But the mere fact that $f^{-1}$ is differentiable at $f(p)$ is not enough. For example, $f(x)=x^{1/3}$ is a bijection of $(-1,1)\subset \mathbb R$ onto i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Trace of the matrix power Say I have matrix $A = \begin{bmatrix}
a & 0 & -c\\
0 & b & 0\\
-c & 0 & a
\end{bmatrix}$.
What is matrix trace
tr(A^200)
Thanks much!
| You may do it by first computing matrix powers and then you may calculate whatever you want. Now question is how to calculate matrix power for a given matrix, say $A$? Your goal here is to develop a useful factorization $A = PDP^{-1}$, when $A$ is $n\times n$ matrix.The matrix $D$ is a diagonal matrix (i.e. entries off... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Show that if $G$ is a group of order 6, then $G \cong \Bbb Z/6\Bbb Z$ or $G\cong S_3$
Show that if $G$ is a group of order 6, then $G \cong \Bbb Z/6\Bbb Z$ or $G\cong S_3$
This is what I tried:
If there is an element $c$ of order 6, then $\langle c \rangle=G$. And we get that $G \cong \Bbb Z/6 \Bbb Z$. Assume there d... | Instead of introducing a new element called $c$, we can use the group structure to show that the elements $ab$ and $ab^2$ are the final two elements of the group, that is $G=\{e,a,b,b^2,ab,ab^2\}$. Notice that if $ab$ were equal to any of $e,a,b$, or $b^{-1}=b^2$, we would arrive at the contradictions $a=b^{-1}$, $b=e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
find the largest perfect number less than $10,000$ in Maple Can anyone tell me how to find largest perfect number less than 10000 in maple?
Actually, I know how to find all the perfect numbers less than or equal to 10000 but I don't know how to find the largest one within the same code?
| Well if you know how to find them all, I suppose you use a loop.
So before your loop add a variable $max=0$. During the loop, for each perfect number $p$ you find, check if $p>max$ and if it is, then do $max=p$.
The value of $max$ after the end of the loop will be the greatest number found ;)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/363424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Which way does the inclusion go? Lemma Let $\mathcal{B}$ and $\mathcal{B'}$ be bases for topologies $\mathcal{T}$ and $\mathcal{T'}$, respectively, on $X$. Then the following are equivalent:
*
*$\mathcal{T'}$ is finer than $\mathcal{T}$.
*For each $x\in X$ and each basis element $B\in \mathcal{B}$ containing $x$, t... | The idea is that we need every $\mathcal{T}$-open set to be $\mathcal{T}'$-open. Since $\mathcal{B}$ is a basis for $\mathcal{T}$, then every $\mathcal{T}$-open set is a union of $\mathcal{B}$-elements (and every union of $\mathcal{B}$-elements is $\mathcal{T}$-open), so it suffices that every $\mathcal{B}$-element is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
High-order elements of $SL_2(\mathbb{Z})$ have no real eigenvalues Let $\gamma=\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \in SL_2(\mathbb{Z})$, $k$ the order of $\gamma$, i.e. $\gamma^k=1$ and $k=\min\{ l : \gamma^l = 1 \}$. I have to show that $\gamma$ has no real eigenvalues if $k>2$.
The eigenvalues of $\gamma$... | Assume there is a real eigenvalue.
Then the minimal polynomial of $\gamma$ is a divisor of $X^k-1$ and has degree at most $2$ and has at least one real root. If its degree is $2$, the other root must also be real.
The only real roots of unity are $\pm1$, so the minimal polynomial os one of $X-1$, $X+1$ or $(X-1)(X+1)=X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Counting binary sequences with $S$ $0$'s and $T$ $1$'s where every pre-sequence contains fewer $1$'s than $0$'s How many $S+T$-digit binary sequences with exactly $S$ $0$'s and $T$ $1$'s exist where in every pre-sequence the number of $1$'s is less than the number of $0$'s?
Examples:
*
*the sequence $011100$, is bad... | This is a famous problem often called Bertand's Ballot Theorem. A good summary is given in the Wikipedia article cited. There are a number of nice proofs.
Note that your statement is the classical one ("always ahead") but the example of a good sequence that you give shows that "never behind" is intended. If that is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
proving that the following limit exist How can I prove that the following limit exist?
$$
\mathop {\lim }\limits_{x,y \to 0} \frac{{x^2 + y^4 }}
{{\left| x \right| + 3\left| y \right|}}
$$
I tried a lot of tricks. At least assuming that this limit exist, I can prove using some special path (for example y=x) that the l... | There are more appropriate ways, but let's use the common hammer. Let $x=r\cos\theta$ and $y=r\sin\theta$. Substitute. The only other fact needed is that $|\sin\theta|+|\cos\theta|$ is bounded below. An easy lower bound is $\frac{1}{\sqrt{2}}$.
When you substitute for $x$ and $y$ on top, you get an $r^2$ term, part of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
If I have $5^{200}$, can I rewrite it in terms of $2$'s and $3$'s to some powers? If I have $5^{200}$, can I rewrite it in terms of $2$'s and $3$'s to some powers? For example, if I had $4^{250}$ can be written in terms of $2$'s like so: $2^{500}$.
| No. This is the Fundamental theorem of algebra: every integer $n\geq 2$ can be written in exactly one way (up to the order of factors) as a product of powers of prime numbers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/363876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Showing that one cannot continuously embed $\ell^\infty$ in $\ell^1$. Is it possible to embed $\ell^\infty$ into $\ell^1$ continuously? I.e. can one find a continuous linear injection $I:\ell^\infty \to \ell^1$.
I have reduced a problem I have been working on to showing that this cannot happen, but I don't see how to p... | Yes, it's possible; for example, you can set
$$
I(a_1,a_2,a_3,\dots):=(\frac{a_1}{1^2},\frac{a_2}{2^2},\frac{a_3}{3^2},\dots).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/363972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Turing Decryption MIT example I am learning mathematics for computer science on OpenCourseWare. I have no clue in understanding below small mathematical problem.
Encryption: The message m can be any integer in the set $\{0,1,2,\dots,p−1\}$; in particular, the message is no longer required to be a prime. The sender ... | The line with the question mark is just a restatement of the explanation above in symbolic form.
We have a message $m$ and encrypted message $m^* = \text{remainder}(mk,p)$. If we are given $m^*$ we can recover $m$ by multiplying by $k^{-1}$ and taking the remainder mod $p$. That is, $\text{remainder}(m^* \cdot k^{-1},p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Calculating $\lim\limits_{n\to\infty}\frac{(\ln n)^{2}}{n}$ What is the value of $\lim\limits_{n\to\infty}\dfrac{(\ln n)^{2}}{n}$ and the proof ?
I can't find anything related to it from questions.
Just only $\lim\limits_{n\to\infty}\dfrac{\ln n}{n}=0$, which I know it is proved by Cesàro.
| We can get L'Hospital's Rule to work in one step. Express $\dfrac{\log^2 x}{x}$ as $\dfrac{\log x}{\sqrt{x}}\cdot\dfrac{\log x}{\sqrt{x}}$. L'Hospital's Rule gives limit $0$ for each part.
Another approach is to let $x=e^y$. Then we want to find $\displaystyle\lim{y\to\infty} \dfrac{y^2}{e^y}$.
Note that for positive ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Intuition behind Borsuk-Ulam Theorem I watched the following video to get more intuition behind Borsuk-Ulam Theorem.
The first part of the video was very clear for me, as I understood it considers only $R^2$ dimension and points $A$ and $B$ moving along the equator and during the video we track the temperature of point... | Creator of the video here.
but along this part we don't move A and B there is no intersection of temperatures as in was in the first part
Yeah I didn't elaborate on this as much as the previous section, but the same thing is happening, just along an arbitrary path of connected opposite points, instead of a great sphe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How to show that $C(\bigcup _{i \in I} A_i)$ is a supremum of a subset $\{C(A_i): i \in I \}$ of the lattice $L_C$ of closed subsets? According to Brris & Sankappanavar's "A course in universal algebra," the set $L_C$ of closed subsets of a set $A$ forms a complete lattice under $\subseteq$. Here, a subset $X$ of $A$ ... | $\bigcup C(A_i)$ is not necessarily closed, and the smallest closed set containing it is $C[\bigcup C(A_i)]$. Now, $\bigcup A_i \subset \bigcup C(A_i)$, thus $C(\bigcup A_i) \subset C[\bigcup C(A_i)]$.
Conversely, $A_i \subset \bigcup A_i$, so $C(A_i) \subset C(\bigcup A_i)$. Therefore, $\bigcup C(A_i) \subset C(\bigcu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Minimal Distance between two curves What is the minimal distance between curves?
*
*$y = |x| + 1$
*$y = \arctan(2x)$
I need to set a point with $\cos(t), \sin(t)$?
| One shortcut here is to note that curves 1, 2 (say $f(x)$, $g(x)$) have a co-normal line passing between the closest two points.
Therefore, since $f'(x) = 1$ for all $x>0$ then just find where $g'(x) = 1$ or
\begin{align}&\frac{2}{4x^2 +1} = 1\\
&2 = 4x^2 + 1\\
&\bf{x = \pm 1/2}\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/364341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to show $\dim_\mathcal{H} f(F) \leq \dim_\mathcal{H} F$ for any set $F \subset \mathbb{R}$ and $f$ continuously differentiable? Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable with continuous derivative. I have to show that for all sets $F \subset \mathbb{R}$, the inequality $$\dim_\mathcal{H} f(F) \leq \dim_\... | Hint: Show that the inequality is true if $f$ is lipschitz. Then, deduce the general case from the following property: $\dim_{\mathcal{H}} \bigcup\limits_{i \geq 0} A_i= \sup\limits_{i \geq 0} \ \dim_{\mathcal{H}}A_i$.
For a reference, there is Fractal Geometry by K. Falconer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/364415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculate eigenvectors I am given the $2\times2$ matrix $$A = \begin {bmatrix} -2&-1 \\\\ 15&6 \ \end{bmatrix}$$
I calculated the Eigenvalues to be 3 and 1. How do I find the vectors? If I plug the value back into the character matrix, I get $$B = \begin {bmatrix} -5&1 \\\\ 15&3 \ \end{bmatrix}$$
Am I doing this right... | Remember what the word "eigenvector" means. If $3$ is an eigenvalue, then you're looking for a vector satisfying this:
$$A = \begin {bmatrix} -2&-1 \\\\ 15&6 \ \end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} = 3\begin{bmatrix} x \\ y\end{bmatrix}$$
Solve that. You'll get infinitely many solutions since every scalar ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solving a set of 3 Nonlinear Equations In the following 3 equations:
$$
k_1\cos^2(\theta)+k_2\sin^2(\theta) = c_1
$$
$$
2(k_2-k_1)\cos(\theta)\sin(\theta)=c_2
$$
$$
k_1\sin^2(\theta)+k_2\cos^2(\theta) = c_3
$$
$c_1$, $c_2$ and $c_3$ are given, and $k_1$, $k_2$ and $\theta$ are the unknowns. What is the best way to solv... | Add and subtract equations $1$ and $3$, giving the system
$$\begin{cases}\begin{align}k_1+k_2&=c_1+c_3\\(k_1-k_2)\sin2\theta&=-c_2\\(k_1-k_2)\cos2\theta&=c_1-c_3\end{align}\end{cases}$$
Then you find $k_1-k_2$ and $2\theta$ by a polar-to-Cartesian transform, giving
$$\begin{cases}\begin{align}k_1+k_2&=c_1+c_3,\\k_1-k_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Find $\arg\max_x \operatorname{corr}(Ax, Bx)$ for vector $x$, matrices $A$ and $B$ This is similar to, but not the same as, canonical correlation: For $(n \times m)$ matrices $A$ and $B$, and unit vector $(m \times 1)$ $x$, is there a closed-form solution to maximize the correlation between $Ax$ and $Bx$ w.r.t. $x$? No... | Here is an answer for the case $m>n$.
Write $x=(x_1,\ldots,x_m)^T,A=(a^{1},\ldots,a^{m}),B=(b^{1},\ldots,b^{m})$, so $Ax=\sum_{i\le m} x_ia^i$, $Bx=\sum_{i\le m} x_i b^i$. Since $m>n$, columns $a^i - b^i$ of the matrix $A-B$ are linearly dependent, i.e. there is $x$ such that $Ax=Bx$. For this $x$ we have ${\rm corr}(A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Graph of an inverse trig function. Which of the following is equivalent to the graph of $arcsin(x)$ ?
(a) Reflecting $arccos(x)$ about the y-axis, then shift down by $\pi /2$ units.
(b) Reflecting $arccos(x)$ about the x-axis, then shift up by $\pi /2$ units.
I think they are both the same thing. Can someone confirm th... | You can look at graphs of all three functions here:
http://www.wolframalpha.com/input/?i=%7Barccos%28-x%29-pi%2F2%2C-arccos%28x%29%2Bpi%2F2%2Carcsin%28x%29%7D
Do they look the same to you?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/364732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to prove that $\|AB-B^{-1}A^{-1}\|_F\geq\|AB-I\|_F$ when $A$ and $B$ are symmetric positive definite? Let $A$ and $B$ be two symmetric positive definite $n \times n$ matrices. Prove or disprove that
$$\|AB-B^{-1}A^{-1}\|_F\geq\|AB-I\|_F$$
where $\|\cdot\|_F$ denotes Frobenius norm. I believe it is true but I have n... | For the Froebenius Norm:
Since $A$ and $B$ are positive definite, we can write $C=AB=QDQ^\dagger$, with $D$ being the a diagonal matrix with the $n$ positive eigenvalues $\lambda_k$ and $Q$ a hermitian matrix ($QQ^\dagger=QQ^{-1}=I$). So we obtain
$$
||C - C^{-1}|| \geq ||C - I||
$$
Since the Froebenius Norm is invari... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Inverse bit in Chinese Remainder Theorem I need to solve the system of equations:
$$x \equiv 13 \mod 11$$
$$3x \equiv 12 \mod 10$$
$$2x \equiv 10 \mod 6.$$
So I have reduced this to
$$x \equiv 2 \mod 11$$
$$x \equiv 4 \mod 10$$
$$x \equiv 2 \mod 3$$
so now I can use CRT. So to do that, I have done
$$x \equiv \{ 2 \tim... | Consider $3 \mod 5$. If I multiply this by $2$, I get $2 \cdot 3 \mod 5 \equiv 1 \mod 5$. Thus when I multiply by $2$, I get the multiplicative identity. This means that I might call $3^{-1} = 2 \mod 5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/364899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Probability of getting exactly 2 heads in 3 coins tossed with order not important? I have been thinking of this problem for the post 3-4 hours, I have come up with this problem it is not a home work exercise
Let's say I have 3 coins and I toss them, Here order is not important
so possible sample space should be
0 H... | The sample space has size $2^3 = 8$ and consists of triples
$$
\begin{array}{*{3}{c}}
H&H&H \\
H&H&T \\
H&T&H \\
H&T&T \\
T&H&H \\
T&H&T \\
T&T&H \\
T&T&T
\end{array}
$$
The events
$$
\begin{align}
\{ 0 \text{ heads} \} &= \{TTT\}, \\
\{ 1 \text{ head} \} &= \{HTT, THT, TTH\},
\end{align}
$$
and I'll let you figure out... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 1
} |
Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$ At this link someone asked how to prove rigorously that
$$
\lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x.
$$
What good intuitive arguments exist for this statement?
Later edit: . . . where $e$ is defined as the base of an exponential fun... | $$ e^x=\lim_{m\rightarrow \infty}\left(1+\frac{1}{m}\right)^{mx} $$
Let $mx=n$, so $m=\frac{n}{x}$
$$e^x=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/365029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 14,
"answer_id": 8
} |
Radicals in a fraction: simplification I cannot for the life of me figure out how this fraction got simplified. Please tell me how the first fraction got simplified into the second one. I've provided initial fraction and its simplified answer:
$$
-\frac{p \cdot (-1 /(2\sqrt{517-p}) )}{\sqrt{517-p}} = \frac{1}{2(517-p)... | $$ -\frac{p \cdot \frac{-1}{2\sqrt{517-p}}}{\sqrt{517-p}} \\\\$$
$$ -\frac{p \cdot -\frac{1}{2\sqrt{517-p}}}{\sqrt{517-p}} \\\\$$
$$ --\frac{p \cdot \frac{1}{2\sqrt{517-p}}}{\sqrt{517-p}} \\\\$$
$$ \frac{p \cdot \frac{1}{2\sqrt{517-p}}}{\sqrt{517-p}} \\\\$$
$$ \frac{\frac{p}{2\sqrt{517-p}}}{\sqrt{517-p}} \\\\$$
$$ \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/365083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Combinatorial Proof I have trouble coming up with combinatorial proofs. How would you justify this equality?
$$
n\binom {n-1}{k-1} = k \binom nk
$$
where $n$ is a positive integer and $k$ is an integer.
| We have a group of $n$ people, and want to count the number of ways to choose a committee of $k$ people with Chair.
For the left-hand side, we select the Chair first, and then $k-1$ from the remaining $n-1$ to join her.
For the right-hand side, we choose $k$ people, and select one of them to be Chair.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/365166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Does the sequence converges? I am trying to prove if the sequence
$a_n=(\root n\of e-1)\cdot n$ is convergent. I know that the sequences $x_n=(1+1/n)^n$ and $y_n=(1+1/n)^{n+1}$ tends to the same limit which is $e$. Can anyone prove if the above sequence $a_n$ is convergent? and if so, find the limit.
My trial was to ... | Let $e^{1/n}-1 = x$. We then have $\dfrac1n = \log(1+x) \implies n = \dfrac1{\log(1+x)}$. Now as $n \to \infty$, we have $x \to 0$. Hence,
$$\lim_{n \to \infty}n(e^{1/n}-1) = \lim_{x \to 0} \dfrac{x}{\log(1+x)} = 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/365237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Does there exist a positive integer $n$ such that it will be twice of $n$ when its digits are reversed? Does there exist a positive integer $n$ such that it will be twice of $n$ when its digits are reversed?
We define $f(n)=m$ where the digits of $m$ and $n$ are reverse.
Such as $f(12)=21,f(122)=221,f(10)=01=1$,so we c... | There is no such integer $n$.
Suppose there is, and let $b = n \bmod 10$ be its units digit (in decimal notation) and $a$ its leading digit, so $a 10^k \leq n < (a+1)10^k$ for some $k$ and $1 \leq a < 10$.
Since $f(n) = 2n$ is larger than $n$, and $f(n)$ has leading digit $b$ and at most as many digits as $n$, we must... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/365307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Every injective function is an inclusion (up to a unique bijection) Let $X$ be a set and let $A$ be a subet of $X$. Let $i:A\longrightarrow X$ be the usual inclusion of $A$ in $X$. Then $i$ is an example of an injective function.
I want to show that every injective function is of this kind.
More precisely: for every ... | No. If $j' \circ g' = f$ then $j'(g'(x)) = f(x)$ for all $x \in X$. But $j'$ is the inclusion of $B'$ in $Y$, so it acts by the identity on elements of $B$, which the $g'(x)$ are, by definition of $g' : X \rightarrow B'$. Hence $g'(x) = f(x)$ for all $x \in X$, so $B' = f(X)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/365358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
When are the sections of the structure sheaf just morphisms to affine space? Let $X$ be a scheme over a field $K$ and $f\in\mathscr O_X(U)$ for some (say, affine) open $U\subseteq X$. For a $K$-rational point $P$, I can denote by $f(P)$ the image of $f$ under the map
$$\mathscr O_X(U) \to \mathscr O_{X,P} \twoheadright... | The scheme $\mathrm{Spec}(k[X])=\mathbf{A}_k^1$ is the universal locally ringed space with a morphism to $\mathrm{Spec}(k)$ and a global section (namely $X$). What I mean by this is that for any locally ringed space $X$ with a morphism to $\mathrm{Spec}(k)$ (equivalently $\mathscr{O}_X(X)$) is a $k$-algebra) and any gl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/365447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
An identity related to Legendre polynomials Let $m$ be a positive integer. I believe the the following identity
$$1+\sum_{k=1}^m (-1)^k\frac{P(k,m)}{(2k)!}=(-1)^m\frac{2^{2m}(m!)^2}{(2m)!}$$
where $P(k,m)=\prod_{i=0}^{k-1} (2m-2i)(2m+2i+1)$, is true, but I don't see a quick proof. Anyone?
| Clearly $P(k,m) = (-1)^k 4^k \cdot (-m)_k \cdot \left(m+\frac{1}{2}\right)_k$, where $(a)_k$ stands for the Pochhammer's symbol. Thus the sums of the left-hand-side of your equality is
$$
1 + \sum_{k=1}^\infty (-1)^k \frac{P(k,m)}{(2k)!} = \sum_{k=0}^\infty 4^k \frac{(-m)_k \left(m+\frac{1}{2}\right)_k}{ (2k)!} = \sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/365499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Given one primitive root, how do you find all the others? For example: if $5$ is a primitive root of $p = 23$.
Since $p$ is a prime there are $\phi(p - 1)$ primitive roots. Is this correct?
If so, $\phi(p - 1) = \phi(22) = \phi(2) \phi(11) = 10$. So $23$ should have $10$ primitive roots?
And, to find all the other ... | The possible powers of 5 are all the $k$'s that $gcd(k,p−1)=1$, so $k$ is in the set {$1, 3, 5, 7, 9, 13, 15, 17, 19, 21$} and $5^k$ is in the set {$5, 10, 20, 17, 11, 21, 19, 15, 7, 14$} which is exactly of length 10.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/365584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Ordered groups - examples Let $G=BS(m,n)$ denote the Baumslag–Solitar groups defined by
the presentation $\langle a,b: b^m a=a b^n\rangle$.
Question: Find $m,n$ such that $G$ is an ordered group, i.e. $G$ is a group on which a partial order relation $\le $ is given such that for any elements $x,y,z \in G$, from $x \... | From Wikipedia:
A group $G$ is a partially ordered group if and only if there exists a subset $H\subset G$ such that:
i) $1 \in H$
ii) If $a ∈ H$ and $b ∈ H$ then $ab ∈ H$
iii) If $a ∈ H$ then $x^{-1}ax ∈ H$ for each $x\in G$
iv) If $a ∈ H$ and $a^{-1} ∈ H$ then $a=0$
For every $n,m$ can you find such a subset? Here's... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/365660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
to get a MDS code from a hyperoval in a projective plane explain how we can get a MDS code of length q+2 and dimension q-1 from a hyperoval
in a projective plane PG2(q) with q a power of 2?
HINT:a hyperoval Q is a set of q+2 points such that no three points in Q are collinear.
you are expected to get a [q+2,q-1,4] bina... | As an addition to the answer of Jyrki Lahtonen:
The standard way to get projective coordinates of the points of a hyperoval over $\mathbb F_q$ is to take the vectors $[1 : t : t^2]$ with $t\in\mathbb F_q$ together with $[0 : 1 : 0]$ and $[0 : 0 : 1]$.
Placing these vectors into the columns of a matrix, in the example $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/365744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Differentiably redundant functions. I am looking for a differentiably redundant function of order 6 from the following.
(a) $e^{-x} + e^{-x/ 2} \cos({\sqrt{3x} \over 2})$
(b) $e^{-x} + \cos(x)$
(c) $e^{x/2}\sin({\sqrt{3x} \over 2})$
I know that (b) has order 4, but I cannot solve for (a) and (c).
It would be a huge was... | "Differentiably redundant function of order $n$" is not a standard mathematical term: this is something that GRE Math authors made up for this particular problem.
Define a function $f(x)$ to be differentiably redundant of order $n$ if the $n$th derivative $f^{(n)}(x)=f(x)$ but $f^{(k)}(x)\ne f(x)$ when $k<n$. Which o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/365791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solving for $f(n+1)$ when $f(k)$ is known for $k=0,1,...,n$ I posted earlier about polynomials but this is different type of problem I think. I seem to have an answer but I mistrust it....
A polynomial $f(x)$ where deg[$f(x)$]$\le{n}$ satisfies $f(k)=2^k$ for $k=0,1,...,n$. Find $f(n+1)$.
So $f(k)=2^k \Rightarrow 2^{... | You are right to mistrust your answer: it's easy to check that it's incorrect in the case $n=1$ (and, for that matter, $n=0$). The mistake you made is in concluding that $2^{-k}f(k) - 1$ must have a certain form; that expression is not a polynomial, so you can't use results about polynomials to categorize it.
In fact, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Disjunction: Why did the inclusive "OR" become the convention? In How to Prove it by Velleman, for defining disjunctions, he gives the difference between exclusive "OR", and inclusive "OR."
Given two events $P$ and $Q$, the disjunction is defined for them as:
*
*Inclusive: One of $P$ and $Q$, or both.
*Exclusive: ... | George Boole, when he originally developed his Laws of Thought to apply mathematics to reasoning, used the exclusive or. However, the system was quite cumbersome in comparison to modern methods.
As others took up his ideas, they found that the inclusive or was far better behaved and easier to work with.
For instance, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
Summation of n-squared, cubed, etc. How do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great.
| In general, if you want to derive a formula for the sum of $1^k+\cdots+n^k,$ let $$f(x)=\sum_{j=0}^{k+1}a_jx^j$$ and solve $$f(x)-f(x-1)=x^k$$ for the coefficients $a_j$. The polynomial that results will have the property that $f(n)=1^k+\cdots+ n^k$ for all positive integers $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/366177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Understanding directional derivative and the gradient I'm having trouble understanding the proof of directional derivative and the gradient. Could someone give me a easy-to-read proof of the directional derivative and explain why does the gradient point to the direction of maximum increase?
Thank you very much for any ... | As for the gradient pointing in the direction of maximum increase, recall that the directional derivative is given by the dot product
$$\nabla f(x)\cdot\textbf{u},$$
where $$\nabla f(x)$$ is the gradient at the point $\textbf{x}$ and $\textbf{u}$ is the unit vector in the direction we are considering. Recall also that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Study the equivalence of these norms I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique.
So I define the following norm
$$\Vert f\Vert=(\Vert g\Vert_{H_1}^2+\Vert h\Vert_{H_2}^2)^{\frac{1}{2}... | Your calculation should be right - it is just the equivalence of the $1$-norm and the $2$-norm on $\mathbb R^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/366296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Computing the homology groups of a given surface Let $\triangle^2=\{(x,y)\in\mathbb{R}^2\mid 0\le x,y\wedge x+y\le1\}$ (that is, a right triangle). Define the equivalence relation $(t,0) \sim (0,t)\sim (t,1-t)$.
I want to compute the homology groups of $\triangle^2/\sim$.
An attempt at doing so was to define $U=\{(x,y)... | Just knowing that sequence is exact is not enough since, for example, $H_2(\Delta^2/\sim) = 0 = H_1(\Delta^2/\sim)$ and $H_2(\Delta^2/\sim) = 0, H_1(\Delta^2/\sim) = \mathbb Z/n\mathbb Z$ both work.
So you need to look at the actual map $H_1(U \cap V) \to H_1(U) \oplus H_1(V) \simeq 0 \oplus H_1(V)$, which is given by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Contour integration to compute $\int_0^\infty \frac{\sin ax}{e^{2\pi x}-1}\,\mathrm dx$ How to show:
$$\int_{0}^{\infty}\frac{\sin ax}{e^{2\pi x}-1}dx=\frac{1}{4}\frac{e^{a}+1}{e^{a}-1}-\frac{1}{2a}$$
integrating $\dfrac{e^{aiz}}{e^{2\pi z}-1}$ round a contour formed by the rectangle whose corners are $0 ,R ,R+i,i$ (th... | For this particular contour, the integral
$$\oint_C dz \frac{e^{i a z}}{e^{2 \pi z}-1}$$
is split into $6$ segments:
$$\int_{\epsilon}^R dx \frac{e^{i a x}}{e^{2 \pi x}-1} + i \int_{\epsilon}^{1-\epsilon} dy \frac{e^{i a R} e^{-a t}}{e^{2 \pi R} e^{i 2 \pi y}-1} + \int_R^{\epsilon} dx \frac{e^{-a} e^{i a x}}{e^{2 \pi x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Intermediate value-like theorem for $\mathbb{C}$? Is there an intermediate value like theorem for $\mathbb{C}$? I know $\mathbb {C}$ isn't ordered, but if we have a function $f:\mathbb{C}\to\mathbb{C}$ that's continuous, what can we conclude about it?
Also, if we have a function, $g:\mathbb{C}\to\mathbb{R}$
,continuo... | Consider $f(x) = e^{\pi i x }$.
We know that $f(0) = 1, f(1) = -1$.
But for no real value $r$ between 0 and 1 is $f(r) = 0$, or even real valued.
Think about how this is a 'counter-example', and what aspect of $\mathbb{C}$ did we use. It could be useful to trace out this graph.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/366502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Geodesics of conformal metrics in complex domains Let $U$ be a non-empty domain in the complex plane $\mathbb C$.
Question: what is the differential equation of the geodesics of the metric
$$m=\varphi(x,y) (dx^2+dy^2)$$ where $\varphi$ is a positive function on $U$ and where
$x,y$ are the usual euclidian coordinate... | You should be worried about the zeroes of $f$; the geodesic equation degenerates at the points where the metric vanishes.
At the points where $f\ne 0$ the local structure of geodesics is indeed simple. Let $F$ be an antiderivative of $f$, that is $F'=f$. The metric $|F'(z)|^2\,|dz|^2$ is exactly the pullback of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proving that $4 \mid m - n$ is an equivalence relation on $\mathbb{Z}$ I have been able to figure out the the distinct equivalence classes. Now I am having difficulties proving the relation IS an equivalence relation.
$F$ is the relation defined on $\Bbb Z$ as follows:
For all $(m, n) \in \Bbb Z^2,\ m F n \iff 4 | ... | 1) reflexivity: $mFm $ since $4|0$
2) simmetry: $mFn \Rightarrow nFm$ since $4|\pm (m-n)$
3) transitivity: if $4|(m-n)$ and $4|(n-r)$ then $4|\big((m-n)+(n-r)\big)$ or $4|(m-r)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/366614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Factorial primes Factorial primes are primes of the form $n!\pm1$. (In this application I'm interested specifically in $n!+1$ but any answer is likely to apply to both forms.) It seems hard to prove that there are infinitely many primes of this form, though Caldwell & Gallot were courageous enough to conjecture that th... | Wilson's Theorem shows there are infinitely many composites. For if $p$ is prime, then $(p-1)!+1$ is divisible by $p$, and apart from the cases $p=2$ and $p=3$, the number $(p-1)!+1$ is greater than $p$.
There are related ways to produce a composite. For example, let $p$ be a prime of the form $4k+3$. Then one of $\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Cylindrical coordinates where $z$ axis is not axis of symmetry. I'm a little bit uncertain of how to set up the limits of integration when the axis of symmetry of the region is not centered at $z$ (this is for cylindrical coordinates). The region is bounded by $(x-1)^2+y^2=1$ and $x^2+y^2+z^2=4$. This is my attempt:
Le... | I prefer to visualize the cross sections in $z$. Draw a picture of various cross-sections in $z$: there is an intersection region for each $z$
You have to find the points where the cross-sections intersect:
$$4-z^2=4 \cos^2{\theta} \implies \sin{\theta} = \pm \frac{z}{2} \ .$$
For $\theta \in [-\arcsin{(z/2)},\arcsi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How many ways are there to add the numbers in set $k$ to equal $n$? How many ways are there to add the numbers in set $k$ to equal $n$?
For a specific example, consider the following: I have infinite pennies, nickels, dimes, quarters, and loonies (equivalent to 0.01, 0.05, 0.1, 0.25, and 1, for those who are not Canadi... | You will have luck googling for this with the phrase "coin problem".
I have a few links at this solution which will lead to the general method. There is a Project Euler problem (or maybe several of them) which ask you to compute absurdly large such numbers of ways, but the programs (I found out) can be just a handful ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Unification and substitutions in first-order logic I am currently learning about first-order logic and various resolution techniques.
When applying a substitution $\theta$ to two sentences $\alpha$ and $\beta$, for unification purposes, aside from
SUBST($\theta$, $\alpha$) = SUBST($\theta$, $\beta$), does the resulti... | The most general unifier $\theta$ is unique in the sense that given any other unifier $\phi$, $\alpha \phi$ can be got from $\alpha \theta$ by a subtitution, and the same for $\beta$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/366860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Combinatorics Example Problem Ten weight lifters are competing in a team weightlifting contest. Of the lifters, 3 are from the United States, 4 are from Russia, 2 are from China, and 1 is from Canada.
Part 1
If the scoring takes account of the countries that the lifters represent, but not their individual identities,... | For the second, you pick the slots (not the people-we said all the people from one country were interchangeable) for the two US people in the bottom ${3 \choose 2}$ ways, but then have to pick which slot the US person in the top is in, which adds a factor ${3 \choose 1}$. Then of the seven left, you just have $\frac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
p-adic modular form example In Serre's paper on $p$-adic modular forms, he gives the example (in the case of $p = 2,3,5$) of $\frac{1}{Q}$ and $\frac{1}{j}$ as $p$-adic modular forms, where $Q = E_4 = 1 + 540\sum \sigma_{3}(n)q^n$ is the normalized Eisenstein series of weight 4 and $j = \frac{\Delta}{Q^3}$ is the $j$-i... | I'm not sure this can quite be correct. The problem is that $Q^{p^m}$ is going to tend to 1, so $Q^{p^m} - 1$ tends to 0, not $1/Q$. I think you may have misread the paper and what was meant was $1/Q = \lim_{m \to \infty} Q^{(p^m - 1)}$; if you're reading Serre's paper in the Antwerp volumes, then this is an easy mista... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Let $f$ be a twice differentiable function on $\mathbb{R}$. Given that $f''(x)>0$ for all $x\in \mathbb{R}$.Then which is true? Let $f$ be a twice differentiable function on $\mathbb{R}$. Given that $f''(x)>0$ for all $x\in \mathbb{R}$.Then which of the following is true?
1.$f(x)=0$ has exactly two solutions on $\mat... | Correct answer for above question is option no.3.Second option is not correct because it has a counter example.$$f(x)=e^x-1$$ f satisfy condition $f(0)=0,f'(0)>0$but $0$ is the only solution of $f(x)=0$.Hence $f(x)=0$ has no positive solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/367059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Cardinality of the set of bijective functions on $\mathbb{N}$? I learned that the set of all one-to-one mappings of $\mathbb{N}$ onto $\mathbb{N}$ has cardinality $|\mathbb{R}|$.
What about surjective functions and bijective functions?
| Choose one natural number. How many are left to choose from?
More rigorously, $$\operatorname{Aut}\mathbb{N} \cong \prod_{n \in \mathbb{N}} \mathbb{N} \setminus \{1, \ldots, n\} \cong \prod_{n \in \mathbb{N}} \mathbb{N} \cong \mathbb{N}^\mathbb{N} = \operatorname{End}\mathbb{N},$$ where $\{1, \ldots, 0\} := \varnothing... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 2
} |
a multiple choice question on monotone non-decreasing real-valued function on $\mathbb{R}$ let $f$ be a monotone non-decreasing real-valued function on $\mathbb{R}$ . Then
$1$. $\lim _ {x \to a}f(x)$ exists at each point $a$.
$2$. If $a<b$ , then $\lim _ {x \to a+}f(x) \le \lim _ {x \to b-}f(x)$.
$3$. $f$ is an unbo... | 1: monotone does not necessarily mean it 's continuous
3: never said it was strictly monotone, could be constant
4: $g$ is strictly decreasing and has a lower bound never reached ($0$). It has an upper bound only if $f$ has a lower bound -> many counter examples
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/367399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
$f: \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2)=5$ The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2) = 5$.
| The conditions allow you to calculate $f(x+1)$ if you know $f(x)$. Try calculating $f(3), f(4), f(5), f(6)$, and looking for a pattern.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/367473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
If $A$ is a diagonalizable $n\times n$ matrix for which the eigenvalues are $0$ and $1$, then $A^2=A$.
If $A$ is a diagonalizable $n\times n$ matrix for which the eigenvalues are $0$ and $1$, then $A^2=A$.
I know how to prove this in the opposite direction, however I can't seem to find a way prove this. Could anyone ... | Write $A = QDQ^{-1}$, where $D$ is a diagonal matrix with the eigenvalues, $0$s and $1$s, on the diagonal. The $A^2 = QDQ^{-1}QDQ^{-1} = QD^{2}Q^{-1}$. But $D^2 = D$, because when you square a diagonal matrix you square the entries on the diagonal and $1^2 = 1$ and $0^2 = 0$. Thus
$$A^{2} = QD^{2}Q^{-1} = QDQ^{-1} = A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Integrating $\frac {e^{iz}}{z}$ over a semicircle around $0$ of radius $\epsilon$ I am trying to find the value of $\int_{-\infty}^{\infty} \frac{\sin (x)}{x}$ using residue theorem and a contour with a kink around $0$. For this, I need to find $\int_{C_\epsilon} \frac {e^{iz}} {z}$ where $C_\epsilon$ is the semicircle... | Note that $\dfrac{e^{iz}}{z}=\dfrac1z+O(1)$. Integrating this counter-clockwise around the semicircle of radius $\epsilon$ is
$$
\begin{align}
\int_\gamma\frac{e^{iz}}{z}\,\mathrm{d}z
&=\int_0^\pi\left(\frac1\epsilon e^{-i\theta}+O(1)\right)\,\epsilon\,ie^{i\theta}\,\mathrm{d}\theta\\
&=\int_0^\pi\frac1\epsilon e^{-i\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What is the closed formula for the following summation? Is there any closed formula for the following summation?
$$\sum_{k=2}^n \frac{1}{\log_2(k)}$$
| There is no closed form as such. However, you can use the Abel summation technique from here to derive the asymptotic. We have
\begin{align}
S_n & = \sum_{k=2}^n \dfrac1{\log(k)} = \int_{2^-}^{n^+} \dfrac{d \lfloor t \rfloor}{\log(t)} = \dfrac{n}{\log(n)} - \dfrac2{\log(2)} + \int_2^{n} \dfrac{dt}{\log^2(t)}\\
& =\dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find the Sum $1\cdot2+2\cdot3+\cdots + (n-1)\cdot n$ Find the sum $$1\cdot2 + 2\cdot3 + \cdot \cdot \cdot + (n-1)\cdot n.$$
This is related to the binomial theorem. My guess is we use the combination formula . . .
$C(n, k) = n!/k!\cdot(n-k)!$
so . . . for the first term $2 = C(2,1) = 2/1!(2-1)! = 2$
but I can't figure... | As I have been directed to teach how to fish... this is a bit clunky, but works.
Define rising factorial powers:
$$
x^{\overline{m}} = \prod_{0 \le k < m} (x + k) = x (x + 1) \ldots (x + m - 1)
$$
Prove by induction over $n$ that:
$$
\sum_{0 \le k \le n} k^{\overline{m}} = \frac{n^{\overline{m + 1}}}{m + 1}
$$
When $n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Evaluating a trigonometric integral using residues Finding the trigonometric integral using the method for residues:
$$\int_0^{2\pi} \frac{d\theta}{ a^2\sin^2 \theta + b^2\cos^2 \theta} = \frac{2\pi}{ab}$$ where $a, b > 0$.
I can't seem to factor this question
I got up to $4/i (z) / ((b^2)(z^2 + 1)^2 - a^2(z^2 - 1)^2 ... | Letting $z = e^{i\theta},$ we get
$$\int_0^{2\pi} \frac{1}{a^2\sin^2\theta + b^2\cos^2\theta} d\theta =
\int_{|z|=1} \frac{1}{iz} \frac{4}{-a^2(z-1/z)^2+b^2(z+1/z)^2} dz \\=
\int_{|z|=1} \frac{1}{iz} \frac{4z^2}{-a^2(z^2-1)^2+b^2(z^2+1)^2} dz =
-i\int_{|z|=1} \frac{4z}{-a^2(z^2-1)^2+b^2(z^2+1)^2} dz.$$
Now the location... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
On a long proof On wikipedia there is a claim that the Abel–Ruffini theorem was nearly proved by Paolo Ruffini, and that his proof spanned over $500$ pages, is this really true? I don't really know much abstract algebra, and I know that the length of a paper will vary due to the size of the font, but what could possibl... | Not only true, but not unique. The abc conjecture has a recent (2012) proposed proof by Shinichi Mochizuki that spans over 500 pages, over 4 papers. The record is the classification of finite simple groups which consists of tens of thousands of pages, over hundreds of papers. Very few people have read all of them, a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "53",
"answer_count": 2,
"answer_id": 0
} |
Difficulties performing Laurent Series expansions to determine Residues The following problems are from Brown and Churchill's Complex Variables, 8ed.
From §71 concerning Residues and Poles, problem #1d:
Determine the residue at $z = 0$ of the function $$\frac{\cot(z)}{z^4} $$
I really don't know where to start with th... | A related problem. Lets consider your first problem
$$ \frac{\cot(z)}{z^4}=\frac{\cos(z)}{z^4\sin(z)}. $$
First, determine the order of the pole of the function at the point $z=0$, which, in this case, is of order $5$. Once the order of the pole has been determined, we can use the formula
$$r = \frac{1}{4!} \lim_{z\to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Calclute the probability? A random function $rand()$ return a integer between $1$ and $k$ with the probability $\frac{1}{k}$. After $n$ times we obtain a sequence $\{b_i\}_{i=1}^n$, where $1\leq b_i\leq k$.
Set $\mathbb{M}=\{b_1\}\cup\{b_2\}\cdots \cup\{b_n\}$.
I want to known the probability $\mathbb{M}\neq \{1, 2\cdo... | Hint: Obviously $n<k$ is trivial. Thereafter, the question becomes equivalent to solving
What fraction of $n$-tuples with the digits $1,\ldots,k$ are in fact $n$-tuples formed from a strict subset of these numbers?
or a surjection-counting problem.
You can find a recursive solution by letting $p_{n,k}$ be the probabi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
How to prove every closed interval in R is compact? Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familiar to us. Here I want to collect the ways to prove $[a,b]$ is compact.
Thanks for your help and any link.
| Perhaps the shortest, slickest proof I know is by Real Induction. See Theorem 17 in the aforelinked note, and observe that the proof is six lines. More importantly, after you spend an hour or two familiarizing yourself with the formalism of Real Induction, the proof essentially writes itself.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/368108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "64",
"answer_count": 12,
"answer_id": 8
} |
Where does the relation $\nabla^2(1/r)=-4\pi\delta^3({\bf r})$ between Laplacian and Dirac delta function come from? It is often quoted in physics textbooks for finding the electric potential using Green's function that
$$\nabla ^2 \left(\frac{1}{r}\right)=-4\pi\delta^3({\bf r}),$$
or more generally
$$\nabla ^2 \lef... | We can use the simplest method to display the results, as shown below : -
$$ \nabla ^2 \left(\frac{1}{r}\right) = \nabla \cdot \nabla \left( \frac 1 r \right) = \nabla \cdot \frac {-1 \mathbf {e_r}} {r^2} $$
Suppose there is a sphere centered on the origin, then the total flux on the surface of the sphere is : -
$$ \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 4,
"answer_id": 3
} |
Method of Characteristics $au_x+bu_y+u_t=0 $ $au_x+bu_y+u_t=0$
$u(x,y,0)=g(x,y)$
solve $u(x,y,t)$
Our professor talked about solving this using Method of Characteristics. However, I am confused about this method. Since it's weekend, I think it might be faster to get respond here. In the lecture, he wrote down the follo... | I think it's easiest just to concisely re-explain the method, so that's what I'll do.
The idea: linear, first-order PDEs have preferred lines (generally curved) along which all the action happens. More specifically, because the differential bit takes the form of $\mathbf f \cdot \nabla u$ where in general $\mathbf f$ v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Rank of matrix of order $2 \times 2$ and $3 \times 3$ How Can I calculate Rank of matrix Using echlon Method::
$(a)\;\; \begin{pmatrix}
1 & -1\\
2 & 3
\end{pmatrix}$
$(b)\;\; \begin{pmatrix}
2 & 1\\
7 & 4
\end{pmatrix}$
$(c)\;\; \begin{pmatrix}
2 & 1\\
4 & 2
\end{pmatrix}$
$(d)\;\; \begin{pmatrix}
2 & -3 & 3\\
2 & ... | Follow this link to find your answer
If you are left with any doubt after reading this this, feel free to discuss.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/368322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Finding the limit of function - exponential one Find the value of $\displaystyle \lim_{x \rightarrow 0}\left(\frac{1+5x^2}{1+3x^2}\right)^{\frac{1}{\large {x^2}}}$
We can write this limit function as :
$$\lim_{x \rightarrow 0}\left(1+ \frac{2x^2}{1+3x^2}\right)^{\frac{1}{\large{x^2}}}$$
Please guide further how to pro... | Write it as
$$\dfrac{(1+5x^2)^{1/x^2}}{(1+3x^2)^{1/x^2}}$$ and recall that $$\lim_{y \to 0} (1+ay)^{1/y} = e^a$$ to conclude what you want.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/368382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Permutations of Symmetric Group of Order 3 Find an example, in the group $S_3$ of permutations of $\{1,2,3\}$, of elements $x,y\in S_3$ for which $x^2 = e = y^2$ but for which $(xy)^4$ $\not=$ e.
| This group is isomorphic to the 6 element dihedral group. Any element including a reflection will have order two. The product of two different such elements will be a pure nonzero rotation, necessarily of order 3.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/368433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Construct matrix Let $B$ any square matrix. Is possible to construct an invertible matrix $Q_B$ such that
$$\|Q_BBQ_B^{-1}\|_2\ \leq\ \rho(B)?$$
Thanks in advance for the help.
Edit: $Q_B$ only need to be invertible, not orthogonal.
| Since $\rho(B)=\rho(Q_BBQ_B^{-1})$, you are essentially asking whether $B$ is similar to some $A$ such that $\|A\|_2=\rho(A)$, Yet, $\|A\|_2=\rho(A)$ if and only if $A$ is the scalar multiple of a unitary matrix. So, if $B$ is not similar to the scalar multiple of a unitary matrix (e.g. when $B$ has at least two eigenv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How do I write a trig function that includes inverses in terms of another variable? It's been awhile since I've used trig and I feel stupid asking this question lol but here goes:
Given:
$z = \tan(\arcsin(x))$
Question:
How do I write something like that in terms of $x$?
Thanks! And sorry for my dumb question.
| $$z = \tan[\arcsin(x)]$$
$$\arctan(z) = \arctan[\tan(\arcsin(x)] = \arcsin(x)$$
$$\sin[\arctan(z)] = \sin[\arcsin(x)] = x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/368603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Uniform limit of holomorphic functions Let $\{f_n\}$ be a sequence of holomorphic functions defined in a generic domain $D \subset \mathbb{C}$. Suppose that there exist $f$ such that $f_n \to f$ uniformly.
My question is: is it true that $f$ is holomorphic too?
| You've already seen an approach using Morera's theorem from the other excellent answers. For a slightly more concrete demonstration of why $f$ is complex differentiable, you can use the fact that every $f_n$ satisfies Cauchy's integral formula, so by uniform convergence $f$ also satisfies Cauchy's integral formula. Thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 4,
"answer_id": 3
} |
Analytic functions of a real variable which do not extend to an open complex neighborhood Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
| If $f$ is real analytic on an open interval $(a,b)$. Then at every point $x_0\in (a,b)$, there is a power series $P_{x_0}(x)=\sum_{n=0}^\infty a_n(x-x_0)^n$ with radius of convergence $r(x_0)>0$ such that $f(x)=P_{x_0}(x)$ for all $x$ in $(a,b)\cap \{x:|x-x_0|<r\}$. Then $f$ can be extended to an open neighborhood $B(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Let A = {a, c, 4, {4, 3}, {1, 4, 3, 3, 2}, d, e, {3}, {4}} Which of the following is true? Let A = {a, c, 4, {4, 3}, {1, 4, 3, 3, 2}, d, e, {3}, {4}}
Which of the following is true? | Yes, you are correct. None of the options are true.
$\{4, \{4\}\}$ is a subset of $A$, since $4 \in A,$ and $\{4\}\in A$, but the set with which it is paired is not a subset of $A$, and none of the items listed as "elements of $A$ are, in fact, elements of $A$.
For example, $\{1, 4, 3\} \not \subset A$ because $1 \not... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
normed division algebra Can we prove that every division algebra over $R$ or $C$ is a normed division algebra?
Or is there any example of division algebra in which it is not possible to define a norm?
Definition of normed division algebra is in here. Thanks!
| Frobenius theorem for (finite-dimensional) asscoative real division algebras states there are only $\mathbb{R},\mathbb{C},\mathbb{H}$, and the proof is elementary (it is given on Wikipedia in fact).
If you don't care about finite-dimensional, then the transcendental field extension $\mathbb{R}(T)/\mathbb{R}$, where he... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Why $f^{-1}(f(A)) \not= A$ Let $A$ be a subset of the domain of a function $f$.
Why $f^{-1}(f(A)) \not= A$.
I was not able to find a function $f$ which satisfies the above equation.
Can you give an example or hint.
I was asking for an example function which is not addressed
here
| Any noninjective function provides a counterexample. To be more specific, let $X$ be any set with at least two elements, $Y$ any nonempty set, $u$ in $X$, $v$ in $Y$, and $f:X\to Y$ defined by $f(x)=v$ for every $x$ in $X$. Then $A=\{u\}\subset X$ is such that $f(A)=\{v\}$ hence $f^{-1}(f(A))=X\ne A$.
In general, for $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Understanding the Hamiltonian function Based on this function:
$$\text{max} \int_0^2(-2tx-u^2) \, dt$$
We know that $$(1) \;-1 \leq u \leq 1, \; \; \; (2) \; \dot{x}=2u, \; \; \; (3) \; x(0)=1, \; \; \; \text{x(2) is free}$$
I can rewrite the function into a hamiltonian function:
$$H=-2tx-u^2+p2u$$
where u(t) maxizmize... | $$
\frac{\partial H}{\partial u} = -2u + 2p \tag{1}
$$
where $u$ is the control variable and $p$ is the costate.
The optimality of $H$ requires (1)=0, where you obtain your $u_t$ expression considering its constraint.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/369096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Positive Outcome! I have a question on Probability.
With two dices, which each have six sides people are making duels with these dices.
the winner is the one who rolls the highest. the chances are of a 50% win as it is a duel between two and the terms are the same, but the person taking bets keeps a 10% commision.
What... | To maximize your long-term profit, you should use Kelly gambling, which says you should bet on a wager proportional to the edge you get from the odds. The intuitive form of the formula is
$$
f^{*} = \frac{\text{expected net winnings}}{\text{net winnings if you win}} \!
$$
If the odds are even (ie. your scenario, but... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/369275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Hyperbolic Functions
Hey everyone, I need help with questions on hyperbolic functions.
I was able to do part (a).
I proved for $\sinh(3y)$ by doing this:
\begin{align*}
\sinh(3y) &= \sinh(2y +y)\\
&= \sinh(2y)\cosh(y) + \cosh(2y)\sinh(y)\\
&= 2\sinh(y)\cosh(y)\cosh(y) + (\cosh^2(y)+\sinh^2(y))\sinh(y)\\
&= 2\sinh(y)(... | Hint 1: Set $\color{#C00000}{x=\sinh(y)}$. Since $0=4\sinh^3(y)+3\sinh(y)-\sinh(3y)$, we have
$$
4x^3+3x-\sinh(3y)=0
$$
and by hypothesis,
$$
4x^3+3x-2=0
$$
So, if $\color{#C00000}{\sinh(3y)=2}$, both equations match. Solve for $x$.
Hint 2: Set $c\,x=\sinh(y)$ for appropriate $c$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/369339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Intuition behind the difference between derived sets and closed sets? I missed the lecture from my Analysis class where my professor talked about derived sets. Furthermore, nothing about derived sets is in my textbook. Upon looking in many topology textbooks, few even have the term "derived set" in their index and many... | The key to the difference is the notion of an isolated point. If $X$ is a space, $A\subseteq X$, and $x\in A$, $x$ is an isolated point of $A$ if there is an open set $U$ such that $U\cap A=\{x\}$. If $X$ is a metric space with metric $d$, this is equivalent to saying that there is an $\epsilon>0$ such that $B(x,\epsil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/369427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Correct way to calculate a complex integral? I have
$$
\int_{[-i,i]} \sin(z)\,dz
$$
Parametrizing the segment $[-i,i]$ I have, if $t\in[0,1]$
$$
z(t) = it + (1-t)(-i) = 2it-i, \quad \dot{z}(t) = 2i.
$$
So
$$
\int_{[-i,i]} \sin(z)\,dz = \int_0^1 \sin(2it-i)2i\, dt = -\cos(2it-i)|_0^1 = 0.
$$
Am I correct?
| Sine is analytic function, so you can also use Cauchy's theorem.
$$
\int_{[-i, i]} \sin z\ dz = -\left . \cos z \right |_{-i}^i = 0
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/369488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.