Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Cyclic groups and generators For each of the groups $\mathbb Z_4$,$\mathbb Z_4^*$ indicate which are cyclic. For those that are cyclic list all the generators.
Solution
$\mathbb Z_4=${0,1,2,3}
$\mathbb Z_4$ is cyclic and all the generators of $\mathbb Z_4=${1,3}
Now if we consider $\mathbb Z_4^*$
$\mathbb Z_4^*$={1,3}
... | Consulting here, you can easy see that in modulo $4$ there are two relatively prime congruence classes, $1$ and $3$, so $(\mathbb{Z}/4\mathbb{Z})^\times \cong \mathrm{C}_2$, the cyclic group with two elements.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/309809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Show $\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$ and another Show that :
$$\sum_{n=1}^{\infty}\frac{\cosh(2nx)}{\cosh(4nx)-\cosh(2x)}=\frac1{4\sinh^2(x)}$$
$$\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$$
| OK, I have figured out the second sum using a completely different method. I begin with the following result (+):
$$\sum_{k=1}^{\infty} e^{-k t} \sin{k x} = \frac{1}{2} \frac{\sin{x}}{\cosh{t}-\cos{x}}$$
I will prove this result below; it is a simple geometrical sum. In any case, let $x=i \pi$ and $t=2 n \pi$; then
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 0
} |
Continuity of the lebesgue integral How does one show that the function, $g(t) = \int \chi_{A+t} f $ is continuous, given that $A$ is measurable, $f$ is integrable and $A+t = \{x+t: x \in A\}$.
Any help would be appreciated, thanks
| Notice that
$$
|g(t+h)-g(t)| \le \int_{(A+t)\Delta A} |f|
$$
so it is enough to prove that
$$
|(A+t)\Delta A| \to 0 \qquad \text{as }t \to 0
$$
where $\Delta$ is the symmetric difference, since
$$
\int_{A_k} f \to 0
$$
if $|A_k|\to 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/309935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Sum of dihedral angles in Tetrahedron I'd like to ask if someone can help me out with this problem. I have to determine what is the lower and upper bound for sum (the largest and smallest sum I can get) of dihedral angles in arbitrary Tetrahedron and prove that. I'm ok with hint for proof, but I'd be grateful for lower... | Lemma: Sum of the 4 internal solid angles of a tetrahedron is bounded above by $2\pi$.
Start with a non-degenerate tetrahedron $\langle p_1p_2p_3p_4 \rangle$. Let $p = p_i$ be one its vertices and $\vec{n} \in S^2$ be any unit vector. Aside from a set of measure zero in choosing $\vec{n}$, the projection
of $p_j, j = 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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application of Lowenheim-Skolem theorem So if minimal model of ZF exists, it is said that it is countable set by Lowenheim-Skolem. So, is Lowenheim-Skolem saying that for any countable theory with existence of infinite model there exists standard model, respecting normal element relation, that is countable infinite?
| No. The existence of standard models is strictly stronger.
It is consistent that there are no standard models, to see this note that the standard models are well-founded, in the sense that there is no infinite decreasing chain of standard models such that $M_{n+1}\in M_n$, simply because $\in$ itself is well-founded an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Baire one extension of continuous functions I struggle with the following comment in Sierpinski's Hypothèse du continu, p. 49.
For every continuous function $f(x):X \rightarrow \mathbb{R}$, where $X \subseteq \mathbb{R}$, there exist a Baire one function $g(x): \mathbb{R} \rightarrow \mathbb{R}$ such that $g(x) = f(x)... | I think it can be proved by the following:
For arbitrary $X\subset\mathbb{R}$, continuous $f:X\longrightarrow\mathbb{R}$ can be extended to a function $F:\mathbb{R}\longrightarrow\mathbb{R}$ such that $F^{-1}(A)$ is a $G_\delta$ set in $\mathbb{R}$ for every closed $A\subset\mathbb{R}$ (a Lebesgue-one function such tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to get as much pie as possible! Alice and Bob are sharing a triangular pie. Alice will cut the pie with one straight cut and pick the bigger piece, but Bob first specifies one point through which the cut must pass. What point should Bob specify to get as much pie as possible? And in that case how much pie can Alice... | You can consider the triangle to be equilateral, as you can make any triangle equilateral with a linear transformation. That transformation will preserve the ratio of areas. The centroid is the obvious point to pick. If Alice cuts parallel to a side, she leaves Bob $\frac 49$ because the centroid is $\frac 23$ of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $k$ and $n$ be any integers such that $k \ge 3$ and $k$ divides $n$. Prove that $D_n$ contains exactly one cyclic subgroup of order $k$ a) Find a cyclic subgroup $H$ of order $10$ in $D_{30}$. List all generators of $H$.
b) Let $k$ and $n$ be any integers such that $k \ge 3$ and $k$ divides $n$. Prove that $D_n$ co... | Let $$D_{2n} = \langle r, s \mid r^n = 1, s^2 = 1, s r = r^{-1}s \rangle$$ be the dihedral group of order $2n$ generated by rotations ($r$) and reflections ($s$) of the regular $n$-gon.
From the presentation it is clear that every element can be put into the form $s^i r^j$ where $i$ is $0$ or $1$. So the cyclic subgrou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Real Analysis Question! Consider the equation $\sin(x^2 + y) − 2x= 0$ for $x ∈ \mathbb{R}$ with $y ∈ \mathbb{R}$
as a parameter.
Prove the existence of neighborhoods $V$ and $U$ of $0$ in $\mathbb{R}$ such that for every
$y ∈ V$ there exists a unique solution $x = ψ(y) ∈ U$. Prove that $ψ$ is a $C\infty$
mapping on $V$... | Implicit differentiation of
$$
\sin(x^2+y)-2x=0\tag{1}
$$
yields
$$
y'=2\sec(x^2+y)-2x\tag{2}
$$
$(1)$ implies
$$
|x|\le\frac12\tag{3}
$$
$(2)$ and $(3)$ imply
$$
\begin{align}
|y'|
&\ge2|\sec(x^2+y)|-2|x|\\
&\ge2-1\\
&=1\tag{4}
\end{align}
$$
By the Inverse Function Theorem, $(4)$ says that for all $y$,
$$
|\psi'(y)|\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Probability of choosing two equal bits from three random bits Given three random bits, pick two (without replacement). What is the probability that the two you pick are equal?
I would like to know if the following analysis is correct and/or if there is a better way to think about it.
$$\Pr[\text{choose two equal bits}]... | Whatever the first bit picked, the probability the second bit matches it is $1/2$.
Remark: We are assuming what was not explicitly stated, that $0$'s and $1$'s are equally likely. One can very well have "random" bits where the probability of $0$ is not the same as the probability of $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/310508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
How to solve the following differential equation We have the following DE: $$ \dfrac{dy}{dx} = \dfrac{x^2 + 3y^2}{2xy}$$
I don't know how to solve this. I know we need to write it as $y/x$ but I don't know how to in this case.
| $y=vx$ so $y'=xv'+v$. Your equation is $xv'+v={{1 \over{2v}}+{{3v} \over {2}}}$.
Now clean up to get $xv'={{v^2+1}\over{2v}}$. Now separate ${2v dv \over {v^2+1}} = {dx \over x}$.
Edit
${{x^2+3y^2} \over {2xy}} = {{{x^2}\over{2xy}}+{{3y^2}\over{2xy}}}={ x \over {2y}}+{{3y}\over{2x}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/310570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
} |
Fractional Derivative Implications/Meaning? I've recently been studying the concept of taking fractional derivatives and antiderivatives, and this question has come to mind: If a first derivative, in Cartesian coordinates, is representative of the function's slope, and the second derivative is representative of its con... | There several approaches to fractional derivatives. I use the Grunwald-Letnikov derivative and its generalizations to complex plane and the two-sided derivatives. However, most papers use what I call "walking dead" derivatives: the Riemann-Liouville and Caputo. If you want to start, don't loose time with them.
There a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 3,
"answer_id": 1
} |
3D - derivative of a point's function, is it the tangent? If I have (for instance) this formula which associates a $(x,y,z)$ point $p$ to each $u,v$ couple (on a 2D surface in 3D):
$p=f(u,v)=(u^2+v^2+4,2uv,u^2−v^2) $
and I calculate the $\frac{\partial p}{\partial u}$, what do I get? The answer should be "a vector tang... | Take a fixed location where $(u,v) = (u_0,v_0)$. Think about the mapping $u \mapsto f(u,v_0)$. This is a curve lying on your surface, which is formed by allowing $u$ to vary while $v$ is held fixed. In fact, in my business, we would say that this is an "isoparametric curve" on the surface. By definition, $\frac{\partia... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
} |
Fubini's theorem for Riemann integrals? The integrals in Fubini's theorem are all Lebesgue integrals. I was wondering if there is a theorem with conclusions similar to Fubini's but only involving Riemann integrals? Thanks and regards!
| To see the difficulties of Fubini with Riemann integrals, study two functions $f$ and $g$ on the rectangle $[0,1]\times[0,1]$ defined by:
(1) $\forall$integer $i\ge0$, $\forall$odd integer $j\in[0,2^i]$,
$\forall$integer $k\ge0$, $\forall$odd integer $\ell\in[0,2^k]$,
define $f(j/2^i,\ell/2^k)=\delta_{ik}$ (here, $\del... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Sub-lattices and lattices. I have read in a textbook that $ \mathcal{P}(X) $, the power-set of $ X $ under the relation ‘contained in’ is a lattice. They also said that $ S := \{ \varnothing,\{ 1,2 \},\{ 2,3 \},\{ 1,2,3 \} \} $ is a lattice but not a sub-lattice. Why is it so?
| The point of confusion is that a lattice can be described in two different ways. One way is to say that it is a poset such that finite meets and joins exist. Another way is to say that it is a set upon which two binary operations (called meet and join) are given that satisfy a short list of axioms. The two definitions ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Good book recommendations on trigonometry I need to find a good book on trigonometry, I was using trigonometry demystified but I got sad when I read this line:
Now that you know how the circular functions are defined, you might wonder how the values are calculated. The answer: with an electronic calculator!
I know a ... | Nothing changed much in basic Trigonometry for a century. Of Loney's book genre is another:
Henry Sinclair Hall, Samuel Ratcliffe Knight, Macmillan and Company, 1893 - Plane trigonometry - 404 pages
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/310980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 11,
"answer_id": 8
} |
Solve $x'(t)=(x(t))^2-t^2+1 $ How can we solve $$x'(t)=(x(t))^2-t^2+1 $$?I have tried to check whether it is Exact, separable, homogeneous, Bernoulli or not. It doesn't resemble to none of them. Who can help me. Thank you. The source of question is CEU entrance examination.
| The non-linear DE $$x'=P(t)+Q(t)x+R(t)x^2$$ is called Ricatti's equation. If $x_1$ is a known particular solution of it, then we can have a family of solutions of the OE of the form $x(t)=x_1+u$ where $u$ is a solution of $$u'=Ru^2+(Q+2x_1R)u$$ or the linear ODE: $$w'+(Q+2x_1R)w=-R,~~~w=u^{-1}$$ Here, as @Ishan noted c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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DNA sequence in MATLAB I am wanting to count how many times synonymous and non- synonymous mutations appear in a sequence of DNA, given the number of synonymous and non- synonymous mutations in each 3 letter codon. ie given that AAA has 7 synonymous and 1 non- synonymous equations, and CCC has 6 and 3 respectively, the... | Assuming you have filtered out the data errors and each time you nicely have three letter, here is one approach:
1) Make your data look like this:
AAA
CCC
ACA
CAC
...
2) Count how many times each of the 64 options occurs.
3) Multiply that found number of times with the corresponding syn and non-sym mutations.
That sh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Ideals in Dedekind domain If I is a non-zero ideal in a Dedekind domain such that $I^m$ and $I^n$ are principal, and are equal to $(a)$ and $(b)$ respectively. How to show that $I^{(m,n)}$ is principal.
Try: $(m,n) = rm +sn$ So, $I^{(m,n)} = (a)^r(b)^s$, where $r$, $s$ can be positive and negative. Both positive case ... | Even in the other cases the argument works, you just have fractional ideals instead. Viewing $I$ as an element $\overline{I}$ of the ideal class group of $A$ your question can be stated as: Suppose $\overline{I}^m=\overline{I}^n=0$ in the ideal class group, then $\overline{I}^{(m,n)}=0$. This statement is true in any g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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How can I calculate the limit of this? What is the limit? $$\lim_{n\rightarrow\infty}\dfrac{3}{(4^n+5^n)^{\frac{1}{n}}}$$
I don't get this limit. Really, I don't know if it has limit.
| Denote the function
$$
f(n) = \frac{3}{(4^n +5^n)^{\frac{1}{n}}}
$$
Recall logarithm is a continuous function, hence denote
$$
L(f(n)) = \log 3-\frac{\log(4^n +5^n)}{n}\\
\lim_{n \to \infty} L(f(n)) = \log 3 - \lim_{n \to \infty}\frac{\log(4^n +5^n)}{n}=\log 3 - \lim_{n \to \infty} \frac{4^n \log 4 + 5^n \log 5}{4^n ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $M(r)$ for $a \leq r \leq b$ by $M(r)=\max\{\frac {r}{a}-1,1-\frac {r}{b}\}$.Then $\min \{M(r):a \leq r \leq b\}=$? I faced the following problem that says:
Let $0<a<b$. Define a function $M(r)$ for $a \leq r \leq b$ by $M(r)=\max\{\frac {r}{a}-1,1-\frac {r}{b}\}$.Then $\min \{M(r):a \leq r \leq b\}$ is which of t... | If $a\leq r \leq \dfrac{2ab}{a+b}$ show that $M(r)=1-\dfrac{r}{b}$ and for $\dfrac{2ab}{a+b}\leq r \leq b$ that $M(r)=\dfrac{r}{a}-1$.
As Did suggested a picture (of $\dfrac{r}{a}-1, 1-\dfrac{r}{b}$) will help. What is $\dfrac{2ab}{a+b}$?
To find $\min \{M(r):a\leq r\leq b\}$ note that $\dfrac{r}{a}-1$ is increasing a... | {
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"timestamp": "2023-03-29T00:00:00",
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getting the inner corner angle I have four points that make concave quad:
now I wanna get the inner angle of the (b) corner in degrees.
note: the inner angle is greater than 180 degree.
| Draw $ac$ and use the law of cosines at $\angle b$, then subtract from $360$
$226=68+50-2\sqrt{50\cdot 68} \cos \theta \\ \cos \theta\approx -0.926 \\ \theta \approx 157.83 \\ \text{Your angle } \approx 202.17$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/311417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Sum of $\prod 1/n_i$ where $n_1,\ldots,n_k$ are divisions of $m$ into $k$ parts. Fix $m$ and $k$ natural numbers. Let $A_{m,k}$ be the set of all partitions divisions of $m$ into $k$ parts. That is:
$$A_{m,k} = \left\{ (n_1,\ldots,n_k) : n_i >0, \sum_{i=1}^k n_i = m \right\} $$
We are interested in the following sum ... | I am not sure I understand the problem correctly, but
$$ g(z) =
\left( \frac{z}{1} + \frac{z^2}{2} +\frac{z^3}{3} + \ldots + \frac{z^q}{q} + \ldots \right)^k =
\left( \log \frac{1}{1-z} \right)^k $$
looks like a good candidate to me, so that
$$ s_{m,k} = [z^m] \left( \log \frac{1}{1-z} \right)^k.$$
This is the exponen... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving the statement using Resolution? I'm trying to solve this problem for my logical programming class:
Every child loves Santa. Everyone who loves Santa loves any reindeer.
Rudolph is a reindeer, and Rudolph has a red nose. Anything which has
a red nose is weird or is a clown. No reindeer is a clown. John
do... | The conclusion is correct.
I will let you tidy this up and fill in the gaps, but you might want to consider the following
W(r) v C(r)
W(r)
~L(j,r)
~L(j,s)
~K(j)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/311531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Exposition On An Integral Of An Absolute Value Function At the moment, I am trying to work on a simple integral, involving an absolute value function. However, I am not just trying to merely solve it; I am undertaking to write, in detail, of everything I am doing.
So, the function is $f(x) = |x^2 + 3x - 4|$. I know th... | Since polynomials and absolute value of continuous functions are continuous , the zeros are
the points for possible sign change.
Then you can write piecewisely your function and integrate..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/311597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Define a domain filter of a function Let $\mathbb{B}, \mathbb{V}$ two sets. I have defined a function $f: \mathbb{B} \rightarrow \mathbb{V}$.
$\mathcal{P}(\mathbb{B})$ means the power set of $\mathbb{B}$,
I am looking for a function $g: (\mathbb{B} \rightarrow \mathbb{V}) \times \mathcal{P}(\mathbb{B}) \rightarrow (\m... | What you denote $g(f,\mathbb S)$ is usually called the restriction of $f$ to $\mathbb S$, and denoted $f|\mathbb S$ or $f\upharpoonright \mathbb S$. Sometimes this notation is used even if $\mathbb S$ is not contained in the domain of $f$, in which case it is understood to be $f\upharpoonright A$, where $A=\mathbb S\ca... | {
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"url": "https://math.stackexchange.com/questions/311641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does every sequence of rationals, whose sum is irrational, have a subsequence whose sum is rational Assume we have a sequence of rational numbers $a=(a_n)$. Assume we have a summation function $S: \mathscr {L}^1 \mapsto \mathbb R, \ \ S(a)=\sum a_n$ ($\mathscr {L}^1$ is the sequence space whose sums of absolute values ... | No; for example, if $(n_i)$ is a strictly increasing sequence of positive integers, then we can imitate the proof of the irrationality of $e$ to see that
$$\sum_{i=1}^\infty \frac{1}{n_1 \dots n_i} \notin \mathbf Q.$$
But every sub-series of this series has the same property (it just amounts to grouping some of the $n_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
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Prove $||a| - |b|| \leq |a - b|$ I'm trying to prove that $||a| - |b|| \leq |a - b|$. So far, by using the triangle inequality, I've got:
$$|a| = |\left(a - b\right) + b| \leq |a - b| + |b|$$
Subtracting $|b|$ from both sides yields,
$$|a| - |b| \leq |a - b|$$
The book I'm working from claims you can achieve this proof... | Hint: If $|a|-|b|<0$, rename $a$ to $b'$ and $b$ to $a'$.
| {
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"url": "https://math.stackexchange.com/questions/311756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional? Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.... | Let $H$ be the null space and take a vector $v$ outside $H$. The point is that $H+\langle v\rangle$ is the whole vector space, this I assume you know (i.e. $H$ has codimension 1). Then $f(v)$ and $g(v)$ uniquely determine the functions $f$ and $v$ and all $x\in X$ can be written as $x=h+tv$ with $h\in H$ so:
$$
f(x) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integrate $\iint_D\exp\{\min(x^2, y^2)\}\mathrm{d}x\mathrm{d}y$ Compute the integral:
\begin{equation}
\iint\limits_{\substack{\displaystyle 0 \leqslant x \leqslant 1\\\displaystyle 0 \leqslant y \leqslant 1}}\exp\left\{\min(x^2, y^2)\right\}\mathrm{d}x\mathrm{d}y
\end{equation}
$D$ is the rectangle with vertices $(0,0... | Note that the derivative of $x\mapsto e^{x^2}$ is $x\mapsto 2xe^{x^2}$, hence by symmetry along the line $x=y$
$$\begin{align} \int_0^1\int_0^1e^{\min\{x^2,y^2\}}\,\mathrm dy\,\mathrm dx &= 2\int_0^1\int_x^1e^{x^2}\,\mathrm dy\,\mathrm dx\\
&=2\int_0^1(1-x)e^{x^2}\,\mathrm dx\\
&=2\int_0^1e^{x^2}\,\mathrm dx-\int_0^12x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Algebraic Number Theory - Lemma for Fermat's Equation with $n=3$ I have to prove the following, in my notes it is lemma before Fermat's Equation, case $n=3$. I was able to prove everything up to the last two points:
Let $\zeta=e^{(\frac{2\pi i}{3})}$. Consider $A:=\mathbb{Z}[\zeta]=\{a+\zeta b \quad|\quad a,b\in \mathb... | For 7), note that 6) tells you that $3 \in (\lambda)$, and since by 6) $\lambda$ is prime, $A \ne (\lambda)$. Moreover $a + \zeta b = a + (1-\lambda) b \equiv a + b \pmod{\lambda}$. So if you want an explicit isomorphism, it is $a + \zeta b + (\lambda) \mapsto a+ b \pmod{3}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Properties of Equivalence Relation Compared with Equality I'm reading about congruences in number theory and my textbook states the following:
The congruence relation on $\mathbb{Z}$ enjoys many (but not all!) of the properties satisfied by the usual relation of equality on $\mathbb{Z}$.
The text then does not go i... | An equivalence relation is the equality relation if and only if its congruence classes are all singletons. Most equivalence relations do not have this characteristic. The equivalence classes of (most) congruence relations on $\Bbb Z$, for example, are infinite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/312044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Two trivial questions in general topology I'd appreciate some guidance regarding the following 2 questions. Number 1 should be clear, and number 2 is more of a discussion:
*
*Let $X$ be a topological space. Let $E$ be a dense subset. Can $E$ be finite without $X$ being finite? Or countable?
*Let $X$ be a topologica... | *
*Let $X = \mathbb{R}$ and define the topology $\mathcal{T} = \{\mathbb{R}, \emptyset, \{0\}\}$. The set $\{1\}$ is dense in $(\mathbb{R}, \mathcal{T})$ (it's closure is clearly all of $\mathbb{R}$, and it's a finite set.
*Consider $X = [0,1] \cup \{2\}$ with the usual Euclidean metric. The ball of radius $1$ abou... | {
"language": "en",
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Understanding Primitive Polynomials in GF(2)? This is an entire field over my head right now, but my research into LFSRs has brought me here.
It's my understanding that a primitive polynomial in $GF(2)$ of degree $n$ indicates which taps will create an LFSR. Such as $x^4+x^3+1$ is primitive in $GF(2)$ and has degree $4... | For a polynomial $p(x)$ of degree $n$ with coefficients in $GF(2)$ to be primitive, it must satisfy the condition that $2^n-1$ is the smallest positive integer $e$ with the property that
$$
x^e\equiv 1\pmod{p(x)}.
$$
You got that right.
For a polynomial to be primitive, it is necessary (but not sufficient) for it to be... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Friends Problem (Basic Combinatorics) Let $k$ and $n$ be fixed integers. In a group of $k$ people, any group of $n$ people all have a friend in common.
*
*If $k=2 n + 1$ prove that there exists a person who is friends with everyone else.
*If $k=2n+2$, give an example of a group of $k$ people satisfying the given co... | The first part of this is just an expansion of Harald Hanche-Olsen’s answer.
For the second part number the $2n+2$ people $P_1,\dots,P_{n+1},Q_1,\dots,Q_{n+1}$. Divide them into pairs: $\{P_1,Q_1\},\{P_2,Q_2\},\dots,\{P_{n+1},Q_{n+1}\}$. The two people in each pair are not friends; i.e., $P_k$ is not friends with $Q_k$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I need to ask a question about vectors and cross product? When you take the determinant on 3 vectors, you calculate and get the volume of that specific shape, correct?
When you take the cross-product of 2 vectors, you calculate and get the area of that shape and you also get the vector perpendicular to the plane, corre... | Kind Of.
When you take the determinant of a set of vectors, you get the volume bounded by the vectors.
For instance, the determinant of the identity matrix (which can be considered as a set of vectors) gives the volume of the solid box in $n$ dimensions. A $3\times3$ identity matrix gives the area of a cube.
However, w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving $\prod((k^2-1)/k^2)=(n+1)/(2n)$ by induction $$P_n = \prod^n_{k=2} \left(\frac{k^2 - 1}{k^2}\right)$$ Someone already helped me see that $$P_n = \frac{1}{2}.\frac{n + 1}{n} $$ Now I have to prove, by induction, that the formula for $P_n$ is correct. The basis step: $n = 2$ is true, $$P_2 = \frac{3}{4} = \frac{... | We do the same thing as in the solution of Brian M. Scott, but slightly backwards, We are interested in the question
$$\frac{n+2}{2(n+1)}\overset{?}{=}\frac{n+1}{2n}\frac{(n+1)^2-1}{(n+1)^2}.$$
The difference-of-squares factorization $(n+1)^2-1=(n)(n+2)$ settles things.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Can we test if a number is a lucky number in polynomial time? I know primality tests exist in polynomial time. But can we test if a number is a lucky number in polynomial time ?
| I doubt you're going to get a satisfactory answer to this question. I'm fairly sure the answer to your question is that there's no known polynomial time algorithm to determine if a number is lucky. The fact that Primes is in P was not shown until 2004, when people have been studying prime numbers and prime number tests... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/312430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Sketch complex curve $z(t) = e^{-1t+it}$, $0 \le t \le b$ for some $b>0$ Sketch complex curve $z(t) = e^{-1t+it}$, $0 \le t \le b$ for some $b>0$
I tried plotting this using mathematica, but I get two curves.
Also, how do I find its length, is it just the integral?
This equation doesn't converge right?
Edit: I forgot ... | (You have received answers for the rest, so let me focus on the length.)
The length of the curve is given by
$$\int_0^b |z'(t)|\,dt = \int_0^b |(-1+i)e^{(-1+i)t}|\,dt = \int_0^b \sqrt 2 e^{-t}\,dt.$$
I think you can work out what happens when $b\to\infty$ now.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How do we plot $u(4−t)$, where $u(t)$ is a step function? How do we plot $u(4−t)$?
$u(t)$ is a step function:
$$u(t)=\begin{cases} 1&\text{ for }t \ge 0,\\
0 & \text{ for }t \lt 0.\end{cases}$$
| $u(4-t) = 1$ for $4-t\ge0$ so for $t\le4$
$u(4-t) = 0$ for $4-t\lt0$ so for $t\gt4$
Here is the plot:
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Cauchy Problem for Heat Equation with Holder Continuous Data This exercise comes from a past PDE qual problem. Assume $u(x,t)$ solves
$$
\left\{\begin{array}{rl}
u_{t}-\Delta u=0&\text{in}\mathbb{R}^{n}\times(0,\infty)\\
u(x,0)=g(x)&\text{on}\mathbb{R}^{n}\times\{t=0\}\end{array}\right.
$$
and $g$ is Holder continuous... | This calls for a scaling argument.
As you noticed, it suffices to consider $x=0$. Replace $g$ with $g-g(0)$; this does not change the derivatives. Now we know that $$|g(x)|\le |x|^\delta\tag1$$
Prove an estimate of the form
$$|u_{t}(0,1)| + |u_{x_ix_j}(0,1)| \le C_n\tag2$$
This requires writing the derivatives
as c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Filling the gap in knowledge of algebra Recently, I realize that my inability to solve problems, sometimes, is because I have gaps in my knowldge of algebra. For example, I recently posted a question that asked why $\sqrt{(9x^2)}$ was not $3x$ which to me was fairly embarrassing because the answer was fairly logical an... | As skullpatrol commented, the Khan Academy has covered a wide range of high school algebra. As you go up the scale, there are many more resources such as the Art of Problem Solving for contest mathematics. Art of Problem Solving has books on high school algebra as well as practice problems: I personally like their stru... | {
"language": "en",
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"source": "stackexchange",
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Graph theory and computer chip design reference Wikipedia says graph theory is used in computer chip design.
... travel, biology, computer chip design, and many other fields. ...
Is there a good reference for that? I can imagine optimal way to draw cpu to chip is to draw shortest hamiltonian cycle in it.
| I don't know about a reference. However, the intuition is that an electrical circuit in a computer chip design is etched into a flat surface. This implies that the graph model of this circuit must be a planar graph. So the theory behind planar graphs is very important in designing such circuits.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/312712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 3
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Applying substitutions in lambda calculus For computing $2+3$, the lambda calculus goes the following: $(\lambda sz.s(sz))(\lambda wyx.y(wyx))(\lambda uv.u(u(uv)))$
I am having a hard time substituing and reaching the final form of $(\lambda wyx.y((wy)x))((\lambda wyx.y((wy)x))(\lambda uv.u(u(uv))))$. Can anyone provid... | Recall that application is left-associative e.g. $w y x = \color{\red}{(}w y\color{red}{)} x$. Then the steps just follow by standard $\beta$-reduction
and $\zeta_1$ (which reduces the left-hand side of an application, i.e. $e_1 \rightarrow e_1' \Rightarrow e_1 e_2 \rightarrow e_1' e_2$.
In the following I have underli... | {
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Exercise 6.1 in Serre's Representations of Finite Groups I am trying to show that if $p$ divides the order of $G$ then the group algebra $K[G]$ for $K$ a field of characteristic $p$ is not semisimple. Now Serre suggests us to consider the ideal
$$U = \left\{ \sum_{s \in G} a_s e_s \hspace{1mm} \Bigg| \hspace{1mm} \sum... | Here's my shot and the problem. I thought of this solution just before going to bed last night.
Suppose that $p$ divides the order of $G$. Then Cauchy's Theorem says that there is an element $x$ of order $p$. Consider $e_x$. If we can find a submodule $V$ such that $K[G] = U \oplus V$ then we can write
$$e_x = u+v$$
f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Normalizer of $S_n$ in $GL_n(K)$ In the exercises on direct product of groups of Dummit & Foote, I proved that the symmetric group $S_n$ is isomorphic to a subgroup of $GL_n(K)$, called the permutation matrices with one 1 in each row and each column.
My question is how can I find the normalizer of this subgroup in $GL... | Edit: I've revised the answer to make it more elementary, and to fix the error YACP pointed out (thank you).
Suppose $X\in N_{GL_n(K)}(S_n)$. Then for every permutation matrix $P\in S_n$ we have $XPX^{-1}\in S_n$, so conjugation by $X$ is an automorphism of $S_n$. If $n\ne 2, 6$, then as YACP noted it must be an inner ... | {
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Prove the limit problems I got two problems asking for the proof of the limit:
Prove the following limit: $$\sup_{x\ge 0}\ x e^{x^2}\int_x^\infty e^{-t^2} \, dt={1\over 2}.$$
and,
Prove the following limit: $$\sup_{x\gt 0}\ x\int_0^\infty {e^{-px}\over {p+1}} \, dp=1.$$
I may feel that these two problems are of t... | Let
$$ f(x)=\ x e^{x^2}\int_x^\infty e^{-t^2} \implies f(x)=\ x e^{x^2}g(x).$$
We can see that $ f(0)=0 $ and $f(x)>0,\,\, \forall x>0$. Taking the limit as $x$ goes to infinity and using L'hobital's rule and Leibniz integral rule yields
$$ \lim_{ x\to \infty } xe^{x^2}g(x) = \lim _{x\to \infty} \frac{g(x)}{\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/313025",
"timestamp": "2023-03-29T00:00:00",
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Help with writing the following as a partial fraction $\frac{4x+5}{x^3+1}$. I need help with writing the following as a partial fraction:
$$\frac{4x+5}{x^3+1}$$
My attempts so far are to factor $x^3$ into $(x+1)$ and $(x^2-x+1)$
This gives me: $A(x^2-x+1)+B(x+1)$.
But I have problems with solving the equation system t... | You need to use one less exponent per factor in the numerator after your factorization.
This leads to:
$$\frac{Ax+B}{x^2-x+1} + \frac{C}{x+1} = \frac{4x+5}{x^3+1}$$
This gives us:
$$Ax^2 + Ax + Bx + B + Cx^2 - Cx + C = 4x + 5$$
This leads to:
$A + C = 0$
$A + B - C = 4$
$B + C = 5$
yielding:
$$A = -\frac{1}{3}, B = \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/313151",
"timestamp": "2023-03-29T00:00:00",
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Integration theory Any help with this problem is appreciated.
Given the $f$ is measurable and finite a.e. on $[0,1]$. Then prove the following statements
$$ \int_E f = 0 \text{ for all measurable $E \subset [0,1]$ with $\mu(E) = 1/2$ }\Rightarrow f = 0 \text{ a.e. on } [0,1]$$
$$ f > 0 \text{ a.e. } \Rightarrow \inf ~... | For $(1)$, we can define the sets $ P:= \{x:f(x)\ge 0\}$ and $ N:=\{x:f(x)\le 0\}$. Then either $\mu(P)\ge \frac{1}{2}$ or $\mu(N)\ge \frac{1}{2}$. Suppose $\mu(P)\ge \frac{1}{2}$, define $SP:= \{x:f(x)> 0\},$ then $\ SP\subset P$. If $ \mu(SP)<\frac{1}{2}$, we can choose a set $E$ such that $$SP \subset E,\ f(x)\ge 0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I prove that a sequence has a given limit? For example, let's say that I have some sequence $$\left\{c_n\right\} = \left\{\frac{n^2 + 10}{2n^2}\right\}$$ How can I prove that $\{c_n\}$ approaches $\frac{1}{2}$ as $n\rightarrow\infty$?
I'm using the Buchanan textbook, but I'm not understanding their proofs at al... | Well we want to show that for any $\epsilon>0$, there is some $N\in\mathbb N$ such that for all $n>N$ we have $|c_n-1/2|<\epsilon$ (this is the definition of a limit). In this case we are looking for a natural number $N$ such that if $n>N$ then
$$\left|\frac{n^2+10}{2n^2}-\frac{1}{2}\right|=\frac{5}{n^2}<\epsilon$$
We ... | {
"language": "en",
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Prove that either $m$ divides $n$ or $n$ divides $m$ given that $\operatorname{lcm}(m,n) + \operatorname{gcd}(m,n) = m + n$? We are given that $m$ and $n$ are positive integers such that $\operatorname{lcm}(m,n) + \operatorname{gcd}(m,n) = m + n$.
We are looking to prove that one of numbers (either $m$ or $n$) must be ... | We may suppose without loss of generality that $m \le n$. If $\text{lcm}(m,n) > n$, then $\text{lcm}(m,n) \ge 2n$, since $\text{lcm}(m,n)$ is a multiple of $n$. But then we have
$\text{lcm}(m,n) < \text{lcm}(m,n)+\gcd(m,n) = m + n \le 2n \le \text{lcm}(m,n)$,
a contradiction. So $\text{lcm}(m,n) = n$.
| {
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Integral Sign with indicator function and random variable I have the following problem. I need to consider all the conditions in which the following integral may be equal to zero:
$$\int_\Omega [p\phi-\lambda(\omega)]f(\omega)\iota(\omega)d\omega$$
Where $p>0$ is a constant. $f$ is a probability density function and $\... | *
*The expectation $\mathbb{E}\left[\iota(p\phi-\lambda)\right]$ could be zero depending on the value of $p\phi-\lambda$ on all $\omega$.
*If there is no $\omega$ for which $\iota(\omega)=1$, again you end up with a zero.
| {
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"timestamp": "2023-03-29T00:00:00",
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How many triangles are formed by $n$ chords of a circle? This is a homework problem I have to solve, and I think I might be misunderstanding it. I'm translating it from Polish word for word.
$n$ points are placed on a circle, and all the chords whose endpoints they are are drawn. We assume that no three chords interse... | The answer to a) isn't $2^n$. It isn't even $2^{n-1}$, which is probably what you meant. Draw the case $n=6$, carefully, and count the regions.
As for $T(n)$, the number of triple-crossings, the problem statement specifically says there are no triple-crossings.
| {
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Clarification Regarding the Tor Functor involved in a Finite Exact Sequence Let $\cdots\rightarrow F_1 \rightarrow F_0 \rightarrow M \rightarrow 0$ be a free resolution of the $A$-module $M$. Let $N$ be an $A$-module. I saw in some notes that we have an exact sequence $0 \rightarrow \operatorname{Tor}(M,N) \rightarrow ... | The tensor functor is right-exact. We have an exact sequence
$F_1\to F_0\to M\to 0$,
implying that the complex
$F_2\otimes N\to F_1\otimes N\to F_0\otimes N\to M\otimes N\to 0$,
is exact except possibly at $F_1\otimes N$. Now how can you calculate $\mathrm{Tor}_1(M,N)$? You should be able to figure it out from here.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $ \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.
Given $a,b,c>0$, prove that $\displaystyle \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.
I e... | We can begin by clearing denominators as follows
$$a^2c+a^2b+b^2a+b^2c+c^2a+c^2b\geq 2a^{5/3}b^{2/3}c^{2/3}+2a^{2/3}b^{5/3}c^{2/3}+2a^{2/3}b^{2/3}c^{5/3}$$
Now by the Arithmetic Mean - Geometric Mean Inequality,
$$\frac{2a^2c+2a^2b+b^2a+c^2a}{6} \geq a^{5/3}b^{2/3}c^{2/3}$$
That is,
$$\frac{2}{3}a^2c+\frac{2}{3}a^2b+... | {
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"answer_id": 2
} |
Derivative of the off-diagonal $L_1$ matrix norm We define the off-diagonal $L_1$ norm of a matrix as follows: for any $A\in \mathcal{M}_{n,n}$, $$\|A\|_1^{\text{off}} = \sum_{i\ne j}|a_{ij}|.$$
So what is $$\frac{\partial \|A\|_1^{\text{off}}}{\partial A}\;?$$
| $
\def\p{\partial}
\def\L{\left}\def\R{\right}\def\LR#1{\L(#1\R)}
\def\t#1{\operatorname{Tr}\LR{#1}}
\def\s#1{\operatorname{sign}\LR{#1}}
\def\g#1#2{\frac{\p #1}{\p #2}}
$Use the element-wise sign() function to define the matrix
$\,S = \s{X}.\;$
The gradient and differential of the Manhattan norm can be written as
$$\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/313704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
$f$ is integrable but has no indefinite integral Let $$f(x)=\cases{0,& $x\ne0$\cr 1, &$x=0.$}$$
Then $f$ is clearly integrable, yet has no antiderivative, on any interval containing $0,$ since any such antiderivative would have a constant value on each side of $0$ and have slope $1$ at $0$—an impossibility.
So does thi... | Just to supplement Emanuele’s answer. If $ I $ is an open subset of $ \mathbb{R} $, then some mathematicians define the indefinite integral of a function $ f: I \to \mathbb{R} $ as follows:
$$
\int f ~ d{x} \stackrel{\text{def}}{=} \{ g \in {D^{1}}(I) ~|~ f = g' \}.
$$
Hence, taking the indefinite integral of $ f $ yie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/313775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
Prove $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{n}} > 2\:(\sqrt{n+1} − 1)$ Basically, I'm trying to prove (by induction) that:
$$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{n}} > 2\:(\sqrt{n+1} − 1)$$
I know to begin, we should use a base case. In this case, I'll use $1$. So we ha... | Mean Value Theorem can also be used,
Let $\displaystyle f(x)=\sqrt{x}$
$\displaystyle f'(x)=\frac{1}{2}\frac{1}{\sqrt{x}}$
Using mean value theorem we have:
$\displaystyle \frac{f(n+1)-f(n)}{(n+1)-n}=f'(c)$ for some $c\in(n,n+1)$
$\displaystyle \Rightarrow \frac{\sqrt{n+1}-\sqrt{n}}{1}=\frac{1}{2}\frac{1}{\sqrt{c}}$..... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/313834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 2
} |
Elementary books by good mathematicians I'm interested in elementary books written by good mathematicians.
For example:
*
*Gelfand (Algebra, Trigonometry, Sequences)
*Lang (A first course in calculus, Geometry)
I'm sure there are many other ones. Can you help me to complete this short list?
As for the level, I'm ... | Solving Mathematical Problems: A Personal Perspective by Terence Tao
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/313980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 21,
"answer_id": 6
} |
Ramsey Number Inequality: $R(\underbrace{3,3,...,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,...3}_k)-1)+2$ I want to prove that:
$$R(\underbrace{3,3,...,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,...3}_k)-1)+2$$
where R is a Ramsey number. In the LHS, there are $k+1$ $3$'s, and in the RHS, there are $k$ $3's$. I really have ... | The general strategy will probably look something like this. (Let $R = R(\underbrace{3,\ldots,3}_k)$ in what follows.)
*
*Take $k+1$ copies of $K_{R-1}$, the complete graph on $R-1$ vertices, and assume that the edges of each one are colored with $k+1$ colors so as to avoid monochromatic triangles.
*Then suppose... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If $f(x_0+x)=P(x)+O(x^n)$, is $f$ $mLet $f : \mathbb{R} \to \mathbb{R}$ be a real function and $x_0 \in \mathbb{R}$ be a real number. Suppose that there exists a polynomial $P \in \mathbb{R}[X]$ such that $f(x_0+x)=P(x)+ \underset{x \to 0}{O} (x^n)$ with $n> \text{deg}(P)$.
Is it true that for $m<n$, $f^{(m)}(x_0)$ exi... | No, these are not true, and classical counterexamples are based on $f(x)=|x|^a\sin(1/|x|^b)$ for $x\ne0$ and $f(0)=0$, considered at $x_0=0$, for well chosen positive $a$ and $b$.
Basically, the idea is that the limited expansion of $f$ at $0$ is $f(x)=O(|x|^n)$ (no polynomial term) with $n$ large if $a$ is large but t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How is this derived? In my textbook I find the following derivation:
$$ \displaystyle \lim _{n \to \infty} \dfrac{1}{n} \displaystyle \sum ^n _{k=1} \dfrac{1}{1 + k/n} = \displaystyle \int^1_0 \dfrac{dx}{1+x}$$
I understand that it's $\displaystyle \int^1_0$ but I don't understand the $\dfrac{dx}{1+x}$ part.
| The sum is a Riemann Sum for the given integral. As $n\to\infty$,
$$
\sum_{k=1}^n\frac1{1+k/n}\frac1n
$$
tends to the sum of rectangles $\frac1{1+k/n}$ high and $\frac1n$ wide. This approximates the integral
$$
\int_0^1\frac1{1+x}\mathrm{d}x
$$
where $x$ is represented by $k/n$ and $\mathrm{d}x$ by $\frac1n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/314171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Limit $a_{n+1}=\frac{(a_n)^2}{6}(n+5)\int_{0}^{3/n}{e^{-2x^2}} \mathrm{d}x$ I need to find the limit when $n$ goes to $\infty$ of
$$a_{n+1}=\frac{(a_n)^2}{6}(n+5)\int_{0}^{3/n}{e^{-2x^2}} \mathrm{d}x, \quad a_{1}=\frac{1}{4}$$
Thanks in advance!
| Obviously $a_n > 0$ for all $n \in \mathbb{N}$. Since
$$
\int_0^{3/n} \exp(-2 x^2) \mathrm{d}x < \int_0^{3/n} \mathrm{d}x = \frac{3}{n}
$$
We have
$$
a_{n+1} < \frac{1}{2} a_n^2 \left(1 + \frac{5}{n} \right) \leqslant 3 a_n^2
$$
Consider sequence $b_n$, such that $b_1 = a_1$ and $b_{n+1} = 3 b_n^2$, then $a_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Sequence Proof with Binomial Coefficient Suppose $\lim_{n\to \infty }z_n=z$.
Let $w_n=\sum_{k=0}^n {2^{-n}{n \choose k}z_k}$
Prove $\lim_{n\to \infty }w_n=z$.
I'm pretty sure I need to use $\sum_{k=0}^\infty{n \choose k}$ = $2^{n}$ in the proof. Help? Thoughts?
| $$w_n-z=\sum_{k=0}^n2^{-n}\binom{n}k(z_k-z)$$
Fix $\epsilon>0$. There is an $m\in\Bbb Z^+$ such that $|z_k-z|<\frac{\epsilon}2$ whenever $k\ge m$. Clearly
$$\lim_{n\to\infty}\sum_{k=0}^m2^{-n}\binom{n}k=0\;,$$
so there is an $r\ge m$ such that $$\sum_{k=0}^m2^{-n}\binom{n}k|z_k-z|<\frac{\epsilon}2$$ whenever $n\ge r$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that $\sigma(n) = \sum_{d|n} \phi(n) d(\frac{n}{d})$ This is a homework question and I am to show that $$\sigma(n) = \sum_{d|n} \phi(n) d\left(\frac{n}{d}\right)$$ where $\sigma(n) = \sum_{d|n}d$, $d(n) = \sum_{d|n} 1 $ and $\phi$ is the Euler Phi function.
What I have. Well I know $$\sum_{d|n}\phi(d) = n$$ I als... | First hint: verify the formula when $n$ is a power of a prime. Then, prove that the function $n \mapsto \sum_{d \mid n} \phi(n) d(n/d)$ is also multiplicative, so it must coincide with $\sigma$. In fact, prove more generally that if $a_n$ and $b_n$ are multiplicative, then $n\mapsto \sum_{d \mid n}a_db_{n/d}$ is multip... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Coordinates for vertices of the "silver" rhombohedron. The "silver" rhombohedron (a.k.a the trigonal trapezohedron) is a three-dimensional object with six faces composed of congruent rhombi. You can see it visualised here.
I am interested in replicating the visualisation linked to above in MATLAB, but to do that I nee... | Use vectors $e_1=(1,0,0)$, $e_2=(\cos{\alpha},\sin{\alpha},0)$ and $e_3=(\cos{\alpha},0,\sin{\alpha})$ as basis.
Then vertices are set of all points with each coordinate $0$ or $1$: $(0,0,0)$, $(0,0,1)$, ..., $(1,1,1)$.
Or $0$, $e_1$, $e_2$, $e_3$, $e_1+e_2$, ..., $e_1+e_2+e_3$.
Multiply coordinates by constant if need... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conditional probabilties in the OR-gate $T=A\cdot B$ with zero-probabilities in $A$ and $B$? My earlier question became too long so succintly:
What are $P(T|A)=P(T\cap A)/P(A)$ and $P(T|B)=P(T\cap B)/P(B)$ if $P(A)=0$ and $P(B)=0$?
I think they are undefined because of the division by zero. How can I specify the con... | Yes, it is undefined in general. It is generally pointless to ask for the conditional probability of $T$ when $A$ occurs when it is known that $A$ almost surely never happens.
But a meaningful specification in your particular case that $T = A\cup B$ is by some intuitive notion of continuity. For $P(A) \neq 0$, if $C\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Help evaluating a gamma function I need to do a calculus review; I never felt fully confident with it and it keeps cropping up as I delve into statistics. Currently, I'm working through a some proof theory and basic analysis as a sort of precursor to the calc review, and I just hit a problem that requires integration. ... | In general:
$$u=t^{x-1}\;\;,\;\;u'=(x-1)t^{x-2}\\v'=e^{-t}\;\;,\;\;v=-e^{-t}$$
so
$$\Gamma(x):=\int\limits_0^\infty t^{x-1}e^{-t}\,dt=\overbrace{\left.-t^{x-1}e^{-t}\right|_0^\infty}^\text{This is zero}+(x-1)\int\limits_0^\infty t^{x-2}e^{-t}=$$
$$=:(x-1)\Gamma(x-1)$$
So you only need to know $\,\Gamma(1)=1\,$ and this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Find the common ratio of the geometric series with the sum and the first term Given: Geometric Series Sum ($S_n$) = 39
First Term ($a_1$) = 3
number of terms ($n$) = 3
Find the common ratio $r$.
*I have been made aware that the the common ratio is 3, but for anyone trying to solve this, don't plug it in as an answer th... | The genreal sum formula in geometric sequences: $$S_n=\frac{a_1\left(r^n-1\right)}{r-1}$$ In your case, $S_n=39$, $a_1=3$, $n=3$. Now you just need to find $r$:
$$39=\frac{3\left(r^3-1\right)}{r-1}$$ $$\ 13=\frac{r^3-1}{r-1}$$ $$\ 13\left(r-1\right)=r^3-1$$ $$\ 13r-13=r^3-1$$ $$\ r^3-13r+12=0$$
$$\ r_1=-4,\ r_2=3,\ r_3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Artinian rings and PID Let $R$ be a commutative ring with unity. Suppose that $R$ is a principal ideal domain, and $0\ne t\in R$. I'm trying to show that $R/Rt$ is an artinian $R$-module, and so is an artinian ring if $t$ is not a unit in $R$.I'm not sure how to begin. please help.
| Hint: Show that the ideals of $R/Rt$ are all principal, and in fact, are in bijection with the divisors of $t$ in $R$ (considered up to multiplication by a unit). Then use that $R$ is a UFD (since it's a PID). Finally, note that the ideals of $R/Rt$ are also precisely its $R$-submodules.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/314644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Suppose that $(x_n)$ is a sequence satisfying $|x_{n+1}-x_n| \leq q|x_n-x_{n-1}|$ for all $n \in \mathbb{N^+}$.Prove that $(x_n)$ converges Let $q$ be a real number satisfying $0<q<1$. Suppose that $(x_n)$ is a sequence satisfying $$|x_{n+1}-x_n| \leq q|x_n-x_{n-1}|$$ for all $n \in \mathbb{N^+}$. Prove that $(x_n)$ is... | Hint: note that for any $N$ and $n>N$, we have $|x_{n+1}-x_n|\leq q|x_n-x_{n-1}|\cdot q^{N-1}|x_2-x_1|$. This can be proven by induction on $N$. If we have $n>m>N$, then how can we bound $|x_n-x_m|$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/314791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Example of a group where $o(a)$ and $o(b)$ are finite but $o(ab)$ is infinite Let G be a group and $a,b \in G$. It is given that $o(a)$ and $o(b)$ are finite. Can you give an example of a group where $o(ab)$ is infinite?
| The standard example is the infinite dihedral group.
Consider the group of maps on $\mathbf{Z}$
$$
D_{\infty} = \{ x \mapsto \pm x + b : b \in \mathbf{Z} \}.
$$
Consider the maps
$$
\sigma: x \mapsto -x,
\qquad
\tau: x \mapsto -x + 1,
$$
both of order $2$. Their composition
$$
\tau \circ \sigma (x) = \tau(\sigma(x)) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 4
} |
Writing direct proofs Let $x$, $y$ be elements of $\mathbb{Z}$. Prove if $17\mid(2x+3y)$ then $17\mid(9x+5y)$. Can someone give advice as to what method of proof should I use for this implication? Or simply what steps to take?
| I would write the equations in $\mathbb{Z}_{17}$, which is a field, because $17$ is prime, so linear algebra applies:
$$ 2x+3y=0 $$
is a linear equation of two variables, and you seek to prove that it implies
$$ 9x+5y=0 $$
which means they're linearly dependent. Two equations are linearly dependent if and only if one i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Klein-bottle and Möbius-strip together with a homeomorphism Consider the Klein bottle (this can be done by making a quotient space). I want to give a proof of the following statement:
The Klein Bottle is homeomorphic to the union of two copies of a Möbius strip joined by a homeomorphism along their boundaries.
I know w... | Present a Klein bottle as a square with vertical edges identified in an orientation-reversing manner and horizontal edges identified in an orientation-preserving manner. Now make two horizontal cuts at one-third of the way up and two-thirds of the way up.
The middle third is one Möbius strip. Take the upper and lower ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/314977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Using Pigeonhole Principle to prove two numbers in a subset of $[2n]$ divide each other Let $n$ be greater or equal to $1$, and let $S$ be an $(n+1)$-subset of $[2n]$. Prove that there exist two numbers in $S$ such that one divides the other.
Any help is appreciated!
| HINT: Create a pigeonhole for each odd positive integer $2k+1<2n$, and put into it all numbers in $[2n]$ of the form $(2k+1)2^r$ for some $r\ge 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/315050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 4,
"answer_id": 0
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Exercise on representations I am stuck on an exercise in Serre, Abelian $\ell$-adic representations (first exercise of chapter 1).
Let $V$ be a vector space of dimension $2$, and $H$ a subgroup of $GL(V)$ such that $\det(1-h)=0$ for all $h \in H$.
Show that in some basis $H$ is a subgroup of either $\begin{pmatrix} 1 ... | By hypothesis, for all $h \in H$ and basis $(e,f)$ one has
$$\det(h(e)-e,h(f)-f) =0.$$
Let $g \in H$ different from the identity.
$\bullet$ Suppose that $1$ is the only eigen-value of $g$. Then in some basis $(e,f)$ the matrix of $g$ is $Mat(g) = \left( \begin{smallmatrix} 1&1\\0&1 \end{smallmatrix}\right)$. Let $h \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
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Dimension of a space I'm reading a book about Hilbert spaces, and in chapter 1 (which is supposed to be a revision of linear algebra), there's a problem I can't solve. I read the solution, which is in the book, and I don't understand it either.
Problem: Prove that the space of continuos functions in the interval (0,1):... | Note that if $f,g$ are continuous and for every rational number $q$ it holds that $f(q)=g(q)$ then $f=g$ everywhere.
This means that $|B|\leq|\mathbb{R^Q}|=|\mathbb{R^N}|=|\mathbb R|$.
Also, $|\mathbb R|$ is not necessarily $\aleph_1$. This assumption is known as the continuum hypothesis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/315193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
functional relation I need to find functions $f : \mathbb{R}_{+} \rightarrow \mathbb{R}$ which satisfies
$$ f(0) =1 $$
$$ f(\max(a,b))=f(a)f(b)$$
For each $a,b \geq 0$.
I have found two functions which satisfy my criteria.
$$ f_1(x)=1$$
$$f_2(x) = \begin{cases}
0, & \text{if } x>0 \\
1, & \text{if } x=0
\end{cases}
$$... | Since $f(a)f(b)=f(b)$ for all $b\ge a$, either $f(b)=0$ or $f(a)=1$. If $f(a)=0$, then $f(b)=0$ for all $b\ge a$. If $f(b)=1$, then $f(a)=1$ for all $a\le b$.
Thus, it appears that for any $a\ge0$, the functions
$$
f_a^+(x)=\left\{\begin{array}{}
1&\text{for }x\le a\\
0&\text{for }x\gt a
\end{array}\right.
$$
and for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Are sets always represented by upper case letters? -- and understanding a bit about equivalance relations I'm trying to solve q question which states:
Let $\leq$ be a preorder relation on a set $X$, and $E=${$(x,y)\in X:x\leq y$ and $y\leq x$} the corresponding equivalence relation on $X$. Describe $E$ if $X$ is the s... | In answer to your second question, this is a major component on the axiomatic set theory. I believe the power theory states that if X is the set of finite subsets of a fixed infinite set U then ∀x∃y∀u(u∈y↔u⊆x). And there are many subsequent theories based on this axiom. http://mathworld.wolfram.com/AxiomofthePowerSet.h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Showing if something is continuous in Topology If $f : X \to \mathbb{R}$ is continuous,
I want to show that $(cf)(x) = cf(x)$ is continuous, where $c$ is a constant.
Attempt: If $f$ is continuous, then we want to show that the inverse image of every open set in $\mathbb{R}$ is an open set of $X$. Choose an open interva... | An alternative answer, that does not require you to prove the composition of continuous functions is continuous:
Let $U\subset\mathbb{R}$ be open. In fact we may assume $U=(a,b)$ by taking the open balls as a base for $\mathbb{R}$.
Now,
$$
(cf)^{-1}(U) = \{ x\in X: \exists y\in(a,b): (cf)(x) = y\}
= \left\{x\in X:\exis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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The Lebesgue Measure of the Cantor Set Why is it that m(C+C) = 2, where C is the Cantor set, but C is itself measure zero? What is happening that makes this change occur? Is it true generally that m(A+B) =/= m(A) + m(B)? Are there cases in which this necessarily holds? I should note that this is all within the usua... | Adding two sets $A$ and $B$ is in general much more than taking the union of a copy of $A$ and a copy of $B$.
Rather, $A+B$ can be seen as made up of many slices that look like $A$ (one for each $b\in B$), and the Cantor set has the same size as the reals, so here we have in fact as many slices as real numbers. Therei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to solve a linear equation by substitution? I've been having a tough time figuring this out. This is how I wrote out the question with substitution.
$$\begin{align}
& I_2 = I_1 + aI_1 (T_2 - T_1) \\
& I_1=100\\
&T_2=35\\
&T_1=-5\\
&a=0.000011
\end{align}$$
My try was $I_2 = 100(1) + 0.000011(100) (35(2)-5(1))$
The ... | The lower temperature is $-5$ and you dropped a sign plus should not be multiplying $T_2$ by $2$. The correct calculation is $I_2=100+0.000011\cdot 100 \cdot (35-(-5))=100.044$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/315507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding a bijection and inverse to show there is a homeomorphism I need to find a bijection and inverse of the following:
$X = \{ (x,y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 4 \}$ with its subspace topology in $\mathbb{R}^2$
$Y = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1$ and $ 0 \leq z \leq 1 \}$ with its subspa... | Hint: $X$ is an annulus, and $Y$ is a cylinder. Imagine "flattening" a cylinder by forcing its bottom edge inwards and its top edge outwards; it will eventually become an annulus. This is the idea of the map; you should work out the actual expressions describing it on your own.
Here is a gif animation I made with Math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
(partial) Derivative of norm of vector with respect to norm of vector I'm doing a weird derivative as part of a physics class that deals with quantum mechanics, and as part of that I got this derivative:
$$\frac{\partial}{\partial r_1} r_{12}$$
where $r_1 = |\vec r_1|$ and $r_{12} = |\vec r_1 - \vec r_2|$. Is there any... | Since these are vectors, one can consider the following approach:
Let
${\bf x}_1 := \overrightarrow{r}_1$, and ${\bf x}_2 : = \overrightarrow{r}_2$, then $r_1 = ||{\bf x}||^{\frac{1}{2}}$, and $r_{12} = ||{\bf x}_1 - {\bf x_2}||^{\frac{1}{2}}$.
Define the following functions:
$g({\bf x}_1) = ||{\bf x}_1 - {\bf x_2}||^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Limit involving probability Let $\mu$ be any probability measure on the interval $]0,\infty[$. I think the following limit holds, but I don't manage to prove it:
$$\frac{1}{\alpha}\log\biggl(\int_0^\infty\! x^\alpha d\mu(x)\biggr) \ \xrightarrow[\alpha\to 0+]{}\ \int_0^\infty\! \log x\ d\mu(x)$$
In probabilistic terms ... | We assume that there is $\alpha_0>0$ such that $\int_0^{+\infty}x^{\alpha_0} d\mu(x)$ is finite. Let $I(\alpha):=\frac 1{\alpha}\log\left(\int_0^{+\infty}x^\alpha d\mu(x)\right)$ and $I:=\int_0^{+\infty}\log xd\mu(x)$.
Since the function $t\mapsto \log t$ is concave, we have $I(\alpha)\geqslant I$ for all $\alpha$.
N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Continuously differentiability of "glued" function I have the following surface for $x,t>0$: $$z(x,t)=\begin{cases}\sin(x-2t)&x\geq 2t\\
(t-\frac{x}{2})^{2}&x<2t \end{cases}$$ How to show that surface is not continuously differentiable along the curve $x=2t$? Truly speaking, I have no idea how to start with this exampl... | When $x=2t$, what is the value of $\frac x2-t$? of $\cos(x-2t)$? Are these equal?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/315837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Armijo's rule line search I have read a paper (http://www.seas.upenn.edu/~taskar/pubs/aistats09.pdf) which describes a way to solve an optimization problem involving Armijo's rule, cf. p363 eq 13.
The variable is $\beta$ which is a square matrix. If $f$ is the objective function, the paper states that Armijo's rule is ... | In general (i.e. for scalar-valued $x$), Armijo's rule states
$$f(x^{new}) - f(x^{old}) \le \eta \, (x^{new}-x^{old})^\top \nabla f(x^{old}).$$
Hence, you need the vectorization of $\beta^{new}-\beta^{old}$ on the right hand side.
(Alternatively, you could replace $\nabla_\beta f$ by the gradient w.r.t. the frobenius i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solve the equation $y''+y'=\sec(x)$ Solve the equation $$y''+y'=\sec(x).$$ By solving the associated homogenous equation, I get complementary solution is $y_c(x)=A+Be^{-x}$. Then by using the method of variation of parameter, I let $y_p(x)=u_1+u_2e^{-x}$ where $u_1,u_2$ satisfy $$u_1'+u_2' e^{-x}=0, \quad -u_2' e^{-x}... | $y''+y'=sec(x)$
$(y'e^x)'=\dfrac{d\int sec(x)e^xdx}{dx}$
$y'=e^{-x}\int sec(x)e^xdx
+Ce^{-x}$
$y=\int e^{-x}\int sec(x)e^xdx dx+ C_1e^{-x}+C_2$
I do not know of any elementary form for the integrals. Wolfram gives an answer
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/316020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Do subspaces obey the axioms for a vector space? If I have a subspace $W$, then would elements in $W$ obey the 8 axioms for a vector space $V$ such as:
$u + (-u) = 0$ ; where $u ∈ V$
| Yes, indeed. A subspace of a vector space is also a vector space, restricted to the operations of the vector space of which it is a subspace. And as such, the axioms for a vector space are all be satisfied by a subspace of a vector space.
As stated in my comment below: In fact, a subset of a vector space is a subspace ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Find all solutions of the equation $x! + y! = z!$ Not sure where to start with this one. Do we look at two cases where $x<y$ and where $x>y$ and then show that the smaller number will have the same values of the greater? What do you think?
| Let $x \le y < z$:
We get $x! = 1\cdot 2 \cdots x$, $y! = 1\cdot 2 \cdots y$, and $z! = 1\cdot 2 \cdots z$.
Now we can divide by $x!$
$$1 + (x+1)\cdot (x+2)\cdots y = (x+1)\cdot (x+2) \cdots z$$
You can easily show by induction that the right side is bigger than the left for all $z>2$. The only cases that remain are $0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 2
} |
Limit point of set $\{\sqrt{m}-\sqrt{n}:m,n\in \mathbb N\} $ How can I calculate the limit points of set $\{\sqrt{m}-\sqrt{n}\mid m,n\in \mathbb N\} $?
| The answer is $\mathbb{R}$, as we can see here, for $x\in (0,\infty)$ and $\epsilon >0$, there are $n_0 , N \in \mathbb{N}$ such that $\sqrt{n_0 +1}-\sqrt{n_0} <1/N<\epsilon /2$. Now we can divide $(0,\infty)$ to pieces of length $1/N$, so there is $k\in \mathbb{N}$ such that $k(\sqrt{n_0 +1}-\sqrt{n_0})\in N_{\epsilon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Number of elements of order $5$ in $S_7$: clarification In finding the number of elements of order $5$ in $S_7$, the symmetric group on $7$ objects, we want to find products of disjoint cycles that partition $\{1,2,3,4,5,6,7\}$ such that the least common multiple of their lengths is $5$. Since $5$ is prime, this allow... | This is your code:
g:=SymmetricGroup(7);
h:=Filtered(Elements(g),x->Order(x)=5);
Size(h);
The size of $h$ is the same as other theoretical approaches suggested. It is $504$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/316307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Method of Undetermined Coefficients I am trying to solve a problem using method of undetermined coefficients to derive a second order scheme for ux using three points, c1, c2, c3 in the following way:
ux = c1*u(x) + c2*u(x - h) + c3*u(x - 2h)
Now second order scheme just means to solve the equation for the second order... | Having a second order scheme means that it's accurate for polynomials up to and including second degree. The scheme should calculate the first order derivative $u_x$, as the formula says.
It suffices to make sure that the scheme is accurate for $1$, $x$, and $x^2$; then it will work for all second-degree polynomials ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Banach-algebra homeomorphism. Let $ A $ be a commutative unital Banach algebra that is generated by a set $ Y \subseteq A $. I want to show that $ \Phi(A) $ is homeomorphic to a closed subset of the Cartesian product $ \displaystyle \prod_{y \in Y} \sigma(y) $. Moreover, if $ Y = \{ a \} $ for some $ a \in A $, I want ... | Note that $\Phi (A)$ is compact in the w$^*$-topology. Also, $\prod \sigma(y)$ is compact Hausdorff in the product topology. For the map $f$ you defined, note that $Ker f = \{0\}$ since $Y$ generates $A$.
To prove continuity, take a net $\{\phi_\alpha\}_{\alpha \in I}$ in $\Phi(A)$, such that $\phi_\alpha \rightarrow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Evaluating the similarity of watermarks I am working on an assignment where we have to use the NEC algorithm for inserting and extracting a watermark from an image file. Using the techniques described in this article. I am at the point where I want to apply the similarity function in Matlab:
$$\begin{align*}
X &= (x_1,... | Well if you talking from an efficient/fastest/concise point of view, then MATLAB is vectorized. Let x1 be the vector of original watermark and let x2 be the vector of the extracted watermark then I would just do something like
dot(x1,x2)/sqrt(dot(x2,x2))
or even
dot(x1,x2)/norm(x2,2). They are identical.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/316572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Linear independence of polynomials from Friedberg In Page 38 Example 4, in Friedberg's Linear Algebra:
For $k = 0, 1, \ldots, n$ let $$p_k = x^k + x^{k+1} +\cdots + x^n.$$
The set $\{p_0(x), p_1(x), \ldots , p_n(x)\}$ is linearly independent in $P_n(F)$. For if $$a_0 p_0(x) + \cdots a_n p_n(x) = 0$$ for some scalars $... | You are taking $n$ as variable, but the variable index is $k$. If you write it out, varying $k$, only $p_0$ has the term $1$ (so only $a_0$ is multiplied by $1$), but all $p_i$ have the term $x^n$. I think if you look at your equations carefully again, you'll see it yourself (visually: in your argument, you make the po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Difference of two sets and distributivity If $A,B,C$ are sets, then we all know that $A\setminus (B\cap C)= (A\setminus B)\cup (A\setminus C)$. So by induction
$$A\setminus\bigcap_{i=1}^nB_i=\bigcup_{i=1}^n (A\setminus B_i)$$
for all $n\in\mathbb N$.
Now if $I$ is an uncountable set and $\{B_i\}_{i\in I}$ is a family ... | De Morgan's laws are most fundamental and hold for all indexed families, no matter the cardinalities involved. So, $$A-\bigcap _{i\in I}A_i=\bigcup _{i\in I}(A-A_i)$$ and dually $$A-\bigcup_{i\in I}A_i=\bigcap _{i\in I}(A-A_i).$$ The proof is a very good exercise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/316699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to evaluate $\int_0^1\frac{\log^2(1+x)}x\mathrm dx$? The definite integral
$$\int_0^1\frac{\log^2(1+x)}x\mathrm dx=\frac{\zeta(3)}4$$
arose in my answer to this question. I couldn't find it treated anywhere online. I eventually found two ways to evaluate the integral, and I'm posting them as answers, but they both ... | I wrote this to answer a question which was deleted (before I posted) because the answers to this question answered that question.
$$
\begin{align}
\int_0^1\frac{\log(1+x)^2}x\,\mathrm{d}x
&=-2\int_0^1\frac{\log(1+x)\log(x)}{1+x}\,\mathrm{d}x\tag1\\
&=-2\sum_{k=0}^\infty(-1)^kH_k\int_0^1x^k\log(x)\,\mathrm{d}x\tag2\\
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 5,
"answer_id": 1
} |
Most efficient method for converting flat rate interest to APR. A while ago, a rather sneaky car salesman tried to sell me a car financing deal, advertising an 'incredibly low' annual interest rate of 1.5%. What he later revealed that this was the 'flat rate' (meaning the interest is charged on the original balance, an... | My rule of thumb to convert APR to Flat or vice versa is as such:
APR = Flat rate x 2 x No. of payments / No. of payments + 1
Example: 4% x 2 x 12 / 12 + 1 = 96 / 13 = 7.38% approx.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/316788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 3
} |
Is an infinite linearly independent subset of $\Bbb R$ dense? Suppose $(a_n)$ is a real sequence and $A:=\{a_n \mid n\in \Bbb N \}$ has an infinite linearly independent subset (with respect to field $\Bbb Q$). Is $A$ dense in $\Bbb R?$
| If $A$ is a linearly independent subset of $\mathbb R$, for each $a\in A$ there is a positive integer $n(a)$ such that $n(a)>|a|$. The set $\left\{\dfrac{a}{n(a)}:a\in A\right\}$ is a linearly independent set with the same cardinality and span as $A$, but it is a subset of $(-1,1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/316866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
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