Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find $\angle DCF$.
Let $ABC$ an isoscel triangle with $AB=AC$ with $\angle{BAC}=100^{\circ}$ and $D\in (BC)$ s.t. $AC=DC$ and $F$ on $(AB)$ with $DF||AC$.
Find $\angle DCF$.
I tried a lot of constructions, to find an inscriptible quadrilateral but I didn't succeed.
| Let $\angle DCF =x$ and $AB =AC=CD=1$. The given $FD || AC$ leads to similar triangles BDF and BCA, and
$\frac{AF}{AC} = \frac 1{2\sin50}$. Then, apply the sine rule to the triangle FCA
$$\frac{\sin\angle ACF}{\sin\angle AFC}=\frac{AF}{AC}
\implies\frac{\sin(40-x)}{\sin(40+x)}=\frac1{2\sin50}$$
or
\begin{align}
\sin ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3666748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is a symmetric matrix positive definite iff $D$ in its LDU decomposition is positive definite? Given
$$A=LDU$$
where
*
*$A$ is a real symmetric matrix
*$L$ is a lower unitriangular matrix
*$D$ is a diagonal matrix
*$U$ is an upper unitriangular matrix
can we say that
$$A>0 \iff D>0$$
?
Edit:
My thinking is tha... | First of all, if $A = LDU$ is symmetric, we must have $U=L^T$ by uniqueness of the decomposition ($LDU = A=A^T = U^T D L^T$). so then $A=LDL^T$.
So if $D>0$ then by for all $x\neq 0$ denoting $y=L^Tx$ (why is $y \neq 0$?) yields
$$x^T A x = x^T L D L^T x = (L^T x)^T D L^T x = y^T D y > 0$$
And if $A>0$ then for all $x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3666885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Can someone explain the definition of primitive mapping?
Definition: If $G$ maps an open set $E \subset R^n$ into $R^n$, and if there is an integer $m$ and a real function $g$ with domain $E$ such that $$G(x)=\sum_{i \neq m} x_i e_i +g(x) e_m,\, (x \in E)$$ then we call $G$ primitive.
Can someone explain this definit... | Since not all notions in the question are defined, we try to guess their meaning. We have that $R^n$ probably is $\Bbb R^n$,
$e_i$ is the standard basis vector of $\Bbb R^n$ such that its $i$-th coordinate is $1$ and the other coordinates are zeroes, given $x\in E\subset \Bbb R^n$, $x=\sum_{i=1}^n x_ie_i$ is the decom... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove an idempotent invertible 2x2 matrix in general linear group $\text{GL}_2(\mathbb{R})$ must be the identity The general linear group of degree $2$ over $\mathbb{R}$ (all $2 \times 2$ invertible real matrices with matrix multiplication), is a group. In any group $G$, if $a \in G$ is idempotent i.e. $aa=a$, then
$$a... | If $a+d=1$, then
$$A = \begin{bmatrix} a & b\\ c & 1-a\\ \end{bmatrix}$$
and $A = A^2$ implies
$$\begin{bmatrix} a^2+bc & b\\ c & 1-a\\ \end{bmatrix}
= \begin{bmatrix} a & b\\ c & a^2+bc-2a+1\\ \end{bmatrix}$$
and we must have $$a^2-a+bc = 0$$
But this implies $\det A = 0$; which is a contraction, since $A$ is invert... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is $p$ necessarily greater than $r$ in this number theory problem? From 1998 St. Petersburg City Mathematical Olympiad, presented in Andreescu & Andrica NT: SEP:
Let $n$ be a positive integer. Show that any number greater than $n^4/16$ can be written in at most one way as the product of two of its divisors having ... | It is implicit in the problem that the divisors in question are positive, because if you allowed negative divisors then you could always get a second factorization by reversing the factors' signs. So I'll take $a,b,c,d$ to all be positive.
Now I'm going to try to simplify the problem by reducing it to the case where th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Spectral sequence of a filtration: a possible mistake $\require{AMScd}$The following is taken from these notes by Daniel Murfet.
Let $ \cdots \subseteq F^{p + 1}(C) \subseteq F^p(C) \subseteq F^{p - 1}(C) \subseteq \cdots$ be a filtration of a complex $C$ in an abelian category.
There is either a mistake or I don't ... | We have
$$\ddot{A^{p,q}_r}=\partial
\newcommand\of[1]{\left({#1}\right)}
\of{
F^{p-r+1}C^{p+q-1}\cap \partial^{-1}
\of{
F^{p+1}C^{p+q}
}
}.
$$
Note that $\partial^{-1}(F^{p+1}C^{p+q})$ is independent of $r$,
and as $r$ increases, the filtration index decreases, and therefore the groups get larger, so you appear to be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3667629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Induction step: $5 + 5n \leq {n}^2$ for $n \geq 6$ Prove by mathematical induction that $5 + 5n \leq {n}^2 $ for all integers $n\geq 6$.
Step 1: Base case
Suppose $n = 6$, hence $5 + 5(6) \leq {6}^2 = 35 \leq 36$
We proved that base case is true as 35 is less than or equal to 36.
Step 2: Induction step
We claim that k ... | \begin{align}
5+5(k+1) &= 5+5k + 5 \\
&\le k^2+5 \\
&\le k^2 + 2k+1 \\
&=(k+1)^2
\end{align}
The second last step is due to $5 \le 2k+1$ which is equivalent to $2 \le k$, we know this is true since $k \ge 6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3667771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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How do we know that the eigenspaces of $T$ are the only $T$-invariant subspaces? Given $T:\mathbb{C}^2 \rightarrow \mathbb{C}^2$, if I know we have two different eigenvalues, let's call them $\lambda_1, \lambda_2$, then we can say $\mathbb{C}^2, \left\{ 0 \right\}, V_{\lambda_1} , V_{\lambda_2}$ are all invariant sub-s... | If there was, it would have to be $1$-dimensional. In other words, it would be equal to $\Bbb Cv$, for some vector $v\ne0$. But asserting that $\Bbb Cv$ is invariant is the same thing as asserting that $v$ is an eigenvector.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3668060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Combinatorial proof of $1+2(\sum_{i=0}^n 3^i)=3^{n+1}$ I have this workbook of proofs that I've been trying to finish for a couple of months now. There is this problem in it that requires me to prove $1+2(\sum_{i=0}^n 3^i)=3^{n+1}$ using combinatorial identities only. This problem has stumped me for several days and I ... | Here's a different solution than in the link:
consider a tournament with $3^{n+1}$ players; divide them into groups of three, with two games per group(every game an elimination), and let the winner proceed to the next round, until we have a single final winner from the last single group of three.
The number of games p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3668233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given $ a_n= 6a_{n-1} -4a_{n-2}$ and initial values, find a closed form for $a_n$ I have a recursive formula:
$$ a_n= 6a_{n-1} -4a_{n-2}$$
with $a_0=1$ and $a_1=3$, and I need to find a closed-form expression of $(a_n)_{n\in \mathbb{N}}$.
I managed to calculate almost everything but at the end I get this expression:
$$... | Note that $3-\sqrt5$ is a little less than $0.764$, so $0<\frac{(3-\sqrt5)^n}2<\frac12$ for all $n\ge 1$. The lefthand side of your final expression must be an integer; call it $m$. Thus,
$$0<m-\frac{(3+\sqrt5)^n}2<\frac12\;,$$
and it follows immediately that
$$m=\left\lceil\frac{(3+\sqrt5)^n}2\right\rceil\;.$$
For thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3668376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence of series by adding parenthesis Given a series $\sum_{n=1}^{\infty}a_n$
If we know that by inserting parenthesis to the sum, while in each pair of parenthesis all elements are of the same sign we get a convergent series, then the original series is also convergent.
My lecturer proved this theorem in a very ... | Let $s_n=\sum_{k=1}^na_n$ and let $S_n$ be the $n$th partial sum of the series that you get after adding the parenthesis. Suppose, say the the first four $a_k$'s are postive, that the five next ones are negative, and then there are some more positive terms. Then $S_1=s_4$. And, after that, $S_2=s_9$. Besides, $s_1\leqs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3668517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Intuition on the proposition: $f:(X,d_{X})\rightarrow(Y,d_{Y})$ is continuous iff the pre-image of an open set is open I am studying continuity on metric spaces and I have been presented to the proposition in the title. My question is: how do we understand the kernel of this definition? I am already acquainted to the $... | Fix an $x \in X$. Note that for any $\epsilon > 0$, the ball of radius $\epsilon$ around $f(x)$ is an open set, so the fact that the pre-image of that ball is open implies that you can find a $\delta$-radius ball around $x$ that maps into the $\epsilon$-ball, which is exactly what is happening with the $\epsilon$-$\del... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How to apply Leibniz rule for this integral? I am struck with the problem of how to evaluate the following integration:
$$\frac{d}{dt}\int_0^{a(t)}{(a (t)-s)^{\alpha-1}f (s) ds}$$
where $f(s)$ is differentiable
and $\alpha\in (0,1)$. The problem is the integrand might not converge when substituting the upper limit
| $$\int\limits_{0}^{a(t)}{(a(t)-s)^{\alpha-1}}f(s)ds = \left|-\frac{(a(t)-s)^\alpha}{\alpha}f(s) \right|_{0}^{a(t)}+\frac{1}{\alpha}\int\limits_{0}^{a(t)}{(a(t)-s)^\alpha}{f^{\prime}(s)}ds$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3669214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Why is calculus needed in this problem involving work and units of energy? Consider this problem:
Suppose it takes k units of energy to lift a cubic meter of water one meter. About how much energy E will it take to pump dry a circular hole one meter in diameter and 100 meters deep that is filled with water?
Here's ho... | You don't have to lift every cup of water out by the whole $100$m. If a given horizontal cross-section of water is a depth $d$ meters, you need to move that cross-section only $d$ meters, not $100$m. So to find the whole amount of work done you need to add up the work done for each thin cross-section, which depends o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3669406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $B^{-1}-A^{-1}$ has r positive eigenvalues when A-B has r positive eigenvalues? Assume that A and B are 2 positive definite matrices of $n\times n$. As is known, $A-B>0$ implies $B^{-1}-A^{-1} >0$. That is to say, $B^{-1}-A^{-1}$ has n positive eigenvalues when A-B has n positive eigenvalues.
My question is :A... | If $B$ is positive definite, then the following argument applies: note that
$$
B^{-1/2}(A - B)B^{-1/2} = B^{-1/2}AB^{-1/2} - I
$$
is a symmetrc matrix with $r$ positive eigenvalues. This occurs if and only if $B^{-1/2}AB^{-1/2} - I$ has $r$ eigenvalues that are greater than $1$. This in turn occurs if and only if $B^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3669561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Polynomial ring is not a UFD
Let $K$ be a field, and consider the ring $R=\{f\in K[x]\mid f'(1)=f''(1)=0\}$. Show that $R$ is not a UFD (Unique Factorization Domain).
My thoughts: I can show that elements such as $(x-1)^3$ and $(x-1)^4$ are irreducible in $R$. Can this be used to show $R$ is not a UFD? I am not sure ... | Silly me. Following the advice of @rschwieb, we have the non-unique factorizations
$$
(x-1)^{12}=(x-1)^3(x-1)^3(x-1)^3(x-1)^3
$$
and
$$
(x-1)^{12}=(x-1)^4(x-1)^4(x-1)^4
$$
into irreducibles. Therefore $R$ is not a UFD. Just to add this bit of detail: $(x-1)^3$ and $(x-1)^4$ are irreducible in $R$ since if they weren't,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3669717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the all positive integer solutions $(a,b)$ to $\frac{a^3+b^3}{ab+4}=2020$.
Find the all positive integer solutions of given equation
$$\frac{a^3+b^3}{ab+4}=2020.$$
I find two possible solutions, namely $(1011,1009)$ and $(1009,1011)$, but the way I solve the equation was messy and I don't know if there are an... | Write for ease $n=2020$ and let $c=a+b$. As $b=c-a$ we get a following quadrtatic equation on $a$: $$(3c+n)a^2-(3c+nc)a+c^3-4n=0$$
So it discriminat must be a perfect square $d^2$ (as it has solution in $\mathbb{Z}$):
$$d^2 = -3c^4+2nc^3+n^2c^2+48nc+16n^2\;\;\;\;\;(*)$$ from here we get $$\boxed{2n\mid d^2+3c^4}$$
Now... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3669889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof that $\epsilon_{ijk}$ is the same as the determinant of the Kronecker delta I just learned about the Kronecker Delta function and $\epsilon_{ijk}$ for the first time, and I still can't wrap my mind around how to prove that $$\epsilon_{ijk} = det\begin{pmatrix} \delta_{i1} & \delta_{i2} & \delta_{i3} \\ \delta_{j1... | A property of the determinant is:
*
*Exchanging two rows while leaving everything else unchanged changes the sign of the determinant, but not the magnitude.
This can be proven from the formula $\det(AB) = (\det A)(\det B)$. The operation of exchanging rows in a matrix $M$ is the same as taking $EM$, where $E$ is t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3670026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that this inequality is true Show that $\frac{2}{3} \cdot \frac{5}{6} \cdot \frac{8}{9} \cdot ... \cdot \frac{999998}{999999} > \frac{1}{100}$.
I tried to take another multiplication $\frac{3}{5} \cdot \frac{6}{8} \cdot \frac{9}{11} \cdot ... \cdot \frac{999996}{999998}$ so that we would have their multiplications... | Let $$A_n=\sqrt[3]{3n+1}\cdot\prod_{k=1}^n\left(1-\frac1{3k}\right).$$
The claim is that
$$\tag1 A_{333333}>1.$$
We compute
$$\begin{align}\left(\frac{A_{n}}{A_{n-1}}\right)^3&=\frac{(3n+1)(3n-1)^3}{(3n-2)27n^3}\\
&=1+\frac{6n-1}{(3n-2)27n^3}\\&>1+\frac{6n-4}{(3n-2)27n^3}\\&=1+\frac2{27n^3}>1\end{align}$$
or $A_n>A_{n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3670193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove $x^3+y^3+z^3\geq 3xyz$ Prove that, $\forall x,y,z\in \mathbb{R}_+\cup \{0\}$,
$$x^3+y^3+z^3\geq 3xyz\text{.}$$
I've been trying for a while now whithout making any significant progress.
| hint
$x\mapsto \ln(x) \; $ is concave at $ (0,+\infty) \;$ since its second derivative $\; x \mapsto \frac{-1}{x^2}\; $ is negative.
$$\frac 13 + \frac 13 + \frac 13 = 1$$
thus
$$\ln\Bigl(\frac 13(x^3+y^3+z^3)\Bigr)\ge$$
$$ \frac 13\Bigl(\ln(x^3)+\ln(y^3)+\ln(z^3)\Bigr)$$
$$(=\ln(xyz))$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3670362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Use mathematical induction to prove n ≥ 3
I have proved for n =3, and assumed S(k) is true already.
I have gotten all the way to the induction step of S(k+1) = 3+4+5+...+(k+1) = ((k+1-2)(k+1+3))/2
I am having trouble proving it past this step, and how to show that what I have added is equal on both sides. Please show... | Assume the formula holds for some $k\ge3$, that is:
$$S(k)= \frac{(k-2)(k+3)}{2}.$$
Now, $S(k+1) = S(k)+k+1$, so using the above formula we get:
$$S(k+1) = \frac{(k-2)(k+3)}{2} +k+1 = \frac{(k-2)(k+3)+2k+2}{2} = \frac{k^{2}+3k-4}{2} = \frac{(k-1)(k+4)}{2} = \frac{((k+1)-2)((k+1)+3)}{2}$$
which is your formula for $n=k+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding $\lim\limits_{n→∞}n\cos x\cos(\cos x)\cdots\underbrace{\cos(\cos(\cdots(\cos x)))}_{n\text{ times of }\cos}$
Find$$\lim_{n→∞}n\cos x\cos(\cos x)\cdots\underbrace{\cos(\cos(\cdots(\cos x)))}_{n \text{ times of } \cos}.$$
I approximated cos(cosx) to cos x, but i don't think it is the proper approach.
I got ans... | Consider the sequence $x_n$ defined by $x_0 = x$ and $x_{n+1} = \cos(x_n)$. Then the sequence in question is
$$a_n = n\prod_{n=1}^\infty x_n.$$
I claim that $a_n \to 0$. Here's a sketch of the proof:
*
*There is a unique point $x^* \in [0, 1)$ such that $\cos(x^*) = x^*$ (the fixed point of $\cos$).
*The sequence o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3670653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is $i$ a positive number? This question came to mind when I saw a way of finding the value of $i^i$ which included transforming it to $e^{i\ln(i)}$ and taking $\ln(i)$ as $i\frac{\pi}{2}$
I understand how we can get $\ln(i)$=$i\frac{\pi}{2}$ geometrically but this got me thinking if $i$ is a positive number as the firs... | The sign can be defined as:
$$
sgn(x)=\begin{cases}
\frac{x}{|x|}, & \text{if $x \neq 0$} \\
0, & \text{if $x = 0$}
\end{cases}
$$
If you extend this to complex numbers, the sign can be any complex unit or zero. In that way $sgn(i)=i\neq0$ so it's not positive.
We usually don't define positive and negative numbers in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3670747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 5
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Number of quarters, dimes and nickles. I'm solving a problem which states the following;
Mary has $3.00 in nickels, dimes, and quarters. If she has twice as
many dimes as quarters and five more nickels than dimes, how many
coins of each type does she have?
I transcribed the relationships as such;
$2q=d, n=d+5$
An... | One nickel is $5$ cents.
The equation is
$$3=0.0\color{red}5n+0.1d+0.25q$$
$
3 = 0.05n+0.1d+0.25q \therefore\\
3=0.05(d+5)+0.1(2q)+0.25q \therefore\\
3=0.05d+0.45q+0.25 \therefore\\
3=0.05(2q)+0.45q+0.25\therefore\\
3=0.55q+0.25\therefore\\
2.75 = 0.55q \therefore\\
5= q
$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3670948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is it possible to find an expression for $\frac{d^n}{dx^n}e^{-x^2}$? I am trying to find a general form to derivatives of the function $e^{-x^2}$. I tried to do it by finding a pattern for the first derivatives but with no success.
Any tip is welcome.
| These are called Hermite polynomials, in one form or another. Let me show you show to "discover them".
Let $p_n$ be your $n$th derivative and $f=p_0$. To simplify matters I will instead consider $f= e^{x^2/2}$ (you can roll back by replacing $x$ by $\sqrt{2}xi$). Then $p_1 = x p_0$. Assume by induction that $p_n$ is of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3671084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding the derivatives using limits. What to do when you have a limit inside a limt I was trying to prove the second derivative formula using limits
\begin{align}f'(x) &= \lim_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h} \\f''(x) &= \lim_{h\rightarrow 0}\dfrac{f'(x+h)-f'(x)}{h} \\ &= \lim_{h\rightarrow 0}\dfrac{\lim_{h\righ... | What you have done is almost fine but you should use different symbols for different limit operators like $$f''(a) =\lim_{h\to 0}\dfrac{\lim_{k\to 0}\dfrac{f(a+h+k)-f(a+h)}{k}-\lim_{l\to 0}\dfrac{f(a+l)-f(a)}{l}}{h}$$ However if you are trying to achieve a definition of $f''$ solely in terms of $f$ as a sort of complic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3671220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A property about uniform convergence Suppose a function series $\sum_{n=1}^{\infty}u_n(x)$ converges on $(a,b)$ and every $u_n(x)$ is continuous on the interval $(a,b)$,but this series diverges at $x=a$ or $x=b$, can we deduce that $\sum_{n=1}^{\infty}u_n(x)$ does not converge uniformly on $(a,b)$?
For example, $\sum... | Consider, $u_n(x) = 0$ in $(a, b)$, $u_n(x) = 1, x=a$ or $x = b$. You get a uniform convergence in $(a, b)$ but divergence at the endpoints.
Continuity Edit:
Let $s_n$ be the partial sum of $u_i$'s up to term n.
Suppose, that the sequence does not converge at $a$, that is,
$$\exists\epsilon > 0, \forall n_0 \exists n,... | {
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"url": "https://math.stackexchange.com/questions/3671376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Motivation for considering the upper numbering of ramification groups Let $L/K$ be a finite Galois extension. We denote by $G_s$ its $s$-th ramification group. Define the Herbrand function $$\eta_{L/K}:[-1,\infty) \to [-1, \infty), \ \eta_{L/K}(s) = \int_0^s \frac{1}{(G_0:G_x)} \ dx.$$ Let $\psi_{L/K} : [-1, \infty) \t... | The short answer is that lower ramification groups behave well when taking subgroups, while upper ramification groups behave well when taking quotients. As a result, the upper numbering can be defined for infinite extensions. I think this is the key motivation for defining them.
Indeed, if $L/K$ is an infinite extensio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing that a quadratic cannot have 3 roots using Determinants Consider the equation $px^2 + qx + r = 0$ and let us assume that a, b and c satisfies the above equation.
So,
$$a^2p + aq + r = 0$$
$$b^2p + bq + r = 0 $$
$$c^2p + cq + r = 0$$
They can be represented using a matrix.
\begin{bmatrix}a^2&a&1\\b^2&b&1\\c^2&c... | $$px^2+qx+r = 0$$
Imagine that $x$ has $3$ roots $a,b,c$
Representing in matrix
$$\begin{bmatrix}a^2&a&1\\b^2&b&1\\c^2&c&1\end{bmatrix}\begin{bmatrix}p\\q\\r\\\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$$
Since the matrix product is $$\begin{bmatrix}0\\0\\0\end{bmatrix}$$
Then it's determinant must be zero
$$\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3671671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $f$ be (Riemann) integrable over $[0,1]$. Show $\sum_{n=0}^{\infty}\int_{0}^{x^n}f(t)dt$ is continuous on $(0,1)$ Problem: Let $f$ be (Riemann) integrable over $[0,1]$. Show $\sum_{n=0}^{\infty}\int_{0}^{x^n}f(t)dt$ is continuous on $(0,1)$.
I know that if $f$ is (Riemann) integrable over $[0,1]$, since $x \in (0,1... | The function $$F(x):=\int_0^x f(t)dt$$ is Lipschitz, as one easily sees, since $|F(x)|\le \int_0^x|f|\le x\underset{[0,1]}{\text{sup}}(|f|)$. Let $K$ be its Lipschitz constant. Then
$$\sum_{n=0}^\infty\left|\int_0^{x^n}f(t)dt\right|=\sum_{n=0}^{\infty}|F(x^n)|\le K\sum_{n=0}^\infty x^n=K\frac{1}{1-x}$$
Since the series... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3671824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this an Expectation Problem? Or is it more complex? Following question is off a probability/statistics review sheet:
Every day a professor leaves their home in the morning and walks to their
office. Every evening they walk home. They take their umbrella with them only if it is raining. If it is raining and they do n... | Follow the hint. In the starting case, with probability $1/3$ it rains, the professor takes the umbrella, and with probability $2/3$, it does not rain when it is time for the professor to return home. So with probability $2/9$ the professor has not walked in the rain but the umbrella is at the office.
Similarly, with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3671937",
"timestamp": "2023-03-29T00:00:00",
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Find the value of $a \in R$ such that $\langle x_n \rangle$ converges to a positive real number when $x_n=\frac{1}{3}\frac{4}{6}...\frac{3n-2}{3n}n^a$ Find the value of $a \in \mathbb{R}$ such that $\langle x_n \rangle$ converges to a positive real number when $x_n=\dfrac{1}{3}\dfrac{4}{6}\cdots\dfrac{3n-2}{3n}n^a$
Her... | Expand $$\log(3n-2) - \log(3n) = \log(1 - \frac23 \frac1n) = - \frac23 \frac1n + \frac49 \epsilon_n \frac1{n^2}$$
where $\epsilon_n$ is a bounded sequence.
Then :
$$\log x_n = -\frac23 \sum_{k = 1}^n \frac1k + a \log n + \frac49 \sum_{k = 1}^n \epsilon_k \frac1{k^2}$$
Le last sum converges : $$\big|\epsilon_k \frac1{k^... | {
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"url": "https://math.stackexchange.com/questions/3672074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closedness of the product of two subgroups Let $G:=\text{SL}(3,\mathbb R)$, equipped with the usual subspace topology, acting on $\mathbb R^3$ by the canonical action. Consider subgroups $\Gamma:=\text{SL}(3,\mathbb Z)$ and $Q_1:=\{g\in G:ge_1=e_1\}$ where $e_1=(1,0,0)$. I wonder how to prove that the product $\Gamma Q... | Let $\pi$ be the projection $G\to G/Q_1$.
Then $\Gamma Q_1$ is closed in $G$ iff the orbit $\Gamma\pi(1)$ is closed in $G/Q_1$. Now observe that the map $g\mapsto ge_1$ induces an identification of $G/Q_1$ with $\mathbf{R}^3\smallsetminus\{0\}$, and in this identification, $\pi(1)=e_1$. Now we see that the orbit of $e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The subgroups of a cyclic group We've got $G=U(\mathbb Z/(27)\mathbb Z)=\langle 2 \rangle$ a cyclic group, and $H=\langle -8, -1 \rangle$ a subgroup of $G$. I've calculated all the subgroups of $G$.
Now I have to indentify $H$ with a subgroup of $G$, without calculating all the elements of $H$.
So I think that I can s... | We have $|G|=27-9=18$, hence $2^9 = -1$.
Now $H=\langle -8,-1 \rangle = \langle -2^3, 2^9\rangle = \langle 2^9\cdot 2^3, 2^9\rangle = \langle 2^{12},2^9\rangle$.
Now thanks to the Bezout Lemma we have $H=\langle 2^{12},2^9\rangle = \langle 2^3\rangle = \langle 8\rangle$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3672569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the value of $\sum _{n=1}^{\infty }\:\frac{a}{n\left(n+a\right)}$ find the value of $\sum _{n=1}^{\infty }\:\frac{a}{n\left(n+a\right)}$ $(a>0)$
I just can analyse $\sum _{n=1}^{\infty }\:\frac{a}{n\left(n+a\right)}=a\left(\frac{1}{1}-\frac{1}{1+a}+\frac{1}{2}-\frac{1}{2+a}+\frac{1}{3}-\frac{1}{3+a}...+\frac{1}{n... | It is $\psi (a + 1) + \gamma$, where $\psi$ is the logarithmic derivative of the gamma function and $\gamma$ is the Euler-Mascheroni constant, cf. http://dlmf.nist.gov/5.7.E6 and http://dlmf.nist.gov/5.5.E2 Using this fact, it follows for example that
$$
\log a + \gamma + \frac{1}{{2a}} - \frac{1}{{12a^2 }} < \sum\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3672902",
"timestamp": "2023-03-29T00:00:00",
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Violation of Faithfulness in Structural Causal Models While reading "Elements of Causal Inference" by Peters et al. I stumbled over something I'm not sure about regarding "Example 6.34 (Violations of Faithfulness)". A link to the book can be found here.
There, a linear Gaussian SCM is defined as
$X:=N_X$, $Y:=aN_X+N_Y$... | Your mistake is in the definition of $Y'$. Actually, $Y'= \tilde{a}X' + \tilde{b}Z' + N_{Y'}$, with $N_{Y'} \sim \mathcal{N}(0, \sigma_{Y'}^2)$.
In order to determine $\tilde{a}, \tilde{b}$ and $\sigma_{Y'}^2$, notice that:
*
*the joint distribution of $X',Y',Z'$ is Gaussian (since the independent $X'$ and $Z'$ are n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3673017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does $\lim_{N\rightarrow \infty} \sum_{n = 1}^{N} \frac{1}{(N+1) \ln (N+1) - n \ln n } = 1$? Question:
Find the limit
\begin{equation}
A = \lim_{N\rightarrow \infty} \sum_{n = 1}^{N} \frac{1}{(N+1) \ln (N+1) - n \ln n }
\end{equation}
The series originated from the asymptotic analysis in this question. I can show that ... | You are right: the limit is $1$.
Here it is the full proof:
For all $n \in \{1, \dots N \}$ we have the following inequality:
$$(N+1) \log (N+1)- n \log n \ge (N+1) \log (N+1)- n \log (N+1) =\\ = \log(N+1) (N+1-n)$$
Thus
$$\sum_{n=1}^N \frac{1}{(N+1)\log(N+1)-n \log n} \le \sum_{n=1}^N \frac{1}{\log(N+1) (N+1-n)} = \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3673227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve $x^{\prime\prime\prime} + 2x^{\prime\prime} - x = e^{-t}\cos(2t)$ using the operator method? I did
$(D^3 + 2D - 1)x = e^{-t}\cos(2t) \Rightarrow x = \frac{1}{D^3 + 2D - 1}e^{-t}\cos(2t) \Rightarrow x= \frac{e^{-t}}{(D-1)^3 + 2(D-1) - 1}\cos(2t) \Rightarrow$
$x =\frac{e^{-t}}{D^3-3D^2+3D-1+2D-2-1}\cos(2t) ... | $$(D^3 + 2D^2 - 1)x = e^{-t}\cos(2t)
$$
$$ \implies x_p = \frac{1}{D^3 + 2D^2 - 1}e^{-t}\cos(2t) $$
You forget the power D for the operator. You have $D^3$ and $D^2$.
$$\implies x_p =e^{-t} \frac{1}{(D-1)^2(D+1) - 1}\cos(2t)$$
$$\implies x_p =e^{-t} \frac{1}{(-2D-3)(D+1) - 1}\cos(2t)$$
$$ x_p =e^{-t} \frac{1}{(-5D+4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3673382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Completely metrizable subspaces and $G_\delta$ in Hausdorff spaces
Theorem: Let $Y$ be a dense subset of a Hausdorff topological space $X$. If $Y$ is completely metrizable, then $Y$ is a $G_\delta$ set in $X$.
This is explained in detail here. The idea is to take the set of points $a\in X$ for which one can find nei... | A very natural example where $Y$ is not open in $X$ is $X=\mathbb{R}$, $Y=\mathbb{R}\setminus\mathbb{Q}$. Here is it not obvious that $Y$ actually is completely metrizable; one way to prove it is to use continued fractions to show that $Y$ is actually homeomorphic to $\mathbb{N}^{\mathbb{N}}$.
For a very simple exampl... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove there is such a triple There are $3n$ colored numbers from $1$ to $3n$ such that each color is represented exactly by $n$ numbers. Prove there is 3-colored triple $a,b,c$ such as $$a+b=c$$
I started by assuming various values for minimum each of color, but didnt come to any conclusions(
| Note that this problem was initially asked and solved by Alekseev and Savchev, in the Kvant journal, 4:23, problem M1040.
Let $A,B,C$ be the three monochromatic subsets of $[3n]$. Without loss of generality let $1,\ldots,k-1$ be in $A$ (i.e. the first $k-1$ integers are in $A$, with $k-1\geq1$), and let $k\in B$.
We ca... | {
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Theorem 29.1 in Munkres's Topology Could someone please explain the highlighted sentence in this proof? I understand that $C$ is contained in $X$, but I don't understand why that implies it is a compact subspace of $X$ given that it is a compact subspace of $Y$. Thanks in advance.
| If $X$ is a subspace of $Y$, the open sets of $X$ are of the form $X\cap U$ where $U$ is an open set in $Y$. Therefore any open covering of $C$ as a subset of $X$ is of the form $\{X\cap U_i\}_{i \in I}$. But then $\{U_i\}_{i \in I}$ is an open covering of $C$ in $Y$, so there is a finite sub-covering.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Lebesgue integral on interval [a,b] Suppose $f$ and $g$ are non negative real functions defined on interval $[a,b]$. $f$,$g$ $\in L^1([a,b])$
If there is a sequence of decreasing measureable sets $...\subset A_2\subset A_{1}$ ,such that $A_n \subset [a,b]$ for all n.
On which $\int_{A_n}f d\lambda = \int_{A_n}g d\lam... | the measure $$\lim_{n \to \infty}\mu(A_n \bigcap A^c)=0$$
$$\lim_{n \to \infty}\int_A (f-g)d\lambda= \lim_{n \to \infty}(\int_{A_n} (f-g)d\lambda - \int_{A_n\bigcap A^c} (f-g)d\lambda)=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3674034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Polynomial $x^3-2x^2-3x-4=0$ Let $\alpha,\beta,\gamma$ be three distinct roots of the polynomial $x^3-2x^2-3x-4=0$. Then find $$\frac{\alpha^6-\beta^6}{\alpha-\beta}+\frac{\beta^6-\gamma^6}{\beta-\gamma}+\frac{\gamma^6-\alpha^6}{\gamma-\alpha}.$$
I tried to solve with Vieta's theorem. We have
$$\begin{align}
\alpha+\b... | The final answer I got is 608.
The answer for this question is provided in the given pdf
By Vieta's theorem and using some special algebraic identities that are prevelant in India..
Click on the link .THE whole values are achieved by some easy calculations
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Follow from $d(X_n, Y_n) \rightarrow 0$ in probability that $\log(n)f_n(X_n) = 0 \implies \log(n)f_n(Y_n) = 0$ Lets consider two sequences of random variables $X_n, Y_n$ with values in a metric space $(M, d)$ and $d(X_n, Y_n) \stackrel{\mathbb{P}}{\longrightarrow} 0$. Now suppose there is a sequence $f_n$ of continuous... | It is false.
Consider $f_n(x) = nx$, and suppose $X_n$ and $Y_n$ are constant functions $X_n \equiv 0$, $Y_n \equiv 1/n$. You have that $d(X_n,Y_n) \to 0$, and $\ln(n)f_n(X_n) = 0$, but $\ln(n)f_n(Y_n) = \ln(n)$ that does not converge to zero.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Presentation of free group I want to prove that
$$<a,b \ | \ aba^{-1}b^{-1},ab^{-1}ab>\cong \mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$$
I already show $a^2=1$, but I don't make sure that $a\neq1$. How can I prove this?
Any help would appreciated. Thanks.
| If $a=1$, then your presentation collapses to the simple presentation $\langle b\ |\ \ \rangle\cong\mathbb Z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3674722",
"timestamp": "2023-03-29T00:00:00",
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Show that $\lim_{x\to 0^+}\int_x^{2x}\frac{\sin t}{t^2}dt=\ln 2.$
Question: Show that $$\lim_{x\to 0^+}\int_x^{2x}\frac{\sin t}{t^2}dt=\ln 2.$$
Solution: Note that $$\lim_{x\to 0^+}\int_x^{2x}\frac{\sin t}{t^2}dt=\ln 2=\lim_{x\to 0^+}\int_x^{2x}\frac{dt}{t}\\\iff\lim_{x\to 0^+}\int_x^{2x}\frac{\sin t-t}{t^2}dt=0.$$
... | Let $ x\in\left(0,1\right] : $
\begin{aligned}\int_{x}^{2x}{\frac{\sin{t}}{t^{2}}\,\mathrm{d}t}&=\ln{2}+\int_{x}^{2x}{\frac{t-\sin{t}}{t^{2}}\,\mathrm{d}t}\\ &=\ln{2}+\int_{0}^{2x}{\frac{t-\sin{t}}{t^{2}}\,\mathrm{d}t}-\int_{0}^{x}{\frac{t-\sin{t}}{t^{2}}\,\mathrm{d}t}\\ &=\ln{2}+\frac{1}{2}\int_{0}^{x}{\frac{2u-\sin{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3674827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 4
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Stability of higher-order fixed points for systems of ordinary differential equations In the book by Strogatz, Nonlinear Dynamics and Chaos (1994), the author discusses examples of higher-order fixed points for systems of ordinary differential equations in polar coordinates:
*
*$\dot{r}=ar^3, \dot{\theta}=1$, $a\ne0... | You can solve these examples explicitly. In the first system, the equations for $\dot{r}$ and $\dot{\theta}$ are decoupled, so you can solve the equations individually. In the second equation, you can solve $\dot{r} = -r$ first, and then plug the result into the second equation to get an explicit function for $\dot{\th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Comparing a ratio of gamma functions to a simple polynomial I am still struggling to build my intuition as far as reasoning with ratios of gamma functions.
Reasoning with factorials is significantly clearer.
Consider this example. I would appreciate if anyone could help me to understand how to complete the following w... | Suppose that we consider the function
$$f(n)=\log \left(\Gamma \left(2n+3-\frac{a}{\log (n)}\right)\right)-\log
(\Gamma (2 n+1))-2 \log (n+1)$$
Using Stirling approxiamtion followed by Taylor series, we have
$$f(n)=-\left(\frac{a \log (2)}{\log (n)}+a-2\log (2)\right)+\frac{a^2-5 a \log (n)-2 \log
^2(n)}{4 n \log... | {
"language": "en",
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"source": "stackexchange",
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Integral of $\frac{|x|}{x}$ Description
In Example 8.6 on my textbook(http://www.math.louisville.edu/~lee/RealAnalysis/), it goes like
Let
$$
f(x) =
\begin{cases}
\frac{|x|}{x}, & x \neq 0\\
0, & x = 0
\end{cases}
$$
And the author just shows that $F(x) = \int^{x}_{-1} f(x) dx = |x| - 1$ without any derivation.
Questio... | If $x\le 0$, then
$$\int_{-1}^xf=\int_{-1}^x(-1)dt=$$
$$\Bigl[-t\Bigr]_{-1}^x=-x-1=-1+|x|$$
If $x\ge 0$, then
$$\int_{-1}^xf=\int_{-1}^0(-1)dt+\int_0^x(1)dt$$
$$=\Bigl[-t\Bigr]_{-1}^0+\Bigl[t\Bigr]_0^x=-1+x=-1+|x|$$
in all cases, it gives $$|x|-1$$
| {
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Finding the dimension of a subspace in $\mathbb R^n$ Given $S_{u}^\bot:=\{v\in \mathbb R^n:v \cdot u=0\}$, I've proved that $S_{u}^\bot$ is a subspace of $\mathbb R^n$. The next part of the question asks me to prove that $S_{u}^\bot$ has dimension of $n-1$.
I assume that it has something to do with the number of elemen... | For any vector $s \in V$, $s - (s \cdot u)u \in S_u^\perp$, by definition of $S_u^\perp$. In particular, $s = (s\cdot u)u + v$ for some $v \in S_u^\perp$.
This shows that every vector can be written as a linear combination of a vector in the span of $u$, and in $S_u^\perp$. In other words, $S_u^\perp + \overline{\{u\}}... | {
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Prove that $\frac{1}{\sqrt{1-x^2}}$ is unbounded for $x\in(-1,1)$ I'm trying to prove that $f(x):=\frac{1}{\sqrt{1-x^2}}$ is unbounded for $x\in(-1,1)$; so I must prove that for all $M\in\mathbb{R}$ is $f(x) > M$ for $x\in(-1,1)$.
With the definition of limit, since
$$\lim_{x \to 1^-} f(x)=\lim_{x \to -1^+} f(x)=\infty... | $\frac{1}{\sqrt{1-x^2}}$ is an even function, it suffices to consider $x \in [0,1).$
Assume $(1-x^2)^{1/2} <M (>0)$, real, for all $x \in [0,1)$.
Then $1/M^2 < 1-x^2$;
Set $y_n:=x_n^2=1-1/n$;
$1/M^2 < 1/n$, for large enough $n$ a contradiction.
| {
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"timestamp": "2023-03-29T00:00:00",
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Tail random variable and extremum Let $(X_n)_n$ be a sequence of real random variables. Is it true that $\limsup_n\frac{1}{n}\max_{1 \leq k\leq n}X_k$ is tail random variable? $\liminf_n \frac{1}{n}\min_{1 \leq k \leq n}X_k$ ? $\limsup_n\frac{1}{n}\min_{1\leq k \leq n}X_k$?
For the first two, I think the answer is corr... | In general: $$\left|\min\left(a,b\right)-\min\left(a',b\right)\right|\leq\left|a-a'\right|$$
So if: $$Z_{n}:=\frac{1}{n}\min_{2\leq k\leq n}X_{k}$$ and: $$Y_{n}:=\frac{1}{n}\min_{1\leq k\leq n}X_{k}=\min\left(\frac{1}{n}X_{1},Z_{n}\right)$$
then: $$\left|Y_{n}-\min\left(0,Z_{n}\right)\right|\leq\frac{1}{n}\left|X_{1}\r... | {
"language": "en",
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The argument of a finite Euler product at a non-trivial zero of $\zeta(s)$. With $p_n$ is the $n$-th prime number, we know that:
$$\arg\left(\zeta(s)\right)=\arg\prod_{n=1}^{\infty} \frac{1}{1-\frac{1}{p_n^s}}=\sum_{n=1}^{\infty} \arg\left(\frac{1}{1-\frac{1}{p_n^s}}\right)\qquad \Re(s) > 1$$
Now define the finite seri... | The Euler product diverges for $\Re(s) < 1$, you need to replace it by the regularized version (valid for $\Re(s)\in (1/2,1)$ assuming the RH is true)
$$\lim_{x\to \infty} -\sum_{p \le x} \log(1-p^{-s}) - pv\int_0^x \frac{t^{-s}}{\log t}dt + \frac{\log(s-1)}{s}$$
If there are infinitely many zeros of real part $\ge \Re... | {
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"timestamp": "2023-03-29T00:00:00",
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What is the domain of $f^2$ if $f(x)=\sqrt{x+2}$ Now, $f(x)$ is defined for all values of $x$ for which $x+2 \geq 0$
$x+2 \geq 0 \implies x \geq -2$
So, $\mathrm{Domain}(f)=[-2,\infty)$ which means $f : [-2,\infty) \longrightarrow \Bbb R$
$f^2(x) = \Big (f(x) \Big )^2=(\sqrt{x+2})^2=x+2$
So, $f^2$ is defined for all v... | You have written $f^2(-5)=(f(-5))^2$. It is correct, however, look at the inside function of the RHS. It is $f(-5)$. Can you define $f(-5)$? No. That means $f^2(-5)$ is undefined. Similarly $f^2$ is undefined for any $x<-2$. So, your book is correct.
| {
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How improper Riemann and Legesgue integral associated? I'm looking for answer to Relation between the Lebesgue integral the improper Riemann integral
.
$$
(L) \int_{a}^{b}f(x)dx \: \: \: \: (R)\lim_{\alpha \to a+}\int_{\alpha}^{b}f(x)dx
$$
I found following theorem:
Let $f$ be a nonnegative continuous function. If $f$ ... | Consider the sequence $g_n = \chi_{[\frac {1}{n},1]}.f$ and apply the monotone convergence theorem.I,of course,assumed the domain to be $[0,1]$ but you get what I mean.
Edit :- You can observe why improper Riemann and Lebesgue integrals might not always be the same.The improper Riemann integral,if you note,is actually ... | {
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Proof that a subset of the plane is an open set. I don't have a specific book that I am learning from because I am trying to learn on my own but my used definition of an open set $A$ is that for every $a\in A$ there exists an $\varepsilon>0$ such that the open ball $B(a,\varepsilon)\subset A$
Let $\boldsymbol{A}=\{(x,y... | You're definitely on the right track. For each of the four inequalities you need to show, you need only to expand the inequality you have so that it ignores the irrelevant stuff. For example, to show $0<x$,
$$a-x<\sqrt{(a-x)^2+(b-y)^2}<\varepsilon<a$$
| {
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Combinatorics task.
Four men and four women shall get in line in a supermarket. In how many ways can they line up, if the line has to alternate between men and women (two men and women can not stand next to each other)?
My solution: $4 \cdot 4 \cdot 3 \cdot 3 \cdot 2 \cdot 2 \cdot 1 \cdot 1 = 576$; Then we multiplica... | You may as well separate them into two lines and call them up one at a time, alternating between the lines. You can arrange each line in $4!$ ways. You can alternate between the lines starting from either the male or female line, so that it $2$ choices. Meaning there are $(4!)^2 \cdot 2= 1152$ total possibilities, agre... | {
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Finding an integration factor $x^ay^b$ to solve an ODE
I have to solve an ODE:$$2(y-3x)dx+x\left(3-\frac{4x}{y} \right) dy=0$$
I am given that I have to use the integrating factor x^a(y^b) where a and b are real numbers in order to turn the problem into a solvable exact ODE. The problem is that I am unsure of how to ... | Multiply by $y$ the DE:
$$2y^2dx-6xydx+3xydy-4x^2dy=0$$
The integrating factor is $\mu (x,y)=xy$
$$y^3dx^2-6(xy)^2dx+3(xy)^2dy-2x^3dy^2=0$$
$$y^3dx^2-2y^2dx^3+x^2dy^3-2x^3dy^2=0$$
Rearrange some terms:
$$(y^3dx^2+x^2dy^3)-2(y^2dx^3+x^3dy^2)=0$$
$$dx^2y^3-2dx^3y^2=0$$
Integration gives us:
$$x^2y^3-2x^3y^2=K$$
To summa... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that if $b_n$ is a subsequence of $a_n$ and $c_n$ is a subsequence of $b_n$, then $c_n$ is a subsequence of $a_n$. Let $(a_{n})_{n=0}^{\infty}$, $(b_{n})_{n=0}^{\infty}$ and $(c_{n})_{n=m}^{\infty}$ be sequences of real numbers. Then $(a_{n})_{n=0}^{\infty}$ is a subsequence of $(a_{n})_{n=0}^{\infty}$. Furthermo... | No steps missing, but there is an error. If $b_n = a_{f(n)}$, then
$$ c_n = b_{g(n)} = a_{f(g(n))} \text{.} $$
You have the composition reversed.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that there are infinitely many odd number can't be written as $pq-p-q$ Let $p$ and $q$ are prime.
Problem
Are there infinitely many odd positive integer $a$, which can't be written as $pq-p-q$ ?
Example $13$ can't be expressed in $pq-p-q$.
Sequence $13,25,33,37,49,53,61,67,73,75,85,93,97,109,...$
It looks there... | Just choose $a=12k+1$ for any $k \in \mathbb{N}$. If $a=pq-p-q \implies a+1 = (p-1)(q-1)$. Clearly, we have $4 \nmid (a+1)$. Since $4 \nmid (p-1)(q-1)$ , WLOG we have $q=2$. This gives:
$$a=p-2 \implies 12k+3=p$$
which is clearly a contradiction as LHS is divisible by $3$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Is there a differentiable function whose derivative is greater than or equal to 1 everywhere but 0 at 0? Does there exist a differentiable function $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that $f^{'}(0)=0$ and $f^{'}(x) \ge 1$ $\forall x \ne 0?$ Prove either way.
I was thinking of coming up with a counterexampl... | Darboux Theroem says that the derivative of any differentiable function has IVP. So no such function exists. Ref: https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proofing That $\int_0^a(f(x) + f^{-1}(x))dx \geq a^2$ for $a > 0$ Hello everyone $f(x)$ is a increasing function that for $f(0) = 0$ and $f^{-1}(x)$ is the opposite function of $f(x)$
How can I proof that $\int_0^a(f(x) + f^{-1}(x))dx \geq a^2$ for $a > 0$
Thanks!
| $\int_{0}^{a} f(x) + f^{-1}(x) dx $
$=\int_{0}^{a} y + f^{-1}(x) dx$
Let, $f^{-1}(a)=a'$
So, $\int_{0}^{a} f^{-1}(x) dx = aa' - \int_{0}^{a'} y dx$
And replacing this to its original expression we get,
$=\int_{0}^{a} y + f^{-1}(x) dx = aa' +\int_{a'}^{a} y dx$
Now, $\int_{a'}^{a} y dx \geq (a-a')f(a')=(a-a')a=a^2-a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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can a complex math equation can create multiple closed area? example : equation of circle can easily represent case A.
can some single complex mathematical equation can create case B ?
| Yes, two curves can be represented with a single equation joining them as their product.Right hand sides of the individual equations should be zero before multiplying.
Multiple curves can be drawn on same x-y plane. We can visualize them together or separately. Draw graphs of two or three curves separately on transpa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3677743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is this really a multiplicative group? [subgroups of $\Bbb Z/15$] When looking at some small multiplicative groups of integers modulo n, such as for 15, I found something that confused me. Due to a miscalculation, I had what looks like another set of integers less than 15 that are closed under multiplication. My questi... | Not sure your second Cayley graph works. $3^2=9\neq1$, so it's not true that <3> acts as a copy of $\mathbb{Z}_2$ mod 15.
Additionally, since I'm trying to prove a subgroup, simply look for $3^{-1}$ which doesn't exist in $\mathbb{Z}_{15}$ (as a ring), and therefore cannot be embedded in any subgroup of the multiplicat... | {
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Having two non-parallel hyperplanes of $ \mathbb{R}^n $, $ S_1 $ and $ S_2 $, prove that $ S_1\cap S_2 \neq\emptyset $ and $\dim(S_1\cap S_2)=n-2$ I've got two affine hyperplanes of $ \mathbb{R}^n $, $ S_1 $ and $ S_2 $, which are non-parallels.
I have to prove that $ S_1\cap S_2 \neq\emptyset $ and $\dim(S_1\cap S_2)=... | What does the non-parallel hypothesis mean ? Try to explore this fact to show the non emptyness of the intersection.
What can you say about the intersection of two linear hyperplanes ? Can you adapt this (maybe by translating the origin ?) to prove what is the dimension of the intersection ?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is this another definition for inverse relation? We know that if $R\colon A\to B$, we can define the inverse relation as follows: $$R^{-1}=\{(y,x)\in B\times A\mid(x,y)\in R\}.$$ Now I want to know if this set, let's call it $R\:'^{-1}$, is the same as $R^{-1}$: $$R\:'^{-1}=\{(x,y)\in R\mid(y,x)\in B\times A\}.$$ I thi... | We have $(y,x)\in B\times A$ whenever $(x,y)\in R\subseteq A\times B$. Hence your $R'^{-1}$ is just $R$ itself
| {
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Solving $\int_0^1 [x^{700}(1-x)^{300} - x^{300}(1-x)^{700}] \, dx$ I am trying to solve the following integral:
$$\int_0^1 [x^{700}(1-x)^{300} - x^{300}(1-x)^{700}] \, dx$$
My intuition is that this integral is equal to zero but I am unsure as to which direction to take to prove this. I was thinking binomal expansion b... | Let $1/2 - x \to x^\prime$ to see that the integral is $0$.
Convert the integrand to
$$(x^\prime /2)^{700} (x^\prime /2)^{300} - (x^\prime /2)^{300}(x^\prime /2)^{700}$$
and put in the right limits...
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Interpolation constraint in Hilbert space The paper "Geodesic Interpolating Splines" made the following comment on an interpolation problem:
Interpolation problem:
Let $\mathcal{H}$ be a Hilbert space, let $f_1, \dots, f_N \in \mathcal{H}$, and $c_1,\dots,c_N \in \mathbb{R}$ be given.
Find $h \in \mathcal{H}$ such tha... | If you can find minimizer $h$ outside of the $\text{span}\{f_1,\cdots,f_N\}$, then since $\langle f_i,h\rangle=0=c_i$, we have $c_1=\cdots=c_N=0$, which may be contradiction (if there exists $c_i\neq0$ given) or such $h$ does not exists (because if $\text{span}\{f_1,\cdots,f_N\}^\perp\neq\emptyset$, then take $a\in\tex... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the sequence $x_n=\dfrac{1}{\sqrt{n}}\left(1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\ldots+\dfrac{1}{\sqrt{n}}\right)$ monotone? Observe that $x_1=1$ and $x_2=\dfrac{1}{\sqrt{2}}\left(1+\dfrac{1}{\sqrt{2}}\right)>\dfrac{1}{\sqrt{2}}\left(\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}\right)=1$.
Thus, $x_2>x_1$. In genera... | A generalization I couldn't pass by. Let $\color{blue}{S_n=\frac1n\sum_{k=1}^{n-1}f\big(\frac{k}{n}\big)}$ where $f:(0,1)\to\mathbb{R}$ is strictly convex: $$f\big((1-t)a+tb\big)<(1-t)f(a)+tf(b)\quad\impliedby\quad a<b,0<t<1.$$ If we put $a=k/(n+1),b=(k+1)/(n+1),t=k/n$ for $0<k<n$ here, we obtain $$f\Big(\frac{k}{n}\Bi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Integral $I(\tau_1,a,b) = \int_{\tau_1}^\infty d\tau_2\ \frac{1}{b^2 + \tau_2^2} \left(\pi - 2 \tan^{-1} \frac{\tau_2}{a} \right)^2$ I am looking at the integral:
$$I(\tau_1,a,b) = \int_{\tau_1}^\infty d\tau_2\ \frac{1}{b^2 + \tau_2^2} \left(\pi - 2 \tan^{-1} \frac{\tau_2}{a} \right)^2, \tag{1}$$
where $\tau_1$ is real... | The given integral can be presented in the form of
$$I(\tau,a,b) = 4\int\limits_\tau^\infty \operatorname{arccot}^2\dfrac{\tau_2}a \,\dfrac{\mathrm d\tau_2}{\tau_2^2+b^2}.\tag1$$
Substitution
$$\varphi=\operatorname{arccot} \dfrac{\tau_2}a,\quad \tau_2 = a\cot\varphi,\quad\mathrm d\tau_2=-a(\cot^2\varphi+1)\,\mathrm d\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Existence of a simple homeomorphism Definitions:
Define $Q=[0,1] \times [0,1]$ with the product topology and $C=\{(s,t) \in Q:s=0\} \cup \{(s,t) \in Q:t=1\} \cup \{(s,t) \in Q:s=1\}$.
Define $Q/C$ the quotient space by the relation:
$a,b \in Q$ satisfies $a \mathscr{R} b$ if and only if $a=b$ or $a,b \in C$.
with the q... | We can find such an $f$ explicitly by constructing a map $g \colon Q \to D^2$ that factors through $Q/C$ and induces the desired homeomorphism.
Such a $g$ must map every horizontal line segment $t = \operatorname{const}$ to a closed Jordan curve with base point $1$, and these loops must sweep the whole disk and be disj... | {
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Minimizing $\| x A - B\|_F^2$ With a Constraint I have previously asked an optimization question Here. I will reiterate the question and simply add a constraint to it:
I have 2 known grayscale images (256×256 matrices) $A$ and $B$ and want to find the unknown scalar variable $x$ so that:
$$\text{Minimize} \quad \|xA-B\... | This can be solved as a linearly-constrained linear least squares problem, for which there are many available numerical solvers, accessible from a variety of computer languages and packages. Alternatively, by not squaring the Frobenius norm, it can be solved as a Second Order Cone Problem (SOCP), again for which there ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The role of topology in continuity Suppose we have two sets $M$ and $N$ endowed with topologies $T_1$ and $T_2$ respectively. Consider a (continuous) map $L: M\to N$. Now if it is possible that we define another topology on M in such a way that the same function becomes discontinuous( is it even possible?), what role d... | Continuity depends on the topology. For example, if N has the trivial topology, or M has the discrete topology, then any map from M to N will be continuous. On the other hand, if N has the discrete topology, then the only continuous functions are locally constant functions. And if M has the trivial topology, the only c... | {
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"timestamp": "2023-03-29T00:00:00",
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What is the expected number of peaks in an array of length $n$ with each number randomly drawed from $[0, 1]$? Suppose we randomly draw a real number from $[0, 1]$ for $n$ times, and get an array: $a_1, a_2, ..., a_n$. If for some integer $i$ such that $2<=i<=n-1$ and $a_{i-1}<a_i$ and $a_i>a_{i+1}$, we call $a_i$ a pe... | Let $A_i =\{\text {Peak at}\ i\}$, and let $Y_i=I_{A_i}$ be the indicator function, where $2\le i\le n-1$. Since the $X_i$ are i.i.d., $P(A_i)=P(\{X_{i-1}\lt X_i\}\cap\{X_i\gt X_{i+1}\})$ is independent of $i$, and an easy calculation or an argument by symmetry shows this probability is $1/3$. The number of peaks is $W... | {
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"url": "https://math.stackexchange.com/questions/3679372",
"timestamp": "2023-03-29T00:00:00",
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Polar set of $S=\{s\in\mathbb{R}^n:1^Ts≤1,\; s≥0\}$ The definition of a polar set, that I am working with, is: $M^{pol}=\{x\in\mathbb{R}^n:m^Tx≤1, \forall m\in M\} $
Now I want to find the polar $S^{pol}$ of $S$, where $S=\{s\in\mathbb{R}^n:$1$^Ts≤1,\; s≥0\}$
S is the set of vectors in $\mathbb{R}^n$ for which each $s_... | $S^{pol}=U:=\{u\in \mathbb R^n \ | \ \max_{u_i\ge 0} u_i\le 1\}$ for $S=\{s\in \mathbb R^n \ | \ 1^Ts \le 1\text{ and } s\ge 0\}$:
(1) $U\subseteq S^{pol}:\quad$
let $u\in U$ and $s\in S$, then by defining $u_+:=\max(u,0)$ and $u_-:=\max(-u,0)$; both componentwise, we have:
$$u^Ts=(u_+-u_-)^Ts=u_+^Ts-u_-^Ts\le u_+^Ts=\... | {
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Finding coefficients of quadratic formula given certain properties Given the quadratic equation $ax^2+bx+c$ , how do you find $a,b$ and $c$ given you know:
the gradient of the curve at the $y$ intercept
the equation of the tangent at point $P$
the gradient of the normal at point $P$
I haven’t included the specific eq... | Let $f(x) = ax^2 + bx + c$ be the curve $C$ in question.
If we know the gradient of $C$ at the $y$-intercept (i.e. where $x=0$) is $m_0$, then that is the same as saying we know that $f'(0) = m_0$.
If we know the equation of the line $L$ which is tangent to $C$ at the point $P$ with co-ordinates $(x_P, y_P)$, then we k... | {
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"url": "https://math.stackexchange.com/questions/3679667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
An intuitive explanation of the multiplication identity of modular arithmetic Is there any intuitive explanation for the validity of this identity?
If $a\equiv b\pmod n$ and $c\equiv d\pmod n$, then
$$a\times c \equiv b\times d \pmod n$$
I want something which appeals to someone's who just beginning to learn number t... | This is not difficult to prove and understand. The idea is to use a simpler result two times.
If $a\equiv b\pmod{n} $ then we have $ka\equiv kb\pmod{n} $.
This should be obvious to understand as it is an immediate consequence of the definition of congruence. Just note that if $a-b$ is a multiple of $n$ then $k(a-b) $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3679822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the value of- $\lim_{x \rightarrow -\infty}\sum_{k=1}^{1000} \frac{x^k}{k!}$
QUESTION: Find the value of- $$\lim_{x \rightarrow -\infty}\sum_{k=1}^{1000} \frac{x^k}{k!}$$
MY ANSWER: Since $x→-\infty$ therefore the summation will look like-
$$ \frac{-\infty^1}{1!}+ \frac{-\infty^2}{2!} + \frac{-\infty^3}{3!} + ... | One may observe that, for $x\neq0$,
$$
\sum_{k=1}^{1000} \frac{x^k}{k!}=\color{red}{\frac{x^{1000}}{1000!}}\times\left(\color{red}1+\frac{1000!}{x\times999!}+\frac{1000!}{x^2\times998!}+\cdots+ \frac{1000!}{x^{1000}\times1!}\right)
$$ giving that
$$
\begin{align}
\lim_{x \rightarrow -\infty}{\sum_{k=1}^{1000} \frac{x^k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3680047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Interesting limit involving hordes of logarithms.
$$\lim_{x\to 0^{+}}\ln (x\ln a)\ln \left(\dfrac{\ln ax}{\ln\frac xa}\right)=6$$
Find the value of $a$.
Answer: $e^3$
This struck me as an interesting problem and wanted to know if there are any more methods to solve it except the one I have used (written as an answe... | $$\frac{\log ax}{\log\dfrac xa}=1+\frac{2\log a}{\log\dfrac xa}$$ so that
$$\log\frac{\log ax}{\log\dfrac xa}\sim\frac{2\log a}{\log\dfrac xa}.$$
As the denominator will simplify with $\log(x\log a))$, we are left with
$$2\log a=6.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3680228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Where does this relation come from? $n^2-1 \approx (n-1)2$ for $n-1 \ll 1$ I came across the relation in the title in a physics textbook and wondered how I get to it.
$$n^2-1 \approx (n-1)2$$
for $$n-1\ll 1$$
Could anybody maybe help me out?
Thanks!
| $n^2-1=(n+1)(n-1)$ so if $n$ is very close to $1$, then $n+1$ is very close to $2$ and $n^2-1$ is very close to $2(n-1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3680378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Show that $\begin{vmatrix} 1+a^2-b^2 & 2ab & -2b \\ 2ab & 1-a^2+b^2 & 2a \\ 2b & -2a & 1-a^2-b^2 \end{vmatrix} = (1+a^2+b^2)^3$
Show that $\begin{vmatrix} 1+a^2-b^2 & 2ab & -2b \\ 2ab & 1-a^2+b^2 & 2a \\ 2b & -2a & 1-a^2-b^2 \end{vmatrix} = (1+a^2+b^2)^3$
Performing the operations $C_1 \rightarrow C_1-bC_3$
and $C_2 ... | $$\begin{aligned}\Delta &=\begin{vmatrix} 1+a^2+b^2&0&-2b \\ \ 0&1+a^2+b^2&2a \\\ b(1+a^2+b^2)&-a(1+a^2+b^2)&1-a^2-b^2 \end {vmatrix}\\
&=(1+a^2+b^2)\begin{vmatrix} 1&0&-2b \\ \ 0&1+a^2+b^2&2a \\\ b&-a(1+a^2+b^2)&1-a^2-b^2 \end {vmatrix}\\
&=(1+a^2+b^2)^2\begin{vmatrix} 1&0&-2b \\ \ 0& 1 &2a \\\ b &-a &1-a^2-b^2 \end {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3680685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solutions to differential equation $\nabla f(x)=f(x)x$ Let us consider the differential equation given by $\nabla f(x)=f(x)x$, where $f:\mathbb{R}^n\to \mathbb{R}$. I have found that $f(x)=K\exp(|x|^2/2)$ is a solution, but are all solution of this form?
| We can use the method of integrating factor to prove that this is the only solution. Moving everything to one side, notice that
$$\exp\left(-\frac{x^2}{2}\right)(\nabla f - x f) = 0 \implies \nabla \left(\exp\left(-\frac{x^2}{2}\right) f\right) = 0$$
which means
$$f(x) = K\exp\left(\frac{x^2}{2}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3680802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Combinatorial problem about $n \times 3$-matrix Let $A$ be an $n \times 3$-matrix so that for each number $k \in \{1,2,3\}$ there are exactly $n$ entries $a_{ij}$ s.t. $a_{ij}=k$. Is it possible to rearrange the entries of each column of $A$, such that in every row every number appears at least once?
I was trying to pr... | The answer is yes!
Proof by construction:
*
*leave the first column as is
*start with row 1 and repeat the following until you can't:
*
*find the two missing numbers for each row in the 2'nd and 3'rd column (order doesn't matter) and move them up.
*move to the next row
If the algorithm ends after the last r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3680939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Inequality of $|e^z - 1|$ I'm doing a problem related with Rouché Theorem and to prove it I need to show the following inequality:
$|e^z - 1| < e - 1$ if $|z| < 1$.
I've seen that $|e^z - 1| \leq e + 1$ and that $|e^z| \leq e$ if $|z| < 1$. But I can't manage to get that $|e^z - 1| < e - 1$.
I would be really grateful ... | $|e^z-1|=|\sum_{n \ge1}\frac{z^n}{n!}| \le \sum_{n \ge1}|\frac{z^n}{n!}| < \sum_{n \ge1}\frac{1}{n!}=e-1$ since $|z| < 1$ so $|z|^n <1$ for all $n$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3681262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determine the real part of $\left(\cfrac{z-1}{z+1}\right)$ Determine the $Re\left(\cfrac{z-1}{z+1}\right)$ if $z = cos\theta + i \, sin\theta$.
I'm not quite sure whether the right approach would be to stick with the polar form and substitute it into $\left(\cfrac{z-1}{z+1}\right)$ and perhaps use $ 1 = sin^2(\theta) +... | Note that $z=\cos\theta + i\sin\theta$ implies $|z|=1$. Now,
$$\frac{z-1}{z+1} = \frac{z\bar z-\bar z}{z\bar z + \bar z} = \frac{1-\bar z}{1+\bar z} = - \overline{ \left(\frac{z-1}{z+1}\right)},$$
so, $\frac{z-1}{z+1}$ is purely imaginary, i.e, the real part is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3681479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Computing an intergral on a sphere I'm struggling with the following integral:
$$\int \limits_{S_t(x)} y_1^2 + y_2^2 + y_3^2 \, dA(y),$$
where $S_t(x) = \{y \in \mathbb{R}^3: \lvert y - x\rvert = t \}$.
I understand the integral above as average value of the integrand over $S_t(x)$. However I don't know how to compute ... | Translating the integral we get that
$$\int_{S_t(x)} y^2\:dA(y) = \int_{S_t(0)} (y+x)^2 \:dA(y)$$
Then use the fact that $dA(y) = t^2d\Omega$
$$\int_{S_t(0)} y^2+2y\cdot x + x^2 \:dA(y) = \int_{S^2} t^4 + x^2t^2\:d\Omega + \int_{S_t(0)} 2y\cdot x \:dA(y) = 4\pi t^2(x^2+t^2)$$
where the second integral vanished because... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3681960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Divisor summatory function for Odd and Even numbers With Dirichlet's Divisor summatory function the mean divisor growth can be determined.
$$D(x)=\frac{1}{x} \sum_{n=1}^x d(x)= \log(x)+2 \gamma -1+ \mathcal{O}\!\left(\frac{1}{\sqrt{x}}\right)$$
Where $\gamma$ is the Euler's constant and $d(n)$ is the divisor function. ... | Not sure if that's the question but if one wants the expressions for $d_{odd}(x)=\sum_{2k+1 \le x}d(2k+1)$ and $d_{even}$, one can easily get them from the recurrence:
$d_{even}(x)=2d_r(x/2)-d_r(x/4), d_{odd}(x)=d_r(x)-2d_r(x/2)+d_r(x/4)$ where $d_r(x)=xD(x)$ the usual divisor sum - these follow from the relation $d(4q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3682506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Confused about the meaning behind making x = 2 in the binomial theorem So the binomial theorem states $(1+x)^n = \sum^{n}_{k=0}$$n \choose k$$x ^ k$.
Now I understand that each term of the sum represents the number of ways to arrange 1 and $x$ out of $n$ choices, so there ends up being $k$ number of $x$'s.
I understand... | Notice that $$\sum _{k=0}^n\binom{n}{k}2^k=\sum _{k=0}^n\binom{n}{n-k}2^{n-k}=\sum _{k=0}^n\binom{n}{k}2^{n-k}=\sum _{k=0}^n\binom{n}{k}\left (\sum _{l=0}^{n-k}\binom{n-k}{l}\right )=\sum _{k=0}^n\sum _{l=0}^{n-k}\binom{n}{k}\binom{n-k}{l}.$$
Check the right hand side of this line, this means choosing the positions for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3682611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding the measure of a set using Lebesgue Integral Let's say I have a Lebesgue integrable function $f: X \to \mathbb{R}$, where $X$ is an arbitrary measure space. Now, suppose I want to find the measure of the set
$$
A = \{x \in X: f(x) \geq c\}
$$
for some $c \in \mathbb{R}$. Is there a way that I can express
$\mu... | Following Cameron Williams, an inequality is given by
$\mu(\{f\geq c\})= \int_ {\{f\geq c\}} 1\, d \mu \leq \frac{1}{c} \int f d\mu,$
since $\frac{f}{c} \geq 1$ on $\{f\geq c\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3682791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Equivalent Definitions of a Compact Subset of a Metric Space Let $X$ be a metric space and let $K \subseteq X$. Please consider the following three statements.
*
*For every collection $\{V_\alpha : \alpha \in \mathcal{A}\}$ of subsets of $X$
which are both open in $X$ and cover $K$,
there exists a finite sub... | This is false. Take $X=\mathbb R$ and $K=\mathbb N$. There is no non-empty open set in $X$ contained in $K$ so 2) is vacuously true. But 1) is false.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3682955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the cubic bezier control points from end points and tangents If i have 2 end points and two unit vectors as tangents at the two end points is it possible to find the cubic bezier curve control points that make the curve ? Is there one solution or many solutions ?
Visual of what i am trying to find:
| There are infinitely many solutions. At the start of the curve, the given point and unit vector define a line. You can place the curve’s second control point anywhere along this line. The same reasoning applies at the end of the curve.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3683203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing that for $x > e^{2.5102}, 0 \le \lfloor\dfrac{1.25506(x+1)}{\ln(x+1)}\rfloor - \lfloor\dfrac{1.25506x}{\ln x}\rfloor \le 1$ Showing that for $x > e^{2.5102}, 0 \le \lfloor\dfrac{1.25506(x+1)}{\ln(x+1)}\rfloor - \lfloor\dfrac{1.25506x}{\ln x}\rfloor \le 1$
Does this argument work:
(1) $\dfrac{1.25506x}{\ln x}$ ... | (1)(2) are correct though you have a typo in (2). It should be $x\gt e$ instead of $x\gt 1$.
(3) is not correct. If $a,b$ are integers such that $0\le a\lt 1$ and $0\le b\lt 1$, then $a=b=0$ for which
$$\left\lfloor\dfrac{1.25506(x+1)}{\ln(x+1)}\right\rfloor - \left\lfloor\dfrac{1.25506(x)}{\ln(x)}\right\rfloor = \dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3683356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Ratio of the heights of triangle is given, determine the sides of triangle. Can someone help me with this excersize?
The ratio of heights to the sides of the triangle is v_{a}:v_{b}:v_{c}=12:5:8. What are the length of the sides of triangle? (a,b,c=?)
Thank you!
I tried using formula for area of triangle, e.g. P=(av_a... | Yes, your idea works!
Since $$S=\frac{ah_a}{2}=\frac{bh_b}{2}=\frac{ch_c}{2},$$ we obtain:
$$\frac{1}{a}:\frac{1}{b}:\frac{1}{c}=12:5:8$$ or
$$a:b:c=\frac{1}{12}:\frac{1}{5}:\frac{1}{8}.$$
We need to check that $$\frac{1}{12}+\frac{1}{8}>\frac{1}{5},$$ of course.
For minimal natural's $a$, $b$ and $c$ we obtain $(10,2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3683537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Retract spaces- Material or book recomendations Good afternoon to all!
I have just done a course on Algebraic Topology and I came across the definition of a retract space. Not much more is mentioned with regards to the topological retraction property in the notes I was using. I wonder what the best material to understa... | The 'Theory of Retracts' that I referenced in the books above is a lot more involved than this (which is why I doubted your professor would introduce it) . Past a little experience, the definition you give is about as in depth as you will need to understand retracts.
Spelling out your definition, let $i:A\hookrightarr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3683683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
relation between indices of a tensor and its vectorization Assume three-mode tensor $A \in R^{n_1 \times n_2 \times n_3}$ and its vectorization as $a=vec(A)$. What is the relation between the index elements of tensor $A$ and vector $a$? Specifically, I would like to know which element of tensor $A$ is correspond to the... | I'll assume that you're following this convention of vectorization. I also assume that your indexing starts at one, which (unfortunately perhaps) is a bit more common in mathematical literature on tensors.
The $(i,j,k)$ entry of the tensor $A$ is mapped to the $p$th vector entry, with
$$
p = 1 + (i-1) + n_1(j-1) + n_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3683853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $b \in \bar{A}$ then there exists a filter $G$ such that $A \in G$ and $G \longrightarrow b$ Let $E=(E,\tau)$ be a topological space and $A \subset E$. I want to prove that if $b \in \bar{A}$ then there exists a filter $G$ in $E$ such that $A \in G$ and $G \longrightarrow b$. For this, note that by definition for th... | A better notation is $$\mathcal{B} = \{A \cap U\mid U \in \mathcal{N}(b) \}$$
where $\mathcal{N}(b)$ is the neighbourhood filter at $b$. Then it's easy to see that $\mathcal{B}$ is a filter base: all sets are non-empty by the assumption $b \in \overline{A}$ and the collection is closed under (finite) intersections, as ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3683955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Structure theorem for modules over Dedekind domains I've come across the structure Theorem for fin. gen. Modules over a Dedekind domain several times now. It was formulated to us the following way:
Let $R$ be a Dedekind domain. For every element $\alpha \in C(R)$, let a representative $I_{\alpha}$ in the group of fract... | To do your exercise, you need one extra piece of information, a theorem of Steinitz:
Let $r$ and $s$ be non-negative integers, and let $I_1,\ldots,I_r$ and $J_1,\ldots,J_s$ be ideals in your Dedekind domain $R$. Then one has an isomorphism $\bigoplus_{m=1}^r I_m\cong \bigoplus_{n=1}^sJ_n$ of $R$-modules if and only if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3684112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Are subgraphs of a minimum spanning tree, also minimum spanning trees? Clarification: MST = minimum spanning tree
Say I take some complete, weighted graph $G$ - I create a MST of the graph then split $G$ into 2 trees $T_1$ and $T_2$ by removing some edge connecting them called $uv$. My question is, that if I were to ta... | The answer is yes; if the subgraphs created by taking the nodes and edges from $T_1$ and $T_2$ had a smaller weight than the subgraphs of the minimum spanning tree, and were combined with the original graph by adding the edge $uv$ back, the resulting graph would be a minimum spanning tree of $G$ but with a smaller weig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3684209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Can a function $f$ have an antiderivative even though its indefinite integral $F(x) = \int_{a}^{x} f(t)\ dt$ is not one? The fundamental theorem of calculus states that if $f:[a,b] \to \mathbb{R}$ is integrable and $F(x) = \int_{a}^{x} f(t)\ dt$, then $F'(x) = f(x)$ at every point $x$ at which $f$ is continuous. This m... | Consider the map$$\begin{array}{rccc}F\colon&\Bbb R&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}x^2\sin\left(\frac1x\right)&\text{ if }x\ne0\\0&\text{ otherwise.}\end{cases}\end{array}$$Then $F$ is differentiable and$$(\forall x\in\Bbb R):F'(x)=\begin{cases}-\cos\left(\frac1x\right)+2x\sin\left(\frac1x\right)&\text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3684379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
} |
Prove $\frac{x^2+yz}{\sqrt{2x^2(y+z)}}+\frac{y^2+zx}{\sqrt{2y^2(z+x)}}+\frac{z^2+xy}{\sqrt{2z^2(x+y)}}\geqq 1$ For $x,y,z>0$ and $\sqrt{x} +\sqrt{y} +\sqrt{z} =1.$ Prove that$:$
$$\frac{x^2+yz}{\sqrt{2x^2(y+z)}}+\frac{y^2+zx}{\sqrt{2y^2(z+x)}}+\frac{z^2+xy}{\sqrt{2z^2(x+y)}}\geq 1$$
My solution$:$
Let $x=a^2,\,y=b^2,\,... | Another way.
Since $$\left(yz-\frac{1}{2}xy-\frac{1}{2}zx, zx-\frac{1}{2}yz-\frac{1}{2}xy,xy-\frac{1}{2}zx-\frac{1}{2}yz\right)$$ and
$$\left(\frac{1}{x\sqrt{2(y+z)}},\frac{1}{y\sqrt{2(z+x)}},\frac{1}{z\sqrt{2(x+y)}}\right)$$ have the same ordering, by AM-GM and Chebyshov we obtain:
$$\sum_{cyc}\frac{x^2+yz}{x\sqrt{2(y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3684495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
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