Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
If one of two sets has larger cardinality, there is a map onto the other set
Let A and B be sets with the cardinality of A less than or equal to B. Show there exists an onto map from B to A.
I am struggling with this proof. I don't know how to show this. Any help would be greatly appreciated
| A not empty
There exists an injective map $f:A\rightarrow B$, there exists an inverse $g:f(A)\rightarrow A$. Let $a\in A$, $h:B\rightarrow A$ defined by $h(x)= g(x), x\in f(A)$ otherwise $h(x)= a$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to evaluate $ \int_0^\infty \frac{\log x}{(x^2+a^2)^2} dx $ Evaluate $$ \int_0^\infty \frac{\log x}{(x^2+a^2)^2} dx $$ $$(a>0) $$
How can I use contour appropriately?
What is the meaning of this integral?
(additionally posted)
I tried to solve this problem.
First, I take a branch $$ \Omega=\mathbb C - \{z|\text{... | I thought that it might be instructive to add to the answer posted by @RonGordon. We note that the integral of interest $I_1(a^2)$ can be written
$$I_1(a^2)=\int_0^\infty \frac{\log^2 x}{(x^2+a^2)^2}\,dx=-\frac{dI_2(a^2)}{d(a^2)}$$
where
$$I_2(a^2)=\int_0^\infty\frac{\log^2x}{x^2+a^2}\,dx$$
Now, we can evaluate the i... | {
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Given a 95% confidence interval why are we using 1.96 and not 1.64? I'm studying for my test and am reviewing the solution to some example problems. The problem is:
You are told that a new standardized test is given to 100 randomly selected third grade students in New Jersey. The sample average score is $\overline Y$ ... | Two reasons:
1) Students are in New Jersey.
2) the $z$-value associated with 95% is z=1.96; this can be seen as part of the 1-2-3, 65-95-99 rule that tells you that values within $ p.m 1-, 2- or 3$- deviations from the mean comprise 65- 95- or 99% of all values in the distribution . Or look it up in a table.
| {
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If $f\in C^{1}(0,1]\cap C[0,1]$ and $f'\not\in L^{1}(0,1)$, then $f$ oscillates at $0$? Please think it easy because it is not an assignment.
Question : Let $f\in C^{1}(0,1]\cap C[0,1]$ and $f'\not\in L^{1}(0,1)$. Then, does $f$ oscillate frequently and $f'$ is unbounded at $0$?
When I asked the similar question befo... | If $f'$ was bounded near $0$, then since $f'$ is continuous on $(0,1]$, then
it would follow that $f'$ is bounded on $[0,1]$ and hence integrable.
Since $\int_x^1 f' = f(1)-f(x)$ for $x>0$, we see that $\lim_{x \downarrow 0} \int_x^1 f'$ exists, hence $f'$ must change sign 'frequently' near zero (if $f'$ does not chang... | {
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Understanding of countable set short theorem. I am struggling to understand what the theorem is stating.
Question 1
Let n equals 2, would that mean that $B_n$ would be $B_2$= $\left\{\left\{a_1\right\},\left\{a_1,a_2\right\}\right\}$?
Question 2
Given n=2 and 3 how would small b, be described?
In the case of $B_{n-1... | My answer for Question 1
Let A = {a,b,c} then B_2 = {(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)}
That's way the proof says B_1 = A
| {
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Prove that the set {→, ¬} is functionally complete I am not sure how to do this question. I have looked at some of the other similar questions but to no avail
I know that for a set of operators to be functionally complete, the set can be used to express all possible truth tables by combining members of the set into a B... | The implication $\to$ is defined by
$$ a\to b \equiv \neg a \vee b. $$
This means, that
$$ \neg a \to b \equiv a \vee b$$
and thus, you can express logical or using $\to$ and $\neg$. Furthermore you know de Morgans rules and you have
$$ \neg(\neg a \vee \neg b) \equiv \neg\neg a\wedge \neg\neg b = a\wedge b.$$
Thus you... | {
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Is $x^6-3$ irreducible over $\mathbb{F}_7$? I know that $\mathbb{F}_7=\mathbb{Z}_7$, and the all possible solutions of $x^6-1=0$ over $\mathbb{Z}_7$ are 1~6, so if we let the root of equation $x^6-3$ as $t $ then the solutions of $x^6-3=0$ is $t$~$6t$, which shows that all the solutions are not in $\mathbb{Z}_7$. Howev... | Assume that you have an irreducible polynomial $P$ of degree $d$ dividing your polynomial. Then taking $r$ to be the class of $X$ in the field :
$$\mathbb{F}_{7^d}:=\frac{\mathbb{F}_7[X]}{(P)} $$
We know that $r^6=3$ in this new field (since $P$ divides $X^6-3$). On the other (because you know Lagrange's theorem) you ... | {
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Subspace topology of matrices I have been given homework in which the $2\times 2$ matrices of determinant $1$ are equipped with the subspace topology of $\mathbb{R}^4$. However, $\mathbb{R}^4$ is a space of 4-tuples, while 2x2 matrices are not n-tuples. How do I get the open sets of this topology?
How is this set of ma... | Any matrix
$$\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}\in\mbox{Mat}^{2\times 2}(\mathbb{Z})$$
can be interpreted as the touple $(a,b,c,d)\in\mathbb{R}^4$, so $\mbox{Mat}^{2\times 2}(\mathbb{Z})$ can be thought of a subspace of $\mathbb{R}^4$ and be given the subspace topology.
| {
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To find ratio of Length and Breadth of a Rectangle Given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of the sheet, what is the ratio of length to breadth of the sheet?
| Let the length be $l$, and the width be $w$. When the sheet is folded so that the diagonals meet, the fold crosses the sheet from one of the long sides to the other, and meets the long sides the same distance from the opposite corners. Let the distance from the intersection of the fold and the long side to the nearest ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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1 Form Integral along the curve Let $\alpha$ be the $1$-form on $D=\mathbb{R}^2-\{(0,0)\}$ defined by,
$$
\alpha=\frac{xdx+ydy}{x^2+y^2},
$$
where $(x,y)$ are cartesian coordinates on $D$.
*
*Evaluate the integral of the $1$-form $\alpha$ along the curve $c$ defined $c(t)=(t\cos\phi,t\sin\phi)$, where $1 \le t \le 2... | In polar coordinates $\alpha=\frac{1}{r}dr=d\ln r$ and the integral along the straight radial line is simply $\ln2-\ln1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Centre of the circle Another approach to the curvature of a unit-speed plane curve $\gamma$ at a point $\gamma (s_0)$ is to look for the ‘best approximating circle’ at this point. We can then define the curvature of $\gamma$ to be the reciprocal of the radius of this circle.
Carry out this programme by showing that the... | $\newcommand{\e}{\epsilon}
\renewcommand{\d}{\pm\Delta s}
\renewcommand{\n}[1][]{\,n\left(s_{0}#1\right)}
\newcommand{\gs}{\gamma\left(s_{0}\right)}
\newcommand{\gd}{\gamma\left(s_{0}\d\right)}
\newcommand{\dg}{\gamma\,'\left(s_{0}\right)}
\newcommand{\ddg}{\gamma\,''\left(s_{0}\right)}
\newcommand{\O}[1][]{\mathcal{O}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
Fourier transform invariant functions other than the bell curve? Are there any functions that are their own Fourier transforms other than $e^{-\pi x^2} $?
| There is a complete characterization of the probability densities that are (modulo a constant factor) their own Fourier transforms (aka characteristic functions) in a paper of K. Schladitz and H.J. Engelbert:
"On probability density functions which are their own characteristic functions",
Theory Probab. Appl., vol. 4... | {
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Help with understanding definition and how to visualize spherical basis vectors? Can someone please help me understand the definition and how I can visualize these spherical basis vectors in 3-space?
https://en.wikipedia.org/wiki/Spherical_basis#Spherical_basis_in_three_dimensions
$e_+ = -\frac{1}{\sqrt{2}}e_x - \frac{... | $a_+$ and $a_-$ are in general not real for real vectors $r$. The allowed values for them to give a real vector is a subspace of $\Bbb C \times \Bbb C$ of two real dimensions. This is inconvenient for ordinary representations of real vectors, but this particular representation is useful in some situations.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Disjunctive Normal Form I need some help understanding how to convert a formula into disjunctive normal form.
Can anybody explain how one would write φ = ((p ∨ q ∨ r) → (¬p ∧ r)) in disjunctive normal form?
Is it possible to use truth tables to help converting to DNF form?
| I think a truth table could work for simpler propositions, but this one in question is relatively complicated - using logical equivalences might be easier.
Consider a conditional proposition $P \implies Q$, this is equivalent to $(\neg P \vee Q)$, which can be verified using a truth table.
Thus considering, $\varphi ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Problem with congruence equation $893x \equiv 266 \pmod{2432}$ I'm trying to solve $893x \equiv 266 \pmod{2432}$.
Firstly, I find the $\operatorname{gcd}(893, 2432)$ using the extended Euclidean Algorithm. When I calculate this, I receive that the gcd is (correctly) $19$, and that $19 = 17(2432) -49(893)$. From this, I... |
Where did I go wrong ?
Nowhere.
How might I go about fixing this ?
There's nothing to fix.
However, I believe this is not correct.
Next time, have more faith in yourself. ;-$)$
| {
"language": "en",
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Finite OUTER measure and measurable set Royden's Real Analysis (4th edition), problem #19 (Chapter 2.5):
Let $E$ have a finite OUTER measure. Show that if $E$ is not measurable, then there is an open set $O$ containing $E$ that has a finite outer measure and for which $m^{*}(O - E) > m^*(O)-m^*(E)$.
My question is how ... | It's probably supposed to say "Let $E$ have finite outer measure".
There's an errata list here. There is no entry for this problem, but the entry for problem 18 on page 43 looks similar to this problem and is supposed to start "Let $E$ have finite outer measure".
| {
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If $ab$ is an element of group $G$, are $a$ and $b$ both elements of group $G$ as well? Obviously, if $a$ and $b$ are elements of group $G$, then $ab$ is in $G$ as well.
Is the converse true?
I tried to think about it in terms of the inverse of the original statement (considering it's the contrapositive of the statemen... | Let $H$ be a subgroup of a group $G$ with the property that whenever $ab \in H$ for some $a,b \in G$, then $a \in H$ or $b \in H$. Then $H=G$. For suppose $H \subsetneq G$. Let $a \in G-H$. And put $b=a^{-1}$. Then $ab=1 \in H$, since $H$ is a subgroup. But then by the property, $a$ or $a^{-1}$ are elements of $H$, whe... | {
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Find the locus of intersection of tangents to an ellipse if the lines joining the points of contact to the centre be perpendicular. Find the locus of intersection of tangents to an ellipse if the lines joining the points of contact to the centre be perpendicular.
Let the equation to the tangent be $$y=mx+\sqrt{a^2m^2+b... | Let $(x_1,y_1)$ be the generic point of the locus.
It is well known that $$\frac {xx_1}{a^2}+\frac {yy_1}{b^2}=1$$ represents the line passing through the points of contact of the tangents from $(x_1,y_1)$.
Now consider the equation $$\frac {x^2}{a^2}+\frac {y^2}{b^2}-\left(\frac {xx_1}{a^2}+\frac {yy_1}{b^2}\right)^2=... | {
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Can we talk about the continuity/discontinuity of a function at a point which is not in its domain? Let us say that I have a function $ f(x)=\tan(x)$ we say that this function is continuous in its domain.
If I have a simple function like $$ f(x)=\frac{1}{(x-1)(x-2)} $$
Can we really talk about its continuity/discontinu... | A function by definition must be defined for all points in the domain. So formally speaking a function like $\tan(x)$ doesn't even know what $\pi/2$ is (other than the codomain maybe). So no, it does not make sense to talk about continuity at points outside the domain.
For example is the function $f\colon \Bbb{R} \to ... | {
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How to find $f(15/2)$ if $f(x+1)=f(x)$? Suppose $f(x+1)=f(x)$ for all real $x$. If $f$ is a polynomial and $f(5)=11$, then $f(15/2)$ is ??
How do I approach questions like this?
| The problem is that how a polynomial can be periodic! Is there any other way than being a constant? So your function is really $f(x)=11$. That's all.
| {
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$a,b\in A$ are selfadjoint elements of $C^*$-algebras, such that $a\le b$, why is $\|a\|\le \|b\|$ Let $A$ be a unital $C^\ast$-algebra with unit $1_A$.
a) Why is $a\le \|a\|1_A$ for all selfadjoint $a\in A$ and
b) If $a,b\in A$ are selfadjoint such that $a\le b$, why is $\|a\|\le \|b\|$?
I would try it with the contin... | For a) you don't need functional calculus. Note that $\|a\|\,1_A-a$ is a selfadjoint with spectrum contained in $[0,\infty)$, so positive ($\sigma(a+k\,1_A)=\{\lambda+k:\ \lambda\in\sigma(a)$).
As stated, b) is not true: $-3<-2$, but $\|-2\|=2<\|-3\|$. It is true for positives, though. For $0\leq a \leq b$, we represen... | {
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limit n tends to infinity for arbitrary non negative real numbers
I tried evaluating the limit taking log and afterwards i couldnt proceed further. I tried solving using numbers instead of variables and did not got any relation from that and help in this matter is appreciated
| Here is a version with L'Hôpital's rule. It's not quite as strong (logically) as the other answers, as we'll explain.
Assume that $a_1 \leq a_2 \leq \dots \leq a_k$. Starting from
$$
L = \lim_{n\to\infty} \left(a_1^n + a_2^n + \dots + a_k^n\right)^{1/n}
$$
we know
$$\begin{split}
\ln L &= \ln \lim_{n\to\infty... | {
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"timestamp": "2023-03-29T00:00:00",
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Compactness of a set in order topology Consider a partial order in $R^2$ given by the relation $(x_1,y_1)<(x_2,y_2)$ EITHER if $x_1<x_2$ OR if $x_1=x_2$ and $y_1<y_2$.
Then in the order topology on $R^2$ defined by the above order, how can I conclude that $[0,1]$×$[0,1]$ is not compact?
My thought:I have two doubts.
... | Hint: $[0,1]^2$ is a subset of $\mathbb R^2$, so it is given a subspace topology (coming from the order topology of $\mathbb R^2$). Note that in this topology, $\{x\} \times [0,1]$ is open as
$$\{x\} \times [0,1] = [0,1]^2 \cap\big( \{x\} \times (-0.1,1.1)\big)$$
and $\{x\} \times (-0.1,1.1)$ is open in the order topo... | {
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Find the number of integer solutions for $x_1+x_2+x_3 = 15$ under some constraints by IEP. For equation
$$
x_1+x_2+x_3 = 15
$$
Find number of positive integer solutions on conditions:
$$
x_1<6, x_2 > 6
$$
Let: $y_1 = x_1, y_2 = x_2 - 6, y_3 = x_3$
than, to solve the problem, equation $y_1+y_2 +y_3 = 9$ where $y_1 < 6,0... | Here is a different way to break it down
$$
x_1\in\{1,2,3,4,5\}
$$
and given $x_1$ we then have $x_1+x_2<15$ and $x_2>6$ combined as
$$
6<x_2<15-x_1
$$
And whenever $x_1$ and $x_2$ are given, the value of $x_3$ follows from them.
For $x_1=5$ we then have $x_2\in\{7,8,9\}$ so three choices for $x_2$. Each time $x_1$ is ... | {
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Proving non compactness of a space I'm trying to show that the space $\mathbb R^p$, endowed with a metric $d'(x,y) = \frac{d_2(x,y)}{1 + d_2(x,y)}$, where $d_2(x,y)$ is the Euclidean distance, is closed and bounded but not compact.
I've had no problem with the first two proof, but I cannot go ahead with the proof of no... | Other answer using sequences.
The sequence $(a_n)$ with $a_n=(n,0, \dots, 0)$ is bounded (by $1$) but has no converging subsequence. Hence applying Bolzano–Weierstrass theorem, our metric space is not compact.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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If $F$ is a field with $\operatorname{char} F = 0$ and $f(x) \in F[x]$ irreducible, all zeros of $f(x)$ in any extension has multiplicity $s = 1$. What I have trying is:
Suppose that $f(x)$ has at least one zero $\alpha$ such that $f(x) = (x - \alpha)^sq(x)$, $s > 1$ in some extension. Then I guess that $(x-\alpha)^{s-... | This isn't quite right, first of all $(x-\alpha)^{s-1}$ might not be a polynomial with coefficients in $F$ when $\alpha\not\in F$. However, you do know in characteristic $0$ that the derivative of a non-constant polynomial is not $0$. If
$$f(x)=(x-\alpha)^s\prod_{i=1}^n (x-\alpha_i)^{e_j}, s>1, e_j\ge 1$$
then we have ... | {
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"source": "stackexchange",
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On partition of integers I came across an example in my textbook where it was asked to find the generating function for the number of integer solutions of:
${2w+3x+5y+7z=n}$ where ${0\le w, 4\le x,y, 5\le z}$
The proposed solution is:
$$\frac1{1-t^2}\cdot\frac{t^{12}}{1-t^3}\cdot\frac{t^{20}}{1-t^5}\cdot\frac{... | Take $x$ as an example; it corresponds to the factor $\dfrac{t^{12}}{1-t^3}$. Now
$$\frac1{1-t^3}=\sum_{n\ge 0}t^{3n}=1+t^3+t^6+t^9+\ldots+t^{3k}+\ldots\;.\tag{1}$$
When you multiply the four series together, a typical term $t^{3k}$ of this series would contribute $3k$ to the total exponent of any term in the product i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1483582",
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In a ring, how do we prove that a * 0 = 0? In a ring, I was trying to prove that for all $a$, $a0 = 0$.
But I found that this depended on a lemma, that is, for all $a$ and $b$, $a(-b) = -ab = (-a)b$.
I am wondering how to prove these directly from the definition of a ring.
Many thanks!
| Proceed like this
*
*$a0 = a(0+0)$, property of $0$.
*$a0 = a0 + a0$, property of distributivity.
*Thus $a0+ (-a0) = (a0 + a0) +(-a0)$, using existence of additive inverse.
*$a0+ (-a0) = a0 + (a0 + (-a0))$ by associativity.
*$0 = a0 + 0$ by properties of additive inverse.
*Finally $0 = a0$ by property of $0$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1483716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Find the inverse of the cubic function What is the resulting equation when $y=x^3 + 2x^2$ is reflected in the line $y=x$ ?
I have tried and tried and am unable to come up with the answer.
The furthest I was able to get without making any mistakes or getting confused was $x= y^3 + 2y^2$. What am I supposed to do after... | If you plot the equation $y = x^3 + 2y^2$, you'll find that it fails the horizontal line test, and thus that it is not a one-to-one function and its inverse is not a function. So user130558's answer must be wrong since it doesn't include any $\pm$ signs.
Unless your textbook/teacher tells you otherwise, they probably e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Two real rectangular matrices $AB=I$ Let $A$ be an $m×n$ real valued matrix and $B$ be an $n×m$ real valued matrix so that $AB$=$I$. Thus we must have
*
*$n>m$
*$m\ge n$
*if $BA=I$ then $m>n$
*either $BA=I$ or $n>m$
What I tried: Either $n<m$ or $n>m$ → either $\operatorname{Rank}(A) = \operatorname{Rank}(B)=n$ o... | Answering my own question for the sake of completeness for the learners in mathematics community, the proof is as follows:
$Rank(AB)=Rank(I_m)=m\implies m\le n$
$\therefore$ either $m<n\ or\ m=n$
In case $m=n$;$AB=I\implies B=A^{-1}\implies BA=I$
Thus, either $m<n\ or\ m=n$ is true.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solving for the radius of the Earth based on distance to horizon problem Lets say you are standing on the top of hill with height $d$ and from the top of this hill you can see the very top of radio tower on the horizon. Your goal is to determine the radius of the Earth, so you drive from the base of the hill, in a stra... | Treating each side of the symmetric situation separately, we have $\cos\phi_1=R/(R+a)$, and thus for $\phi\ll1$ (which holds for realistic hills and radio towers) $1-\frac12\phi_1^2\approx R/(R+a)$. Solving for $\phi_1$ yields $\phi_1\approx\sqrt{2a/(R+a)}\approx\sqrt{2a/R}$. Likewise, $\phi_2\approx\sqrt{2d/R}$. Then ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$f \in L^p \cap L^q$ implies $f \in L^r$ for $p\leq r \leq q$. Let $(X, \mathfrak{M}, \mu)$ is a measure space. Let $f:X \rightarrow \mathbb{C}$ be a measurable function. Prove that the set $\{1 \le p \le \infty \, | \, f \in L^p(X)\}$ is connected.
In other words prove that if $f \in L^p(X) \cap L^q(X)$ with$ 1\le ... | Alternative:
$$\int\left|f\right|^{r}d\mu=\int_{\left\{ \left|f\right|\leq1\right\} }\left|f\right|^{r}d\mu+\int_{\left\{ \left|f\right|>1\right\} }\left|f\right|^{r}d\mu\leq\int_{\left\{ \left|f\right|\leq1\right\} }\left|f\right|^{p}d\mu+\int_{\left\{ \left|f\right|>1\right\} }\left|f\right|^{q}d\mu$$$$\leq\int\left|... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The partial fraction expansion of a 4x4 matrix The partial fraction expansion of a matrix is given by $$(I\xi-A)^{-1}=\sum_{i=1}^{N}\sum_{j=1}^{n_{i}}T_{ij}\frac{1}{(\xi-\lambda_{i})^{j}}$$,
$T_{ij}\in\mathbb{R}^{n\times n}$, $\lambda_{i}$ the eigenvalues of the matrix $A$ and $n_{i}$ the multiplicity of the respective... | In your case, the multiplicity of each eigenvalue is one so you have
$$ (I\xi-A)^{-1}=\sum_{i=1}^{N} T_{i}\frac{1}{(\xi-\lambda_{i})}. $$
The matrix $A$ is diagonalizable so you have a basis $(v_1, \ldots, v_N)$ of eigenvectors of $A$ with $Av_i = \lambda_i v_i$. Let us multiply the equation above by $v_j$:
$$ \frac{v_... | {
"language": "en",
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Can you integrate by parts with one integral inside another? From the definition of the Laplace transform:
$$\mathcal{L}[f(t)]\equiv \int_{t=0}^{\infty}f(t)e^{-st}\mathrm{d}t$$
where $s \in \mathbb{R^+}$.
$$\mathcal{L}\left[\int_{u=0}^{t}f(u)\mathrm{d}u\right]=\int_{t=0}^{\infty}e^{-st}\mathrm{d}t\int_{u=0}^{t}f(u)\ma... | 1) $t$
2) No: The $f(t)$ there should be thought of as ${d\over dt}\int_0^t f(u)\,du$, in accordance with the integration by parts.
3) No, $\mathcal L[f]$ is correct.
| {
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SHORTEST method for finding the third vertex of an equilateral triangle given two vertices? I know the usual method of calculating third vertex by using distance formula , forming quadratic and solving and stuff , but i was wondering if there was a shortcut method for finding it without much havoc ?
Eg: Equilateral t... | midpoint of $AB$ = $(4, 1.5)$
slope of $AB = -\frac{1}{2} $
right bisector of $AB$ ... $(y-1.5)=2(x-4)$
parametrize bisector ... $$\vec \ell(t)= (4, 1.5) + \frac{t}{\sqrt 5}(1,2) $$ where I have put in the factor of $\sqrt 5$ so that the distance from (4, 1.5) is given by $|t|$
now the altitude of an equilateral trian... | {
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How many telephone numbers that are seven digits in length have exactly five 6's? How many phone numbers that are seven digits in length, have exactly five 6's?
My attempt: {{6,6,6,6,6}{ , }}
$(5(top) 5(bottom)) * (18(top) 2(bottom)) = 153$
my reasoning is that the first subset containing the 6's must all be 6, so when... | The locations of the $6$'s can be chosen in $\binom{7}{5}$ ways. For each such choice, the remaining two slots can be filled in $9^2$ ways.
| {
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Question about possible relations Hello I have a question about possible equivalence relations.
I know that a relation can be Reflexive, Symmetric , Transitive.
But my question is, is there any strict limitations one has on the other.
For example if we had a relation then there are eight possible combinations of the ab... | What about a relation on a set where nothing relates to anything? This is symmetric and transitive, but not reflexive.
| {
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How many non-isomorphic simple graphs are there on n vertices when n is... How many non-isomorphic simple graphs are there on n vertices when n is 2? 3?
4? and 5?
So I got...
$2$ when $n=2$
$4$ when $n=3$
$13$? (so far) when $n = 4$ But I have a feeling it will be closer to 16.
I was wondering if there is any sort of ... | There is no nice formula, I’m afraid. What you want is the number of simple graphs on $n$ unlabelled vertices. For questions like this the On-Line Encyclopedia of Integer Sequences can be very helpful. I searched in on the words unlabeled graphs, and the very first entry returned was OEIS A000088, whose header is Numbe... | {
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What is the purpose of showing some numbers exist? For example in my Analysis class the professor showed $\sqrt{2}$ exists using Archimedean properties of $\mathbb{R}$ and we showed $e$ exists. I want to know why it's important to show their existence?
| You believe $\sqrt{2}$ is a number... so the question is whether or not it's a real number.
If the real numbers didn't have a number whose square is 2, that would be a rather serious defect; it would mean that the real numbers cannot be used as the setting for the kinds of mathematics where one wants to take a square r... | {
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Show that a group of order 7 has no subgroups of order 4. I can do this with the use of Lagrange's theorem but my professor says its possible without it. I can't find how to go about solving it. Any hints would be appreciated.
| Okay, suppose your subgroup is $G=\{1,a,b,c\}$, and your big group is $H=G\cup \{d,e,f\}$.
Now, what can $ad$ be? Not $1$, since $a^{-1}\in G$ and $d\notin G$. Not $a$, since $d\neq 1$. Not $b$ (resp. $c$) since $a^{-1}b\in G$ (resp. $a^{-1}c\in G$). Also not $d$, since $a\neq 1$. Hence $ad\in \{e,f\}$. By simil... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Very fascinating probability game about maximising greed? Two people play a mathematical game. Each person chooses a number between 1 and 100 inclusive, with both numbers revealed at the same time. The person who has a smaller number will keep their number value while the person who has a larger number will halve their... | As @Greg Martin pointed out, you can solve such games using linear programming under the assumption that the goal is to win by the largest margin over the other player.
I used an online zero-sum game solver to find the following solution; I'm not sure if this optimal strategy is unique.
Never choose numbers in $[1,25]... | {
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Prove that Riemann Sum is larger Let $ f:[a,b] \rightarrow \mathbb{R}$ be a differentiable function. Show that if $ P = (x_0, x_1, ..., x_n) $ is a partition of [a,b] then
$$ U(P, f') := \sum_{j=1}^n M_j \Delta x_j \ge f(b) - f(a) $$ where $$ M_j := sup\{f'(t) : t \in [x_{j-1}, x_j] \}$$
How do I go about proving this?... | If $f$ is continuously differentiable, then by fundamental theorem of calculus we have $\int_{a}^{b}f' = f(b) - f(a)$; the result follows from the fact that Darboux integration is equivalent to Riemann integration and that $\inf U(f'; P) = \int_{a}^{b}f'$ by definition.
If $f$ is simply differentiable, then by mean-va... | {
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If $f(x)$ is positive and $f'(x)<0$, $f"(x)>0$ for all $x>0$, prove that $\frac{1}{2}f(1)+\int_1^nf(x)dx<\sum_{r=1}^nf(r)$
If $f(x)$ is positive and $f'(x)<0$, $f"(x)>0$ for all $x>0$, prove that $$\frac{1}{2}f(1)+\int_1^nf(x)dx<\sum_{r=1}^nf(r)<f(1)+\int_1^nf(x)dx$$
Since function is decreasing, then area of rectang... | $f'' > 0$ means that $f$ is strictly convex, so the graph of $f$ restricted to each interval $[r, r+1]$ lies below the straight line connecting
$(r, f(r))$ and $(r+1, f(r+1))$:
$$
f(x) \le (r+1-x) \, f(r) + (x-r) \, f(r+1)
$$
with equality only at the endpoints of the interval.
The integral of a linear function over a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Lyapunov invariant set for affine systems Given a linear system $\dot{x}=Ax$ such that the real part of every eigenvalue of $A$ is less than $0$, Lyapunov's equation $A^T P + P A = -Q$ with $Q$ being any suitably sized positive definite matrix gives us an invariant ellipsoid $x^T P x \leq 1$, i.e. for any initial state... | As Evgeny suggested, it is enough to translate the coordinate system, i.e. for $x=y-A^{-1} b$ we have $\dot{y} = \dot{x} = A x + b = A (y - A^{-1} b) + b = A y - b + b = A y$ and thus the invariant ellipsoid is $y^T P y = (x+A^{-1}b)^T P (x+A^{-1}b) \leq 1$. Note that $A^{-1}$ exists since we demanded that the real par... | {
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Why is "similarity" more specific than "equivalence"? At least regarding matrices, we have
*
*$A$ is similar to $B$ if $\exists S: B=S^{-1}AS$
*$A$ is equivalent to $B$ if $\exists P,Q: B=Q^{-1}AP$
I am confused about the usage of the terms "similar" and "equivalent". I would have thought that equivalence is more... | $A$ and $B$ are similar if they represent the same endomorphism $f$ of a finite dimensional $K$-vector space $E$ in different bases. Thus similar matrices are equivalent, but the converse is false:
They're similar if and only if they have the same Jordan normal form, and also if and only if they have the same simila... | {
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Trying to understand a remark from Atiyah and Macdonald's *Introduction to Commutative Algebra* Can someone explain the following remark from page 52 of Atiyah and Macdonald's Introduction to Commutative Algebra book.
The names isolated and embedded come from geometry. Thus if $A=k[x_1,...,x_n]$, where $k$ is a field,... | Let $\mathfrak a=\mathfrak q_1\cap\cdots\cap\mathfrak q_n$ be a reduced primary decomposition. Then $V(\mathfrak a)=V(\mathfrak q_1)\cup\cdots\cup V(\mathfrak q_n)$. Let $\mathfrak p_i=\sqrt{\mathfrak q_i}$. Now we have $X=V(\mathfrak p_1)\cup\cdots\cup V(\mathfrak p_n)$. If $\mathfrak p_i\subset\mathfrak p_j$, then $V... | {
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Is a coordinate system a requirement for a vector space? In other words, can a vector space exist, and not have a coordinate system?
I'm asking because in the definitions that I've seen of a vector space, there's no mention of a coordinate system.
| For any finite dimensional vector space $V$, there exists a coordinate system (it is not unique). Indeed, let $ n := \dim V$. Then you can find a linearly independent set $\mathcal{B} = \{u_1, \dots, u_n\}$ of $n$ vectors that will generate $V$, and $\mathcal{B}$ can be taken as a coordinate system if your order its el... | {
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Limit of a the function $f(x,y)=x \log(y-x)$ at $(0,0)$ Let $D=\{(x,y) \in \mathbb{R}^2 :y>x\}$ and $f:D\to \mathbb{R}$ with $f(x,y)=x \log(y-x)$.
Does the limit $\lim_{(x,y)\rightarrow(0,0)} f(x,y)$ exist?
I consider $E=\{(-t,t) : t>0\}$ so $f\vert_{E}=-t\log(2t) \to 0 $ for $t \to 0$. So the limit is null?
| Hint: Approach $(0,0)$ along a path with $x$ positive and $y=x+e^{-1/x}$.
| {
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How many participants required?
A test consisting of 20 problems is given at a math competition. Each correct answer to each problem gains 4 points; each wrong answer takes away 1 point, and each problem left without an answer gets 0 points. What is the lowest possible number of participants in the competition that is... | I understand the problem as follows: What is the minimum number of participants such that we can be sure to see two participants with the same score.
Denote by $c$, $w$ the number of correct, resp., wrong answers of some participant, and by $s$ his score. Then $$c\geq 0,\quad w\geq0,\quad c+w\leq20,\quad s=4c-w\ .$$
Gi... | {
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Evaluate the limit of $\lim_{n\to\infty}\frac{n^{1/3}\sin(n!)}{1+n}$ I cannot seem to wrap my head around this problem due to its complexity. The answer seems to be that the limit goes to zero for the following reasons:
*
*$\sin(x)$ will never be greater than $1$, and thus will only lessen the value of the numerator... | Use the sandwich/squeeze theorem + algebra of limits + as you already mentioned the bounded property of the sine function if you want a more rigorous way to show the limit. Alternatively you could formally prove that it has the limit you suggest using the definition of a limit.
| {
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Easiest way in general to find the sin, cos, arcsin, arccos, of "not so easy" angles/values without using a calculator? I was wondering if there are any easy ways in general to find the sin, cos, arcsin, arccos, of "not so easy" angles/values without using a calculator. By "not so easy" I mean just not things like 0, $... | Since $\pi/4$ is an angle of a $1:1:\sqrt{2}$ right-angled triangle, and $\pi/6$ and $\pi/3$ are angles of a $1:\sqrt{3}:2$ right-angled triangle, the $\sin / \cos / \tan$ of these and their multiples are easy to calculate on the fly.
$\sin 6\pi/5$ or something similar would be harder.
| {
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$L^q(X) \subset L^p(X)$ if $p\leq q$ and $\mu(X) < \infty$. Let $(X,\mathfrak{M},\mu)$ be a measure space. Show that if $\mu(X) < \infty$ then $L^q(X) \subset L^p(X)$ for $1\le p \le q \le \infty$.
Is it enough to define $$E_0 = \{x \in X \, : \, 0 \leq |f(x)| < 1\}$$ and
$E_1 = X \setminus E_0$ so that $\mu(E_0), \,... | Your proof is fine. You can also try to show that if $p<q$ then $$\lVert f\rVert_p \leqslant \lVert f\rVert_q \mu(X)^{1/p-1/q}$$
by using Hölder's inequality. This gives a bit more information than your proof. In particular for $X$ a probability space gives $p\mapsto \lVert f\rVert_p$ is increasing.
| {
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How can I get eigenvalues of infinite dimensional linear operator? What I want to prove is that for infinite dimensional vector space, $0$ is the only eigenvalue doesn't imply $T$ is nilpotent.
But I am not sure how to find eigenvalues of infinite dimensional linear operator $T$. Since we normally find eigenvalues by f... | There's no general way to find eigenvalues of an operator in an infinite dimensional space. Actually, some operators don't have any eigenvalue.
You may consider the vector space $k[X]$ of polynomials over the field $k$, and the operator $D : P \mapsto P'$. This operator has only $0$ as an eigenvalue, but is not nilpote... | {
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Proof about the subgroup of injective functions being the subgroup of group of even permutations I have the following problem:
Let $H$ be the subgroup of $S_n$, where $S_n$ denotes the group of injective functions mapping a set to itself, and the order of $H$ is odd. I need to prove that $H$ is a subgroup of $A_n$ wher... | Have you seen in your course (or maybe an earlier one) that if $AB$ are subgroups of a finite group $G$, then the set $AB$ (which need not be a subgroup in general) has cardinality $\frac{|A||B|}{|A \cap B|}$. If you already know that $[S_{n}:A_{n}] = 2$, you can then check that when $H$ is a subgroup of $S_{n}$ of odd... | {
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Proving $\Gamma (\frac{1}{2}) = \sqrt{\pi}$ There are already proofs out there, however, the problem I am working on requires doing the proof by using the gamma function with $y = \sqrt{2x}$ to show that $\Gamma(\frac{1}{2}) = \sqrt{\pi}$.
Here is what I have so far:
$\Gamma(\frac{1}{2}) = \frac{1}{\sqrt{\beta}}\int_0^... | It's using the Gaussian integral or the fact that the area under normal density is 1. Since you have the statistics tag, I'm guessing the latter should be familiar already.
| {
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Trying to compute an integral using Dirichlet's problem solution I want to compute for $r < 1$ that
$$ r \cos \phi = \frac{1}{2 \pi} \int_0^{2 \pi} \frac{ (1-r^2) \cos \theta }{1 - 2r \cos( \phi - \theta) + r^2 } d \theta $$
In my notes, it says that the way to show this is by solving the Dirichlet Problem directly fo... | Note that if in general $f(e^{i\theta})$ is a function defined on $\{|x| = 1\}$ in $\mathbb R^2$, then the function
$$u(re^{i\phi}) = \frac{1}{2\pi} \int_0^{2\pi} \frac{1 - r^2}{1 - 2r \cos(\phi - \theta) + r^2} f(e^{i\theta}) d\theta $$
defined in $\{|x| \le 1\}$ solves the Dirichlet's problem. That is, $u$ satisfies... | {
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The page numbers of an open book totalled $871$. At what pages was the book open? So far I have $x + (x-1) = 871$. This has led to one page being $435$ and the other, $436$. Is there anything needing alteration, or is this equation workable?
| This equation (the equation you wrote) works.
x+ (x−1) = 871.
However, the equation x + (x + 1) = 871 also works, since these two numbers are also consecutive numbers.
Edit: The solution to the second equation yields the exact same answer.
| {
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"url": "https://math.stackexchange.com/questions/1487075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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If $x$ divides $x-z$, then $x$ divides $z$
For any integer x and z , if $x|(x-z)$ then $x|z$
My attempt: suppose $x|(x-z),$ let $y= x-z$
$x|y $ means there is any integer r such that $y=r*x$
So $ x-z=rx $, which equals $(x-z)/(x) =r $ this is where I get stuck.
| You con write:
$$\frac {x-z}{x}=k$$ with $k$ integer. This expression si equivalent: $$\frac {x}{x}-\frac {z}{x}=k$$ but you can continue $$1-\frac {z}{x}=k$$ $k$ is integer therefore $x|z$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Give an example to show that converse is not true: I know that if $ T: X \rightarrow Y$ be a continuous then $Gr(T)=\{(x,Tx):x\in X\}$ is closed in $X\times Y$.
Since let $({x_n,T(x_n)}) \subseteq Gr(T)$ such that $(x_n,T(x_n)) \rightarrow (x,y) \in $$X\times Y$.Thus $x_n\rightarrow x$ , $T(x_n) \rightarrow y$.
But $... | It seems that you're not necessarily looking for a linear map.
Consider the example $f:[-1,1] \to \Bbb R$ given by
$$
f(x) =
\begin{cases}
1/x & x \neq 0\\
0 & x = 0
\end{cases}
$$
Note that the graph of the function is closed, but the function is discontinuous.
Interesting fact: functions on a compact metric space ar... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can you disprove the statement $4=5$? I know this sounds insane. But bear with me, my friend said this to me with a straight face:
Can you disprove the statement: $4=5$?
And I was like is that even a question, thats obvious, $5$ is $4+1$ and $5$ comes after $4$.
He was like: but you haven't still disproved my st... | Subtract 4 from both sides of $4=5$, yielding $0=1$. Now $0$ is defined to be the number such that $0+x=x$ for every $x$ and $1$ the number such that $1\cdot x = x$ for every $x$. But with that definition of $0$ we get that $0\cdot x = (0+0)\cdot x = 0\cdot x + 0\cdot x$, so $0\cdot x =0$ for every $x$. So since $1$ is... | {
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"source": "stackexchange",
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Proving homogenous quadratic inequalities Okay, I'm having trouble proving this:
$$5x^2-4xy+6y^2 \ge 0, \text{ where } x,y \in \mathbb{R}$$
I have tried a few values of $x$ and $y$ and I find that is true. EX: $x=y=0$ which make the equation $= 0$
I have tried factoring it but i find that the factors will involve imagi... | $$
5x^2-4xy+6y^2 = 2(x-y)^2 + 3x^2 + 4y^2,
$$
where each term is non-negative
| {
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How do I find an integer value for which an expression is non-prime? I've just begun Robert S. Wolf's, Proof, Logic and Conjecture. At the end of the first chapter there are some exercises to warm you up for the proof techniques he will eventually introduce. I only mention this so that you are aware that I have yet to ... | How about $n=41$?
In general, if you choose $n$ so that all of the terms in a sum are divisible by the same number, then the whole sum will be divisible by that number.
Edit: My understanding is that your approach was to set $n^2-n+41 = 2n+1$ and look for integer solutions. But this is quite a strong condition: you're ... | {
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An example showing that van der Waerden's theorem is not true for infinite arithmetic progressions One of the possible formulations of Van der Waerden's theorem is the following:
If $\mathbb N=A_1\cup \dots\cup A_k$ is a partition of the set $\mathbb N$, then one of the sets $A_1,\dots,A_k$ contains finite arithmetic ... | Let's build an example with just two sets in the partition, $A$ and $B$.
Enumerate the set $\mathbb{N}\times\mathbb{N}^{>0}$ as $\{(n_k,d_k)\mid k\in \mathbb{N}\}$ (think of $n$ as the starting point of an arithmetic progression and $d$ as the constant difference between the terms). We start with $A = B = \emptyset$ an... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the pmf of rolling a die until obtaining three consecutive $6$s? I am trying to find the pmf of rolling a die until 3 consecutive 6s turn up. I am able to find the expected value using a tree diagram, but the pmf is not obvious to me.
Let A be the event of not rolling 6, and let B be the event of rolling a 6.
T... | The probabilities can be calculated recursively by $p(1)=p(2)=0$, $p(3)={1\over 216}$ and for $n>3$, $$p(n)={5\over 6}\,p(n-1)+{5\over 36}\,p(n-2)+{5\over 216}\,p(n-3).\tag1$$
I suspect that an explicit formula for $p(n)$ will be too complicated to be useful, but you may get useful information via the probability gener... | {
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Why is $h(z)$ tending to $0$ $?$ To prove the uniqueness of Laurent's Decomposition :
$f(z)$ is analytic in the region $\rho\lt |z-z_0|\lt \delta $ .It is decomposed as $$f(z)=f_0(z)+f_1(z)$$ where $f_0(z)$ is analytic for $|z-z_0|\lt \delta $ and $f_1(z)$ is analytic for $|z-z_0|\gt \rho$ .
L... | The theorem you are citing has another condition:
We normalize the decomposition so that $f_1(\infty) = 0$
Also, the proof of the claim says
And $h(z)$ tends to $0$ as $z\to\infty$.
So, and I don't mean this in a mean way, but all your problems originated from sloppy reading. A very common and fatal flaw in mathema... | {
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Continuous function $\geq 0$ on dense subset I'd like to use this fact, but I couldn't find out, how to prove it. Assume we have a continuous function $f : X \rightarrow \mathbf{R}$ with $X$ a topological space and $f \vert_D \geq 0$ on a dense subset $D \subset X$. Can I conclude that $f \geq 0$ on the whole space $X$... | Take $x\in X$. If $x\in D$, you done. If not, there exists $(x_n)\in D^{\mathbb{N}}$ converging to $x$, since $\bar{D}=X$. Then $$0\leq f(x_n)\xrightarrow[n\to\infty]{}f(x)$$
using continuity of $f$.
Edit: not sufficient for general topological spaces, cf. comment below. I'm leaving this answer here exactly because of ... | {
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A concentration problem Consider $N$ variables $X_1,X_2,\ldots,X_N$ with $\Pr(X_i=a_i)=\Pr(X_i=-a_i)=1/2$. Here $a_1,a_2,\ldots,a_N\in [0,1]$.
Does there exist some concentration result about $\sum_{i=1}^N X_i$?
| If $a_i$ are deterministic, than you can think about new random variables $Y_i=\frac{X_i}{a_i}$ and you already have the lower bound for this.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determine the convergence of a recursive sequence Determine the convergence of the sequence defined by $\sum_{n \in \mathbb{N}}a_n$ such as
$$a_1=1,\quad a_{n+1}=\frac{2+\cos(n)}{\sqrt{n}}a_n$$
By the test of the general term, we have
$$\frac{1}{\sqrt{n}} \le \frac{2 + \cos{(n)}}{\sqrt{n}} \le \frac{3}{\sqrt{n}} $$
S... | For $n\geq 16 $ we have $|a_{n+1}|=(|2+\cos n|/\sqrt n)|a_n|\leq(3/4)|a_n|$ so for $n\geq 16$ we have $|a_{n+1}|\leq (3/4)^{n-15}|a_{16}|.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Isomorphism between two groups mapping the same elements I need to show that there is an isomorphic function $f:\Bbb Z_6 \to U(14)$ that $f(5)=5$
I can show that $U(14)$ is isomorphic to $\Bbb Z_6$ by showing that $U(14)$ is cyclic of order 6, but I do not know whether this would imply that $\Bbb Z_6$ is isomorphic to ... | Note that if there exists an isomorphism f: G ----> G', f^-1 is an isomorphism from G' to G. (If f is a bijection, then it inverse is as well, and t is easy to check that the condition for isomorphisms hold for inv(f).
| {
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meaning of $B \in \left \{ \mathcal P(A)|A \in \mathcal F \right \}$
$$B \in \left \{ \mathcal P(A)|A \in \mathcal F \right \}$$
I am having a hard time understanding what this is saying. I know how to rewrite this to $\exists A \in \mathcal F(B=\mathcal P(A))$, which is the same meaning, but still I am confused ab... | $B$ is not an element of the powerset of $A,$ where $A\in \mathcal F.$ Rather, $B$ is a powerset of $A$ for some $A\in\mathcal F.$ As you wrote yourself, there exists $A\in\mathcal F$ such that $B=\mathcal P(A).$
When we say that $B\in\{\mathcal P(A)\mid A\in \mathcal F\},$ we're saying that $B$ is an element of the se... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Overview of Linear Algebra Very new student tackling this course, and I've never been this terrified from Math before. I cannot grasp the meaning of things in Linear Algebra, most of what's stated is either obscure, meaningless, and abstract when I am first tackling them. What sort of advice can you give?
I'll be more ... | First of all, while it is good to have an intuition and visualization for what mathematical objects are, you cannot remove yourself from the formal definitions. When things get too complicated to visualize, knowing and being able to apply definitions will save you.
That being said, recall that a vector space $X$ is a s... | {
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Why proximal gradient instead of plain subgradient methods for LASSO? I was thinking to solve LASSO via vanilla subgradient methods. But,I have read people suggesting to use Proximal GD. Can somebody highlight why proximal GD instead of vanilla subgradient methods be used for LASSO?
| OK, since you liked my comments... :-)
*
*Better performance. That's it.
*Longer, but more handwaving, answer: proximal methods allow you to use a lot more information about the function at each iteration. Also, proximal gradient methods take into account a much larger neighborhood around the initial point, enablin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Largest number among the given numbers How to find the largest number among the following numbers.
$(a)\;3^{210}$
$(b)\;7^{140}$
$(c)\;17^{105}$
$(d)\;31^{84}$
Any help will be greatly appreciated.
| If you are allowed a calculator, take the logs and compute them. $\log 3^{210}=210 \log 3$ and so on
If you must do it by hand, you need to be clever. To compare a and b, note that the exponents are both divisible by $70$, so you are comparing $(3^3)^{70}$ and $(7^2)^{70}$. You may know $3^3$ and $7^2$, or they ar... | {
"language": "en",
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How do I isolate $y$ in $y = 4y + 9$? $y = 4y + 9$
How do I isolate y?
Can I do
$y = 4y + 9$
$\frac{y}{4y} = 9$ etc
Also some other questions please:
*
*$\frac{5x + 1}{3} - 4 = 5 - 7x$
In the above (1), if I want to remove the '3' from the denominator of the LHS, do I multiply everything on the RHS by 3? What abo... | You have the right idea to get $y$ by itself on the left. However, instead of dividing by $4y$, you would subtract $4y$ from both sides. This will give you
$$y - 4y = 9.$$
Now, if you combine the like $y$-terms on the left. What would $y - 4y$ turn into? You can think of this as: $1y - 4y$.
For your next problem, if y... | {
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An inequality involving the $L^1$ norm and a Fourier transform Can someone please tell me where this inequality comes from:
$$\int_{\infty}^{\infty} |g(\xi)|\cdot | \hat{\eta}(\xi) |^2 \;d\xi \leq \frac{1}{2\pi} \| \eta \|_{L^1(\mathbb{R})}^2 \int_{-\infty}^{\infty} |g(\xi)| \; d\xi$$
where the hat denotes the Fourier ... | Starting with Hölder, we have
$$ \def\norm#1{\left\|#1\right\|} \norm{g \hat \eta^2}_{L^1} \le \norm{g}_{L^1} \norm{\hat \eta^2}_{L^\infty} $$
By definition of $\hat \eta$, we have
\begin{align*}
\def\abs#1{\left|#1\right|}\abs{\hat \eta(\xi)} &=\abs{ \frac 1{(2\pi)^{1/2}} \int_{\mathbf R} \eta(x) \exp(-ix\xi)\, dx}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Write $n^2$ real numbers into $n \times n$ square grid Let $k$ and $n$ be two positive integers such that $k<n$ and $k$ does not divide $n$. Show that one can fill a $n \times n$ square grid with $n^2$ real numbers such that sum of numbers in an arbitrary $k \times k$ square grid is negative, but sum of all $n^2$ numbe... | This can be done so that all rows are the same. Here's the idea of my solution: let the first column all have a value of $a > 0$, to be chosen later. Let the next $k - 1$ columns all have $-1$ in them. Then repeat this pattern: one column of $a$'s, $k-1$ column's of $-1$'s. Then every $k \times k$ square has exactl... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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representation of a complex number in polar from
Write the following in polar form: $\frac{1+\sqrt{3}i}{1-\sqrt{3}i}$
$$\frac{1+\sqrt{3}i}{1-\sqrt{3}i}=\frac{1+\sqrt{3}i}{1-\sqrt{3}i}\cdot\frac{1+\sqrt{3}i}{1+\sqrt{3}i}=\frac{(1+\sqrt{3}i)^2}{1^2+(\sqrt{3})^2}=\frac{1+2\sqrt{3}i-3}{4}=\frac{-2+2\sqrt{3}i}{4}=-\frac{1... | HINT:
$$\frac{1+i\sqrt{3}}{1-i\sqrt{3}}=$$
$$\left|\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right|e^{\arg\left(\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)i}=$$
$$\frac{|1+i\sqrt{3}|}{|1-i\sqrt{3}|}e^{\left(\arg\left(1+i\sqrt{3}\right)-\arg\left(1-i\sqrt{3}\right)\right)i}=$$
$$\frac{\sqrt{1^2+\left(\sqrt{3}\right)^2}}{\sqrt{1^2+\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1489339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $\ \limsup_{n\to \infty}(\frac{2^n}{n!}) $ Find$\ \limsup_{n\to \infty}(\frac{2^n}{n!}) $
The following 2 facts have already been proven.
*
*$\ 2^n < n! $ for $n \ge 4$ (Proof by induction)
*$\ \frac{2^{n+1}}{(n+1)!} < \frac{2^n}{n!} $ for $\ n\ge 4 $
It would be sufficient to show that $\ \lim_{n\to \infty} \... | This might be a little simpler: For large $n,$
$$\frac{2^n}{n!} = \frac{2}{n}\frac{2}{n-1}\cdots \frac{2}{3}\frac{2}{2}\frac{2}{1}\le \frac{4}{n(n-1)}\cdot 2 = \frac{8}{n(n-1)}\to 0.$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What would this region even look like and how can I sketch it? I need to sketch the following region in $\mathbb{R^3}$.
$$D=\{(x,y,z) : 0 \leq z \leq 1-|x|-|y|\}$$
I really have no idea how to go about sketching this and I can never visualise in 3D.
I have tried letting $x,y,z=0$ and seeing if that can help me see what... | If you are comfortable sketching functions in the plane, first think in horizontal planes at different heights. This amounts to fixing $z = h$ (a given height) and letting $x$ and $y$ vary.
At height $h$ what you see is the plot of the function $h = 1 - |x| - |y|$, that is, you should plot
$$|y| = 1-h -|x|$$
$$y = \pm... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Number of words with length 3 with certain restrictions In today's quiz (about combinatorics) there was a question I was not able to solve mathematically:
Find the number of words with length 3 over an alphabet $ \Sigma = \{1,2,...,10\} $ where $x \lt y \le z$. ($x$ is the first letter, and so on).
How can I solve this... | Think about it like this: If we had to choose $a,b,c$ from $\{1,2,\ldots,10\}$ so that $a < b < c$, we could just choose a subset of $3$ elements from $\{1,2,\ldots,10\}$ and sort them, giving $a < b < c$. The number of such subsets is $\binom{10}{3}$. Now, since we have $x < y \leq z$, we have $x < y < (z + 1)$; we ... | {
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"timestamp": "2023-03-29T00:00:00",
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I know R is closed and open but is it a closed and open interval? I know $\mathbb{R}$ is open and closed but is this the same as saying it's an open interval and a closed interval?
| This really depends on your working definition of "interval" — that is, your textbook's definition or your teacher's.
If "interval" is defined as:
*
*$I \subseteq \mathbb{R}$ is an interval $\iff$ there are $x, y \in \mathbb{R}$ such that $I$ is one of $[a,b], [a,b), (a,b]$ or $(a,b)$
then No, $\mathbb{R}$ ... | {
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Represent $ f(x) = 1/x $ as a power series around $ x = 1 $ As stated on the title, my question is: (a) represent the function $ f(x) = 1/x $ as a power series around $ x = 1 $. (b) represent the function $ f(x) = \ln (x) $ as a power series around $ x = 1 $.
Here's what I tried:
(a) We can rewrite $ 1/x $ as $ \frac{... | $$
\begin{align}
\frac1x
&=\frac1{1+(x-1)}\\
&=1-(x-1)+(x-1)^2-(x-1)^3+\dots\\
&=\sum_{k=0}^\infty(-1)^k(x-1)^k
\end{align}
$$
(a) You are correct; your series is the same as mine, however, usually we expand in powers of $(x-a)^n$.
(b) integrating $\frac1t$ between $t=1$ and $t=x$ gives
$$
\begin{align}
\log(x)
&=\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Taylor Series Expansion of $ f(x) = \sqrt{x} $ around $ a = 4 $ guys.
Here's the exercise: find a series representation for the function $ f(x) = \sqrt{x} $ around $ a = 4 $ and find it's radius of convergence.
My doubt is on the first part: I can't seem to find a pattern.
$ \circ f(x) = \sqrt x \rightarrow f(4) = 2 \\... | You were on the right track. We have for $n\ge 2$
$$\begin{align}
f^{(n)}(x)&=(-1)^{n-1}\frac12 \frac12 \frac32 \frac52 \cdots \frac{2n-3}{2}x^{-(2n-1)/2}\\\\
&=(-1)^{n-1}\frac{(2n-3)!!}{2^n}x^{-(2n-1)/2}\tag 1
\end{align}$$
where $(2n-3)!! = 1\cdot 3\cdot 5\cdot (2n-3)$ is the double factorial of $2n-3$. Evaluating ... | {
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"url": "https://math.stackexchange.com/questions/1490190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the probability that $A$ occurs before $B$ Given an experiment has $3$ possible outcomes $A, B$ and $C$ with respective probabilities $p, q$ and $r$ where $p + q + r = 1.$ The experiment is repeated until either outcome $A$ or outcome $B$ occurs. How can I show that $A$ occurs before B with probability $\dfrac{... | Let $a$ denote the given probability. Then by independence, we see that $$a = p + ra,$$
which yields
$$a = \frac p{1-r}= \frac p{p+q}. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find the area of the triangle There are two points $N$ and $M$ on the sides $AB$ and $BC$ of the triangle $ABC$ respectively. The lines $AM$ and $CN$ intersect at point $P$. Find the area of the triangle $ABC$, if areas of triangles $ANP, CMP, CPA$ are $6,8,7$ respectively.
|
From the hypothesis, it follows that
*
*$CN:CP = area(\triangle CAN):area(\triangle CPA)=13:7$
*$MP:MA =area(\triangle CMP):area(\triangle CMA)=8:15$
Applying Menelaus' Theorem to $\triangle APN$ with $B, M,C$ we have that
$$\frac{BN}{BA}\cdot \frac{CN}{CP}\cdot \frac{MP}{MA}=1,\tag{3}$$
(note that we use non-di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Does this integral have a closed form? $\int^{\infty}_{-\infty} \frac{e^{ikx} dx}{1-\mu e^{-x^2}}$ The original integral contains $\sin [n(x+a)] \sin [l(x+a)]$ but I think this form is simpler:
$$\int^{\infty}_{-\infty} \frac{e^{ikx} dx}{1-\mu e^{-x^2}}$$
$\mu <1$, so I can use the Taylor expansion to calculate the int... | We have:
$$\begin{eqnarray*}\int_{-\infty}^{+\infty}\frac{e^{ikx}}{1-\mu e^{-x^2}}\,dx = \int_{-\infty}^{+\infty}\frac{\cos(kx)}{1-\mu e^{-x^2}}\,dx &=& 2\sum_{r=1}^{+\infty}\mu^r\int_{0}^{+\infty}\cos(kx)e^{-rx^2}\,dx\\&=&2\sum_{r\geq 1}\mu^r \sqrt{\frac{\pi}{r}}\,e^{-\frac{k^2}{4r}}\\&=&2\sqrt{\pi}\sum_{r\geq 1}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why E is the same for hypergeometric and binomial laws I'm confused a bit, not sure if it's tiredness or something but...
Why do the mathematical expectation of the hypergeometric law is the same as the binomial one?
After all, the hypergeometric one correspond to sampling without replacement, while for binomial it's w... | This can be regarded as a consequence of linearity of expectation, even for non-independent random variables. The marginal (i.e. unconditional) probability for each ball remains the same even without replacement.
For example, the first ball has a probability $\dfrac{N}{N+M}$ of being black and $\dfrac{M}{N+M}$ of bein... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Using $dx$ for $h$. Is it mathematically correct to write
$$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx},$$
rather than
$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}?$$
If not, what is the difference? If so, why isn't this notation used from the beginning? My feeling for the latter is that it would align the derivative mor... | No it doesn't matter, as pointed out what you write doesn't matter, as long as the concept gets across, $dx$, $f$, $john$ $\oplus$, you can use any of them, $h$ is just the common one and used because everyone recognises it but in certain context, to make clear, you must use other letters as you're doing more than one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
if $(X,A)$ has homotopy extension, so does $(X \cup CA,CA)$ I guess I could use the property: The homotopy extension property for $(X,A)$ is equivalent to $X\times \{0\}\cup A\times I\ $ being a retract of $X\times I$.
Then there is a retraction:$$X\times I\rightarrow X\times \{0\}\cup A\times I$$
I need to show$$(X\cu... | Putting it more generally, if $(X,A)$ has the HEP, and $f:A \to B$ is a map, then $(B\cup_f X,B)$ has the HEP. The proof is analogous to that of Stefan, which is the case $B=CA$.
You need the method of defining homotopies on adjunction spaces, which again uses the local compactness of $I$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proof of the hockey stick/Zhu Shijie identity $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$ After reading this question, the most popular answer use the identity
$$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1},$$
or, what is equivalent,
$$\sum_{t=k}^n \binom{t}{k} = \binom{n+1}{k+1}.$$
What's the name of this ident... | This is essentially the same as the induction answer already mentioned, but it brings a pictorial perspective so I thought to add it as an alternative answer here.
Here's a restatement of the identity (which you can verify to be equivalent easily): On Pascal's triangle, start from a number (one of the $1$s) on the left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "73",
"answer_count": 18,
"answer_id": 15
} |
Finding a triangle ratio.
In the triangle ABC, the point P is found on the side AB. AC = 6 cm, AP = 4 cm, AB = 9 cm. Calculate BC:CP.
For some reason, I cannot get this even though I tried for half an hour.
I got that, $BC/CP < 17/10 = 1.7$ by the triangle inequality. $AP/PB = 4/5$
But that does not help one but, I'm... | Let $\angle{BAC}=\theta$. Then, by the law of cosines,
$$\begin{align}BC:CP&=\sqrt{9^2+6^2-2\cdot 9\cdot 6\cos\theta}:\sqrt{4^2+6^2-2\cdot 4\cdot 6\cos\theta}\\&=\sqrt{9(13-12\cos\theta)}:\sqrt{4(13-12\cos\theta)}\\&=\color{red}{3:2}\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Probabilistic approach to prove a graph theory theorem A theorem in graph theory is as the following,
Let $G=(V,E)$ be a finite graph where $V$ is the set of vertexes and $E$ is the set of edges. Then there exists a vertex subset $W$, i.e. $W \subset V$, such that the number of edges connecting $W$ and $W^C$ is at lea... | Take the graph $G = (V,E)$ and for each vertex, randomly select whether it will lie in $W$ with independent probability $p = \frac{1}{2}$. For each edge $e_i \in E$, define the random variable $X_i$, which is $1$ if $e$ connects $W$ to $W^C$, and $0$ otherwise. Then, $\mathbb{P}(X_i = 1) = \frac{1}{2}$, and $\mathbb{E}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1491000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Is Abelian Group Subset consisting of elements of finite order a Subgroup? I know this question has asked and answered before, so my apologies for posting a duplicate query. Instead of the solution, I'm seeking a "nudge" in the right direction to complete the proof on my own. So please, hints or criticism only. Than... | HINT: Suppose that $a,b\in H$; you need to show that $ab\in H$. There are positive integers $m,n$ such that $a^m=b^n=e$; use the fact that $G$ is Abelian to show that $(ab)^{mn}=e$ and hence that $ab\in H$. Basically this requires showing by induction that $(ab)^k=a^kb^k$ for $k\ge 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1491092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding the value of a summation Question:
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1)=1$, $c(2n)=c(n)$, and $c(2n+1)=(-1)^nc(n)$. Find the value of $\displaystyle\sum_{n=1}^{2013}c(n)c(n+2)$.
To solve this question, I tried to figure out some patterns.
we are given $c(1)=1$ bu... | Rewrite the sum as
$$S = \sum_{m=1}^{1006}c(2m)c(2m+2) + \sum_{m=0}^{1006}c(2m+1)c(2m+3).$$
The first sum covers the even $n$; the second covers the odd $n$.
Then replace:
$$S = \sum_{m=1}^{1006}c(m)c(m+1) + \sum_{m=0}^{1006}(-1)^m c(m) (-1)^{m+1} c(m+1)$$
$$S = c(1)c(3) + \sum_{m=1}^{1006} [1 + (-1)^m(-1)^{m+1}] c(m)c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1491200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find $2^{3^{100}}$ (mod 5) and its last digit The question is to find $2^{3^{100}} \pmod 5$ and its last digit.
I think we have to find $2 \pmod 5$ and $3^{100}\pmod 5$ separately, right?
$$2 = 2 \pmod 5$$
$$3^4 = 1 \pmod 5$$
$$3^{100} = 1 \pmod 5$$
$$2^{3^{100}} = 2^1 = 2 \pmod 5$$
Is this solution correct? And to fin... | Yes, you solve modulo $10$ for the last digit. Note that the number is even, so its last digit is taken from $2, 5, 6, 8$--it's not divisible by $5$ so the last digit cannot be $0$. Your original solution is a bit off, though $3^4\equiv 1\mod 5$, but in the exponent it's every $4$ which gives $1$, so we want $3^2\equiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1491279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Intuitive understanding of the -b in this definition of a plane I am learning about SVMs in computer science. The book I'm reading defines a hyperplane with an "intercept term", b
$\vec{w}^T \vec{x}= -b$
What does this intercept mean, intuitively? From Khan Academy, I understand how the dot product of a vector on a pla... | That's not too complicated. Let us imagine $n=3$. Suppose we are looking for plane whose unit normal is ${\bf{n}}$ and passes through the point ${{{\bf{x}}_0}}$. Now the equation of all point lying on this plane will be
$$\eqalign{
& {\bf{n}}.\left( {{\bf{x}} - {{\bf{x}}_0}} \right) = 0 \cr
& {\bf{n}}.{\bf{x}} - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1491370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $C(a) $ ={$r \in R : ra = ar$} is a subring of $R$ containing $a$. For a fixed element $a \in R$ define $C(a)$ ={$r \in R : ra = ar$}. Prove that $C(a)$ is a subring of $R$ containing $a$.
attempt: Recall by definition , $B$ is a subring of $A$ if and only if $B$ is closed under subtraction and multiplicati... | Let $c\in C (a)$. Then $ac=ca$. Now we have $-ca=-(ca)=-(ac)=-ac=a (-c)$. This shows that $-c\in C (a)$, and the above argument completes the proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1491470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What does $P(A \triangle B)$ mean? I know its a simple question but its hard to google these symbols. Anyways, anyone care to explain what $P(A \triangle B)$ means? I haven't seen this symbol before and I'm not sure on how to interpret it.
| The triangle is almost certainly being used to for symmetric difference:
$$A\mathrel{\triangle}B=(A\setminus B)\cup(B\setminus A)=(A\cup B)\setminus(A\cap B)\;.$$
The $P$ might be the power set operator, in which case the whole thing is the set of all subsets of $A\mathrel{\triangle}B$:
$$P(A\mathrel{\triangle}B)=\{X:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1491702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
(Combinatorial) proof of an identity of McKay Lemma 2.1 of this paper claims that for integer $s>0$ and $v \in \mathbb{N}$, we have
$$
\sum_{k=1}^s \binom{2s-k}{s} \frac{k}{2s-k} v^k (v-1)^{s-k}
= v \sum_{k=0}^{s-1} \binom{2s}{k} \frac{s-k}{s} (v-1)^k
$$
The author gives a combinatorial interpretation of the left hand ... | Suppose we seek to verify that
$$\sum_{k=1}^n {2n-k\choose n} \frac{k}{2n-k} v^k (v-1)^{n-k}
= v \sum_{k=0}^{n-1} {2n\choose k} \frac{n-k}{n} (v-1)^k.$$
Now the LHS is
$$\frac{1}{n}\sum_{k=1}^n {2n-k-1\choose n-1} k v^k (v-1)^{n-k}.$$
Re-write this as
$$\frac{1}{n}\sum_{k=1}^n {2n-k-1\choose n-k} k v^k (v-1)^{n-k}.$$
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1491768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Correct Terminology for Semi Inverse Mapping Suppose we have two finite sets $X$ and $Y$ and a many to one mapping $f:X\rightarrow Y$.
Now let me define another mapping $g:Y\rightarrow\mathcal{P}(X)$ where $\mathcal{P}$ denotes the power set. We define $g(y)=\{x|x\in X, f(x)=y\} \forall y$.
Can we call $g$ as an inve... | More generally, there is a function $$f^{-1}:\mathcal{P}(Y) \to \mathcal{P}(X):S \mapsto \{x \in X \ \vert \ f(x) \in S\}$$ called the inverse image (or preimage) of $f$.
The map you described is the special case where you only consider singletons. This could be called the fiber map (this is not usual terminology thou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1491864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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