Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
find sigma Algebra of X If the function $X:\mathbb{R}\to\mathbb{R}$ defined by $X:=I_{[0,1]}+4I_{[2,4]}.$
What is $\sigma(X)$
Could it be
$\{B_1\cup B_2\cup B_3:B_1\subseteq\{[0,1]\},B_2\subseteq\{[2,4]\},
B_3\subseteq \{(-\infty,1)\cup(1,2)\cup(4,\infty)\}\}$ ??
| $\sigma (X)$ is same as $\sigma ([0,1],[2,4])$. In general $\sigma (A,B)=\{ \varnothing, \mathbb R, A,B,A^{c},B^{c}, A\cap B, B\cup A, B^{c}\cap A, B\cap A^{c},B^{c}\cup A, B\cup A^{c}, B^{c}\cup A^{c},B^{c}\cap A^{c}\}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Visualizing Lagrange multipliers Sorry if this seems like a very basic question but I am having trouble visualizing Lagrange multipliers. Particularly the equation:
$ \nabla f = \lambda * \nabla g $
f = function to maximise. g = constraint.
I don't understand why equating the gradients in such a way produces the extrem... | Suppose $x_0$ is an extremum, then the constraint contour $g(x) = 0$ and cost contour $f(x)=f(x_0)$ pass through the point $x_0$.
If the contours are not 'parallel' at $x_0$, that is suppose they cross. Then you can see that there are points on the contour $g(x) = 0$ that have higher and lower values of $f$,
which cont... | {
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Assume $f$ is uniformly continuous and unbounded on $[0,\infty)$. Whether the following statement is true? Assume $f$ is uniformly continuous and unbounded on $[0,\infty)$. Is it true that either $\displaystyle\lim_{x\to\infty}f(x)=+\infty$ or $\displaystyle\lim_{x\to\infty}f(x)=-\infty$
Please give a hint for the pro... | Intuitively, $x\sin(x)$ was on the right the track but the function has to keep forcing itself to $0$ every $\pi$, so if only there was a way to slow down the oscillation as the function grows unbounded.
For a pretty counterexample, use
$$f(x) = \begin{cases} x\sin(\log|x|) & x\neq 0 \\ 0 & x=0 \\ \end{cases}$$
The d... | {
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What is $\arctan(k\cdot\tan(x))$? Is there any simplification for this? What is $\arctan(k\cdot\tan(x))$ if $k$ is a real number? Is there any simplification for this? I want solve it interms of $x$. I would be happy if it was $x\cdot\arctan(k)$ but it is not. So a similar solution would help.
Thanks in advance.
| There may not be an elementary answer, but you could Taylor expand to however many terms you need. For example, if $|k\cdot \tan (x)| < 1$ then
$$\arctan(k\cdot \tan(x)) \approx k\cdot \tan (x) - \frac{(k\cdot \tan (x))^3}{3} + \cdots$$
converges rather quickly. If instead $k\cdot \tan (x) > 1$, then use $\arctan(x) =... | {
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Equations like Taylor Formula for polynomials
Consider the linear maps $D: P_n(\mathbb{R}) \to P_n(\mathbb{R})$ given by
$$(Dp)(x) = \sum_{i = 1}^{n}ic_i x^{i-1}$$
and $A: P_n(\mathbb{R}) \to P_n(\mathbb{R})$ given by
$$(Ap)(x) = \sum_{i=0}^{n}c_i (x+1)^i,$$
where $p(x) = \sum_{i=0}^{n}c_i x^i$
Prove that
$$1 + \frac{... | Both $\ A\ $ and $\ D: P_n(\mathbb{R}) \to P_n(\mathbb{R})\ $
are linear. Hence
$$ \frac I{0!}+\frac D{1!}+\ldots+ \frac{D^n}{n!} $$
which formally is $\ \sum_{k=0}^n \frac{D^k}{k!},\ $ is linear too.
Thus, it is enough to verify that
$$ \forall_{m=0}^n\qquad
\left(\sum_{k=0}^n\frac{D^k}{k!}\right)\left(x^m\right)\... | {
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Can a Closed interval be a basis for Usual topology on$R$? I was thinking if $B$$=${$[a,b]$ , $a$ $\neq$ $b$ and a,b belong to $R$} is a basis for Usual topology on $R$.
I could not find a problem in this basis satisfying the two properties of basis.
Edit: if the answer to the above question is yes, then that fundament... | The set $B$ will not form a basis for any topology on $\mathbb{R}$. Recall that a basis must have the property that for any $B_1, B_2 \in B$ and $x \in B_1 \cap B_2$, there exists $B_3 \in B$ such that $x \in B_3 \subset B_1 \cap B_2$. This is not satisfied by $B$, since $[a,b] \cap [b,c] = \{b\}$, which contains no el... | {
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How is it that this partial derivative is equal to 0? In my probability textbook are the following equations:
$Z=X/Y$ and $X=W$.
There is a Jacobian as follows:
$$J =
\begin{bmatrix}
\frac{\partial x}{\partial z} &
\frac{\partial x}{\partial w} \\[1ex] % <-- 1ex more space between rows of matrix
\frac{\partia... | Think of $x$ as a function of two variables $z$ and $w$. Then $x(z,w)=w$. Since $z$ doesn't appear in this expression, you have $\partial x/\partial z = 0$.
The point is to express $x$ and $y$ as functions of $z$ and $w$. Then regarding $x$ and $y$ as the dependent variables, and $z$ and $w$ as the independent variable... | {
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How do I prove that for an nxn matrix M, with all eigenvalues being positive that (M+Id) is or is not invertible? So far I've tried creating a regressive definition for the determinant of an nxn matrix to create an inequality. I know that Det(M) is positive and I think Det(I+M) is positive, I've also tried creating cou... | The eigenvalues of $A$ are the values of $\lambda$ that make $A-\lambda I$ a singular matrix. All other values of $\lambda$ make $A-\lambda I$ invertible.
If all of the eigenvalues of $A$ are positive, then $A+d I$ must always be invertible because the eigenvalues all correspond to negative values of $d$.
I have assume... | {
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Matrix equality with identity
Given two matrices $A$ and $B$. To show:
Part 1: $(I+AB)^{-1}=I-A(I+BA)^{-1}B$, provided $BA$ and $AB$ exist.
Part 2: Let $a$ and $b$ be scalars. Using part 1, to show $(aI_{k}+bJ_{k})^{-1}=I_{k}/a -bJ_{k}/\{a(a+kb)\}$, where $I_p$ is the $p\times p$ identity matrix, and $J_p$ is the $p\t... | You write $B(I+AB)=B+BAB=(I+AB)B$, but the last term should be $(I+BA)B$ (the order of $A$ and $B$ matters since $AB \neq BA$ in general).
I can't comment on the second part because I'm not sure what is meant by $J_p$ and $1_p$.
| {
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Boolean algebras and measures without complements A (concrete) Boolean algebra $\mathcal{B}$ on a set $X$ is any nonempty set of subsets of $X$ that is closed under union, intersection, and complements relative to $X$. With some redundancy, this implies that it is characterized by the following axioms:
*
*$\varnoth... | A Boolean algebra is a distributive lattice with negation. Dropping the negation part, you are just left with the distributive lattice part. So for your first question: a (concrete) distributive lattice would be the right name.
| {
"language": "en",
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How many ways can a tennis player be scheduled to play $4$ matches in $8$ days, with at most one match per day? Should I use combinations or permutations?
How many ways can a tennis player be scheduled to play $4$ matches in $8$ days, with at most one match per day?
Currently, I work mostly in combinatorics to catch ... | The question asks for the number of ways it is possible to select the days he plays a tennis match. There are only
$$\binom{8}{4} = \frac{8!}{4!4!} = 70$$
ways to select four of the eight days for him to play a tennis match.
It does not say anything about his opponents. In particular, it does not specify that he is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3346078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the smallest positive integer $m$ such that there exists positive integer $n$ that satisfies...
Find the smallest positive integer $m$ such that there exists positive integer $n$ that satisfies that $\lvert {n\over m} - {2\over 5} \rvert\le {1\over100}$
I tried to simplify and turn it into the following
$$\lvert... | $|\frac{n}{m}\,-\,\frac{2}{5}|\,\leq\frac{1}{100} $
$$ |\frac{100n}{m}\,-\,40|\, \leq 1. $$
39$\leq\frac{100n}{m}\,\leq41$
Therefore $\frac{n}{m}=\frac{2}{5}$ Hence 5n=2m hence min m will be 5 when n=2.
| {
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Conversion from Ito equations to Stratonovich I have the stochastic equations
$$
dx = pdt + \beta (x^2 + p)dV \\
dp = xdt - \gamma (x^3 + p^2)dW \\
$$
where $dV$ and $dW$ are mutually independent Wiener processes. I am asked to calculate the corresponding Stratonovich equations.
I know how to go from an Ito-integral t... | Hint:
Assume $n$-dimensional Ito process $\mathbf{X}$ which has the following dynamics:
\begin{equation}
d\textbf{X}_t = \textbf{b}(\textbf{X}_t,t)dt + \textbf{B}(\textbf{X}_t, t)d\textbf{W}_t
\end{equation}
where $\mathbf{b}$ : $\mathbb{R}^n\to\mathbb{R}^n$ and $\mathbf{B}: \mathbb{R}^n\to\mathbb{R}^{n\times m}$... | {
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Urn without Replacement, all balls the same There's an urn with 12 balls: 5 white, 7 black. 4 balls are picked without replacement.
What are the odds that all balls picked are white?
I know part of the answer involves the term $\frac{7}{12}*\frac{6}{11}*\frac{5}{10}*\frac{4}{9}$, but I also know that since order doesn... | We may temporarily imagine that every ball has a unique number on it.
Now... we have the option to treat our sample space as the ways in which we draw four balls in sequence and we record the numbers of the balls that we drew, in which case we have $12\times 11\times 10\times 9$ possible equally likely results...
Or...... | {
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Bijection and Isomorphim in Set Theory Let A and B be two well-ordered sets. Is it ture that A and B are isomorphic if A and B are bijective?
Motivation:
I am studying Bourbaki's Elements of Methematics and have just reached Sec 3.2, Chap 3 where the concept of $\mathbb{N}$ have not appeared yet but Bourbaki seems to h... | The statement above is false. For example, consider:
*
*The natural numbers with the usual ordering.
*The set $\{1-{1\over n}: n\in\mathbb{N}_{>0}\}\cup\{1\}$ with the usual ordering.
(The corresponding ordinals are $\omega$ and $\omega+1$, respectively.)
These sets are each countable, so in bijection with each oth... | {
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What can I conclude from this inequality? I was solving an equation and reached this inequality
$$t^2−1≤\sin^2\alpha$$
*
*What conclusions can I draw about $t$ from the above inequality (given that $\alpha \in \mathbb R$)?
*Is it correct to use the minimum value of RHS (zero) and proceed?
Here's the complete contex... | Since the minimum value of $\sin^2 \alpha$ is $0$, this means that for any real $t$ in $-1 ≤ t ≤ 1$, the inequality will hold regardless of the value of $\alpha$.
Similarly, the maximum value of $\sin^2 \alpha$ is $1$. This means that for any $t$ that satisfy $t < -\sqrt{2}$ or $t > \sqrt{2}$, the inequality will not h... | {
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Use Z-transform on the sequence $x(n) = \frac 1{(n+1)(n+2)...(n+p)}$ Use Z-transform on the sequence $x(n) = \frac 1{(n+1)(n+2)...(n+p)}, $ $n\geq 0$. I know that the Z-transform on a sequence is:
$$Z\{x(n)\}=\sum_{n=0}^{\infty}x(n)z^{-n}$$
But with this exercise I should arrive at something like:
$$Z\{x(n)\}=\frac {(1... | Note that $x(n)$ can be expressed in terms of the Falling and Rising Factorial
as
$$
x(n) = {1 \over {\left( {n + 1} \right)^{\overline {\,p\,} } }} = n^{\,\underline {\, - \,p\,} }
$$
The Falling and Rising Factorials have a simple expression for the finite difference and summation
$$
\left\{ \matrix{
\Delta _{\,n}... | {
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Finding Injectivity in function I have a doubt with the following problem. I have to find if the function is injective, surjective and if bijective find the inverse function
$$f : \mathbb{Z} \longrightarrow \mathbb{N},\qquad
f(a) =
\begin{cases}
2a & \text{if } a >0, \\
1 - 2a & \text{if } a \le 0.... | The function is injective if you draw the graph. It looks like a ''V''.
But it is not surjective, since $0$ doesn't lie in the image: If $a>0$, then $f(a)=2a>0$ and if $a$ is non-positive, the function has the form: $f(0) = 1$, $f(-1)=3$, $f(-2)=5$ and so on.
| {
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Proving $\sqrt{2}$ is irrational I had my first lecture the University of Toronto, MAT157 with professor Meinrenken. I am struggling with proof #3 which he briefly walked us through. I do not understand the method of thinking and there are a lot of missing lines. Can someone guide me through how this proof works? I hav... | Welcome to Math StackExchange!
So, the third from last line. This is saying that $x = \sqrt{2}-1$, or $x + 1 = \sqrt{2}$. Now, if $x > 1$, then $x+1 > 2$. But, $x +1 = \sqrt{2}$, so that would imply $\sqrt{2} > 2$, or $4 > 2$, which is a contradiction. Therefore, $x<1$. Likewise, if $x<0$, $x+1 <1$, so $\sqrt{2} < 1$,... | {
"language": "en",
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Divisibility of $(x+2y)^{2n}-x^{2n}$ by $2$ Let $n$ be a positive integer. What is the largest integer $f(n)$ such that there exists $p\in \mathbb{Z}[x, y]$ such that $$(x+2y)^{2n}-x^{2n}=2^{f(n)} p?$$
I have computed it for some small values and my conjecture is that $f(n)$ is the order of divisibility of $n$ by $2$ p... | By the binomial theorem, the left-hand side is simply
$$\sum_{i=1}^{2n} \binom{2n}{i} 2^i y^i x^{2n-i},$$
so $f(n)$ is just
$$\min_{1 \leq i \leq 2n} v_2\left(\binom{2n}{i} 2^i\right) = \min_{1 \leq i \leq 2n} \left(i + v_2 \binom{2n}{i}\right),$$
where $v_2(x)$ is the number of times $2$ divides $x$. So, as noted by a... | {
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Vector Spaces problems and axioms I'm currently stuck in two vector space problems. I feel as though the signs and my lack of knowledge throw me off. I am not aware of what $\boxplus, \boxdot$ mean in the first problem (should I just think of them as addition and multiplication). Same happens with problem 2. Should I a... | The sign $(x_1,y_1)\boxplus (x_2,y_2)$
between the two pairs of numbers defines an new kind of addition
same with the sign $\boxdot$ in $\alpha\boxdot(,)=(,0)$a new kind of multiplication by a scalar is defined, you have to find out if with these new definitions in $\mathbb{R}^2$ you have still a Vector space, if not... | {
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Proving $\Bbb C$ is algebraically closed by analytic approach. I want to prove that $\Bbb C$ is algebraically closed in the following way. Let us first take a non-constant polynomial $F$ with coefficients from $\Bbb R.$ Then $F \in \Bbb R[X] \subset \Bbb C[X].$ Now consider the polynomial function $$F : \Bbb C \longri... | You use the open mapping theorem, which states that every analytic non-constant map from an open connected subset of $\mathbb C$ into $\mathbb C$ is open.
| {
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How to interpret this combination formula
If you have $m$ and $n$ number of distinct items, and you get to choose $p$ and $q$ things respectively out of them, then the number of ways you can permute $p+q$ items is as follows:
$m$ items can be chosen for $p$ places like $^mC_p$
$n$ items can be chosen for $q$ places l... | Here's a way to think of things. In the ${}^mC_p{}^nC_q(p+q)!$ way of counting things, you are picking $p$ out of the $m$ objects from the first set, then picking $q$ out of the $n$ objects from the second set, then taking these $p+q$ objects and lining them up in a row, while in the second way of counting things, you ... | {
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Intersection coordinates of the two polar curves I am learning pre-calculus and I am not able to answer this question from the textbook:
Find the rectangular coordinates of all the points of intersection of the two polar curves ${\sqrt 3}\sin\theta=r$ and $\cos\theta=r$
| Dealing with intersections of polar curves can be tricky. It would be awesome if we could just set the two equations equal to each other and solve for $\theta$.
$$\sqrt3\sin\theta=\cos\theta\\
\tan\theta=\frac1{\sqrt 3}\\
\theta=\frac\pi6,\frac{7\pi}6\\
(r,\theta)=\left(\frac{\sqrt3}2,\frac\pi6\right),\left(\frac{-\sq... | {
"language": "en",
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If you draw two cards in consecutively in a standard deck of 52 cards, what is the probability of getting black on the second draw? This is my thought process:
$P(2^{nd}\ \text{black}) = P(2^{nd}\ \text{black} \mid 1^{st}\ \text{red}) P(1^{st}\ \text{red}) + P(2^{nd}\ \text{black} \mid 1^{st}\ \text{black}) P(1^{st} \ ... | You are correct... unless the colour of the first card is known, in which case it would be either:
$\dfrac{25}{51}$ if the first card was black, or
$\dfrac{26}{51}$ if the first card was red
| {
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Table seating question for real life use! I am trying to figure out how to rotate 75 people between 10 tables for 3 rotations with the people not overlapping when they rotate. Goal: for the attendess to meet new people at each table rotation. Thanks in advance!
| For 81 guests
A neat way without requiring calculation is to arrange 81 names (or numbers) in a 9x9 grid.
The tables for one arrangement are given by the rows.
The tables for the next arrangement are given by the columns.
The tables for the final arrangement are given by diagonals (wrapping around as if a tube has bee... | {
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Interesting integral $\int_0^{2\pi}\tan(\cos(x))dx=0$ Consider the integral $$I=\int_0^{2\pi}\tan(\cos(x))dx$$
I would like to show this integral is $0$ via elementary methods (excluding complex analysis, special functions, series representations).
The bounds of integration suggest some kind of symmetry argument to s... | Let $t=\cos x
\implies \sqrt {1-t^2}dt=dx$
Limits change from $0$ to $2\pi $
To $1$ to $1$
Limit $1$ to $1$ is obviously $0$
| {
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Implication (if I can observe that Q is true sometimes and false sometimes) I got a question for implication, P implies Q is true when P is true, Q is true at the same time. P implies Q is false when P is true, but Q is false.
My question is if P is true, but sometimes Q is true and sometimes is false, then, can I say ... | According to the definition, a proposition P is defined as a sentence which is either True or False, but not both. And also it doesn't depend on the time. The truth value of a proposition is true if it is a true statement, and false if it is a false statement. For example, $1+3=5$ is a proposition because one can deter... | {
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"answer_id": 3
} |
Can you find $\lim_{x\to0}x^2\sin\left(\frac{1}{x}\right)$ without sandwich theorem? The lesson I'm doing is solving
$\lim_{x\to0}x^2\sin\left(\dfrac{1}{x}\right)$
with sandwich theorem like so :
$$\displaystyle\lim_{x\to0}-x^2\leq\displaystyle\lim_{x\to0}x^2\sin\left(\dfrac{1}{x}\right)\leq\displaystyle\lim_{x\to 0}x... | I think It make sense. see whatever is in sine function it will always give a value from [-1,1] and if we multiply it by a number which is almost zero we will get zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3348957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Showing the nth roots of unity satisfy two properties Assuming $ω_0,ω_1,...,ω_n$ are the $n^{th}$ roots of unity, I am asked to show $$(x−ω_0)(x−ω_1)···(x−ω_{n−1}) =x^n−1$$ and $$\sum^{n−1}_{a=0}ω_a= 0$$
I understand that by definition, the $n^{th}$ roots of unity are the roots of the polynomial $x^n-1$. I'm not sure ... | Since omegas are the roots of $x^n-1$
You can factor $x^n-1$ to get $$x^n-1=(x−ω_0)(x−ω_1)···(x−ω_{n−1}) $$
The sum of roots of a monomial of degree $n$ is the opposite of the coefficient of $x^{n-1}$ which in this case is $0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3349101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Prove that $\lim\limits_{x\to0^+}\frac{f(x)}{f'(x)}=0$.
Let $f:(0,\infty)\to\mathbb{R}$ be a twice differentiable function with $f''$ continuous and let $\lim\limits_{x\to0^+}f'(x)=-\infty$ and $\lim\limits_{x\to0^+}f''(x)=+\infty$. Prove that:
$$\lim_{x\to0^+}\frac{f(x)}{f'(x)}=0.$$
My problem is not a proof of t... | There's c so that when $0\lt x \lt c$, function $f'(x)$ is monotonic increasing.
Rewrite $f(x)=f(c)-\int_x^c f'(x) dx$
For any $\epsilon \gt 0$, there must be $\delta_1, \delta_2$ so that when $0 \lt x\lt \delta_1$, $\left|\frac{f(c)}{f'(x)}\right| \lt \frac{\epsilon}3$ and when $0\lt x\lt \delta_2$, $\left|\frac{f'(\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3349260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Normal vector of a cone pointing outward Consider a cone with height $h$ and maximum radius $R$.
Assume that the vertex of the cone is in $(0,0,0)$ and its axis is the $z$-axis.
For $(x,y,z)$ lying in the outside of the cone I want to determine the normal vector pointing away from the cone.
My plan is to use adjusted c... | Cylindrical coordinates are hopefuly not the most appropriate in this case.
It looks to me simpler to work with cartesian coordinates.
For this, we need the cartesian equation of this cone, which is :
$$f(x,y,z) := x^2+y^2-(Rz/h)^2=0$$
(why that ? If such a cone is cut by plane $z=z_0$, its horizontal "slices" are circ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3349384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question on point-wise/uniform convergence of a sequence of functions Let $\{f_n(\cdot)\}_{n \in \mathbb{N}}$ be a sequence of real-valued functions defined on $[0, k]$ for some $0 < k < 1$. Suppose that for every fixed $x \in [0,k]$, we have the point-wise convergence
\begin{equation}
\lim_{n \to \infty} f_n(x) = 0.... | No, let $f_{n}:[0,k]\rightarrow\mathbb{R}$ be given by
$$f_{n}(x)=\begin{cases}n&\text{ if }x=\frac{1}{n}\\0&\text{ else}\end{cases}.$$
Clearl pointwise $f_{n}$ converges to $0$, but $$\lim_{n\rightarrow\infty}f_{n}(\frac{1}{n})=\lim_{n\rightarrow\infty}n=\infty.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3349542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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All mappings from $\mathbb{R}^N$ to $\mathbb{R}^N$ where the image of any line segment is a line segment I know that a projective transformation maps a line segment to a line segment. Are there mappings other than projective transformations where the image of any line segment is a line segment in $\mathbb{R}^N$?
Thanks... | According to the Wikipedia article on projective transformations, all collineations in real projective spaces are projective transformations. A collineation is a one-to-one mapping of lines.
You could think of a collineation as preserving collinearity of points. I.e. any three collinear points are mapped to three poi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving a sequence of functions converging pointwise doesn't converge uniformly on $(0,\infty)$ but does in $[a,\infty)$ where $a>0$ Let $f_n : (0,\infty) →\mathbb{R} , \,f_n(x) = \frac{1}{nx}$:
i) Prove that the $\lim_{n\rightarrow \infty} f_n(x)$ exists $\forall x > 0$.
ii )Prove that the convergence is not uniform i... | Suppose $f_n$ converges to $f\equiv0$ uniformly on $(0,\infty)$. Then given $\varepsilon=\frac12$, there exists $N\in \mathbb N$ such that $|f_n(x)|\lt\frac12$ for $n\geq N$ and for all $x\in(0,\infty)$. In particular, this should hold for $x=\frac1N\in(0,\infty)$, a contradiction.
Now consider $[a,\infty)$ with $a\gt ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many topological spaces with finitely many points are path-connected? Here's a set-point topology question I thought of.
How many topological spaces with $n$-points are path-connected?
| See this text by May for a proof that for finite topological spaces, path-connectedness and connectedness are equivalent.
I also believe there is no known formula for the number (non-homeomorphic or total) of such topologies on $n$ points as a formula in $n$. A search on the online encyclopedia didn't yield promising ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Good approximation of a partial sum Let $a \in (0,1)$ and consider the partial sum
\begin{equation}
S_{n,a} = \sum_{i=1}^n \frac{1}{(i+1) i^{1+a}}.
\end{equation}
Is there a good approximation of $S_{n,a}$ in terms of $n$ and $a$? This sum is roughly $\sum_{i=1}^n 1 / i^2$ for small $a$. I am trying to use integral-tes... | Note that we have
$$\frac1{k+1}=\frac1k\frac1{1+\frac1k}=\sum_{m=1}^\infty\frac{(-1)^{m+1}}{k^m}\tag{$|k|>1$}$$
and hence we can derive bounds such as
$$S_{n,a}-\frac12<\sum_{k=2}^n\frac1{k^{a+2}}<\int_1^n\frac{\mathrm dx}{x^{a+2}}=\frac{1-n^{-(a+1)}}{a+1}\underset{{a=1/n\\n\to\infty}}\longrightarrow1$$
and likewise,
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3350009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does there exist a circle, on the surface of a torus, that is not coplanar with the axis of revolution? I have got two closely related questions regarding Tori.
1.Does there exist a circle, on the surface of a torus, that is not co planar with the torus' axis of revolution?
2.Can the centre of a circle that is lying o... | Both are true, so the easy way is to exhibit them:
Picture a standard torus with the axis of revolution along $z$ (i.e. rotated in the $x$-$y$ plane). Horizontal cuts are not coplanar with the $z$ axis (indeed, are perpendicular to it.) These are circles (and must be because they are parallel to the plane of rotation... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3350116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I show this predicate is true? Given a while loop and a predicate, show that if the predicate is true before entry to the loop, then it is also true after exit from the loop.
predicate: m^3 > n^2
while (m ≥ 0 and m ≤ 100) do
m := 3 · m
n := 5 · n
end while
I tried to plug the $m$ and $n$ values int... | Well, that's on the right track: every time you go through the loop, $m$ gets $3$ times as large, and $n$ gets $5$ times as large. So, if beforehand we have that $m^3 > n^2$, then afterwards we have that $(3m)^3= 27m^3 > 27n^2 > 25 n^2 = (5n)^2$. That is: the new $m$ raised to the power of $3$ will still be bigger than... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding the plane that contains both a given point and every point of a line. I'm not sure how to begin with this question as I've just started with linear algebra and its starting to become overwhelming.
Given this problem:
Let L denote the line consisting of all points of the form (t, 0,2), where t can be any real nu... | All of the answer choice planes contain $v$ so that isn't an issue.
For the line, we know that $c$ is not the correct option because it doesn't contain $y=0$.
We also know that $a$ is not the correct option because the line is parallel to the $x$ axis, and this plane only contains one $x$-value.
The correct answer is $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3350344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof that each of the three cevians is divided in the ratio $1:3:3$ Points $D,E,F$ are the first trisection points of $BC,CA,AB$ respectively. Let $[ABC]$ denotes the area of triangle $ABC$. If $[ABC]=1$, find $[GHI]$, the area of the shaded triangle. (the two images are from "The Art and Craft of Problem Solving" -Pa... | We may and do assume that the area $[ABC]$ is one.
Following the arguments of the OP, we already have the following information regarding the areas cut / separated from $[ABC]$ by two cevians, $AF$ and $BD$:
In particular we have $2:1=CD:DB=\color{red}{7y}:\color{blue}{3x+2y}$, which implies $7y=2(3x+2y)$, i.e. $y=2x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 2
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Is following tautology: $((A∧B)→C)≡((A→C)∨(B→C))$ I came across following problem:
Is following tautology:
$((A ∧ B) → C) ≡ ((A → C) ∨ (B → C))$
I solved it like this:
$(A\wedge B)\rightarrow C$
$=\neg(A\wedge B)\vee C$
$=(\neg A\vee \neg B) \vee C$
$=\neg A\vee \neg B \vee C$
$=\neg A\vee \neg B \vee C \vee C$ $\hs... | There is no mistake. The order of the rows in the tables are different, that's all.
The one False entry in the last column is when $A=T, B=T, C=F$ in both tables.
Look here
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3350614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Tensor product of irreducible representations of semisimple Lie algebras Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be semisimple Lie algebras over $\mathbb{C} $. Let $V_1$ and $V_2$ be simple modules over $\mathfrak{g}_1$ and $\mathfrak{g}_2$, respectively. They can be infinite dimensional.
$V_1 \otimes V_2$ is a $\m... | This is not such a trivial problem. The solution I know uses the Jacobson density theorem:
Let $M$ be a simple $R$-module, $D = \mathrm{End}_R(M)$. If $\varphi \in \mathrm{End}_D(M)$, $x_1,\ldots, x_n \in M$, then there exists $r \in R$ such that $rx_i = \varphi(x_i)$ for all $i$. I will show how this reduces to the ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3350849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Finding an upper bound for the approximation error of a series. Suppose that $$s=\sum_{n=1}^{\infty}\frac{\cos^2(n)}{n^5+2n^2}$$ and $$s_n=\sum_{i=1}^{n}\frac{\cos^2(i)}{i^5+2i^2}$$, then the approximation error is defined as $R_n=s-s_n$. I need to find a higher bound for this error and also find a $n$ for which the ap... | Note that
$$|s-s_n|=\sum_{k=n+1}^\infty\frac{\cos^2(k)}{k^5+2k^2}<\sum_{k=n+1}^\infty\frac1{k^5}<\int_n^\infty\frac{\mathrm dx}{x^5}=\frac1{4n^4}$$
so it suffices to take $n\ge71$ to get $|s-s_n|<10^{-8}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many $14$-digit even numbers can be formed using $0,1,1,2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5$? What is a good way to approach this question?
How many $14$-digit even numbers can be formed using $0,1,1,2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5$?
I know that I can list the possible outcomes with different ending numbers and then ... | There are three even numbers, so they make the three cases.
Case 1. Last digit is $0$:
$$\frac{13!}{(2!)^2(3!)^3}$$
Case 2. Last digit is $2$:
$$\frac{13!-12!}{2!(3!)^3}$$
Case 3. Last digit is 4:
$$\frac{13!-12!}{(2!)^3(3!)^2}$$
Hence, adding the three results in $40{,}471{,}200$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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For $A \in \mathcal{M}_3(\mathbb{C})$, does $\mathrm{tr}(A^2) = \mathrm{tr}(A^3) \in \mathbb{Q}$ imply $\mathrm{tr}(A^4) \in \mathbb{Q}$? Let $A \in \mathcal{M}_3(\mathbb{C})$ such that $\mathrm{tr}(A^2) = \mathrm{tr}(A^3) \in \mathbb{Q},$ where $\mathrm{tr}(A)$ is the trace of $A.$ It is possible to prove that $\mathr... | We fix some $t\in \Bbb C$. Let $a,b,c$ be the complex roots of the equation
$$
X^3-6tX^2+18t^2 X -36 t^3\ .
$$
Let $e_n$ be the elementary symmetric polynomial of degree $n$ in $a,b,c$.
Let $p_n$ be the Newton symmetric polynomial of degree $n$ in $a,b,c$.
By Vieta,
$$\left\{
\begin{aligned}
e_1 &= a+b+c = 6t\ ,\\
e_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3352424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Find chord length given equations for circle and line Is there any way to find the length of a chord if you are given the equations for the chord and the circle? Say you have a circle with an equation of $x^2 + y^2 = 4$ and the line $y = 3x - 4$. Is there any way to get the chord length from this?
I would assume this m... | Another way can be found to calculate the intersection points:
$$x^2+(3x-4)^2=4$$
For your work: We find
$$\left\{\left\{x\to \frac{1}{5} \left(6-\sqrt{6}\right),y\to \frac{1}{5} \left(-2-3
\sqrt{6}\right)\right\},\left\{x\to \frac{1}{5} \left(6+\sqrt{6}\right),y\to \frac{3
\sqrt{6}}{5}-\frac{2}{5}\right\}\right\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3352532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $(B_t)$ is a Brownian motion and adapted to the filtration $(\mathcal F_t)$ does $B_t-B_s$ is independent of $\mathcal F_s$? Let $(B_t)$ a Brownian motion adapted to the filtration $(\mathcal F_t)$.
Q1) If $t> s$, does $B_t-B_s$ is necessarily independent of $\mathcal F_s$ or $(\mathcal F_t)$ must be the natural fil... | The filtration $(\mathcal{F}_t)_{t \geq 0}$ does not need to be necessarily the natural filtration, but you can't pick just any filtration.
For instance, consider the filtration $$\mathcal{F}_t := \sigma(B_r; r \geq 0).$$ Note that $\mathcal{F}_t$ actually does not depend on $t$. It is trivial that $(B_t)_{t \geq 0}$ i... | {
"language": "en",
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Meaning of choose function with a negative number I have encountered an expression for an inverse Laplace transform, and now I am wondering whether it is correct or maybe there is some error. More likely I don't understand it. so here goes:
$$
g(p)(e^{ap}+\beta)^{-c} \Leftrightarrow \sum_{0\leq n\leq t/a -c}\left(\begi... | The definition of $\binom xn$ is
$$\binom xn=\frac{x(x-1)(x-2)\cdots(x-n+1)}{n!}$$
for any nonnegative integer $n$ and any complex number $x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3352713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Whether each dense linear subspace of a separable Banach space has finite co-dimension? This question is a sort of converse to this post. If $E$ is an infinite-dimensional separable Banach space and $F$ is a dense (infinite-dimensional) (linear) subspace, then is $E/F$ of finite co-dimension? (I know it will not be H... | Let $E=\ell_2$, $F=\ell_{2f}$, that is a set of sequences $(x_n)_{n\in\Bbb N}\in\ell^2$ which are eventually zero. Then $F$ is dense in $E$, but a vector space $E/F$ is not finitely-dimensional, because $E\ne A+F$ for each finite dimensional subspace $A$ of $F$. Let us prove the last claim. We have $F=\bigcup F_n$, whe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3352848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Ceva's theorem when the point of concurrency lies on the side of the triangle $S$ tend towards Point $D$ along $AD$">
In the figure suppose the point $S$ tends to $D$ along Line $AD$ .
According to Ceva
$$ \frac{AF}{FB} \cdot\frac{BD}{DC}\cdot\frac{CE}{EA} =1$$
as we see if S tends to D $ \frac{AF}{FB} $ increases ... | It's not valid when $S=D$, because what that formula would then contain is a division by $0$, the length of $FB$. That's nonsensical. On the other hand, what would hold is
$$
AF\cdot BD\cdot CE=FB\cdot DC\cdot EA
$$
For any other case is exactly the same result, but it doesn't turn into garbage if $S$ lies on one of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3352963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Induction on $(1+\frac{1}{n})^n \geq 2$ $$(1 + \frac{1}{n})^n \geq 2$$
Base base: $n = 1$
LHS: $(1+ 1)^1 \leq 2$ RHS: 2
$2 \leq 2$ True.
Inductive step:
$$(1 + \frac{1}{n+1})^{n+1} \geq 2$$
$$(1 + \frac{1}{n+1})^n (1 + \frac{1}{n+1}) \geq 2$$
Im stuck
| Hint: Use Bernoulli's inequality:
$$
(1+x)^n \ge 1 + nx \quad \mbox{for $x\ge 0$}
$$
This can be proved by induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3353103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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confused about the mean of negative binomial distribution Let X be a random variable of the negative binomial distribution with parameters $r$ and $p$,
$$P(X = n)~=~{n-1\choose r-1} p^r(1-p)^{n-r} .$$
The textbook says the mean is ${r(1-p)\over p}$, which confuses me because I always consider the negative binomial di... | There are two different conventions for the geometric distribution. One counts the number of trials, has support $\{ 1,2,\dots \}$, and has mean $1/p$. The other counts the number of failures, has support $\{ 0,1,\dots \}$, and has mean $(1-p)/p$.
Summing iid copies of these gives two different kinds of negative binom... | {
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"timestamp": "2023-03-29T00:00:00",
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Showing Associativity and Commutativity of a binary operation given by a Cayley table Let $*$ be a binary operation on the set $S:=\{0,1\}$ given by the following Cayley table:
\begin{array}{c|cc}
* & 0 & 1\\\hline
0 & 0 & 1\\
1 & 1 & 0
\end{array}
If I wish to show that this table is associative, how many different co... | As an alternative to case-by-case checking (which does require $8$ checks), you could note that $a*b \equiv a+b \pmod{2}$. Then you get that $*$ is associative because modular addition is associative.
| {
"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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What is the bound on a function $f(x)$ given that $|f'(x)| \leq |g'(x)|$? Suppose you know that $|f'(x)| \leq |g'(x)|$ for some interval on the real line. Is it possible to infer a bound on $|f(x)|$ in terms of $g(x)$?
Maybe you can use the Mean Value Theorem to do so? If so, what additional assumptions do you need abo... | If $f'$ is integrable, you can say
$$ \lvert f(x)-f(y) \rvert = \left\lvert \int_y^x f' \right\rvert \leq \int_y^x \lvert f' \rvert \leq \int_y^x \lvert g' \rvert = V_y^x(g) , $$
the total variation of $g$. You can see this in play in a pair with $g'$ taking on positive and negative values, $f' = \lvert g' \rvert$. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3353535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Example of a reflexive canonical module I want an example of a Cohen-Macaulay local ring $R$ with the canonical module $\omega_R$ such that $\omega_R$ is reflexive (that is $\omega_R \cong {\rm Hom}_R ({\rm Hom}_R(\omega_R,R),R)$).
| As a simple example, take $R=k[[x^3,x^2y,xy^2,y^3]]$. It is a two dimensional CM local ring with an isolated singularity. It is also not Gorenstein. So, $\omega_R$ has depth 2 and the natural map $\omega_R\to\omega_R^{**}$ is injective and the cokernel is finite length, since $R$ has an isolated singularity. This force... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3353706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Weak convergence + pointwise convergence on dense subspace of $L^2$ Let $H$ be a dense subspace of $L^2(X,\mu)$ ($X$ being a separable and complete metric space with finite measure $\mu$) and $(f_n)_{n\in\mathbb N}$ be a sequence in $H$ that converges to $f\in L^2(X,\mu)$ in the following sense:
$$
\langle f_n - f, h \... | Another counterexample for $L^2(-\pi,\pi)$:
Lemma If $Z$ is an inner product space, $x_1,\dots,x_n\in Z$ are independent, and $c_1,\dots,c_n$ are scalars then there exists $x\in Z$ with $\langle x,x_j\rangle=c_j$ for $j=1,\dots, n$.
Proof: By linearity (and symmetry) we can assume $c_1=\dots=c_{n-1}=0$, $c_n=1$. Le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3353912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Help in a mathematical induction problem involving a chess board. We are given a chess game desk and some identical L-shaped figures each of which can cover exactly three fields on the desk (see the figure attached).
enter image description here
1) Can the desk be fully covered by the figures, i.e., the figures are pla... | The answer to the first part of the question is no because $64$ is not divided by 3.
As for the second part the answer is yes and here's an algorithm to fill the chess board with the L shaped pieces:
Firstly split the chess board into $4$ squares of size $4*4$. The 3 squares that do not contain the gray square form a b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3354033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding an eigenvector for a matrix such that the entries of each column sum 1 Let $A \in \mathbb{R}^{n \times n}$ a matrix such that each entrie of $A$ (I denoted them by $\left[ A \right] _{ij}$) satisfies that $\left[ A \right] _{ij} \geq 0$ and if you sum the entries of each column, the result is $1$ (i.e. $\left[ ... | Your $A$ is column-stochastic. If it is also positive, or in general, if it is primitive, then $x$ is the steady-state distribution associated with the Markov chain defined by $A$, i.e. $\lim_{k\to\infty}A^k=xe^T$,
where $e=(1,\ldots,1)^T$. In turn,
$$
x=\lim_{k\to\infty}A^kp
$$
where $p$ is any vector whose entries su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3354137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$3$-Sylow in $\mathbb Z/ 6\mathbb Z$ In $\mathbb Z/6\mathbb Z$, by the definiton of Sylow $p$-subgroups, we can talk about Sylow $2$-subgroups or Sylow $3$-subgroups. The textbook asserts that $\langle 2\rangle$ is a 3-Sylow and the only one.
I understand that $2^3\equiv1 \pmod 6$ (and $3$ is the smallest natural num... | $\langle 2\rangle$ and $\langle 4\rangle$ are the same subgroup, which (as a set), is the classes of $\{0,2,4\}$ in $\mathbb Z/6\mathbb Z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3354253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
joint pdf of $(X,X)$ Let $X$ be a continuous random variable with uniform distribution on $[0,1]$, i.e, the probability density function of $X$ is $f(x)=1$ on $[0,1]$. Let $Y=X$. Then what is the joint probability density function $h(x,y)$ of $X$ and $Y$? It seems that $h$ is supported on the line $y=x$, which has $0$ ... | $(X,X)$ does not have a density w.r.t. Lebesgue measure on $\mathbb R^2$. Indeed if $D=\{(x,x), x\in [0,1]\}$, then $P_{(X,X)}(D)=1$ whereas $\lambda_2(D)=0$.
Besides, $P_{(X,X)}(\{(a,b)\})=0$ for any $(a,b)\in \mathbb R^2$, so the distribution of $(X,X)$ is not discrete either.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3354374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Which among the following is true for a Wronskian Of a Differential Equation Let $u$ and $v$ be two solutions of the differential Equation
$y^{"} + P(x)y^{'} + Q(x)y = 0$ on $[a,b]$, Let $W(u,v)$ denote the Wronskian Of $u$ and $v$ Then
(a) $W(u,v)$ vanishes at point $x_{0} \in [a,b]$ $\implies$ $u,v$ are Linearly De... | From the definition
$W(u, v) = \det \begin{bmatrix} u & v \\ u' & v' \end{bmatrix} = uv' - vu' \tag 1$
we find
$W'(u, v) = u'v' + uv'' - v'u' - vu'' = uv'' - vu''; \tag 2$
using the fact that $u$ and $v$ are both solutions of
$y'' + P(x)y' + Q(x)y = 0, \tag 3$
we may transform (2) to
$W'(u, v) = u(-P(x)v' - Q(x)v) - v(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3354468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For $x$ and $k$ real numbers, determine the values of $k$ for which the graphs of $f(x)=x^2-4$ and $g(x)=2|x|+k$ do not intersect For $x$ and $k$ real numbers, determine the values of $k$ for which the graphs of $f(x)=x^2-4$ and $g(x)=2|x|+k$ do not intersect.
For $x$ and $k$ real numbers, determine the values of $k$ f... | Set the following equations from $f(x)=g(x)$,
$$ x^2 -4 = 2x+k$$
$$ x^2 -4 =- 2x+k$$
Or,
$$ x^2 -2x -(4+k)= 0$$
$$ x^2 +2x -(4+k)= 0$$
Their discriminant is
$$\Delta =b^2-4ac = 20+4k$$
If they do not intersect, meaning above equations have no solution, we set $\Delta <0$. So, $k <-5$.
If they have two intersections, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3354578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Proving inequality $|\sin(n\theta) |\le n\sin\theta$ I was trying to prove following inequality:
$$|\sin n\theta| \leq n\sin \theta \
\text{for all n=1,2,3... and } \
0<\theta<π $$
I succeeded in proving this via induction but I didn't get "feel" over the proof. Are there other proof for this inequality?
| You can show that $|\sin(x)|$ is subadditive, i.e.
$$|\sin(x + y)| \le |\sin(x)| + |\sin(y)|.$$
To prove this, simply expand the left side:
\begin{align*}
|\sin(x + y)| &= |\sin(x)\cos(y) + \sin(y)\cos(x)| \\
&\le |\sin(x)| \cdot |\cos(y)| + |\sin(y)| \cdot | \cos(y)| \\
&\le |\sin(x)| + |\sin(y)|,
\end{align*}
as $|\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3354684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Set of Vector Space Homomorphism Definition V, W are vector spaces over F and we have hom (V,W) as the set of homomorphisms of linear spaces. Define addition and scalar multiplication on Hom(V, W) so it's vector space over F.
I assume i would define it the same way as a vector space homomorphism which is:
Let V an... | You need to define addition of two elements of $\operatorname {Hom}(V,W)$, and also scalar multiplication.
So given $f,g\in \operatorname {Hom}(V,W)$, define $(f+g)(v):=f(v)+g(v)$.
Then define $(c\cdot f)(v):=c\cdot f(v)$.
The axioms should go through rather easily.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3354786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
A combinatorics puzzle related to Stars and bars problem The problem goes like:
A store sells 11 different flavors of ice cream. In how many ways can a customer choose 6 ice cream cones, not necessarily of different flavors?
I tried thinking for 2 days then I gave up and seen solution. The answer was somewhat connecte... | Yes, Stars and bars technique is the right tool here. Note that you have to count the non-negative integer solutions of
$$x_1+x_2+\dots+x_{11}=6$$
where $x_i$ is the number of icecreams taken from the "box" containing "identical" icecreams all of the $i$-th flavour.
Can you take it from here? What is the final answer?... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3354885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
a function or a factor to scale a list of real numbers from one range to another Edit:
I made a mistake or someone did. The a and b and max and min is swapped in the formula I used. Upon switching the a for min and b for max, the algorithm worked as I expected.
Previously I had
a = -180, b= 180, and then min -100, m... | So basically you have a list (list_sum) and you would like to scale it so that the minimum is some fixed number $a$ and the maximum is another fixed number $b$. In other words, you would like a function $f(x)$ that has the properties
$$
f(\text{min})=a \quad \text{and}\quad f(\text{max})=b
$$
We can make this with a li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3355027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Hard inequality :$\Big(\frac{1}{a^2+b^2}\Big)^2+\Big(\frac{1}{b^2+c^2}\Big)^2+\Big(\frac{1}{c^2+a^2}\Big)^2\geq \frac{3}{4}$ I have a hard problem this is it :
Let $a,b,c>0$ such that $a^ab^bc^c=1$ then we have :
$$\Big(\frac{1}{a^2+b^2}\Big)^2+\Big(\frac{1}{b^2+c^2}\Big)^2+\Big(\frac{1}{c^2+a^2}\Big)^2\geq \frac{3}... | First, I will ignore the constraint $a^ab^bc^c=1$ to sketch a possible solution for the aforementioned inequality, under an alternative constraint:
Set $x=\frac{1}{a^2+b^2}$ and $x=\frac{1}{b^2+c^2}$ and $z=\frac{1}{c^2+a^2}$.
From the aritmetic-geometric inequality
$$
x^2+y^2+z^2\geq 3(x^2y^2z^2)^{\frac{1}{3}},
$$
it ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3355161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
} |
Dynamical Systems / Differentiatial Equations Problem I need help with my Dynamical systems homework, I honestly don't even really know where to start on it. Any help would be appreciated
Show that if $x(t)$ is a solution to $x'(t) = x(t)^2 - 1 ,$ then so is $z(t) = -x(t)^{-1}$ and $y(t) = -x(-t).$ What type of symmet... | As a Riccati equation, you can directly solve this via $x(t)=-\frac{u'(t)}{u(t)}$ where
$$
u''(t)-u(t)=0\implies u(t)=Ae^{-t}+Be^t
\implies
x(t)=\frac{Ae^{-t}-Be^{t}}{Ae^{-t}+Be^{t}}
$$
Then
$$
z(t)=-y(-t)=\frac{Be^{-t}-Ae^{t}}{Be^{-t}+Ae^{t}},
$$
has the same form, and so does
$$
y(t)=x(t)^{-1}=\frac{Ae^{-t}-(-B)e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3355273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Train stops at $n$ stations This is a problem from a mathematical contest I was unable to answer.
I am placing this problem with both the recreational math and calculus tags because I think derivatives are needed.
The problem is:
A train stops in all $n$ stations along its route (including the initial and last ones). A... | Derivatives are not required, nor useful, really.
At station $k$, $n-k$ people get on the train, and $k-1$ people get off, so the net increase in passengers is $n-2k+1$. The number of passengers on the strain as it pulls away from station $s$ is $$\sum_{k=1}^s(n-2k+1)=(n+1)s-2\sum_{k=1}^sk=(n+1)s-s(s+1)=s(n-s),$$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3355421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
For homeomorphism $f:X\to Y$, show that $f^{-1}(\mathcal{B}_{Y})=\mathcal{B}_{X}$
My question is that if a function $f: X\to Y$ is homeomorphism, show that $f^{-1}(\mathcal{B}_{Y})=\mathcal{B}_{X}$.
where $f^{-1}(\mathcal{B}_{Y}):=\{f^{-1}(A): A\in \mathcal{B}_{Y}\}$.
I try to use the continuity of $f$ but I am stu... | Prove that $\{f^{-1}(B): B \in \mathcal{B}_Y\}$ is a sigma algebra on $X$ and contains the open sets.
Every open set $A$ in $X$ has the form $A=f^{-1}(B)$ where $B$ an open subest of $Y$ since $f$ is a homeomorphism, so we have $\mathcal{B}_X \subset f^{-1}(\mathcal{B}_Y)$ because $\mathcal{B}_X$ is the smallest sig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3355514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Almost sure convergence of a sum of independent random variables Let $\{X_n\}_{n\geq1}$ be a sequence of centered independent random variables such as $E(X_n^2)=2n$
and $$Y_n=\frac1{n^\alpha}\sum_{i=1}^{i=n}X_i\quad\quad\alpha\geq1$$
I am trying to prove that for $\alpha > \frac32$, $Y_n \rightarrow 0$ almost surely.
... | Let us recall:
(1): Kronecker's Lemma: If $(x_n)_{n \in \mathbb N}$ is a sequence such that $\sum_{n} x_n $ converges, $(b_n)_{n \in \mathbb N}$ is increasing positive sequence such that $\lim_{n \to \infty} b_n = +\infty$, then $\lim_{n \to \infty} \frac{1}{b_n} \sum_{j=1}^n x_jb_j = 0$
(2): Kolmogorov's two-series th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3355660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Uniform continuity of a $ |\sin x| $ in $ [0,\infty) $
Given a function $ |\sin (x)| $ over the interval $ [0, + \infty) $ we have to check whether the function is uniformly continuous or not ?
So I begin by letting $ |x-y| < \delta $ then if I consider $ | |\sin x|-|\sin y| | $ then by triangular inequality we have ... | In such a proof, you are given $\epsilon$ and need to choose $\delta$. It seems you’ve tried to do the opposite: you seem to be trying to choose a large enough $\epsilon$, when in reality $\epsilon$ should be fixed and your goal is to choose a small enough $\delta$.
Regardless, an easier line of reasoning: by the mean... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3355922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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To construct a polynomial using Bézier Curves. Given a polynomial equation (can be of any degree) how do we find the control points for the Bézier curve that follows the polynomial equation?
Note:
*
*I know that doing this is counter-intuitive because the polynomial
equation gives us the points of the curve and he... | Let $F(t)$ be the given polynomial. I'm going to assume that $F$ is cubic (the most common case). Other degrees can be handled using the same techniques, but the notation is much more cumbersome.
One approach is to find a symmetric multi-affine function $G(u,v,w)$, called the blossom or polar form of $F$, such that $G(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3356084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Countinous map $F:X \to Y$, $Y$ Hausdorff. If $F$ has a continuous left inverse, then it's proper. $F:X \to Y$ is a countinous map between topological spaces, $Y$ is Hausdorff.
If $F$ has a continuous left inverse (i.e. $\exists$ continuous map $G: Y \to X\ $ s.t. $\ G \circ F = id_X$), then it's proper (preimage of c... | I have a better idea and a generalized conclusion:
If $X$ and $Z$ are topological space, $Y$ is Hausdorff space, $f:X\to Y$ is continuous, $g:Y\to Z$ is continuous, and $g\circ f$ is proper, then $f$ is proper.
Proof:
If $K \subset Y$ is compact, then $K \subset g^{-1}(g(K))$ and $f^{-1}(K)\subset f^{-1}(g^{-1}(g(K)))＝... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3356178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Why any countable subset of $\mathbb{R}→\mathbb{R}$ is generated by a finite set under composition? Given a sequence of functions $\{g_k\}$, where $g_k: \Bbb R\to\Bbb R$ for all $n\in \mathbb N$. Prove that there exists a finite set of functions
$$f_1,f_2,\ldots,f_n$$ such that any function $g_k$ can be expressed as a ... | More generally:
Theorem. For any infinite set $S$ and any countable set $F$ of functions $f:S\to S$, there are two functions $g,h:S\to S$ such that $F$ is contained in the semigroup generated by $\{g,h\}$ under composition. (On the other hand, if $S$ is a finite set with more than two elements, then three selfmaps of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3356332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
"answer_id": 2
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Solving $\int\frac{1}{\cos x}$ without knowing anything about secant! As the title says, I am trying to solve this integral without using anything related to secant, it might sound stupid but I want to know if it can be done.
I have tried the following approach but it didn't lead me anywhere:
$$ \int\frac{1}{\cos x}... | You can start out like this:
$$
\int \frac{dx}{\cos x}
= \int \frac{\cos x \, dx}{\cos^2 x}
= \int \frac{\cos x \, dx}{1-\sin^2 x}
.
$$
Then let $u = \sin x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3356423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Does $\left\{\frac{p^a}{q^b}:p,q\in\mathbb{Z}\right\}=\left\{\frac{j^{\gcd(a,b)}}{k^{\gcd(a,b)}}:j,k\in\mathbb{Z}\right\}$ for $a,b\in\mathbb{N}$? How to prove $$\left\{\frac{p^a}{q^b}:p,q\in\mathbb{Z}\right\}=\left\{\frac{j^{\gcd(a,b)}}{k^{\gcd(a,b)}}:j,k\in\mathbb{Z}\right\}$$ for $a,b\in\mathbb{N}$?
First, is this i... | Yes, it is true.
The containment $\subseteq$ is easy to show.
For $\supseteq$, take any $a,b,i,j$. Let $x_1,x_2,y_1,y_2$ be such that $ax_1+by_1=ax_2+by_2=d=\operatorname{gcd}(a,b)$ and $x_1,y_2\geq 0\geq y_1,x_2$. Take $p=i^{x_1}j^{-x_2}$, $q=i^{-y_1}j^{y_2}$. Then $p^aq^{-b}=i^{ax_1+by_1}j^{-ax_2-by_2}=i^dj^{-d}$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3356530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Using conditional expectation with MSE function I'm trying to understand a derivation step in applying conditional expectation to the following starting function:
$f_* = E_{P_{(x,y)}} [(y-f(x))^{2})]$
and how, after applying conditional expectation, you end up with this:
$f_* = E_{P_{(x)}} [E_{P_{(y|x)}}[(y-f(x))^{2}|x... | Formally, we are given two random variables $X$, $Y$ and we let $f^* = E[(Y-f(X))^2]$.
By the law of iterated expectations, $f^* = E[E[(Y-f(X))^2|X]]$.
Let $\mu$ be a probability kernel such that $\forall A\in \mathcal B(\mathbb R), P(Y\in A|X) = \mu(X,A)$. Then $$E[(Y-f(X))^2|X] = \int (y-f(X))^2d\mu(X,y)$$
Your $E_{P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3356632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Do I understand the concept of $x^{0.84}$ correctly? I'm trying to understand the concept of $x^{0.84}$ that Jeffrey Lagarias found for Collatz Conjecture. If I'm wrong, please correct me with an answer. I understand such that,
Suppose the interval $\left[1, 2^{1\,000\,000} \right]$ is given. In this interval we have a... | $x^{0.84} = x^\frac{21}{25} = (x^{21})^\frac{1}{25}$, which is the unique solution $y$ of $y^{25} = x^{21}$. For irrational $y$, $x^y = \sup_{z\in\mathbb{Q}\cap(-\infty,y)}x^z = \inf_{z\in\mathbb{Q}\cap(y,\infty)} x^z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3356741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
partial fraction decomposition in integral I have a question that asks to calculate the following integral:
$$ \int_0^\infty {\frac{w\cdot \sin w}{4a^2+w^2}dw} $$
In the official solution they used partial fraction decomposition in order to later use Plancherel's identity:
$$ \frac{w\cdot \sin w}{4a^2+w^2} = $$
$$ \fr... | $$1=\frac{w^2}{w^2+4a^2}+\frac{4a^2}{w^2+4a^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3356858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How many triplets $(p,q,r)$ of integers between $1$ and $1000$ satisfy $p^2\sin^2x+q\sin2x+r^2\cos^2x>1$, if $\sin x$ and $\cos x$ are non-zero?
We have an inequality of the form
$$p^2\sin^2 x + q\sin2x + r^2\cos^2x > 1$$
for integers $p$, $q$, $r$ with $1\leq p,q,r \leq 1000$, and such that $x$ cannot be any val... | We must find a condition on $p,q,r$ such that, for all $\theta \in \mathbb{R}$ :
$$f(\cos \theta,\sin \theta):=p^2\sin^2 \theta + 2q\sin \theta \cos \theta + r^2\cos^2 \theta \ \color{red}{> 1}\tag{1}$$
(1) can be transformed into the following issue :
$$\begin{cases}\text{minimization of } \ f(x,y):=r^2 x^2 + 2qxy + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3357097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
General stochastic integration by parts. For $W_t$ the Brownian motion,
Let $X_t$, $Y_t$ be two diffusions, then since $$d(X_tY_t) = X_tdY_t+Y_tdX_t+dX_tdY_t$$ the generic stochastic integration by parts formula is given by $$\int_a^bX_t\,dY_t=X_bY_b-X_aY_a-\int_a^bY_t\,dX_t - \int_a^bdX_t\,dY_t$$
May I clarify for $\i... | Technically, you have written the following equation in a different form :
\begin{align}
\int_0^T \frac{1}{1+W_t^2}\,dW_t &= \frac{W_T}{1+W_T^2}+2\int_0^T\frac{W_t^2}{(1+W_t^2)^2}\,dW_t + 3\int_0^T\frac{W_t}{(1+W_t^2)^2}\,dt-4\int_0^T\frac{W_t^3}{(1+W_t^2)^3}\,dt
\end{align}
to
\begin{align}
\int_0^T\frac{1}{1+W_t^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3357238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Show that $\mathbb{F}_p[x,y]/\langle x^py-xy^p\rangle$ is not finitely generated over $\mathbb{F}_p[x-ay]$ [The current bounty is for a specific answer which I can reward in 24 hours from now. Please ignore this bounty. An answer has been already given.]
Let $p$ be a prime. Prove that $\forall a\in\mathbb{F}_p$ the al... | First consider the ring automorphism $\varphi:\mathbb{F}_p[X,Y]\to \mathbb{F}_p[X,Y] $, $(X,Y)\mapsto (X+aY,Y)$. We observe $\varphi(X^pY-XY^p)=(X+aY)^pY-(X+aY)Y^p=X^pY-XY^p$ because of $a^p=a$ and $\varphi(\mathbb{F}_p[X-aY])=\mathbb{F}_p[X]$. Therefore we only need to show that $K:=\mathbb{F}_p[X,Y]/\left<X^pY-XY^p\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3357356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Which version of the Pigeonhole principle is correct? One is far stronger than the other One version of the Pigeonhole principle says that if the cardinality of a set $A$ is greater than that of a set $B$, then there can be no one-to-one function that maps from $A$ to $B$.
Another version says: If $n$ elements are par... | Besides the two good answers already present, I would like to mention that the second version is also a trivial corollary of the first: indeed, if you take $m$ disjoint subsets $A_1,\ldots,A_m$ of $A$, each with less than $\lceil n/m\rceil$ elements, then the union $\bigcup_{j=1}^m A_j$ has at most $m\lfloor n/m\rflo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3357414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 4,
"answer_id": 2
} |
How do I conclude from $E[X_{+}1_{Y\geq 0}]=0$ that $P(X > 0, Y \leq 0)=0$ Let $X_{+}$ be the part of $X$ that is strictly positive.
How do I conclude from $E[X_{+}1_{Y\geq 0}]=0$ that $P(X > 0, Y \leq 0)=0$?
My idea
$0=E[X_{+}1_{Y\geq 0}]=E[X1_{X>0,Y\leq0}]$ and since $X\neq 0$ on $1_{X>0}$ it follows that $P(X>0,Y\l... | Your claim is wrong. Consider $X$ a random variable with exponential distribution and let $Y=-X$. Then $X_{+}1_{Y\geq 0} = 0$ a.s., thus $E(X_{+}1_{Y\geq 0} )=0$. However $$P(X>0\cap Y\leq 0) = P(X>0)=1 $$
The only thing you can derive from $E(X_{+}1_{Y\geq 0} )=0$ is $X_{+}1_{Y\geq 0}$ a.s., which is equivalent to $(X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3357532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Soccer and general probability distribution? Three teams (Team 1, Team 2, and Team 3) are playing in a soccer championship. In each game, two teams play each other and the third team does not play. The winner of any given game $n$ plays again in game $n+1$ against the team that did not play in game $n$, and the loser o... | Hint: there are three states of game play: 1 vs. 2, 1 vs. 3, and 2 vs 3. For each state, you know the probability of which state the next game will be. Given that, what would the states of games have to be to have the fourth game be the same state as the first game?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3357615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Why are the following not logically equivalent? Why are the following statements not logically equivalent?
$$((\forall x\in D,P(x))\vee(\forall x\in D, Q(x))$$
and $$\forall x\in D, (P(x)\vee Q(x)$$
I have thought for several hours about this and I can't think of any reason why they shouldn't be equivalent. May someon... | For example, let $D=\mathbb R$ and $P(x)$ be $x\ge0$ and $Q(x)$ be $x<0$. Then your first statement is not true, because both $\forall x\in D,P(x)$ and $\forall x\in D, Q(x)$ are false, but your second statement is true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3357723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Probability of birthday month Question: In a class of $12$ students, what is the probability that each student was born in a different month?
Approach: would it be $\left(\frac{1}{12}\right)^{12}$?
| It would be $\frac{11!}{12^{11}}$
Reason: For the first person, he can have any month as their birthday, of which the probability = $1$. For the second person, they need to have their birthday in a month that was the previous person's birthday month, of which probability is $\frac{11}{12}$. Using the same logic, you ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3357791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Proving $m(S_i\cap S_j)=0\;\forall i\ne j \Rightarrow m(\cup_iS_i)=\sum_im(S_i)$ Let $(E,\Sigma,m)$ be a measure space and let $S_1,S_2,\ldots\in\Sigma$ so that $m(S_i\cap S_j)=0\;\forall i\ne j$.
How to prove that $m(\cup_iS_i)=\sum_im(S_i)$?
I know that since $m$ is a measure and therefore countably additive, then $... | Let $N= \cup_{i,j} (S_i \cap S_j)$ and note that $mN = 0$.
Let $S'_k = S_k \setminus N$. Note that $m(S_k') = m(S_k)$ and $m( \cup_k S_k ) = m (\cup_k S_k')$.
Since the $ S_k'$ are disjoint, we have the desired result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3357872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Number of maximum degree vertices in a tree Let $T$ be a tree of order $n$, where $T$ has $\Delta(T) \geq 3$ leaves. Show that $T$ has exactly one vertex of maximum degree $\Delta(T)$.
I am stuck on how to prove this statement.
| Suppose that there is 2 or more vertices of maximum degree $\Delta (T)$, then the number of leaves will be at least $\Delta (T) -2$, since (I'll consider it a directed graph now just to use the terms indegree and out degree, but it also works in ordinary graphs) every vertex in a tree has its indegree $1$ or $0$, so it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3358109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to convert the equation to a standard form (paraboloid)? I am given the equation: $9x^2 + 4y^2 + z = 3$
The standard equation of a paraboloid parallel to z-axis is: $$\frac{z-z_0}{c} = \frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2}$$
I think the given equation should be a paraboloid due to the $z$ component, then I ... | Note $1 - \frac{z}{3} = \frac{3 - z}{3} = \frac{z - 3}{-3}$. Thus, $z_0 = 3$ and $c = -3$, along with $x_0 = y_0 = 0$, $a = \frac{1}{\sqrt{3}}$ and $b = \frac{\sqrt{3}}{2}$, allows your equation to be in standard form.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3358196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there a canonical way of lifting a vector field $X$ on $M$ to its tangent bundle $TM$? Is there a canonical way of lifting vector field $X$ on $M$ to its tangent bundle $TM$?
I came up with this question while studying tangent bundle formalism of Lagrangian mechanics. In Lagrangian mechanics, a vector field on $X$ o... | Let $M$ be a differentiable manifold (in the context of lagrangian mechanics, you can see $M$ as the space of all possible positions of the system). Suppone that on $M$ lagrangian coordinates $q=(q^j)$ are defined.
A vector field $X$ on $M$ can be written as
$$X = X^j(q) \frac{\partial}{\partial q^j}$$
where $X^j$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3358513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to find the averages of any consecutive numbers in a sequence? The practical purpose of this is I want to identify users that use my app on a weekly basis. For each user I can generate a sequence of times of their interactions, and from that I can generate a sequence of the length of time between each interaction.
... | I think you have to investigate every contiguous sublist, so there's no way around an order $N^2$ algorithm. If you can bound the length of the sublist then you can do this in order $N$.
You can probably speed up the individual computations by noting that if you have an average $A$ of $k$ numbers you can update the val... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3358631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Zero element of tensor product of $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$ Let $\bar{a} \otimes 1 \in \mathbb{Z}/n\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}/m\mathbb{Z}$ where $m,n \neq \pm 1$.
Question: Does $\bar{a} \otimes 1 = 0$ imply that $\bar{a} = 0$?
Approach:
*
*I know that $\bar{a} \otimes 1 =... | No, because $\;\mathbf Z/n\mathbf Z\otimes_{\mathbf Z}\mathbf Zm\mathbf Z\cong \mathbf Z/(m,n)\mathbf Z $.
So it only means that $a\in(m,n)$, i.e. $a$ is divisible by $\gcd(m,n)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3358745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Number of rational functions with low-degree numerator and denominator The problem
Let $\mathbb F$ be a finite field and $d < | \mathbb F |$. (Can think of the setting $d \approx |\mathbb F|/2$)
Let
$$ A = \left\{ \frac{p(x)}{q(x)} \ \colon \ p,q \in \mathbb F[x] \text{ and } \mathrm{deg}(q), \mathrm{deg}(p) \leq d \r... | Let $M_d$ be the set of all monic polynomials over $\mathbb{F}$ of degree (exactly) $d$, and let $$M_{\leqslant d}:=\bigcup_{k=0}^{d}M_d,\qquad G_d:=\{(p,q)\in M_{\leqslant d}^2 : \gcd(p,q)=1\}.$$ As any pair $(p,q)\in M_{\leqslant d}^2$ is uniquely represented by $(p'g,q'g)$, where $g=\gcd(p,q)$ is a monic polynomial,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3358846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determine if the following sequence is monotonic Consider the recursion $$ X_1 = 10 \quad \&\quad X_{n+1} = \sqrt {3+2X_n}$$
Is it monotonic or not?
problem
What I have done so far.
$X_1 = 10$
$X_2 = \sqrt {23}$
$X_3 = \sqrt {2 \sqrt{23} +3}$
We can clearly see that the first term is bigger than the second and so on,... | HINT: Note that $$x_{n+2}^2-x_{n+1}^2=2(x_{n+1}-x_n).$$ Now factor the LHS and since the sequence is clearly positive both $x_{n+2}-x_{n+1}$ and $x_{n+1}-x_n$ has the same parity for all $n\in \mathbb{N}.$ Then inductively $x_2-x_1$ must have the same parity, check it. For that comparing $x_2^2$ and $x_1^2$ is enough. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3359083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Suppose that $f(x,y) \leq f(y,x)$ for all $(x,y)\in\textbf{R}^{2}$. Show that $f(x,y) = f(y,x)$ for all $(x,y)\in\textbf{R}^2$ The question is: Suppose that $f(x,y) \leq f(y,x)$ for all $(x,y)$ in $\textbf{R}^{2}$. Show that $f(x,y) = f(y,x)$ for all $(x,y)\in\textbf{R}^{2}$.
I tried searching this question, but all th... | If $f(x,y) \leq f(y,x)$ for all $(x,y)$ in $\textbf{R}^{2},$
then taking $(x,y)=(a,b)$ we have $f(a,b)\le f(b,a),$
and taking $(x,y)=(b,a)$ we have $f(b,a)\le f(a,b)$.
So for all $a,b\in \textbf R, $ we have $f(a,b)\le f(b,a)\le f(a,b)$.
Now if $f(a,b)<f(b,a)$,
then we would have $f(a,b)<f(a,b),$ which is absurd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3359222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Some relations between multi-trace combinations I recently checked a relation between three $2 \times 2$ matrices $\mathrm{A,B,C}$.
$$\text{tr (ABC)}+\text{tr (ACB)}+\text{tr(A)tr(B)tr(C)}=\text{tr(A)tr(BC)}+\text{tr(B)tr(AC)}+\text{tr(C)tr(AB)}$$
Each component of the above relation makes all the possible multi-trace ... | Yes. Here is a generalization:
Definition. Let $K$ be a field, and let $m$ and $n$ be two nonnegative integers. Let $A_1, A_2, \ldots, A_n$ be $n$ arbitrary $m\times m$-matrices over $K$. Let $\sigma$ be a permutation of $\left\{1,2,\ldots,n\right\}$. Write $\sigma$ as a product of disjoint cycles (including $1$-cycle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3359302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Contradiction in proof by contraposition and counterexample $\forall x\in Q (\forall y \in R(y \notin Q \to \frac{x}{y} \notin Q))$
counterexample: let $x=0, y=\sqrt2$, then $\frac{0}{\sqrt2} = 0 = \frac{0}{1} \in Q$,
but by contrapositive:
$\forall x\in Q (\forall y \in R(\frac{x}{y} \in Q \to y \in Q ))$
Suppose $\f... |
Suppose $\frac{x}{y}$ is rational, then $x \in Z$ and $y \in Z$ by definition of rational
False. If $x=y=\frac12$, then $\frac{x}{y}$ is rational, but $x\notin \mathbb Z$ and $y\notin\mathbb Z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3359428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
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