Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Eigenvalues of an "Half-Kronecker "Product The Problem:
Given a 2 by 2 matrix $C$(the matrix elements of C are given), and two other
2 by 2 matrices $A$ and $B$(the matrix elements of A and B are given).
Now we can construct a new matrix $D$, which is given by the direct product
of (the first row of $C$) and $A$, the ... | It seems very doubtful that there could be a simple closed form for the
eigenvalues in general (i.e. simpler than explicitly taking the characteristic
polynomial and solving this quartic polynomial in radicals).
Case in point: take $$a_{{1,1}}=-3,a_{{1,2}}=3,a_{{2,1}}=0,a_{{2,2}}=1,b_{{1,1}}=-3,b_{{1,2
}}=-1,b_{{2,1}}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/701377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence in distribution of random variables question $X, X_1, X_2,\ldots $ are real random variables with $\mathbb{P}(X_n\leq x)\to \mathbb{P}(X\leq x)$ whenever $\mathbb{P}(X=x)=0$.
Why does $X_n\stackrel{L}{\to} X$? At the least, where would I begin?
| A sequence $X_1,X_2,\ldots$ of random variables is said to converge in distribution, or converge weakly, or converge in law to a random variable $X$ if
$$
\lim_{n\to\infty}F_n(x)=F(x)
$$
for every number $x\in\mathbb R$ at which $F$ is continuous, where $F_n(x)=\mathbb P(X_n\le x)$ and $F(x)=\mathbb P(X\le x)$.
Thus, w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Prove the statement for definite integral We have positive continuous function $f(x)$ defined on $\mathbb{R}$,
such as $\int_{-\infty}^{+\infty} f(x) dx = 1$
let $\alpha \in (0,1)$ and $[a,b]$ is an interval
of minimal length amongst intervals for those holds: $\int_{a}^{b} f(x) dx = \alpha$.
Task is to prove that $f(... | If $f(a)\neq f(b)$, say $f(a)$ is larger. Then we can always shift the interval towards the direction of $a$ to make it shorter (since $f$ is continuous).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to determine the eigenvectors of this matrix? I have some problems to determine the eigenvectors of a given matrix:
The matrix is:
$$
A = \left( \begin{array}{ccc}
1 & 0 &0 \\
0 & 1 & 1 \\
0 & 0 & 2
\end{array} \right)
$$
I calculated the eigenvalues first and got $$ \lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 1$$
T... | Update: I have undeleted my answer because I think it is fixed now.
You got $$ V_{\lambda_2} = \left(\begin{array}{ccc} 0 \\ 1 \\ 1 \end{array} \right) $$
correct but then copied it down wrongly.(I think..)
Then you correctly wrote down the case $\lambda_1$. From
$$ \left(\begin{array}{ccc } 0 & 0 & 0 \\ 0 & 0 & 1 \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/701673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is this complex function harmonic? Let us consider the following convergente series in the set $0<x<1$ and all real $y$:
$$h(x+iy)=∑_{n=2}^{∞}(-1)ⁿ⁻¹((n^{2x-1}-1)/n^{x})n^{iy}$$
My question is: Is this complex function harmonic?
| Look at the terms of the series. Ignoring the $(-1)^n$ for the moment, we have
$$\frac{n^{2x-1}-1}{n^x}n^{iy} = n^{x-1}n^{iy} - n^{-x}n^{iy} = n^{z-1} - n^{-\overline{z}}.$$
The first term is holomorphic, and hence harmonic. The second term is antiholomorphic, and hence harmonic. Thus the difference of the two terms is... | {
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dirichlet and gcds Let (a,b)=1 and c>0. Prove that there is an integer x such that (ax+b, c)=1.
Right now, I have the following approach:
Let's assume that for every x, (ax+b,c)$\neq$ 1. Then $\exists$ d, where d/c that also d/ax+b=> ax$\equiv$-b (mod d). I'm not sure how to continue from here on.
| The critical thing is that $d|ax+b$ for all $x$. So take two different $x$'s and get $d|a$, then $d|b$ and you are done.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to calculate lim inf and lim sup for given sequence of sets Let the indicator function be defined as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and $I_{\nu}(x) \in [0,1]$ be a continuous approximation of the $I(x)$ such that $$\lim_{\nu \rightarrow \infty} I_{\nu}(x) = I(x... | Let $I_n(x)$ be $0$ for $x \le -1/n$ and $1$ for $x \ge 1/n$. On $[-/n,0]$ define $I_n(x)$ as
$$f_n(x)=n^n(1-1/n)(x+1/n)^n,$$
and on $[0,1/n]$ define it as
$$g_n(x)=1-n^{n-1}(1/n-x)^n.$$
One can check that $f_n(0)=g_n(0)=1-1/n$ and that $f(-1/n)=g(1/n)=0,$ so that the rules are not in conflict at interval endpoints and... | {
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Prove that the intersection of two subgroups is a subgroup. In more detail, if $G$ is a group and $H_1$, $H_2$ are subgroups of G then $H_1 \cap H_2$ is a subgroup of G.
Next, give an example of a particular group $G$ (any one you like), and two different subgroups $H_1$, $H_2$ of $G$ , compute the intersec... | Theorem Let $G$ be a non-trivial finite group. Then the following are equivalent.
(a) For each pair of subgroups $H_1$ and $H_2$ of $G$, $H_1 \cup H_2$ is a subgroup
(b) $G$ is cyclic of prime-power order.
Proof (b) $\Rightarrow$ (a) follows from the fact that in a cyclic group there is a unique subgroup of order $d$ f... | {
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Proof involving matrix equation $A$ and $B$ are $(n\times n)$ matrices and $AB + B + A = 0$. Prove that then $AB=BA$.
How should I approach this problem?
| Adding the identity matrix $I$ on both sides, we find $(A+I)(B+I) = I$. Hence $A+I$ and $B+I$ are inverses of each other. It follows that $(B+I)(A+I) = I$ as well. Expanding gives $BA + B + A = 0$, hence $AB = BA$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Question about proof of FTA I read a short proof of the Fundamental Theorem of Algebra, as follows:
Assume $p(z)$ is a nonconstant polynomial with no roots. Then $1/p$ is an analytic function on $\mathbb{C}$. Also, $1/p \to 0$ as $z \to \infty$, so $p$ is bounded. By Liouville's theorem, any bounded analytic function i... | Absolute value of polynomial tends to infinity for $\left|z\right|\to\infty$. That is, for each $M>0$, there exists $R>0$ such that for $\left|z\right|>R$ we have $\left|p(z)\right|>M$. Take sufficiently large closed disk, so that $\left|p(z)\right|>1$ for $z$ outside the disk. The disk is compact, so it's image by $\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What Is Exponentiation? Is there an intuitive definition of exponentiation?
In elementary school, we learned that
$$
a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times})
$$
where $b$ is an integer.
Then later on this was expanded to include rational exponents, so that
$$
a^{\frac{b}{c}} = \sqrt[c]{a^b}
$$
From t... | $a^b$ refers to the "multiplicative power" of performing b multiplications by a. This is intuitively obvious with positive integer 'b's, but still holds for fractional and negative values when you put a little brain grease into considering what it means to do 'half a multiplication' or a 'negative multiplication'.
$9^\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/702414",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "70",
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"answer_id": 2
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Solve $\sin x - \cos x = -1$ for the interval $(0, 2\pi)$ We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals.
How to solve
$$\sin x - \cos x = -1$$
for the interval $(0, 2\pi)$.
| Method $\#1$
Avoid squaring which immediately introduces extraneous roots which demand exclusion
We have $\displaystyle\sin x-\cos x=-1$
$$\iff\sin x=-(1-\cos x)\iff2\sin\frac x2\cos\frac x2=-2\sin^2\frac x2$$
$$\iff2\sin\frac x2\left(\cos\frac x2+\sin\frac x2\right)=0$$
If $\displaystyle \sin\frac x2=0,\frac x2=n\pi... | {
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A question regarding a prefix code Let $C=\{ c_1, c_2, \dots, c_m \}$ be a set of sequences over an alphabet $\Sigma$ and $|\Sigma|=\sigma$. Assume that $C$ is a prefix-free code, in the sense that no codeword in $C$ is a prefix of another codeword in $C$, with $|c_i|= n_i\ \forall i$. Prove that $\sum_{h=1}^m \sigma^{... | @mnz has given a prefectly rigorous answer. There is another proof that works for any uniquely decodable scheme, i.e. given a sequence of letters from the alphabet, there is at most one way to separate the letters such that each subsequence is in $C$. It's clear that prefix-free codes can be uniquely decoded.
We define... | {
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Assuming ${a_n}$ is a convergent sequence, prove that the lim inf of $a_{n+1}$ is equal to the lim inf of $a_n$ I'm aware that you have to use the definition of a limit of a sequence, which is:
$\lim\limits_{n \to \infty} a_n = L $ if for every $E > 0$, there is an $N$ such that if $n >N $then $|a_n - L | < E$
I just h... | First of all you should understand intuitively why this holds. The limit of a sequence $a_n$ tells us about the long-term behavior of the sequence. But $a_n$ and $a_{n+1}$ have the same long-term behavior, they are merely indexed differently; so they should have the same limit.
For the formal proof, suppose $a_n$ conve... | {
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Find basis so Transformation Matrix will be diagonal $e_1,e_2$ will be basis for $V$. $W$ has a basis $\{e_1+ ae_2,2e_1+be_2\}$. Choose an $a,b$ s.t. that the basis for $W$ will have a transformation matrix $T$ will be in diagonal form.
$T(e_1) = 1e_1+5e_2$
$T(e_2) = 2e_1+4e_2$
$V$ and $W$ are linear spaces of dimensio... | In the basis for $V$,
$$T_{[V]} = \left(\begin{matrix}
1 & 2 \\
5 & 4 \end{matrix} \right)$$
If you want the transformation $T$ written in $W$'s basis to be diagonal, then you want each basis vector of $W$ to be mapped to some multiple of itself:
$$T_{[W]} = \left(\begin{matrix}
\lambda_1 & 0 \\
0 & \lambda_2 \end{matr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Probability of two digit number sequence in series of numbers Given a random sequence (say $15$) of numbers I want to find the odds of finding '$90$' and '$09$' in the sequence. Looking at just two numbers in the sequence you have a $\dfrac{2}{10}$ chance of getting a '$9$' or '$0$' as the first digit, followed by $\df... | Your question is not totally clear, but working from your calculation that the probability of success with a two digit string is $\frac{1}{50}$ then you seem to be looking for the probability of either $09$ or $90$ (or both) in your string.
This will be easier to consider as the complement of neither $09$ nor $90$ appe... | {
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Equation with an infinite number of solutions I have the following equation: $x^3+y^3=6xy$. I have two questions: 1. Does it have an infinite number of rational solutions?
2. Which are the solutions over the integers?($ x=3 $ and $ y=3 $ is one)
Thank you!
| Wolfram Alpha says that there are no rational solutions except the one you noted, $x=y=3$ although.
It seems that it chose to skip the trivial $x=y=0$ though. The link has some irrational solutions too, if you need them.
| {
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prove a subset of $l^2$ is closed? Let $\{f_i\}$ be a sequence of (nice) functions in $L^p[0,1],$ and $p>1, \frac{1}{p}+\frac{1}{q}=1.$
Define a subset $A$ of the space $l^2$ as
$$A=\left\{(a_1, a_2, \ldots)\in l^2: \text{ such that } a_i=\int_0^2g(x)f_i(x)dx, \text{ for }g\in L^q[0,1] \text{ and } \|g\|_q\leq1\right\}... | I will assume the limits of the integral should be $0$ and $1$, not $0$ and $2$. I also won't really comment on the requirements of the sequence $f_i$; the set you've provided is well-defined in any case. If we can establish that this set is always closed, regardless of the condition that $(a_1, a_2, \ldots) \in l^2$ f... | {
"language": "en",
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Find the limit of $\lim_{x\to 0}\frac{\sqrt{x^2+a^2}-a}{\sqrt{x^2+b^2}-b}$ Can someone help me solve this limit?
$$\lim_{x\to0}\frac{\sqrt{x^2+a^2}-a}{\sqrt{x^2+b^2}-b}$$
with $a>0$ and $b>0$.
| No need for L'Hopital - we simply multiply and divide by the conjugate radical expression:
\begin{align}
\frac{\sqrt{x^2+a^2}-a}{\sqrt{x^2+b^2}-b}&=\left(\frac{\sqrt{x^2+a^2}-a}{\sqrt{x^2+b^2}-b}\cdot\frac{\sqrt{x^2+a^2}+a}{\sqrt{x^2+b^2}+b}\right)\cdot\frac{\sqrt{x^2+b^2}+b}{\sqrt{x^2+a^2}+a}
\\ &=\frac{x^2+a^2-a^2}{x... | {
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"timestamp": "2023-03-29T00:00:00",
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Proof: $a^2 - b^2 = (a-b)(a+b)$ holds $\forall a,b \in R$ iff R is commutative We want to show that for some ring $R$, the equality $a^2 - b^2 = (a-b)(a+b)$ holds $\forall a,b \in R$ if and only if $R$ is commutative.
Here's my proof --- I'm not sure if the first part stands up to examination. I'd be grateful if someo... | What you did is correct, as far as I can tell.
Perhaps a shorter solution is that $(a^2 - b^2) - (a-b)(a+b) = ba - ab$. This is $0$ iff the $a$ and $b$ commute. So, the expression $(a^2 - b^2) - (a-b)(a+b)$ is identically $0$ iff the ring is commutative.
| {
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Where is the mistake in this limit calculation? I got this limit:
$$\lim_{x\to1}\frac{\sqrt[3]{x}-1}{\sqrt[4]{x}-1} \implies \lim_{x\to1}\frac{\frac{x-1}{\sqrt[3]{x²}+\sqrt[3]{x}+1}}{\sqrt[4]{x}-1} \implies \lim_{x\to1}\frac{x-1}{\sqrt[3]{x²}+\sqrt[3]{x}+1}*\frac{1}{\sqrt[4]{x}-1}*\frac{\sqrt[4]{x}+1}{\sqrt[4]{x}+1} \i... | That's probably because $\dfrac{x-1}{\sqrt[4]x-1}=\sqrt[4]{x^3}+\sqrt[4]{x^2}+\sqrt[4]{x}+1$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Why square the result of $x_1 - \bar{x}$ in the standard deviation? I don't understand the necessity of square the result of $x_1 - \bar{x}$ in $$\sqrt{\frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N-1}}$$. In fact I don't understand even why is $N - 1$ on the denominator instead of just $N$. Someone could explain it or re... | Squaring the Deviations
The variance of a sample measures the spread of the values in a sample or distribution. We could do this with any function of $|x_k-\bar{x}|$. The reason that we use $(x_k-\bar{x})^2$ is because the variance computed this way has very nice properties. Here are a couple:
$1$. The variance of the ... | {
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"timestamp": "2023-03-29T00:00:00",
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Solving $x^2 - 16 > 0$ for $x$ This may be a very simple question, but I can't figure out the correct reason behind it.
If $x^2 - 16 >0$, which of the following must be true?
a. $4 < x$
b. $-4 > x > 4$
c. $-4 > x < 4$
d. $-4 < x < 4$
I know the answer but I didn't get how they figured out the direction.
| Since $(-4)^2=4^2=16,$ and for $x^2-16>0$ to be true, $x$ has to be strictly greater than $4$ and strictly less than $-4,$ then from this I think you can tell which answer must be correct
| {
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Why doesn't this calculation work? I want to find some closed form for $\gcd(x^3+1,3x^2 + 3x + 1)$ but get $7$ which is not always true.
| With your procedure you found that the GCD between the two polynomials $x^3+1$ and $3x^2+3x+1$ in $\mathbb{Q}[x]$ is $7$, or equivalently $1$, because the GCD of polynomials is defined up to constants (every scalar value $c$ divides any polynomial $p(x)\in\mathbb{Q}[x]$).
Thus there is not contradiction in your statem... | {
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If $X_1,X_2\sim U(0,1)$ and $Y$ is one of them who is closest to an end point, find distribution of $Y$. Let $X_1$ and $X_2$ be independent, $U (0, 1)$-distributed random variables, and
let $Y$ denote the point that is closest to an endpoint. Determine the distribution of $Y$.
It's a question in chapter of "order stati... | We derive the density function directly, in a way analogous to the way one finds the distribution of order statistics. The only interesting value of $y$ are between $0$ and $1$. We find $f_Y(y)$ for $0\lt y\lt \frac{1}{2}$.
The probability that $y$ is between $y$ and $y+dy$, for "small" $dy$, is approximately $f_Y(y)\... | {
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variation of the binomial theorem Why does:
$$ \sum_{k=0}^{n} k \binom nk p^k (1-p)^{n-k} = np $$ ?
Taking the derivative of:
$$ \sum_{k=0}^{n} \binom nk p^k (1-p)^{n-k} = (1 + [1-P])^n = 1 $$
does not seem useful, since you would get zero. And induction hasn't yet worked for me, since – during the inductive step –... | This is saying that the average number of coin flips if you flip $n$ fair coins each with heads probability $p$ is $np$ (you are calculating the mean of a Binomial(n,p) distribution).
You can use linearity of expectation - if $Y = \sum_i X_i$ where $X_i$ is Bernoulli(p) and there are $n$ such $X_i$, then $Y$ is Binomi... | {
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Doing take aways I am reviewing take aways. I am having trouble
How do I do $342 - 58$?
For the ones column I made the $2$ into a $12$ so I can do $12 - 8 = 4$ but I must take away one tens. So I get $3 - 5$ in tens column but I cant do $3 - 5$. What do I do now do I borrow something else?
| You lower the hundreds spot by one, and increase the tens spot by 10. Example:
300 + 40 + 2 = 200 + 140 + 2
Now that you do this, you get 13-5 and your answer is an 8 in the tens place, and with a 2 in the hundreds place.
As a whole, it looks like this
342 - 58
300 + 40 + 2 - 50 - 8
200 + 130 + 12 -50 - 8 (1 ... | {
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"url": "https://math.stackexchange.com/questions/703771",
"timestamp": "2023-03-29T00:00:00",
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Big Oh and Big Theta relations confirmation I just want to confirm these statements,
I know that Big O, and Big theta, are partial order and equivalence relations respectively, all positive integers, but not sure on these restrictions.
$f:N \rightarrow R^+$
where $f$ R $g$ is and only if $f(n) = O(g(n))$
This is still... | You are correct that Big-Oh is a partial order, and Big-Theta is an equivalence relation. One can say that $f < g$ if $f \in O(g)$ (or $f = O(g)$, alternate notation-wise).
Notice though that $\sin(n) \in O(n)$ and $\cos(n) \in O(n)$, so that with this order we have $\sin(n) < n$ and $\cos(n) < n$, but we have neither ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solution of definite integrals involving incomplete Gamma function The solution of the integral $$\int_0^{\infty}e^{-\beta x}\gamma(\nu,\alpha \sqrt x)dx $$ is given as $$2^{-\frac{1}{2}\nu}\alpha^{\nu}\beta^{\frac{1}{2}\nu-1}\Gamma(\nu)\exp(\frac{\alpha^2}{8\beta})D_{-\nu}(\frac{\alpha}{\sqrt{2\beta}})$$
[Re $\beta>0$... | $\int_0^ae^{-\beta x}\gamma(\nu,\alpha\sqrt x)~dx$
$=-\int_0^a\gamma(\nu,\alpha\sqrt x)~d\left(\dfrac{e^{-\beta x}}{\beta}\right)$
$=-\left[\dfrac{e^{-\beta x}\gamma(\nu,\alpha\sqrt x)}{\beta}\right]_0^a+\int_0^a\dfrac{e^{-\beta x}}{\beta}d\left(\gamma(\nu,\alpha\sqrt x)\right)$
$=-\dfrac{e^{-\beta x}\gamma(\nu,\alpha\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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justification of a limit I encountered something interesting when trying to differentiate $F(x) = c$.
Consider: $\lim_{x→0}\frac0x$.
I understand that for any $x$, no matter how incredibly small, we will have $0$ as the quotient. But don't things change when one takes matters to infinitesimals?
I.e. why is the functio... | A function isn't just an expression, but you can think whether a single expression can be applied to an argument. The expression $0^{-1}$ is rather meaningless, so you don't know how to get the behavior of the function $f(x)=0\cdot x^{-1}$ at $x=0$ from the expression.
Limits are just a way to describe the behavior (if... | {
"language": "en",
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Prove that $f_n$ converges uniformly on $[a,b]$ Let $f_n$ be a sequence of functions defined on $[a,b]$. Suppose that for every $c \in [a,b]$, there exist an interval around $c$ in which $f_n$ converges uniformly. Prove that $f_n$ converges uniformly on $[a,b]$
I know that since for every $c \in [a,b]$, there exist an ... | You can use a trick similar to mathematical induction but on real line:
Let $I = \{ x \in [a,b]: f_n \mbox{ converges uniformly on } [a,x] \}$
We know:
For every $c \in [a,b]$, there exist an interval around $c$ in which
$f_n$ converges uniformly
Take $c=a$ in the above statement, and then there exists an interval ... | {
"language": "en",
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How does this equation create this chart?
I am trying to understand this formula from the chart above.
For example, from the middle graph, How does h(x) = 0.5x get the coordinates 2,1?
Any explanations on the other graphs would be helpful, too.
Edit: How would I find theta 0 and theta 1 given the following graph?
| The horizontal axis is for values of $x$ and the vertical one is for the values of a function $h_{\theta}(x)$. So if we have $h_{\theta}(x)=\theta_0+\theta_1x$ where $\theta_0=0$ and $\theta_1=0,5$ than you have the function $h_{\theta}(x)=0+0,5x=0,5x$ and for $x=2$ you have h_{\theta}(2)=0,5*2=1$ so if in the horizon... | {
"language": "en",
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Use L'Hopital's Rule to Prove Let $$f: \mathbb R\rightarrow \mathbb R$$ be differentiable, let a in $\mathbb R$. Suppose that $f''(a)$ exists. Prove that $$\lim_{h\rightarrow0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}=f''(a) $$
Suppose further that $f''(x)$ exists for all $x$, and that $f'''(0)$ exists. Prove that $$\lim_{h\righ... | Question 1
$$
\lim_{h\rightarrow0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2} \\
$$
when $h = 0$ is substituted numerator and denominator reduce to $0$. So, applying L'Hopital's rule (differentiate wrt $h$)
$$
\lim_{h\rightarrow0}\frac{f'(a+h)-f'(a-h)}{2h} \\
\lim_{h\rightarrow0}\frac{f'(a+h)-f'(a)-f'(a-h)+f'(a)}{2h} \\
\lim_{h\ri... | {
"language": "en",
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Number of groups of a given order In general, for what $n$ do there exist two groups of order $n$? How about three groups of order $n$?
I know that if $n$ is prime, there only exists one group of order $n$, by Lagrange's Theorem, but how do you classify all other such $n$ that have $2, 3, 4, ...$ groups?
This question ... | Exactly 2 groups. There is a paper, which claims to classify "Orders for which there exist exactly two groups".
This link contains the text of the paper in text (!) format. I didn't find a pdf.
Disclamer: I didn't check if the proofs in the paper are correct. I also don't know, if the paper was published in any peer-re... | {
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Poisson Distribution more than 2 raindrops will fall on the square inch Assume that raindrops fall on a particular square inch of a city block according to a Poisson process will an average of 2 raindrops per second. Find the probability that more than 2 raindrops will fall on the square inch during a 5-second time int... | Indeed $$P(Y>2)= 1 - [P(Y=0)+P(Y=1)+P(Y=2)]$$
Then if you mean $$1-\left[\frac{(5\cdot2)^2}{2!}e^{-2\cdot5} + \frac{(5\cdot2)^1}{1!}e^{-2\cdot5} + \frac{(5\cdot2)^0}{0!}e^{-2\cdot5}\right]$$ your answer is correct (be careful with the parentheses in your expression).
| {
"language": "en",
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Solving a differential equation $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$ Let $\alpha$ be a regular curve in $\mathbb{R}^3$ such that $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$ for $w$ a constant vector. How can we determine $\alpha$ ?
$\displaystyle w \times\alpha$ : cross product
Any hint would be a... | You can write $w\times\alpha$ as $\Omega\alpha$, where $\Omega$ is an (antisymmetric) matrix. Then the problem reduces to a linear ODE.
| {
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ambidextrous mathematician. combinations problem Please help me solve this problem. At first it seemed to be easy, but I got stuck.
An ambidextrous mathematician with a very short attention span keeps two video game credit cards, one in each of her two front pockets. One game card has credit for 5 games. The other gam... | Hint: The problem is a natural for using a division into cases.
Without loss of generality we may assume that the $4$-game card is in the left pocket, and the $5$-game card in the right pocket.
Either (i) there are $4$ credits left on the $4$-card or (ii) $4$ left on the $5$-card.
Event (i) has probability $\frac{1}... | {
"language": "en",
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Linear algebra questions that a high-schooler could explore Are there any deep/significant concepts in linear algebra that are not overly complicated that a high schooler could explore in depth?
| A few ideas:
(1) Numerical Stuff:
Look at various methods of solving linear systems or inverting matrices. Study performance (the number of operations involved), and what sorts of things can go wrong numerically. Show that the naive textbook methods don't work very well in practice. See the linear system example in "Wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/704718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Modular Arithmetic - Pirate Problem I was reading an example from my book, and I need further clarification because I don't understand some things. I'm just going to include the $f_1$ part in full detail because $f_2$ and $f_3$ are identical.
Consider the following problem.
Once upon a time, a band of seven pirates se... | The innocent way. Reduce your work with the $\mathrm{\color{Red}{red}}$ equivalence between congruence systems.
$$\begin{cases}
x\equiv 2\pmod 7 \\
x\equiv 5 \pmod 6 \\
x \equiv 2 \pmod 5
\end{cases}\color{Red}{\iff}\begin{cases}
x\equiv 2\pmod {35} \\
x\equiv 5 \pmod 6
\end{cases}$$
From the first congruence, $x=35t+... | {
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Finding value of x in logarithms?
Q) Find the value of $x$ in $2 \log x + \log 5 = 2.69897$
So far I got:
$$2 \log x + \log 5 = 2.69897$$
$$\Rightarrow \log x^2 + \log 5 = 2.69897 $$
$$\Rightarrow \log 5x^2 = 2.69897 $$
What should I do next?
Note: In this question $\log(x) \implies \log_{10}(x)$ , it is therefore im... | Raise the base of the logarithm to both sides. Then, you get $5 x^2 = b^{2.69897}$ where $b$ is the base of the logarithm (probably $b=10$). Then, solve for $x$ by dividing by $5$ and taking square roots.
| {
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"timestamp": "2023-03-29T00:00:00",
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Why does $\sum_{k=0}^\infty\frac{r^k(k+n)!}{k!}=\frac{n!}{(1-r)^{n+1}}$? When I put the following series in Mathematica, I get an answer:
$$\sum_{k=0}^\infty\frac{r^k(k+n)!}{k!}=\frac{n!}{(1-r)^{n+1}}$$
Here $0<r<1$ and $n$ is a non-negative integer.
My question is: how does one arrive at this solution (without the use... | Hint: $\dfrac{(k+n)!}{k!}=n!\cdot\dfrac{(k+n)!}{n!\cdot k!}=n!\cdot\Large{n+k\choose k}$. Now see binomial series.
| {
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How to prove this inequality $ x + \frac{1}{x} \geq 2 $ I was asked to prove that:
$$x + \frac{1}{x}\geqslant 2$$
for all values of $ x > 0 $
I tried substituting random numbers into $x$ and I did get the answer greater than $2$. But I have a feeling that this is an unprofessional way of proving this. So how do I prov... | Hint: $x^2 -2x + 1 = (x-1)^2 \ge 0$. If you do not see it, divide the inequality through by $x$.
| {
"language": "en",
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"source": "stackexchange",
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Cauchy's Theorem and maximum modulus principle Suppose $f\in H(\Omega), \Gamma$ is a cycle in $\Omega$ such that $Ind_{\Gamma}(\alpha)=0$,for all$\alpha \notin \Omega$，$|f(\zeta)|\leq 1$ for every $\zeta \in \Gamma$,
and $Ind_{\Gamma}(z) \neq 0 $.Prove that $|f(z)|\leq 1$
Cauchy's Theorem implies:
$$ |f(z)\cdot Ind_{\G... | Oh,I found the answer.
$Ind_{\Gamma}(\alpha)=0$,for all $\alpha \notin \Omega$ implies:
The union of $\Gamma$ and the components of the complement of E which $Ind_{\Gamma}\neq 0 $ is
$ \subset \Omega$.This is the point of the question. Then the boundary of the component contained z is $\subset \Gamma$ and maximum modu... | {
"language": "en",
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Decomposing real line as a union of a nullset and a set of first category $\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category!
can anybody prove this?
I guess $A$ must be an $G_{\delta}$ set which is dense in $\Bbb R$ and obviously $B=\Bbb R-A$.
| Enumerate the rational numbers as a sequence $\{ r_n;\; n\in\mathbb N\}$. For each $n\in\mathbb N$ and all $j\in\mathbb N$, set $$I_{n,j}:=\left] r_n-\frac{1}{j}\, 2^{-n}, r_n+\frac{1}j\, 2^{-n}\right[\, .$$
Then define
$$O_j:=\bigcup_{n\in\mathbb N} I_{n,j}\, , $$
and
$$ A:=\bigcap_{j\in\mathbb N} O_j\, .$$
Each $O... | {
"language": "en",
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Open and connected in $R^n$ revised I am trying to understand the following: If we have an open and connected set in $R^n$ then it can be connected with line segments parallel to the axes.
I managed to prove this:
If a set $U$ is open and connected in $\mathbb{R}^n$ then we can prove it is polygonally connected(there ... | First note that for any cube $C=[-r,r]^n\subseteq\mathbb R^n$ any point $c=(c_0, c_1,\ldots c_{n-1})\in C$ is polygonally connected to the center of $C$ along the axes.
$$(0,0,0,\ldots,0)\to(c_0,0,0, \ldots, 0)\to(c_0,c_1,0,\ldots0)\to\ldots\to(c_0,c_1,\ldots c_{n-1})$$
Let $G$ be any nonempty open connected set in $\m... | {
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Multiplication of infinite series Why multiplication of finite sums $(\sum_{i=0}^n a_i)(\sum_{i=0}^n b_i)=\sum_{i=0}^n (\sum_{j=0}^ia_jb_{i-j})$ (EDIT: This assumption was shown to be false) does not work in infinite case? I have constructed proof which shows it does but it must hase some flaw which I can not find. Her... | As @Claude has stated for the simpler cases here is it for 4 elements in $a$ and in $b$. The correct sum is the sum over all elements of the ("outer"(?)) product $C$ of the two vectors
$$ A^T \cdot B=C= \small \begin{array} {r|rrrr}
& b_0 & b_1 & b_2 & b_3 \\
\hline
a_0 & a_0b_0 & a_0b_1 & a_0b_2 & a_0b_3 \\
a... | {
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How to tell if a code is lossless Consider the following code mapping:
$$a \mapsto 010, \quad b\mapsto 001, \quad c\mapsto 01$$
It's easy to see that the code isn't lossless by observing the code $01001$, which can be translated to "ac" or "cb".
Given a general code, how can you tell if it's lossless or not? I don't ... | Thats kind of hard for a general code but you can use the Sardinas-Patterson algorithm.
The algorithm generates all possible "dangling suffixes" and checks to see if any of them is a codeword. A dangling suffix is the bits that are "left over" when you compare two similar sequences of your code. If you want your code t... | {
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Prove that {$a ∈ ℤ : a ≤ k$} has a greatest element How can I prove that the set {$a ∈ ℤ : a ≤ k$}, where $k∈ℝ$, has a greatest element?
I have tried using the Well-ordering theorem in order to get a contradiction but I'm having trouble with my approach.
Thanks.
| Hint: The floor function: $\lfloor k \rfloor$. For example, if $k$ were $2.1$, then the greatest element of your set would be $2$.
| {
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Tangent spaces and $\mathbb{R}^n$ The tangent space of a circle is a line.
The tangent space of a sphere (in every point) can be thought of as a plane.
Is this a general thing? I mean, having an $n$ dimensional Riemannian manifold, can the tangent space in every point be thought as $\mathbb{R}^n$?
If the answer is ye... | If $M$ is a smooth $n$-dimensional manifold, then for each $p\in M$ the tangent space $T_p M$ is an $n$-dimensional real vector space. This tangent space is therefore isomorphic to $\mathbb R^n$ as a real vector space, though not in a "natural" way, in the sense that $T_pM$ does not have a distinguished basis correspon... | {
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Find the center of a specific group The group $G$ is generated by the two elements $\sigma$ and $\tau$, of order $5$ and $4$ respectively. We assume that $\tau\sigma\tau^{-1}=\sigma^2$.
I have shown the following:
* $\tau\sigma^k\tau^{-1}=\sigma^{2k}$ and $\tau^k\sigma\tau^{-k}=\sigma^{2^k}$.
* $\langle\sigma\rangle$ i... | You have a normal subgroup of order $5$. Your calculations are already sufficient to show that the elements of this group other than the identity don't commute with $\tau$, or indeed any of its powers. So $\tau, \tau^2, \tau^3$ are not in the centre. $1=\tau^0=\tau^4$ is of course in the centre.
Suppose we have an elem... | {
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$\mathbb Z[\sqrt {-5}]$ is Noetherian I'm trying to prove that $\mathbb Z[\sqrt {-5}]$ is Noetherian. I already know that $\mathbb Z[X]$ is Noetherian and I'm trying to find a surjective map
$$\varphi: \mathbb Z[X]\to \mathbb Z[\sqrt{-5}]$$
with $\ker\varphi=(X^2+5)$.
If I could find this map I could prove that $\math... | Define $\varphi:\mathbb Z[X]\to\mathbb Z[\sqrt{-5}]$ by $\varphi(f)=f(\sqrt{-5})$. It's that simple.
| {
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Find the greatest common divisor (gcd) of $n^2 - 3n - 1$ and $2$ Find the greatest common divisor (gcd) of $n^2 - 3n - 1$ and $2$ considering that $n$ is an integer. Thanks.
| Hint $\ $ One of $\,\ n,\,\ n\!-\!3\,$ is even so $\ n(n\!-\!3)-1\,$ is odd, so coprime to $\,2.$
Alternatively $\,2\nmid f(n)=n^2-3n-1\,$ since $f$ has no roots mod $\,2\!:\ f(0)\equiv 1\equiv f(1),\,$ which is a special case of the Parity Root Test.
| {
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A function vanishing at infinity is uniformly continuous If $f\in C_0(\mathbb{R})$ (i.e. $f$ continuous and for all $\varepsilon>0$ there is $R>0$ such that $|f(x)|<\varepsilon$ whenever $|x|>R$), then why is $f$ uniformly continuous? I know that we should somehow use that $f$ is "small" outside a compact interval (on ... | We shall use the fact that a continuous function in a closed interval is uniformly continuous.
Let $f\in C_0(\mathbb R)$ and $\varepsilon>0$. We shall find a $\delta>0$, such that $\lvert x-y\rvert<\delta$ implies that $\lvert f(x)-f(y)\rvert<\varepsilon$.
As $\lim_{\lvert x\rvert\to\infty}f(x)=0$, there exists an $M>0... | {
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Proving if $\gcd(c,m)=1$ then $\{x\in \Bbb Z \mid ax\equiv b \pmod m\} =\{x\in \Bbb Z \mid cax\equiv cb \pmod m\}$ Okay so I'm confused on how to approach this question.
If $\gcd(c,m)=1$, then $S=T$ where $S=\{x\in \Bbb Z \mid ax\equiv b \pmod m\}$ and $T=\{x\in \Bbb Z \mid cax\equiv cb \pmod m\}$.
I know that since... | It might be worth stating and proving Bezut here. Bezuts lemma says that if $(a,b)=1$, then $\exists x,y$ st $ax+by=1$. To prove this, consider the set $S:=\{d>0|\exists x,y,ax+by=d\}$. Let $d_0$ be the minimal element of this set and use the division algorithm on $a$ and $d_0$ to find out that $d_0|a$ (Specifically, t... | {
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Find to which $( \forall x)$ , each occurrence of x belongs to. (logic) Find to which $( \forall x) $, each occurrence of x belongs to.
$$ (\forall x)((\forall x)(\forall y)\ x < y \lor x > z ) \rightarrow (\forall y)\ y=x $$
Is it right that the third and fourth occurrence of x belongs to the second occurrence of $ \f... | With this particular notation, there are two conventions (let the example be $(\forall x)\phi \star \psi$)
*
*quantifier binds as far as it can (the example becomes $(\forall x)(\phi \star \psi)$),
*quantifier binds only the closest subexpression (while here it is $\big((\forall x)\phi\big) \star \psi$).
You are ... | {
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Given $p(x)$ is a polynomial with integer coefficients and that $p(a)=1$ for some integer $a$ prove that $p(x)$ has no more than two integral roots. Given $p(x)$ is a polynomial with integer coefficients and that $p(a)=1$ for some integer $a$ prove that $p(x)$ has no more than two integral roots.
I've attempted a proof... | For this to be true, we need to specify that $p$ has integer coefficients: without this assumption, $p(x) = \frac16x(x-1)(x+1)$ is a counterexample, with roots at $-1,0,$ and $1$, but $p(2)=1$.
Suppose a polynomial $p(x)$ with integer coefficients has three or more distinct integral roots. This means that $p(x) = (x-a_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How is this trig identity equal? I do not understand how this is equal.
$$
{cos\theta(cos\theta-1)\over 1-cos\theta} = -cos\theta
$$
What simplification step am I missing? Thanks.
| $${\cos\theta(\cos\theta-1)\over 1-\cos\theta} = -\cos\theta $$
$$\iff \cos\theta(\cos\theta-1)=-\cos\theta(1-\cos\theta)$$
$$\iff \cos^2\theta - \cos\theta = \cos^2\theta - \cos\theta $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/706231",
"timestamp": "2023-03-29T00:00:00",
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What is the probability that $HH$ occurs before $TH$ in an infinte sequence of coin flips? This is one of the questions of a set of exam review questions that don't have solutions to them. I can't get my head around this but it seems so simple.
By flipping a fair coin repeatedly and independently, we obtain a sequence... | First flip is either heads or tails. If the second flip is heads we have a winner no matter what. If the second flip is tails we have no winner, but it follows that Player 2 must win. Why?
Flip three is either heads or tails. If it is heads, player 2 wins. Tails, no one wins. Flip four and each afterward either results... | {
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"timestamp": "2023-03-29T00:00:00",
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Calculating the limit $\lim((n!)^{1/n})$ Find $\lim_{n\to\infty} ((n!)^{1/n})$. The question seemed rather simple at first, and then I realized I was not sure how to properly deal with this at all. My attempt: take the logarithm,
$$\lim_{n\to\infty} \ln((n!)^{1/n}) = \lim_{n\to\infty} (1/n)\ln(n!) = \lim_{n\to\infty... | Let a $n\in \Bbb N$. By definition $$[\frac n2]\leq \frac n2<[\frac n2]+1.$$
Then $n!=1\cdot 2\cdot ...\cdot[\frac n2]\cdot ([\frac n2]+1)\cdot...\cdot n>(\frac n2)^{n-[\frac n2]+a}>(\frac n2)^{\frac n2 +a}$, so $(n!)^{\frac 1n}>(\frac n2)^{\frac 12 + \frac an}\to \infty$, thus $(n!)^{\frac 1n}\to \infty.$ We set $a:=0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/706461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
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Proving that $\dim(\mathrm{span}({I_n,A,A^2,...})) \leq n$ Let $A$ be an $n\times n$ matrix. Prove that $\dim(\mathrm{span}({I_n,A,A^2,...})) ≤ n$
I'm at a total loss here...
Can someone help me get started?
| The following observations suffice to prove the statement:
*
*A power $A^k$ is in the span of lower powers $A^0,\ldots,A^{k-1}$ if and only if there exists a (monic) polynomial$~P$ of degree$~k$ with $P[A]=0$.
*If this happens for some $k=m$, it also happens for all $k>m$, so that by an immediate induction argument... | {
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Evaluating $\int_{0}^{\infty}\frac{1-e^{-t}}{t}\sin{t}\operatorname d\!t$ Find this integral
$$I=\int_{0}^{\infty}\dfrac{1-e^{-t}}{t}\sin{t}\operatorname d\!t$$
I know this
$$\int_{0}^{\infty}\dfrac{\sin{t}}{t}\operatorname d\!t=\dfrac{\pi}{2}$$But I can't find this value,Thank you
| Since you know that
$$\int_0^\infty \frac{\sin t}tdt=\frac\pi2$$
so it suffices to find
$$\int_0^\infty\frac{e^{-t}}t\sin tdt$$
so let
$$f(x)=\int_0^\infty\frac{e^{-t}}t\sin (xt)dt=\int_0^\infty h(x,t)dt$$
so using Leibniz theorem and since
$$\left|\frac{\partial h}{\partial x}(x,t)\right|\le e^{-t}\in L^1((0,\infty))
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Optimization for fat line intersecting most points Let's say I have a bunch of $(X,Y)$ points in 2D space. I want to find the line $y=mx+b$ which intersects the most points. We can add some kind of buffer (a delta) so if the line $y=mx+b$ is within delta of the point, then it also intersects the point.
I've never taken... | Your optimization problem can be formulated as
$$\min_{m,b \in \mathbb{R}} \|Y-mX-b\|_0,$$
where $\|\cdot\|_0$ is the $\ell^0$ semi-norm defined as $\|v\|_0 = \#\{v_i \neq 0\}$ (See here).
The only drawback is that $\|\cdot\|_0$ is not convex, thus you cannot employ the classical convex optimization tools.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Invertibility of a function in Z/m - Does what I have written work? Ok, so I know when $(a, m) = 1$, by Euler's Theorem, $a^{\phi(m)} \equiv 1 \mod m$. Since $\phi(323) = 288$, $a^{288} \equiv 1 \mod m$ when $(a, 323) = 1$. However, there are some elements $a$ such that $(a, 323) \not= 1$ and $a^{288} \not\equiv 1 \mod... | Note that $f$ being invertible doesn't mean that all elements of $\mathbb{Z} / 323\mathbb{Z}$ are invertible. That $f$ is an invertible map just means that there is an inverse map $g$ such that
$$\begin{align}
f \circ g &= 1 \\
g \circ f &= 1.
\end{align}
$$
That is: $f(g(x)) = x$ for all $x \in \mathbb{Z} / 323\mathbb... | {
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Composition relation of P∘P Consider the following relation P on the set B = {a, b, {a, b}}:
P = {(a, a), (a, b), (b, {a, b}), ({a, b}, a)}.
Answer questions 6 to 8 by using the given relation P.
Question 6
Which one of the following alternatives represents the domain of P (dom(P))?
*
*{a, b}
*{{a, b}}
*a, ... | Question 6 :
4
domain of P is the entire set B
Question 7:
2
you just need to write it down and compose!
Question 8:
1
when $p . p \subseteq p$ then p is transitive.
| {
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Real part of a quotient Is there some fast way to know the real part of a quotient?
$$\Re\left(\frac{z_1}{z_2}\right)$$
$z_i\in \mathbb{C}$
| If you don't have $z_1$ and $z_2$ in a "nice" form eg some of the values of $e^{i\theta}$ for various $\theta$, you could use $1/2(z+\bar{z})=\Re{(z)}$.
So you'd get:
$\Re(\frac{z_1}{z_2})=\frac{z_1\bar{z_2}+\bar{z_1}z_2}{2z_2\bar{z_2}}$
for
$z_1=r_1e^{i\theta_1}$
$z_2=r_2e^{i\theta_2}$
$\Re(\frac{z_1}{z_2})=\frac{r_1}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Use Fourier transform to find Fourier series coeficcients I understand that the Fourier Transform can be seen as a generalisation of the Fourier Series, where the period $T_0 \to \infty$ . Now I have encountered this strange question (in an engineering course on signal analysis):
Given a periodic function $x(t)$, find... | Suppose we have a function $\tilde x(t)$ that is zero except on the interval $[-T_0/2,T_0/2]$ (on which $\tilde x(t) = x(t)$) and whose Fourier transform is given by
$$
\widehat x(\omega) = \int_{-\infty}^\infty \tilde x(t) e^{-i\omega t}dt
= \int_{-T_0/2}^{T_0/2} x(t) e^{-i\omega t}dt
$$
Using $\widehat x(\omega)$, w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Lowest product of pair multiplication This is kind of an algebra question, and I am interested in an algebric proof to it.
Suppose we have $k$ natural numbers that are all greater than $0$.
We would like to arrange them in multiplication-pairs of two, such that the sum of each pair's product is the lowest possible.
For... | ?:
A={1,2,3}
Suppose you have made one pair (1,2) and removed it from the set. How do you make another pair?
Your reasoning requires additional conditions.
Saying set to be containing even number of elements would be better.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Showing that a series in $l_{\infty}$ converges weakly, given a boundedness condition. I'm trying to understand the following:
Let $\{x_k\}_{k=1}^{\infty}$ be a sequence of elements in $l_{\infty}$ such that for some constant $K$, $$\|\sum_{1}^n \lambda_k x_k \|\leq K\sup_k |\lambda_k| \quad \textrm{for all} \ \{\lamb... | Let $M=\sup_{k\ge 1} |\lambda_k|$. By assumption, the partial sums $s_n=\sum_{k=1}^n \lambda_k x_k$ satisfy $\|s_n\|_\infty\le MK$ for all $n$. For bounded sequences, weak* convergence in $\ell_\infty$ is precisely the coordinate-wise convergence. Thus, we only need to check that for each fixed index $j$ the numeric ... | {
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Is $d(x,y) = \sqrt{|x-y|}$ a metric on R? For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$.
Is this a metric on $\mathbb{R}$?
It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality seems to hold for all values I have tested, but I have not found this function a... | Sure looks like it. It's translation invariant, so to prove the TE for $x \le y \le z$, adjust everything so that the lowest, $x$, of the three values is at $0$ (i.e., add $-x$ to all three numbers). Then you need to show that
$$
\sqrt{y} + \sqrt{z} \ge \sqrt{y+z}
$$
for any nonnegative $y$ and $z$, which is true (by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/707468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Separation of variables won't work
Find all solutions on $\mathbb{R}$ of the differential equation $y'=3|y|^{2/3}.$
Why wouldn't separation of variables method work for this differential equation? Why does the initial condition have to be nonzero?
| I do not understand your statement "the separation of variables method will only work if the initial condition is nonzero".
I have problem with the absolute value so I shall only help you solving $$y'=3 y^{2/3}$$ You can separate the variables and integrate both sides. This leads to $$y^{1/3}=x+c$$ and then $$y=(x+c)... | {
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"timestamp": "2023-03-29T00:00:00",
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Showing independence of random variables When proving $\bar x$ and $S^2$ are independent in my noted it says that "functions of independent quantities are independent ". Can someone tell me how functions of independent quantities are independent happen?
Also let $X_1,X_2,X_3,\dots,X_n$ be a random sample.And suppose we... | For your first question, suppose $X$ and $Y$ are independent random variables.
The statement is that for any Borel measurable functions $f$ and $g$, $f(X)$ and $g(Y)$ are independent. In fact, independence of $X$ and $Y$ is equivalent to
(and in some formulations is defined as) the events $X \in A$ and $Y \in B$ being... | {
"language": "en",
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Circle Area formula question Take a peek at the following proof
Everything makes sense but one thing: how did they determine that $\sqrt{\cos^2\theta}$ was positive and not negative? Thanks.
| Remember that $\sqrt{\phantom{x}}$ denotes the positive square root, so
$$\sqrt{\cos^2\theta}=|\cos\theta|\ .$$
In the paper you linked, this occurs in an integral where $\theta$ goes from $0$ to $\pi/2$. For these $\theta$ values, $\cos\theta$ is positive, so $|\cos\theta|=\cos\theta$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using Newton's Method with a Piecewise Defined Function Using a Natural Cubic Spline approximation, I've generated an approximation polynomial to six points of data. Using the Cubic Spline approximation polynomial, I now need to use Newton's method to find a root of it (the spline approximation). I'm unsure of how to t... | I assume you have (or can write) functions that return the value and first derivative of your spline at any given argument value. If so, you can just use Newton's method directly -- no changes are needed to handle piecewise-defined functions.
Saying this another way ...
The fact that the spline is defined piecewise is ... | {
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Markov processes and semimartingales Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. More generally semimartingales with independent increments are Markov. I'm interest... | If $(B_t)_{t \geq 0}$ is a Brownian motion, then $(|B_t|^{\frac{1}{3}})_{t \geq 0}$ is not a semimartingale (see this question) but a Markov process.
| {
"language": "en",
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Height unmixed ideal and a non-zero divisor
Let $R$ be a commutative Noetherian ring with unit and $I$ an unmixed ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an unmixed ideal?
Background:
A proper ideal $I$ in a Noetherian ring $R$ is said to be unmixed
if the heights of it... | Let $R$ be a noetherian integral domain and $I=(0)$. If $\dim R=2$ and $R$ is not Cohen-Macaulay, then there is $x\in R$, $x\ne 0$, such that $xR$ is not unmixed. (For more details look here.)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve inequality: $\frac{2x}{x^2-2x+5} + \frac{3x}{x^2+2x+5} \leq \frac{7}{8}$ Rational method to solve $\frac{2x}{x^2-2x+5} + \frac{3x}{x^2+2x+5} \leq \frac{7}{8}$ inequality?
I tried to lead fractions to a common denominator, but I think that this way is wrong, because I had fourth-degree polynomial in the numerator... | HINT:
As $\displaystyle x^2\pm2x+5=(x\pm1)^2+4\ge4>0$ for real $x$
we can safely multiply out either sides by $(x^2+2x+5)(x^2-2x+5)$
Then find the roots of the Quartic Equation
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $G$ be a group and let $H,K$ be subgroups of $G$ where $|H|=12$ and $|K|=5$. Show that $H\cap K = \{e\}$.
Let $G$ be a group and let $H,K$ be subgroups of $G$ where $|H|=12$ and $|K|=5$. Show that $H\cap K = \{e\}$.
I used LaGrange's theorem to show that $|H|||G|$ and $|K|||G|$ so $12||G|$ and $5||G|$, and that... | Let $x\in H\cap K$ then by Lagrange theorem the order of $x$ divides the two coprime orders: $|H|$ and $|K|$ so $o(x)=1$ and then $x=e$. Conclude.
| {
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Show a group of order 351 is NOT simple. I have read a bunch of answers around the web, but they perform a jump which i can't follow.
I have determined that there must be $12*27=324$ elements of order 13 in G, but when i try to count the amount of elements in G of order 3 i run into some problems, i don't get the contr... | If $n_3=1$, we are done. Suppose not, so there are 13 Sylows for 3. If there is one 13 Sylow there is nothing to prove, so suppose also there are 27 Sylows for 13. If $P$, $Q$ are two 13-subgroups then they are either disjoint or equal for every nontrivial element in them is a generator, since 13 is prime. This justifi... | {
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Solving $\frac{x}{1-x}$ using definition of derivative I was trying to find the equation of the tangent line for this function. I solved this using the quotient rule and got $\frac{1}{(x-1)^2}$ but I can't produce the same result using definition of derivatives. Can someone show me how to do it? I tried looking it up o... | Using the definition of derivatives, we have
$f(x)'=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}$
Thus, the derivative of $\frac{x}{1-x}$ is
$\large f(x)'=\lim_{h\rightarrow0}\frac{1}{h}(\frac{x+h}{1-x-h}-\frac{x}{1-x})=\lim_{h\rightarrow 0}\frac{1}{h}(\frac{(x+h)(1-x)-x(1-x-h)}{(1-x-h)(1-x)})=\lim_{h\rightarrow0}\frac{1}... | {
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$\lim_{x\rightarrow 0^+} \frac {\ln(x)}{\ln( \sin x)}$ without l'Hôpital's rule How to calculate $\displaystyle
\lim_{x\rightarrow 0^{+}}\frac{\ln x}{\ln (\sin x)}$ without l'Hôpital's rule please?
If anybody knows please help
I don´t have any idea :-(
I´m looking forward your helps
| $\frac{\ln x}{\ln \sin x }=\frac{\ln x-\ln \sin x}{\ln \sin x }+1=\frac{\ln \frac{x}{\sin x}}{\ln \sin x }+1\rightarrow \frac{\ln 1}{-\infty}+1=1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is there a (f.g., free) module isomorphic to a quotient of itself? My question is as in the title: is there an example of a (unital but not necessarily commutative) ring $R$ and a left $R$-module $M$ with nonzero submodule $N$, such that $M \simeq M/N$?
What if $M$ and $N$ are finitely-generated? What if $M$ is free? M... | One keyword which should bring up many useful results: Leavitt álgebras.
| {
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"timestamp": "2023-03-29T00:00:00",
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Positive Definite and Hermitian Matrices If we know that $C$ is positive definite and Hermitian, how can we prove that there exists a matrix $Q$ such that $Q^∗CQ=I$. Where, $Q^∗$ is complex conjugate.
The definition of positive definiteness for a Hermitian Matrix I am using is if all principal minors are positive.
I am... | Sylvester's Law of Inertia tells you that two hermitian matrices are congruent if, and only if, they have the same inertia. (Wikipedia only deals with the reals, but everything works out the same over $\mathbb C$).
Since $C$ is positive definite, its eigenvalues are all positive, thus it has the same inertia as the ide... | {
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How can I explain to my professor his argument invokes the AC? This is not the standard definition, but my topology professor restricted contexts in metric spaces.
Definition
An open set $U$ in a metric space $X$ is a subset of $X$ such that the interior of it and $U$ Itself are identical. (Interior point $x$ of a sub... | My suggestion is not to bother your professor with this very much.
There are theorems whose choiceless proofs are very different than their choice-using proofs. There are theorems whose usual proof can be easily modified by showing that the arbitrary choice has some easy canonical choice instead.
When it is the former ... | {
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What is the exact coefficient of $x^{12}$ in $(2+x^3+x^6)^{10}$? What is the coefficient of $x^{12}$ in $(2+x^3+x^6)^{10}$?
I figure you need to pick $x^3$ 4 times so $C(10,4)$...but what happens with the other numbers/variables???
Can someone explain to me how this is done properly?
Thanks.
EDIT:
$(x + y)^n = C(n,k) \... | Hint 1: $x^{12} = x^6 x^6 = x^3 x^3 x^3 x^3 = x^3 x^3 x^6$
How many ways to pick $x^6 x^6$? Everything that's not an $x$ term is a multiplier of $2$. This would be $2^8 {10 \choose 2}$ for a total of $10$ elements.
For four $x^3$ there would be $10 \choose 4$ ways to pick.
Hint 2: For $x^3 x^3 x^6$ there are $10$ way... | {
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Sum of Prime Factors - TopCoder Recently in Topcoder, I faced a problem which stated as follows:
"You have a text document, with a single character already written in it. You are allowed to perform just two operations - copy the entire text (counted as 1 step), or paste whatever is in the clipboard (counted as 1 step).... | At every step, the number of characters copied to be pasted then an arbitrary number of times has to be a divisor of $n$(because no matter how many times we paste them then, the resulting amount of characters will be a multiple of the number of characters we copied).
So let $\xrightarrow{a}$ mean that I copy the charac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/708976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
how to compute the last 2 digits of $3^{3^{3^{3}}}$ to n times? Input $n$, output the last $2$ digits of the result.
n=1 03 3=3
n=2 27 3^3=27
n=3 87 3^27=7625597484987
n=4 ?? 3^7625597484987=??
Sorry guys, the formula given is $T_n=3^{T_{n-1}}$, I have updated the example.
I was asked this question during an inter... | Notice $$3^{100} = 515377520732011331036461129765621272702107522001 \equiv 1 \pmod{100}$$
If we define $p_n$ such that $p_1 = 3$ and $p_n = 3^{p_{n-1}}$ recursively and
split $p_n$ as $100 q_n + r_n$ where $q_n, r_n \in \mathbb{Z}$, $0 \le r_n < 100$, we have
$$p_n = 3^{p_{n-1}} = 3^{100 q_{n-1} + r_{n-1}} \equiv 1^{q_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/709187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Calculate ratio $\frac{p^{3n}}{(\frac{p}{2})^{3n}}$ How do I calculate this ratio? I do not know even where to begin.
$$\frac{p^{3n}}{(\frac{p}{2})^{3n}}$$
Thanks
| Regarding the original question:
$$
\frac{p^{3n}}{\frac{p^{3n}}{2}} =
\frac{p^{3n}}{\frac{p^{3n}}{2}} \cdot 1 =
\frac{p^{3n}}{\frac{p^{3n}}{2}} \cdot \frac{2}{2} =
\frac{p^{3n}\cdot 2}{\frac{p^{3n}}{2} \cdot 2} =
\frac{p^{3n}\cdot 2}{p^{3n}\cdot 1} =
\frac{p^{3n}}{p^{3n}} \cdot \frac 2 1 =
1\cdot \frac 2 1 = 2.
$$
In ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/709277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit as $x$ approaches 2 is undefined? Does following function have a limit if x approaches 2. Calculate what the limit is and motivate why if it is missing.
$$
\frac{(x-2)^2}{(x-2)^3} =\frac{ 1 }{ x-2}.
$$
I answered $\frac{1 }{ 0 }= 0 $ undefined is that correct?
| It looks like you are considering the function
$$
f(x) = \frac{(x-2)^2}{(x-2)^3} = \frac{1}{x-2}.
$$
You want to consider what happens to this function when $x$ approaches $2$. Note that the numerator is just the constant $1$ and when $x$ approaches $2$, then $x - 2$ approaches $0$. So you have something that approache... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/709395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Reference request: Nonlinear dynamics graduate reference There are already a number of requests for textbooks detailing nonlinear stability theory, chaos theory etc. but many of them are more introductory (e.g. Strogatz - Nonlinear Dynamics and Chaos)
I've covered all this material before but I'm prone to forgetting th... | "Nonlinear Oscillations, Dynamical Systems ,and Bifurcation of Vector Fields" by John Guckenheimer and Philip Holmes comes to mind. I took a class on Dynamical Systems with the first author many years ago and this was the text. I see people using the book by Strogatz and always feel that it is just not at the same leve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/709463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Expansion of lower incomplete gamma function $\gamma(s,x)$ for $s < 0$. The lower incomplete gamma function for positive $s$ is defined by the integral
$$
\gamma(s,x)=\int_0^{x} t^{s-1} e^{-t} dt.
$$
Taylor expansion of the exponential function and term by term integration give the following expansion
$$
\gamma(s,x)=\s... | From http://dlmf.nist.gov/8.7.E3 we have the series expansion
$$\Gamma(s,x) = \Gamma(s) - \sum_{n=0}^\infty \frac{(-1)^n x^{n+s}}{n! (n+s)}, \qquad s \ne 0, -1, -2, \ldots
$$
Combine this with the relation for the gamma functions (http://dlmf.nist.gov/8.2.E3)
$$\gamma(s,x) + \Gamma(s,x) = \Gamma(s).$$
Therefore the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/709556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Factorising $X^n+...+X+1$ in $\mathbb{R}$ How can factorize this polynom in $\mathbb{R}$:
$X^n+...+X+1$
I already try to factorize it in $\mathbb{C}$ but I couldn't find a way to turn to $\mathbb{R}$
| We have
$$\sum_{k=0}^n x^k=\frac{x^{n+1}-1}{x-1}$$
hence
$$\sum_{k=0}^n x^k=\prod_{k=1}^{n}\left(x-e^{2ik\pi/{n+1}}\right)$$
so if $n$ is odd say $n=2p+1$ then
$$\sum_{k=0}^{2p+1}=\prod_{k=1}^{2p+1}\left(x-e^{2ik\pi/{2p+2}}\right)=(x+1)\prod_{k=1}^{p}\left(x-e^{2ik\pi/{2p+2}}\right)\prod_{k=p+2}^{2p+2}\left(x-e^{2ik\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/709693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
The best symbol for non-negative integers? I would like to specify the set $\{0, 1, 2, \dots\}$, i.e., non-negative integers in an engineering conference paper. Which symbol is more preferable?
*
*$\mathbb{N}_0$
*$\mathbb{N}\cup\{0\}$
*$\mathbb{Z}_{\ge 0}$
*$\mathbb{Z}_{+}$
*$\mathbb{Z}_{0+}$
*$\mathbb{Z}_{*}$
... | Wolfram Mathworld has $\mathbb{Z}^*$.
Nonnegative integer
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/709802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 5,
"answer_id": 0
} |
Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets? Is there a name for a topological space $X$ which satisfies the following condition:
Every closed set in $X$ is contained in a countable union of compact sets
Does Baire space satisfy this condition... | This property is equivalent to $\sigma$-compactness, which says that the space itself is a countable union of compact subsets. If your property holds for a space $X$, then since $X$ is a closed subspace of itself, it is contained in a countable union of compact subsets. Conversely, if $X$ is $\sigma$-compact, then you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/709880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Differentiation of improper integrals defined on the whole real line. I am considering improper Riemann integrals (not Lebesgue integrals, mind you) of the form $$\int_{-\infty}^\infty f(t,x)dt,$$
with $f:\mathbf{R}\times\Omega\rightarrow\mathbf{R}$ continuous ($\Omega$ an open set in $\mathbf{R}$). What are sufficent ... | A sufficient condition is that the integral $\int_{-\infty}^\infty \frac{\partial}{\partial x}f(t,x)\,dt$ is uniformly convergent with respect to parameter $x$ (in some neighborhood of the point $x$ that you are interested in). This means you can bound the tail of integral by $\epsilon$ using the same size of tail for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/710000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Volume of solid region bounded by $z=4x$, $z=x^2$, $y=0$, and $y=3$ as an iterated integral
Suppose R is the solid region bounded by the plane $z = 4x$, the surface $z = x^2$, and the planes $y = 0$ and $y = 3$. Write an iterated integral in the form below to find the volume of the solid R.
$$\iiint\limits_Rf(x,y,x)... | the volume comes out to be:
$$\int_{0}^4 \int_{0}^{3} \int_{x^2}^{4x} 1dzdydx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/710370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Probability of 5 cards drawn from shuffled deck
Five cards are drawn from a shuffled deck with $52$ cards. Find the probability that
a) four cards are aces
b) four cards are aces and the other is a king
c) three cards are tens and two are jacks
d) at least one card is an ace
My attempt:
a) $\left(13*12*\binom{4}{4}*\... | deepsea gave a complete and clear answer.
I'd just add that you could see straight away that the answers for (b) and (a) cannot be the same, because the requirement in (b) is so much more specific. Many hands that satisfy (a) do not satisfy the conditions for (b).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/710456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
If $G$ is non-abelian group of order 6, it is isomorphic to $S_3$ Let $G$ be a non-abelian group of order $6$ with exactly three elements of order $2$. Show that the conjugation action on the set of elements of order $2$ induces an isomorphism.
I just need to show that the kernel of the action is trivial. Not sure how... | Hint: Suppose $x\in G$ is an element of the kernel of the action, i.e. fixes the three involutions under conjugation. What do you know about the group generated by the three involution, and what does that tell you about $x$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/710524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Two Dimensional Delta Epsilon Proof I was dawdling in some 2D delta epsilon examples, and I was wondering how to prove that the limit of $x^2+2xy+y^2$ has limit 3 as $(x,y)\rightarrow(1,1)$, using epsilon delta.
| Let $\epsilon>0$ and we look for $\delta>0$ such that $|x^2+2xy+y^2-4|<\epsilon$ whenever $$||(x,y)-(1,1)||=\sqrt{(x-1)^2+(y-1)^2}<\delta\;(*)$$
We have
$$|x^2+2xy+y^2-4|=|(x+y)^2-4|=|(x+y-2)(x+y+2)|\le|(x+y-2)(|x|+|y|+2)|$$
Now let $\delta<1$ and with $(*)$ we have $|x|,|y|<\delta+1<2$ so
$$|(x+y-2)(|x|+|y|+2)|<6|(x+y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/710576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Checking some Regular Expression problems I'm given the alphabet $$ \Sigma = {\{a,b}\} $$
I tried to write a regular expressions for presenting the following sets:
All strings in $$\Sigma ^ *$$
with:
a-) number of 2s divisible by 4
b-) exactly one occurrence of 122
c-) exactly one or two 1s
Well I tried to find their... | a). Words with number of $2$s divisible by 4 are words made of subwords that contain exactly 4 $2$s with arbitrary number of $1$s between them.
$$(1^*21^*21^*21^*21^*)^*$$
b). Note that $(21^*+11(1+2)+121)^*$ is the complementary of $122$.
$$(21^*+11(1+2)+121)^*122(21^*+11(1+2)+121)^*$$
c). In a similar concept as in c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/710704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Generate random sample with three-state Markov chain I have a Markov chain with the transition matrix
$$\pmatrix{0 & 0.7 & 0.3 \\ 0.8 & 0 & 0.2 \\ 0.6 & 0.4 & 0}$$
and I would like to generate a random sequence between the three states (such as $1, 2, 1, 3, \dots, n$). How do I get there while making sure the transitio... | Well, suppose you have a sequence of iid r.v. $U_t$ uniform on $[0,1]$.
Then define $$n_1(u) = 2\times 1_{u<0.7} + 3\times 1_{u\ge 0.7}\\
n_2(u) = 1\times 1_{u<0.8} + 3\times 1_{u\ge 0.8}\\
n_3(u) = 1\times 1_{u<0.6} + 2\times 1_{u\ge 0.6}\\
X_{t+1} = n_{X_t}(U_t)
$$
Then $X$ is a realisation of your Markov chain start... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/710798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
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