Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
$IS \mid JS$ implies that $I\mid J$ where $I$ and $J$ are ideals of a number ring contained in another number ring $S$ $K$ and $L$ are number fields and $K \subset L$ . Now, if $I$ and $J$ are two ideals in number ring of $K$ and $IS \mid JS$ then we have to show that $I \mid J$. Here, $S$ is number ring of $L$.
T... | Let $I = \mathfrak p_1^{e_1}\dots\mathfrak p_s^{e_s}$, $J = \mathfrak q_1^{k_1}\dots\mathfrak q_r^{k_r}$ in $R$, the ring of integers of $K$. Let $P_1$ be a prime ideal in the factorization of $\mathfrak p_1S$
Then we have $P_1 \mid \mathfrak p_1S \mid IS \mid JS$. So as $P_1$ is a prime ideal and the factorzation in p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is there an efficient way to compute the square root of modulo prime? Is there a reasonably simple way to find the square root of $a$ modulo $p$ where $p$ is an odd prime?
If the odd prime is small number it seems you can do this by brute force. HOwever if we want to solve something like x^2=2 mod 103 (whose solution i... | Let $p$ be an odd prime and we can say whether an integer $a$, $a\ne0(\mbox{mod})p,$ has a square root mod $p$ or not by
$$\mbox{a is }\begin{cases}\mbox{a quadratic residue if $a^{\frac{p-1}{2}}\equiv 1(\mbox{mod }p)$} \\\mbox{a quadratic nonresidue if $a^{\frac{p-1}{2}}\equiv -1(\mbox{mod }p)$}\end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Recurrence of $\{0,1,2\}^n$ tuples that don't contain $2$ followed immediately by $0$
I'm doing part (a) and need some hints with it.
My approach is to divide members of $\{t_n\}$ into 2 sets:
$\bullet$ n-tuples start with $0$, i.e. $(0,\_ \ ,\_ \ ,\_ \ ,\_ \ ,\_ \ ,...)$: there are $t_{n-1}$ of those (i.e. we fill in... | The first of the sets is a good start, but why isn't your other set simply the valid $n$-tuples that start with $1$ or $2$?
To extend a tuple from the first class, you can stick either a $0$ or an $1$ in front of it.
To extend a tuple from the second class, you can stick $0$, $1$, or $2$ in front of it.
This gives you ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $d(x,0)$ not a norm? If $\|x\|$ is a norm, then we can define $d(x,y):=\|x-y\|$ and it will be a metric. Now, if $d$ is a metric, why is $\|x\|:= d(x,0)$ not a norm? I think it fail for the sub-linearity, but I don't see how.
| Take for example $d$ as the discrete distance, then for $x\not=0$ and $|\lambda|\not=0,1$, we
$$1=d(\lambda x,0)=\|\lambda x\|\not=|\lambda|\|x\|=|\lambda|d(x,0)=|\lambda|$$
and the absolute homogeneity property does not hold.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why is $2$ double root of the derivative?
A polynomial function $P(x)$ with degree $5$ increases in the interval
$(-\infty, 1)$ and $(3, \infty)$ and decreases in the interval
$(1,3)$. Given that $P'(2)= 0$ and $P(0) = 4$, find $P'(6)$.
In this problem, I have recognised that $2$ is an inflection point and the de... | Since $2$ is an inflection point, we know that $P'(2)=0$ and $P''(2)=0$. (Convince yourself of this. What happens if $P''(2) \neq 0$?)
$P''(x)$ is then $-15ax^2 + 60ax - 55a + 20x^3 - 90x^2 + 110x - 30$. Plug in $2$ to get
$$P''(2) = 5 a - 10$$
And so $P''(2)$ is $0$ when $a=2$.
EDIT: mechanodroid gives us a really cl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is probability determined by perspective? My question: is probability determined by perspective?
The scenario that raised the question for me:
Initial condition: The Monty Hall problem. We know the contestant’s original choice of door #1 (of 3 total) is only correct 33% of the time. After Monty Hall reveals door #... | Of course it is. And honestly, although I love to discuss Bayesian vs. frequentist interpretation and related issues any other day (being a decided frequentist myself), I think that we do not need any sophisticated approach here. It helps to frame it in expected values rather than probabilities. Just change the scenari... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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How to know if some series doesn't converge to a rational function I was looking into a previous exam from 2011 of a course I am taking of Complex Analysis, and they ask
Which of the following series converge to a rational function in some domain?
$$\sum_{k=0}^\infty \frac{1}{k!+k^2+k}z^{k^2+2k}\\\sum_{k=0}^\infty \... | The first series converges uniformly to some $f$ in $\overline {\mathbb D},$ and diverges for all other $z.$ Thus $f$ is analytic in $\mathbb D.$ Suppose $f=R$ in some domain $U,$ where $R$ is a rational function. Then $U\subset \mathbb D.$ Let $P$ be the set of poles of $R.$ Then $f,R$ are both analytic in $\mathbb D\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Eigenvalue of an $n \times n$ real symmetric matrix with rank 2 Below is a question from the GATE Exam.
$\text{Let A be an $n \times n$ real valued square symmetric matrix of rank 2 with}$
$\text{$\sum_{i=1}^{n} \sum_{j=1}^{n}A_{ij}^2=50$. Consider the following statements}$
$\quad\text{(I) One Eigenvalue must be in $... | If $A = \begin{bmatrix} -5 & 0 \\ 0 & 5 \end{bmatrix} $ then $\|A\|_{F}^{2}\neq 50.$ If $A $ is a diagonal matrix then $ \|D\| = \max_{1 \leq i \leq n } |d_{i}|$ so $ \|A \|_{F}^{2} = 5$
A = [-5,0;0,5];
my = norm(A);
display(my)
my =
5
Also, the eigenvalues of a triangular matrix are the diagonal. And diagnonal ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2853116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to calculate the sine manually, without any rules, calculator or anything else? I want to know how to calculate the value of sin, not using table values or calculator.
I found this $\frac{(e^{ix})^2-1}{2ie^{ix}}$, but how to deal with $i$ number, if it's $\sqrt{-1}$?
| As for how computer actually evaluate sin(x) and other trig / transcendental functions, rather than using the Taylor series, which can converge rather slowly at times, the method usually used is a Chebyshev Polynomial. It should be noted that the whooshing sound you can hear is the mathematics on that page going clean... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2853310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Final step of homotopy lemma In proving Homotopy lemma in Milnor Topology from the differential viewpoint we consider $V_1 \cap V_2$ where $V_1$ is a neighbourhood of $y$ in which $card f^{-1}( y)$ is constant. Similarly $V_2$ s a neighbourhood of $y$ in which $card g^{-1}( y)$ is constant. If $F$ is smooth homotopy be... | $V_1 \cap V_2$ is non-empty (since $y$ is in both terms) and open. If there were no regular values for $F$ there, you'd have a set of positive measure containing only critical values. This contradicts Sard. (See also Brown's corollary on page 11 of the Princeton Landmarks version.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2853428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to rotate relative points in degrees? I have 4 points in range [0.0, 1.0] representing the top-left and bottom-right corners of a bounding box.
For example: [0.25 0.33 0.71 0.73]
In other words, the first pair (in (y, x) format) means that the point is 25% down the top of the image, and 33% from the left. The seco... | Based off Martin Roberts answer, here's my complete solution:
// values in absolute pixels
box = [y_min, x_min, y_max, x_max];
// Make points relative to image
pct = [box[0] / height, box[1] / width, box[2] / height, box[3] / width];
//^
//|
//+--------+
// |
// v
rot90 = [pct[1], 1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove: $\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$
Prove that for every $x$,$y$ greater than $1$:
$$\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$$
What I've tried has got me to:
$$\frac{\log_y(x)}{\log_y(2)}+\log_x(y)+3\log_y(2)\geq \sqrt[3]{81}$$
I didn't really get far.. I can't see where I can go from her... | Two things:
$\log_a b = \frac 1{\log_b a}$ and $\frac {\log_b c}{\log_b a} = \log_a c$ so $\log_a b\log_b c = \frac {\log_b c}{\log_b a} = \log_a c$.
And AM-GM says $\frac {a + b+ c}3 \ge \sqrt[3]{abc}$.
So.....
$\frac {\log_2 x + \log_x y + \log_y 8}3 \ge \sqrt[3]{\log_2 x \log_x y \log_y 8} = \sqrt[3]{\log_2 8}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2853732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluating a nested log integral
Question:$$\int\limits_0^1\mathrm dx\,\frac {\log\log\frac 1x}{(1+x)^2}=\frac 12\log\frac {\pi}2-\frac {\gamma}2$$
I’ve had some practice with similar integrals, but this one eludes me for some reason. I first made the transformation $x\mapsto-\log x$ to get rid of the nested log. The... | By the dominated convergence theorem we have
$$ \mathfrak{I} = \lim_{r \nearrow 1} I(r) \, ,$$
where for $r \in (0,1)$ we have defined
$$ I(r) = \int\limits_0^1 \mathrm{d} x\,\frac {\log\log\frac 1x}{(1+r x)^2} \, . $$
With this regularisation interchanging summation and integration is actually justified and your calcu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2853819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Why does changing the operator in $\lim_{h\to0}$ alter the result of this function? Let $f(x) = |x|$.
Attempting to differentiate $f(x)$ at 0 will fail because the limit does not exist at 0 as the left and right side are unequal.
$$\lim_{h\to0}\dfrac{f(0+h)-f(0)}{h}$$
However, my problem is about understanding why we g... | For the right side limit $h \gt 0,$ so $|0+h|=h$. For the left side limit, $h \lt 0,$ so $|0+h|=-h$. It is just the result of applying the absolute value. I think we are prone to intuitively think variables are positive, but that is not the case for the left side limit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2853917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\left(1+\frac1 n\right)^n > 2$ I'm trying to demonstrate that $\left( 1+\frac1 n \right)^n$ is bigger than $2$. I have tried to prove that $\left( 1+\frac1 n \right)^n$ is smaller than $\left( 1+\frac1{n+1} \right)^{n+1}$ by expanding
$\left( 1+\frac1n \right)^n = \sum\limits_{i=0}^n \left( \frac{n}{k} \ri... | Another way is to prove first that your sequence is monotonically increasing like has been done here:
I have to show $(1+\frac1n)^n$ is monotonically increasing sequence
... and since your first term is $2$, it follows that the subsequent ones are larger than $2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find out the subsequential limits of a sequence
Let $$x_n=(-1)^n \left(2+\frac{3^n}{n!}+\frac{4}{n^2}\right)$$ and find the upper and lower limits of the sequence $\{x_n\}_{n=1}^\infty$.
We put $n=1, 2, 3 \dots $ then $x_1=-9, x_2 = \frac{15}{2}, \dots $ after some stage we see that limit superior is $x_2$ and inferi... | Notice
$\underset{n \to \infty}{\lim \sup}\, x_n = \inf \{x_{2n} : n \in \mathbb{N}\} $,
and
$\underset{n \to \infty}{\lim \inf}\, x_n = \sup \{x_{2n-1} : n \in \mathbb{N}\} $.
Moreover, $2$ is a lower bound for the set $\left\{2+\frac{3^{2n}}{(2n)!}+\frac{1}{n^2} : n \in \mathbb{N}\right\}$, and $-2$ is an upper boun... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The spectral norm of real matrices with positive entries is increasing in its entries? Suppose that I restrict myself to $M_{n \times n}(\mathbb{R}_+)$ the set of real matrices with positive entries that are square and have size $n$, and I denote by $\|\cdot\|_2$ the spectral norm of members of this set. I am wondering... | First note that $$||A||_2=\sup_{||x||_2=1}||Ax||_2$$if we denote the entries of $A$ by $a_{ij}$ we conclude that$$||A||_2=\sup_{||x||_2=1}\sqrt{\sum_{i=1}^{n}(a_{i1}x_1+a_{i2}x_2+\cdots +a_{in}x_n)^2}$$since all the entries of $A$ are non-negative a supremum is achieved when $x_{i}\ge 0$ for all $i$ (or $x_{i}\le 0$ fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2854261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating ${\Large\int} _{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sin(x)+\cos(x)}{\sin^4(x)-4}dx$ This integral is giving me hard times, could anyone "prompt" a strategy about? I tried, resultless, parameterization and some change of variables.
$${\Large\int} _{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sin(x)+\cos(x)}{\sin^... | Remove the odd part and then set $t = \sin x$.
$$\int_{-\frac\pi2}^{\frac\pi2}\frac{\cos x}{\sin^4(x)-4}\,dx = \int_{-1}^1 \frac{dt}{t^4-4} = \frac14\int_{-1}^1 \frac{dt}{t^2-2} - \frac14\int_{-1}^1 \frac{dt}{t^2+2}$$
Can you finish?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$f_n:=\int _I h(x,y)f_{n-1}(y)dy$ uniformly convergent Proposition:$h(x,y)$ is $C^1$ function on $[0,1]^2$ and $f_0(x)$ is continuous on $I:=[0,1]$.
Let $f_n:=\int _I h(x,y)f_{n-1}(y)dy$ $(n=1,2,\cdots)$
Suppose $M:=\sup_n \max_x |f_n(x)| <\infty $ and for all continuous $g(x)$ on $I$, $\int_I f_n(y)g(y)dy$ convergent... | In the metric space $C[0,1]$ with the supremum metric the sequence $\{f_n\}$ is relatively compact. If a subsequence $\{f_{n_{k}}\}$ converges to a function $h$ then $\int f_{n_{k}} g \to \int hg$. If $h'$ is the limit of another subsequence then we get $\int hg =\int h'g$ for all $g \in C[0,1]$ and this implies $h=h'$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2854439",
"timestamp": "2023-03-29T00:00:00",
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Difficult integration by parts in deriving Euler-Lagrange equations I am doing some reading about the calculus of variations and I am finding it really difficult to see how the integrals are being manipulated. I sense it is due to an application of integration by parts (or some multivariable calculus) but I've been sta... | It's all integration by parts: $$\int_{a}^b (\underbrace{\eta F_y}_{\text{first}}+\underbrace{\eta' F_{y'}}_{\text{second}}+\underbrace{\eta'' F_{y''}}_{\text{third}})dx$$
let's study the second and third integral with integration by parts
*
*(Second integral) Let $f'=\eta'$ and $g=F_{y'}$ then $$\int_a^b\eta' F_ydx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2854693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Question of whether two given spaces are homeomorphic. Let $D^2$ be the closed disk on the plane.
First we pick an arbitrary point $x\in bd(D^2)$ on $D^2$, and define $X = D^2-\{x\}$.
Then define another space $Y$ by removing a homeomorphic image of the closed interval $I$ from the boundary, that is, $Y = D^2-h(I)$.(T... | The answer of @PaulFrost is awesome, and should be the accepted answer. I thought I'd just give an explicit description of his approach.
We'll write the unit disk as
$$ \mathbb{D} = \{re^{\pi it}\mid 0\le r\le1,\ -1\le t\le1\}$$
And define the lower hemisphere as $H=\{e^{\pi it}\mid -1\le t\le0\}$, and the point $q=-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2854913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove that $\sum\limits_{i=1}^{n} a_i\geq n^2$. A hint can be helpful, but not a whole solution.
The Problem (conjecture):
Given a natural number $n \geq 1$ and a sequence of natural numbers $(a_i)_{1 \leq i \leq n}$ in which for every pair $(i,j)$ with $i \neq j,$ we have
$$\gcd(a_i,a_j)\nmid i-j$$
prove that $\... | Just some ideas: if $a_n \geq 2n - 1$, you can proceed by induction. Hence you may assume that $n \leq a_n \leq 2n - 1$. In this case your second observation becomes quite a bit stronger. Indeed, for any prime $p$ dividing $a_n$ we may take $i = n$ and $j = n - \frac{a_n}{p}$. Note that $1 \leq j < n$ exactly by our co... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Multi sports tournament $10$ teams $6$ sports simultaneous I'm hosting a bar sports tournament with $10$ teams and $6$ different sports (pool, darts, table tennis, foosball, beer pong and cornhole). Trying to get the fixtures as fair as possible so that each team plays each sport twice and playing against the same team... | Label the teams $0$ to $9$ and the sports $0$ to $5$. Here $k\,\%\,6$ denotes the remainder of $k$ modulo $6$.
First let each team $i$ play each other team $j$ in the sport $i+j\,\%\,6$:
$$
\matrix{&1&2&3&4&5&0&1&2&3}\\
\matrix{1&&3&4&5&0&1&2&3&4}\\
\matrix{2&3&&5&0&1&2&3&4&5}\\
\matrix{3&4&5&&1&2&3&4&5&0}\\
\matrix{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2855142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Example of two spaces indisinguishable by their homology modules (with $\mathbb{Z}$ coefficients) but with different cohomology rings I'm running a student seminar on cohomology (for masters students) and would like to motivate the dualisation of homology by talking about cup products. So I'm looking for an example of ... | A similar example to yours is $\mathbb CP^n$ and $\bigvee\{S^i:0<i\leq 2n$ with $i$ even$\}$, as the cohomology ring of $\mathbb CP^n$, with coefficients in $\Bbb Z$, is
$$\mathbb Z[\alpha]/\alpha^{n+1},\text{ } deg(\alpha)=2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2855215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find the number of ways in which a list can be formed of the order of the 24 boats There are 15 rowing clubs;two of the clubs have each 3 boats on the river;5 others have each 2 and the remaining eight have each 1;find the number of ways in which a list can be formed of the order of the 24 boats,observing that the seco... | HINT: You can interpret the question as the following:
In the image above, suppose we have a stack of piles and we can't take the pile $3$ without taking $1$ and $2$; and we can't take the pile $2$ without taking $1$ (because pile $3$ is under pile $1$ and $2$ and pile $2$ is under pile $1$). In this case, notice that... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $ \left\lfloor{\frac xn}\right\rfloor= \left\lfloor{\lfloor{x}\rfloor\over n}\right\rfloor$ where $n \ge 1, n \in \mathbb{N}$
Prove that $ \left\lfloor{\frac xn}\right\rfloor= \left\lfloor{\lfloor{x}\rfloor\over n}\right\rfloor$ where $n \ge 1, n \in \mathbb{N}$ and $\lfloor{.}\rfloor$ represents Greatest I... | Consider by the archimedian principal there are a unique integers $k,m$ so that $kn \le kn + m\le x< kn + m + 1 \le (k+1)n$.
So $\frac {[x]}n = k + \frac mn$
So $\{\frac {[x]}n\} = \frac mn\le \frac {n-1}n$ and $\frac {\{x\}}n < \frac 1n$ so $\{\frac {\{x\}}n\} = \frac {\{x\}}n < \frac 1n$.
So $\{\frac {[x]}n\} + \{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2855619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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a tough sum of binomial coefficients
Find the sum: $$\sum_{i=0}^{2}\sum_{j=0}^{2}\binom{2}{i}\binom{2}{j}\binom{2}{k-i-j}\binom{4}{k-l+i+j},\space\space 0\leq k,l\leq 6$$
I know to find $\sum_{i=0}^{2}\binom{2}{i}\binom{2}{2-i}$, I need to find the coefficient of $x^2$ of $(1+x)^4$ (which is $\binom{4}{2}$). But I fa... | I would add some comments following the given solution. First at all we need a variable change $l'=l+4$ to bring the GF into the real world.
Suppose we have to fill the structure above with $l'$ identical white balls and $k$ identical black balls, white in the upper row, black in the lower row. Then there is a rule th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Newton's method for a vector field Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be $C^2$ and let $f(x^*)=0$. Since
$$f(x^*) \approx f(x) + Df(x) (x^* - x)$$
we can have the iterative procedure
$$x_{k+1} = x_k - Df(x_k)^{-1} f(x_k)$$
Is $G(x): = x - Df(x)^{-1} f(x)$ invertible near $x=x_0$? Are there any results on the conv... | This even doesn't hold for 1-D function since $$DG(x_0)=1-\left(\dfrac{f(x)}{f'(x)}\right)'|_{x_0}=1-\dfrac{f'^2(x_0)-f(x_0)f''(x_0)}{f'^2(x_0)}=0$$which is singular. To show that for higher dimensions let's define $$Df^{-1}(x)=[a_{ij}(x)]\\Df^{-1}(x)f(x)=[c_{i}(x)]$$and $$D(Df^{-1}(x)f(x))=[b_{ij}(x)]$$therefore $$c_{... | {
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Show that preimage is an embedded surface I have given a function
$$f:\mathbb{R}^4 \to \mathbb{R}^3: (x,y,z,u) \mapsto (xz-y^2, yu-z^2,xu-yz)$$
and I want to show that $f^{-1}(0)\setminus\{0\}$ is an embedded surface.
If it would be an embedded curve I could verify that $f$ is an submersion to get a 1-dim submanifold. ... | I saw this problem from Amann's Analysis II, page 257.
Note that
$$\begin{aligned}
y^2&=xz\\
z^2&=yu\\
yz&=xu.
\end{aligned}$$
If $yz\neq 0$, then
$$\begin{aligned}
y^2\cdot yz=xz\cdot xu&\implies y=x^{2/3}\cdot u^{1/3}\\
z^2\cdot yz=yu\cdot xu&\implies z=x^{1/3}\cdot u^{2/3}.
\end{aligned}$$
In the case $yz=0$, then i... | {
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Let $a, b, c \in Z$ such that $\gcd(a,c) = d$ for some integer $d$. Prove if $a\mid bc$ then $a\mid bd$. Here is what I have tried.
If $\gcd (a,c) = d$ then you can pick $x, y$ such that $d = ax + cy$
So to show $bd = la$, multiply $b$ into above to get $bd = bax + bcy$
And since $bc = ma$, $bd = bax + may$
Is this suf... | This is a sufficient proof. You have shown $bd = a(bx+my)=al$, which is what you wanted.
| {
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Suppose $x$ is an integer such that $3x \equiv 15 \pmod{64}$. Find remainder when $q$ is divided by $64$. Suppose $x$ is an integer such that $3x \equiv 15 \pmod{64}$. If $x$ has remainder $2$ and quotient $q$ when divided by $23$, determine the remainder when $q$ is divided by $64$.
I tried a couple things. By divisio... | Continuing from where you stopped:
$$\begin{align} 3(23q + 2) &\equiv 15 \pmod{64} \Rightarrow \\
69q+6 &\equiv 15 \pmod{64} \Rightarrow \\
69q &\equiv 9 \pmod{64} \Rightarrow \\
64q+5q &\equiv 9 \pmod{64} \Rightarrow \\
5q &\equiv 9 \pmod{64} \Rightarrow \\
5q\cdot 13 &\equiv 9\cdot 13 \pmod{64} \Rightarrow \\
65q &\e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
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Estimation of $f'(z)$ on the unit circle This is an old problem from Ph.D Qualifying Exam of Complex Analysis.
Let $f$ be a holomorphic function in the open disc $D(0,2)$ of radius 2 centered at the origin and suppose that $|f(z)|=1$ whenever $|z|=1$, and $f(0)=0$. Prove that $|f'(z)|\ge 1$ if $|z|=1$.
My attempt: By... | Let $z_0$ in the unit circle, since we know that $f$ is holomorphic :
$$f'(z_0)=\lim_{\lambda \to 0^+}\frac {f(z_0(1-\lambda))-f(z_0)}{-\lambda z_0}.$$
Let $1>\lambda >0$ :
$$\left|\frac {f(z_0(1-\lambda))-f(z_0)}{-\lambda z_0}\right|=\left|\frac {f(z_0(1-\lambda))-f(z_0)}{\lambda}\right| \ge\left|\frac {\left|f(z_0(1-... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that $|z_{1}+z_{2}|^2\leq (1+c)|z_{1}|^2+\bigg(1+\frac{1}{c}\bigg)|z_{2}|^2$
If $z_{1},z_{2}$ are two complex numbers and $c>0.$ Then prove that
$\displaystyle |z_{1}+z_{2}|^2\leq (1+c)|z_{1}|^2+\bigg(1+\frac{1}{c}\bigg)|z_{2}|^2$
Try: put $z_{1}=x_{1}+iy_{1}$ and $z_{2}=x_{2}+iy_{2}.$ Then
from left side
$$(x_... | By the AM-GM Inequality,
$$c|z_1|^2+\frac{1}{c}|z_2|^2\geq 2|z_1||z_2|\,.$$
Thus, $$(1+c)|z_1|^2+\left(1+\frac{1}{c}\right)|z_2|^2\geq \big(|z_1|+|z_2|\big)^2\geq |z_1+z_2|^2\,,$$
where the last inequality follows from the Triangle Inequality. Note that the inequality $$(1+c)|z_1|^2+\left(1+\frac{1}{c}\right)|z_2|^2 \... | {
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Minimum of $\left(a + b + c + d\right)\left(\frac{1}{a} + \frac{1}{b} + \frac{4}{c} + \frac{16}{d}\right)$ If $a$, $b$, $c$, $d$ are positive integers, find the minimum value of
$$P = \left(a + b + c + d\right)\left(\frac{1}{a} + \frac{1}{b} + \frac{4}{c} + \frac{16}{d}\right)$$
and the values of $a$, $b$, $c$, $d$ whe... | If you want to use the AM-GM Inequality, it can be done as follows. Observe that
$$a+b+c+d=a+b+2\left(\frac{c}{2}\right)+4\left(\frac{d}{4}\right)\geq 8\sqrt[8]{ab\left(\frac{c}{2}\right)^2\left(\frac{d}{4}\right)^4}$$
and that
$$\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}=\frac{1}{a}+\frac{1}{b}+2\left(\frac{2}{... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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Solving second order ODE oscillator I'm currently just taking my first ODE course and one of the questions is,
A projectile of mass $m$ is fired from the origin at speed $v_0$ and angle $\theta$. It is attached to the origin by a spring with spring constant
$k$ and relaxed length zero. Find $x(t)$ and $y(t)$.
Here's ho... | In fact, the equation can be integrated in vector form.
$$\ddot{\vec r}+\omega^2 \vec r=\vec g$$
has the homogeneous solution
$$\vec r=\vec{c_c}\cos\omega t+\vec{c_s}\sin\omega t$$ and the particular solution $$\vec r=\vec g.$$
Now with the initial conditions,
$$\vec r_0=\vec{c_c}+\vec g=\vec 0,
\\\vec v_0=\omega\vec c... | {
"language": "en",
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"question_score": "1",
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"answer_id": 4
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Laplace method (or other integral asymptotic) with near-corner Consider the integral
$$\int_{-\infty}^\infty \exp(-\sqrt{h^2+M^2x^2}) dx.$$
Here $h$ is a small positive parameter and $M$ is a large positive parameter. I would like to obtain a "reasonably uniform" asymptotic approximation for this integral in the limit ... | As noticed, a simple Laplace method cannot be used here, as 2 scales are involved. A uniform asymptotic expansion should be found. Alternatively, in this case, we can recognize a modified Bessel function. Indeed, changing $x=\frac{h}{M}\sinh t$, the integral can be written as
\begin{align}
I&=\int_{-\infty}^\infty \ex... | {
"language": "en",
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"source": "stackexchange",
"question_score": "8",
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What is the meaning of having imaginary solutions to a differential equation I am trying to solve this equation
$$x+e^{xt}=0$$
for different values of x and see when there is no real solution. Plotting this on a graph and assuming x is real gives me the different values of t. I want to find the value of t when the abov... | I don't know about "real life", but one place your equation can come up is in solving a PDE such as
$$ \dfrac{\partial u}{\partial t} = \dfrac{\partial^2 u}{\partial x^2} $$
with boundary condition
$$ \dfrac{\partial u}{\partial x}(0,t) + u(L,t) = 0$$
If you assume a solution of the form $u(x,t) = \exp(r x + r^2 t)$ yo... | {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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Does $\int{\frac{x}{\cos(x)}dx}$ have an elementary solution? I need to solve the following integral: $\displaystyle\int\frac{x}{\cos(x)}\,dx$. My procedure is the following:
\begin{align*}\int\frac{x}{\cos(x)}\,dx &= \int x\sec(x)\,dx\\
&=x\ln(\tan(x)+\sec(x))-\int\ln(\tan(x)+\sec(x))\,dx.
\end{align*}
But, I'm stuck ... | Perhaps the expanding of $e^x$ be useful, well
\begin{align}
\int\dfrac{x}{\cos x}dx
&= \int\dfrac{2xe^{-ix}}{1+e^{-2ix}}dx \\
&= \int 2xe^{-ix}\sum_{n\geq0}e^{-2inx}dx \\
&= \sum_{n\geq0}\int 2xe^{-i(2n+1)x}dx \\
&= \sum_{n\geq0}2e^{-i(2n+1)x}\left(\dfrac{ix}{2n+1}+\dfrac{1}{(2n+1)^2}\right) \\
\end{align}
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Existence of faithful normal state Does there always exist faithful normal state on center of von Neumann algebra?
Further, in type II$_{1}$ von Neumann algebras are tracial states coming from the center valued trace?
| The first question is the same as asking if every abelian von Neumann algebra has a faithful normal state. The answer is easily no, by taking for instance the example in this answer.
As for all tracial states coming from the center-valued trace, it is not entirely clear to me what you mean. If you have
$$
M=\bigoplus... | {
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"source": "stackexchange",
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find all the points on the line $y = 1 - x$ which are $2$ units from $(1, -1)$ I am really struggling with this one. I'm teaching my self pre-calc out of a book and it isn't showing me how to do this. I've been all over the internet and could only find a few examples. I only know how to solve quadratic equations by ... | You have
$2x^2 - 6x + 1 = 0$
This looks good.
then I would say
$x = \frac {3\pm\sqrt 7}{2}$ is simpler than $x = 1.5 \pm \sqrt {1.75}$
Plug each value of $x$ into $y = 1-x$ to find $y.$
$y = 1 - \frac {3+\sqrt 7}{2} = \frac {-1-\sqrt 7}{2}\\y = 1 - \frac {3-\sqrt 7}{2} = \frac {-1+\sqrt 7}{2}$
| {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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How to show a simple closed curve is not nullhomotopic in the complement of another simple closed curve Let $J,L$ be the two simple closed curves in the solid torus $T$ as shown below.
I am trying to show that $J$ is not null-homotopic in the complement of $L$. The hint given was to consider the universal cover of $T$... | The method described in the hint works (start by drawing a picture of the universal cover and the chain of lifts of L and J, then look at two adjacent components and show they are not homotopically unlinked. If there are lifts that are homotopically linked, you can use the lifting property to show the originals must b... | {
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"source": "stackexchange",
"question_score": "2",
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Lemma 11.42 Rotman's algebraic topology
This is lemma 11.42, pg 355, of Rotman's Algebraic Topology.
The context is we are trying to determine an explicit map for "connecting homomorphism".
where $i:A \rightarrow X$ is inclusion. This is Rotman's proof.
Then there is calculations here giving explicit map $\... | All right, I made a mess a bit in the comments. Let me make it clear with diagrams.
Note that by iterating as many as you want, we can just consider the adjunction $[S^{n+1},X]\to [S^n,\Omega X]$ instead of iterated adjunction as in Rotman. Actually, this does not have exactly the form of adjunction; instead it is a co... | {
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"source": "stackexchange",
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Check proof that $\sin\sqrt{|x|}$ is not periodic I want to:
Prove that $\sin\sqrt{|x|}$ is not periodic
A similar question had already been asked here. But the accepted answer uses definition of the derivative. And i'm trying to do it in a "pre-calculus" manner.
Here is my try. By definition of periodic functions:
$... | You are working at $x\ge T$, so, how can you suppose that exist some $x<T$? It doesn't make sense.
A suggestion could be use the identity:
$$\sin p-\sin q=2\sin\left(\frac{p-q}{2}\right)\cos\left(\frac{p+q}{2}\right).$$
In your case you have $p=\sqrt{x}$ and $q=\sqrt{x-T}$, for $x\ge T.$ So,
$$\frac{\sqrt{x}-\sqrt{x-T}... | {
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Fibonacci-type Sequence with Complex Numbers I have been playing around with Fibonacci-type of sequence that involve complex numbers. I have stumbled upon the following sequence, which seemed interesting to me:
$$0,1,2i,-3,-4i,5,6i,...$$
so $F_n = 2iF_{n-1} + F_{n-2}$. These look like a sequence of natural numbers (e... | The characteristic polynomial for your recursion is $$x^2-2ix-1=(x-i)^2$$
Visibly, this has a double root at $x=i$. Thus the general form of the solution to the recursion is $$F_n=Ai^n+Bni^n$$ Using your initial conditions it is easy to specify the solution to your case.
| {
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"source": "stackexchange",
"question_score": "2",
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If $f$ and $1/f$ are harmonic then $f$ is holomorphic or antiholomorphic I have this problem.
Let $f:D\to \mathbb{C}$ be a function such that $f$ and $1/f$ are harmonic (Their real and imaginary parts are harmonic). Then $f$ is holomorphic or antiholomorphic.
I tried to solve it by computing the laplacian of real and i... | Possible idea. Let $f=f(x,y)$. If $f$ and $1/f$ are harmonic then
$$
\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0
$$
$$
\frac{\partial^2 1/f}{\partial x^2} + \frac{\partial^2 1/f}{\partial y^2} = 0
$$
where
$$
\frac{\partial 1/f}{\partial x} = -\frac{\partial f}{\partial x} \frac{1}{f^2}... | {
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"source": "stackexchange",
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Find $\log _{24}48$ if $\log_{12}36=k$ Find $\log _{24}48$ if $\log_{12}36=k$
My method:
We have $$\frac{\log 36}{\log 12}=k$$ $\implies$
$$\frac{\log 12+\log 3}{\log 12}=k$$ $\implies$
$$\frac{\log3}{2\log 2+\log 3}=k-1$$ So
$$\log 3=(k-1)t \tag{1}$$
$$2\log 2+\log 3=t$$ $\implies$
$$\log 2=\frac{(2-k)t}{2} \tag{2}$$
... | You can convert all to the smallest common base $2$:
$$\log_{12}36=\frac{\log_{2}36}{\log_2 12}=\frac{2+2\log_2 3}{2+\log_2 3}=k \Rightarrow \log_2 3=\frac{2k-2}{2-k}.$$
Hence:
$$\log _{24}48=\frac{\log_2 48}{\log_2 24}=\frac{4+\log_23}{3+\log_23}=\frac{4+\frac{2k-2}{2-k}}{3+\frac{2k-2}{2-k}}=\frac{6-2k}{4-k}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2857929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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Question about substitutions in the double integral $\int_0^1\int_0^1 \frac{-\ln xy}{1-xy} dx\, dy = 2\zeta(3)$ \begin{align}
\int_0^1\int_0^1 \frac{-\ln xy}{1-xy} dx\, dy = 2\zeta(3) \tag{1}
\end{align}
Since,
\begin{align}
\int_0^1 \frac{1}{1-az} dz = \frac{-\ln (1-a)}{a}
\end{align}
and putting $a = 1-xy$, $(1)$ bec... | By geometric series, your integral equals
$$-\int^1_0\int^1_0(\ln x+\ln y)\sum^\infty_{k=0}(xy)^kdydx$$
Due to the symmetry in $x$ and $y$, this equals
$$-2\int^1_0\int^1_0(\ln x)\sum^\infty_{k=0}(xy)^kdydx$$
Since the series converges uniformly, we can integrate it termwise to obtain
$$-2\sum^\infty_{k=0}\int^1_0 x^k\... | {
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"source": "stackexchange",
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How to prove this inequality ???
Let $a,b,c$ are positive real numbers such that $a+b+c=3$.
Prove that
$$\sum\frac{(7a^{3}+3)(b+c)}{7a+3} \geq 6 $$
I try to prove $LHS \geq \sum\frac{9}{5}a+\frac{1}{5}$ but don't succeed
| Yes, you are right, the TL method does not help here.
But $uvw$ helps.
Indeed, let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.
Hence, we need to prove that
$$\sum_{cyc}(7a^3+3u^3)(3u-a)(7b+3u)(7c+3u)\geq6u^3\prod_{cyc}(7a+3u)$$ and we see that our inequality it's $f(w^3)\geq0,$ where
$$f(w^3)=-343\cdot3w^6+A(u,v^2)w^3+B... | {
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Prove: $\sin\frac{\pi}{20}+\cos\frac{\pi}{20}+\sin\frac{3\pi}{20}-\cos\frac{3\pi}{20}=\frac{\sqrt2}{2}$
Prove: $$\sin\frac{\pi}{20}+\cos\frac{\pi}{20}+\sin\frac{3\pi}{20}-\cos\frac{3\pi}{20}=\frac{\sqrt2}{2}$$
ok, what I saw instantly is that:
$$\sin\frac{\pi}{20}+\sin\frac{3\pi}{20}=2\sin\frac{2\pi}{20}\cos\frac{\pi... | Let $A,B,C,\ldots,T$ be points on a circle with diameter $1$ dividing it into $20$ equal arcs. Let $RG$ intersect $BO, BK, KF$ at $U,V,W$, respectively.
It is easy to get the following equalities
$$\angle URO = \angle OUR = \angle BUV = \angle UVB = \angle WVK = \angle KWV = \angle FWG = \angle WGF = \frac 25 \pi,$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2858498",
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"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 2
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Linear Transformations between 2 non-standard basis of Polynomials
If
$$
A = \begin{pmatrix} 1 & -1 & 2 \\
-2 & 1 &-1 \\ 1 & 2 & 3 \end{pmatrix}
$$
is the matrix representation of a
linear transformation $T : P_3(x) \to P_3(x)$ with respect to
bases $\{1-x,x(1-x),x(1+x)\}$ and $\{1,1+x,1+x^2\}$. Find T.... | I also suspect the real question is to find the image of $a+bx+cx^2$ in the canonical basis.
I would do it in a formal way first: denote $X, Y$, &c. the column vectors of polynomials in the canonical basis, $X_1, Y_1$, &c. their column vectors in the first basis and $X_2, Y_2$, &c. their column vectors in the second b... | {
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"source": "stackexchange",
"question_score": "2",
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My solution to inhomogeneous $\frac{d^2y}{dx^2} + y = \sin{x}$ does not conform to my book's solution! I need help with the solution of this particular equation:
$$\frac{d^2y}{dx^2} + y = \sin{x}$$
Due to me having to go to work, I cannot display all my work in mathjax, my shift starts in 5 min...but my solution is: ... | With characteristic equation $\lambda^2+1=0$ we know that the general solution is of the form $y_g=C_1\sin x+C_2\cos x$. Since the right side of the equation is $\sin$, one of the general answer, then the particular solution is of the the form $y_p=Ax\sin x+Bx\cos x$, after substitution we have $A=0$ and $B=-\dfrac12$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2858675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Taking the derivative of $\sum_\limits{n=1}^{\infty}\arctan(\frac{x}{n^2})$
Problem: Study the possibility of taking the derivative of the following series:
$$\sum_\limits{n=1}^{\infty}\arctan\left(\frac{x}{n^2}\right)\:\:,x\in\mathbb{R}$$
I have studied the following theorem:
Theorem: Suppose that $\sum_\limits{n=k... | The standard theorem is this:
If $\sum_{n=1}^{\infty}f_n(x)$ converges at least at one point, and the series of derivatives $\sum_{n=1}^{\infty}f_n^{'}(x)$ converges uniformly in some interval $I$, then in that interval the series can be differentiated term by term, meaning that the sum of the derivatives converges to... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are the facets of the Birkhoff Polytope when $n=2$? I've read in several sources that the number of facets of the Birkhoff polytope $\mathcal{B}(n)$ is $n^2$.
Is this supposed to hold when $n=2$? Since $\mathcal{B}(2)$ has dimension $1$, the facets would be the two $0$-dimensional vertices, which are the two perm... | No, it doesn't apply to $n=2$; your sources (like this one) apparently failed to treat this special case.
The generally $n^2$ facets correspond to the non-negativity constraints for the $n^2$ entries of the matrix. But for $n=2$, the $4$ non-negativity constraints form two pairs of identical constraints if you restric... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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description of an ideal generated by the projections in a $C^*$ algebra If $A$ is a $C^*$ algebra,$P_i$ are projections in $A$,$I$ is the ideal generated by the projections.I think $I$ is the $C^*$ algebra generated by $P_iAP_i$.How to charectarize $I$,is there a precise description of $I$?
| It is not the C$^*$-algebra generated by $P_iAP_i$. For instance take $A=M_2(\mathbb C)$, $P=E_{11}$. The ideal generated by $P$ is $A$, and not $PAP=\mathbb C P$.
The ideal generated by elements $x_1,\ldots,x_m$ in $A$ is the C$^*$-subalgebra generated by $$\{ ax_jb:\ a,b\in A,\ j=1,\ldots,m\}.$$ I don't think you ca... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find an "upper bound" for a given sequence. Let $a_1=5$ and let $$a_{n+1}=\frac{a_n^2}{a_n^2-4a_n+6}$$
Find the biggest integer $m$ not bigger than $a_{2018}$, that is $m\leq a_{2018}$.
My go:
Apparently the limit must satisfy $$l=\frac{l^2}{l^2-4l+6}\Leftrightarrow l=0\vee l=3\vee l=2$$ Computing first few terms i see... | Suppose we compute the fixed points of the iteration $x\mapsto\frac{x^2}{x^2-4x+6}$; those fixed points are 0, 2 and 3. Now analyse the stability of those fixed points; a fixed point $x_0$ of $x\mapsto f(x)$ is stable (attractive) if $|f'(x_0)|<1$.
The numerator of the derivative of the given map is
$$2x(x^2-4x+6)-x^2(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2859118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is Modular Arithmetic Notation Good? When we write
$$a = b\, (\mathrm{mod}\, n)$$
we mean $a$ and $b$ belong to the same equivalence class. This is a symmetric property, so writing $$a \equiv_n b$$ is intuitive. On the other hand, we also treat $(\mathrm{mod}\, n)$ like an operator: that is, given some $b$, we write $b... | Nobody uses $a\mod b$ as an operator. That's an exaggeration, but basically that notation is not as popular as you seem to think. It is non-standard and discouraged. Programmers sometimes use it because they're used to the % operator in many programming languages, but I don't think I've ever seen someone write $a\mod b... | {
"language": "en",
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What is $\log(n+1)-\log(n)$? What is gap $\log(n+1)-\log(n)$ between log of consecutive integers? That is what precision of logarithms determines integers correctly?
| This is rather similar to previous answers, but I think it's still worth pointing out. You're asking about the slope of a chord of the graph of $\log x$, the chord joining $(n,\log n)$ to $(n+1,\log(n+1))$. By the mean value theorem, this equals the slope of the tangent line, $1/x$, at some $x$ between $n$ and $n+1$. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Partial sums of the series $\sum\limits_{k\geq1}\frac{1}{\sqrt{2k+\sqrt{4k^2-1}}}$ The series $\sum\limits_{k=1}^{\infty}\frac{1}{\sqrt{2k+\sqrt{4k^2-1}}}$ is divergent. I am interested in its partial sums to do some computations based on them.
I tried to multiply $\sqrt{2k+\sqrt{4k^2-1}}$ by $\sqrt{2k-\sqrt{4k^2-1}}$ ... | Using the identity
$$\sqrt{a+\sqrt{b}}=\sqrt{\frac{1}{2} \left(a+\sqrt{a^2-b}\right)}+\sqrt{\frac{1}{2} \left(a-\sqrt{a^2-b}\right)}$$
where: $a = 2 k$ and $b=4 k^2-1$, along with evaluating a telescoping series, we find that
$$\begin{align}
\color{red}{\sum _{k=1}^n \frac{1}{\sqrt{2 k+\sqrt{4 k^2-1}}}}&=\sum _{k=1}^n... | {
"language": "en",
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How is it possible to solve a second degree polynomial combined with a modulo? I have two equations: a second degree polynomial and one with a modulo. There are two variables: $ x, y \in \mathbb{R}$; $ x, y \geq 0 $, and four constants, which have a known value: $a, b, c, d \in \mathbb{R}$. The two equations:
$$ y = ax... | You need $a \gt 0$ to make the parabola open upward. We can start by ignoring the constraint from $d$ and find the vertex of the parabola, which gives the minimum $y$ on it. Now round up to the next $y$ above the minimum that satisfies the $d$ constraint. Unless the vertex satisfies the $d$ constraint, there will be... | {
"language": "en",
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Cantor set with pairs of points identified Consider the middle-thirds Cantor set in $[0,1]$. I want to identify points in the following way.
First, identify the two points $1/3$ and $2/3$.
Then, idenfity $1/9$ with $2/9$, and also identify $7/9$ with $8/9$.
At the third step there will be four pairs of points: $1/... | Yes, it is. You can see this very neatly with an explicit map. Let $K$ denote the Cantor set. Given an element $$x=\sum_{n=1}^\infty\frac{2a_n}{3^n}\in K$$ where $a_n=0$ or $1$ for each $n$, let $$f(x)=\sum_{n=1}^\infty\frac{a_n}{2^n}.$$ That is, $f$ takes the ternary expansion of $x$ using $0$s and $2$s, replaces ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2859732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to get the shaded region of the rectangle? I have this problem:
So my development was:
Denote side of rectangle with: $2a, 2b$.
So, $4ab= 64, ab = 16$
Denote shaded region with $S$
Denote area of triangle $DGH = A_1$ and triangle $FBE = A_2$.
So, $A_1 + A_2 + S = 64$
$S = 64 - A_1 - A_2$
The triangles $A_1, A_2$ ... |
If you notice that if you combine two right triangles then they occupy the area of $\dfrac14$ of the total area.
So, the area of the shaded region is $=64-\dfrac14(64)=64-16=48$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How many elements have to verify the associativity property in a group? If this is a duplicate please mark it down.
We know that if $(G,\ast)$ is a group then it must verify the associative property, that is, $$\forall x,y,z\in G:\quad x\ast(y\ast z)\quad=\quad(x\ast y)\ast z\,.$$
My question is how many elements have ... | In “Verification of Identities” (1997), Rajagopalan and Schulman discuss associativity testing for binary operations on a finite set of $n$ elements. For certain types of operations, a random algorithm works with high probability. However:
This random sampling approach does not work in general. For every $n≥3$, t... | {
"language": "en",
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"source": "stackexchange",
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What is wrong with the reasoning in $(-1)^ \frac{2}{4} = \sqrt[4]{(-1)^2} = \sqrt[4]{1} = 1$ and $(-1)^ \frac{2}{4} = (-1)^ \frac{1}{2} = i$? $$(-1)^ \frac{2}{4} = \sqrt[4]{(-1)^2} = \sqrt[4]{1} = 1$$
$$(-1)^ \frac{2}{4} = (-1)^ \frac{1}{2} = i$$
Came across an interesting Y11 question that made pose this one to my sel... | The existing answers are not addressing the real issue, and hence are in my opinion misleading. The problem has nothing to do with multiple roots.
When you write "$a^b$", this is a notation that indicates exponentiation. In modern mathematics, this is typically understood as the exponentiation function applied to the t... | {
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Effectively reconstructing all original 5-tuples from a subset of their respective 4-tuples Let's take integer numbers from [1..36]. We can generate 376992 different (order is not important) five-number-combinations like (1,3,5,7,12), etc. Such five-number-combinations always have five distinct (unique) numbers. Each s... | Sort the tuples lexicographically: always keep the items within in sorted order, so {1,2,3,4} is okay but {1,2,4,3} is not.
Similarly, sort the list of tuples lexicographically: {1,2,4,5} should come after {1,2,3,6} should come after {1,2,3,5}.
Now: Consider pairs of tuples with the same first three elements and differ... | {
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Why use the term 'models' to interpret the double turnstile symbol? A $\vDash $B can be read in words as:
*
*A entails B
*B is a semantic consequence of A
*A models B
The first two are fine. But the third one seems a bit counter-intuitive to me. Somehow, I can't reconcile the term 'model' (used colloquially) wi... | The 'reading' #3 is a different thingy from reading #2 and #1.
As far as I know, the double turnstile ($\vDash$) has two uses.
In your reading #3, $A$ is some 'structure', which can be called a 'model', an interpretation, or an 'assignment', while $B$ is a propositional formula. The notation is more like $\mathcal{A}\v... | {
"language": "en",
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Proof that every prime has a primitive root. So I encountered this proof on a Number Theory book, I will link the pdf at the end of the post (proof at page 96), it says: "Every prime has a primitive root,
proof: Let p be a prime and let m be a positive integer such that:
p−1=mk for some integer k. Let F(m) be the numb... | There are $p-1$ positive integers less than $p$, namely $1, 2, ..., p-1$.
Each of these will have some multiplicative order modulo $p$. So if we count all those of order $1$, all those of order $2$, all those of order $3$, etc then the total count is $p-1$. There are $F(1)$ of order $1$, $F(2)$ of order $2$, etc, so:
$... | {
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about $C([0,1])$ with sup metric Define the space $C([0,1])$ as the space of continuous functions $f : [0,1] \mapsto \Bbb R$ with $C([0,1])$ $$ d(f,g) = \sup _{x \in [0,1]}{|f(x)-g(x)|} , $$
so let $$A= \left\{f \in C([0,1])\ \middle| \ 0 <\int_0^1 f(x) \ \mathrm{d}x < 1\right\}$$ now is $A$ open , close , bound... | $A$ is not bounded
The piecewise linear map defined by $f_n(0)=f_n(1/n)=f_n(1)=0$ and $f_n(1/(2n))=n$ is such that $\int_0^1 f_n = 1/2$ but $d(f_n,0) = n$ is unbounded.
$A$ is connected
$A$ is connected because it is convex.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Taylor's formula with remainder for vector-valued functions Let $f: \mathbb{R}^n \to \mathbb{R}^n $. Does there exist a generalization of Taylor's Theorem with Lagrange Remainder for such a vector-valued function?
| The short answer is "yes". The multivariable arguments to f require partial derivatives of f to determine the coefficients of the polynomial terms. The vector valued output leads to vectors of polynomials. Combining results in a mathematical structure analogous to the Taylor Polynomial, but called a "jet". The jet is u... | {
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If $0Let $U\subseteq\mathbb{R}^n$ be convex and closed. I'm trying to find conditions for which the following happens:
If $0<r\neq 1$ then $\partial U\cap\partial (rU)=\emptyset$.
It seems to me that this is not true if $0$ is in the frontier of $U$, for instance $U=\{(0,y):y\in\mathbb{R}\}$ or $U=\{(x,y):y\ge |x|\}$... | WLOG assume that $r>1$ (if $0<r<1$ then let $U' = rU$ and $r' = \frac{1}{r}$, so $r'U' = U$ and $r'>1$). Suppose that the intersection is nonempty. Then, there is some $x\in \partial (rU)$ such that $x\in\partial U$. Since $x\in \partial (rU)$, $y = r^{-1}x\in \partial U$. Thus, both $x$ and $y$ are on the boundary of ... | {
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Help needed understanding Iterated Limits $\lim_{n\rightarrow \infty}(\lim_{m \rightarrow \infty} a_{mn})$ and ... Let $a_{m,n}= \frac{m}{m+n}.$
Compute the iterated limits $$\lim_{n\rightarrow \infty}(\lim_{m \rightarrow \infty} a_{mn})$$ and $$\lim_{m\rightarrow \infty}(\lim_{n \rightarrow \infty} a_{mn})$$
My attem... | There is a general theory that considers iterated limits in relation to double limits and, among other things, explains under what conditions
$$\tag{*}A=\lim_{n,m \to \infty} a_{mn} = \lim_{m\rightarrow \infty}(\lim_{n \rightarrow \infty} a_{mn}) = \lim_{n.\rightarrow \infty}(\lim_{m \rightarrow \infty} a_{mn})
$$
The ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Understanding $\lvert z-1 \rvert+\lvert z+1 \rvert=7$ graphically $\lvert z-1 \rvert+\lvert z+1 \rvert=7$ is a circle of radius 3.5 if we use a computer algebra system to draw it. One can take $z:=x+iy$ and get the equation
$$\sqrt{(x-1)^2+y^2}+\sqrt{(x+1)^2+y^2}=7$$
Then we can square both sides and get another expres... | Although your example is mistaken, I still think the general question is useful.
One observation is that $|z-z_0|$, for any constant complex value $z_0$, represents the distance of $z$ from $z_0$. For instance $|z-1|$ represents the distance of a complex number (in the complex plane) from $1$. So the equation
$$
|z-1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Computing the spectral norm of a projection matrix I was reading a paper in which there was an argument as trivial, but could not make myself sure about it. It is said that given a full row-rank matrix $A$, the norm (probably $\ell_2$-induced matrix norm) of $A^T(AA^T)^{-1}A$ is one. Is that trivial, and correct for an... | We recognize in that expression a projection matrix onto $\operatorname{Row(A)}$ and since $A$ is a full row rank we have that
$$P=A^T(AA^T)^{-1}A \implies \sup\left\{\frac{\|Ax\|}{\|x\|},\, x\neq 0\right\}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2861176",
"timestamp": "2023-03-29T00:00:00",
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How many fair dice of this kind exist? I am not talking about the shape of the dice here, I am talking about another type. You will see what I mean soon.
For example, when there are 1 dice, a normal dice is a fair dice, because the probability of getting each number is the same by $ \frac{1}{6} $
When 2 normal dice are... | Let $n$ be any integer.
Dice 1: $\left[\matrix{1+n\\ 2+n\\ 3+n\\ 4+n\\ 5+n\\ 6+n}\right]$
$\qquad $Dice 2: $\left[\matrix{0-n\\ 6-n\\ 12-n\\ 18-n\\ 24-n\\ 30-n}\right]$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2861275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why am I getting a wrong answer on solving $|x-1|+|x-2|=1$ I'm solving the equation,
$$|x-1| + |x-2| = 1$$
I'm making cases,
$C-1, \, x \in [2, \infty) $
So, $ x-1 + x-2 = 1 \Rightarrow x= 2$
$C-2, \, x \in [1, 2) $
$x-1 - x + 2 = 1 \Rightarrow 1 =1 \Rightarrow x\in [1,2) $
$C-3, \, x \in (- \infty, 1)$
$ - x + 1 - x... | Alternative Solution
By the Triangle Inequality,
$$|x-1|+|x-2|=|x-1|+|2-x|\geq \big|(x-1)+(2-x)\big|=1\,.$$
The inequality becomes an equality if and only if $2-x=0$, or $x-1=\lambda(2-x)$ for some $\lambda\geq0$ (which gives, by the way, $x=\frac{2\lambda+1}{\lambda+1}\in[1,2)$). It follows immediately that $[1,2]$ i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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probability that he is NOT killed in $20$ years Question
$\text{suppose that the probability of being killed in a single flight is
}P_{c}=\frac{10^{-6}}{4} \text{based}$
$\text{ on available statistics. Assume that different flights are independent. If a businessman}$ $\text{takes 20 flights per year, what is the ... | There is another approach to this problem.
Consider $P(n)$ be the probability that the businessman dies in his $n$th flight.
$p_c$ is the probability of the person being killed in any flight.
Now, $P(i) = {(1-p_c)}^{i-1} \dot p_c$ i.e. he survived $i-1$ flights and died in $i$th flight.
He will take total $20 \times 2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can a definite condition be considered as an object in $ZF$? Can a definite condition be considered as an object in $ZF$?
This question arose from the following question:
For every class $A$, prove that $A$ is a set if for some class $B, A \in B$
Where $A \in B$ is defined to be equivalent to the fact that $B$ is a set... | Well. Not quite exactly, but almost.
For example, the empty set is really the class $\{x\mid x\neq x\}$. But really there is an axiom which states $\exists x(\forall y(y\in x\leftrightarrow y\neq y))$.
So a condition which defines a class is not a set per se, but it can be extensionally equivalent to a set.
For another... | {
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Determining the last column so that the resulting matrix is an orthogonal matrix
Determine the last column so that the resulting matrix is an orthogonal
matrix $$\begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{6}} & ? \\ \dfrac{1}{\sqrt{2}} & -\dfrac{1}{\sqrt{6}} & ? \\ 0 & \dfrac{2}{\sqrt{6}} & ? \end{bmatrix... | A matrix is orthogonal if all the column vectors are unit vectors and any two columns have dot product zero. Now write the unknown last column as $(x\ y\ z)^T$. You will get two equations when when you insist it be orthogonal to the known first and second columns.
So solve a system of 2 equations an $x,y,z$. There wil... | {
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Value of $\lim_{n \to \infty} \left({\frac{(n+1)(n+2)(n+3)...(3n)}{n{^{2n}}}}\right)^{1/n}$ I was asked to evaluate the following expression:
$\lim_{n \to \infty} \left({\frac{(n+1)(n+2)(n+3)...(3n)}{n{^{2n}}}}\right)^{1/n}$
My first step was to assume that the limit existed, and set that value to $y$.
$ y = \lim_{n \... | Consider
$$\int_0^2f(x)\,dx$$
where
$$f(x)=\ln(1+x).$$
Splitting $[1,2]$ into $2n$ intervals of length $1/n$ gives a Riemann
sum
$$\frac1n\sum_{k=1}^{2n}f(k/n)=\frac1n\sum_{k=1}^{2n}\ln\left(
1+\frac kn\right)$$
which is exactly yours.
Alternatively you could use Stirling's formula.
| {
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Does $\sin ^n x$ converge uniformly on $[0,\frac{\pi}{2})$?
Does $f_n(x)=\sin ^n x$ converge uniformly on $[0,\frac{\pi}{2})$ ?
I know $f_n(x) \rightarrow 0$ pointwise, since $\vert \sin ^n x \vert< 1$. How about the uniform convergence?
Any hint?
| Uniform convergence of $(f_n)$ to $f(x) = 0$ would require that
$$
M_n = \sup \{ |f_n(x) - f(x) | : 0 \le x < \frac \pi 2 \}
$$
converges to zero. However, for each $n$
$$
M_n \ge \lim_{x \to \pi /2} f_n(x) = 1 \, .
$$
| {
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"url": "https://math.stackexchange.com/questions/2862140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to determine if a set of five $2\times2$ matrices is independent $$S=\bigg\{\left[\begin{matrix}1&2\\2&1\end{matrix}\right], \left[\begin{matrix}2&1\\-1&2\end{matrix}\right], \left[\begin{matrix}0&1\\1&2\end{matrix}\right],\left[\begin{matrix}1&0\\1&1\end{matrix}\right],
\left[\begin{matrix}1&4\\0&3\end{matrix}\ri... | As the others have said, this set of $5$ must be linearly dependent because the dimension of the space of all $2\times 2$ matrices is $4$.
More generally, how do you show that a set of vectors is linearly dependent or independent? Create a linear combination of the vectors, set it equal to $0$, and try to solve it. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2862389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Find the number of ways to choose N non-negative numbers that sum up to $S$ and are in strictly increasing order? More formally, find the number of ways of dividing a sum $S$ among $N$ numbers — $a_1, a_2, a_3, \dots, a_N$,
such that they are in strictly increasing order i.e. — $a_1 < a_2 < a_3 < \dots < a_n$,
given t... | We will first assume $0$ is not allowed as one of the numbers. We will cover $0$ at the end.
You can write a recurrence. If $A(S,N)$ is the number of ways of writing $S$ as a strictly increasing sum of $N$ numbers greater than $0$, we can look at whether $1$ is one of the numbers. If it is, we need to express $S-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2862475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Strongly convergence and Uniformly convergence in Banach space Let $X,Y$ be Banach spaces, $T_n:X\to Y$ are bounded linear operators and $S:X\to Y$ is compact operator. Suppose for all $x\in X$, $||T_nx||\leq ||Sx||$ and $T_n$ strong convergent to $T$(bounded linear operator) . Then $T_n$ convergent uniformly.
My idea... | Firstly, observe that we may assume that $T=0$.
Reason: For any $x\in X$, we have $||T_{n}x||\leq||Sx||$. Letting
$n\rightarrow\infty$, we have $||Tx||\leq||Sx||$. Now
\begin{eqnarray*}
||(T_{n}-T)x|| & \leq & ||T_{n}x||+||Tx||\\
& \leq & ||Sx||+||Sx||\\
& = & ||(2S)x||.
\end{eqnarray*}
$\{T_{n}-T\mid n\in\mathbb{N}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Inverse Laplace transform of $K_0 \left(r \sqrt{s^2-1}\right)$ This question is about inverse Laplace transform $\mathscr{L}^{-1}:s\rightarrow t$. Although I was not able to find appropriate contour to invert $K_0 \left(r s\right)$, I somehow know that
$$\mathscr{L}^{-1}\{K_0 \left(r s\right)\}=\frac{\theta(t-r)}{\sqr... | The result given by Mariusz can be verified as follows. The substitution $t = r \cosh \tau$ gives
$$F(s) = \int_0^\infty
\frac {\cosh \sqrt {t^2 - r^2}} {\sqrt {t^2 - r^2}}
\theta(t - r) e^{-s t} dt =
\int_0^\infty e^{-r s \cosh \tau} \cosh(r \sinh \tau) d\tau, \\
\operatorname{Re} s > 1.$$
Converting $\cosh(r \sinh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2862697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Image and pseduo-inverse of an operator Let $\mathcal{H}$ be a Hilbert space and $(e_n)_{n\in \Bbb N}$ be an orthonormal basis for $\mathcal{H}$.
Define the surjective operator $T\in B(\mathcal{H})$ such that $Te_{2n-1}=\frac{1}{2^n}e_1$ and $Te_{2n}= e_{n+1}$ for each $n\in\Bbb N$.
There are several questions:
$\bul... | The answer to the previous version of the question, which was is $e_{2k-1} \in R(T^*)$, is no.
We have
\begin{align}
T^*x &= \sum_{n=1}^\infty \langle T^*x, e_n\rangle e_n \\
&= \sum_{n=1}^\infty \langle x, Te_n\rangle e_n \\
&= \sum_{n=1}^\infty \langle x, Te_{2n-1}\rangle e_{2n-1} + \sum_{n=1}^\infty \langle x, Te_{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2862814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What exactly is a function? This remark appears in Terence Tao's Analysis I
Remark 3.3.6. Strictly speaking, functions are not sets, and sets are not functions; it does not make sense to ask whether an object $x$ is an element of a function $f$, and it does not make sense to apply a set $A$ to an input $x$ to create a... | I like and suggest the following definition:
A function $f:A\to B$ is a triple
*
*a first set $A$ (domain)
*a second set $B$ (codomain)
*a law (i.e. a rule, a relationship, etc.) such that at each element of $A$ is associated one and only one element of $B$ that is
$$\forall x\in A \quad \exists ! y\in B:\,y=f(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2862927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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"answer_id": 0
} |
Is it ok this CNF of a Boolean function? I have to find out the CNF of $$\begin{matrix} f(x,y,z)&=&(x\wedge y)\vee(x\wedge z),\end{matrix}$$ where $f$ is a Boolean function.
$$\begin{matrix}&f(x,y,z)&=&(x\wedge y)\vee(x\wedge z)&1\\
&&=&x\wedge(y\vee z)&2\\
&&=&(x\vee0_B\vee0_B)\wedge(y\vee z\vee 0_B)&3\\
&&=&(x\vee(y... | In general, formulas do not have a unique conjunctive normal form (CNF), see an example here. So, in general, the fact that you get another CNF does not imply automatically that you are wrong. The important thing is that all CNF's you get should be equivalent.
In this particular case, you should notice that actually in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Sufficient condition for a matrix to be diagonalizable and similar matrices my question is about diagonalizable matrices and similar matrices.
I have a trouble proving a matrix is diagonalizable.
I know some options to do that:
Matrix $A$ $(n \times n)$, is diagonalizable if:
*
*Number of eigenvectors equals to num... | First a comment
The wording Number of eigenvectors equals to number of eigenvalues... is confusing. If $A$ has a non zero eigenvector then $A$ has an infinite number of eigenvectors (providing you work in $\mathbb R$ or $\mathbb C$ for example). A proper wording would be $A$ has a basis of eigenvectors.
If a matrix has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Are (quasi-)regular polytopes uniquely determined by their edge graph? I consider polytopes $P\subset\Bbb R^n,n\ge 2$ of arbitrary dimension (intersection of finitely many halfspaces, therefore convex), which are vertex- and edge-transitive (also called quasi-regular).
Question: Can there exist two different such pol... | (Quasi-)regular polytopes surely are not uniquely defined by their edge graphs. Just consider the icosahedron x3o5o and the great dodecahedron x5o5/2o. In fact the latter is an edge-faceting of the former (i.e. respecting the same edge graph).
But as soon as you add the (true) convexity constraint, you enforce the edge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Problem while using divergence theorem Evaluate $$\iint_S \mathbf A\cdot n \ \mathrm d S$$
where $\mathbf A=y\mathbf i+2x\mathbf j-z\mathbf k$ and $S$ is the surface of the plane $2x+y=6$ in the first octant cut off by the plane $z=4$
I was doing this problem then $\operatorname{div} A = -1$ . So I got eventually integ... | The divergence theorem applies to the flux through a closed surface. You have only one piece of a plane (not a closed surface). You would need to create a volume instead. Add surfaces where calculating the integral might be easier (such as the $xy, xz, yz$ sides), to create a closed surface.
There are two ways to solve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Which one of the following are true?
Consider the function $e^{-z^{-4}}$ for $z≠0$ and $f(0)=0$. Then,
(A) $f$ is not analytic.
(B)$f$ is not differentiable at $z=0.$
(c)$f$ does not satisfy the C-R(Cauchy-Riemann) equation.
(d)$f$ satisfies the C-R(Cauchy-Riemann) equation and not analytic.
(A) $f$ is not analytic.... | Consider $\zeta_4$ a $4$-th root of $-1$ and for $\epsilon >0$, $z_\epsilon = \zeta_4 \epsilon$. You have $$
\lim\limits_{\epsilon \to 0}f(z_\epsilon) = \lim\limits_{\epsilon \to 0}e^{-z_\epsilon^{-4}} = \lim\limits_{\epsilon \to 0}e^{1/\epsilon^4} = \infty.$$
Therefore $f$ is not even continuous at $0$.
Hence:
*
*$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there a perfect square that is the sum of $3$ perfect squares? This is part of a bigger question, but it boils down to:
Is there a square number that is equal to the sum of three different square numbers?
I could only find a special case where two of the three are equal? https://pir2.forumeiros.com/t86615-soma-de-tr... | There are infinitely many: for any positive integer $n$ we have
$$n^2(n+1)^2+n^2+(n+1)^2=(n(n+1)+1)^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 5
} |
Integrate $\int \frac{1}{1+ \tan x}dx$ Does this integral have a closed form?
$$\int \frac{1}{1+ \tan x}\,dx$$
My attempt:
$$\int \frac{1}{1+ \tan x}\,dx=\ln (\sin x + \cos x) +\int \frac{\tan x}{1+ \tan x}\,dx$$
What is next?
| Notice that $$\int \frac{\tan{x}}{1+\tan{x}}= \int 1-\frac{1}{1+\tan{x}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 1
} |
Combinatorics problem in walking left and right In this game, there will be $n$ turns. For the sake of clarity, let us say $n = 4$.
We have three options in this game: we can either stay in place ($O$), we can step to the left ($L$), or we can step to the right ($R$). However, the ability to step left only occurs on t... | The count of sequences is the count of ways to select $k$ from $\lfloor n/2\rfloor$ places for R and to select $k$ from $\lceil n/2\rceil$ places for L, for every $k$ that is an integer in $\{0,..,\lfloor n/2\rfloor\}$.
$$X_n~{=\sum_{k=0}^{\lfloor n/2\rfloor}\dbinom {\lfloor n/2\rfloor}k\dbinom{\lceil n/2\rceil}k\\ =\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
An exercise applying "The method of Distribution Functions" I'm not understanding something about this question:
A process for refining sugar yields up to 1 ton of sugar per day, but the actual amount produced, $Y$ is a random variable because of the machine breaking down and other slow downs. Suppose that $Y$ has de... | Because $f_Y(y) = \begin{cases}2y &:& {0\leqslant y\leqslant 1}\\0&:& \text{elsewhere}\end{cases}$, therefore: $$F_Y(y) = \begin{cases} 0 &:& \qquad y < 0 \\ y^2 &:& ~0\leqslant y < 1\\ 1 &:& ~1\leqslant y\end{cases}$$
Then we have that for $g(u)=\dfrac{u+1}{3}$.
$$\begin{align} F_U(u) &= F_Y(g(u))\\[1ex] & =\begin{cas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2863999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If matrix $A$ is totally unimodular, then matrix $\begin{bmatrix} A &\pm A\end{bmatrix}$ is also totally unimodular
Given a totally unimodular matrix $A \in\{-1,0,1\}^{m\times n}$, show that the matrix $$\begin{bmatrix} A &\pm A\end{bmatrix}$$ is also totally unimodular.
I want to prove that exchanging any two column... | The goal is to show that if a square submatrix of $[A,\pm A]$ is nonsingular then it has determinant $\pm 1$. Observe that any nonsingular square submatrix of $[A,\pm A]$ is (up to permutation and sign of the columns) a nonsingular square submatrix of $A$ (justified later). Hence since permutation changes determinants ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2864087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that Laplace density (standard) lies in Sobolev space of order $\leq 3/2$ I'm trying to prove, that the function $f(x) = e^{-|x|}$ lies in $H^{s}(\mathbb{R}^2)$ for $s\leq \frac{3}{2}$.
Therefore I calculate the functions sobolev norm $$\|f\|_s^2 = \frac{1}{4\pi^2}\int_{\mathbb{R}^2}(1+\| u\|)^s |\mathscr{F}f(u)|... | So, finally, through the help of a fellow student I realized that transforming to polar coordinates would make sense.
This yields $$\int_0^{2\pi}\int_0^\infty r\;(1+r^2)^{s-2}drd\theta.$$
Assuming $s<2$ (otherwise, the integral would obviously diverge) and using $r\leq 1+r$, we find $$\|f\|_s^2 \leq 2\pi \int_0^\infty ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2864202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
How to evaluate $ \prod_{1\le i While doing my research on elementary number theory, I came across the following problem which I cannot overcome: Let $p$ be an odd prime, $g$ be any primitive of $p$. Define
$$f(p)=\prod_{1\le i <j\le\frac{p-1}{2}}j^2-i^2\pmod p$$ and
$$h(p,g)=\prod_{1\le i <j\le\frac{p-1}{2}}g^{2j}-g^{... | I presume you are doing the calculations modulo $p$. In the first case
the admissible $i^2$ are the quadratic residues modulo $p$.
In the second case
the admissible $g_{2i}$ are also the quadratic residues modulo $p$.
Both products have the form $\prod_{1\le i<j\le m}(a_j-a_i)$
where $a_1,\ldots,a_m$ ($m=\frac12(p-1)$)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2864445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Show that $\sum_{n=1}^\infty 2^{2n}\sin^4\frac a{2^n}=a^2-\sin^2a$ Show that $$\sum_{n=1}^\infty 2^{2n}\sin^4\frac a{2^n}=a^2-\sin^2a$$
I am studying for an exam and I bumped into this question. It's really bothering me because I really don't have any clue what to do. Does it have anything to do with the Cauchy condens... | Solution
Notice that
\begin{align*}
2^{2n}\sin^4\frac a{2^n}&=2^{2n}\cdot\sin^2\frac a{2^n}\cdot\sin^2\frac a{2^n}\\&=2^{2n}\cdot\sin^2\frac a{2^n}\cdot\left(1-\cos^2\frac a{2^n}\right)\\&=2^{2n}\cdot\sin^2\frac a{2^n}-2^{2n}\cdot\sin^2\frac a{2^n}\cos^2\frac a{2^n}\\&=2^{2n}\cdot\sin^2\frac a{2^n}-2^{2n-2}\sin^2\frac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2864538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Is there a standard notation for the multiplicative group generated by the primes $p\in P$? Is there a standard notation for the multiplicative group generated by the primes $p\in P$?
Let $P$ be some set of primes e.g. $P=\{2,3\}$
Then $G_P$ is the multiplicative group generated by these primes so e.g. $G_{\{2,3\}}$ is... | When $P = \{2, 3\}$ the elements of $G_P$ have a unique representation of the form $2^i3^j$ for $i, j \in \Bbb{Z}$ and this is easily checked to give an isomorphism between $G_P$ and the sum $\Bbb{Z}^2$ of two copies of the additive group of integers. In general, up to isomorphism, $G_P$ depends only on the cardinality... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2864630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For all integers $n$: if $7n+4$ is even, then $5n+6$ is even. I am still new to the proof game so please be kind! This is my third time attempting a proof. Any feedback would be greatly appreciated. Thank you in advance.
Claim: For all integers $n$: if $7n+4$ is even, then $5n+6$ is even.
Proof: Assume $5n+6$ is odd. I... | Modulo $2$ we have by assumption
$$
0\equiv 7n+4\equiv 1\cdot n+0=n,
$$
so that
$$
5n+6\equiv 5\cdot 0+0\equiv 0.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2864713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 6
} |
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