Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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No group of following property. Is this true? Let $p$ be a prime greater than 3 and $G$ be group of order $p^5$. Is it true that there is no group $G$ of order $p^5$ such that the order of frattini subgroup is $p^3$ and the order of center is $p^2$?
If the answer is yes, how to prove it.
| The class $3$ quotient of the Burnside group $B(2,p)$ has these properties. It has the presentation
$\langle a,b,c,d,e \mid a^p=b^p=c^p=d^p=e^p=1, [b,a]=c, [c,a]=d, [c,b]=e, d,e\ {\rm central}\ \rangle$
The Frattini subgroup is $\langle c,d,e \rangle$ and the centre is $\langle d,e\rangle$.
| {
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A second order recurrence relation problem I was asked by a friend with this problem but I can't solve it. Can anyone help?
We have a sequence $\left\{a_n\right\}$ that satisfies $a_1=1$, $a_2=2$,
$$a_n+\frac{1}{a_n} =\frac{a_{n+1}^2+1}{a_{n+2}}$$
where $n$ is a positive integer.
Prove that
*
*$a_{n+1}=a_n+\frac{1}... | (a)
By induction, assume $a_{k+1} = a_k + \frac1{a_k}$. Then to get the result for $n=k+1$, consider
$$\begin{align*}
a_k+\frac1{a_k} =& \frac{a_{k+1}^2+1}{a_{k+2}}\\
a_{k+1}=&\frac{a_{k+1}^2+1}{a_{k+2}}\\
a_{k+2} =& a_{k+1}+\frac1{a_{k+1}}
\end{align*}$$
Also prove the base case for $n=1$ holds.
(b)
Using $a_{n+1}^2 ... | {
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Graphs with both Eulerian circuits and Hamiltonian paths Which graphs have both Eulerian circuits and Hamiltonian paths, simultaneously?
Honestly I don't know the level of the question. One of my friens asked me to put the question on math.stackexchange.
Thanks.
| Actually there is no such specific class of graphs. You can always find examples that will be both Eulerian and Hamiltonian but not fit within any specification. The set of graphs you are looking for is not those compiled of cycles.
For any $G$ with an even number of vertices the regular graph with, $$ degree(v) = \fr... | {
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munkres analysis integration question Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2 \to \mathbb{R}$ be defined by setting $f(x,y)=0$ if $y \neq x$, and $f(x,y) = 1$ if $y=x$. Show that $f$ is integrable over $[0,1]^2$.
|
Consider the partition $P$ like the above picture.
$U(f, P) - L(f, P) = \frac{3 n - 2}{n^2}$.
$\frac{3 n - 2}{n^2} \to 0 (n \to \infty)$.
So, for any $\epsilon > 0$, there exists a partition $P$ such that $U(f, P) - L(f, P) < \epsilon$.
By Theorem 10.3 (p.86), $f$ is integrable over $[0, 1] \times [0, 1]$.
| {
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Why does this assumption change the formula this way I am working through some notes and I cannot understand why the following assumption changes the formula as such.
The formula is basically referring to a right angled triangle of base $ L $ and height $ \frac{D}{2} $. The difference between the hypotenuse and the ba... | Let us rewrite the second formula this way:
$$
\Delta\theta = \frac{2\pi}{\lambda} \cdot L \cdot \left[ \left(1 + \frac{D^2}{4L^2}\right)^{1/2} - 1\right] = \frac{2\pi}{\lambda} \cdot L \cdot \left[ \left(1 + x \right)^{1/2} - 1\right],
$$
where $x = \frac{D^2}{4L^2}$.
Then, since $D << L$, we can approximate $\left(1 ... | {
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An example of a $P$-primary ideal $I$ satisfying $I^2 = IP$ Give some examples of a $P$-primary ideal $I \not=P $ in a noetherian domain $R$ such that $I^2=PI $.
| Let $R = k [[t^3, t^4, t^5]]$, $P = (t^3, t^4, t^5)$, and $I = (t^3, t^4)$. Then $$IP = I^2 = (t^6, t^7, t^8). $$
| {
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If $\,\,\frac{x^2}{cy+bz}=\frac{y^2}{az+cx}=\frac{z^2}{bx+ay}=1,$ then show that .... I am stuck on the following problem that one of my friends gave me:
If $\,\,\frac{x^2}{cy+bz}=\frac{y^2}{az+cx}=\frac{z^2}{bx+ay}=1,$ then show that $$\frac{a}{a+x}+\frac{b}{b+y}+\frac{c}{c+z}=1$$. I did a problem which was similar t... | HINT:
We have $$x^2=cy+bz\iff ax+cy+bz=x^2+ax=x(x+a)$$
$$\implies\frac1{a+x}=\frac x{ax+cy+bz}\implies \frac a{a+x}=\frac{ax}{ax+cy+bz}$$
| {
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Mathematical and Theoretical Physics Books Which are the good introductory books on modern mathematical physics? Which are the good advanced books?
I read Whittaker's Analytical Dynamics, and I am reading Arnold's Mathematical Methods of Classical Mechanics. However, I am not very interested in books on classical mecha... | If Lectures on Quantum Mechanics for Mathematical Student work, then you should check Quantum Mechanics for Mathematicians written to provide a somewhat more modern and thorough exposition of Quantum Mechanics for Mathematical Student. (The author is a student of one of the authors of the former book).
| {
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"question_score": "28",
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One-to-one correspondence between these two problems?
How many 3-digit positive integers are there whose middle digit is
equal to the sum of the first and last digits?
I noticed that the solution to this problem, $45$, is the same as the solution to the problem
How many 3-digit positive integers are there whose mi... | Note that in both problems, the middle digit (if valid) is determined by the first and last digits.
There are 90 possible pairs of first and last digits $(f,\ell)$, since $1\le f\le 9$ and $0\le\ell\le9$. These can be grouped into 45 pairs $\{ (f,\ell), (10-f,9-\ell) \}$.
Note that exactly one member of each such pair ... | {
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limit of $\frac{2xy^3}{7x^2+4y^6}$, different answers. Good evening, everyone I've tried my possible best to evaluate the limit as $(x,y) \to (0,0)$ but Using sage the answer is 0 either ways but one textbook is saying the limit doesn't exist. Could the 2 answers be correct?
| My guess is the problem is that sage cannot test every approach curve, so it probably tests a small grid of points about the origin. Since the place where the limit is bad is shaoed like a curved cubic, the grid will miss the curve and give the wrong answer.
Note that Wolfram Alpha has the same problem: http://m.wolfra... | {
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What is the value of $\lim_{x\to0}\frac{x^2\sin\left(\frac{1}{x}\right)}{\sin x}$? So this is the question in my text book $$\lim_{x\to0}\frac{x^2\sin\left(\frac{1}{x}\right)}{\sin x}$$
here what i have done
It can be rearranged as $$\lim_{x\to0}\left(\frac{x}{\sin x}\right)x\sin\left(\frac{1}{x}\right)$$$$\Rightarrow\... | $$\lim_{x\to0}\frac{x^2\sin\frac1x}{\sin x}=\lim_{x\to0}\frac{x\sin\frac1x}{\frac{\sin x}x}=\lim_{x\to0}\frac{x\sin\frac1x}1=0.$$
| {
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Proof that the derivative of the prime counting function is the probability of prime? The derivative of the estimation of the prime counting function, $\frac{x}{ln(x)}$, is $\frac{ln(x)-1}{ln(x)^2}$, which is approximately $\frac{1}{lnx}$ for large values of $x$. According to the prime number theory, $\frac{1}{lnx}$ is... | You are describing Cramer's model of the primes, which is pretty good. However, Maier showed about 1985 that it gave incorrect estimates for short intervals. See if i can find it, it's a famous episode.
Maier's theorem
| {
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small generalization of a linear algebra exercise I came aross the following exercise and was wondering if the conditions are actually necessary or if we could generalize this. Here Hom$(\mathbb R, \mathbb R)$ is the Vectorspace of linear functions from $\mathbb R$ to $\mathbb R$
let $F$ be a function Hom$(\mathbb R,\... | Yes, it works much more generally. If $V$ is a $K$-vector space ($K$ any field), and $X$ any set, for every $x\in X$, the evaluation map
$$\operatorname{ev}_x \colon \mathscr{F}(X,V) \to V;\quad \operatorname{ev}_x (f) = f(x)$$
is $K$-linear. That indeed follows directly from the definition of addition and scalar multi... | {
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Simple limit of function How do I show that $\lim_{x \rightarrow \infty } \frac {\log(x^{2}+1)}{x}=0$?I was able to do that using L'Hôpital's rule. But is there any other way?
| $$\lim_{x\to\infty}\frac{\ln(x^2+1)}x\sim\lim_{x\to\infty}\frac{\ln(x^2)}x=\lim_{x\to\infty}\frac{2\cdot\ln x}x=2\cdot\lim_{t\to\infty}\frac t{e^t}=2\cdot0=0.$$
| {
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Other than $\models$, is there standardized notation for semantic consequence? It is common practice to use $\models$ both for the satisfaction relation between models and sentences, and for the corresponding semantic consequence relation.
Question. Suppose I don't want to use $\models$ for semantic consequence (person... | In Ben-Gurion University, where I did my B.Sc. and M.Sc. we used $T\implies\varphi$ to denote logical implication, which was really a semantic property:
$T\implies\varphi$ if and only if for every interpretation for the language $M$ and assignment $s$ for $M$, such that every formula in $T$ is true under that assignm... | {
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$\mathbb{Q}$ is not isomorphic to $\mathbb{Q}^+$ How can we show that $\mathbb{Q}$ as an additive group is not isomorphic to $\mathbb{Q}^+$ as a multiplicative group?
Both have a countable number of elements, neither is cyclic, neither has an element $x\neq e$ such that $x^2=e$, both are abelian ... I don't know what t... | If there were an isomorphism $f:(Q^+,\times)\to (Q,+)$ then $f(2)$ would be the sum $a+a$ where $a=f(2)/2$, but then $a$ could not have an inverse image under $f$.
| {
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Is $f=0$ if the integral is zero If $f: [0,1]\to \mathbb R$ is continuous and for all $s\in [0,1]$
$$ \int_0^s f(t)dt = 0$$
does it then follows that $f=0$? I can show it for $f \ge 0$ but I am wondering if it is also true if $f$ not positive.
| Define $F(x) = \int_0^x f(s)\,ds$. By the Fundamental Theorem of Calculus, $F'=f$. But by your assumption, $F(x) = 0$ for all $x \in [0,1]$. So $f \equiv 0$.
| {
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Is this set convex ?2 Is this set convex for every arbitrary $\alpha\in \mathbb R$?
$$\Big\{(x_1,x_2)\in \mathbb R^2_{++} \,\Big|\, x_1x_2\geq \alpha\Big\}$$
Where $\mathbb R^2_{++}=[0,+\infty)\times [0,+\infty)$.
| Yes, it is. You should separate the case $a<0$ (when it is the quadrant) and $a>0$ when it is bounded by an arc of a hyperbola.
| {
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Sum of $\sum\limits_{i=1}^\infty (-1)^i\frac{x^{2i+1}}{2i+1}$ Can someone help me with this series? It was on my exam and I don't know how to do it.
For $|x| < 1$ determine the sum of
$$\sum\limits_{i=1}^\infty (-1)^i\frac{x^{2i+1}}{2i+1}$$
| The derivative of the given sum is the geometric sum
$$\sum_{n=1}^\infty (-x^2)^n=-\frac{x^2}{1+x^2}=-1+\frac{1}{1+x^2}$$
so the given sum which vanish at $0$ is
$$\int_0^x\left(-1+\frac{1}{1+t^2}\right)dt=-x+\arctan x$$
| {
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Find DNF and CNF of an expression I want to find the DNF and CNF of the following expression
$$ x \oplus y \oplus z $$
I tried by using
$$x \oplus y = (\neg x\wedge y) \vee (x\wedge \neg y)$$
but it got all messy.
I also plotted it in Wolfram Alpha, and of course it showed them, but not the steps you need to make to... | For DNF:
*
*look at each row where $p = 1$
*encode a proposition from the atoms $p_i$ for row $i$ (that gives $p$ is 1) that has $a_i$ if that atom is 1 in the truth table and $\neg a_i$ if it's 0. You are using an and to combine the atoms so that only this terms is 1 when you are on that row. You can think of this... | {
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Probability that the second roll comes up yellow given the first roll was purple. A bag contains $20$ dice. $5$ of the dice have entirely purple sides, $7$ of the dice have $2$ purple and $4$ yellow sides, and $8$ of the dice have $3$ purple and $3$ yellow sides. If you randomly pick a die, roll it, and observe that th... | Let $P$ be the event the first roll gave purple, and $Y$ the event the second roll gave yellow. We want $\Pr(Y|P)$. By the definition of conditional probability, we have
$$\Pr(Y|P)=\frac{\Pr(P\cap Y)}{\Pr(P)}.$$
You calculated $\Pr(P)$ using the correct approach. I have not checked the arithmetic. We need $\Pr(P\cap Y)... | {
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An element does not belong to an ideal
How can I prove that the element $x-5$ does not belong to the ideal $(x^2-25,-4x+20)$ in $\mathbb Z[x]$.
I tried to show that by proving $x-5\neq(x^2-25)f(x)+(-4x+20)g(x)$ for all $f,g$. Any help?
| ${\rm mod}\ \color{#c00}2\!:\ (x^2\!-25) f -\color{#c00}2\, g\ $ is either $\,0\,$ or of degree $\ge 2,\,$ so is $\,\not\equiv\, x-5$
| {
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How do I make pi = 3? This question emerges from a discussion on quora which concluded that if a circle was drawn on the surface of a sphere, the ratio of radius (from the circle's centre as projected to the sphere's surface, measured over the surface of the sphere) to the circumference could be made to equal exactly 1... | If you take the unit sphere $r=1$, denoted by $S^2$ and take the circle's center to be the north pole $n=(0,0,1)^T$, you want to know the diameter of the circle to be such that $\pi \cdot d = 3$ so $d = \frac3\pi$. From that you can compute backwards the height of the hyperplane $H:= \{x\in\mathbb R^3, x_3 = h\}$ such ... | {
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Find slope of a curve without calculus Is it possible to find the slope of a curve at a point without using calculus?
| Slope of a curve at a specific point MUST be a limit, which I am not sure whether you classify as calculus or not. Slope is by definition a function of two distinct points, and the only interpretation of "slope of a curve at a point" is that of two points approaching each other along the curve.
If you are not allowed t... | {
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evaluation of series Evaluate :
a.$$\frac{1}{2\cdot3}+\frac{1}{4\cdot5}+\frac{1}{6\cdot7}+\cdots$$
For this I looked at
$$x+x^3+x^5+\cdots=\frac{x}{1-x^2} \text{ for }|x|<1$$
Integrating the above series from $0$ to $t$ yields
$$\frac{t^2}{2}+\frac{t^4}{4}+\frac{t^6}{6}+\cdots=\int_{0}^{t}\frac{x}{1-x^2} \, dx$$
Again... | Using $\displaystyle \frac1{n(n+1)}=\frac1n-\frac1{n+1}$
$$\frac1{2\cdot3}+\frac1{4\cdot5}+\frac1{6\cdot7}+\cdots=\frac12-\frac13+\frac14-\frac15+\frac16-\frac17+\cdots$$
Now use Convergence for log 2 or Taylor series for $\log(1+x)$ and its convergence
| {
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How are the values $3\sqrt{2}$ and $\sqrt{2}$ determined? Those values $\sqrt{2}$ and $3\sqrt{2}$
How do they suppose to match with $MB=BN=2$?
| Draw the bottom square $ABCD$!
I've added the point $T$: the centre of the square $ABCD$.
The computation is essentially not different from the one given by mathlove in the earlier answer: the square $MBNT$ has sides of length $2$, its diagonal $TB$ has length $2 \sqrt{2}$, and $S$ is in the middle, so $SB = \sqrt{2}$... | {
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The Limit of $x\left(\sqrt[x]{a}-1\right)$ as $x\to\infty$. How to find the limit of:
$$ \lim_{x\to \infty }x\left(\sqrt[x]{a}-1\right)$$
Without using L'hôspital rule.
I've tried to bound the term and use the squeeze theorem but I couldn't find the right upper bound. I've also tried to convert $a^\frac{1}{x}$ to $e^{\... | Setting $\displaystyle \frac1x=h$
$$\lim_{n\to\infty}x(\sqrt[x]a-1)=\lim_{h\to0}\frac{a^h-1}h=\ln a\lim_{h\to0}\frac{e^{h\ln a}-1}{h\ln a}=\cdots$$
Use Proof of $ f(x) = (e^x-1)/x = 1 \text{ as } x\to 0$ using epsilon-delta definition of a limit
| {
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Calc expected value of 5 random number with uniform distribution Assume we have a random numbers $\sim U(0,100)$.
Then the expected value of that number will be: $\int_{0}^{100} \frac{x}{100}$ = 50.5
Now assume we have 5 random numbers $\sim U(0,100)$.
How can I calculate what would be the expected value of the maximal... | You need to learn about order statistics:
https://en.wikipedia.org/wiki/Order_statistics
The maximum of five independent observations is the fifth order statistic (of that sample). In your case, that will have a certain (scaled) beta distribution. You can find the datails in wikipedia above. In your case it will be (10... | {
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Why - not how - do you solve Differential Equations? I know HOW to mechanically solve basic diff. equations. To recap, you start out with the derivative $\frac{dy}{dx}=...$ and you aim to find out y=... To do this, you separate the variables, and then integrate.
But, can someone give me a some context? A simple e... | In the introduction of Arnold's book, talking about differential equations:
[...]
Newton considered this invention of his so important that he encoded it as an anagram whose meaning in modern terms can be freely translated as follows:
“The laws of Nature are expressed by differential equations.”
| {
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Solve $12x\equiv9\pmod{15}$ Question:
Solve $12x\equiv9\pmod{15}$
My try:
$\gcd(12,15)=3$ so it has at least $3$ solutions.
Now
$15=12\times1+3\\
3=15-12\times 1\\
3=15+2\times(-1)\\
\implies9=15\times3+12\times(-3)\\
\implies12\times(-3)\equiv9\pmod{15}$
So $x\equiv-3\pmod{15}$
Am I correct?
| Hint: $12x\equiv 9 \pmod{15}$ if and only if $4x\equiv 3\pmod{5}$, this is easy to verify by definition. Now, everything is co-prime to the modulus, so the problem is trivial.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Gramian matrix test Are there some test to know if a matrix $M$ is gramian ? $M$ is gramian if it exists a matrix W such $M=W^HW$. Also if it is possible to determine $W$.
Thanks
| Gramian matrix is positive-definite. So it's possible to find square root. It will be symmetrical so this it is possible to write such decomposition for any positive definite matrix. The solution of $M=WW^{T}$ it is not unique, because starting from nonsymmetric matrix $A$ one can construct matrix $B=AA^T$ and then a s... | {
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Prove that the limit definition of the exponential function implies its infinite series definition. Here's the problem: Let $x$ be any real number. Show that
$$
\lim_{m \to \infty} \left( 1 + \frac{x}{m} \right)^m = \sum_{n=0}^ \infty \frac{x^n}{n!}
$$
I'm sure there are many ways of pulling this off, but there are 3 ... | Recall that
$$\binom{m}{n}=\frac{m!}{n! (m-n)!}$$
By Stirling's formula you get for large $m$
$$m!\approx \sqrt{2\pi m}\left(\frac{m}{e}\right)^{m}\\
(m-n)!\approx \sqrt{2\pi(m-n)}\left(\frac{m-n}{e}\right)^{m-n} $$
Then note
$$ \left(\frac{m-n}{e}\right)^{m-n}=e^{(m-n)\ln(m-n)-m+n}=e^{(m-n)\ln(m)+(m-n)\ln(1-\frac{n}{m... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $\lim_{x\rightarrow\infty}(f(x+1)-f(x))=L$ prove that $f=O(x)$. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$\lim_{x\rightarrow\infty}(f(x+1)-f(x))=L$$
Prove that $$\lim_{x\rightarrow \infty}\dfrac{f(x)}{x}=L$$
This was an exam question that I was given and got nowhere on it. Going back now, I don't think I'... | You need to impose that $f$ maps bounded intervals onto bounded intervals:
By the way, the proof by contraposition seems to be the more appropriate. But first, we recall the basic identity
$$ (f(x+1)-f(x)-L)^2=\left([f(x+1)-L(x+1)]+[Lx-f(x)]\right)^2= \\ \ \\
=(f(x+1)-L(x+1))^2+(f(x)-Lx)^2- \\ \ \\ -2(f(x)-Lx)(f(x+1)-L... | {
"language": "en",
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"source": "stackexchange",
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Hardy-Littlewood maximal function weak type estimate Show that if $f\in L^1(\mathbb{R}^d)$ and $E\subset \mathbb{R}^d$ has finite measure, then for any $0<q<1$,
$$\int_E |f^{*}(x)|^q dx\leq C_q|E|^{1-q}||f||_{L^1(\mathbb{R}^d)}^{q}$$
where $C_q$ is a positive constant depending only on $q$ and $d$.
Here the function $... | Indeed, the weak type estimate is useful. Using Fubini's theorem, we have
$$\int_E|f^{*}(x)|\mathrm dx=q\int_0^\infty t^{q-1}\lambda\{|f^*(x)|\chi_E\geqslant t\}\mathrm dt.$$
Notice that $$\lambda\{|f^*(x)|\chi_E\geqslant t\}\leqslant \min\left\{|E|;\frac{3^d}t\lVert f\rVert_{\mathbb L^1}\right\},$$
hence cut the inte... | {
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find the domain of root of a logarithmic function I'm a little confused about this question since output of a logarithmic function varies from $ -\infty $ to $\infty$ .I should find the domain of this function: $ y=\sqrt{\log_x(10-x^2)} $ . How can I find the interval that makes $\log_x(10-x^2)$ greater than zero?
| Recall for $a>0, a\neq 1,b>0$:
$$\log_ab=\frac{\ln b}{\ln a}$$
Thus we have
$$f(x)=\log_x(10-x^2)=\frac{\ln(10-x^2)}{\ln x}$$
Then $f(x)\geq 0$ if and only if $10-x^2\geq 1,x>1$ or $0<10-x^2\leq 1,0<x<1$. Since the later case cannot happen, then we must have $10-x^2\geq 1$ and $x>1$, which gives $1<x\leq3$
| {
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What is a series? This question is rather pedantic, but it is something that has been bothering me for some time.
Summing up infinitely many terms of a sequence is something that is done in pretty much every subfield of mathematics, so series are right at the core of mathematics. But strangely, I have never seen a form... | I think I remember that when I first learned about this, my professor said that this is the first 'abuse of notation' that we would encounter- the symbol $\sum_{n=0}^\infty a_n$ is both used for the sequence and its limit.
One way to answer your original question could be to think of a series as a pair of sequences $(a... | {
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Find $\int_0^{+\infty}\cos 2x\prod_{n=1}^{\infty}\cos\frac{x}{n}dx$ Evaluate the following integral
$$\int_0^{+\infty}\cos 2x\prod_{n=1}^{\infty}\cos\frac{x}{n}dx$$
I was thinking of a way which do not need to explicitly find the closed form of the infinite product, since I don't have any idea to tackle that. Any hints... | The integral
$$g(y)={1\over \pi}\int_0^\infty \cos(xy)\prod_{n=1}^\infty\cos{x\over n}\,dx$$
is the density function of a random variable that I call the Random Harmonic Series.
The value $g(2)$ is particularly interesting as it is almost, but not quite equal, $1/8$.
To fifty decimal places, it is
$$g(2)=.1249999999... | {
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Probability of getting two pair in poker I was looking at this website http://www.cwu.edu/~glasbys/POKER.HTM and I read the explanation for how to calculate the probability of getting a full house. To me, the logic basically looked like you figure out the number of possible ranks and multiply by the number of ways to ... | I find permutation more intuitive to follow for this kind of problems. For people like me:
We have five slots to fill: - - - - - . The first slot can take all 52 cards. The second slot can take only three cards so that they can make a pair. Similarly, the third and fourth slots can take 48 and 3 cards, respectively. Th... | {
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Number of Irreducible Factors of $x^{63} - 1$ I have to find the number of irreducible factors of $x^{63}-1$ over $\mathbb F_2$ using the $2$-cyclotomic cosets modulo $63$.
Is there a way to see how many the cyclotomic cosets are and what is their cardinality which is faster than the direct computation?
Thank you.
| Note that $x^{p^n}-x\in\mathbb{Z}_p[x]$ equals to product of all irreducible factors of degree $d$ such that $d|n$. Suppose $w_p(d)$ is the number of irreducible factors of degree $d$ on $\mathbb{Z}_p$, then we have
$$p^n=\sum_{d|n}dw_p(d)$$
now use Mobius Inversion Formula to obtain
$$w_p(n)=\frac1{n}\sum_{d|n}\mu(\f... | {
"language": "en",
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How to prove that $\lim_{(x,y) \to (0,0)} \frac{x^3y}{x^4+y^2} = 0?$
How to prove that $\lim_{(x,y) \to (0,0)} \dfrac{x^3y}{x^4+y^2} = 0?$
First I tried to contradict by using $y = mx$ , but I found that the limit exists.
Secondly I tried to use polar coordinates, $x = \cos\theta $ and $y = \sin\theta$,
And failed .... | Observe that $x^4 + y^2 \geq |x^2y|$ (for instance, because $x^4+y^2+2x^2y = (x^2+y)^2\geq0$ and $x^4+y^2-2x^2y = (x^2-y)^2 \geq0$). Hence $\displaystyle \left|\frac{x^2y}{x^4+y^2}\right| \leq 1$ when $(x,y)\neq (0,0)$ and thus $$\lim_{(x,y)\rightarrow (0,0)} \left|\frac{x^3y}{x^4+y^2}\right| \leq \lim_{(x,y)\rightar... | {
"language": "en",
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How do I calculate the intersection between two cosine functions?
$f(x) = A_1 \cdot \cos\left(B_1 \cdot (x + C_1)\right) + D_1$
$g(x) = A_2 \cdot \cos\left(B_2 \cdot (x + C_2)\right) + D_2$
Is it possible at all to solve this analytically? I can start doing this but I get stuck half way.
$A_1 \cdot \cos\left(B_1 \cd... | Substituting $\xi:=B_1(x+C_1)$ brings it to the fundamental form
$$ \cos (\xi) = p\cos (a\xi+b)+q$$
with $p=\frac{A_2}{A_1}$, $q=\frac{D_2-D_1}{A_1}$, $a=\frac{A_2}{B_1}$ and $b=\frac{C_2-C_1}{B_1}$. As far as I know, this form cannot be solved analytically in general.
| {
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Why these points are isolated points in this exercise This is an exercise from a calculus book I'm reading:
I can do the exercise but I don't understand the $(\dots)$ in $(c)$. The $x$ are in $\mathbb R$ and $\mathbb R$ does not have isolated points (an isolated point of a set is a point that is not an accumulation po... | Isolated in this context means isolated within the set: that there are not points arbitrarily close. If I define the set $\{\frac 1n: n \in \Bbb N\}$ each point is isolated because for each point $x_n$ I can find an $\epsilon$ so that there are no other points of the set within $(x_n-\epsilon,x_n+\epsilon)$
| {
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Probability of a path of a given length between two vertices of a random graph Suppose that in random graph $G$ on $n$ vertices any $2$ vertices can be connected by an edge with probability $\dfrac{1}{2}$, independently of all other edges. What is the probability $P_n(k)$ that two arbitrary vertices are connected by... | It's an old question, but I was thinking about the same problem today, and here's a solution for a special case.
Let's say that there are $n+2$ total vertices and the edge-probability is $p$. Edge-probability is the probability that there exists a direct edge between two vertices. Let's say we want to find the probabil... | {
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When do parametric equations constitute a line? The given equations specifically are
$x=3t^3 + 7$
$y=2-t^3$
$z=5t^3 + 3$
And
$x=5t^2-1$
$y=2t^2 + 3$
$z=1-t^2$
| For the first case, $r=(7,2,3)+t^3(3,-1,5)$. As $t$ varies through $\mathbb R$, $t^3$ varies through $\mathbb R$, so we have a line.
For the second case, $r=(-1,3,1)+t^2(5,2,-1)$. As $t$ varies through $\mathbb R$, $t^2$ varies through the non-negative reals, so we have a ray.
| {
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Why doesn't this work in Geogebra I've got a really simple equation that I want GeoGebra to plot:
$\sqrt {2x}-\sqrt {3y} =2$
It says it's an illegal operation so I try:
$3y=2x-4\sqrt{2x}+4$
When this doesn't try, I try changing $\sqrt{2x}$ to $(2x)^{1/2}$ and it informs me that exponents can only be integers. Since whe... | The original equation in this thread now plots correctly as well as several others that previously would not plot such as $\ \sin (x+y) = x/y \ $. Even simple ones like $ \ \sin y = x \ $ would not plot a year ago and now work with no problem. Nice work GeoGebra.
| {
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Union of Chain of Ideals I'm writing a project in a "Rings and Modules" course, and I've come across the following proposition, stated without proof:
Proposition 1.2.
In a commutative ring R , the product of ideals is commutative and associative, and distributes sums and unions of chains.
Generally, whenever one of the... | I would interpret it as saying
$I \sum_{\alpha\in \mathcal{A}} J_\alpha = \sum_\alpha IJ_\alpha$
whenever $\{J_\alpha : \alpha \in \mathcal{A}\}$ is a set of ideals,
(product distributes over sums)
and
$I \cdot \bigcup_\alpha J_\alpha = \bigcup_\alpha IJ_\alpha$
whenever $\{J_\alpha : \alpha \in \mathcal{A}\}$ is a ch... | {
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Orbit space of S3/S1 is S2
I'm having trouble finishing this homework assignment. I did the first part by showing that the orbits are invariant: every element from the same $(S^1(z_1, z_2) \in S^3/S^1)$ is mapped to the same point in $\mathbb{R}$.
For the second part I found the following equations. For $(z_1, z_2), ... | To do part $(2),$ note that if you know the values of the first two functions, you know $\Re z_1 z_2$ and $\Im z_1 z_2,$ so you know $z_1 z_2 = C.$ So, you know that $z_2 = C/z_1.$ you can then solve $|z_1|^2 + |z_2|^2 = 1; |z_1|^2 - |z_2|^2 = D$ (where $D$ is what ever $\tilde{h}$ tells you) for $|z_1|$ so, you know $... | {
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Satisfying the inequality of a bounded derivative Hi I am having an issue of proving this inequality.
The problem:
Suppose: $G(x) = |x|^2 + 1$
Show: $\left|\frac{d}{dx}G(x)\right| \leq cG$ (A bounded derivative)
My initial attempt would use the lipschitz (or Gronwalls Lemma?) However, I am unsure how to finish the pr... | First we know that
$$ |\frac{d}{dx} G(x)| = |2x| = 2|x|$$
but when $x \geq 1$ you know that $|x| \leq |x|^2$, moreover this implies that it is always true that $|x| \leq |x|^2 +1$ but we know that $ \frac{1}{c}|\frac{d}{dx} G(x)| = |x|$ ( in our case the constant $1/c =1/2$), so from this we get:
$$ \frac{1}{c} |... | {
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Quaternion identity proof If $q \in \mathbb{H}$ satisfies $qi = iq$, prove that $q \in \mathbb{C}$
This seems kinda of intuitive since quaternions extend the complex numbers. I am thinking that $q=i$ because i know that $ij = k , ji = -k$, which is expand to all combinations of $i,j,k,$ which I think means that I have ... | If you put $\;q=a+bi+cj+dk\;$ , then
$$\begin{align*}qi=ai-b-ck+dj\\
iq=ai-b+ck-dj\end{align*}$$
Well, what do you deduce about the coefficients $\;a,b,c,d\in\Bbb R\;$ above?
| {
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Why isn't $\int\sin(ix)~dx$ equal to $i\cos(ix)+C$ ? I was playing around with imaginary numbers, and I tried to solve $$\int\sin(ix)~dx$$ and ended up getting $$i\cos(ix)+C$$
But apparently the answer is $$i\cosh(x)+C$$
I was just wondering, is this correct? And what does the "$h$" stand for/mean? Where did it even c... | Here are two useful definitions / relations
$$\cosh(x) = \frac{e^x + e^{-x}}{2}$$
$$\cos(x) = \frac{e^{ix} + e^{-ix}}{2}$$
Using these definitions you can see that
$$\cos(ix) = \frac{e^{i(ix)} + e^{-i(ix)}}{2} = \frac{e^{-x} + e^x}{2} = \cosh (x)$$
So you did get the same answer, but you just had it in a different form... | {
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Prove that the Lebesgue measure of a particular set is zero. I am doing revision and got extremely stuck with the following exercise, which appeared in an exam from the previous year.
Consider the measure space $(\mathbb R, \mathbb B, \lambda)$ where $\mathbb B$ is the Borel sigma algebra and $\lambda$ the Lebesgue mea... | For any finite $K > 0$, consider the set
$$
V(K) := \left\{x \in \mathbb R: (\lvert x\rvert < K)\land \left( \exists \text{ infinitely many } p \in \mathbb Z, \, q \in \mathbb N \text{ where } \left|x - \frac{p}{q}\right| \le \frac{1}{q^{2+\delta}}\right)\right\}.
$$
Proving that each $V(K)$ has measure $0$ suffices, ... | {
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The sum of areas of 2 squares is 400and the difference between their perimeters is 16cm. Find the sides of both squares. The sum of areas of 2 squares is 400and the difference between their perimeters is 16cm. Find the sides of both squares.
I HAVE TRIED IT AS BELOW BUT ANSWER IS NOT CORRECT.......CHECK - HELP!
Let sid... | Well, your mistake came about in the step $$\sqrt{400-x^2}=20-x,\tag{$\star$}$$ which isn't true in general. But why can't we draw this conclusion? Observe that if we let $y=-x,$ then $y^2=x^2,$ so $400-y^2=400-x^2.$ But then we can use the same (erroneous) reasoning to conclude that $$20-y \overset{(\star)}{=} \sqrt{4... | {
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Homomorphism of modules and Tensor Product. Let $\phi: A \rightarrow B$ be a ring homomorphism. Let $M$ be an $A$-module. We can think $B$ as $A$-module via the map $\phi$ defined by $\phi:A\times B \rightarrow B$, $(a,b)\mapsto\phi(a)\cdot b$.
So we can construct the $A$-module $M \otimes_A B$. Furthermore, $M \otime... | Hint: Given an $A$-linear map $f : M \to N$, define $B \times M \to N$ by $(b,m) \mapsto b \, f(m)$. Check that this is $A$-bilinear, hence lifts to an $A$-linear map $h : B \otimes_A M \to N$ characterized by $h(b \otimes m)= b \, f(m)$. Check that it is actually $B$-linear. Conversely, given a $B$-linear map $h : B ... | {
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Give an example of Euclidean space. In the question it is asking what will be if we take out one condition of theorem.
$\textbf{Theorem}$: Let {$\phi_{n}$} be orthonormal system in a complete Euclidean Space R. Then {$\phi_{n}$} is complete if and only if R contains no nonzero element orthogonal to all elements of { ... | Take separable Hilbert space $H$ with basis $e_1, ..., e_n, ...$ Now take the subspace generated (algebraically) by $e_2,e_3, ..., e_n, ...$ and the vector $e_1 + 1/2 e_2 + ... + 1/n e_n + ...$. This is a Euclidean space. The system $e_2, ... , e_n, ...$ is maximal orthonormal, but it is not complete, because the subs... | {
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Locally path-connected implies that the components are open If $X$ is a locally path-connected space, then its connected components are open.
I am trying to prove this, but for some reason it doesn't seem right to me, knowing that components are always closed. If the statement is true, wouldn't it be the case the comp... | Hints:
1) If $X$ is locally path-connected, then path components of $X$ are open
2) If $X$ is locally path-connected, then path components and connected components coincide
| {
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Solving a differential equation with natural log I am given:
$x\dfrac{dy}{dx}=\dfrac{1}{y^3}$
After separating and integrating, I have:
$y^4/4=\ln x+C$
I am supposed to solve this equation, but I'm stuck here. Should I solve explicitly so I can keep $C$?
EDIT:
A solution I came up with last night was:
$y=(4\ln x+C)^{1... | Try differentiating to see if you got the correct solution!
You can compute
$$
\frac{dy}{dx} = \frac{1}{4}\left(4 \ln x + C\right)^{-3/4}\left(\frac{4}{x}\right) = \frac{1}{x}\left(4 \ln x + C\right)^{-3/4}
$$
so
$$
x\frac{dy}{dx} = \left(4 \ln x + C\right)^{-3/4}.
$$
Is this equal to $\dfrac{1}{y^{3}}$?
| {
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Infinite direct product of rings free. Let $A$ be a commutative ring (viewed as an $A$-module over itself) that is not a field. Are there some conditions that guarantee that $\prod_{k=0}^\infty A$ is free? What if $A=\mathbf{Z}$ or more generally any pid?
| Well, if $A$ is a field, then $\prod_\mathbb{N} A$ is certainly free. I claim that the direct product $\prod_\mathbb{N} \mathbb{Z}_4$ is also free, where $\mathbb{Z}_4 = \mathbb{Z}/4\mathbb{Z}$.
To prove this let
$\{c_i\}_{i\in\mathcal{I}}$ be a basis for $\prod_\mathbb{N}\mathbb{Z}_2$, where $\mathcal{I}$ is some ind... | {
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Pre-cal trigonometric equation problem am i correct to factor out a 2 first?
$$2\sin^2 2x =1$$
$$2(\sin^2 x-1) =1$$
$$2\cos^2 x =1$$
$$\cos^2 x ={1\over2}$$
$$\cos x =\pm{\sqrt{2}\over 2 }$$
i'm only looking for solutions from $$0≤ x ≤ 2\pi $$
$$x = {\frac{\pi}{4}},{\frac{7\pi}{4}},{\frac{3\pi}{4}},{\frac{5\pi}{4}}$$
t... | Hint: let $y = \sin 2x$ and solve for $y$ first using algebra. Then figure out what value(s) of $x$ would make the equation true.
| {
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Equality case of triangle inequality with functions If $f, g: \mathbb{R} \rightarrow \mathbb{R}$ such that $\left|f(x) + g(x)\right| = \left|f(x)\right| + \left|g(x)\right|$ for all $x \in \mathbb{R}$, then must $f = cg$ for some $c \gt 0$?
| No. Consider $f(x)=x^2$, $g(x)=x^4$. $f+g$ is always positive, and so are $f$ and $g$. So the equality holds, but your condition is false.
| {
"language": "en",
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Check this is a hilbert norm: $ \ell^2 $ with norm $\| \cdot \| := \| \cdot \|_{\ell^2} + \| \cdot \|_{\ell^p}$ Clearly $ p \geq 2 $ so it gains sense calculating the $\ell^p $-norm.
According to my calculation this norm is equivalent to the $\ell^2 $ norm, in fact given a cauchy sequence w.r.t $\| \cdot \| $ it his a ... | In order for a norm, of a normed space to come from an inner product, it has to satisfy the parallelogram identity:
$$
\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2.
$$
Your norm does not satisfy such an identity, i.e.,
$$
x=e_1,\,\,y=e_2,\quad \|x\|=\|x\|_2+\|x\|_p=2=\|y\|,\,\,
$$
$$
\|x-y\|=\|x+y\|=\sqrt{2}+\sqrt[p]{2}
$$
T... | {
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Please explain this simple rule of logarithms to me Right I know this one is simple and I know that I just need a push to make it sink in in my head..
I am studying control systems and in one of the tutorial examples the tutor says
Show that
$$20\log(1/x) = -20\log(x)$$
I know that when you have a divide or a multiply... | You have : $0 = 20\times\log(1) = 20\times\log(x\times\frac1{x})$
But $\log(x\times \frac1{x})=\log(x)+\log(\frac1{x})$.
So $0=20(\log(x)+\log(\frac1{x}))$ and you have your result.
| {
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Condition for $N! > A^N$ I am given $A$ , I need to find minimum value of $N$ such that the condition $N! > A^N$ holds.
EXAMPLE : If $A=2$ then minimum $N=4$ and similarly if $A=3$ then minimum $N=7$.
How to solve this problem?
| Solve
$$({\frac{n}{e})}^n\sqrt{2\pi n}-A^n = 0$$
with numerical methods, for example the bisection method.
The approximation of n! is good enough even to solve the A=2-case.
The uprounded result is the desired number.
Since
$$n^n > n!$$
for all $n>1$, the desired number must be >A .
So, by induction you can easily prov... | {
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How to determine if $2+x+y$ is a factor of $4-(x+y)^2$? I know it is a factor but how could have I determined that it was? Feel free to link whatever concept is needed than solve it. Studying for clep and it's one of the practice problems. When I expand it I get nonsense.
| The solution's already in the other answers, but in many cases, as in this one, you can try some substitution:
$$t:=x+y\implies\;\text{is}\;\;2+t\;\;\text{a factor of}\;\;4-t^2\;?$$
and now all depends on you remembering the high school algebra's slick formula, namely difference of squares:
$$4-t^2=(2^2)-t^2=(2-t)(2+t)... | {
"language": "en",
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Unique combination of sets We start with a finite number of $N$ sets, $\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N$, each containing a finite number of integers. The sets do not in general have the same number of elements. The goal is to find all possible unique combination that you can get by taking the union of some of ... | I would generate the partitions of $N$. For $4$, they are $4, 3+1, 2+2, 2+1+1, 1+1+1+1$, then assign that many sets to each partition. The $4$ corresponds to your combination $15$. For $3+1$ there are four ways to choose the $3$, giving your combinations $11,12,13,14$. When you have multiple partitions of the same ... | {
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Consequences of Schur's Lemma Schur's Lemma states that given two irreducible representations $(\rho,V)$ and $(\pi,W)$ of a finite group $G$ (V and W are linear vector spaces over the same field), and a homomorphism $\phi\colon V\to W$ such that $\phi(\rho(g)\mathbf{v}) = \pi(\phi(\mathbf{v}))$ for all $g$ in $G$ and a... | It doesn't imply anything about $V$ and $W$, just that $\phi$ is the zero map; there is always a zero map between any two representations.
The point of Schur's Lemma is that any non-zero map between irreducible representations is an isomorphism. So if you have a non-zero map $V\to W$, then it is an isomorphism, and $V$... | {
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Again, improper integrals involving $\ln(1+x^2)$ How can I check for which values of $\alpha $ this integral $$\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}\,dx $$ converges?
I managed to do this in $0$ because I know $\ln(1+x)\sim x$ near 0.
I have no idea how to do this in $\infty$ where nothing is known about $\ln... | HINT
I suggest you develop $\log(1+x^2)$ as an infinite series (Taylor); divide each term by $x^a$, compute the anti-derivative and look where and when arrive the problems when you compute the integral between zero and infinity.
| {
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Show that $f(x)=0$ for all $x\geq0$ I have been struggling with this problem..
Q. Let $f(x)$, $x\geq 0$, be a non-negative continuous function, and let $F(x)=\int_0^x f(t) dt$, $x\geq0$. If for some $c>0$, $f(x)\leq cF(x)$ for all $x\geq 0$, then show that $f(x)=0$ for all $x\geq0$ .
I have tried everything in my ab... | Let $$\phi(t) = e^{-ct} F(t)$$ Then $\phi(0) = 0$, and $\phi(t) \ge 0$ for all $ t \ge 0$.
Furthermore, $$\phi'(t) = e^{-ct}(F'(t) - c F(t)) \le 0$$hence $\phi(t) = \int_0^t \phi'(\tau) d \tau \le 0$, and so $\phi(t) =0 $ for all $t \ge 0$.
If $ϕ(t)=0$ for all $t≥0$ , then $F(t)=0$ for all $t≥0$ . Since $f$ is conti... | {
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Prove that $4^{2n+1}+3^{n+2} : \forall n\in\mathbb{N}$ is a multiple of $13$ How to prove that $\forall n\in\mathbb{N},\exists k\in\mathbb{Z}:4^{2n+1}+3^{n+2}=13\cdot k$
I've tried to do it by induction. For $n=0$ it's trivial.
Now for the general case, I decided to throw the idea of working with $13\cdot k$ and try to... | Since you can use congruence arithmetic, you can exploit it to the hilt to prove it more simply
$\qquad\qquad\qquad \begin{eqnarray}{\rm mod}\ 13\!:\,\ 4^{2n+1}\!+3^{n+2} &=\,& 4\cdot \color{#0a0}{16}^n +\, \color{#c00}9\cdot 3^n \\ &\equiv\,& 4\,\cdot\, \color{#0a0}3^n\, \color{#c00}{-\,4} \cdot 3^n\equiv 0\quad {\bf ... | {
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Position and nature of singularities of an algebraic function (Ahlfors) I want to solve the following exercise, from Ahlfors' Complex Analysis text, page 306:
Determine the position and nature of the singularities of the algebraic function defined by $w^3-3wz+2z^3=0.$
Here is my solution attempt. I would appreciate y... | Regarding 1, yes.
Regarding 2, we can show the following.
*
*$w$ cannot be bounded as $z$ tends to infinity, by dividing both sides of $w^3-4wz+3z^3=0$ by $z^3$.
*$\frac{w}{z}$ must be bounded as $z$ tends to infinity, by dividing both sides of $w^3-4wz+3z^3=0$ by $w^3$.
Therefore $z=\infty$ is an algebraic pole of... | {
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Integration Techniques - Adding [arbitrary] values to the numerator. Suppose you wanted to evaluate the following integral.
Where did the 4 come from? I understand that it makes the solution but how would you make an educated guess to put a 4? And how in the future would I solve similar questions?
| We want to have a fraction with the form
$$\frac{f'}f$$
so since the derivative of the denominator $x^2+4x+13$ is $2x+4$ so we write the numerator
$$x-2=\frac 1 2 (2x-4)=\frac 1 2 (2x+4)-4$$
| {
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Comparison of notation for sets Different authors use different notation, no question here...but doesn't this make the study of maths a little more difficult, always chasing different definitions of how a set is represented? I ask this because I've studied linear algebra from two different sources; Axler and Cooperste... | I'm a second year graduate student and I used to wonder the same thing. There are commonly used symbols, but by no means is there agreement on what symbol to use in every case. I think it mainly comes down to style and the fact there are only a finite number of symbols out there to represent the ideas you want to get... | {
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Room for computational geometry in advanced algorithms course I am currently putting together an independent study in advanced algorithms and because of my interest in (computational) geometry, wanted to include as many interesting algorithms from this field as possible. Does anyone have any suggestions for material t... | Given a bunch of ink blobs on a plane. Find the shortest path between any two given points out of the ink and not touching the ink.
| {
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"timestamp": "2023-03-29T00:00:00",
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There is a positive integer $y$ such that for a polynomial with integer coefficients we have $f(y)$ as composite Show that if $f(x)=a_nx^n+\cdots a_1x+a_0$ with $a_i \in \mathbb{Z}$, then there is a positive integer $y$ such that $f(y)$ is composite.
To prove this, we suppose that $f(x)=p$. Then for $f(x+kp)$ we have ... | Look at $(x+kp)^n$
$$
(x+kp)^n = x^n + {n \choose 1} x^{n-1} k p + {n \choose 2} x^{n-2} k^2 p^2 + \cdots \\ =
x^n + \alpha_n p \text{ where $\alpha_n$ is an integer} $$
Similarly for other powers. This gives
$$ f(x+kp) = a_n (x^n + \alpha_n p) + a_{n-1} (x^{n-1} + \alpha_{n-1} p) \cdot \\=
a_n x^n + a_{n-1} x^{n-1} +... | {
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Integers divide several solutions to Greatest Common Divisor equation I'm not sure about the topic's correctness but my problem is following:
Suppose $u_1,v_1$ and $u_2,v_2$ are two different solutions for $au_i + bv_i = 1$, then $a \mid v_2-v_1$ and $b\mid u_1-u_2$.
Well, I have tried to prove this without success, bu... | If $u_1,v_1$ and $u_2,v_2$ are solutions to $au_i+bv_i=1$, then $$0=au_1+bv_1-au_2-bv_2=a(u_1-u_2)+b(v_1-v_2).$$
Thus $a \mid b(v_1-v_2)$. Now use $(a,b)=1$ to conclude $a \mid (v_1-v_2)$.
| {
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How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$. How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$.
Note. $x$,$y$ and $z$ are integer values and $z$ is $4096$ bits integer or even more.
For all $z>1$.
And if equation be $z$=$2^x$.$3^y$ then what is your answer?
| If $z = 2^x \ 3^y$, then define $a_k = {\log_2 (z) \over{2^k}}$ rounded to the nearest integer.
Let $k = 2$ and $\beta = a_1$
If $ z \equiv_{2^\beta} 0$, $\beta \to \beta + a_k$
Else, $\beta \to \beta - a_k$
Regardless, $k \to k+1$
Repeat until $a_k = 0$, then $x \in \{\beta-1, \beta, \beta+1 \}$ (simple to check which... | {
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(Highschool Pre-calculus) Solving quadratic via completing the square I'm trying to solve the following equation by completing the square:
$x^2 - 6x = 16$
The correct answer is -6,1. This is my attempt:
$x^2 - 6x = 16$
$(x - 3)^2 = 16$
$(x - 3)^2 = 25$
$\sqrt(x -3)^2 = \sqrt(25)$
$x - 3 = \pm5$
$x =\pm5 - 3$
$x = -8,... | $x^2 - 6x$ isn't the same as $(x-3)^2$, so your equation $(x-3)^2 = 16$ is wrong.
$(x-3)^2$ is actually $x^2 - 6x + 9$, so you should write
$$x^2 - 6x + 9 = 25$$
and then
$$(x-3)^2 = 25$$.
So your third equation is correct, even though your second wasn't.
I don't agree with your equation $\sqrt{(x-3)^2} = \sqrt{25}$... | {
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Inequality proof involving series I have a question which starts like this:
Show that for $n>m$ we have $S_m < S_n\leq S_m+\frac{1}{m}\frac{1}{m!}$
Where $S_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+ ... +\frac{1}{n!}$
I have tried using induction on n but that doesn't work for me, can somebody just point me in the rig... | Hint.
First note that the required bound is independent of $n$. So you won't need to use $n$ in an essential way. In the worst case, $n$ will be very large, so you might as well write down the infinite sum $S = \frac{1}{(m+1)!} + \frac{1}{(m+2)!} + \cdots$. You want to prove that we have $S \leq \frac{1}{m} \frac{1}{m!... | {
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Most efficient way to solve this combination problem I have 4 very large list (l1, l2, l3, l4) of components (approx 9 million items in each list). Each list item has a cost and a value.
I want to know how I can achieve the maximum combined value for an agreed combined total cost:
So, over all combinations in the 4 li... | This is not the knapsack problem... But going through every possibility would take $O(n^4)$ time. This can be reduced to $O(n^3\log(n))$ by sorting the last list and going through every possible triple from the first 3 lists and finding by binary search the best object to take from the last list.
In practice you can pr... | {
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Does dividing by zero ever make sense? Good afternoon,
The square root of $-1$, AKA $i$, seemed a crazy number allowing contradictions as $1=-1$ by the usual rules of the real numbers. However, it proved to be useful and non-self-contradicting, after removing identities like $\sqrt{ab}=\sqrt a\sqrt b$.
Could dividing ... | If $\,0\,$ has an inverse then $\ \color{#c00}1 = 0\cdot 0^{-1} = \color{#c00}0\,\Rightarrow\ a = a\cdot \color{#c00}1 = a\cdot \color{#c00}0 = 0\,$ so every element $= 0,\,$ i.e. the ring is the trivial one element ring.
So you need to drop some ring axiom(s) if you wish to divide by zero with nontrivial consequences... | {
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Is the set of all invertible $n \times n$ matrices a vector space? I'm studying Algebra and I'm asked to prove or disprove "Is the set of all invertible $n \times n$ matrices a vector space?" I assume with respect to the usual matrix-sum and scalar multiplication. I found that is true, but I'm not sure how to prove it... | The set of all invertible $n\times n$ matrices of real numbers is NOT a vector space.
Let for example $I$, the unit matrix is invertible and so is $-I$. But their sum $I+(-I)=0$ is definitely not invertible!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $f(p(X))=p(X)-p'(X)$, where p is a polynomial, is bijective Let $V_d$ be the vector space of all polynomials with real coefficients of degree less than or equal to $d$. The linear map $f: V_3 \rightarrow V_3$ is given by: $f(p(X)) := p(X)-p'(X)$. Show that $f$ is bijective.
To show that $f$ is injective I co... | Your linear map $f:V_3\to V_3$ is equal to $f=1-L$ with $L:p\in V_3\mapsto p'\in V_3$.
Notice that $L$ is nilpotent, that is, that some power of $L$ is zero.
It follows by a calculation then that $g=1+L+L^2+\cdots+L^k$, for any $k$ sufficiently large so that $L^{k+1}=0$, is an inverse map to $g$: indeed, you can simpl... | {
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Field containing sum of square roots also contains individual square roots
Let $F$ be a field of characteristic $\neq 2$. Let $a\neq b $ be in $F$. Suppose $\sqrt{a}+\sqrt{b}\in F$. Prove that $\sqrt{a}\in F$.
We have $(\sqrt{a}+\sqrt{b})^2=a+b+2\sqrt{ab}\in F$, so $\sqrt{ab}\in F$. Then $\sqrt{ab}(\sqrt{a}+\sqrt{b})... | Hint $\ $ If a field $F$ has two $F$-linear independent combinations of $\ \sqrt{a},\ \sqrt{b}\ $ then
you can solve for $\ \sqrt{a},\ \sqrt{b}\ $ in $F.\,$ For example, the Primitive Element Theorem
works that way, obtaining two such independent combinations by
Pigeonholing the infinite set $\ F(\sqrt{a} + r\ \sqrt{b}... | {
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Why is that an automorphism that preserves $B$ and $H$ an automorphism of $\Phi$ that leaves $\Delta$ invariant? Let $L$ be a semisimple finite dimensional Lie algebra, $H$ its CSA and $\Phi$ its root system with base $\Delta$ and $B = B(\Delta) = H\bigoplus_{\alpha \succ 0}L_\alpha$. If we have an automorphism of $L$ ... | $\Phi$ is determined by the pair $(L,H)$, so if the automorphism preserves both $L$ and $H$ it must preserve $\Phi$.
Again, $\Delta$ is determined by the set of all positive roots. But from borel you can reconstruct positive roots (they are eigenvalues of $ad$ on $B$).
| {
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Galois group command for Magma online calculator? I need to test if a family of 7th deg and 13 deg equations are solvable. I'm new to Magma, so my apologies, but what would I type in,
http://magma.maths.usyd.edu.au/calc/
to determine the Galois group of $x^5+5x-12=0$ (for example)?
|
> P< x >:=PolynomialAlgebra(Rationals());
> f:=x^5+5*x-12;
> G:=GaloisGroup(f);
> print G;
Symmetric group G acting on a set of cardinality 5
Order = 120 = 2^3 * 3 * 5
Although the permutation group on [1..Degree($f$)] is permutationally isomorphic to the Galois group, the bijection with the set of roots of your sepa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/642329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
Is there an entire function which is not a polynomial such that $\lim_{z\to\infty}\frac{f\left(z\right)}{z}=\infty $ I'm wondering if there's an entire function $f$ which is not a polynomial such that $\lim_{z\to\infty}\frac{f\left(z\right)}{z}=\infty$?
Thanks in advance!
| Your hypotheses imply that $f(z)/z$ is everywhere meromorphic on the Riemann sphere, and is thus a rational function.
The only affine singularity is at zero, so the denominator of $f(z) / z$ must be a power of $z$.
But because $f(z)$ is entire, the denominator of $f(z) / z$ cannot have more than one factor of $z$, and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/642433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $n,k\in\mathbb N$, solve $2^8+2^{11}+2^n=k^2$. If $n,k\in\mathbb N$, solve $$2^8+2^{11}+2^n=k^2$$
It's hard for me to find an idea. Some help would be great. Thanks.
| HINT:
We have $\displaystyle2^8+2^{11}=2304$
Now, $2^8+2^{11}+2^0=2304+1\ne k^2$
$\displaystyle 2^8+2^{11}+2^1=2304+2\equiv2\pmod8$, but $a^2\equiv0,1,4\pmod8$
So, $n\ge2$ let $n=m+2$ where $m\ge0$
$\displaystyle 2^8+2^{11}+2^{m+2}=4(576+2^m)\implies 576+2^m$ must be perfect square
Like either method $m\ge2$ let $m=r+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/642564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Calculate the pseudo inverse of the matrix The subject is to calculate the pseudo inverse if matrix $\begin{equation*}
\mathbf{A} = \left(
\begin{array}{ccc}
1 & 0 \\
2 & 1 \\
0 & 1 \\
\end{array}
\right)
\end{equation*}$
My answer is as follows: (SVD decomposition)
First, $\begin{equat... | Recall, for $\mathbf{\Sigma}$ we take the square roots of the non-zero eigenvalues and populate the diagonal with them, putting the largest in $\mathbf{\Sigma}_{11}$, the next largest in $\mathbf{\Sigma}_{22}$ and so on until the smallest value
ends up in $\mathbf{\Sigma}_{mm}$.
$$\begin{equation*}
\mathbf{\Sigma} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/642647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Solve $x^3 = 27\pmod {41}$ I don't know how to approach this problem. Can anyone give me a hint? If it matters, the first part of the question was to find the order of $5$ in the field $\mathbb Z_{41}$ (the field mod $41$), which I did, but I'm not sure how it relates to the second part.
Thanks for your help.
| We shall use the lemma that if $a^r\equiv a^s\pmod{n}$, then $r\equiv s\pmod{\operatorname{ord}_n(a)}$. Note that $6$ is a primitive root $\pmod{41}$, and that $6^5\equiv 27\pmod{41}$. We can write $x^3 = 6^{3k}$ for some $k$, so that $$6^{3k}\equiv 6^5\pmod{41}\implies 3k\equiv 5\pmod{\varphi(41)=40}.$$ It is straight... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/642746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Question regarding specific limits I was going through some past final exams for my "Analysis 1" class and I came across the following problem, which I've been so far unable to solve.
Let $f''$ be continuous in $(-1,1)$; with $f(0)=0$, $f'(0)=3$ and $f''(0)=5$.
$$
\lim_{h\rightarrow 0}\frac{f(h)+f(-h)}{h^{2}}
\\
\lim_{... | For the second limit, note that $f$ is continuous, and hence so is $|f|$. So, there exists $m,M$ such that $m \leq |f| \leq M$. So, by using these bounds, we can show that the second limit is $0$. The first limit does not exist, as $lim_{h \rightarrow 0}\frac{1}{h}$ does not exist. (for $h \rightarrow 0+$the limit is $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/642844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Binary concatenation In decimal, I have the numbers 4 and 5 for example. I want to concatenate them into the number 45, but then in binary. In decimal, it's just a matter of sticking the numbers together, but I need to do that in binary. I'm not using a programming language, I'm using logical electronic circuitry. But ... | $(45)_{10}$ (i.e., in decimal) is just
$$(45)_{10}=5_{10}((4)_{10}+(5)_{10})$$
$(4)_{10}+(5)_{10}$ is in binary, $(100)_{2}+(101)_2=(1001)_2$. Multiply this by $(101)_2$ to get
$$(1001)_2\times (101)_2=(1001)_2+(100100)_2$$
This on addition of course gives $(101101)_2=(45)_{10}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/643002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do I deal with sines or cosines greater than $1$? When I'm solving trigonometric equations, I occasionally end up with a sine or cosine that's greater than $1$ -- and not on the unit circle. For example, today I had one that was $3 \tan 3x = \sqrt{3}$, which simplified to $\tan 3x = \frac{\sqrt{3}}{3}$, which simpl... | Hint: If you are solving $\tan x = a/b$, you have to realize $a/b$ can be written as $(a/c)/(b/c)$ for any nonzero $c$. You have to pick a value of $c$ before you can try to do that. Look at $c= \pm \sqrt{a^2+b^2}$ and see if either of those values gets you further.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/643156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Powers of 10 in binary expansion I noticed an interesting pattern the other day. Let's take a look at the powers of 10 in binary:
*
*$10^0$ = 1 = 1 b
*$10^1$ = 10 = 10 10 b
*$10^2$ = 100 = 1100 100 b
*$10^3$ = 1000 = 111110 1000 b
Basically, it seems that $10^n$ for any non-negative integer $n$ written out in b... | Multiplication of a number in binary by $2^{n}$ adds $n$ zeroes to the expression. $10^{n}=2^{n}5^{n}$, so as $2^{n}$ divides $10^{n}$, when expressed in binary you are adding $n$ zeroes to the number $5^{n}$ expressed in binary form. This is directly analogous to the base $10$ case.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/643244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve Bessel equation with order $\frac{1}{2}$ using Frobenius method Consider the Bessel equation of index $n= \frac{1}{2}$
$x^2y''(x) +xy'(x)+(x^2-\frac{1}{4})y(x) = 0$ $, x>0$
$(i)$ Show that $y(x) = u(x)x^{\frac{-1}{2}}$ solves the equation above if and only if $u$ satiesfies a familiar differential eq... | For (i), what do you get when you substitute $y(x) = u(x) x^{-1/2}$ in the differential equation?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/643311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is the probability that no letter is in its proper envelope? Five letters are addressed to five different persons and the corresponding
envelopes are prepared. The letters are put into the envelopes at random.
What is the probability that no letter is in its proper envelope?
| Let $1,2,3,4,5$ be the letters and let $A,B,C,D,E$ be the proper envelope resepctively.
1) In case of five proper envelopes, only $1$ pattern.
2) In case of only four proper envelopes, the last one is in the proper envelope, so $0$ pattern.
3) In case of only three proper envelopes, you have $\binom{5}{3}\times 1=10$ p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/643434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
} |
If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$ Assume that the matrices $A,\: B\in \mathbb{R}^{n\times n}$ satisfy
$$
A^k=0,\,\, \text{for some $\,k\in \mathbb{Z^+}$}\quad\text{and}\quad
AB=BA.
$$
Prove that $$\det(A+B)=\det B.$$
| Here's an alternative to the (admittedly excellent) answer by Andreas:
Since $A$ and $B$ commute, they are simulaneously triangularizable. However, since $A$ is nilpotent, for any basis in which $A$ is triangular, the diagonal entries of $A$ will in fact be $0$. Thus in the chosen basis, the diagonal entries of $A + B$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/643531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 3,
"answer_id": 0
} |
Looking for an elementary solution of this limit I was collecting some exercises for my students, and I found this one in a book: compute, if it exists, the limit
$$
\lim_{x \to +\infty} \int_x^{2x} \sin \left( \frac{1}{t} \right) \, dt.
$$
It seems to me that this limit exists by monotonicity. Moreover, since $\frac{2... | $$
\int_x^{2x}\sin\left(\frac{1}{t} \right)\,dt=\int_{\frac{1}{2x}}^{\frac{1}{x}}\frac{\sin(u)}{u^2}\,du,
$$
$$
\sin(u)=u+o(1)u,
$$
if $u$ is around $0$,
(here we use the elemetary $\lim_{u\to0}\frac{\sin(u)}{u}=1$)
so
$$
\int_{\frac{1}{2x}}^{\frac{1}{x}}\frac{\sin(u)}{u^2}\,du=\int_{\frac{1}{2x}}^{\frac{1}{x}}\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/643612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 2
} |
Non-homotopy equivalent spaces with isomorphic fundamental groups I know that if two spaces $X,Y$ have the same fundamental group : $\pi_1(X,a) \cong \pi_1(Y,b) $
does not imply that they are homotopy equivalent.
But I can't find an example.
I was thinking about the Moebius strip and the circle, as both of them have th... | The standard way to prove two spaces are not homotopy equivalent is to find some homotopy invariant that distinguishes them. Since they are going to have the same fundamental group, the obvious candidates are homology groups and higher homotopy groups (either of which will tell you a sphere is not homotopy equivalent t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/643692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Function approximating this product Is there any function approximating, for large values of $p$, the quotient between the product of all primes and the product of all primes $-1$?
Basically: $2/1 \cdot 3/2 \cdot 5/4 \cdot 7/6 \cdot 11/10 \cdots $
Can that be approximated, for large values of $p$, with some known funct... | To find the behavior of
$$
\prod_{p \leq x} \frac{p}{p-1} = \prod_{p \leq x} \left(1 - \frac{1}{p}\right)^{-1}
$$
We begin by taking its logarithm, which we then rewrite as
$$
\log \prod_{p \leq x} \left(1 - \frac{1}{p}\right)^{-1} = - \sum_{p \leq x} \log\left(1 - \frac{1}{p}\right).
$$
Now $\log(1-1/p) \sim -1/p$ for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/643783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find all even natural numbers which can be written as a sum of two odd composite numbers. Find all even natural numbers which can be written as a sum of two odd composite numbers.
Please help in in solving the above problem.
| Let $n\geq 100$ an even number. Consider the quantities $n-91$, $n-93$ and $n-95$, one of these is a multiple of 3, and not exactly 3 cause $100-95>3$, then is a composite odd number.
Observing thet 91,93 and 95 are composite, you conclude that every $n\geq 100$ works. Now check directly the remaining numbers, and you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/643847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Showing that $\int \frac{ \sinh (az)}{\sinh (\pi z)} \, e^{ibz} \, dz $ vanishes along three sides of a rectangle in the upper half-plane One of several ways to evaluate $$\int_{0}^{\infty} \frac{\sinh (ax)}{\sinh (\pi x)} \, \cos (bx) \, dx \, , \quad \, |a|< \pi,$$ is to sum the residues of $$ f(z) = \frac{\sinh (... | So that my question doesn't remain unanswered, I'm going to post an answer using the hints that Marko Riedel provided in the comments.
On the right side of the rectangle (and similiarly on the left side of the rectangle), we have
\begin{align} \left| \int_{0}^{N + \frac{1}{2}} \frac{\sinh\big(a(N+it)\big)}{\sinh\big... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/643944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
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