Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to calculate modulo of high power of 2 I know there are other such topics but I really can't figure how to calculate the following equation: 2^731 mod 645.
Obviously I can't use the little theorem of Fermat since 645 is not a prime number and I can't also do the step by step rising of powers(multiplying by 2) since... | $645 = 15\cdot 43\,$ so we can compute $\,2^{\large 731}\!$ mod $15$ and $43,\,$ then combine them (by CRT or lcm).
${\rm mod}\ 15\!:\,\ 2^{\large\color{#c00} 4}\equiv 1\,\Rightarrow\, 2^{\large{731}}\equiv 2^{\large 3}\,$ by $\,731\equiv 3\pmod{\!\color{#c00}4}$
${\rm mod}\ 43\!:\,\ 2^{\large 7}\equiv -1\,\Rightarrow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2116939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find all prime solutions of equation $5x^2-7x+1=y^2.$ Find all prime solutions of the equation $5x^2-7x+1=y^2.$
It is easy to see that
$y^2+2x^2=1 \mod 7.$ Since $\mod 7$-residues are $1,2,4$ it follows that $y^2=4 \mod 7$, $x^2=2 \mod 7$ or $y=2,5 \mod 7$ and $x=3,4 \mod 7.$
In the same way from $y^2+2x=1 \mod 5$ w... | Try working mod $3$ and mod $8$. Assuming $x, y>3$, we have $x,y = \pm 1$ mod $3$. Since $x, y$ are odd we have $x^2, y^2=1$ mod $8$, so
$$x^2, y^2 = 1 \text{ mod } 24.$$
Substituting in the equation gives $$x = 24k+11 $$ for some integer $k$.
Rearranging the original equation we get
$$x(5x-7)=(y-1)(y+1), \tag{1}$$
the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2117054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Squaring a Pareto random variable: Is there a proper name for this transformation? Let $Y= X^2$ be a function of a random variable where $X$ has a Pareto Type II (Lomax) distribution with parameters $\alpha = 8$, $\theta = 4000$, $x > 0$. Find the distribution of $Y$.
I basically went through the ropes as typical for t... | The Pareto II (Lomax) CDF is given by $$F_X(x) = 1 - \left(1 + \frac{x}{\sigma}\right)^{-\alpha}, \quad x > 0.$$ Your parametrization is slightly different, but the calculations are comparable.
The Pareto IV CDF is given by $$F_Y(y) = 1 - \left(1 + \left(\frac{y-\mu}{\sigma}\right)^{1/\gamma} \right)^{-\alpha},$$ so i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2117160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$m_a(x) = x^n + b_1x^{n−1} + \cdots + b_n$ be the minimal polynomial of $a$ over $E$. Prove that $E = K(b_1, \ldots , b_n)$ Let $L = K(a)$ be an algebraic extension. Let $E \subset L$ be a sub-field containing $K$. Let $m_a(x) = x^n + b_1x^{n−1} + \cdots + b_n$ be the minimal polynomial of $a$ over $E$. Prove that $E =... | Let $f(x)$ be the minimal polynomial of $a$ over $K(b_1,\ldots,b_n)$. Since $f(x)\in E[x]$ vanishes at $a$, we have that $m_a(x)$ divides $f(x)$. On the other hand, we also have that $m_a(x) \in K(b_1,\ldots,b_n)[x]$ vanishes
at $a$. Thus $f(x)$ divides $m_a(x)$, and then $f(x)=m_a(x)$.
Therefore $$[L: K(b_1,\ldo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2117278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Knock out tournament 1 8n players $P_1$, $P_2$, $P_3$, .....$P{_8}{_n}$ play a knock out tournament. It is known that all players are of equal strength. The tournament is held in three rounds where the players are paired at random in each round. If it is given that $P_1$ wins in the third round then what is the conditi... | First Round = 8n men. Second Round or Semi-Final = 4n men. 3rd Round or Final = 2n men. The tournament Winner is $P_{_{1}}$.The Probability that he is the winner is given by $\frac{1}{Totalmen -1 } = \frac{1}{8n -1 }$ . But $P_{_{2}}$ lost in the Second Round. Now In Second Round, there are 4n men = 2n Losers + 2n Win... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2117383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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Find the limit $\lim_{x\to 1}\frac{\sin{(\pi\sqrt x)}}{\sin{(\pi x)}}$ Find the following limit:
$$\lim_{x\to 1}\frac{\sin{(\pi\sqrt x)}}{\sin{(\pi x)}}$$
My attempt:
$$t:=x-1,\ x \rightarrow 1 \Rightarrow t\rightarrow 0,\ x=t+1$$
$$\lim_{x\to 1}\frac{\sin{(\pi\sqrt x)}}{\sin{(\pi x)}}=\lim_{t\to 0}\frac{\sin{(\pi\sqr... | Your mistake is here $$\lim _{ t\to 0 } \frac { \sin { \left( \pi \left( t+1 \right) \right) } }{ \pi \left( t+1 \right) } \neq 1\\ \lim _{ t\to 0 } \frac { \sin { \left( \pi \sqrt { \left( t+1 \right) } \right) } }{ \pi \sqrt { \left( t+1 \right) } } \neq 1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Every group of order $56$ has a proper non - trivial normal subgroup? I tried getting $n_{2}$ and $n_{7}$ , denote the number of Sylow-2-Subgroups and Sylow-7-Subgroups respectively.
I got two cases for $n_{2} = 1 , 7 $ and for $n_{7} = 1 , 8$ , i noticed that if $n_{2} = 1$ and $n_{7}= 1$ ,then we are done since they... | Count the total number of elements. Suppose neither $n_2$ nor $n_7$ is equal to $1$. Then $n_7=8$, which makes $8\cdot 6$ elements which belong to a $7$-Sylow subgroup, and to no other Sylow subgroup.
On the other hand, since $n_2=7$, each $2$-Sylow subgroup contains at least $4$ elements which belong to no other Sylow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2117566",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Sums of squares are closed under division Surprisingly, we got only one question for our 2-hour exam and I think nobody solved it. Here is the problem:
Assuming that $K$ is a field, show that $S$ is stable under addition, multiplication and division, where $S$ is defined as follow:
$$S=\left\{\sum_{i=1}^{n}{x_i}^2 ... | $$\frac{1}{x_1^2+x_2^2+x_3^2}=\frac{x_1^2}{(x_1^2+x_2^2+x_3^2)^2}+\frac{x_2^2}{(x_1^2+x_2^2+x_3^2)^2}+\frac{x_3^2}{(x_1^2+x_2^2+x_3^2)^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2117719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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permutations without any successive digits I need some help with the following problem.
Consider the permutations of the set $ \{1, \dots , n \}$. What is the probability to find a permutation in which the digits $1$, $2$, $3$ are not successive. In other worlds what is the probability to occur a permutation of the fo... | As you found, the focus set $\{1,2,3\}$ can be arranged $3!$ ways, the remainder $(n-3)!$ ways and the two part-sets can be interleaved by choosing $3$ of the $n{-}2$ gaps between and around the remainder elements in $\binom {n-2}3$ ways, giving the probability: $$ 3!(n-3)!\binom {n-2}3 \frac 1 {n!} = \frac {3!(n-3)!... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2117815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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When is the derivative of an inverse function equal to the reciprocal of the derivative? When is this statement true?
$$\dfrac {\mathrm dx}{\mathrm dy} = \frac 1 {\frac {\mathrm dy}{\mathrm dx}}$$
where $y=y(x)$. I think that $y(x)$ has to be bijective in order to have an inverse and let the expression $\dfrac {\mathrm... | Assume $g(f(x))=x$. Then
$$g'(f(x))f'(x)=1$$ and then
$$g'(f(x))=\frac1{f'(x)}$$
Note that we need also that $f'(x)\neq 0$. All the conditions (the injectivity and the differentability of $f$ and that $f'$ does not vanish) must meet in a neighbourhood of the point where you are differentiating, that is, this works loca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2117928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
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Coefficient of $x^2$ in $(x+\frac 2x)^6$ I did $6C4 x^2\times (\dfrac 2x)^4$ and got that the coefficient of $x^2$ is $15$, but the answer is $60$, why? Did I miss a step?
| $\binom{6}{4}x^2\times(\frac{2}{x})^4$ would give the coefficient for $\frac{1}{x^2}$. What you want instead is $$\binom{6}{2}x^4\times(\frac{2}{x})^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2118037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is volume of a high-D ball concentrated near its surface? I came across the following sentence while reading a book on applied math:
Volume of a high dimensional unit ball is concentrated near its
surface and is also concentrated at its equator.
This is from book's introduction, and I believe the sentence will ... | The volume of an n dimensional sphere is proportional to $r^n$. For example the area of a circle (2-sphere) is $\pi r^2$ and the volume of a 3-sphere is $\frac43 \pi r^3$.
This means that the volume grows with a high power of radius, and there is more of that volume near the boundary than near the centre. For a circle,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2118098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to factorize this cubic equation? In one of the mathematics book, the author factorized following term
$$x^3 - 6x + 4 = 0$$
to
$$( x - 2) ( x^2 + 2x -2 ) = 0.$$
How did he do it?
|
Note: I understand that there is already an accepted answer for this question, so this answer may be useless, but regardless, I'm still posting this to spread knowledge!
A simple way to factorize depressed cubic polynomials of the form$$x^3+Ax+B=0\tag1$$
Is to first move all the constants to the RHS, so $(1)$ becomes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2118291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 2
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Sequence Of Primes Hello I have a basic number theory question.
*
*I want to find a list of primes of the form
a, a + d, a + 2d, ... , a + 5d
*
*So a sequence of at least 6 or greater if I want to select a = 101
then what would I choose as my d to get a sequence of primes (they
don't have to be consecutive).
*I... | A well-known sequence $a_n:=a+nd$ producing at least $6$ primes with $3$-digit $a$ (this was your requirement here) is $a_n=199+210n$, i.e.,
$$
199, 409, 619, 829,1039,1249,1459,1669,1879,2089,
$$
so that the first consecutive $10$ sequence members are prime numbers. With $a=101$ you can search for such a $d$ by compu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2118394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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two models of ZFC such that there is a isomorphism between their ordinals if two models of ZFC have their ordinals isomorphic then there is a isomorphism between their constructibles?
| The answer is no.
It's not hard to show - similarly to the situation with respect to PA - that if $M$ is a countable model of ZFC such that $\omega^M$ is non-well-founded, then $ON^M$ has ordertype $(\omega+\zeta\cdot\eta)\cdot(1+\eta)$, where $\zeta$ and $\eta$ represent the ordertypes of $\mathbb{Z}$ and $\mathbb{Q}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2118514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $p^{2k+1} | m^2$, show $p^{k+1} | m$ Question: If $p^{2k+1} | m^2$, show $p^{k+1} | m$.
Answer that was provided: If $p^k$ is the largest power of $p$ that divides $m$, then $p^{2k}$ is the largest power of $p$ that divides $m^2$. Hence if a power of $p$ larger than $2k$ can divide $m$, then $p^{k+1}$ surely also di... |
If $p^k$ is the largest power of p that divides m, then $p^{2k}$ is the largest power of p that divvies $m^2$.
The confusion here lies with (re)using the same variable name for $k$.
The statement would have been easier to follow if it said: "let $p^a$ be the largest power of $p$ that divides $m$, then $p^{2a}$ is the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2118638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding recursive function Range A function $f: \Bbb{N^+} \rightarrow \Bbb{N^+}$ , defined on the set of positive integers $\Bbb{N^+}$,satisfies the following properties:
$$f(n)=\begin{cases} f(n/2) & \text{if } n \text{ is even}\\
f(n+5) & \text{if } n \text{ is odd} \end{cases}$$
Let $R=\{ i \mid \exists{j} : f(j)=i ... | Suppose that $f(1) = a$ and $f(5) = b$. It is clear that $$f(5n) = b$$ for all $n$. We'll prove by induction that for all $n \ne 5k$, $f(n) = a$. First note that
$$f(2) = f(\frac{2}{2}) = f(1) = a,$$
$$f(3) = f(3+5) = f(8) = f(4) = f(2) = a,$$
$$f(4) = f(2) = a.$$
Now suppose $n = 5k + r$, where $0 < r < 5$, and for al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2118739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Example of an $\mathbb R$-linear map from $\mathbb C^2$ to $\mathbb C^2$ that is not $\mathbb C$-linear Example of an $\mathbb R$-linear map from $\mathbb C^2$ to $\mathbb C^2$ that is not $\mathbb C$-linear. One class of examples that I can think of is the conjugate linear maps, are there any other importatnt examples... | All you have to do is completely forget about the $\mathbb{C}$ structure, and pick your favorite endomorphism of $\mathbb{R}^4$. For example, $(a+bi, c+di)\rightarrow (d+ai, b+ci)$ is not $\mathbb{C}$-linear, like most endomorphisms you pick out.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2118828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\frac {\sec (16A) - 1}{\sec (8A) - 1}=\frac {\tan (16A)}{\tan (4A)}$ Prove that:$$\frac {\sec (16A) - 1}{\sec (8A) - 1}=\frac {\tan (16A)}{\tan (4A)}$$.
My Attempt,
$$L.H.S= \frac {\sec (16A)-1}{\sec (8A)-1}$$
$$=\frac {\frac {1}{\cos (16A)} -1}{\frac {1}{\cos (8A)} -1}$$
$$=\frac {(1-\cos (16A)).(\cos (8A)... | $\frac{\sec 16A -1}{\sec 8A -1}$
= $\frac{\frac{1}{\cos 16A}-1}{\frac{1}{\cos 8A}-1}$
= $\frac{\frac{1 - \cos 16A}{\cos 16A}}{\frac{1 - \cos 8A}{\cos 8A}}$
= $\frac{2 \sin^2 8A}{\cos 16A} × \frac{\cos 8A}{2 \sin^2 4A}$
= $\frac{2 \sin 8A \cos 8A}{\cos 16A} × \frac{\sin 8A}{2 \sin^2 4A}$
= $\frac{\sin 16A}{\cos 16A} × \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2118974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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How to solve $\frac{dy}{dt}=\alpha y-\beta y^n$ for $n\geq 2,\beta>0$? Is there a way to solve $\frac{dy}{dt}=\alpha y-\beta y^n$ for $n\geq 2,\beta>0$?
I know how to solve it for $n=2$ but I am not sure how to solve for $n\geq 2$?
| $$\text{y}'\left(t\right)=\alpha\cdot\text{y}\left(t\right)-\beta\cdot\text{y}\left(t\right)^\text{n}\space\Longleftrightarrow\space\int\frac{\text{y}'\left(t\right)}{\alpha\cdot\text{y}\left(t\right)-\beta\cdot\text{y}\left(t\right)^\text{n}}\space\text{d}t=\int1\space\text{d}t\tag1$$
Now, use:
*
*Substitute $\text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2119049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How to solve this limit: $\lim\limits_{x \to 0}\left(\frac{(1+2x)^\frac1x}{e^2 +x}\right)^\frac1x$ $$\lim\limits_{x \to 0}\left(\frac{(1+2x)^\frac{1}{x}}{e^2 +x}\right)^\frac{1}{x}=~?$$
Can not solve this limit, already tried with logarithm but this is where i run out of ideas. Thanks.
| Using L'Hospital rule twice we get $$\lim _{ x\to 0 } \left( \frac { (1+2x)^{ \frac { 1 }{ x } } }{ e^{ 2 }+x } \right) ^{ \frac { 1 }{ x } }=~ { e }^{ \lim _{ x\rightarrow 0 }{ \frac { 1 }{ x } \ln { \left( \frac { (1+2x)^{ \frac { 1 }{ x } } }{ e^{ 2 }+x } \right) } } }={ e }^{ \lim _{ x\rightarrow 0 }{ \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2119157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 1
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Find the length of $EF$.
$AB = 60\text{ cm}$
$CD:DB = 3:5$
$AF:AD = 4:5$
$E$ is the midpoint of $AC$
Find $EF$.
I have this following problem. I tried solving it, but it doesn't make sense alot. Can I have a hint or a guide for solving this question. Thanks!
| First we notice that $CD=\frac{3CB}{8}$ and $DB=\frac{5CB}{8}$
Draw line $l$ parallel to AB from F.Let G be the point of intersection of $l$ and CB, then we have $DG=\frac{CB}{8}$ But then $CG=CD+DG=\frac{4CB}{8}$ and thus G is the mid point of CB and thus line $l$ passes through E.Now cause of similar triangles we hav... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2119284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Quadratic equation, find $1/x_1^3+1/x_2^3$ In an exam there is given the general equation for quadratic: $ax^2+bx+c=0$.
It is asking: what does $\dfrac{1}{{x_1}^3}+\dfrac{1}{{x_2}^3}$ equal?
| If $x_{1}$ and $x_{2}$ are the roots then
$x_{1} + x_{2} = -\frac{b}{a}$ and $x_{1} \cdot x_{2}=\frac{c}{a}$, now $\frac{1}{x_{1}}+\frac{1}{x_{2}} = -\frac{b}{c}$ and $$\frac{1}{x_{1}^{3}}+\frac{1}{x_{2}^{3}} = \left(\frac{1}{x_{1}}+\frac{1}{x_{2}}\right)^3- 3\cdot\frac{1}{x_{1}.x_{2}}\left(\frac{1}{x_{1}}+\frac{1}{x_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2119399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Is the sum of permutations of disjointed sets larger than the count of permutations in their union? My wife and I are having a bit of a disagreement.
Concerning eight-digit passwords like the kind most secure websites require, she believes you could generate more unique password combinations by using only one of the fo... | There is a context in which the wife is right, which may explain her intuition. Suppose we split the permitted symbols into two sets, $S$ and $T$, of size $s=|S|$ and $t=|T|$. Suppose further that our password must be exactly two letters long.
The wife's strategy yields $s^2+t^2$ passwords, either two from $S$ or two... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2119493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Exercise on Manifolds: Transition Maps
Let $M\;$ be a differentiable manifold of distance $3$, $p\in M\;$ and
two charts $(U,φ=(x_1,x_2,x_3))\;\;,(U,ψ=(y_1,y_2,y_3))\;$ of $M$
near $p\;$ with $φ(p)=(1,1,-2)\;$ such that:
$y_1=x_1\;,\;y_2=x_2-{x_1}^3\;,\;y_3=x_3+3x_1 {x_2}^2 \;$ in $U$
Find transition maps : $ψο{φ... | Note that $\varphi, \psi \colon U \rightarrow \mathbb{R}^3$ and you have no idea what $U$ is (it is some open subset of some manifold $M$).
The equations
$$y_1 = x_1, y_2 = x_2 - x_1^3, y_3 = x_3 + 3x_1 x_2^2$$ already give you (practically by definition)
$$(\psi \circ \varphi^{-1})(x_1,x_2,x_3) = (y_1,y_2,y_3) = (x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2119722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Stuck on this differential equation I want to solve this differential equation with the power-series method: $$ x^{2}\cdot y''(x) +(1+x^{2}) y'(x) + y(x) =0$$ where $$y(0)=1$$
They want the solution given in elementary functions. I managed to get the recursive formula for $$j\ge 2$$$$a_{j+1} = \frac{-((j(j-1)+1)a_{j} +... | I agree with you're recurrence relation $(i+1)a_{i+1}=-((i^2-i+1)a_i+(i-1)a_{i-1})$. & the first few values $a_0=1, a_1=-1, a_2=1/2, a_3=-1/6$ from this I guess the general formulae $a_i=(-1)^i/i!$. You can easily show (by induction) that this is indeed true. Now the solution is obviously $y=e^{-x}$ & this does indeed ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2119822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
How to intuitively explain that if kernel(L) equal to the set containing the zero vector, then L is one-to-one? Whenever I try to learn about a relationship, I try to reason intuitively why a theorem or lemma should make sense. I know often times this is increasingly difficult to achieve. However, I have the following:... | We know that the kernel is a subvectorspace. Hence if $x_1, x_2$ are (two different elements, nonzero) in the kernel, we have that $x_1 - x_2$ is in the kernel, so we find that
$$0 = L(x_1 - x_2) = L(x_1) - L(x_2)$$
by linearity. This would mean that $L$ is not one-to-one, since we have two different points with the sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2119929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
$A+B=AB$ does it follows that $AB=BA$? If $A$, $B$ are two normal operators such that: $A+B=AB$ does it follow that $AB=BA$?
| First, note that if $C$ is a normal operator which has a left inverse $D$ (so $DC=I$), then $C$ is invertible (and thus $D$ is its inverse and $CD=I$ as well). This follows from the spectral theorem: you can identify your Hilbert space with $L^2(X)$ for some semifinite measure space $X$ and $C$ with multiplication by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Find a ring that contain $\mathbb{Q}$ as a group and has solution for $x^2 \equiv 2$ but no solution to $x^2 \equiv 3$ I have the following question :
Find a ring that contain $\mathbb{Q}$ as a group and has solution for $x^2 \equiv 2$ but no solution to $x^2 \equiv 3$
Hint : Start from $\mathbb{Q}[x]$
I really don't k... | You have the right idea. The ideal $(x^2 - 2) \subseteq \mathbb{Q}[x]$ is maximal and so $R = \frac{\mathbb{Q}[x]}{(x^2-2)}$ is a field.
Note that $R \cong \{ a + b \sqrt{2} \ | \ a,b \in \mathbb{Q} \}$ and $\mathbb{Q} \subseteq R$.
Suppose that $\sqrt{3} \in R$. Then there exist $a,b \in \mathbb{Q}$ such that
$$\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to use finite differences to determine an equation of a polynomial given consecutive integer $x$ and corresponding $y$ coordinates of the graph? This chart is given:
for $x=-3$, $y=-9$
for $x=-2$, $y=3$
for $x=-1$, $y=3$
for $x=0$, $y=-3$
for $x=1$, $y=-9$
for $x=2$, $y=-9$
for $x=3$, $y=3$
I found the finite diffe... | Treat your numbers as a sequence with $g(0)$ being the first term, corresponding to $f(-3)$,
$$f(x-3):=g(x) : \color{green}{-9},3,3,-3,-9,-9,3$$
$$\Delta g=g(x+1)-g(x) : \color{green}{12},0,-6,-6,0,12$$
$$\Delta^2 g : \color{green}{-12},-6,0,6,12$$
$$\Delta^3 g : \color{green}{6},6,6$$
Now assume all else is $0$. To ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Please help solving this inital-value problem The initial-value problem
$$\begin{cases}(ye^{xy} + cos(x))dx + (xe^{xy})dy = 0\\y(\dfrac{\pi}{2}) = 0\end{cases}$$
From my calculations it seems the integrating factor depends on y and x, but I am unsure how to find the correct integrating factor.
Please help
| Hint:$$m.dx+n.dy=0 \\$$
$$\begin{cases}(ye^{xy} + cos(x))dx + (xe^{xy})dy = 0\\y(\dfrac{\pi}{2}) = 0\end{cases}\\
\frac{\partial (ye^{xy} + cos(x)) }{\partial y}=1.e^{xy}+xy.e^{xy}+0\\
\frac{\partial (xe^{xy})}{\partial x}=1.e^{xy}+yxe^{xy}\\so \\
\frac{\partial n}{\partial x}=\frac{\partial m }{\partial y}\\exact \spa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Top does not satisfy the axiom of choice I am trying to think about the axiom of choice from a categorical point of view.
I found out in some book that one possible formalization of this axiom is obtained by stating: "a category $\mathcal{C}$ satisfies the axiom of choice if all epics in $\mathcal{C}$ are split".
In th... | Top indeed does not satisfy the axiom of choice. To show this it suffices to exhibit a fiber bundle without a section, and for example the unit tangent bundle of $S^2$ has this property by the hairy ball theorem. Various other kinds of examples are possible; see, for example, this blog post.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Find a basis for the given subspace: confirmation I would like to have a confirmation if possible!
Let $V$ be the subspace of $\mathbb{R}^3$ given by the solution of the system
\begin{equation}
\begin{cases} x+6y-3z=0\\2x+12y-6z=0 \end{cases}
\end{equation}
Find a basis for $V$.
By solving the homogeneous system I can ... | Yes, all what you have done is correct !
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Be $f:[0,1]\longrightarrow[0,1]$ a continuous function, prove that exists $x\in[0,1]$ so that $f(x)=x$ . Be $f:[0,1]\longrightarrow[0,1]$ a continuous function, prove that exists $x\in[0,1]$ so that $f(x)=x$ .
I am studying mathematical analysis in functions of one variable, and looking through my notes I can't find an... | Hint.
Consider the function
$$g(x)=f(x)-x$$
and use the intermediate value theorem (this is possible because $f$ is continuous) to prove that there exists $x\in [0,1]$ such that $g(x)=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Fourier Series/ fourier transform What is the Fourier series of the following piece-wise function?
$$
f(x) = \begin{cases}
0 & -1 \leq x < -0.5 \\
\cos (3 \pi x) & -0.5 < x < 0.5 \\
0 & 0.5 \leq x < 1
\end{cases}
$$
| Given f(x) = \begin{cases}
0 & -1 \leq x < -0.5 \\
\cos (3 \pi x) & -0.5 < x < 0.5 \\
0 & 0.5 \leq x < 1
\end{cases}
Its nth Fourier polynomial is $S_n(x)=\sum_{v=-n}^{n}\alpha_ve^{ivx}$, where $\alpha_v=\int_{-\pi}^{\pi}f(x)e^{ivx}dx=\int_{-1/2}^{1/2} \cos(3\pi x)e^{ivx}dx$. Notice that $\dfrac{e^{3\pi ix}+e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Lexicographic Product I found out that the lexicographic product of a complete graph of order m and a cycle of order n is singular iff n is divisible by 4. But I am having a hard time on how to prove it using eigenvalues. Can someone help me? Thank you!
| The eigenvalues of the second graph (if regular) are also the eigenvalues of the product (lexicographic) graph. Further, if the second graph is $C_n$, then its eigenvalues are given by $2\cos\left(\frac{2\pi j}{n}\right),$ for $j=0, 1, \ldots, n-1$. Now if $n=4k$, for some $k$, then choosing $j=k$, we get $0$ as an eig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2120950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate a limit involving definite integral Evaluate the following limit:
$$\lim_{n \to \infty} \left[n - n^2 \int_{0}^{\pi/4}(\cos x - \sin x)^n dx\right]$$
I've tried to rewrite the expression as follows:
$$\lim_{n \to \infty} \left[n - n^2 \sqrt{2}^n \int_{0}^{\pi/4}\sin^n \left( \frac{\pi}{4} - x \right) dx\right]... | Writing
$$ (\cos x - \sin x)^n = \color{blue}{\frac{\cos x - \sin x}{\cos x + \sin x}} \cdot \color{red}{ (\cos x + \sin x)(\cos x - \sin x)^{n-1}} $$
and applying integrating by parts, we have
$$ \int_{0}^{\frac{\pi}{4}} (\cos x - \sin x)^n \, dx
= \frac{1}{n} - \frac{2}{n} \int_{0}^{\frac{\pi}{4}} \frac{(\cos x - \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2121035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Homogeneous ODE I am a bit confused with the definition and the solutions ways for homogeneous ODE.
I understand that they are 2 different definitions for homogeneous ODE
*
*The ODE is a function of $y$ and its derivatives such that $F(y,y',y'',...,y^{(n)})=0$
*the ODE has the same order of homogeneous such that $... | setting $$y=xu$$ in your equation then we get with $$y'=u+xu'$$
$$u-u^2-u-xu'=0$$ and you will get $$-\frac{dx}{x}=\frac{du}{u^2}$$ which is easy to solve
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2121130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Finding a geometric interpretation I recently solved a question of complex numbers which was this:
$A\left( \frac{2}{\sqrt{3}} e^\frac{i\pi}{2}\right)$, $B\left( \frac{2}{\sqrt{3}} e^\frac{-i\pi}{6}\right)$, $C\left( \frac{2}{\sqrt{3}} e^\frac{-5i\pi}{6}\right)$ are the vertices of an equilateral triangle. If $P$ be a ... | The sides of your triangle are $2$. E.g. the distance of $A$ and $B$ is the length of $$\frac{2}{\sqrt{3}}(e^{-\frac{i\pi}{6}}-e^{\frac{i\pi}{2}})=\frac{2}{\sqrt{3}}(\frac{\sqrt{3}}{2}-\frac{3}{2}i)=1-\sqrt{3}i.$$
We can calculate $PA^2+PB^2+PC^2$ using elementary methods, if $P$ is a point on the incircle of an equila... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2121205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Evaluation of $\int_{0}^{\infty}\frac{x^{a-1}-x^{b-1}}{1-x}dx$
For $0<a,b<1.$
Evaluation of $$\int_{0}^{\infty}\frac{x^{a-1}-x^{b-1}}{1-x}dx$$
$\bf{My\; Try::}$ Let $$I = \int_{0}^{1}\frac{x^{a-1}-x^{b-1}}{1-x}dx+\int_{1}^{\infty}\frac{x^{a-1}-x^{b-1}}{1-x}dx$$
Now how can i proceed further, Help required, Thanks
| You have to study what happens near $x=1$ and when $x\to \infty$, as I suppose you have notice.
Defining $f(x)=\frac{x^{a-1}-x^{b-1}}{1-x}$, $\lim_{x\to 1} f(x)$ exists and it's finite (why?). So you can define $f(1)$ by continuity, and $\int_0^1 f(x)\ dx$ exists and it's a number.
I do not know how to proceed from he... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2121311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Projecting from volume to surface area of a sphere Does it make sense to, and is there a formula for the sphere whose surface consists of all the points inside another sphere?
I realize that I'm trying to compare cubed units with squared units here but I think the assumption I'm making is that these spheres are more li... | The region between two spheres of radius $R_{1} < R_{2}$ has volume $\frac{4}{3}\pi (R_{2}^{3} - R_{1}^{3})$. If it were filled with sand grains of side length $r \ll R_{1}$, and if these grains were rearranged into a spherical shell of radius $R$ and thickness $r$, the shell would have volume (roughly, to within a fac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2121553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Can Atlas on S^1 only contain one chart? The smallest atlas for the $S^1=\{(x,y)\in \Bbb R^2|\,x^2+y^2=1\}$ must contain two charts. How to prove it?
My route is first to prove that it can only be homeomorphic to $
\Bbb R^1$ by invariant of domain (dimension). Then it restricts on mapping from $S^1$ into $\Bbb R$.
Se... | If $U$ is any (nonempty) open subset of $\mathbb{R}$, then $U$ minus any point is disconnected. However, $S^1$ minus a point is always still connected.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2121658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How many distinct groups of 5 possible from 10 seniors and 6 freshman? A school is forming a group of 5. There are 10 freshman and 6 seniors, and the group must have at least 2 freshman and at least 1 senior. How many distinct groups are possible?
My approach (that I think is definitely flawed) -
This would be a combi... | As you say the order doesn't matter. Then you have to think of all allowed groups, that is 2 freshmen 3 seniors, 3 freshmen 2 seniors, and 4 freshmen 1 senior.
2 freshmen 3 seniors:
$\dfrac{10 \cdot 9}{2!} \cdot \dfrac{6 \cdot 5 \cdot 4}{3!} = \dfrac{10!}{2!(10-2)!}\cdot \dfrac{6!}{3!(6-3)!} = 900 $ possible combinat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that a subring of a division ring must be a domain.
Show that a subring of a division ring must be a domain.
Let $S$ be a subring of $R$ and let $R$ be a division ring.
Can we just say that since every element has an inverse in $R$, then every element also has an inverse in $S$, then we can deduce that since for... | Hint $\ $ If $\,ax=0\,$ has unique root $\,x=0\,$ in $R\,$ then the same holds true in every subring.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2121863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Is L' context free language? Given context free grammar : G=(V,T,P,S).
and L'={a $\in$ (V $\cup$T)* | S$\Rightarrow$*a} when $\Sigma$=V $\cup$ T
Is L' context free language ?
I think no because for the grammer:
G=(S,{a,b},P,S) ,P={ S$\rightarrow$aSb|ab}
we will get that L' is:
L'={$a^nS^nb^n$}$\cup${$a^nb^n$} an... | Given context-free grammar $G = (V, T, P, S)$, define $G' = (V, T \cup V', P \cup P', S)$, such that $V'$ has one element for each element of $V$,
$$ V' = \{ v' \mid v \in V \} \enspace, $$
and, for each element $v \in V$, $P'$ contains the production $v \rightarrow v'$. The language of $G'$ is context-free, and is is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2122132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Dimension of an invariant subspace Let $V$ be a vector space of dimension $k$ and $n$ a positive integer. Let $a: V^n \to V$ be the map sending $(v_1,\ldots,v_n)$ to $v_1 + \cdots + v_n$. The symmetric group $S_n$ acts on $V^n$ by permuting the factors and the subspace $K := ker(a)$ is stable under this action.
What is... | UPDATE: this answer responded to an earlier version of the question, see comments below.
Any element of $V^{\oplus n}$ (this includes elements of $K$) that is invariant under $S_n$ is of the form $(v, v, \ldots, v)$. Now for such an element to lie in $K$ we must have $v + v + \ldots + v = 0$ ($n$ terms). If the charact... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2122212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determinant of 5x5 matrix with letters I want to find the determinant of the following $5 \times 5$ matrix
\begin{bmatrix} 2 & 0 & p & 0 & q \\r & 2 & s & 1 & 2 \\0 & 0 & 1 & 0 & 0\\u & 1 & v & 1 & w\\ 0 & 0 & x & 0 & 2 \\\end{bmatrix}
I know I have to do some row/column operations and expansions but I really don't get... | Let given matrix is A.
\begin{bmatrix} 2 & 0 & p & 0 & q \\r & 2 & s & 1 & 2 \\0 & 0 & 1 & 0 & 0\\u & 1 & v & 1 & w\\ 0 & 0 & x & 0 & 2 \\\end{bmatrix}
Expand it by $A_{33}$
= 1 × \begin{bmatrix} 2 & 0 & 0 & q \\r & 2 & 1 & 2 \\u & 1 & 1 & w\\ 0 & 0 & 0 & 2 \\\end{bmatrix}
Expand it by $A_{44}$
= 1 × 2 × \begin{bmatrix... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2122314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the value of $x$ that satisfy the equation: $3^{11}+3^{11}+3^{11} = 3^x$ I have this question:
$$3^{11}+3^{11}+3^{11} = 3^x$$
Find the value of $x$
| More simply:
$$3^{11}+3^{11}+3^{11}=3\times 3^{11}=3^{12}$$
so the answer you are looking for is $x=12$ (because $x\mapsto 3^x$ is injective).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2122390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Show that $\mathbb{Q_p} $ is locally compact Suppose $\mathbb{Q_p} $ is the fraction field of $\mathbb{Z_p}$ ($p$-adic integers) i.e.
$$\mathbb{Q_p} = \left\lbrace\frac{x}{y} \space \bigg{|} \space x,y \in \mathbb{Z_p} , y\neq 0 \right\rbrace$$
Now with respect to the topology defined by $d(x,y) = e^{-v_p(x-y)}$ ($v... | Every point has a fundamental system of neighborhoods given by $\{x+p^n\Bbb Z_p\}_{n\in\Bbb N}$ which are compact.
In essence this is just the fact that $p^n\Bbb Z_p$ is a fundamental system of compact neighborhoods of $0$: since we are in a vector space--really a topological group is enough, but not everyone is famil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2122493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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2 Answers for Derivative of $x^2 \sin 1/x$ Derivative of $x^2\sin(1/x)$ :
if we use the derivative rules to take the derivative of this function we conclude that It is not defined at x =0,
but if we use the derivative definition (lim x -> a), we get this:
$$x\sin(1/x)$$
which has a derivative of 1 at 0.
Which is the ... | The derivative by product rule and chain rule for $x\ne0$ gives us
$$\frac d{dx}x^2\sin(1/x)=2x\sin(1/x)-\cos(1/x)$$
While it is true that this is undefined at $x=0$, this is simply because we are not able to apply chain rule at $x=0$. At $x=0$, we must apply the limit definition:
$$\frac d{dx}\bigg|_{x=0}x^2\sin(1/x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2122599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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why $\int_\Omega S(t) div(w) = \int_\Omega div(w)$ ? where $S(t)$ is a heat semigroup with neumann condition In this work
https://www.dropbox.com/s/sygzebrr87ma99z/cao_2015_publicado.pdf?dl=0
page 1984, The author claims that
Let $w \in (C^{\infty}_0(\Omega))^N$ then $\int_{\Omega}S(t)div(w)=\int_{\Omega}div(w)$ See,... | The divergence term is not important here. What is important is that solutions of the heat equation with homogeneous Neumann boundary conditions preserve total heat, so that
$$\int_\Omega S(t) f \, dx = \int_\Omega f \, dx$$
for all $t$. To show this, simply note that $u(x,t)=S(t)f(x)$ solves the heat equation $u_t = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2122720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the determinant of order $100$ Find the determinant of order $100$:
$$D=\begin{vmatrix}
5 &5 &5 &\ldots &5 &5 &-1\\
5 &5 &5 &\ldots &5 &-1 &5\\
5 &5 &5 &\ldots &-1 &5 &5\\
\vdots &\vdots &\vdots &\ddots &\vdots &\vdots &\vdots\\
5 &5 &-1 &\ldots &5 &5 &5\\
5 &-1 &5 &\ldots &5 &5 &5\\
-1 &5 &5 &\ldots &5 &5 &5
\en... | you start with a $2×2$ matrix and see the eigenvalues they are $4,6$.
Then see for $3×3$ matrix the eigenvalues are $9,6,-6$
for $4×4$ they are $14,6,-6,6.$.
Hence whenever the order is even the eigenvalues 6 exceeds the eigenvalue $-6$ by 1 in multiplicity.
and hence for even $n$ the snswer is
$det= (5(n-1)-1).6^{n/2}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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Non identity character of an Abelian group How does one show that for any non-trivial abelian group, there exists some non-trivial character?
After looking up for a while, the best solution I could find was Pontrayagin duality, which seems like too heavy a machinery for this problem.
| For arbitrary abelian groups something non-trivial has to be used, it seems. For finite abelian groups, though, we have a very elementary Lemma:
Lemma: Let $G$ be a finite abelian group and $a,b$ distinct elements in $G$. Then there exists a character $\chi$ of $G$ such that $\chi(a)\neq \chi(b)$.
For a proof see here,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $\sum g_n$ converges uniformly , then does $(g_n)$ converge uniformly to $0$? If $\sum g_n$ converges uniformly , then does $(g_n)$ converge uniformly to $0$?
I think that it does basically from the fact that the convergence of numerical series implies that the numerical sequence of terms goes to $0$. But I feel I m... | Note that
$$g_n(x) = \left(\sum_{k=1}^{n} g_k(x) - g(x)\right) - \left(\sum_{k=1}^{n-1} g_k(x) - g(x) \right)$$
where both series converge uniformly to the same function $g$ on some set $D$.
Using the triangle inequality we have
$$|g_n(x)| \leqslant \left|\sum_{k=1}^{n} g_k(x) - g(x)\right| + \left|\sum_{k=1}^{n-1} g_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2123069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Beth number - justify existence of Beth omega I read through the answers to previous questions regarding Beth numbers and was unable to find the answer to my question, so I hope this isn't a duplicate.
I am studying the definition of Beth numbers, specifically:
$\beth_0:=\aleph_0$
$\beth_{\alpha+1}:=2^{\beth_\alpha}$
... | To expand on my comment, recall the following version of the Recursion Theorem reads, where $\phi$ is a formula in the language of set theory:
Suppose that $\forall x \exists! y\phi(s,y)$, and define $G(s)$ to be the unique $y$ such that $\phi(s,y)$ (note the use Replacement). Then we can define a formula $\psi$ for w... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Methods for choosing $u$ and $dv$ when integrating by parts? When doing integration by parts, how do you know which part should be $u$ ?
For example,
For the following:
$$\int x^2e^xdx$$
$u = x^2$?
However for:
$$\int \sqrt{x}\ln xdx$$
$u = \ln x$?
Is there a rule for which part should be $u$ ? As this is confusing.
| My general principle is "Which bit gets nicer to work with when differentiated than when integrated?", which roughly lines up with the LIATE approach:
*
*The derivative of $\ln x$ is $\frac{1}{x}$, which will interact nicely with polynomial terms, whereas the integral is some weird rubbish on the order of $x \ln x$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2123271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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$A=\emptyset $ if and only if $B = A \bigtriangleup B$ Is this true?
If $A$ and $B$ are sets, then $A=\emptyset $ if and only if $B = A \bigtriangleup B$.
If $A=\emptyset$ then $B=\emptyset$ too?
Could someone help me please?
| $A \Delta B$ is the set of things that belong to exactly one of $A$ and $B$.
So, if $A = \emptyset$, then $A\Delta B = (A\backslash B) \cup (B\backslash A) = B$.
If $A\Delta B = B$, then $A\backslash B = \emptyset$, or $A\subset B$, but $B\backslash A = B$ then $A = \emptyset$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2123377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Basic Question on the real and imaginary part of a complex number I was introduced to the use of Re and Im to denote the real and imaginary parts of complex numbers today, but I'm still feeling unclear on how exactly they work.
I think understand them for a basic example: $$Re(a+ib)=a$$
But we were given this equality,... | Recall $|z|^2=z\overline{z}$ where $\overline{z}$ is the complex conjugate of $z$ (that is, if $z=a+ib$. then $\overline{z}=a-ib$).
Then use $$ \frac{1}{a+ib}=\frac{1}{a+ib}\frac{a-ib}{a-ib}=\frac{a-ib}{a^2+b^2}=\frac{a}{a^2+b^2}-i\frac{b}{a^2+b^2}.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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find the least value of $n$ such $2017^{{2017}^{2017}}~~|~n!$
Let $n$ be positive integer. Find the minimum of the $n$ such
$$2017^{{2017}^{2017}}~~|~n!$$
[Note that 2017 is a prime]
use this formula:
$$v_{2017}(n!)=\dfrac{n-S_{2017}(n)}{2016}$$so find least $n$ such $$ n-S_{2017}(n)\ge 2017^{2017}$$
$S_{p}(n)$ den... | idea
If $n=2017^{k}$, then
$$\nu_{2017}(n!)=\sum_{j=1}^{k}\left\lfloor\frac{n}{2017^j}\right\rfloor=1+\dotsb +2017^{k-1}=\frac{2017^k-1}{2016}$$
Now perhaps we want $k$ such that
$$\frac{2017^k-1}{2016} \geq 2017^{2017}.$$
Note: this is just an idea which may require a bit more massaging to get the least value of $n$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prove: If $x+y+z=xyz$ then $\frac {x}{1-x^2} +\frac {y}{1-y^2} + \frac {z}{1-z^2}=\frac {4xyz}{(1-x^2)(1-y^2)(1-z^2)}$ If $x+y+z=xyz$, prove that:
$$\frac {x}{1-x^2} +\frac {y}{1-y^2} + \frac {z}{1-z^2}=\frac {4xyz}{(1-x^2)(1-y^2)(1-z^2)}$$.
My Attempt:
$$L.H.S=\frac {x}{1-x^2}+\frac {y}{1-y^2}+\frac {z}{1-z^2}$$
$$=\f... | Continuing from where you left, expressing terms of the numerator as:
$$-xz^2-x^2z+x^2yz^2 =-xz (z+x-xyz) =-xz (-y) $$
$$-xy^2-x^2y+x^2y^2z =-xy (x+y-xyz) =-xy (-z) $$
$$-yz^2-y^2z+xy^2z^2 =-yz (y+z-xyz)=-yz (-x) $$
$$x+y+z=xyz $$
Now add everything and the result follows. Hope it helps.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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finding limit with $\cos$ function occur $n$ times Finding $\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos(1-\cos(1-\cos(1-\cdots \cdots (1-\cos x))))}{x^{2^n}}$
where number of $\cos$ is $n$ times
when $x\rightarrow 0$ then $\displaystyle 1-\cos x = 2\sin^2 \frac{x}{2} \rightarrow 2\frac{x}{2} = x$
so $1-\cos (1-\cos... | $$\lim_{x\rightarrow 0}\frac{1-\cos x}{x^2}=\frac12$$
thus $1-\cos x\to\dfrac12x^2$ with substantiation
$$\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos(1-\cos(1-\cos(1-\cdots \cdots (1-\cos x))))}{x^{2^n}}$$
$$=\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos(1-\cos(1-\cos(1-\cdots \cdots (\dfrac12x^2))))}{x^{2^n}}$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why is $\int_0^1 \frac{t^x - 1}{\log(t)} dt$ finite for $x \in [0,1]$? I want to show that the following integral
$$\int_0^1 \frac{t^x - 1}{\log(t)} dt$$
is finite for all $x \in [0,1]$. But I struggle to come up with a good argument for that. Thanks in advance :)
| IF we differentiate under the integral sign we have
\begin{array}
$ I(x)&=&\displaystyle\int_0^1\frac{t^x-1}{\log t}dt\\
I'(x)&=&\displaystyle\int_0^1\frac{t^x\log t}{\log t}dt\\
&=&\displaystyle\int_0^1 t^x dt\\
&=&\displaystyle\frac{1}{1+x}, \quad x\neq -1
\end{array}
Next integrating we have $I(x)=\log(1+x)+c$. Sinc... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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why $x^{4} + y^{4} - 4 x y + 1$ graph differently in my textbook and mathematica? why this function $x^{4} + y^{4} - 4 x y + 1$ graph differently between mathematica and my textbook? did I make some error with my mathematica syntax?
| If $f(x,y)=x^4+y^4-4xy+1$, then
$$
D_1f(x,y)=4x^3-4y\\
D_2f(x,y)=4y^3-4x
$$
which has critical points at $(0,0)$, $(-1,-1)$ and $(1,1)$.
The Hessian is $H(x,y)=16(9x^2y^2-1)$; since $H(0,0)<0$ and $H(-1,-1)=H(1,2)>0$, we have that $(0,0)$ is a local maximum, while $(-1,-1)$ and $(1,1)$ are local minimum.
Now $f(0,0)=0$... | {
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Middle school problem - Percentage of students passing an exam This is a middle school problem that my nephew got, the teacher asked to solve it without using proportions:
The $\frac23$ of boys and the $\frac34$ of girls have passed an exam. Knowing that the
number of boys enrolled in the exam is three times the num... | Obviously you can't solve the problem without using facts about proportions, since all the input data as well as the answer are themselves proportions.
But here's an attempt to do it without performing any division operation,
which is the usual way of getting a proportion.
Instead, we use a probabilistic argument.
Let ... | {
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Identifying random variables from moment generating functions and using characteristic functions Suppose that $X$ and $Y$ have moment generating functions $M_X(t),M_Y(t)$ respectively and $U$ has uniform distribution on $[0,1]$.
What is a random variable that is a function of X,Y and U such that it's moment generating ... | Suppose $W=\begin{cases} 1 & \text{with probability } 1/2, \\ 0 & \text{with probability } 1/2, \end{cases}$
$\vphantom{\dfrac11}$and $Z=WX + (1-W)Y,$ so that $Z=X$ if $W=1$ and $Z=Y$ if $W=0.$ Then
$$
M_Z(t) = \operatorname{E}(e^{tZ}) = \operatorname{E}(\operatorname{E}( e^{tZ}\mid W)).
$$
Oberserve that
$$
M_{Z\,\m... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the least number $x$ such that $ 11$ divides $x$ and sum of its digits $S(x)$ is $27$. Find the least number $x$ such that $ 11$ divides $x$ and sum of its digits $S(x)$ is $27$.
Since $S(999)=27 $ it is clear that the number of digits $n>3.$ Let $x_i$ be digits then we have two equations
\begin{cases}
x_1+x_2+\c... | Since $\sum x_i=27$, $\sum (-1)^{i-1}x_i=27-2(x_2+x_4+\cdots)$ must be odd and divisible by $11$, and hence either $11$ or $-11$ (it must be between $-27$ and $27$.)
If $\sum (-1)^{i-1}x_i=11$ then $x_1+x_3+\cdots = 19, x_2+x_4+\cdots = 8$. The only way to get $19$ is with three digits, $(x_1,x_3,x_5)=(1,9,9)$ yieldin... | {
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"source": "stackexchange",
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Show that $I$ is an interval iff any function $f:I\rightarrow \{0,1\}$ is continuous. Let $I\in R$ be a nonemplty set. Show that $I$ is an interval iff any function $f:I\rightarrow \{0,1\}$ is continuous.
I've no idea how to handle this problem. Thanks in advvance to anyone who comes up with a step - solution.
| Can't be true.
Let $I$ be any set, interval or not, then $f(x) = 0$ is a continuous function whether or not $I$ is an interval. And $I$ = $A\cup B$ where $A$ and $B$ are separated then if $f(A) = 0$ and $f(B) = 1$, $f$ is continuous.
Likewise on an interval one can easily find an non-continuous function to $\{0,1\}$. ... | {
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Determining function is onto a. Consider the set $ℝ^+ = \{x∈ℝ|x>0\}$ together. Let $f:ℝ^+→ℝ^+$ be the function given by $f(x) = x^2.$ Is $f$ onto?
b. Consider the set $ℚ^+ = \{x∈ℚ|x>0\}$ together. Let $f:ℚ^+→ℚ^+$ be the function given by $f(x) = x^2.$ Is $f$ onto?
Workings:
a. Let $y \in \mathbb{R}^+$. Let $x = \sq... | Your complete proof for (a) should be as follows (the red is the bit you left out).
Let $y\in{\Bbb R}^+$. Let $x=\sqrt y$. Then $\color{red}{x\in{\Bbb R}^+}$ and we have
$$f(x)=x^2=(\sqrt y)^2=y\ .$$
Therefore f is onto.
The corresponding proof for (b) would be:
Let $y\in{\Bbb Q}^+$. Let $x=\sqrt y$. Then $\co... | {
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Determining Limits
The Question above states determine the limits, for problem a) I evaluated for when lim x->2 instead of 4. This was marked wrong im simply asking for a clarification on what the question is asking me to do.
| [2, 3, 6] might be the number of points given for each correct solution. In any case, it doesn't mean that you should calculate $\lim_{x\to 2}$ instead of $\lim_{x\to 4}$.
So what you should do for a) is
$$\lim_{x\to 4}\sqrt{4x+\sqrt x}=\sqrt{4\cdot 4+\sqrt 4}=\sqrt{18}=3\sqrt 2$$
The first answer is 2 points because y... | {
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"timestamp": "2023-03-29T00:00:00",
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On a characterization of the material conditional $\to$ I understand the usual motivation behind the truth table for the logical connective $\to$.
However, I would like to know if there is a more fundamental reason for that truth table. Something that would have to do with arguments and validity.
A.G.Hamilton writes in... | From the statements of $\{A{\to}B, A\}$ we can infer something new; that $B$ is true.
However, from the statements of $\{A{\to}B, \neg A\}$ we cannot infer anything new.
That is all.
| {
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"url": "https://math.stackexchange.com/questions/2125136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Triple integral bounds
*
*Let $W$ be the region bounded by the planes
$x = 0$, $y = 0$, $z = 0$, $x + y = 1$, and $z = x + y$.
*$(x^2 + y^2 + z^2)\, \mathrm dx\, \mathrm dy\, \mathrm dz$; $W$ is the region bounded by $x + y + z = a$ (where $a > 0$), $x = 0$, $y = 0$, and $z = 0$.
$x,y,z$ being $0$ is throwing me off... | 1) give you nothing to integrate. But we can still find the limits
$z = 0$ lower limit for $z$
$z = x+y$ upper limit for $z$
$y = 0$ lower limit for $y$
$y = 1-x$ upper limit for $y$
$x = 0$ lower limit for $x$
$x = 1$ upper limit for $x$
For the last one, because that is where $y = 0$ intersects $x+y = 1$
2)
$\int_0^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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First order logic to English statement? Assume
$A(x) = x$ is an American.
$D(y) = y$ is a dream.
$H(x,y) = x$ has $y$.
Then, Convert below first order logic to English statements :
*
*$∀x ∃y \left ( A(x)\rightarrow D(y) ∧ H(x,y) \right )$
I tried to translate this as "Every American has his own set of dreams".
*... | The first one is pretty close. I would say "Every American has a dream," because we only know that there exists one $y$.
The second one is weird. If there exists any y that is not a dream, then it is vacuously true. Literally it's something like "For every American there is a thing y such that if y is a dream, then the... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the greatest common divisor of $2^{2004}-1$ and $2^{2002}-1$. Find the greatest common divisor of $2^{2004}-1$ and $2^{2002}-1$.
Using Euclidean algorithm:
$$2^{2004}-1=4(2^{2002}-1)+3$$
$$2^{2002}-1=x\cdot 3+y$$
The solution manual says that $2^{2002}-1$has the remainder $0$ when divided by $3$, that is $y=0$ so... | We have,
$2^{2002} - 1$
$= (2^4)^{500}.2^2 - 1$
Now $2^4 \equiv 1 (\mod 3)$
From above,
$= (1)^{500} .2^2 - 1$
$= 1.2^2 - 1$
$= 4 - 1 = 3$
Divisible by 3.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solution of a equation .
Solve the equation:
$\frac{a}{ax-1}$+$\frac{b}{bx-1}$=$a+b$
It is my problem.I simply evaluate the equation and I found $ab(a+b)x^2-[(a+b)^2+2ab]x+2(a+b)=0$.I use Sridhar Achharya's theorem but it became complicate . Somebody please help me to solve the equation.
| $ab(a+b)x^2-[(a+b)^2+2ab]x+2(a+b)=0$
$(abx-a-b)(ax+bx-2)=0$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Understanding an example of equivalence relation On $\mathbb{N}\times\mathbb{N}$, Let $\left( a,b\right) \equiv \left( c,d\right) \Leftrightarrow a+d=c+b$.
Show that it is equivalence relation. Find equivalence of class of this.
My answer.
For all $a,b\in\mathbb{N}$, we have $a+b=a+b$. Clear. So, The relation is reflex... | Note that $a+d=c+b$ is true iff $a-b=c-d$. Let us interpret a pair such as $(a,b)$ as: $a$ is the amount of income this month, $b$ as the expenditure this month for a single person. We say two persons are equivalent if the money they saved this month (that is income minus the expenses) is the same for these two person... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational. Generalise this. I'm reading R. Courant & H. Robbins' "What is Mathematics: An Elementary Approach to Ideas and Methods" for fun. I'm on page $60$ and $61$ of the second edition. There are three exercises on proving numbers irrational spanning these pages, the last... | Using the straightforward fact that the sum of two (or more) algebraic integers is an algebraic integer (see here), one has the following very short argument: since $\sqrt{2},\sqrt{3},\sqrt{5}$ are algebraic integers, their sum $A=\sqrt{2}+\sqrt{3}+\sqrt{5}$ must be also. Hence if $A$ is rational, it must be an integer... | {
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"timestamp": "2023-03-29T00:00:00",
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Let $a,b,c$ be positive real numbers such that $abc =1$ Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$a^2+b^2+c^2\geq a+b+c$$.
Also, state the condition for equality.
My Attempt,
$a,b,c$ are real and positive numbers, then
$$(a-1)^2+(b-1)^2+(c-1)^2\ge 0$$
$$a^2-2a+1+ b^2-2b+1+c^2-2c+1\ge 0$$
$... | By AM-GM $$6(a^2+b^2+c^2)=\sum_{cyc}(4a^2+b^2+c^2)\geq6\sum_{cyc}\sqrt[6]{a^8b^2c^2}=6(a+b+c)$$
and we are done!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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What real numbers do algebraic numbers cover? Hardy and Wright mention ( though don't give a proof ) that any finite combination of real quadratic surds is an algebraic number. For example
$\sqrt{11+2\sqrt{7}}$. Are all finite combinations of cube root, fourth root ... $n^{th}$ root also algebraic ? such as $\sqrt[3]{2... | Yes, this holds in general. Recall that an extension $F/\Bbb Q$ is algebraic iff for each $\alpha\in F$ we have
$$[\Bbb Q(\alpha):\Bbb Q]=\dim_{\Bbb Q} \Bbb Q(\alpha)< \infty$$
that is, if every element in it generates a finite extension. Similarly any finite extension is algebraic since if $\beta\in F$ then $\Bbb Q(... | {
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"timestamp": "2023-03-29T00:00:00",
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Given $K \subset L^2(\mathbb{R})$, how to determine $K^{\perp}$? Problem: Define $K \subset L^2(\mathbb{R})$ given by $$ K = \left\{f \in L^2(\mathbb{R}) \mid f(-x) = 2f(x) \ \text{for almost all} \ x \geq 0 \right\}. $$
I need to give an explicit expression for $K^{\perp}$.
My attempt: By definition $$K^{\perp} = \l... | $g \in K^\perp$ iff
$$
0= \int_{0}^{\infty}g(t)f(t)dt+\int_{-\infty}^{0}2g(t)f(-t)dt \\
=\int_{0}^{\infty}\{g(t)+2g(-t)\}f(t)dt.
$$
This must hold for all $f \in K$. However $f\in K$ is an arbitrary $L^2$ function on $(0,\infty)$, which forces $g(-t)=-\frac{1}{2}g(t)$ for $t > 0$. This checks because
$$
... | {
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"timestamp": "2023-03-29T00:00:00",
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} |
Two inequalities involving the rearrangement inequality Well, there are two more inequalities I'm struggling to prove using the Rearrangement Inequality (for $a,b,c>0$):
$$
a^4b^2+a^4c^2+b^4a^2+b^4c^2+c^4a^2+c^4b^2\ge 6a^2b^2c^2
$$
and
$$a^2b+ab^2+b^2c+bc^2+ac^2+a^2c\ge 6abc
$$
They seems somewhat similar, so I hope ... | by AM-GM we get $$\frac{a^4b^2+a^4c^2+b^4a^2+b^4c^2+c^4a^2+c^4b^2}{6}\geq \sqrt[6]{a^{12}b^{12}c^{12}}=a^2b^2c^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2126270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Calculate $\int \frac{1}{\sqrt{4-x^2}}dx$ Calculate $$\int \dfrac{1}{\sqrt{4-x^2}}dx$$
Suppose that I only know regular substitution, not trig.
I tried to get help from an integral calculator, and what they did was:
$$\text{Let u = $\frac{x}{2}$} \to\dfrac{\mathrm{d}u}{\mathrm{d}x}=\dfrac{1}{2}$$
Then the integral beca... | $$\int \frac{\text{d}x}{\sqrt{4-x^2}}=\int \frac{2 \ \text{d}u}{\sqrt{4-(2u)^2}}=\int \frac{2 \ \text{d}u}{\sqrt{4(1-u^2)}}=\int \frac{2 \ \text{d}u}{2\sqrt{1-u^2}}=\int \frac{ \text{d}u}{\sqrt{1-u^2}} $$
Why especially this substitution: Notice that
$$\int \frac{\text{d}x}{\sqrt{4-x^2}}=\int \frac{\text{d}x}{\sqrt{4\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2126378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Decomposition Theorem for Posets There is in module theory the following (krull-shmidt) theorem:
If $(M_i)_{i\in I}$ and $(N_j)_{j\in J}$ are two families of simple module such that $\bigoplus\limits_{i\in I} M_i \simeq \bigoplus\limits_{j\in J} M_j$, then there exists a bijection $\sigma:I\rightarrow J$ such that $M_i... | The question concerns identifying a class of partial orders whose products have unique factorizations. I will give positive and negative partial answers, both (co)authored by Hashimoto.
The positive answer may be found in
Hashimoto, Junji
On direct product decomposition of partially ordered sets.
Ann. of Math. (2) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2126482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to evaluate an infinite sum involving remainders I've been trying to evaluate the sum
$$\sum_{k=0}^\infty \frac{m^k\bmod n}{m^k}$$
where $m$ and $n$ are positive integers greater than $1$ and $a\bmod b$ is the remainder when $a$ is divided by $b$. This came up in a combinatorics problem I was doing, and I know how ... | The numerators must repeat because only finitely many possible remainders exist. Suppose the repeating part starts after the first $K$ terms, so you have
\begin{align}
& \sum_{k=1}^K \frac{m^k\bmod n}{m^k} + \sum_{k=K+1}^\infty \frac{m^k\bmod n}{m^k} \\[10pt]
= {} & \sum_{k=1}^K + \sum_{k=K+1}^{K+R} + \sum_{k=K+R+1}^{K... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2126554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
In a regular 12-sided polygon, how many triangles can be formed using vertices of the polygon that do not share a side with the polygon? I think is is 12 choose 3 minus 12, because that is the number of triangles that share a side with the polygon. Is that correct?
| You're not quite right. For each side of the 12-gon, there are 12-4=8 non-adjacent vertices which you can use to form a triangle, so there are 12*8=96 such triangles. There are also those triangles which share two sides with the 12-gon, of which there are 12 (one for each vertex of the 12-gon--choose the two sides ad... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2126691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Geometry Derivative Show that the normals to the curve $y=4x^2$ from points the same distance on either side of the y-axis intersect on the y-axis.
My attempt, I differentiated so it becomes $8x$, but I don't understand the question. Can anyone explain it to me ? Thanks in advance:
| That parabola is symmetric with respect to $y$-axis, hence the normal lines issued from two symmetric points (as in your case) are corresponding one another through a reflection across $y$-axis. Unless they are both parallel (which is not possible if they are issued from different points) they must therefore meet on th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2126773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to prove that $\sum_{n \, \text{odd}} \frac{n^2}{(4-n^2)^2} = \pi^2/16$? The series:
$$\sum_{n \, \text{odd}}^{\infty} \frac{n^2}{(4-n^2)^2} = \pi^2/16$$
showed up in my quantum mechanics homework. The problem was solved using a method that avoids evaluating the series and then by equivalence the value of the serie... | First, the partial fraction of the summand can be written
$$\begin{align}
\frac{n^2}{(4-n^2)^2}&=\frac14\left(\frac{1}{n-2}+\frac{1}{n+2}\right)^2\\\\
&=\frac14 \left(\frac{1}{(n-2)^2}+\frac{1}{(n+2)^2}+\frac{1/2}{n-2}-\frac{1/2}{n+2}\right)
\end{align}$$
Second, we note that
$$\begin{align}
\sum_{n\,\,\text{odd}}\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2126888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
find maximum and minimum value of $|z|$ If $a,b,c$ are complex number of equal magnitude and satisfy $az^2+bz+c=0,$
then finding maximum and minimum value of $|z|$
with the help of triangle inequality $|az^2+bz+c|\leq |az^2|+|bz|+|c|=|a||z|^2+|b||z|+|c|$
now let $|a|=|b| = |c| = k>0$
so $|az^2+bz+c|\leq k(|z|^2+|z|+1)... | We can divide across by $a$ and get $z^2+bz+c= 0$ with $|b|=|c| = 1$. Solving
the quadratic gives
$z = {1 \over 2} (-b \pm \sqrt{b^2 -4c} ) $. From this we get
$\sqrt{5}-1 \le 2 |z| \le \sqrt{5}+1$, and by choosing $b=1,c=-1$ we see
that these bounds are attained.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2126974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proving det(B) = -det(A) theorem - am I on the right track? Theorem
Let $A$ be some $n \times n$ matrix, and $B$ be an $n \times n$ matrix
which is the result of performing a row swap operation on $A$. Column swaps won't be touched on for brevity's sake.
Let $b_{ij}$ be an element of $B$ at row $i$ column $j$.
Let $C_... | To swap the i and j'th row of A, multiply A on the right side such that $M_{i,j} = 1, M_{j,i} = 1, M_{k,k} = 1$ if $k\ne i$, or $k\ne j$ and $0's$ elsewehere.
$MA = B\\
\det MA = \det M \det A\\
\det M = -1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2127091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to decide whether an equation in index notation is valid. I am given the following equation in index notation: $k_{ijkl} = a_{i}b_{kl}c_{njm}d_{mn} + e_{ik}e_{jn}f_{n}$. I am told that this is a valid equation, but can anyone explain why? It doesn't violate the summation convention, and there's no obvious illegal c... | Hint: Check if all non-free indices are on both sides.
Left side we have $i,j,k,l$. On the right-hand side, we have for the first term $i,k,l,j$ ($m$ and $l$ are repeated indices, which by itself is not valid as far as I remember) and for the second term we have $i,k,j$ ($n$ is a repeated index).
Hence, this expression... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2127199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluate $\lim_{x\to 0^+}\left(\frac{\sin x}{x}\right)^{\frac{1}{x}}$ I need to find the following limit: $$\lim_{x\to 0^+}\left(\frac{\sin x}{x}\right)^{\frac{1}{x}}$$
I started this way: $$\left(\frac{\sin x}{x}\right)^{\frac{1}{x}}=e^{\frac{1}{x}\cdot \ln\left[\frac {\sin x}{x}\right]}$$
So it's enough to find: $$\l... | There is an interesting massive overkill: we may prove that in a right neighbourhood of the origin we have
$$1-\frac{x^2}{6}\leq\frac{\sin x}{x}\leq \exp\left(-\frac{x^2}{6}\right) \tag{1}$$
hence the wanted limit is trivially $\exp(0)=1$. By the Weierstrass product for the sine function:
$$ \log\left(\frac{\sin x}{x}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2127280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
If:$(\sqrt{x + 9})^{\frac{1}{3}} - (\sqrt{x-9})^{\frac{1}{3}} = 3$, Find $x^2$
If:$$\sqrt[3]{(x + 9)} - \sqrt[3]{(x-9)} = 3$$
Find $x^2$
I can't seem to solve this question. Any hints or solutions is welcomed.
| If $(9+x)^\frac 13 + (9-x)^\frac 13 = 3$,
then there is a $y$ such that $(9+x)^\frac 13 = (\frac 32+y)$ and $(9-x)^\frac 13 = (\frac 32-y)$.
Taking cubes of both equations you get $9 \pm x = (\frac {27} 8+\frac 92y^2) \pm (\frac {27} 4 +y^2)y$,
and so $9 = \frac {27} 8+\frac 92y^2$ and $x = (\frac {27} 4 +y^2)y$.
This ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2127400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Algorithm for Generating Well-Formed Formulas in Polish Notation I am trying to write an algorithm that constructs only well-formed formulas in PN. I have some list of symbols for binary connectives, unary connectives, and propositional variables (trying to make program robust for any length).
Here Polish Notation ment... | Here's one solution in Python that generates formulae up to a certain depth. To keep clutter to a minimum, no arguments are passed to the script, but adding such parameters and passing in signature and depth should be easy. This version seems to work with both Python 2.7 and 3.5.
""" Enumerates wffs in Polish notatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2127518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to find length of a part of a curve? How can I find the length of a curve, for example $f(x) = x^3$, between two limits on $x$, for example $1$ and $8$?
I was bored in a maths lesson at school and posed myself the question:
What's the perimeter of the region bounded by the $x$-axis, the lines $x=1$ and $x=8$ and ... | Your idea is along the right lines.
To find the arc length from $x=1$ to $x=8$ we first split the interval $[1,8]$ into sub-intervals $[x_0,x_1], \ldots [x_{n-1},x_n]$ where $$1=x_0 < x_1 < x_2 < \ldots < x_n = 8$$
This is called a partition of $[1, 8]$. Like you suggested in your question, we can then superimpose tria... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2127642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 7,
"answer_id": 2
} |
Alternative to Maplesoft I want to use MapleSoft for small project (few days) and want don't to buy it. Since, it doesn't provide any free trial period, I am looking for some alternative. I am planning to use Maple to solve some sequence and series that it seems solves seamlessly.
| *
*Mathematica has a free trial that you could take a look at (and in my opinion is the most powerful CAS)
*Sage is also pretty good, but may be more difficult to get work on all platforms
*Personally, I think the easiest one to use for quick stuff is SymPy; it has virtually no learning curve if you already know Pyt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2127727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Evaluating supremum and infinum I have no verified solution for this question.
I had a few questions regarding this:
$X = [1,3]$
$Y = (1,3]$
$X-Y = \{ x-y| x\in X, y\in Y\}$
The two questions that I have are that:
$a)$ Find the value of: $X-Y$.
$b)$ Are $\sup(X-Y)$ and $\inf(X-Y)$ elements of $X-Y$?
Firstly is the answ... | The solution is [-2,2) let me know if you need further clarification.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2127819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Find all conditions for $x$ that the equation $1\pm 2 \pm 3 \pm 4 \pm \dots \pm n=x$ has a solution. Find all conditions for $x$ so that the equation $1\pm 2 \pm 3 \pm 4 \pm \dots \pm 1395=x$ has a solution.
My attempt: $x$ cannot be odd because the left hand side is always even then we have $x=2k(k \in \mathbb{N})$ al... | Given that the $1$ in the series is not $\pm1$, I can see one other condition, namely that $x$ cannot have the values $Max-2$ or $Min + 2$, where Max and Min are the maximum and minimum values of the series.
The Max occurs when all signs are positive. The only way to get a sum of $Max-2$ is by changing the sign of $1$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2127922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 9,
"answer_id": 3
} |
What is the Mathematical Property that justifies equating coefficients while solving partial fractions? The McGraw Hill PreCaculus Textbook gives several good examples of solving partial fractions, and they justify all but one step with established mathematical properties.
In the 4th step of Example 1, when going from:... | Lemma: If $px+q=0$ for all values of $x$, then $p=q=0$.
Proof: In particular, $p(0)+q=0$, which means that $q=0$. So $px=0$ for all $x$, which means that $p(1)=0$, and so $p=0$.
Theorem: If $px+q$ and $rx+s$ are equal for all values of $x$, then $p=r$ and $q=s$.
Proof: If $px+q$ and $rx+s$ are equal for all values of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2128048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
} |
Prob. 18, Sec. 2.3, in I.N. Herstein's TOPICS IN ALGEBRA, 2nd ed: For any $n > 2$ construct a non-abelian group of order $2n$ Here is Prob. 18, Sec. 2.3, in the book Topics in Algebra by I.N. Herstein, 2nd edition:
For any $n > 2$ construct a non-abelian group of order $2n$.
Herstein gives the following hint to this ... | I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein.
This problem is Problem 18 on p.36.
I solved this problem as follows:
Let $\phi:=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$.
Let $\psi:=\begin{pmatrix}\cos\frac{2\pi}{n}&-\sin\frac{2\pi}{n}\\\sin\frac{2\pi}{n}&\cos\frac{2\pi}{n}\end{pmatrix}$.
Let $D:=\{I_2,\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2128175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
A closed form for a triple integral with sines and cosines $$\small\int^\infty_0 \int^\infty_0 \int^\infty_0 \frac{\sin(x)\sin(y)\sin(z)}{xyz(x+y+z)}(\sin(x)\cos(y)\cos(z) + \sin(y)\cos(z)\cos(x) + \sin(z)\cos(x)\cos(y))\,dx\,dy\,dz$$
I saw this integral $I$ posted on a page on Facebook . The author claims that there i... | Another approach to break down the last integral might be to consider the integral of $\displaystyle \frac{\log^3 (1-iz)}{z^2}$ along a positively oriented semi-circular contour $\gamma_R = [-R,R]\cup Re^{i[0,\pi]}$ in the upper half-plane. (We choose the branch of logarithm $\log (1-iz)$ in the lower half-plane along ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2128300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 3,
"answer_id": 0
} |
How to find the point on the sphere that is closest to a plane? Consider the plane $x+2y+2z=4$, how to find the point on the sphere $x^2+y^2+z^2=1$ that is closest to the plane?
I could find the distance from the plane to the origin using the formula $D=\frac{|1\cdot 0+2\cdot 0+2\cdot 0-4|}{\sqrt{1^2+2^2+2^2}}=\frac43$... | A little different:
Plane: $f(x,y,z) = x + 2y + 2z - 4 = 0 $; Circle $ g(x,y,z) = x^2 + y^2 + z^2 - 1 = 0 $.
Problem:
Point on a sphere with minimum distance to the plane.
Normal vector to the plane, $n_p$:
$\nabla f = (\frac {\partial f}{\partial x},\frac {\partial f}{\partial y},\frac {\partial f}{\partial z})$.
We g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2128416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
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