Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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convergence of series: $ \sum_{n=1}^\infty(\sqrt{n+1}-\sqrt{n})\cdot(x+1)^n $ I would like to prove the convergence of series: $$ \sum_{n=1}^\infty(\sqrt{n+1}-\sqrt{n})\cdot(x+1)^n $$ for x $\in \mathbb{R}$. I am a bit lost on this one. I guess I would be interested in
*
*$x<-1$
*$x = -1$
*$x > -1$
Any help would... | We have $\dfrac{\sqrt{n+2}-\sqrt{n+1}}{\sqrt{n+1}-\sqrt{n}}\dfrac{|x+1|^{n+1}}{|x+1|^n}\to|x+1|$. By the ratio test, the series conveges in $(-2,0)$ and diverges in $(-\infty,-2)\cup(0,\infty)$.
Now, $\sum_{k=1}^n(\sqrt{k+1}-\sqrt{k})=\sqrt{n+1}-1\to\infty$. Thus series diverges for $x=0$.
Can you solve for $x=-2$?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\int \delta(x + xy/u - a)\delta(y + xy/v - b)f(x,y)dxdy$? I need help evaluating the following integral:
$$\int \delta(x + uxy - a)\delta(y + vxy - b)p(x,y)dxdy$$
where $\delta(x)$ is Dirac-delta function, and $p(x,y)$ is some sufficiently well behaved function. The parameters $a,b,u,v$ are all real.
I'd know how to d... | There is a property of the Dirac Delta function one can use:
$$\delta\left(f\left(x,y\right)\right)\delta\left(g\left(x,y\right)\right)=\frac{\delta\left(x-x_{0}\right)\delta\left(y-y_{0}\right)}{\left|\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial g}{\partial x}\frac{\partial f}{\partial y}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1609872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $a_n=\frac{1}{n}\sum_{k=1}^n\frac{\varphi(k)}{k}$ convergent? Let $(a_n)_{n\in\mathbb{N}}$ be defined as $a_n=\frac{1}{n}\sum_{k=1}^n\frac{\varphi(k)}{k}$ where $\varphi$ is the euler totient function. Is $(a_n)$ convergent. If so, what is its limit?
I have checked it numerically; it seems to converge to the value
$... | Here you can find that the value you're looking for is $ \frac{6}{\pi^2} $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1609980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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othogonality of chebychev polynomials Let the chebyshev polynomials be defined as :
with zeros :
My goal is to show that the family of polynomials :
are orthogonal with respect to
where :
To achieve this we show :
However there is something wrong with the proof, the last expression
fails for m odd and doesn't y... | Note that in this question, $m=k\pm l$ for distinct $k,l\in\{0,1,..n\}$.
When they compute the sum, they don't put the $e^{\frac{im\pi}{2(n+1)}}$ back in before taking real parts. The interior ought to end up as $\Re(e^{\frac{im\pi}{2}}\frac{\sin(\frac{m\pi}{2})}{\sin\frac{m\pi}{2(n+1)}})$, which does vanish for odd $m... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Let $f(x) = (x^n-1)/(x-1)$. Why does $f(1)=n$? In the back of De Souza (Berkeley Problems in Mathematics, page 305), it says:
For $x \neq 1$,
$$
f(x) = (x^n-1)/(x-1) = x^{n-1} + \cdots + 1
$$
so $f(1) = n$.
The expansion for $x \neq 1$ obviously follows from the definition of a partial geometric series. But since... | so must be replaced by and, then we can say "[...] so that $f$ is continuous everywhere.".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1610106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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How to calculate sum of vector subspaces How do you sum these given subspaces? $$S_1=\{(x,y) \in R^2 | x=y\}$$$$S_2=\{(x,y) \in R^2 | x=-y\}$$
The book that I am currently learning from gives the answer to be $R^2$, but how do you get there?
Why does $S_1+S_2=R^2$? It also says that the sum is a direct one. What does t... | Since $S_1$ is generated by $b_1=(1,1)$ and $S_2$ by $b_2=(1,-1)$ and both are linearly independent in $\Bbb R^2$ then $S_1+S_2=\Bbb R^2$.
For $S_3$ the restriction $3x-2y=0$ implies that
$$(x,y)=(x,\frac{3}{2}x)$$
after solving for $y$. The geometrical meaning is a parameterization (in terms of $x$) of a line which ... | {
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"question_score": "1",
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Every open set in $\mathbb{R}$ is a disjoint union of open intervals: I'm struggling to follow the disjoint constraint The general idea of the proof our professor showed us was like this: let $O \subset \mathbb{R}$ be an open set, and take any $x \in O$. Now $O$ is bounded so any interval contained in $O$ has an infimu... | To see that $a$ and $b$ need not be $\sup O$ or $\inf O$, note that $O$ need not be 'connected'. That is, consider the open set $O=(0,1) \cup (2,3)$. Then $(0,3)$ is not a subset of $O$, is it? In this case, $b=1$ if $x \in (0,1)$ and $b=3$ if $x \in (2,3)$. similarly for $a$.
Now, let $x \ne y$ be in $O$. We want to... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Choose a composition from the previous composition Problem:In how many ways can one choose a composition $ \alpha $ of n, and then choose a composition of each part of $ \alpha $?
My attempt:
Consider the dot-and-bar argument on a row.
Let the final result be the composition $ \beta $ of n. Suppose $ \beta $ has k p... | HINT: This is an expansion of Michael Lugo’s hint in the comments. Suppose that you use start with $n$ dots and use some number of copies of $|_1$ to split these dots into the composition $\alpha$ of $n$. Then you use copies of $|_2$ to break each block of $\alpha$ into a composition. (Note that you need not break up a... | {
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What is the complexity of first order logic? I would say that first-order-logic has a data complexity and a formula complexity.
Data complexity: fix the theory and let the structure vary and measure
complexity in the size of the domain of the structure. Complexity is exponential.
Formula complexity: fix the structure a... | If the domain contains $n$ elements and the formula contains $m$ variables, then the truth table should be no larger than $n^m$. And if this is what you mean by this question, this would imply that your second statement makes sense and the first statement is under question.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1610524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Evaluation of Real-Valued Integrals (Complex Analysis) How to get calculate the integration of follwing:
$$\int_{0}^{2\pi} \frac{dt}{a + cos t} (a>1)$$
My attempt:
let, $z=e^{it}$
$\implies dt = \frac{dz}{it}$
and
$$\cos t = \frac{z + \frac{1}{z}}{2}$$
On substituting everything in the integral I got:
$$\frac{2}{i}\in... | HINT: The denominator: $z^2+2az+1$ can be easily factorize using quadratic formula as follows $$z^2+2az+1=(z+a-\sqrt{a^2-1})(z+a+\sqrt{a^2-1})$$
hence, $$\frac{1}{z^2+2az+1}=\frac{A}{z+a-\sqrt{a^2-1}}+\frac{B}{z+a+\sqrt{a^2-1}}$$
$$=\frac{1}{2\sqrt{a^2-1}}\left(\frac{1}{z+a-\sqrt{a^2-1}}-\frac{1}{z+a+\sqrt{a^2-1}}\righ... | {
"language": "en",
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Probability that $2^a+3^b+5^c$ is divisible by 4
If $a,b,c\in{1,2,3,4,5}$, find the probability that $2^a+3^b+5^c$ is divisible by 4.
For a number to be divisible by $4$, the last two digits have to be divisible by $4$
$5^c= \_~\_25$ if $c>1$
$3^1=3,~3^2=9,~3^3=27,~3^4=81,~ 3^5=243$
$2^1=2,~2^2=4,~2^3=8,~2^4=16,~2^5=... | Observe that
$$2^a+3^b+5^c \equiv 2^a+(-1)^b+1 \pmod{4}$$
So for this to be $0 \pmod 4$, we have the following scenarios
*
*$a \geq 2$, $b$ is odd and $c$ is any number.
*$a=1$, $b$ is even and $c$ is any number.
The number of three tuples $(a,b,c)$ that satisfy the first case =$(4)(3)(5)=60$ and the number of th... | {
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How do you find the value of $N$ given $P(N) = N+51$ and other information about the polynomial $P(x)$? Problem:
Let $P(x)$ be a polynomial with integer coefficients such that $P(21)=17$, $P(32)=-247$, $P(37)=33$. If $P(N) = N + 51$ for some positive integer $N$, then find $N$.
I can't think of anyway to begin this qu... | Alternative approach: Polynomial Remainder Theorem (kind of).
$$p(x) = k(x)(x-21)(x-32)(x-37)+r(x)$$
$r(x)$ is the remainder after division by a degree 3 polynomial, so $r(x)$ is at most degree 2:
$$r(x)=ax^2+bx+c$$
$$p(21)=17=r(21) \therefore 17 = 17^2a+17b+c$$
$$p(32)=-247=r(32) \therefore -247 = 32^2a+32b+c$$
$$p(37... | {
"language": "en",
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Conjugacy classes, irreducible representations, character table of $D_{10}$ (order 20) $D_{10}=\langle r,s \rangle$ is the dihedral group of order 20 . I have been struggling a bit with this question, particularly c, regarding values in the character table
(a). Find the conjugacy classes of $D_{10}$
Attempt at (a): G=... | (a) You have $r^{10}=1$, $s^2=1$ (so $s=s^{-1}$), and $rsrs=1$ (so $rs=sr^{-1}$ and $sr=r^{-1}s$).
For conjugates of $r^k$: Notice that $r^jr^kr^{-j}=r^k$ and $(r^js)r^k(r^js)^{-1}=r^jsr^ksr^{-j}=r^{j-k}ssr^{-j}=r^{-k}$. So $r^k$ is conjugate to itself and $r^{-k}$. We get conjugacy classes: $\{1\}$, $\{r,r^9\}$, $\{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1610835",
"timestamp": "2023-03-29T00:00:00",
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Uniform convergence of the series on unbounded domain
1 $$\sum_{n=1}^\infty (-1)^n \frac{x^2 + n}{n^2} $$
Is the series converges uniformly $\mathbb R$
I have tired by this result
*
*if $\{f_n(x)\}$ is a sequence of a function defined on a domain $D$ such that
*
*$f_n(x) \geq 0$ for all $x \in D$ and fo... | For the second problem, we can use the inequality
$$\sin\left(\sqrt{\frac{x}{n}}\right)\ge \sqrt{\frac{x}{x+n}}$$
for $0\le \sqrt{x/n}\le \pi/2$.
Then, we have
$$\begin{align}
\sum_{n=N}^\infty \frac{x\,\sin\left(\sqrt{\frac{x}{n}}\right)}{x+n}&\ge \sum_{n=N}^\infty \left(\frac{x}{x+n}\right)^{3/2}\\\\
&\ge \int_N^\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1610975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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How to proof that two lines in cube are perpendicular, without use of vectors Given: Cube $ABCDA_1B_1C_1D_1$
Prove that $BD$ is perpendicular to $AC_1$
I don't have any idea how to proof this. Also I can't use vectors(we didn't study them in school). I can use all theorems from the stereometry(I think another name for... | Perpendicular means if you translate $BD$ so that it begins at $A$ instead, the resulting lines are perpendicular. So translate $ABCD$ over to the left to get a square in the same plane, say $A'ADD'.$ Note that $C_1 D' = \sqrt{5}, AC_1 = \sqrt{3},$ and $AD' = \sqrt{2},$ so this is a right triangle.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $a\mid b^2, b^2\mid a^3,\ldots ,a^n\mid b^{n+1},b^{n+1}\mid a^{n+2},\ldots$ then $a=b$ I'm stuck with this problem :
Let $a,b$ positive integers such that
$$a\mid b^2, b^2\mid a^3,\ldots ,a^n\mid b^{n+1},b^{n+1}\mid a^{n+2},\ldots$$
Show that $a=b$.
If were $ b > a $ then $\lim_{n \to \infty}\frac{b^n}{a^n}=0 $ choo... | Pick a prime $p$, we must show $v_p(a)=v_p(b)$.
From the divisibility relations we have, for each $n\in \mathbb Z^+$:
$v_p(a)\leq \frac{v_p(b)2n}{2n-1}\implies v_p(a)\leq v_p(b)$.
We also have:
$v_p(b)\leq \frac{v_p(a)2n+1}{2n}\implies v_p(b)\leq v_p(a)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1611144",
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"source": "stackexchange",
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How many times should I roll a die to get 4 different results? What is the expected value of the number $X$ of rolling a die until we obtain 4 different results (for example, $X=6$ in case of the event $(1,4,4,1,5,2)$)?
I'm not only interested in technical details of a solution---I can solve it to some extent, see belo... | This is essentially the collector's problem.
You want to model each unique face as a geometric distribution.
$X_i\sim\text{Geom}\left(p = \frac{7-i}{6}\right)$ on $\{1,2,3,\dotsc\}$ for $i = 1,2,3,4$ denotes the number of rolls until the $i$th unique face. In the typical collector's problem, we are interested in $i$ fr... | {
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Integrating $\int_{-\infty}^0e^x\sin(x)dx$ I ask if anyone could solve the following:
$$\int_{-\infty}^0e^x\sin(x)dx=?$$
I can visually see that it will converge and that it should be less than $1$:
$$\int_{-\infty}^0e^x\sin(x)dx<\int_{-\infty}^0e^xdx=1$$
But I am unsure what its exact value is.
Trying to find the defi... | $$
\int_{-\infty}^0e^x\sin(x)dx =
\mbox{Im}\int_{-\infty}^0 e^{x(1+i)}dx
=\mbox{Im}\left.\frac{1}{1+i}e^{x(1+i)}\right|_{-\infty}^0
=\mbox{Im}\frac{1}{1+i}=\mbox{Im}\frac{1-i}{2}=-\frac{1}{2}.
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Dirichlet Convolution of Mobius function and distinct prime factor counter function. Let us define an Arithmetical function $\nu(1)=0$. For $n > 1$, let $\nu(n)$ be the number of distinct prime factors of $n$.
I need to prove $\mu * \nu (n)$ is always 0 or 1.
According to my computation, if $n$ is prime, it is 1. If $n... | $$F(a,s) = \prod_p (1+a \sum_{k=1}^\infty p^{-sk}) = \sum_{n=1}^\infty n^{-s} a^{\nu(n)}$$
$$\frac{\partial F(a,s)}{\partial a}|_{a=1} = \sum_{n=1}^\infty n^{-s} \nu(n)$$
$$G(a,s) = \frac{F(a,s)}{\zeta(s)} = \prod_p (1-p^{-s}) (1+a \sum_{a=1}^\infty p^{-sk}) = \prod_p (1+(a-1)p ^{-s})
$$
$$\frac{\partial G(a,s)}{\parti... | {
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Integration of Associated Legendre Polynomial I am interested in the following integral $$I=\int_{-1}^1P_\ell^2(x)P_n(x)\mathrm{d}x,$$
where $P_n(x)$ is Legendre Polynomial of $n$th order, and $P_\ell^2$ is Associated Legendre Polynomial. Any one has any idea on how to proceed?
| This is not an answer but it is too long for a comment.
Considering $$I_{n,l}=\int_{-1}^{+1} P_l^2(x) P_n(x)\,dx$$ this integral seems to show interesting patterns I give you below (this is just based on numerical evaluation and observation).
For positive values of $n,l$
*
*for $l<n \implies I_{n,l}=0$
*for $l=n+(2k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1611556",
"timestamp": "2023-03-29T00:00:00",
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Probability to win a chance game The game is quite simple, let's put it in marbles terms :
There's a bag of 25 marbles, 1 is white.
Each user picks one, and doesn't put it back.
I've figured the probability of each pick to be the winning pick, but I'm struggling to figure the probability of a game to be won after N cli... | The game is equally likely to end at Pick $1$, Pick $2$, Pick $3$, and so on. So the probability it ends at or before Pick $10$ is $\dfrac{10}{25}$.
To see that the game is equally likely to end at any pick, imagine that the balls are arranged in a line at random, with all positions for the white equally likely. Then ... | {
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Find prob. that only select red balls from $n$ (red+blue) balls There are 4 blue balls and 6 red balls(total 10 balls). $X$ is a random variable of the number of selected balls(without replacement), in which
$$P(X=1)=0.1$$
$$P(X=2)=0.5$$
$$P(X=3)=0.2$$
$$P(X=4)=0.1$$
$$P(X=10)=0.1$$
Then, what is probability of only se... | Since there is $6$ red balls, if $10$ balls are selected, probability of selecting only red balls is $0$ and we only have to consider selecting $1,2,3,4$ balls.
Let $R$ be number of red balls selected.
$$\begin{align}P(R=i|X=i)&=\frac{_6P_i}{_{10}P_i}\\
P(R=X)&=\sum\limits_{i=1}^4P(X=i)\cdot P(R=i|X=i)\\
&=0.1\cdot \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1611774",
"timestamp": "2023-03-29T00:00:00",
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Multivariable Laurent Series Is it possible to Laurent Expand over two complex variables? for example
$\frac{w+\tilde{w}}{(w\tilde{w})^{3}}$
where $w=i\sqrt{2}z+\hat{d}x+i\hat{e}y$ and $\tilde{w}=i\sqrt{2}z-\hat{d}x-i\hat{e}y$
Can someone point me in the right direction? i don't seem to be able to find much for more t... | It is possible, depending on the domain of the function.
In your example, if you take a point on the smooth part of the pole divisor, you can find a product of annuli of small radii where the function is holomorphic. Then construct the Laurent series the same way as in one variable, by Cauchy's integral.
I'd refer to ... | {
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Eigenvalues of inverse matrix to a given matrix How to calculate the eigenvalue of the inverse of a matrix given matrix is $A= \begin{bmatrix} 0&1&0\\ 0&0&1 \\4&-17&8\end{bmatrix}$
Is there any fast method?
| $$
A^{-1} x = \lambda x \iff \\
x = \lambda A x \Rightarrow \\
A x = (1/\lambda) x
$$
So the non-zero eigenvalues of $A^{-1}$ and $A$ are
related, are the multiplicative inverse to each other.
Looking for the eigenvalues of $A$ we get the characteristic polynomial
$$
(-\lambda)((-\lambda)(8-\lambda)+17) - 4 = \\
(-\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1611955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof $a + b \le c $ implies $\log(a) + \log(b) \le 2\log(c) - 2$ I have to proof the following statement:
Assume $a, b$ and $c$ are natural numbers that are different of $0$.
If $a + b \le c $, then $\log(a) + \log(b) \le 2\log(c) - 2$.
All $\log$ functions are the second $\log$ functions, thus $\log2$.... | Start from $c \geq a+b$ so :
$$2\log_2 c -2 \geq 2 \log_2(a+b)-2$$
If you can show that :$$2 \log_2(a+b)-2 \geq \log_2 a +\log_2 b$$ then you're done .
This is equivalent with :
$$2^{2 \log_2(a+b)-2} \geq 2^{\log_2 a +\log_2 b}$$ or :
$$\frac{1}{4} (a+b)^2 \geq ab$$ (note that I used the obvious fact that $2^{\log x} ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why is $\lim_\limits{x\to 0}\frac{\sin(6x)}{\sin(2x)} = \frac{6}{2}=3$? Why is $\lim_\limits{x\to 0}\frac{\sin(6x)}{\sin(2x)} = \frac{6}{2}=3$?
The justification is that $\lim_\limits{x\to 0}\frac{\sin(x)}{x} = 1$
But, I am not seeing the connection.
L'Hospital's rule? Is there a double angle substitution happening?
| $$\lim_{x\to0}\frac{2x}{\sin2x}\cdot\frac{\sin6x}{6x}=1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 7
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product of likelihoods vs PMF I am trying to understand better how the binomial PMF relates to likelihood. My understanding is that the the product of likelihoods from many trials is equal to the overall likelihood of observing all of the trials:
$\ell = \prod_{i=1}^nP(k_i, p)$
where $k_i$ is the success or failu... | You need to remember that there are different results for individual trials that all result in a total of $k$ successes. For your example, the LHS = 0.125, while the RHS = 0.375 = 3*LHS. Why the factor of 3? We really have:
binopdf(1, 1, 0.5) * binopdf(1, 1, 0.5) * binopdf(0, 1, 0.5)
+ binopdf(1, 1, 0.5) * binopd... | {
"language": "en",
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if $\ f(f(x))= x^2 + 1$ , then $\ f(6)= $? I want to know how to solve this type of questions. How can I find $\ f(x)$ from $\ f(f(x))$
Suppose, $\ f(f(x)) = x$ , then $\ f(x)=x$ or $\ f(x)=\dfrac{(x+1)}{(x-1)}$
how to find these solutions..
I have found one.. $\ f(6) = \sqrt{222} $
Is this correct?
| Here is a javascript function that conforms to the original poster's requirement that $f(f(x)) = x^2+1$. The original poster asked what is $f(6)$. This program computes that
$f(6)=12.813735153397387$
and
$f(12.813735153397387)=37$
QED we see $f(f(6)) = 6^2+1=37$
function f(x) {
x<0 && (x=-x);
return x==x+1 ? Math.p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1612308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Making an infinite generating function a finite one If we have some generating function $G(x)$ that generates terms indefinitely, is there a way to translate it to be a finite generating function?
For example if I only want to generate the first $k$ terms of a sequence, can I do $G(x) - x^kG(x)$ or something similar? T... | Edited Jan 27 2018. Answer by M.Scheuer is sufficient.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A circle inscribed in a rhombus. A circle is inscribed (i.e. touches all four sides) into rhombus ABCD with one angle 60 degree. The distance from centre of circle to the nearest vertex is 1. If P is any point on the circle, then value of
$|PA|^2+|PB|^2+|PC|^2+|PD|^2$ will be?
Can something hint the starting approach f... | Hint:
If $\angle DAB=\angle DCB=60°$ than The triangles $DAB$ and $DCB$ are equilateral, so $\angle ADB=60°$. Let $O$ the center of the circle, than the radius of the circle is $r=AO\sin 60°=\frac{\sqrt{3}}{2}$ and the distance $AO=\sqrt{3}$.
Now you can use a coordinate sistem with center $O$, a point on the circle ha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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integral of a square root function by substitution. A practice problem:
$$\int \sqrt{x^2+9}\ dx $$
So what I did was to substitute $x$ with $\tan \theta$, which yields
$$\int \sqrt{9\tan^2\theta+9}\ dx $$
Then I brought the 9 out
$$\int 3\sqrt{\tan^2\theta+1}\ dx $$
Using trig identity I simplified it to:
$$\int... | The solution to your problem is that we have $x = \tan\theta$. The differential of this would be $dx = \sec^2\theta\,d\theta$. You replace this with $dx$. You do this because you want your entire integral in terms of $\theta$. You don't need to solve for $\theta$ and get $d\theta$. Doing this causes $x$'s to appear in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1612585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating combinations within constraints Building out a web dev portfolio. For a web app I am working with lottery probabilities.
How many combinations are there if I only choose combinations within observed maximum and minimum values? Here are the stats:
The lottery I'm starting with is Mega Millions. 5 numbers d... | Your first calculation of the total number of combinations is close but may not be correct. I would guess that the sixth number cannot match any of the first five. In that case, you need to multiply first draws that have one number $1-15$ by $14$, those that have two by $13$, etc. Easier is to pick the sixth number fir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1612759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability with flipping the coins I flip a coin for $N$ times. I stop the flipping until I get 4 consecutive heads. Let $X=P(N\leq6)$.
On the other hand, I flip the coin for exactly 6 times. Once I finish all the flips, I check whether I got 4 consecutive heads. Let $Y=P(4$ consecutive Heads in 6 Flips$)$.
Is $Y=X?$
... | The one direction: If $N\le 6$ occurred, this implies that "$4$ consecutive Heads in the first $6$ Flips" indeed occurred! This shows (in your notation) that $$X\ge Y$$
The converse direction: If "$4$ consecutive Heads in the first $6$ Flips" occurred this implies that $N\le 6$ occurred. This shows that $$X\le Y$$ Putt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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how to find out coset representation of the following subgroup of the given group? $G$ is a group of $2\times 2$ matrices= $SL(2,\mathbb{Z})/\{I_2,-I_2\}$ where $SL(2,\mathbb{Z})$ is invertible matrix with entries in $\mathbb{Z}$ and determinant $1$.
then I know that G is generated by following two matrices
$$\begin{bm... | Well, I would do it the following way :
$$G\rightarrow PSL(2,\mathbb{F}_3) $$
$$A\mapsto A\text{ mod }3$$
Is clearly a group morphism whose kernel is $H$. Since $PSL(2,\mathbb{F}_3)$ is of cardinal $12$ and the coset representatives you found are different modulo $3$, it follows that it is surjective. Hence :
$$G/H=PSL... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Bound of solutions of autonomous linear ODEs Given the linear system $\dot{z} = Az$.
(a) Assume all the eigenvalues of $A$ have negative real part. Give a counterexample to this statement: every solution of $\dot{z} = Az$ satisfies $|z(t)|\leq |z(s)| \ \forall\ t> s $
(b) Assume A is symmetric and all the eigenvalues ... | The only way to get an initial increase is to have eigenvalues with multiplicity greater than 1. So try
$$
A=\begin{bmatrix}-1 & N\\ 0 & -1\end{bmatrix}
$$
and make $N$ large enough (positive or negative) to produce that initial increase.
One can probably also construct examples where the eigenvectors are not "orthogo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Explain $\langle \emptyset \rangle=\{1\},\langle 1 \rangle=\{1\}. H\leq G \implies \langle H\rangle=H.$ On page $61$ of the book Algebra by Tauno Metsänkylä, Marjatta Näätänen, it states
$\langle \emptyset \rangle =\{1\},\langle 1 \rangle =\{1\}. H\leq G \implies \langle H \rangle =H$
where $H \leq G$ means that H is... | The notation $H \leq G$ means that $H$ is a subgroup of $G$. Your proposed counterexample fails because $\emptyset$ is not a subgroup of $G$ (it doesn't contain the identity element).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Relation between ratio and percentage I would like to know easiest method to solve following:
Que:
If $p$ is $128$% of $r$, $q$ is $96$% of $r$ and $r$ is $250$% of $s$, find the ratio of $p$:$q$:$s$.
My Approach:
Step 1:
$p =\frac{128r}{100}$
$q = \frac{96r}{100}$
$r = \frac{250s}{100}$
So, $s = \frac{100r}{250}$
I ... | Notice, $p, q, r$ all depend on $s$ as follows $$r=\frac{250}{100}\times s=\frac{250s}{100}$$
$$q=\frac{96}{100}\times r=\frac{96}{100}\times \frac{250s}{100}$$
$$p=\frac{128}{100}\times r=\frac{128}{100}\times \frac{250s}{100}$$
hence, $$\color{red}{p:q:s}=\left(\frac{128}{100}\times \frac{250s}{100}\right):\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1613232",
"timestamp": "2023-03-29T00:00:00",
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Compute this integral (Is there a trick hidden to make it eassier?) I need some tips to compute this integral:
$$ \int\,\dfrac{\sqrt{x^2-1}}{x^5\sqrt{9x^2-1}}\,dx $$
What I did was express the denominator in the following form:
$$ \int\,\dfrac{\sqrt{x^2-1}}{x^5\sqrt{9x^2-1}}\,dx = \int\,\dfrac{\sqrt{x^2-1}}{x^5\sqrt{8x... | You could try a partial fractional decomposition of the discriminant, and see if that approach is of any use.
$$\frac{x^2-1}{9x^2-1} = \frac{1}{9}\cdot\left(1+\frac{4}{3x+1}-\frac{4}{3x-1}\right) \ .$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the number of bicycles and tricycles Help for my son. My math is a bit rusty and I'm trying to remember how to go about answering this question: "There are 3 times as many bicycles in the playground as there are tricycles. There is a total of 81 wheels. What is the total number of bicycles and tricycles in the... | \begin{align}
&\text{Number of bicycles} =3 \times 2x\text{ wheels}\\
&\text{Number of tricycles} =3x\text{ wheels}\\
&3 \times 2x\text{ wheels}\ +\ 3x\text{ wheels}=81\text{ wheels}\\
&9x=81\\
&x=9\\
&\text{Number of bicycles} =3 \times 9\\
&\text{Number of tricycles} =9\\
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1613461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 5
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General solution for the series $a_n = \sqrt{(a_{n-1} \cdot a_{n-2})}$ Hey I'm searching a general solution for this recursive series:
$a_n = \sqrt{(a_{n-1}\cdot a_{n-2})}$
$\forall n \geq 2$
$a_0 = 1$,
$a_1 = 2$
| Elaborating on what Wojowu has mentioned,
$$a_n^2=a_{n-1}\cdot a_{n-2}$$
$$a_n^2\cdot a_{n-1}=a_{n-1}^2\cdot a_{n-2}$$
That is, $a_n^2\cdot a_{n-1}=$ constant is invariant.
Hence $$a_n^2\cdot a_{n-1}=a_{n-1}^2\cdot a_{n-2}= \ldots = a_1^2a_0 = 4$$
or, $$a_n^2=\frac{4}{a_{n-1}}=\frac{4}{\frac{2}{\sqrt{a_{n-2}}}}=2\sqrt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simplifying nested square roots ($\sqrt{6-4\sqrt{2}} + \sqrt{2}$) I guess I learned it many years ago at school, but I must have forgotten it. From a geometry puzzle I got to the solution
$\sqrt{6-4\sqrt{2}} + \sqrt{2}$
My calculator tells me that (within its precision) the result equals exactly 2, but I have no idea h... | Hint:
If the expression under the first radical is a perfect square, the double product $4\sqrt2$ factors as $2\cdot2\cdot\sqrt2$. Then you indeed have $6-4\sqrt2=2^2-2\cdot2\cdot\sqrt2+(\sqrt2)^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1613686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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"answer_id": 2
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Prove $\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{a+b}{a+c}+\frac{b+c}{b+a}+\frac{c+a}{c+b}.$
Prove that for all positive real numbers $a,b,$ and $c$, we have $$\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a} \geq \dfrac{a+b}{a+c}+\dfrac{b+c}{b+a}+\dfrac{c+a}{c+b}.$$
What I tried is saying $\dfrac{a}{b}+\dfrac{b}{c}+\dfra... | Without Loss of Generality, Let us assume that $c$ is the maximum of $a,b,c$
Notice that $$\sum _{ cyc }^{ }{ \frac { a }{ b } } -3=\frac { a }{ b } +\frac { b }{ a } -2+\frac { b }{ c } +\frac { c }{ a } +\frac { b }{ a } -1=\frac{(a-b)^2}{ab}+\frac{(c-a)(c-b)}{ac}$$
However, since $(a-b)^2,(c-a)(c-b)\ge 0$, $$\frac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Polyakov action in complex coordinates Let $\Sigma$ be a compact $2$-manifold with riemannian metric $g$ and $f:\Sigma \to \mathbf{R}^n$ given locally by $f_1(x_1,x_2),\dots,f_n(x_1,x_2)$. Define
$$
S(f,g) = -\frac{1}{2\pi\alpha'}\int_{\Sigma}\left(\sum_{i=1}^n\sum_{k,j=1}^2g^{jk}(x)\frac{\partial f_i}{\partial x_j}\... | You should remember that the sum $ g^{jk} \; \partial_j f_i \; \partial_k f_i $ should be understood as $ g^{jk} \langle \partial_j f, \partial_k f \rangle $, where $ \langle , \rangle $ is the metric on the target space. In the case of $ \mathbb{R}^n $, it is just Euclidean. In the case the target space is $ \mathbb{C... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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When doesn't a supremum exist? Other than ∞, is there another case where a supremum (or an infimum for that matter) doesn't exist?
| Within the extended line $[-\infty,+\infty] = \mathbb R\cup \{\pm\infty\}$ every subset has a supremum and an infimum. Within the line $(-\infty,+\infty) = \mathbb R$ every subset has a supremum and and infimum except when the supremum or infimum within the extended line is $-\infty$ or $+\infty$. For example
$$
\sup... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Differential Equations to solve the changing radius of a drop of liquid This is the question:
"Your lab partner leaves a drop of bleach on the lab bench, which takes the shape of a hemisphere. The drop initially has a radius of 1.6mm, and evaporates at a rate proportional to its surface area. After 10 minutes, the... | Your suggestion is correct, since it turns out that $\frac{dr}{dt}$ is constant. This is not completely obvious, so we do the calculation.
The (curved) surface area $A$ is $2\pi r^2$, where $r$ is the radius, and the volume $V$ is $\frac{2}{3}\pi r^3$.
We are told that $\frac{dV}{dt}=-kA=-2k\pi r^2$, where $k$ is a pos... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The two definitions of a compact set
*
* In general, $A$ is compact if every open cover of $A$ contains a finite subcover of $A$.
* In $R$, $A$ is compact if it is closed and bounded.
The second is very easy to understand because I can easily come up with an example like $[0,1]$ which is both closed and bounded so i... | Let $X \subset R$
1) Compact => bounded.
I find it easy to just do this. For every $x \in X$ let $V_x = (x-1/2, x + 1/2)$. $V_x$ is open and $X \subset of \cup V_x$. So {$V_x$} is an open cover. So it has a finite subcover. So there is a lowest interval and there is a greatest interval in the finite subcollection... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Unique factorization theorem in algebraic number theory Consider the set $S = a + b \sqrt {-6}$, where $a$ and $b$ are integers. Now, to prove that unique factorization theorem does not hold in set $S$, we can take the example as follows:
$$
10 = 2 \cdot 5 = (2+\sqrt {-6}) (2-\sqrt {-6})
$$
"Thus we can conclude that ... | $2$ is Irreducible but not Prime.
In fact if $2=cd$, then $N(c)=2$ but there is no solution to $a^2 + 6b^2 = 2$ reducing modulo 6. Thus $2$ is Irreducible.
$2 |10 = (2+\sqrt {-6}) (2-\sqrt {-6})$, but if $2 |(2+\sqrt {-6})$ then $2 |\sqrt{-6}$ which is impossible since $2(a+b\sqrt{-6})=\sqrt{-6}$ has no solutions. Sam... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to find Laurent expansion I have been presented with the function $g(z) = \frac{2z}{z^2 + z^3}$ and asked to find the Laurent expansion around the point $z=0$.
I split the function into partial fractions to obtain $g(z) = \frac{2}{z} - \frac{2}{1+z}$, but do not know where to go from here.
|
The function $$g(z)=\frac{2}{z}-\frac{2}{1+z}$$ has two simple poles at $0$ and $-1$.
We observe the fraction $\frac{2}{z}$ is already the principal part of the Laurent expansion at $z=0$. We can keep the focus on the other fraction.
Since we want to find a Laurent expansion with center $0$, we look at the other pol... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Probability of a draft without replacement There is an urn with $N_1$ balls of type $1$, $N_2$ of type $2$ and $N_3$ of type $3$. I want to show that the probability of picking a type $1$ ball before a type $2$ ball is $N_1/(N_1+N_2)$. (without replacement = when you pick a ball you don't put it back in the urn, you ke... | You can ignore the type 3 balls, as picking one leaves you with the same number of type 1 and 2 balls. Take all the type 3 balls out and pick one ball. It is type 1 with probability $\frac {N_1}{N_1+N_2}$ as you say.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1614526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does "the subgroups of $G$ form a chain" mean? I am being asked to show that: $G$ is a cyclic $p$-group $\iff$ its subgroups form a chain.
What does "its subgroups form a chain" mean?
Please keep in mind that I am just asking for the meaning of that phrase.
| The subsets of $G$ form a partially ordered set with respect to the inclusion $\subseteq$; the subgroups are a subset of this partially ordered set.
For any partiall ordered set $(S,\leq)$ a subset $C \subseteq S$ is called a chain if $C$ is totally ordered with respect to $\leq$.
So the subgroups of $G$ forming a chai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1614738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A improper integral Can someone tell me what is wrong with this I cant seem to find the error
\begin{eqnarray*}
\int_{-\infty}^{\infty} \frac{2x}{1+x^{2}} dx &=& \displaystyle\int_{-\infty}^{0} \frac{2x}{1+x^{2}} dx + \int_{0}^{\infty} \frac{2x}{1+x^{2}} dx \\
&=& \lim_{a \rightarrow -\infty}\int_{a}^{0} \frac{2x}{1+... | You have discovered conditional convergence. You will also find that for this function,
$$
\lim_{a\to\infty} \int_{-a}^a \ne \lim_{a\to\infty} \int_{-a}^{2a\quad \longleftarrow \text{ “}2a\text{'', not “}a\text{''}}.
$$
This sort of thing happens only if the integral of the absolute value is infinite. One has
$$
\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1614947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Cardinality of set of fractional sums What is the cardinality of the set $S_2$:
$$ \frac{1}{a_1^n} + \frac{1}{a_2^n}, 1 \leq a_1,a_2 \leq k \in N$$
for different values of $n$?
I suspect there is an $n_0$ for which $|S_2| = \binom{k+1}{2}, \forall n \geq n_0$.
Is there such an $n_0$ for all sets $S_m$:
$$ \frac{1}{a_1... | Proposition. For each $m, k\ge 2$, $|S_m(k,n)|={k+m-1\choose m}$ for each $n\ge \log_{\frac k{k-1}} m-1$.
Proof. We shall follow A.P.’s comment. Let $\mathcal S$ be a family of non-decreasing sequences $(a_1,\dots, a_m)$ of natural numbers between $1$ and $k$. For each $S=(a_1,\dots, a_m)\in \mathcal S$ put $\Sigma S=... | {
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"timestamp": "2023-03-29T00:00:00",
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How to find a basis for $W = \{A \in \mathbb{M}^{\mathbb{R}}_{3x3} \mid AB = 0\}$ $B = \begin{bmatrix}
1 & 2 & 1 \\
1 & 3 & 1 \\
1 & 4 & 1
\end{bmatrix}$ and I need to find a basis for $W = \{A \in \mathbb{M}^{\mathbb{R}}_{3x3} \mid AB = 0\}$ .
I know that $AB = A\cdot\begin{bmatrix} 1\\1\\1 \end{bmatrix... |
Then I can conclude that (assume $A_1,...,A_n$ are columns of $A$):
1) $A_1 + A_2 + A_3 = 0$
2) $2A_1 + 3A_2 +4 A_3 = 0$
Close! Instead of rows they should be columns because matrix multiplication would take the dot products of the row vectors of the first matrix with the column vectors of the second.
By elementa... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is a proof of this limit of this nested radical? It seems as if $$\lim_{x\to 0^+} \sqrt{x+\sqrt[3]{x+\sqrt[4]{\cdots}}}=1$$
I really am at a loss at a proof here. This doesn't come from anywhere, but just out of curiosity. Graphing proves this result fairly well.
| For any $2 \le n \le m$, let $\phi_{n,m}(x) = \sqrt[n]{x + \sqrt[n+1]{x + \sqrt[n+2]{x + \cdots \sqrt[m]{x}}}}$. I will interpret the expression we have as following limit.
$$\sqrt{x + \sqrt[3]{x + \sqrt[4]{x + \cdots }}}\;
= \phi_{2,\infty}(x) \stackrel{def}{=}\;\lim_{m\to\infty} \phi_{2,m}(x)$$
For any $x \in (0,1)$,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closed form or approximation of $\sum\limits_{i=0}^{n-1}\sum\limits_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)} (i + 2x)(j +2x)$ During the solution of my programming problem I ended up with the following double sum:
$$\sum_{i=0}^{n-1}\sum_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)}\cdot (i + 2x)(j +2x)$$
where $... | Using Maple, I get
$$ (n+1) H(n) (n+4x-2)(x-1/2) + 2 n (n+2) x + \frac{n^3-3n}{2} $$
where $$H(n) = \sum_{k=1}^n 1/k = \Psi(n+1) + \gamma$$
As $n \to \infty$,
$$ H(n) \sim \ln(n) + \gamma + \dfrac{1}{2n} - \dfrac{1}{12n^2}
+ \dfrac{1}{120 n^4} - \dfrac{1}{252 n^6} + \dfrac{1}{240 n^8} + \ldots $$
| {
"language": "en",
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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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The roots of equation $x3^x=1$ are I have to find roots of equation
$x3^x=1$
A.Infinitely many roots
B.$2$ roots
C.$1$ root
D. No roots\
How do i start? Thanks
| For a simple answer plot the graphs of $y_1=3^x$ and $y_2=\frac{1}{x}$, (that are elementary), and see that these graph intersects only at a point $x_0$ such that $0<x_0<1$ because:
1) $3^0=1$ and $ \frac{1}{x} \to +\infty$ for $x \to 0^+$
2) $y_1(1)=3^1=3>y_2(1)=\frac{1}{1}=1$
3) the two functions are continuous in $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I show that $\text{End}_R\mathbb{Z}^n\cong\mathbb{Z}$ for $R=\text{Mat}_n(\mathbb{Z})$? Let $R=\text{Mat}_n(\mathbb{Z})$ and $M=\mathbb{Z}^n$ the (left) $R$-module with action the matrix multiplication.
How do I prove that $\text{End}_RM\cong\mathbb{Z}$?
Should I find an explicit isomorphism?
| To rephrase your question, the elements of $End(M_R)$ are those elements of $R$ which commute with all other elements of $R$.
So, you are just looking for the center of a matrix ring.
Here are breadcrumbs to follow to get to this idea:
For any $S$-module $M$, we have $S\subseteq End(M_\Bbb Z)$ in a natural way (multipl... | {
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Calculate: $\lim_{x\to 0} \frac{f(x^2)-f(0)}{\sin^2(x)}$. Let $f(x)$ be a differentiable function. s.t. $f^\prime(0)=1$.
calculate the limit: $$\lim_{x\to 0} \frac{f(x^2)-f(0)}{\sin^2(x)}.$$
SOLUTION ATTEMPT: I thought that because $f$ is differentiable its also continuous, then we can say: $\lim_{x\to 0} f(x^2)=f(0)$... | HINT: $f(0)$ stands for a constant function in your limit. What's the derivative of a constant function?
| {
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Prove that there is such A, such that for each $x \in [0,2]:$ $|f(x)-x| \le A(x-1)^2 $. Let $f$ have continuous derivatives until second order in the interval $[0,2]$, s.t. $f(1)=f^\prime (1)=1$.
Prove that there is such A, such that for each $x \in [0,2]:$
$$|f(x)-x| \le A(x-1)^2 $$
SOLUTION ATTEMPT:
I know that:
*... | Hint: Expand to a first order Taylor polynomial about the point $x_0 = 1$ with the (Lagrange) mean value remainder form.
| {
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lcm of orders of reduced residue classes of $1001$ By Euler's theorem, I know that all orders of reduced residue classes of $1001$ must divide $\phi(1001) = 720$. However, by a computer program, I know that the lcm of the orders of all $720$ reduced residue classes is $60$ (and, therefore, $1001$ has no primitive roots... | We have $1001=7\cdot 11\cdot 13$. Since $7$, $11$, and $13$ are primes, there are objects of order respectively $6$, $10$, and $12$ modulo $7$, $11$, and $13$. Thus by the Chinese Remainder Theorem there is an element of order (modulo $1001$) equal to the lcm of $6$, $10$, and $12$, which is $60$.
It is easy to show th... | {
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Evaluate $\lim_{x\rightarrow 0}\frac{1}{\sqrt{x}}\exp\left[-\frac{a^2}{x}\right]$ I am interested in the following limit:
$$\lim_{x\rightarrow 0}\frac{1}{\sqrt{x}}\exp\left[-\frac{a^2}{x}\right]$$
Does this limit exist for real $a$?
Edit: I am only interested in the case when $x$ is non-negative. Thanks for reminding.
| The left hand limit is not defined because of the square root in the denominator. So, you just need to check the value of the function at $f(0^+)$ and $f(0)$. If these two are equal, then limit is defined at $x=0$.
| {
"language": "en",
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If a continuous path $\Xi$ in $A\subset\mathbb{R}^2$ starts and end on $\partial(A)$, show that $A-\Xi$ is disconnected If a continuous path $\Xi$ in a closed and bounded subset $A\subset\mathbb{R}^2$ starts and end on $\partial(A)$, show that $A-\Xi$ is disconnected
To make things formal let $T=[0,1]$ and say that $\X... | I don't think this is true. Take $A = [0,1]\times [0,2]$ and let $(x,y) : T \to [0,1]^2$ be surjective, i.e.a space filling curve, which we can easily take to start and end in $(0,0) \in \partial A$.
In this case, $A \setminus \Xi = [0,1]\times (1,2]$ is connected.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Volume of 1/2 using hull of finite point set with diameter 1 It's easy to bound a volume of a half. For example, the points $(0,0,0),(0,0,1),(0,1,0),(3,0,0)$ can do it. The problem is harder if no two points can be further than 1 apart. Bound a volume of 1/2 with a diameter $\le 1$ point set.
With infinite points at ... | Just for fun, I got down to 162 points and a volume of .5058 by starting with a triangulation of a Icosahedron and subdividing each triangle into 4 smaller triangles twice.
I improved my own first try by using a Fibonacci Sphere for n points I than calculated the volume for 100 points up to 150 poimts. At 128 points, ... | {
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Error in the CLRS book for analyzing time complexity?
4.3-8 Using the master method in Section 4.5, you can show that the solution to the recurrence $T(n) = 4T(n/2) + n^2$ is $\Theta(n^2)$.
Wouldn't it be $\Theta(n^2 \log n)$?
| You are right, this is the second case of the Master Theorem (in its generic form) with $c=\log_b a = \log_2 4 =2$ and $k=0$. Indeed, we have
$$
T(n) = aT\left(\frac{n}{b}\right) + f(n)
$$
where $a=4$, $b=2$, and $f(n)=n^2$.
Setting $k=0$, since $f(n) \in \Theta(n^c\log^k n) = \Theta(n^2)$, we get $T(n) = \Theta(n^c\lo... | {
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Mathematical models in psychology Do you know examples of application of mathematics in psychology besides statistical data processing?
For example, do there exist mathematical models of addiction to Internet sites?
| Actually there are a few models using a different approach.
Have a look at Abelson 1967 and more recently
Agent-Based Modeling: A New Approach for Theory Building in Social Psychology
Eliot R. Smith and Frederica R. Conrey
Pers Soc Psychol Rev 2007; 11; 87
DOI: 10.1177/1088868306294789
I published also some models, see... | {
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how to bring the PDE $u_{tt}-u_{xx} = x^2 -t^2$ to the canonical form How to bring to the canonical form and solve the below PDE?
$$u_{tt}-u_{xx} = x^2 -t^2$$
I recognize that it is a hyperbolic PDE, as the $b^2-4ac=(-4(1)(-1))=4 > 0$.
I don't know how to proceed further to get the canonical form.
I know how to deal wi... | Proceed as for $u_{tt}-u_{xx}=0$. You will find $\xi=x+t$, $\eta=x-t$. Then
$$
x^2-t^2=(x+t)(x-t)=\xi\,\eta.
$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Which functions are $C^k$, but not $C^{k+1}$ on $ \mathbb{R} $ The only functions I can think of that fulfill this property are the polynomials of degree k.
However this does not necessarily imply, that this are the only functions that are in such a space. So I am curious, if there exist some further characterization ... | Most (in the sense of Baire category) continuous functions are nowhere differentiable. See Most Continuous Functions are Nowhere Differentiable.
Given any continuous nowhere differentiable function ($C^0$ and not $C^1$), any primitive will be $C^1$ but not $C^2$...
| {
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To find the sum of the series $\,1+ \frac{1}{3\cdot4}+\frac{1}{5\cdot4^2}+\frac{1}{7\cdot4^3}+\ldots$ The answer given is $\log 3$.
Now looking at the series
\begin{align}
1+ \dfrac{1}{3\cdot4}+\dfrac{1}{5\cdot4^2}+\dfrac{1}{7\cdot4^3}+\ldots &=
\sum\limits_{i=0}^\infty \dfrac{1}{\left(2n-1\right)\cdot4^n}
\\
\log 3 ... | HINT... consider the series for $\ln(1+x)$ and $\ln(1-x)$
| {
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What can we learn from the sign structure of the Jacobian matrix? I am studying a $4 \times 4$ Jacobian matrix. I know the sign structure (that is, I know whether each element is positive, negative or zero), but I do not know magnitudes of each elements (i.e. their numerical size).
\begin{align}
J = \begin{bmatrix}
... | I think you might be looking for the Routh-Hurwitz stability criterion, which is closely related to the eponymous theorem. Basically, this relates the sign of subdeterminants of the matrix to the sign of the real parts of the eigenvalues of the original matrix -- which is quite relevant for stability analysis. For your... | {
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Minimal cyclotomic field containing a given quadratic field? There was an exercise labeled difficile (English: difficult) in the material without solution:
Suppose $d\in\mathbb Z\backslash\{0,1\}$ without square factors, and $n$ is the smallest natural number $n$ such that $\sqrt d\in\mathbb Q(\zeta_n)$, where $\zeta_... | As you've surmised, one can simply count all index-$ 2 $ subgroups of the Galois group, which we know how to compute using the Chinese remainder theorem. We have the following general result: let $ n = 2^{r_0} \prod_{i=1}^n p_i^{r_i} $ be the prime factorization of $ n $. We have the following for $ r_0 = 0, 1 $:
$$ \t... | {
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Derivative test for contraction mapping on open sets I have a question on the derivative test for showing that a function is a contraction. The proposition that I know is in this version:
Let $I$ be a closed bounded set of $\mathbb{R}^l$ and $f : I → I$
a differentiable function with $||f'(x)|| \leq \beta$,
$0<\b... | This is not true. Consider the function that maps an open ball (say, the unit ball) to $c$ for some constant $c$
| {
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Proving an integration equality I am interested in why $$\int_0^1\dfrac{\ln(1+x^n)}{x}dx=\frac{\pi^2}{12(n)}$$
This is what WA gives me http://www.wolframalpha.com/input/?i=integral+of+ln%281%2Bx%5En%29%2Fx
Is there a way to prove this?
| First, note the following:
$$\ln(1+x)=\sum_{k=1}^{\infty}\frac{x^k(-1)^{k+1}}{k}\implies\ln(1+x^n)=\sum_{k=1}^{\infty}\frac{x^{kn}(-1)^{k+1}}{k}$$
Now, since our limits are from $0$ to $1$ we are fine to proceed with integrating. Therefore, we now have:
$$\int_0^1 \sum_{k=1}^{\infty}\frac{x^{kn}(-1)^{k+1}}{k}\,dx=\sum_... | {
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Existence of bounded analytic function on unbounded domain?
Given any proper open connected unbounded set $U$ in $\mathbb C$.Does there always exist a non constant bounded analytic function $ f\colon U \to \mathbb C$ ?
Edit: $U$ is any arbitrary domain. I don't have idea to do it. Please help.
| Take $f(z) = {1 \over z} $ on $U=\{z \mid |z|>1 \}$.
This example can be extended to any $U$ such that $U^c$ contains an open set.
| {
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Ordered sets $\langle \mathbb{N} \times \mathbb{Q}, \le_{lex} \rangle$ and $\langle \mathbb{Q} \times \mathbb{N}, \le_{lex} \rangle$ not isomorphic I'm doing this exercise:
Prove that ordered sets $\langle \mathbb{N} \times \mathbb{Q}, \le_{lex} \rangle$ and $\langle \mathbb{Q} \times \mathbb{N}, \le_{lex} \rangle$ are... | Suppose there is an isomorphism $\phi:\mathbb{Q} \times \mathbb{N} \to \mathbb{N} \times \mathbb{Q} $.
Let $(n,q) = \phi((0,0))$ and let $(a,b) = \phi^{-1}((n,q-1))$. Note that
we must have
$a<0$ since $(n,q-1) < (n,q)$.
Note that we have $(a,b) < (a,b+1) < (0,0)$.
Note that there are no elements in $\mathbb{N} \time... | {
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calculate the the limit of the sequence $a_n = \lim_{n \to \infty} n^\frac{2}{3}\cdot ( \sqrt{n-1} + \sqrt{n+1} -2\sqrt{n} )$ Iv'e been struggling with this one for a bit too long:
$$
a_n = \lim_{n \to \infty} n^\frac{2}{3}\cdot ( \sqrt{n-1} + \sqrt{n+1} -2\sqrt{n} )$$
What Iv'e tried so far was using the fact that the... | Keep on going... the difference between the fractions is
$$\frac{\sqrt{n-1}-\sqrt{n+1}}{(\sqrt{n+1}+\sqrt{n})(\sqrt{n-1}+\sqrt{n})}$$
which, by similar reasoning as before (diff between two squares...), produces
$$\frac{-2}{(\sqrt{n-1}+\sqrt{n+1})(\sqrt{n+1}+\sqrt{n})(\sqrt{n-1}+\sqrt{n})}$$
Now, as $n \to \infty$, th... | {
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How do you plot a graph where $y$ increases in short bursts and not linearly? I am wondering if it's possible to plot this sort of graph with one equation.
Note: the application of what I am doing is for video animation, but I
am just asking for the mathematical explanation; software has nothing
to do with this.
... | Hint:
You can use a function like
$$
y=ax-b|\sin (cx+d)|
$$
adjusting the constants $a,b,c,d$.
| {
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Find the number of elements in each factorgroup.
(a) and (c) are 2 and 3 and I don't think I have a problem with those.
However for (b) I get 11 cosets but they are not disjoint. According to theory there should only be 6. So what is happening here?
| Consider [b]. The element $8$ in $\mathbb{Z}/12$ has order $12/\gcd(8,12) = 3$: indeed we have $8+8+8 = 24 \equiv 0 \mod 12$. Thus the order of $(\mathbb{Z}/12)/\langle 8 \rangle$ is $12/3 = 4$. Explicitly, the cosets are $\langle 8 \rangle = \{0,8,4\}$, $1+\langle 8 \rangle = \{1,9,5\}$, $2+\langle 8\rangle = \{2,10,6... | {
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Existence of an injective continuous function $\Bbb R^2\to\Bbb R$? Let's say $f(x,y)$ is a continuous function. $x$ and $y$ can be any real numbers. Can this function have one unique value for any two different pairs of variables? In other words can $f(a,b) \neq f(c,d)$ for any $a$, $b$, $c$, and $d$ such that $a \neq ... | Assume $f : \mathbb{R}^2 \to \mathbb{R}$ is continuous and injective. Then for each fixed $y$, the function $x \mapsto f(x,y)$ is monotonic. Its image is some interval, and in particular contains a rational number. None of these points can be re-used for some other $y$. So $y$ can't be drawn from an uncountable set, si... | {
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Real values of $x$ satisfying the equation $x^9+\frac{9}{8}x^6+\frac{27}{64}x^3-x+\frac{219}{512} =0$
Real values of $x$ satisfying the equation $$x^9+\frac{9}{8}x^6+\frac{27}{64}x^3-x+\frac{219}{512} =0$$
We can write it as $$512x^9+576x^6+216x^3-512x+219=0$$
I did not understand how can i factorise it.
Help me
| If this problem can be solved without computation it is reducible; we assume this.
$f(x)=512x^9+576x^+216x^3-512x+219=0$ has two change-sign and $f(-x)$ has three ones so $f(x)$ has at least $9-5=4$ non real roots. We try to find a quadratic factor using the fact that $219=3\cdot73$ and $512=2^9$; this factor could co... | {
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The number of ordered pairs $(x,y)$ satisfying the equation The number of ordered pairs $(x,y)$ satisfying the equation $\lfloor\frac{x}{2}\rfloor+\lfloor\frac{2x}{3}\rfloor+\lfloor\frac{y}{4}\rfloor+\lfloor\frac{4y}{5}\rfloor=\frac{7x}{6}+\frac{21y}{20}$,where $0<x,y<30$
It appears that $\frac{x}{2}+\frac{2x}{3}=\fra... | Note that
$$\Bigl\lfloor\frac x2\Bigr\rfloor\le\frac x2\ ,$$
and likewise for the other terms. So we always have $LHS\le RHS$, and the only way they can be equal is if
$$\Bigl\lfloor\frac x2\Bigr\rfloor=\frac x2\ ,$$
and likewise for the other terms. So
$$\frac x2\ ,\quad \frac{2x}3\ ,\quad \frac y4\ ,\quad\frac{4y}5... | {
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How to show that $i^m-i(i-1)^m+\frac{i(i-1)}{1.2} (i-2)^m-...(-1)^{i-1}.i.1^m=0$?
How to show the following? $$i^m-i(i-1)^m+\frac{i(i-1)}{1.2}
(i-2)^m-...(-1)^{i-1}.i.1^m=0$$ (if $i>m$)
This seems really complicated.Can't spot any pattern as such :\ .Someone help me out!
P.S: I don't think the question means $i$ is ... | I suppôse that $m\geq 1$.
Your sum seems to be
$$S=\sum_{k=0}^{i} { i \choose k}(i-k)^m (-1)^k$$
Putting $i-k=j$, this becomes
$$ S=(-1)^i \sum_{j=0}^{i} { i \choose j}(j)^m (-1)^j=(-1)^i T$$
We have
$$\sum_{j=0}^i {i \choose j}(-1)^j x^j=(1-x)^i=P_i(x)$$
Let $\tau =x\frac{d}{dx}$. It is easy to see by induction that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1617965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Show that $2^n-(n-1)2^{n-2}+\frac{(n-2)(n-3)}{2!}2^{n-4}-...=n+1$
If n is a positive integer I need to show that
$2^n-(n-1)2^{n-2}+\frac{(n-2)(n-3)}{2!}2^{n-4}-...=n+1$
My guess: Somehow I need two equivalent binomial expression whose coefficients I need to compare.But which two binomial expressions? I know not!
P.S... | Here is a generating function approach
$$
\begin{align}
\sum_{n=0}^\infty a_nx^n
&=\sum_{n=0}^\infty\sum_{k=0}^n(-1)^k\binom{n-k}{k}2^{n-2k}x^n\tag{1}\\
&=\sum_{k=0}^\infty\sum_{n=k}^\infty(-1)^k\binom{n-k}{k}2^{n-2k}x^n\tag{2}\\
&=\sum_{k=0}^\infty\left(-\frac14\right)^k\sum_{n=k}^\infty\binom{n-k}{k}(2x)^n\tag{3}\\
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Which book to use in conjunction with Munkres' TOPOLOGY, 2nd edition? Although Topology by James R. Munkres, 2nd edition, is a fairly easy read in itself, I would still like to know if there's any text (or set of notes available online) that is a particularly good choice to serve as an aid to Munkres' book, in case one... | Two recently published books that I have used (actually instead of Munkres) include:
*
*Topology by Manetti: http://www.springer.com/gp/book/9783319169576.
*Topology: An Introduction by Waldmann: http://www.springer.com/gp/book/9783319096797.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 1
} |
Existence of a special kind of continuous injective function $f\colon A \to \mathbb R$, where $A$ is countable, relating to connectedness Let $A \subseteq \mathbb R$ be a countable set ($A$ induced with usual subspace topology), then does there necessarily exist a continuous injective function $f\colon A \to \mathbb R$... | Look at $ℚ \subset ℝ$, there is no continuous injective map $f:ℚ →ℝ$ which fulfills your conditions.
Lemma: If $f^{-1}(S) = \{a \in ℚ\}$ and $f(a), f(a)+ε \in S$ for $ε > 0$ and $S$ connected then $f(A) \subset (-∞, f(a)]$.
Proof: As both $(-∞, a)\cap ℚ$ and $(a, ∞)\cap ℚ$ are connected, their images must each lay in o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 2
} |
Why can't you have more turning points than the degree? I get that each degree can correspond to a factor. $x^5=(x+a)(x+b)(x+c)(x+d)(x+e)$ and that results in 4 turning points, so the graph can "turn around" and hit the next zero. Why can't a curve have more turning points than zeros?
In the graph below, there ... | The problem is that you are confusing real zeros of a polynomial with the degree. These are not the same. The degree of a single variable polynomial is the highest power the polynomial has.
Your hand drawn graph has only 4 real roots, but if it was a polynomial it must have more complex roots. You could not make all... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
How do I factorise this difficult quadratic without a calculator? On one of the UKMT maths challenge past papers(Team challenge) it asks you this question:
Factorise $120x^2 + 97x - 84$
That is the whole question.
I used a calculator and found that you factorise it into $(40x-21)(3x+4)$
Bearing in mind that a calculat... | In this context, "factorise" obviously means that the two bracketed linear terms will have integer coefficients.
Start by listing the possible factor pairs for the first and last terms of the quadratic:
$$\left(\begin{matrix}120\\1\end{matrix}\right),\left(\begin{matrix}60\\2\end{matrix}\right),\left(\begin{matrix}40\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find general solution to the equation $xy^{'}=2x^2\sqrt{y}+4y$ This is Bernoulli's differential equation:
$$xy^{'}-4y=2x^2y^{\frac{1}{2}}$$
Substitution $y=z^{\frac{1}{1-\alpha}},\alpha=\frac{1}{2},y^{'}=z^{'2}$ gives $$xz^{'2}-4z^2=2x^2z$$
Is this correct? What is the method for solving this equation?
| $$xy'(x)=2x^2\sqrt{y(x)}+4y(x)\Longleftrightarrow$$
$$xy'(x)-4y(x)=2x^2\sqrt{y(x)}-4y(x)\Longleftrightarrow$$
$$\frac{y'(x)}{2\sqrt{y(x)}}-\frac{2\sqrt{y(x)}}{x}=x\Longleftrightarrow$$
Let $v(x)=\sqrt{y(x)}$; which gives $v'(x)=\frac{y'(x)}{2\sqrt{y(x)}}$:
$$v'(x)-\frac{2v(x)}{x}=x\Longleftrightarrow$$
Let $\mu(x)=e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Not getting the same solution when using the rule sin(x)\x=1 on a limit There is a rule in limits that when $x$ approaches zero:
$$\frac{\sin\left(x\right)}{x}=1$$
So I used this rule on the following exercise:
Evaluate
$$
\lim _{x\to 0}\:\frac{x-\sin\left(x\right)}{\sin\left(2x\right)-\tan\left(2x\right)}
$$
I... | When taking limits of an expression you cannot arbritraily replace parts of the expression in isolation. You need to calculate the limit of the entire expression.
In your case you could use l'Hopital three times to get:
$$\begin{align}
\lim_{x\to0}\:\frac{x-\sin(x)}{\sin(2x)-\tan(2x)}&=\lim_{x\to0}\:\frac{1-\cos(x)}{2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
How to prove that all odd powers of two add one are multiples of three
For example
\begin{align}
2^5 + 1 &= 33\\
2^{11} + 1 &= 2049\ \text{(dividing by $3$ gives $683$)}
\end{align}
I know that $2^{61}- 1$ is a prime number, but how do I prove that $2^{61}+1$ is a multiple of three?
| $2^2=4\equiv1\pmod 3$, so $4^k\equiv1\pmod3$ for all integers $k$. And so for any odd number $2k+1$, we get $2^{2k+1}+1 = 4^k\cdot 2+1\equiv 2+1\equiv0\pmod3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 11,
"answer_id": 2
} |
Is it possible to get a closed-form for $\int_0^1\frac{(1-x)^{n+2k-2}}{(1+x)^{2k-1}}dx$? It is know that
$$H_n=-n\int_0^1(1-t)^{n-1}\log (t)dt,$$
see [1], where $H_n=1+1/2+\ldots+1/n$ it the nth harmonic number. Then I believe that can be used for $x>0$
$$\frac{1}{2}\log(x)=\sum_{k=1}^{\infty}\frac{(-1)^{2k-1}}{2k-1}\... | (More of a comment, since I think you suggest you know this, but maybe this is useful to somebody else answering.)
According to Mathematica if $k,n\in\mathbb{Z}$ and $2 k+n>1$,
$$I=\frac{\, _2F_1(1,2 k-1;2 k+n;-1)}{2 k+n-1}$$
Where $F$ is the hypergeometric function on MathWorld.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$. Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$.
I tried use the Eisenstein Criterion (with a change variable) but I have not succeeded.
Thanks for your help.
| $f(1)=1$; $f(-1)=-1$; $f(2)=29$; $f(-3)=-251$; $f(4)=1009$; $f(6)=7741$; $f(10)=99901$.
The above seven examples of $f(x)=$prime or $1$, shows that $f$ is irreducible because if not $f(x)=g(x)h(x)$ and f(x) can not be prime seven times.Do you see why? If not, try it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Which of the numbers $300!$ and $100^{300}$ is greater Determine which of the two numbers $300!$ and $100^{300}$ is greater.
My attempt:Since numbers starting from $100$ to $300$ are all greater than $100$. But am not able to justify for numbers between $1$ to $100$
| Using Ross Millikan's suggestion and Ivoirians's idea, let us consider $$f(n)=\log_{100}(n!)-n$$ Now, let us use Stirling approximation for $n!$; this gives $$f(n) =-n+\frac{n (\log (n)-1)}{\log (100)}+\frac{\log (2 \pi n)}{2 \log (100)}+O\left(\sqrt{\frac{1}{n}}\right)$$ So, $$f'(n)\approx \frac{\log (n)}{\log (100)}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1618992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 1
} |
Is it possible to have multiple Conjugate Priors? In Bayesian probability theory, can a probability distribution have more than one conjugate prior for the same model parameters?
I know that the Normal distribution has another Normal distribution or the Normal-Inverse-Gamma distribution has conjugate priors, depending... | If you have a conjugate prior density $f(\theta)$, then $h(\theta)f(\theta)/\int h(\tilde{\theta})f(\tilde{\theta})\,d\tilde{\theta}$ is another conjugate prior for any positive $h$ that is integrable wrt $f$. One therefore typically speaks of a conjugate prior rather than the conjugate prior.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1619077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
To evaluate the given determinant Question: Evaluate the determinant
$\left|
\begin{array}{cc} b^2c^2 & bc & b+c \\
c^2a^2 & ca & c+a \\
a^2b^2 & ab & a+b \\
\end{array}
\right|$
My answer:
$\left|
\begin{array}{cc} b^2c^2 & bc & b+c \\
c^2a^2 & ca & c+a \\
a^2b^2 & ab & a+b \\
\end{array}
\right|= \left|
\begin... | $F=\left|
\begin{array}{cc} b^2c^2 & bc & b+c \\
c^2a^2 & ca & c+a \\
a^2b^2 & ab & a+b \\
\end{array}
\right|$
$=\dfrac1{abc}\left|
\begin{array}{cc} ab^2c^2 & abc & a(b+c) \\
c^2a^2b & bca & b(c+a) \\
a^2b^2c & abc & c(a+b) \\
\end{array}
\right|$
$=\left|
\begin{array}{cc} ab^2c^2 &1& a(b+c) \\
c^2a^2b &1& b(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1619175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Disprove bijection between reals and naturals Coming across diagonalization, I was thinking of other methods to disprove the existence of a bijection between reals and naturals. Can any method that shows that a completely new number is created which is different from any real number in the list disprove the existence ... | Your idea for a falsification is correct, however your method has to show that your new number is not already part of your list.
If you were right with your example, then there would also be no bijection between rational numbers and natural numbers, but there is (which can be proved by diagonalization).
Your example of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1619386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Does $\sum_{k=1}^{\infty}\frac{k!}{k^k}$ converge? I have tried using ratio test:
$$P =\lim_{k\rightarrow\infty}\left|\frac{(k+1)!}{(k+1)^{k+1}}\cdot\frac{k^k}{k!}\right|$$
$$ P=\lim_{k\rightarrow\infty}\left|\frac{(k+1)\cdot k^k}{(k+1)^{k+1}}\right|$$
$$ P=\lim_{k\rightarrow\infty}\left|\frac{k^{k+1}+ k^k}{(k+1)^{k+1}... | You should proceed as follows:
$$P =\lim_{k\rightarrow\infty}\left|\frac{(k+1)!}{(k+1)^{k+1}}\cdot\frac{k^k}{k!}\right|$$ or
$$ P=\lim_{k\rightarrow\infty}\left|\frac{(k+1)\cdot k^k}{(k+1)^{k+1}}\right|$$ or
$$ P=\lim_{k\rightarrow\infty}\left|\frac{k^k}{(k+1)^{k}}\right|$$ or
$$ P=\frac{1}{\lim_\limits{k\rightarrow\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1619509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 0
} |
How to solve the following system of partial differential equations? I have a system of partial differential equations:
\begin{align}
& u(a,b,c) \frac{\partial y}{\partial c} = \frac{4}{3} ab, \\
& u(a,b,c) \frac{\partial y}{\partial b} = \frac{2}{3} ac + 2 b^2, \\
& u(a,b,c) \frac{\partial y}{\partial a} = \frac{4}{3}... | The command "pdsolve([pde]);" can solve this system of partial differential equations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1619596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Prove that $2\sin^{-1}\sqrt x - \sin^{-1}(2x-1) = \frac{\pi}{2}$. Prove that $2\sin^{-1}\sqrt x - \sin^{-1}(2x-1) = \dfrac\pi2$.
Do you integrate or differentiate to prove this equality? If so, why?
| alternative to differentiating, let $$\phi=2\sin^{-1}\sqrt{x}$$
$$\implies x=\sin^2(\phi/2)=\frac 12(1-\cos \phi)$$
$$\implies \cos\phi=1-2x$$
$$\implies\phi=\cos^{-1}(1-2x)=\frac{\pi}{2}-\sin^{-1}(1-2x)=\frac{\pi}{2}+\sin^{-1}(2x-1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1619719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Difference between Gamma and partial Omega to describe a domain boundary Is there a difference between referring to the boundary of a domain $\Omega$ as $\Gamma$ or $\partial \Omega$ ? Or is this just preference or synonyms of the same thing? From my experience, they seem to be used arbitrarily, but I feel like I might... | It depends on the definition of $\Gamma$ in your context. At least in finite element literature $\Gamma$ is often defined to be the whole boundary of the domain $\Omega$, in other words the same as $\partial\Omega$, but not always. Scott and Brenner, for instance, occasionally defines $\Gamma$ as a part of the boundary... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1619813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Why $e^x$ is always greater than $x^e$? I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$
I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the reason behind the above inequality? I know this is math community, but I'd apprecia... | Here is a different approach, just for the sake of variety, which could be made more rigorous:
Consider the tangent to the curve $y=\ln x$ at the point $x=e,y=1$ is $$y-1=\frac 1e(x-e)\implies y=\frac xe$$
We know that the curve is concave so lies below the tangent except at the tangent point.
Therefore, $$\ln x\leq\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1619911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 8,
"answer_id": 2
} |
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