Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to guess that $ \sum_{i=1}^{n}3^i = \frac{3}{2}(3^n - 1)$ As in title how do you guess that $ \sum_{i=1}^{n}3^i = \frac{3}{2}(3^n - 1)$?
I have homework about solving recurrence relations and using iterate method I can find that http://www.wolframalpha.com/input/?i=RSolve%5B%7Ba%5Bn%5D+%3D+a%5Bn-1%5D+%2B+3%5En%2C+a... | Well... When I look deep inside of me and try to really see how I would guess it, it comes to this anecdote:
An infinite number of mathematicians walk into a bar.
The first asks the bartender for a beer.
The second asks for half a beer. The third one says he wants a fourth of a beer.
The bartender interrupts,... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 2
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Find the expressions for the common difference and common ratio
The first 2 terms of a geometric progression (first term $a$ and common ratio $r$) are the same as the first 2 terms of an arithmetic progression (first term $a$ and common difference $d$). The third term of the geometric progression is twice as big as th... | I'm afraid that you've done nothing wrong. (There's an odd thing to say.)
It's worth noting that if $a=0,$ then we get $d=0$ immediately, but $r$ can take literally any value. Since the problem expects two possible values for $r,$ then we clearly must assume $a\neq0.$
Since $a\ne 0,$ then $r^2-4r+2=0,$ and from there, ... | {
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"url": "https://math.stackexchange.com/questions/532119",
"timestamp": "2023-03-29T00:00:00",
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Expressing a product in a Dihedral group Write the product $x^2yx^{-1}y^{-1}x^3y^3$ in the form $x^iy^j$ in the dihedral group $D_n$.
I used the fact that the dihedral group is generated by two elements $x$ and $y$ such that: $y^n=1$, $x^2=1$ and $xy=y^{-1}x$
and I found that $x^2yx^{-1}y^{-1}x^3y^3=y^5$
Is it correct ... | Yes, I think, except that we can also write
$$ x^2yx^{-1}y^{-1}x^3y^3 = 1yxy^{-1}xy^3 = yxxyy^3 = y1y^4 = y^5 = y^{(5 \mod n)}.$$ Here we have used the fact that $x^2=1$ and so $x^{-1} = x$, and $5 \mod n$ denotes the remainder when $5$ is divided by $n$ in accordance with the Euclidean division algorithm.
| {
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"timestamp": "2023-03-29T00:00:00",
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Rearrangement of double infinite sums Lets say I have a double infinite sum $\sum\limits_{n=0}^\infty a_n\sum\limits_{k=n}^\infty b_k$ If I know that the sum is absolutely convergent and that $\sum\limits_{k=n}^\infty b_k$ is absolutely convergent, what kind of rearrangements are valid?
For example I would like expand... | If the series $\sum_{n=1}^\infty |a_n| \sum_{k=n}^\infty |b_k| $ converges, then you can do whatever you want with $a_n b_k$; any rearrangement will converge to the same sum. More generally: if $I$ is an index set and $\sum_{i\in I} |c_i|<\infty$, then any rearrangement of $c_i$ converges to the same sum. This is becau... | {
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How to solve equations of this form: $x^x = n$? How would I go about solving equations of this form:
$$
x^x = n
$$
for values of n that do not have obvious solutions through factoring, such as $27$ ($3^3$) or $256$ ($4^4$).
For instance, how would I solve for x in this equation:
$$x^x = 7$$
I am a high school student, ... | The solution involves a function called Lambert's W-function. There is a Wikipedia page on it.
As Oliver says, it is not a nice neat form. It is an entirely new function. Its definition is $$ye^y=z, W(z)=y$$
Can you use logs to turn $x^x=n$ into $ye^y=f(n)$, for $y$ equal to some function of $x$?
| {
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How to analyze the asymptotic behaviour of this integral function? Based on the asymptotic analysis of correlation functions at large distence in Physics, now I get a math question. Let the function $$f(x)=\int_{-1}^{1}\sqrt{1-k^2}e^{ikx}dk.$$
Without working out the explicit form of this function, how do we know the a... | I going to manipulate the integral into a form that I can analyze the method of stationary phase. Let $k=\cos{\theta}$ and the integral becomes
$$\begin{align}I(x) &= \int_0^{\pi} d\theta \, \sin^2{\theta} \, e^{i x \cos{\theta}}\\ &= \int_0^{\pi} d\theta \, e^{i x \cos{\theta}} - \int_0^{\pi} d\theta \, \cos^2{\thet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/532394",
"timestamp": "2023-03-29T00:00:00",
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How do you factor this using complete the square? $6+12y-36y^2$ I'm so embarrassed that I'm stuck on this simple algebra problem that is embedded in an integral, but I honestly don't understand how this is factored into $a^2-u^2$
Here are my exact steps:
$6+12y-36y^2$ can be rearranged this way: $6+(12y-36y^2)$ and I k... | Here is the first step:
$$6+12y-36y^2 = -\left((6y)^2-2\cdot (6y)-6\right)$$
A complete square would be
$$\left(6y-1\right)^2 = (6y)^2-2\cdot (6y) + 1.$$
Ignoring the overall minus for a moment and adding and subtracting 1 to the r.h.s. of the first equation above, we get (using $a^2-b^2=(a-b)(a+b)$ formula:
$$(6y)^2-2... | {
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Are there any open mathematical puzzles? Are there any (mathematical) puzzles that are still unresolved? I only mean questions that are accessible to and understandable by the complete layman and which have not been solved, despite serious efforts, by mathematicians (or laymen for that matter)?
My question does not ask... | In his comment, user Vincent Pfenninger referred to a YouTube video that, amongst other fascinating, layman accessible puzzles, discusses packing squares problems proposed by Paul Erdős. I thought I'd include it among the answers (as a community wiki).
How big a square do you need to hold eleven little squares?
We do... | {
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Simple Polynomial Interpolation Problem Simple polynomial interpolation in two dimensions is not always possible. For example, suppose that the following data are to be represented by a polynomial of first degree in $x$ and $y$, $p(t)=a+bx+cy$, where $t=(x,y):$
Data: $f(1,1) = 3, f(3,2)=2, f(5,3)=6$
Show that it is not... | Since you want a polynomial of degree $\leqslant 1$, you have the three equations
$$\begin{align}
a + b + c &= 3\tag{1}\\
a + 3b + 2c &= 2\tag{2}\\
a + 5b + 3c &= 6\tag{3}
\end{align}$$
Subtracting $(1)$ from $(2)$ yields $2b + c = -1$, and subtracting $(2)$ from $(3)$ yields $2b + c = 4$. These are incompatible.
| {
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Use the relation of Laplace Transform and its derivative to figure out $L\left\{t\right\}$，$L\left\{t^2\right\}$，$L\left\{t^n\right\}$ If $F(s) = L\left\{f(t)\right\}$, then $F'(s) = -L\left\{tf(t)\right\}$
Use this relation to determine
$(a)$ $L\left\{t\right\}$
$(b)$ $L\left\{t^2\right\}$
$(c)$ $L\left\{t^n\right\}$ ... | Induction. You know how to write down $F'(s)$. Now, $\mathscr L\{t\cdot t^n\}=\cdots$?
For example, $\mathscr L\{t^3\}=\mathscr L\{t\cdot t^2\}=\mathscr L\{tf(t)\} $ with $f(t)=t^2$. If you knew that $\mathscr L\{t^2\}=\dfrac{2!}{s^3}$ then $\mathscr L\{t^3\}=-F'(s)=-\dfrac{d}{ds}\left(\dfrac{2!}{s^3}\right)=\dfrac{3... | {
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Differential Equation $y(x)'=(y(x)+x)/(y(x)-x)$ can someone give me some tips on how to solve this differential equation.
I looked at the Wolfram solution which substituted $y(x)=xv(x)$. I'd know how to solve from there, but I have know idea why they did it in the first place, well why the algorithm did it in the first... | Let's assume a equation scaling $x \to \alpha x$ and $y \to \beta y$. Under such scaling the equation becomes
$$
y'
=
{\alpha \over \beta}\,
{y + \left(\alpha/\beta\right)x \over y - \left(\alpha/\beta\right)x}
$$
Then, we can see the equation doesn't change its form whenever $\alpha = \beta$. It means $y/x$ doesn't ch... | {
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How do symbolic math software work? As the answer to a question of mine I was referred to a website (see here please)
How can WolframAlpha do it like humans?
| For the specific system (Mathematica) that you mentioned, there are descriptions of its internals on these web pages and these. But Mathematica is a commercial system, so its internal workings are proprietary, which is why the descriptions don't provide much detail.
While these systems might appear to work "the same w... | {
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Probability : A bag contains 12 pair socks . Four socks are picked up at random. Find the probability that there is at least one pair. Probability : A bag contains 12 pair socks . Four socks are picked up at random. Find the probability that there is at least one pair.
My approach :
Number of ways of selecting 4 sock... | The error is in the second line. The number of ways to select four different pairs must be multiplied by the number of ways to select just one sock from each of those four selected pairs.
This is: (12C4 × 24)
| {
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My conjecture on almost integers. Here when I was studying almost integers , I made the following conjecture - Let $x$ be a natural number then For sufficiently large $n$ (Natural number) Let $$\Omega=(\sqrt x+\lfloor \sqrt x \rfloor)^n$$ then $\Omega$ is an almost integer . The value of $n$ depends upon the differe... | Well, we have that $$(\sqrt{x}+\lfloor\sqrt{x}\rfloor)^{n} + (-\sqrt{x}+\lfloor\sqrt{x}\rfloor)^{n} $$
is an integer (by the Binomial Theorem), but $(-\sqrt{x}+\lfloor\sqrt{x}\rfloor)^{n}\to 0$ if $\sqrt{x}$ was not already an integer.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Proving an identity We define $\|x\|_A^2:= x^TAx$ and $(x,y)_M := y^TMx$ for a symmetric positive definite matrix $A$ and an invertible matrix $M$.
I want to show the following identity for the errors of Richardson's method for solving an equation system $Ax=b$:
$$\frac{\|e_k\|^2_A - \|e_{k+1}\|_A^2}{\|e_k\|_A^2} = \fr... | I guess this actually does hold. Not for the (preconditioned) Richardson method but for the (preconditioned) steepest descent method.
In the preconditioned SD method, you update your $x_k$ such that the $A$-norm of the error of $x_{k+1}$ is minimal along the direction of the preconditioned residual $y_k=M^{-1}r_k$, tha... | {
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If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ .... Problem :
If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ then the value of $(1-\alpha_1)(1-\alpha_2)(1-\alpha_3)(1-\alpha_4)$ is
(a) 2$\sqrt{3}$
(b) 5
(c) 1
(d) 4
My... | Let give a polinom $p_4(x)=a_4 x^4+b_3x^3+c_2x^2+d_1x+e$. Use formula:
$$x_1+x_2+x_3+x_4=-\frac{b}{a}$$
$$x_1x_2+x_1x_3+x_2x_3+x_1x_4+x_2x_4+x_3x_4=\frac{c}{a}$$
$$x_1x_2x_3+x_1x_2x_4+x_1x_3x_4+x_2x_3x_4=-\frac{d}{a}$$
$$x_1x_2x_3x_4=\frac{e}{a}$$
where $x_1,x_2,x_3,x_4$ is rrots of the given polinom.
For the given exa... | {
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Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$? For any $C^1$ function defined in $(0,1)$, is it true that
$$ 2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}
$$
If it is true, how to prove it?
| If $x,y \in (0,1)$ then $$|f(x)| \le |f(y)| + |f(x) - f(y)| \le |f(y)| + \int_x^y |f'(t)| \, dt$$ so that $$ |f(x)| \le |f(y)| + |x-y|^{1-1/p} \left( \int_x^y |f'(t)|^p \, dt\right)^{1/p}$$ by e.g. Holder's inequality. This leads to $$|f(x)|^p \le 2^p |f(y)|^p + 2^p |x-y|^{p-1} \int_x^y |f'(t)|^p \, dt \le 2^p |f(y)|^p... | {
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Set theory relation: irreflexive and transitive Which of the following relations on $T = \{1, 2, 3\}$ is irreflexive and transitive.
*
*$\{(2, 1), (2, 3)\}$
*$\{(1, 1), (2, 1), (3, 2)\}$
*$\{(2, 1), (1, 2), (3, 2), (2, 3)\}$
*$\{(1, 1), (2, 2), (3, 3), (2, 1), (1, 2)\}$
From my understanding 2 and 4 are exclude... | $(1)$ is transitive, because the condition of transitivity is vacuously satisfied. There are no elements related in such a way for transitivity to fails, hence, by default, the relation is transitive.
$(3)$ is not transitive, because $(3, 2), (2, 3) \in R$ but $(3, 3)\notin R$.
| {
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How to solve the inequality $n! \le n^{n-2}$? The inequality is $n! \le n^{n-2}$. I used Stirling's approximation for factorials and my answer was $n \le (e(2\pi)^{-1/2})^{2/5}$ but this doesn't seem right. Any help would be much appreciated.
| Here's an elementary proof, with some details left out to be filled in by you :).
Taking logarithms, you want to show that
$$
S_n = \log 2 + \log 3 + \dots \log n \le (n-2) \log n \, .
$$
Note that there are $n-1$ terms on the left, each less than $\log n$. So you only have to "squeeze out" an additional term $\log n$.... | {
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Is $\mathbb{Q}[α]=\{a+bα+cα^2 :a,b,c ∈ \mathbb{Q}\}$ with $α=\sqrt[3]{2}$ a field? I'm making some exercises to prepare for my ring theory exam:
Is $\mathbb{Q}[α]=\{a+bα+cα^2 :a,b,c ∈ \mathbb{Q}\}$ with
$α=\sqrt[3]{2}$ a field ?
If $(a+bα+cα^2)(a'+b'α+c'α^2)=1$, then (after quite some calculation and noticing that ... | There is also a low-brow approach.
It will suffice that the three equations you wrote out always have a solution $a',b',c'$ for any $a,b,c$, except $0,0,0$.
This is a linear system of equations, with the corresponding matrix $$A = \begin{bmatrix} a & 2c & 2b \\ b & a & 2c \\ c & b & a
\end{bmatrix}.$$
So, to check tha... | {
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Multiplication of projection matrices Let $A$ and $B$ be two projection matrices having same dimensions. Then does the following hold
\begin{equation}
AB\leq I,
\end{equation}
where $I$ is the identity matrix. In other words is it true that $I-AB$ is positive semi-definite.
| It's not clear to me what you mean by matrix inequality, but assuming you mean an inequality componentwise, it's clearly false, for instance let
$$A = B = \frac{1}{2} \left[\begin{array}{cc}1 & 1\\1 &1\end{array}\right]$$
be projection onto the vector $(1,1)$.
Then $AB = A$ has positive off-diagonal entries so violates... | {
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What finite fields are quadratically closed? A field is quadratically closed if each of its elements is a square.
The field $\mathbb{F}_2$ with two elements is obviously quadratically closed.
However, testing some more finite fields with this property, I didn't find any more. Hence my question is:
Which finite fields ... | Consider the squaring map from the multiplicative group of a finite field $F$ to itself. The kernel is $\{\pm1 \}$, i.e., it is trivial if and only if the characteristic of $F$ is $2$. Since this map is surjective if and only if it is injective, every element of $F$ is a square if and only if the characteristic of $F$ ... | {
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A criterion for weak convergence of probability measures
Let $\mathbb P_n$ and $\mathbb P$ be probability measures. We have that $\mathbb P_n$ converges weakly to $\mathbb P$ if for each continuous bounded function, $\int f(x)\mathrm d\mathbb P_n\to\int f(x)\mathrm d\mathbb P$. Show that $\mathbb P_n$ weakly converges... | The first part in contained in the statement of portmanteau theorem. One direction is not specially hard (approximate pointwise the characteristic function of a closed set by a continuous function in order to get $\limsup_n\mathbb P_n(F)\leqslant \mathbb P(F)$).
For the second one, the question reduces to the following... | {
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Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly $4$ distinct real roots Problem :
Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly four distinct real roots, then which of the following options are correct :
(a) $f(x) =0$ has all three real roots
(b) $f(x) =0$ has exactly... | Both $f(x)=(x-1)(x-4)^2$ (two real roots including one double root), and $f(x)=(x+1)(x-1)(x-4)$ (three real roots) fulfit the conditions of the problem, with $f(x^2)=0$ having roots in $x=\pm1$ and $x=\pm2$.
| {
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Prove this limit as $x\to\infty$ Let $f:(a,\infty)\to\mathbb{R}$ be a function. Suppose for each $b>a$, $f$ is bounded on $(a,b)$ and $\lim_{x\to\infty}f(x+1)-f(x)=A$. Prove
$$\lim_{x\to\infty}\frac{f(x)}{x}=A.$$
Here we assume $f$ to be arbitrary and no further conditions. $f$ could be continuous,discontinuous,as long... | We have that $$\lim_{x\to\infty}f(x+1)-f(x)=A$$ which means that for any $\varepsilon\in\mathbb R^+$ there is an $n\in\mathbb N$ such that for each $x>n$ then $A-\varepsilon<f(x+1)-f(x)<A+\varepsilon$. (if we find some $n<a$ for this condition then we can chose $n=\lceil a\rceil$ for the remind of the proof.)
For simp... | {
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A $2 \times 2$ matrix $A$ such that $A^n$ is the identity matrix So basically determine a $2 \times 2$ matrix $A$ such that $A^n$ is an identity matrix, but none of $A^1, A^2,..., A^{n-1}$ are the identity matrix. (Hint: Think geometric mappings)
I don't understand this question at all, can someone help please?
| Hint: Rotate through $\frac{2\pi}{n}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Map induced by $O(n)\hookrightarrow U(n)$ on homotopy groups There is an inclusion $O(n)\hookrightarrow U(n)$ which views an $n\times n$ orthogonal matrix as a unitary matrix. It is also a theorem, sometimes called Bott periodicity, that we have the following homotopy groups:
$\pi_{4i-1}(O(n)) = \mathbb{Z}$ and $\pi_{4... | I had this question myself a few days ago and found this question, but I now have an answer with a hint from my supervisor.
The key seems to be in a theorem (3.2) in 'Homotopy theory of Lie Groups' by Mimura, which can be found in the Handbook of Algebraic Topology. The theorem gives a weak homotopy equivalence
$$BO\ri... | {
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Is this also a homotopy If $F$ is a path at a point $x$ then the following defines a homotopy from the path $FF^{-1}$ to the constant path $e$:
$$ \begin{array}{cc}
H(t,s) = F(2t) & s \ge 2t \\
H(t,s) = F(s) & s \le 2t \land s \le -2t + 2 \\
H(t,s) = F(2-2t) & s \ge -2t + 2
\end{array}$$
Is it possible that the followi... | As far as I can see, you haven't described what happens when $t\geq s$ for instance what is $H(\frac{3}{4},\frac{1}{2})$? And so no, this does not define a homotopy as it is not even a well defined function.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to compute probability of winning The producer places at random a valuable prize behind one of the three doors, numbered 1 to 3, and nothing behind the remaining two and each door is equally likely to hide the prize. If you have selected a door that hides a prize, the host will always open the smaller-numbered one ... | If the prize was behind door $1$, you lose or win with probability $\frac12$ each.
If the prize was behind door $2$, you lose, since you never switch to door $2$.
If the prize was behind door $3$, you win, since the host opens door $1$ and you stick with door $3$.
Thus you win with probability $\frac13\left(\frac12+0+1... | {
"language": "en",
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Solve $x^2 = I_2$ where x is a 2 by 2 matrix I tried a basic approach and wrote x as a matrix of four unknown elements $\begin{pmatrix} a && b \\ c && d \end{pmatrix}$ and squared it when I obtained $\begin{pmatrix} a^2 + bc && ab + bd \\ ca + dc && cd + d^2\end{pmatrix}$ and by making it equal with $I_2$ I got the fo... | The second and third equalities are
$$b(a+d)=0$$
$$c(a+d)=0$$
Split the problem in two cases:
Case 1: $a+d \neq 0$. Then $b=0, c=0$, and from the first and last equation you get $a,d$.
Case 2: $a+d=0$. Then $d=-a$ and $bc=1-a^2$. Show that any matrix satisfying this works.
| {
"language": "en",
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"source": "stackexchange",
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Laurent series in all possible regions of convergence Let $f=\frac{2}{z}-\frac{3}{z-2}+\frac{1}{z+4}$ . Find the Laurent series in all possible regions of convergence about z=0 and read the residuum.
I am not sure if I have to consider all 3 regions (inside circle pf radius 2,
inside annulus $2<r<4$ and ouside circl... | Note that for $z\neq -4,0,$ we can write $$\frac1{z+4}=\cfrac1{4-(-z)}=\frac1{4}\cdot\cfrac1{1-\left(-\frac{z}{4}\right)}$$ and $$\frac1{z+4}=\cfrac1{z-(-4)}=\frac1{z}\cdot\cfrac1{1-\left(-\frac{4}{z}\right)}.$$
Now, one of these can be expanded as a multiple of a geometric series in the disk $|z|<4,$ and the other can... | {
"language": "en",
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Evaluate this power series Evaluate the sum
$$x+\frac{2}{3}x^3+\frac{2}{3}\cdot\frac{4}{5}x^5+\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}x^7+\dots$$
Totally no idea. I think this series may related to the $\sin x$ series because of those missing even powers. Another way of writing this series:
$$\sum_{k=0}^{\infty}\fra... | In this answer, I mention this identity, which can be proven by repeated integration by parts:
$$
\int_0^{\pi/2}\sin^{2k+1}(x)\;\mathrm{d}x=\frac{2k}{2k+1}\frac{2k-2}{2k-1}\cdots\frac{2}{3}=\frac{1}{2k+1}\frac{4^k}{\binom{2k}{k}}\tag{1}
$$
Your sum can be rewritten as
$$
f(x)=\sum_{k=0}^\infty\frac1{(2k+1)}\frac{4^k}{\... | {
"language": "en",
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How to I find the distribution of $\log p(X)$ when $X$ is distributed under $p$? I have a feeling there's no general solution to this problem, but I'll ask anyway.
I have a multivariate PDF $p$ and, given a random vector $X\sim p$, I'd like to find the the PDF of $\log p(X)$.
For example, if I have a simple 2-dimension... | I think this is achieved as follows:
*
*augment $\log(p)$ to form a vector of the same dimension of $X$ (call this vector $Y$);
*partition the support of $Y$ into regions where $Y = g(X)$ is a one-to-one function of $X$ (hence invertible);
*use the standard results on transformations of multivariate probability d... | {
"language": "en",
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Computing the total curvature Let $C$ be the curve in $\Bbb{R}^2$ given by $(t-\sin t,1-\cos t)$ for $0 \le t \le 2 \pi$. I want to find the total curvature of $C$.
I found it brutally by finding the curvature $k(t)$, and then reparametrize it by arc-length $s$, and then $\int_0^Lk(t(s))ds$, where $L=8$ is the lenght o... | A simpler answer is as follows: given the natural equation for a curve in the form of $\kappa(s)$, it can be shown that the tangent angle is given by
$$\theta=\int \kappa (s) ds \ \ \ \text{or} \ \ \ \kappa (s)=\frac{d\theta}{ds}$$
Thus from the definition of curvature used here, we obtain
$$K=\int_{s_1}^{s_2}\kappa(s... | {
"language": "en",
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What's the limit of this sequence? $\lim_{n \to \infty}\frac{1}{n}\bigg(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\cdots + 1 \bigg)$
My attempt:
$\lim_{n \to \infty}\frac{1}{n}\bigg(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\cdots + 1 \bigg)=\lim_{n \to \infty}\bigg(\frac{\sqrt{1}}{\sqrt{n^3}}+\frac{\sqrt{2}}{\sqrt{n^3}}+\cdot... | $\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\isdiv}{\,\left.\right\vert\,}%
\newcommand{\ds}[1]{\displaystyle{#1}}%
\newcommand{\equalby}[1]{{#1 \atop... | {
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Product / GM of numbers, with fixed mean, increase as numbers get closer to mean. I am trying to prove a statement which goes like this.
Let $a_i$ and $b_i$ be positive real numbers where $i = 1,2,3,\ldots,n$; where $n$ is a positive integer greater than or equal to $2$, such that,
$$0\lt a_1\le a_2 \le \ldots\le a_n \... | If we only 'move' two elements of the sequence $(a_k)_k$, letting $b_i:=a_i$ except for $i=j,k$ ($j<k$) when $b_j:=a_j+\varepsilon$ and $b_k:=a_k-\varepsilon$, then we have
$$b_jb_k=(a_j+\varepsilon)(a_k-\varepsilon) = a_ja_k+\varepsilon\,(a_k-a_j\, -\varepsilon) \ > \ a_ja_{k}$$
using that $a_k>a_j+\varepsilon$. (Actu... | {
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Quotient $K[x,y]/(f)$, $f$ irreducible, which is not UFD Does anyone know an irreducible polynomial $f \in K[x,y]$ such that the quotient $K[x,y]/(f)$ is not a UFD? Is it known when this quotient is a UFD?
Thanks.
| Yes, I think that the standard example is to take $f(x,y)=x^3-y^2$. Then when you look at it this is the same as the set of polynomials in $K[t]$ with no degree-one term. That is, things that look like $c_0+\sum_{i>1}c_it^i$, the sum being finite, of course. You prove this by mapping $K[x,y]$ to $K[t]$ by sending $x$ t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Preparedness for a graduate course in Complex Analysis I am entering graduate school next year without any background in Complex Analysis. I have, however, taken 2 semesters of Real Analysis and a reader course in Measure Theory (using Bartle's Elements of Integration and Lebesgue Measure). I can, of course, brush up o... | When you start a course of complex analysis you want to be sure that you have a rock solid basis on complex numbers, de Moivre and solving basic complex equations. You also may want to practice Completing the Square on quadratic equations with complex coefficients. I found that preparation very useful when I did a cour... | {
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number theory proof Does this proof work?
Prove or disprove that if $\sigma(n)$ is a prime number, n must be a power of a prime.
Since $\sigma(n)$ is prime, $n$ can not be prime unless it is the only even prime, $2$, since $\sigma(n)$ for prime $n=p+1$ which will always be even and therefore not a power of a prime.
Now... | Not quite. $\sigma$ is a multiplicative function, as you said, which means that if $\textrm{gcd}(a,b)=1$, then $\sigma(a)\sigma(b)=\sigma(ab)$. But with $a=b=p$, we clearly don't have $\textrm{gcd}(a,b)=1$. By looking at the contrapositive of that statement, you can see that you're being asked to show is that if $n$ is... | {
"language": "en",
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"source": "stackexchange",
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Cyclic groups question Show that $\mathbb Z_{35}^\times$ is not cyclic.
I assume that I need to show that no element of $\mathbb Z_{35}$ has a particular order, indicating it is not cyclic, but I'm not sure how to do this.
| Your group $G=\mathbb Z_{35}^{\times}$ has order $24$, right? So all you have to do is show that none of the $24$ elements of $G$ has order $24$. Let's just do it.
Let's start by picking an element of $G$ at random and computing its order. I picked $2$, and computed: $2^1=2$, $2^2=4$, $2^3=8$, $2^4=16$, $2^5$=32, $2^6=... | {
"language": "en",
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"source": "stackexchange",
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Orthonormal Sets and the Gram-Schmidt Procedure
What my problem in understanding in the above procedure is , how they
constructed the successive vectors by substracting? Can you elaborate
please?
| Let $|w_1\rangle=|u_1\rangle$. Let us find $|w_2\rangle$ as a linear combination of $|w_1\rangle$ and $|u_2\rangle$. Suppose
$$
|w_2\rangle = \lambda |w_1\rangle + |u_2\rangle.
$$
Now we wish that $\langle w_1 | w_2\rangle =0$ (since $|w_1\rangle$ and $|w_2\rangle$ must be orthogonal). Multiply the above equality by ... | {
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The existence of inequalities between the sum of a sequence and the sum of its members Let $(a_n)_{n=m}^\infty$ be a sequence of positive real numbers. Let $I$ denote some finite subset of $M := \{m, m+1, \cdots \}$, i.e., $I$ is the index of some points of $(a_n)_{n=m}^\infty$.
Does there exist a real number $r$ such ... | No. Since each $a_n$ is positive you have $$\sum_{n=m}^\infty a_n = \sup_I \sum_{i \in I} a_i$$ where the supremum is taken over all finite subsets $I$ of $M$. This is a consequence of the monotone convergence theorem for series.
| {
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How find the integral $I=\int_{-R}^{R}\frac{\sqrt{R^2-x^2}}{(a-x)\sqrt{R^2+a^2-2ax}}dx$ Find the integral:
$$I=\int_{-R}^{R}\dfrac{\sqrt{R^2-x^2}}{(a-x)\sqrt{R^2+a^2-2ax}}\;\mathrm dx$$
My try:
Let $x=R\sin{t},\;t\in\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$
then,
$$I=\int_{-\pi/2}^{\pi/2}\dfrac{R\cos{t}}{(a-R... | The second line of the OP is
$$I=\int_{-\pi/2}^{\pi/2} \frac{\cos^2 t}
{(\alpha -\sin t)\sqrt{1+\alpha^2-2\alpha \sin t}}dt$$
$$=\int_{-\pi/2}^{\pi/2} \frac{1-\sin^2 t}
{(\alpha -\sin t)\sqrt{1+\alpha^2-2\alpha \sin t}}dt$$
where $\alpha\equiv a/R$.
Partial fraction decomposition yields
$$=\int_{-\pi/2}^{\pi/2} [\sin t... | {
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Compare this topology with the usual topology I have to compare the following topology with the usual one. Which of them is finer?
$\tau= \{U\subseteq \mathbb{R}^2:$ for any $(a,b) \in U$ exists $\epsilon >0 $ where $[a,a+\epsilon] \times [b-\epsilon, b+\epsilon]\subseteq U\}$
By definition, $\tau\subseteq\tau_u $ if ... | *
*If $U \in \tau_u$, then for any $(a,b) \in U$, there is a basic open set
$$
(a-\delta, a+\delta)\times (b-\delta', b+\delta') \subset U
$$
you can take
$$
\epsilon = \min\{\delta/2, \delta'/2\}
$$
then
$$
[a,a+\epsilon]\times [b-\epsilon,b+\epsilon] \subset U
$$
Hence $U \in \tau$. So
$$
\tau_u \subset \tau
$$
*Th... | {
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prove that quadrangle is isosceles trapezoid How to prove that quadrangle $ABCD$ is a isosceles trapezoid?
where $AB$ is parallel to $CD$
| You are given two facts:
1. A qualrilateral ABCD is inscribed in a circle.
2. AB is parallel to CD
Because of the parallel lines, qualrilateral ABCD is, by definition, a trapezoid.
To prove that trapezoid ABCD is isosceles, you need to show that the non-parallel sides BD and AC have equal lengths. This can be... | {
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how to find center/radius of a sphere Say you have an irregular tetrahedron, but you know the (x,y,z) coordinates of the four vertices; is there a simple formula for finding a sphere whose center exists within the tetrahedron formed by the four points and on whose surface the four points lie?
| Given four points, a, b, c, and d, you can find the center by setting the following determinant to zero and solving it:
$$
\begin{vmatrix}
(x^2 + y^2 + z^2) & x & y & z & 1 \\
(ax^2 + ay^2 + az^2) & ax & ay & az & 1 \\
(bx^2 + by^2 + bz^2) & bx & by & bz & 1 \\
(cx^2 + cy^2 + cz^2) & cx & cy & cz & 1 \\
(dx^2 + d... | {
"language": "en",
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"source": "stackexchange",
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Dense subspace of linear functionals We know that any real measurable function can be approximated by increasing simple functions. So, integral of real valued measurable function can be written as a limit of integrals of simple functions. We can observe that the integral of simple functions is just a linear combination... | Note that you need to require your functions to be bounded for the norm to make sense. So $L=\ell^\infty(X)$.
Regarding your question, let us take $X=\mathbb N$. It is well-known that there are functionals that annihilate all finitely supported functions. Concretely, you take any free ultrafilter $\omega\in\beta\mathb... | {
"language": "en",
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An interesting (unknown) property of prime numbers. I don't know if this is the right place to ask this question. Please excuse my ignorance if it is not.
I like to play with integers. I have been doing this since my childhood. I spend a lot of time looking up new integer sequences on OEIS. Last week I stumbled upon a ... | I believe that posting your observation here will clarify that the observation is yours.
Also as Bill Cook mentions http://arxiv.org/ would be the best place to post your discovery.
But if you don't tell us which is the ''unknown property of prime numbers'' you will never know if you are a genius, or someone who has i... | {
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Proof that $\int_0^{2\pi}\sin nx\,dx=\int_0^{2\pi}\cos nx\,dx=0$
Prove that $\int_0^{2\pi}\sin nx\,dx=\int_0^{2\pi}\cos nx\,dx=0$ for all integers $n \neq 0$.
I think I'm encouraged to prove this by induction (but a simpler method would probably work, too). Here's what I've attempted: $$\text{1.}\int_0^{2\pi}\sin x\,... | I do not know if you can use $e^{ix}=\cos x+i\sin x$ but here is one solution:
$$
\int_0^{2\pi}e^{inx}\,dx=\frac 1{in}e^{inx}\left|_0^{2\pi}\right.=0
$$
| {
"language": "en",
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"source": "stackexchange",
"question_score": "8",
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Proof of Aristarchus' Inequality Does anyone know how to prove that if $0<\alpha<\beta<\frac{\pi}{2}$ then $\frac{\sin\alpha}{\alpha}>\frac{\sin\beta}{\beta}$. Any methods/techniques may be used.
| The function $f(x)=\frac{\sin(x)}{x}$ is decreasing on $[0,\pi/2]$. Since its derivative is
$$
f^{'}(x)=\frac{\cos(x)x-\sin(x)}{x^2},
$$
we've reduce the problem to seeing that $\cos(x)x-\sin(x)\leq 0$. For small values of $x$, we have $\sin(x)\approx x$, so $\cos(x)x-\sin(x)\approx x(\cos(x)-1)$, which is negative. To... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Pullback of a Volume Form Under a Diffeomorphism. I have an exercise here, which I have no idea how to do.
Problem: Let $ U $ and $ V $ be open sets in $ \mathbb{R}^{n} $ and $ f: U \to V $ an orientation-preserving diffeomorphism. Then show that
$$
{f^{*}}(\text{vol}_{V})
= \sqrt{\det \! \left( \left[
\left\lan... | I don't have enough for a comment (my original account was wiped out), but this is my impression:I think this is just the change-of-basis theorem/result; if $U,V$ are open balls, a diffeomorphism is basically a coordinate change map, and so it transforms according to the (determinant of the ) Jacobian.
| {
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What is the relationship between Fourier transformation and Fourier series? Is there any connection between Fourier transformation of a function and its Fourier series of the function? I only know the formula to find Fourier transformation and to find Fourier coefficients to find the corresponding Fourier series.
| Given a locally compact abelian group $G$, one can define the character group of $G$ as the group of continuous homomorphisms $G \to S^1$. (It should actually land in $\mathbf C^\times$, but for the purpose at hand, this is good enough.)
The character group of the circle $S^1$ is isomorphic with $\mathbf Z$ (the charac... | {
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Group covered by finitely many cosets This question appears in my textbook's exercises, who can help me prove it?
If a group $G$ is the set-theoretic union of finitely-many cosets, $$G=x_1S_1\cup\cdots\cup x_nS_n$$ prove that at least one of the subgroups $S_i$ has finite index in $G$.
I think that the intersection o... | For completeness, the Neumann proof is roughly as follows.
Let $r$ be the number of distinct subgroups in $S_1,S_2,\dots,S_n$. We will prove by induction on $r$.
If $r=1$, then $G=\bigcup_{i=1}^n x_iS_1$ and $S_1$ thus has finite index in $G$.
Now assume true for $r-1$ distinct groups, and assume $S_1,\dots,S_n$ has $... | {
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Do Hyperreal numbers include infinitesimals? According to definition of Hyperreal numbers
The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form
1 + 1 + 1 + ...... + 1.[this definition has been extracted from wiki encyclopedia Hyperr... | Yes, they do: if $x\in{}^*\Bbb R$ is greater than any ordinary integer, then $\frac1x$ is necessarily a positive infinitesimal. There is no contradiction: the first statement doesn’t mention the infinitesimals explicitly, but the very next sentence does:
Such a number is infinite, and its reciprocal is infinitesimal.
... | {
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Show that $\frac{(m+n)!}{m!n!}$ is an integer whenever $m$ and $n$ are positive integers using Legendre's Theorem Show that $\frac{(m+n)!}{m!n!}$ is an integer whenever $m$ and $n$ are positive integers using Legendre's Theorem.
Hi everyone, I seen similar questions on this forum and none of them really talked about ho... | For a prime $p$, denote by $v_p(r)$ the exponent of $p$ in the prime factorization of $r$. So for example, $v_2(12) = 2$. Legendre's theorem states that for any prime $p$ and integer $n$, we have
$$v_p(n!) = \sum_{i = 1}^\infty \left\lfloor \frac{n}{p^i} \right\rfloor $$
Note that $v_p(rs) = v_p(r) + v_p(s)$. Also $r$... | {
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"url": "https://math.stackexchange.com/questions/536644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fermat's Little Theorem and polynomials We know that in $F_p[y]$, $y^p-y=y(y-1)(y-2)\cdots (y-(p-1))$. Let $g(y)\in F_p[y]$. Why is it valid to set $y=g(y)$ in the above equation to obtain $g(y)^p-g(y)=g(y)(g(y)-1)\cdots (g(y)-(p-1))$. This is done in Theorem 1 of Chapter 22 of A Concrete Introduction to Higher algebra... | The identity
$$
z^p-z=z(z-1)(z-2)\cdots (z-p+1)
$$
holds for all elements $z$ of any commutative ring $R$ of characteristic $p$.
This follows from the corresponding identity in the polynomial ring by the universal property of univariate polynomial rings.
In this example the selections $R=F_p[y]$, $z=g(y)$ were made.
| {
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How often in years do calendars repeat with the same day-date combinations (Julian calendar)? E.g. I'm using this formulas for calculating day of week (Julian calendar):
\begin{align}
a & = \left\lfloor\frac{14 - \text{month}}{12}\right\rfloor\\
y & = \text{year} + 4800 - a \\
m & = \text{month} + 12a - 3
\end{align}
... | I can directly say 28 years. It's because 7 is a prime number, thus is relatively prime with every natural number smaller than itself. How does this information helps us solve the problem. This way:
You know that $365 \equiv 1 (mod\, 7)$ and $366 \equiv 2 (mod\, 7)$. As 1 year in each $4$ year contains $366$ days. Then... | {
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Countable basis of function spaces Show that the space of functions $f: \mathbb{N} \to \mathbb{R}$ does not have a countable basis.
I really don't know where to start with this one! Could anyone help me?
Thanks
| Try this. Since $\mathbb N$ and $\mathbb Q$ have the same cardinal, it is the same question to ask whether the set of functions $f : \mathbb Q \to \mathbb R$ has a countable basis. For $x \in \mathbb R$, define $g_x : \mathbb Q \to \mathbb R$, by
$$
g_x(s)=1\quad\text{if }s<x,\qquad g_x(s)=0\quad\text{if }s\ge x
$$
... | {
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''min $c^tx$ subject to $x^tAx=1$'': is is possible to solve it with Lagrange multiplier or in the scope of KKT? I find a problem:
Minimize $c^tx$ subject to $x^tAx=1$, where $A$ is a positive semidefinite symmetric matrix.
But the question obligates to use KKT but I am trying to apply simple Lagrange multiplier solu... | If $A$ is a positive definite matrix then you can use usual Lagrangian methods but if $A$ is only positive semi-definite matrix then you have some constraint on the $x_s$ that is missing from your original question. For example, if $A=\left(\begin{array}{cc}0 & 0 \\0 & 1\end{array}\right)$ and $c=(-1,2)$ then there is ... | {
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$A \oplus B = A \oplus C$ imply $B = C$? I don't quite yet understand how $\oplus$ (xor) works yet. I know that fundamentally in terms of truth tables it means only 1 value(p or q) can be true, but not both.
But when it comes to solving problems with them or proving equalities I have no idea how to use $\oplus$.
For ex... | Hint: $A\oplus(A\oplus B)=(A\oplus A)\oplus B = B$.
And of course $A\cup B=A\cup C$ does not imply $B=C$ (consider the case $B=A\ne \emptyset = C$). And $A\cap B=A\cap C$ does not imply $B=C$ either (consider the case $A=\emptyset$)
| {
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Why does knowing where two adjacent vertices of regular $n$-gon move under rigid motion determines the motion? I am reading the book Abstract Algebra by Dummit and Foote.
In the section about the group $D_{2n}$ (of order $2n$) the authors
claim that knowing where two adjacent vertices move to, completely
determine the ... | An Euclidean movement is determined by the effect on three (noncollinear) points. If $A\mapsto A'$, $B\mapsto B'$, $C\mapsto C'$, we can consider the movement composed of
*
*the translation along $\vec{A'A}$
*the rotation about $A$ that maps the translated image of $B'$ to $B$
*the reflection at $AB$ if necesar... | {
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Proof of $10^{n+1} -9n -10 \equiv 0 \pmod {81}$ I am trying to prove that $10^{n+1} -9n -10 \equiv 0 \pmod {81}$. I think that decomposing into 9 and then 9 again is the way to go, but I just cannot get there. Any help is greatly appreciated.
\emph{edit} I originally posted this a $9^n$ not $9n$. Apologies.
| Without anything more thatn the geometric sum formula:
It's certainly true mod $9$, since it is easy to reduce the left side by $9$:
$$1^{n+1}-0-1\equiv{0}\mod{9}$$
Now dividing the left side by $9$ gives $$10\frac{10^n-1}{9}-n$$ which is $$10\frac{(10^{n-1}+10^{n-2}+\cdots+10^1+1)(10-1)}{9}-n$$ which is $$10(10^{n-1... | {
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Reduction of Order: $t^{2}y''+3ty'+y=0$, $\quad t>0$; $\quad y_{1}(t)=t^{-1}$ I am working an exercise from Elementary Differential Equations and Boundary Value Problems Ninth Edition by Boyce and Diprima, and I think there is mistake\typo. On page 173 Section 3.4 exercise 25.
The book is correct I dropped the minus ... | For the sake of simplicity it is better to write $y_2=vt^{-1}$ so that $y_2'= v't^{-1}-vt^{-2}$ and $y''_2= v''t^{-1}-2v't^{-2}+2vt^{-3}$. Then
$t^{2}y_2''+3ty_2'+y_2=tv''+v'=0$. This implies $(tv')'=0$. Integrating this we get $tv'=c_1$. Thus integrating once more we obtain $v=c_1\ln t+c_2$. Since we wish a particula... | {
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Computing some differential forms using complex coordinates I was computing some things in the Poincaré disk $\mathbb{H}^2$ in complex coordinates and then I tried to show that $\sigma_r(z) = \frac{r^2}{z}$ is an isometry. However $d\sigma_r = \frac{-r^2}{z^2}dz$ and $g = \frac{4 dz\otimes d\overline{z}}{ (1 - z\overli... | $d\sigma_r = \frac{-r^2}{z^2}dz$ and $g = \frac{4 dz\otimes d\overline{z}}{ (1 - z\overline{z})^2} $, then $d\sigma_r(\partial z) = \frac{-r^2}{z^2} \partial z$, and not $d\sigma_r(\partial z) = \frac{-r^2}{z^2}$, therefore $g(d\sigma_r(\partial z), d\sigma_r(\partial z)) = g (\partial z, \partial z)$. The problem is t... | {
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Determining where the derivative of a function equals zero, will always produce absolute max/min values. T/F? Is this true of false and why?
I believe it is false since when the derivative equals zero, it produces local max/min.
Also endpoints will also give you absolute max/min thus you must check that.
thoughts?
| It is not true that where the derivative is zero the is a local extreme. What is true is the opposite: if there is a local extreme and the function is differentiable around the point, then the derivative is zero.
Think $f(x)=x^3$ at $x=0$.
| {
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Efficient division using binary math I'm writing code for an FPGA and I need to divide a number by $1.024$. I could use floating and/or fixed point and instantiate a multiplier but I would like to see if I could do this multiplication more efficiently.
I noticed that $2^0$ + $2^-$$^6$ + $2^-$$^7$ = $1.0234375$ which i... | $$\frac{1}{1.024} = \frac{1024-24}{1024} = \left(\frac{1024 - 16 - 8}{1024}\right)$$
So to divide N,
$$ N*\left(\frac{1000}{1024}\right) = ((N << 10) - (N << 4) - (N << 3)) >> 10 $$
You need 2 adders. Shift operation will be free in FPGA as all are powers of 2. If you want to use fraction of the result, simply use low... | {
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The probability that a randomly chosen grain weighs less than the mean grain weight If Y has a log-normal distribution with parameters $\mu$ and $\sigma^2$, it can be shown that
$E(Y)=e^\frac{\mu + \sigma^2}{2}$ and
$V(Y)=e^{2\mu +\sigma^2}(e^{\sigma^2}-1)$.
The grains composing polyerstalline metals tend to have weigh... | It can also be shown that for a log-normal distribution, the cumulative distribution function is
$$F(x)=\Phi \left( \frac{ \log x - \mu} {\sigma}\right).$$
Here, $\Phi$ is the cumulative distribution function of the standard normal. So the answer is
$$
\Phi \left(\frac{\frac{19}{2} -e^{19/2} }{e^{11}\sqrt{e^{16} -1}} ... | {
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Derivative: chain rule applied to $\cos(\pi x)$
What is the derivative of the function $f(x)= \cos(\pi x)$?
I found the derivative to be $f^{\prime}(x)= -\pi\sin(\pi x)$. Am I correct? Can you show me how to find the answer step by step?
This is a homework question:
What is $x$ equal to if $-\pi\sin(\pi x)=0$?
| To find the derivative of this function you need to use the chain rule.
Let u = $\pi x$, and we know $\frac{d}{du}(\cos(u)) = -\sin(u)$.
Then $$\frac{d}{dx}\cos(\pi x)=\frac{d\cos(u)}{du} \frac{du}{dx}$$
simplifies to your answer that you found.
| {
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Inequality. $\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}+\frac{c}{\sqrt{a}}\geq3$ Let $a,b,c \in (0, \infty)$, with $a+b+c=3$. How can I prove that:
$$\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}+\frac{c}{\sqrt{a}}\geq3 ?$$.
I try to use Cauchy-Schwarz rewriting the inequality like :
$$\sum_{cyc}\frac{a\sqrt{b}}{b} \geq \frac{(\... | With homogenizazion: We need to prove for all $a,b,c>0$: $$\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}+\frac{c}{\sqrt{a}}\geq\sqrt{3}\sqrt{a+b+c}.$$ By squaring this is equivalent to $$\frac{\sum_{cyc} a^3c+2\sum_{cyc} a^2 b^\frac32 c^\frac12}{abc}\geq \frac{3\sum_{cyc} a^2bc}{abc}$$ which follows immediately from AM-GM, for... | {
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How to recognize removable singularity and how to remove it I don't understand the idea of a removable singularity yet. Can someone explain me how to recognize a removable singularity and how to remove it?
Example: $g(z)=f(z)/z$. Is $z=0$ then a removable singularity and if yes, how would I remove it?
| If both $f$ and $g$ are holomorphic at $z_0$, they have Taylor expansions $$f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k,\quad g(z)=\sum_{k=0}^\infty b_k(z-z_0)^k$$
Let $m$ be the smallest index for which $a_k\ne 0$, and $n$ be the smallest index for which $b_k\ne 0$. Then
$$
\frac{f(z)}{g(z)} = z^{m-n}\frac{a_{m }+a_{m+1}(z... | {
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What on earth is (B|A)? I'm stumped and getting nowhere with this question:
Description:
The Air Pollution and Mortality data of 60 cities were collected in a study. 11 can be considered to have high hydrocarbon pollution potential levels. Suppose that two cities are picked at random from the list. (That is, two cities... | If event A is true, then the first city picked is polluted, therefore the other 59 cities contain 10 polluted cities, therefore P(B|A)=10/59 .
If event A' is true, then the first city picked is clean, therefore the other 59 cities contain 11 polluted cities, therefore P(B|A')=11/59 .
So, you were correct in both your g... | {
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Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq f(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant. Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq u(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.
I think i should use Liouville's t... | Since $u$ is entire, and $\mathbb{C}$ is simply connected, there is an entire holomorphic function $f \colon \mathbb{C}\to\mathbb{C}$ with $u = \operatorname{Re} f$. Then
$$g(z) = \frac{f(z)-1}{f(z)+1}$$
is an entire bounded function, hence $g$ is constant. Therefore $f$ is constant, and hence also $u$.
| {
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Integration of $\int_{x(0)}^{x(t)}\frac{1}{\sqrt{|y|} }dy$ for $y<0$ I tried to solve $x'(t)=\sqrt{|x(t)|}$ by using separation of variables.
So I did $$\int_{0}^{t}\frac{x'(s)}{\sqrt{|x(s)|}}ds$$
and used the substituion $y=x(s)$, which gave me the integral
$$\int_{x(0)}^{x(t)}\frac{1}{\sqrt{|y|} }dy$$
If we have $y>0... | Do you have any more information about the initial condition? For instance, $x'$ is always non-negative so if $x(0)>0$, then $x(t)>0$ for all $t>0$, and so $y$ is always positive as well.
It's also perhaps worth noting that this differential equations is one of the typical 'pathological' examples from existence and uni... | {
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Does $\cos(x+y)=\cos x + \cos y$? Find the value using a calculator: $\cos 75°$
At first I thought all I need is to separate the simpler known values like this:
$\cos 75^\circ = \cos 30°+\cos45° = {\sqrt3}/{2} + {\sqrt2}/{2} $
$= {(\sqrt3+\sqrt2)}/{2} $ This is my answer which translates to= $1.5731$ by calculator
Ho... | You can simply plot $cos(x+y)-(cos(x)+cos(y))$ to have your answer:
| {
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Prime Factors of Cyclotomic Polynomials I'm trying to show that if $q$ is a prime and $f_{q}(x)$ is the $q$-th cyclotomic polynomial, then all prime divisors of $f_{q}(a)$ for some fixed $a \neq 1$ either satisfy $p \equiv 1\, \text{mod}\; q$ or $p = q$.
Clearly $f_{q}(a) \equiv 1\, \text{mod}\; q$, because $1 + a + \... | Suppose $1+x+x^2+ \ldots + x^{q-1} \equiv 0 \pmod p$ for some integer $x$.
If $x \equiv 1 \pmod p$, then $q \equiv 0 \pmod p$, and because $p$ and $q$ are prime, $p=q$.
If $x \neq 1 \pmod p$, then $p$ divides $(1+x+x^2+ \ldots + x^{q-1})(x-1) = x^q - 1$, which means that $x$ is a nontrivial $q$th root of unity modulo ... | {
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Holomorphic functional Calculus in Dunford and Schwartz I am currently studying the spectral theory for bounded operators as described in the book "Linear Operators" by Dunford and Schwartz because I would like to obtain a better understanding of the functional Calculus that uses the Cauchy formula
$$
f(T) = \frac{1}{2... | It looks like what you need is Hermite Interpolation. It requires you to prescribe the same number of derivatives at all points; but you are dealing with a finite number of points, so you just take the bigger $m$ and make up the values for the missing derivatives.
| {
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If $\displaystyle x= \frac{e^{3z}}{y^4}$ then $z(x,y)$? If $\displaystyle x= \frac{e^{3z}}{y^4}$ then $z(x,y)$ ?
I know that we subtract powers in fractions but how do we solve it when there is a variable $3z$? And what is what is $z(x,y)$?
This question is supposed to be easy and it irritate me that I can't solve it. ... | It seems like they are asking for $z$ in terms of $x$ and $y$. That is what $z(x,y)$ means. (It's not entirely clear from just what you've posted, but that would be my best guess.)
So you need to solve for $z$. First multiply by $y^4$, then take the natural log, then divide by 3.
| {
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Find the remainder when $2^{47}$ is divided by $47$ So I am trying to work out how to find remainders in several different way.
I have a few very similar question,
1.) Find the remainder when $2^{47}$ is divided by 47
So i have a solution that says
$$2^{24} \equiv 2$$
$$2^{48} \equiv 4$$
$$2^{47} \equiv 2$$ Since $(2,... | Fermat's Little Theorem gives you the answer at once: since $\;47\; $ is prime, we get
$$2^{47}=2\pmod {47}$$
which means that the wanted residue is $\;2\;$ .
| {
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2 regular graphs and permutations I have found this question on MSE before but I didn't find the answer satisfactory and it is so old I doubt anyone is still following it.
Let $f_{n}$ be the number of permutations on $[n]$ with no fixed points or two cycles. Let $g_{n}$ be the number of simple, labeled two regular gra... | HINT: Each permutation of $[n]$ with no fixed points and no $2$-cycles corresponds in an obvious way to a unique labelled two-regular directed graph in which each cyclic component is a directed cycle. Given such a graph $G$, we can split the components into two sets, $\mathscr{C}_I(G)$ and $\mathscr{C}_D(G)$, as follow... | {
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I want to prove that $m||v||_1\le ||v||$ Let $(V,||\cdot ||)$ a finite-dimensional real normed space, and $\{v_1,...,v_n\}$ a basis. We define $||v||_1=\sqrt{x_1^2+...+x_n^2}$, where $x_1,...,x_n$ are the coordinates of $v$.
I already proved that there exists $w\in S:=\{v\in V:||v||_1=1\}$ such that $$||w||\le ||v||$$ ... | Hint: First note, that $w \ne 0$, let $m := \|w\| >0$. Now let $v \in V$. Then $v':= v/\|v\|_1 \in S$, hence $m = \|w\| \le \|v'\|$. Now use the defition of $v'$ and homogenity of $\|\cdot \|$.
| {
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How to prove that the multi-period market satisfies Non-arbitrage given that the single-period market admits Non-arbitrage Here is the question:
Let $(Ω,\mathscr F,\mathbb P,\mathbb F= (\mathscr F_k)_{k=0,...,T})$ be a filtered probability space and $S=(S_k)_{k=0,...,T}$ a discounted price process. Show that the follow... | No-arbitrage is equivalent to existence of state prices or stochastic discount factor. The proof usually is based on separating hyperplane theorem (see Duffie's Dynamic Asset Pricing Theory for example) but the conclusion is that prices do not admit arbitrage from date $t_1$ to date $t_2$ if and only if there exist sta... | {
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Prove that $f(n)$ is the nearest integer to $\frac12(1+\sqrt2)^{n+1}$? Let $f(n)$ denote the number of sequences $a_1, a_2, \ldots, a_n$ that can be constructed where each $a_i$ is $+1$, $-1$, or $0$.
Note that no two consecutive terms can be $+1$, and no two consecutive terms can be $-1$. Prove that $f(n)$ is the nea... | Let $f_0(n)$ be the number of such sequences ending by $0$ and $f_1(n)$ be the number of such sequences ending by $1$ (number of such sequences ending by $-1$ is $f_1(n)$ too due to symmetry reason). Then $f_0(n) = f_0(n-1) + 2f_1(n-1)$, $f_1(n) = f_0(n-1) + f_1(n-1)$ and $f(n) = f_0(n) + 2f_1(n) = f_0(n+1)$. Some tran... | {
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Proof of $(\forall x)(x^2+4x+5 \geqslant 0)$ $(\forall x)(x^2+4x+5\geqslant 0)$ universe is $\Re$
I went about it this way
$x^2+4x \geqslant -5$
$x(x+4) \geqslant -5$
And then I deduce that if $x$ is positive, then $x(x+4)$ is positive, so it's $\geqslant 5$
If $ 0 \geqslant x \geqslant -4$, then $x(x+4)$ is also $\geq... | Just note that $(x+2)^2=x^2+4x+4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/539375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 0
} |
Family with three children. Chances of at least one boy and girl? Here's my exercise:
A family has three children. What's the probability of event $A \cup C$ where:
$A$- Family has children of both sexes.
$C$- Family has at most one girl.
Well I see two ways to look at this conundrum:
The first one would be to differen... | You distinguish between four possibilities: $BBB, BBD, BDD, DDD$. In three out of four cases the event $A \cup C$ occurs. However this only implies $P(A \cup C) = \frac{3}{4}$ when each of the four cases are equally likely, which is not the case here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/539441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Mapping homotopic to the identity map has a fixed point Suppose $\phi:\mathbb{S}^2\to\mathbb{S}^2$ is a mapping, homotopic to the identity map. Show that there is a fixed point $\phi(p)=p$.
| Have you seen homology and degree of continuous mappings? If so, this is pretty short:
Since $f$ is homotopic to the identity, $\deg f = \deg \text{id}_{S^2} = +1$. On the other hand, if a continuous map $g:S^n\to S^n$ has no fixed points, then $\deg g = (-1)^{n+1}$. This shows that $f$ must have a fixed point, since o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/539506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
area of a triangle using double integral I can find the area of a triangle with known vertices but the problem here is that the question is general: I have to use double integral to prove that the area of the triangle is:
$$A_{\text{triangle}}=\frac {\text{base}\times\text{height}}{2}$$
I assume that the width is $b$ ... | Here is an approach. Just assume it has the three vertices $(0,0),(0,b)$ and $(c,h)$ such that $b,c >0$. Now, you need to
1) find the equation of the line passing through points $(0,0),(c,h)$
2) find the line passing through the points $(0,b), (c,h) $
3) consider a horizontal strip; i.e. the double integral
$$ \iint_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/539592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Volume of $n$-ball in terms of $n-2$ ball Let $V_n(R)$ be the volume of the ball with radius $R$ in $\mathbb{R}^n$. This page says
$$V_n(R)=\int_0^{2\pi}\int_0^RV_{n-2}(\sqrt{R^2-r^2})r\,dr\,d\theta$$
I don't really understand the explanation given in there. Could someone explain it in the case $n=3$ (so $n-2=1$) ho... | Let's do the $n=2$ case first. What is a zero-dimensional sphere, you ask? Well, it's nothing but a point, and $0$-dimensional volume is, I claim, just cardinality. So $V_0 (R) = \lvert\{\ast \}\rvert= 1$.
Now, how do we calculate $V_2 (R)$, the $2$-volume (i.e. area) of a $2$-ball (i.e. disc)? Well, we want to add... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/539727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Use a change of contour to show that $\int_0^\infty \frac{\cos{(\alpha x)}}{x+\beta}dx = \int_0^\infty \frac{te^{-\alpha \beta t}}{t^2 + 1}dt$ A problem from an old qualifying exam:
Use a change of contour to show that
$$\int_0^\infty \frac{\cos{(\alpha x)}}{x+\beta}dx = \int_0^\infty \frac{te^{-\alpha \beta t}}{t^2 + ... | Yes, it really looks like you should go for $z^2$. Let's try $x = \beta t^2$. Then we have $ dx = 2\beta tdt$ and hence:
$$ \int_0^\infty\frac{\cos(\alpha x)}{x+\beta}dx = \int_0^\infty\frac{t\cos(\alpha \beta t^2)}{t^2+1}dt $$
Maybe you should note that $\cos(x) = \frac{e^{-ix} + e^{ix}}{2}$. This would also explain t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/539807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Find $\int \frac {\mathrm dx}{(x + 1)(x^2 + 2)}$ I'm supposed to find the antiderivative of
$$\frac {1}{(x + 1)(x^2 + 2)}$$
and I'm completely stumped. I've been trying substitutions with $u = x^2$ and that's led me nowhere. I don't think I can use partial fractions here since I have one linear factor and one quadrat... | Consider $$\int {1\over (x+1)(x^2+2)}dx.$$
Using the method of partial fraction decomposition we have $${1\over (x+1)(x^2+2)} = {A\over x+1}+{Bx+C\over x^2+2}.$$ Solving for constans $A$, $B$,and $C$ we obtain $${1\over 3(x+1)}+{1-x\over 3(x^2+2)}.$$ So we must integrate $$\int{1\over 3(x+1)}+{1-x\over 3(x^2+2)}dx.$$ R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/539906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Reflexivity of a Banach space without the James map The reflexivity of a Banach space is usually defined as having to be enforced by a particular isometric isomorphism. Namely the map that takes each element to the evaluation, which is already an injective linear isometry from a Banach space to its double dual. It ju... | Here are answers to your questions.
1) Robert C. James provided the first example of non-reflexive Banach space isomorphic to its double dual. Also, see this MO post. Thanks to David Mitra for the reference.
2) Banach space is reflexive iff $X^*$ is reflexive. See this post for details.
3) Reflexivity does not bound th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/539974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find the coefficient of a power of x in the product of polynomials - Link with Combinations? I came across a new set of problems while studying combinatorics which involves restrictions to several variables and use of multinomial theoram to evaluate the number of possible combinations of the variables subjected to thos... | Here's one way to think of it. Let's rewrite as $$(r^0+r^1+r^2+r^3)(g^0+g^1+g^2+g^3+g^4)(w^0+w^1+w^2+w^3+w^4+w^5),$$ for a moment. When we expand this, every single one of the terms will be of the form $r^ag^bw^c.$ This gives us a one-to-one correspondence with the various ways in which the balls can be drawn--where $r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/540082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Is $\mbox{lcm}(a,b,c)=\mbox{lcm}(\mbox{lcm}(a,b),c)$? $\newcommand{\lcm}{\operatorname{lcm}}$Is $\lcm(a,b,c)=\lcm(\lcm(a,b),c)$?
I managed to show thus far, that $a,b,c\mid\lcm(\lcm(a,b),c)$, yet I'm unable to prove, that $\lcm(\lcm(a,b),c)$ is the lowest such number...
| The "universal property" of the $\newcommand{\lcm}{\operatorname{lcm}}\lcm$ is
if $\lcm(a,b) \mid x$, and $c \mid x$, then $\lcm(\lcm(a,b),c) \mid x$.
Make a good choice for $x$, for which you can prove the hypothesis and/or the conclusion is useful.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/540135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Path connected spaces with same homotopy type have isomorphic fundamental groups I was try to understand the following theorem:-
Let $X,Y$ be two path connected spaces which are of the same homotopy type.Then their fundamental groups are isomorphic.
Proof: The fundamental groups of both the spaces $X$ and $Y$ are indep... | I suppose $σ$ is a path from $x_0$ the a point $x_1$ and the map $\sigma_\#$ is defined as:
$$σ_\#:\pi_1(X,x_0)\to π_0(X,x_1)\\σ_\#([p])=[σ\cdot p\cdot\barσ]$$ where $σ⋅p$ is the path which first traverses the loop $p$ and then $σ$, and $\bar σ$ is the reversed path.
This is a bijection because it has the inverse $\bar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/540227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
probability question: find min{a,b,c} X, Y, Z: discrete random variables
Probability of X, Y, Z have distinct values is 1.
Let $a=P(X>Y), b=P(Y>z), c=P(Z>X)$
(i) Find min{a,b,c}.
(Note: in an election, it's possible for more than half of the voters to prefer candidate A to candidate B, more than half B to C, and m... | Since any equality has probability $0$, we only have to look at six possibilities
*
*$p_1 = P(X \gt Y \gt Z)$
*$p_2 = P(X \gt Z \gt Y)$
*$p_3 = P(Y \gt Z \gt X)$
*$p_4 = P(Y \gt X \gt Z)$
*$p_5 = P(Z \gt X \gt Y)$
*$p_6 = P(Z \gt Y \gt X)$
and we have $p_1+p_2+p_3+p_4+p_5+p_6=1$ with
*
*$a=p_1+p_2+p_5$;
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/540367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\int_{-\infty}^{0}xe^{x}$ diverge or converge How would one find whether the following improper integral converge or diverge.
$\int_{-\infty}^{0}xe^{x}$
I did the following.
$t\rightarrow\infty$
$\int_{t}^{0}xe^x$
I did the integration by parts.
$u=x$
$dv=e^x$
$xe^x-\int 1e^x$
$xe^x-1xe^x$
$(0)(e^0)-e^0(0)-te^t-te^t$
... | Step 1: set the limit $\lim_{t \rightarrow -\infty} $
Step 2: find integral (i got $xe^x - e^x$)
Step 3: plug in the limits and ftc, so u get
$\lim_{ t-> -\infty} [(0-e^0)-(te^t-e^t)]$
*key: $e^\infty = \infty$,
$e^-\infty = 0$
step 4: evaluate
answer: convergent sum $-1 $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/540450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\sqrt{2n^2+2n+3} $ is irrational I have proven this by cases on $n$.
I would like to see a neater proof. One similar to the proof of the fact that $\sqrt{2}$ is irrational.
| Assume by contradiction that $\sqrt{2n^2+2n+3}$ is rational. Then
$$\sqrt{2n^2+2n+3} =\frac{p}{q}$$
with $gcd(p,q)=1$. Squaring we get $q^2|p^2$ which implies $q=1$. Thus
$$2n^2+2n+3=p^2$$
It follows that $p$ is odd. Let $p=2k+1$
Then
$$2n^2+2n+3=4k^2+4k+1 \Rightarrow n^2+n=2k^2+2k-1$$
P.S. Here is the same argument, i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/540555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit of e with imaginary number Important part:
$\lim\limits_{x\to\infty} e^{-ix} - e^{-x} $
This is suppose to approximate to "$1$" but the way I see it we have $0 - 0$ ...
| Expand the term with the imaginary exponent using Euler's formula:
$$\lim_{x\to\infty} \left[e^{-ix} - e^{-x}\right] = \lim_{x\to\infty} \left[\cos(-x) + i\sin(-x) - e^{-x}\right]=\lim_{x\to\infty} \cos(x) - i\lim_{x\to\infty} \sin(x)$$
which does not converge to either 0 or 1, but instead oscillates around the unit ci... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/540723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Radical of a homogeneous ideal If $S=\bigoplus_{d\ge 0} S_d$ is a graded ring and $\mathfrak a\subset S$ a homogenous ideal, I'm trying to prove this implication:
$\sqrt {\mathfrak a}=S_+=\bigoplus_{d\gt 0}S_d\implies S_d \subset \mathfrak
a$ for some $d\gt 0$.
My attempt of solution
I know that $x_i^d\in \sqrt {\ma... | I think you are supposed to be assuming Noetherian here, which is what the other users are pointing out.
Let me assume that $\mathfrak{a}\subseteq S:=k[x_0,\ldots,x_n]$ and let you generalize. If $\sqrt{\mathfrak{a}}=S_+$, then for all $i$ there exists $m_i$ such that $x_i^{m_i}\in\mathfrak{a}$. Let $M=(n+1)\max\{m_i\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/540794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Complex integral of an exponent divided by a linear ($\int \frac{e^u}{u-1}$) Here is the question I'm working on:
Evaluate the following integral:
$$ \oint_{|z+1|=1} \frac{\sin \frac{\pi z}{4}}{z^2-1}dz$$
I've tried along the following line. Substitute $sin(z) = \frac{e^z-e^{-z}}{2i}$:
$$ \frac{1}{2i} \int \frac{e^... | Use the Residue Theorem:
$$\oint\limits_{|z+1|=1}\frac{\sin\frac{\pi z}4}{z^2-1}dz=\frac12\left(\overbrace{\oint\limits_{|z+1|=1}\frac{\sin\frac{\pi z}4}{z-1}dz}^{=0}-\oint\limits_{|z+1|=1}\frac{\sin\frac{\pi z}4}{z+1}dz\right)=$$
$$=\left.-\frac{2\pi i}2\sin\frac{\pi z}4\right|_{z=-1}=\frac{\pi i}{\sqrt2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/540888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Describing co-ordinate systems in 3D for which Laplace's equation is separable Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in many areas of science. One of the usual ways... | The Helmholtz equation is separable only in ellipsoidal coordinates (and degenerations like polar coordinates, and cartesian of course). For Laplace, there are a couple more; see the MathWorld article about Laplace's equation.
A good book source on this subject is Chapter 5 of Morse & Feshbach, part I.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/540955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
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