Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
The significance of the composition of an operator and its adjoint As I read the literature, I have noticed that the composition $T^*T$ of a linear operator $T:H\to H$ and its adjoint frequently turns up in all kind of places. I am aware that it is Hermitian (at least when $T$ is bounded), that $||Tx||^2=\langle Tx,Tx\... | Considering it as an operator we have that its self adjont i.e $(T^{*}Tx,y)=(x,T^{*}Ty)$, and thus diagonalisable if its compact. One significant fact allowing this to happen is that the orthagonal complement of an eigenvector of the operator is invariant under the operator, atleast in Hilbert spaces. Hence once you pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1599104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Why is there only one term on the RHS of this chain rule with partial derivatives? I know that if $u=u(s,t)$ and $s=s(x,y)$ and $t=t(x,y)$ then the chain rule is $$\begin{align}\color{blue}{\fbox{$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial s}\times \frac{\partial s}{\partial x}+\frac{\partial u}{\partial ... | It comes from the divergence operator $\nabla$. Let $f$ be a scalar valued function then $\nabla f \equiv \partial \left\langle \dfrac{\partial f}{\partial x}, \dots \right\rangle$ vectorizes $f$. If you picture $f$ as the height of a hill and its parameters the coordinates of each point of the hill on the Earth, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1599205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Can exist an even number greater than $36$ with more even divisors than $36$, all of them being a prime$-1$? I did a little test today looking for all the numbers such as their even divisors are exactly all of them a prime number minus 1, to verify possible properties of them. These are the first terms, it is not inclu... | I made a program that calculates numbers, that you are searching, but only found $$198,26118,347\color{brown}58,49338,67698,79038,109818,...$$
which has only $6$ '$prime-1$' divisors like $36$. The range was $10^8$.
It shows that a number with even, $7$ '$prime-1$' divisors like these does not exist or they are very bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1599295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 2
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Line integral of a vector field $$\int_{\gamma}ydx+zdy+xdz$$
given that $\gamma$ is the intersection of $x+y=2$ and $x^2+y^2+z^2=2(x+y)$ and its projection in the $xz$ plane is taken clockwise.
In my solution, I solved the non-linear system of equations and I found that $x^2+y^2+z^2 = 4$. Given the projection in the $... | Alternatively, you can use the Stokes theorem here, it simplifies calculations. Let $\vec{F}=(y,z,x)$. Since $\gamma$ is a closed curve, the following equality holds:
$$
\oint_{\gamma}\vec{F}\cdot d\vec{r} = \iint_{S}\nabla\times\vec{F}\cdot d\vec{S}
= \iint_{S}\nabla\times\vec{F}\cdot\vec{n}\;dS,
$$
and this is not a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1599393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Sparse Matrix format CRS I am study the format CRS of an sparse matrix. I have a doubt respect to the pointer row_ptr. What happend if the matrix has a row with all entries zero. Could you help to describe the row_ptr vector in the next case
\begin{bmatrix}
2 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1
\end{bmatrix}
| The row vector is
$$1,2,2,3$$
according to http://www.math.tamu.edu/~srobertp/Courses/Math639_2014_Sp/CRSDescription/CRSStuff.pdf.
Based on the reference, if row $i$ contains entirely zeros, you store in row-ptr($i$) the same as in row-ptr($i+1$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1599479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Example of a module which is free at an isolated point I'm looking for the most simple example of a quasicoherent sheaf $\mathcal{F}$ over a scheme $X$ (preferably affine for simplicity) which has a free stalk $\mathcal{F}_x$ at a point $x \in X$ and yet for every open neighborhood $x \in U$ the restricted sheaf $\math... | I'm surprised no one has mentioned the following very simple example. Let $A=\mathbb{Z}$ and let $M=\mathbb{Q}$. Then $M$ is free at the generic point of $\operatorname{Spec} A$, but is not free in any open neighborhood. Or, if you want $x$ to be a closed point, you can instead take $M=\mathbb{Z}_{(p)}$ for some pri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1599577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to show that a function is computable?
Is the following function $$g(x) = \begin{cases} 1 & \mbox{if } \phi_x(x) \downarrow \mbox{or } x \geq 1 \\ 0 & \mbox{otherwise } \end{cases}$$ computable?
Please note that $\phi_i(x) \downarrow$ means that the function with index $i$, converges on input $x$.
Let there exis... | This function is computable. Clearly (by the definition of $g$) we have $g(x) = 1$ for all $x \geq 1$. Also $g(0)$ can be either $0$ or $1$. Regardless, this function is computable: in the latter case it is the constant function $g(x) \equiv 1$ and in the first case it is the charateristic function of the set of positi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1599729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $0 \le a \le b \le c$ and $a+b+c=1$. Show that $a^2+3b^2+5c^2 \ge 1$
Let $0 \le a \le b \le c$ and $a+b+c=1$. Show that $a^2+3b^2+5c^2
\ge 1$.
My solution: since $a+b+c=1$ we have to show that $a^2+3b^2+5c^2\ge1=a+b+c$
Since $a,b,c \ge 0 $ the inequality is true given that every term on the left hand side of ... | By means of rearrangement inequality, it can be shown that, is $a_1\le a_2\le\cdots\le a_n$ and $b_1\le b_2\le\cdots\le b_n$, then $\dfrac{a_1b_1+a_2b_2+\cdots+a_nb_n}{n}\ge \dfrac{a_1+a_2+\cdots+a_n}{n}\dfrac{b_1+b_2+\cdots+b_n}{n}$
Then, $\dfrac{a^2+3b^2+5c^2}{3}\ge\dfrac{a^2+b^2+c^2}{3}\dfrac{1+3+5}{3}$. Thus $a^2+3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1599853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivative of multivariate normal distribution wrt mean and covariance I want to differentiate this wrt $\mu$ and $\Sigma$ :
$${1\over \sqrt{(2\pi)^k |\Sigma |}} e^{-0.5 (x-\mu)^T \Sigma^{-1} (x-\mu)} $$
I'm following the matrix cookbook here and also this answer . The solution given in the answer (2nd link), doesn't... | If you're trying to find the derivative with respect to $\mu\in\mathbb R^{n\times 1}$ and $\Sigma\in\mathbb R^{n\times n}$, then I don't think the answer could possibly be $(\Sigma^{-1} + \Sigma^{-T}) (x-\mu)$. I'm wondering if what you're trying to do is find the values of $\mu$ and $\Sigma$ that maximize the express... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1599966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Set of the vertex sets to make connected graph into disjoint sets of vertices? Suppose a non-directed graph G with vertices V and paths P. What is the name for the vertex sets to make break the graph by removal of some vertices?
| I list below the relevant things in Mathematics. In comparison, the computing perspectives and visualisation more here.
Theorems and Lemmas
*
*Planar separator theorem
*Tutte theorem related to edge cuts
*Minimal vertex separator lemma
*
*Menger's theorem has vertex-connectivity version and edge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1600059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the common integer solutions to $a + b = c \cdot d$ and $a \cdot b = c + d$ I find nice that $$ 1+5=2 \cdot 3 \qquad 1 \cdot 5=2 + 3 .$$
Do you know if there are other integer solutions to
$$ a+b=c \cdot d \quad \text{ and } \quad a \cdot b=c+d$$
besides the trivial solutions $a=b=c=d=0$ and $a=b=c=d=2$?
... | First, note that if $(a, b, c, d)$ is a solution, so are $(a, b, d, c)$, $(c, d, a, b)$ and the five other reorderings these permutations generate.
We can quickly dispense with the case that all of $a, b, c, d$ are positive using an argument of @dREaM: If none of the numbers is $1$, we have
$ab \geq a + b = cd \geq c +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1600332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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Finding the sum of the infinite series whose general term is not easy to visualize: $\frac16+\frac5{6\cdot12}+\frac{5\cdot8}{6\cdot12\cdot18}+\cdots$ I am to find out the sum of infinite series:-
$$\frac{1}{6}+\frac{5}{6\cdot12}+\frac{5\cdot8}{6\cdot12\cdot18}+\frac{5\cdot8\cdot11}{6\cdot12\cdot18\cdot24}+................ | Let us consider $$\Sigma=\frac{1}{6}+\frac{5}{6\times 12}+\frac{5\times8}{6\times12\times18}+\frac{5\times8\times11}{6\times12\times18\times24}+\cdots$$ and let us rewrite it as $$\Sigma=\frac{1}{6}+\frac 16\left(\frac{5}{ 12}+\frac{5\times8}{12\times18}+\frac{5\times8\times11}{12\times18\times24}+\cdots\right)=\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1600414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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If $a^2 + b^2 = 1$, show there is $t$ such that $a = \frac{1 - t^2}{1 + t^2}$ and $b = \frac{2t}{1 + t^2}$ My question is how we can prove the following:
If $a^2+b^2=1$, then there is $t$ such that $$a=\frac{1-t^2}{1+t^2} \quad \text{and} \quad b=\frac{2t}{1+t^2}$$
| Hint At least for $(a, b) \neq (-1, 0)$, which is not realized by any value $t$, draw the line through $(-1, 0)$ and $(a, b)$ in the $xy$-plane. Writing the point $(a, b)$ of intersection of the line and the unit circle $x^2 + y^2 = 1$ in terms of the slope $t$ of the line gives exactly $$(a, b) = \left(\frac{1 - t^2}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1600499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Geometric progression in an inequality Problem: Show that if $a>0$ and $n>3$ is an integer then $$\frac{1+a+a^2 \cdots +a^n}{a^2+a^3+ \cdots a^{n-2}} \geq \frac{n+1}{n-3}$$
I am unable to prove the above the inequality.
I used the geometric progression summation formula to reduce it to proving $\frac{a^{n+1}-1}{a^2(... | The inequality holds trivially for $a = 1$, and it is invariant
under the substitution $a \to 1/a$. Therefore it suffices to
prove the inequality for $\mathbf{a > 1}$.
The hyperbolic sine is a convex function on $[0, \infty)$,
so for $a > 1$ the function
$$
f(x) = 2 \sinh \bigl(\log a \cdot \frac x2 \bigr)
= a^{x/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1600626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Ring of Invariants of symmetric group The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the regular representation of $S_n$, the basis of the vector space is indexed by the eleme... | The elementary symmetric polynomial do not generate the ring if $n>2$, let $g_1,g_2$ be two distinct elements of $S_n$, consider $G.(g_1,g_2)=\{gg_1,gg_2),g\in G\}$ the polynomial $\sum X_{gg_1}X_{gg_2}$ is invariant by $G$ but not invariant by $S_{n!}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1600724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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How to estimate ln(1.1) using quadratic approximation? So the general idea for quadratic approximation is assuming there a function $Q(x)$ we want to estimate near $a$:
$Q_a(a) = f(a)$
$Q_a'(a) = f '(a)$
$Q_a''(a) = f ''(a)$
But then how do you derive the function $Q_a(x) = f(a) + f '(a)(x-a) + f ''(a) (x-a)^2/2$?
Or ... | What you wrote is the formula for quadratic approximation, which is derived from Taylor series.
In your case, you need to set $f(x)=\ln x$ and $a=1$, then use the formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1600911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simulation of the variance of a typical waiting time W(q) in a queue Write a computer programme that by means of stochastic simulation finds
an approximation of the variance of a typical waiting time W(q) (in the queue) before
service for a typical customer arriving to a steady-state M(1)/M(2)/1/2 queuing system.
(In o... | I read the logic to be that at the start of each loop, we are in the state where the queue is empty and a customer is being serviced. Then $s1$ is his service time and $t1$ the time till the next customer arrival.
If $s1\lt t1$ then the time to wait for the next customer will be $0$ - so you have that right.
If $t1\lt ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1601008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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what are the geodesics in the hyperbolic upper half plane? In the upper half-plane $H = \{(x, y) \in \mathbb{R}^2 \mid y > 0\}$.
The distance between the two points (a,A) and (b,B) is set by the shortest curvature in metric $F(y) = \int_a^b \frac{\sqrt{1 + (y')^2}}{y}dx,\; y(a) = A, y(b) = B$.
What is the geodesic in t... | Hyperbolic metric is :
$$ ds^2 = \dfrac{dx^2 + dy^2}{y^2} $$
Hyperbolic distance
$$ = \int_a ^b \frac{\sqrt{1+y'^{2}} }{y} dx $$
that integrates to semi-circle geodesics centered on x-axis. It is $ \log \tan ( \pi/2 + \phi/2) $ reckoned to a point slope $\phi$ from top.
This is Poincaré half-plane model, geodesics have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1601098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence and sum of a series How could I prove that the following series does converge?
$$\sum_{k = 1}^{+\infty}\ \frac{e^{k!}}{k^{k!}}$$
And how to determine its total sum?
I think that that series has to converge, because I took ratio test, $n$-th root test and so on.My problem is to compute the whole sum. Any i... | For convergence, use the root test:
$$
\sqrt[k]{ \frac{e^{k!}}{k^{k!}} }=\left( \frac{e}{k} \right) ^{(k-1)!} \to 0
$$
as for $k >7$ you have
$$
0< \left( \frac{e}{k} \right) ^{(k-1)!} < \left( \frac{1}{2} \right) ^{(k-1)!} < \left( \frac{1}{2} \right) ^{k-1}
$$
For the limit, it would surprise me if we can calculate ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1601204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Rank = trace for idempotent nonsymmetric matrices If $A$ is idempotent and symmetric, one can show that the rank of $A$ equals its trace. Is such equality preserved in general if we only know that $A$ is idempotent and not necessarily symmetric?
| If $A^2=A$ then $A$ is the identity on the image of $A$ (and of course zero on the kernel), hence with respect to a suitable basis, $A$ has $\operatorname{rank}A$ ones and otherwise zeroes on the diagonal, so $\operatorname{rank}A=\operatorname{tr}A$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1601297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Indefinite integral $\int x\sqrt{1+x}\mathrm{d}x$ using integration by parts
$$\int x\sqrt{1+x}\mathrm{d}x$$
$v'=\sqrt{1+x}$
$v=\frac{2}{3}(1+x)^{\frac{3}{2}}$
$u=x$
$u'=1$
$$\frac{2x}{3}(1+x)^{\frac{3}{2}}-\int\frac{2}{3}(1+x)^{\frac{3}{2}}=\frac{2x}{3}(1+x)^{\frac{3}{2}}-\frac{2}{3}*\frac{2}{5}(1+x)^{\frac{5}{2}}+c... | You're correct.
$u = x$, $dv=\sqrt{x+1}dx$ $\Rightarrow$ $v=\frac{2}{3}(x+1)^{3/2}$, $du=dx$.
&
$$\int udv=uv-\int vdu$$
$$\Downarrow$$
SOLUTION:
$$\int x \sqrt{x+1}dx=x\frac{2}{3}(x+1)^{3/2}-\int\frac{2}{3}(x+1)^{3/2}dx=\bbox[5px,border:2px solid #F0A]{x\frac{2}{3}(x+1)^{3/2} -\frac{4}{15}(x+1)^{5/2}+C}$$
VERIFIC... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1601372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove that $|-x| = |x|$ Using only the definition of Absolute Value:
$\left|x\right| = \begin{cases} x & x> 0 \\
-x & x < 0 \\
0 & x = 0,\end{cases}$
Prove that $|-x| = |x|.$
This seems so simple, but I keep getting hung up. I use the definition insert $-x$ into the definition, but I end up with:
$\left|-x\right| = \... | $\left|-x\right| = \begin{cases} -x & \bbox[5px,border:2px solid #F0A]{-x> 0} \\
-(-x) & \bbox[5px,border:2px solid #F0A]{-x < 0} \\
0 & \bbox[5px,border:2px solid #F0A]{-x = 0}\end{cases}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1601430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 3
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Finding $\lim_{x \to \infty}\int_0^x{e^{-x^2+t^2}}\,dt$ If we aren’t able to solve the integral $\int e^{-x^2}\,dx$, then how is it possible to find the $\lim_{x \to \infty}\int_0^x{e^{-x^2+t^2}}\,dt$? This was given to me by my prof, and I asked him multiple times if it was able to be solved. He said yes, but I’m just... | Note that
$$\int_{0}^{x}{e^{-x^2+t^2}}\,dt=e^{-x^2}\int_{0}^{x}e^{t^2}\,dt=\frac{\int_{0}^{x}e^{t^2}\,dt}{e^{x^2}}$$
Now, to solve the limit, use Fundamental Theorem of Calculus and L'Hospital's Rule:
$$\lim_{x\to\infty}\frac{\int_{0}^{x}e^{t^2}\,dt}{e^{x^2}}=\lim_{x\to\infty}\frac{e^{x^2}}{2xe^{x^2}}=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1601608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Convergence of Newton Iteration For $a>0$, I want to compute $\frac{1}{a}$ using Newton's iteration by finding a zero of $f(x)=a-\frac{1}{x}$. Newton's iteration formula reads $$x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}=2x_k-ax_k^2$$
By the Banach Fixed Point Theorem, I can conclude that this Netwon iteration converges for st... | $a$ is an unessential parameter. Indeed
$$x_{k+1}=2x_k-ax_k^2$$
is equivalent to
$$ax_{k+1}=2ax_k-a^2x_k^2$$
i.e. by setting $t=ax$,
$$t_{k+1}=2t_k-t_k^2.$$
Then notice that the function $f(t)=2t-t^2$ maps $[0,2]$ to $[0,1]$, and values outside this range to negative.
As
$$0<t<1\implies t<f(t)<1$$
and
$$f(0)=0,f(1)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1601717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that the only subfields of $\mathbb{Q}(i, \sqrt{5})$ is $\mathbb{Q}, \mathbb{Q}(i),\mathbb{Q}(\sqrt{5}), \mathbb{Q}(i \sqrt{5})$ and itself? I'm reading Stewart's Galois Theory and encountered this exercise in Chapter 8. I want to show this by contradiction: Assuming there exists a proper subfield $\mathbb{Q}(\alp... | I'll give you two different ways to do this as I don't know if you know the main theorem yet but even if you don't you can revisit this once you do learn it.
Method 1:
The simplest method is to use the Fundamental Theorem/ Galois Correspondence Theorem which says the intermediate fields are in bijection with subgroups ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1601819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Are projections with the same kernels the same? This question is related to another question I just asked.
I thought I figured it out but I got confused again. Given two projections $k^n\rightarrow k^n$, represented by $n\times n$ matrices $A$ and $B$, if they have the same range $H$ which is a subspace $H\subset k^n$... | As already pointed out in the comments, two projections with the same kernel must not be the same; consider for example $p_1, p_2 \colon k^n \to k^n$ with
$$
p_1(x,y) = (x,0)
\quad\text{and}\quad
p(x,y) = (x,x).
$$
In the case of your earlier question you have the additional property that all projections you conside... | {
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"url": "https://math.stackexchange.com/questions/1601936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Did I prove correctly that $f:\mathbb E\to \mathbb N;\quad f(x)=\frac12 x$ is surjective? Suppose we have two infinite sets, $\mathbb{N}$ (the set of natural numbers) and $\mathbb{E}$ (the set of even natural numbers). Give an example of a surjective function $\mathbb{E}\rightarrow\mathbb{N}$.
$f:\mathbb{E}\rightarrow\... | Your argument is not entirely correct. In proofs like this one, phrases like "for all" and "there is" are very important. As it is currently written, I would not actually say that we have a solid proof.
If you want to prove that $f$ is surjective, you must show/stress that for all $y\in\mathbb N$, there is an $x\in \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Variant of "prisoners and hats" puzzle with more than two colors There are $n$ prisoners and $n$ hats. Each hat is colored with one of $k$ given colors. Each prisoner is assigned a random hat, but the number of each color hat is not known to the prisoners. The prisoners will be lined up single file where each can see t... | Label the colors $\{1,2,3\dots k\}$. The first one says the sum of the hats in front of him $\bmod k$ (the last $n-1$ persons). After this the second one can deduce which number corresponds to the color of his hat (by subtracting the sum that he can see minus the sum previously said).
The third person, having heard all... | {
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"source": "stackexchange",
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Why do I get an imaginary result for the cube root of a negative number? I have a function that includes the phrase $(-x)^{1/3}$. It seems like this should always evaluate to $-(x^{1/3})$. For example, $-1 \cdot -1 \cdot -1 = -1$, so it seems that $(-1)^{1/3}$ should equal $-1$.
When I plug $(-1)^{(1/3)}$ into somethin... | Use $e^{i\pi}=-1$. Then $(e^{i\pi})^{1/3}=e^{i\pi/3}=-1^{1/3}$
De Moivre's gives $e^{i\theta}=\cos(\theta)+i\sin(\theta)$
If $\theta=\frac{\pi}{3}$, then it follows that
$(-1)^{1/3}=e^{i(\pi/3)}=\frac12+\frac{i\sqrt3}{2}\approx0.5 + 0.866025i$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1602160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Solve $\lim_\limits{x\to 0}\frac {e^{3x}-1}{e^{x}-1} $ I have problem with $$\lim_{x\to 0}\frac {e^{3x}-1}{e^{x}-1} $$
I have no idea what to do first.
| Hint:
Use equivalents:
$$\mathrm e^{ax}-1\sim_0 ax,\quad\text{hence}\quad \frac{\mathrm e^{ax}-1}{\mathrm e^x-1}\sim_0 \frac{ax}x=a.$$
Alternative hint:
$$\frac{\mathrm e^{ax}-1}{x}\xrightarrow[x\to0]{}(\mathrm e^{ax})'\,\Big\lvert_{x=0}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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is $7^{101} + 18^{101}$ divisible by $25$? I am not able to find a solution for this question. I am thinking in the lines of taking out some common element like $(7\cdot 7^{100}) + (18\cdot18^{100})$
but couldn't go anywhere further.
| You might want to check Euler's theorem:
$a^{\phi(n)} \equiv 1 \mod n$ for each $a$ which is coprime with n.
$\phi(25) = \phi(5^2) = 5^2-5=20$
So $7^{101} = 7^{100}*7 = (7^5)^{20}*7\equiv7 \mod 25$
In the same way $18^{101} \equiv 18 \mod 25$, so their sum is $17+8=25=0 \mod 25$, which means that the given expression i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 5
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How to know what type of cross section is it going to be? A plane intersects a right rectangular pyramid. Producing a cross section. The plane is parallel to the base. What shape is the cross section?
I thought it would be triangle cause triangle cut is going to be triangle right? Also I don't really understand what c... | Hint:
You cut the pyramid with a knife, horizontallly at height $z \le h$, what shape is the cut part that remains if the top is removed? Or in other words: what is the intersection, the set of common points, of cutting plane and pyramid? If $h$ is the height of the pyramid, how does that intersection, here refered as... | {
"language": "en",
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"source": "stackexchange",
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Binomial Coefficient Inequality, prove $\binom{n}{0} < \binom{n}{1} < \binom{n}{2}< ... <\binom{n}{\left \lfloor {\frac{n}{2}}\right \rfloor}$ I don't know how to prove this inequality
$$\binom{n}{0} < \binom{n}{1} < \binom{n}{2}< ... <\binom{n}{\left \lfloor {\frac{n}{2}}\right \rfloor}$$
Knowing that
$$
(n-2k)\binom{... | Applying the equality to $n+1$ and $k < \frac{n+1}{2}$, you get
$$
(n+1)\left(\binom{n}{k} - \binom{n}{k-1}\right) = (n+1-2k)\binom{n+1}{k} > 0
$$
so $\binom{n}{k} - \binom{n}{k-1} > 0$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Not sure why this is true about matrices, but this isn't if they are commutative Let $A$, $B$ and $C$ be three matrices.
Although for general matrices $PQ \ne QP$, in this particular case I am told that $AC=CA$.
I am also told that $A(B+C) \ne BA + CA$.
If I am being told in part of the question that the matrices are c... | You are given that $AC = CA$ i.e. that $A$ and $C$ commmute. You are not told that $A$ commutes with every matrix. Therefore, we cannot conclude that $A(B+C) = BA + CA$. In fact, the claim that $A(B+C) \ne BA + CA$ is perfectly consistent with the given data. Indeed, distributivity gives $$A(B+C) = AB + AC$$
and then c... | {
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There is an $m$ such that $M^{m}-I_{n}$ is not invertible for all $M \in GL({n,q})$ We have a general linear group over a finite field. I need to show that for every $M$ in my group I can find an integer $m$ such that $$M^{m}-I_{n}$$ is not invertible.
I know this happens because of finite field, since after finding th... | We can actually do better than this: Since the group $G := GL(n, q)$ is finite, for every $M \in G$ we have $M^{|G|} = I_n$, and so $M^{|G|} - I_n$ is actually the zero matrix. Hence, we may take $m$ to be
$$|G| = (q^n - 1) (q^n - q) \cdots (q^n - q^{n - 1}) .$$ Probably one can improve on this in general. For $n = q =... | {
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L'Hospital's rule's hypothesis that the right hand limit should exist Why does the l'Hospital's rule assume that the right hand limit should exist? How does it work for x ln(-x) as x tends to 0 from the the left hand side?
| For $x\ln(-x)$, we can re-write this as $$\frac{\ln(-x)}{\frac{1}{x}}$$
Now L'Hospitals rule actually states that if we take the derivative of the top and bottom and get a finite number (or $\pm\infty$), then the limit is the same as the original limit.
However, if we take the derivatives, we get
$$\frac{\frac{1}{x}}{\... | {
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What's an example of an infinitesimal? If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give to the students? What I am asking for are specific techniques for explaining infinitesimals to students, geometrically, algebraically, or analytically.
Note 1. This page is r... | My pedagogical answer is to explain one over a generic natural number.
We cannot explicitly write down a generic natural number just as we cannot explicitly write a generic (non-constructible) irrational number. Like a non-constructible irrational number, it is an abstraction. We do know that it is larger than any fixe... | {
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Construct a triangle with b, c and $|\angle B - \angle C|$ How can we construct a triangle with given b, c and $|\angle B - \angle C|$?
| Let $B-C=x$ we know $B+C=180-A$ so we get a relation between B and A its $B=90-A/2+x/2$ so $C=90-A/2-x/2$ so now you said you know the value of $x$ ie $B-C$ so you know $A+B+C=180$ now you have all three angles in terms of A so get it. Then find the remaining side if you want use Sine rule ie $\frac{a}{sinA}=\frac{b}{s... | {
"language": "en",
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Existence of solutions to first order ODE The fundamental theorem of autonomous ODE states that if $V:\Bbb R^n\to\Bbb R^n$ is a smooth map, then the initial value problem
$$
\begin{aligned}
\dot{y}^i(t) &= V^i(y^1(t),\ldots,y^n(t)),&i=1,\ldots,n \\
y^i(t_0) &= c^i, &i=1,\ldots,n
\end{aligned}\tag{1}
$$
for $t_0\in\Bbb ... | The answer to the reformulation is negative. Consider the problem
\begin{equation}
\begin{cases}
y'=y^2 \\
y(0)=c
\end{cases}
\end{equation}
Its solution is $y(t)=\frac{1}{c^{-1}-t}$ and it is defined on $J_c=(-\infty, c^{-1})$. So, for example, the solution with initial datum $c=1$ is defined on $(-\infty, 1)$ while ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the general equation of lines going through 'a' particular point? I want to know the general equation of lines going through a single point where there will be arbitary constants which will change and cause the line to rotate in a circle and consequently the center will be the given point
| Its known as family of lines its $$(ax+by+c)+\lambda(dx+ey+f)=0$$ where $\lambda$=arbitrary constant.
EDIT
if we know the angle we can use rotation matrix to get new equation which is
$$\left(\begin{matrix} \cos\theta& \sin\theta\\-\sin\theta & \cos\theta\end{matrix}\right)$$
| {
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Filter without cluster point, then the clopen members have empty intersection Consider a topological space $(X,\tau)$ and a filter $F$ on $X$ with no cluster point. The set $C$ of all clopen members of $F$ has the finite intersection property. Why has the intersection $\bigcap_{x \in C} x$ to be empty?
I cannot find a ... | Your original question was already answered by Henno Brandsma. I will try to respond to the new version of your question.
The key here is probably the fact that $2^I$ is zero-dimensional. I.e., it has a base consisting of clopen sets. So let us check whether the claim holds in such spaces.
Suppose that $\mathcal F\subs... | {
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$AB$ is any chord of the circle $x^2+y^2-6x-8y-11=0,$which subtend $90^\circ$ at $(1,2)$.If locus of mid-point of $AB$ is circle $x^2+y^2-2ax-2by-c=0$ $AB$ is any chord of the circle $x^2+y^2-6x-8y-11=0,$which subtend $90^\circ$ at $(1,2)$.If locus of mid-point of $AB$ is circle $x^2+y^2-2ax-2by-c=0$.Find $a,b,c$.
The... | Use polar coordinate.
Let $(x_1,y_1)=(3+6\cos\theta_1,4+6\sin\theta_1), (x_2,y_2)=(3+6\cos\theta_2, 4+6\cos \theta_2)$.
Then by your equation, we have
$$(2+6\cos \theta_1)(2+6\cos \theta_2)+(2+6\sin\theta_1)(2+6\sin\theta_2)=0$$
Using sum and difference formula, this gives us
$$18\cos(\theta_2-\theta_1)=-4-3(\cos\the... | {
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What is the ratio of empty to filled volume of the glass? The base diameter of a glass is $20$% smaller than the diameter at the rim. The glass is filled to half of the height. Then what is the ratio of empty to filled volume of the glass ?
| I know it is an old question. But I think my way of solving the problem is different from the others and an easy to understand.
Solution: Let radius of the rim be $10$ unit.
Now since the base diameter of a glass is $20~\%$ smaller than the diameter at the rim, so the radius. Hence the radius of the base is $10-\left(... | {
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Simplify Implication Expression (Predicate/Prop Logic) I'm trying to do some past paper questions for revision and find myself perplexed on some of the expressions that need normalized/simplified which involves an implies.
For example:
(A ∧ ¬B) → B ∨ C ∨ ¬ (A ∧ ¬C)
Now I know that A → B can be normalized to ¬A or B, b... | The rule that you cite:
$$A \rightarrow B = \neg A \vee B$$
also works when $A$ is a compound expression. (All of these rules do). In your example, the simplification would go like this:
$$
\begin{aligned}
(A \wedge \neg B) &\rightarrow B \vee C \vee \neg(A \wedge \neg C)\\
\neg(A \wedge \neg B) &\vee (B \vee C \vee \n... | {
"language": "en",
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How to compute $\lim _{x\to 0}\frac{x\bigl(\sqrt{3e^x+e^{3x^2}}-2\bigr)}{4-(\cos x+1)^2}$? I have a problem with this limit, I don't know what method to use. I have no idea how to compute it. Is it possible to compute this limit with the McLaurin expansion? Can you explain the method and the steps used? Thanks. (I pref... | To give another approach:
You can compute it by splitting it up:
$$
\lim_{x\to 0}\left(\frac{x\left(\sqrt{3e^x+e^{3x^2}}-2\right)}{4-(1+\cos(x))^2}\right)=\lim_{x\to 0}\left(\frac{\sqrt{3e^x+e^{3x^2}}-2}{x}\right)\lim_{x\to 0}\left(\frac{x^2}{4-(1+\cos(x))^2}\right)
$$
If you define $f(x)=\sqrt{3e^x+e^{3x^2}}$ and use ... | {
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An Analogous Riemann Integral $$1=\sum_{n=2}^\infty (\zeta (n)-1)$$
is a fairly well known result
W|A validates this result
Is there a closed form to the analogous integral: $$\text{?}=\int_2^\infty \text{d}x \, (\zeta(x)-1)$$
I have managed to prove that the integral converges, but can get nowhere beyond a numeric app... | Since $\zeta(x) - 1 = \sum_{n=2}^\infty n^{-x}$, your integral is
$$ \sum_{n=2}^\infty \int_2^\infty n^{-x}\; dx = \sum_{n=2}^\infty \dfrac{1}{n^2 \ln n}$$
I don't think this has a closed form.
| {
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Find the maximum number of students the class can contain. A pair of students is selected at random from a class.The probability that the pair selected will consist of one male and one female student is $\frac{10}{19}$.Find the maximum number of students the class can contain.
Let the class has $x$ boy students and $y... | Let $x$ and $y$ be the number of boys and girls.We find max $x+y$.we have $\dfrac{2xy}{(x+y)(x+y-1)}=\dfrac{10}{19} \Rightarrow \dfrac{5}{19} \leq \dfrac{m^2}{4m(m-1)}$.Can you solve this inequality? Here we have $m=x+y$.
| {
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Minimal polynomial over $\mathbb Q(\sqrt{-2})$
Find the minimal polynomial for $\sqrt[3]{25} - \sqrt[3]{5} $ over $\mathbb Q$ and $\mathbb Q(\sqrt{-2})$.
I have done the first part of this, over $ Q$, and have a polynomial. But I do not know how to do this over $ Q \sqrt{-2}$
| Put $\theta=\sqrt[3] 5$ so $$x=\theta(\theta-1)\Rightarrow x^3=\theta^3(\theta^3-3\theta^2+3\theta-1)\Rightarrow x^3+15x-20=0$$ This is the minimal polynomial (irreducible by Einsenstein with $p=5$) of $\theta(\theta-1)$ over $\mathbb Q$ and also the m. p. of the same element $\theta(\theta-1)$ over any $K=\mathbb Q(t... | {
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What is $\int_0^3 x^2e^{-x}\ dx$? Getting a different answer. So I was solving some papers and I came across this problem. The answer is supposed to be $2-17/e^3$, but I'm getting $1/e^3 + 2$. I'm not familiar with the formatting and am in a hurry so please excuse the poor formatting. Thanks in advance!
| Integrating your expression gives $-e^{-x}(x^2+2x+2) + c$ Then substitute 3 and subtract by substituting 0.
| {
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Permuted action of the ramified covering Let $f:E\rightarrow S^2$ be a ramified covering of degree n, and let
$t_1,t_2,..t_m$ be all its points of ramifications. Pick a point $t\in S^2$ distinct from all $t_i$ and connect it with the points $t_i$ by a smooth non intersecting segment say $\gamma_i$. Then $\gamma_i$ act... | Since the degree of the covering is $n$, the pre-image $f^{-1}(t)$ consists of $n$ points $\{s_1,\dots,s_n\}$, since $t$ is not a ramification point.
Instead of $\gamma_i$ being a segment from $t$ to $t_i$, instead make $\gamma_i$ a loop based at $t$ that "goes around" the point $t_i$. You can think of the path $\gamma... | {
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Physical meaning of the various types of boundary conditions for a vibrating string I wonder what is the physical meaning of Dirichlet, Neumann and Robin boundary conditions for a vibrating string? Or link to other applications?
| Let $u(x,t)$ be the transverse position of the string in question, and denote $\frac{du}{dx}$ by $u_x$.
Dirichlet
Dirichlet boundary conditions are ones in which the value of $u$ itself is given at the ends of the string. Many times $u$ on the boundaries will be specified as a constant value. In this case, the Dirichle... | {
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What are some math concepts which were originally inspired by physics? There are a number of concepts which were first introduced in the physics literature (usually in an ad-hoc manner) to solve or simplify a particular problem, but later proven rigorously and adopted as general mathematical tools.
One example is the ... | The notion of $\color{red}{\text{Derivatives}}$ and more generally calculus.
*
*The ancient Egyptians and Greeks (in particular Archimedes) used the notions of infinitesimals to study the areas and volumes of objects.
*Indian mathematicians (in particular Aryabhatta) used infinitesimals to study the motion of moon ... | {
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Prove: $\forall$ $n\in \mathbb N, (2^n)!$ is divisible by $2^{(2^n)-1}$ and is not divisible by $2^{2^n}$ I assume induction must be used, but I'm having trouble thinking on how to use it when dealing with divisibility when there's no clear, useful way of factorizing the numbers.
| No need to have a sledgehammer to crack a nut: a simple induction proof will do. We have to show $\;v_2\bigl((2^n)!\bigr)=2^n-1$.
For $n=0$, it is trivial since it means $\;v_2(1!)=2^0-1=0$.
Now suppose, by induction hypothesis, that $\;v_2\bigl((2^n)!\bigr)=2^n-1$, and let us prove that $\;v_2\bigl((2^{n+1})!\bigr)=2... | {
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Functions where the pre-image of convex sets is convex For functions $f:\mathbb R\to\mathbb R$, I've noticed an interesting property:
$f$ is monotonous exactly if the pre-images of convex sets are convex.
Now the latter condition can of course be defined for any map between real linear spaces. Obviously there it won't ... | This is rather a comment.
Actually, your observation in the scalar case is equivalent to "the monotone functions are exactly the quasilinear functions", see https://en.wikipedia.org/wiki/Quasiconvex_function.
Further, it is not hard to see that
*
*For all convex $C \subset \mathbb{R}^n$, the preimage $F^{-1}(C)$ is ... | {
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Exam Question - language recognizition I was at the Math-exam yesterday, and I am a bit unsure, if i solved a math problem correctly.
The question was something like this:
Draw a automata that recognise the following language:
$$ L = \{w : (0 | 1)^* \text{and } w \text{ ends with } 00 \} $$
See the image below whe... | Your automaton should not accept the string $1001$. It needs transitions $(q_3,0)\rightarrow q_3$ and $(q_3,1) \rightarrow q_1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1605002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simple explanation of the differentiation of $\ln(f(x))$ Could somebody explain why the derivative of $\ln[f(x)]$ = $f'(x)/f(x)$ .
Why is it not simply $1/f(x)$ as is the case for the derivative of $\ln(x)$ being $1/x$?
| The derivative of $\log(f(x))$ with respect to $x$ is given by the chain rule. Let $y=f(x)$.
Noting that the derivative of $\log(y)$ with respect to $y$ is $\frac1y$, we can write
$$\begin{align}
\frac{d}{dx}\log(f(x))&=\left(\left.\frac{d\log(y)}{dx}\right)\right|_{y=f(x)}\\\\
&=\left(\left.\frac{d\log(y)}{dy}\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1605083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Argument of $\pi e^{-\frac{3i\pi}{2}}$
Find the argument of $\displaystyle \pi e^{-\frac{3i\pi}{2}}$
I thought the formula for the argument was $\arg{z} = i\log\frac{|z|}{z}$
In this case $|z|= \pi$, so it turns out that $-i \log(e^{-3i\pi/2}) = \frac{\pi}{2}.$
However, I don't understand why $-i \log(e^{-3i\pi/2}) =... | The argument is the angle that the vector makes with respect to the positive real axis.
Using that $e^{i \pi} = \cos \pi + i \sin \pi$ and that $-\frac{3 \pi } {2} = \frac{\pi}{2}$, we see then that
$$ \textrm{exp} \left(\frac{3\pi i } {2}\right) = \textrm{exp} \left(\frac{\pi i}{2}\right) = \cos \frac{\pi}{2} + i \si... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate the value $-te^{-t} - e^{-t}$ at t $\rightarrow\infty$ I am trying to evaluate the following equation at $t \rightarrow \infty$ and $t \rightarrow 0$:
$-te^{-t} - e^{-t}$
I am trying to use L'Hopital's Rule to evaluate $-te^{-t}$ at $\infty$, so it becomes $\frac{e^{-t}}{-t^{-1}}$, but it does not work for ... | Try this:
$$-te^{-t}-e^{-t}=\frac{-t-1}{e^t}$$
Using l'hospitals
$$\frac{-1}{e^t}\to 0$$
Note that at $t=0$ we don't need l'hospitals. We simply get $\frac{-1}{1}=-1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Real image the complex polynomial Find $z \in \mathbb{C}$ such that $p(z)=z^4-6z^2+25<0$
I think that the solution for this problem is the following:
$$p(z)=z^4-6z^2+25=(z^2-3)^2+16$$
Thus,
$$p(z)<0 \Leftrightarrow (z^2-3)^2+16<0 \Leftrightarrow (z^2-3)^2<-16$$
But, I see not how find $z$.
| $$z^2-3=4ki$$,any $$|k|>1,z=\sqrt{(3-4ki)}$$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many sequences of rational numbers converging to 1 are there? I have a problem with this exercise:
How many sequences of rational numbers converging to 1 are there?
I know that the number of all sequences of rational numbers is $\mathfrak{c}$. But here we count sequences converging to 1 only, so the total numbe... | For each real number $x$ define
$$a_n(x):= 1+ \frac{ \lfloor xn \rfloor}{n^2}$$
Show that this is a one-to-one function from $\mathbb R$ to the set of sequences converging to $1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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"answer_id": 5
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Solve $y'\cos x + y \sin x= x \sin 2x + x^2$ Given a differential equation as below
$$y'\cos x + y \sin x= x \sin 2x + x^2.$$
I need some tips on how to start solving. What do I have to determine?
Homogenity, linearity, or exactness?
| Writing your DE as $$y'+y\tan x=\frac{x\sin 2x+x^2}{\cos x},$$
the integrating factor is $$I=e^{\int\tan x dx}=\sec x$$
Therefore the solution is given by $$y\sec x=\int\frac{x\sin 2x+x^2}{\cos^2 x}dx$$
$$=\int2x\tan x+x^2\sec^2 x dx$$
$$\Rightarrow y\sec x=x^2\tan x+c$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Prove a function is uniformly continuous
Prove the function $f(x)=\sqrt{x^2+1}$ $ (x\in\mathbb{R})$ is uniformly continuous.
Now I understand the definition, I am just struggling on what to assign $x$ and $x_0$
Let $\epsilon>0$ we want $|x-x_0|<\delta$ so that $|f(x)-f(x_0)|<\epsilon$
Could anyone help fill in th... | You have $$\begin{aligned}\vert f(x)-f(y) \vert &= \left\vert \sqrt{x^2+1}-\sqrt{y^2+1} \right\vert \\
&= \left\vert (\sqrt{x^2+1}-\sqrt{y^2+1}) \frac{\sqrt{x^2+1}+\sqrt{y^2+1}}{\sqrt{x^2+1}+\sqrt{y^2+1}} \right\vert \\
&= \left\vert \frac{x^2-y^2}{\sqrt{x^2+1}+\sqrt{y^2+1}} \right\vert \\
&\le \frac{\vert x-y \vert (\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that if $f(x)=a/(x+b)$ then $f((x_1+x_2)/2)\le(f(x_1)+f(x_2))/2$ This exercise :
If $f(x)=a/(x+b)$ then : $$ f((x_1+x_2)/2)\le(f(x_1)+f(x_2))/2$$
was in my math olympiad today (for 16 years olds).
I proved this by saying this is true due to Jensen's inequality.
Is this an acceptable answer?(with leaving as... | You can prove that f is convex by finding it second derivative, which is positive.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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integrate $\int \cos^{4}x\sin^{4}xdx$
$$\int \cos^4x\sin^4xdx$$
How should I approach this?
I know that $\sin^2x={1-\cos2x\over 2}$ and $\cos^2x={1+\cos2x\over 2}$
| Continuing from where you left off:
$$\sin^2x\cos^2x=\left({1-\cos2x\over 2}\right)\left({1+\cos2x\over 2}\right)$$
$$\sin^2x\cos^2x=\frac{1}{4}(1 - \cos^2(2x))$$
$$\sin^2x\cos^2x=\frac{1}{4}(\sin^2(2x))$$
Squaring both the sides:
$$\sin^4x\cos^4x=\frac{1}{16}(\sin^4(2x))$$
This can be integrated in two ways:
Method 1:... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find the Area of the lens of 2 overlapping circles I'm trying to find the area of a lens of 2 overlapping circles of the same size.
The circles are both $4$ feet diameter (Radius $2$ feet) and the distance between both radii is $2.75$ feet. After using the formula posted for $2$ circles of the same radii (found here) ... | So you say that the two radii are equal. Then $R = r = 2$ feet. The offset is $d = 2.75$ feet. Thus, if I apply the same formula, I get
\begin{align*}
A &= 2^2\cos^{-1}\left(\frac{2.75^2+2^2-2^2}{2(2.75)(2)}\right)+2^2\cos^{-1}\left(\frac{2.75^2+2^2-2^2}{2(2.75)(2)}\right)\\
&\qquad-\frac{1}{2}\sqrt{(-2.75+2+2)(2.75+2-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why do these seemingly logically equivalent combinations give different answers? I'm having trouble figuring out why these two different ways to write this combination give different answers. Here is the scenario:
Q: Choose a group of 10 people from 17 men and 15 women, in how many ways are at most 2 women chosen?
Solu... | Consider a simpler problem. How many ways are there to choose ten men from a set of ten men? By your second reasoning, choose eight first, then choose another two. That gives a total of:
$$\binom{10}{8}\binom{2}{2}=45$$
Why is that reasoning problematic?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Is there an error in my textbook?
The weird thing is, the last sentence says, for the case where $x_1$, "since $\Delta x_1$ approaches zero as $\max \Delta x_k \rightarrow 0...$". But given that $\Delta x_1$ is formed between $x_0$ and $x_1$, and if $x_1$ is equal to $0$ and the question says $[0,1]$, wouldn't $\Delta... | If the partition is $\{0,\ 1/6,\ 3/6,\ 4/6,\ 5/6,\ 1\}$ then $\Delta x_1= \dfrac 1 6 - 0 = \dfrac 1 6 \ne 0$.
In your first case, $x_1^* \ne 0$ and in your second case $x_1^*=0$. But in either case $\Delta x_1 = x_1 - x_0$ need not be $0$. However, if it is $0$, then the statement that it approaches $0$ would still b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Product of diagonal and symmetric positive definite matrix. Let $C$ be an $n \times n$ diagonal matrix with positive diagonal entries, and let $G$ be an $n \times n$ symmetric positive definite matrix. What can we say about $CG + GC$? For example, is it non-singular? Is it positive definite? What restrictions on $C... | *
*Notation: $\Re_+$ denotes the set of positive scalar real numbers
Let diagonal positive definite matrix be: $C=diag(c_1, c_2, \cdots, c_n)$ with $c_i\in\Re_+$, for $i=1,2,\cdots,n$. Also, let the positive definite matrix $G$ be denoted as
$G=\left[\begin{matrix} g_{11} & g_{12} & \cdots & g_{1n} \\ \vdots & \vdots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1606401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Modular arithmetic problem (mod $22$) $$\large29^{2013^{2014}} - 3^{2013^{2014}}\pmod{22}$$
I am practicing for my exam and I can solve almost all problem, but this type of problem is very hard to me. In this case, I have to compute this by modulo $22$.
| fermat's little theorem (and euler's theorem) is your friend.
22 is coprime to 29 so $29^{\phi(22)} = 29^ {10} \equiv 1 \mod 22$. So $29^{2013^{2014}} \equiv 29^k \mod 22$ where $2013^{2014} \equiv k \mod 10$.
As 10 and 2013 are co-prime $2013^{\phi (10)} = 2013^4 \equiv 1 \mod 10$ so$2013^{2014} \equiv 2013^2 \equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1606511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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The quadric contains the whole line I am looking at the following exercise:
Show that, if a quadric contains three points on a straight line, it contains the whole line.
Deduce that, if $L_1$, $L_2$ and $L_3$ are nonintersecting straight lines in $\mathbb{R}^3$, there is a quadric containing all three lines.
$$$$
A... | For the first part, you are on the good track. Plug in $\gamma(t)$ in the equation of a quadric. As an equation in $t$ it is quadratic and by the assumption it has 3 different zeros. That means it is identically zero, or in other words every point on a line is on a quadric too.
| {
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"url": "https://math.stackexchange.com/questions/1606636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integral of square root with quadratics, trouble with substitution due to 1/(2x) I have a following case to integrate:
$$\int{\sqrt{1+\left(\frac12x-\frac1{2x}\right)^2}dx}$$
I tried following the steps that are suggested for integrating square roots with enclosed sum of quadratics, but I am having trouble with the su... | hint: for the integrand we get $$\frac{x^4+2x^2+1}{4x^2}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving Wiener's attack on RSA: help understanding what is meant by a "classic approximation relation"? I am researching Wiener's attack on the RSA cryptosystem. The theorem, found here beginning on page 4, is as follows:
Let $N=pq$ with $q < p < 2q$. Let $d < \frac{1}{3}N^\frac{1}{4}$. Given $(e, N)$ with $ed = 1\bmo... | By Legendre's theorem in Diophantine approximations, if $|\alpha - \frac{k}{d}|< \frac{1}{2d^2}$, then $k/d$ has to be a convergent $p_m/q_m$ of continued fraction expansion of $\alpha$. Since the denominators $q_m$ grow exponentially ($q_m \geq F_m$, where $F_m$ is $m$-th Fibonacci number), there are less then $\log_2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Subsequence of a sequence
If $\lim_{n\rightarrow\infty}{\langle a_n \rangle} = a$ and $\langle a_{in} \rangle$ is any subsequence of $\langle a_n \rangle$, then $\lim_{n\rightarrow\infty}{\langle a_{in} \rangle} = a$, but the opposite is not (necessarily) true.
I am trying to understand the above theorem. However, I ... | That means , if a sequence converges to a limit then all sub-sequence of it converges to the same limit. But if two or more sub-sequences are convergent then the sequence need not be convergent.
For example , consider the sequence $\{x_n\}=\{(-1)^n\}$. It has two sub-sequences : $\{1,1,\cdots\}$ which converges to $1$ ... | {
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"timestamp": "2023-03-29T00:00:00",
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Limit of: $\lim_{n \to \infty}(\sqrt[3]{n^3+\sqrt{n}}-\sqrt[3]{n^3-1})\cdot \sqrt{(3n^3+1)}$ I want to find the limit of: $$\lim_{n \to \infty}(\sqrt[3]{n^3+\sqrt{n}}-\sqrt[3]{n^3-1})\cdot \sqrt{(3n^3+1)}$$
I tried expanding it by $$ \frac{(n^3+n^{1/2})^{1/3}+(n^3-1)^{1/3}}{(n^3+n^{1/2})^{1/3}+(n^3-1)^{1/3}} $$
but it ... | With the expansions
$$
(n^3+\sqrt{n})^{1/3}\sim n+(1/3)n^{-3/2}+...
$$
and
$$
(n^3-1)^{1/3}\sim n-(1/3)n^{-2}+...
$$
and
$$
(3n^3+1)^{1/2}\sim \sqrt{3}n^{3/2}+...
$$
as $n\to\infty$, the result follows immediately.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1607031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove if $2|x^{2} - 1$ then $8|x^{2} - 1$ I have seen this question posted before but my question is in the way I proved it. My books tells us to recall we have proven
if $2|x^{2} - 1$ then $4|x^{2} - 1$
Using this and the fact $x^{2} - 1 = (x+1)(x-1)$ and a previous proof in which we have shown if$a|b$ and $c|d$ then... | To answer your question without resorting to any explanation beyond what you have provided:
No. This part is seriously off "Assume 2|x−1 and 4|x+1" without any explanation. When you say "assume this", "assume that", you have to cover all options.
2 is prime so it must divide either (x+1) or (x-1), that essential part i... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Underdetermined Linear Systems I'm working through an introductory linear algebra textbook and one exercise gives the system
$2x+3y+5z+2w=0$
$-5x+6y-17z-3w=0$
$7x-4y+3z+13w=0$
And asks why, without doing any calculations, it has infinitely many solutions. Now, a previous exercise gives the same system without the fourt... | Obviously , $x=y=z=w=0$ is a solution. The rank of matrix $A$ is at most $3$, but we have $4$ columns.
If the solution would be unique, the rank of $A$ would have to be equal to the number of columns, which is not the case. Hence, there are infinite many solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1607250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Deciding if a statement is true or false for given sets Given the sets:
$A = \{a,b,c\},\enspace B = \{a,b,A,C\},\enspace C = \{a,c\},\enspace D = \{A,B,C\},\enspace and\enspace G = \{A,B,C,D\}$
How can I determine if the following statement is true or false:
$∃x ∈ D : ∃y ∈ x : y ∉ B ∪ x$
First of all, please c... | The notation is correct. maybe that the use of a capital $X$ for the sets can be better, but it's not necessary.
The statement is false: $y \in x \Rightarrow y \in B\cup x$
| {
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Let $H_n=1+1/2+..+1/n=p_n/q_n$. Find all $n$ such that $3|p_n$ Problem: Let $H_n=1+1/2+..+1/n=p_n/q_n$ with $\gcd(p_n,q_n)=1$. Find all $n$ such that $3|p_n$.
Observations:
Note that $H_n=(n!/1+n!/2+...+n!/n)/n!$. If $3|p_n$, the numerator of this fraction must have 3-adic valuation at least $3^t$.
Let $v_3(n!)=t$, an... | I have found a solution using the idea from my edit in the question details. Call $n$ good if $3|p_n$.
We can write
$H_n=(1/3)H_{\lfloor n/3 \rfloor}+(1/1+1/2+1/4+1/5+..+1/(3k+1)+1/(3k+2))$
for an appropriate $k$. The sum in the parentheses can be written
$3/1*2+9/4*5+15/7*8+..+(6k+3)/[(3k+1)*(3k+2)]$
The numerator of... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Irreducibility of $x^4-5$ over $\mathbb{Z}_{17}$ Obviously, it'd be hard to try all the $17$ elements to see if there is some root, and even if there is none, it'd be necessary to verify if it can't be factored into two irreducible 2 degree polynomials. The Eisenstein criterion will just be able to say about irreducibi... | If $x^4\equiv 5\pmod{17}$, then $x^{16}\equiv 5^4\equiv 13\pmod{17}$, but this contradicts Fermat's Little theorem.
More strongly, $y^2\equiv 5\pmod{17}$ has no solution. By Quadratic Reciprocity:
$$\left(\frac{5}{17}\right)=\left(\frac{17}{5}\right)=\left(\frac{2}{5}\right)=-1$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Dummy Variables and variables that change a funtion I am reading notes for my Calculus 2 class and am confused why these two functions are not equivalent
"
$x$ is not a dummy variable, for example,
$$\int 2xdx = x^2+C$$
and
$$\int 2tdt = t^2 + C$$
are functions of different variables, so
they are not equal.
"
Is it b... | Either $x$ or $t$ is just a token representing numbers, so they are equivalent.
Let's take the expression $x+1$ (or $t+1$) for instance: Letting $x=1.5$ is equivalent to letting $t=1.5$, because both of them make the expression evaluate to $1.5+1=2.5$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solution of a given legendre equation I was trying to solve following question :
$$ (1-x^2)y''-2xy'+n(n+1)y=0 $$ has the $n^{th}$ degree polynomial solution $y_n(x)$ such that $y_n(1) =3$. We are given
$\int_{-1}^{1} [y_n^2(x) + y_{n-1}^2(x)] dx = 144/15$. Find the value of $n$.
My attemnpt:
We know that $\int_{-1}^{... | The general solution of the differential equation $$(1-x^2)y''-2xy'+n(n+1)y=0$$ is given by $$y=c_1 P_n(x)+c_2 Q_n(x)$$ where appear Legendre polynomials of first and second kind (see here) but the second ones are not polynomials (so, I suppose that $c_2=0$ is a requirement).
On the other side, as you wrote, for any $n... | {
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Extension of idempotent ideals Let $R$ be a Noetherian commutative ring with $1$. If $R[[x]]$ denotes the ring of formal power series over $R$ and $I$ is an idempotent ideal of $R$ I want to know whether the extension of $I$ in $R[[x]]$ is also idempotent.
Since $R$ is Noetherian, one has $I[[x]]=IR[[x]]$. So, if a ser... | Every finitely generated idempotent ideal in any commutative ring is generated by a single idempotent element. Indeed, by Nakayama's lemma, $I^2=I$ implies there is $r\in R$ such that $1-r\in I$ and $rI=0$. Then $r(1-r)=0$, so $r=r^2$ is an idempotent. For any $i\in I$, $i=i-ri=(1-r)i$, so $I$ is generated by the id... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1607796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Calculate $\lim_{n\to \infty} \frac{\frac{2}{1}+\frac{3^2}{2}+\frac{4^3}{3^2}+...+\frac{(n+1)^n}{n^{n-1}}}{n^2}$ Calculate
$$\lim_{n\to \infty} \frac{\displaystyle \frac{2}{1}+\frac{3^2}{2}+\frac{4^3}{3^2}+...+\frac{(n+1)^n}{n^{n-1}}}{n^2}$$
I have messed around with this task for quite a while now, but I haven't succ... | Combining $$\frac{1}{1-1/n}>1+\frac1n \ \ \ n>1 $$ with the fact that for all positive $n$ we have $$ \left(1+\frac1n\right)^n<e<\left(1+\frac{1}{n}\right)^{n+1}, $$ and after rewriting your sequence $a_n$ as $\frac{1}{n^2}\sum_{k=1}^n k(1+1/k)^k$, we find the general term $s_k$ of the sum satisfies $$k\left(e-\frac ek... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1607951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
probability density of the maximum of samples from a normalized uniform distribution Suppose
$$X_1, X_2, \dots, X_n\sim Unif(0, 1), iid$$
and suppose
$$\hat\theta = \max\{X_1, X_2, \dots, X_n\} / \sum_i^nX_i$$
How would I find the probability density of $\hat\theta$?
I know the answer if it's iid. But I don't know how ... | Note that $$\hat\theta_n\sim \frac 1 {1+\sum_{i=1}^{n-1}U_i} $$
for i.i.d. standard uniforms $U_i$. Now see this and this
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Prove that at a wedding reception you don't need more than $20 \sqrt{mn}$ of ribbon to adornate the cakes.
At a wedding reception,$n$ guests have assembled into $m$ groups to
converse.(The groups are not necessarily equal sized.)The host is
preparing $m$ square cakes,each with an ornate ribbon adorning its
perim... | You are almost there
$$s_i^2=25 x_i \implies \sum_{i=1}^ms_i^2=25 n$$
$$L= 4 \sum_{i=1}^m s_i$$
$$20 \sqrt{mn}=20 \sqrt{m \frac{\sum_{i=1}^m s_i^2}{25}}= 4 \sqrt{m \sum_{i=1}^m s_i^2} $$
Hence we want to prove that
$$ \sum_{i=1}^m s_i \le \sqrt{m \sum_{i=1}^m s_i^2} $$
or
$$ \left(\frac{\sum_{i=1}^m s_i}{m}\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the sum $\sum_{k=2}^n \frac{n!}{(n-k)!(k-2)!}.$ Find the following sum $$\sum_{k=2}^n \frac{n!}{(n-k)!(k-2)!}.$$I found that , $$\sum_{k=2}^n \frac{n!}{(n-k)!(k-2)!}=n!\left[\frac{1}{(n-2)!0!}+\frac{1}{(n-3)!1!}+\cdots +\frac{1}{0!(n-2)!}\right]$$From here how I proceed ?
| Outline: By the Binomial Theorem we have
$$(1+x)^n=\sum_{k=0}^n \frac{n!}{(n-k)!k!}x^k.$$
Differentiate twice and set $x=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Formulae for Catalan's constant. Some years ago, someone had shown me the formula (1).
I have searched for its origin and for a proof.
I wasn't able to get true origin of this formula but I was able to find out an elementary proof for it.
Since then, I'm interested in different approaches to find more formulae as (1).
... | A more natural proof.
\begin{align}
\beta&=\sqrt{3}-1\\
J&=\int_0^1 \frac{\arctan\left(\frac{x(1-x)}{2-x}\right)}{x}dx\\
&\overset{\text{IBP}}=\left[\arctan\left(\frac{x(1-x)}{2-x}\right)\ln x\right]_0^1-\int_0^1 \frac{(x^2-4x+2)\ln x}{(x^2+\beta x+2)(x^2-(\beta+2)x+2)}dx\\
&=-\int_0^1 \frac{(x^2-4x+2)\ln x}{(x^2+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
How can I find the coefficient of x when the power is greater than the powers of 2 brackets using binomial expansion? I have been given this question:
Find the coefficient of $x^{13}$ in the expansion of $(1 + 2x)^4(2 + x)^{10}$.
I know how I would find $x^4$ or lower degrees, but I am unsure how to approach this, as n... | If you expand the two terms, the first one will give you terms from $1$ to $16x^4$. The second will give you terms from $2^{10}$ to $x^{10}$. When you multiply them, the only two ways to get $x^{13}$ is to use the $x^3$ term from the first and the $x^{10}$ term from the second or to use the $x^4$ from the first and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
How to prove $\cos \theta + \sin \theta =\sqrt{2} \cos\theta$. I'm learning Trigonometry right now with myself. I'm stuck in a problem from sometimes (If $ \cos \theta - \sin \theta =\sqrt{2} \sin\theta $, proof that $ \cos \theta + \sin \theta =\sqrt{2} \cos\theta $) . I don't know what to do next. Please have a look... | $$\begin{align}
\cos{\theta}-\sin{\theta} &=\sqrt{2}\sin{\theta} \\
\cos{\theta} &= (1+\sqrt{2})\sin{\theta} \\
(1-\sqrt{2})\cos{\theta} &=(1-\sqrt{2})(1+\sqrt{2})\sin{\theta}=(1-2)\sin{\theta} \\
\cos{\theta}+\sin{\theta} &= \sqrt{2}\cos{\theta}
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Can $f\in W^{k,p}(U)$ be extended to a function in $W^{k,p}(\mathbb{R}^n)$ in general? Let $U$ be a nonempty open subset of $\mathbb{R}^n$. Suppose $f\in L^p(U)$ for some $p$ with $1\leq p<\infty$. By extending $f$ to be identically zero outside $U$, one has $f\in L^p(\mathbb{R}^n)$.
My question is: if one replaces $L... | You can't just choose the function to be identically zero outside $U$ in general, because this can cause the weak derivative to fail to be $L^p$ because of "bad regularity" at the boundary. For instance when $n=1$, all $W^{k,p}$ functions are actually continuous, so extending $f(x)=1$ on $(0,1)$ to be identically zero ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integral Problem: $\int_{0}^{1} \sqrt{e^{2x}+e^{-2x}+2}$ I am having trouble with this definite integral problem
$$\int_{0}^{1} \sqrt{e^{2x}+e^{-2x}+2} \, dx$$
I know that the solution is
$$e - \dfrac{1}{e}$$
I checked the step by step solution from Wolfram Alpha and it says because $$ 0 < x < 1 $$ the integral can be... | Another option:$$e^{2x}+e^{-2x}+2=2\frac{e^{2x}+e^{-2x}}{2}+2=2\cosh(2x)+2=2\left[2\cosh^2(x)-1\right]+2=4\cosh^2(x)$$Hence,$$\int\limits_{0}^{1}\sqrt{e^{2x}+e^{-2x}+2}\text{d}x=\int\limits_{0}^{1}\sqrt{4\cosh^2(x)}\text{d}x=2\int\limits_{0}^{1}\cosh(x)\text{d}x=2\Big.\sinh(x)\Big\vert_{0}^{1}=2\sinh(1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Finding the limit $\lim_{n\to \infty}\frac{n^k}{n\choose k}$ I am confused with solving problems with combinatorial limits. My question here is that for $k\in N$, what is the $$\lim_{n\to \infty}\frac{n^k}{n\choose k}$$
| Note that:
As n $\to \infty$,
$$\frac{n^{k}}{\binom{n}{k}}=\frac{n^{k} k!}{n(n-1)(n-2)\cdot\ldots \cdot(n-k+1)}=\frac{k!}{1\cdot\left(1-\frac{1}n\right)\left(1-\frac{2}n\right)\cdot\ldots\cdot\left(1-\frac{k-1}{n}\right)} \to k! $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Question about a Symmetric random walk, Problem 4.1.1 in Durrett I am working on the following problem:
Let $X_1, X_2, \dots \in \mathbb{R}$ be i.i.d. with a distribution that is symmetric about $0$ and nondegenerate, i.e. $P(X_i=0)<1$. Show that $-\infty = \liminf S_n < \limsup S_n=\infty$. Here $S_n=X_1 + \dots +X_n... | If $S_n\to\infty$ a.s. then for $\epsilon>0$ there is $N$ s.t. for $n>N$, $P[S_n>1]>1-\epsilon$, say. But you know for any finite $n$, $P[S_n<0]=P[S_n>0]$ (E.g. induction. Given symmetric independent RVs $U$ and $V$, $P[U+V<0]=\int_x F_U(-x)dF_V(x)=\int_x(1-F_U(x))dF_V(-x)=P[U+V>0]$, ignoring continuity issues.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1608960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If n is positive integer, prove that the prime factorization of $2^{2n}\times 3^n - 1$ contains $11$ as one of the prime factors I have: $2^{2n} \cdot 3^{n} - 1 = (2^2 \cdot 3)^n - 1 = 12^n - 1$.
I know every positive integer is a product of primes, so that,
$$12^n - 1 = p_1 \cdot p_2 \cdot \dots \cdot p_r. $$
Also, ... | By the Binomial theorem,
$$12^n-1=(11+1)^n-1\\
=11^n+n\,11^{n-1}+\frac{n(n-1)}211^{n-2}+\cdots+\frac{n(n-1)}211^2+n\,11+1-1,$$
which is a multiple of $11$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1609070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
I've shown $\lim_{n\to\infty}\sqrt[n]{n}=e$, but it should be $1$. Where's the mistake? $$\lim_{n\to \infty} \sqrt[n] n = \lim_{n\to \infty} n^{\frac{1}{n}} = \lim_{n \to \infty} \{(1+(n-1))^{\frac{1}{n-1}}\}{^{(n-1)\frac{1}{n}}} = \lim_{n\to \infty} e^{\frac{n-1}{n}} = e$$
But this clearly isn't true as the actual lim... | You can do it using another way $$A=n^{\frac 1n}\implies \log(A)=\frac {\log(n)}n$$ and you know that $\log(n)$ is "slower" than $n$. So, $\log(A)\to 0$ and then $A\to 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1609190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Extreme of $\cos A\cos B\cos C$ in a triangle without calculus.
If $A,B,C$ are angles of a triangle, find the extreme value of $\cos A\cos B\cos C$.
I have tried using $A+B+C=\pi$, and applying all and any trig formulas, also AM-GM, but nothing helps.
On this topic we learned also about Cauchy inequality, but I have... | If $y=\cos A\cos B\cos C,$
$2y=\cos C[2\cos A\cos B]=\cos C\{\cos(A-B)+\cos(A+B)\}$
As $A+B=\pi-C,\cos(A+B)=-\cos C$
On rearrangement we have $$\cos^2C-\cos C\cos(A-B)+2y=0$$
As $C$ is real, so will be $\cos C$
$\implies$ the discriminant $$\cos^2(A-B)-8y\ge0\iff y\le\dfrac{\cos^2(A-B)}8\le\dfrac18$$
The equality occu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1609327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 8,
"answer_id": 0
} |
Show that $[-1,1] \times [-1,1]$ is a closed set. Show that $A = [-1,1] \times [-1,1]$ is a closed set.
I know that I have to show $A^c$ is open. So I have to find $\epsilon > 0$
sufficiently small such that for all $x \in A^c$, $B(x,\epsilon) \subset A^c$. I am a bit blocked at this point. I think I have to use the ... | It is an easy exercise to show that the following are open subsets of $\mathbb{R}^2$ for any $r\in\mathbb{R}$:
\begin{align*}
(r,\infty)\times\mathbb{R} &&
(-\infty,r)\times\mathbb{R} &&
\mathbb{R}\times(r,\infty) &&
\mathbb{R}\times(-\infty,r) &&
\end{align*}
Now notice that $([-1,1]\times[-1,1])^C$ can be described a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1609393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What are the disadvantages of non-standard analysis? Most students are first taught standard analysis, and the might learn about NSA later on. However, what has kept NSA from becoming the standard? At least from what I've been told, it is much more intuitive than the standard. So why has it not been adopted as the main... | NSA is an interesting intellectual game in its own right, but it is not helping the student to a better understanding of multivariate analysis: volume elements, $ds$ versus $dx$, etcetera. The difficulties there reside largely in the geometric intuition, and not in the $\epsilon/\delta$-procedures reformulated in terms... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1609463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
} |
Prove that $\mathrm{span}(S) = S$ for a subspace $S$. Prove that if $S$ is a subspace of a vector space $V$, then $\mathrm{span}(S) = S$.
What I tried: I considered using the properties of vector spaces or maybe using an example where $S \subseteq \mathbb{R}^2$, but those strategies didn't amount to much. Any thoughts?... | Note that the span of a set $S\subset V$ is the intersection of all subspaces of $V$ that contain $S$, i.e. $$\operatorname{Span}(S) = \bigcap_{S\ \subset\ T\ \leqslant V}T. $$
The intersection of subspaces is again a subspace, so $$\bigcap_{S\ \subset\ T\ \leqslant V}T$$ is a subspace containing $S$, which implies tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1609596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Conditional probability - Proof I interpret this as the statement that if A is true then B is more likely.
I have
$P(B \mid A) > P(B)$
Now I want to formulate and prove:
*
*If not-B is true, then A becomes less likely
*If not-A is true, then B becomes less likely
If I get help with the first one I am sure I will be... | What we want to show for the first is $P(A)>P(A|B^c)$. Let's start by using the law of total probability, as suggested by Alex. The law states that
$$P(A) = P(B^c)P(A|B^c)+P(B)P(A|B).$$
Now, we can solve for $P(A|B^c)$ and we get
$$P(A|B^c) = \frac{P(A)-P(B)P(A|B)}{P(B^c)}.$$
Noting that $P(B)P(A|B) = P(A \cap B) = P(A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1609666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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