Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find period of the following function Find period of :
$$f(x)=| \sin(x) + \cos(x) |$$
Attempt:
Let $$f(x)=|\sin x +\cos x|\tag{I}$$
Let $t$ be the required period,
So,
$f(x)=f(x+t)$
Putting $x=0$ we get,
$f(0)=f(t)$
f(0)=1 (from 'I')
Now we have the following,
$1=| \sin t + \cos t |$
If we put $t=\pi/2$ ,that would ... | Let's assume you do not know the shape of the graphs but want to find the period. We will assume that you can evaluate the functions and know the sum and difference formulas for sine and cosine.
\begin{eqnarray}
|\sin x+\cos x|&=&|\sin(x+t)+\cos(x+t)|\\
&=&|\sin x\cos t+\cos x\sin t+\cos x\cos t-\sin x\sin t|\\
&=&|\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2787865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Evaluate integral for $\int \sin^2 (x+\frac{\pi}{6}) dx$ Can someone walk me through how to evaluate the integral $$\int \sin^2 (x+\frac{\pi}{6}) dx?$$
I get as far as
$$\int \frac {1 - \cos(x + \frac{\pi}{6})}2dx,$$
but I am not sure how to proceed.
| Note that in the particular case of $\sin^2(x)$ and $\cos^2(x)$ there is a trick you can use.
$\begin{cases}\cos^2(x)+\sin^2(x)=1\\\cos^2(x)-\sin^2(x) = \big(\sin(x)\cos(x)\big)' \end{cases}$
Thus if we call $C=\int\cos^2$ and $S=\int\sin^2$ we get the system below
$\begin{cases}C(x)+S(x)=x+cst\\C(x)-S(x)=\sin(x)\cos(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2787969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Eigenvalue problem corresponding to a univariate differential operator Let $Af=f''(x)-xf'(x)$ and $\mu$ be the Gaussian measure, i.e, $\,d\mu(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. We consider the following eigenvalue problem $$Af=\lambda f,$$ where $f\in C^\infty(\mathbb{R})\cap L^p(\mu)$ for all $p>1$. Show that any so... | The function $f$ will be a polynomial when $\lambda\in\mathbb N$. That can be seen from the fact that if we substitute a series $y=\sum_0^\infty a_nx^n$, we get the recursions
$$\tag1
a_{2n+2}=\frac{(2n-\lambda)(2n-2-\lambda)\cdots(2-\lambda)\lambda}{(2n+2)!}\,a_0,
$$
$$\tag2
a_{2n+1}=\frac{(2n-1-\lambda)(2n-3-\lambd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2788064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integrating $\int_0^1 \frac {\log(1-x)\log^2(1+x)}x \mathrm{d}x$
Question: Integrate$$\int\limits_0^1dx\,\frac {\log(1-x)\log^2(1+x)}x=-\frac {\pi^4}{240}$$
I'm curious as to if there is a way to integrate this. I've tried using integration by parts to get$$I=-\frac {\pi^2}6\log^22+2\int\limits_0^1dx\,\frac {\operato... | The integral is hard to tackle directly (without using Euler sums), but there is a nice trick (which is literally the same as posed above).
Let $$I = \int_0^1 {\frac{{\ln (1 - x){{\ln }^2}(1 + x)}}{x}dx} \qquad J = \int_0^1 {\frac{{{{\ln }^2}(1 - x)\ln (1 + x)}}{x}dx} $$
We have
$$\begin{aligned} 3I + 3J + \int_0^1 {\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2788232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
} |
Why can we not find n+ 2 vectors in $R^n$ so that the dot product of any two of them is negative? Take for example 3-space, where you can arrange 4 vectors so that the dot product of any two is negative, but when you add a fifth vector there is a least one non-negative dot-product.
I have seen examples of proofs using... | It might be better to just bite the bullet and try to understand the inductive proof. It's much simpler (and, once you understand it, more intuitive) than any proof you will get based on angles. In outline, in the inductive proof, you basically show that you can fix one of the vectors $x$ and sequentially modify anot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2788391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Determinate $\lambda\in R$ so that the following equation has 2 real,distinct solutions. Determinate $\lambda\in R$ so that the following equation has 2 real,distinct solutions. $$2x+\ln x-\lambda(x-\ln x)=0$$
I think this should be solved using Rolle property for finding intervals with solutions.So i calculated $f^|(x... | Hint: defining $$f(x)=2x+2\ln(x)-\lambda(x-\ln(x))$$ then $$f'(x)=2+\frac{2}{x}-\lambda\left(1-\frac{1}{x}\right)$$ then $$f'(x)=0$$ if $$x_E=\frac{\lambda+2}{\lambda-2}$$ and $$f''(x_E)=-\frac{(-2+\lambda)^2}{\lambda+2}$$ then must hold
$$f''(x_E)<0$$ and $$f(x_E)=(\lambda+2)\left(\ln(x_E)-1\right)>0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2788479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solution to Dirichlet boundary value problem on upper halve plane using Green's function Currently I am studying for an exam about partial differential equations. While looking through some of the exercises concerning Green's function on the plane, I came across a rather impenetrable-seeming integral connected to a Dir... | I'll assume you made a typo in the equation, since the Green's function you have is for $\nabla$, not $-\nabla$
The inhomogeneous part only depends on $y$, so you can guess a solution of the form $u(x,y) = g(y)$, where
$$ -g''(y) = \frac{1}{1+y}, \ g(0) = 0 $$
Then integrating twice gives
$$ g(y) = (1+y)\ln(1+y) - y + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2788617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find $\lim \mathbb{E} \frac{X_{n}^{2}}{\log(1+X_{n}^{2})}$ Consider $Z_{n} = \frac{X_{n}^{2}}{\log(1+X_{n}^{2})}$, in which $X_n$ denotes a Gaussian random variable with zero expected value and variance equals $1/n$. Now we want find
$\lim_{n \to \infty} \mathbb{E} \frac{X_{n}^{2}}{\log(1+X_{n}^{2})}$.
I thought about... | I think it could be cracked by real analysis method. For convenience note $X_n$ by $f_n$. Then $EX_n^2=\int f_n^2dP=1/n$, and $f_n\overset{P}{\to}0$.
For $\delta>0$, set $$M(\delta)=\sup_{|x|\leq\delta}\frac{x^2}{\log(1+x^2)}$$ $$m(\delta)=\inf_{|x|\leq\delta}\frac{x^2}{\log(1+x^2)}$$
Note that $$\lim_{\delta\to0}M(\de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2788765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Solving a 2 variable integral with a delta function How can this integrals that include the dirac delta function be solved?
$$\int_{-\infty}^{+\infty}dq\int_{-\infty}^{+\infty}dp\cdot p^n\cdot \delta(p^2+q^2-E)$$
| Note that the the integral is zero by symmetry if $n$ is odd. Assume from now on that $n\geq 0$ is even.
One idea is to use polar coordinates $$(p,q)~=~(r\cos\theta,r\sin\theta).\tag{1}$$
Then
$$I~:=~ \iint_{\mathbb{R}^2}\! \mathrm{d}p~\mathrm{d}q~p^n~\delta(p^2+q^2-E)
~=~ \int_{\mathbb{R}_+}\! \mathrm{d}r~r^{n+1}\delt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2788885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Permutation Problem - Seating with Empty Chairs There are 3 men and 3 women to be seated in a row of 10 chairs. In how many different ways can they be seated if one man must be seated at each end of the row?
I began by calculating $_3P_2 = 6$ for the possible combinations for the end seats. The book gives the answer as ... | Ignore "permutation" formulas and just do this directly via rule of product. (It is afterall from the rule of product that we get the permutation formulas in the first place).
*
*Choose which man sits at the far left end (three options)
*Choose which man sits at the far right end (two remaining options)
*Choose w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2789152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$AA^*A=A$ with eigenvalues $1$ and $0$, prove that $A$ is unitarily diagonalizable.
Let $A$ be a $2$ by $2$ complex matrix such that $AA^*A=A$, and has eigenvalues of $1$ and $0$. Prove that $A$ is unitarily diagonalizable.
Well, if a matrix is Hermitian, then it is unitary diagonalizable. It also has real eigenvalu... | Here's a proof that works:
Every matrix is unitarily triangularizable. So, there exists a unitary $U$ such that $A = UTU^*$, where
$$
T = \pmatrix{1&t\\0&0}
$$
If we show that $t$ is necessarily zero, then we may conclude that $A$ is unitarily diagonalizable (since $T$ would then be diagonal).
With that goal in mind, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2789304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Stochastically dominating variables with same expectation Consider the random variables $X, Y$ with distributions $F_X, F_Y$ which satisfy $F_X(t) \leq F_Y(t), \; \forall t$. Additionally, we know $X, Y \geq 0$. The aim is to show that $\mathbb{E}(X) = \mathbb{E}(Y) \Rightarrow F_X(t) = F_Y(t), \forall t$.
My attempt:
... | $g(t)=F_Y(t)-F_X(t)$ is a non-negative measurable function whose integral over $\mathbb R^{+}$ is zero. This implies that $g(t)=0$ almost everywhere. In turn this implies that it is zero on a dense subset of $\mathbb R^{+}$. Since $g$ is right continuous it follows that $g(t)=0$ for all $t$. [Details: if $A$ has Lebesg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2789442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do I solve $\int_1^{2}\frac{(\ln{x})^2}{x^3} dx$? $\int_1^{2}\frac{(\ln{x})^2}{x^3} dx$
This seemed like an integration by parts problem, so I used it and got:
$\int_1^{2}\frac{(\ln{x})^2}{x^3} dx = [-\frac{1}{2}x^{-2}(\ln{x})^2 | _{1}^{2}] - \int_1^{2}\frac{2\ln{x}}{x^3}dx$
But I can't get anywhere with the $\int_... | Hint :
Put $\ln x=t$ and hence $dx=e^t dt$
The integral then changes to $$\int_{0}^{\ln 2} t^2e^{-2t}dt$$
Now apply repeated integration by parts with $u=t^2$ and $dv=e^{-2t}dt$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2789541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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How to solve this nonlinear least square optimization? Problem description
Given data at many time instance $t$,
$$\min _{\alpha, \xi, \beta} \lVert y(t) - \alpha e^{\Lambda t} \beta \rVert_F$$
with $$ \lVert \alpha \rVert_F^2 = 1 $$
where $y(t) \in \mathbb{R}^{n \times M}$, $\Lambda \in \mathbb{R}^{r\times r}$ is a di... | First, note that the problem is non-convex, and I am not aware of a convex reformulation. That is, we resort to methods which might get stuck in a non-optimal point instead of converging to the optimum, including the naive gradient descent.
The first thing I would try is alternating minimization. Note that the Frobeniu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2789635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A quick way, say in a minute, to deduce whether $1037$ is a prime number So with $1037 = 17 \cdot 61$, is there a fast method to deduce that it's not a prime number?
Say $1037 = 10^3+6^2+1$. Does $a^3 + b^2 + 1$ factorize in some way?
As part of their interviews, a company is asking whether a number is prime. I have n... | $u\mid v$ is read as "$u$ divides $v$" and $u\nmid v$ is read as "$u$ does not divide $v$."
Obviously $2\nmid 1037$ since it has an odd last digit.
By the "Divisibility by $3$ Rule," it follows that $3\nmid 1037$ since $3\nmid 1+0+3+7=11$.
Obviously $5\nmid 1037$ since the last digit is not $0$ nor $5$.
If $7\mid 1037... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2789794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
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"answer_id": 3
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Discrete Maths: Ways of sitting 3 exams during a 13 week semester. I had a discrete maths test today and this question was the last one and it really threw me off.
During a 13 week semester ( no breaks in between ), a student must sit 3 exams >for a particular course.
However, he must sit them in order, i.e 1st -> ... | (Edited the answer to fit all criterias.)
Choose the 3 weeks out of 13 on which the student writes the exam. On the first chosen week the first exam, the second chosen week the second exam, third week the third.
There is a known mathematical formula for choosing $k$ instances from $n$, which is calculated by
$${n\choos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2789889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given that $u$ is harmonic. Prove $\Delta v \geq 0$ where $v = |\nabla u |^2$ Let $\Omega \subset \Re^2$ be open, and let $B_r(x) \subset \Omega$ be any open disc in $\Omega$.
Assume that $u \in C^2(\Omega)$ is harmonic in $\Omega$
Let $v = |\nabla u |^2$. Prove $\Delta v \geq 0$ in $\Omega$
My Attempt:
So what I hav... | Useful general formula: If $u,v\in C^2,$ then
$$\Delta (uv) = u\Delta v + v\Delta u + 2 \langle \nabla u, \nabla v\rangle.$$
Thus if $u$ is harmonic, then $\Delta (u^2) = 2 |\nabla u|^2.$ In your problem, apply this to each $(D_k u)^2,$ and recall that if $u$ is harmonic, then so is $D_ku.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2790004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Area in $uv$ plane corresponding to two areas in $xy$? The inequalities $1 \leq x^2 - y^2 \leq 4$ and $3 \leq xy \leq 5$ describe the area $D$. See picture below.
The excerice is to calculate the integral: $\iint_D 2(x^4 - y^4) \,dx\,dy$
I would do a vaiable substitution where $u = x^2 - y^2$ and $v = xy$ where $1 \le... | To integrate we need also to express the intagrand function $2(x^4 - y^4)$ in terms of $u$ and $v$
We can proceed as follow
*
*$x^4 - y^4=(x^2-y^2)(x^2+y^2)=u\sqrt{u^2+4v^2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2790195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Without-loss-of-generality question Going through solutions of IMO'09. Bumped into a without-loss-of-generality assumption that I can't comprehend.
Here's the statement of the problem:
Let $a,b,c$ be positive real numbers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=a+b+c$. Prove that
$$
\frac{1}{(2a+b+c)^2}+\frac{1}... | There is no loss of generality because the inequality
$$
\frac{(a+b+c)^2}{(2a+b+c)^2}+\frac{(a+b+c)^2}{(2b+c+a)^2}+\frac{(a+b+c)^2}{(2c+a+b)^2}\leq\frac{3}{16}(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)
$$
is homogenized. This means: if you multiply $a,b$ and $c$ with the same constant $r$, the inequality... | {
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"url": "https://math.stackexchange.com/questions/2790370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Having problems with the actual proof step of epsilon-N sequence convergence proofs I'm having problems actually completing epsilon-N proofs for the convergence of sequences, namely the part where, once I've found an N, I have to work backward to derive the original statement. I almost always up ending in a situation w... | $a_n = \frac {2n-1}{4n^2}$
If $n>1$ then $0<a_n< \frac {1}{2n}$
For any epsilon, if $N > \frac {1}{2\epsilon}$ then $n>N \implies |a_n| < \epsilon$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2790489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $|a_n-a_{n+1}|\to 0$ imply $(a_n)$ is Cauchy? My textbook has this problem as a kind of "concept check", where one is supposed to find a counterexample to the following statement:
A sequence of real numbers is cauchy iff.
$$ \forall \epsilon>0, \, \exists N \in \mathbb{N}, \, \forall n \geq N: |a_n-a_{n+1}|< \ep... | $a_n = \ln n$ will do the job.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2790561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Which manifold is formed by the set of resolutions of the identity operator into orthogonal projectors? A resolution of the identity operator $I$ on $\mathbb{R}^n$ or $\mathbb{C}^n$ is a decomposition
$$I = \sum_{i=1}^n P_i,$$
where the $\{P_i\}$ are a set of orthogonal rank-one projection operators. What is the manifo... | Yes, it's just $U(n)/U(1)^n$. This is not a quotient group because $U(1)^n$ is not a normal subgroup. This manifold is known as the (complete) flag variety of $\mathbb{C}^n$, and much is known about it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2790669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Kolmogorov like Maximal inequality with exponential expected value Let $X_i$ a countable collection of independent random variables with symmetric distribution i.e. $P(X_i\in A)=P(X_i \in -A)$ for all $i\geq 1$. If $\lambda\in\mathbb{R}$ is such that $E(e^{\lambda X_i})$ is finite for all $i$ I want to prove the follow... | Using extrapolation I am able to prove your statement but only for $t$ large relative to $\lambda^{-1}$ and with the stronger condition of $E[e^{\lambda |X_i|}] < \infty$ (maybe you were missing an absolute value). Let us denote by $S$ the random variable
$$
S = \sum_{j = 1}^\infty X_j
$$
and by $\Sigma_n$ the $\sig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2790767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Value of Solving Unsolved / Edge-case Mathematical Problems I just came across Unsolved Problems in Group Theory, of which there are 100's of very specific, detailed problems, such as these:
...
15.68. Does there exist an infinite finitely generated 2-group (of finite exponent) all of whose proper subgroups are locall... | Each of these problems comes with the name of the person who sent it. If one of them strikes you, it could be worthwhile to have a look at the kind of research this person does, either via their published papers or their personal webpage, to see what kind of mathematics they are doing and specifically what problems the... | {
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"url": "https://math.stackexchange.com/questions/2790866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sine of angle between direct common tangents of two circles? This is a question I found: Given two circles intersecting orthogonally having the length of common chord 24/5 units the radius of one of the circles is 3 units then what is the sine of the angle between the direct common tangents? The answer given is 4√6/25.... | To make things simpler. There's this formula you can memorise or derive as you may like by similarity and Pythagoras.
Alpha{angle between DCT}= 2sin^-1[|r2-r1|/ d]
Where R1 and R2 are radii and d is distance between centres.
Here, sin a/2 comes out to be 1/5.
Thus sin a = 2 sin a/2 cos a/2
= 4✓6/25
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2790926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Which of the following statements is FALSE?
Which of the following statements is FALSE? There exists an integer $x$ such that
$1.$ $x \equiv 23$ mod $1000$ and $x \equiv 45$ mod $6789$.
$2.$ $x \equiv 23$ mod $1000$ and $x \equiv 54$ mod $6789$.
$3.$ $x \equiv 32$ mod $1000$ and $x \equiv 54$ mod $9876$.
$4.$ $x \equi... | The first and second case clearly work out, since $6789$ and $1000$ are co prime, so one can solve the given equations simultaneously.
If $x \equiv 32 \mod 1000$ then $x$ is a multiple of $8$, since $x = 1000k + 32$ so it is a sum of multiples of $8$.
However, $9876 \equiv 4 \mod 8$, so for any $x = 9876k + 54$ , it le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2791044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many spheres are needed to shield a point source of light?
How many spheres are needed to shield a point source of light?
I read this from a mathematical puzzle book. And book says the answer is six without explanation.
From the geometric point of view, I'm thinking of a tetrahedron where the point source is loc... | The minimal number of spheres required to shield the origin is $4$.
4 spheres is sufficient.
Consider following $4$ points on unit sphere $S^2$,
$$
v_0 = \frac{1}{\sqrt{3}}( -1,-1, -1),
v_1 = \frac{1}{\sqrt{3}}( -1, 1, 1),
v_2 = \frac{1}{\sqrt{3}}( 1,-1, 1),
v_3 = \frac{1}{\sqrt{3}}( 1, 1, -1)
$$
they are formi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2791186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
} |
Complement of a projective two sided ideal is two sided again? If a finite dimensional algebra $A$ over a field $\mathbb{k}$ is semisimple then any two sided ideal of $A$ is generated (as a left module) by a central idempotent, so its (unique) complement is a two sided ideal again. In general, this observation is wrong... | If $A$ is any finite dimensional algebra and $e$ any idempotent, then $A=Ae\oplus A(1-e)$ as a direct sum of left ideals. The endomorphism ring of $A$ as a left module is $A^{op}$ acting by right multiplication, so the condition that $Ae$ is a two sided ideal is equivalent to $\text{Hom}_A\left(Ae,A(1-e)\right)=0$.
But... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2791284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\lim_{n\rightarrow\infty}\sum_{k=1}^n\arcsin(\frac k{n^2})$
Compute $$\lim_{n\to\infty}\sum_{k=1}^n\arcsin\left(\frac k{n^2}\right)$$
Hello, I'm deeply sorry but I don't know how to approach any infinite sum that involves $\arcsin$, so I couldn't do anything to this question. Any hints/tips would be appreci... | Another way to look at this is to observe that
$$\arcsin{\left ( \frac{k}{n^2} \right )} = \frac{k}{n^2} \int_0^1 \frac{du}{\sqrt{1-\frac{k^2}{n^4} u^2}} $$
Then you can reverse order of summation and integration and get that the sum equals
$$\int_0^1 du \, \frac1{n} \sum_{k=1}^n \frac{(k/n)}{\sqrt{1-\frac{k^2}{n^4} u^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2791378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
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Is a small linear invertible perturbation of a linear isomorphism also an isomorphism? Suppose we have a linear isomorphism $T_{0}: \mathbb{R}^{n} \mapsto \mathbb{R}^{n}$, and $T_{\epsilon}$ is a small linear perturbation of it, i.e.
$$T_{\epsilon} = T_{0} + \epsilon T_{1} + O(\epsilon^{2}),$$
where $T_{1}$ is also lin... | The answer is yes and it does not depend on $T_1$ being an isomorphism.
The reason is that the map $$\mathcal{L}(\Bbb R^n\to\Bbb R^n)\to\Bbb R: T\mapsto \det(T)$$ is continuous, and the set of isomorphisms is the preimage of $\Bbb R\setminus \{0\}$ which is an open set, hence the set of isomorphisms is open itself.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2791489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Leading terms in asymptotic expansion of modified bessel function of the first kind Show that leading and next-to-leading terms in an asymptotic expansion for large $x>0$ of the modified Bessel functions of the first kind $I_0(x)$ and $I_1(x)$ are:
$$I_0(x) \sim \frac{e^x}{\sqrt{2\pi x}}\left(1+\frac{1}{8x}\right) \ \ ... | From
$$ I_n(x)
~=~\frac{1}{\pi}\int_0^{\pi} \! \mathrm{d}\theta ~\exp\left(x\cos\theta\right)\cos n\theta, \qquad n\in~\mathbb{N}_0,\tag{A} $$
we calculate
$$\begin{align}
\sqrt{x}\pi e^{-x}I_n(x)
&~~=~\sqrt{x} \int_0^{\pi} \! \mathrm{d}\theta ~\exp\left(- x(1-\cos\theta)\right)\cos n\theta \cr
&\stackrel{t=\sqrt{x}\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2791588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Find all continuous functions $ f : \mathbb{R} \to \mathbb{R} $ such that $(f(x)g(x))' = f'(x)g'(x) $
Find all continuous functions $ f : \mathbb{R} \to \mathbb{R} $ such that $(f(x)g(x))' = f'(x)g'(x), f,g \neq const $
My solution: $ f'(x)g(x)+g'(x)f(x) = f'(x)g'(x) \\
g(x)+g'(x)f(x)/f'(x) = g'(x) \\
f(... | $$f'(x)g(x)+f(x)g'(x)=f'(x)g'(x)$$
$$\frac{f(x)}{f'(x)}+\frac{g(x)}{g'(x)}=1$$
$$\frac{1}{[\ln f(x)]'}+\frac{1}{[\ln g(x)]'}=1$$
call $\ln f(x)=p(x)$ and $\ln g(x)=q(x)$
$$p'(x)=\frac{q'(x)}{q'(x)-1}=1+\frac{1}{q'(x)-1}$$
$$p(x)=x+\int\frac{1}{q'(t)-1}dt+c$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2791725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Prove that $Z-(Y-X)=X\cup(Z-Y)$ if $X\subset Y\subset Z$
Let $X\subset Y\subset Z$. Prove that $Z-(Y-X)=X\cup(Z-Y)$.
Here, $A-B$ is the complement of $B$ in $A$.
To establish equality, I need to show that $Z-(Y-X)\subset X\cup(Z-Y)$ and $X\cup(Z-Y)\subset Z-(Y-X)$. Here's how I start the proof:
Let $\alpha\in Z-(Y-X... | You can also use directly the De Morgan laws.
$\begin{align}
Z-(Y-X)&=Z\cap(Y-X)^\complement & \text{subtraction formula : } A-B=A\cap B^\complement\\
&=Z\cap(Y\cap X^\complement)^\complement & \text{subtraction formula} \\
&=Z\cap(Y^\complement\cup X) & \text{inversion : } (A\cap B)^\complement=(A^\complement\cup B^\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2791890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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To what number does $ n^{-2} \times \sum_{m=1}^{n-1} n \bmod m$ converge, as $n$ gets large? I evaluated the expression $$ n^{-2} \times \sum_{m=1}^{n-1} n \bmod m$$ for "large" $n$ values ($10^3$, $10^4$, $10^5$, $...$) and it seems to converge to the number approximately $0.17753188$. I tried to search for this numbe... | Write $n\bmod m = n -m\cdot\left\lfloor\frac{n}m\right\rfloor$ and hence your partial sums are
\begin{align}
s_n
&=
\sum_{m= 1}^{n-1}\,\frac1n - \frac{m}{n^2}\cdot\left\lfloor\frac{n}m\right\rfloor
\\&=
1 - \frac{1}{n} -\frac1{n^2} \left( \sum_{m= 1}^{n-1}\,m \cdot \left\lfloor\frac{n}m\right\rfloor \right).
\end{align... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2792028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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Homotopy equivalence of pairs Let $ I = [0,1] \subseteq \mathbb{R}$. I want to prove that the pair $(I^n,\partial I^n)$ is homotopy equivalent to $(\mathbb{R}^n,\mathbb{R}^n\setminus\{ 0,0,...,0 \})$, but I have a problem with the definition itself.
Can someone please state the definition of homotopy equivalence of pa... | Let $A\subset X$ and $B\subset Y$ be CW-pairs (or any pairs of topological spaces such that the inclusions are cofibrations, see at the end).
The map $f:X\to Y$ is a homotopy equivalence between the pairs $(X,A)$ and $(Y,B)$ if there is a map $g:Y\to X$ such that:
*
*$f(A)\subset B$
*$g(B)\subset A$
*There is a h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2792166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that if $p$ is a prime such that $p|(2^{64}+1)$ then $p \equiv 1 $ (mod 128) I'm not sure if I'm on the right track with this problem. So far I've said: $2^{64} = (2^{32})^2 \equiv -1$ (mod p). Then by Fermat's two square theorem $p = 2$ or $p \equiv 1$ (mod 4). We know $p \not = 2$ because $p|(2^{64}+1)$. Then $p... | Edit: I misread and thought you wrote $p|2^{2^{64}}+1$. The result happens to hold as well, and it is actually the number Euler was trying to factor when he proved the theorem below.
Hint: You are on the right track since what you need is a generalization of Fermat's two-square theorem, due to Euler who used it to pro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2792268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that there does not exist $g\in C_{\Bbb R}([-1,1])$ such that $f(0)=\langle f,g\rangle$ for every $f\in C_{\Bbb R}([-1,1])$ . Let $C_{\Bbb R}([-1,1])$ be the vector space of continuous real valued functions on the interval $[-1,1]$ with inner product given by $\langle f,g\rangle=\int _{-1}^1f(x)g(x)\,dx$
for $f,g\... | Take $f(x)=x^2g(x)$, you get $$\int_0^1(xg(x))^2dx=0.$$
The continuity and nonnegativity of the integrand imply then that $xg(x)=0$ for all $x\in [-1,1]$. Thus $g(x)=0$ for all $x\in [-1,1]\setminus\{0\}$. But, since $g$ is contiuous we conclude that $g\equiv0$.
Now, testing with the constant function $f\equiv1$ we ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2792367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is it true that every compact subset of $\mathbb{R}$ (with a Euclidean metric) is homeomorphic to a closed and bounded interval? Could anyone please give me a hint about how to start to prove that or a counter-example if it's false? It sounds true, since both sets are compact, but it is hard to construct a bijection be... | Every interval $[a,b]$ for $a< b$ is homeomorphic to $[0,1]$ via $x\mapsto (x-a)/(b-a)$. And therefore any two closed, bounded and non-singleton intervals are homeomorphic.
Now not every compact subset of $\mathbb{R}$ is homeomorphic to a closed and bounded interval. The simpliest example is a subset of two points, e.g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sum of $2008$ consecutive positive integers
The sum of $2008$ consecutive positive integers is a perfect square. What is the
minimum value of the largest of these integers?
I understand this means that I need the sum of numbers, where $n=2008$ and I believe that the nearest perfect square might be $1936$ or $2025$.... | The sum is:
$$\sum_{k=0}^{2007} (n+k)=x^2 \Rightarrow 2008n+1004\cdot 2007=x^2.$$
Note that $x=2k$. Then:
$$502n+251\cdot 2007=k^2 \Rightarrow 251\cdot (2n+2007)=k^2.$$
Since $251$ is a prime number, then $k=251m$. Then:
$$2n+2007=251m^2.$$
The smallest $m=3$ and $n=126$. Hence, the minimum of the largest of them is $1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2792585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can we view a vector space as a field extention? We can view field extentions as vector spaces over a field (an idea that in my experience has not really been explained but I get it more or less).
But can any vector space over a field also be seen as an extention of that field? I think no, since vector spaces are not f... | A vector space over a field can be regarded as an extension of that field as long as you can equip the vector space with an appropriate concept of product that satisfies the field axioms.
For example, equipping $\mathbb{R}^2$ with the product $(a,b) \times (c,d) = (ac-bd, ad+bc)$ creates a field which is an extension o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2792699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding the smallest n satisfying $S_n > 10$ Let $S_n = 1 + \frac 12 + \frac 13 + \cdots + \frac 1n$, where $n \in \{ 1,2,3,\cdots\}$ Find the smallest $n$ satisfying $S_n > 10$.
Sorry, it's my first time asking and I don't know how to format this thing. I still don't see anything even after staring at this for really... | Should be near
$$ \lfloor{ \frac{e^9}{2} }+1 $$
I use
$$ \int_k^{k+1} \frac{1}{t} dt \leq \frac{1}{k} \leq \int_{k-1}^{k} \frac{1}{t} dt $$
You sum and integrate which give you log then you solve for upper and lower bound and verify.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2792815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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Properties of a polynomial with zero discriminant In Wikipedia it says[1] that
"The discriminant of a polynomial over an integral domain is zero if and only if the polynomial and its derivative have a non-constant common divisor."
How does one prove this fact?
[1] https://en.wikipedia.org/wiki/Discriminant#Zero_discr... | If $P(x)$ is a polynomial with degree $n$ with coefficients in an integer domain $D$, if $K$ is the algebraic closure of the ring of fractions of $D$ and if $r_1,\ldots,r_n\in K$ are the roots of $P(x)$ (there may be repeated roots, of course), then the discriminant $\Delta$ of $P(x)$ is $\left(\prod_{k=1}^n(r_i-r_j)\r... | {
"language": "en",
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Find all positive integers $x,y$ such that $\frac{x^2-1}{xy+1}$ is a non-negative integer. QUESTION: Find all positive integers $x,y$ such that $\frac{x^2-1}{xy+1}$ is a non-negative integer.
ATTEMPT (AND SOME SOLUTIONS): So, for a positive integer $x$, $x^2-1\ge0$. In the case $x^2-1=0$ i.e $x=1$ (since it's a positiv... | $x^2-1=(x-1)(x+1)$
$\frac{x^2-1}{xy+1}$ can be an integer if: | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2792995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A compact in a measurable set The problem is as follows:
Given a Lebesgue-measurable set $M\subset\mathbb{R^n}$ find a compact $C_\epsilon$ and an open $A_\epsilon$, $C_\epsilon\subset M\subset A_\epsilon$, such that $\lambda(A_\epsilon)-\lambda(C_\epsilon)<\epsilon$.
$\lambda$ is the Lebesgue-measure, in particular,... | If $M$ is unbounded and $\lambda(M)<\infty$, start by taking a $k>0$ such $\lambda(M)-\lambda\bigl(M\cap[-k,k]\bigr)<\frac\varepsilon2$.
If $\lambda(M)=\infty$, then you are right: there is no such compact set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2793081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What's the intuition behind using Law of Excluded Middle in Natural Deduction? I've recently started learning First-Order Logic and I have been doing some Natural Deduction exercises. I understand the principles behind most of the Inference Rules but when it comes to applying Classical Rules such as the law of excluded... | As per @Sudix's answer above, the intuition behind the use of the Law of Excluded Middle in the proof of :
$(φ → ∃x ψ) ⊢ ∃x (φ → ψ)$
is to apply a "case analysis".
(i) Assume that $φ$ does not hold, i.e. assume $¬φ$.
This means (by the truth-table for the conditional) that $φ → ψ$ is TRUE, and thus also $∃x (φ → ψ)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2793186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding $n$-th power of transition matrix Is there any shortcut to find $P^{n}=\begin{pmatrix} 1-p & p \\ q & 1-q \end{pmatrix}^n$ quickly and elegantly?
This type of matrix often comes up while dealing with Markov chains. Diagonalization takes way too much time. I need to find eigenvalues, eigenvectors and all that..... | Let $S$ be the matrix
$$
S=
\begin{bmatrix}
1&p\\1&-q
\end{bmatrix}
\ ,
$$
which has on the columns the eigenvectors to the eigenvalues $1, 1-p-q$. Then
$$
\begin{aligned}
S^{-1}PS &=
\begin{bmatrix}
1&0\\0&1-p-q
\end{bmatrix}
\ ,
\\
P &=
S
\begin{bmatrix}
1&0\\0&1-p-q
\end{bmatrix}
S^{-1}\ ,
\\[2mm]
&\qquad\text{ so }... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2793274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Group homology Künneth Formula I am trying to understand Weibel's proof of Künneth Formula for group homology (Prop. 6.1.13 from An Introduction to Homological Algebra), and I am struggling with following statement:
Let $G,H$ be groups. Let $P\rightarrow\mathbb{Z}$ and $Q\rightarrow\mathbb{Z}$ be free resolutions of $\... | We have, by Künneth formula,
$$H_n(P_{\cdot} \otimes_{\Bbb Z} Q_{\cdot}) \cong
\left(
\bigoplus_{p+q=n} H_p(P_{\cdot}) \otimes_{\Bbb Z} H_q(Q_{\cdot})
\right) \oplus
\bigoplus_{p+q=n-1} \mathrm{Tor}_1^{\Bbb Z}(H_p(P_{\cdot}), H_q(Q_{\cdot}))$$
When $r>0$, we know that $H_r(P_{\cdot}) = H_r(Q_{\cdot}) = 0$, since thes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2793385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $f(x) = \frac{(x+1)\sin x}{x}$ is bounded at $(0,1)$. Is it a valid proof? Proof:
We assume that $f(x)$ is unbounded at $(0, 1)$. So for any $M\in \mathbb{R}$
there exists an $x\in (0,1)$ such that:
$|f(x)| = |\frac{(x+1)\sin x}{x}| > M \Rightarrow^{(1)} \frac{(x+1)\sin x}{x}>M
\Rightarrow x\sin x+\sin x >... | Here's a direct proof, similar to yours, but perhaps a little simpler . . .
On the interval $(0,1)$, we have $0 < \sin x < x$, hence
$$0 < \frac{(x+1)\sin x}{x} < \frac{(x+1)\,x}{x} = x+1 < 1 + 1 = 2$$
so on the interval $(0,1)$, we have $0 < f(x) < 2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2793498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Does the sequence of r.v. given by the law of large numbers converge almost surely to the mean in an oscillatory fashion? I state the very famous strong law of large numbers in it's simplest form:
Given an IID sequence of random variables $\{ X_n\}_{n \in N}$ then
$$ \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i= 1... | We consider the case $E(X_1^2)<\infty$. We reformulate the problem, by defining $U_i=X_i-E(X_i)$. So, we wish, to show that $\sum_{i}^nU_i$ oscillates from zero.
We know that $\limsup_{n}\frac{U_n}{\sqrt{n}}=\infty$, almost surely. One way to see this is by using the CLT theorem and Kolmogorov's zero-one law.
By symme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2793583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\lim_{n\rightarrow\infty}\frac{a_{n}}{n}=1$ implies that $\lim_{n\rightarrow\infty}\sup_{0\leq k\leq n}\frac{|a_{k}-k|}{n}=0$ I'm trying to prove the following:
If $$\displaystyle\lim_{n\rightarrow\infty}\frac{a_{n}}{n}=1$$ then $$\displaystyle\lim_{n\rightarrow\infty}\sup_{0\leq k\leq n}\frac{|a_{k}-k|}{n}=0.$$
So,... | Let $\varepsilon>0$ be arbitary. Then there exists $N_{1}\in\mathbb{N}$
such that $\left|\frac{a_{n}-n}{n}\right|<\varepsilon$ whenever $n\geq N_{1}$.
Let $M=\max_{0\leq k<N_{1}}|a_{k}-k|$. Choose $N_{2}\in\mathbb{N}$
such that $\frac{M}{N_{2}}<\varepsilon$. Let $N=\max(N_{1},N_{2})$.
Let $n\geq N$ be arbitrary. Let $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2793708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Clarification needed: Smallest subfield of a field
Q: Given a field does there exist a smallest subfield ? | Once you prove that the intersection of all subfields of a field is itself a field, you see that every field has a single smallest subfield. The function that Lubin suggests gives you a way to understand the structure of that smallest field. Lubin is defining the function $\varphi: \mathbb{Z} \rightarrow F$ (where $F$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2793816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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multiplication in $\mathbb{Z}_p$ I'm not even sure what would be a good title for my question, so feel free to edit it (if you can).
I want to prove (or dispute) the correctness of the following lemma:
Let $p$ be a prime number, and let $i,j,c \in \mathbb{Z}_p \setminus
\{0\}$ such that $i \neq j$. Is it true that $... | Recall that an integral domain is a commutative ring such that $ab=0$ implies $a=0$ or $b=0$.
The essential property of primes I'm going to use is the fact that if $p$ is prime and $p|ab$ then $p|a$ or $p|b$.
Lemma. $\mathbb{Z}_n$ is an integral domain if and only if $n$ is prime.
Proof. "$\Rightarrow$" assume $n$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2793951",
"timestamp": "2023-03-29T00:00:00",
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Identity Theorem: Extending from $\mathbb{R}$ to $\mathbb{C}$ Suppose we have $f_1, f_2: \mathbb{C} \rightarrow \mathbb{C}$ holomorphic, and $f_1 = f_2$ on $\mathbb{R}$. Can we then say $f_1 = f_2$ identically on $\mathbb{C}$?
This appears to be true by the uniqueness of analytic continuations, but the identity theorem... | The identity theorem does not require that the functions must agree on an open set !
Let $A:=\{z \in \mathbb C:f_1(z)=f_2(z)\}$ .
If $A$ has an accumulation point in $ \mathbb C$, then , by the identitiy theorem, $f_1=f_2$ on $ \mathbb C$.
In your case we have $A = \mathbb R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2794024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Finding the sum of $\cos\frac{\pi}{7}$, $\cos\frac{3\pi}{7}$, $\cos\frac{5π}{7}$ by first finding a polynomial with those roots
Without using tables, find the value of $$\cos\frac{\pi}{7}+\cos\frac{3\pi}{7}+\cos\frac{5\pi}{7}$$
This is a very common high school trigonometric problem, and the usual way to solve this i... | $\text{Using Complex Number}$
Let $z=e^{\frac{\pi i}{7}},$ so $z^7=-1.$ Let $Q$ be the desired quantity. Then
$$2Q=z+\frac{1}{z}+z^3+\frac{1}{z^3}+z^5+\frac{1}{z^5} = \frac{z^{10}+z^8+z^6+z^4+z^2+1}{z^5} = \frac{z^{12}-1}{z^5(z^2-1)}$$
$$=\frac{-z^5-1}{z^7-z^5} = \frac{-z^5-1}{-1-z^5} = 1$$ $$\therefore Q=\frac{1}{2}\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2794176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Identity for triple integrals I found an identity but I haven't been able to prove it.
This is the identity
$\int _{ 0 }^{ x }{ \int _{ 0 }^{ y }{ \int _{ 0 }^{ z }{ f(t)dtdzdy\quad =\quad \frac { 1 }{ 2 } \int _{ 0 }^{ x }{ (x-t)^{ 2 }f(t)dt } } } } $
A teacher told me that the first step to achieve the result i... | The region $z\in[0,y]$ and $t\in[0,z]$ is a triangular region in the $t$-$z$ plane bounded by $t=0$, $z=y$, and $z=t$.
By changing the order of integration we can write
$$\int_0^y\int_0^zf(t)\,dt\,dz=\int_0^y \int_t^yf(t)\,dy\,dt=\int_0^y(y-t)f(t)\,dt$$
Can you proceed by evaluating the integral $\int_0^x \int_0^y(y-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2794308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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reference for special relativity I got a student finishing its first year of Math and she would like to study a bit of special relativity from a mathematics point of view. I know the subject quite well but I don't know any basic references. what I look for her his some basic on Minkowsky? and if possible the Maxwell eq... | There is a really lovely book by David Bressoud called Second Year Calculus: From Celestial Mechanics to Special Relativity. He is an excellent writer and the book is a joy to read. It gives the gentlest introduction to differential forms and special relativity that you could hope for.
https://www.springer.com/us/boo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2794461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Evaluate $\int_0^1x(\tan^{-1}x)^2~\textrm{d}x$
Evaluate $\int\limits_0^1x(\tan^{-1}x)^2~\textrm{d}x$
My Attempt
Let, $\tan^{-1}x=y\implies x=\tan y\implies dx=\sec^2y.dy=(1+\tan^2y)dy$
$$
\begin{align}
&\int\limits_0^1x(\tan^{-1}x)^2dx=\int\limits_0^{\pi/4}\tan y.y^2.(1+\tan^2y)dy\\
&=\int\limits_0^{\pi/4}\tan y.y^2d... | $$\int_ {0}^{1}x(arctan(x))^2dx$$
First solve the integral without boundaries
$$\int x(arctan(x))^2dx$$
Apply Integration By Parts, where $u=\arctan^2(x),v^{\prime}=x$
$$=\arctan^2(x)\frac{x^2}{2}-\int \frac{2\arctan(x)}{1+x^2} (\frac{x^2}{2})dx$$
$$=\frac12 x^2\arctan^2(x)-\int \frac{x^2\arctan(x)}{x^2+1}dx$$
Note tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2794560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 4
} |
Is $\mathbb{F}_{p^r}/\mathbb{F}_{p}$ a Galois extension? Let $r>0$. Is $\mathbb{F}_{p^r}/\mathbb{F}_{p}$ a Galois extension? If so, why?
I know that it is a finite extension, with $[\mathbb{F}_{p^r}:\mathbb{F}_{p}]=r$. To show that it is a Galois extension, it suffices to show that $|Aut(\mathbb{F}_{p^r}/\mathbb{F}_{p}... | Every field $K$ of characteristic $p$ has the Frobenius endomorphism
$F:x\mapsto x^p$. This is a homomorphism of fields, and so is injective.
If $K$ is finite, then $F$ must be bijective, so an automorphism.
On $\Bbb F_p$, $F$ acts trivially.
The fixed points of $F^t$ are the solutions of $x^{p^t}-x$.
Every element of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2794655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability: How to get to a percentage? Exercice: We have a class of $16$ students with $2$ of them not having phones. Every class in the school is like that ($14$ with phones, $2$ without). How many students in the school do I have to pick for the probability of having at least one student without a phone to be at $9... | The question asks for at least one student without a phone, it could be 1 student, 2 student ,... So for solving this, we can say if the probability of having at least one student without a phone is 99.9% (or higher) then the probability of ALL having phone must be 0.1% or less. If you choose one student it has $\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2794808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Optimizing singular Rayleigh quotient subject to linear constraint I want to numerically solve
$$\min_x \frac{x^TAx}{x^TBx} \quad \mathrm{s.t.}\quad Cx=0,$$
where $A$ and $B$ are large sparse matrices, $A$ is positive semi-definite, $B$ is positive-definite, and $C$ is sparse.
One approach would be to find a basis $N$ ... | Answering my own question as I've found a rather satisfactory solution to this problem.
From the KKT conditions of the original problem one sees that the minimizer is the $x$-part of the eigenvector with smallest eigenvalue in the generalized eigenvalue problem
$$\begin{bmatrix} A & C^T \\ C & 0\end{bmatrix}\begin{bmat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2794900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is there a pattern to the last digits of square numbers? I was programming and I realized that the last digit of all the integer numbers squared end in $ 0, 1, 4, 5, 6,$ or $ 9 $.
And in addition, the numbers that end in $ 1, 4, 9, 6 $ are repeated twice as many times as the numbers that end in $ 0, 5$
I checked th... | Consider:
$$(x+k)^2=(x+k)(x+k)=x^2+2xk+k^2$$
In your case, $x=10z, z\in \Bbb Z$, and $0\le k\le9, k\in \Bbb Z$. Thus it becomes:
$$(x+k)^2=100z^2+20zk+k^2$$ for which the only possible unit is the unit from $k^2$, and so the facts that:
$$1^2,9^2\space\text{end in}\space 1$$
$$2^2,8^2\space\text{end in}\space 4$$
$$3^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2795029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 7,
"answer_id": 2
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equidimensional equation For $x^2y''+axy'+by=0$ where $a,b\in\mathbb{R}$, find conditions on $a,b$ such that all solutions are bounded as $x\to0$.
First write $y''+\frac{a}{x}y'+\frac{b}{x^2}y=0$ and guess $y=x^r$. Plug it in and simplify I got $r^2+(a-1)r+b=0$, thus $r=\frac{1-a}{2}\pm\sqrt{\left(\frac{1-a}{2}\right)^... | I would assume that your solution is required to be real.
Does your solution need to be well defined at $x = 0$? Nothing was stated about the range of validity.
Your statement that if one is positive then it is partially bounded is suspect. If one of the roots is less than zero then one term will be complex for val... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2795086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimal sufficient statistic for normal distribution with known variance Let $X_1, ..., X_n$ be a random sample from the $N(\theta,1)$ distribution. Find a minimal sufficient statistic for $\theta$.
Now, I can find a sufficient statistic using the factorisation theorem ($\sum X_i$), but I don't think that this statisti... | By the factorization criterion
$$
\mathcal{L}(\theta)=\frac{1}{(2\pi)^{n/2}}\exp\{-\sum_{i=1}^nX_i^2/2 +\bar{X}_n \theta -n\theta^2/2\}
$$
$$
\qquad = \exp\{\bar{X}_n\theta-n\theta^2/2\}\times(2\pi)^{-n/2}\exp\{-\sum X_i^2/2\}.
$$
So $\bar{X}_n$ is sufficient statistic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2795184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determining the points at constant distance $d$ from parabola $y = x^2$ My friends recently learned about Locus and how parabola are the points equdistant from a fixed point (focus) and a straight line (directrix). However, now we're trying to find an equation or set of equations for the points at a constant distance f... | We have $f(x) = x^2$. A tangent vector at each point $P(x)= (x, f(x))$ is $$
t(x)=(1, f'(x)).
$$
The normal vectors are orthogonal to the tangent vectors which gives
$$
n_i(x) = (\mp f'(x), \pm 1)
$$
and satisfies
$$
t(x) \cdot n_i(x)
= 1 \cdot (\mp f'(x)) + f'(x) \cdot (\pm 1)
= 0.
$$
As we want to specify a distance... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2795313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Probability that $z$ is EVEN satisfying the equation $x + y + z = 10$ is Question
Three randomly chosen non negative integers $x, y \text{ and } z$ are found to satisfy the equation $x + y + z = 10$. Then the probability that $z$ is even, is"?
My Approach
Calculating Sample space -:
Number of possible solution for... | Using computer power:
sage: allCases = len( [ S for S in cartesian_product( [ [0..10], [0..10], [0..10] ] ) if sum(S) == 10 ] )
sage: goodCases = len( [ S for S in cartesian_product( [ [0..10], [0..10], [0..10] ] ) if sum(S) == 10 and S[2] % 2 == 0 ] )
sage: allCases
66
sage: goodCases
36
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2795440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
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If $\frac{\cos(\alpha -3\theta)}{\cos^3 \theta}=\frac{\sin(\alpha -3\theta)}{\sin^3 \theta}=m$ prove that $\cos\alpha=\frac{2-m^2}{m}$
If $$\frac{\cos(\alpha -3\theta)}{\cos^3 \theta}=\frac{\sin(\alpha -3\theta)}{\sin^3 \theta}=m$$
prove that
$$\cos\alpha=\frac{2-m^2}{m}$$
My approach:
$$\cos^2(\alpha-3\theta)+... | Hint for your last equation:
$$\sin(2\theta)=2sin(\theta)\cos(\theta)$$
so $$\sin(2\theta)^2=4\sin^2(\theta)\cos^2(\theta)$$
and you can eliminate $\theta$
So you get $$4\left(1-\cos(\theta)^2\right)\cos(\theta)^2=\frac{4}{3}\frac{m^2-1}{m}$$
Solve this for $\cos(\theta)$
I know this, when you get $\theta$ then you can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2795570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Divisibility of Sum of Equally Spaced Binomial Coefficients According to a numerical calculation I did for small values of $k$, it appears that the following is true.
$$4|\left[\sum_{j=1}^{n-1}\binom{3n}{3j}\right]$$ or $$\sum_{j=1}^{n-1}\binom{3n}{3j}=4p, p\in\mathbb{Z}$$
Ex.
If $n=2, \binom{6}{3}=20=4\cdot 5$
If $n... | Let $$S=\sum_{j=1}^{n-1}\binom {3n}{3j}$$
Adding terms for $j=0$ and $j=n$ gives
$$\begin{align}
S+2
&=\sum_{j=0}^n \binom {3n}{3j}\\
&=\sum_{r=0}^{3n}\binom {3n}r-\left[\sum_{j=0}^{n-1}\binom {3n}{3j+1}+\binom {3n}{3j+2}\right]\\
&=2^{3n}+2\Re\left[\sum_{j=0}^{n-1}\binom {3n}{3j+1}e^{i2\pi/3}+\binom {3n}{3j+2}e^{i4\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2795730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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inverse Fourier transform of product of two functions What is the inverse Fourier transform of $i\omega f(\omega)g(\omega)$?
is it just $\frac{d}{dt}(f(t)\cdot g(t))$ or will I end up with some kind of convolution?
| It will be a convolution by the convolution theorem for inverse Fourier transforms:
$\mathcal{F}^{-1}\left(i\omega \hat{f}(\omega)\hat{g}(\omega)\right)(t)=\frac{d}{dt}\mathcal{F}^{-1}\left(\hat{f}(\omega)\hat{g}(\omega)\right)(t)=\frac{d}{dt}\mathcal{F}^{-1}\left(\widehat{(f*g)}(\omega)\right)(t)=\frac{d}{dt}(f*g)(t)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2795851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Subspace $\ell^2$ is of first category
Let $\ell^2$ be the Hilbert space of square summable sequences, and $\mathcal{H}$ be the subspace consisting of sequences $\{x_n\}$ with $\sum_{n=1}^\infty n^2|x_n|^2<\infty$
Show that $\mathcal{H}$ is of the first category.
I unsure how to start this problem.
| Hint: Consider $E_M=\{(x_n)\in l^2: \sum n^2x_n^2\le M\}.$ Show $E_M$ is closed in $l^2.$ (Fatou's lemma). Then show $E_M$ has no interior.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2795914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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How does GIMPS work and what are these iterations? I downloaded GIMPS today just out of curiosity and have been running it. On my machine it is checking $M_{52898149}=2^{52898149}-1$.
From what I could find on Wikipedia I suppose that GIMPS uses Lucas-Lehmer primality test which means it will pick $s_{52898147}$ ter... | Each one of those iterations is just squaring the number then put it in modulo of whatever your prime exponent is. Iteration 160000/52898149 simply means that your computer system has completed 160000 such iterations. ms/iter counts how many milliseconds are needed on average for the last 10000 iterations, and ETA esti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2796155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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change in unit radial vector and unit polar angle due to change in polar angle In cylindrical coordinates, the set B of basis vectors is
$B=\left \{ \vec{e}_{r},\vec{e}_{\theta} \right \}$.
It is clear, geometrically, that
$\frac{\partial e_{r}}{\partial r}=0=\frac{\partial e_{\theta}}{\partial r}$
However, I am unabl... | This is for polar coordinates, cylindrical would have a third basis vector $e_z$, but regardless, as the angle $\Delta\theta$ goes to $0$ in the ccw direction, the differential angle makes an appearance resulting in a differential change in the direction of the radial vector
So wherever $e_r$ was pointing, it now has t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2796406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finitely-generated abelian groups in a long exact sequence I learned here that if the outer groups in a short exact sequence are finitely-generated, then the middle group is, too.
Question. Is there a generalization of this to long-exact sequences?
Julian Kuelshammer mentioned the horseshoe lemma but I'm not really s... | Well, if you have an exact sequence $$A\stackrel{f}\to B\stackrel{g}\to C$$ where $A$ and $C$ are finitely generated, then so is $B$, This follows from the short exact sequence $$0\to \ker(g) \to B\to \operatorname{im}(g)\to 0$$ where $\ker(g)=\operatorname{im}(f)$ and $\operatorname{im}(g)$ are finitely generated bec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2796523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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argument and atan/arctan As I have to works with sine voltages and currents, I often have to use complex numbers.
I know that $\tan\arg(Z)=\left(\frac{\Re(Z)}{\Im(z)}\right)^{-1}$
but, how to prove that we need to add $ \pi$ in case $\Re(Z)\leq0$ ?
I know that $\arctan(x) + \arctan(\frac{1}{x})=\pm \frac\pi2, \forall ... | That is because, when you want to determine the argument of a complex number $z=x+iy$, really you have to solve, not a single trigonometric equation, but a system of trigonometric equations:
$$\begin{cases}\cos\theta=\dfrac x{\sqrt{x^2+y^2}}\\[1ex] \sin\theta=\dfrac y{\sqrt{x^2+y^2}}\end{cases} $$
This system implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2796641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Why can't we write $\sin x$ as $\prod_{n=0}^{\infty}\left(x^2-n^2\pi^2\right)$? Why can't we write $\sin x$ as $\prod_{n=0}^{\infty}\left(x^2-n^2\pi^2\right)$?
Since $ n\pi$ where $n \in \mathbb{N}$ are all the roots of $\sin{x}$, then by the fundamental theorem of arithmetic it may be written as
$$\sin x=\prod_{n=0}^{... | The answers so far (which essentially say: "sine is not a polynomial") are ok. However, they fail on the heuristic side. Thinking like this Euler would have never solved the Basel problem.
Here is my attempt:
There are two main problems with your answer:
1) sin only has a single root at $x=0$ whereas your expression ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2796722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $\sum _{n=1}^{\infty }\left(-1\right)^{n+1}\left(1-\cos\left(\frac{1}{\sqrt{n}}\right)\right)$ converges conditionally? I'm trying to understand whether the following series converges absolutely, conditionally or diverges.
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n+1}\left(1-\cos\left(\frac{1}{\sqrt{n}}\right)\righ... | Note that
$$1-\cos\left(\frac{1}{\sqrt{n}}\right)= \frac1{2n}+O\left(\frac1{n\sqrt n}\right)$$
then the given series doesn’t converge absolutely by limit comparison test with $\sum \frac1{n}$ and then
$$
\sum _{n=1}^{\infty }\left(-1\right)^{n+1}\left(1-\cos\left(\frac{1}{\sqrt{n}}\right)\right)=\sum _{n=1}^{\infty }\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2796827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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image of homomorphism contains free group, then so does domain This question is a question which I had on my abstract algebra exam, I could not solve it:
True or false: suppose $f:G \to H$ is a group homomomorphism and assume that $\operatorname{Im}(f)$ contains a free group of rank $2$, then $G$ contains a free group... | Your proof is most of the way there.
Let $F = \langle x_1, \dots, x_n\rangle$ be a copy of the free group of rank $n$, and write $H = \langle g_1, \dots, g_n\rangle$ for your unknown group. Then the map $x_i \mapsto g_i$ is a surjective group homomorphism $F\to H$.
What's its kernel? Well, whatever it is, it must be co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2796925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Dividing concentric super ellipses to equal area slices How the concentric super ellipses as shown in the figure can be divided into parts containing equal area such that the total area of the superellipse A = A1 + A2 + ... An where n = 60 in the shown figure given its semi-major axis a and semi-minor axis b. The origi... | Solve the problem for concentric circles in the unit circle centered at the origin. Then stretch the $x$-axis by $a$ and the $y$-axis by $b$ to turn the circles into ellipses. All the areas will be scaled by the same stretch factor.
This is essentially the same question you asked here: Dividing an ellipse into equal a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Manipulating a factorial equation How do I convert $\dfrac{1\cdot2\cdot3\cdots2n}{n!}$ to $(n+1)\cdot(n+2)\cdot\cdots\cdot(2n)$?
I don't understand how the $n + x$ terms have appeared in this equation.
| You can write :
$$\begin{align}
1\cdot 2\cdot 3\cdots (2n) &= (1\cdot 3\cdot 5\cdots (2n-1))\cdot(2\cdot 4\cdot 6\cdots (2n))\\ &= (1\cdot 3\cdot 5\cdots (2n-1))\cdot 2^n\cdot(1\cdot 2\cdot 3\cdots n)\\ &= (1\cdot 3\cdot 5\cdots (2n-1))\cdot 2^n\cdot n!
\end{align}$$
Note that this goes beyond the question, as it show... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of $n(1-x)^n$ as $n\to\infty$ when $0Wolfram alpha gives that this is $0$, but I'm not sure how to show it.
I Tried writing as $\frac{(1-x)^n}{1/n}$ and using L'Hopital's rule, but a new $n$ term shows up every time I take the derivative.
I also try to set $h=\frac1n$ and then write as
$$
\lim_{n\to\infty} n(1-x)... | You can solve this by taking the logarithm of each side. Let $y = \lim_{n \rightarrow \infty} n(1-x)^n$, and then we see that $$\ln(y) = \ln(\lim_{n \rightarrow \infty} n(1 -x)^n) = \lim_{n\rightarrow \infty } \ln(n (1-x)^n)$$ by using the continuity of the natural log function. Now, we use log rules and see $$\ln(y) =... | {
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"url": "https://math.stackexchange.com/questions/2797318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Number of ways to flip a coin 10 times with no consecutive heads The problem statement is as follows: A fair coin is to be tossed $10_{}^{}$ times. Let $i/j^{}_{}$, in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j_{}^{}$.
My solution was to consider the sequence of flips as a... | Your $89$ is presumably $1+9+28+35+15+1$
It should be $1+10+36+56+35+6$
Since your $89$ would be correct for the numerator with nine coin tosses, you have presumably missed all those starting with heads, or all those finishing with heads
It is not a coincidence that $89$ and $144$ are consecutive Fibonacci numbers, an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2797448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Suppose we are dealt five cards from an ordinary 52-card deck. What is the probability that Suppose we are dealt five cards from an ordinary 52-card deck. What is the
probability that
(c) we get no pairs (i.e., all five cards are different values)?
(d) we get a full house (i.e., three cards of a kind, plus a different ... |
Suppose we are dealt five cards from an ordinary $52$-card deck. What is the probability that we get no pairs?
We must select cards from five of the thirteen ranks. For each selected rank, we must select one of the four suits. Hence, the number of favorable cases is
$$\binom{13}{5}4^5$$
Since there are $\binom{52... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2797737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Upper bound in an integral with exponential I'm looking to find an upper bound on the following integral
$$\int_0^\infty K(u)S(t-u)du\,, $$
where
$$ K(u) = e^{-u}(u-u^2/2), $$
and $ S(t) < C$ for some constant $C$.
Could someone help?
| Observe the fact that since $S(t)<C$, we can use the estimate that $$\int_{0}^\infty K(u)S(t-u)du<|C|\,|\int_0^\infty e^{-u}(u-u^2/2)du|\leq|C|\int_0^\infty |K(u)|du$$
Now to finish your proof, you have to show that $\int_{0}^\infty |e^{-u}(u-u^2/2)|du$ exists and is finite.
Edit: Consider the derivative of $\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2797871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Limit $ \sum_{k=0}^∞ \left( \sum_{j=0}^k \binom{k}{j} \left(-\frac{1}{3}\right)^j \right) $ I have to find the limit of the following series:
$$ \sum_{k=0}^∞ \left( \sum_{j=0}^k \binom{k}{j} \left(-\frac{1}{3}\right)^j \right) $$
I don't even know how to approach this... Any help would be very appreciated
| Since,
by the binomial theoram,
$ \sum_{j=0}^k \binom{k}{j}x^j
=(1+x)^k
$,
$\begin{array}\\
\sum_{k=0}^∞ \left( \sum_{j=0}^k \binom{k}{j}x^j \right)
&=\sum_{k=0}^∞ (1+x)^k\\
&=\dfrac{1}{1-(1+x)}
\qquad\text{geometric series with ratio }1+x\\
&=\dfrac{-1}{x}\\
\end{array}
$
Putting $x=-\frac13$,
this gives
$\dfrac{-1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2797945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Let $\varphi(x) = \frac{1 - e^{-ax}}{1 + e^{-ax}}$, proof that $\varphi'(x) = \frac{a}{2}(1-\varphi^2(x))$ I am trying to find the required steps to reach that derivative, but I am not finding the right way for that. My current development has the following steps:
$\varphi(x) = \dfrac{1 - e^{-ax}}{1 + e^{-ax}}$, then
$... | You can approach the problem through basic differential equation theory by going "backwards". Suppose that you have the separable ODE
$$
\tag{$\star$}
\frac{\varphi'}{1-\varphi^2}=\frac{a}{2}
$$
with boundary condition $\varphi(0)=0$ given by the form of $\varphi$ (just calculate it in $0$).
By integrating on both sid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2798076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Rademacher theorem for 2nd order derivative The (simplest form of the) Rademacher theorem reads as follows:
Any Lipschitz continuous function $f: \mathbb{R} \to \mathbb{R}$ is Lebesgue-almost everywhere differentiable.
In other words: If the finite difference $\Delta_h^1[f](x) := f(x+h)-f(x)$ satisfies $$|\Delta_h^1[... | I believe the answer is yes. Writing $D$ for the derivative in the sense of distributions, and $f'$ for the pointwise derivative:
It's easy to see that $\frac1{h^2}\Delta_h^2 f\to D^2f$ in the sense of distributions (or $2D^2f$ or $\frac 12D^2f$ or whatever it is). So the hypothesis implies $$D^2f\in L^\infty.$$This im... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2798170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Exponential equation with double radical I'm trying ti solve this exponential equation:
$(\sqrt{2+\sqrt{3}})^x+(\sqrt{2-\sqrt{3}})^x=2^x$.
Here my try:
$\sqrt{2+\sqrt{3}}=\sqrt{\frac{3}{2}}+\sqrt{\frac{1}{2}}$ and
$\sqrt{2-\sqrt{3}}=\sqrt{\frac{3}{2}}-\sqrt{\frac{1}{2}}$
So i get this relation:
$\sqrt{\frac{3}{2}}+\sqr... | Hint:
Write $s= \sqrt{2+\sqrt{3}}$, then $\sqrt{2-\sqrt{3}} = {1\over s}$
Then $$s^x+({1\over s})^x\geq 2 \implies x\geq 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2798236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Compatibility of a Kähler form On a complex manifold $M$ a Kähler form is a symplectic form $\omega$ which is compatible with the canonical almost complex structure $J$ in the following sense
$$\omega({}\cdot{},J{}\cdot{})$$
is a Riemannian metric tensor, i.e. symmetric and positive definite.
From $J^*\omega=\omega$ ... | You are not wrong. You can double check this as follows:
$$g(\partial_z,\partial_z) = \frac{1}{4}g(\partial_x-i\partial_y,\partial_x-i\partial_y) = \frac{1}{4}(1-1)=0.$$
The metric $g$ I use here is the Riemannian metric on your manifold extended by $\mathbb{C}$-linearity on the complexified tangent bundle. This sugges... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2798376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the base of $\log x$? I've seen "$\log x$" being used in some papers (and by Wolfram|Alpha), and I was confused because so far I have only ever seen the $\log$ used with a base ( so e.g. $\log_y x$).
Am I correct that $\log x = \log_e x = \ln x$?
*
*If so, why is $\log x$ used over $\ln x$? Isn't the letter ... | On a standard scientific calculator, the log button denotes the "common logarithm", i.e. $\log_{10}$. This is consistent with the common usage in engineering and the natural sciences; for example, the pH scale used for measuring acidity, the Richter scale used for measuring earthquake intensity, and the decibel scale ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2798541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove that a strongly convex function imples $2c(F(w)-F_*) \leq ||\nabla F(w)||_2^2$
The proof is given as follows:
My question is why is the unique minimizer $\bar{w}_* = w - \frac{1}{c} \nabla F(w)$?
| There is an easier argument for quadratic functionals: completing the square. The following identity is just as easy to prove as the 1D version you know from highschool:
$$ \alpha ‖x‖^2 + \beta · x = \alpha\left\|x+\frac1{2\alpha}{\beta} \right\|^2 - \frac{‖\beta‖^2}{4\alpha}$$
In your case, this yields (remember, $w$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2798749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Hypothesis testing - Critical region and confidence level Exercise :
For the estimation of the unknown rate of votes $p$ that a political group $A$ will gather in the following elections, suppose we selected a random sample of $n=15$ voters. Suppose that you want to check the null hypothesis $H_0 : p = 0.5$ against the... | So $\alpha$ represents the probability of making a Type I error, that is, rejecting $H_0$ when $H_0$ is true, that is why we say "we are 95% confident that $H_0$ is true", because there is a $1-95\%=\alpha$ probability of this conclusion not being true. Therefore:
\begin{align}
\alpha&=Pr(\text{Reject } H_0 | H_0) \\ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2798882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find a Basis for $W=\{p(x)\in V: p(1)=p'(1)=0\}$
Let $V=\mathbb{P_4}$ and $W=\{p(x)\in V: p(1)=p'(1)=0\}$. Assuming that $W$ is a subspace of $V$, find a basis for $W$ and thereby determine the dimension of $W$.
I think that $\dim(W)=3$ as there are two restrictions enforced upon $W$ $($ $p(1)=1$ and $p'(1)=0$$)$ and... | Consider the linear map $F\colon\mathbb{P}_4\to\mathbb{R}^2$ defined by
$$
F(p)=\begin{bmatrix} p(1) \\ p'(1) \end{bmatrix}
$$
Then $W=\ker F$. The matrix of $F$ with respect to the standard basis $\{1,x,x^2,x^3,x^4\}$ is
$$
\begin{bmatrix}
1 & 1 & 1 & 1 & 1 \\
0 & 1 & 2 & 3 & 4
\end{bmatrix}
$$
which has the RREF
$$
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2799016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Evaluate the limit with exponents using L'Hôpital's rule or series expansion Evaluate the limit$$\lim_{x\to 0}\dfrac{\left(\frac{a^x+b^x}{2}\right)^{\frac{1}{x}}
-\sqrt{ab}}{x}$$ It is known that $a>0,b>0$
My Attempt:
I could only fathom that $$\lim_{x\to 0}\left(\frac{a^x+b^x}{2}\right)^{\frac{1}{x}}=\sqrt{ab}$$
| just use the l-H hospital for it,a tip for that is you can write the $((a^x+b^x)/2)^{1/x}$ to this inform: $e^{[\ln((a^x+b^x)/2)]/x}$,so you can use chain rule to differeitiate it,here you are,the answer leaves for you
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2799146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
What is the integral of the indicator function of VITALI? For a positive function $f$, the Lebesgue integral is the supremum of the integral of all simple functions below $f$. So if $f$ is the indicator function for the VITALI set, the lebesgue integral for it must exist. But in general, the integral of an indicator fu... | Any measurable subset $U$ of Vitali's set $V$ has measure zero. Therefore, if you take the supremum of the integrals of simple functions not larger than the indicator of vitali's set, their integrals are zero, and so is the supremum.
If $U\subset V$ is measurable, then for an enumeration $q_i$ of the rationals we have ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2799289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Can a quotient of polynomials be simplified before been analyzed? Having the function $f(x) =\frac{X^3+3X^2}{2X^2+4x}$, why it is not the same to analyze $\frac{X^2+3X}{2X+4}$, if it verifies $\frac{X^3+3X^2}{2X^2+4x}=\frac{X^2+3X}{2X+4}$ ? In this case, the first one has only one root, while the second one has another... | Hardy is fairly robust on this issue (and while possibly slightly at variance with current set-theoretic dogma, nevertheless quite consistent with the standard calculus interpretations):
The function $\frac{x^2-1}{x-1}$ has no value for $x=1$; for $x=1$, $\frac{x^2-1}{x-1}$ is strictly and absolutely meaningless. The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2799425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Is it a contraction map? I have that map
$$
f:(\mathbb{R}^2,d_1)\to(\mathbb{R}^2,d_1)\\
(x,y)\mapsto \left(y-\frac13 \tanh(x)+\frac14 Argsh(y),\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 4x-\tanh(y)+\frac43 Argsh(x)\right)
$$
where $d_1((x,y),(x',y'))=|x-x'|+|y+y'|$
I calcultate $$ d_1(f(x,y),f(x',y'))\leq \frac{43}{12... | You work with the metric $d_1((x,y),(x',y') = \lvert x - x' \rvert + \lvert y - y' \rvert $ (typo in your question!). Your inequality
$$d_1(f(x,y),f(x',y')) \le 43/12\lvert y - y' \rvert + 13/3\lvert x - x' \rvert$$
does not help because you cannot be sure that it best possible.
It suffices to look at special values. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2799521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why does divergence represent expansion or contraction? Why does $\mathrm{div}\ V$ represent how much $V$ is expanding or contracting? By its definition I know that diverging means deviating from its original path, but what about $\mathrm{div}\ V$ makes it so $V$ expands or contracts, is there a $\mathrm{div}$ formula ... | Let $D$ be a small spherical region centered at point $P$.
By the divergence theorem,
$$(\nabla\cdot{\bf V})_P \approx \frac{1}{\mathrm{vol}(D)} \iint_S {\bf V}\cdot{\bf n}\, dS.$$
But $\iint_S {\bf V}\cdot{\bf n}\, dS$ is the net flux of ${\bf V}$ through the surface $S$ of $D$.
Thus, $\nabla\cdot{\bf V}$ is a meas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2799582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Proving $\lim\limits_{n\to\infty}\int_0^{\pi/4} \tan^n{x}\,dx=0$ How would you prove that $\displaystyle\lim\limits_{n\to\infty}\int_0^{\pi/4} \tan^n{x}\,dx=0$.
It is obvious if you see the graph of $\tan^n{x}$ on $(0, \pi/4)$ as $n$ increases but i'm looking for a more algebraic way.
This result is for connecting the... | Squeezing is straightforward:
$$ 0\leq \int_{0}^{\pi/4}\tan^n(x)\,dx \stackrel{x\mapsto\arctan u}{=}\int_{0}^{1}\frac{u^n}{1+u^2}\,du \leq \int_{0}^{1}u^n\,du = \frac{1}{n+1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2799740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Inner circle of torus of revolution is calibrated I'm working on the following problem from Lee's "Introduction to Smooth Manifolds":
Let $D \subseteq \mathbb R^3$ be the surface obtained by revolving the circle $(r-2)^2 + z^2 = 1$ around the z-axis, with the induced Riemannian metric from $\mathbb R^3$, and let $C \... | Assume that $\omega (x)= g(x,\frac{\partial_\theta}{|\partial_\theta|^2})$.
Then $d\omega (\partial_t ,\partial_\theta )=0$ so that it is a closed
form.
If $c$ represents $C$ and $ c\sim \alpha$ and $\alpha$ has a unit
speed, then $$ {\rm length\ of}\ c=\int_0^{2\pi}\ \omega ( c' )=
\int_0^{2\pi}\ \omega(\alpha')\ dt \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2799872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Use $ \ \nabla f(3,2) \ $ to find a vector normal to the curve at $ \ (3,2)\ $ View the curve $ \ (y-x)^2+2=xy-3 \ $ as a contour of $ \ f(x,y) \ $
Use $ \ \nabla f(3,2) \ $ to find a vector normal to the curve at $ \ (3,2)\ $
Answer:
Let $ \ f(x,y)=(y-x)^2-xy+5=0 \ $
Then,
$ \nabla f(x,y)=\left\langle f_x,f_y \right... | Here is a graph of your curve, plotted by Maple.
It shows clearly that the curve is horizontal at $(3,2)$, so the normal is vertical, so your answer is correct and the software marking it is wrong.
The only suggestion I could make is that any vertical vector is normal to the curve at this point, that is, any vector $(0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2799962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Given a set $U=\{1,2,3,...n\}$. How to partition elements in $U$ into sets $A$ and $B$ such that the sum of the elements in $A$ and $B$ is in minimum. I came across a particular problem wherein I think the answer would necessitate me to partition $U=\{1,2,3,...n\}$ into subsets $A$ and $B$ such that the sum of the elem... | Split them into consecutive groups of $4$ starting with the largest ( so the first block is $\{n-3,n-2,n-1,n\}$ ) and in each group $\{a,a+1,a+2,a+3\}$ split them so $a$ and $a+3$ are in $A$ and $a+1$ and $a+2$ are in $B$.
If the number of elements is a multiple of $4$ we are done.
If the remainder is $1$ put $1$ in $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2800037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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