Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to find the maximum of the value $\frac{\sin{x}+1}{\sqrt{3+2\cos{x}+\sin{x}}}$ Find the maximun of the value
$$f(x)=\dfrac{\sin{x}+1}{\sqrt{3+2\cos{x}+\sin{x}}}$$
I use wolframpha this found this maximum is $\dfrac{4\sqrt{2}}{5}$,But How to prove and how to find this value?(without derivative)
idea 1
let $\tan{\df... | We need to minimize $$\dfrac{3+2\cos x+\sin x}{(1+\sin x)^2}$$
Now WLOG let $x=\dfrac\pi2-2y$ to get $$\dfrac{3+2\sin2y+\cos2y}{(1+\cos2y)^2}$$
Using Weierstrass substitution, writing $\tan y=t$
we get $$2f(t)=(t^2+1)(t^2+2t+2)$$
Now use Second derivative test, to find the minimum value of $f(t)$ occurs at $-\dfrac12$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2277893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Natural rational map from a variety to the section ring of a divisor Let $X$ be a variety over $\mathbb{C}$. Let $D$ be an effective divisor on $X$. I heard there is a natural rational map $X\dashrightarrow Proj (R)$ where $R=\oplus_{n=0}^\infty H^0(X,nD)$.
My question:
*
*How is this map defined?
*When is this a ... | *
*The map is defined by sending $x \in X$ to the homogeneous ideal of sections vanishing at $x$. (Note that it could be the case that every nonconstant section vanishes at $x$, in which case the ideal is the irrelevant ideal, and the map is not defined at $x$.)
*The map is birational precisely when $D$ is big. Depen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Writing a ring $\mathbb Z[\alpha]$ as a quotient ring Context.
If have an algebraic element $\alpha$ over $\mathbb Q$, and I want to write $\mathbb Z[\alpha]:=\{a+\alpha b,\ a,b\in \mathbb Z\}$ as a quotient ring of the form $\mathbb Z[X]/I$.
Is the following approach correct?
Let $\pi$ be an irreducible of $\mathbb ... | The way you have defined it, it is not even a ring
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Proving the Bessel function formula by expanding its generator function. I am trying to show that the Bessel functions $J_n(x)$ have the form
$$J_n(x)=\sum_{k=0}^{\infty}\frac{(-1)^k}{k!(n+k)!}\bigg( \frac{x}{2} \bigg)^{n+2k},$$ from its generator function
$$G(x,t)=\sum_{n=-\infty}^{\infty}t^nJ_n(x)=e^{\frac{x}{2}(t-\... | You want to identity the coefficients of $t^n$. Your formula includes
$t^{m-2k}$. So you want to set $m-2k=n$. Eliminating $m$ your formula
is the sum of
$$\frac{(-1)^k}{k!(n+k)!}t^n\left(\frac x2\right)^{n+2k}$$
over $n$ and $k$ with $0\le k\le n+2k$, that is $k\ge\max(0,-n)$.
So for $n\ge0$ you get
$$J_n(x)=\sum_{k=0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differential equation with homoegeneous coefficient, solution other than in book I have a differential equation:
$$ x \frac{dy}{dx} - y - x\sin\left(\frac{y}{x}\right) = 0. $$
I'm multiplying both sides by $dx$ and I'm obtaining:
$$ x\,dy - y\,dx - x \sin\left(\frac{y}{x}\right)\, dx = 0. $$
Next, after simplification... | The book's solution is correct, as you can easily check by substituting it in to the differential equation: note that
$$ \eqalign{\sin(2 \arctan(cx)) &= 2 \sin(\arctan(cx)) \cos(\arctan(cx))\cr &= 2 \tan(\arctan(cx)) \cos^2(\arctan(cx))\cr & = \frac{2 \tan(\arctan(cx)}{1+\tan^2(\arctan(cx))}\cr &= \frac{2 c x}{1 + c^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Let $y_1, y_2, ....$ be a sequence such that $0\leq y_n \leq 1$ and $\sum_{n=1}^\infty y_n=\infty$. Prove that $\prod_{n=1}^\infty (1-y_n)=0.$ Let $y_1, y_2, ....$ be a sequence such that $0\leq y_n \leq 1$ for all $n$, and
$\displaystyle\sum_{n=1}^{\infty} y_n=\infty$. Prove that $\displaystyle\prod_{n=1}^{\infty} (1... | I don't see any reason to introduce exponential or logarithm functions. Assume that $0\le x<1$. Then $1/(1-x) \ge (1+x)$. Assume $0\le y<1$. Then $(1+x)(1+y)\ge 1+(x+y)$.
Now if $0\leq y_n < 1$ for all $n$, we get that $$\prod_{n=1}^\infty \frac{1}{1-y_n} \ge 1+\sum_{n=1}^\infty y_n$$ and since we are given that the su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Equation of circumcircle From a point $(a,b)$ two tangents $\overset{\leftrightarrow}{PQ}$ and $\overset{\leftrightarrow}{PR}$ are drawn to a circle
$x^2 + y^2 - a^2=0$
Find the equation of the circumcircle of $\triangle PQR$.
My attempt:
The circumcircle and the given circle have a common chord $\overline{QR}$.
Apart... | HINT:
The equation of tangent at $P(a\cos2t,a\sin2t)$ is $$x\cos2t+y\sin2t=a$$
Now if this passes through $(a,b),$
$$a\cos2t+b\sin2t=a\iff2b\sin t\cos t=a(2\sin^2t)\implies$$
either $\sin t=0\implies P_1(a,0)$
or $\tan t=\dfrac ba\implies P_2\left(a\cdot\dfrac{a^2-b^2}{a^2+b^2},\dfrac{2a^2b}{a^2+b^2}\right)$
We already... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Probability problem 2 There are 10 boxes each containing 6 white and 7 red balls. Two random
boxes are chosen, one ball is drawn simultaneously at random
from each and transferred to the other box. Now a box is again chosen
from the 10 boxes and a ball is chosen from it. Then the probability
that this ball is white is ... | Hint:
There are $130$ balls in total that have equal chances to be drawn. $60$ of them are white.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Having trouble using my usual method of partial fraction decomposition for $\frac{9 + 3s}{s^3 + 2s^2 - s - 2}$. I'm having trouble using my usual method of partial fraction decomposition for $\dfrac{9 + 3s}{s^3 + 2s^2 - s - 2}$.
We can factor such that $$\dfrac{9 + 3s}{s^3 + 2s^2 - s - 2} = \dfrac{A}{s - 1} + \dfrac{B}... | You've got the right expression
$$
9 + 3s = A(s + 1)(s + 2) + B(s - 1)(s + 2) + C(s + 1)(s - 1).
$$
Now you can think like this: it is a polynomial equation of the kind $p(s)=0$. A nonzero polynomial cannot have more than the finite number of zeros. Since this equation has infinitely many solutions (all $s$ except a fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Show that if $f$ is differentiable then...
Show that if $f$ is differentiable at $a$ then one may expand $f$ around $a$ as $$f(x)=f(a)+(x-a)f'(a) +(x-a)E(x)$$
where $E(x) \to 0$ as $x \to a$
If $f$ is differentiable at $a$ then we have $$f'(a)= \lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}$$
We can multiply both sides by $(x... | My suggestion would be to define $E(x)$ (for $x \neq a$) as
$$ E(x) = \frac{f(x) - f(a)}{x - a} - f'(a),$$
(so that, if you rearrange this, you'll get the equation you wrote down).
It only remains to show that $\lim_{x \to a} E(x) = 0$. For this step, just remind yourself of the definition of the derivative $f'(a)$, an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Minimize $m+n$ given $\frac{2016}{2017}<\frac mn<\frac{2017}{2018}$ Given:
$$\dfrac{2016}{2017}<\dfrac mn<\dfrac{2017}{2018}$$
Find the smallest value possible of the sum of the denominator and the numerator, i.e. $m+n$.
I don't know how to spot the very peculiar fraction with the minimum values of $m$ and $n$ in the d... | $$\frac{1}{2016}>\frac{m}{n}-1>\frac{1}{2017}$$
$$\frac{1}{2016}>\frac{m-n}{n}>\frac{1}{2017}$$
$$2016<\frac{n}{m-n}<2017$$
So $n\ge 2\cdot2016+1=4033$ and $m-n\ge 2$. Hence $m\ge 4035$
The smallest value is $8068$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
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monomial ideals I was reading the book $\textit{Ideals, Varieties, and Algorithms}$ by Cox, Little, and O'Shea, and on Chapter 2 page 71 the have the following lemma
Lemma 3: Let $I$ be a monomial ideal, and let $f \in k[x_1, \ldots, x_n]$. Then the following are equivalent:
i) $f \in I$
ii) Every term of $f$ lies in $... | Let
$$f = \sum_{\mathbf{u} \in \mathbf{N}^n} a_\mathbf{u} \mathbf{x}^\mathbf{u}$$
and suppose that for each $\mathbf{u} \in \mathbf{N}^n$,
$$a_\mathbf{u} \mathbf{x}^\mathbf{u} = \sum_{\mathbf{v} \in \mathbf{N}^n} p_\mathbf{u,v}(\mathbf{x}) \mathbf{x}^\mathbf{v} $$
where $p_\mathbf{u,v}(\mathbf{x}) \in k[\mathbf{x}]$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Formula to create a Reuleaux polygon The Wikipedia articles for Reuleaux triangle and curve of constant width do a good job of describing the properties of a Reuleaux polygon, but they don't give a straightforward formula for computing or drawing such a figure, except in terms of the manual compass-and-straightedge con... | As you can see from the diagram below, if $L$ is the length of a side of the regular polygon, $n$ (odd) the number of its sides and $W$ its width, then:
$$L=2W\sin{\pi\over 2n}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
$\sqrt{x^2-x+1}+\sqrt{y^2-y+1}=\sqrt{x^2+xy+y^2}$ fine $x,y$ :
$$\sqrt{x^2-x+1}+\sqrt{y^2-y+1}=\sqrt{x^2+xy+y^2} \ \ \ : x ,y \in \mathbb{Z}$$
My Try :
$$\sqrt{x^2-x+1}+\sqrt{y^2-y+1}=\sqrt{x^2+xy+y^2} \\ x^2-x+1+y^2-y+1+2\sqrt{(x^2-x+1)(y^2-y+1)}=x^2+xy+y^2 \\ 2\sqrt{(x^2-x+1)(y^2-y+1)}=xy +x+y-2$$
Now ?
| You can simplify as follows
$$\begin{align}
2\sqrt{(x^2-x+1)(y^2-y+1)}&=xy +x+y-2\\
4(x^2-x+1)(y^2-y+1)&=x^2y^2+x^2+y^2+4+2x^2y+2xy^2-4xy\\&\,\,\,\,\,+2xy-4x-4y\\
4x^2y^2-4x^2y+4x^2-4xy^2+4xy+4y^2&=x^2y^2+2x^2y+x^2+2xy^2-2xy+y^2\\
3x^2y^2-6x^2y+3x^2-6xy^2+6xy+3y^2&=0\\
x^2y^2-2x^2y+x^2-2xy^2+2xy+y^2&=0\\
(xy-x-y)^2&=0
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Optimization with Constraints using Alternating Direction of Method of Multipliers I have an optimization problem of the form:
\begin{align*}
\text{minimize} &\quad f(x) + g(x) \\
\text{such that} & \quad Ax = b
\end{align*}
Where $f,g$ are both convex (but not differentiable), and I know the proximal operators for $f,... | One approach using the Douglas-Rachford method (which can be viewed as being a special case of ADMM) is to reformulate your problem as minimizing $$
\underbrace{f(x) + g(y)}_{F(x,y)} + I_C(x,y),$$ where $C=\{(x,y)\mid Ax=b, x=y\}$.
You can now minimize $F(x,y) + I_C(x,y)$ using the Douglas-Rachford method. Because $F$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Rewrite $\|AXBd -c \|^2$ as $\|Qx -c \|^2$ to solve it using standard solvers I need to solve a quadratic problem that I have formulated as $\|AXBd -c \|^2$, where $X$ is a matrix of unknowns and $A$, $B$ constant square matrices and $d$ a vector of constants.
How can I rewrite such problem so I get a system of the typ... | Denoting by $\operatorname{vec}:\mathbb{R}^{m\times n} \rightarrow \mathbb{R}^{mn}$ the operator transforming an $m\times n$ matrix into a column vector of $mn$ elements by "stacking" the matrix columns, it holds
$$
\operatorname{vec}(\mathbf{L} \mathbf{X} \mathbf{R}) = \left(\mathbf{R}^T \otimes \mathbf{L} \right) \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Probability of forming a polygon from a stick with $n$ breaks We have a stick of length one which is broken into $n$ pieces. What is the probability that a quadrilateral can be formed from the broken pieces?
I know the answer for $n=4$ as it is the probability that none of the pieces are above $0.5$ in length, that i... | The probability you can form a closed polygon is the probability that no segment has length greater than $1/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2280002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Stuck on an integral $4x\sin (x^2)\cos (x^2)$ I'm trying to get my head around this integral but it just doesn't click $$4x\sin (x^2)\cos (x^2) $$
I have tried substitution but I am confusing myself!
Do I substitute $\cos (x^2)$ getting $du=- 2x\sin (x^2) $ but I'm not sure where to go from there.
| Let
$$
u = x^{2} \qquad \Rightarrow \qquad du = 2x dx
$$
Then the primitive becomes
$$
\int 4 x \sin \left(x^2\right) \cos \left(x^2\right) \, dx \Rightarrow
\int 2 \sin u \cos u \, du =
-\frac{1}{2} \cos (2 u) \, du \Rightarrow
-\frac{1}{2} \cos \left(2 x^2\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2280090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Tribonacci Sequence Term
A tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its seeds For example, if the three seeds of a tribonacci sequence are $1,2$,and $3$, it's 4th terms is $6... | In this paper, the authors show that, if $S_1,S_2,S_3$ are the seeds, then
$$S_n=T_{n-2}\,S_1+(T_{n-2}+T_{n-3})\,S_2+T_{n-1}\,S_3$$ where $T_k$ is the "usual" Tribonacci number.
Applied to the seeds you give, this generates the following values
$$\left(
\begin{array}{ccc}
n & T_n & S_n \\
1 & 0 & 6 \\
2 & 1 & 19 \\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2280243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
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How to evaluate the integral $ \int_{0}^{2\pi}|2e^{it}-1|^2 2ie^{it} dt $? How to evaluate the integral $$\int_{0}^{2\pi}|2e^{it}-1|^2 2ie^{it} dt $$
I have attempted it replacing $e^{it}$ with $\cos t +i\sin t$ but it doesn't seem to be working. How would I go about this?
| $\int_{0}^{2\pi}|2e^{it}-1|^2 2ie^{it}dt=\\
\int_{0}^{2\pi} |2\cos(t)+i\sin(t)-1|^2 2e^{it}dt=\\
\int_{0}^{2\pi} ((2\cos(t)-1)^2+(2\sin(t))^2) 2ie^{it}dt=\\
\int_{0}^{2\pi} (4\cos(t)^2+4\sin(t)^2+1-4\cos(t)) 2ie^{it}dt=\\
\int_{0}^{2\pi} (5-4\cos(t))2i(\cos(t)+i\sin(t)) dt=...$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2280450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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The integral $\int J_0(x) \cos(x) dx$ I am trying to show that
$$ \int J_0(x) \cos(x) dx = x J_0 \cos(x) + x J_1 \sin(x) +C$$
where $J_n$ is the Bessel function of the first kind, using integration by parts. However, both obvious factors lead nowhere.
$$ \int J_0(x) \cos(x) dx = J_0 \sin(x) + \int \sin(x) J_1(x) dx +... | As Daniel Fischer pointed on the comment above, the integral can be computed as follows:
$\int J_o(x) \cos(x) dx=x J_0(x) \cos(x) + \int x J_1(x) \cos(x) dx + \int x J_0 \sin(x) dx +C= J_0(x) \cos(x) + \int x J_1(x) \cos(x) dx+x J_1(x) \sin(x) - \int x J_1(x) \cos(x) dx$
which produces the desired result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2280600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Probability that 2 of t strings of length n are equal Given t bit-strings of length n that are generated randomly.
What is the probability that at least 2 of these strings are equal.
I've seen someone who wrote that the probability is $ \le \frac{t^2}{2^n}$
The reasoning is that you have $t$ options to choose the first... | I think this probability is $1-\frac{\binom{2^n}{t}}{\binom{2^n -1 +t}{2^n -1}}$. Essentially this means 1- probability to have all strings different. The numerator is all different strings. The denominator is all cases of selecting t out of $2^n$ with repetitions.
EDIT:
OK @MishaLavrov's solution $\frac{\binom{2^n}{t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2280875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
What is the meaning of the $\vdash$ symbol? As seen in the following:
$$\large \lambda=(3,2)\vdash 5$$
I looked it up and in logic the symbol means that what is on the right is provable by what is on the left, but what does it mean in the mathematical context? Does it mean "corresponds to"?
| I hope it can help you:
*
*A partition of a positive integer n
is a multiset of positive integers that sum to n.
We denote the number of partitions of n
by $p_{n}$
*We use Greek letters to denote partitions often $\lambda$,$\mu$ and $\nu$
We’ll write:
$λ:n=n_{1}+n_{2}+···+n_{k}$ or $λ⊢n$.
The notation λ ⊢ n means ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2281004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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$\lim_{n\to \infty} \frac{3}{n}(1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+...+\sqrt{\frac{n}{4n-3}})$ fine the limits :
$$\lim_{n\to \infty} \frac{3}{n}(1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+...+\sqrt{\frac{n}{4n-3}})$$
My Try :
$$\lim_{n\to \infty} \frac{3}{n}(1+\frac{1}{\sqrt{1+\frac{3}{n}}}+\frac{1}{\sqrt{1+... | $$\frac3n\sum_{k=0}^{n-1}\sqrt{\frac n{n+3k}}=\frac3n\sum_{k=0}^{n-1}\sqrt{\frac1{1+\frac{3k}n}}\xrightarrow[n\to\infty]{}3\int_0^1\frac1{\sqrt{1+3x}}\,dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2281138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Unit of a quotient ring. I'm having trouble showing whether this is true or false and why:
The class $X^3+1+(X^3+2X+1)$ is a unit in the quotient ring $\Bbb Z_3[X]/(X^3+2X+1)$
I think I need to show that it has a multiplicative inverse but not sure if that's right or how.
Any help would be greatly appreciated.
| Yes, that is what you need to show.
Hint: Note that $[X^3 + 1] = [X]$ (using brackets to denote residue classes), and also $$[1] = [2X^3 + X] = [X][2X^2 + 1] = [X^3 + 1][2X^2 + 1].$$ Therefore...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2281227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Integrating a time derivative let $x$ be a function of time $$x = x(t)$$
what is the integral of $$\int \frac{\dot x}{ \sqrt{1+ \dot x^2}} \, dt$$
I have tried using trigonometric substitution of $$ \dot x = \tan(\theta) $$
but I end up at the same point but with trigonometric terms
| You can integrate by parts,
$$\int \frac{\dot x}{\sqrt{1+\dot x^2}}\mathrm d x=\frac{x}{\sqrt{1+\dot x^2}}+\int \frac{\dot x^2 \ddot x}{(1+\dot x ^2)^{\frac32}} \mathrm d x.$$
Since $\mathrm{d} \dot x = \ddot x \; \mathrm{d} t$, plug into the previous integral,
$$\int \frac{\dot x}{\sqrt{1+\dot x^2}}\mathrm d x=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2281333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An urn contains n balls numbered $2^{(n-1)}$. Expected value of sum of random selections Here's the problem:
An urn contains n balls numbered $1, 2, 4, 8$, etc. up to $2^{(n-1)}$ where $n > 1$. If a person selects
1 ball from the urn, then replaces it and selects a second ball, what is the expected value of the sum of... | You are right. The $\frac 1n$ can be pulled out of the sum, leaving you with a geometric series to sum.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Inequality with 2-norm of matrix-vector product With the matrix $A \in \mathbb{R}^{m \times n}$, and vector $x \in \mathbb{R}^n$, I'm trying to figure out this inequality:
$$\|Ax\|^2 \leq L\|x\|^2$$ with $L \in \mathbb{R}$, where $L$ is proposed to be the sum of the squares of the entries in the matrix.
I tried expandi... | This is the Cauchy–Bunyakovsky–Schwarz inequality. For any given $i$, your equation becomes
$$\left(\sum_{j=1}^n a_{ij}x_j\right)^2\le\sum_{j=1}^n a_{ij}^2\sum_{k=1}^n x_k^2$$
Following wikipedia, we can start with a quadratic polynomial
$$0\le(a_{i1}y+x_1)^2+(a_{i2}y+x_2)^2+...+(a_{in}y+x_n)^2=y^2\sum_{j=1}^n a_{ij}^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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cyclotomic polynomial Let $w$ be a primitive 10th root of unity. Find the irreducible polynomial of $w+w^{-1}$.
I know that the cyclotomic polynomial of $g_{10}(x)=x^4-x^3+x^2-x+1$ but I can't apply the same techniques used in this question (Cyclotomic polynomials and Galois groups
) where $w$ is a 7th root of unity i... | First off, note that the $10$th roots of unity and $5$th roots of unity both define the $5$th cyclotomic field, $Q(ζ_5)$. If $w$ is a $10$th root of unity, then the minimal polynomial of $w$ is:
$x^4-x^3+x^2-x+1$
Note that this is true for $w^5$ as well.
Now $w^{-1}+w$ is $1/w$ + $w$, or using the equivalent definitio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2281618",
"timestamp": "2023-03-29T00:00:00",
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express $a$ in terms of $b$ and $c$
Given that $$c=\frac{\sqrt{a+3b}}{a-3b}$$ express $a$ in terms of $b$ and $c$
My attempt,
\begin{align}c^2(a^2-6ab+9b^2)&=a+3b\\
c^2a^2+(-6bc^2-1)a+9b^2c^2-3b&=0\\
a&=\frac{-(-6bc^2-1)\pm \sqrt{(-6bc^2-1)^2-4c^2(9b^2c^2-3b)}}{2c^2}\\
a&=\frac{6bc^2+1\pm \sqrt{24bc^2+1}}{2c^2}\end{a... | A slightly different take on Yves Daoust's approach:
Let $\sqrt{a+3b}=u\ge0$, so that $a=u^2-3b$. Then
$$c={u\over u^2-6b}\implies cu^2-u-6bc=0$$
The quadratic formula gives
$$u={1\pm\sqrt{1+24bc^2}\over2c}$$
as the formal solutions. It is convenient to rewrite them as
$$u={1+\sqrt{1+24bc^3}\over2c}\qquad\text{and}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2281781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Birthday paradox combinatorics Here's a problem in Harvard's STAT 110 Probability course:
"3. A college has 10 (non-overlapping) time slots for its courses, and blithely assigns courses to time slots randomly and independently. A student randomly chooses 3 of the courses to enroll in (for the PTP, to avoid getting fine... | I was using the calculations that disregard order, but order matters in this question because the classes are obviously different from one another. Thus,
$$
P(conflict)=\frac{10^3-P(10,3)}{10^3}=0.28,
$$
which is the correct answer.
| {
"language": "en",
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A problem related to locus in circles Two circles intersect at $A$ and $B$; $PQ$ is a straight line through $A$ meeting the circles at $P$ and $Q$. Find the locus of midpoint of $PQ$.
I know the locus is a circle. But I am unable to prove it.
| Angles $\angle BPA$ and $\angle BQA$ don't depend on the positions of $P$ and $Q$, while they vary on the outer parts of the circles. That means that all triangles $BQP$ have the same angles, independent of the positions of $P$ and $Q$, and that is true even if $P$ and $Q$ lie on the inner parts of the circles.
If $M$... | {
"language": "en",
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Finite dimensional division algebra over $\mathbb{C}$ must be equal to $\mathbb{C}$ Let $A$ be a finite dimensional $\mathbb{C}$-algebra such that for any $0 \neq a \in A,$ there exists $b\in A$ such that $ab = ba =1.$ I want to show that $A = \mathbb{C}.$
First, let $U$ be a nonzero $A$-submodule of $A$. (So consideri... | I know this has been answered but I can't help giving another proof as it is purely topological.
The map $A^* \to A^*, z \mapsto z^2$ is a covering of degree $2$, where $A^* := A \backslash \{0\}$. Now the total space of this covering is connected so this covering can't be trivial. But the base space is simply connect... | {
"language": "en",
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"source": "stackexchange",
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Is there a simple example of a transitive vector field on the three sphere? I was just thinking about the transitive field on the torus and the possibility of defining a vector field without singularities on $S^3$ and this idea popped up.
Edit: I think my question was misunderstood. I am specifically asking about trans... | You can define a nowhere-zero vector field on any odd-dimension sphere. For instance, one is given by embedding the $(2n-1)$-sphere as the unit sphere in $\Bbb R^{2n}$, and at the point $x = (x_1, x_2, x_3,\ldots,x_{2n})\in S^{2n-1}$ define the tangent vector
$$
v_x = (x_2, -x_1, x_4, -x_3, \ldots, x_{2n}, -x_{2n-1})
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2282214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Central limit theorem: Poisson equals Normal? Tell me where I'm wrong We just covered the Central Limit theorem in class, and I stumbled upon the following reasoning that makes me think I am missing some key part of... well, something. So, here goes:
Let $X_1$ and $X_2$ be independent Poisson random variables with para... | Suppose $X_1,X_2,X_3,\ldots\sim \operatorname{i.i.d.~Poisson(1)}.$ Then
$$
\frac{(X_1+\cdots+X_n)-n}{\sqrt n} \overset{\text{distribution}} \longrightarrow N(0,1) \text{ as } n\to\infty. \tag 1
$$
More generally, if $X\sim\operatorname{Poisson}(\lambda)$ then
$$
\frac{X-\lambda}{\sqrt\lambda} \overset{\text{distributio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2282342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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logarithm proof I'm trying to prove the following inequalities:
\begin{align}
\frac{b+c}{b} \geq \frac{\log(\frac{a}{b})}{\log(\frac{a+c}{b+c})}, c \in (0,1)
\end{align}
I know that $\log(\frac{a}{b}) > \log(\frac{a+c}{b+c})$ for $ c > 0$, but I'm stuck as to how to proceed with the proof. This is not a homework and is... | Let $a,b,c \in \mathbb{R}^+$ and $a>b$.
Assume
$$\left(\frac{a+c}{b+c}\right)^{b+c}>\left(\frac{a}{b}\right)^{b}$$
Then
$$\log\left(\left(\frac{a+c}{b+c}\right)^{b+c}\right)>\log\left(\left(\frac{a}{b}\right)^{b}\right)$$(as both sides are positive and $\log$ is an increasing function)
$$\implies (b+c)\log\left(\frac{a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Do I condition a constant while computing the conditional expectation? I have this event $A$ and I know $P(A)$.
I also have a r.v. $T$, which is exponential with a given $\lambda$.
I want to compute ${\bf E}[T+5|A]$.
I remember that unconditionally, ${\bf E}[T+b]={\bf E}[T]+b$. But does it mean that ${\bf E}[T+b|A]={\... | Yes, the Linearity of Expectation holds for a conditional expectation too.
When $T$ is a random variable, $A$ is an event, and $b$ a constant, then:
$$\mathsf E(T+b\mid A) ~=~ \mathsf E(T\mid A)+b$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can an open set be covered by proper open subsets? Let $(X, \tau)$ be a topological space, let $U\in\tau$. An open cover of $U$ is a set $\{U_i \ |\ i\in I\}$ (of open sets $U_i$) whose union $\bigcup U_i$ contains $U$.
If $U\subsetneq X$, then $U$ admits an open covering by open sets $\{U_i \ |\ i\in I\}$, where an ar... | A space is called $T_1$ if, for any two points in the space, each has an open neighborhood that misses the other. This is one of many separation axioms that measure how strongly we can separate points in the space. For some mathematicians, topological spaces aren't even worth considering until they are at least $T_2$ (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2282623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Extrapolate a sum using partial sums at powers of two In an online textbook for MIT OCW 18.013a, Calculus with Applications, the author uses residue calculus to derive the well-known formula
$$\sum_{n>0} n^{-2} = \frac{\pi^2}{6}$$
(See Some Special Tricks)
He then writes:
You can actually sum the first 128 (or 1024) t... | This follows from the Euler–Maclaurin formula, we have:
$$\sum_{n = N}^{\infty} f(n) = \int_{N}^{\infty}f(x) dx + \frac{1}{2}f(N) -\sum_{k=1}^{M}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(N) + R_M$$
where the $B_{2k}$ are the Bernoulli numbers and $R_M$ is a remainder term. In case of $f(n) = \dfrac{1}{n^2}$, this yields:
$$\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2282740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing the following two integrals are equal I would like to show that
$$\int_{0}^{\infty} \frac{e^{-t}}{\sqrt{t+x}} dt = 2\int_{0}^{\infty}e^{-t^{2}-2t\sqrt{x}}dt,\quad x>0.$$
I haven't been able to have very much success with this integral. So far I have made a couple observations:
1) I think there is an obvious st... | This is not an answer, so please don't down-vote this. Nevertheless, here's how this problem can be solved using computer algebra. In Mathematica:
Assuming[Re[x] > 0,
Integrate[Exp[-t]/Sqrt[t + x], {t, 0, \[Infinity]}]] ==
Assuming[Re[x] > 0,
2 Integrate[Exp[-t^2 - 2 t Sqrt[x]], {t, 0, \[Infinity]}]]
(* True... | {
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How do I solve this system of 2 equations? I need to solve for variables $u$ and $v$ in this system of equations:
$(x+u)^2+(y+v)^2=1$
$u^2+v^2=k$
How do I isolate $u$ and $v$ to get them both in terms of $x$, $y$, and $k$?
| If you expand the squares in the first equation, you can use the second to eliminate the $u^2,v^2$ terms. That leaves you with one linear equation and one quadratic. Solve the first for $u$ and substitute into the second. That gives you a quadratic in $v$ which you can solve, getting two roots. Plug them into the f... | {
"language": "en",
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a subset of a zero Lebesgue measure set is measurable? This question is related to What's the quickest way to see that the subset of a set of measure zero has measure zero? But I'm specifically concerned about Lebesgue measure, $m$, on a real interval, $X=[a,b]$, and specifically about why a subset, $A$, of zero measu... | Suppose $\lambda(E)=0$, and that $A\subseteq E$. Then as you said, $\lambda^*(A) = \lambda(E)$. In order for $A$ to be measurable, it must be the case that for each $B\subseteq [a,b]$, we have that
$$\lambda^*(A) = \lambda^*(A\cap B)+\lambda^*(A\cap B')$$
where $B' = [a,b]\backslash B$. Indeed, since $A\cap B, A\cap B'... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof of: a commutator subgroup is a normal subgroup The proof of the statement "a commutator subgroup $G'$ is a normal subgroup" goes like this:
1) Show that $g^{-1}[a,b]g \in G'$
2) Show that $g^{-1}[a_1,b_1][a_2,b_2]...[a_n,b_n]g \in G'$
I don't understand why step 1 doesn't suffice itsel
| Step $1$ doesn't suffice becaue the commutator subgroup is not the set of all commutators (as this set doesn't usually form a group), rather, it is the subgroup generated by the set of commutators. If you only prove step $1$, it could still be the case that, say, $g^{-1}[a,b][c,d]g \notin G'$ for some $a,b,c,d,g\in G$.... | {
"language": "en",
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Finding a point on an ellipsoid Find a point on the ellipsoid $x^2+4y^2+z^2=9$ where the tangent plane is perpendicular to the line with parametric equations \begin{align}x&=2+2t\\y&=1+2t\\z&=3-t\end{align}
The answer to this question is: $$\left(\frac{6\sqrt{13}}{13}, \frac{6\sqrt{13}}{13}, -\frac{3\sqrt{13}}{13}\righ... | Obviously, the points that you mentioned are not the answer, since they don't even belong to the ellipsoid.
The points where the tangent plane is perpendicular to the given line are $\pm\left(\sqrt6,\sqrt{\frac38},-\sqrt{\frac32}\right)$, since these are the points of the ellipsoid such that the gradient of $x^2+4y^2+z... | {
"language": "en",
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Why isnt product of nth roots of unity always 1 I know product of nth roots of unity is 1 or -1 depending whether n is odd or even.
But in this way I am getting 1. Where am I wrong?
$ \text{Let }\alpha = \cos \frac{2 \pi}{n} + \iota \sin \frac{2 \pi}{n} \text{ be a root of }x^n=1 \\
\text{Then product of nth roots will... | Hint:
$$\alpha^{n(n-1)/2}=(\alpha^{n/2})^{n-1}$$
Now $\alpha^{n/2}=\cdots=-1$
| {
"language": "en",
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Exercise X.3.2 Mac Lane CWM (Kan extension of representable functors) I found an exercise on Mac Lane CWM , pg.240 ex.2 :
If A= $Set$, and M, C have small hom-sets, show that the left Kan extension of $M$(m,-) is $C$(Km,-) with unit $\eta$:$Id_M$ $\rightarrow$$C$(Km,K-) given by $\eta$m=$1_{Km}$ (with K:M$\rightarrow... | Here is a sketch of the proof:
(I). Define $\eta _{[m,m']}:[m,m']\to [Km,Km']$ in the obvious way $f\mapsto Kf$ and show it is natural.
(II). Suppose $\alpha:[m,-]\overset{\cdot }{\rightarrow} A^KS=SK$ is a given natural transformation. Then, Yoneda applies to show that there is a unique $a\in SKm$ such that $\alpha_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2283532",
"timestamp": "2023-03-29T00:00:00",
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How to evaluate the closed form for $\int_{0}^{\pi/2}{\mathrm dx\over \sin^2(x)}\ln\left({a+b\sin^2(x)\over a-b\sin^2(x)}\right)=\pi\cdot F(a,b)?$ Proposed:
$$\int_{0}^{\pi/2}{\mathrm dx\over \sin^2(x)}\ln\left({a+b\sin^2(x)\over a-b\sin^2(x)}\right)=\pi\cdot F(a,b)\tag1$$
Where $a\ge b$
Examples:
Where $F(1,1)= \s... | Let $u=\cot{x}$. Then $du = dx/\sin^2{x}$ and $\sin^2{x}=1/(1+u^2)$, and the limits become $\infty$ and $0$, so the integral becomes
$$ \int_0^{\infty} \log{\left( \frac{a+b/(1+u^2)}{a-b/(1+u^2)} \right)} \, du = \int_0^{\infty} \log{\left( \frac{a+b+au^2}{a-b+au^2} \right)} du $$
One can now integrate this by parts t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Every planar graph with n vertices have at least n/27 non adjacent pairs of vertices of degree $ \leq 8$ I wont lie - this is for homework.
I am struggling with this task for hours now. I know it has something to do with $k \leq 3n - 6$ but I am just unable to find the connection.
Thanks in advance!
| Let us prove this by induction on the number of vertices in our graph, $N$. If $N=0$, this is clearly true. Let our claim hold for all $N <n$.
Given a graph $G$ with at $n$ vertices and $m$ edges, we know that $m\leqslant 3n - 6$, as $G$ is planar. This means that there exists at least one vertex $v$ of degree less tha... | {
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Are Inverse Trig functions a different form of log? When studying complex analysis, we realize that trigonometric functions are nothing but exponentials, and we can define real trigonometric functions in terms of complex exponentials.
I was wondering if we can apply this logic to define inverse trigonometric functions... | Yes absolutely! See here, where the logarithmic forms of $\arcsin$ etc are given. There may be issues defining the domain, since the complex log function has a fair amount of subtlety in its definition. In a related but slightly simpler vein, the inverses of the hyperbolic trig functions can also be written in terms ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Calculate $\lim_{n\rightarrow \infty} \int_{-\infty}^{\infty}\frac{1}{1+\frac{x^{4}}{n}}dx$ if it exists. Calculate $$\lim_{n\rightarrow \infty} \int_{-\infty}^\infty \frac{1}{1+\frac{x^4}{n}} \, dx$$
if it exists. If this limit does not exist, show why it does not exist.
My attemp: Consider $f_n(x):=\frac{1}{1+\frac{x... | $$
\int_{-\infty}^{\infty}\frac{1}{1+\frac{x^4}{n}}\mathrm{d}x=\pi\frac{\sqrt[4]{n}}{\sqrt{2}}
$$
$$
\lim_{n\to\infty}\int_{-\infty}^{\infty}\frac{1}{1+\frac{x^4}{n}}\mathrm{d}x=\infty
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2284004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Function Composition and Expected Value I was presented with the question
The function $f(x)$ is equal to $x^2$ and the function $\chi(x)$ gives a random real number from 0 to $x$. Which usually has a greater expected value, $(f \circ \chi)(x)$ or $(\chi \circ f)(x)$ For what values of $x$?
I am almost entirely unexp... | If we take a given number $x_0$, then our random number is a uniform random variable $X$ on $[0,x_0]$. Its probability density function is:
$$ f_{X}(x) = \begin{cases} \frac{1}{x_0} & 0\leq x \leq x_0 \\ 0 & \text{else} \end{cases} $$
If we generate a random number from this distribution, and then square it, our expect... | {
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Given two machines break down, what is probability one is from each company? A factory has four machines, of which two are imported from Country A and two are imported from Country B.
The probability that the machine imported from Country A breaks down is $0.3$.
The probability that the machine imported from Country B ... | What you tried to calculate is the probability of exactly one machine from each country breaking down, but that is indeed not what is being asked. they ask for the conditional probability that this is the case given that two machines break down. More formally, if we let $AB$ be the event that one machine from each cou... | {
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Point inside a triangle that is the same distance from each vertex Let's say we have three exact locations on the Earth. Let's say people from those three locations want to meet in a point that is between all three locations, but also the same distance from each location.
Center of mass/triangle center doesn't work bec... | The point you want is called the circumcenter, it is the intersection of all three perpendicular bisectors of the sides of the triangle. The other common important points of triangles are probably:
*
*center of mass: point of intersection of the three medians (always inside triangle)
*orthocenter: point of intersec... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Mathematical notation $\max$ with simple example for non-mathematician First, let me start off by saying I'm not a mathematician so I'm going to need this explained to me at a pretty basic, almost intuitive level. I've taken Calculus but it's been some time so I do have some math background.
I was reading a book tonig... | $$\max_{\theta\in\Theta}R_T(\theta)$$means the maximum value that $R_T(\theta)$ can take as $\theta$ varies over all possible values in $\Theta$. This quantity is still dependent on $T$, and its least value for all possible values of $T$ is $$\min_{T}\max_{\theta\in\Theta}R_T(\theta).$$ Here the possible values for $T$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2284569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
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Do Homotopy Groups commute with generalized filtered colimits? I know that if X is a topological space such that $X= \underset{i}{\bigcup} X_i$ where $X_0 \subset X_1 \subset ... \subset X_n \subset ...$, where $X_i$ are all hausdorff, then the functor $\pi_n(\_)$ commutes with the colimit:
$$\varinjlim \pi_n(X_i, x_0... | The answer is no, not even $\pi_1$ does:
If $X$ is a metric space and $Y$ any topological space, then a map $f\colon X\to Y$ is continuous if and only if for every convergent sequence $(x_n)_{n\in\mathbb{N}}$ in $X$, the sequence $(f(x_n))_{n\in\mathbb{N}}$ converges with limit point $f(\lim_n x_n)$.
Furthermore, if $X... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Global optimization of non-smooth function I have a number of functions (see for example two of them down below), and I need to find their global optimum for each of them. They are non-smooth, but they are always funnel-shaped, exhibiting a large minimum. If you zoom out, (e.g. when the x range is 0-100), the function ... | One possible suggestion:
First, use your global optimization to define some small set $I$ (e.g. interval) in which to work.
Then, use an optimization algorithm that can handle noise without gradient information. Even simple grid search on $I$ could work.
Another idea could be a evolutionary algorithm, like differenti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2284794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Let $x \sim y$ if and only if $x-y\in \mathbb{Q}$.Show that the quotient topology on $\mathbb{R}/\sim$ is the indiscrete topology. Consider $\mathbb{R}$ with standard (Euclidean) topology.
Let
$x \sim y$ if and only if $x-y\in \mathbb{Q}$.
Show that $\mathbb{R} /\sim$ is uncountable but that the quotient topology on $\... | A different attempt:
*
*Recall that a set in $\mathbb{R} / \sim$ is open iff its preimage under the projection is open.
*Show that every open set in $\mathbb{R}$ contains a representative of every residue class.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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Fractal curves area I was reading about the Sierpinski curve on the wiki page and it says that, considering the sequence of Sierpinski curves $S_n$ such that $\lim\limits_{n\rightarrow \infty}S_n$ completely fills the unit square, the limit of the area enclosed by those curves is of $\frac{5}{12}$.
I cannot think how... | When we say that
$$\lim_{n\to\infty} S_n = I^2 \text{ (the unit square)},$$
we are referring to a particular notion of distance. In particular, the set $S$ is "close" to the set $T$ if every point of $S$ is close to some point of $T$ and vice-versa. (This is really an intuitive description of the Hausdorff metric.)
Th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there a non-homogeneous ideal whose radical is homogeneous? Let $I$ be an ideal, and let $\sqrt{I}$ denote its radical (intersection of all primes containing $I$).
One has that (1)(2): $I$ homogeneous $\implies $ $\sqrt{I}$ is homogeneous.
Question: Does one also have that: $\sqrt{I}$ homogeneous $\implies$ $I$ is ... | Take $R=k[x,y]$, polynomial ring in two variables with the usual grading and let $I=(x+y^2, y^3)$. Then $I$ is not homogeneous, but $\sqrt{I}=(x,y)$ is.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Picture about what's going on in a proof for knot theory In Lickorish's 'An Introduction to Knot Theory' after Proposition 6.3 one reads:
Now suppose that $F$ is a Seifert surface for an oriented link $L$ in $\mathbb{S}^3$, so that $\partial F=L$. Let $N$ be a regular neighbourhood of $L$, a disjoint union of solid to... | Take $L$ to be the unit circle sitting in some plane in $\mathbb{R}^3$, and $F$ to be the unit disk in that plane. Then $N$ is a solid torus (thickened $L$) which "cuts" into $F$ near the boundary $\partial F = L$. That's the visual to see how the collar neighborhood of $\partial F$ sits inside $N$. Because $N$ has a r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Method to solve $|x| + |2-x| \leq x+ 1$ Even if it seems really easy, I'm struggling to solve $$|x| +|2-x |\leq x+1.$$
The book says that $ x \in [1,3] $.
I first rewrote as $x+(2-x)\leq x+1$
with $x\geq 0$ and $-x-(2-x)\leq x+1$ with $x<0$. Then I solved.
For the first, I got $1\leq x$ and for the 2nd, $-3\leq x$
$$x... | The mistake you have made is that,you have taken
$|2-x| = 2-x \forall x\ge 0$, which is not true.
For $x \gt 2$, $|2-x|$ is $ = x-2$.
Hence, you must take intervals as three cases, as George Law points out,
$ (i)x\le 0$,
$(ii) 0\lt x\le2$
$(iii) x>2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2285369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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A linear operator on the complex Lebesgue space $L^2$ Let $X$ be the complex Lebesgue space $L^2(0,1)$. Let $T:X\to X$ be
$(Tf)(x)=x\int_0^1 \int_0^r f(s)\ ds\ dr-\int_0^x\int_0^r f(s)\ ds\ dr$
Show that $|Tf(x)|\leq \lVert f \rVert$ for any $0\leq x \leq 1$ and $f\in X$.
By writing $|Tf(x)|\leq |\int_0^1 \int_0^r f(s)... | We can write
$$Tf(x) = (x-1)\int_0^x \int_0^r f(s)\,ds\,dr + x\int_x^1\int_0^r f(s)\,ds\,dr.$$
To estimate each part, we use the Cauchy-Schwarz inequality, giving
$$\Biggl\lvert \int_0^r1 \cdot f(s)\,ds\Biggr\rvert \leqslant \sqrt{r}\cdot \lVert f\rVert,$$
and hence
\begin{align}
\Biggl\lvert(x-1)\int_0^x\int_0^r f(s)\... | {
"language": "en",
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Solve Quadratic Congruence Equation How to solve $3x^2 - 5x + 5 \equiv 0 \pmod 7$? In general, how to approach this kind of problem? Any help is appreciated.
| As the characteristic isn't two you can directly use the quadratic formula:
$$\Delta=(-5)^2-4\cdot3\cdot5=25-60=-35=0\pmod 7\implies\text{ there's a single double root:}$$
$$x_{1,2}=\frac{5}{2\cdot3}=\frac{5}{-1}=-5=2\pmod 7$$
Another way: Factor out the quadratic coefficient and try to complete the square:
$$3x^2-5x+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Does $\int_{-\infty}^{\infty} \sin(t) \,dt $ converge? Does $\int_{-\infty}^\infty \sin(t) \,dt $ converge or diverge? How would I prove it?
Should I use 'principle value' to do:
$$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt$$
| $$\int_{-a}^a \sin(t)\,dt=0$$
for $a>0$ since sine is an odd function. Hence
$$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt=0.$$
| {
"language": "en",
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"source": "stackexchange",
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Jacobian question -
I am stuck on a question in my calc III class which is shown above (part a). I completely understand how to find the Jacobian; however I don't understand why the relationship shown is true. How do I find the inverse transformation?
In terms of the Jacobian, I got $$J = \frac{1}{2u}$$.
Thanks for yo... | Hint: To find J(u, v) you wrote x and y in terms of u and v in preparation to do the differentiation.
For the inverse you first need to figure out how to express u and v in terms of x any y.
| {
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Differentiating $e^x + e^y = e^{x + y}$ Differentiating with respect to x, $e^x + e^y = e^{x + y}$, could anyone give me a hint? I do not know even how to start, taking the ln of both sides does not solve the problem.
| You didn't provide enough information, so I am going to assume that you are taking the derivative with respect to $x$, that $y = y(x)$, and that are familiar with the chain rule.
Let $f(x) = e^x$, so $f'(x)= e^x $
$\frac{d}{dx}f(y(x)) = f'(y(x))y'(x) = e^y \frac{dy}{dx} $
$\frac{d}{dx}f(x+y(x)) = f'(x+y(x))(1 + y'(x))... | {
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Proof verification of an elementary exercise in abstract algebra Problem: let $x\in G$.
$|x|=n$, the order of $x$, is an odd number.
Now prove that $x^i\neq x^{-i}$ for all $i=1,2,\ldots, n-1$.
My attempt: Let's assume $|x|=n=2k+1$ for some integer $k>0$. Suppose otherwise, that $x^i=x^{-i}$ for some $i=1,2,\ldots,$ or... | That works. A slightly faster (but essentially the same) way to see it is as follows. Suppose $|x| = n$ odd, and suppose $x^i = x^{-i}$ for some $i \in \{1,\dots,n-1\}$. Then $x^{2i} = 1$. Now since $|x| = n$ we must have $2i > n$ (since $n$ is odd, we can't have equality). But in addition, $2i < 2n$ by assumption, so ... | {
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Find number of zeros of $\sin \pi x$ on a domain $D=\{|z-3-4i|<6\}$. I am going to take written exam in complex analysis in a week.
Among sample problems I found the following one:
Find $\lim \limits_{n \to \infty}N_{P_n}(D)$ where $N$ is number of zeros of $P_n$ on a domain, $D=\{|z-3-4i|<6\}$ and $$P_n = \sum_{k=0}... | Notice that $\sin(\pi x)$ only has zeroes when $x$ is a real integer (source), so you just need to figure out how many real integers are in the open disk you've mentioned.
Similarly for $\cos(\pi x)$ have a zero only when $x+\frac12$ is an integer, so you need to check how many of these are in the disk.
For $e^{ax}$ ju... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Most elegant way to proof that the $\ell_1$-norm of a unit vector is larger equal the $\ell_2$-norm of it I was wondering which proof method can be used best to show that the $\ell_1$-norm of a unit vector is larger equal the $\ell_2$-norm of it?
| Consider
$$ \sum_{k=1}^n|x_n| = \left|\sum_{k=1}^n|x_n|\right| = \left(\left(\sum_{k=1}^n |x_n|\right)^2\right)^{1/2} \geq \left(\sum_{k=1}^n |x_n|^2\right)^{1/2} = \left(\sum_{k=1}^n x_n^2\right)^{1/2}, $$
because $y^2 = |y|^2$. If you expand the expression for $\left(\sum_{k=1}^n |x_n|\right)^2$ you get all of $\sum... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to calculate this complex number expression How do I calculate this term?
$|(\frac{\sqrt{3}}{2}-\frac{1}{2}i)^{15}|$
I've started by transforming in into polar form:
$|{e^{i(\frac{11}{6}\pi)}}^{15}|$
How do I go on from here?
| As @5xum stated, your calculation method is inefficient. However, it is still a valid way to do so.
Continuing from your progress, I think you meant
$$|{e^{i(\frac{11}{6}\pi)\cdot 15}}|=|{e^{i(\frac{55}{2}\pi)}}|=^*|{e^{-i(\frac{\pi}{2})}}|=|-i|=?$$
where $(*)$ follows from the fact that $e^{i\theta}$ is $2\pi-$perio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find the derivative of $y=x^x$. Find the derivative of $y=x^x$.
My Attempt:
$$y=x^x$$
Taking $\textrm {ln}$ on both sides, we get:
$$\textrm {ln} y= \textrm {ln} x^x$$
$$\textrm {ln} y = x \textrm {ln} x$$
How do I procees further?
| Here is an alternative way to do it, without the use of implicit differentiation:
One can start by writing:
$$y=x^x=(e^{\ln{x}})^x=e^{x\ln{x}}$$
Hence, it now becomes extremely obvious to apply the chain rule to obtain:
$$\frac{dy}{dx}=\frac{d}{dx}(e^{x\ln{x}})=\color{green}{\frac{d}{dx}(x\ln{x})}\cdot e^{x\ln{x}}$$
N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2286600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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1999 Putnam A-5 as a direct result of Archimedean Property I was working on question A-5 from the 1999 Putnam Exam:
Prove that there is a constant C such that, if $p(x)$ is a polynomial of degree 1999, then
$|p(0)|≤C\int_{-1}^1|p(x)|dx$
This seemed like a direct consequence of the Archimedean Property of the reals to m... | One way to do this is to let $\mathcal P$ denote the set of polynomials of degree $\le 1999.$ Then $\mathcal P$ is a vector space (say over $\mathbb R$) of dimension $2000.$ On this vector space we can introduce two norms:
$$\|p\|_1 = \sum_{k=0}^{1999} |p(k)|,\,\, \|p\|_2 = \int_{-1}^1 |p(x)|\, dx.$$
It's easy to check... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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exisitance of inf Let $f:[0, \infty] \rightarrow [0, \infty]$ be a strictly increasing (therefore one-to-one) but not onto function, with $f(0)=0$. For some $c >0$, I want to prove that $\inf\{ x \mid f(x) \ge c\}$ always exists. How do I prove this? We know that 0 is not in the set $\{ x \mid f(x) \ge c\}$ but how can... | If $\{ x \mid f(x) \ge c\} \neq \emptyset$ then its infimum exists, since it is a bounded-below subset of $R$.
Note that $x=0$ is a lower bound for the set.
| {
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Reasoning in an integration substitution
Evaluate: $\displaystyle\int_{0}^{1}\frac{\ln(x+1)}{x^2+1}\,\mathrm{d}x$
So I did this a completely different way than what the answer key states. I used integration by parts and some symmetry tricks and got the correct answer. However the answer key says:
Make the substitut... | Here's how I would do it. Let $$f(\alpha)=\int_0^1 \frac{\log(1+\alpha x)}{1+x^2}\ dx$$ where $f(1)$ is the integral we seek to evaluate.
By differentiation under the integral sign we have $$f'(\alpha)=\int_0^1\frac{\partial}{\partial\alpha}\frac{\log(1+\alpha x)}{1+x^2}\ dx\\ =\int_0^1\frac{x}{(1+x^2)(1+\alpha x)}\ dx... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Integral limit without using Taylor expansion I’m trying to compute
$$\lim_{y \to 1-} (1-y + \ln (y))\int_0^y \frac{dx}{(x-1) \ln(x)}$$
I was able to show with Taylor series that this converges to 0, but it was tedious .
Is there a more elegant way to do this perhaps using upper and lower bounds?
Thank you.
|
Herein, we present a solution that relies only on elementary inequalities and the squeeze theorem. To that end we now proceed.
Let $I(y)$ be the integral given by
$$I(y)=\int_0^y \frac{1}{(x-1)\log(x)}\,dx\tag 1$$
for $y\in (0,1)$.
ASIDE:
In THIS ANSWER, I showed using only the limit definition of the exponential... | {
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partial differential equation with polar coordinate i have difficultises to resolve the following problem. Thank you for the help.
We consider the heat equation
$$
\dfrac{\partial u}{\partial t}= c^2 (\dfrac{\partial^2 u}{\partial x^2}+ \dfrac{\partial^2 u}{\partial y^2})
$$
1. Write this equation using the polar coord... | The first question boils down to express the laplacian in polar coordinates:
$$\triangle=\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial\theta^2}.$$
In order to establish that simply recall that the change of variable is the following:
$$(r,\theta)\mapsto (... | {
"language": "en",
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Origin of vectors Background
I was reviewing notes of physics, and i realized that something about the mathematics of vectors was wrong in my head.
Example-problem
Suppose a vector is $A=5\textbf{i} + 3\textbf{j}$, and other $B=7\textbf{i}+3\textbf{j}$. Then $A-B=C=-2\textbf{i}$.
The question is: Why C is not placed ... | Notice the calculation for $B-A$ is not correct in the question.
$$B-A = (7i -3j) - (5i + 3j) = (7-5)i + (-3 -3)j = 2i - 6j$$
So $i$ represents a unit vector in $x$ direction, and $j$ represents a unit vector in $y$ direction.
So what $B-A$ means is that it has component in $x$ with $2$ of unit vectors, and component i... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Evaluating $\frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3-x}\right)$ I am given that $\dfrac{d^{100}}{dx^{100}}\left(\dfrac{p(x)}{x^3-x}\right) = \dfrac{f(x)}{g(x)}$ for some polynomials $f(x)$ and $g(x).$ $p(x)$ doesn't have the factor $x^3-x$ and I need to find the least possible degree of $f(x)$.
My Attempt: I am de... | Wrong. For example, if $p(x) = 1$, $f(x)$ has degree $200$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2287350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Local maximum definition What is the Analyse definition of local maximum point
I am not talking about this :
If a function f is defined for all $
\hspace{0.33em}{x}\mathrm{\in}{I}
$ Not indeed the full definition set
/ I mean ${I}$ can be a subset of the definition set of the function f/
then if $
{f}{\... | Let $f:D\rightarrow \mathbb{R}$ be a function with domain $D$.
When we say that $x_0$ is a local maximum (minimum) of $f$, intuitively, we want a neighborhood of $x_0$ satisfying that $f(x_0)$ gives the maximum value in that neighborhood. So here we need to think about what a neighborhood means. From the point of view ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find the direction vector of a ball falling off an ellipsoid?
A tiny ball is placed in top of an ellipsoid $3x^2+2y^2+z^2=9$ at $(1,1,2)$. Find the three-dimensional vector $\underline u$ in whose direction the ball will start moving after the ball is released.
I feel this problem involves usage of gradients b... | I will assume that z-axis is oriented vertical upwards.
$f(x,y,z) = 3x^2+2y^2+z^2 - 9 = 0$
Vector $\vec{N}$ perpendicular to the surface $f$ at point $(x,y,z)$ is defined by function gradient:
$\vec{N} = \frac{\partial f}{\partial x} \vec{i} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial y} \vec{j} + \fra... | {
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Evaluating the integral $\int_0^\infty \frac{e^{-(t+\frac{1}{t})} dt}{t}$ How can the closed form of the following integral be evaluated?
$$\int_0^\infty \frac{e^{-(t+\frac{1}{t})} dt}{t}$$
I could not find any substitution nor factorisation that can be applied here.
I am not getting any idea regarding the evaluation... | To elaborate on the answer in the comments, substitute $t = e^u$, giving
$$\int_0^\infty \frac{e^{-(t+\frac 1t)}}{t} dt = \int_{-\infty}^\infty e^{-(e^u + e^{-u})} du \\ = 2\int_0^\infty e^{-2\cosh u} du \\ = 2K_0(2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2287729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
Find the general solution of recurrence relation of order four Find the general solution of the recurrence relation.
$a_n = 4a_{n - 1} - 5a_{n - 2} + 4a_{n - 3} - 4a_{n - 4}$
I've found the roots, but I don't really understand how does general solution look like.
$$r^n = 4r^{n - 1} - 5r^{n - 2} + 4r^{n - 3} - 4r^{n - 4... | You have a root of $2$ with multiplicity $2$, one root of $-i$, and one root of $i$. Therefore:
$$a_n = c_1 2^n + c_2 n 2^n + c_3 (-i)^n + c_4 i^n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2287921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Confused by combinatorical reasoning (n functional antennas, m defective problem) This is an example question and solution straight out of "A First Course in Probability" by Sheldon Ross, on page 6 (fyi: all that's covered till this point in the book is the basic and generalized principles of counting, counting orderin... | Suppose there are a total of $7$ antennas, $2$ of which are bad. That means $n=7$, $m=2$, so the number of good antennas is $n-m=5$. We line up the five good ones, and ask where the bad ones can go:
__ G __ G __ G __ G __ G __
Each of the places where there's a line is an available spot for one (and no more than one!) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which goes through $(3,1)$ and has the minimal area?
How to find the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which goes through $(3,1)$ and has the minimal area? Ellipse area is given as $\pi ab$.
My approach is to use Lagrange method where the constrain... | We have the general equation of an ellipse:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$
whose area is:
$$ f(a,b) = \pi a b$$
We want to minimize the area $f(a,b)$ subject to the constraint that the ellipse passes through the point $(3,1)$, that is:
$$ g(a,b) = \frac{9}{a^2} + \frac{1}{b^2} = 1 $$
Applying the method of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is $E\left(\frac{X}{X+Y}\right)=\frac12$ when $X$ and $Y$ follow the same probability distribution? I had an exercise regarding random variables, and I tried to figure if the following is true:
If $X$ and $Y$ follow the same probability distribution, then $E(\frac{X}{X+Y})=E(\frac{1}{2})=\frac{1}{2}$.
I'm skeptical a... | Even if $X$ and $Y$ are independent, the result does not necessarily hold.
As a simple example, if $X$ and $Y$ are IID standard Normal, then $E[X]=E[Y]=0$, but $E[\frac{X}{X+Y}]$ is not equal to $\frac12$ (the expectation does not converge).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Some inequality with complex numbers Is there a straightforward way to prove the following inequality:
$$|1 + k\big(\exp(it)-1\big)|\leq 1 $$
where $k\in(0,1)$ and $t \in \mathbb{R}$ (correction, see dxiv answer) with $|t| \leq 1$, other than writing the quantity into its real and imaginary parts and checking that they... | The inequality does not hold true for $t \in \mathbb{C}$ with $|t| \leq 1\,$ in general, take for example $t=-i$ then $\,e^{it}=e\,$ and $\,|1 + k (e-1)|=1+k(e-1) \gt 1\,$ since $\,e \gt 1$ and $k \gt 0\,$.
Assuming $t \in \mathbb{R}\,$, instead, let $z=e^{it}\,$, then $|z|=1\,$ and:
$$
\begin{align}
|1 + k (z-1)|^2 &=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to express result of derivative of a matrix? For example, if we are asked to find $f'(x)$ of $f(x) = e^{x_1 + x_2} , x \in \mathbb{R}^{2} $ then our answer would have two partial derivatives. Would we say $\frac{\partial}{\partial x_1}$ = .. and $\frac{\partial}{\partial x_2}$ = .. ? What if $x \in \mathbb{R}^{d}$ ... | If you have a map $f: \mathbb{R}^m \to \mathbb{R}^n$, the derivative will be the Jacobian, which is a $n$ by $m$ matrix, where the $i,j$-th coordinate contains the partial derivative of the $i$-th coordinate of $f$ with respect to the $j$-th variable.
If $n=1$, you get the gradient (which is a vector).
An aside: Matri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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There exists a $2 \times 2$ matrix $R$ such that $r = R v$ for all 2-dimensional vectors $v$. Find $R$. For a vector $v$, let $r$ be the reflection of $v$ over the line
$$x = t \begin{pmatrix} 2 \\ -1 \end{pmatrix}.$$
There exists a $2 \times 2$ matrix $R$ such that
$$r = R v$$
for all 2-dimensional vectors $v$. Find $... | If $u$ is the projection of $v$ onto $w$, the reflection of $v$ over $w$ is given by $2u-v$. See the diagram below.
Hence $$v'=2u-v=2\begin{pmatrix}\frac{4}{5}&-\frac{2}{5}\\-\frac{2}{5}&\frac{1}{5}\end{pmatrix}v-\begin{pmatrix}1&0\\0&1\end{pmatrix}v = \begin{pmatrix}\frac{3}{5}&-\frac{4}{5}\\-\frac45&-\frac35\end{pma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show, without invoking the Pythagorean theorem, that the $3-4-5$ triangle is right The ancient Egyptians knew the $3-4-5$ triangle was a right triangle, but they did not possess the Pythagorean theorem or any equivalent theory. Can it be shown that the $3-4-5$ triangle is a right triangle without using the Pythagorean... | Take a stick length $3$ and another length $4$. Place these at right-angles to each other. Create a stick to measure the diagonal. Demonstrate that this stick plus the $3$-stick is the same length as two $4$-sticks. (You can create a $3$-stick from a $4$-stick by bisecting a $4$-stick, appending, and bisecting again).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Prove that, for $p> 1$, $\lim_{n\to\infty} \|f_n\|_p = +\infty$ I need some help with this one, a hint would be greatly appreciated:
Let $f_n \geq 0$ and $f_n \in L^1\quad \forall n \in \mathbb{N}$ such that $\|f_n\|_1 =1$ for all $n$.
Suppose also that for each $\delta >0$: $$\lim_{n \to \infty} \int_{\{|t|>\delta\}}... | You can start from here:
let $\delta > 0$ be fixed. Then
$$
1 = \int f_n = \int_{\{|t|\leq \delta\}} f_n + \int_{\{|t| > \delta\}} f_n
\leq (2\delta)^{1/p'} \left(\int_{\{|t| \leq \delta\}} |f_n|^p\right)^{1/p} +
\int_{\{|t| > \delta\}} f_n
$$
$$
\leq
(2\delta)^{1/p'} \|f_n\|_p +
\int_{\{|t| > \delta\}} f_n.
$$
If $L :... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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convergence of square of a geometric series Given
$ \sum_{j=0}^\infty [2.67(0.8^j) - 1.67(0.5^j)] $
Does the above series converge because the terms are a sum of two convergent geometric series?
How can I prove the square is convergent as well?
$ \sum_{j=0}^\infty [2.67(0.8^j) - 1.67(0.5^j)]^2 < \infty $
Do I expand... | Since the series $$\sum_{j=0}^\infty2.67(0.8^j)$$
is absolutely convergent same applies to
$$\sum_{j=0}^\infty1.67(0.5^j)$$
So is their sum. Also since $$\sum_{j=0}^\infty1.67^2(0.25^j)$$
is absolutely convergent and
$$\sum_{j=0}^\infty(-2)\cdot2.67\cdot 1.67(0.4^j)$$
is also absolutely convergent. And for last
$$\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2288973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Determine $\lim_{n\to \infty} \frac{n}{\sqrt[n]{n!}}$ and $\lim_{n\to \infty}[ \frac{n+\sum_{k=1}^{n-1}\log(k)}{\log(n)} +1 -n]$ I want to use Stirlings Formula $\lim_{n\to \infty} {n!} \sim \lim_{n \to \infty} \frac{n^n}{e^n}\cdot \sqrt{n}$ to evaluate the following limits:
$$\lim_{n\to \infty} \frac{n}{\sqrt[n]{n!}}$... | If you are unsure about what you are allowed to do with equivalents, then go back to the equivalent characterization as first-order Taylor expansion (in particular, when you write Stirling's formula, you should not have $\lim$ on either side: this is mixing two different notions; you are also missing a constant):
$$
n!... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How to integrate a squared derivative for the swing equation? While going to some literature of Power System Stability for generators I came across a derivation of the power angle when a fault occurs from the swift equation.
The Swing Equation is:
$\mathbf{\frac {2H}{w_s}}\frac{d^2\delta}{dt^2}=P_m-P_e $
Where:
$ H$ i... | $$\int{\frac{df(t)}{dt}}dt=f(t)+C,$$
where $C$ is some constant, is the meaning of the integral sign - it is the "anti-derivative", i.e., it "cancels" the operation of differentiation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Forming 4 groups from 16 people How many ways can $16$ students split up into $4$ study groups of size $4$ if (i) each group studies a different topic? (ii) all groups study the same topic?
My Idea : (i) Since with respect to topics we can form groups we see for the first topic there are $\binom{16}{4}$ ways next there... | You are very close. The difference between (i) and (ii) is that the order of the four groups no longer matters (since they are all studying the same topic), so you need to divide by another 4!. Hence the answer to (ii) is $\dfrac{\binom{16}{4}\binom{12}{4}\binom{8}{4}}{4!}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Division notation in ring Let $a,b\in R$ where $R$ is a ring.
Is there any problem/ambiguity with this notation $\frac{ab}{a}=b$, where $a\neq 0$?
What are the minimal conditions (e.g. commutative, UFD, etc) that we need such that the above makes sense?
Thanks for any help.
| In domains (for instance polynomial rings over domains) this is used quite frequently: one often finds an expression such as
$$
\frac{X^n-1}{X-1}
$$
to denote $X^{n-1}+X^{n-2}+\dots+X+1$ or, more generally,
$$
\frac{f(X)}{X-a}
$$
where $f(X)$ is a polynomial and $a$ is a root thereof, so this denotes a well defined pol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How many different ways can we place 4 identical rooks on the following chess board How many different ways can we place 4 identical rooks on the following chess board so that no two of them attack each other?
I know when the board is $n \times n$ and we have $n$ towers then the solution is simply $n!$
However I am no... | Just to verify that Bram28's answer is correct, we can count it up by building the graph whose 32 vertices are the positions on the board, with edges for rook attacks. Then count the number of independent sets of size four.
Here's Sage code to do that:
from sage.graphs.independent_sets import IndependentSets
posns = [... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $T(n) = T(n/3) + T2n/3) + 5n$ is $O(n log n)$ I'm doing some research about time complexity of algorithms and stumbled upon the following problem that I'm not able to solve:
Let $T(n) = T(n/3) + T(2n/3) + 5n$. prove that $T(n) = O(n log n)$
First, I made a recursion tree, which is the same as the one in the ... | If you wish to use the recursive tree approach instead:
First level work: $5n$
Second level work: $5n/3 + 10n/3 = 5n$
Third level work: $5n/9 + 10n/9 + 10n/9 + 20n/9 = 5n$
And so on and so on.
In other words, each complete level has total work $5n$. Every leaf in the recursion tree has depth between $\log_3(n)$ and $\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Is the cardinality of the union of a chain of sets of cardinality $2^{\aleph_0}$ still $2^{\aleph_0}$? I have a simple question that I haven't been able to prove but I think is true, I hope you can help me. Suppose I have the POSET $(B,\subseteq)$ where each element of B has cardinality of $2^{\aleph_0}$. If $C$ is a c... | Nope; this is false with any infinite cardinal in place of $2^{\aleph_0}$. Here's a quick and easy proof using Zorn's lemma. Let $\kappa$ be an infinite cardinal and let $X$ be any set of cardinality greater than $\kappa$. Let $B$ be the set of all subsets of $X$ of cardinality $\kappa$. By Zorn's lemma, the poset ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Extreme Points of a Convex Hull Given an arbitrary set $A\subset\mathbb R^n$, why do all extreme points of the convex hull $\operatorname{conv}(A)$ lie in $A$?
(An extreme point of a convex set is defined as one that cannot be written as a strictly convex combination of two distinct points of this set.)
| An element $x\in\operatorname{conv}A$ if and only if there are a positive natural number $n$, elements $x_1,\cdots,x_n\in A$ and non-negative real numbers $t_1,\cdots,t_n$ such that $$\begin{cases}x=\sum\limits_{k=1}^n t_kx_k\\ \sum\limits_{k=1}^n t_k=1\end{cases}$$
We can assign to each $x\in\operatorname{conv}A$ the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
to show this function does not belong to the uniform closure of A Problem no: $21$
Chapter $7$: Sequence and series if functions
Book: Principles of mathematical analysis
Writer : W. Rudin
Let $K$ be the unit circle in the complex plane and $A$ be the algebra of all functions of the form
$f( e^ {ia}) = \sum_{n=0}^{... | Prove that if $f$ is an element of the uniform closure $A$ then $\int_0^{2\pi}f(e^{it})e^{it}\,dt=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2289999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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