Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Countably additive property of Lebesgue measure I am farily new to the world of measure theory, and I am working on the Billingsey (third edition).
At page 23 the field $B_0$ is defined as the field of finite disjoint union of subintervals of $\Omega$. However, to prove that the Lebesgue measure is countably additive ... | The operative phrase here is suppose that. In general $B_0$ will not be closed under taking countable unions; it's not a $\sigma$-algebra and nobody is claiming that it is. But it could happen that there is some sequence of disjoint sets $A_1, A_2, \dots$ from $B_0$, such that their union $A = \bigcup_{k=1}^\infty A_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1514798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Pointwise convergence implies $L^p$ Simply, why is it that convergence pointwise, $u_j \rightarrow u$, implies convergence in $L^p$ if $|u_j(x)| \le g(x)$ for some $g$ in $L_+^p$?
| Simply, Lebesgue's dominated convergence theorem. "Domination" by $g$ is necessary: a traveling square wave $f_n=\chi_{[n,n+1]}$ converges pointwise to zero but has norm 1 for all $p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1514956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Riemann like sums for Lebesgue integrable function Let $f(x)$ be a non-negative function on $\mathbb{R}$ such that $\int_{-\infty}^{\infty}f(x) \ dx=1.$ Actually, f(x) is a probability density function for a continuous random variable.
Can I justify that
$$
\lim_{n \rightarrow \infty} \sum_{m \in\mathbb{Z}} \frac{1}{n... | This is not true even for continuous nonnegative $f.$ For $N=1,2,\dots,$ we define $f$ on the disjoint intervals $[N-1/(N^2e^N),[N+1/(N^2e^N)]$ to be an isosceles triangular spike of height $e^N$ centered over this interval. Define $f$ to be $0$ everywhere else. Then $f$ is continuous everywhere, and
$$\int_{-\infty}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1515086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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proving the existence of a complex root If
$x^5+ax^4+ bx^3+cx^2+dx+e$ where $a,b,c,d,e \in {\bf R}$
and
$2a^2< 5b$
then the polynomial has at least one non-real root.
We have $-a = x_1 + \dots + x_5$ and $b = x_1 x_2 + x_1 x_3 + \dots + x_4 x_5$.
| We will prove the slightly stronger statement
Claim: Let
$$
p(x) = x^n + ax^{n-1} + bx^{n-2} + \cdots + c
$$
be a polynomial with real coefficients. If $n=2,3,4,5$ and $2a^2 < 5b$ then $p$ has at least one non-real zero.
If $n \geq 6$ then $p$ may have all real zeros regardless of whether $2a^2 < 5b$ or $2a^2 \geq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1515227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Any finite-dimensional vector space is the dual space of another Any finite-dimensional vector space is the dual space of another?
This is from a true/false section in my book and the statement is supposedly true. I can't say I see the reasoning behind this and any hint/direction is greatly appreciated. My intuition sa... | For a finite dimensional inner-product space $V$, we can define any linear functional $f \in V^{\ast}$ as $f(w) = \langle v,w\rangle$ for a unique $v \in V$, this is often written as $f = \langle v,-\rangle$. For example, given the basis $\{e_1,\dots,e_n\}$ (the standard basis) we have the dual basis: $\{\pi_1,\dots,\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1515352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does cancelling change this equation? The equation
$$2t^2 + t^3 = t^4$$
is satisfied by $t = 0$
But if you cancel a $t^2$ on both sides, making it
$$2 + t = t^2$$
$t = 0$ is no longer a solution.
What gives? I thought nothing really changed, so the same solutions should apply.
Thanks
| No such thing as cancelling actually exists. What you do in those moments is to divide both sides of an equation by the same value. This means, cancelling like this:
$2t^2 + t^3 = t^4$ to $2t + t^2 = t^3$
Is actually a shortcut for:
$2t^2 + t^3 = t^4$ to $(2t^2 + t^3)/t = t^4/t$ to $2t + t^2 = t^3$
And you can only do ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1515492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
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"answer_id": 2
} |
Prove that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides Prove,by vector method,that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides.
My Attempt:
Let the... |
The trapezium's diagonals intersect at $X.$ We want to show that $B,X,\text{ and }R$ are collinear.$$ $$
Triangle $XOP$ is similar to triangle $XQA.$ So $$\frac{XO}{XQ}=\frac{OP}{QA}\\=\frac1\lambda.$$
Now, $$\vec{BX}=\vec{OX}-\vec{OB}\\=\frac1{\lambda+1}\vec{OQ}-\vec{OB}\\=\frac1{\lambda+1}\left(\mathbf{a}+2\lambda\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1515600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Solve equations like $3^x+4^x=7^x$ How can I solve something like this?
$$3^x+4^x=7^x$$
I know that $x=1$, but I don't know how to find it. Thank you!
| Defining $f(x):=\mathrm{log}_7\left(3^x+4^x\right)$, you want to search for fixed points of $f$. But
$$
f^\prime(x)=\frac{1}{\ln 7}\cdot \frac{3^x\ln 3+4^x \ln 4}{3^x+4^x}>0
$$
and
$$
f^{\prime\prime}(x)=\frac{1}{\ln 7}\cdot \frac{3^x4^x}{(3^x+4^x)^2} \cdot ((\ln 4)^2+(\ln 3)^2-2\ln 3 \ln 4)
$$
which is positive too by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1515776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
on a continous function dominated by a certain function and taking certain values In a book by Charalambos D. Aliprantis and Owen Burkinshaw, Positive Operators, in page 13, Example 1.16, there is a statement which I cannot see why it is always true. The statement is as follows:
Take $0<f∈C[−1, 1]$, and let $0<c<2π$. A... | Let $h_n$ be any non-negative continuous function such that $h_n(\sin(c+t_n))=0$ and $h_n(\sin c)=f(\sin c)$, for instance
$$
h_n(x)=\frac{|x-\sin(c+t_n)|}{|\sin c-\sin(c+t_n)|}\,\,f(\sin c).
$$
Then take $g_n(x)=\min(f(x),h_n(x))$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1515875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Solve $ \left|\frac{x}{x+2}\right|\leq 2 $ I am experiencing a little confusion in answering a problem on Absolute Value inequalities which I just started learning. This is the problem: Solve:
$$
\left|\frac{x}{x+2}\right|\leq 2
$$
The answer is given to be $x\leq-4$ or $x \geq-1$
This is my attempt to solve the probl... | the case $x+2>0$, you do not need to solve $-1/2\le 1/(x+2)\leftarrow x+2>0$.
the case $x+2<0$, the inequality you solve it's not the original one.
It should be $-1/2 \le 1/(x+2) \le 3/2$ and in case $x+2<0\rightarrow 1/(x+2)<3/2$, so it's just $-1/2 \le 1/(x+2)$.
Think it simply.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1515986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Prove that taking the orthogonal projection of a vector is continuous Let $Y$ be a closed subspace of the Hilbert space $X$, and define $T:X\to Y$ as
$$Tx = Proj_Y x$$
Then I want to check that $T$ is continuous. So I did the following, I took a converging sequence $(x_n) \in X$ with limit $x$, then I want to prove tha... | Hint (for a complete proof of the result).
I suppose that you define $Tx = Proj_Y x=p_Y(x)$ as the point at smallest distance of $x$ belonging to $Y$. By the Hilbert projection theorem, $p_Y(x)$ exists and is uniques as $Y$ is supposed to be closed.
Now you can prove that a point in $Y$ is equal to $p_Y(x)$ if and only... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1516084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Optimization of a split cone This was a thought that I've been pondering on for several days now, however I can not seem to optimize using calculus correctly in relation to the following statement. Any help would be greatly appreciated.
Assume you have 3 dimensional cone with a perfectly circular base.
Let the height o... | The problem itself is not very difficult. I still don't have enough reputation to comment, so I'm going to suggest this as an answer: define surface area as a function of $h$ and use pythagoras' theorem; also, consider that if, and I'm taking a guess here because it is unclear in your statement, the ring touches the ed... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1516220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is it possible that $A\subseteq A\times B$ for some non empty sets $A,B$? I was wondering if there exist two non empty sets $A,B$ such that
$$A\subseteq A\times B.$$
I know that always exists a subset of $A\times B$ with the same cardinality of $A$, but i'm requesting here that $A$ is a subset of $A\times B$ without us... | An element of $\Bbb N^{\Bbb N}$ is an infinite sequence of natural numbers. Buy what do we mean by a sequence. $n_0,n_1,\ldots n_k,\ldots$ is really nothing more than a function $f:\Bbb N\to\Bbb N$. For example, $f(k)=n_k$.
So, $\Bbb N^{\Bbb N}=\{f:\Bbb N\to\Bbb N\}$. So, what is an element of $A\times\Bbb N$? It ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1516538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Permutations of bit-sequence(discrete math) How many bit-squences with length 8 has 1 as it's first bit and 00 as the two last bits(e.g $1011 1100$)
I thought the solution to this problem would be $1 * 2 * 2 * 2 * 2 * 2 * 1 * 1 = 2^5$, but my teacher proposed the following solution:
amount that starts with 1: $1 * 2 *... | Yes, the answer is simply $2^5$. Your teacher is calculating some other count.
OK, your teacher is calculating the count of those sequences which start with 1 or end with 00. And she's using the inclusion exclusion principle for counting them.
Inclusion-Exclusion Principle
Maybe you misunderstood the problem. You t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1516640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Twelve people travelling in three cars - clarification on existing answer Please refer this question
Twelve people travel in three cars, with four people in each car. Each
car is driven by its owner. Find the number of ways in which the
remaining nine people may be allocated to the cars. (The arrangement
within... | Yes, you are right.
The mistake I did was to take extra care of the dissimilarity of the cars, thus causing repetitions.
Please check the updated answer, thanks :-)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1516754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How many bit strings of length 10 either begin with three 0s or end with two 0s? The question : How many bit strings of length 10 either begin with three $0$s or end with two $0$'s?
My solution : $0$ $0$ $0$ X X X X X $0$ $0$ = $2^5 = 256$
editing** I noticed the word"or" so I changed the solution to
$2^7$ (three $0$'... | These are the binary words $000x$ ($2^7$ many $x$) and $y00$ ($2^8$ many $y$) minus $000z00$ ($2^5$ many $z$). Looks good.
But I calculate $352$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1516842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Solve $z^3=(sr)^3$ where $r,s,z$ are integers? Let $x,y,z$ be 3 non-zero integers defined as followed:
$$(x+y)(x^2-xy+y^2)=z^3$$
Let assume that $(x+y)$ and $(x^2-xy+y^2)$ are coprime
and set $x+y=r^3$ and $x^2-xy+y^3=s^3$
Can one write that $z=rs$ where $r,s$ are 2 integers?
I am not seeing why not but I want to be s... | Yes.
If $z^3 = r^3s^3$ we can take cube roots of both sides to get $z = rs$. It's valid to do this because $n$ is the only solution to $({n^3})^{1/3}$ for real $n$. If we go to complex numbers we have to be more careful because cube roots have 3 solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1516925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Complex integral computation Evaluate$$ \oint f\left( z \right )dz$$
where C is the unit circle
$$\left | z \right |=3$$
$$f\left ( z \right )=\frac{1}{z^{3}+2z^{2}}$$
Let
$$z=3e^{it}$$
$$z^{3}=27e^{i3t}$$
$$dz=3ie^{it}$$
$$2z^{2}=18e^{i2t}$$
Substituting everything into the initial equation and reducing, we get $$\i... | One can use the Residue Theorem to show that the value of the integral is zero. If one wishes to proceed directly, then we have
$$\begin{align}
\oint_{|z|=3} \frac{1}{z^3+2z^2}\,dz&=\frac14 \oint_{|z|=3}\left(\frac{1}{z+2}+\frac2{z^2}-\frac{1}{z}\right) \,dz\\\\
&=\frac i4\int_0^{2\pi}\frac{3e^{it}}{3e^{it}+2}\,dt+\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1517002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Estimates for a product involving primes Is $$\prod_{p\leq z}\bigg(1 + \frac{p}{(p-1)}\frac{\log z}{\log p}\bigg) = O(z)?$$ where $p$ ranges over the primes less than $z$. I believe I can show that it is $O(z\log \log z)$.
| Let $f(z)=\exp\left((\log z)^{1/2}\right),$
and consider$$\prod_{p\leq f(z)}\left(1+\frac{\log z}{\log p}\right).$$
This is clearly a lower bound for your product, and this is bounded below by $$\prod_{p\leq f(z)}\left(\frac{\log z}{\log f(z)}\right)=\left(\log z\right)^{\pi(f(z))/2}=\exp\left(\frac{1}{2}\pi(f(z))\lo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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At what points does the function $f(z)=Arg(z)$ , $ z \in \mathbb Z-\{0\}$ have a limit? At what points does the function $f(z)=Arg(z)$, $z \in \mathbb Z-\{0\} $ have a limit ?
I know that $-\pi<Arg(z) \leq\pi$. Then how can I use that property to find points where limit exists ?
A where is $f(z)$ is continuous ?
| Note that,
$$
\mathrm{Arg}(z)=\mathrm{Im}\,\log z.
$$
The function $\log z$ is definable in every simply connected domain $\Omega$ which does not contain $z=0$. Hence, $\mathrm{Arg}(z)$ is continuous (and indeed real analytic) in every such $\Omega$.
Conversely, if $\mathrm{Arg}(z)$ is defined, as a continuous function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1517161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Adjoint of an operator on an inner product space Find the adjoint of
$$\left ( Lf \right )\left ( t \right )=t^{2}\frac{df}{dt}+tf$$ where $$f\left ( 0 \right )=1$$ and$$ f'\left ( 1 \right )=2$$
under $$\left \langle f,g \right \rangle=\int_{0}^{1} f\left (t \right )g\left ( t \right )dt$$
Working out I get
$$\int_... | The first term is given by integration by parts, the second one by noticing that
$$(tf(t))g(t) = f(t)(tg(t)).$$
Therefore,
$$\langle tf(t),g(t)\rangle = \langle f(t),tg(t)\rangle$$
so that the operator $f(t)\mapsto tf(t)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1517265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Collection of all open sets whose closure is contained in a cover is a cover (of a regular space) Let $X$ be a regular space. Let $\mathcal{A}$ be an open cover of $X$. We define $\mathcal{B}$ as the collection of all open sets $U$ such that $\overline{U}$ is contained in an element of $\mathcal{A}$. How to prove that ... | For $x\in X$ there is some $V\in\mathcal A$ with $x\in V$ or equivalently $x\notin V^c$.
Set $V^c$ is closed so the regularity of $X$ ensures the existence of open sets $U,W$ s.t. $x\in U$ and $V^c\subseteq W$ and $U\cap W=\varnothing$.
Set $W^c$ is closed, so $U\subseteq W^c$ implies that $\overline{U}\subseteq W^c\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1517352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find complete integral of $(y-x)(qy-px) = (p-q)^{2}$
Find complete integral for partial differential equation $(y-x)(qy-px) = (p-q)^{2}$ where $p={ \partial z \over \partial x},q={ \partial z \over \partial y}$.
My attempt:
The given equation is f(x,y,z,p,q) = $(y-x)(qy-px) - (p-q)^{2}$
The Charpit's auxilary equ... | Hint:
Let $\begin{cases}u=x+y\\v=x-y\end{cases}$ ,
Then $\dfrac{\partial z}{\partial x}=\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial z}{\partial v}\dfrac{\partial v}{\partial x}=\dfrac{\partial z}{\partial u}+\dfrac{\partial z}{\partial v}$
$\dfrac{\partial z}{\partial y}=\dfrac{\partial... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1517446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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How can I rearrange a formula containing both cot and cosec? I want to rearrange the following:
$$\cot{\theta}-\csc{\theta}=\dfrac{2B\mu}{bmg}$$
to get $\theta$.
I am unsure how to reduce a cot and cosec sum. Are there any maths tricks or identities that I am missing?
| $$\cot\theta - \csc \theta = k$$
$$\cot\theta \pm \sqrt{\cot^2 \theta+1} = k$$
$$\pm \sqrt{\cot^2 \theta+1} = k-\cot\theta$$
$$\cot^2 \theta+1 = (k-\cot\theta)^2$$
$$\cot^2 \theta+1 = k^2-2k\cot\theta+\cot^2\theta$$
$$1 = k^2-2k\cot\theta$$
$$\cot\theta = \frac{k^2-1}{2k}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1517518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Show $(m^2+n^2)(o^2+p^2)(r^2+s^2) \geq 8mnoprs$ Let $m,n,o,p,r,s$ be positive numbers. Show that:
$$(m^2+n^2)(o^2+p^2)(r^2+s^2) \geq 8mnoprs$$
I tested an example,
let $m,n,o,p,r,s=1,2,3,4,5,6$ respectively. Then:
$$(1+4)(9+16)(25+30)=(5)(25)(55) \geq 8123456$$
but I get $6875$ which is not greater than the right side.... | Yes, you misunderstood the problem, for $8mnoprs$ has tacit multiplication signs, it does not mean concatenation. Therefore, your test case is $$(1 + 4)(9 + 16)(25 + 30) = 6875 \geq 8 \times 1 \times 2 \times 3 \times 4 \times 5 \times 6.$$ But the first case I would test is the minimal case $m = n = o = p = r = s = 1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1517726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Find $dy/dx$ if $xe^{9y}+y^4\sin(4x)=e^{8x}$ If $xe^{9y}+y^4\sin(4x)=e^{8x}$ implicitly defines $y$ as a function of $x$ then what is $\displaystyle \frac{dy}{dx}$?
So far, I have made the following steps:
1) Get all parts of the equation onto one side
2) Find the derivative of the whole equation
3) This equals $9e^{9y... | Implicitl differentiation means that we need to treat $y$ as a function of $x$ and apply Chain Rule. You also forgot to apply Product Rule to the first term. The derivative of the first term is:
\begin{align*}
\frac{d}{dx}[xe^{9y}]
&= x\left[\frac{d}{dx}e^{9y}\right] + \left[\frac{d}{dx}x\right]e^{9y} \\
&= xe^{9y}\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1517818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Does series with factorials converge/diverge: $\sum\limits_{n=1}^\infty \frac{4^n n!n!}{(2n)!}$? $$\sum_{n=1}^\infty {{4^n n!n!}\over{(2n)!}}$$
I tried the ratio test but got that the limit is equal to 1, this tells me nothing of whether the series diverges or converges. if I didn't make any errors when doing the ratio... | Note that $$\frac{(2n)!}{n!n!}\leq\sum_0^{2n}\binom{2n}{k}=(1+1)^{2n}=4^n$$
So $$\frac{4^n n!n!}{(2n)!}\geq 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1517927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Define $g :\ell_2 \to \mathbb R$ by $g(x)= \sum_{n=1}^{\infty} \frac{x_n}n$. Is $g$ continuous? Define $g :\ell_2 \to \mathbb R$ by
$$g(x)= \sum_{n=1}^{\infty} \frac{x_n}n $$
Is $g$ continuous?
I need to solve this but I could not see how to tackle it? any hints or suggestion?
| Hint: $g$ is linear. A linear operator is continuous if and only if it is continuous at $x=0$, if and only if it is bounded.
Try to find a constant $C>0$ such that for all $x$:
$$
|g(x)|\leq C \|x\|
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1518031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluate the limit of $(f(2+h)-f(2))/h$ as $h$ approaches $0$ for $f(x) = \sin(x)$. If $f(x)=\sin x$, evaluate $\displaystyle\lim_{h \to 0} \frac{f(2+h) - f(2)}{h}$ to two decimal places.
Have tried to answer in many different ways, but always end up getting confused along the way, and am unable to cancel terms.
Any ... | Remember that $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
Here we see that $f(x) = \sin x$ and $x = 2$. Therefore, all we have to do is find $f'(2)$. If you know that the derivative of the sine function is $\cos x$ then you immediately have the answer as $\cos (2)$ (Remember that this is in Radians... all of Cal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1518107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Finding derivative with integral
Came to this problem on my test study guide. I thought for finding the derivative you just took the antiderivative, which would be -cos(theta). How is it sin(x)? Or did I come across another error? At this point In this class I never know anymore.
| $$g(x)=\int_\pi^x\sin\theta\,d\theta=-\cos x+\cos\pi=-\cos (x)-1$$then
$$g'(x)=[-\cos (x)-1]'=\sin x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1518340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Answering Part of Durrett Exercise 3.4.13 4th edition I am trying to prove part (iii) of Durrett exercise 3.4.13
Suppose $P(X_j=j)=P(X_j=-j)=\frac{1}{2j^\beta}$ and $P(X_j=0)=1-j^{-\beta}$ where $\beta>0$. Show that if $\beta=1$ then $\frac{S_n}{n}\implies \aleph$ where $E(\exp(it\aleph))=\exp\left(-\int\limits_0^1 x^... | It is a well-known theorem of Levy that if $\{X_n\}$ is a collection of random variables and $Y$ is another random variable then $X_n \Rightarrow Y$ iff $\phi_{X_n}(t) \rightarrow \phi_Y(t)$ as $n \rightarrow \infty$ and $\phi_Y$ is continuous at $t = 0$. Moreover, by properties of Fourier transforms, $\phi_{S_n/n}(t)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof that group of even order contains element of order 2 I tried to prove the following claim:
If $G$ is a group of even order then $G$ contains an element of order $2$.
Please could someone check my proof and tell me if it is correct?
My proof:
Let $|G|=2n$ for some $n$ and let $\{k_1, \dots, k_{2n}\}$ be the set ... | Consider the set $G-\{e\}$. Its cardinality must be odd because $|G|$ is even.
If no element has order $2$ in $G$ then for every $g \in G-\{e\}$ there exists its inverse $g^{-1} \in G$ with $g \neq g^{-1}$. But then we will have an even number of elements in $G-\{e\}$. A contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1518577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
how to measure a circle. As we all know we can measure a line with a scale or any instrument but right now I have studied circles and was wondering if there was any way , instrument or method to measure circle i.e the circumferrence without the formula $2πr$.
| To measure the circumference of a circle without using $C = 2πr $ you need to find a way to make it linear. This can be done by simply wrapping a string around the circumference, cutting it to make it the correct length, then measuring the length of the string.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to compute this finite sum $\sum_{k=1}^n \frac{k}{2^k} + \frac{n}{2^n}$? I do not know how to find the value of this sum:
$$\sum_{k=1}^n \frac{k}{2^k} + \frac{n}{2^n}$$
(Yes, the last term is added twice).
Of course I've already plugged it to wolfram online, and the answer is $$2-\frac{1}{2^{n-1}}$$
But I do not kn... | Let $$S=\sum_{k=1}^{n}\frac{k}{2^k}$$
or, $$S=\frac{1}{2^1}+\frac{2}{2^2}+\frac{3}{2^3}+\frac{4}{2^4}+\frac{5}{2^5}+....+\frac{n}{2^n}$$
and $$\frac{S}{2}= \frac{1}{2^2}+\frac{2}{2^3}+\frac{3}{2^4}+\frac{4}{2^5}+....+\frac{n-1}{2^n}+\frac{n}{2^{n+1}}$$
Subtracting we get,
$$\frac{S}{2}= \frac{1}{2^1}+\frac{1}{2^2}+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1518757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
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Functional separation of regular open sets of a topological space Let $\langle X,\mathscr{O}\rangle$ be a topological space. We say that disjoint $A,B\in 2^X$ are functionally separated iff there exists a continuous function $f\colon X\to [0,1]$ such that:
*
*if $x\in A$, then $f(x)=0$
*if $x\in B$, then $f(x)=1$
... | If $x\in -(C\cdot D)$,
$$\begin{split}\Rightarrow & x \in \text{Cl}(-C)\text{ or }x\in \text{Cl}(-D) \\
\Rightarrow & f(x) =1\text{ or }g(x)=1 \\
\Rightarrow &h(x) = 1
\end{split}$$
as $f, g$ has range $[0,1]$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1518838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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how to find an infinity limit in a fraction I don't understand how to find the limits of this expression when $x\to\infty$ and $x\to-\infty$:
$$\left(\frac{3e^{2x}+8e^x-3}{1+e^x}\right)$$
I've searched for hours. How to compute these limits?
| $$\lim_{x\to\infty}\frac{3e^{2x}+8e^x-3}{1+e^x}=$$
$$\lim_{x\to\infty}\frac{-3e^{-x}+8+3e^x}{1+e^{-x}}=$$
$$\lim_{x\to\infty}\left(8+3e^x\right)=$$
$$\lim_{x\to\infty}8+\lim_{x\to\infty}\left(3e^x\right)=$$
$$\lim_{x\to\infty}8+3\lim_{x\to\infty}e^x=$$
$$\lim_{x\to\infty}8+3\exp\left(\lim_{x\to\infty}x\right)=$$
$$8+3\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1518925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 5
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Prove that $\tau=\{ \mathbb{R},\emptyset\} \cup \{(-t,t)|t\in\mathbb{I}^+\}$,is not a topology on $\mathbb{R}$. Prove that $\tau=\{ \mathbb{R},\emptyset\} \cup \{(-t,t) \,\, | \,\, t\in\mathbb{I}^+\}$ is not a topology on $\mathbb{R}$, where $\mathbb{I}^+$ denotes the positive irrational numbers.
Any pointers on how to... | Assuming that your topology is $\tau = \{ \mathbb R , \emptyset \} \cup \{ (-t,t) : t \in \mathbb Q^c_+ \}$
Let $t_n$ be an increasing sequence of positive irrational numbers converging to a rational number, say $a$. (You should find such a sequence)
Then it is easy to see that although $(-t_n \ ,t_n)$ is in your topol... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Comments on my proof of the transitive property of subsets? Because I am in an advanced Calc III course that is quite proof-based, my university does not require that I take the "introduction to formal proofs" course before proceeding to higher level math classes because they assume the class already knows everything. ... | When someone says write a formal proof, what they usually mean is you do things strictly by the definitions. By definition, $X \subseteq Y$ if and only if for any $x \in X$, we have $x \in Y$.
Let $x \in A$. Since $A \subseteq B$, we have $x \in B$. Since $B \subseteq C$, we have $x \in C$. We showed that for any... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find $n$ if $n!$ ends in exactly $100$ zeroes and $n$ is divisible by $8$. This question was in school maths challenge.
I dont know how to approach this one.. any help would be appreciated.
| Ends in 100 zeros means that $5^{100}$ is a divisor. If $n \ge 5^im$ then $n > 2^im$ so if $5^{100}$ is a divisor so is $2^{100}$ so is $10^{100}$. so $5^{100}$ divisor is sufficient to end in 100 zeros.
To end in exactly 100 zeros means that in {1,2, ...., n} there are precisely 100 occurrences of 5 and added powers... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1519251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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prove polynomial division for any natural number Show that for any natural numbers $a$, $b$, $c~$ we have $~x^2 + x + 1|x^{3a+2} + x^{3b+1} + x^{3c}$.
Any hints on what to use?
| We have
$$x^3-1=(x-1)\left(x^2+x+1\right)$$
This means that
$$\begin{align}x^3-1&\equiv0\pmod{x^2+x+1}\\x^3&\equiv1\pmod{x^2+x+1}\end{align}$$
Now, substitute it
$$\begin{align}x^{3a+2}+x^{3b+1}+x^{3c}&\equiv x^2\left(x^3\right)^a+x\left(x^3\right)^b+\left(x^3\right)^a\\&\equiv x^2\cdot1^a+x\cdot1^b+1^c\\&\equiv x^2+x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1519336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Use of the symbol $\lneq$ This answer uses kind of a "proper less equal" symbol: $\lneq$
I would have expected that $<$ is sufficient, because it seems to be the same relation.
For $\subset$ vs $\subsetneq$ some authors interpret $\subset$ allowing self-inclusion $A \subset A$ to be true, while others do not, so the ex... | Yes $<$ and $\lneq$ are the same relation.
However, be careful about set inclusion: older texts, and some current mathematicians, use $\subset$ to mean $\subseteq$, and to indicate strict inclusion they might use $\subsetneq$ or $\subsetneqq$. Other authors who use both $\subset$ and $\subseteq$ will use $\subset$ to m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1519389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Let $V$ be a linear space s.t. $\dim\, V=n$ and $f_1,f_2,\dots,f_m$ be linear functionals ($m
Let $V$ be a linear space s.t. $\dim\, V=n$ and $f_1,f_2,\dots,f_m$ be
linear functionals ($m<n$). Show that $\exists x\neq 0$ s.t.
$f_i(x)=0$ $\forall i$.
We can suppose that these $f_i$ are linearly indepent.
Let $\{e_i\... | $F = (f_1,\ldots,f_m)$ defined by $F(x) = (f_1(x),\ldots,f_m(x))$ is a linear map from $V$ to $F^m$.
$\dim(\ker(F)) + \dim(\operatorname{im}(F)) = \dim(V)$.
We know that the right hand side equals $n$. The image of $F$ has dimension at most $m < n$. So the dimension of $\ker(F)$ cannot be $0$, so has some vector $x \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1519464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to Find the Linearization of $f(x) = 8\cos^2(x)$ at the Point Where $x = \pi/4$ I am looking to find the linearization of $f(x) = 8\cos^2(x)$ at the point where $x = \pi/4$
Now, I know that the general formula for finding linearization is $L(x) = f(x) + f'(x) (x-a)$. So, I have made the following steps to get my an... | First find the derivative:
$f'(x)=16\cos(x)(-\sin(x))=-16\sin(x)\cos(x)$.
Then $f'(\frac{\pi}{4})=-16\cos(\frac{\pi}{4})\sin(\frac{\pi}{4})=-16\frac{\sqrt2}{2}\cdot\frac{\sqrt2}{2}=-16\cdot\frac{2}{4}=-8$.
Then find the tangent: $L=f(\frac{\pi}{4})-8(x-\frac{\pi}{4})=4+2\pi-8x$
EDIT: I think your confusion is in the wa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1519580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Show that $\sqrt{x-1}+\sqrt{y-1}\leq \sqrt{xy}$ How can one show that $\sqrt{x-1}+\sqrt{y-1}\leq \sqrt{xy}$
Assuming that :
$\sqrt{x-1}+\sqrt{y-1}\leq \sqrt{xy}$
So
$(\sqrt{x-1}+\sqrt{y-1})^2\leq xy$
$\sqrt{(x-1)(y-1)} \leq xy-x-y+2$
$ (y-1)(x-1)+3 \leq \sqrt{(x-1)(y-1)}$
Here I'm stuck !
| There are some errors in your calculation, e.g. a missing factor 2 in
$$
(\sqrt{x-1}+\sqrt{y-1})^2 = x - 1 + y - 1 + 2\sqrt{x-1}\sqrt{y-1}
$$
and in the last step the inequality sign is in the wrong direction and
the number $3$ is wrong.
For $x \ge 1$, $y \ge 1$ you can square the inequality (since both
sides are non-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1519682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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How to treat small number within square root guys.I am reading a math book. It has a equation shown as follows,
$\sqrt{(1+\Delta^2)}$
And then,since $\Delta$ is very small, it can be written as,
$\sqrt{(1+\Delta^2)} = (1+\frac12\Delta^2)$
What is the theory behind this equation?
| Square both sides to get $1+\Delta^2\approx 1+\Delta^2+\frac{1}{4}\Delta^4$
which is a close approximation because when $\Delta$ is very small, $\frac{1}{4}\Delta^4$ is close to $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1519750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Specific conditional entropy $H(X|Y=y)$ is not bounded by $H(X)$? Suppose that $P(Y=y)>0$ so that
$$
H(X|Y=y)=-\sum_{x} p(x|y) \log_{2} p(x|y)
$$
makes sense. I've assumed for a long time that $H(X|Y=y)\le H(X)$, but then it seems that the wiki article claims that $H(X|Y=y)$ is not necessarily bounded by $H(X)$.
It i... | It's important to distinguish $H(X \mid Y)$ from $H(X \mid Y=y)$. The first is a number, the second is a function of $y$. The first is the average of the second -averaged over $p(y)$:
$$H(X \mid Y) = E_y \left[H(X \mid Y=y)\right]$$
Hence, beacuse it's an average, in general we'll have $H(X \mid Y=y)>H(X \mid Y)$ for s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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properties of a norm I am quite bothered by this problem in Royden's 4th ed. Show that $||f||=\int_{a}^{b} {x^2 |f|}$ is a norm given that $f\in L^1$ on $[a,b]$.
I want to show that $||f||=0$ iff $f=0$.
Am I doing this right?
if suppose $||f||=0$ then $||f||=\int_{a}^{b} {x^2 |f|}=0$
that is, $0=||f||=|f| \int_{a}^{b} ... | The triangle inequality:
$||f+g|| = \int_a^b x^2 |f(x)+g(x)|dx \leq \int_a^b x^2 (|f(x)| + |g(x)|)= \int_a^b x^2 |f(x) + \int_a^b x^2 |g(x)| = ||f|| + ||g||$
Therefore
$||f+g|| \leq ||f|| + ||g||$
Positive Homogeneity:
$||\alpha f||= \int_a^b x^2 |\alpha f(x)| dx =|\alpha| \int_a^b x^2 |f(x)| =|\alpha| ||f||$
Therefore... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Basic algebra inequality mistake (I probably didn't break algebra) I'm not sure what I have done here. I'm guessing it's something to do with the absolute values but I really have no idea what it is.
$$\begin{array}{ccc}
y>\frac1y & & y>\frac1y \\
y^2>1 & & 1>\frac1{y^2} \\
|y|>1 & & 1>\frac1{|y|} \\... | The error is on the left.
You went from
$$|y|>1$$
to
$$\left|\frac{1}{y}\right|>1$$
This is incorrect. You should have divided both sides by $|y|$, leaving
$$1>\frac{1}{|y|}$$
This agrees with the right-hand result.
However, something happened with the absolute value. Try the case where $y=-2$. In this case, the final ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1520118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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If $f$ is continuously differentiable periodic, then $n\int_{0}^{1} f(x) \sin (2\pi nx) \mathrm dx \to 0 $ If $f: \mathbb R \to \mathbb R$ is continuously differentiable periodic function of period $1$, then
$$n\int_{0}^{1} f(x) \sin(2\pi nx)\mathrm dx \to 0 $$
as $ n\to\infty$.
| Hint. You may integrate by parts:
$$
\begin{align}
&n\int_{0}^{1} f(x) \sin(2\pi n x)\: dx\\\\
&=\left. -f(x)\frac{\cos(2\pi n x)}{2\pi}\right|_0^1+\frac1{2\pi}\int_0^1f'(x)\cos(2\pi n x)\:dx\\\\
&=\frac{f(0)-f(1)}{2\pi}+\frac1{2\pi}\int_0^1f'(x)\cos(2\pi n x)\:dx\\\\
&=0+\frac1{2\pi}\int_0^1f'(x)\cos(2\pi n x)\:dx \to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1520214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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I randomly assign $m$ objects into one of $n$ sets. How do I compute expected value of the number of non-empty sets? To put it into more colorful terms, let's say I have $m$ balls and $n$ boxes. I select one of the $m$ balls and randomly place it into one of the $n$ boxes. I do this with each ball until I have none l... | For $i=1,2,\cdots{},n$ define $X_i$ as $X_i=1$ if box $i$ ends up with non-zero balls and $X_i=0$ if not. Set $$X=X_1+X_2+\cdots{}X_n.$$
Notice that $X$ is the total number of boxes which are non-empty. Therefore,
$$\mathbb{E}(X)=\mathbb{E}\left(X_1+X_2+\cdots{}+X_n\right)=\mathbb{E}(X_1)+\mathbb{E}(X_2)+\cdots{}+\math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1520327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Simple proof that Automorphisms preserve definable subsets? I've been looking all over for a proof of this result, but I haven't really found anything. Is anyone aware of a particularly simple or elegant one?
| Unfortunately, at this basic level in model theory, you're going to have to get your hands dirty with induction on the complexity of terms and formulas, there's no way around it. Fortunately, once you do a proof or two like this, you'll find they become routine.
"Automorphisms preserve definable subsets" means that if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1520420",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Proof that finite group contains an element of prime order I tried to prove the following claim but it seemed a bit too easy:
Let $G$ be a finite group. Then $G$ contains an element of prime order.
Please could someone tell me if my proof is correct or if I'm missing something?
Let $g$ be any non-identity element of ... | No the proof is not correct. It is possible that $g^{\frac{|g|}{p}}=e$. The theorem you are stating is commonly called Cauchy's theorem. There is an elementary proof here using the class equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1520513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Contraposition of continuity Function f is continuous on a metric space (M,d) if
$$\forall x\ \forall \epsilon >0\ \exists \delta >0\ \forall y\ \ d(x,y)< \delta \Rightarrow d(f(x),f(y))<\epsilon $$
I want to find a contrapositive of this. I know that I have to change $\exists$ to $\forall$ and $\forall$ to $\exists$ b... | The contraposition of the implication is: we have $y \in M$ and $d(f(x),f(y)) \geq \varepsilon$ only if $d(x,y) \geq \delta$.
The negation of the whole statement is: there are some $\varepsilon > 0$ and some $x \in M$ such that $\delta > 0$ only if there is some $y \in M$ such that $d(x,y) < \delta$ and $d(f(x), f(y)) ... | {
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If $x(t)$ is a solution to $ \dot{x}(t)=f(x)$ then $x(t)$approaches infinity or a fixed point, and if $x(t_1)=x(t_2)$ then $x(t)$ is constant This is an easy problem, but I would like to know a formal solution using calculus and not the Picard–Lindelöf theorem, and also without using the (obvious) fact that solutions a... | 1.) $x(n+1)-x(n)=x'(n+\theta_n)=f(x(n+\theta_n))$, $θ_n\in(0,1)$, gives you the derivative in the limit, since the first difference tends to $0$.
2.) You got two solutions passing through the same point. By local uniqueness, this is impossible if they are not identical. You can get the local uniqueness also from the Gr... | {
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How to prove a combinatoric statement?
From Number 10B with PICTURE. Suppose there are n plates equally spaced around a circular table. Ross wishes to
place an identical gift on each of k plates, so that no two neighbouring plates have
gifts. Let f(n, k) represent the number of ways in which he can place the gifts... | We will fix the plates in their position and number them $1,2,3,\ldots$ in clockwise order.
Let $A_{n,k}$ denote the set of valid arrangements of $k$ gifts on $n$ plates. Then $f(n,k) = \vert A_{n,k}\vert$ and we will prove the claim by setting up a bijection (graphically) from $A_{n-2,k-1}\cup A_{n-1,k}$ to $A_{n,k}$,... | {
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Limit of a sum of infinite series How do I find the following?
$$\lim_{n\to\infty} \frac{\left(\sum_{r=1}^n\sqrt{r}\right)\left(\sum_{r=1}^n\frac1{\sqrt{r}}\right)}{\sum_{r=1}^n r}$$
The lower sum is easy to find. However, I don't think there is an expression for the sums of the individual numerator terms... Nor can I... | A way is to use Abel's summation to get $$\sum_{r\leq n}\sqrt{r}=\sum_{r\leq n}1\cdot\sqrt{r}=n^{3/2}-\frac{1}{2}\int_{1}^{n}\frac{\left\lfloor t\right\rfloor }{\sqrt{t}}dt
$$ where $\left\lfloor t\right\rfloor
$ is the integer part of $t
$ and using $\left\lfloor t\right\rfloor =t+O\left(1\right)
$ we have $$\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1520866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Additivity & one-point Continuity of $f \in \mathbb{R}^\mathbb{R}$ imply there is $\alpha \in \mathbb{R}$ s.t. $f(\alpha x) = \alpha x$ I am looking for hints/help concerning a proposition I found self-studying Carothers:
Suppose that $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(x+y) = f(x)+f(y)$ for every $x, y \in \... | Hint: $f(x)=f(x+x_0-x_1)-f(x_0-x_1)$. Use that to show that $f(x)$ is continuous at any $x_1$ if it is continuous at $x_0$.
Next, let $\alpha = f(1)$ and show that $f(x)=f(1)x$ first for $x$ an integer, then $x$ a rational, and then, by continuity, all the reals.
| {
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For matrices $A, B$, show that $BA = 0 \Rightarrow (AB)^2 = 0$ I need to prove the following statement for all matrices A, B:
$$A_{n \times n}, B_{n \times n} \text{ over } \mathbb{R}$$
$$BA = 0_{n \times n} \implies (AB)^2 = 0_{n \times n}$$
| $(AB)^2=AB.AB=A(BA)B=0$ since $BA=0$
| {
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Proof of indefinite integral of inverse function
Assume that $f$ has a inverse on its domain.
a. Let $y = f^{-1}(x)$ and show that
$\int f^{-1}(x)dx=\int yf'(y)dy$
b. Use part a to show that
$\int f^{-1}(x)dx= yf(y)-\int yf(y)dy$
c. Use part b to evaluate $\int ln(x)dx$
d. Use part b to evaluate $\int sin^{-1}(x... | $f^{-1}(x)=y \implies f(y)=x$. Differentiating both sides of $f(y)=x$ gives $ f'(y) \cdot \frac{dy}{dx}=1 \implies f'(y) dy=dx$ Now just substitute.
| {
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Applying L'Hopital on $\lim_{x\to1^+}\left( \frac{1}{\ln(x)} - \frac{1}{x - 1} \right)$ I am trying to apply L'Hopital's rule here:
$$\lim_{x\to1^+}\left( \frac{1}{\ln(x)} - \frac{1}{x - 1} \right)$$
The indeterminate form here is $\infty - \infty$, so I need to somehow shape this limit to have a quotient instead. My ... | If you make a single fraction, you get
$$
\frac{x-1-\ln x}{(x-1)\ln x}
$$
Both the numerator and the denominator are continuous at $1$, so this limit is of the form $\frac{0}{0}$. No need to consider signs, at this point.
You can avoid long applications of l'Hôpital with a simple observation:
$$
\lim_{x\to1^+}\frac{x-1... | {
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Deducing equivalence between norms from simple condition Let $||\cdot||,\;||\cdot||'$ be norms on $V$. Suppose for some $a,b>0$ we have:
$$||x||<a\Rightarrow ||x||'<1\Rightarrow ||x||<b$$
Show that $||\cdot||,\;||\cdot||'$ are Lipschitz equivalent.
I don't think this should be hard, but I am completely stuck. My idea ... | Hint.
For $x \in V$, what is a collinear vector $y$ to $x$ such that $\Vert y \Vert <a$? What happens then to $\Vert y \Vert^\prime$?
And vice versa?
| {
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how to prove that the sum of the power two of three reals is less than 5? $a, b$ and $c$ are three real number such that: $(\forall x \in [-1,1]): |ax^2+bx+c|\leq 1$ .
Prove that:
1) $|c|\leq 1$ .
2) $-1\leq a+c\leq 1$ .
3)Deduce that : $a^2+b^2+c^2\leq 5$.
The first and second questions are easy to prove ( just take... | $x=0$ gives us $|c| \le 1$. Similarly set $x = \pm1$ to get $|a \pm b+c| \le 1$. So we have the inequalities:
$$-1 \le c \le 1 \iff -1 \le -c \le 1\tag{1}$$
$$-1 \le a+b+c \le 1 \iff -1\le -a-b-c \le 1 \tag{2}$$
$$-1 \le a-b+c \le 1 \iff -1 \le -a+b-c \le 1\tag{3}$$
Note the right side of the above three inequalities... | {
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proof that there is a unique isomorphism between two free modules I'm trying to figure out if there's any way to shorten the following proof of Corollary 7 in Dummit & Foote (p. 355) so that I don't have to write down the expressions of any specific elements like $x = r_1a_1 + \dots + r_na_n$. Is it possible to write o... | You can use the universal property applied to $A \hookrightarrow F_1$. The function $\Phi: F_1 \rightarrow F_1$ is the identity on $A$, so it satisfies the universal property for the map $A \hookrightarrow F_1$. In other words $\Phi$ extends $A \hookrightarrow F_1$ to a map $F_1 \rightarrow F_1$. But there is already s... | {
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Are there any situations in which L'Hopital's Rule WILL NOT work? Today was my first day learning L'Hopital's Rule, and I was wondering if there are any situations in which you cannot use this rule, with the exception of when a limit is determinable.
| Here's a similar example to Ittay Weiss's, but more nefarious, since in Ittay's example the limit was easily seen to be equal to 1:
$$\lim_{x \rightarrow 0}\frac{e^{-\frac{1}{x^2}}}{x}$$
Both the numerator and denominator go to zero. But applying l'Hospital gives:
$$\lim_{x \rightarrow 0}\frac{2e^{-\frac{1}{x^2}}}{x^3}... | {
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Why 1 is not a cluster point of $\frac1n$ I came across this definition of cluster point. Let $S \subset R$ be a set. A number $x \in \mathbb R$ is called a cluster point of $S$ if for every
$\varepsilon > 0$, the set $(x−ε, x+ε)∩S \setminus \{x\}$ is not empty.
For the set $\{1/n:n∈\mathbb N\}$ why 1 is not a cluster ... | If we take $\epsilon$ to be $.25$ and we have $S = \{\frac{1}{n} : n \in \mathbb{N}\}$, then $(1-\epsilon,1+\epsilon) \cap S = (.75,1.25) \cap S = 1,$ so $1$ is not a cluster point.
| {
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Proof for Product of Congruences mod P Consider $Z_p$, with $p$ prime. Prove that $[x][x] = [y][y]$ if and only if $[x] =[y]$ or $[x] =[-y]$.
I think this question comes down to the fact that $Z_p$ is cyclic but am not sure.
| Hint: One direction is easy, and does not require primality.
From a number-theoretic point of view, the harder direction boils down to the fact that $p$ divides $x^2-y^2$ if and only if it divides $(x-y)(x+y)$.
And if a prime divides a product, it divides (at least) one of the terms.
| {
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Convergent sequence on unit sphere Suppose $x_n$ is a bounded sequence in a vector space $V$ with norm $||\cdot||$. Show that if:
$$\hat{x}_n=\frac{x_n}{||x_n||}\;\;\text{converges}\Rightarrow x_n\;\text{has a convergent subsequence}$$
Thoughts: Writing the limit on the sphere as $\hat{x}=x/||x||$ I would like to show... | Hint: The sequence of numbers $\| x_n\|$ is real and bounded, so it must contain a convergent subsequence by the Bolzano–Weierstrass theorem.
| {
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Show that $M_a\neq M_b$. Let $f\in \mathbb Z[x]$ be a non-constant polynomial .Show that as $a$ varies over $\mathbb Z$,the set of divisors of $f(a)$ includes infinitely many different primes.
My try:
Let $f(x)=a_0+a_1x+a_2x^2+\cdot+\cdot +a_nx^n$,
$f(a)=a_0+a_1a+a_2a^2+\cdot\cdot\cdot\cdot +a_na^n$.Suppose that $f(a... | Hint: Suppose that the set of prime divisors of the $f(a), a\in \mathbb{Z}$ is finite. Then there exists $a\in \mathbb{Z}$ such that $f(a)\not =0$ has the maximum of prime divisors possible. Fix such an $a$, and put $f(a+x)=\sum_{j=0}^n b_jx^j$, with the $b_j\in \mathbb{Z}$ ($b_n\not = 0$, and $b_0=f(a))$. Now put $x=m... | {
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Evaluation of the Definite Integral $\int_{\zeta=0}^{2x} \lvert \sin(\zeta)\rvert \mathrm{d}\zeta$ I need help understanding how to evaluate the following integral:
$$\int_{\zeta=0}^{2x} \lvert \sin(\zeta)\rvert \mathrm{d}\zeta$$
I am a bit lost so if someone could help out that would be great.
| Sketch the graph of $|\sin x|$. Suppose $x>0$. Let $k=\lfloor 2\,x/\pi\rfloor$ be the unique non-negative integer such that $k\,\pi\le2\,x<(k+1)\pi$. Then
$$
\int_0^{2x}|\sin\zeta|\,d\zeta=\int_0^{k\pi}|\sin\zeta|\,d\zeta+\int_{k\pi}^{2x}|\sin\zeta|\,d\zeta=k\int_0^\pi\sin\zeta\,d\zeta+\int_0^{2x-k\pi}\sin\zeta\,d\zeta... | {
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Find all real numbers $x$ such that $x[x[x[x]]]=88$ where $[\cdot]$ denotes floor function.
Question: Find the all real numbers $x$ such that $x[x[x[x]]]=88$ where $[\cdot ]$ denotes floor function.
Attempt: $$x[x[x[x]]]=88\implies [x[x[x]]]=\frac{88}{x}.$$
Since left side of equation always gives an integer value, $... | So, the given equation basically implies $x^4\approx88$, and we know $\frac{88}x$ is an integer. Now, if $x^4\approx88$, then $x\approx\sqrt[4]{88}=3.063$, and then $\frac{88}x\approx28.732$. So I'd guess either $\frac{88}x=28$ or $\frac{88}x=29$. That is, I'd guess either $x=\frac{88}{28}=\frac{22}7$ or $x=\frac{88}{2... | {
"language": "en",
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Geometry question on triangle Let ABC be an isosceles triangle with AB = AC and let Γ denote its circumcircle. A point D is on the arc AB of Γ not containing C and a point E is on the arc AC of Γ not containing B such that AD = CE. how can I prove that BE is parallel to AD.?
|
Note $|AB|=|D'E'|$.
Let $|AF|=|AD'|$, then $\angle AD'B=\angle AFC$.
Rotate $A\to D'$ and so $\angle AD'B=\angle D'AE'$.
$|D'B|=|AE'|$ so $AD' \parallel BE'$.
| {
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Does double negation distribute over disjunction intuitionistically? Does the following equivalence
$$\lnot \lnot (A \lor B) \leftrightarrow (\lnot \lnot A \lor \lnot \lnot B)$$
hold in propositional intuitionistic logic? And in propositional minimal logic? (In propositional classical logic this is obvious since $A \le... | $\lnot \lnot (A \lor B) \to (\lnot \lnot A \lor \lnot \lnot B)$ is not intuitionistically acceptable. One way of seeing this is by considering the Heyting algebra whose elements are the open subsets of the unit interval $[0, 1] \subseteq \Bbb{R}$ under the subspace topology, with $A \lor B = A \cup B$, $A \to B = \math... | {
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Is my proof for $A\cap B=A\cap C\Longleftrightarrow B=C$ correct? Prove or disprove: for every $3$ sets $A,B,C,: A\cap B=A\cap C\Longleftrightarrow B=C$
Proof: This proof is by case analysis.
Case I: Assuming $B=C$, I will prove $A\cap B=A\cap C$
1) $A\cap B=A\cap C$ (because $B=C$)
2) $A\cap C=A\cap B$ (because $B=C$)... | $\textbf{Hint}$: Consider the case $A= \varnothing$.
You proved $B=C \Rightarrow A \cap C = A \cap B$ correctly. Just note that step 1 and 2 are equivalent.
For proving that $B=C$, you have to prove $B \subseteq C$ and $C \subseteq B$, i.e. take an element in $B$ and prove it is also in $C$ (and conversely).
Step 1 in ... | {
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Stable subspace, unstable subspace, centre subspace: Of which kind of stability are we talking? Let $x'(t)=Ax(t)$ be a linear ODE system.
Then the span of the eigenvectors belonging to eigenvalues with negative real-part is called the stable subspace, the subspace spanned by the eigenvectors of eigenvalues of positive ... | *
*Globally asymptotically stable: solutions with initial values anywhere in this subspace converge to $0$.
*Negation of any kind of stability: the solutions with initial values in this subspace are unbounded, eventually leaving every compact set.
*Center is (globally) Lyapunov stable, not asymptotically stable.
All... | {
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Is there a name for $2^n$ in combinatorics? $2^n$, with $n$ corresponding to the number of elements, is an almost unavoidable calculation involved in counting. At the root of its usefulness there lies - I presume - the identity:
$$\sum_{0\leq{k}\leq{n}}\binom nk =2^n$$
that explains its application to the calculation o... | I don't think there is a short name for $2^n$. However, I think that "the number of subsets of a set with $n$ elements" (or cardinal of the powerset, of course) is already nice enough. For instance, it fits very well with your exemple of +'es and -'es: just select the subset of positive terms.
| {
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What is the Fourier transform of $1/|x|$? I looked it up in several tables and calculated it in Mathematica and Matlab. Some tables say that the answer is simply $$\frac{1}{|\omega|}$$ and in other table it is $$-\sqrt{\frac{2}{\pi}}\ln|\omega|$$
and in Mathematica and Matlab (mupad) it is
$$-\sqrt{\frac{2}{\pi}}\ln|\o... | To be fully rigorous, we should concede that $1/|x|$ is not directly a distn on $\mathbb R$. But it does arise as an even tempered distn $u$ such that $x\cdot u=sign(x)$.
Up to a constant, Fourier transform gives ${d\over dx}\hat{u}=PV{1\over x}$. This equation has at least solution $\hat{u}=\log|x|$. The associated ho... | {
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Convergence of $\frac{n^{n+1}}{e^n\left(n+1\right)!}$ Determine whether the series $\displaystyle \sum\frac{n^{n+1}}{e^n\left(n+1\right)!}$ is convergent or divergent. I can't use the integral test.
I can't use condensation test either, since the series isn't decreasing. Any hints/ideas on how to pursue this exercise? ... | The series is divergent, I will provide a complete answer below, but before that, I want to make sure you know what the 'Monotone Convergence Theorem' is, as well as one of the definitions of '$e$', as they will be relevant in order to understand the answer that will be posted below. The ratio test yields an inconclusi... | {
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Prove $ \forall x >0, \quad \sqrt{x +2} - \sqrt{x +1} \neq \sqrt{x +1}-\sqrt{x}$
I would like to prove
$$ \forall x >0, \quad \sqrt{x +2} - \sqrt{x +1} \neq \sqrt{x +1}-\sqrt{x}$$
*
*I'm interested in more ways of proving it
My thoughts:
\begin{align}
\sqrt{x+2}-\sqrt{x+1}\neq \sqrt{x+1}-\sqrt{x}\\
\fr... | By the mean value theorem - MVT, for all $x >0$, it exists $\zeta_x \in (x, x+1)$ such that $$\frac{1}{2\sqrt{\zeta_x}}((x+1)-x)=\frac{1}{2\sqrt{\zeta_x}}=\sqrt{x+1}-\sqrt{x}$$ As $y \to \frac{1}{2\sqrt{y}}$ is stricly decreasing, you have $$\frac{1}{2\sqrt{\zeta_{x+1}}}=\sqrt{x+2} -\sqrt{x+1} < \sqrt {x+1}-\sqrt{x}=\f... | {
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Finding the basis with given transition matrix \begin{equation}
P = \begin{bmatrix}1 & 1 & 0 \\ 0 & 1 & 3 \\ 3 & 0 & 1
\end{bmatrix}
\end{equation}
a) P is the transition matrix from what basis B to the standard
basis S = {e1, e2, e3} for R3?
b) P is the transition matrix from the standard basis
S = {e1, e2, e3} to wha... | Your idea looks correct. Since S is the standard basis, i.e. S is the identity matrix, in a) your basis B are the columns of $P^{-1}S = P^{-1}$.
In b) B equals P: $B = PS = P$.
| {
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Using the Law of the Mean (MVT) to prove the inequality $\log(1+x)If $x \gt0$, then $\log(1+x) \lt x$.
My attempt at the proof thus far...
Let $f(x) = x-\log(1+x)$, then $f'(x) = 1-\frac{1}{1+x}$ for some $\mu \in (0,x)$ (since $x>0$)
MVT give us $f(x) = f(a) + f'(\mu)(x-a)$
So plug in our values we get:
$$x-\log(1+x) ... | You're almost there. Have a look at your statement $$\log(1 + x) = \frac{x}{1+\mu}.$$
No matter what $\mu$ is, $1 + \mu$ is bigger than $1$, hence $$\frac{1}{1+\mu} < 1.$$
Multiplying this final inequality through by $x$ gives you the result you want. In words, dividing $x$ by something bigger than $1$ gives you someth... | {
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Prove that $[0,1]$ isn't homeomorphic to $\mathbb{R}$ Prove that $[0,1]$ isn't homeomorphic to $\mathbb{R}$
My first thought is that there can not be a continuous bijection, $f$, from $[0,1]$ to $\mathbb{R}$ because a continuous function that maps $[0,1]\rightarrow \mathbb{R}$ must be bounded so there can't be a surjec... | Here's a way to use the IVT: Suppose $f: \mathbb{R}\rightarrow [0, 1]$ is a continuous bijection. Let $a\in\mathbb{R}$ be such that $f(a)=0$; now look at $x=a-1$ and $y=a+1$. Since $f$ is injective, we have $f(x), f(y)\not=0$; let $0<c<\min\{f(x), f(y)\}$. By IVT, we have $f(x')=c$ for some $x<x'<a$ and $f(y')=c$ for s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1523521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The element that is an associate of everything. Suppose I have an integral domain $R$ containing an element $a \in R$ with the following property:
$$(\forall r \in R)\, a \text{ is an associate of } r.$$
Is it true that the ring must be either the zero ring or the ring $\{0,1\}$?
"Associates" are defined as follows: ... | $R$ must be the zero ring. Here's a proof.
By definition, if $a$ and $b$ are associates then each is a unit multiple of the other. In particular for this special $a$ we have that $b$ can be anything. Take $b=0$. Then $a=cb=0$ for some unit $c$, hence $a=0$. This uses the fact that $a$ is a multiple of $b$.
To finish th... | {
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Prove that in every sequence of 79 consecutive positive numbers written in decimal system there is a number whose sum of the digits is divisible by 13
Prove that in every sequence of $79$ consecutive positive numbers written in decimal notation there is a number the sum of whose digits is divisible by $13$.
I tried t... | For $x\in{\mathbb N}$ denote by $r(x)$ the remainder modulo $13$ of the decimal representation of $x$. If $x$ is not divisible by $10$ then $r(x)=r(x-1)+1$. If $x$ is divisible by $10$, but not by $100$, then $$r(x)=r(x-1)-9+1=r(x-1)+5\ .\tag{1}$$ If $x$ is divisible by $100$, things are more complicated; see below.
Co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1523694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 2
} |
Berkeley Problems in Mathematics 7.5.22 This is the problem: Let $A$ be a real symmetric $n \times n$ matrix with non negative entries. Prove that $A$ has an eigenvector with non-negative entries
I looked at the answer key and don't quite understand it. In the expression containing max, why should it correspond to the ... | $A$ is real symmetric, so it has $n$ independent eigenvectors with real eigen-values $\lambda_1\geq \cdots \geq\lambda_n$.
Let $v_1,v_2,\cdots, v_n$ be the corresponding independent eigenvectors, and we can assume that their length is $1$ (i.e. they form orthonormal basis).
In max expression, you considered $x$ with $\... | {
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"url": "https://math.stackexchange.com/questions/1523783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Definition: honest distance function What does honest distance function mean? In the context of metric spaces.
| This is not a mathematical term but a figure of speech. One says that A is an honest B to assert that A satisfies the definition of B. Usually the reason to emphasize honest is that A was informally called "B" earlier in the text. This is associated with a somewhat conversational style of writing.
Specifically for dis... | {
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"source": "stackexchange",
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Writing a sentence as a proportion I tried to write the following sentence as a proportion:
6 printers is to 24 computers as 2 printers is to 6 computers:
${6\,\,printers\over 2\,\,printers} = {24\,\,computers\over 6\,\,computers} $
But when I try to check if the statement is true, it is not. Therefore, it's not a pr... | If you need $6$ printers to $24$ computers so $2$ printers for $\color{blue}8$ computers
$$\frac{24}{6}=\frac{8}{2}$$
$$\frac{24}{6}\color{red}\neq\frac{6}{2}$$
$\Longrightarrow1$ printer for $4$ computers, so if you have $6$ computers so you need $2$ printers
| {
"language": "en",
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Why is the dimension of a kernel with the basis {[0,0,0]} equal to zero What is the dimension of a kernel with the basis {[0,0,0]}?
I'm confused because the definition of the dimension is number of vectors in a basis. So there is 1 vector here which is [0,0,0].
Why does my professor say that the dimension of kernel is... | A basis is defined as a set of linearly independent vectors that span a space. For the row space (kernel) of your problem, it clearly only includes the zero vector. However the zero vector, v, is linearly indepedent (as there exists a scalar, C != 0, such that Cv = 0), therefore the basis of the row space has no dimens... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1524025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that if each row of a matrix sums to zero, then it has no inverse. Could anyone help me with this proof without using determinant? I tried two ways.
Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity ... | Hint: You can sum the elements of a row by multiplying this row with a vector of $1$'s. Can you find now a matrix $X$ (with appropriate columns) such that $AX=Ο$?
| {
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"timestamp": "2023-03-29T00:00:00",
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Why are the coefficients of the equation of a plane the normal vector of a plane?
Why are the coefficients of the equation of a plane the normal vector of a plane?
I borrowed the below picture from Pauls Online Calculus 3 notes:
http://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx
And I think the explanat... | Given a point $O=(x_0,y_0,z_0)$ and a vector $\vec n=\langle a,b,c\rangle$, we can describe a plane as the set of points $P=(x,y,z)$ such that $\vec{OP}\cdot \vec n=0$. In other words, the set of vectors, perpedicular to $\vec n$.
$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0\implies ax+by+cz=ax_0+by_0+cz_0.$$
Letting $d=ax_0+by_0+c... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I solve limit with absolute value $\mathop {\lim }\limits_{x \to -\infty } {{x} \over {9}}|{\sin}{{6} \over {x}}|$ I try to evaluate this limit by L'Hopitals rule, but I don't know how affect limit absolute value. Can someone give me some advice please?
$$\mathop {\lim }\limits_{x \to -\infty } {{x} \over {9}}|... | L'Hôpital is not a magical thing to be used all the time. Thinking is advantageous.
If you want to consider $x\to-\infty$, you will only be dealing with $x<0$. So the numbers $6/x$ are going to be negative and close to zero. In that zone, the sine is negative, so
$$
\left|\sin\frac6x\right|=-\sin\frac6x
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1524372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Property of a morphism inherited by the fibers Let $T$ be a scheme over an algebraically closed field $k$ and let $f:X\to Y$ be a $k$-morphism over $T$.
For which properties P is it true that if $f_t:X_t\to Y_t$ has P for all geometric points $t\in T(k)$, then $f$ has P? Is it true for open immersions, for instance? W... | Suppose $X,Y$ are flat and of finite presentation over $T$.
Then, let $P$ be a property in $\{$flat, smooth, etale, open immersion, isomorphism, flat and a relative complete intersection morphism$\}$, then $f$ has property $P$ if and only if each $f_t$ has $P$ for all geometric points $t$ in $T$ (Here $T$ doesn't have ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In a group of 4 people, is it possible for each person to have exactly 3 friends? In a group of 4 people, is it possible for each person to have exactly 3 friends? Why?
My solution
n Let G be a graph with 4 vertices, one vertex representing each person in the group.
Join two vertices u and v by an edge if and only if u... | Easy solution. They're all friends with each other but not themselves. That's all, no graphs.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1524540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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inverse derivative shortcut and chain rule I know that when you have the equation y=ln(x), and you need to find the derivative, you can use the shortcut y'= 1/x. My question is why, when using the shortcut, do you have to multiply by the derivative of x? I'm aware it has something to do with the chain rule but I don'... | If $y = \ln x$ then $\dfrac{dy}{dx} = \dfrac 1 x$. That's not a "shortcut"; that is the derivative of this function.
Recall that $\ln(5x) = \ln 5 + \ln x$, so $\dfrac d {dx}\ln(5x) = \dfrac d{dx} \ln 5 + \dfrac d {dx} \ln x$, and $\dfrac d {dx}\ln 5 = 0$ since $\ln 5$ is a constant, i.e. $\ln 5$ does not change as $x$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $e^{At}B = B e^{At}$ if $AB= BA$ Let $A$ and $B$ be two matrices. How can I show that $e^{At}B = B e^{At}$ if $AB= BA$ and then conclude that $(d/dt) e^{At} e^{Bt} = (A+B) e^{At} e^{Bt}$? I have no idea how to do this.
| By definition
$$e^{At} = \sum_{n=0}^\infty \frac 1{n!} t^n A^n$$
thus term by term
$$B e^{At} = \sum_{n=0}^\infty \frac 1{n!} t^n BA^n = \sum_{n=0}^\infty \frac 1{n!} t^n A^nB = e^{At} B$$
Now apply this lemma for the second part.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1525146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Can someone help me understand the Euclidean metric? A Euclidean metric is defined as:
$g_{euclid} = g_{ij} = \delta_{ij} dx^i \times dx^j = dx^1dx^1 + \ldots + dx^ndx^n$
Can someone explain the following:
*
*why do we use $dx^i$ instead of $x^i$ which is a coordinate in $R^n$
*Why oes tensor product $\times$ gets ... | *
*Consider the properties of the $dx^i$ compared to the $x^i$. As linear functionals, the basis one-forms $dx^i$ are...well, linear on their arguments. Inner products are bilinear, but we are using two one-forms in each linearly independent term.
*Notational convention. My opinion? It's very misleading. I would almo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1525245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How can I prove every positive real number has square root? Does it ask that we can express any positive real number as square root of something? like 4 is equal to square root of 16?
| Hint: Consider the set $\{x\in \Bbb{R}_{\geq 0} \colon x^2<y\}$. By the completeness axiom, there is a least upper bound of this set, call it $x_0$.
If $x_0^2<y$, then by density of rationals, there is some rational number $q$ in $(x_0^2,y)$. Does this make sense?
If $x_0^2>y$. What happens?
So there's only one choic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1525346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
} |
What is the method for finding $\int\frac {x^3 + 5x^2 +2x -4}{(x^4-1)}\mathrm{d}x$? $$
\int\frac {x^3 + 5x^2 +2x -4}{(x^4-1)}dx
$$
A bit confused with how to integrate this question. I though it was partial fractions but was unsure about the what to do after that.
| Partial Solution
Well, assuming you did the partial fraction decomposition already (as you said you did), you should get the following integral
$$\int\left(\frac{9-x}{2(x^2+1)}+\frac{1}{x-1}+\frac{1}{2 (x+1)}\right) dx$$
$$=\frac{9}{2}\int\frac {dx}{x^2+1} - \frac{1}{2}\int \frac {x}{x^2+1}dx + \int \frac{dx}{x-1}+ \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1525385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Find the equation of the locus of R if the chord $PQ$ passes through $(0,a)$ The parabola is $x^2=4ay$
Information given:
Points of P$(2ap, ap^2)$, Q$(2aq,aq^2)$, and R $(2ar, ar^2)$ lie on the parabola $x^2=4ay$.
The equation of the tangent at P is $y=px-ap^2$
The equation of the tangent at Q is $y=qx-aq^2$
The equ... | The line passing through $P$ and $Q$ is
$$y-aq^2=\frac{aq^2-ap^2}{2aq-2ap}(x-2aq),$$
i.e.
$$y=\frac{p+q}{2}x-apq.$$
If $PQ$ passes through $(0,a)$, then we have
$$a=\frac{p+q}{2}\cdot 0-apq\quad\Rightarrow \quad pq=-1.$$
Here note that
$$R_x=-apq(p+q)=a(p+q)\quad\Rightarrow \quad p+q=\frac{R_x}{a}$$
and that
$$R_y=a(p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1525597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $\lim_{x \to \infty} (f(x)-g(x)) = 0$ then $\lim_{x \to \infty} (f^2(x)-g^2(x)) = 0\ $: False?
If $\lim_{x \to \infty} (f(x)-g(x)) = 0$, then $\lim_{x \to \infty} (f^2(x)-g^2(x)) = 0$
My teacher said that the above is false, but I can't find any example that shows that it's false! Can someone explain to me why it'... | Take $f(x)=x$ and $g(x)=x+\dfrac{1}{x}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1525710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Derivative of $e^{e^{e^x}}$ $$\Large e^{e^{e^x}}$$ Derivative of this function needs to be found out but I am not getting any substitution. Any help/hint would do . Thanks!
| Step 1: Think of the function as $e^{f(x)}$, where $f(x)=e^{e^x}$.
Step 2: Apply the chain rule. At some point you will need $f'(x)$, which will need the chain rule again.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1525957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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