Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$ A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help.
Prove:
$$\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\... | If we let $x\mapsto e^{-x}$ and then use the series of $\log(1-e^{-x})$, we obtain that
$$\underbrace{\int_0^{\infty} \sum_{n=1}^{\infty}\log(4 \pi^2) \frac{e^{-y n}}{n} \ dy}_{\displaystyle \zeta(2) \log(4 \pi^2)} -\underbrace{\sum_{n=1}^{\infty}\int_0^{\infty} \log(4 \pi^2+y^2) \frac{e^{-y n}}{n} \ dy}_{\displaystyle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "179",
"answer_count": 4,
"answer_id": 3
} |
Florida lottery selection Florida lottery consists of 6 numbered balls from a bin containing 53 numbered balls. If you buy one ticket, you win a share of the grand prize if the 6 numbers you picked, in any order, matches the 6 chosen by the lottery official. What is the probability of winning a share of the grand prize... | There are $\dbinom{53}{6}$ ways for the Lottery Corporation to pick $6$ numbrs from the $53$. If the randomizing procedure is good, all these ways are equally likely.
Thus if you hold $1$ ticket, the probability your numbers match the Corporation's numbers is $\dfrac{1}{\binom{53}{6}}$.
This is $\dfrac{(6)(5)(4)(3)(2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Three fives dice toss If four dice are tossed, find the probability that exactly 3 fives will show ( answer to the nearest thousandth in the for 0.xxx)?
| HINT: Imagine rolling the dice one at a time. Say that a set of four rolls is good if exactly $3$ of the $4$ are fives. Then there are $4$ kinds of good roll: the non-five can come on the first die, the second, the third, or the fourth. Suppose, for example, that it comes on the second roll; let’s calculate the probabi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What are the differences between Jacobson's "Basic Algebra" and "Lectures in Abstract Algebra"? Nathan Jacobson's books "Basic Algebra I, II" and "Lectures in Abstract Algebra - Volumes I, II, III (GTM 30, 31, 32)". What are the differences between these two books?
1) The subject.
The material of the two books overlap... | The Basic texts are much more modern in content and organization than the Lectures. The Lectures are based on Jacobson's graduate lectures on algebra at John Hopkins and Yale in the 1940's and early 1950's-consequently, the style is far more classical and categorical/homological methods are nearly completely missing. B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
} |
ellipse boundary after rotation Assume I have this vertical ellipse with a certain major axis $a$ and minor axis $b$.
If we take the center of the ellipse to be at $(0,0)$, then the top right small red circle will be at $(b,a)$.
Then I rotate it (say by an arbitrary angle $\theta$) about its center:
My question is th... | $$
r(t)=(a\,\cos (t), b\, \sin(t))
$$
After rotation,
$$
r_2(t)=R_\theta.r(t)= (a\,cos(t)\cos(\theta)+b\sin(t)\sin(\theta),-a\,cos(t)\sin(\theta)+b\sin(t)\cos(\theta))
$$
So you need to find the maximum of $ a\,cos(t)\cos(\theta)+b\sin(t)\sin(\theta)$ and $-a\,cos(t)\sin(\theta)+b\sin(t)\cos(\theta)$.
Can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
understanding 'p∈ (n, succ)' I understand that this may be a stupid question to some, but I've come to my wit's end trying to understand this condition:
if p ∈ (n, succ) then
I keep running across this in some pseudo code that I've been reading for the past 16 hours. I understand that '∈' typicaly symbolizes a set, ... | I'm not entirely sure that this answer is correct, but it seems reasonable and makes (some) intuitive sense.
Let us assume that we have some comparison method $\prec$ with respect to which we want to find successors and predecessors. For example, we could say that $p \prec n$ if $p$ "precedes" $n$.
So the predecessor ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving that $n!≤((n+1)/2)^n$ by induction I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this:
$V(1): 1≤1 \text{ true}$
$V(n): n!≤((n+1)/2)^n$
$V(n+1): (n+1)!≤((n+2)/2)^{(n+1)}$
and I've got : $(((n+1)/2)^n)\cdot(n+1)≤((n+2)/2)^{(n+1)}$ $((n+1)^n)n(... | It is more easy to prove this inequality without induction. Really $$0 < i\cdot (n + 1 - i) = \left(\frac{n+1}2 + \frac{2i - n - 1}2\right)\left(\frac{n+1}2 - \frac{2i - n - 1}2\right) = \left(\frac{n+1}2\right)^2 - \left(\frac{2i - n - 1}2\right)^2 \le \left(\frac{n+1}2\right)^2.$$
Multiply this inequalities for all $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Given a distribution find the probability. There are 4 elevators. So far...
Elevator 1 opened 45.455% of the time (5/11).
Elevator 2 opened 27.273% of the time (3/11).
Elevator 3 opened 18.182% of the time (2/11).
Elevator 4 opened 9.091% of the time (1/11).
Given this information, what is the probability of getting ea... | Two approaches:
1) Your lifts could be appearing with equal probability and you are just seeing this pattern because of your small sample. It slightly depends on how you measure distance from what might be expected
If you take a sum of squares approach, I think you may find that the probability of getting as extreme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$? I am interested in finding a closed form for the sum $\sum_{k=2}^N {N \choose k} \frac{k-1}{k}$. Does anyone know if there is some Binomial identity that might be helpful here?
Thank you.
| You might consider expanding $$(1+x)^N - \int \frac{(1+x)^N}{x} dx$$ and then letting $x=1$. This will need some slight adjustment as it has a few extra terms compared with your sum.
The problem is the integral as it involves a hypergeometric function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/523685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
The Banach-Steinhaus theorem for seminormed spaces Assume that we have a vector space $X$ over reals with a countable sequence of seminorms $p_n$ on $X$ such that:
$$
p_n(x)\leq p_{n+1}(x) \textrm{ for } n\in \mathbb N, x\in X,
$$
$$
\textrm{ for } x\in X\setminus \{0\} \textrm{ there is } n\in \mathbb N \textrm{ su... | Yes, this is equicontinuous. Since the topology is defined by a sequence of seminorms, so you can define the equicontinuity by the $p_k$.
This is the problem about Banach-Steinhaus theorem on Frechet space (or metric linear spaces.)
Maybe you can see Rolewicz'book "Metric Linear Spaces" or goolge for more about this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Any hints on how to graph this piece wise function? I am supposed to sketch the graph of
y= |x-1| if 0≤x≤2
|x-3| if 2≤x≤4
and specify any x or y intercepts
i'm just confused about graphing it because of the absolute value signs. Any help or ideas?
| You can sketch the functions $x-1$ on $0\leq x\leq2$ nad $x-3$ on $2\leq x\leq4$, then you can get absolute from of your graph and obtain main graph.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/523869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that if $A$ and $B$ are sets, then $A \subseteq B$ if and only if $A \cap B = A$. I'm working through a real analysis textbook, and it starts out with set theory. The first exercise is
Show that if $A$ and $B$ are sets, then $A \subseteq B$ if and only if $A \cap B = A$.
I think I proved it correctly but I'm not... | Since you are just starting, I would suggest to be verbose instead of pulling everything in a single sentence.
To prove $A \subseteq B$ iff $A \cap B = A$, you have to
*
*show $A \cap B = A$ given $A \subseteq B$. That is to
*
*show $A \cap B \subseteq A$.
*show $A \subseteq A \cap B$.
*show $A \subseteq B$ giv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/523944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A set is infinite iff there is a one-to-one correspondent with one of its proper subsets? Maxwell Rosenlicht claims in "Introduction to analysis" that a set is infinite if and only if it may be placed into one-to-one correspondence with a proper subset of itself.
He says this is self-evident because a finite set cannot... | This is known as
the Dedekind definition
of a set being infinite.
Here is more:
http://en.wikipedia.org/wiki/Dedekind-infinite_set
As an exercise,
you might try to show that
this is equivalent
to the definition
stating that
the set,
or some subset of it,
can be placed into a
1-1 correspondence with
the positive intege... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 8,
"answer_id": 4
} |
Finding minimum $\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}$ I would appreciate if somebody could help me with the following problem
Q. Finding maximum minimum
$$\frac{x+y}{z}+\frac{x+z}{y}+\frac{y+z}{x}(\text{where} ~~x,y,z>0)$$
| Note that if $F(x,y,z) = \frac{x+y}{z}+\frac{x+z}{y} + \frac{y+z}{x}$, then $F(kx,ky,kz)=F(x,y,z),\ k>0$. So we will use
Lagrange multiplier method. Let $g(x,y,z)=x+y+z$. Constraint is $x+y+z=1$.
$$\nabla F = (\frac{1}{z}+\frac{1}{y} - \frac{y+z}{x^2},\frac{1}{z}+\frac{1}{x} - \frac{x+z}{y^2},\frac{1}{y}+\frac{1}{x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
limit of summation Using Riemann integrals of suitably functions, find the following limit
$$\lim_{n\to \infty}\sum_{k=1}^n \frac{k}{n^2+k^2}$$
Please help me check my method:
Let $$f(x)=\frac{x}{1+x^2}$$
For each n$\in$ $\Bbb N$, let partition $$P_n=({\frac{k}{n}:0\le k\le n})$$ and $$\xi^{(n)}=(\frac{1}{n},\frac{2}{n... | Your approach is fine and Riemann sums are definitely the way to go.
Anyway, I will show you an interesting overkill. Since:
$$ \frac{2k}{k^2+n^2}=\frac{1}{k+in}+\frac{1}{k-in}=\int_{0}^{+\infty}e^{-kx}\left(e^{-inx}+e^{inx}\right)\,dx $$
we may write the original sum as:
$$\begin{eqnarray*} S_n=\sum_{k=1}^{n}\int_{0}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Linear Algebra : Eigenvalues and rank 1) A $4\times4$ square matrix has distinct eigenvalues $\{0, 1, 2, 3\}$. What is its rank?
2) Let $a,b\in\mathbb{R}^n$ be two non-zero linearly independent vectors, and let $\alpha,\beta\in\mathbb{R}$ be two non-zero scalars.
i) What is the rank of the matrix $M = \begin{bmatrix}a&... | *
*Your answer is correct (but it seems for the wrong reason; see below). The equation $A\mathbf{x}=0\mathbf{x}$ (implying $\mathbf{x}$ is an eigenvector with eigenvalue $0$, or $\mathbf{x}=\mathbf{0}$) is the same as $A\mathbf{x}=\mathbf{0}$ (implying $\mathbf{x}$ is in the null space of $A$). In other words:
The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$a+b+c=0;a^2+b^2+c^2=1$ then $a^4+b^4+c^4$ is equal to what?
$a+b+c=0;a^2+b^2+c^2=1$ then $a^4+b^4+c^4$ is equal to what?
I tried to solve this problem, and I get $a^4+b^4+c^4 = 2(a^2b^2 + 1)$ but I'm not sure if it's correct
| $$a+b+c=0\Rightarrow c=-(a+b)$$
$$1=a^2+b^2+(a+b)^2=2a^2+2ab+2b^2=2(a^2+ab+b^2)\Rightarrow a^2+ab+b^2=\frac12$$
$$\begin{align*}
a^4+b^4+c^4 &= a^4+b^4+(a^4+4a^3b+6a^2b^2+4ab^3+b^4)\\
&= 2(a^4+2a^3b+3a^2b^2+2ab^3+b^4)\\
&= 2(a^2(a^2+ab+b^2)+b^2(a^2+ab+b^2)+ab(a^2+ab+b^2))\\
&= a^2+ab+b^2=\frac12
\end{align*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/524335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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For each of series find the smallest $k$, that $a_n = O(n^k)$ Hey I need you to check my solutions:
a) $a_n = (2n^{81.2}+3n^{45.1})/(4n^{23.3}+5n^{11.3})$
This one is done from $\sum_{i=1}^{k} O(a_i(n)) = O(max\lbrace a_i,..,a_k \rbrace )$
So it's $n^{81.2}/n^{23.3} = n ^ {57.9}$, thus $k = 57.9$
b) $a_n =5^{\log_2(n)}... | b) $5^{\log_2n}=2^{\log_25\log_2n}=(2^{\log_2n})^{\log_25}=n^{\log_25}$
c) Try to show that no matter what $k$ is, $a_n$ is not $O(n^k)$.
d) There is no smallest $k$, but try to show that $k=1$ doesn't work, but that for every positive $\epsilon$, $k=1+\epsilon$ works.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/524406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Inequality $|z_1+z_2|^2 \le (1+|z_1|^2)(1+|z_2|^2)$ I have a problem to prove this inequality
$|z_1+z_2|^2 \le (1+|z_1|^2)(1+|z_2|^2)$ $\forall (z_1, z_2)\in \mathbb{C}$.
I tried to take the right hand set and subtract the lfs and after simplification I got this:
$1+(ax)^2+(by)^2 -2(ax+by)+(ay)^2+(bx)^2$ and I couldn... | Put $z_k=r_k(\cos\theta_k+i\sin\theta_k),k=1,2$
$(z_1+z_2)^2=r_1^2+r_2^2+2r_1r_2\cos(\theta_1-\theta_2)$
$(1+|z_1|^2)(1+|z_2|^2)=(1+r_1^2)(1+r_2^2)=1+r_1^2+r_2^2+r_1^2r_2^2$
this will be $\ge r_1^2+r_2^2+2r_1r_2\cos(\theta_1-\theta_2)$
$\iff 1+r_1^2r_2^2\ge 2r_1r_2\cos(\theta_1-\theta_2)$
But $1+r_1^2r_2^2\ge 2r_1r_2$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Prove property of floor function (one with square roots) I want to prove that:
$$\lfloor\sqrt{x}\rfloor=\lfloor\sqrt{\lfloor x\rfloor}\rfloor$$
It's true that (by definition of floor operation):
$$\lfloor\sqrt{x}\rfloor\leq\sqrt{x}<\lfloor\sqrt{x}\rfloor+1$$
$$\lfloor\sqrt{\lfloor x\rfloor}\rfloor\leq\sqrt{\lfloor x\rf... | Let $x=i+f$ where $i\in \Bbb{Z},f\in[0,1)$ .
$\therefore\sqrt x=\sqrt{i+f}$
Now $\exists g\in[0,1)$ such that $0<g<f$ and $f=2\sqrt ig+g^2$
Also $\exists h\in[0,1)$ such that $h>f$ and $f=-2\sqrt ih+h^2$
So $$\sqrt x=\sqrt i+g=\sqrt i-h$$
However $[\sqrt i]\leq [\sqrt i+g]\le [\sqrt i]+1$ and $[\sqrt i]-1\leq [\sqrt i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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find $\lim_{n \to \infty} \frac{(n^2+1)(n^2+2) \cdots (n^2+n)}{(n^2-1)(n^2-2) \cdots (n^2-n)}$
Find $$\lim_{n \to \infty} \frac{(n^2+1)(n^2+2) \cdots (n^2+n)}{(n^2-1)(n^2-2) \cdots (n^2-n)}$$
I tried to apply the squeeze theorem, yet none of my attempts led me to the solution.
| Let $f(n)$ be defined by
\begin{align*}
f(n) &= \frac{(n^2+1)(n^2+2)\cdots(n^2+n)}{(n^2-1)(n^2-2)\cdots(n^2-n)}\\
&= \frac{(1 + {1 \over n^2} )(1 +{2 \over n^2})\cdots(1 + {n \over n^2})}{(1 - {1 \over n^2} )(1 - {2 \over n^2})\cdots(1 - {n \over n^2})}
\end{align*}
Then
$$\ln f(n) = \sum_{k=1}^n \ln\left(1 + {k \ove... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Proving that a statement about $<$ is valid I need to do assignment for my homework, in which I need to prove that the following statement is valid.
$$
(s<t \text{ and } t<u)\implies(s<u)
$$
I need to do this assignment using the laws and definitions of inequality.
The problem is that I don't know how to do it.
Can som... | Well actually you define real numbers $\mathbb{R}$ a set for which your inequality is valid. Is called transitive property. As stated in previous comments.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/524646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
neighborhood space metric Let $M$ be a metric space and $a \in M$. We say that $V \subseteq M$ is a
neighborhood of $a$ when $a \in \operatorname{Int}(V)$.
Show that if $(x_n)$ is a sequence in $M$, then the following are equivalent:
*
*$\lim x_n = a$;
*For every neighborhood $V$ of $a$ there is $n_{0} \in \mathbb{... | I assume that your definition of convergence is as follows:
If $M$ is a metric space with metric $d,$ and $a\in M$ and $(x_n)$ is a sequence in $M,$ then we say that $\lim x_n=a$ if for all real $\epsilon>0,$ there is some $n_0\in\Bbb N$ such that $d(x_n,a)<\epsilon$ whenever $n\ge n_0$.
Here's what I recommend, then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If n is such that every element $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is a root of $x^2-1$. Prove that $n$ divides 24. I have a hard time formulating proofs. For this problem, I can see that if $n$ is equal to $8,$ this statement is true. $(\mathbb{Z}/8\mathbb{Z})^{\times}$ includes elements: $1,3,5,7$, and all of th... | $\,\overbrace{{\rm if}\,\ 5\!\nmid\! n\,\ {\rm then}\,\ n\,|\, \color{#90f}{24}\!=\! 5^2\!-\!1}^{\large \text{by hypothesis}},\,$ else $\,5\!\mid\! n =\!\! \overbrace{\color{#0a0}2^{\large j}\color{#c00}k}^{\large {\rm odd}\ \color{#c00}k}\!, \,$ & $\,(\color{#0a0}2\!+\!5\color{#c00}k)^{\large 2}\!\not\equiv 1\bmod{5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 3
} |
Does there exist a symmetric tridiagonal matrix with zero determinant? I will like to know whether there exists a symmetric tridiagonal matrix with zero determinant? I will refer the definition of a tridiagonal matrix to the one found in Wikipedia:
"A tridiagonal matrix is a matrix that has nonzero elements only on th... | The trivial example is
$$
\pmatrix{0 & 0 \\ 0 & 0}.
$$
Or you could consider:
$$
\pmatrix{1 & 1 \\ 1 & 1} \quad\text{or} \quad\pmatrix{1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1}.
$$
Otherwise if you only want non-zero entries of the diagonals, then how about
$$
\pmatrix{1 & a & 0 \\ a & 1 & a \\ 0 & a & 1}
$$
where $a = \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Show that $\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}$ Based on the definition of $e: = \lim_{x\to\infty} \left(1+\frac1x \right)^x$, how can we show that
$$\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}?$$
So far I've tried changing variables, $\eta = \frac{-x}{\lamb... | $\newcommand{\abs}[1]{\left\vert #1\right\vert}$
When $\lambda = 0$, the result is trivially true: $1 = {\rm e}^{0}$. Let's consider the cases $\lambda \not= 0$:
*
*$\large\lambda < 0$
$$
\lim_{x\to \infty}\left(1 - {\lambda \over x}\right)^{x}
=
\lim_{x\to \infty}\left[%
\left(1 + {\abs{\lambda} \over x}\right)^{x/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/524957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Elements of order $10$ in $\Bbb Z_2 \times \Bbb Z_{10}$
How many elements in the group $\mathbb Z_2 \times \mathbb Z_{10}$ are of order $10$?
I think the easiest way to answer this question might be to write them out, but I'm not sure how to write them out.
| As you noted in a comment, the order of a group means one thing, and the order of an element means entirely another.
If $(A,\circ_A)$ and $(B,\circ_B)$ are groups, then the direct product $(A,\circ_A)\times(B,\circ_B)$ (often written just $A\times B$) is defined as the group $(A\times B,\circ)$, where $(a_1,b_1)\circ(a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trigonometry and differential equations I have the expression $A\cos(wt)+B\sin(wt)$ and I need to write it in the form $r\sin(wt−\theta)$.
I then have to determine $r$ and $\theta$ in terms of $A$ and $B$. If $R\cos(wt - \delta) = r\sin(wt - \theta)$, determine the relationship among $R, r, \theta,$ and $\delta$.
I th... | Just use the sine addition theorem:
$$r \sin{(\omega t-\theta)} = r \sin{\omega t} \, \cos{\theta} - r \cos{\omega t} \, \sin{\theta}$$
Comparing...we get
$$A = r \cos{\theta}$$
$$B = r \sin{\theta}$$
so that
$$r^2 = A^2+B^2$$
$$\theta = \arctan{\frac{B}{A}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/525125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Lambda Calculus: Reducing to Normal Form I'm having trouble understanding how to reduce lambda terms to normal form. We just got assigned a sheet with a few problems to reduce, and the only helpful thing I've found is the following example in the book:
(λf.λx.f(fx))(λy.y+1)2
->(λx.(λy.y+1)((λy.y+1)x))2 //How'd it get ... | Basically, the function $(\lambda f.\lambda x.f(fx))$ is applied to the argument $\lambda y.y+1$. This step is also called beta reduction.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to Project a Symmetric Matrix onto the Cone of Positive Semi Definite (PSD) Matrices How would you project a symmetric real matrix onto the cone of all positive semi-definite matrices?
| If you merely want to find a projection $\pi$ such that $\pi(S)$ is positive semidefinite for some fixed real symmetric matrix $S$, you may first orthogonally diagonalise $S$ as $QDQ^\top$ and then define $\pi: M\mapsto Q\Sigma Q^\top M$, where $\Sigma$ is a 0-1 diagonal matrix whose $i$-th diagonal entry is $1$ if the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
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Linear Algebra: Geometric means What is the geometric means of
$$M=\begin{pmatrix}\cos \theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$$
I would like to show that its eigenvector is not real.
| This matrix rotates the plane by $\theta$ degrees anti-clockwise. To find its eigen values, simply solve the equation
$$
det(A-\lambda I) = 0
$$
You will get
$$
(\cos(\theta) - \lambda)^2 +\sin^2(\theta) = 0
$$
$$
\Leftrightarrow 1 - 2\cos(\theta)\lambda + \lambda^2 = 0
$$
$$
\Leftrightarrow \lambda = \cos(\theta) \pm ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Reference for Gauss-Manin connection I wish to understand the notion of ``Gass-Manin connection''. I have some understanding of differential geometry, topology and algebraic geometry. Where should I begin? IF the sources are freely available, that will be good. Even better will be if someone can give some little motiva... | Let $π : X \to T$ be a smooth algebraic family of complex projective manifolds of
dimension d such that the parameter space $T$ is a nonsingular variety. Consider the local
systems $R^kπ_∗\mathbb C$, $0 ≤ k ≤ 2d$, and the associated vector bundles
$\mathcal H^k:= (R^kπ_∗\mathbb C)\otimes_\mathbb C \mathcal O_T$
over $... | {
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"url": "https://math.stackexchange.com/questions/525575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
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Word Problem (Inverse Proportions) This is the problem:
There is a job that has to be made in 60 days by a group of workers. After one day of work five workers more are added to the original group and they work one day. After this second day of work another 5 workers are added to the group, and then they work together ... | One day for the whole group was saved by adding $5 \cdot 58 + 5 \cdot 57$ days by one worker. So there were originally $5 \cdot 58 + 5 \cdot 57 = 675$ workers.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Basis for recurrence relation solutions So, I have a question: Imagine a recurrence relation $U(n+2) = 2U(n+1) + U(n)$.
How do I determine the dimension (and the vectors that constitute the basis) of a vector space which contains all sequences that satisfy that rule?
| The sequences that result from such a recurrence relation are determined by the initial conditions, which would ordinarily be prescribed as $U(0)$ and $U(1)$. It should be clear that any values can be assigned for these first two values, and that once that is done the rest of the sequence is fully determined by repeat... | {
"language": "en",
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"source": "stackexchange",
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Solving equations where the solution is an operator Ok, so here's some context.
Solving regular equations we might have something like this:
$2 + x = 5$, solving for $x$ we get 3. We might even have an equation like $x + y = 5$ where there are multiple solutions.
But what's in common with all these equations is that th... | Equations where "operations" (really just a funny notation for functions) are the unknowns are called functional equations. Solving them can range from the trivial (like the examples you give) to very complex. Some techiques are given in the Wikipedia link above.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Intermediate fields between $\mathbb{Z}_2 (\sqrt{x},\sqrt{y})$ and $\mathbb{Z}_2 (x,y)$ Let $K=\mathbb{Z}_2 (x,y)$, where $x,y$ are independent, and $L$ be a splitting field extension of $(X^2 - x) (X^2 - y)$, then $[L:K] = 4$ and $L = K(\sqrt{x},\sqrt{y})$ where $\sqrt{x},\sqrt{y}$ are roots of $X^2-x$, $X^2 - y$ resp... | I don't like writing square roots, so let's just pick $K = \Bbb F_2(X^2,Y^2)$ and $L = F_2(X,Y)$.
Since $K \subset L$ is of degree $4$, any intermediate field is of degree $2$. As you said, every element of $L$ square to something in $K$ (in fact, the map $x \in L \mapsto x^2 \in K$ is an isomorphism of fields), so tho... | {
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How find the sum $\sum_{n=1}^{\infty}\frac{\binom{4n-4}{n-1}}{2^{4n-3}(3n-2)}$ today,I see a amazing math problem:
show that
$$\sum_{n=1}^{\infty}\dfrac{\binom{4n-4}{n-1}}{2^{4n-3}(3n-2)}=\dfrac{\sqrt[3]{17+3\sqrt{33}}}{3}-\dfrac{2}{3\sqrt[3]{17+3\sqrt{33}}}-\dfrac{1}{3}$$
This problem is from here.
But I consider s... | The transformed series in Marin's post seems suspiciously like Taylor's expansion of Bring radical
$$\begin{aligned}\sum_{n=1}^{\infty}\dfrac{\binom{4n-4}{n-1}}{2^{4n-3}(3n-2)} &= \sum_{n=0}^{\infty} \binom{4n}{n} \frac{1}{(3n + 1)2^{4n + 1}} \\ &= \frac{16^{1/3}}{2} \sum_{n=0}^{\infty} \binom{4n}{n} \frac{\left ( 16^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/526072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 2
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Irreducible subsets of a topological space I found this definition on Hartshorne, Algebraic geometry, page 3...
Definition A nonempty subset $Y$ of a topological space $X$ is irreducible if it cannot be expressed as the union $Y=Y_1\cup Y_2$ of two proper subsets, each one of which is closed in $Y$. The empty set is no... | How about this:
Generate a topology on $\mathbb{R} \times \mathbb{R}$ by defining closed sets to be the horizontal and vertical lines. (This gives a subbasis whose open sets are complements of lines; therefore all finite intersections of said complements yield a topology.) The closed sets are lines, finite collections ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A question about extreme points If the extreme points of the unit ball of $C[0, 1]$ are $\pm{1}$, where $C[0, 1]$ is the Banach space of all continuous real-valued functions on the unit interval, then what would the extreme points of the unit ball be if we considered all continuous complex-valued functions on the unit ... | The extreme points would be all continuous functions $f$ with $|f(t)| = 1$ for all $t \in [0,1]$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What CI should I use when finding Margin of Error? I've begun an assignment where I collect my own data. The assignment is now asking me to find margin of error with the collected data. I have nearly everything I need for the equation with the exception of the Confidence Interval.
Do you think it would matter what I ch... | It depends how accurate you want your results to be. An $x$% C.I. for the population mean is such that if you repeat the experiment / data collection $100$ times, the mean should lie within the C.I. $90$ times. So for any given experiment / set of data, there's a $90$% chance the population mean lies within the C.I.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivative of the $\sin(x)$ when $x$ is measured in degrees So a classic thing to derive in calculus textbooks is something like a statement as follows
Is $\frac{d}{dx}\sin(u)$ the same as the derivative of $\frac{d}{dx}\sin(x)$ where $u$ is an angle measured in degrees and $x$ is measured in radians? and of course the... | Yes, you're describing the same object $x$, but units matter. That is, the units you use to measure that object matter in describing what you mean. Like, 1 meter and 3.2808 feet are practically the same length, but are described through different units of measure.
And in Calculus, radians is a unit of measure that will... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Tax inclusive pricing I have a system where is user can enter a price (without tax) and a tax rate. I then calculate the total cost of the item.
Example:
Price:100.00
Tax percent: 10%
Final price: 110.00 = (100 + (100* (10/100))
I have got a request to work backwards and start with the final price and tax and determine... | I'm assuming you only need your output price accurate to two decimals. The meaning of "X is accurate to $n$ decimals" is that X is an approximation, but the difference between it and the true value is less than $\displaystyle\frac{5}{10^{n+1}}$ (we want to say $1/10^n$ but we have to account for rounding).
Technical d... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is b in the span of {w_1, .. , w_p}? Suppose b is in the span of { v_1 , ... , v_n }, and that each v_i is in the span of { w_1, ... , w_p }. Is b, then, in the span of { w_1, ... , w_p } ? If not, how could you modify the proposition so that it is true?
Not sure at all how to approach this.
| If $b$ is in the span of $\{v_1, \dots, v_n \}$, then
$$b = \sum_{i=1}^n \alpha_i v_i,\tag{1}$$
for some $\alpha_i$. Similarly, if each $v_i$ is in the span of $\{w_1, \dots, w_p\}$, then
$$v_i = \sum_{j=1}^p \beta_{ij} w_j,\tag{2}$$
for some $\beta_{ij}$. Now, substitute $v_i$ in $(1)$ with what you have in $(2)$. Now... | {
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How do I prove a "double limit"?
Prove $$\lim_{b \to \infty} \lim_{h \to 0} \frac{b^h - 1}{h} = \infty$$
I have never worked with double limits before so I have no idea how to approach the problem. Please don't use "$e$" in your solutions, since the above limit is part of the derivation of "$e$", so for all purposes ... | Let $f_b$ be the function defined by $f_b(x) = b^x$ . Then: $$\lim_{b \rightarrow \infty} \lim_{h \rightarrow 0} \frac{b^h - 1}{h} = \lim_{b \rightarrow \infty} f_b'(0) = \lim_{b \rightarrow \infty} \log(b) = \infty$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Recursively defining the set of bit strings set having more zeros than ones Question:
Recursively define the set of bit strings that have more zeros than ones.
I tried it this way:
$\Sigma\subset \{0,1\}^*$
Basis step: $0 \in \Sigma$
Recursive step: For any $x\in \Sigma$, $00x1\in L$
Is it a valid answer?
| It might be a little late to answer but this would be another way to answer it.
Basic Step: $0 \in S$.
Recursive Step: If $x, y \in S$ then $xy \in S$, $x1y \in S$, $1xy \in S$, $xy1 \in S$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Defining an ideal in the tensor algebra In the wikipedia article about exterior algebra:
The exterior algebra $Λ(V)$ over a vector space $V$ over a field $K$ is defined as the Quotient algebra of the tensor algebra by the two-sided Ideal $I$ generated by all elements of the form $x \otimes x$ such that $x \in V$.
I ... | If $S$ is a subset of a (non commutative) ring $R$, the ideal generated by $S$ consists of all elements of the form
$$
\sum_{i=1}^{n} a_i s_i b_i
$$
where $n$ is an arbitrary integer, $a_i,b_i\in R$ and $s_i\in S$.
This set is obviously closed under addition (by construction) and contains $0$; it's also closed by left ... | {
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"timestamp": "2023-03-29T00:00:00",
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Derivative of Gamma function In Computing the integral of $\log(\sin x)$, user17762 provided a solution which requires differentiating $\displaystyle \frac{\Gamma(2z+1)}{4^z\Gamma^2(z+1)}\frac{\pi}{2}$ with respect to $z$. How is this done?
I had actually got $\displaystyle\int_0^{\pi/2}\sin^{2z}(x)dx = \frac{\Gamma(z+... | How is the derivative taken? If you have
$$
\int^{\pi/2}_0 \! \sin^{2z} (x) \ \mathrm{d}x = \frac{\pi}{2}\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1)
$$
then differentiating both sides with respect to $z$ gives
\begin{align}
2\int^{\pi/2}_0 \! \sin^{2z}(x) \log(\sin(x)) \ \mathrm{d}x =
\frac{\pi}{2}&\left\{ 2\psi(2z+1)4^{-z}\... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $ \left( \frac1n \sum_{i=1}^n x_i^{p+1} \right) \geq \left( \frac1n \sum_{i=1}^n x_i \right)^{p+1} $ true? Let us suppose that $0 \leq p \leq 1$. All variables are assumed to be non-negative.
The function $x \mapsto x^{p+1}$ is strictly convex upwards, so $$ \left( \frac1n \sum_{i=1}^n x_i^{p+1} \right) \geq \l... | Check out Jensen's inequality. For a convex function $f$ we have
$$f\left(\frac{x_1 + \dots + x_n}{n}\right) \le \frac{f(x_1) + \dots + f(x_n)}{n}$$
and the inequality sign is reversed for concave functions.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the integral of this gaussian I want to know what is the following integral
$\int e^{-(y-\mu)^T \Lambda(y-\mu) } dy$
I am trying to see the properties of gaussian integral but I couldn't find anything for this one. Any help guys?
I want to know how given
where Z(x) is the partition function
This is the paper
... | For the integral to exist, $\Lambda$ has to be positive definite. If $\Lambda$ is positive definite, then we can decompose $\Lambda$ as $\Lambda = R^TR$ (for instance, Cholesky decomposition).
We then have $$I = \int_{\mathbb{R}^n} \exp \left(-(y-\mu)^T R^T R (y- \mu)\right)dy$$
Let $x = R(y -\mu)$. We then have $dx = ... | {
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Differentiation answer check $$f(x)=\arcsin \sqrt{\frac{x}{x+1}} + \arctan \sqrt{x} \mbox{.}$$
$$f'(x) = \frac{1}{\sqrt{1-\frac{x}{x+1}}} \cdot \frac{1}{2}\sqrt{\frac{x+1}{x}} \cdot -x^{-2} + \frac{1}{1+x} \cdot \frac{1}{2\sqrt{x}} = -\frac{\sqrt{x+1}}{2\sqrt{1-\frac{x}{x+1}}{\sqrt{x}}\frac{1}{x^2}} + \frac{1}{2\sqrt{x... | HINT:
Let $\displaystyle\arcsin\sqrt{\frac x{x+1}}=y$
$$\implies \sqrt{\frac x{x+1}}=\sin y\implies \frac x{x+1}=\sin^2y$$
$$\implies x=\tan^2y\implies \sqrt x=\tan y \implies y=\arctan \sqrt x$$
Alternatively, let $\displaystyle\arctan \sqrt x=u\implies \sqrt x=\tan u\implies x=\tan^2u$
$\displaystyle\implies \frac x... | {
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Calculate a ratio from part of a range. I'm building a smartphone javascript application but my question today is really only math related. To give you a quick idea of what I'm doing, my code checks the smartphone's gyroscope to determine how much it is rotated. Based on that rotation, it's supposed to pan a background... | You want a line that goes through $(140,0)$ and $(220,-100)$. The two point form says it is $y-0=\frac {-100-0}{220-140}(x-140)$. $x$ is the gyro, $y$ the output. I left the constants so you could match them up with the equation in case you want to change them in the future.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/527421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing rational numbers are algebraic A polynomial with integer coefficients is an expression of the form:
$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$
where $a_n$, $a_{n-1}, \ldots, a_1, a_0$ are integers and $a_n$ is not equal to $0$.
a zero of the polynomial is a $c \in \mathbb{R}$ such that $f(c)=0$
A... | The rational number $5/7$ is a zero of the polynomial $7x+(-5)$. We have $n=1$, $a_1=7$, $a_0=-5$.
So try showing that works with every rational number.
| {
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"source": "stackexchange",
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Solve the triple integral $\iiint_D (x^2 + y^2 + z^2)\, dxdydz$ How does one go about solving the integral:
$$
\iiint_D (x^2 + y^2 + z^2)\, dxdydz,
$$
where
$$
D=\{(x,y,z) \in \mathbb{R}^3: x^2 + y^2 + z^2 \le 9\}.
$$
I believe I am supposed to convert to spherical coordinates but I would need some help with how th... | A quick way to evaluate it is to note that the volume of the spherical shell from radius $r$ to radius $r + \Delta r$ is approximately $4\pi r^2 \Delta r$, so your result should be
$$\int_0^3 r^2 (4\pi r^2) \,dr$$
$$= {4 \over 5} \pi r^5\bigg|_{r=0}^3$$
$$= {4 \over 5} 3^5 \pi$$
$$={972 \pi \over 5}$$
To do it properly... | {
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"timestamp": "2023-03-29T00:00:00",
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Prime ideals in $C[0,1]$ Are there any prime ideals in the ring $C[0,1]$ of continuous functions $[0,1]\rightarrow \mathbb{R}$, which are not maximal?
Perhaps, I duplicate smb's question, but this is an interesting problem!
Could you give me any hint or give a link to some literature?
| If $R$ is a reduced commutative ring, then the following statements are equivalent:
*
*$\dim(R)=0$
*Every prime ideal of $R$ is maximal.
*For every $a \in R$ we have $(a^2)=(a)$.
*For every $a \in R$ there is some unit $u \in R$ such that $ua$ is idempotent.
In that case, $R$ is called von Neumann regular. The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/527658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 3
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The Game of Chess In how many ways can the first four moves (two from each side) be made in a game of chess?
I've seen and solved one on the first two moves. Now I wonder what the answer will look like for the first four moves
| I bielive it's better to ask computer (implement brute force algorithm) than to try to calculate this number by hand.
For any possible (or imaginary) position there are no more than 121 moves for each side.
Let count roughly: 2 for each pawn (en passant is possible to be the third or the fourth move of the pawn only an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/527730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How would I computationally find a generating functions coefficient? More specifically $a_n=(1,5,10,25,100,500,1000,2000,10000)$
$G(x)=\Pi_{n=0}^8 \sum_{i=0}^{\infty}x^{a_ni}$
So when $a_n=1$ the series = $1+x+x^2+x^3+...$
$a_n=5, 1+x^5+x^{10}+x^{15}+...$
$a_n=10, 1+x^{10}+x^{20}+x^{30}+...$
etc.
| Here is a start.
You want
$G(x)=\Pi_{n=0}^8 \sum_{i=0}^{\infty}x^{a_ni}$,
so look at the sums, which are
just geometric series.
$\sum_{i=0}^{\infty}x^{a_ni}
=\sum_{i=0}^{\infty}(x^{a_n})^{i}
=\dfrac{1}{1-x^{a_n}}
$
so
$G(x)
=\Pi_{n=0}^8 \dfrac{1}{1-x^{a_n}}
= \dfrac{1}{\Pi_{n=0}^8(1-x^{a_n})}
$.
At this point,
I'd prob... | {
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"source": "stackexchange",
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Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$. Let $f$ be a continuous, nonnegative, real-valued function and $$\int_0^{\infty}f(x)dx<\infty.$$ Prove that $$\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0.$$
A start: If $\lim\limits_{n\to\infty}\int_0^{n}xf(x)dx$ is finite, then it's obvious. Otherwise, perf... | $$\forall n\geqslant k,\qquad0\leqslant\frac1{n}\int_0^{n}xf(x)\mathrm dx\leqslant\frac1n\int_0^{k}xf(x)\mathrm dx+\int_k^{\infty}f(x)\mathrm dx$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
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Semigroups (ideals of a semigroup)
How many ideals are there in the $\mathbb Z_{28}$?
$\mathbb Z_{28}=\{0, 1, 2, 3, 4, 5, ..., 27\}$ is a semigroup under multiplication modulo 28.
| As a ring, $\mathbb{Z}_{28}$ is isomorphic to $\mathbb{Z}_{7} \times \mathbb{Z}_{4}$.
The multiplicative structure of $\mathbb{Z}_{7}$ is a cyclic group of order $6$ plus a zero. It contains two (nonempty) ideals: $0$ and $\mathbb{Z}_{7}$. The multiplicative structure of $\mathbb{Z}_{4}$ is a semigroup with a chain of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/528075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Translating English into First Order Logic Translate the following into a formula of first-order logic. "A language L that is regular will have
the following property: there will be some number N (that depends on L) such that if s is a string
in L (a string is a sequence of characters) whose length is at least N then s... | Before putting everything into symbols, try to rewrite the sentence in a more logical way. For example:
For all $L$, if $L$ is a language and $L$ is regular, then there exist $N$ and $s$ such that if
*
*$N \in \mathbb{N}$, and
*$s \in L$, and
*$\mathrm{length}(s) \geq N,$
then there exist $x$, $y$ and $z$ such tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/528213",
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"source": "stackexchange",
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general solution for a 4th order PDE I have a fourth order partial differential equation of motion of a tube, with clamped boundary conditions, I don't know what would be the general solution for $W$:
$$EI \frac{d^4 w(x,t)}{dx^4} + MU^2 \frac{d^2 w(x,t)}{dx^2} + 2MU\frac{d^2 w(x,t)}{dx\,dt} +M \frac{d^2 w(x,t)}{dt^2}=... | If you want a single Fourier mode, you're looking for a solution of the form
$$W=e^{ikx+\omega t}$$
The only thing you're missing is the dispersion function $\omega(k)$. If you plug the above solution into your equation, you get the condition
$$EI k^4-M (k U-i \omega )^2=0\ ,$$
which has the solution
$$\omega = i \left... | {
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"source": "stackexchange",
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Linear span proof
Let $\mathbb{F}$ be a field, $A \in \cal{M}_{n\times n}(\mathbb{F})$
and $W=\left\{B \in {\cal{M}}_{n\times n}(\mathbb{F})|AB=BA \right\}$.
Suppose there exists a column vector $v \in \mathbb{F}^n$ such that
$\left\{v,Av,A^2v,..., A^{n-1}v\right\}$ is a basis for
$\mathbb{F}^n$. Prove that $... | Hint
$$Bx = \sum_{k=0}^{n-1} \lambda_k A^k v$$
For any $x$ (since we have a basis). Now check $ABx$ and $BAx$ and compare coefficients.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Uniform convergence of $\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n}}{n!}$ on $\mathbb{R}$? For every $r>0,$ the series $f(x)=\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n}}{n!}$ converges uniformly on $[-r,r]$. May I know how to prove/disprove that $\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n}}{n!}$ converges uniformly on $\mathbb{R}$ ?
... | Hint: This Taylorseries is recognizable as function $e^{-x^{2}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/528468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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on two dimensional graded vector spaces I read the following statement:
Consider a graded vector space $V$ with basis $\{a, b\}$ such that $a \in V^2$
and $b \in V^5$.
Does this mean that $V=\bigoplus_{i\geq 0}V^i$ such that all $V^i$ are $0$ except $V^2$ and $V^5$ hence we can simply write $V=V^2\oplus V^5$ and if... | We can write $V = V^2\oplus V^5$, but writing as in the second part of your question does not respect the grading. If we just write $V$ is two dimensional, or $V = A \oplus B$ where $A$ and $B$ are one-dimensional we have lost information about the grading and the homogeneous elements of $V$ (that is the multiples of $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding matrix for parallel transport map. Consider the surface $S$ given by the patch $$\sigma (u, v) = (u, v, 0)$$ and the points $$p(0, 0, 0),\ q(1, 1, 0) \in S.$$ Choose bases for $T_pS$ and $T_qS$ and write down the matrix for the parallel transport map $P_{\gamma_{p,q}}$ where $\gamma(t) = (t, t, 0)$.
| The surface that you have is just a flat plane. In that case, parallel transport is just good, old-fashioned translation, i.e. to parallel transport a vector you slide it around the plane while keeping it parallel to, and pointing in the same direction as, the original.
Hence, the matrix would be the two-by-two ident... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Estimating $\sum_{k=1}^N a_kb_k$ given the means $\bar a_k,\bar b_k$ and determining the error I need to calculate the following expression:
$$\sum_{k=1}^N a_k b_k$$
I know the average values of $a_k$ , defined as $\overline {a_k} = {\sum_{k=1}^N a_k \over N } $ and $b_k$ , defined as $\overline {b_k} = {\sum_{k=1}^N b... | Mathematically, the largest the product can be (assuming all then numbers are positive and "reasonable size") is $N^2\overline{a_k}\overline{b_k}$ when all but one of the $a_k,b_k$ are zero. The smallest it can be is zero if $a_k$ can be zero. In an engineering sense, you probably have some information on how much va... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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why check all primes under the root of an interger? I am in high school and I need to factorize numbers. My teacher told me to check all numbers which are smaller than the root of the number I want to factorize. This seems to work just fine, but I do not know why it works and neither does my teacher.
Is there someone h... | First, you need only check the prime numbers less than $\sqrt n$, for the number $n$ that you are trying to factorize.
Why? If you find a composite number that goes into $n$, all of that composite number's prime factors will also go into $n$ (basically, if $p$ divides $a$ and $a$ divides $n$, then $p$ divides $n$).
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Convergence almost everywhere and convergence in measure Let $(\mathbb{R},\mathcal{L},m)$, let $f_{n}(x)=n\chi_{[0,\frac{1}{n}]}$ then the sequence converges to $0$ everywhere except at $x=0$ thus $f_{n}$ converges a.e.
Then in my book (Folland) we have that if $f_{n}\to f$ a.e and $|f_{n}|\le g\in L^{1}$ then $f_{n}\t... | This works always, for example $(\mathbb R, \mathcal B, \lambda)$ forms a $\sigma$-finite but infinite measure space on which this works.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Fourier Series: Shifting in time domain I am reading "Fourier Transformation for Pedestrians" from T. Butz. He speaks about what happens to the Fourier coefficients when the function is shift in time. I have copied the equation I have a problem with:
I don't understand the logic behind going from $f(t-a)$ in the first... | He is doing $t'=t-a$, note that he is also shifting the interval of the integral, and also $t=t'+a$, replaced in the exponent and applying the exponent rule.
For the non-complex, he is shifting from $\mathbb C$ to $\mathbb R^2$, but just by notation.
$$C_k=A_k+iB_k=\{A_k;B_k\}\\
e^{i\theta}=\cos\theta+i\sin\theta=\{\co... | {
"language": "en",
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Can someone explain the solution to this statement? Say C: set of courses
P(x,y): 'x is a prerequisite for course y'
statement: 'some courses have several prerequisites'
symbolically:
∃ x ∈ C, ∃ y ∈ C, ∃ z ∈ C, P(y, x) ∧ P(z, x) ∧ y ≠ z
I don't really understand how you get the symbolic expression from the verbal ex... | The statement
$$\exists x\in C\,\exists y\in C\,\exists z\in C\Big(P(y,x)\land P(z,x)\land y\ne z\Big)$$
says that
there is a course $x$ such that there are courses $y$ and $z$ that are prerequisites for $x$ and are different courses.
In less convoluted language, this says that there is a course $x$ that has at leas... | {
"language": "en",
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$\sigma\mathcal C$ is the $\sigma$-algebra generated by $\mathcal C$. Show $\sigma\mathcal C\subset\sigma\mathcal D$ if $\mathcal C\subset\mathcal D$. If $\mathcal{C}$ and $\mathcal{D}$ are two collections of subsets of $E$. How do I prove the following:
$$\mathcal{C}\subset\mathcal{D}\implies\sigma\mathcal{C}\subset\s... | You could also use the fact that $\sigma\mathcal{A}$ is the smallest $\sigma$-algebra containing $\mathcal{A}$. If you know this fact then you can derive the desired result in a few easy steps:
*
*$\sigma\mathcal{D}$ is the smallest $\sigma$-algebra containing $\mathcal{D}$;
*$\mathcal{C}$ is contained in $\mathca... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Are power series in a normal matrix themselves normal? Are (convergent) power series in a normal matrix themselves normal? I have looked around for this result, and not found it. How might we prove it?
| Yes, because a matrix is normal if and only if it is unitarily diagonalizable, we can simultaneously diagonalize a matrix and analytic functions of that matrix, given that said function is analytic in a domain containing the spectrum of the matrix. More concretely, if $X = S \Lambda S^{-1}$, where $\Lambda$ is the diag... | {
"language": "en",
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If I weigh 250 lbs on earth, how much do I weigh on the moon? One of my homework questions is to determine how much a 250 lb person weighs on the moon. I first googled a calculator for this and found that the weight is 41.5 lbs. So I tried to derive it myself and I cannot seem to get the correct answer.
Here is what ... | $1112$ N is the force on earth: it’s (approximately) $$113.5\text{ kg}\cdot 9.8\frac{\text{m}}{\text{s}^2}\;.$$ To get the force on the moon you want
$$113.5\text{ kg}\cdot 1.625\frac{\text{m}}{\text{s}^2}\;,$$
which you’ll then have to convert to pounds. Of course you could simply multiply $250$ by the ratio of gravit... | {
"language": "en",
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What is the length of the bar needed to represent 75 kilometers( in centimeters)? In a bar graph, 1 centimeter represents 30 kilometers. What is the length of the bar needed to represent 75 kilometers( in centimeters)?
| HINT: $1$ cm represents $30$ km, so $2$ cm reprsents $60$ km, and $3$ cm represents $90$ km; clearly the answer is between $2$ cm and $3$ cm. How many $30$ km segments can you fit into a $75$ km stretch of road? (The answer won’t be a whole number.) That’s the number of $1$ cm segments that you’ll need to represent tha... | {
"language": "en",
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evaluating norm of sum of roots of unity let $l_1,...,l_n$ be roots of unity.
I want to prove that the norm(the product of all conjugates)of $a=l_1+...+l_n$ is not greater than $n$, not smaller than $-n$.
how can I do to prove this?
| I don't know why you might think the field norm would be at most $n$ in absolute value. For example, if $x=e^{2\pi i/11}$ then I calculate that the norm from ${\bf Q}(x)$ to $\bf Q$ of $x+x^2+x^4$ is 23.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Duality discrete math problem This is the only answer I got wrong on my HW and the prof does not want to give us the correct answers before our midterm
The dual of a compound proposition that contains only the logical operators $\lor$ , $\land$ , and
$\neg$ is the compound proposition obtained by replacing each $\lo... | You did more as you should. Forming the dual just wants you to replace $p$ by $\neg p$ for each literal $p$, $\lor$ by $\land$ and vice versa and $T$ by $F$. You did more than that, in dualising (2), one obtains
$$ \neg p \lor \bigl( \neg q \land (\neg r \lor F)\bigr) $$
(you missed a $\neg$ in front of $r$). We have ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What distribution do the rows of the Stirling numbers of the second kind approach? In wikipedia about the Pascal triangle:
Relation to binomial distribution
"When divided by 2n, the nth row of Pascal's triangle becomes the binomial distribution in the symmetric case where p = 1/2. By the central limit theorem, this dis... | I think this is what you are looking for:
Stirling Behavior is Asymptotically Normal,
L. H. Harper,
The Annals of Mathematical Statistics
Vol. 38, No. 2 (Apr., 1967), pp. 410-414
Link
| {
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What is the most likely codeword sent? I have a question that asks me: Consider the code C for which the parity check is:
$M =$ \begin{bmatrix}
1 & 1 & 0 \\
1 & 1 & 0\\
1 & 0 & 1 \\
0 & 1 & 1\\
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}
What is the most likely codeword sent if we recei... | Let $t$ be the codeword that was originally transmitted; then $tM = 0$. Suppose that over transmission some error vector $e$ was added to $t$, so that $w = t + e$. Then $wM = (t+e)M = 0+eM = eM$. This value is called the syndrome. I'll denote it $s$.
Let's calculate the syndrome of your received word:
$$s = wM = \begin... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help with differentiable function $f:\mathbb{R^2} \to \mathbb{R}$ I need help with a question that appeared in my test.
True or False:
Let $f$ be a function $f:\mathbb{R^2} \to \mathbb{R}$ not differentiable at (0,0), then $f^2$ is not differentiable at (0,0).
I answered False. I gave an example of $\sqrt{x+y-1}$ wh... | The function you specified is not defined on $\mathbb{R}^2$.
Try $f(x,y) = \sqrt{|xy|}$. Then $f$ is not differentiable at $(0,0)$, but $f(x,y)^2 = |xy|$ is differentiable at $(0,0)$ (since $|xy| \le \frac{1}{2} (x^2+y^2)$).
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Understand Picard-Lindelöf Proof I am trying to understand the Picard-Lindelöf from my book which uses the fixed point theorem.
The task is trying to find $x \in C(a,b)$ in open interval $(a, b)$ containing $t_0$ such that it satisfies the integral equation $x(t) = x_0 + \int_{t_0}^{t}f(s, x(s))ds$ for all $t \in (a, b... | The big picture:
*
*A certain subset $X\subset C[a,b]$ is introduced; we'll be looking for a solution of the integral equation in $X$.
*$X$ is shown to be closed in $C[a,b]$. Hence, $X$ is a complete metric space with the induced metric.
*The restriction of the integral operator to $X$ is shown to be a contractio... | {
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Borsuk-Ulam theorem for $n = 1$ I'm thinking about the Borsuk-Ulam theorem for $n = 1$. How can I show that every continuous map $f : S^{1} \to \mathbb{R}$ has some $x \in S^{1}$ such that $f(x) = f(-x)$?
My first idea was: I consider the new function $g = f(x) - f(-x)$. Let's now calculate the zero of $g$, i.e. $f(x)... | Yes, it is. Now note that if $g(-x)=-g(x)$ and use the connectedness of $S^1$.
| {
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Probability recursion Im trying to find the recursive relation that find the probability that when tossing a coin over and over the pattern tth show for the first time on the nth trial. I'm really stumped on this, I tried using the law of the total probability on the first outcome(head or tails ) to no avail and defin... | This might help: OEIS sequence A000071 counts "the number of 001-avoiding binary words of length $n-3$."
To expand on this, the probability of getting TTH for the first time on the $n$th toss is clearly ${1\over8}p(n-3)$, where $p(n-3)$ is the probability of avoiding TTH in a string of length $n-3$. Now a TTH-avoidin... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The sequence of prime gaps is never strictly monotonic I have an assignment question that asks me to show that the sequence of prime gaps is never strictly monotonic. I'm also allowed to assume the Prime Number Theorem.
I've managed to show that it cannot be strictly decreasing by considering the numbers $N!+2, N!+3,..... | It looks as if you're on a good track.
Suppose that after $p_n$, the gaps $d_n := p_{n+1} - p_n$ are strictly increasing, so that surely $d_{n+k} \geq k$. We can estimate $p_{n+k}$:
$$p_{n+k} > d_{n+k-1} + \dots + d_{n} \geq (k-1) + (k-2) + \dots + 1 = \frac{k(k-1)}{2} \gg k^2.$$
But this is a problem, because $\pi(p_{... | {
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} |
Reference for integration Does anyone have a good reference for a book that already assumes knowledge of calculus/analysis and whose main focus is computing more difficult integrals? I'm looking for something which will have a lot of worked examples for differentiation under the integral, tricky substitutions, unusual ... | You might be interested in The Handbook of Integration by Zwillinger. It appears to be the standard reference on integration methods for scientists and engineers.
The downside is that it probably doesn't contain the "tricky" techniques you are looking for. Hopefully someone can find a more math contest-oriented book f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/530394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
How to reason about congruences? If $x^2 \equiv a$ (mod $m$) and $y^3 \equiv a$ (mod $m$), then $\gcd(a,m) = 1$ Generally, I have no high level conception of what is going on in my number theory class. It feels like a loose collection of theorems and techniques that you can use on some problems, but I have trouble link... | The original statement is false (which did not have the $z^6$ restriction). Let $a=2^6=64$, and $m$ be arbitrary and even, so long as $m>a$. Then $4^3=64=8^2$, and this is also true modulo $m$. Yet $gcd(m,a)\ge 2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/530481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to find the second derivative of an implicit function? We know from multivariable calculus that if $y(x)$ is a function given implicitly by the equation $F(x,y) = 0$, then
$$
\frac{dy}{dx} = -\frac{F_x}{F_y} \tag{1}
$$
This is quickly proved by applying the multivariable chain rule to $\frac{d}{dx}F(x,y(x))=0$.
The... | I think it's pretty clear, despite the question in Tanner Swett's comment, that $F(x, y)$ is a sufficiently smooth function of the two variables $x$ and $y$ that the equation $F(x, y) = 0$ defines $y(x)$ as an implicit function of $x$; that is, $F(x, y(x)) = 0$. Of course lurking behind such a definition of $y(x)$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/530573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
computing probability of pairing Alice has $n$ pairs of socks with $n$ colors ranging in shades of grey enumerated from $1$ to $n$. She takes the socks out of the drier and pairs them randomly. We will assume in each pair,the right and the left socks are identical.
Assume that Alice finds the pairing acceptable if all ... | The question assumes you can't tell the difference between right and left socks.
So there are $2n-1$ possible socks to match with the first sock, not $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/530626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Induced isomorphism from a group $G$ to $\{ (x_1,x_2) \in G_1 \times G_2 | f_1(x_1) = f_2(x_2) \}$ so here is what I have:
$G$ is a group. For $i \in \{1,2\}: p_i:G \twoheadrightarrow G_i$ is a surjective morphism of groups. $H_i = ker(p_i), H_1 \cap H_2 = \{1\}$
In the other parts of the question I found out that:
$p... | The solution from class in the end was this:
$ p: G\rightarrow L=\{(x_1,x_2) \in G_1 \times G_2 | f_1(x_1)=f_2(x_2)\} \subset G_1\times G_2$
because we know that $p$ is an injective morphism of groups and for
$g\in G: p(g)=(p_1(g),p_2(g))$ and $f_1(p_1(g)) = \bar{p_1}^{-1} \circ \pi (p_1(g)) = \pi(g) = f_2(p_2(g)) \Ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/530794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
"If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer" If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer.
I found this question in RMO 1992 paper !
Can anyone help me to prove... | We have, $ab=(a+b)c$. Then $$a+b=(a/m)(b/n)=a^\prime b^\prime$$ where $a=ma^\prime ,b=nb^\prime$ and $c=mn$.
Now, let a prime $p\mid a^\prime$. Then, $p\mid a$ and hence $p\mid b$. If $p\mid n$, then $p\mid c$ but this is a contradiction. Hence $p\mid b^\prime$. Similarly, if a prime $p\mid b^\prime$ then $p\mid a^\pri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/530915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 6,
"answer_id": 4
} |
u | u + v Coding G1 is generator of C1 and G2 is generator of C2. If C is c1 || c1 + c2 then how do you find the generator and parity-check of C?
I have tried two examples and I see a pattern in the G and H of C. I think I am able to generalize it as well. I have also tried with row and column variables but cannot get ... | A generator matrix of the $(C_1 \mid C_1 + C_2)$ code is given by
$$
\begin{pmatrix}
G_1 & G_1 \\
0 & G_2
\end{pmatrix}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/530980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Examples of famous problems resolved easily Have there been examples of seemingly long standing hard problems, answered quite easily possibly with tools existing at the time the problems were made? More modern examples would be nice. An example could be Hilbert's basis theorem. Another could be Dwork's p-adic technique... | Van der Waerden's conjecture on the permanent of a doubly stochastic matrix, proved independently by Falikmann & Egorychev. It turned out to be an easy consequence of an inequality which had been known for a long time.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/531070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "51",
"answer_count": 14,
"answer_id": 4
} |
Dimension of a subspace of finite-dimensional product space $V$ equals $\dim V - 1$ Suppose $w$ is a nonzero vector in a finite dimensional inner product space $V$. Let $P = \{ v \in V | \langle v,w\rangle = 0\}$. Show that $\dim P = \dim V - 1$ where $P$ is a subspace of $V$.
| Since $P$ is the kernel of the linear mapping $v\mapsto \langle v,w\rangle$, it can be described (in any coordinates) as a nullspace of the matrix $w^T$. Now use what you know about matrix row reduction to finish the problem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/531128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
} |
Convergence of $ x_n = \left( \frac{1}{2} + \frac{3}{n} \right)^n$ I need to show that the sequence $ x_n = \left( \frac{1}{2} + \frac{3}{n} \right)^n$ is convergent.
Using calculus in $ \mathbb{R}$, we could see that $ \lim _{n \to \infty} \left( \frac{1}{2} + \frac{3}{n} \right)^n = \lim_{n\to \infty} e^{n \ln \left... | Hint: $\dfrac12+\dfrac3n \le \dfrac78$, when $n\ge8$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/531208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Greedy algorithm Prove or disprove that the greedy algorithm for making change always uses the fewest coins possible when the denominations available are pennies (1-cent coins), nickels (5-cent coins), and quarters (25-cent coins).
Does anyone know how to solve this?
| After giving out the maximal number of quarters, there will be $0-24$ cents remaining. Then there will be at most 4 nickels to give out. After giving out nickels greedily, there will be $0-4$ cents remaining, so there will be at most 4 pennies to give out. Now, can you prove that we cannot rearrange our change to us... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/531342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\mathbb{F}_5[X]/(X^2+3)$ is a field with 25 elements Prove that the ring shown above is a field containing 25 elements.
Research effort:
$$\mathbb{F}_5[X]/(X^2+3)\cong (\mathbb{Z}[X]/5\mathbb{Z}[X])/ \overline{(X^2+3)} \cong \mathbb{Z}[X]/(5,X^2+3)$$
Modifying the term $3$, I tried to find $k$ such that $5k... | You probably saw a theorem that states that for any field $F$ and irreducible polynomial $P(X)\in F[X]$, the quotient ring $F[X]/(P(X))$ is a field. So, you need to establish that your polynomial $X^2+3$ is irreducible over the field $F=\mathbb F_5$. Since the polynomial has degree $2$, all you need to do is verify tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/531405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Finding the equation for a tangent line at a certain point I would like to find the equation for the tangent line of $y=(x^3-25x)^8$ at point $(-5,0)$.
I know that you have to find the first derivative, but I don't know where to go from there.
| Once a value for the slope $m$ is known from the derivative at the given point, use the parameterized definition of a line to get values for x and y in terms of the parameter t and the tangent point (-5,0):
$x = 1 * t - 5$
$y = m * t + 0$
If desired, eliminate t from this system of equations to obtain a standard equa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/531499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Modular arithmetic How do I prove the following inequality with modular arithmetic? (No use of Fermat's last theorem is allowed.)
$$3987^{12} + 4365^{12} \neq 4472^{12}$$
| Note that $3|3987,3|4365,3\not| 4472$ so we have $3^{12}a^{12}+3^{12}b^{12}=4472^{12}\implies 3|4472,\text{ and }3\not| 4472\implies 3987^{12}+4365^{12}\ne 4472^{12}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/531602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Proof that if $x, y \in \mathbb{Z}$ then $xy \in \mathbb{Z}$ How do you prove that the product of two integers is an integer?
| Let's say you assume that $\mathbb{Z}$ is a group under addition, then it is just induction : If $x \in \mathbb{Z}$, then it suffices to prove that $nx \in \mathbb{Z}$ for all $n \in \mathbb{N}$, since if $nx \in \mathbb{Z}$, then $-nx \in \mathbb{Z}$.
To that end, say $n=2$, then
$$
nx = x + x \in \mathbb{Z}
$$
so as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/531679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is $\max(f)$ well defined if $f$ is finite? Say I have a function that's finite almost everywhere within an interval $[a,b]$. Does that mean that it has an upper bound if I ignore those points on which it is infinite? i.e. is:
$$\text{max}_{[a,b]}(f)$$
well defined?
| No, for one of two possible reasons:
1) The function may still be unbounded in $[a,b]$. Bounded and finite are not the same concept. Bounded means there is a number $M<\infty$ such that $|f(x)|<M$ for $x\in[a,b]$, whereas finite means that $|f(x)|<\infty$ for $x\in[a,b]$. This doesn't change if you say these properties... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/531764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
What is the series $\sum_{i=n}^{\infty}\frac{i}{2^i}$ When computing the expected value for a random variable I reached the following series:
$$\sum_{i=n}^{\infty}\frac{i}{2^i}$$
I am confident it is convergent, but have no idea how to compute it.
| There are a few standard tricks that one can use. For example:
Let $\displaystyle f(x) = \sum_{i=n}^\infty i x^{i-1} = \frac{d}{dx} \sum_{i=n}^\infty x^i = \frac{d}{dx} \frac{x^n}{1-x} = \frac{nx^{n-1}(1-x) + x^n }{(1-x)^2}$.
So $\displaystyle \sum_{i=n}^\infty \frac{i}{2^i} = \frac{1}{2} \sum_{i=n}^\infty i \left(\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/531869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$ I'm undergraduate student of mathematics.
I need to prove:
$$\sum_{k=0}^{n} \binom{2n}{2k}= 2^{2n-1}$$
Can you please help me
| $$\sum_{k=0}^{n} \binom{2n}{2k} = \sum_{k=0}^n \left[\binom{2n - 1}{2k} + \binom{2n - 1}{2k - 1}\right] = \sum_{k=-1}^{2n} \binom{2n - 1}{k} = \sum_{k=0}^{2n - 1} \binom{2n - 1}{k} = 2^{2n-1} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/531961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
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