Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Information theory applied to a dice (intuition about information theory) Let's say I have a die that has the values 1 to 6 written on it, and I don't know the probability to get each value, I only know that when I throw the die many times I get an average value of 3.5, just like a fair die.
According to information th... | In general, for a given mean $m$, we have the restrictions $\sum_{i=1}^6 p_i=1$ and $\sum_{i=1}^6 p_i i = m$. Applying Lagrange multipliers we get for the critical point:
$ -1 - \log p_i + \lambda i +\beta=0$
Hence the extrema is given by a (truncated) exponentional family $$p_i = a \exp({-b i}) $$ where the constants ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2189234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
relation between union and intersection A and B are any two sets.
if A U B ≠ B
is it also true that A ∩ B ≠ A ?
and if so then why?
This is just a step I am using for a bigger proof, if the above is not true then I'll have to search for a different direction.
Thanks in advance.
| It's easier to prove this the other way round, i.e. you can show that
if $A\cap B=A$, then $A\cup B=B$.
This is easy to show. The fact that $A\cap B = A$ implies that $A\subseteq B$, which also directly means that $A\cup B = B$.
Or, if you want the traditional long way round:
*
*Let $b\in A\cup B$.
*
*Then, if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2189365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How should I write down the alternating group $A_3$? I didn't understand how to write down the alternating group A3.
Is this the group consisting of only the even permutations? Also, what familiar group is this isomorphic to?
| Yes, $A_3$ is the set of all even permutations in $S_3 = \{id, (12), (13), (23), (123), (132)\}$.
Remember that an even permutation can be written as the product of an even number of transpositions. The identity of any symmetric group is even, because id can be written as the product of two transposition. In this ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2189495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Mobius image of a line I've been looking at this question;
Let gamma be the mobius transformation defined by:$$\gamma(z) = \frac{2iz-2}{z+1}, z\ne -1$$
Show that $\gamma$ maps the line $\{z: Im z = Re z - 1\}$ into the circle $C(i,1)$.
I've gone about this in a similar method to an example in my notes however I'm not s... | Let $$w=\frac{2iz-2}{z+1}\implies z=-\frac{w+2}{w-2i}$$
Now write $z=x+iy$ and $w=u+iv$
Then after a couple of lines of algebra, we get $$x=-\frac{u^2+v^2-2v+2u}{u^2+(v-2)^2}$$ and $$y=\frac{-2u+2v-4}{u^2+(v-2)^2}$$
Now substitute these into the given line equation $y=x-1$ and after some simplification we get $$u^2+(v-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2189575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Surds question grade 10 I am a student and need help answering this question. I need a step by step solution.
Simplify:
$ \sqrt {18}$ - $ \sqrt {9}$
What I tried:
($ \sqrt {9}$ × $ \sqrt {2}$) - 9
= (3$ \sqrt {2}$ ) - 9
I don't know what to do next.
Thank you and help is appreciated.
| this is $$\sqrt{2\cdot 9}-\sqrt{9}=3\sqrt{2}-3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2189712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Epsilon-delta proof that $ \lim_{x\to 0} {1\over x^2}$ does not exist I'd like to see an epsilon delta proof that the $\lim: \lim_{x\to 0} {1\over x^2}$ does not exist and an explanation of the exact reason it does not exist, because I am not so sure I believe that the limit does not, in fact, exist, so I need to be pr... | An epsilon-delta proof is used to show that the limit exists and is $L$, not usually to show that no limit exists. We can see where it fails. Suppose we claim that $\lim_{x \to 0}\frac 1{x^2}=L$ If somebody gives us an $\epsilon \gt 0$ we have to find a $\delta \gt 0$ such that $|x| \lt \delta \implies |f(x)-L|=|\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2189990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Calculating of PI from sin() function in Java Ok... So I'm trying to finish my school project with processing and I'm wondering if there is a way to calculate PI from sin() like this. But I don't know how to use sin() function with degrees in Java or how to write my own. The problem with radians is that I need to conve... | Sorry, but this is not a good idea. The formula that you saw essentially expresses that $$\sin x\approx x$$ when $x$ is small, and the smaller $x$ the more exact the approximation. It is valid for angles in radians.
When the angles are in degrees, this relation becomes
$$\sin°x\approx \frac{\pi x}{180}$$ where $\sin°$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solving Differential Equations For Equilibrium: Is a Different Answer Format Required? I'm going to include the exact wording of the question here, because I think it's relevant:
"First solve the equation $f(x)=0$ to find the critical points of the given autonomous differential equation $\frac{dx}{dt} = f(x)$. Then ana... | Both are fine since $C$ is just arbitrary. Try experimenting with different values of $x_0$ and you'll see. When given the initial condition $x_0$ you will still be able to calculate $C$, so it wouldn't matter.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving a differential equation with wronskians So I am asked to find a solution to this ODE here and I feel like I am missing something very obvious.
I am asked to find the general solution of:
$x^2y"-3xy'+4y=\frac{x^2}{ln(x)}, y>1$
So I first tried to find the homogeneous solution which was just a cauchy Euler equati... | Yes, you made a mistake.
Note that the coefficient of $y''$ in your differential equation is $x^2$, but you're using a formula intended for a d.e. where the coefficient of $y''$ is $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
To what extent is pi non-repeating? I've been told that pi is an irrational (infinite and non-repeating) number.
But to what extent is it non-repeating?
It obviously repeats individual numbers, and I find it hard to believe that it doesn't repeat 2-3 digit sections eventually.
|
I've been told that pi is an irrational ,(infinite and non repeating), number. But to what extent is it non repeating. It obviously repeats individual numbers, and i find it hard to believe that it doesn't repeat 2-3 digit sections eventually.
$\pi$ certainly does repeat 2-3 digit sections eventually. There are only ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Prove that $ x^4 - 2 $ is irreducible over $ \mathbb{Z}[i] $
How do I prove that $ p(x) = x^4 - 2 $ is irreducible over $ \mathbb{Z}[i] $?
This seems very elementary yet I'm not sure how to do it.
Someone suggested using Eisenstein and $ p = 1+i $, but this doesn't seem right because $ (1+i)^2 = 2i $ is an associate... | You can also note that $x^4-2$ is irreducible over $\mathbb F_5$ and $\mathbb F_5 = \mathbb Z[i]/(2+i)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Definite Integral Problem using substitution $t=\tan(x/2)$ Solve $$ \int_0^\pi \frac {x} {1+\sin x} dx $$ using $\sin x = \frac {2\tan(x/2)} {1+\tan^2(x/2)}$ and the substitution $t = \tan(x/2)$.
I tried doing this but I got to a point where my integral limits were $0$ to $\infty
$. This happened when I substituted for... | After the main symmetry trick, another symmetry trick and a rationalization:
$$ \int_{0}^{\pi}\frac{du}{1+\sin u}=2\int_{0}^{\pi/2}\frac{1-\sin u}{\cos^2 u}\,du =2\left[\tan u-\frac{1}{\cos u}\right]_{0}^{\pi/2}=\color{red}{\large2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
solution to an equation involving natural numbers Suppose $a,b$ are real numbers and we have
$$ 1 = b + an \; \; \; \; \forall n \in \mathbb{N} $$
My book says that only solution is $b=0$, $a=1$. But, this does not make sense to me since if we put $n = 1$, we have
$$ 1 = b + a $$
And $a=b=1/2$ is a solution. What is... | For all $n$ implies that we have:
$$1=b+a$$
$$1=b+2a$$
So $a=0$ and $b=1$ is the only solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\frac{n}{2^n}$ is a null sequence from $\epsilon$ definition of limit I am trying to prove that $\frac{n}{2^n}$ is a null sequence using the $\epsilon$ definition of a limit. Now I chose to use the fact that $2^n > n^2$ for $n > 4$. I said let $\epsilon > 0$ be given. Then for $n > 4$, $\vert\frac{n}{2^n}\... | A null sequence
eventually gets as small as you want.
Initial terms do not matter.
So you can start at $n=4$
or
$n=10000$
or
$n=10^{10^{10}}$.
All that matters is that,
for any $c > 0$,
there is a $N(c)$
such that
$|a_n| < c$
for $n > N(c)$.
Once you have shown that
there is a $m$ such that
$2^n > n^2$
for all $n > m$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
path connectedness of a preimage Suppose $p:X \rightarrow Y$ is a fibration such that $Y$ is path connected and $p^{-1}\{y\}$ is path connected for some $y \in Y$. Could anyone please show me that with all these conditions that $X$ is also path-connected. Thank you all for helping
| I will use more common notation. So let $\pi:E\to B$ be a fibration with $B$ path connected and $b\in B$ be such that $\pi^{-1}(b)$ is path connected.
Pick $x, y\in E$ and consider $\lambda:I\to B$ a path such that $\lambda(0)=\pi(x)$ and $\lambda(1)=\pi(y)$. Such path exists because $B$ is path connected. Let $\{*\}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2190908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A proof that a countable product of countable sets is non-empty that does not use the axiom of choice. Is the proof correct? Let $I$ be some non-empty set of indexes and for every $i \in I$ let $A_i$ be a set with cardinality $\aleph_0$.
Is the following proof that the cartesian product $\prod_{i \in I}A_i$ is non-empt... | Your argument does invoke choice, albeit in a subtle way: when you choose a family of bijections $\{f_i: i\in I\}$. Just because, for each $i$, the set $F_i$ of bijections from $A_i$ to $\mathbb{N}$ is nonempty, doesn't mean that you can pick one for each $i$; this is exactly the axiom of choice applied to the family $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
How do you find the elements of $\mathbb{F} _5 [x] / (x^2 + 2)$? I am new to the field $\mathbb{F} _5 [x] / (x^2 + 2)$. How would I find all the elements present in this field?
Additionally, I know that the order of the element $x$ is 8 and the order of element $(1+x)$ is 1 (it is the generator), but how would I prove... | To add to @Ethan Bolker's answer, I want to address your question about the order of $x$ and $(x+1)$. You are correct about the order of $x$, since
$$x^8=(x^2)^4=(-2)^4=16=1.$$
However, the order of $x+1$ is not one. This would imply that $(x+1)^2=(x+1)$, which is not true. It should be noted that the order of a gener... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Integration of complex numbers I was solving some integration questions, then one question arised in my mind that "Do the integration of complex no. possible?"
If yes, then what is $\int i dx$?
Definite integration is the area under curve of the graph but the above graph cannot be plotted on real plane.
I want someo... | The simple answer is that $i$ is a constant, so $\int i \,dx = i x + C$
The more complete answer is that "area under curve of the graph" doesn't really make sense for what you are doing when you say $\int i \,dx$, you can integrate a complex variable $z=x +iy$ over a contour. I would check out contour integration here,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Question About the Logic of my Proof Okay, I am working on the following relatively simple problem:
Let $f(x) = |x-3| + |x-1|$ for all $x \in \mathbb{R}$. Find all $t$ for which $f(t+2) = f(t)$.
So, if $f(t+2)=f(t)$, the is equivalent to $|t+1| = |t-3|$. Thus, if this holds, one can square both sides and arrive at $t... | When squaring both sides of an equation you can't lose solutions you could only get extra false solutions. Since $t=1$ satisfies $|t+1|=|t-3|$ that means $t=1$ is not a false solution and is the only solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
searching for $f(x)$ when knowing $f(2^{2x})$ I have a function that works for powers of 2
This is only for INTEGERS
is there a way to calculate any integer x?
$$f(2^{2x})=\frac{4^x+2}3.$$
$f(x)$=?
here are the first 100 values of f(x)
1, 2, 2, 2, 2, 4, 4, 6, 6, 8, 8, 6, 6, 8, 8, 6, 6, 8, 8, 6, 6, 8, 8, \ 14, 14, 16, 1... | Note that $4^x=2^{2x}$, so you can simply swap them both for $x$ and get
$$
f(x)=\frac{x+2}{3}
$$
This new description is only valid for positive real $x$, and even then only if the original description was valid for any real $x$. If the original expression was only valid for integer $x$, for instance, then it would pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Volume preserving mean curvature flow preserving uniformly convex Let $(M_t,g_t)$ be a Riemannian manifold evolve by volume preserving mean curvature flow. So , for second fundamental form, we have
$$
\partial_t h_{ij}=\Delta h_{ij}-2H h_{im}h^m_j+hh_{im}h^m_j + |A|^2 h_{ij}
$$
$H=g^{ij}h_{ij}$ is mean curvature, $|A|... | Use Theorem 9.1 of Hamilton's THREE-MANIFOLDS WITH POSITIVE RICCI CURVATURE, which tells you that if a time-dependent symmetric tensor field $h$ satisfies $$\partial_t h_{ij} = \Delta h_{ij} + N_{ij}$$
with the reaction term $N$ satisfying the null-eigenvector condition $$h_{ij} v^i = 0 \implies N_{ij}v^iv^j \ge 0,$$
t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Reducing $x^3-\sqrt{2}$ over $\mathbb{Q}(\sqrt{2})[x]$ I think it can be shown that $x^3-\sqrt{2}$ is irreducible by arguing that $x=\sqrt[6]{2}$, which is the only real solution, is not in $\mathbb{Q}(\sqrt{2})$, so the polynomial has no solutions over this field, which implies that the polynomial cannot be expressed ... | Let $K=\mathbb{Q}(\sqrt{2})$, and let $f = x^3 - \sqrt{2}$.
Since $f \in K[x]$ is cubic, to show $f$ is irreducible in $K[x]$, it suffices to show $f$ doesn't have a root in $K$.
Suppose otherwise. Thus, suppose $r^3=\sqrt{2}$, for some $r \in K$.
Since $r \in K$, we can write $r = a + b\sqrt{2}$, for some $a,b \in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is Completeness and Compactness not equivalent in Normed Spaces? Given a complete normed space $X=(X,\|\cdot\|)$. Every Cauchy sequence converges in it. I am not able to understand why we can't show that every bounded sequence in $X$ will have a convergent subsequence.
Please give an example to clarify why complet... | Your question seems to be about the local compactness of Banach spaces.
Look at a sequence of sequences with general term $A_n=(\delta_{i,n})_{i>0}$ which is in $l^p$. What can you say about the norm of its term and the norm of the "potential" limit? By the way there is compactness in a weak sense, but you might need H... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2191968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Prove that a finite non empty set of real numbers in bounded . I'm trying to solve an exercise that is as follow:
let $S = \{a_1, a_2, a_3, ......, a_n\}$ be a finite nonempty set of real numbers.
show that S is bounded.
I know that to prove a set is bounded you need to prove that it is bounded from above and below, bu... | You might let $M = |a_1|+|a_2|+\cdots+|a_n|$. Then for any $i$, you have
$$a_i \leq |a_i| \leq M,$$
so $M$ is an upper bound of $S$. Similarly $-M$ is a lower bound.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2192062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Faster way to solve a equation. Solve the equation:
$\sqrt[3]{x-2} + \sqrt[3]{x} + \sqrt[3]{x+2} = 0$
$f(x) = \sqrt[3]{x-2} + \sqrt[3]{x} + \sqrt[3]{x+2}$
Firstly I check the amount of solutions.
*
*Graph of the function starts at the bottom and ends at the top.
*The derivative is always greater than 0, so the func... | Observe that for the given function, $f(a)=-f(-a)$ for any value of $a$.
Now put $a=0$.
Thus $f(0)=-f(0)$.
=> $2f(0)=0$
=> $f(0)=0$
=> $x=0$ is a solution.
Also, $f'(x)>0$ for all real numbers $x$.
So, the function is always increasing and thus $x=0$ is the only solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2192212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can a $C^r$ differentiable function on a arbitrary set be extend to a $G_\delta$ set? Let $S\subseteq\mathbb R^n$ be a arbitrary set, $0\le r\le\infty$.
For $r<\infty$, call a function $f:S\to\mathbb R$ is $C^r$, if there are continuous functions $\{f_\alpha:S\to\mathbb R\}_{0\le|\alpha|\le r}$ such that $f(x+h)=\sum_{... | Suppose that the closure of $S$ has non-empty interior $V$. If $f$ is continuous on the set $S\cap V$ then it has a continuous extension to a $G_\delta$ set in the closure of $S\cap V$ (countable intersection of open dense set). Conversely, given a $G_\delta$ dense set $\Omega$ in Euclidean space you may construct a co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2192298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are these sets representing the subspace topology equal? I have a question regarding the representation of the subspace topology.
Let $(\Omega, \mathcal T)$ be a topological space, $\Omega' \subseteq \Omega$ and $\mathcal T' \subseteq \mathcal T$ the subspace topology with respect to $\mathcal T$. Then, by definiton $$... | The second definition is not correct in general.
The subspace topology $\Omega '$ is composed by the intersection of the open sets of $\Omega$ with $\Omega'$, hence is possible that there are $U\in\mathcal T$ that are not subsets of $\Omega'$ and that at the same time we need them to define the subtopology (the open se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2192462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
show that for $n=1,2,...,$ the number $1+1/2+1/3+...+1/n-\ln(n)$ is positive show that for $n=1,2,...,$ the number $1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}-\ln(n)$ is positive, that it decreases as $n$ increases, and hence that the sequence of
these numbers converges to a limit between $0$ and $1$ (Euler's constant)... | Note that the sequence in question, let's call it $\alpha_n$, can be written as
$$\alpha_n = \Big(\sum^n_{k=1} 1/k\Big) - ln(n).$$
Note that
\begin{align}
ln(n) &:= \int^n_1 \frac {1} {t} dt \\\ &= \int_1^2 \frac {1}{t} dt + \int_2^3 \frac {1}{t} dt + \dots + \int^n_{n-1} \frac {1}{t} dt \\\
& \le (2-1) \cdot \frac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2192605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
CDF of a random variable Consider a random variable $Y_{n}$ such that,
$P$($Y_{n}$ = $\frac{i}{n}$) = $\frac{1}{n}$ , $i$ = 1, 2,..., $n$.
Is the cdf of $Y_n$ for every integer $n \ge 1$ simply $\sum_{k=1}^i \frac{1}{n}$ = $\frac{i}{n}$?
Also, how do you show that for any $u$ $\in$ $R$ the $\lim_{n\to\infty} P($$Y_{... | The cdf of $Y_n$ is given by
$$
P(Y_n\le y)=\frac{\lfloor ny\rfloor}n
$$
for $0\le y\le 1$, where $\lfloor\cdot\rfloor$ is the floor function. We have that $\lfloor ny\rfloor/ny\to1$ as $n\to\infty$. Hence, $P(Y_n\le y)\to y$ as $n\to\infty$ which is the cdf of the continuous uniform distribution on $[0,1]$.
Alternativ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2192675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Junior olympiad question: Minimum value of 3 digit number divided by sum of its digits I recently had a maths competition where we were given this problem. I solved the question, but I narrowed down the possibilities then did more of a guess and check method. I was hoping someone else could help me get the answer to th... | Let the number be $\overline{abc}=100a+10b+c\,$ with digits $1 \le a,b,c \le 9\,$. Then:
$$
\begin{align}
\frac{100a+10b+c}{a+b+c} & = 1 + 9\cdot\frac{11a+b}{a+b+c} \\[3px]
& \ge 1 + 9\cdot\frac{11a+b}{a+b+\color{red}{9}} \quad\quad\quad\quad\text{(*)}\\[3px]
& = 1 + 9 + 9 \cdot \frac{10a-9}{a+b+9} \\[3px]
& \ge 1 + 9 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2192758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Some curious binomial coefficient identities I was playing around with some polynomials involving binomial coefficients, and inadvertently proved the following two identities:
(1) For all $p, q \in \mathbb{N}$ and $i \in \left[ 0, \min\{p, q\} \right]$:
$$
\begin{pmatrix}p \\ i\end{pmatrix} = \sum_{j=0}^i (-1)^j \begin... | The first one is just inclusion-exclusion in the following way:
Take the set $[p+q]=\{1,\cdots ,p+q\},$ so you want to take $i$ elements from those such that they all belong to $[p].$ By definition you just restrict yourself to the set $[p]$ and hence there are $\binom{p}{i}$, but on the other hand it is the same as th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2192865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Partial Fraction decomposition (degrees) Let us say that I want to decompose the fraction
$$\frac{x-3}{x^2 +6x+5}$$ into partial fractions.
I know that we have to factor the denominator and write it as $$\frac{A}{x+1} + \frac{B}{x+5}.$$ Then we get
$$x - 3 = A(x+5) + B(x+1).$$
Select convenient values for $x$ and sol... | If $f(x)=\frac{P(x)}{Q(x)}$ is a rational function and $\deg(P(x))\ge\deg(Q(x))$ we call $f$ an Improper rational function, it is said to be "top-heavy".
Conversely, if $\deg(P(x))\lt\deg(Q(x))$ then $f$ is a Proper rational function.
When $\deg(P(x))\ge\deg(Q(x))$, we use polynomial long division to turn a top-heavy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2192932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How to compose two functions? I have three functions:
$$f(x) = x+1 ,\; g(x) = x - 1 ,\; h(x) = 2x$$
I want to find $g\circ f$ such that $g(f(x))$, which equals $(x+1)-1=x$ but how?
I don't understand the steps.
Also, I want to find $h\circ f$ which equals $h(f(x)) = 2x-1$ but I dont know how to get that answer eith... | The function $f$ will take an input and return an output equal to one more than the input.
$f(\underbrace{\color{red}{x}}) = \underbrace{\color{red}{x}}+1$
Similarly $f(\underbrace{\color{red}{55}})=\underbrace{\color{red}{55}}+1$ and $f(\underbrace{\color{red}{8x^2-3}})=\underbrace{\color{red}{8x^2-3}}+1$
The function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Determining whether the series $\sum_{n=1}^{\infty} \frac{\sqrt{n}+\sin(n)}{n^2+5}$ is convergent or divergent by comparison test I am given the series:
$$\sum_{n=1}^{\infty} \frac{\sqrt{n}+\sin(n)}{n^2+5}$$
and I am asked to determine whether it is convergent or not. I know I need to use the comparison test to determ... | Since $\sin(n) \leq 1$,so $\sum_{n=1}^{\infty} \frac{\sqrt{n}+\sin(n)}{n^2+5}\leq \sum_{n=1}^{\infty} \frac{\sqrt{n}+1}{n^2+5}$,
Now for large $n$, $(n^{2} + 5)$ can be taken to be $n^{2}$ ,
so the series becomes $\sum_{n=1}^{\infty} \frac{\sqrt{n}+1}{n^2} = \sum_{n=1}^{\infty} \frac{1}{n^{1.5}}+\sum_{n=1}^{\infty} \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the generating function or closed form for the recurrence relation $a_n = a_{n-1} + 4a_{n-2} + 2a_{n-3}$ I was trying to solve this recurrence relation using generating function
$a_n = a_{n-1} + 4a_{n-2} + 2a_{n-3} \qquad : \quad a_0 =1,a_1 =1,a_2 =5, $
I did in the following way
$
\begin{align*}
&G(x) = \sum_{n... | Here are two variants to derive $a_n$. The first one gives a closed form, the other one an explicit expression, which results in a nice binomial identity.
First variant: Partial fractions
In case it's easy to derive the zeros of the denominator of
\begin{align*}
G(x) = \frac{1}{1-x-4x^2-2x^3}
\end{align*}
the p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Showing a linear map is injective if and only if kernel is {$ {0} $} So my prof gave me this proof:
$f(x) = f(y) ⇐⇒ f(y − x) = 0 ⇐⇒ y − x ∈ Ker f.$
I dont see why this proof is enough, this only says $y-x \in Ker f$
| First suppose $f$ is injective.
Since $f$ is linear, $f(0) = 0$, hence $0 \in \text{ker}(f)$.
But if $x$ is any element of $\text{ker}(f)$, then
\begin{align*}
&x \in \text{ker}(f)&&\\[4pt]
\implies\; &f(x) = 0&&\\[4pt]
\implies\; &f(x) = f(0)&&\text{[since $f(0) = 0$]}\\[4pt]
\implies\; &x = 0&&\text{[since $f$ is i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
} |
Partial sum of order statistics of exponential r.v.'s and $\chi^2$ Suppose $X_i \sim Exp(\frac{1}{\lambda}), i = 1,\cdots,n$, where
$f(x) = I_{(0,\infty)}\frac{1}{\lambda}e^{-\frac{x}{\lambda}}$
is the p.d.f. of $X_i$'s.
And we have a positive integer $r$, and the order statistics
$$X_{(1)}\leq X_{(2)} \leq \cdots \le... | Your question was actually answered eight decades ago by P. V. Sukhatme. I'll explain how to prove the problem assuming that you are familiar with the fundamentals of probability theory and the relationships between the exponential, gamma and $\chi^2$ distributions. First, if you reviewed any textbook that addresses or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Modal logic: justification of the rule of necessitation I'm studying some lecture notes about modal logic and I'm now reading a paragraph which goes as follows:
The rules of inference of system K are modus ponens and the rule of necessitation:
NEC: if A is a theorem, then ◻A is a theorem.
This rule is legitimate in th... | You write "This rule is legitimate in that it preserves truth in a world in any model".
This is not true. Necessitation preserves truth in a model but not truth in a world in a model. It is easy to see that what is true in a world need not be necessarily so in that world, hence adding the box operator may not preserve ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Orthonormal columns and rows The assignment:
a) Prove that square-matrix A is orthogonal if and only if A has orthonormal columns.
b) Prove that square-matrix A is orthogonal if and only if A has orthonormal rows.
So I know that A matrix has orthonormal columns if and if only $A^TA=I$.
But how about orthonormal rows? S... | For (b): Let me denote the matrix $A$ as follows:
$$\begin{pmatrix}
- & a_1 & -\\
- & a_2 & -\\
& \vdots & \\
- & a_n & -
\end{pmatrix}$$
where the $a_i$ are row vectors and I emphasised this by adding '-'. We know that a matrix $A$ is orthogonal if $AA^T = I$. We want to show that the rows of $A$ form an orthonormal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Transitivity of a relation on a set.
My book says the answer is B. But how. I understand that the relation is reflexive as (a,a) belongs to R for all a belonging to the given set. Furthermore I understand that the relation is not reflexive as (a,b) belongs to R does not imply that (b,a) belongs to R for all a,b belong... | For it not to be transitive, you need an a,b,c such that (a,b) and (b,c) are in R, but (a,c) is not (a,b,c need not be different). Are seeing any instance of this? Let's see. Here are all pairs of pairs (a,b) and (b,c) in R:
(1,1) and (1,1)
(1,1) and (1,2)
(1,1) and (1,3)
(1,2) and (2,2)
(2,2) and (2,2)
(3,2) and (2,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Norm of a Positive definite matrix is the largest eigenvalue Let $A$ be a positive definite symmetric matrix. I need to show that
$$\lambda_n=\max \{\frac{\|Ax\|}{\|x\|}: x\ne 0\}$$ is the largest eigen value of $A$. My try is $\frac{\|Ax\|}{\|x\|}\le \lambda,\forall \lambda$ as a not eigen value of $A$, and the equali... | By the spectral theorem, there is an orthonormal basis of eigenvectors of $A$, say $x_1,\dots,x_n$. WLOG, assume $0 < \lambda_1 \leq \dots \leq \lambda_n$, where $Ax_i = \lambda_i x_i$. For any non-zero $x$, write $x = c_1x_1 + \cdots + c_nx_n$. Then
$$ \frac{\lVert{Ax\rVert}^2}{\lVert{x\rVert}^2} = \frac{\lVert{c_1\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to solve $\lim\limits_{x \to \infty}\left(\frac{x}{x+1}\right)^x$ How to solve $$\lim_{x \to \infty}(\dfrac{x}{x+1})^x$$
The answer is $\dfrac{1}{e}$
I can factor the $x$ out to get:
$$\lim_{x \to \infty}\left(\dfrac{x(1)}{x(1+1/x)}\right)^x = \lim_{x \to \infty}\left(\dfrac{1}{1+1/x)}\right)^x$$
How do I further s... | You almost got it:
$$\left(\frac1{1+\frac1x}\right)^x=\frac1{\left(1+\frac1x\right)^x}\xrightarrow[x\to\infty]{}\frac1e$$
where the limit is gotten using arithmetic of limits...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2193982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Bilinear operator norm question. So I wrote this down a week ago and cannot figure out what I was thinking. Not sure if this is correct.
Context we have a bilinear operator $B:X\times Y\to \mathbb{K}$.
Is it true that
$$\sup_{x\in X, y\in Y} \|B(x,y)\|< \infty \implies |B(x,y)|\leq K \|x\|\|y\|$$
So the absolute value... | $$\sup_{x\in X, y\in Y} \|B(x,y)\|< \infty \implies B=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2194073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the closed form for a recurrence relation I'm having trouble finding a closed form for a geometric recurrence relation where the term being recursively multiplied is of the form (x+a) instead of just (x).
Here's the recursive sequence:
$a_{n} = 4a_{n-1} + 5$ for $n \geq 1$ with the initial condition $a_{0} = 2... | This sequence is an affine recursion of the form $a_{n+1} = \lambda a_n + \mu$, with $\lambda=4\neq 1$ and $\mu=5$. Its $n$th term is given by
$$a_n = \lambda^n (a_0 - \rho) +\rho \, ,$$
where $\rho= \frac{\mu}{1-\lambda}$. The formula for the $n$th term can be obtained by setting $b_n = a_{n+1}-a_n$, which is a geomet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2194219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Subset of normed Linear Space is closed The problem in question is as follows:
"Show that the set $P$ of all polynomials on the segment $[a,b]$ is a linear space. For $P$ considered as a subset of the normed linear space $C_[a,b]$ with the norm
$||f(x)||$ = $max_{a{\leq}x{\leq}b}$ $|f(x)|$
Show that $P$ fails to be cl... | Are you familiar with the Weierstrass Approximation theorem?
http://www.mast.queensu.ca/~speicher/Section14.pdf
This pdf does a good job explaining it. But the statement of the theorem should be enough alone, for your purposes: there are continuous functions which are not polynomials, but by the theorem, are the unifor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2194287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Intuition difference between derivatives of $\exp (x) $ and $\log (x)$ It is well known that the exponential function $\exp (x)$ has the derivative $$\frac{d}{dx} \exp (x) = \exp (x).$$ However, its inverse, the (natural) logarithm, $\log (x)$, in fact changes after derivation: $$\frac{d}{dx} \log (x) = \frac{1}{x}. $$... | We can look at it geometrically.
Let $f(x)=e^x$, so $f^{-1}(x)=g(x)=\log(x)$.
Any function's inverse should look like its reflection over $y=x$. Let's consider a point $(x,f^{-1}(x))=(x,y)$ on the graph of log.
By definition, $f(y)=x$. As you know, the tangent line to $f$ has slope $e^x$, so the tangent line to $f$ ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2194427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 2
} |
Show basis for a topology A Subbasis $\mathcal R$ of a set $X$ is defined to a collection of subsets of $X$ whose union equals $X$. Show that the collection $\mathcal B$ of all finite intersections of element of $\mathcal R$ is a topological basis of $X$.
Attempt :
Let $X$ be a set, let $\mathcal B$ a collection of su... | $R$ satisfies $1$ because its union is $X$. The closure over finite intersections satisfies $2$ (choose $B_3$ equal to $B_1$ intersection $B_2$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2194531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why do all elementary functions have an elementary derivative? Considering many elementary functions have an antiderivative which is not elementary, why does this type of thing not also happen in differential calculus?
| The short answer is that we have differentiation rules for all the elementary functions, and we have differentiation rules for every way we can combine elementary functions (addition, multiplication, composition), where the derivative of a combination of two functions may be expressed using the functions, their derivat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2194769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "52",
"answer_count": 6,
"answer_id": 3
} |
Notation for juxtaposition operation on matrices Does there exist a fairly standard notation for horizontal and vertical juxtaposition operations on matrices? (Vertical juxatposition can be called "stacking.")
For example, juxtaposing horizontally a matrix of the size $m\times n_1$ with a matrix of the size $m\times n... | Just draw a matrix of matrices:
$$M=\begin{pmatrix}A&B\\C&D\end{pmatrix}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2194893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Possibilities for the first four positions at the World Cup $2014$
$32$ nations participated in the World Cup $2014$.
How many possibilities were there for the order of the first four positions?
In general,
$$32 \choose 4$$
gives us the possibilities to choose $4$ nations out of $32$ nations. This doesn't include the... | I believe your reasoning is correct for the number of possibilities for the first four positions.
The distinction you make between switching $k$ and $n$ seems arbitrary. We could equivalently see the problem as distributing 4 positions among 32 nations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Connected categories and connected limits Let $C$ be a category together with an equivalence relation $∼$ on the objects by $x ∼ y$ whenever there is a morphism $f: x \to y$.
*
*What does $(Ob(C)/∼) \cong 1$ mean? What kind of isomorphism is this?
*What does it mean for a category to have all small connected limit... | First of all, note that this relation is in general not symmetric, so we have to take the equivalence relation generated by this relation. That is, $x\sim y$ if and only if there is a zigzag of morphisms $x\to x_1\leftarrow x_2\to \dots \leftarrow x_n \to y$ in $C$. Therefore, we can think of an equivalence class as a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=1$, $\lim_{n\to\infty}\frac{a_{2n}}{a_n}=\frac{1}{2}$ then $\lim_{n\to\infty}\frac{a_{3n}}{a_n}=\frac{1}{3}$ Let $\{a_n\}$ be a decreasing sequence and $a_n>0$ for all $n$.
If $\displaystyle\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=1$ and $\displaystyle\lim_{n\to\infty}\frac{a_{2n}}... | This is not an answer, but a longer train of thoughts / conjectures that might lead to a full answer.
It might be an idea to use $\displaystyle\lim_{n\to\infty}\frac{a_{2n}}{a_n}=\frac{1}{2}$ and apply it $r$ times. This gives
$$
\displaystyle\lim_{n\to\infty}\frac{a_{2^rn}}{a_n}=\frac{1}{2^r}
$$
which holds for all $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 2,
"answer_id": 0
} |
Reverse mode differentiation vs. forward mode differentiation - where are the benefits? According to Wikipedia forward mode differentiation is preferred when $f: \mathbb{R}^n \mapsto \mathbb{R}^m$, m >> n. I cannot see any computational benefits. Let us take simple example: $f(x,y) = sin(xy)$. We can visualize it
as gr... | An analogy might help. Let $\bf A$, $\bf B$, and $\bf C$ be matrices with dimensions such that $\bf ABC$ is well defined. There are two obvious ways to compute this product, represented by $(\bf AB)\bf C$ and $\bf A(\bf BC)$. Which of those will require fewer multiplications and additions depends on the dimensions of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 2,
"answer_id": 1
} |
Calculating a Point's X position on an Ellipse, given pos Y How can I calculate for the X coordinate given the Y value of the position? Which the Y position is 10 units, as seen in the image below.
We know the X diameter is 200 and the Y diameter is 150.
| If you have an ellipse, you can use the standard equation for an ellipse:
$$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 = 1,$$
where $a = \frac{200}{2}$ and $b = \frac{150}{2}$. Then you can find the $x$ value simply by substituting your $y = 10$ value and solving the above equation algebraically for $x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluating the limit: $\lim _{x\to \infty }\left(2^x\sin\left(\frac{b}{2^x}\right)\right)$ I need to find the following limit :
$$\lim_{x\to \infty}\left(2^x\cdot \sin\left(\frac{b}{2^x}\right)\right)$$
I have tried it but I keep getting stuck, so any help would be helpful!
Thank you!
| By substituting $u = 2^x$, this is $$\lim_{u \to \infty} u \sin \left(\frac{b}{u} \right)$$
You can do this by L'Hôpital: $$\lim_{u \to \infty} \frac{\sin \left(\frac{b}{u}\right)}{1/u}$$
which takes us to $$\lim_{u \to \infty} b \cos \left( \frac{b}{u} \right)$$
which I'm sure you can finish.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is a fiber in algebra the same as a fiber in topology I am reading Dummit & Foote, and they describe a fiber in algebra as property of a homomorphism. So if $\phi$ is a homomorphism from a group $G$ to a group $H$, then the fibers of $\phi$ are the sets of elements in $G$ that map to a single element in $H$.
Now I kno... | Fibre is one of those catch-all words that abounds throughout mathematics.
The general setup is you have some collection of 'objects' such as groups/rings/topological spaces/manifolds and a class of 'nice' maps between them $-$ group-homomorphisms/ring-homomorphisms/continuous maps/differentiable maps.
In all cases th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Is this limit valid/defined and if so, what is the value of it? I was wondering if the following limit is even defined/valid; if it makes any sense. If so, what is the value of it? If not, why is it not defined/valid?
Define $n$:
$$ab=n$$ for $ a\rightarrow \infty$ and $ b\rightarrow 0$
Sure, this might be a weird ques... | There are many limits we run across that are of the form $0 \cdot \infty$. This is known as an indeterminate form and needs closer study, which requires a clearer definition than you have supplied. Each of $a$ and $b$ normally comes with a formula. Often those depend on a parameter that is common to them. For examp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
If $A$ is an empty set, how should I understand $\forall x\in A$? It might look quite stupid, but I had become little confused when understanding empty functions. Anyway, my question is,
If there is a statement $P(x)$ starting with "for $\forall x\in A$,..." and $A$ is an empty set, should I understand this as because ... | Imagine this:
Everytime I have played the lottery I won the jackpot!
Why is this true? Am I the luckiest person on the planet? No. I just have never played the lottery.
That is: the set of all times T that I played the lottery is empty ... which is exactly why the claim $\forall t \in T: Jackpot!(t)$ is true.
And yes,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $f'(0)$ does not exist for the given function $f(x)$ If we want to show this we must show that $f$ is not differentiable at $x=0$.
The function is defined as follows:
$$f(x)=
\begin{cases}
x\sin{\frac{1}{x}}, & \text{if $x\ne0$} \\
0, &\text{if $x=0$}
\end{cases}
$$ which is a piecewise function.
I ... | Short answer:
$$\lim_{h\to0}\frac{h\sin\dfrac1h-0}h=\lim_{h\to0}\sin\dfrac1h=\lim_{t\to\infty}\sin t$$
doesn't exist (because for any $L$, $\max(|\sin t-L|)\ge 1$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Writing in Cartesian form and converting to polar form Question: Write $\frac{u+v}{w}$ in the form $re^{i\theta}$ when $u$=$1$, $v$=$\sqrt3i$, and $w$=$1+i$I added $u$ and $v$ for the Cartesian form because it is easier to do.After adding $u$ and $v$, I get $\frac{1+\sqrt3i}{1+i}$. Need help converting to polar form pl... | Hint: Write $u+v$ and $w$ in polar form and then divide...
you will get something like
$\frac{r_1 e^{i \theta_1}}{r_2 e^{i \theta_2}} \\\ $
Write this as $\frac{r_1}{r_2} e^{i(\theta_1-\theta_2)}$
Can anyone change this to a spoiler please :)?
$\frac{2 (\frac{1}{2} +\frac{\sqrt{3}}{2}i)}{\frac{2}{\sqrt{2}}(\frac{\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
If $a$ and $b$ are relatively prime integers then prove that ($a$ ,$b^2$) =1 From the title I've stucked in this question for half an hour. Could anyone help me?
| Consider the prime factors of $a$ and those of $b$. The fact that $a$ and $b$ are relatively prime means that $a$ and $b$ have no common prime factors. But then $a$ and $b^2$ have no common prime factors either; that is, $a$ and $b^2$ are relatively prime. QED.
(We have used the observation that $b^2$ has the same prim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
When would a number $p$ be divisible by $p-k$, where $p$ and $k$ are positive integers? When would a number $p$ be divisible by $p-k$, where $p$ and $k$ are positive integers?
Suppose we then set a constant value for $k$, then what would be condition satisfying which, $p-k$ would be a factor of $p$.
| For the first question : when $k= p - d$ such that $d | p$ , for example : $p=15$ and $d= \{1,3,5,15\}$ so $k = \{14,12,10,0\}$ and we can exclude $0$ to make $k$ positive.
For the second question : let $d|k$ then $p=\frac{k(d+1)}{d}$ for example : $k=18$ then $d = \{1,2,3,6,9,18\}$ so $p = \{36,27,24,21,20,19\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing compactness of operator We consider $X=l^p(\mathbb{Z},\mathbb{C})$ for $1\leq p\leq\infty$ and define $T((x_n)_{n\in\mathbb{Z}})= (a_n x_n)_{n\in\mathbb{Z}} :=(\frac{1}{n^2+1}x_n)_{n\in\mathbb{Z}}$. Let $a_n^N = a_n$ if $n\leq N$ and $a_n^N = 0$ else and $T_N(x_n) = (a_n^N x_n)$. Then $rk (T_N)\leq N$, thus $T_... | Your idea is good ! But you should explain in detail why $T_N \to T$ in $L(X)$.
Hence give a proof for $||T_N -T|| \to 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Matrix Equation - Make X the Subject I'm having a complete mind blank here even though i'm pretty sure the solution is relatively easy.
I need to make X the subject of the following equation:
$$AB - AX = X $$
All i've done so far is:
$$A(B-X) = X$$
$$B-X = A^{-1} X$$
Not sure if thats right?
Thanks in advance.
| Whatever you have written is correct if inverse of A exists.
Hint for another way of writing an expression for $X$: $AB = (I+A)X.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Writing a probability in terms of the edges of a simplex I describe the problem dimension $2$, but it could be generalized to $n$ dimensions.
So we have $X_{1}, X_{2}, X_{3}$ three $iid$ random variables of continuous law $F(x,y)$ on $\Bbb R^2$. Let's denote by $S[X_{1}, X_{2}, X_{3}]$ the simplex generated by those ra... | This is based on the barycentric coordinates. Applying Cramer's rule or using suitable software, one obtains
\begin{aligned}
a_1 &= \frac{x_2 y_3 - y_2 x_3 + x_3 y - x y_3 - x_2 y + y_2 x}{-x_2 y_1 + x_2 y_3 - x_1 y_3 + y_1 x_3 + y_2 x_1 - y_2 x_3} \, , \\
\\
a_2 &= \frac{x_1 y - x_1 y_3 - x_3 y + y_1 x_3 + x y_3 - y_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
are the partial derivatives must be continuous in the chain rule? Let $g(t)=(x(t),y(t))$ and suppose that $g'(t)$ exist in $t=t_0$. By the chain rule, if the partial derivatives of a function $f(x,y)$ are continuous in an open neighborhood of $g(t_0)=P$, then
$$
(f\circ g)'(t_0)=f_x(P)x'(t_0)+f_y(P)y'(t_0).
$$
Are the... | Let $A\subset\mathbb{R}^{n},\ B\subset\mathbb{R}^{m}$ open sets and let $f:A\to\mathbb{R}^{m},\ g:B\to\mathbb{R^{l}}$ be functions such that:
*
*$f$ is differentiable at $a\in A$
*$g$ is differentiable at $f(a)$
*$f(A)\subseteq B$
Then, $g_{o}f$ is differentiable at $a$ and $(g_{o}f)'(a)=g'(f(a))f'(a)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the last two digits of $47^{89}$ Find the last two digits of the number $47^{89}$
I applied the concept of cyclicity
$47\cdot 47^{88}$
I basically divided the power by 4 and then calculated $7^4=2401$ and multplied it with 47 which gave me the answer $47$ but the actual answer is $67$. How?
| Your argument would work fine for finding the last two digits of $7^{89}$. But just because $7^4 = 2\,401$ ends in $01$ doesn't mean that $47^4$ will. (In fact, $47^4 = 4\,879\,681$.)
It takes a lot longer for the last digits two of $47^n$ to cycle. You'll at least be able to do calculations with smaller numbers if you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2196910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Prove $f(c) = c$ under a condition.
Suppose $f(x)$ is continuous on $[0,1]$ and $f(0) = 1, f(1) = 0$. Prove that there is a point $c$ in $(0, 1)$ such that $f(c) = c$.
let $l(x) = x$ and $d(x) = f(x) - l(x)$.
We have $d(1) = f(1) - l(1) = -1$ and
$d(0) = 1$.
We divide $[-1, 1]$ in $H$ equal parts , where $H$ is a in... |
$\DeclareMathOperator{\st}{st}$This is indeed the correct and usual approach to proving the intermediate value theorem (and hence this question) under nonstandard analysis.
The fundamental idea is if $f\colon[a,b]\to\mathbb{R}$ is a continuous function and $u\in\mathbb{R}$ satisfies $f(a)<u<f(b)$ (or $f(b)<u<f(a)$) th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2197020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Show that PTIME and PSPACE is closed under Klenee star how to show that PSPACE and PTIME are closed under Kleene star ? I can only show that NP is closed, but it is easy because we can use non-determinism to guess partition of word. In these two cases I don't have idea how to attack it.
Edit
Using @sdcvvc's hint.
I... | Hint: Consider a graph $G$ where vertices are positions in the word, and there is an edge $i \to j$ if the subword $w[i..j-1]$ is in $L$. Show that this graph can be computed in $PTIME$ (or $PSPACE$). Can you make a connection between this graph and Kleene star $L^{\ast}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2197148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A monoid where square of all elements are 1 is abelian The following problem gives me a very hard time:
Let $M$ be a monoid with $a^2 = 1$ for $a \in M$. Show that $M$ is abelian.
It looks so simple as a monoid only needs to be associative and must have a neutral element (here $1$). So there are not much things to tr... | $1=abab\to b=ababb \to b=aba \to ab =aaba \to ab =ba $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2197240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving an equation where the unknown appears in under the exponent and as a constant I encountered a problem trying to find a solution to the following equation (the problem 9.45 from the wonderful textbook on Applied Calculus by Hoffman et.al (2013, p. 719)):
$$ A(t) = \frac{3}{k} (1 - e^{-kt}) $$
According to the ex... | The "analytical" answer is $$k = \frac{30}{23} + W\left(-\frac{30}{23} e^{-30/23}\right) $$
where $W$ is the Lambert W function.
But I doubt that Hoffman et al intended you to solve it this way.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2197357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why don't we take clopen maps as morphisms of the category of topological spaces? Here, by clopen maps I mean a function mapping an open set into an open set and a closed set into a closed set. We say continuous maps are the morphisms of the category of the topological spaces. But isn't it more natural to consider clop... | Because then $f : \mathbb{R} \to \mathbb{R}$, $x \mapsto e^x$ would not be a morphism (its image is not closed). Recall that almost every notion in mathematics is motivated by examples, and this includes the definition of a category and explicit examples of categories. You don't want to just play around with the axioms... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2197604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 0
} |
Describe and draw the half-strip $R$ under the mapping $f(z) = z^2$ Describe mathematically and draw what happens to the half-strip $R = \{z=x+iy: 0 \leq x \leq 1, y \geq 0 \}$ under the mapping $f(z) = z^2$
I need help with describing and drawing the mapping.
solution:
For, $z = x+iy$ and $w=f(z)=z^2$
$\Rightarrow w=... | The strip is the thing that should be drawn on the $x-y$ plane and what it goes to on the $u-v$ plane. It's probably best to consider the boundary first. The first part is the positive $y$ axis with $x=0.$ This goes $iy\to (iy)^2 = -y^2$ so its image is the negative real axis. Next do the segment on the $x$ axis betwee... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2197682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that the Gram Matrix G(B) is Positive Definite
Suppose we have $\boldsymbol P_{\leqslant 1}=\operatorname{span}\{1,x\}$ that is an inner product space with respect to $\int^1_0p(x)q(x)dx$.
Consider the basis $B=\left\{b_1 = 1, b_2 = x\right\}$.
Finding the Gram matrix $G(B)$ I would have
$G(B)=\begin{bmatrix}\in... | There are a lot of ways to prove that a matrix is positive definite, but sometimes working from the definition $x^TAx > 0$ if $x$ nonzero is easiest. In this case you'll see that the Gramian being positive-definite is very general, much more so than looking at monomials.
Let $\langle \cdot, \cdot\rangle $ be your inn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2197821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Extend triple of coprime numbers to summation of coprime numbers I apologize if this has already been answered or is common knowledge. If either of these is the case a reference will suffice.
Given $a, b, c$ positive integers which are pairwise relatively prime, do there exist positive integers $x,y,z$ such that
$$a x... | Yes. In fact, we can always do this with $x = 1$.
We start by just finding an arbitrary solution to the identity. Choose $y_0$ such that $by_0 \equiv -a \pmod c$; this is possible because $b$ has an inverse modulo $c$. Then we have $a + by_0 = cz_0$, and all is right with the world, except that these might not be relat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2198072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that for all $n\ge9$, there exist natural numbers $x,y$ such that $n=2x+5y$. How would you use induction to prove this?
| If $n=9$ then $n = 2\cdot 2 + 5\cdot 1$; taking $x:= 2$ and $y:=1$ suffices.
If $n \geq 9$ is an integer such that
$n-1= 2x + 5y$ for some integers $x,y > 0$, then
$n = n-1 + 1 = 2x+5y + 1 = 2x' + 5y'$.
Note that $2x+5y+1 = 2x+5y+(5-4) = 2(x-2) + 5(y+1)$.
So the preceding equalities are equivalent to
$$
n = 2x' + 5y' =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2198173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
find equation of a curve that represents the sum of f(x) Given a straight line with equation f(x)=2x where x belongs to {0,1,2,3,4,5},
how do i find g(x) (a curve?) which represents the sum over f(x)
In plain english if the price of an object doubles every time I purchase it, what will be the total cost if I purchased... | Your plain English question makes much more sense than your attempt to turn it into algebra.
You should be able to guess the pattern here:
number purchased total price
1 P
2 P + 2P = 3P
3 3P + 4P = 7P
4 7P + 8P = 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2198287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Suppose $p>q$, $X\sim \text{Bernoulli}(p)$, $Y\sim \text{Bernoulli}(q)$. Couple $X$ and $Y$ to maximise $P(X=Y)$. My professor's solution to this is as follows: "Create a 2 x 2 matrix with the first row (corresponding to $X=0$) summing to $P(X=0)=1-p$, the second row summing to $P(X=1)=p$, the first column ($Y=0$) summ... | The first diagonal entry is the probability that $X=Y=0$, and we know that both of the following statements are true:
*
*Since $\Pr[X=Y=0] \le \Pr[X=0]$, it is at most $1-p$.
*Since $\Pr[X=Y=0] \le \Pr[Y=0]$, it is at most $1-q$.
However, $p>q$, so the first constraint is stronger, and we can forget about the se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2198454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Order of a permutation, how to calculate I know this is a basic question however I am slightly confused what to do if the permutation contains the same element twice in cycle notation.
For example: the permutation $(1 2 3)(2 4 1)$, how would I calculate the order when $2$ maps to $3$ and $4$? Is it just the same? D... | First you'll need to express $(123)(241)$ in terms of the product of disjoint cycles.
$(123)$ and $(241)$ are not disjoint cycles, as you note, since both share the elements $1, 2$.
To do so, you start from the right cycle, and compose with the left cycle. So, in the right-hand cycle, we have $1\mapsto 2$ and in the l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2198574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Compact operators satisfying a certain relation must be finite-rank Let $H$ be an infinite-dimensional Hilbert space, equipped with a given Hilbert basis $(e_i)_{i \in \mathbb{N}}$.
Consider the following introductory problem : can we find a compact operator $A$ in $H$ that satisfies the relation
$$ \sum \limits_{k=0}... | Denote $i$ the smallest index such that $c_i\ne0$. Then we can factorize the polynomial
$$
\sum_{k=i}^n a_k t^k = c_i t^i \prod_{k=i+1}^n (t-\lambda_k)
$$
This implies
$$
c_i \left( \prod_{k=i+1}^n (A-\lambda_k I) \right) A^i=0.
$$
Since $A$ is compact, the null spaces of all operators $A-\lambda_k I$
are finite-dime... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2198744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Spivak Chapter 2 Problem 7 Question: Use the method of Problem 6 to show that $\sum_{i=0}^n k^p$ can always be written in the form $\frac{n^{p+1}}{p+1} +An^p + Bn^{p-1} + Bn^{p-2} + Cn^{p-3} + ..... $
The method in 6 they are talking about is the telescoping method.
I have tried to derive the solution for some while an... | Hint:
If you aren't familiar with the notation, when I write $\mathcal{O}(k^r)$, I basically mean "terms involving $k^r$". Then it should be clear that $\mathcal{O}(k^r) = \frac{1}{p+1} \mathcal{O}(k^r)$.
In this case, we have $r < p$, so we have
$$(k+1)^{p+1} - k^{p+1} = (p+1)k^p + \mathcal{O}(k^r),$$
so
$$\frac{(k+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2198842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the point where the angle of incidence is equal to the angle of reflection Say I have two points, $$A=(-1,1)\\B=(2,1)$$ and a line at $$y=0$$ How do I find the point on the line that makes A and B have the same angle?
|
In order to have the same angle it is enough to have
$$PD=PC=\frac{3}{2}\to 2-x=\frac{3}{2}\to x=\frac{1}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2198925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Help to Understand the Proof of Extreme Value Theorem Below are sceen shots taken from Pugh's Book Real Mathematical Analysis.
My question mainly is from the proof below. How does it follow that $M<M$ from $b=c$? Thanks for your help.
| *
*Notice that during the first part of Case 2, we prove that the least upper bound of $V_c < M$, which means that the least upper bound of $f$ on $[a,c]$ is strictly less than $M$.
*But recall that at the beginning, we defined $M$ as the least upper bound of $f$ on the entire interval $[a,b]$. Using the same notati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2199046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
does a rectangular matrix have an inverse? I know all square matrices have easily to identify inverses, but does that continue on with rectangular matrices?
| Actually, not all square matrices have inverses. Only the invertible ones do. For example, $\begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}$ does not have an inverse.
And no, non-square matrices do not have inverses in the traditional sense.
There is the concept of a generalized inverse. To very briefly summarize the l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2199172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Find all the positive integers a, b, and c such for which $\binom{a}{b} \binom{b}{c} = 2\binom{a}{c}$ I tried an the following equivalence from a different users post from a while back
that stated the following.
$\binom{a}{b} \binom{b}{c} = \binom{a}{c} \binom{a-c}{b-c}$
where does this equivalence come from?
After app... | You've reduced the equation to finding solutions to
$$\binom{a-c}{b-c} = 2. $$
Since $a,b,c$ are positive integers, and the only binomial coefficient equal to $2$ is $\binom{2}{1}$, the equalities
$$ a - c = 2$$ and $$b-c = 1$$ are forced.
Thus the solution set consists of consecutive triples of positive integers
$$(a,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2199257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Algebra Precalculus
$a> 0 < b$ For all the integer value of $a$ and $b$
$X= (a^2 + ab)-(ab^2-b)/(2a^2+b^2 -ab)$
Quantity I: $x $
Quantity II: $1.5$
(a) Quantity I $\lt$ Quantity II
(b) Quantity I $\gt$ Quantity II
(c) Quantity I $\ge$ Quantity II
(d) Quantity I $=$ Quantity II
(e) No relation
$(x^a)^c = x^c$
$x^... | I assume $a = 1$
Then,
$$x^{2b}/x^a = (x^{5a}) * (x^d)*(x^b) \implies x^{2b}/x = (x^{5}) * (x^d)*(x^b) \implies x^{2b} = (x^{6}) * (x^{d+1})*(x^{b+1}) \implies x^{2b - b - 1} = (x^{6}) * (x^{d+1})\implies x^{b - 1} = (x^{d+7}) \implies x^{b - 1 - d - 7} = 1 \implies x^{b- d - 8} = 1 \implies b = d+8$$
Therefore $b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2199433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Boundedness of $a+\frac 1a$ when iterated Here's something I was wondering...
Is $$a + \frac 1a$$ for any positive real number $a$ bounded when iterated?
For example,, if we start at $a=1$, continuing gives us $a= 1+ \frac 11=2$, then $a=2+\frac 12=2.5$ and so on. A quick program shows that it seems to grow without... | The function
$$
f(x)=x+\frac1x
$$
is strictly increasing for $x\ge1$ (i.e. if $x>y$, then $f(x)>f(y)$). Also, we have that
$$
f(x)=x+\frac1x>x
$$
for $x\ge1$. Hence, $f(f(x))>f(x)$. However, the function might be bounded, i.e. $f(x)\le M$ for all $x\ge1$. But we have that $f(M)>M$, which is a contradiction. So it grows... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2199561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
finding moment generating function I am having a bit of trouble finding the moment generation function for
$f(x)=(\frac{1}{2})^{x+1}$ for $x=0,1,2,3,\ldots$
I know that $M(x)=\sum e^{tx}(\frac{1}{2})^{x+1}$ which I have rearranged to make $\frac{1}{2} \sum (\frac{1}{2}e^t)^x$ but I am not sure how to simplify this fur... | Indeed, the moment generating function is defined as
$$E\left[e^{tX}\right]=\sum_{i=0}^{\infty}e^{ti}P(X=i)=\sum_{i=0}^{\infty}e^{ti}\frac1{2^{i+1}}=\frac 12+\frac12\sum_{i=1}^{\infty}\left(\frac {e^t}2\right)^i.$$
This is a geometrical series with $q=\frac {e^t}2.$ This series is convergent if $0\le q<1$: $t<\ln(2).$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2199719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Boxplot Skewness I do know there are some rules about boxes and whiskers to determine the skewness in a boxplot, but I am confused with some rules in this particular case:
Keeping in mind the rules, in this boxplot the median falls to the right of the center of the box, thus its distribution is negatively skewed. But ... | You are correct that 'indications' of right-skewness of a sample from a boxplot
may be that (a) the median is left of center inside the box and (b) a longer
whisker to the right than to the left. However, boxplots are best used for
samples of moderate or large size.
Of course, I don't know for sure, but
I would guess ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2199835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Show that $f(x,y)=\frac{xy^2}{x^2+y^4}$ is bounded
Let $f\colon\mathbb R^2\to\mathbb R$ be a function given by:
$$
f(x,y)=\begin{cases}\frac{xy^2}{x^2+y^4}&\text{if }(x,y)\neq(0,0),\\
0&\text{if }(x,y)=(0,0).
\end{cases}
$$
I need to show that $f$ is bounded on $\mathbb R^2$, so I need to show that there exists $M>... | *
*Consider $(z+1)^2 \geq0$. This leads to $\Large \frac{z}{z^2+1}$$\geq-$$\Large\frac{1}{2}$.
*Now take $(z-1)^2 \geq0$ This yields $\Large \frac{z}{z^2+1}$$\leq$$\Large\frac{1}{2}$.
Therefore $\large f $ is bounded.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2199970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Calculate area of Ellipse without calculus? I like the way integration works, but the final formula $\pi ab$ is too simple.
I know there is a more deeper way to derive it. I just don't like to use calculus here, too many equations.
I'd like to use simple math, which does offer deeper insight into it.
| Consider the unit disk (bounded by the circle of radius $1$, centered at the origin). Now, to construct an ellipse whose axes are $a$ along the $x$-axis and $b$-along the $y$-axis. This corresponds to the application of the linear transformation
$$
\begin{bmatrix}a&0\\0&b\end{bmatrix}.
$$
We can confirm that this is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2200113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Chern class of $E \times F \to M \times N$ Let $M$ and $N$ be two complex manifolds and $E \to M$, $F \to N$ be two holomorphic vector bundles.
1) Is there a way to define vector bundles $G \to M \times N$ such that a fiber over $(x,y) \in M \times N$ is, eg, $E_x \oplus F_y$ or $E_x \otimes F_y$ ?
2) How are the Cher... | If $p_1:M\times N\to M$ and $p_2:M\times N\to N$ are the projections, then you can pull back to get bundles $p_1^*E\to M\times N$ and $p_2^*F\to M\times N$. Then we can define the "box product" $E\boxtimes F\to M\times N$ by
$$E\boxtimes F=p_1^*E\otimes p_2^*F.$$
Now, if $(x,y)\in M\times N$ then since $(p_1^*E)_{(x,y)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2200200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Show that $S(x) = \int_{h(x)}^{g(x)}f(t)dt \implies S'(x) = f(g(x))g'(x)-f(h(x))h'(x)$ let $g(x), h(x)$ be derivatible functions in $R$ and let $f(x)$ be a continuous function in $R$.
$S(x) = \int_{h(x)}^{g(x)}f(t)dt$.
How can I prove:
$S'(x) = f(g(x))g'(x)-f(h(x))h'(x)$.
| Not very rigorous, but useful to help to remember the rule:
Let $F(x)=\int f(t)\mathrm dt$, so is, a primitive of $f$, then $F'(x)=f(x)$
$S(x)=\int_{h(x)}^{g(x)}f(t)dt=[F(t)]_{h(x)}^{g(x)}=F(g(x))-F(h(x))$
$S'(x)=F'(g(x))g'(x)-F'(h(x))h('(x)=f((g(x))g'(x)-f(h(x))h'(x)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2200304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
An open covering of $\mathbb{Q} \cap [0,1]$ that does not contain any finite subcovering Consider the topological subspace $\mathbb{Q} \cap [0,1]$ endowed with the usual topology of $[0,1]$, since $[0,1]$ is Hausdorff and that $\mathbb{Q} \cap [0,1]$ is not closed, we conclude that $\mathbb{Q} \cap [0,1]$ is not compac... | Pick your favorite irrational number $\xi\in (0,1)$, and consider the open cover
$$\Big\{\Big[0,\xi-\frac{1}{n}\Big)\cup\Big(\xi+\frac{1}{n},1\Big]\Big\}_{n=n_0}^{\infty}$$
where $n_0$ is chosen large enough that $\xi-\frac{1}{n_0}>0$ and $\xi+\frac{1}{n_0}<1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2200414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
if it exist the limit of $f(z_n)$ then there exist the the limit of $z_n$ Let $D \subset \mathbb{C}$ a closed compact subset of the complex plane.
Let $f:D \to \mathbb{C}$ a continuous function.
Let $\{z_n\}_{n \in \mathbb{N}} \in D$ a sequence of $D$.
Is it true that
$$
\lim_{n \to \infty} f(z_n)=L \Longrightarrow \li... | No. Take $f$ a constant function and $\{z_n\}$ any divergent sequence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2200512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Differential equation in $\mathcal{S}'$, Fourier method I have to solve that equation with Fourier method: $y'-iy=1+\delta'(x)$
Fourier transform is defined like this:
$F[\varphi](k)=\int\limits_{-\infty}^{\infty}\varphi(x)e^{ikx}dx$
$F^{-1}[\varphi](x)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\varphi(k)e^{-ikx}dk$... | $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2200658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove this sequence takes every rational number Given the sequence $a_1 = 0$ and $a_{n+1} = \dfrac{1}{2 \cdot\lfloor{a_n}\rfloor-a_n+1}$ and $p,q\in \mathbb N$ and coprime find $x$ so that $a_x = \dfrac{p}{q}$. I do not even know where would you start with a problem like this.
| Observation: $a_k<1$ iff $k$ is odd.
Lemma: If $a_{2n}$ = $a_n$+1.
Proof: By induction. $a_2 = 1 = 1+a_1$. Further suppose $a_{2(n-1)}=a_{n-1}+1$. Denote $x=2\lfloor a_{n-1}\rfloor-a_{n-1}+1$. Then
$$a_n=\frac 1x,$$
$$a_{2n-1} = \frac 1{x+1},$$
$$a_{2n} = \frac 1{2\cdot0-\frac1{x+1}+1} = \frac1{\frac{x}{x+1}}=\frac{x+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2200740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Notation for partial derivatives I thought that the meaning of
$$
\frac{\partial f(x, y, z)}{\partial x}
$$
is differentiation on $x$ with fixed $y$ and $z$. So $(x, y, z)$ in the numerator is just saying which variables are fixed. If I need to indicate where the derivative is evaluated, I write it in the right of a ve... | *
*Yes, $\frac{\partial f(x,y,z)}{\partial x}$ is derivative w.r.t. $x$ at fixed $y,z$.
*$\frac{\partial f(0,0,0)}{\partial x}$ is not standard notation. Strictly speaking, it should be zero, because $f(0,0,0)$ is a constant which does not depend on $x$. Sometimes, yes, it is used as a shorthand for $\frac{\partial f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2200982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 6,
"answer_id": 2
} |
How find this maximum of the value $\sum_{i=1}^{6}x_{i}x_{i+1}x_{i+2}x_{i+3}$? Let
$$x_{1},x_{2},x_{3},x_{5},x_{6}\ge 0$$ such that
$$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}=1$$
Find the maximum of the value of
$$\sum_{i=1}^{6}x_{i}\;x_{i+1}\;x_{i+2}\;x_{i+3}$$
where
$$x_{7}=x_{1},\quad x_{8}=x_{2},\quad x_{9}=x_{3}\,.$... | For $x_i=\frac{1}{6}$ we get $\frac{1}{216}$.
We'll prove that it's a maximal value.
Indeed, let $x_1=\min\{x_i\}$, $x_2=x_1+a$, $x_3=x_1+b$, $x_4=x_1+c$, $x_5=x_1+d$ and $x_6=x_1+e$.
Hence, $a$, $b$, $c$, $d$ and $e$ are non-negatives and we need to prove that:
$$216\sum_{i=1}^6x_ix_{i+1}x_{i+2}x_{i+3}\leq\left(\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
What's the fastest way to determine all the subgroups of the additive group $\mathbb{Z}_{24}$ Question is as in title.
I know that all of the subgroups of $\mathbb{Z}_{24}$ (under addition) must be cyclic, and I could find them by finding the generating groups for each element of $\mathbb{Z}_{24}$ - but surely there is... | What is important has already been said in comments and other answers: for each $d \mid n$, there is a single subgroup of order $d$ in $\Bbb Z_n$, and it is isomorphic to $\Bbb Z_d$. It can be explicitly described as
$$\left\{ \widehat {\frac {kn} d} \Bigg| 0 \le k \le d-1 \right\} .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2201185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.