Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Problem with a proof where algebraic extensions are assumed to be finite extensions I'm reading the article "Integration in Finite Terms" by Maxwell Rosenlicht and I have a problem with one step in a proof. Rosenlicht wants to prove the following: If $F$ is a differential field of characteristic zero and $K$ an algebr... | Let $K/F$ be an algebraic extension and let $\mathscr F$ denote the set of finite subextensions of $K/F$ that's the set of subfields $E$ of $K$ containing $F$ such that $E/F$ is a finite extension.
For all $E\in\mathscr F$ your text proves the existence of one and only one derivation $d_E:E\to E$ extending that given o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2889191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to minimize $f(x) = \|Ax-b\|$ Solve the problem of minimizing $f(x) = ||Ax-b||$. Consider all the cases and interpret geometrically.
If we write
$$\|Ax-b\| = (a_{11}x_1 + \cdots + a_{1n}x_n - b_1)^2 + \cdots + (a_{n1}x_1 + \cdots + a_{nn}x_n - b_1)^2$$
then
$$\frac{\partial \|Ax-b\|}{\partial x_j} = 2(a_{11}x_1 + ... | \begin{eqnarray*}
\left \lVert Ax-y\right\rVert^{2} &=& x^{T} A^{T} A x -2 y^{T} A x + y^{T} y
\end{eqnarray*}
Gradient w.r.t $x$ to $0$ translates to
\begin{eqnarray*}
\nabla_{x} \left \lVert Ax-y\right\rVert^{2} &=& 2 A^{T} A x - 2A^{T} y =0
\end{eqnarray*}
yields, the normal equations. The solution is (the well kno... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2889288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Show that the ring of polynomials with coefficients in a field, and in infinitely many variables, is not Noetherian Show that the ring of polynomials with coefficients in a
field, and in infinitely many variables, is not Noetherian, that is, $R = k [x_i: i\geq1]$ is not Noetherian.
I know that I need to exhibit an idea... | A ring is Noetherian if and only if it satisfies the ascending chain condition, i.e. every increasing chain of ideals terminates. Now you have a chain
$$(x_1)\subsetneq(x_1,x_2)\subsetneq(x_1,x_2,x_3)\subsetneq\cdots$$
that never terminates, so $k[x_i: i\ge 1]$ is not Noetherian.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2889434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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A "concrete" example of a left Hopf algebra I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition.
To be more precise, let $\Bbbk$ be a field and $(B,\mu,\eta,\Delta,\varepsilon)$ a $\Bbbk$... | I learn this "obvious" example from Peter Shawenburg. It doesn't give you a direct answer but is very closed.
Take the $A=k\{a,b\}$ free algebra on 2 generators $a$ and $b$. Declare them to be group-like elements, so you have $A=$the semigroup algebra on the free monoid on 2 generators, that is a bialgebra.
Consider th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2889576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Integrating linearly I came across this question and just want to make sure my understanding is correct.
I need to find the general solution of:
$$
\frac{dx}{dt} = a(1 - x)
$$
In this case, I'm finding the how $x$ changes with respect to $t$ so I'm integrating with respect to $t$. Does that mean the answer is $at - xa... | You can't say that, simply because the function $x$ has a dependence in $t$ !
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2889665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Question about continuous function.
Question. Which of the following statements are true:
*
*If $f \in C[0,2]$ is such that $f(0)=f(2)$, then there exist $x_1$ and $x_2$ in $[0,2]$ such that $x_1-x_2=1$ and $f(x_1)=f(x_2)$.
*Let $f$ and $g$ be continuous real valued functions on $\mathbb{R}$ such that for ... | As always in this kind of problems, to use intermediate values theorem, you have to convert your "equation" in the form $g(x)=0$. Here you have $f(x+1)=f(x)$, so let $g:x\mapsto f(x+1)-f(x)$ be defined on interval $[0,1]$.
You have $g(0)=f(1)-f(0)$, $g(1)=f(2)-f(1)$, so $g(0)+g(1)=f(2)-f(0)=0$.
Either $g(0)$ and $g(1)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2889761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Calculate, $f\bigg(\frac{1}{1997}\bigg)+f\bigg(\frac{2}{1997}\bigg)+f\bigg(\frac{3}{1997}\bigg)\ldots f\bigg(\frac{1996}{1997}\bigg)$
If $$f(x)=\frac{4^x}{4^x+2}$$
Calculate,
$$f\bigg(\frac{1}{1997}\bigg)+f\bigg(\frac{2}{1997}\bigg)+f\bigg(\frac{3}{1997}\bigg)\ldots
f\bigg(\frac{1996}{1997}\bigg)$$
My Attempt:
I was... | I would say your method is practically speaking what I would also do.
Maybe I would rephrase it as follows:
Claim: $f(a)+f(1-a)=1$.
Then write $S$ for the sum in question, and then $2S$ can be written as $f(1/1997+1996/1997) + \cdots$ (the Gauss trick), which is $1996$ by the claim, so $S=998$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2889855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Sufficient Condition for Positivity of Matrix with Operator-valued Entries Let $\mathcal{H}$ be some Hilbert space, let $B(\mathcal{H})$ denote the bounded linear operators acting on $\mathcal{H}$, and let $M_n(B(\mathcal{H}))$ denote the $n \times n$ matrices with operator-valued entries. Let $A = [a_{ij}]$ be one suc... | Yes. Let $\xi\in \mathcal H^n$. Choose $r $ such that $\|\xi_r\|\geq\|\xi_j\|$ for all $j $.
For each $j$, let $x_j\in B(\mathcal H)$ be a contraction such that $x_j\xi_r=\xi_j $. By the argument in this answer, there exists a unitary with $u_j\xi_r=\xi_j$. Then
$$
\langle A\xi,\xi\rangle=\sum_{k,j}\langle a_{kj}\xi_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2889948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving a function $f:\mathbb{Z}_{nm} \rightarrow \mathbb{Z}_n \times \mathbb{Z}_m$ is an isomorphism
Let $n,m$ be two coprime numbers. Prove that the function $f:\mathbb{Z}_{nm} \rightarrow \mathbb{Z}_n \times \mathbb{Z}_m$ such that $f(\overline{r})=(\overline{r},\overline{r})$ is an isomorphism between rings.
I'v... | Surjectivity, in particular, $n,m$ are relatively prime. $um+vn=1$ implies that $r=anv+bmu$.
Injectivity, $(\bar r,\bar r)=(\bar 0,\bar 0)$ implies that $r(um+vn)=r=0$ mod $m$ this implies that $rvn=0$ mod $m$ and $m$ divides $rv$, we deduce that $m$ divides $r$, since $m$ cannot divides $v$ since $um+rv=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2890045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Calculus/ Analysis books I am looking for some titles. Not looking for basic textbooks nor advanced, I am craving for real stuff.
More in detail I would like some book that covers calculus in one variable from a more mature perspective ( such as the one that a phd student should have).
Something that may be helpful in ... | "Introduction to measure theory" by Tao is great, it's also online
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2890174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 5
} |
Ultrafilters on finite boolean algebras I am asked to prove a special case of Stone duality, namely that $B\cong \mathcal{P}(\text{Ult}(B))$ by the map $\phi:B\to \mathcal{P}(\text{Ult}(B))$ given by the homomorphism
$$
\phi(x)=\{V\in \text{Ult}(B) \;|\; x\in V\},
$$
where $B$ is a finite boolean algebra and $\text{Ult... | Here's an easier way to show surjectivity: since $\mathcal{P}(\text{Ult}(B))$ is generated by singletons, it suffices to show that every singleton is in the image of $\phi$. In other words, given an ultrafilter $U$ on $B$, we want to find $x\in B$ which is in $U$ and no other ultrafilters.
I encourage you to try to fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2890280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Hint on a measure theory question Let $(X, \mathcal{A}, \mu)$ be a measure space, and define
$\mu^\bullet: \mathcal{A} \to [0, +\infty]$ by
$$
\mu^\bullet (A) =
\sup
\left\{
\mu(B) : B \subseteq A,
B \in \mathcal{A}, \mu(B) < + \infty
\right\}
$$
I'm trying to show that $\mu^\bullet$ is countably additive. One thi... | Hint: fix $\epsilon>0$ and for each $n$ take $B_n$ with $\mu(B_n)+\epsilon/2^n>\mu(A_n)$, then try to use the fact that $\sum_n\epsilon/2^n=\epsilon$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2890387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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$\mathbf{H}(3)$ is diffeomorphic to $\mathbf{SL}\left( 2,\mathbf{C}\right) \mathbf{/SU}\left( 2\right) $ I'm reading the book from Jensen's "Surfaces in Classical Geometries".
Could anyone help me understand why $\mathbf{H}(3)$ is diffeomorphic to $\mathbf{SL}\left( 2,\mathbf{C}\right) \mathbf{%
/SU}\left( 2\right) $?
... | This follows from the general statement that if a lie group $G$ acts transitively on a space $X$, and if given $x\in X$ we define $G_x:=\{g\in G \mid gx=x \}$, then $X\cong G/G_x$. This can be seen as a generalization of the orbit-stabalizer theorem, as when $X$ is a finite set and $G$ is a finite group, $|G/G_x|=|G|/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2890508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Is it possible to assume the existence of “Dominating Turing Machines”? Consider three-tape (tape $1$ for the input, tape $2$ for the computation, tape $3$ for the output) two-symbol (blank symbol and non-blank symbol) Turing machines.
Let $F(x, y)$ denote the minimal natural number greater than number of non-blank c... | I can tell you that if Dominating machines exist, at most they have 14 states. Checking all small machines for a property as complicated as yours will be too laborious without some idea for why it is important or interesting.
Let's look at a concept of Universal Turing machine. There is 2-symbol, 1-tape Universal Turin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2890703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given GCD of two numbers is 42 and their product is 15876. How many possible sets of members can be found? Given GCD of two numbers is 42 and their product is 15876. How many possible sets of numbers can be found?
I have no idea. I can only evaluate the lcm. Don't know how to get the answers.
| Use $\gcd(x,y) \times \operatorname{lcm}(x,y) = xy$
\begin{array}{rrr}
xy &= &15876 \\
\operatorname{lcm}(x,y) &= &378 \\
\hline
\gcd(x,y) &= &42
\end{array}
Assume $x < y$. Use $\gcd(p^a, p^b) = p^{min(a,b)}$ and
$\operatorname{lcm}(p^a, p^b) = p^{max(a,b)}$ when $p$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2890832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to prove that infinite number of pentagons exist satisfying the given requirements A given convex pentagon ABCDE has the property that the area of each of the 5 triangles ABC, BCD, CDE, DEA and EAB are equal. How can I prove that there exist infinitely many non-congruent pentagons having the above property? I tried... | The regular pentagon has this property. Any invertible affine transformation
of the plane has the property that equal areas are sent to equal areas. So
for instance we can stretch the regular pentagon by a given scale factor in the
$x$-direction (but leave the scale in the $y$-direction unchanged) to get an infinite fa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2890977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Solve: $\lim_{x\to -\infty} (\sqrt {4x^2+7x}+2x)$ Solve: $$\lim_{x\to -\infty} (\sqrt {4x^2+7x}+2x)$$
My attempt:
Rationalizing:
$$\lim_{x\to -\infty} (\sqrt {4x^2+7x}+2x) *\frac{\sqrt {4x^2+7x}-2x}{\sqrt {4x^2+7x}-2x}$$
$$=\lim_{x\to -\infty} \frac{4x^2+7x-4x^2}{\sqrt {4x^2+7x}-2x}$$
$$=\lim_{x\to -\infty}\frac{7x}{\s... | hint When $x$ goes to $-\infty,$ it becomes negative .
on the other hand,
we have $$\boxed{\sqrt{x^2}=|x|}$$
the mistake you made can be corrected by $$\sqrt{(-x)^2}=-x$$.
In the denominator, factor out by $(-x)^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2891096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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How to solve the given integral avoiding infinite series sum? Question: How to solve the following integral?
$$I = \int_0^\infty \dfrac{x^{N_a + N_b - 1}}{(p \Omega_1 + \Omega_2 x)^{N_a + 1}} \ln (1 + Qx) \, _2F_1\left( N_b + 1, N_b; N_b +1; \dfrac{-\Omega_3}{\Omega_4}x\right)dx, \tag{1}$$
where $N_a, N_b \in \mathbb Z... | Here is an alternate way to solve.
\begin{align}
& \int_{0}^{\infty} \dfrac{x^{N_a + N_b - 1}}{(p \Omega_1 + \Omega_2 x)^{N_a + 1}} \ln(1 + Qx) \, _2F_1\left( N_b + 1, N_b,; N_b +1 , \dfrac{-\Omega_3}{\Omega_4}x\right)dx \\
= & (p\Omega_1)^{-(N_a + 1)} \int_{0}^{\infty} x^{N_a + N_b - 1} \left( 1 + \dfrac{\Omega_2}{p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2891227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Finding the range of a function without using inverse So I'm fairly close to beginner level in calculus and have usually used the inverse of a function to find its range however I'm not sure what to do when dealing with this particular function. $$ h(t) = \frac{t}{\sqrt{2-t}}$$ I found the domain to be $(-\infty, 2)$ b... | We have that $h(t)$ is a continuos function defined for $t<2$ and
$$\lim_{t \to -\infty} h(t)=-\infty$$
$$\lim_{t \to 2^-} h(t)=\infty$$
therefore by IVT the range is $\mathbb{R}$.
Morover we have
$$h'(t)=\frac{4-t}{2\sqrt{(2-t)^3}}>0$$
therefore $h(t)$ is also injective and the inverse exists from $\mathbb{R}\to (-\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2891340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Determining Whether the Number $11111$ is Prime. Used Divisibility Tests. I am asked to determine whether the number $11111$ is prime. Upon using the divisibility tests for the numbers 1 to 11, I couldn't find anything that divides it, so I assumed that it is prime. However, it apparently isn't prime. So what is the pr... | Here is a list for test of prime factor of less than $50$.
Test for divisibility by $41$. Subtract four times the last digit from the remaining leading truncated number. If the result is divisible by $41$, then so was the first number. Apply this rule over and over again as necessary.
$$1111-4(1)=1107$$
$$110-4(7)=110... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2891615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Thinking of a cube as $\mathbb Z_6\times\mathbb Z_4$ I was wondering if it makes sense to think of a cube as an additive group in this way. For example $(4,1)$ corresponds to face $4$ edge $1$.
I am a beginner in group theory and I hope this makes sense!
If I added together face $4$ edge $1$ to face $3$ edge $0$: $(4,1... | I don't see how faces and edges are related here. I makes sense, but it doesn't seem to have anything to do with the cube. For instance, you could take four types of pens in six different colors. Then define addition of pen i with color j to be like the group $\mathbb{Z}_6 \times \mathbb{Z}_4$. ... If it helps you thin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2891724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Factoring a quadratic polynomial (absolute beginner level), are both answers correct? I'm following video tutorials on factoring quadratic polynomials. So I'm given the polynomial:
$$x^2 + 3x - 10$$
And I'm given the task of finding the values of $a$ and $b$ in:
$$(x + a) (x + b)$$
Obviously the answer is:
$$(x + 5)(x ... | .Yes, you are correct. Since $(x+5)(x-2) = (x-2)(x+5) = x^2 + 3x-10$, we note that $a$ and $b$ may either take the values $(5,-2)$ or $(-2,5)$.
I would consider providing just one of the two solutions to be insufficient, since the question itself ask for the values of $a$ and $b$, but nowhere mentions that they are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2891852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Prove or disprove that base of space is base of subspace Let vectors $v_1,v_2,v_3,v_4$ is base of space $V$, and if $W$ is subspace of $V$ such that $v_1,v_2\in W$ and $v_3,v_4\not \in W$ then $v_1,v_2$ is base of $W$?
My Professor said that you can not make a base of subspace from base of space, but you can make a bas... | Since $v_1,v_2,v_3,v_4$ is base of space $V$, $W$ is subspace of $V$ and $v_1,v_2\in W$
we know that $\dim W\ge 2.$
But it is possible to have $\dim W=3.$ For example, assume $v_3+v_4\in W.$ That is, consider $W=span\{v_1,v_2,v_3+v_4\}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2891938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Span of 3 linearly dependent vectors The vectors are $(1,1,1)$,$(1,2,0)$, and $(2,3,1)$. I have shown that they are linearly dependent but don't really know how to find their span. (Note: my lecturer just literally defined what a span is and didn't get to the part where we actually calculate spans, so I'm completely lo... | Given a set of vectors their span is given by the set of all linear combinations of those vectors.
In that case the span is
$$a(1,1,1)+b(1,2,0)+c(2,3,1)$$
Since the three vectors are linearly dependent but $(1,1,1)$ and $(1,2,0)$ are linearly independent the span is also given by
$$a(1,1,1)+b(1,2,0)$$
or by any other p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2892027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Nature of infinite series $ \sum\limits_{n\geq 1}\left[\frac{1}{n} - \log(1 + \frac{1}{n})\right] $ $$\sum\limits_{n\geq 1}\left[\frac{1}{n} - \log\left(1 + \frac{1}{n}\right)\right]$$
Is it convergent or divergent?
Wolfram suggests to use comparison test but I can't find an auxiliary series.
| We may take the series $\sum_{n\geq 1}\left[\frac{1}{n}-\log\left(1+\frac{1}{n}\right)\right]$ as an equivalent definition of the Euler-Mascheroni constant $\gamma=\lim_{n\to +\infty}\left(H_n-\log n\right)$, where $H_n=\frac{1}{1}+\frac{1}{2}+\ldots+\frac{1}{n}$ is the $n$-th harmonic number.
Over the interval $(0,1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2892148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
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Prove $\lim_{n\to \infty} n\int_0^1x^nf(x) \,\text dx=0 $ Assume $f(x)$ is continuous on $[0,1]$ , and $f(1)=0$. Prove
$$\lim_{n\to \infty} n\int_0^1x^nf(x)\,\text dx=0 $$
I already know that $\lim_{n\to \infty}\int_0^1x^nf(x) \, \text dx=0$. Is this helpful in the question above?
| Fix some $\epsilon>0$. Then there is a $\delta>0$ (smaller one) so that on the interval $[1-\delta,1]$ we have $|f|<\epsilon$.
Now we can easily estimate:
$$
\begin{aligned}
0
&\le
\left|n\int_0^1 x^n\; f(x)\; dx\right|
\\
&\le
\int_0^1 (n+1)x^n\; |f(x)|\; dx
\\
&=
\int_0^{1-\delta}(n+1)x^n\; |f(x)|\; dx
+
\int_{1-\del... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2892340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Standard Coordinate Charts On A Sphere Below are excerpts from Lee's Introduction To Smooth Manifolds for the context of my question:
What I am confused about is the part where he talks about $\phi_i^+ \circ (\phi_i^-)^{-1} = \phi_i^- \circ (\phi_i^+)^{-1} = Id_{\mathbb{B}}$. There seems to be a mismatch of domain... | Ugh. Example 1.31 is entirely messed up. Someone pointed this out to me several months ago, but I was too busy at the time and forgot to get back to it. I've now added a correction to my errata list.
Thanks for pointing it out.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2892366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Asymptotic for $y'' + \frac{\epsilon y'}{y^2} - y' = 0$, $y(-\infty) = 1$, $y(+\infty) = \epsilon$.
Asymptotic for $y'' + \frac{\epsilon y'}{y^2} - y' = 0$, $y(-\infty) = 1$, $y(+\infty) = \epsilon$.
I started with a regular expansion for
$$y^2y'' + \epsilon y' - y^2 y' = 0$$
and
$$y = y_0 + \epsilon y_1 + O(\epsil... | The differential equation has an exact solution in implicit form
$$ x - x_0 = \frac{\ln \left( cy + y
^{2}+\epsilon \right)}{2} +{\frac {c}{\sqrt {{c}^{2}-4\, \epsilon}}{\rm arctanh}
\left({\frac {2\,y +c}{\sqrt {{c}^{2}-4\, \epsilon}}}\right)
}
$$
Actually it's better (changing the constant $x_0$) to write thi... | {
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"source": "stackexchange",
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If we restrict cosine to only where it satisfies a linear property, will it create ellipses? On another forum, someone asked if cosine was linear. I remarked, of course not! We know already that
$$\cos(x+y) = \cos x \cos y - \sin x \sin y$$
If it were linear, we would need $$\cos(x+y) = \cos x + \cos y$$ So, I decided ... | The graph suggests that we may simplify the problem by rotating the coordinate axes in the anti-clockwise direction (or clockwise, but let's just choose to go anti-clockwise).
(Indeed, note that for every solution $(a, b)$, $(b, a)$ is also a solution. So the graph is symmetric along the line $x=y$. Rotating thus makes... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
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Are the sets of a power set considered "elements?" I'm trying to review some set theory. The question I'm encountering is "How many elements are in a power set of a set?" I know the answer if my interpretation of the question is correct. If the original set A has n elements, the power set of A will have 2^n new sets... | Short Answer: Yes
Long Answer:
I know why this is confusing, but always think of it this way: a set can contain any kind of objects, but the term elements exclusively refers to the objects that are members of the set. For example, if I say $x$ is an element of $y$, then $x\in y$, regardless of what $x$ is, even if it i... | {
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"source": "stackexchange",
"question_score": "3",
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Finite dimensional division algebras over the reals other than $\mathbb{R},\mathbb{C},\mathbb{H},$ or $\mathbb{O}$ Have all the finite-dimensional division algebras over the reals been discovered/classified?
The are many layman accessible sources on the web describing different properties of such algebras, but all the ... | Real division algebras have not been classified. The best known result is the classification of flexible division algebras, which is found in this paper by Darpo. Some more details can be found in this survey paper (also by Darpo), where he states that the general classification is not known.
| {
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"source": "stackexchange",
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Improper integral $\int_{0}^{\infty}\left(\frac{1}{\sqrt{x^2+4}}-\frac{k}{x+2}\right)\text dx$
Given improper integral $$\int \limits_{0}^{\infty}\left(\frac{1}{\sqrt{x^2+4}}-\frac{k}{x+2}\right)\text dx \, ,$$
there exists $k$ that makes this integral convergent.
Find its integration value.
Choices are $\ln 2$... | There are no integration issue in a right neighbourhood of the origin, but when $x\to +\infty$ we have that the integrand function behaves like $\frac{1-k}{x}+O\left(\frac{1}{x^2}\right)$, so a necessary and sufficient condition for the integrability is $k=1$. In such a case
$$\begin{eqnarray*} \int_{0}^{+\infty}\left[... | {
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If $ x,y ∈\Bbb{Z} $ find $x$ and $y$ given: $2x^2-3xy-2y^2=7$ We are given an equation:
$$2x^2-3xy-2y^2=7$$
And we have to find $x,y$ where $x,y ∈\Bbb{Z}$.
After we subtract 7 from both sides, it's clear that this is quadratic equation in its standard form, where $a$ coefficient equals to 2, $b=-3y$ and $c=-2y^2-7$. T... | Notice $$2x^2-3xy-2y^2=(2x+y)(x-2y)=7.$$
Therefore, we have the four cases:
1) $2x+y=1, x-2y=7.$ Thus $x=\dfrac{9}{5},y=-\dfrac{13}{5}$, which are not integers.
2) $2x+y=7, x-2y=1.$ Thus $x=3,y=1$, which is a group of proper solution.
3) $2x+y=-1, x-2y=-7.$ $x=-\dfrac{9}{5},y=\dfrac{13}{5}$, which are not integers.
4... | {
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"source": "stackexchange",
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How to reduce a polynomial congruences Consider the Legendre Symbol
(2|p)
which give the congruences
$2^{\frac{p-1}{2}} = (-1)^{\frac{p^2-1}{8}} mod p$.
Now ${\frac{p^2-1}{8}}$ is odd if is equal to 2k+1 with k integer
that gives $p^2 = 16 k + 9$ and brings to the polynomial congruences
$p^2 \equiv 9 (mod \,\,\,16)$.... | If $2^{m+2}|(p-a)(p+a),m\ge1$ and $a$ odd
As $p+a,p-a$ have the same parity, both must be even
$\implies2^m|\dfrac{p-a}2\cdot\dfrac{p+a}2$
As the two multipliers have opposite parities
$2^m$ will divide exactly one of them
| {
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$X$ is an admissible variation of $\mathbf{x}$ I'm reading this subject as a hobby. Could someone help me solve this problem, since I have been doing some geometry for some time?
Let $\mathbf{e}_{3}$ be smooth unit normal along the immersion $\mathbf{x} \colon M \to \mathbb{R}^3$ compatible with the orientation of $M... | This would be correct if you had no function $g$ in the variation, i.e., if $g=1$ everywhere. You'll need to divide your $\epsilon$ by the maximum of $|g|$ on $M$. Let's call that quantity $C$.
Then the result is easy enough to prove. To simplify things, assume there are no umbilic points and let $\mathbf e_1,\mathbf e... | {
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Notation of the Taylor Polynomial with Lagrange Remainder I have this Theorem in my book:
Consider $f: \mathbb{R^n} \rightarrow \mathbb{R}$ a function of class $C^1$ and $\overline{x}, d \in \mathbb{R^n}$. If $f$ is twice differentiable in the segment $(\overline{x}, \overline{x}+d)$, then exist $t \in (0,1)$ such th... | What you wrote doesn't make sense. I hope your book isn't writing the last term like that, or the author is using a strange notation.
What you should have is something like this $${1\over 2} d^T H_f(\overline{x}) d$$ where $H_f(\overline{x})$ is the Hessian matrix which is the matrix of all the second order partial de... | {
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Truthtelling/Lying question (If B is lying then I'm lying) I'm stuck at a question about truthtelling/lying. Although similar problems have been posted I couldn't find a case similar to mine.
So a person is either a truthteller or a liar. Let's say we have two persons A and B. If I ask A: Are any of you telling the tr... | The last one is not true. If $B$ tells the truth, then the implication $A$ is stating is a true implication. Hence, $A$ told the truth. Do you understand?
| {
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Walter Rudin Real and Complex Analysis Chapter 2 Walter Rudin Real and Complex Analysis Chapter 2 2.14 Riesz representation theorem the last step.
Why did he put the absolute value of $a$ ?
Is not it sufficient to assume $f$ is positive?
Proof.
Clearly, it is enough to prove this for real $f$.
Also, it is enough t... | If you use $a$ instead of $|a|$ you cannot go from the second to the third line, since not knowing that sign of $a$ precludes you from knowing if you keep the inequality $\sum_i\Lambda_i\geq\mu(K)$.
And, if you do the proof just for $f\geq0$, you only get the inequality $\Lambda f\leq\int_Xf\,d\mu$, and not equality.... | {
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Convergence of sequence where every subsequence of specific type converges Let $(x_n)$ be a sequence in $\mathbb{R}$. Suppose that every infinite subsequence of $(x_n)$ which omits infinitely many terms of $(x_n)$ converges.
Does this imply that $(x_n)$ converges?
I have failed so far to come up with a counterexample. ... | Suppose the sequence $x_n$ has this property, but does not converge.
The subsequence $x_{2n}$ (omitting all the odd-numbered terms) must converge, let's say to $L$. But since $x_n$ does not converge to $L$, there is some $\epsilon > 0$
such that infinitely many $x_n$ have $|x_n - L| > \epsilon$. Those $x_n$ form a su... | {
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Can any one prove for me $\ln(1+x) = \large{G}_{2,2}^{1,2}\left( x \left| \begin{array}{cc} 1,1 \\ 1,0 \end{array} \right. \right).$ I am a PhD student in Wireless Communications and recently I found a paper about the use of "The generalized upper incomplete Fox’s H function".
I think that in order to understand this ... | We have to use certain conventions for these cases.
$$
\prod_{j=3}^2 w_j = 1,\quad\text{known as an "empty product"}
$$
and similarly
$$
\prod_{j=2}^1 w_j = 1.
$$
The $\Gamma$ function is defined by an integral for positive arguments, but may be extended to other arguments. The functional equation
$$
\Gamma(z+1) = z\G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2893832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fermat's Last Theorem Resources Are there any resources which describe FLT in a very tangible way which will motivate students to be interested in this subject?
| About ten years ago I wrote a monograph directed at students with a high school competency. It goes through many proofs of intermediate results that preceded the Wiles proof, but no higher analysis. It focuses on the mathematics, and not the history. I don't know whether it is still in print. See:
https://www.amazon.co... | {
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A holomorphic function with infinitely many zeros in the unit disc
Prove that if $f$ is holomorphic in the unit disc, bounded and not identically zero, and $z_1, z_2, z_3, \dotsc, z_n, \dotsc$ are its zeros ($\vert z_k \vert$ $\lt1$ ),then
$$\sum_{k=1}^\infty (1-\vert z_k \vert) \lt \infty$$
[Hint:Use Jensen's for... | Of course this is a theorem instead of an exercise in many complex books, so we may as well add MSE to the list of places one can look it up...
Don't pull out the $r$ from $\log(|z_k|/r)$. Instead look at it this way: Define $$\log^+(t)=\begin{cases}\log(t),&(t>1),
\\0,&(0<t\le1).\end{cases}$$
Note that $$\sum_k\log^+(... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Discontinuity - Unsure If Piecewise Equation(s) Have Them I have a question on whether the functions following have a discontinuity, and if not, what are the points where two functions meet.
First, the piece wise equation :
\begin{align*}
f(x)=
\begin{cases}
\sin(x), &\text{ if } 0 \leq x \leq 2\pi;\\
0, &\text{ if } ... | I don't know if the points where the components of piece wise functions join have a name, but checking whether they are discontinuities, or non-differentiable points is not difficult.
Lets begin with discontinuities. In order for the point $p$ not to be a discontinuity, $$\lim_{x\to p^+} f(x) = \lim_{x\to p^-} f(x) = f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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G-principal bundle and homotopy retract Suppose that $f:X\rightarrow Y$ a continuous map between (connected) CW-complexes such that there exists a continuous map $g:Y\rightarrow X$ with the property that $g\circ f$ is homotopy equivalent to $id_{X}$ i.e., $X$ is homotopy retract of $Y.$ Let $P$ be a $G$-principal bundl... | I'm not sure if you have made a mistake with your question, but if you mean 'is $f^*P$ a retract of $P$' then the answer is false.
For instance take $X=S^2$, $Y=S^2\times S^2$ and let $P=S^2\times S^3$ be the product of the trivial bundle and the Hopf bundle. Let $f=in_1:S^2\rightarrow S^2\times S^2$, $x\mapsto (x,\ast... | {
"language": "en",
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A spherical snowballs radius is decreasing by 4% per second. Find the percentage rate at which its volume is decreasing. For this question I have to find the rate at which the volume of the sphere decreases, $\frac{dV}{dt}$. I already have $\frac{dr}{dt}$, the rate at which the radius decreases, which is $-\frac{4}{100... | After one second,
$$\frac{V'-V}{V\cdot1}=\frac{(0.96r)^3-r^3}{r^3\cdot1}=-0.115264.$$
After one millisecond
$$\frac{V'-V}{V\cdot0.001}=\frac{(0.99996r)^3-r^3}{r^3\cdot0.001}=-0.11999520\cdots.$$
After infinitesimal time,
$$\frac{V'-V}{V\cdot \theta}=\frac{((1-0.04\theta)r)^3-r^3}{r^3\theta}=-0.12+0.0048\theta-0.000064\... | {
"language": "en",
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If $X$ is exponentially distributed with parameter $1$, prove that $\exp(-X)$ is uniformly distributed on $[0,1]$. This is what I have so far:
The PDF of $X$ is $$f_X(x)=e^{-x}$$ when $x\geq0$ and $0$ otherwise. The CDF of $X$ is $$P(X\leq x)=F_X(x)=1-e^{-x}$$ when $x\geq 0$ and $0$ otherwise. I know that I want to en... | You're almost there. You only need to observe that $$\Pr (-\ln y \le X) =1 - \Pr(X < -\ln y) = 1 - \left( 1 - e^{\ln y}\right) = y.$$
Hence, $F_Y(y) = y$, and $f_Y(y) = 1$.
| {
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Solve for a variable with a variable exponent. I'm working on a video-game and have come up with the following equation:
$$B^S={B-1\over R} +1$$
I need to solve for B in terms of R and S (which will be supplied at runtime) but for the life of me I can't seem to simplify it. I even tried an online variable solver and it... | Except for very few specific cases, you cannot get explicit solutions and you need numerical methods.
Consider that you look for the zero(s) of function
$$f(B)=B^S-\frac{B-1}{R}-1$$
$$f'(B)=S B^{S-1}-\frac{1}{R}$$
$$f''(B)=(S-1) S B^{S-2}$$
The first derivative cancels at a point
$$B_*=\left(\frac{1}{R S}\right)^{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2894601",
"timestamp": "2023-03-29T00:00:00",
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permutation with repeated identical elements First of all I do know the solution to below problem I'm asking different way!!
The problem is like this:
consider the word $AABBB$ how many 3 letter words can be written using the given word?
clearly this is a permutation problem, my problem is can we find the answer only ... | In a case like this, it may be easier to start from all words that can be made using the letters A and B, then subtract those which don't work. So there are $2^3=8$ three-letter words using only As and Bs (since each letter independently has $2$ choices), and the only one which doesn't fit into AABBB is AAA, leaving yo... | {
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Why is $\sup_{n \in \mathbb{N}}\Big|\frac{n-1}{n}z_n\Big|\le\sup_{n \in \mathbb{N}}\Big|\frac{n-1}{n}\Big|\sup_{n \in \mathbb{N}}\Big|z_n\Big|$ $\sup_{n \in \mathbb{N}}\Big|\frac{n-1}{n}z_n\Big|\le\sup_{n \in \mathbb{N}}\Big|\frac{n-1}{n}\Big|\sup_{n \in \mathbb{N}}\Big|z_n\Big|$
when $z_n$ is a bounded sequence $\in \... | Let $a_n:=\frac{n+1}{n}$ and $c_n:=a_n|z_n|$. We want to estimate $\sup c_n$. Now, since $a_n\ge 0$,
$$
c_n\le (\sup_m a_m)|z_n|\le (\sup_m a_m)(\sup_k |z_k|), $$
so taking the $\sup$ of both sides yields
$$
\sup_n c_n \le (\sup_m a_m)(\sup_k |z_k|),$$
as we wanted.
| {
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Why is the "greater than" or "less than" symbol referred to as operators? My understanding of operators is it works on elements of a set and produces another element of the same set.
I don't see how or why the "$>,≥,<,≤$" would be referred to as "operators" on some pages as it doesn't map to another element. (I think I... | In mathematics, you generally won't see inequality signs referred to as "operators" at all.
In programming languages, "operator" means generally any syntactic construct that can be used to build expressions from other simpler expressions. In most programming languages, inequality signs count as operators, because they ... | {
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How to solve simultaneous linear equations with only two possible values per variable? I am trying to solve a system of simultaneous linear equations whose unknowns have only two possible values. How do I approach this, or what area of mathematics do I employ inorder to arrive at the exact solution. e.g.
\begin{align}... | You have linear system $Ax = b$ and want to know if there is a solution such that all coordinates of $x$ are either $4$ or $6$. Instead of that, consider system $Ay = b/2$ where all coordinates of $y$ have to be either $2$ or $3$. If there exists a solution to that system, there must be a solution to the same system mo... | {
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A statement following from the law of excluded middle Does the statement ~~$A\equiv A$ follow from the law of excluded middle?
According to my book which is not on logic it does, but I do not know how to use the law of excluded middle for this simple tautology.
| This question isn't clear. Is the question "Can (~~A≡A) follow from the law of the excluded middle" the answer is 'yes'. That follows immediately from every single tautology implying every other tautology. Or equivalently, "all tautologies imply every other tautology."
However, if the question is "does the law o... | {
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Differentiate $\frac{x^3}{{(x-1)}^2}$
Find $\frac{d}{dx}\frac{x^3}{{(x-1)}^2}$
I start by finding the derivative of the denominator, since I have to use the chain rule.
Thus, I make $u=x-1$ and $g=u^{-2}$. I find that $u'=1$ and $g'=-2u^{-3}$. I then multiply the two together and substitute $u$ in to get:
$$\frac{d}... | It's better to use the quotient rule:
$$\frac{d(\frac fg)}{dx}=\frac{f'g-g'f}{g^2}$$
$$f=x^3\to f'=3x^2$$
$$g=(x-1)^2\to g'=2(x-1)$$
$$\to\frac{d(\frac {x^3}{(x-1)^2})}{dx}=\frac{3x^2(x-1)^2-2x^3(x-1)}{(x-1)^4}=\frac{x^2(x-3)}{(x-1)^3}$$
| {
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"source": "stackexchange",
"question_score": "2",
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Differentiate $\tan^3(x^2)$
Differentiate $\tan^3(x^2)$
I first applied the chain rule and made $u=x^2$ and $g=\tan^3u$. I then calculated the derivative of $u$, which is $$u'=2x$$ and the derivative of $g$, which is
$$g'=3\tan^2u$$
I then applied the chain rule and multiplied them together, which gave me
$$f'(x)=2... | $$u'=2x$$
$$g'=3\tan^2u \cdot sec^2u$$
$$f'(x)=2x \cdot 3\tan^2(x^2)\sec^2(x^2) = 6x\tan^2(x^2)\sec^2(x^2)$$
| {
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Proof by Deduction $\sqrt{xy} ≤ \frac{x+y}{2}$ I want to ask a question about proof of deduction.
I sat my Pure Mathematics Exam more than $3$ years ago but decided to return to the subject for a refresher.
Proofs were not a requirement for my course but as my younger siblings are studying it, I decided to give it a ... | $\sqrt{xy} ≤ \frac{x+y}{2} \iff 2\sqrt{xy} ≤ {x+y} \iff x-2\sqrt{xy}+y \geq 0$ $ \iff (\sqrt{x}-\sqrt{y})^2 \geq 0 $.
Which is true
| {
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"url": "https://math.stackexchange.com/questions/2895436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inequality for a function 4 Let $u:[0,+\infty)\to\mathbb R^+$ be a bounded positive function such that
$$u(t)\leq \int_0^t\left(-\frac{1}{\sqrt N}u(s)+\frac{1}{N}\right)ds +\frac{1}{N^{\frac{1}{4}}}$$ for every $t\geq 0$, where $N\in\mathbb N$.
Is it correct that
$$u(t)\leq\frac{1}{\sqrt N}+\frac{1}{N^{\frac{1}{4}}}$... | The inequality does not hold. As a counterexample, consider
$$ u(t) = \begin{cases}1/2, &0\leq t < 8 \\ 3, &8\leq t<9 \\ 1/2, &9\leq t<\infty\end{cases} $$
Then one can show that $u$ satisfies the first inequality for $N = 1$:
$$ \int_0^t\left(-\frac{1}{\sqrt N}u(s)+\frac{1}{N}\right)ds +\frac{1}{N^{\frac{1}{4}}} = 1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Homomorphism: How do we get the equality? Let $Z = (\mathbb{Z},+)$ the additive group of integers and $G = (M,\star )$ an arbitrary group.
I want to show that for all $a \in G$ the map $\phi_a : Z \rightarrow G$ defined by $\phi_a(k) = a^k$ is an homomorphism from $Z$ to $G$.
Let $m,n\in Z$. Then we have that $\phi_a... | Yes we see that as the multiplication between elements of group.
Since $a^m$ and $ a^n$ are elements of your group, $a^m \star a^n = a^{m+n}$ is given as the associative property of the group operation.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does $f(x)= x \sin(\frac{\pi}{x})$ behave? I think this function is increasing for $x>1$ but wanted to find the reason. So I thought about taking the derivative:
$f(x)= x \sin(\frac{\pi}{x})$
Aplying the chain an the product rule, we get:
$f'(x)= \sin(\frac{\pi}{x})-\frac{\pi}{x} \cos (\frac{\pi}{x})$
The function ... | Try the Maclaurin series for tangent:
$$\tan x = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \ldots$$
All terms are positive when $x > 0$, so $\tan x > x$. This proves that $\tan \frac{\pi}{x} > \frac{\pi}{x}$ for $0 < x < \frac{\pi}2$.
Alternatively, define $f(x) = \tan x - x$. Then $f(0) = 0$ and $f'(x) = \sec^2 x - 1 > 0... | {
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Determine the point where function output will go from positive to negative I have a function that is like:
f(x) = c - x^2 (c = some constant positive integer, x = +ve integer >= 0)
The output of this function, goes from positive to negative as x -> +infinity.
*
*Is there a way to directly figure out the x which p... | If we solve the quadratic equation $c-x^2=0$ for a real number $x$, we get $x=\sqrt c$ or $x=-\sqrt c$. Since we are looking at the branch $x\ge 0$, the required solution is the largest integer not exceeding $\sqrt c$. For example, if $c=2$ then $\sqrt c=1.414\ldots$ and the solution would be $1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Product of a decreasing sequence and a diverging series Suppose we have a monotonic decreasing sequence $a_n$ converging to $0$.
Does there always exist a non-negative series $b_n$, such that
$$\sum_{n=1}^\infty b_n = \infty$$ but
$$\sum_{n=1}^\infty a_n b_n < \infty$$?
Edit: yes, as answered below. What if we also i... | Choose $n_1<n_2<...$ such that $a_{n_{k}} <\frac 1 {2^{k}}$ and define $b_n=1$ if $n \in \{n_1,n_2,...\}$, $b_n=0$ otherwise. Note that $b_n=1$ for infintely many $n$. Hence $b_n$ does not tend to $0$ which implies $\sum b_n =\infty$.
| {
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Last digit of sequence of numbers
We define the sequence of natural numbers
$$
a_1 = 3 \quad \text{and} \quad a_{n+1}=a_n^{a_n},
\quad
\text{ for $n \geq 1$}.
$$
I want to show that the last digit of the numbers of the sequence $a_n$ alternates between the numbers $3$ and $7$. Specifically, if we symbolize wit... | The mistake you are making is that if $a_n \equiv 2 \pmod 5$ it's not true that $a_n^{a_n} \equiv 2^2 \pmod 5$. The reason behind this is that the exponents aren't repeating in blocks of $5$, but instead in blocks of $\phi(5) = 4$, in your case. Indeed by Fermat's Little Theorem we have that $a_n^4 \equiv 1 \pmod 5$. T... | {
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Probability of events with retries? Let's say I want to roll $n$ 20-sided dice, and I want none of those dice to be a 1. I figure that the probability at least one die will be a 1 is $\frac{19}{20}^n$. But now let's say that we will re-roll each individual die that is a 1 up to $r$ times. I want to know 2 things:
*
... | In order for a die to end up being $1$, it has come up $1$ a total of $r+1$ times in a row. Therefore, each die ends up being one with probability $(1/20)^{r+1}$, so
$$
P(\text{at least one die is }1)=1-P(\text{no dice are }1)=\boxed{1-\bigg(1-\frac1{20^{r+1}}\bigg)^n.}
$$
To compute the expected number of rolls, we c... | {
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$C^1$ function with limit decay at infinity Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuously differentiable function (i.e. $f \in C^1(\mathbb{R})$). Assume that $$\lim_{x \rightarrow \infty} xf'(x) = 0 \ \mbox{and} \ \ \lim_{n \rightarrow \infty} f(2^n) = 0.$$ Then I would like to show that $\lim_{x \righta... | Assume that $f(x)$ does not tend to zero as $x\to\infty$. Then there exists $\varepsilon>0$ and an infinite set of natural numbers $I$ such that for every $n\in I$ you can find a number $x_n$ in the interval $[2^n,2^{n+1})$ for which $|f(x_n)|>\varepsilon$. (If this were not the case, then for every $\varepsilon>0$ the... | {
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If $p(x) = x^4-4x^3+6x^2-4x+1$ is the Taylor polynomial of $f$ around $x=1$, then $1$ is a local minimum
Consider $f:\mathbb{R} \to \mathbb{R} \in C^4$. Show that $p(x) =
x^4-4x^3+6x^2-4x+1$ is the Taylor polynomial of order $4$ of $f$ around $x=1$, then
$1$ is a local minimum.
I'm not sure how to proceed. I know t... | For $0 \le k \le 4$, let $f^{(k)}$ denote the $k$-th derivative of $f$.
Since the Taylor polynomial of degree $4$ for $f$ at $x=1$ is
$$x^4-4x^3+6x^2-4x+1=(x-1)^4$$
it follows that $f^{(k)}(1)=0$, for $0\le k\le 3$, and $f^{(4)}(1)=24$.
For brevity, let
*
*"holds near $x = 1$ on the left" mean "holds in some open... | {
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Binomial sum formula for $(n+1)^{n-1}$ Has anybody seen a proof for
$$
(n+1)^{n-1}=\frac{1}{2^n}\sum_{k=0}^n
C_n^k(2k+1)^{k-1}(2(n-k)+1)^{n-k-1} ?
$$
There are lots of reasons to think that this is true. In particular the formula holds for $n=0,1,2,3,4,5$.
| A (not straight) proof.
Using Bürmann-Lagrange formula we expand the regular at $z=0$ solution $w(z)$ of the equation $w=ze^{aw^2}$ as the following series
$$
w(z)=\sum_{n=0}^{\infty}\frac{a^n(2n+1)^{n-1}} {n!}z^{2n+1}
$$
(see problem 26.07 in СБОРНИК ЗАДАЧ ПО ТЕОРИИ АНАЛИТИЧЕСКИХ ФУНКЦИЙ, Под редакцией М. А ЕВГРАФОВА... | {
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On Infinite Limits I am currently learning about infinite limits in Calculus, basically determining the limit of a function as x approaches infinity. However, I am struggling to understand the method being used to find it.
Let’s take the function above. The method above seems to be to ignore all the terms with a lowe... | For 2 and 3, it seems like you are making an algebra mistake.
$$(\sqrt x+10)^2=x+20\sqrt x+100\ne x+100$$
For example, if $x=10000, \sqrt x=100$. $\sqrt{10100}<101$. Indeed, your problem shows $\sqrt{100+x}-\sqrt x=\dfrac{100}{\sqrt{100+x}+\sqrt x}$. As x increases, the denominator increases without bound while the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2896895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Question about limits. I am currently doing a physics project and have two expressions of two versions of a length $L$ of the form
$$L_h=2\pi N\sqrt{\left\langle R \right\rangle^2 +\left(\dfrac{h}{2\pi N}\right)^2}$$
and
$$L_v=\pi N \sqrt{\left\langle R \right\rangle^2+\left(\dfrac{h}{\pi N}\right)^2} +\frac{\pi^2 N^... | $h$ seems to be some kind of a length-scale of your system, thus you may try to rescale $L_\nu$ by it, i.e. devide all your equations by $h$ and you have $u$-s all around. As for the rest: expand the root and the log consecutively in a power-series, then you may easily carry out the limit. For instance:
$$
\sqrt{\langl... | {
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How to take second derivative implicitly Let $$y^4 + 5x = 21.$$
What is the value of $d^2y/dx^2$ at the point $(2, 1)$?
I’m stuck at trying to work out the second implicit derivative of this function. As far as I can work out, the first derivative implicitly is
$$\dfrac {-5}{4y^{3}}$$
How do you take the second derivat... | There's a typo in your problem statement. The point $(2,1)$ is not on the curve. They probably meant $(1,2)$.
I would not have solved for the first derivative. Just differentiate the equation again, remembering that both $y$ and $y'$ are functions of $x$, and so the chain rule applies:
$$y^4+5x=21$$
$$4y^3y'+5=0$$
$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$A^3 = A^2$ How can $A$'s minimal polynomial look like?
Let $K$ be a field and $A \in K^{n \times n}$ a matrix with $A^3 = A^2$.
How can $A$'s minimal polynomial $\mu_A$ look like?
The only possibilities I could think of are
*
*$A = 0$. Then the characteristic polynomial is $P_A(t) = -t^n$.
*$A = E$, where $E$ ... | Let $m(x)$ be the minimal polynomial. Because of Hamilton-Cayley's theorem, we have $m(x) | x^3-x^2$.
So $m(x)$ can be $x,x-1$ (as you said) , but also
*
*$x(x-1)$ for example in $$\begin{bmatrix}
1 & 0 \\
0 & 0
\end{bmatrix}$$
*$x^2$ for example in $$\begin{bmatrix}
0 & 1 \\
0 ... | {
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Multi-variable chain rule with multi-variable functions as arguments What is the chain rule of a multi-variable function with arguments that are also multi-variable functions?
Suppose $x$, $y$, $z$ are independent variables. I mean changing $x$ won't change $y$ and $z$.
Is the general form of multi-variable chain rule ... | That's nearly right, but you left off the third term that accounts for $t$. So you should have
$$\frac{\partial w}{\partial x} = \frac{\partial w}{\partial u}\cdot\frac{\partial u}{\partial x} + \frac{\partial w}{\partial v}\cdot\frac{\partial v}{\partial x} + \frac{\partial w}{\partial t}\cdot\frac{\partial t}{\partia... | {
"language": "en",
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Number of integer solutions combinatorics problem what is the number of integer solutions to
$$x_1+x_2+x_3+x_4+x_5=18$$ with $$x_1\ge1\;\;\;x_2\ge2\;\;\;x_3\ge3\;\;\;x_4\ge4\;\;\; x_5\ge5$$
I know I have to use this formula $$\frac{(n+r-1)!}{(n-1)!\;r!}= {{n+r-1}\choose r}$$
My instinct says that I should use $n=18-1-2... | We can change the problem from its original form
$$x_1+x_2+x_3+x_4+x_5=18$$
$$x_1\ge1,x_2\ge2,x_3\ge3,x_4\ge4,x_5\ge5$$
to
$$y_1+y_2+y_3+y_4+y_5=(x_1-1)+(x_2-2)+(x_3-3)+(x_4-4)+(x_5-5)=3$$
$$y_1\ge0,y_2\ge0,y_3\ge0,y_4\ge0,y_5\ge0$$
and use generating functions approach, giving the following function
$$(1+y+y^2+...+y^k... | {
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"timestamp": "2023-03-29T00:00:00",
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Vector Space with unusual addition? I'm studying before my class starts in a few weeks and I encountered this question in one of the practice problems:
The addition it has given me is defined as,
$(a,b)+(c,d)= (ac,bd)$
It's asking me if this is a vector of space and I am stuck after proving this,
There is an elem... | As you have the neutral element $o=(1,1)$ you need to make sure your inverses are relative to that. Assuming $V=\{(a,b): a,b\in\mathbb{R}, a,b>0\}$ or something of that kind you could use $(a,b)+(\frac1a,\frac1b)=(1,1)$.
What you still need is to tell us how your base field acts on $V$.
| {
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"timestamp": "2023-03-29T00:00:00",
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$f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2$ , find $f(x)$
Find $f(x)$ if $f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2$, where $x, f(x)\in (-\infty , \infty)$ and $f(x)$ is continuous.
| $y=mx^2+c$ is one solution where $c\in R$
$$mx^2-2\cdot m \frac{x^2}{4}+m \frac{x^2}{16}=x^2$$
$$m=\frac{16}{9}$$
$$y=\frac{16}{9}x^2+c$$
As mentioned by @lulu if $f(x)$ is a solution than $f(x)+c$ is also a solution.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question regarding basis and dimension of vector space of polynomials
Let $V_n$ be the vector space consisting of all polynomials of the form
$$f(x,y)=\sum_{i=0}^n\sum_{j=0}^n a_{i,j}x^iy^j$$
where $a_{i,j}\in\mathbb{R}$.
(a) State the dimension of $V$, and give a basis for $V$.
(b) Let $U\leq V$ be the subspace o... | You're correct about $(a)$.
For part $(b)$, note that $U$ is the kernel of the linear map $$V\to \Bbb R^{2n+1}:\sum_{i=0}^n\sum_{j=0}^n a_{i,j}x^iy^j\mapsto \left(\sum_{i+j=N} a_{i,j}\right)_{N=0\ldots 2n+1}$$
This map is surjective, which gives you the dimension of $U$. For a basis, fix the value of $N$, set $a_{i,j}... | {
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$I_n = \int_{0}^{\frac{\pi}{2}}(\cos t)^n \ dt$ converges to 0? How one can prove that the sequence $\left ( I_n \right )$ defined as $$ I_n = \int_{0}^{\frac{\pi}{2}}(\cos t)^n \ dt, $$ $n \in \{ 0,1,2,...\}$ converges to $0$?
Is easy to show, by the way, that the sequence is decreasing because, for $t \in (0, \pi/... | With integation by parts one may show that
$$I_{n}=\dfrac{n-1}{n}I_{n-2}$$
then from $I_0=\dfrac{\pi}{2}$ and $I_1=1$, both odd and even terms go $0$ as $n\to\infty$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability of program equality based on samples Program p implements a side-effect-free function f that accepts k1 bytes as input and produces k2 bytes of output.
Suppose we take N samples (tuples of input/output pairs where p(i) = o), where the inputs are perfectly random. Program q satisfies these samples (q(i) = o)... | Suppose there are $n$ possible inputs to the function. In your case, assuming a byte can have $256$ values and the input consists of $k_1$ bytes, we have $n = 256^{k_1}$.
The most difficult situation to detect is when only one of the inputs results in an error and all the other inputs are processed correctly. In th... | {
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Is this proof for if $0 < a < b$ then $a^2 < b^2$ correct? I'm reading the book 'How to prove it' from Daniel Velleman which he presents a proof for the following; if $0 < a < b$ then $a^2 < b^2$ as;
Proof. Suppose $0 < a < b$. Multiplying the inequality $a < b$ by the positive number $a$ we can conclude that $a^2 < a... | In the proof from the book you presented, you've assumed $0<a<b$ and deduced $a^2<b^2$.
In your presented proof, you've essentially assumed $a^2<b^2$ and deduced $a<b$. Why? The hypothesis $0<a<b$ is never used and by taking the square root of $a^2<b^2$, you implicitly assume that statement, deducing $a<b$ from it.
Thu... | {
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What is isogonal family of a given family of curves? I searched in Wikipedia isogonal trajectories about the definition but I do not understand what does it mean by fixed "angle". Angle with the tangents of the curves? Clockwise angle? Orientated Angle? Thanks in advance.
| I think that for two curves $y=f(x)$ and $y=g(x)$ which intersect at $(x_0,y_0)$, they are defining the angle between the curves to be $\ \mathrm{arctan}(f'(x_0)) - \mathrm{arctan}(g'(x_0))$.
| {
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if $W$ is a subspace of an inner product space $V$, which of the following statements is true? if $W$ is a subspace of an inner product space $V$, which of the following statements are true?
$1)$ there is a unique subspace $W'$ such that $W' + W = V$
$2)$ there is a unique subspace $W'$ such that $W'\oplus W = ... | (1) and (2) are certainly false; in fact they're false if $V=\Bbb R^2$ and $W=\{x,0)::x\in\Bbb R\}$.
(3) and (4) are true if $V$ has finite dimension (or if $V$ is a Hilbert space and $W$ is a closed subspace), but they're also false in a general inner-product space.
For example, let $V$ be the space of sequences $x=(x... | {
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$A\in \mathbb{R}^{n\times n}$ has eigenvalues in $\mathbb{Z}$ with at least 3 different eigenvalues. $\det(A)^n = 5^4$, find $A$'s eigenvalues $\newcommand{\adj}{\text{adj}}$
The question as it appeared in the first place: $A\in \mathbb{R}^{n\times n}$ such that all $A$'s eigenvalues are in $\mathbb{Z}$ and $A$ has at ... | Since the eigenvalues are all integers, their product $\det(A)$ is an integer. What are the divisors of $5^4$?
EDIT: Use the fact that $n \ge 4$.
| {
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Aristarchus' Inequality - algebraic proof While looking for trigonometric inequalities, I stumbled upon Aristarchus' inequality, which states that for $0<\alpha<\beta<\pi/2$
$$\frac{\sin(\beta)}{\sin(\alpha)}<\frac{\beta}{\alpha}<\frac{\tan(\beta)}{\tan(\alpha)}.$$
In this post
(Proof of Aristarchus' Inequality) use... | I managed to find an easy proof:
For $0<\alpha<\beta<\pi/2$ we have
$1>\cos(\alpha)>\cos(\beta)>0$
and thus (the orange chain was known)
$$0<\sin(\alpha)\cos(\alpha)<\color{orange} {\sin(\alpha)<\alpha<\tan(\alpha)}=\frac{\sin(\alpha)}{\cos(\alpha)}<\color{red}{\frac{\sin(\alpha)}{\cos(\beta)}}.$$
Consequently, using t... | {
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"question_score": "2",
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"answer_id": 0
} |
Prove that the directional derivative is the dot product of the gradient and the vector. I looked for few proofs online but was looking for alternate, more direct proofs. The one on Khan Academy used the Linear Approximation and one used the chain rule of multivariable functions. Are there any alternate methods to prov... | The property holds for differentiable functions, indeed by definition of differentiability we have that
$$\lim_{\vec h\to \vec 0} \frac{ f(\vec x_0+\vec h)-f(\vec x_0)-\nabla f(\vec x_0)\cdot \vec h}{\| \vec h\|}=0 \iff f(\vec x_0+\vec h)-f(\vec x_0)=\nabla f(\vec x_0)\cdot \vec h+o(\| \vec h\|)$$
and assuming $\vec h ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2898777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding a set of continuous functions with a certain property 2 I need help finding the set of continuous functions $f : \Bbb R \to \Bbb R$ such that for all $x \in \Bbb R$, the following integral converges:
$$\int_0^1 \frac {f(x+t) - f(x)} {t^2} \ \mathrm dt$$
I think it might be the set of constant functions but i ha... | Although Rigel’s answer solved the matter brilliantly, I would like to present an alternative solution to this:
Consider the sets $A_{\varepsilon,x} =\{ u > x, \, |f(u)-f(x)| < \varepsilon |u-x|\}.$
Notice that these sets are clearly open, by the continuity of $f$. Also, these sets are nonempty for every $x \in \mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2898983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Limit as $(x,y,z)\to (0,0,0)$ of $f(x,y,z) = \dfrac{xy+yz+xz}{\sqrt{x^2+y^2+z^2}}$ To find this limit, I converted to spherical coordinates and rewrote:
$$\lim_{r\to 0} \dfrac{r^2(\sin^2\theta \cos\phi \sin \phi + \sin\theta \cos \theta \sin \phi + \sin\theta \cos \theta \cos \phi)}{r} = 0$$
Is this method alright? Our... | Using the full spherical coordinates is overkill here. Let $r=\sqrt{x^2+y^2+z^2}$.
Then $|x|\le r$, $|y|\le r$, $|z|\le r$. So
$$|xy+xz+yz|\le|xy|+|xz|+|yz|\le 3r^2$$ and so
$$\left|\frac{xy+xz+yz}{\sqrt{x^2+y^2+z^2}}\right|\le 3r.$$
As $\lim_{(x,y,z)\to(0,0,0)}r= 0$ then
$$\lim_{(x,y,z)\to(0,0,0)}\left|\frac{xy+xz+yz}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How to compute the levy path integral with zero potential? In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as
$$K(x_b,t_b|x_a,t_a)=\int_{x_{t_a}=x_a,x_{t_b}=x_b}Dx(t)\exp\left\{-\frac{i}{h}\int_{t_a}^{t_b}dtV(x(... | It turns out that for the Levy path integral, the calculation of the amplitude of a quantum particle uses the Fourier translation of the probability density function, since this representation is an integral of an exponential function which is easy to compute.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove that a sum of degrees in a path between two vertices is smaller than $3n$ Let $G$ be a simple graph with $n$ vertices. Let $P$ be the shortest path between any two vertices. Prove that: $$\sum_{v\in P}deg(v)\leq 3n$$
Let the sum of degrees be bigger than $3n$. If so, there is a vertex on a path that has degree bi... | Hint. Let $v_0,v_1,v_2,\dots,v_n$ be a path of minimum length from $v_0$ to $v_n.$ Can you show that $\deg v_0+\deg v_3+\deg v_6+\cdots+\deg v_{\lfloor n/3\rfloor}\le n?$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Finding sum of a geometric series I am asked to find the summation of $1/3^n$ from $n=5$ to infinity.
I have done the calculation: $1/(1-r)$, for $r=1/3$, and received $1.5$. As this summation starts from $5$, I subtracted $3^0, 3^-1, 3^-2, 3^-3$ and $3^-4$ from $1.5$ and got $6.17e-3$. However, apparently this answer ... | Your method is fine indeed
$$\sum_{k=5}^\infty \frac1{3^k}=\sum_{k=0}^\infty \frac1{3^k}-\sum_{k=0}^4 \frac1{3^k}=\frac32 - \sum_{k=0}^4 \frac1{3^k}=\frac32-1-\frac13-\frac19-\frac1{27}-\frac1{81}=\frac1{162}$$
As an alternative, following the clever method suggested by lulu in the comment, we have
$$\sum_{k=5}^\infty ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Confusion with the proof that the Cantor set is closed I have encountered the definition of Cantor set and its property.
Cantor set is constructed by removing the middle third open set of each interval .So each time we get some union of closed sets.
As $F_0=[0,1]$
$F_1=[0,1/3]\cup [2/3,1]$
$F_2=[0,1/9]\cup [2/9,1/3]\cu... | For any large $n$, $F_n$ is the union of finitely many closed intervals so it is closed.
As you mentioned the intersection of an arbitrary family of closed sets is closed so $F=\cap_{n\to \infty} F_n$ is closed.
Note that for each $n$ we have finitely many closed sets, no matter how large $n$ is, so the union is close... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Lebesgue Dominated Convergence Application I want to compute the integral
$$\lim_{n\to \infty}\int_0^\infty \frac{\sin\left(\frac{x}{n}\right)}{(1+x/n)^n}\,\mathrm{d}x$$
Since
$$\left| \frac{\sin\left(\frac{x}{n}\right)}{\left(1+\frac{x}{n}\right)^n}\right | \le \frac{1}{\left|\left(1+\frac{x}{n}\right)^n\right|}\le \... | The reasoning is not correct because $$ \int_0^{+\infty} \frac{1}{1+x} = + \infty $$
And thus $\frac{1}{1+x}$ is not integrable.
Hint for a correct reasoning:
For $n \geq 2$ we have $(1+y)^n \geq 1 + n \cdot y + \frac{n \cdot (n-1)}{2} \cdot y^2 \space \space \space \forall y \geq 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$C_{[a,b]} \rightarrow \mathbb{R}$, $x(t) \mapsto f(x) = \int_a^b x(t)dt$ continuous?
Problem: Prove that $f:C_{[a,b]} \rightarrow \mathbb{R}$, $x(t) \mapsto f(x) = \displaystyle\int_a^b x(t)dt$ is continuous.
My opinion:
Let $x, x' \in C_{[a,b]}$, so they are bounded and we can choose a $\delta >0$ such that $d(x,x'... | You have to define $d(x,x')$ in order to be able to find your $\epsilon$ accordingly.
For example if you define $$d(x,x')= \max _{t\in [a,b]} \{ |x(t)-x'(t)|\}$$ then $$|f(x)-f(x')|\le (b-a)d(x,x')$$
so you can find your $\delta$ if an $\epsilon$ is given.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Gauss Jordan elimination reduces to row-echelon form always? I am reading this text:
and I'm wondering if gauss-jordan elimination always leads to an identity matrix on the left? If so, that helps me understand this passage:
I'm trying to figure out why [A 0] can be rewritten as [I 0]. Why is this?
| Gauss-Jordan eliminition works if and only if an inverse exists. It doesn't work for the null element matrix (matrix of zeros).
They said that [A 0] can be rewritten as [I 0] using elementary row operations so there are operations so that works.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Combinatorics distribution problem indistinguishable items in distinguishable boxes In how many ways can you put $10$ identical gold coins into four colored boxes so that at least $1$ goes into the blue box, at least $1$ into yellow, at most $2$ into red and at least $3$ into green?
The way I solved this was by writing... | There is an easier way to solve this problem which generalizes better to when there are several boxes which may have at most $N$ coins, where $N$ is large. I will illustrate with your example.
First, count the number of ways to put all the coins into the boxes without the $R\le 2$ restriction. This is $\binom{5+4-1}5$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2899966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How can i make the following change to this infinite series? $$ e^z - 1 = \sum_{n=1}^\infty \frac {z^n}{n!} $$
Given the above function and its corresponding series expansion, is there anything i could do to the left side of the equation so that the infinite series looks like this instead???
$$ \sum_{n=1}^\infty (\fr... | If you put a series to a certain power, it doesn't mean it's equal to the series of each term to that power.
I don't think there is a general expression for this series except when $a$ is 0 or 1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2900067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is the following Area of Crescent all right? In the figure below. There are two overlapping circles and the area of Crescent in Red that I have found is $A_{C} = \frac{\pi rw}{2}$, where $w$ is the shift from center $'X'$ in blue to $'X'$ in red.
Details:
$$A_C = \frac{A_{elipse} - A_{circle}}{2} = \frac{[\pi r^2 + \pi... | WLOG, assume both centers lie on the $x$-axis. You can use this diagram afterwards:
Since the area of the circle is $A=\pi r^2$, then the area of the crescent should be:
$$A_{\text{crescent}}=\pi r^2-(2A_{\text{sector }EAF}-2A_{\triangle AEF})$$
This is because $A_{\text{sector }EAF}=A_{\text{sector }ECF}$, and so doe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2900164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Qualitative inspection of solutions to $x^{4}-2x+1=0$ Consider the following polynomial
$$
x^{4}-2x+1=0
$$
Is it possible to check if there is or there is not a solution in $x\in\left]0,1\right[$ without explicitly evaluating the expression? What other tests are there to qualitatively classify the solutions for this po... | Since $x=1$ works,
$$
x^4 - 2x + 1 = (x-1)p(x)\quad [p \in \mathbb R[x]_3].
$$
Now
$$
x^4 - 2x +1 = x^2(x^2 -1) + (x-1)^2 = (x-1)(x-1+x^2(x+1)) = (x-1)(x^3 + x^2 + x - 1).
$$
Then $p(x) = x^3 + x^2 + x-1$. Since
$$
p(1)= 2 >0, p(0) = -1 < 0,
$$
by intermediate value theorem, $p$ has a root in $(0,1)$, hence so does ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2900232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
When does a bijection of topologies induce a homeomorphism of spaces? If two topological spaces $(X, \tau)$ and $(Y, \tau')$ are homeomorphic, we have a bijective correspondence between $\tau$ and $\tau'$ via $U \in \tau \mapsto f(U) \in \tau'$ where $f: X \to Y$ is a homeomorphism.
Are the reasonable conditions to be... | So, let me first impress on you how outrageously weak the mere existence of such a bijection $\Gamma$ is. It means solely that $X$ and $Y$ have the same number of open sets. When dealing with infinite sets, a cardinality statement like this says extremely little. For instance, if $X$ is any infinite separable metric... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2900320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Name convention for functor and natural transformation composition If there are functors $H: D \to C; F,G: C \to D$ and $K: B \to C$ and a natural transformation $\alpha: F \xrightarrow{.} G$ then we can construct 2 new natural transformations:
Aka "left composition"
$$H \alpha : H F \xrightarrow{.} H G $$
and "right c... | It's called whiskering; you can show that it is the same as the horizontal composition of $\alpha$ with $1_H:H\Rightarrow H$ (for your "left composition") or with $1_K:K\Rightarrow K$ (for your "right composition").
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2900427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$I$ is infinite, $A_k$ is countably infinite, and $A_i$ is countable for all $i \neq k$. Is $\prod\limits_{i\in I}A_i$ countable? The Cartesian product of a family $(A_i\mid i\in I)$ is defined as $$\prod\limits_{i\in I}A_i=\{f:I\to\bigcup A_i\mid f(i)\in A_i \text{ for all } i \in I\}$$
Let $(A_i \mid i \in I)$ be a ... | The Cartesian product will be countable if any of the sets $A_i$ is empty, since then the Cartesian product will also be empty. Let us assume that for all $i$, $A_i\neq \varnothing$.
If there exists a finite subset $J$ of $I$ such that $A_i$ is a singleton for all $i\in I\setminus J$, then the Cartesian product will b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2900657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Generally accepted notation for referencing function without defining it. Let $F\subseteq (\mathbb R \to\mathbb R)$ be some space of functions, and let $G:F\to \mathbb R$ be a functional. I have a statement of the following form:
$$\begin{align}\text{Let } &f^*(x):=x^2. \quad\quad\quad
\text{Then }\\
&f^*\in \arg\max_... | I don't understand either of your statements, so both of them are too concise to be readable.
Do you mean that $f^*$, which is $\underset{f \in F}{\operatorname{argmax}} G(f)$, turns out to be the function defined by $f^*(x) = x^2$? If so, for the sake of comprehensibility rather than brevity, you should write this o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2900728",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Does a proper class have arbitrary large subsets? Assume that we are working in ZFC, that we have a well-formed formula $P(x)$, that $x$ is the only free variable of $P(x)$, that there is no set $S$ such that
$$
\forall x\ (x\in S\iff P(x)),
$$
and that $\alpha$ is a cardinal.
Is there necessarily a set $T$ of cardin... | This is true.
Assume to the contrary that there is an ordinal $\alpha$ such that no set equinumerous to $\alpha$ is contained in $P$. Then there is a smallest such $\alpha$. However this means that for each $\beta<\alpha$ we can choose an $A_\beta$ such that $A_\beta \subseteq P$ and $|A_\beta|=|\beta|$ (employing Scot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2900825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
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