Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Area of triangle using double integrals I have one (rather simple) problem, but I'm stuck and can't figure out what I'm constantly doing wrong. I need to calculate area of triangle with points at $(0,0)$, $(t,0)$, $(t,\frac{t}{2})$. In other words triangle under function $y=\frac{x}{2}$, for $x\in [0,t]$ I thought it i... | The integral that you actually computed corresponds to the area of a rectangle. You should actually compute$$\int_0^t\int_0^{\frac x2}\,\mathrm dy\,\mathrm dx.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3025762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Number of solutions of the equation $\cos(\pi\sqrt{x-4})\cos(\pi\sqrt{x})=1$
Find the number of solutions of the equation $\cos(\pi\sqrt{x-4})\cos(\pi\sqrt{x})=1$
\begin{align}
2\cos(\pi\sqrt{x-4})&.\cos(\pi\sqrt{x})=2\\\implies\cos\Big[\pi(\sqrt{x-4}+\sqrt{x})\Big]&+\cos\Big[\pi(\sqrt{x-4}-\sqrt{x})\Big]=2\\
\implie... | I suppose $x$ is real in the following and that $\cos$ is the function $\cos:\Bbb R\to\Bbb R$. (There is an other function $\cos:\Bbb C\to\Bbb C$, to use it i have to ask for the branch of the square root(s) first.)
The two $\cos$ functions in the product (evaluated at those two places) must have (in a correlated way... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3025913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Expected values of squares Question A fair coin is tossed three times. Let Y be the random variable that denotes the square of the number of heads. For example, in the outcome HTH, there are two heads and Y = 4. What is E[Y]?
My answer:
possible outcomes to toss a coin three times : 0, 1, 2, 3
possible outcomes of Y :... | In general for $n$ tosses
$$
\mathbf{E} = \sum_{i=0}^n{n \choose i}\left(\frac1{2}\right)^{\!\!n} i^2
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3026066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
On the definition of locally connectedness. Why don’t we just define the locally connectedness the same way we define locally compactness, that is, every point has a connected neighborhood?
On the wiki page the weak locally connectedness and connectedness are proved to be “almost identical”, but it does not mention the... | I think the crucial fact is that (reference)
A space is locally connected if and only if for every open set U, the connected components of U (in the subspace topology) are open.
With the other definition (the weak one), this would be false.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3026183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Is this a sufficient proof for "For all $k \in\Bbb R$, if $k$ is odd, then $4k + 7$ is odd Can I just say 4k is obviously even and even + odd is always odd, so odd?
Or is that too simple, am I missing something?
| Yes, its simple like that. But if you are making a test or something like that you have to be sure that you can use the fact that "even+odd is odd" without proving it... otherwise just prove it! (Not so difficult)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3026283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculating convergence of a sum I have the sum:
$$S=\sum_{n=1}^\infty\frac{n!}{n^n}$$
and I am using D'Alembert's test for convergence which states for some sum:
$$\sum_{n=a}^\infty u_n\,\,(a\neq\pm\infty)$$
that it is convergent if:
$$\lim_{n\to\infty}\left|\frac{u_{n+1}}{u_n}\right|<1$$
so to begin with I know that:... | Yes that's correct, more simply form here
$$\frac{n^n}{(n+1)^n}=\frac1{\left(1+\frac1n\right)^n} \to \frac1e$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3026425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is $det(A)=0$ a good indicator to say that a matrix is not invertible? In finite elements, for example, appears huge sparce (CRS) matrices (matrices with a lot of zeros). It is possible that matlab (or some other program) calculates $det(A)=0$ even though the matrix is invertible?
| Computing determinant of a matrix is quite sensitive to round-off. On top of that, it is easy to obtain a zero or infinite determinant as output of computational procedures due to floating precision underflow or overflow.
Consider, e.g., $A_n=0.1\times I_n$, where $I_n$ is the $n\times n$ identity matrix. We have $\de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3026547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Show that $\text{sin}(\bar z)$ is not holomorphic using uniqueness theorem. I want to show that $\text{sin}(\bar z)$ is not analytic using the uniqueness theorem.
The theorem essentially states that if we have a series $z_n$ such that non-constant $f(z_n)$ is zero for each $n$, then the function is not holomorphic if t... | If $\sin (\overline {z})$ is holomorphic then it must coincide everywhere with $\sin \, z$ because these two holomorphic functions are equal on the real line (which has limit points). This is a contradiction because these functions are not equal when $z=i$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3026675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $\mathbb E[X\mid Y]$ can be seen as a projection why $\mathbb E[X\mid Y]=\frac{\mathbb E[XY]}{\mathbb E[Y^2]}Y$ is not always true? We know that $\mathbb E[XY]$ is a scalaire product on $L^2(\mathbb P)$. In a book (an introduction to stochastic differential equation of Evans) page 30-31, it's written that $\mathbb E... | Well if you take $X$ and $Y$ to be independent, the equation reads $$1 = \frac{Y\mathbb{E}(Y)}{\mathbb{E}(Y^2)},$$ which seems a little bit weird.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3026847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Does exist a compact connected $K' \subset U$ such that $K \subset K'$, if $U$ is an open connected and $K$ in $U$ a compact? Let $U$ be an open and connected set in $\mathbb{R}^n$.
Suppose $K \subset U$ is a compact set. Is it true that there exists a compact and connected set $K' \subset U$ such that $K \subset K'$?
... | For any $x\in K$ there is $\epsilon >0$ such that $B(x,\epsilon)\subset U$. Let $A_x=B(x,\epsilon/2)$. Then the closure of $A_x$ is a compact connected set contained in $U$.
Since $\{A_x, x\in K\}$ is an open conver of $K$, by compactness it has a finite sub-cover, say $A_1,\dots,A_n$. Choose $x_0\in U$ and connect ea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3026934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solve for the exponent of a matrix we discussed matrices in class and had the following task: Given
$$U=\begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0\\1 & \frac{1}{15} & 0 & 0 & 0 &0\\0 & \frac{8}{15} & \frac{3}{15} & 0 & 0 & 0\\0 & \frac{6}{15} & \frac{9}{15} & \frac{6}{15} & 0 & 0\\ 0 & 0 & \frac{3}{15} & \frac{8}{15} & \frac... | Hint :
Using this you get the diagonalization of $U=PDP^{-1}$, then $U^x = PDP^{-1} \dots PDP^{-1} = P D^x P^{-1}$ where $D$ is the matrix with the elements of $D$ elevated to power $x$. Using this you can solve for the last element being greater to $0.99$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3027061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A counterexample to the epsilon-delta criterion for Absolute Continuity of Measures Let $p>0$, and let $\mu$ be a Borel measure on $[0,\infty)$ defined by $\mu(E)=\int_Ex^pd\lambda$ where $\lambda$ denotes Lebesgue measure. Show that $\mu$ is absolutely continuous with respect to $\lambda$, but $\mu$ does not meet the... | Hopefully the case $p=1$ will help:
$$\lambda([a,b])= b-a$$
and
$$
\mu([a,b]) = \frac{1}{2}(b^2-a^2) = \frac{1}{2}(b-a)(b+a)
$$
Taking, for example, $a=3^n$ and $b=3^n+\frac{1}{2^n}$ will give you sets $E_n$ which you can show the desired properties of.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3027194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Recurrence and Fibonacci: $a_{n+1}=\frac {1+a_n}{2+a_n}$ For the recurrence relation
$$a_{n+1}=\frac {1+a_n}{2+a_n}$$
where $a_1=1$, the solution is $a_n=\frac {F_{2n}}{F_{2n+1}}$, where $F_n$ is the $n$-th Fibonacci number, according to the convention where $F_1=0, F_2=1,\cdots
$
This can easily be proven by substitu... | Yes, it can be derived directly, assuming some familiarity with the Fibonacci numbers. I am using the initial conditions $F_1=F_2=1$ for the Fibonacci numbers, which impies that
$$
a_{n}=\frac{F_{2n-1}}{F_{2n}}
$$
There is a nice property involving functions of the form
$$
f(x) = \frac{ax+b}{cx+d}
$$
If you compose su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3027273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 1
} |
Group Isomorphism regarding Sylow Subgroups Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of Syllow subgroups of these are also same then "are $G_1,G_2$ isomorphic"? I ha... | This is easily seen to fail for abelian groups, since all abelian groups of a given order have the same number of Sylow subgroups. For a nonabelian example, consider two distinct nonabelian groups of order $p^n$ for some prime $p$ and integer $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3027502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ if and only if $d$ divides $n$ I am trying to solve the following exercise of Dummit and Foote Book(page # 551).
Let $a>1$ be an integer. Prove for any positive integers $n,d$ that $d$ divides $n$ if and only if $a^d-1$ divides $a^n-1$. Conclude in particular that $\mathbb{F... | If $\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$, then $n=[\mathbb{F}_{p^n}:\mathbb{F_p}]=[\mathbb{F}_{p^n}:\mathbb{F}_{p^d}][\mathbb{F}_{p^d}:\mathbb{F_p}]=[\mathbb{F}_{p^n}:\mathbb{F}_{p^d}]d$, and so $d$ divides $n$.
If $d$ divides $n$, then $\mathbb{F}_{p^d}^\times$ is a subgroup of $\mathbb{F}_{p^n}^\times$ and so $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3027600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Invertibility of $(\textbf{A}^T\textbf{A}+\epsilon \textbf{I})$? I'm given a problem:
$\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_r$ are the nonzero singular values of $\textbf{A}\in\mathbb{R}^{M\times N}$. If $\epsilon \neq 0$ is a real scalar, s.t. $|\epsilon| < \sigma^{2}_r$, show that $(\textbf{A}^T\textbf{A}+\eps... | You can show that $A^\top A$ is positive semi-definite (specifically, that it has nonnegative eigenvalues $\sigma_1^2, \ldots, \sigma_r^2, 0, \ldots, 0$).
[It is not always invertible. Specifically, if some of its eigenvalues are zero, then it is not invertible.]
Knowing this fact about $A^\top A$, can you explicitly w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3027674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What do brackets mean for mod operation? I'm solving equation 5 = (6 * 8 + 9 * b)(mod 10). I tried to use wolframalpha and it gives me answer b = 3. But if I remove brackets around mod 5 = (6 * 8 + 9 * b) mod 10 it makes a plot, and doesn't give me any real answer. I have no idea how to solve this without guessing the ... | Okay.
$\pmod n$ means we are doing modulo arithmetic on equivalence classes.
$5 \equiv (6*8 + 9*b)\pmod {10}$ means to find which modulo class $b$ belongs to.
Perhaps a less confusing notation is $5 \equiv_{10} (6*8+9b)$. The $\pmod {10}$ isn't something you do. It's a statement about what "universe" of arithmetic yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3027798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Why are open sets denoted $U$, $G$, and measurable sets $E$? Why are open sets usually denoted by $U$?
Is there a reference about this?
Sometimes open set uses the letter $G$, such as $G_{\delta} $ set.
I also wonder the meaning of $G$.
Additional question: Why do we use or who first used $E$ to denote a subset in meas... | For $G_\delta$ set and $F_\sigma$ set, each of this is from german word Gebiet and french word fermé respectively.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3028215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
$\Bbb Q(\sqrt 2)$ and $\Bbb Q(\sqrt 3)$ are not isomorphic How to prove that $\Bbb Q(\sqrt 2)$ and $\Bbb Q(\sqrt 3)$ are not isomorphic. I thought that they are but I got this problem in Dummit Foote Section 14.1. Question no 4. As they extension over $\Bbb Q$ by the polynomials $x^2-2$ and $x^2-3$ resp.
| Hint:
Take $\;w=a+b\sqrt2\in\Bbb Q(\sqrt2)\;$ s.t. $\;\phi w=\sqrt3\in\Bbb Q(\sqrt3)\;$ , with $\;\phi\;$ an isomorphism. This means that
$$3=\phi w^2=\phi(a^2+2b^2+2ab\sqrt2)=a^2+2b^2+2ab\phi\sqrt2\implies\phi\sqrt2\in\Bbb Q$$
and now get a contradiction...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3028343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is the division symbol $\div$ acceptable based on international standards? The division symbol $\div$ is found in almost all calculators; however, I seldom see it in any formal writing. It seems people almost exclusively prefer $\frac{a}{b}$, $a/b$ or $ab^{-1}$ to $a\div b$. Is the symbol $\div$ considered outdated tod... | The $\div$ symbol is outdated and should be avoided.
Quoting from Florian Cajori's book A History of Mathematical Notations.
(Volume I, Chapter III, Part B, Paragraph $243$ A critical Estimate of $:$ and $\div$ as Symbols)
In 1923 the National Committee on Mathematical Requirements voiced the following opinion:
"Sinc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3028499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Can a linear isometry always be expressed in terms of an orthogonal matrix? Is the following true?
Let $S: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation such that $||S(v)|| = ||v|| \ \text{for all} \ v \in \mathbb{R}^n$, where $||\cdot||$ denotes the Euclidean norm. Then, for some $A \in O(n)$ and for all ... | Let $A$ be the matrix of $S$. Since $\langle u,v\rangle=\frac14\lVert u+v\rVert^2-\frac14\lVert u-v\rVert^2$, we have that $\langle Au,Av\rangle=\langle u,v\rangle$ for all $u,v$. Id est, $u^t(A^tA)v=(Au)^t(Av)=u^tv=u^tIv$ for all $u,v$. Since $e_i^tXe_j=X_{ij}$, that identity implies $A^tA=I$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3028687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Weird thing about z-transform and difference here is my doubt: we were told that the ROC of the Z-transform of the sum of two sequences is the intersection of the respective ROCs as the two of them are limited only if both of them are. Now I had to solve an exercise where I had to compute the z-transform of the differe... | You are right. The ROC of the sum of $u[n]$ and $-u[n-10]$ is the full plane except $z=0$. And the ROC of each summand is $|z|>1$.
we were told that the ROC of the Z-transform of the sum of two sequences is the intersection of the respective ROCs
Not quite. More precisely: given the ROCs of two signals, the ROC of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3028842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Every inner product space is a metric space. Show that every inner product space is a metric space.
To show this should I set the distance metric as $d(x,y) = <x-y,x-y>$, then show properties of being metric space such as d(x,y) = d(y,x) etc.?
If so the point I do not understand is why we set metric as $d(x,y) = <x-y,x... | That is wrong. It should be $d(x,y)=\sqrt{\langle x-y,x-y\rangle}$ because the map $x\mapsto\sqrt{\langle x,x\rangle}$ is a norm. And, whenever you have a norm $\lVert\cdot\rVert$, the map $(x,y)\mapsto\lVert x-y\rVert$ is a distance.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3028948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Sigma notation for iterating through number of members of a set with constant expression Say I have a graph G and I want to sum some constant C (like the minimum degree of the graph) for every vertex. Can I use the following notation?
$$\sum_{x \in V(G)}C $$
Is this an appropriate way to use sigma notation?
There is a... | Indeed, you are correct, you can use $\sum_{x \in S} C$ for any set $S$ and constant $C$, and since $C$ does not depend on $x$, this simplifies to
$$
\sum_{x \in S} C = C \cdot |S|,
$$
for any finite set $S$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3029105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\lim_{x\to \infty}(\frac{x}{x-1})^x$ I am going over a solution given to solving the follow limit,
$$\lim_{x\to \infty}(\frac{x}{x-1})^x$$
The solution continues as follows,
Consider raising the function to $e^{ln\cdots}$
We can find the limit as follows,
$$\lim_{x\to \infty} x \ln(\frac{x}{x-1}) = \lim_{x\... | L'Hopital is rarely the method of choice. In this case, let $y = x-1$. Then
$$
\left(\frac{x }{x-1}\right)^x
=
\left(\frac{y+1}{y}\right)^{y+1}
=
\left(1 + \frac{ 1}{y}\right)^{y }\left(1 + \frac{ 1}{y}\right)^{ 1}.
$$
Now you can recognize the limit as $y \to \infty$ as $e \times 1 = e$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3029295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
${x^4}$ as "tesseracting" a number $x$ So, this strange thought popped up into my head. You know how we call ${x^2}$ squaring due to the fact that what you're essentially doing is finding the area of a square with side length $x$? The same goes for cubing. Saying ${x^3}$ is really going to give you the volume of a cube... | I think this is a really good way to bring geometric intuition into exponentiation. I'm not sure "tesseract" is universally the term for the "4-cube," but "4-cubing" seems great.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3029414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Show that $\mathbb{Q}(\sqrt{3},\sqrt[4]{3}, \sqrt[8]{3},...)$ is algebraic over $\mathbb{Q}$ but not a finite extension.
Show that $\mathbb{Q}(\sqrt{3},\sqrt[4]{3}, \sqrt[8]{3},...)$ is algebraic over $\mathbb{Q}$ but not a finite extension.
I think for the algebraic part, since for every simple extension, each of th... | Any element $\alpha$ of $F$ is a rational expression in the numbers adjoined. As such, it can involve only finitely many of the $\sqrt[2^k]3$. If in such an expression, $\sqrt[2^n]3$ is one with maximal $k$, then all other $\sqrt[2^k]3$ are powers of $\sqrt[2 k]3$. It follows that $\alpha\in\Bbb Q(\sqrt[2^n]3)$ and $\a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3029568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Roots of a polynomial with holomorphic coefficients Let $f_1,f_2,\dots,f_n : \mathbb{D} \to \mathbb{C}$ be holomorphic functions and consider the polynomial
$$ w^n + f_1(z)w^{n-1} + \dots + f_n(z). $$
Suppose, I happen to know that
*
*For each $z$, the roots of the above polynomial are all in $\mathbb{D}$.
*For ea... | It is true. Here's the proof. Let $\zeta=e^{\frac{2\pi i}{n}}$ be the $n$-th root of unity. Note that by the assumption, for each $z\in \mathbb{D}$, there is $k\in \{0,1,\ldots,n-1\}$ such that
$$
p(z^n, z\zeta^k) = 0
$$ where $$
p(z,w) = w^n + f_1(z)w^{n-1} + \cdots + f_n(z).
$$ Let $D_k$ be the set of all $z\in \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3029748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Spectrum of the product of two bounded operators
If $T$ is not invertible normal operator and $S$ is a bounded operator. Why $TS$ and $ST$ have the same spectrum?
Proof: Assume that $T$ is not invertible normal operator, then $0 \in \sigma(T)$. Since $0$ is in the approximate point spectrum of $T$, it is clear that ... | Without the assumption of normality, we can at least prove the following:
If $T,S\in \mathcal{B}(\mathcal{H})$, then $\{0\}\cup\sigma(ST)=\{0\}\cup\sigma(TS)$.
(This holds more generally in unital Banach algebras.) The second case is saying that $0$ is already in $\sigma(ST)$ and $\sigma(TS)$.
EDIT
This follows from... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3029923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Show that $\lim_{n\to\infty}\frac{\ln(n!)}{n} = +\infty$
Show that
$$
\lim_{n\to\infty}\frac{\ln(n!)}{n} = +\infty
$$
The only way i've been able to show that is using Stirling's approximation:
$$
n! \sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n
$$
Let:
$$
\begin{cases}
x_n = \frac{\ln(n!)}{n}\\
n \in \Bbb N
\end{ca... | The Cesaro-Stolz criterion is your easiest and cleanest way out of this. It states that given sequences $x, y \in \mathbb{R}^{\mathbb{N}}$ such that $y$ is strictly increasing and unbounded and the sequence of successive increments converges in the extended real line
$$\lim_{n \to \infty} \frac{x_{n+1}-x_{n}}{y_{n+1}-y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3030037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
$a = d$ implies $a^b = d^b$
Prove that $a = d$ implies $a^b = d^b$, where $a, d$ are arbitrary
nonnegative integers and $b$ is any positive integer.
If I could use division I think it could be something like that:
$a^b / d^b = a ^{b-b} = a^0 = 1$ (assuming $a = d$), but I'm trying to figure out how to prove this ... | There must be something I don't understand. Are you sure you have asked the question you intended?
The equals sign in the assumption $a=d$ means that "$a$" and "$d$" are essentially just different names for the same number. So you can substitute one for the other in any formula. There is nothing to prove.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3030168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that $r(A)=\operatorname{tr}(A^2)$ Let $A\in M_n(\mathbb{C})$. Show that if $A^3=A$, then $r(A)=\operatorname{tr}(A^2)$.
Since $A^3=A$, the possible eigenvalues are $0,1,-1$. I don't know from here how to compute the rank of $A$.
Edited
Since eigenvalues of $A$ are $0,1,-1$ SO eigenvalues of $A^2=0,1$ So $r(A^2)... | The rank is the number of nonzero eigenvalues and as their squares are 0 or 1, this number is just the same as the sum of the squares.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3030278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating the limit using Taylor Series We're asked to find the following limit by using Taylor expansions $$\lim_{x\to{}0}\frac{e^{3x}-\sin(x)-\cos(x)+\ln(1-2x)}{-1+\cos(5x)}$$
My Attempt:
Expressing $e^{3x}$, $\sin(x)$, $\cos(x)$, $\ln(1-2x)$ and $\cos(5x)$ in their respective taylor expansions yielded the following... | We need only quote the numerator and denominator up to $x^2$ terms: $$\lim_{x\to 0}\frac{1+3x+\color{blue}{9x^2/2}-x-1+\color{blue}{x^2/2}-2x\color{blue}{-2x^2}+O(x^3)}{-1+1\color{blue}{-25x^2/2}+O(x^3)}.$$You'll find only $x^2$ terms survive in each.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3030377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
de Rham cohomology of doubly punctured torus Let $T^2=S^1\times S^1$. I'd like to know all de Rham cohomology groups of $M=T^2-\{a,b\}$ but I couldn't find a result. So I want to compute it and I'm thinking of using Mayer Vietoris sequence. I need two open sets whose union covers $M$. I'm having difficulty choosing the... | Its easy to visualize when looking at the fundamental domain of the torus:
$U$ is a neighbourhood of $x$, $V$ is a neighbourhood of $y$. Removing $x$ and $y$ from the torus is homotopy equivalent to removing the whole neighbourhoods $U$ and $V$. Further we can choose $U,V$ so large, that they fill the entire triangle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3030496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Can every connected reductive group over a char $p$ field be defined over $\mathbb F_p$? If I have a connected reductive group $G$ over a field with characteristic $p>0$, can it always be defined over $\mathbb F_p$? For split groups like $GL_n, GSp_{2n}$ it's trivial, how about general case?
| No. Here's a simple example. Let $T:=\mathsf{Res}^1_{\mathbb{F}_{p^4}/\mathbb{F}_{p^2}}\mathbb{G}_{m,\mathbb{F}_{p^4}}$. Then, $T$ is a non-split one-dimensional torus over $\mathbb{F}_{p^2}$ which does not have a model over $\mathbb{F}_p$. Indeed, to say that $T$ has a model over $\mathbb{F}_p$ would mean that there w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3030607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
If the square of every element of a ring is in the center, must the ring be commutative? Let $R$ be a ring with identity such that the square of any element belongs to the center of $R$. Is it necessary true that $R$ is commutative?
(I can show that for any $x,y\in R$, $2(xy-yx) =0 $ but I cannot prove commutativity of... | Here's a counterexample. Consider the $\mathbb{F}_2$-algebra $R$ generated by two elements $x,y$ modulo relations that $x^2=y^2=0$ and every word of length $3$ formed by $x$ and $y$ is $0$. Explicitly, $R$ has $\{1,x,y,xy,yx\}$ as a basis and any product of basis elements that would give a word not in this set is $0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3030793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
How to prove that there is a differentiable function $f$ such that $[f(x)]^{153}+f(x)+x=0$ for all $x$ Prove that there is a differentiable function $f$ such that $[f(x)]^{153}+f(x)+x=0$ for all $x$. Furthermore, find $f'$ in terms of $f$.
To me I just write $y$ instead of all $f(x)$ and find that $x=-(y^{153}+y)$ If I... | HINT:
If $x = -[f(x)]^{153} - f(x)$,
then $g(x) = -x^{153} - x$ is the inverse function of $f(x)$.
Then if you prove $g(x)$ is differentiable, what does it tell about its inverse function ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3030957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to prove that the set-theoretic difference operation $\setminus$ cannot be defined through $\cap$ and $\cup$ How does one go about proving that the set-theoretic difference operation $\setminus$ cannot be defined through the operations $\cap$ and $\cup$?
My thoughts: I first assumed $A$ and $B$ are two non-disjoint... | All you need is a counterexample.
Let $A = B = \{ 0 \}$. Then all of the sets
$$A,~ B,~ A \cup A,~ A \cap A,~ A \cup B,~ A \cap B,~ B \cup A,~ B \cap A,~ B \cup B,~ B \cup B$$
are equal to $\{ 0 \}$, and so any set built out of $A$, $B$ and the operations $\cup$ and $\cap$ is equal to $\{ 0 \}$.
But $B \setminus A = \v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3031087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Does a sequence of random variables constructed in a certain manner converge in distribution to a Gaussian? Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of of IID random variables taken for simplicity with mean zero and variance one.
The Central Limit Theorem give us that
$$
\frac{X_1 + \dots + X_n}{\sqrt{n}} \xrigh... | Let $\left(X_i\right)_{i\geqslant 1}$ be an i.i.d. sequence and let $f\colon \mathbb R^2\to \mathbb R$ be a function such that such that the random variable $Y_i:=f(X_i,X_{i+1})$ is centered and square integrable.
Let $n$ be a fixed integer and $q\in\left\{1,\dots,n\right\}$. We write
\begin{align}
\sum_{i=1}^nY_i&= \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3031285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find all positive integers $a$ and $b$ such that $(1 + a)(8 + b)(a + b) = 27ab$. Here's the problem I'm having difficulties with:
Find all positive integers $a$ and $b$ such that $$(1 + a)(8 + b)(a + b) = 27ab\,.$$
Does anyone have an idea how to do this? Any detailed solution is welcome! :)
| Using Hölder's inequality,
$$27ab = (a+1)(8+b)(b+a) \geqslant \left(2\sqrt[3]{ab}+\sqrt[3]{ab} \right)^3=27ab$$
Hence we are looking for the equality case for Hölder, which is when $a:8:b=1:b:a \implies (a, b)=(2, 4)$.
In fact, this is the only solution among positive reals, not just positive integers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3031422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Using Wallis' product to derive $\sqrt\pi$ Recall Wallis' product:
$$\lim_{n\to\infty}\Big(\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdots\frac{2n}{2n-1}\cdot\frac{2n}{2n+1}\Big)=\frac{\pi}{2}$$
We have to show that $$\lim_{n\to\infty}\frac{(n!)^22^{2n}}{(2n)!\sqrt n}=\sqrt\pi$$
The hi... | As you noticed, $P_n \to \frac{\pi}{2}$ since it's the inside of limit of the L.H.S of the Wallis Product Formula multiplied by $1$. Since continuous maps preserve limits, this implies $\sqrt{2P_n} \to \sqrt{\pi} $ and note that
$$\lim_{n \to \infty}\sqrt{2P_n} = \lim_{n \to \infty}\frac{\sqrt{2}(n!)^2 2^{2n}}{(2n)!\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3031546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Positive divisors of n = $2^{14} \cdot 3^9 \cdot 5^8 \cdot 7^{10} \cdot 11^3 \cdot 13^5 \cdot 37^{10}$ How do I find positive divisors of n that are perfect cubes that are multiples of 2^10 * 3^9 * 5^2 * 7^5 * 11^2 * 13^2 * 37^2
The answer is (1)(1)(2)(2)(1)(1)(3) = 12
I don't understand though because I would have don... | For $d$ to be a divisor of $n$, $d$ must be of the form $2^a \cdot 3^b \cdot 5^c \cdot 7^d \cdot 11^e \cdot 13^f \cdot 37^g$, where $0 \leq a \leq 14, 0 \leq b \leq 9, 0 \leq c \leq 8, 0 \leq d \leq 10, 0 \leq e\leq 3, 0\leq f \leq 5, 0 \leq g \leq 10$.
Now we want $d$ ot be a multiple of the number given, that means $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3031641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What does $\min\sum\dots$ indicate? What does "min" indicate infront of a sigma sign?
$$\min \sum_{e\in E} c_e x_e$$
Source: https://www.math.unipd.it/~luigi/courses/metmodoc1718/m08.01.TSPexact.en.pdf
| The $c_e$ are some constants, the $x_e$ are variables that satisfy some constraints and you are minimizing the sum $\sum_{e\in E} c_e x_e$ over all feasible choices of the $x_e$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3031729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Cone with height $9$ cm and radius $3$ cm is filled at a rate of $1.2~\text{cm}^3$. Find the rate of change when $h=3$. A cone with radius $3$ cm and height $9$ cm is filled with water at a rate of $1.2~\text{cm}^3$. Find the rate of change of the height of the water when the height of the water is $3$ cm.
I differenti... | Much simpler is to note that when $h=3$ cm the radius is $1$ cm. The area of the water is then $\pi$ cm$^2$ so the rate of rise is $\frac {1.2}\pi$ cm/sec
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3031830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How many obtuse angles can be formed from the 15 rays on a single point on a same plane? Consider 15 rays that originate from a point. What is the maximum number of obtuse angles they can form, assuming that the angles between two rays is less than or equal to 180 degrees?
| Call two rays near if they form a non-obtuse angle. If there are $n$ ordered pairs of near rays, there are exactly ${15\choose 2}-\frac n2$ obtuse angles among the rays. Hence we want to minimize $n$.
Claim. $n\ge 60$.
Proof. Suppose one of the rays (wlog the positive $x$ axis) is near $a\le 2$ other rays (i.e., we ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3031951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Correlation between parallel lines and volumes of tetrahedrons. Given 4 parallel lines $d_1$, $d_2$, $d_3$, $d_4$ , no more than 2 of which can be on a same plane. Plane (P) intersects the 4 lines at 4 points A, B, C, D. Plane (Q) ( not identical to plane (P)) intersects the 4 lines at $A_1$, $B_1$, $C_1$, $D_1$. Proof... | It is not explicitly stated that one of the given four parallel lines passes through both $D$ and $D_1,$ but I will assume this is given since otherwise
the two tetrahedra might have different volumes.
If you know that a shear transformation of three-dimensional Euclidean space preserves volume, you can construct a pl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3032077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Distance from eigenspace of matrix In linear algebra, is there a separate name / concept for the notion of distance between linear vector subspaces?
I'm asking this because I'm considering a problem in numerical linear algebra where a Krylov subspace iterative method is used. Since for every subsequent $n$ a Krylov su... | The common notion of distance is to consider an orthogonal projection $P$ onto the first linear subspace $V$, and an orthogonal projection $Q$ onto the other subspace $W$.
At this point we can define
$$d(V,W) = \| P - Q \|$$ as the distance between these subspaces, where the norm used is the operator norm. For properti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3032192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
The minimum edge cover of a tree is at least the maximum degree Let $T$ be a tree with maximum degree $\Delta(T)$, and let $\beta'(T)$ denote the size of the minimum edge cover of $T$. The question is to prove that $\beta'(T) \ge \Delta(T)$.
I started by proving that each tree has at least $\Delta(T)$ using induction o... | Let $v$ be a vertex of a maximal degree $\Delta$ of the tree $T$. Let $N(v)$ be a set of neighbors of the vertex $v$. If any edge of $T$ is incident to at least two vertices $v,w$ of $N(v)$ then $u-v-w-u$ is a cycle, which cannot occur in a tree. So each edge cover need at least $|N(v)|=\Delta$ distinct vertices to co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3032549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $a+\sqrt{a^2+1}= b+\sqrt{b^2+1}$, then $a=b$ or not? It might be a silly question but if $$a+\sqrt{a^2+1}= b+\sqrt{b^2+1},$$ then can I conclude that $a=b$? I thought about squaring both sides but I think it is wrong! Because radicals will not be removed by doing that! Can you help me with proving that $a=b$ or not?... | Alternatively, for $a,b\in \Bbb R$,
\begin{align}a+\sqrt{a^2+1}&=b+\sqrt{b^2+1}\\&\implies \frac{1}{a+\sqrt{a^2+1}}=\frac{1}{b+\sqrt{b^2+1}}\wedge a+\sqrt{a^2+1}=b+\sqrt{b^2+1}\\
&\implies \sqrt{a^2+1}-a=\sqrt{b^2+1}-b \wedge a+\sqrt{a^2+1}=b+\sqrt{b^2+1}\\
&\implies \sqrt{a^2+1}-a=\sqrt{b^2+1}-b \wedge a+\sqrt{a^2+1}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3032720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
What are some advanced books on metric space? Metric space, with the additional notion of “distance between points”, has properties that are more “concrete” than a topological structure. After a basic study I saw a number of strange and interesting results which depend heavily on the metric structure. So I wonder if th... | I can recommend the following books which deal mainly with metric spaces:
*
*Heinonen: Lectures on Analysis in Metric spaces This is a short and well written book talking about various topics and well suited for self-study.
*Bridson, Martin R., Häfliger, André : Metric Spaces of Non-Positive Curvature A more advanc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3032835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Can distinct elements of a $C^*$-algebra be separated by a maximal left ideal? Let $A$ be a $C^*$-algebra, and let $f\neq g\in A$. Does there exist a maximal ideal $J\trianglelefteq A$ with $f+J\neq g+J$?
I'm particularly interested in the case of $A=B(\mathcal H)$, and why things aren't obvious. In this case, $f\neq g... | Your ideal is maximal when $A=B(H)$. Let $J=\{T:\ Th=0\}$. Let $S\in B(H)\setminus J$. Then $Sh\ne0$. Choose $k$ with $\langle Sh,k\rangle=1$, and let $Wx=\langle x,k\rangle\,h$. Put $R=I-WS$. Then
$$ Rh=h-WSh=h-h=0, $$ so $R\in J$. Then $$ I=R+WS\in J+\mathbb C\,WS,$$ and $J$ is maximal (its codimension is 1).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3033276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove $(b-a) \cdot \int_a^b \alpha(x) \beta(x) dx \ge \int_a^b \alpha(x) dx \cdot \int_a^b \beta(x) dx$.
Suppose $\alpha: [a, b] \to (-\infty, \infty)$ and $\beta:[a,b] \to (-\infty, \infty)$ are both nondecreasing. Then,
$$(b-a) \cdot \int_a^b \alpha(x) \beta(x) dx \ge \int_a^b \alpha(x) dx \cdot \int_a^b \beta(x)... | Proof.
Let $D = [a,b]^2$. Then
$$
\int_a^b 1 \int_a^b \alpha\cdot \beta- \int_a^b \alpha \int_a^b \beta\stackrel{(1)}= \iint_D( \alpha(y)\beta(y)-\alpha(x)\beta(y) )\,\mathrm dx \,\mathrm dy \stackrel{(2)}= \iint_D (\alpha (x)\beta(x)-\alpha(y)\beta(x))\,\mathrm dx\,\mathrm dy = \frac 12 \iint_D (\alpha(x)\beta(x)+\alp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3033379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding $\frac{1}{2\pi}\int_{0}^{2\pi}\phi^\prime(x) dx$, where $\phi(x)=\arctan\frac{3\cos x}{4(\cos x+\sin x)}$. Why isn't it $\phi(2\pi)-\phi(0)$? I'm tasked with the following problem:
Evaluate
$$I_C=\frac{1}{2\pi}\int_{0}^{2\pi}\left(\frac{d}{d\theta}\phi(\theta)\right) d\theta,\quad\text{where}\; \phi(\theta)=\a... | Hint: what happens at $\theta =3\pi/4, 7\pi/4$? See the graph if required.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3033554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $\int_{0}^{\pi/6} {\cos (x^2)}\mathrm{d}x\ge\frac12$.
Prove that $\displaystyle\int_{0}^{\frac\pi 6} {\cos ({x^2)}\mathrm{d}x\ge\dfrac12}$.
I know this is a Fresnel integral but without going into advanced calculus is there a way to show that this is true? using calculus 1 knowledge, I tried Riemann's sum t... | For $0 < x \le \frac \pi 6 < 1$ we have $x^2 < x$ and therefore
$$
\int_{0}^{\pi/6} \cos (x^2) \, dx > \int_{0}^{\pi/6} \cos (x) \ dx
= \sin( \frac \pi 6) = \frac 12
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3033674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Compute $\sum\limits_{n=0}^\infty a_nx^n$ if $a_0=3$, $a_1=5$, and $na_n=\frac23a_{n-1}-(n-1)a_{n-1}$ for every $n>1$
Assume that $a_0=3$, $a_1=5$, and, for arbitrary $n>1$ , $na_n=\frac{2}{3}a_{n-1}-(n-1)a_{n-1}$. Prove that, when $|x|<1$, the series $\sum\limits_{n=0}^\infty a_nx^n$ converges, and compute its sum.
... | $na_n=\frac{5}{3}a_{n-1}-na_{n-1}$
Multiply by $x^n$ both sides and sum from $n=1$ to $\infty$
$\sum\limits_{n=1}^\infty na_n x^n=\frac{5}{3}\sum\limits_{n=1}^\infty a_{n-1}x^n-\sum\limits_{n=1}^\infty na_{n-1}x^n$
Reindex of the RHS:
$\sum\limits_{n=1}^\infty na_nx^n=\frac{5}{3}x\sum\limits_{n=0}^\infty a_{n}x^n-x\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3033773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that there are infinitely many prime numbers $p$ such that $\left(\frac{a}{p}\right)=1$ for fixed $a$. I already proved this is true for all prime numbers and clearly see how this is true for all perfect squares, I'm just having trouble expanding it to any prime factorization. If we let $a$ have prime factorizati... | Through quadratic reciprocity and Dirichlet's theorem we have a straightforward proof: for any $a\in\mathbb{N}^+$ there is some prime $p$ such that $p\equiv{1}\pmod{4}$ and $p\equiv 1\pmod{a}$. For such a prime
$$ \left(\frac{a}{p}\right)=\left(\frac{p}{a}\right)=\left(\frac{1}{a}\right)=1.$$
Yet another overkill: by C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3033908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is this Diophantine problem solvable without invoking Fermat's Last Theorem? Let $a,b,c,n$ be positive integers with $a<b<c$ and $n\geq 3$ odd. Given that $a^n + b^n < 2c^n$, can one prove that $a^{n+2}+b^{n+2}\neq c^{n+2}$ without invoking Wiles' theorem ? Or is this actually equivalent to Fermat's Last Theorem ?
| As stated in the comments, $a<b<c$ implies trivially $a^n+b^n<2c^n$, hence this condition is extraneous. The equation also immediately implies $a,b<c$ so that is not useful either. Finally the equation is symmetric in $a,b$ so we may WLOG assume $a\leq b$, so your condition just becomes $a\neq b$. In fact, $a\neq b$ is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3034070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Review on my method for $Number$ $of$ $diagonals$ in a regular $n$-gon is $\frac12n(n-3)$ I have an assignment on permutations and combinations topics. In that there is a question-
The number of interior angles of a regular polygon is $150^\circ$ each. The number of diagonals of the polygon is _____.
Attempt
I don't kn... | You can also use the Handshake Lemma from graph theory. Let $G(V,E)$ be a graph on $n$ vertices, where the vertices form a regular $n$-gon and the edges are the diagonals of the $n$-gon. Then, prove that each vertex of $G$ has degree $n-3$. By the Handshake Lemma, $G$ has
$$|E|=\frac{1}{2}\,\sum_{v\in V}\,\deg(v)=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3034176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How to solve this system of equations systematically? This might seem a trivial problem, but I have some trouble in arranging the data. So suppose you are given $f(x,y)=x^2y^2(1+x+2y)$ and you want to find it's critical points. Thus we find
$$\frac{\partial f}{\partial x}(x,y)=xy^2(2+3x+4y)\textrm{ and }\frac{\partial ... | Use the converse of the distributive property:
$((x=0)\lor(y=0)\lor(3x+4y=-2))\land((x=0)\lor(y=0)\lor(x+3y=-1))\\\equiv(x=0)\lor(y=0)\lor[(3x+4y=-2)\land(x+3y=-1)]$
$3x+4y+2=0=x+3y+1$ is just a pair of straight lines (linear equations) intersecting at $(-2/5,-1/5)$. Therefore, you have $(x=0)\lor(y=0)\lor(x=-2/5\land ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3034418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Prove inverse of strictly monotone increasing function is continuous over the range of original function Let $f:[a,b] \rightarrow \Bbb R$ be a strictly monotone increasing. Then $f$ has an inverse function $g:[c,d]\rightarrow \Bbb R,$ where $[c,d]$ is the range of $f$. I'm trying to prove that $g$ is continuous at d.
... | Let $R = f([a,b])$ be the range of $f$. Since $f$ is strictly increasing, we have $R \subset [f(a),f(b)]$, but in general $R \ne [f(a),f(b)]$. For example, let $f : [0,2] \to \mathbb{R}, f(x) = x$ for $x \in [0,1)$, $f(1) = 2$, $f(x) = x + 2$ for $x \in (1,2]$.
But although $R$ is general no interval, the usual definit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3034525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proof about inequalities Let $a,b,c \in \mathbb{R}$. Prove that if for $a<c$ $\forall$ $c>b$, then $a \leq b$.
My attempted proof was that if we take the contrapositive of the statement, then we get the inequality $c \leq b$ $\forall$ $a \geq c$. Then if we add the the given inequality to this we get $a+c \leq b+c$ wh... | To take the contrapositive of a statement you need to be extra careful about quantifiers.
The contrapositive of $$"a<c \text{ holding for all } c>b \text{ implies } a \le b."$$ is instead
$$"a > b \text{ implies that there exists } c>b \text{ such that }a \ge c."$$ Note that the 'for all $c>b$' turned into a 'there e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3034729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
how to integrate $\frac{-1+e^{-i k x}}{x^2}$ How do I integrate the following?$$\int_{-\infty }^{\infty } \frac{-1+e^{-i k x}}{x^2} \, dx$$
I am not very familiar with complex analysis, but I did try to use contour integral to do this but I couldn't get any success with that.
| Another way to do it if you want to use contour integration is to use an indented semi circle in the upper half plane going around the singularity at $z=0$. For this contour, the function $f(z)=(e^{-ikz}-1)/z^2$ will integrate to 0 by Cauchy's theorem since it is holomorphic in the region enclosed by the indented semi ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3034826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Homotheties: Let $A$ and $B$ be distinct points of a circle $o$. What is the set of possible centroids of triangles $ABC$ with $C\in o$? Question: Let $A$ and $B$ be distinct points of a circle $o$. What is the set of possible centroids of triangles $ABC$ with $C\in o$?
Here is what I have:
The angle at $C$ will always... | Here's a moderately obnoxious idea:
If you use complex numbers and set your circle to be the unit circle, then the centroid of the triangle determined by $a,b,c$ is simply $\frac{a+b+c}{3}$. Thus, the locus of possible centroids, as $A$ and $B$ are fixed and $C$ varies, is simply the circle with radius $\frac{1}{3}$ ce... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3034991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to calculate area of an ellipse based on its formula? How can I determine the area of a half-ellipse if all that is given is $y = \sqrt{1-n^2x^2}$? I have tried both geometry and calculus, but without convincing results…
Thank you
| With the use of generalized polar coordinates
\begin{aligned}x&=ar\cos t\\
y&=br\sin t\end{aligned}
where $a=\frac 1n,\; b=1,\; t \in [0,\pi]\; \text{and}\; r\in [0,1]$ in the given case. The Jacobian is ${r\over n}$ and the area
$$\cal{A}=\int_0^{\pi} \int_0^1 1\cdot {r\over n}\;dr \;dt=\frac{\pi}{2n}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3035202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Question on graph theory regarding roads connecting a pair of cities I have the following question with me from "Problem Solving Strategies" by Arthur Engel, given as an example in page 44
"Every road in Sikinia is one-way. Every pair of cities is connected exactly by one direct road. Show that there exists a city whic... | Considering the way the question was put, the answer is “no”, since the roads are one-way. But I suppose you need help with the problem, so there’s my proof below.
I’ll proceed with a proof that is, basically, an inductive algorithm. In graph theory this is very common, so take a moment to understand what I did.
Take... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3035351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Checking if a vector is in a subspace? I have
v1= $$
\begin{pmatrix}
1 \\
0 \\
-1\\
\end{pmatrix}
$$
v2=
\begin{pmatrix}
2 \\
1 \\
3\\
\end{pmatrix}
v3=\begin{pmatrix}
4 \\
2 \\
6\\
\end{pmatrix}
and
w=\begin{pmatrix}
3 \\
1 \\
2\\
\end{pmatrix}
I need to check whether w is in the subspace sp... | The dimension of the space of columns of a matrix is the maximal number of column vectors that are linearly independent.
In your example, both dimensions are $2$, as the last two columns can be written as a linear combination of the first two columns.
An example where the dimensions are not equal can be given by
$$\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3035454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $\int_{0}^{\frac{\pi}{4}} \ln(\sec x)dx$ Evaluate $$P=\int_{0}^{\frac{\pi}{4}} \ln(\sec x)dx$$
My try: I tried using its complimentary integral:
Let $$Q=\int_{0}^{\frac{\pi}{4}} \ln(\csc x)dx$$
Adding both we get:
$$P+Q=\int_{0}^{\frac{\pi}{4}}\ln(\sec x\csc x)dx$$ $\implies$
$$2P+2Q=\int_{0}^{\frac{\pi}{4}}\l... | In fact
\begin{eqnarray*}
P-Q&=&\int_0^{\frac{\pi}{4}}\ln(\tan t)dt\\
&=&\int_0^1\frac{\ln u}{1+u^2}du\\
&=&\int_0^1\ln u\sum_{n=0}^\infty(-1)^nu^{2n}du\\
&=&\sum_{n=0}^\infty(-1)^n\int_0^1u^{2n}\ln udu\\
&=&-\sum_{n=0}^\infty(-1)^n\frac{1}{(2n+1)^2}\\
&=&-C
\end{eqnarray*}
where $C$ is the Catalan constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3035569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Conditional probability - mistake in interpretation? Assume 3 events $A, B, C$ with success rates $p_1, p_2, p_3$. Let $X$ define an event, where exactly one of the three events had a success.
For me it's clear that $P(X) = P(X|A)P(A) + P(X|B)P(B) + P(X|C)P(C)$.
Further, given that we know that $X$ holds (exactly one ... | Careful! By your same formula we have that
$$P(A|X)=\frac{P(A\cap X)}{P(X)} = \frac{P(X|A)P(A)}{P(X|A)P(A) + P(X|B)P(B) + P(X|C)P(C)}$$
which is not equal to $p_1(1-p_2)(1-p_3)$.
The denominator in the above formula is exactly what you would interpret as "given that we know that $X$ holds". Indeed the denominator cont... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3035739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Differential of Hopf's map Let $$h : \mathbb{C^2} \rightarrow \mathbb{C \times R} $$
$$h(z_1, z_2) = (2z_1z_2^*, |z_1|^2-|z_2|^2)$$
How do you find the differential of $h$ and show it is onto/surjective?
I know that I can express $h$ as $\mathbb{R^4}$ instead of $\mathbb{C^2}$, but then how do I proceed? Do I just diff... | In fact you must understand $h$ as a map from $\mathbb{R}^4$ to $\mathbb{R}^3$ and the derivative of $h$ as a the derivative in the sense of real multivariable calculus. You have done this almost correctly (I corrected a typo), and you are right that the Jacobian $Jh(x)$ of $h$ at $x$ is a $3 \times 4$-matrix. With res... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3035823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why is 2 $\cdot$ $\sin(\alpha)$ $\cos(\alpha)$=$\sin(2\alpha)$? I am solving a physics problem involving a 2-dimensional throw. However, I have hit a bump trying to understand
$$2\cdot \sin(\alpha)\cdot\cos(\alpha)=\sin(2\alpha)$$
I have googled and searched on Stack Exhange but found nothing. I hope someone can explai... |
Consider the semicircle over the diameter $[AB]$ of radius 1 and center $E$. Let $C$ be an arbitrary point on the semicircle and $D$ the foot of the altitude of the side $[AB]$ in the triangle $\Delta ABC$. Denote furthermore by $\alpha$ the angle $\angle BAC$. It follows that $\angle BEC=2\alpha$.
By the definition ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3035934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Need help with some basic math/exponential rules I don't understand. How do I get from a to be here?
See image. I've looked up just about every rule I can find and I can't figure out how I am supposed to arrive at that answer. Can someone explain to me what has been done step by step here?
| $$0.5=\frac12$$
$$0.5-1=-0.5$$
$$x^{-n}=\frac1{x^n}$$
That should be enough to figure it out
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3036017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Why doesn't a 20 degree rotation change the slopes of $y=x$ and $y=\frac{x}{2}$ by the same amount? It seems that if I rotate different lines (lying in the same quadrant) the same number of degrees they move different amounts (in terms of their slope). (where the rotation is such that all the lines do not enter a diffe... | Slope is $\tan x$ , $x$ being the angle the line makes with $x$ axis. $x$ is 45 in the line $y=x$ and around 26 when $2y=x$. As you know that $\tan$ approaches infinity when $x$ approaches 90, so as we add 20 to 45, the value of $\tan$ will increase at a greater rate than when we add 20 to ~26.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3036144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
multivariable calculus and architecture It's often said that architecture involves a lot of multivariable calculus, and for my (high school) Multivariable calculus project, I wanted to do further research on that. However, so far I haven't been able to exactly determine what specific maths architects use and how they u... | *I should clarify that I might be missing the point of the question entirely, and if so, please excuse me.
Source: I have worked professionally as an Architect for the past 5 years, and have completed my Master of Architecture degree.
Outside of the profession, the perceived level of mathematics is often overestimate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3036266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime. Let $p=2^{2^k}+1$ be a prime where $k\ge1$. Prove that the set of quadratic non-residues mod $p$ is the same as the set of primitive roots mod $p$. Use this to show that $7$ is a primitive root mod $p$.
I've already shown the theorem to be true. T... | Powers of 2 are congruent to $1, 2, $ or $4$ modulo $7$ according as the power is congruent to $0, 1$ or $2$ modulo $3$ (as $3$ is the order of $2$ modulo $7$). As $3\nmid 2^k,$ $p\equiv 3\ or\ 5 \pmod 7.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3036390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Evaluate the limit of the sequence: $\lim_{n_\to\infty}\frac{\sqrt{(n-1)!}}{(1+\sqrt{1})\cdot(1+\sqrt{2})\cdot (1+\sqrt{3})\cdots (1+\sqrt{n})}$ Evaluate the limit of the sequence:
$$\lim_{n\to\infty}\frac{\sqrt{(n-1)!}}{(1+\sqrt{1})\cdot(1+\sqrt{2})\cdot (1+\sqrt{3})\cdots (1+\sqrt{n})}$$
My try:
Stolz-cesaro: The li... | Consider:
$$
(1+\sqrt{1})\cdot(1+\sqrt{2})\cdot (1+\sqrt{3})\cdots (1+\sqrt{n})
$$
Take the root from each pair of parentheses and multiply them, then:
$$
(1+\sqrt{1})\cdot(1+\sqrt{2})\cdot (1+\sqrt{3})\cdots (1+\sqrt{n}) > \sqrt{n!} \iff \\
\iff \frac{1}{(1+\sqrt{1})\cdot(1+\sqrt{2})\cdot (1+\sqrt{3})\cdots (1+\sqrt{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3036510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
prove there exist postive integers $a,b$ such $p^2|a^2+ab+b^2$ Problem 1: Let prime $p\equiv 1\pmod 3$.show that:there exist postive integers $a\le b<p$ such
$$p^2|a^2+ab+b^2$$
I have only prove there $a,b$ such $$p|a^2+ab+b^2$$
Problem 1 from this:
Problem 2.3 (Noam Elkies). Prove that there are infinitely many triple... | This relies on the following statement: if some residue class $a$ is a square mod $p$, then it is a square mod $p^2$.
Indeed, if $b^2=a+tp [p^2]$, then $(b+kp)^2=a+(t+2k)p [p^2]$.
So you know that there is some $x$ such that $p|x^2+x+1$, thus $-3=(2x-1)^2$ is a square mod $p$, thus mod $p^2$, and if $y^2=-3[p^2]$, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3036675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Average distance between point in a disc and line segment What is the average distance between a (randomly chosen) point in a disc of radius r and a line segment of length $a < 2r$ whose midpoint is at the center of the disc? ["Distance" here being the shortest distance to any point on the line segment.]
|
We devote our calculations to only the first quadrant because all the other quadrants are symmetrical in terms of calculations.
Dividing the quadrant into three regions, we will use the concept of expectation value of a variable denoted as follows $$\lt x\gt = \frac{\int xP(x)dx}{\int P(x)dx}$$
where $P(x)$ is the nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3036812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Find $\lim_{n\to\infty} \cos(\frac{\pi}{4}) \cos(\frac{\pi}{8})\ldots \cos(\frac{\pi}{2^n}) $ I already know that $$ a_n = \cos\left(\frac{\pi}{2^{n+1}}\right) = \overbrace{\frac{\sqrt{2+\sqrt{2+\ldots + \sqrt{2}}}}{2}}^{n\text{ roots}}$$
Also I know that $$\lim_{n\to\infty} 2\cos\left(\frac{\pi}{2^n}\right) = 2
\tex... | What you are trying to proof is Viete's formula. What he did was trying to compare area's of regular polygons that are inscribed in a unit circle. The area of a regular polygon with $n$ sides is given by
$$ A_n = \frac12 n \sin\left(\frac\pi n\right)$$
If you compute now the ratio between two regular polygons, one with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3036917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
Determine the Expected Value of Uniformly random elements of sets
Answer is D
The way I attempted this was that for X = MAX(a,b), the random variable X is equivalent to the max value of a and b. So, from the 2 sets, the probability of getting k from set {1,2...100} is $\frac{k}{100}$ and for the second set we have to ... | Your general logic is fine, although you calculate the probability incorrectly. Note that, if $\max(a,b) = k$, then we have one of three cases:
*
*$a = k$ and $b < k$.
*$a < k$ and $b = k$.
*$a = b = k$
It should be readily apparent that these cases are disjoint, so to find the probability $P(X = k)$, we can add t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3037061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is $A \& B \multimap A$ derivable? Intuitively, the sentence $A \& B \multimap A$ seems to mean "Using a choice between $A$ and $B$, get an $A$." This feels like it should be derivable for any $A$ and $B$, but I haven't found any way to derive it from the definition of $\&$. Is it possible to establish this in linear l... | $\DeclareMathOperator{\par}{\unicode{8523}}$
Yes, $A \& B \multimap A$ is provable in linear logic sequent calculus, but the derivation in the answer above is wrong, because there is no rule that allows one to derive $\vdash (A^\bot\oplus B^\bot) \par A$ from $\vdash A^\bot \par A$ (an inference rule in the sequent cal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3037188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is continuous increasing function in $H^1([0,1])$ Consider a function $f(x):[0,1]\rightarrow \mathbb{R}$. If $f(x)$ is continuous and increasing, is $f(x)$ in $H^1(\Omega)$, the Sobolev space with norm $\sqrt{\int_0^1 (|f(x)|^2 + |Df(x)|^2) dx}$?
| As in the comments, the Devil's staircase, $\mathcal D=\mathcal D(x)$ is a non-decreasing function with no weak derivative; therefore it cannot lie in any Sobolev space. Since the OP wants a function that is increasing, one can consider $f(x) := x + \mathcal D(x)$ instead.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3037295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A function that is continuous almost everywhere is Lebesgue measurable If $f: E \to \mathfrak{M}$ (where $\mathfrak{M}$ is the Lebesgue measurable sets) is continuous a.e., is it true that $f$ is Lebesgue measurable?
I know that continuous functions on $E \in \mathfrak{M}$ are Lebesgue measurable, but I am wondering i... | The first set is open, hence measurable.
Edit: indeed, the first set is not open.
However, let us denote $S_1$ the first set, $S_2$ the second one, $S=S_1 \cup S_2$. Then $S_2$ has null measure and $S_1 \subset S’ \subset S=S_1 \cup S_2$ where $S’$ is the interior of $S$.
So $S$ has symmetric difference of null meas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3037477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Looking for reference on cup and cap product without invoking acyclic model theorem I am looking for reference on cup and cap product without invoking acyclic model theorem. To me, acyclic model theorem is very strange phenomena though I could understand it but I do not see direct construction.
$\textbf{Q:}$ Is there ... | The cup product is graded commutative in homology, but not on the chain level. Maps witnessing higher non-commutativity in a coherent way are known as $i$-cup products, and were introduced by N. Steenrod in this paper. Computations there are very explicit. The fundamental result for (usual) cup products is that if $a$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3037622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $(u(x,y))^2+u(x,y)v(x,y)$ has a local maximum or minimum in $D$, then $f$ must be constant?
Let $f(z) = u(x,y)+iv(x,y)$ be an analytic function on a connected open set $D$ with $u(x,y)$ and $v(x,y)$ being the real and imaginary parts of $f(z)$, respectively. If $(u(x,y))^2+u(x,y)v(x,y)$ has a local maximum or minim... | I didn't think about it much, so I don't know if it helps for the answer, but we can show that $(u(x,y))^2+u(x,y)v(x,y)$ is constant.
For seeing this you should first calculate the Laplace operator of $f$, i.e. $\triangle f$ and see that $-\triangle f\leq 0$ everywhere. You need to use the Cauchy-Riemann-equations for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3037768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solving independent linear equations
\begin{align}
&{-}2y+2z-1=0 \tag{4} \\[4px]
&{-}2x+4y-2z-2=0 \tag{5} \\[4px]
&\phantom{-2}x-y+3/2=0 \tag{6}
\end{align}
Equation (6) is the sum of (4) and (5). There are only two independent equations.
Putting $z=0$ in (5) and (6) and solving for x and y, we have
\begin{align}... | You can also think of as follows.
*
*A single variable linear equation represents a single point.
*A two-variable linear equation represents a line. If there are two two-variable linear equations, their solution is the intersection point of them as long as both equations are independent.
*A three-variable linear e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3037882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
When for given $n$ and $m$ we get $p^n-1=k(p^m-1)$ provided that $k$ is an even number Consider $p$ is an odd prime number.
Assume that $n$ and $m$ are two positive integer numbers
provided that $m \mid n$ which results in $p^m-1 \mid p^n-1$.
Therefore, we get $p^n-1=k(p^m-1)$ where $k$ is a positive integer number. ... | You can write $$\frac{p^n-1}{p^m-1}=\frac{(p^{m})^l-1}{p^m-1}=\sum_{i=0}^{l-1}(p^m)^i$$ There are $l$ terms on the RHS all being odd, so you need $l$ to be even, that is $\frac{n}{m}$ must be even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3038024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Let $f(x) = \ln x - 5x$, for $x > 0$. I’m in IB Math and we are working on some calculus problems but I wanted to get extra practice so this is a problem in my book. The number in parenthesis next to the parts are the “marks” we get for the question if we get it right. So, usually that’s about how much work we have to ... | Point "a)" is ok.
For point "b)" recall that
$$\frac1x = x^{-1} \implies \frac{d}{dx}(x^{-1})=-x^{-2}=-\frac1{x^2}$$
Note also that for point "c)" in nay case
$$\frac1x - 5 = \frac1x \iff -5=0$$
that would mean that the equation has not solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3038155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Why don't we always consider complete measure spaces? Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and let $X=(X_t)$ and $Y=(Y_t)$ two stochastic processes. I know for example that $X$ and $Y$ are indistinguishable if there is a set $N$ of measure $0$ s.t. for all $\omega \notin N$ we have $X_t=Y_t$ for al... | The main advantage I can think of is that the composition of two $(\mathcal B,\mathcal B)$-measurable functions are also $(\mathcal B,\mathcal B)$-measurable whereas the composition of two $(\mathcal L,\mathcal B)$-measurable functions need not be $(\mathcal L,\mathcal B)$-measurable. Of course, here $\mathcal B$ is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3038312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Extra factor of 2 when evaluating an infinite sum using fourier series and parseval's theorem. I'm asked to find the fourier series of the $2 \pi $ periodic function f(x) which is $sin(x)$ between $0$ and $\pi$ and $0$ between $\pi$ and $2\pi$
I use the complex form to proceed and get $$\frac{1}{2\pi}\int_{0}^{\pi}sin(... | I can't check your calculations since you haven't included them, but it is clear that the Fourier series you found is not the Fourier series of $f$. Your function is not even, so it cannot have a Fourier cosine series. For example,
$$ b_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \sin(x) \, dx = \frac{1}{\pi} \int_0^{\pi} \si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3038460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Closed from of $\int_0^{\infty} \frac{e^{iax}}{x^{n}+1}dx$? I've been trying to find the general form of a certain group of integrals of the form$$I(a,n)=\int_0^{\infty} \frac{e^{iax}}{x^{n}+1}dx$$
I know that the real part of $I(a,2)$ can be calculated using Fourier Transform or residues, and $I(a,1)$ reduces to a for... | Since the integrand is a product of two Meijer G-functions and the integration range is $[0, \infty)$, there is a closed form, but it involves the Fox H-function:
$$\int_0^\infty \frac {e^{i a x}} {x^n + 1} dx =
\int_0^\infty
G_{0, 1}^{1, 0} {\left(- i a x \middle| { - \atop 0} \right)}
G_{1, 1}^{1, 1} {\left(x^n \mi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3038558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Given $f(x)$ is integrable on $[0, 1]$ and $0 < f(x) < 1$, prove that $\int_{0}^{1} (f(x))^{n} \mathop{dx}$ converges to $0$.
Given $f(x)$ is integrable on $[0, 1]$ and $0 < f(x) < 1$, prove that
$\int_{0}^{1} (f(x))^{n} \mathop{dx}$ converges to $0$.
I understand why the statement is true intuitively because as $n... | You may use the following theorem due to Arzelà :---
Let $\{f_n\}$ be a sequence of Riemann integrable Functions on $[a,b]$ and converges point-wise to $f$, also there is a positive number $M$ such that $|f_n(x)|≤M,\forall x\in [a,b],\forall n\in \Bbb N$. Now if $f$ is Riemann integrable over $[a,b]$ then , $$\lim_{n\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3038847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
What is the convex hull of $\text{conv}(u_1,u_2,\cdots,u_p)+\text{conv}(v_1,v_2,\cdots,v_s)$? Let $u_i, i= 1,\cdots,p$ and $v_j, j= 1,\cdots,s$ be finitely many vectors in $\mathbb{R}^n$. Show that
$$
\text{conv}(u_1,u_2,\cdots,u_p)+\text{conv}(v_1,v_2,\cdots,v_s)=\text{conv}\{u_i+v_j \mid i= 1,\cdots,p, \,\, j= 1,\cd... | For two sets
$$\operatorname{co}(S_1 \times S_2) = \operatorname{co}(S_1) \times \operatorname{co}(S_2)$$
the inclusion from left to right is clear, for the other way notice
the equality
$$(\sum \lambda_i u_i, \sum \mu_j v_j) = \sum \lambda_i \mu_j (u_i , v_j)$$
if $\lambda_i$, $\mu_j$ positive with sum $1$. Now apply ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3038918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How to show polyhedral cone of nonnegative vectors contains finitely generated cone? Let $P=\{x \in \mathbb{R}^n \mid Ax \geq b, x \geq 0 \}$ be a nonempty polyhedron for matrix $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$.
According to Minkowski-Weyl theorem $P$ can be written as
$$
P=\text{conv}(v_1,\cdo... | We know that $P$ can be written as
$$
P=\operatorname{conv}(v_1,\cdots,v_p)+ \operatorname{cone}(d_1,\cdots,d_l)=V+D.
$$
The set $D$ is a cone, hence, for every $v\in V$ and $d\in D$ we have that $v+td\in P$, $\forall t\ge 0$. That is
$$
A(v+td)\ge b,\quad v+td\ge 0,\quad\forall t\ge 0.
$$
Now divide by $t$ and let $t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Upper bound of expected maximum of weighted sub-gaussian r.v.s Let $X_1, X_2, \ldots$ be an infinite sequence of sub-Gaussian random variables which are not necessarily independent.
My question is how to prove
\begin{eqnarray}
\mathbb{E}\max_i \frac{|X_i|}{\sqrt{1+\log i}} \leq C K,
\end{eqnarray}
where $K=\max_i \|X_... | Without loss of generality, assume that $K = c$ (the constant in the exponent of subgaussian tail).
\begin{eqnarray}
\mathbb{E}\max \frac{|X_i|}{\sqrt{1+\log i}} &=& \int_0^\infty \mathbb{P}\left(\max \frac{|X_i|}{\sqrt{1+\log i}} > t \right) dt\\
&\leq& \int_0^2 \mathbb{P}\left(\max \frac{|X_i|}{\sqrt{1+\log i}} > t \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Show that if $(A+2I)^2=0$, then $A+\lambda I$ is invertible for $\lambda \ne 2$.
Show that if $(A+2I)^2=0$, then $A+\lambda I$ is invertible for $\lambda \ne 2$.
I tried to solve this by treating $(A+\lambda I)v=0$ as linear equation system, and proving that $v$ must be $0$ (trivial solution) therefore $A+\lambda I$ ... | Have you heard about eigenvalues? YOur equation for $A$ shows that $-2$ is the only eigenvalue of A(because any eigenvalue would satisfy $(x+2)^2=0$, the equation satisfied by $A$), so no other $\lambda$ could be an eigenvalue for $A$, qed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Apply Rolle's theorem to find real roots
Suppose the function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ such that $f(a)=f(b)=0$. Prove that there exist a point $c\in(a,b)$ such that
$$f(c)-f'(c)=0$$
From the question above, or otherwise, show that the equation
$$1+x+\frac{x^2}{2!}+\cdots+\frac{x... | The answer above addressed the second part of your question, so I guess I'll answer the third. We break it into cases for $n$ even and $n$ odd.
Let $f(x)=e^x-x^n$, so that $f'(x)=e^x-nx^{n-1}$
First, assume $n$ is odd, so that $x<0\rightarrow x^n<0$; this along with the fact that $0$ is not a root of $f$ means our r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$?
Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly
on $(a, b)$ then is $f$ analytic on $(a, b)$?
Intuitively, I think that the answer is no. I know that the statement holds for in... | Yes, you are correct. Just consider$$\begin{array}{rccc}f_n\colon&(-1,1)&\longrightarrow&\mathbb R\\&x&\mapsto&\sqrt{x^2+\frac1{n^2}}.\end{array}$$The sequence $(f_n)_{n\in\mathbb N}$ is a sequence of analytic functions that converges uniformly to the absolute value functions, which isn't differentiable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$\lim_{n \to \infty}(1+\frac{1}{n^2})(1+\frac{2}{n^2})...(1+\frac{n}{n^2})=e^{\frac{1}{2}}$. Here is the beginning of a proof:
Suppose $0<k \leq n$,
$1+\frac{1}{n}<(1+\frac{k}{n^2})(1+\frac{n+1-k}{n^2})=1+\frac{n+1}{n^2}+\frac{k(n+1-k)}{n^4}\leq 1+\frac{1}{n}+\frac{1}{n^2}+\frac{(n+1)^2}{4n^4}$.
I'm confused by the sec... | Hint: By AM-GM Inequality
$$\frac{k+ n+1-k}{2} \geq \sqrt{k(n+1-k)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
In what dimensions are PL and Diff equivalent? I have read in different places, that for $n\leq 4$, PL and Diff are equivalent (passing through PDIFF). I believe Milnor gave an example of the inequivalence for $n=7$. What about for $n=5$ or $n=6$? Are there examples of PL manifolds which are not smoothable?
| A more qeneral question is References on the relations between Top, Diff and PL. It gives a reference to https://mathoverflow.net/q/96670 which gives a complete (positive) answer for $n =5, 6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find all matrices which satisfy $M^2-3M+3I = 0$ I am trying to find all matrices which solve the matrix equation
$$M^2 -3M +3I=0$$
Since this doesn't factor I tried expanding this in terms of the coordinates of the matrix. It also occurs to me to put it into "vertex" form:
$$M^2 - 3M + \frac{9}{4}I+\frac{3}{4}I=0$$
$... | Minimal polynomial of $M, m_M(x),$ is a factor of $x^2-3x+3=[x-(\frac{3+i\sqrt3}2)][x-(\frac{3-i\sqrt3}2)]$
Either $m_M(x)=x-(\frac{3+i\sqrt3}2)\implies M=[\frac{3+i\sqrt3}2]$
or $m_M(x)=x-(\frac{3-i\sqrt3}2)\implies M=[\frac{3-i\sqrt3}2]$
or $m_M(x)=x^2-3x+3\implies$ the eigenvalues of $M$ are $\frac{3\pm i\sqrt3}2$
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3039910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.