Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
I need help with this divisibility problem. I need help with the following divisibility problem.
Find all prime numbers m and n such that $mn |12^{n+m}-1$ and $m= n+2$.
| You want to solve $p(p+2)|(12^{p+1}-1)(12^{p+1}+1)$.
Hint: First exclude $p=2,3$, so we have
$$\eqalign{
12^{p+1}-1 \equiv 143 &= 11 \cdot 13 &\pmod p,\\
12^{p+1}+1 \equiv 145 &= 5 \cdot 29 &\pmod p,
}$$
and deduce that $p$ must be one of $5,11,29$.
Edit: I'll just add more details: We want that $p$ divides $(12^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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On solvable quintics and septics Here is a nice sufficient (but not necessary) condition on whether a quintic is solvable in radicals or not. Given,
$x^5+10cx^3+10dx^2+5ex+f = 0\tag{1}$
If there is an ordering of its roots such that,
$x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_5 + x_5 x_1 - (x_1 x_3 + x_3 x_5 + x_5 x_2 + x_... | This problem is old but quite interesting. I have an answer to (I) which depends on some calculations in $\textsf{GAP}$ and Mathematica. I haven't thought about (II).
Suppose an irreducible septic has roots $x_1,\ldots,x_7$ that satisfy
$$
x_1 x_2 + x_2 x_3 + \dots + x_7 x_1 – (x_1 x_3 + x_3 x_5 + \dots + x_6 x_1) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
} |
Can a non-proper variety contain a proper curve Let $f:X\to S$ be a finite type, separated but non-proper morphism of schemes.
Can there be a projective curve $g:C\to S$ and a closed immersion $C\to X$ over $S$?
Just to be clear: A projective curve is a smooth projective morphism $X\to S$ such that the geometric fibr... | Sure, $\mathbb{P}_k^2-\{pt\}$ is not proper over $Spec(k)$ and contains a proper $\mathbb{P}_k^1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/129713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show the existence of a complex differentiable function defined outside $|z|=4$ with derivative $\frac{z}{(z-1)(z-2)(z-3)}$ My attempt
I wrote the given function as a sum of rational functions (via partial fraction decomposition), namely
$$
\frac{z}{(z-1)(z-2)(z-3)} = \frac{1/2}{z-1} + \frac{-2}{z-2} + \frac{3/2}{z-3... | The key seems to be that the coefficients $1/2$, $-2$ and $3/2$ sum to 0. So if you choose branch cuts for the three logarithm such that the cuts coincide through the $|z|>4$ region, then jumps at the cuts will cancel each other out (the amount each raw logarithm jumps is always $2\pi i$) and leave a single continuous ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Interpreting $F[x,y]$ for $F$ a field. First, is it appropriate to write $F[x,y] = (F[x])[y] $? In particular, if $F$ is a field, then we know $F[x]$ is a Euclidean domain. Are there necessary and sufficient conditions for when $F[x,y]$ is also a Euclidean domain?
| In most constructions of polynomial rings (e.g., as almost null sequences with values in the ground ring), the rings $F[x,y]$, $(F[x])(y)$, and $(F[y])[x]$ are formally different objects: $F[x,y]$ is the set of all functions $m\colon \mathbb{N}\times\mathbb{N}\to F$ with $m(i,j)=0$ for almost all $(i,j)$; $(F[x])[y]$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Spread out the zeros in a binary sequence Suppose I have a machine that processes units at a fixed rate. If I want to run it at a lower rate, I have to leave gaps in the infeed. For example, if you want to process 3/4, then you would feed in 3, leave a gap, then repeat. This could be encoded as a binary sequence: 11... | Consider the following algorithm: let $a_i$ be the digit given on the $i$-th step; $b_i = b_{i-1}+a_i$ ($b_0 = 0$) the number of ones given until $i$-th step (and $i-b_i$ the number of zeros give until $i$-th step); define $a_{i+1}$ to be a zero iff $\frac{b_i}{i+1} > p$ (where $p$ is given probability); one otherwise.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Specify the convergence of function series The task is to specify convergence of infinite function series (pointwise, almost uniform and uniform convergence):
a) $\displaystyle \sum_{n=1}^{+\infty}\frac{\sin(n^2x)}{n^2+x^2}, \ x\in\mathbb{R}$
b) $\displaystyle\sum_{n=1}^{+\infty}\frac{(-1)^n}{n+x}, \ x\in[0;+\infty)$
c... | First observe that each of the series converges pointwise on its given interval (using standard comparison tests and results on $p$-series, geometric series, and
alternating series.
Towards determining uniform convergence, let's first recall the Weierstrass $M$-test:
Suppose $(f_n)$ is a sequence of real-valued fun... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
If "multiples" is to "product", "_____" is to "sum" I know this might be a really simple question for those fluent in English, but I can't find the term that describes numbers that make up a sum.
The numbers of a certain product are called "multiples" of that "product".
Then what are the numbers of a certain sum call... | To approach the question from another direction: A "multiple" of 7 is a number that is the result of multiplying 7 with something else.
If your try to generalize that to sums, you get something like: A "__" of 7 is a number that is the result of adding 7 to something. But that's everything -- at least as long as we all... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Combinations of lego bricks figures in an array of random bricks I have an assignment where a robot should assemble some lego figures of the simpsons. See the figures here: Simpsons figures!
To start with we have some identical sized, different colored lego bricks on a conveyor belt. See image.
My problem is to find ou... | The problem of minimizing the number of unused blocks is an integer linear programming problem, equivalent to maximizing the number of blocks that you do use. Integer programming problems are in general hard, and I don’t know much about them or the methods used to solve them. In case it turns out to be at all useful to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating conditional probability for markov chain I have a Markov chain with state space $E = \{1,2,3,4,5\}$ and transition matrix below:
$$ \begin{bmatrix} 1/2 & 0 & 1/2 & 0 & 0 \\\ 1/3 & 2/3 & 0 & 0 & 0 \\\ 0 & 1/4 & 1/4 & 1/4 & 1/4 \\\ 0 & 0 & 0 & 3/4 & 1/4 \\\ 0 & 0 & 0 & 1/5 & 4/5\ \end{bmatrix} $$
How would I ... | The notation $p^{(2)}_{15}$ is not to be confused with the square of $p_{15}$ since it stands for the $(1,5)$ entry of the square of the transition matrix. Thus,
$$
p^{(2)}_{15}=\sum_{i=1}^5p_{1i}p_{i5}.
$$
Likewise for $p^{(3)}_{11}$, which is the $(1,1)$ entry of the cube of the transition matrix, that is,
$$
p^{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Critical points of $f(x,y)=x^2+xy+y^2+\frac{1}{x}+\frac{1}{y}$ I would like some help finding the critical points of $f(x,y)=x^2+xy+y^2+\frac{1}{x}+\frac{1}{y}$. I tried solving $f_x=0, f_y=0$ (where $f_x, f_y$ are the partial derivatives) but the resulting equation is very complex. The exercise has a hint: think of $f... | After doing some computations I found the following (lets hope I didn't make any mistakes). You need to solve the equations
$$f_x = 2x + y - \frac{1}{x^2} = 0 \quad f_y = 2y + x -\frac{1}{y^2} = 0$$
therefore after subtracting and adding them as in the hint we get
$$\begin{align}
f_x - f_y &= x - y - \frac{1}{x^2} + \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Improving Gift Wrapping Algorithm I am trying to solve taks 2 from exercise 3.4.1 from Computational Geometry in C by Joseph O'Rourke. The task asks to improve Gift Wrapping Algorithm for building convex hull for the set of points.
Exercise: During the course of gift wrapping, it's sometimes possible to identify point... | Hint: You're already computing the angles of the lines from the current point to all other points. What does the relationship of these angles to the angle of the line to the starting point tell you? (This also doesn't improve the time complexity of the algorithm.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/130338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the minimal polynomials of trigonometric expressions quickly If if an exam I had to calculate the minimal polynomials of, say, $\sin(\frac{2 \pi}{5})$ or $\cos(\frac{2 \pi}{19})$, what would be the quickest way to do it? I can use the identity $e^{ \frac{2 i \pi}{n}} = \cos(\frac{2 \pi}{n}) + i\sin(\frac{ 2 \pi... | $\cos(2\pi/19)$ has degree 9, and I doubt anyone would put it on an exam. $\sin(2\pi/5)$ is a bit more reasonable. Note that $\sin(4\pi/5)=2\sin(2\pi/5)\cos(2\pi/5)$, and also $\sin(4\pi/5)=\sin(\pi/5)$, and $\cos(2\pi/5)=1-2\sin^2(\pi/5)$, and with a few more identities like that you should be able to pull out a formu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Residue integral: $\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx$ with $0 \lt a \lt 1$. I'm self studying complex analysis. I've encountered the following integral:
$$\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx \text{ with } a \in \mathbb{R},\ 0 \lt a \lt 1. $$
I've done the substitution $e^x = y$. What ki... | there is a nice question related with this problem , and we can evaluate it by using Real analysis
$$I=\int_{-\infty }^{\infty }\frac{e^{ax}}{1-e^{x}}dx,\ \ \ \ \ \ \ \ (*)\ \ \ \ \ 0<a<1\\
\\
let \ x\rightarrow -x \ \ then\ \ I=\int_{-\infty }^{\infty }\frac{e^{-ax}}{1-e^{-x}}\ \ \ \ \ (**)\\
\\
adding\ (*)\ and\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Is there a computer program that does diagram chases? After having done many tedious, robotic proofs that various arrows in a given diagram have certain properties, I've begun to wonder whether or not someone has made this automatic. You should be able to tell a program where the objects and arrows are, and which arrow... | I am developing a Mathematica package called WildCats which allows the computer algebra system Mathematica to perform category theory computations (even graphical computations).
The posted version 0.36 is available at
https://sites.google.com/site/wildcatsformma/
A much more powerful version 0.50 will be available at... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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Probability - Coin Toss - Find Formula The problem statement, all variables and given/known data:
Suppose a fair coin is tossed $n$ times. Find simple formulae in terms of $n$ and $k$ for
a) $P(k-1 \mbox{ heads} \mid k-1 \mbox{ or } k \mbox{ heads})$
b) $P(k \mbox{ heads} \mid k-1 \mbox{ or } k \mbox{ heads})$
Releva... | By the usual formula for conditional probability, an ugly form of the answer is
$$\frac{\binom{n}{k-1}(1/2)^n}{\binom{n}{k-1}(1/2)^n+\binom{n}{k}(1/2)^n}.$$
Cancel the $(1/2)^n$. Now the usual formula for $\binom{a}{b}$ plus a bit of algebra will give what you want.
We can simplify the calculation somewhat by using the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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All partial derivatives are 0. I know that for a function $f$ all of its partial derivatives are $0$.
Thus, $\frac{\partial f_i}{\partial x_j} = 0$ for any $i = 1, \dots, m$ and any $j = 1, \dots, n$.
Is there any easy way to prove that $f$ is constant? The results seems obvious but I'm having a hard time expressing i... | It's not hard to see that given $\textbf{p,q}\in\mathbb{R}^M$:
$\hspace{2cm}\textbf{f(p)}-\textbf{f(q)}=\textbf{f}\bigl(f_1(\textbf{p})-f_1(\textbf{q}), f_2(\textbf{p})-f_2(\textbf{q}), ... f_N(\textbf{p})-f_N(\textbf{q})\bigr)$
$\hspace{4.5cm} =\bigl(\int_{\gamma} \nabla f_1(\mathbf{r})\cdot d\mathbf{r}, \int_{\gamma}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Notation for modules. Given a module $A$ for a group $G$, and a subgroup $P\leq G$ with unipotent radical $U$, I have encountered the notation $[A,U]$ in a paper. Is this a standard module-theoretic notation, and if so, what does it mean.
In the specific case I am looking at, it works out that $[A,U]$ is equal to the s... | In group theory it is standard to view $G$-modules $A$ as embedded in the semi-direct product $G \ltimes A$. Inside the semidirect product, the commutator subgroup $[A,U]$ makes sense for any subgroup $U \leq G$ and since $A$ is normal in $G \ltimes A$, we get $[A,U] \leq A$ and by the end we need make no reference to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130802",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Where's my mistake? This is partially an electrical engineering problem, but the actual issue apparently lays in my maths.
I have the equation $\frac{V_T}{I_0} \cdot e^{-V_D\div V_T} = 100$
$V_T$ an $I_0$ are given and I am solving for $V_D$.
These are my steps:
Divide both sides by $\frac{V_T}{I_0}$:
$$e^{-V_D\div V_... | You are confusing resistance with incremental resistance, I think. The incremental resistance only matters for small signal analysis. The problem is to set the operating point so that the incremental resistance will be the required value. This involves computing $V_D, I_D$. However, you cannot use the incremental resis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/130909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Can there be a scalar function with a vector variable? Can I define a scalar function which has a vector-valued argument?
For example, let $U$ be a potential function in 3-D, and its variable is $\vec{r}=x\hat{\mathrm{i}}+y\hat{\mathrm{j}}+z\hat{\mathrm{k}}$.
Then $U$ will have the form of $U(\vec{r})$.
Is there any pr... | That's a perfectly fine thing to do. The classic example of such a field is the temperature in a room: the temperature $T$ at each point $(x,y,z)$ is a function of a $3$-vector, but the output of the function is just a scalar ($T$). $\phi^4$ scalar fields are also an example, as are utility functions in economics.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/130953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof that $\sum\limits_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$ regarding $\zeta(3)$ and Apéry's proof I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$
$$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$
and ... | Formally, the first identity is repeated application of the rewriting rule
$$\dfrac 1 x = \dfrac 1 {x+a} + \dfrac {a}{x(x+a)} $$
to its own rightmost term, first with $a = a_1$, then $a=a_2$, then $a=a_3, \ldots$
The only convergence condition on the $a_i$'s is that the $n$th term in the infinite sum go to zero. [i.e. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding all reals such that two field extensions are equal. So we want to find an $u$ such that $\mathbb{Q}(u)=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$. I obtained that if $u$ is of the following form: $$u=\sqrt[6]{2^a5^b}$$Where $a\equiv 1\pmod{2}$, and $a\equiv 0\pmod{3}$, and $b\equiv 0\pmod{2}$ and $ b\equiv 1\pmod{3}$. T... | If we take $u = \sqrt{2} + \sqrt[3]{5}$, such a $u$ almost always turns out to work. In fact let's try if a rational linear combination of $\sqrt{2}$ and $\sqrt[3]{5}$ will work. Let us now write $u$ as $u = a\sqrt{2} + b\sqrt[3]{5}$ for rationals $a$ and $b$.
Clearly we have that $\Bbb{Q}(u)\subseteq \Bbb{Q}(\sqrt{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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A more general case of the Laurent series expansion? I was recently reading about Laurent series for complex functions. I'm curious about a seemingly similar situation that came up in my reading.
Suppose $\Omega$ is a doubly connected region such that $\Omega^c$ (its complement) has two components $E_0$ and $E_1$. So i... | I'll suppose both $E_0$ and $E_1$ are bounded. Let $\Gamma_0$ and $\Gamma_1$ be disjoint positively-oriented simple closed contours in $\Omega$ enclosing $E_0$ and $E_1$ respectively, and $\Gamma_2$ a large positively-oriented circle enclosing both $\Gamma_0$ and $\Gamma_1$. Let $\Omega_1$ be the region inside $\Gamm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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graph theory connectivity
This cut induced confuses me.... I dont really understand what it is saying...
I am not understanding what connectivity is in graph theory. I thought connectivity is when you have a tree because all the vertices are connected but the above mentions something weird like components could someo... | Connectivity and components
Intuitively, a graph is connected if you can't break it into pieces which have no edges in common. More formally, we define connectivity to mean that there is a path joining any two vertices - where a path is a sequence of vertices joined by edges. The example of $Q_3$ in your question is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Probability theory. 100 items and 3 controllers 100 items are checked by 3 controllers. What is probability that each of them will check more than 25 items?
Here is full quotation of problem from workbook:
"Set of 100 articles randomly allocated to test between the three controllers. Find the probability that each con... | Let $N_i$ denote the number of items checked by controller $i$. One asks for $1-p$ where $p$ is the probability that some controller got less than $k=25$ items. Since $(N_1,N_2,N_3)$ is exchangeable and since at most two controllers can get less than $k$ items, $p=3u-3v$ where $u=\mathrm P(N_1\lt k)$ and $v=\mathrm P(N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Reduction of a $2$ dimensional quadratic form I'm given a matrix $$A = \begin{pmatrix}-2&2\\2&1\end{pmatrix}$$ and we're asked to sketch the curve $\underline{x}^T A \underline{x} = 2$ where I assume $x = \begin{pmatrix}x\\y \end{pmatrix}$. Multiplying this out gives $-2 x^2+4 x y+y^2 = 2$.
Also, I diagonalised this m... | A quadratic form is "equivalent" to it's diagonalized form only in the sense that these quadratic forms share an equivalence class, but the resulting polynomials are of course different (as you showed), otherwise we wouldn't need the equivalence classes!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/131355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Is $2^{2n} = O(2^n)$? Is $2^{2n} = O(2^n)$?
My solution is:
$2^n 2^n \leq C_{1}2^n$
$2^n \leq C_{1}$,
TRUE.
Is this correct?
| If $2^{2n}=O(2^n)$, then there is a constant $C$ and an integer $M$ such that for all $n\ge M$, the inequality $2^{2n}\le C 2^n$ holds.
This would imply that $2^n\cdot 2^n\le C 2^n$ for all $n\ge M$, which in turn implies $$\tag{1} 2^n\le C \quad {\bf for\ all } \quad n\ge M. $$ Can such $C$ and $M$ exist? Note th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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The graph of Fourier Transform I am trying to grasp Fourier transform, I read few websites about it, and I think I don't understand it very good. I know how I can transform simple functions but there is few things that are puzzling to me.
Fourier transform takes a function from time domain to a frequency domain, so now... | You can refer the following link. Here you can find intuitive explanation to Fourier Transform.
The Frequency Domain values after Fourier Transform represent the contribution of that each Frequency in the signal
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/131469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Various types of TQFTs I am interested in topological quantum field theory (TQFT). It seems that there are many types of TQFTs. The first book I pick up is "Quantum invariants of knots and 3-manifolds" by Turaev. But it doesn't say which type of TQFT are dealt in the book. I found at least two TQFTs which contain Turae... | Review of a recent Turaev book at http://www.ams.org/journals/bull/2012-49-02/S0273-0979-2011-01351-9/ also discussing earlier volumes to some extent.
Here we go, the book you ask about: http://www.ams.org/journals/bull/1996-33-01/S0273-0979-96-00621-0/home.html
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Positive series problem: $\sum\limits_{n\geq1}a_n=+\infty$ implies $\sum_{n\geq1}\frac{a_n}{1+a_n}=+\infty$ Let $\sum\limits_{n\geq1}a_n$ be a positive series, and $\sum\limits_{n\geq1}a_n=+\infty$, prove that: $$\sum_{n\geq1}\frac{a_n}{1+a_n}=+\infty.$$
| We can divide into cases:
*
*If a(n) has limit zero : It is lower than 1 for all n bigger than n0, then we can compare with a(n)/2 which is lower than a(n)/(1+a(n)).
*If a(n) has limit different to zero , also a(n)/1+a(n) and then the series diverges
*If a(n) is not bounded it ha a subsequence that converges to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 6,
"answer_id": 4
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Surface Element in Spherical Coordinates In spherical polars,
$$x=r\cos(\phi)\sin(\theta)$$
$$y=r\sin(\phi)\sin(\theta)$$
$$z=r\cos(\theta)$$
I want to work out an integral over the surface of a sphere - ie $r$ constant. I'm able to derive through scale factors, ie $\delta(s)^2=h_1^2\delta(\theta)^2+h_2^2\delta(\phi)^... | I've come across the picture you're looking for in physics textbooks before (say, in classical mechanics). A bit of googling and I found this one for you!
Alternatively, we can use the first fundamental form to determine the surface area element. Recall that this is the metric tensor, whose components are obtained by... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 6,
"answer_id": 0
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Deriving the exponential distribution from a shift property of its expectation (equivalent to memorylessness). Suppose $X$ is a continuous, nonnegative random variable with distribution function $F$ and probability density function $f$. If for $a>0,\ E(X|X>a)=a+E(X)$, find the distribution $F$ of $X$.
|
About the necessary hypotheses (and in relation to a discussion somewhat buried in the comments to @bgins's answer), here is a solution which does not assume that the distribution of $X$ has a density, but only that $X$ is integrable and unbounded (otherwise, the identity in the post makes no sense).
A useful tool he... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Similarity between $I+N$ and $e^N$ when $N$ is nilpotent Let
$$
N=\begin{pmatrix}0&1&&\\&\ddots&\ddots&\\&&0&1\\&&&0
\end{pmatrix}_{n\times n}
$$
and $I$ is the identity matrix of order $n$. How to prove $I+N\sim e^N$?
Clarification: this is the definition of similarity, which is not the same as equivalence.
Update:
I... | By subtracting $I$ this is equivalent to asking about the similarity class of a nilpotent square matrix of size $N$. The similarity type of $N$ is determined by the dimensions of the kernels of powers of $N$. In the upper triangular case the list of dimensions of $\ker N^i$ is $1,2,3,4,...,n$ for both of the matric... | {
"language": "en",
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"source": "stackexchange",
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Integrating a spherically symmetric function over an $n$-dimensional sphere I've become confused reading over some notes that I've received. The integral in question is $\int_{|x| < \sqrt{R}} |x|^2\, dx$ where $x \in \mathbb{R}^n$ and $R > 0$ is some positive constant. The notes state that because of the spherical sy... | The $\omega_n$ is meant to be the volume of the unit $n$-sphere. See http://en.wikipedia.org/wiki/N-sphere for notation.
Also, you are correct that $r = |x|$, and the $r^{n-1}$ comes from the Jacobian of the transformation from rectangular to spherical coordinates. (The $\omega_n$ also comes from this transformation).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/131925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is the axiom of choice separated from the other axioms? I don't know much about set theory or foundational mathematics, this question arose just out of curiosity. As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms with the axiom of choice. But the last axiom seems to be the mo... | The basic axiom of "naive set theory" is general comprehension: For any property $P$, you may form the set consisting of all elements satisfying $P$. Russell's paradox shows that general comprehension is inconsistent, so you need to break it down into more restricted types of comprehension.
The other axioms of ZF (exce... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 4,
"answer_id": 0
} |
simple-connectedness of convex and star domains Let $D$ be a convex domain in the complex plane, and is domain $D$ a simply connected domain? What about when $D$ is a star domain?
| Yes star-domains are simply connected, since every path is homotopy equivalent to one going through the center.
The disc with one point removed is not simply connected, but also not convex.
Open convex sets are among the star-domains.
All that is not special to the $\mathbb{C}$, but any $\mathbb{R}^d$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to expand undifferential function as power series? If a function has infinite order derivative at $0$ and $\lim_{n\to \infty}(f(x)-\sum_{i=1}^{n} a_{n}x^n)=0$ for every $x \in (-r,r)$,then it can be expand as power series$\sum a_{n}x^n$,
My question is if this function is not differential at $0$,how to expand it as... | If $\sum_{j=0}^\infty a_j x^j$ converges for every $x$ in an interval $(-r,r)$, then the radius of convergence of the series is at least $r$, and the sum is analytic in the disk $\{z \in {\mathbb C}: |z| < r\}$. So if $f(x)$ is not analytic in $(-r,r)$, in particular if it is not differentiable at $0$, there is no w... | {
"language": "en",
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"source": "stackexchange",
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Can asymptotic of a Mellin (or laplace inverse ) be evaluated? I mean, given the Mellin inverse integral $ \int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $, can we evaluate this integral, at least as $ x \rightarrow \infty $?
Can the same be made for $ \int_{c-i\infty}^{c+i\infty}dsF(s)\exp(st) $ as $ x \rightarrow \infty $?... | yes we can evaluate above integral but it depends on F(s).what is your F(s).then we can see how to solve it.above integral is inverse mellin transform.Some times it is v difficult to find inverse,it all depends on what F(s) is.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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convergence with respect to integral norm but not pointwise I want to give an example of a sequence of functions $f_1 \dots f_n$ that converges with respect to the metric $d(f,g) = \int_a^b |f(x) - g(x)| dx$ but does not converge pointwise.
I'm thinking of a function $f_n$ that is piecewise triangle, whose area converg... | You can suppose that $g(x) = 0$, because your metric is translation invariant ; $d(f_n,g) = d(f_n-g,0)$. Think of the sequence $f_n : [0,1] \to \mathbb R$ defined by $f_n(x) = x^n$ if $n$ is odd, and $(1-x)^n$ if $n$ is even. Therefore,
$$
d(f_{2n+1},0) = \int_0^1 (1-x)^{2n+1} \, dx = \frac 1{2n+2} \underset{ n \to \in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/132238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Finding the singularities of affine and projective varieties I'm having trouble calculating singularities of varieties: when my professor covered it it was the last lecture of the course and a shade rushed.
I'm not sure if the definition I've been given is standard, so I will quote it to be safe:
the tangent space of a... | You are confusing $\mathbb P^5$ and $\mathbb A^6$.
The calculation you did is valid for the cone $C\subset \mathbb A^6$ with equation $ x_{12} x_{34} - x_{13}x_{24}+ x_{14}x_{23}=0$.
It is of codimension $1$ (hence of dimension $5$), and smooth outside of the origin, as your jacobian matrix shows.
The image $\mathbb ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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nilpotent ideals
Possible Duplicate:
The set of all nilpotent element is an ideal of R
An element $a$ of a ring $R$ is nilpotent if $a^n = 0$ for some positive integer $n$. Let $R$ be a
commutative ring, and let $N$ be the set of all nilpotent elements of $R$.
(a) I'm trying to show that $N$ is an ideal, and that th... | The first question on why the set of all nilpotent elements in a commutative ring $R$ is an ideal (also called the nilradical of $R$, denoted by $\mathfrak{R}$) has already been answered numerous times on this site. I will tell you why $R/\mathfrak{R}$ has no nilpotent elements.
Suppose in the quotient we have an elem... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find the Laplace transform? Jobs arrive to a computer facility according to a Poisson process with rate $\lambda$ jobs / hour. Each job requires a service time $X$ which is uniformly distributed between $0$ and $T$ hours independently of all other jobs.
Let $Y$ denote the service time for all jobs which arrive ... | The general form of a Poisson process (or a Levy process) can be defined as; the number of events in time interval $(t, t + T]$ follows a Poisson distribution with associated parameter λT. This relation is given as
\begin{equation}
P[(N(t+T)-N(t))=k] = \frac{(\lambda T)^k e^{- \lambda T}}{k!}
\end{equation}
Where $N(t ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/132430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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What is the runtime of a modulus operation Hi I have an algorithm for which I would like to provide the total runtime:
def foo(x):
s = []
if(len(x)%2 != 0):
return false
else:
for i in range(len(x)/2):
//some more operations
return true
The loop is in O(n/2) but what is O(... | There are two meanings for "run time". Many people have pointed out that if we assume the numbers fit in one register, the mod operaton is atomic and takes constant time.
If we want to look at arbitrary values of $x$, we need to look at the model of computation that is used. The standard way of measuring computational ... | {
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"source": "stackexchange",
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Two linearly independent eigenvectors with eigenvalue zero What is the only $2\times 2$ matrix that only has eigenvalue zero but does have two linearly independent eigenvectors?
I know there is only one such matrix, but I'm not sure how to find it.
| Answer is the zero matrix obviously.
EDIT, here is a simple reason: let the matrix be $(c_1\ c_2)$, where $c_1$ and $c_2$ are both $2\times1$ column vectors. For any eigenvector $(a_1 \ a_2)^T$ with eigenvalue $0$, $a_1c_1 + a_2c_2 = 0$. Similarly, for another eigenvector $(b_1 \ b_2)^T$, $b_1c_1 + b_2c_2 = 0$. So $(a_... | {
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Inequality ${n \choose k} \leq \left(\frac{en}{ k}\right)^k$ This is from page 3 of http://www.math.ucsd.edu/~phorn/math261/9_26_notes.pdf (Wayback Machine).
Copying the relevant segment:
Stirling’s approximation tells us $\sqrt{2\pi n} (n/e)^n \leq n! \leq e^{1/12n} \sqrt{2\pi n} (n/e)^n$. In particular we can use th... | First of all, note that $n!/(n-k)! \le n^k$. Use Stirling only for $k!$.
${n \choose k} \le \frac{n^k}{k!} \le \frac{n^k}{(\sqrt{2\pi k}(k/e)^k)} \le \frac{n^k}{(k/e)^k} = (\frac{en}{k})^k$
| {
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"timestamp": "2023-03-29T00:00:00",
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What does $2^x$ really mean when $x$ is not an integer? We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so please let me know what you think.
| If this helps:
For all n:
$$2^n=e^{n\log 2}$$
This is a smooth function that is defined everywhere.
Another way to think about this (in a more straightforward manner then others described):
We know
$$a^{b+c}=a^ba^c$$
Then say, for example, $b=c=1/2$. Then we have:
$$a^{1}=a=a^{1/2}a^{1/2}$$
Thus $a^{1/2}=\sqrt{a}$ is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Existence of a sequence. While reading about Weierstrass' Theorem and holomorphic functions, I came across a statement that said: "Let $U$ be any connected open subset of $\mathbb{C}$ and let $\{z_j\} \subset U$ be a sequence of points that has no accumulation point in $U$ but that accumulates at every boundary point o... | Construct your sequence in stages indexed by positive integers $N$. At stage $N$, enumerate those points $(j+ik)/N^2$ for integers $j,k$ with $|j|+|k|\le N^3$ that are within distance $1/N$ of the boundary of $G$.
EDIT: Oops, not so clear that this will give you points in $G$. I'll have to be a bit less specific. ... | {
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give a counterexample of monoid If $G$ is a monoid, $e$ is its identity, if $ab=e$ and $ac=e$, can you give me a counterexample such that $b\neq c$?
If not, please prove $b=c$.
Thanks a lot.
| Lets look at the endomorphisms of a particular vector space: namely let our vector space $V := \mathbb{R}^\infty$ so an element of $V$ looks like a countable (not necessarily convergent) sequence of real numbers. The set of linear maps $\phi: V \rightarrow V$ form a monoid under composition(prove it!). Let $R: V \right... | {
"language": "en",
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Capelli Lemma for polynomials I have seen this lemma given without proof in some articles (see example here), and I guess it is well known, but I couldn't find an online reference for a proof.
It states like this:
Let $K$ be a field and $f,g \in K[x]$. Let $\alpha$ be a root of $f$ in the algebraic closure of $K$. The... | Let $\theta$ be a root of $g-\alpha$. From $g(\theta)=\alpha$ we get that $f(g(\theta))=0$. Now all it is a matter of field extensions. Notice that $[K(\theta):K]\le \deg (f\circ g)=\deg f\deg g$, $[K(\theta):K(\alpha)]\le \deg(g-\alpha)$ $=$ $\deg g$ and $[K(\alpha):K]\le\deg f$. Each inequality becomes equality iff t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Correct precedence of division operators Say i have the followingv operation - $6/3/6$, i get different answers depending on which division i perform first.
$6/3/6 = 2/6 = .33333...$
$6/3/6 = 6/.5 = 12$
So which answer is correct?
| By convention it's done from left to right, but by virtually universal preference it's not done at all; one uses parentheses.
However, I see students writing fractions like this:
$$
\begin{array}{c}
a \\ \hline \\ b \\ \hline \\ c
\end{array}
$$
Similarly they write $\sqrt{b^2 - 4a} c$ or $\sqrt{b^2 - 4}ac$ or even $... | {
"language": "en",
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Probability of a specific event at a given trial Recently I have founded some problems about probabilities, that ask to find the probability of an event at a given trial.
A dollar coin is toss several times until ones get "one dollar" face up. What is the probability to toss the coin at least $3$ times?
I thought to ... | Hint: It is the same as the probability of two tails in a row, because you need to toss at least $3$ times precisely if the first two tosses are tails. And the probability of two tails in a row is $\frac{1}{4}$.
Remark: For fun, let's also do this problem the hard way. We need at least $6$ tosses if we toss $2$ tails... | {
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Splitting field of $x^6+x^3+1$ over $\mathbb{Q}$ I am trying to find the splitting field of $x^6+x^3+1$ over $\mathbb{Q}$.
Finding the roots of the polynomial is easy (substituting $x^3=t$ , finding the two roots of the polynomial in $t$ and then taking a 3-rd root from each one). The roots can be seen here [if there i... | You've got something wrong: the roots of $t^2+t+1$ are the complex cubic roots of one, not of $-1$: $t^3-1 = (t-1)(t^2+t+1)$, so every root of $t^2+t+1$ satisfies $\alpha^3=1$). That means that you actually want the cubic roots of some of the cubic roots of $1$; that is, you want some ninth roots of $1$ (not of $-1$). ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find x that defined in the picture?
$O$ is center of the circle that surrounds the ABC triangle.
$|EF| // |BC|$
we only know $a,b,c$
$(a=|BC|, b=|AC|,c=|AB|)$
$x=|EG|=?$
Could you please give me hand to see an easy way to find the x that depends on given a,b,c?
| This can be done using trigonometry.
Let $D$ be the foot of perpendicular from $O$ to $BC$.
Then we have that $\angle{BOD} = \angle{BAC} (= \alpha, \text{say})$.
Let $\angle{CBA} = \beta$.
Let the radius of the circumcircle be $R$.
Let $I$ be the foot of perpendicular from $G$ on $BC$.
Then we have that $DB = R\sin \al... | {
"language": "en",
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Explaining Horizontal Shifting and Scaling I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect.
For example, $f(x+1)$ is a horizontal shift to the left (a shift toward the negative side ... | What does the graph of $g(x) = f(x+1)$ look like? Well, $g(0)$ is $f(1)$, $g(1)$ is $f(2)$, and so on. Put another way, the point $(1,f(1))$ on the graph of $f(x)$ has become the point $(0,g(0))$ on the graph of $g(x)$, and so on. At this point, drawing an actual graph and showing how the points on the graph of $f(x)$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why this element in this tensor product is not zero? $R=k[[x,y]]/(xy)$, $k$ a field. This ring is local with maximal ideal $m=(x,y)R$. Then the book proves that $x\otimes y\in m\otimes m$ is not zero, but I don't understand what's going on, if the tensor product is $R$-linear, then $x\otimes y=1\otimes xy=1\otimes 0=0$... | For your first question, $1$ does not lie in $m$, so $1 \otimes xy$ is not actually an element of $m \otimes m$.
$R$-linearity implies $a(b\otimes c)=(ab)\otimes c$, but (and this is important), if you have $(ab)\otimes c$, and $b$ does not lie in $m$, then "$ab$" is not actually a factorization of the element inside $... | {
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Techniques for (upper-)bounding LP maximization I have a huge maximization linear program (variables grow as a factorial of a parameter). I would like to bound the objective function from above. I know that looking at the dual bounds the objective function of the primal program from below.
I know the structure of the c... | In determining the value of a primal maximization problem, primal solutions give lower bounds and dual solutions give upper bounds. There's really only one technique for getting good solutions to large, structured LPs, and that's column generation, which involves solving a problem-specific optimization problem over the... | {
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Proving that a limit does not exist Given the function
$$f(x)= \left\{\begin{matrix}
1 & x \gt 0 \\
0 & x =0 \\
-1 & x \lt 0
\end{matrix}\right\}$$
What is $\lim_{a}f$ for all $a \in \mathbb{R},a \gt 0$?
It seems easy enough to guess that the limit is $1$, but how do I take into account the fact that $f(x)=-1$ ... | Here $\lim_{x\to 0^+}f(x)=1$ & $\lim_{x\to 0^-}f(x)=-1$ both limit are not equal.
therefore limit is not exists
| {
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"url": "https://math.stackexchange.com/questions/133534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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When is $X^n-a$ is irreducible over F? Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a$ in $F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show that $x^n -a$ is irreducible over $F$?
| I will assume "$m \geq 1$", since otherwise $a \in F(\omega)$, but $F(\omega)$ is $(n-1)$th extension and not $n$th extension, so $x^n-a$ must have been reducible.
Let $b^n=a$ (from the algebraic closure of $F$).
$x^n-a$ is irreducible even over $F(\omega)$. Otherwise $$f= \prod_{k=0}^n (x-\omega^k b) = (x^p + \cdots +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 2,
"answer_id": 0
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Question about the independence definition. Why does the independence definition requires that every subfamily of events $A_1,A_2,\ldots,A_n$ satisfies $P(A_{i1}\cap \cdots \cap A_{ik})=\prod_j P(A_{ij})$ where $i_1 < i_2 < \cdots < i_n$ and $j < n$.
My doubt arose from this: Suppose $A_1,A_2$ and $A_3$ such as... | $P(ABC)=P(A)P(B)P(C)$ does not imply that $P(ABC^C)=P(A)P(B)P(C^C)$, which it seems you're using. Consider, for instance, $C=\emptyset$.
However, see this question.
Another example:
Let $S=\{a,b,c,d,e,f\}$ with $P(a)=P(b)={1\over8}$, and $P(c)=P(d)=P(e)=P(f)={3\over16}$.
Let $A=\{a,d,e\}$, $B=\{a,c,e\}$, and $C=\{a,c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Problem of finding subgroup without Sylow's Thm. Let $G$ is a group with order $p^n$ where $p$ is prime and $n \geq 3$.
By Sylow's Thm, we know that $G$ has a subgroup with order $p^2$.
But, I wonder to proof without Sylow's Thm.
| Well, we could apply the fact that the center of such a group $G$ is nontrivial (proof). Since the center is nontrivial, it either has order $p$ or $p^m$ for $1<m\leq n$. In the former case, $G/Z(G)$ is of order $p^{n-1}$ so $Z(G/Z(G))$ is nontrivial, hence has order a power of $p$. Since $Z(G/Z(G))$ is abelian, it i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
limits of the sequence $n/(n+1)$ Given the problem:
Determine the limits of the sequnce $\{x_n\}^
\infty_{
n=1}$
$$x_n = \frac{n}{n+1}$$
The solution to this is:
step1:
$\lim\limits_{n \rightarrow \infty} x_n = \lim\limits_{n \rightarrow \infty} \frac{n}{n + 1}$
step2:
$=\lim\limits_{n \rightarrow \infty} \frac... | Divide the numerator and denominator by $n$. Why is this legal, in other words, why does this leave your fraction unchanged?
Because $$\frac {\frac a n} {\frac b n}=\frac {a \cdot \frac 1 n} {b \cdot \frac 1 n}=\frac a b$$ where the last equality is because $\dfrac 1 n$'s get cancelled.
Further, remember the fact t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Integral related to $\sum\limits_{n=1}^\infty\sin^n(x)\cos^n(x)$ Playing around in Mathematica, I found the following:
$$\int_0^\pi\sum_{n=1}^\infty\sin^n(x)\cos^n(x)\ dx=0.48600607\ldots =\Gamma(1/3)\Gamma(2/3)-\pi.$$
I'm curious... how could one derive this?
| For giggles:
$$\begin{align*}
\sum_{n=1}^\infty\int_0^\pi \sin^n u\,\cos^n u\;\mathrm du&=\sum_{n=1}^\infty\frac1{2^n}\int_0^\pi \sin^n 2u\;\mathrm du\\
&=\frac12\sum_{n=1}^\infty\frac1{2^n}\int_0^{2\pi} \sin^n u\;\mathrm du\\
&=\frac12\sum_{n=1}^\infty\frac1{2^{2n}}\int_0^{2\pi} \sin^{2n} u\;\mathrm du\\
&=2\sum_{n=1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Why use absolute value for Cauchy Schwarz Inequality? I see the Cauchy-Schwarz Inequality written as follows
$$|\langle u,v\rangle| \leq \lVert u\rVert \cdot\lVert v\rVert.$$
Why the is the absolute value of $\langle u,v\rangle$ specified? Surely it is apparent if the right hand side is greater than or equal to, for e... | I assumed that we work in a real inner product space, otherwise of course we have to put the modulus.
The inequality $\langle u,v\rangle\leq \lVert u\rVert\lVert v\rVert$ is also true, but doesn't give any information if $\langle u,v\rangle\leq 0$, since in this case it's true, and just the trivial fact that a non-neg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Changing the bilinear form on a Euclidean space with an orthonormal basis. I'm having trouble getting my head around how Euclidean spaces, bilinear forms and dot product all link in with each other. I am told that on a Euclidean space any bilinear form is denoted by $$\tau(u,v) = u\cdot v$$ and in an orthonormal basis ... | Orthonormal is defined with respect to $\tau$. That is, with no positive definite symmetric bilinear form $\tau$ around, the statement "$\{v_1,...,v_n\}$ is a an orthonormal basis" is meaningless.
Once you have such a $\tau$, then you can say $\{v_1,...,v_n\}$ is an orthonormal basis with respect to $\tau$." This mea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
Summing an unusual series: $\frac {x} {2!(n-2)!}+\frac {x^{2}} {5!(n-5)!}+\dots +\frac {x^{\frac{n}{3}}} {(n-1)!}$ How to sum the following series
$$\frac {x} {2!(n-2)!}+\frac {x^{2}} {5!(n-5)!}+\frac {x^{3}} {8!(n-8)!}+\dots +\frac {x^{\frac{n}{3}}} {(n-1)!}$$ n being a multiple of 3.
This question is from a book, i... | Start with
$$ (1 + x)^n = \sum_{r=0}^{n} \binom{n}{r} x^r$$
Multiply by $x$
$$ f(x) = x(1 + x)^n = \sum_{r=0}^{n} \binom{n}{r} x^{r+1}$$
Now if $w$ is a primitive cube-root of unity then
$$f(x) + f(wx) + f(w^2 x) = 3\sum_{k=1}^{n/3} \binom{n}{3k-1} x^{3k}$$
Replace $x$ by $\sqrt[3]{x}$ and divide by $n!$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/134148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding an ON basis of $L_2$ The set $\{f_n : n \in \mathbb{Z}\}$ with $f_n(x) = e^{2πinx}$ forms an orthonormal basis of the complex space $L_2([0,1])$.
I understand why its ON but not why its a basis?
| It is known that orthonormal system $\{f_n:n\in\mathbb{Z}\}$ is a basis if
$$
\operatorname{cl}_{L_2}(\operatorname{span}(\{f_n:n\in\mathbb{Z}\}))=L_2([0,1])
$$
where $\operatorname{cl}_{L_2}$ means the closure in the $L_2$ norm.
Denote by $C_0([0,1])$ the space of continuous functions on $[0,1]$ which equals $0$ at p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Irreducibility of polynomials This is a very basic question, but one that has frustrated me somewhat.
I'm dealing with polynomials and trying to see if they are irreducible or not. Now, I can apply Eisenstein's Criterion and deduce for some prime p if a polynomial over Z is irreducible over Q or not and I can sort of ... | $t^3-2=(t-\sqrt[3]{2})(t^2+(\sqrt[3]2)t+\sqrt[3]4)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/134408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
A basic estimate for Sobolev spaces Here is a statement that I came upon whilst studying Sobolev spaces, which I cannot quite fill in the gaps:
If $s>t>u$ then we can estimate:
\begin{equation}
(1 + |\xi|)^{2t} \leq \varepsilon (1 + |\xi|)^{2s} + C(\varepsilon)(1 + |\xi|)^{2u}
\end{equation}
for any $\varepsilon > 0$
... | Let $f(x)=(1+x)^{2(t-u)}-\varepsilon(1+x)^{2(s-u)}$. We have $f(0)=1-\varepsilon$ and since $s-u>t-u$ and $\varepsilon>0$, we know $f(x)\to-\infty$ as $x\to\infty$. Hence $C(\varepsilon):=\displaystyle\sup_{0\leq x<\infty}f(x)<\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/134473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Are groups algebras over an operad? I'm trying to understand a little bit about operads. I think I understand that monoids are algebras over the associative operad in sets, but can groups be realised as algebras over some operad? In other words, can we require the existence of inverses in the structure of the operad? S... | No, there is no operad whose algebras are groups. Since there are many variants of operads a more precise answer is that if one considers what are known as (either symmetric or non-symmetric) coloured operads then there is no operad $P$ such that morphisms $P\to \bf Set$, e.g., $P$-algebras, correspond to groups.
In g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
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A finitely presented group
Given the presented group
$$G=\Bigl\langle a,b\Bigm| a^2=c,\ b(c^2)b,\ ca(b^4)\Bigr\rangle,$$
determine the structure of the quotient $G/G'$,where G' is the derived subgroup of $G$ (i.e., the commutator subgroup of $G$).
Simple elimination shows $G$ is cyclic (as it's generated by $b$)... | Indeed, the group $G/G'$ is generated by $bG'$: let $\alpha$ denote the image of $a$ in $G/G'$ and $\beta$ the image of $b$. Then we have the relations $\alpha^4\beta^2 = \alpha^3\beta^4 = 1$; from there we obtain
$$\beta^2 = \alpha^{-4} = \alpha^{-1}\alpha^{-3} = \alpha^{-1}\beta^{4},$$
so $\alpha = \beta^{2}$. And th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
solution to a differential equation I have given the following differential equation:
$x'= - y$ and $y' = x$
How can I solve them?
Thanks for helping!
Greetings
| Let $\displaystyle X(t)= \binom{x(t)}{y(t)}$ so
$$ X' = \left( \begin{array}{ccc}
0 & -1 \\
1 & 0 \\
\end{array} \right)X .$$
This has solution $$ X(t)= \exp\biggr( \left( \begin{array}{ccc}
0 & -t \\
t & 0 \\
\end{array} \right) \biggr) X(0)= \left( \begin{array}{ccc}
0 & e^{-t} \\
e^t & 0 \\
\end{array} \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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If p then q misunderstanding? The statement $P\rightarrow Q$ means: if $P$ then $Q$.
p | q | p->q
_____________
T | F | F
F | F | T
T | T | T
F | T | T
Lets say: if I'm hungry $h$ - I'm eating $e$.
p | q | p->q
_______________________
h | not(e) | F
not(h) | not(e) | T
h | e | T
not(h) ... | Rather than your example about food — which is not very good, there is a lot of people starving and not eating —, let consider a more mathematical one : if $n$ equals 2 then $n$ is even.
I — How to interpret the truth table ?
Fix $n$ an integer.
Let $p$ denote the assertion “$n = 2$”, and $q$ the assertion “$n$ is even... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Taking the derivative of $y = \dfrac{x}{2} + \dfrac {1}{4} \sin(2x)$ Again a simple problem that I can't seem to get the derivative of
I have $\frac{x}{2} + \frac{1}{4}\sin(2x)$
I am getting $\frac{x^2}{4} + \frac{4\sin(2x)}{16}$
This is all very wrong, and I do not know why.
| You deal with the sum of functions, $f(x) = \frac{x}{2}$ and $g(x)= \frac{1}{4} \sin(2 x)$. So you would use linearity of the derivative:
$$
\frac{d}{d x} \left( f(x) + g(x) \right) = \frac{d f(x)}{d x} + \frac{d g(x)}{d x}
$$
To evaluate these derivatives, you would use $\frac{d}{d x}\left( c f(x) \right) = c \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
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Can someone check my work on this integral? $$
\begin{align}
\int_0^{2\pi}\log|e^{i\theta} - 1|d\theta
&= \int_0^{2\pi}\log(1-\cos(\theta))d\theta \\
&= \int_0^{2\pi}\log(\cos(0) - \cos(\theta))\,d\theta\\
&= \int_0^{2\pi}\log\left(-2\sin\left(\frac{\theta}{2}\right)\sin\left(\frac{-\theta}{2}\right)\right)\,d\theta\... | You can use Jensen's Formula, I believe: http://mathworld.wolfram.com/JensensFormula.html
Edit: Jensen's formula seems to imply that your integral is zero...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/134902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Morphism between projective schemes induced by surjection of graded rings Ravi Vakil 9.2.B is "Suppose that $S \rightarrow R$ is a surjection of graded rings. Show that the induced morphism $\text{Proj }R \rightarrow \text{Proj }S$ is a closed embedding."
I don't even see how to prove that the morphism is affine. The ... | I think a good strategy could be to verify the statement locally, and then verify that the glueing is successful, as you said. Let us call $\phi:S\to R$ your surjective graded morphism, and $\phi^\ast:\textrm{Proj}\,\,R\to \textrm{Proj}\,\,S$ the corresponding morphism. Note that $$\textrm{Proj}\,\,R=\bigcup_{t\in S_1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/134964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
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How to show that if a matrix A is diagonalizable, then a similar matrix B is also diagonalizable? So a matrix $B$ is similar to $A$ if for some invertible $S$, $B=S^{-1}AS$. My idea was to start with saying that if $A$ is diagonalizable, that means $A={X_A}^{-1}\Lambda_A X_A$, where $X$ is the eigenvector matrix of $A$... | Hint: Substitute $A = X_A^{-1} \Lambda X_A$ into $B = S^{-1} A S$ and use the formula $D^{-1}C^{-1} = (CD)^{-1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/135020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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what's the ordinary derivative of the kronecker delta function? What's ordinary derivative of the kronecker delta function? I have used "ordinary" in order not to confuse the reader with the covariant derivative. I have tried the following:
$$\delta[x-n]=\frac{1}{2\pi}\int_{0}^{2\pi}e^{i(x-n)t}dt$$
but that doesn't w... | May be it is already too late, but I will answer. If I am wrong, please correct me.
Let's have a Kronecker delta via the Fourier transform getting a $Sinc$ function:
$$\delta_{k,0} = \frac{1}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}} e^{-\frac{i 2 \pi k x}{a}} \, dx = \frac{\sin (\pi k)}{\pi k}$$
This function looks like:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
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How to verify the following function is convex or not? Consider function
$$f(x)=\frac{x^{n_{1}}}{1-x}+\frac{(1-x)^{n_{2}}}{x},x\in(0,1)$$
where $n_{1}$ and $n_2$ are some fixed positive integers.
My question: Is $f(x)$ convex for any fixed $n_1$ and $n_2$?
The second derivation of function $f$ is very complex, so I wis... | In mathematics, a real-valued function defined on an interval is called convex (or convex downward or concave upward) if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Probability that a coin lands on tails an odd number of times when it is tossed $100$ times
A coin is tossed 100 times , Find the probability that tail occurs odd number of times!
I do not know the answer, but I tried this, that there are these $4$ possible outcomes in which tossing of a coin $100$ times can unfold. ... | There are only two possible outcomes: Either both heads and tails come out an even number of times, or they both come out an odd number of times. This is so because if heads came up $x$ times and tails came up $y$ times then $x+y=100$, and the even number 100 can't be the sum of an even and an odd number.
A good way to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
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Convergence to the stable law I am reading the book Kolmogorov A.N., Gnedenko B.V. Limit distributions for sums of independent random variables.
From the general theory there it is known that if $X_i$ are symmetric i.i.d r.v such that $P(|X_1|>x)=x^{-\alpha},\, x \geq 1$, then $(X_1+\ldots+X_n)n^{-1/\alpha}\to Y$, wh... | I yr integral make the change of variables $z = \frac y {n^{\frac 1 {\alpha}}}$. This brings a factor $\frac 1n$ out front. The write $cos(tz) = 1 + (cos(tz) -1)$. Integrate the 1 explicitly, and the integral invlving $cos(tz)-1$ converges because it is nice at zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/135401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Newton polygons This question is primarily to clear up some confusion I have about Newton polygons.
Consider the polynomial $x^4 + 5x^2 +25 \in \mathbb{Q}_{5}[x]$. I have to decide if this polynomial is irreducible over $\mathbb{Q}_{5}$.
So, I compute its Newton polygon. On doing this I find that the vertices of the p... | There is no vertex at $(2,1)$. In my opinion, the right way to think of a Newton Polygon of a polynomial is as a closed convex body in ${\mathbb{R}}^2$ with vertical sides on both right and left. A point $P$ is only a vertex if there's a line through it touching the polygon at only one point. So this polynomial defini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Find all connected 2-sheeted covering spaces of $S^1 \lor S^1$ This is exercise 1.3.10 in Hatcher's book "Algebraic Topology".
Find all the connected 2-sheeted and 3-sheeted covering spaces of $X=S^1 \lor S^1$, up to isomorphism of covering spaces without basepoints.
I need some start-help with this. I know there is ... | A covering space of $S^1 \lor S^1$ is just a certain kind of graph, with edges labeled by $a$'s and $b$'s, as shown in the full-page picture on pg. 58 of Hatcher's book.
Just try to draw all labeled graphs of this type with exactly two or three vertices. Several of these are already listed in parts (1) through (6) of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Using the definition of a concave function prove that $f(x)=4-x^2$ is concave (do not use derivative).
Let $D=[-2,2]$ and $f:D\rightarrow \mathbb{R}$ be $f(x)=4-x^2$. Sketch this function.Using the definition of a concave function prove that it is concave (do not use derivative).
Attempt:
$f(x)=4-x^2$ is a down-facin... | If you expand your inequality, and fiddle around you can end up with
$$
(\lambda u-\lambda v)^2\leq (\sqrt{\lambda}u-\sqrt{\lambda}v)^2.
$$
Without loss of generality, you may assume that $u\geq v$. This allows you to drop the squares. Another manipulation gives you something fairly obvious. Now, work your steps bac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Evaluate the $\sin$, $\cos$ and $\tan$ without using calculator?
Evaluate the $\sin$, $\cos$ and $\tan$ without using calculator?
$150$ degree
the right answer are $\frac{1}{2}$, $-\frac{\sqrt{3}}{2}$and $-\frac{1}{\sqrt{3}} $
$-315$ degree
the right answer are $\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{2}}$ and $1$.
| You can look up cos and sin on the unit circle.
The angles labelled above are those of the special right triangles 30-60-90 and 45-45-90. Note that -315 ≡ 45 (mod 360).
For tan, use the identity $\tan{\theta} = \frac{\sin{\theta}}{\cos \theta}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/135698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Classical contradiction in logic I am studying for my logic finals and I came across this question:
Prove $((E\implies F)\implies E)\implies E$
I don't understand how there is a classical contradiction in this proof.
Using natural deduction, could some one explain why there is a classical contradiction in this proof?... | This proof follows the OP's attempt at a proof. The main difference is that explosion (X) is used on line 5:
The classical contradictions are lines 4 and 8. According to Wikipedia,
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions.
For line 4, the two lines th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Combinatorics - An unproved slight generalization of a familiar combinatorial identity One of the most basic and famous combinatorial identites is that
$$\sum_{i=0}^n \binom{n}{i} = 2^n \; \forall n \in \mathbb{Z^+} \tag 1$$
There are several ways to make generalizations of $(1)$, one is that:
Rewrite $(1)$: $$\sum_{a_... | Induction using Vandermonde's identity
$$
\sum_{i_1+i_2=m}\binom{n_1}{i_1}\binom{n_2}{i_2}=\binom{n_1+n_2}{m}\tag{1}
$$
yields
$$
\sum_{i_1+i_2+\dots+i_k=m}\binom{n_1}{i_1}\binom{n_2}{i_2}\dots\binom{n_k}{i_k}=\binom{n_1+n_2+\dots+n_k}{m}\tag{2}
$$
| {
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"url": "https://math.stackexchange.com/questions/135813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Differential operator - what is $\frac{\partial f}{\partial x}x$? given $f$ - a smooth function, $f\colon\mathbb{R}^2\to \mathbb{R}$.
I have a differential operator that takes $f$ to $\frac{\partial f}{\partial x}x$, but I am unsure what this is.
If, for example, the operator tooked $f$ to $\frac{\partial f}{\partial ... | Let $D$ be a (first order) differential operator (i.e. a derivation) and let $h$ be a function. (In your example $h$ is the function $x$ and $D$ is $\frac{\partial}{\partial x}$). I claim that there is a differential operator $hD$ given by
$$
f \mapsto hDf.
$$
(This means multiply $h$ by $Df$ pointwise.) For this claim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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moment generating function of exponential distribution I have a question concerning the aforementioned topic :)
So, with $f_X(t)={\lambda} e^{-\lambda t}$, we get:
$$\phi_X(t)=\int_{0}^{\infty}e^{tX}\lambda e^{-\lambda X}dX =\lambda \int_{0}^{\infty}e^{(t-\lambda)X}dX =\lambda \frac{1}{t-\lambda}\left[ e^{(t-\lambda)X}... | You did nothing wrong. The moment generating function of $X$ simply isn't defined, as your work shows, for $t\ge\lambda$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/136009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Prove that $|e^{a}-e^{b}|<|a-b|$
Possible Duplicate:
$|e^a-e^b| \leq |a-b|$
Could someone help me through this problem?
Let a, b be two complex numbers in the left half-plane. Prove that $|e^{a}-e^{b}|<|a-b|$
| By mean value theorem,
$$ |e^a - e^b| \leqslant |a-b|\max_{x\in [a,b]} e^x $$
But $a$ and $b$ have a negative real part, and then all $x$ in $[a,b]$ also have a negative real part. Hence the $\max$ is less than one. And thus
$$ |e^a - e^b| <|a-b|. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/136075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Finding $\frac{dy}{dt}$ given a curve and the speed of the particle Today I was doing some practice problems for the AP Calculus BC exam and came across a question I was unable to solve.
In the xy-plane, a particle moves along the parabola $y=x^2-x$ with a constant speed of $2\sqrt{10}$ units per second. If $\frac{dx}... | As you noticed, $dy=(2x-1)dx$. The speed is constant, so that $\dot{x}^2+\dot{y}^2=40$. So you get the system $$\begin{cases} \dot{y}=(2x-1)\dot{x} \\\ \dot{x}^2+\dot{y}^2=40 \end{cases}.$$ By substitution, $\left[1+(2x-1)^2 \right]\dot{x}^2 = 40$, whence, at $x=2$, $\dot{x}^2=4$. Since $\dot{x}>0$, you find $\dot{x}=2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/136143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Wedge product $d(u\, dz)= \bar{\partial}u \wedge dz$. How to show that if $u \in C_0^\infty(\mathbb{C})$ then
$d(u\, dz)= \bar{\partial}u \wedge dz$.
Obrigado.
| Note that $$d(u\,dz)=du\wedge dz+(-1)^0u\,ddz=du\wedge dz=(\partial u+\bar{\partial} u)\wedge dz.$$
Since $$\partial u=\frac{\partial u}{\partial z}dz\hspace{2mm}\mbox{ and }\hspace{2mm}\bar{\partial} u=\frac{\partial u}{\partial \bar{z}}d\bar{z},$$
we have
$$d(u\,dz)=\frac{\partial u}{\partial z}dz\wedge dz+\frac{\par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/136202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to show that every connected graph has a spanning tree, working from the graph "down" I am confused about how to approach this. It says:
Show that every connected graph has a spanning tree. It's possible to
find a proof that starts with the graph and works "down" towards the
spanning tree.
I was told that a p... | Let G be a simple connected graph, if G has no cycle, then G is the spaning tree of itself. If G has cycles, remove one edge from each cycle,the resulting graph will be a spanning tree.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/136249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 3
} |
How to prove this equality $ t(1-t)^{-1}=\sum_{k\geq0} 2^k t^{2^k}(1+t^{2^k})^{-1}$?
Prove the equality $\quad t(1-t)^{-1}=\sum_{k\geq0} 2^k t^{2^k}(1+t^{2^k})^{-1}$.
I have just tried to use the Taylor's expansion of the left to prove it.But I failed.
I don't know how the $k$ and $2^k$ in the right occur. And this ... | This is all for $|t|<1$. Start with the geometric series
$$\frac{t}{1-t} = \sum_{n=1}^\infty t^n$$
On the right side, each term expands as a geometric series
$$\frac{2^k t^{2^k}}{1+t^{2^k}} = \sum_{j=1}^\infty 2^k (-1)^{j-1} t^{j 2^k}$$
If we add this up over all nonnegative integers $k$, for each integer $n$ you g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/136333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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What is so interesting about the zeroes of the Riemann $\zeta$ function? The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$
The Riemann hypothesis asserts that all the non-tri... | Here is a visual supplement to Eric's answer, based on this paper by Riesel and Göhl, and Mathematica code by Stan Wagon:
The animation demonstrates the eventual transformation from Riemann's famed approximation to the prime counting function
$$R(x)=\sum_{k=1}^\infty \frac{\mu(k)}{k} \mathrm{li}(\sqrt[k]{x})=1+\sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/136417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "72",
"answer_count": 5,
"answer_id": 2
} |
Find the area of a surface of revolution I'm a calculus II student and I'm completely stuck on one question:
Find the area of the surface generated by revolving the right-hand
loop of the lemniscate $r^2 = \cos2 θ$ about the vertical line through
the origin (y-axis).
Can anyone help me out?
Thanks in advance
| Note some useful relationships and identities:
$r^2 = x^2 + y^2$
${cos 2\theta} = cos^2\theta - sin^2\theta$
${sin \theta} = {y\over r} = {y\over{\sqrt{x^2 + y^2}}}$
${sin^2 \theta} = {y^2\over {x^2 + y^2}}$
These hint at the possibility of doing this in Cartesian coordinates.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/136486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q \implies \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $ Assume that $a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q$ are rational，prove $\sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q$,are rational.
I know that can be proved, would like to know that there is no easier... | [See here and here for an introduction to the proof. They are explicitly worked special cases]
As you surmised, induction works, employing our prior Lemma (case $\rm\:n = 2\:\!).\:$ Put $\rm\:K = \mathbb Q\:$ in
Theorem $\rm\ \sqrt{c_1}+\cdots+\!\sqrt{c_{n}} = k\in K\ \Rightarrow \sqrt{c_i}\in K\:$ for all $\rm i,\:$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/136556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
$\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
| Using $\text{AM} \ge \text{GM}$
$$ \frac{1 + \frac{1}{2} + \dots + \frac{1}{n}}{n} \ge \sqrt[n]{\frac{1}{n!}}$$
$$\sqrt[n]{n!} \ge \frac{n}{H_n}$$
where $H_n = 1 + \frac{1}{2} + \dots + \frac{1}{n} \le \log n+1$
Thus
$$\sqrt[n]{n!} \ge \frac{n}{\log n+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/136626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "58",
"answer_count": 12,
"answer_id": 4
} |
How to explain integrals and derivatives to a 10 years old kid? I have a sister that is interested in learning "what I do". I'm a 17 years old math loving person, but I don't know how to explain integrals and derivatives with some type of analogies.
I just want to explain it well so that in the future she could remembe... | This is a question for a long car journey - both speed and distance travelled are available, and the relationship between the two can be explored and sampled, and later plotted and examined. And the questions as to why both speed and distance are important can be asked etc
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/136664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
Reference request for "Hodge Theorem" I have been told about a theorem (it was called Hodge Theorem), which states the following isomorphism:
$H^q(X, E) \simeq Ker(\Delta^q).$
Where $X$ is a Kähler Manifold, $E$ an Hermitian vector bundle on it and $\Delta^q$ is the Laplacian acting on the space of $(0,q)$-forms $A^{0,... | There is a proof of Hodge theorem in John Roe's book, Elliptic Operators, topology, and asymptotic expansion of heat kernel. The proof is only two page long and very readable. However he only proved it for the classical Laplace operator, and the statement holds for any generalized Laplace operator. Another place you ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/136742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Cardinality of the power set of the set of all primes Please show me what I am doing wrong...
Given the set $P$ of all primes I can construct the set $Q$ being the power set of P.
Now let $q$ be an element in $Q$. ($q = \{p_1,p_2,p_3,\ldots\}$ where every $p_n$ is an element in $P$.)
Now I can map every $q$ to a number... | Many (most) of the elements $q$ have an infinite number of elements. Then you cannot form the product of those elements. You have shown that the set of finite subsets of the primes is countable, which is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/136799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
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