Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Lebesgue integration of simple functions Define $f : [0,1] \to \Bbb R$ by $f(x) := 0$ if $x$ is rational, and
$f(x) := d^2$ if $x$ is irrational, where $d$ is the first nonzero
digit in the decimal expansion of $x$. Show that $\int_{[0,1]} f d m
= 95/3$. Here $m$ is the Lebesgue measure
I know that $f$ is simple,... | If you want, you can think about this integral geometrically. The function you described is equal almost everywhere to the following step function (or should I say, function of infinite repeating narrower staircases; intervals are not drawn to scale):
As you can see in the picture, what we get for the total area unde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Proving that a positive derivative means the function is smaller "to the left" and larger "to the right" for certain values I was trying to prove that if $g$ is differentiable on an open interval $I$ with $a\in I$ and $g'(a)>0$ then we can find $x<a$ for which $g(x)<g(a)$ and $y>a$ for which $g(y)>g(a)$, I think I und... | You have $$-\frac{K}{2}<\frac{f(x)-f(a)}{x-a}-K<\frac{K}{2}.$$ Add $K$ to everything to see the difference quotient is positive for this range of $x.$ How could that quotient be positive to the right of $a$ unless the numerator is positive? Same to the left of $a.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1246981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$ Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable.
Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$
Consider $g(x):= f(x)-Acos(5x)-Bsin(5x)$ where $A$ and $B$ are chosen so ... | In your expression involving $\frac {25}2g^2+\frac 12 g'^2=C$ you can set $x=0$ to determine the constant (this is where the bit you have been given is important). Then you have the sum of two squares equal to [?].
Note: you get the first part - the values of $A$ and $B$ from the values of $f(0)$ and $f'(0)$ if these ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1247087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
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Expected number of cards drawn before drawing a $4$ or $5$ I'm working on the following problem:
Compute the number of expected cards drawn from a standard 52 card
deck (without replacement) until a $4$ or $5$ is drawn.
I tried to model it using a geometric distribution, but am running into problems since the proba... | The number of ways you can draw $n$ cards from $52$ is $P(n,52)=\frac{52!}{(52-n)!}$. Note that order matters here. The number of ways all of those are not $4,5$ will be $P(n,44)=\frac{44!}{(44-n)!}$. Here $P$ stands for permutations.
Now let $P(n)$ (not the same meaning of $P$) denote the probability that we fail the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1247170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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symplectic manifolds I know sometimes they use advanced methods to prove a given 4-manifold is not symplectic. for instance by Seiberg-Witten theory. But for a manifold to be symplectic we just need to check that there is an element in the second cohomology which is closed and when we cup product with itself gives us a... | On Mike Miller's suggestion, I've turned my comments into an answer:
From a more analytic perspective, you're asking for a global solution $\omega \in \Omega^2(M)$ to the differential equation $d\omega = 0$ on $M^{2n}$ which also satisfies $\omega^n \neq 0$. In general, I don't think it's easy to know when a differenti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1247247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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the lim of sum of sequence I have to calculate the following:
$\lim_{n \to \infty} (\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} )$
I managed to understand that it is
$\lim_{n \to \infty} (\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} ) = \lim_{n \to \infty} \sum_{i=1}^n\frac{n}... | Hint: $$\sum \frac{n}{n^2+i^2}=\frac{1}{n}\sum \frac{1}{1+(\frac{i}{n})^2}$$
then use calculus.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1247341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Prove that $\| f \| := \max_{x \in [0,1]} |f(x)|$ is a norm on the vector space $\mathcal{C}[0,1]$.
Let $\mathcal{C}[0,1]$ be the set of all continuous functions $f: [0,1] \to \mathbb{R}$.
Prove that $\| f \| := \max_{x \in [0,1]} |f(x)|$ is a norm on the vector space $\mathcal{C}[0,1]$.
I know a norm has to satisf... | For condition $(1)$ it is clear that $\|f\| \geq 0$ for all $f$. And: $$\|f\| = \sup_{x \in [0,1]}|f(x)| = 0 \implies |f(x)| = 0,\quad\forall\,x\in[0,1]\implies f(x) = 0,\quad\forall\,x\in[0,1].$$ For $(3)$, you take the supremum in the right order: $$|f(x)+g(x)|\leq |f(x)|+|g(x)| \leq \|f\|+\|g\|,\quad \forall\,x\in[0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1247485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I finally understand simple congruences. Now how to solve a quadratic congruence? Now that I have plain old congruences, $19x\equiv 4 \pmod {141}$ for example, I am trying to wrap my brain around quadratic ones. My textbook shows how to tackle the aforementioned congruences, but not quadratic.
$$15x^2 + 19x\equiv 5 \pm... | $0\equiv 4x^2\!+\!8x\!-\!5\equiv \overbrace{(2x)^2\!+\!4(2x)\!-\!5}^{\large X^2\ +\,\ 4\,X\,\ -\,\ 5\!\!\! }\equiv\overbrace{(2x\!+\!5)(2x\!-\!1)}^{\large (X\ +\ 5)\ (X\ -\ 1)}\,$ so $\,\begin{align} &2x\equiv\color{#c00}{-5}\equiv6\\ &2x\equiv\,\color{#c00} 1\,\equiv 12\end{align}\ $ so $\ x\equiv \ldots$
Or, complet... | {
"language": "en",
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"source": "stackexchange",
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Does this limit exist finitely? Find the value of the following (if it exists):
$$\lim_{n\to\infty}n\sin(2\pi en!)$$
Does it exist? I think that it doesn't exist but I can't prove it. Please help me.
| Here is a complete proof:
For given $n\geq2$ one has
$$e\cdot n!=n!\sum_{t=0}^\infty{1\over t!}=n!\left(\sum_{t=0}^n{1\over t!}+\sum_{t=n+1}^\infty{1\over t!}\right)=m_n+r_n$$
with $m_n\in{\mathbb Z}$ and
$${1\over n+1}<r_n={1\over n+1}+{1\over (n+1)(n+2)}+\ldots<{1\over n}+{1\over n^2}+\ldots<{1\over n-1}\ .$$
Since
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding $b$ such that $e^{5B_t - bt}$ is a martingale I have $X_t = e^{5B_t}$ and
Where $B_t$ is brownian motion at time $t$.
$M_t = X_t \cdot e^{-bt}$
I need to find a value for $b$ such that $M_t$ is a martingale.
I am encountering difficulty, however.
$$\mathbb{E}[ e^{5B_t}e^{-bt} | \mathcal{F}_s] \space \text{for}... | Let $f(t,x)$ be the function defined by $f(t,x) = e^{5x - bt}$. We observe that $\partial_t f(t,x) = -b f(t,x)$ and $\partial_x^2 f(t,x) = 25 f(t,x)$. For $(X_t)_{t \geq 0}$ to be a local martingale, we require that $\partial_t f(t,x) = -\frac{1}{2} \partial_x^2 f(t,x)$. This tell us that for $(X_t)_{t \geq 0}$ to be a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1247837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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solving a second order nonlinear pde I would like to solve the following PDE,
$$f_{y}^{2} = 2 f f_{yy}$$
where $f= f(x,y)$ is a real function of two variables $x,y$.
My solution
: derivative of $f_{y}^{2}$ with respect to $y$ is itself, so $f_{y}^{2}= c(x)e^{y}$ and
$$f_{y} = c(x) e^{\frac{y}{2}}$$
so,
$f(x,y) = 2 c(... | This is really an ODE, it suffices to make two arbitrary integration constants in the general solution arbitrary functions of $x$.
To solve this ODE, rewrite it as
$$\frac{f_{yy}}{f_y}=\frac12\frac{f_y}{f}\qquad \left(\ln f_y\right)_y=\frac12 \left(\ln f\right)_y,$$
which gives $\ln f_y=\frac12\ln f+ \mathrm{const}$ o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Radius of convergence of a derivative of a power series. The power series $\sum_{0}^{\infty}k_n(x-b)^n$ and $\sum_{1}^{\infty}nk_n(x-b)^{n-1}$ have the same radius of convergence, however would it be true to say that $\sum_{1}^{\infty}k_n(x-b)^n$ and $\sum_{1}^{\infty}nk_n(x-b)^{n-1}$ have the same radii of convergence... | No, it does not affect the radius of convergence. The convergence of a series does not change if a finite number of terms are omitted. That is
$$
\sum_{n=0}^\infty a_n\text{ converges if and only if }\sum_{n=N}^\infty a_n\text{ converges,}
$$
where $N$ is a positive integer.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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To find analytic function with given condition How to find all analytic function on the disc $\{z:|z-1|<1 \}$ with $f(1)=1$ and $f(z)=f(z^2)$ ?.
| For any $z = re^{i\theta}, r \gt 0$ in the disc, the sequence $z_n = r^{1 \over 2^n} e^{i \theta /2^n}$ will converge to $1$, and $f(z_n) = f(z_n^{2^n}) = f(z)$. Since $f$ is continuous, $f(z) = \lim_{n \rightarrow \infty} f(z_n) = f(\lim_{n \rightarrow \infty} z_n) = f(1) = 1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proofs involving endomorphisms on the space of polynomials Define endomorphisms $D$ and $E$ on the space of polynomials with rational co-efficients $ \mathbb{Q}[x] $ such that $ D(x^n)= nx^{n-1}, E(x^n) = \frac{1}{n+1}x^{n+1} $
We must show that $ DE = I $ but $ ED \neq I $
I attempted to write as a composition or as 2... | So the best way to see that $ED\neq I$ is to show that it is not injective, or that it has a nontrivial kernel.
Note: let $f(x) = 1$ be the constant polynomial. Then $D(f(x)) = 0$ (this should be part of the definition of $D$), so $ED(f(x)) = 0 \neq f(x)$. Thus, $ED\neq I$.
In terms of your other questions, pursue thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What are all the integral domains that are not division rings? A commutative division ring is an integral domain. But what are all the integral domains that are not division rings?
The examples I currently know are the following: $\mathbb{Z}$, $\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt 2]$, $\mathbb{Z}[\sqrt k]$ where $k$ is n... | *
*For any integral domain $D$, the polynomial ring $D[x_1, \ldots, x_k]$, $k > 0$.
*For any connected, open subset $U \subseteq \mathbb{C}$, the ring of holomorphic functions on $U$. (Note that the space of merely smooth functions on $U$ is not an integral domain.)
(Since all finite domains are fields, all example... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that $\int _E f=0$ for each subset $E $ of $\mathbb R $ of finite Lebesgue measure Let $ f : \mathbb R \rightarrow \mathbb R$ be a bounded Lebesgue measurable function such that $\int_a^b f =0$ for all real $a,b.$
Show that $\int _E f=0$ for each subset $E $ of $\mathbb R $ of finite Lebesgue
measure
Actually I a... | Hint: We have, $$f = f^+-f^-$$ where $f^+$ and $f^-$ denote the positive and negative part of $f$, respectively. By assumption, the ($\sigma$-finite) measures
$$\nu(dx) := f^+(x) \, dx \qquad \mu(dx) := f^-(x) \, dx$$
satisfy
$$\mu((a,b)) = \nu((a,b)).$$
Conclude from the uniqueness of measure theorem that $\mu = \nu$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\lim_{n\to \infty} n\cdot r^n=0$ where $(0\leq r <1)$ without using ratio test $\lim_{n\to \infty} n\cdot r^n=0$, where $0\leq r <1$, can be obtained by vanishing condition (considering $\sum^{\infty}_{n=1}n\cdot r^n$, which converges, using ratio test).
Is there a direct way of finding this limit without u... | We can prove this by contradiction. Suppose that the sequence does not go to zero, then there is an $\epsilon >0$ for which given any $N \in \mathbb{N}$ there is an integer $n > N$ for which $n r^n > \epsilon$. Therefore we have $$r > \epsilon^{1/n} \left( \frac1n \right)^{1/n}.$$
Since we can make $n$ as large as we l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find Functions That Can Be Inverted from Their Sums I have the following situation:$$
f_1(x_1) + f_1(x_2) + f_1(x_3) + \cdots + f_1(x_n) = c_1\\
f_2(x_1) + f_2(x_2) + f_2(x_3) + \cdots + f_2(x_n) = c_2\\
\vdots\\
f_n(x_1) + f_n(x_2) + f_n(x_3) + \cdots + f_n(x_n) = c_n
$$These formulae are evaluated at a particular... | The idea of setting $f_i(x)=x^i$ actually works quite well:
Observation 1: The $n$ sums $$c_i=\sum_{j=1}^n f_i(x_j)$$ with $1\leq i\leq n$ can be used to express all the elementary symmetric polynomials $$e_k(\vec{x})=\sum_{\substack{A\subseteq \{x_1,x_2,\ldots,x_n\} \\ |A|=k}}\prod_{x\in A}x$$ with $0\leq k\leq n$.
Ob... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find point of intersection between quadratic and exponential equation How can I find the point of intersection between a function like $2^x$ and $x^2$?
I know you have to equate them but I don't know what to do after that.
| The equation : $2^x=x^2$ has two obvious soutions for $x=2$ and $x=4$.
We can verify that for $x>4$ the two function $f(x)=2^x$ and $g(x)=x^2$ are such that and $f'>g'$ so they have no other intersections for $x>4$.
For $-1\le x \le 0$ we can see that $f(-1)=1/2$, $g(-1)=1$, $f(0)=1$ $g(0)0$, so, since the two functio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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find the value of $\lim_{x\to0}\frac{e-(1+x)^{\frac{1}{x}}}{x}$ find the value of $$\lim_{x\to0}\frac{e-(1+x)^{\frac{1}{x}}}{x}$$I use hospital law and can't find answer
| Without using l'Hospital (I tend to avoid it as much as possible, as it always looked like a heavy hammer to me):
$$
(1+x)^{1/x} = e^{\frac{1}{x}\ln(1+x)} = e^{\frac{1}{x}(x-\frac{x^2}{2}+o(x^2))} = e^{1-\frac{x}{2}+o(x))} = e(1+\frac{x}{2}+o(x)) = e-\frac{xe}{2}+o(x)
$$
using the Taylor expansions of $\ln(1+x)$ and $e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Basis for the vector space P2 I am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial
$
ax^2 + bx + c
$
with the matrix
$
A = \begin{bmatrix}
1,0,0 \\
0,1,0 \\
0,0,1 \\
\end{bmatrix}
$
and the vector
$
\begin{bmatrix}
1 \\
x \\
x^2 \\
\end{bmatrix}
$
what corresponds to $... | I think you need to be clear about what you mean by "representing" the polynomial. You can if you like make the assignments
$$
x^2 \;\; \to \;\; \left [ \begin{array}{c}
0\\
0\\
1\\
\end{array} \right ] \hspace{2pc} x \;\; \to \;\; \left [ \begin{array}{c}
0\\
1\\
0\\
\end{array} \right ] \hspace{2pc} 1 \;\; \to \;\; ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1248933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Compute the integral $\int_{0}^{\infty} \frac{(1 + x + x^2)}{(1+x^4)} dx $ with a residue on suitable contour. I believe that I could try to compute the same integral with limits from $-\infty$ to $\infty$ using residue on a half circle and then let the radius tend off to infinity, and once I have that value I should b... | We will use the contour $\gamma$ avoiding the positive real axis,
Change variables $x\mapsto x^{1/4}$ to get
$$
\int_0^\infty\frac{1+x+x^2}{1+x^4}\,\mathrm{d}x=\frac14\int_0^\infty\frac{x^{-3/4}+x^{-1/2}+x^{-1/4}}{1+x}\,\mathrm{d}x\tag{1}
$$
Then, as in this answer, handle the three integrals
$$
\int_0^\infty\frac{x^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to Find the Function of a Given Power Series? (Please see edit below; I originally asked how to find a power series expansion of a given function, but I now wanted to know how to do the reverse case.)
Can someone please explain how to find the power series expansion of the function in the title? Also, how would you... | $$x\left(\frac{1}{1-(x+x^3)}\right)$$ will do it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 5
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Is it possible for integer square roots to add up to another? I initially was wondering if it were possible for there to be three $x,y,z \in \mathbb{Q}$ and $\sqrt{x},\sqrt{y},\sqrt{z} \notin \mathbb{Q}$ such that $\sqrt{x} + \sqrt{y} = \sqrt{z}$. I had suspected not, but then I found $x = \dfrac{1}{2}, y = \dfrac{1}{2... | If you square both sides, you have $x + y + 2\sqrt{xy} = z$ which can be satisfied if and only if $\sqrt{xy}$ is an integer (it can't be a half integer). So, that's the condition, if $xy$ is a square then you're fine, otherwise, its unsolvable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 3,
"answer_id": 1
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Basic question about modular arithmetic applied to the divisor sum function $\sigma(n)$ when $n=5p$ While studying the divisor sum function $\sigma(n)$ (as the sum of the divisors of a number) I observed that the following expression seems to be true always (1):
$\forall\ n=5p, p\in\Bbb P,\ p\gt 5,\ if\ d(5p)=4\ then\... | Note that if $k=3$, then $\sigma(15) =1+3+5+15 = 24$ is congruent to $6$ modulo $9$. This is the only counterexample, as we shall show.
Note that the equation $6k \equiv 3, 6 \pmod{9}$ is equivalent to saying that $3$ divides $6k$, but $3^2$ does not. Clearly the former is true, and the latter is true for all $k \ne ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A math contest question related to Ramsey numbers In a group of 17 nations, any two nations are either mutual friends,
mutual enemies, or neutral to each other. Show that there is a
subgroup of 3 or more nations such that any two nations in the
subgroup share the same kind of relationship.
I am kind of guessing that yo... | You will indeed use the pigeonhole principle: if $n = km + 1$ objects are distributed among $m$ boxes, then the pigeonhole principle asserts that one of the box will contain at least $k + 1$ objects.
To expand a bit about my comment, you can even actually use the result for $6$ vertices an $2$ colors, to solve for $17$... | {
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"url": "https://math.stackexchange.com/questions/1249491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Techniques to solve nonlinear first-order ODEs I am trying to solve the following nonlinear ODE:
$$\frac{dy}{dx} = \frac{1}{x(ayx-b)},$$
where $a, b$ are constants and $a>0$. Moreover, you may assume that $b \neq 0$ if necessary.
This equation is part of a system of two differential equations I need to solve in order ... | What I should do is to rewrite the differential equation as $$\frac{dx}{dy} = x(ayx-b)$$ which looks slightly better. Now, changing variable $x=\frac 1z$, the equation write $$\frac{dz}{dy}-b z+ay=0$$which looks much better. It is easy now to get $$z=\frac{a (1+b y)}{b^2}+c_1 e^{b y}$$ $$x=\frac{b^2}{a(1+ b y)+ c_1 e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The $L^p(\mathbb R)$ norm is increasing as a function of $p$ (Update: It's false!) Update: This is false. See the answers for a counterexample.
Let $C\ge 1$ be a constant. Fix $f\in L^p(\mathbb R)$ for $p\ge C$. Show that $$p\rightarrow \left( \int |f|^p
\right)^{1/p}$$ is non-decreasing.
Comments: I'm posting this ... | This is not true: Consider the function $f(x)=\chi_{\mathbb{R}\setminus(-1,1)}(x)|x|^{-1/2}$. Then for $p\leq 2$ we have $\| f\|_{p}=\infty$ and for $p>2$ we get
$$
\| f\|_p = \left(\frac{4}{p-2}\right)^{1/p}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $x$ has order $5$. let $ x \in G$ such that $(a^{-1})*(x^2)*(a) = x^3$ for some self inverse $a.$ Prove that $x$ has order $5.$
I don't know how to start this proof. Seems really difficult.
| $a^{-1}x^2a=x^3 \implies a^{-1}x^4a=x^6 \implies a^{-1}x^6a=x^9$ but $x^6=x^3x^3=(a^{-1}x^2a)(a^{-1}x^2a)=a^{-1}x^4a \implies a^{-1}(a^{-1}x^4a)a=x^9 \implies a^{-1}a^{-1}x^4aa=x^9$ now using $a^2=a^{-2}=e$ we have $x^4=x^9$ so $x^5=e$ and because $5$ is prime $x=e$ or $x$ has order $5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1249730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Find area of rhombus Given the following rhombus, where points E and F divide the sides CD and BC respectively, AF = 13 and EF = 10
I think the length of the diagonal BD is two times EF = 20, but i got stuck from there.
| "Straightforward" algebraical solution, in which one don't have to think:
Consider a vector basis $AB$,$AD$ and $A$ to have coordinates $(0,0)$,
$AE=AB+BC/2$ , $EF=1/2AD - 1/2BC$, we're given
$AE^2=(AB+BC/2)^2=AB^2+(AB,BC)+BC^2/4=13^2$,
$EF^2=1/4AB^2 - 1/2(AB,BC) + 1/4 BC^2 = 10^2$.
These are linear equations over $AB^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How does $\frac{-1}{x^2}+2x=0$ become $2x^3-1=0$? Below is part of a solution to a critical points question. I'm just not sure how the equation on the left becomes the equation on the right. Could someone please show me the steps in-between? Thanks.
$$\frac{-1}{x^2}+2x=0 \implies 2x^3-1=0$$
| Multiply by $x^2$ both sides of the equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1250132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Fundamental Theorem of Calculus, application I want to derive the function $$F(x)=\int_a^{x^2}\sin^3t\,dt$$ with the fundamental theorem of calculus, but I dont know how to handle the $x^2$.
Maybe with subsitution I think
Fundamental theorem of calculus: Let $f:[a,b]\to\mathbb{R}$ a continuous function. Then the funct... | one form of the fundamental theorem of calculus is $$d\left(\int_a^b f(t) \, dt \right) = f(b)\, db - f(a) \, da \tag 1$$ if you keep the lower limit fixed at $a$ and the upper limit is a variable $x^2,$ then $(1)$ becomes
$$d\left(\int_a^{x^2} \sin^3 t\, dt \right) = \sin^3\left(x^2\right)\, d\left(x^2\right) - \sin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inconsistent Matrices I'm teaching myself Linear Algebra and am not sure how to approach this problem:
Let A be a 4×4 matrix, and let b and c be two vectors in R4. We are told that the system Ax = b is inconsistent. What can you say about the number of solutions of the system Ax = c?
Bretscher, Otto (2013-02-21). Lin... | The definition you quote is for a system of equations to be inconsistent. At the end you are talking about individual matrices being inconsistent. You can't make that leap. The reduced row echelon form of the augmented matrix could have a row that looks like the row you display. In this case the system of equations... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show $\mathbb{E}(X \mid Y,Z) = \mathbb{E}(X \mid Y)$ if $Z$ is independent of $X$ and $Y$
Let $X,Y,Z$ be random variables, $X$ integrable, $Z$ independent of $X$ and $Y$. Then we have $E[X\mid Y,Z]=E[X\mid Y]$. Why is only assuming $Z$ independent of $Y$ not enough.
I was able to verify this for random variables that... | Hints:
*
*The $\sigma$-algebra $\sigma(Y,Z)$ is generated by sets of the form $$\{Y \in A\} \cap \{Z \in B\}$$ for Borel sets $A,B \in \mathcal{B}(\mathbb{R})$.
*By step 1 and the definition of conditional expectation, it suffices to show that $$\int_{\{Y \in A\} \cap \{Z \in B\}} \mathbb{E}(X \mid Y) \, d\mathbb{P... | {
"language": "en",
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Borel $\sigma$-algebra definition question So I am studying measure theory and I have found myself struggling to fully understand the concept of the Borel $\sigma$-algebra in depth. We know that the Borel $\sigma$-algebra is the smallest $\sigma$-algebra containing all open sets. The part that I cannot clearly grasp is... | Lets consider $\Omega=\{1,2,3,4\}$
$\sigma(\{1\})$ is the smallest $\sigma$-algebra which contains $1$.
So we must take any other elements of $P(\Omega)$ such that the conditions for being a $\sigma$-algebra are fulfilled.
It does clearly contains $1$. Also $1^C=\{2,3,4\}$. And $\Omega,\emptyset$.
So we have $\sigma(\{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there another terminology to designate this? Let $R$ be a principal ideal domain.
Let $M$ be a finitely generated $R$-module. Then there exists a free $R$-submodule $F$ of $M$ such that $M=Tor(M)\oplus F$ and the ranks of such $F$'s are the same.
Lang, in his text, defines the rank of $F$ as the rank of $M$. But thi... | If you look careful at $M=t(M)\oplus F$ can notice that $F\simeq M/t(M)$. But finitely generated torsion-free modules over a PID are free, so $M/t(M)$ is free and thus has a (finite) rank. That's why Lang calls this the rank of $M$ which seems a reasonable choice. (Bourbaki also calls this the rank of $M$.) But as you ... | {
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Show that: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 +1 } )$ could someone Please give me some hint of how to do this question thanks
| Hint:
Assuming we already know these hyperbolic functions are invertible:
$$\sinh(\log(x+\sqrt{x^2+1})):=\frac12\left(e^{\log(x+\sqrt{x^2+1})}-e^{-\log(x+\sqrt{x^2+1})}\right)=$$
$$=\frac12\left(x+\sqrt{x^2+1}-\frac1{x+\sqrt{x^2+1}}\right)=\frac12\left(\frac{x^2+x^2+1+2x\sqrt{x^2+1}-1}{x+\sqrt{x^2+1}}\right)=$$
$$=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1250881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the ten's digit of $7^{7^{7^{7^7}}}$ What is the ten's digit of $\zeta=7^{7^{7^{7^7}}}$. I got this question while doing binomial theorem. I think that $7^4=2401$ and we only need $\zeta\pmod{100}$. All I could think of is already presented (thought it is nothing actually), would you help?
| Note that
$$7^2=49,\ 7^3=343,\ 7^4=2401,\ 7^5=168\color{red}{07}.$$
So, all we need is to find $7^{7^{7^{7}}}$ in mod $4$.
Now
$$7^{7^{7^{7}}}\equiv (-1)^{7^{7^{7}}}\equiv -1\equiv 3\pmod 4.$$
Hence, the answer we want is the same as the ten's digit of $343$, i.e. $\color{red}{4}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that markov chain is recurrent I have the following markov chain :
$S=\{0,1,2,3\}$
$p_{i,0} = q$ (if we are in one of the states $0,1,2,3$ we can return to $0$ with probability $q$)
$p_{i,i+1} = 1-q , i\in\{0,1,2\}$ (if we are in state $i$ we can move to state i+1 with probability $1-q$)
$p_{3,3} = 1-q$ (state ... | Define $\tau_{00} = \inf \{ n>0 : X_n = 0 | X_0=0 \}$ (with the usual convention $\inf \emptyset = +\infty$). Then either you stay at $0$ the first step, in which case $\tau_{00}=1$, or you leave $0$ in the first step, in which case $\tau_{00}=1+T$, where $T$ is geometric with parameter $q$. So $\mathbb{E}[\tau_{00}] \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What other types of distributivity are there? When I say ‘Distributivity,’ I mean the way a number $x$ can be ‘Put in to’ some other function or the like.
For example, to distribute $x$ into $\text{id}_y$, you simply have to multiply $y$ by $x$ to get $\text{id}_{xy}$ (normal distribution). At first, I thought that thi... | There is no real function such that $f(x+y) = xf(y)$
If this was true, you would have for all $x$,
$$x^2f(y) = xf(x+y) = f(x+x+y) = 2xf(y)$$
| {
"language": "en",
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Find the image of the unit circle under the transformation $f(z)=\frac{z+1}{2z+1}$. How Do I approach these questions? Find the image of the unit circle under the transformation $f(z)=\frac{z+1}{2z+1}$. How Do I approach these questions?
I tried writing $z$ as $e^{i \phi}$, but I didn't know how to continue from there.... | You know $|z| = 1$. Let $A = \frac{z+1}{2z+1}$.
$$(2z+1)A = z + 1 \\
z(2A-1) = 1 - A\\
z = \frac{1-A}{2A-1}\\
1 = \frac{|1-A|}{|2A-1|}\\
|1-A| = |1 - 2A|\\
|1-A| = 2 \left|\frac{1}{2} - A \right|$$
The set of all such $A$ is always a circle.
Next, find two polar opposite points on the circle; that gives you the centre ... | {
"language": "en",
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Is a pattern proof? Let's say I want a formula that takes any number and makes it into 170, and I come up with a formula that I think does it. If I plug 1 into it, 2 into it, 3 into it, etc. up to a pretty large number and it works, can I call the formula proven?
| Here's a very contrived example: $$f(n) = \Big\lfloor \frac{\pi(n)}{10^7} \Big\rfloor + 170,$$ where $\pi(x)$ is the prime counting function. You'd have to go above $n = 179424672$ to find a counterexample. Of course "$f(n) = 170$ always" is easy enough to disprove if you know that there are infinitely many prime numbe... | {
"language": "en",
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$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ $f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically?
Using integration by parts I got the form:
$\int_a^bg(x)f(x)-g... | In particular, putting $g(x)=(x-a)(b-x)f(x)$, we have $\int_a^b{(x-a)(b-x)f(x)^2dx}=0$. The integrand is non-negative in $[a,b]$, so $f = 0$ almost everywhere. As $f$ is continuous, $f$ must be identically zero.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I make a series expansion of $F(x) = \int_0^x \exp -{(t^2)}\ dt$? $$F(x) = \int_0^x \exp -{(t^2)}\ dt$$
We need to find the series expansion for $F(x)$.
I tried differentiating $F(x)$ but couldn't establish certain pattern so that Taylor series formation may help..
Kindly help !!
| You know the expansion for $\exp(t)$, hence the expansion for $\exp(-t^2/2)$. Plug in, integrate, and by happy: $$\exp(t) = \sum_{n \geq 0} \frac{t^n}{n!} \implies \exp\left(-\frac{t^2}{2}\right) = \sum_{n \geq 0}(-1)^n \frac{t^{2n}}{2^nn!},$$ so we get: $$F(x) = \sum_{n \geq 0}(-1)^n\frac{x^{2n+1}}{(2n+1)2^nn!}.$$
Edi... | {
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How to find a function that is the upper bound of this sum? The Problem
Consider the recurrence
$ T(n) =
\begin{cases}
c & \text{if $n$ is 1} \\
T(\lfloor(n/2)\rfloor) + T(\lfloor(n/4)\rfloor) + 4n, & \text{if $n$ is > 1}
\end{cases}$
A. Express the cost of all levels of the recursion tree as a sum over the cost of e... | Suppose we start by solving the following recurrence:
$$T(n) = T(\lfloor n/2 \rfloor) + T(\lfloor n/4 \rfloor) + 4n$$
where $T(1) = c$ and $T(0) = 0.$
Now let $$n = \sum_{k=0}^{\lfloor \log_2 n \rfloor} d_k 2^k$$
be the binary representation of $n.$
We unroll the recursion to obtain an exact formula for $n\ge 2$
$$T(n)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What the's contradiction in showing the regular representation is indecomposable in characteristic $p$? Suppose $G$ is a nontrivial $p$ group, and $F$ is a field of characteristic $p$. The group ring $FG$ is a module over itself affording the regular representation $g\cdot g_i=gg_i$.
Why is $FG$ indecomposable? I read... | The contradiction is because in the first case $FG$ has two district trivial submodules, whereas in fact there is a unique trivial submodule spanned by the sum you mention.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1251932",
"timestamp": "2023-03-29T00:00:00",
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Proving $ \neg ( \neg \alpha \wedge \neg \neg \alpha )$ I'm training to prove this statement , but first I need to know if this statement can be proved in :
1 - both in classical and Intuitionistic logic ( in this case i need to provide demonstration in Intuitionistic logic )
2 - classical logic but not Intuitionist... | This statement can be proved in minimal logic. When you rewrite the negations as implications in the usual way, the statement is
$$
((\alpha \to \bot) \land ((\alpha \to \bot) \to \bot) \to \bot
$$
which is really of the form
$$
(X \land X \to Y) \to Y
$$
which is just a form of modus ponens. The provability of the sta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Epsilon and Delta proof of $\lim_{x\to0} \frac{2-\sqrt{4-x}}{ x}$ I need to prove $\lim_{x\to0} \frac{2-\sqrt{4-x}}{ x}$
I first found the limit to be $\frac{1}{4}$ by using l'hopital's rule.
By definition i need to find a $\delta > 0$ for every $\epsilon >0$
Then i will have $|x-0|<\delta$ and
$$|\frac{2-\sqrt{4-x}}... | It gets much simple when you write
$$
\frac{2−\sqrt{4−x}}x = \frac{(2-\sqrt{4−x})(2+\sqrt{4−x})}{x(2+\sqrt{4−x})}
=\frac1{2+\sqrt{4−x}}
$$
then:
\begin{align}
\left|
\frac1{2+\sqrt{4−x}} - \frac 14
\right|
&= \frac{|2 - \sqrt{4-x}|}{4(2+\sqrt{4−x})}
\le \frac{|2 - \sqrt{4-x}|}8
\\ &=
\frac{|2 - \sqrt{4-x}... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluate the limit $\displaystyle\lim_{x \to 0}\frac{(e-\left(1 + x\right)^{1/x})}{\tan x}$. How to evaluate the following limit
$$\displaystyle\lim_{x\to 0} \dfrac{e-\left(1 + x\right)^{1/x}}{\tan x}$$
I have tried to solve it using L-Hospital's Rule, but it creates utter mess.
Thanks for your generous help in advance... | This is not possible to do without the use of differentiation (i.e. L'Hospital or Taylor) or some amount of integration to obtain the inequalities for logarithm function. The best approach seems to be L'Hospital's Rule.
We proceed as follows
\begin{align}
L &= \lim_{x \to 0}\frac{e - (1 + x)^{1/x}}{\tan x}\notag\\
&= \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simple generator modules Let a ring $R$ with identity element be such that the category of left $R$-modules has a simple generator $T$.
My question: "Is $T$ isomorphic with any simple left $R$-module $M$?"
I tried the meaning of generation by $T$, i.e. there exists an $R$-epimorphism $f$ from a direct sum $⊕T$ to $M$, ... | If I understand your question correctly, then the question is whether it follows that all simple modules are isomorphic provided that one has a simple module which is a generator.
I think the answer is yes:
Suppose $S$ is another simple module. Then there is an epi $T^{(I)}\twoheadrightarrow S$.
However, one has the f... | {
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How find $\max _{z: \ |z|=1} \ f \left( z \right)$ for $f \left( z \right) = |z^3 - z +2|$ Let $f : C \mapsto R $, $f \left( z \right) = |z^3 - z +2|$. How find $\max _{z: \ |z|=1} \ f \left( z \right)$ ?
| i am going to parametrize the unit circle by $z = \cos t + i \sin t.$ we have $$\begin{align}|z^3 - z + 2|^2 &= (\cos 3t - \cos t + 2)^2 +(\sin 3t - \sin t)^2 \\&=\cos^2 3t + \cos^2t+4-2\cos 3t \cos t+4\cos 3t-4\cos t \\ &+\sin^2 3t + \sin^2 t-2\sin 3t \sin t\\
&=6-2\cos 2t+4\cos 3t-4 \cos t\end{align}$$
the critical ... | {
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$\int_a^b f(x) g'(x) dx = 0$ implies $f$ is constant Given $f$ is continuous on $[a,b]$, $\forall g$ which is a continuously differentiable function on $[a,b]$, with $g(a)=g(b)=0$, the following equation is satisfied:
$\int_a^b f(x) g'(x) dx = 0$.
I want to show that $f$ is a constant.
This is a question similar to thi... | Let $I=\int_a^b f(s)ds$.
Let also $$g(x) = \int_a^x \left(f(s) - \frac{I}{b-a}\right)ds.$$
Clearly, $g(a)=g(b)=0$. On top of that $g'(x) = f(x)-\frac{I}{b-a}$.
$$\int_a^b f(x)g'(x)dx = \int_a^bf^2(x)dx-\frac{I^2}{b-a},$$or
$$\int_a^bf^2(x)dx = \frac{1}{b-a}\left(\int_a^bf(x)dx\right)^2.$$
By the Cauchy-Schwarz inequali... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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probability over 3 values with dependency At the exercise, there is no information that B and C are independent, but with logical reasoning, there must be a pendency.
The problem is, I can not create a connection with depency of B and C, is that even possible and if yes, how and I calculate this?
My current approach:
... | Assuming this is a Bayesian network problem, events $B,C$ are dependent, but given $A$, are conditionally independent. So you can find $P(A\mid \bar B\cap C)$ as follows:
\begin{eqnarray*}
P(A\mid \bar B\cap C) &=& \dfrac{P(A\cap\bar B\cap C)}{P(\bar B\cap C)} \\
&=& \dfrac{P(A\cap\bar B\cap C)}{P(A\cap\bar B\cap C) + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Area of an equilateral triangle
Prove that if triangle $\triangle RST$ is equilateral, then the area of $\triangle RST$ is $\sqrt{\frac34}$ times the square of the length of a side.
My thoughts:
Let $s$ be the length of $RT$. Then $\frac s2$ is half the length of $\overline{RT}$. Construct the altitude from the $S$ t... | From AoPS wiki,
Method 1: Dropping the altitude of our triangle splits it into two triangles. By HL congruence, these are congruent, so the "short side" is $\frac{s}{2}$. Using the Pythagorean theorem, we get $s^{2}=h^{2}+\frac{s^{2}}{4}$, where $h$ is the height of the triangle. Solving, $h=\frac{s \sqrt{3}}{2}$. (no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Are elements of a $C^*$-Algebra strictly positive iff their spectrum is strictly positive? Let $A$ be a $C^*$-Algebra. An element $a\in A$ is said to be positive iff $a=a^*$ and the spectrum $\sigma(a)$ is nonnegative, ie. $\sigma(a)\subset[0,\infty)$. This is equivalent to $\varphi(a)\ge 0$ for all positive linear fun... | As you note, the notion of strictly positive is irrelevant on unital C$^*$-algebras. But it is a different notion in the non-unital case. Consider the algebra of compact operators $K(H)$ on a separable Hilbert space. It has a strictly positive element, because every separable C$^*$-algebra has one; but no compact opera... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Probability of a pair of red and a pair of white socks among five chosen
In the box are $7$ white socks, $5$ red socks and $3$ black socks.
$2$ socks are considered a pair if they have the same color.
$5$ arbitrary socks are selected at random from the box.
Find the probability that among the selected socks there are ... | Your answer is correct if the question is to be interpreted literally. It would include situations where the fifth sock is also perhaps red or white. If the questioner intended that there should be precisely two red socks and two white socks, with the third therefore being black, then replace the 11 in the numerator wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1252978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Proving convergence/divergence via the ratio test Consider the series
$$\sum\limits_{k=1}^\infty \frac{-3^k\cdot k!}{k^k}$$
Using the ratio test, the expression $\frac{|a_{k+1}|}{|a_k|}$ is calculated as:
$$\frac{3^{k+1}\cdot (k+1)!}{(k+1)^{k+1}}\cdot \frac{k^k}{3^k\cdot k!}=\frac{3}{(k+1)^{k}}\cdot {k^k}=3\cdot \frac{... | Seems you have problem on $\lim_{k\rightarrow \infty}\frac{k}{k+1}^k$
Let
$$y=\lim_{k\rightarrow \infty}\frac{k}{k+1}^k$$
Take $\ln$ on both sides,
$$\ln y = \lim_{k\rightarrow \infty} \frac{\ln k - \ln (k+1)}{k^{-1}}$$
Use L'Hopital's rule,
$$\lim_{k\rightarrow \infty} \frac{\ln k - \ln (k+1)}{k^{-1}} = \lim_{k\righta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
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Kronecker's 1870 paper on finite Abelian Groups?? Could anyone please provide me with the exact bibliographic reference for Kronecker's 1870 work on finite Abelian groups? If you could provide me with his exact formulation (or even with a acanned copy of it from the book) it would be great.
Thnak in advance.
| L. Kronecker, Auseinandersetzung einiger Eigenschaften der Klassenzahl idealer complexer Zahlen, Monatsbericht der Königlich-Preussischen Akademie der Wissenschaften zu Berlin, 1870, pp. 881-889. The entire volume is freely available at Archive.org in a variety of formats. You can read it online or download PDFs of ind... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove that any non-zero-divisor of a finite dimensional algebra has an inverse Let $A$ be a finite dimensional algebra. Prove that an element of $A$ is invertible iff it is not a zero divisor.
Let $a$ be an invertible element, then there exists an element $b$ such that $ab=1$ and assyme that $a$ is a zero divisor, then... | In any ring, invertible elements (units) cannot be zero divisors; for let $a$ be invertible in the ring $R$; then we have $b \in R$ such that $ab = ba = 1$. If $a$ is a zero divisor, then we have either $ac = 0$ or $ca = 0$ for some $0 \ne c \in R$. But $ac = 0$ implies
$c = 1c = (ba)c = b(ac) =0, \tag{1}$
likewise, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 3
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How do I find the radius and interval of convergence of $\sum_{n=1}^\infty {(-1)^n(x+2)^n \over n} $ $$\sum_{n=1}^\infty {(-1)^n(x+2)^n \over n} $$
I used the ratio test to test for absolute convergence, but I'm sort of stuck on:
$$n(x+2) \over n+1$$
| Hint, the ratio needs to be such that
$$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|<1 $$
So, what values of $x$ will satisfy this condition after the limit has been taken?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Real analysis: simple second order ODE I'm studying real analysis at the moment (just covered the mean value theorem, constancy theorem, applications to DEs etc.) and have run across this question that I'm stuck on. Any help would be much appreciated - I have no idea where to start!
"$f$ is an infinitely differentiabl... | This can be done without assuming a certain solution (which would have to be proven as unique first).
Consider $v=y'$ and write
$$ \frac{\mathrm{d}v}{\mathrm{d}x} =-y \\
\frac{\mathrm{d}v}{\mathrm{d}y} v =-y \\
\int v \mathrm{d}v = -\int y \mathrm{d}y \\
\frac{1}{2} (y')^2 = \frac{1}{2} y^2 + \frac{C}{2} \\
y' = \sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically I am working through the exercises and have spent half a day on one problem so I decided to get some help because I can't figure it out.
Show that if $n$ is a positive integer at most equal to $m$, then
$$
\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}... | Proof using formula:
$$
{{m\choose n} := {m(m-1)\ldots(m-n+1)\over n!}}
$$
$$
\eqalign{{m\choose n}+{m\choose n-1}&=
{m(m-1)\ldots(m-n+1)\over n!}+{m(m-1)\ldots(m-(n-1)+1)\over (n-1)!}\cr
&={m(m-1)\ldots(m-n+2)\cdot(m-n+1)\over n!}+{m(m-1)\ldots(m-n+2)\over (n-1)!} \cdot {n \over n}\cr
&={m(m-1)\ldots(m-n+2)\cdot[(m-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Eggs in a Basket (Remainders) I'm working on a problem:
A woman has a basket of eggs and she drops them all. All she knows is that when she puts them in groups of 2, 3, 4, 5, and 6, there is one left over. When she puts them into groups of 7, there are none left over. What is the minimum number of eggs she could have i... | The first half is a constant-case optimization of CRT: $\ 2,3,4,5,6\mid x\!-\!1\!\iff\! 60\mid x\!-\!1,\,$ since $\,{\rm lcm}(2,3,4,5,6)=60.\,$ So $\, x = 1\!+\!60t.\,$ Further $\, x\equiv 0\pmod 7\ $ so, by Easy CRT
$\qquad\qquad\qquad\! {\rm mod}\ 7\!:\ 0 \equiv x \equiv 1\!+\!60\color{}t\equiv 1\!-\!3t\iff 3t\equiv1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $\nabla \cdot (f\textbf r)$ and $\nabla \times (f\textbf r)$ of the function $f(x,y,z) = (x^2+y^2)\log(1-z)$ I have been given the function $f(x,y,z) = (x^2+y^2)\log(1-z)$ and I need to find the divergence $\nabla \cdot (f\textbf r)$ and curl $\nabla \times (f\textbf r)$ where $\textbf r$ is the position vector.
I... | You multiply the function by the position vector and then just calculate the derivatives...
[\begin{array}{l}
\vec r = \left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right)\\
f(x,y,z) = \left( {{x^2} + {y^2}} \right)\log (1 - z) = {x^2}\log (1 - z) + {y^2}\log (1 - z)\\
f(x,y,z)\vec r = \vec F = \left( {\begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Determining whether $f(z)=\ln r + i\theta$ (with domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$) is analytic
Define $$f(z)=\ln r + i\theta$$ on the domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$.
This domain is just a punctured disk of radius $\ln r$, correct?
How does one determine whether this is analytic, I can't see how ... | The domain is the disk of radius $r$ with the closed positive real half-axis removed; in particular this is not a punctured disk.
A good way to check analyticity of $f$ is to write the C.-R. equations in polar coordinates. (The alternative is to write $f$ in rectangular coordinates, but it is slightly tricky to handle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How does the existence of Platonic graphs imply the existence of Platonic solids? I will use the following definitions
Platonic graph: A 3-connected planar graph with faces bounded by the same number of edges and vertices having the same number of incident edges.
(remark: the faces of a 3-connected planar graph are we... | Consider the docecahedron as an example: Between a minuscule regular pentagon on $S^2$ having angles slightly over $108^\circ$ at the vertices and a hemispherical regular pentagon with angles $180^\circ$ at the vertices there is a regular pentagon on $S^2$ with angles $120^\circ$ at the vertices. Twelve such pentagons ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Bourbaki and the symbol for set inclusion Which notation ($\subset$ or $\subseteq$) was preferred by Bourbaki for set inclusion (not proper)?
A side question: Was the notation for subset one of the many notations invented by Bourbaki?
| See :
*
*Nicolas Bourbaki, Théorie des ensembles (2nd ed 1970) :
Définition 1 (L'inclusion). La relation désignée par $(\forall z)((z \in x) \implies (z \in y))$ dans laquelle ne figurent que les lettres $x$ et $y$, se note de l'une quelconque des manières suivantes : $x \subset y, y \supset x$,
« $x$ est cont... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1253991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does $\tan x\cdot \cos x$ equal $\sin x$? Is it true that $\tan x\cdot \cos x = \sin x$? If I put $x=30$ in my calculator then I don't get the same answer as $\sin 30$, why is this? Don't the two cosines cancel out? I'm probably missing something really stupid here.
| As tan(x)≡ Sin(x)/Cos(x), you are right in that Tan(x) * cos(x) ≡ Sin(x). This is by definition of the Tan function, which is defined as Sin(x) / Cos(x). If it helps consider the right angle triangle from the unit circle, where cos(x) = Hypotenuse / adjacent and Sin(x) = opposite / hypotenuse, so Tan(x) as equalling o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1254100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Mean age among employees in a company. In a company there are 32 men and 59 women. Male mean age is 48.5 and female 39.2. One of the women (47 years old) ended working at the company and was replaced with a 23 year old man.
Calculate the employees average mean age.
32 men + 59 women = 91 employees.
48.5+39.2=87.7 (ag... | First of all, an important thing is that it is impossible to know the answer unless you know the age of the woman that stopped working.
The total age is not $48.5 + 39.2$. That is just the sum of two ages. In fact, $48.5$ is the mean age of all male employees, i.e. before the replacement, there are $32$ men with an av... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1254191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $ i \equiv j \pmod{p-1}$ and $p\nmid n$ then $n^j \equiv n^i \pmod p$
Let $p$ be prime. Show that $ i \equiv j \pmod{p-1}$ and $p\nmid n$ then $n^j \equiv n^i \pmod p$
I know that $i$ and $j$ have the same remainder when divided by $p-1$, and that's pretty much it. Any hints?
| Write $i = k(p-1) + j$ for some $k \in \mathbb{Z}$. Then $n^i = n^{k(p-1) + j} = (n^{p-1})^{k}\cdot n^j$. Using Fermat's little theorem, $(n^{p-1})^{k} \equiv 1$ (mod $p$), so $n^i \equiv n^j$ (mod $p$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1254284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $T\colon V\to V$ be a linear transformation such that $\dim(V)=n<\infty$. Prove that $T$ is bijection >iff T is injective.
Let $T\colon V\to V$ be a linear transformation such that $\dim(V)=n<\infty$. Prove that
(a)$T$ is bijection
iff
(b)T is injective.
Solution:
*
*show $(a)\implies(b)$
If $T$ is bijection, t... | I think it is essentially correct, but I would write things a bit differently. You are right on using rank-nullity. But then I would say that the only vector $z$ with $T(z)=0$ is the zero vector. Then the kernel is 0-dimensional, i.e., nullity is $0$, so rank is $n$, and then $T(V)$ is an n-dimensional subspace of $V$,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to solve equation $ x=W(a+bx^{n})+1 $? How i can resolve the equation $x=W(a+b x^n)+1$, where $W$ is the Lambert $W$ function?
thanks
| I don't think it will be easy to solve. Since $x = W(y)$ if and only if $y \ge - \dfrac 1e$ and $y = xe^{x}$, you will have to solve $$a+bx^n = (x-1)e^{x-1},\quad a + bx^n \ge - \dfrac 1e.$$
Are you looking for an exact solution?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1254482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding all solutions: $a^2 + b^2 = c^2 + d^2$ I want to find all solutions to the problem of two squares equaling two other squares. $$a^2 + b^2 = c^2 + d^2 \qquad b \le N$$Clearly, without loss of generality, I can assume that
$$gcd(a,b,c,d) = 1$$
and $c\le a \le b \le d$. But after that, I'm a bit stuck. I can see a... | This is easily solved by rearranging and factoring:
\begin{align}
a^2-c^2 &= d^2-b^2 \\
(a-c)(a+c) &= (d-b)(d+b).
\end{align}
If one of the factors is zero, than we have a trivial solution; one of the factors on the other side must also be zero, etc.
Assuming $a \ne \pm c$, we can find $g=\gcd(a-c,d-b)\ge 1$, say $a-c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1254603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is $ (-1)^n(x-a)^n = (a-x)^n?$ If not, why? I came across this during an attempt at a Taylor series expansion (which I'm not very good at yet), and assumed this would be true because $(ab)^n = a^nb^n$. Plugged it into Wolfram Alpha, though, and it returns false. Can't figure out why this might be.
| Wolfram is probably assuming $n$ is potentially complex, in which case it gets trickier - it is not true in general that $a^nb^n=(ab)^n$ if $n$ is complex. Indeed, it is already problematic with $n$ rational, say.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1254684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the slope of the line that goes through the given points I know the formula for this type of problem is the second y coordinate subtracted from the first y coordinate over the second x coordinate subtracted from the first x coordinate but for the numbers given to me for this problem (-9, - 5) and (-7, 5) it says t... | The problem is you aren't subtracting in the denominator.
As you said, the slope of the line passing through $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is $\dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}$.
You have the points $(-9, -5)$ and $(-7,5)$.
You have been calculating this as $\dfrac{-5 - 5}{-9 + -7} = \dfrac{-10}{-16}$. Notice... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1254832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Equivalence of category of subsets and subobjects I'm trying to show that the categories $\mathcal{P}(X)$ and $Sub(X)$ are equivalent.
According to Steve Awodey's "Category Theory" I need to find two functors
*
*$ E: \mathcal{P}(X) \to Sub(X)$
*$F: Sub(X) \to \mathcal{P}(X)$
and a pair of natural isomorphisms
*... | Well, what are the arrows in $P(X)$?
I assume it is the poset category, ie. the arrows are just the inclusions $Y_1\subseteq Y_2$.
Basically $E$ is the identity, also on arrows: for $Y_1\subseteq Y_2\subseteq X$, $\ E$ maps it to the identical inclusions $Y_1\hookrightarrow Y_2\hookrightarrow X$.
On the other hand, f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1254936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determine whether the set $\{v_1 + v_2 - v_3, 2v_1 + 2v_3, -v_1 + v_2 - 3v_3\}$ is linearly dependent or independent. We had a question on our last test that was very similar to this and I only got $2$ points of $6$ and I want to make sure I do it right this time. Here's my solution to that one:
Let $v_1, v_2,$ and $v... | i think you can explain what you are doing in words. you have not made clear that you understand what linear dependence of vectors mean. true, you are showing some row reduction, but why and how does it relate to the question.
one way to do is to explain how you determine linear independence by supposing $$a(v_1+v_2-v_... | {
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"timestamp": "2023-03-29T00:00:00",
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How is an infinitesimal $dx$ different from $\Delta x\,$? When I learned calc, I was always taught
$$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$
But I have heard $dx$ is called an infinitesimal and I don't know what this means. In particular, I gather the validity of treating a ratio of ... | The term "infinitesimal" was used by Leibniz. This was at a time before the concept of limits, as we know it today. He still thought of $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ as a quotient with $\mathrm{d}y$ & $\mathrm{d}x$ being very small.
Today $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ is not a quotient but is notation for the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1255294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Show that the series is absolutely convergent The series is
$$\sum^\infty_{n=2} \frac{(-1)^n}{n(\ln(n))^3}$$
I tried the ratio test which did not do anything. I also tried the root test which gave me
$$\frac{-1}{\sqrt[n]{n}\cdot (\ln(n)^3-n)}$$
which I don't think is right. Is their another test I can do to confirm t... | $\textbf{Integral Test}$
$$\sum_{n=2}^{\infty} \frac{1}{n (\ln(n))^3} \ \ \ \text{converges} \iff \lim_{R\to \infty} \int_{2}^{R} \frac{1}{x(\ln(x))^3}\ \text{dx}\ \ \text{converges}$$
Let $u =\ln(x), du = \frac{1}{x} \ \text{dx}\ $ ... (go from here!)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1255379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Correction factor for the variance In an exercise they asked me: "Why could we use the following correction factor? $\text{varianceX} = \frac{n-1}{n}*\text{varianceY}$
What I said was basically, because the unbiased sample variance have a factor of $\frac{n}{n-1}$, we could multiply the variance by $\frac{n-1}{n}$ to c... | Well, after reading a lot on wikipedia I think this is the answer:
In calculating the expected value of the sample variance a factor equal to $\frac{n-1}{n}$ is obtained, which underestimates the expected variance so usually when calculating the sample variance we multiply it by the factor $\frac{n}{n-1}$ which is gene... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1255469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Help with discrete math proof I'm having trouble with the following:
$\ a_1=1$ and $a_n=1+\sum_{i=1}^{n-1} a_i$ for $n>1$
How should I go about proving the below? Any hints?
$a_n = 2^{n-1}$
| Induction is the obvious choice in these kinds of problems.
*
*For $n=1$, the statement is obviously true.
*Now, we should prove the statement for $n+1$ by assuming it is true for all values $k\leq n$. That is, we know that $a_k = 2^{k-1}$ for all values $k\leq n$. Then, we have $$a_{n+1} =1 + \sum_{i=1}^n a_i = 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1255587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Do finite products commute with colimits in the category of spaces? Let $X$ be a topological space. The endofunctor $\_\times X$ of the category of all topological spaces does in general not possess a right adjoint, since the category is not cartesian closed.
*
*Is it nevertheless true in general, that $colim(A_i)\t... | Here is a counterexample for the directed limit:
Let $X_n$ be a wedge of $n$ circles $C_1,\dots,C_n$. We have a sequence of closed inclusions $X_1\hookrightarrow X_2\hookrightarrow\dots$ whose colimit is $X$, a wedge of countably many circles. There is a continuous bijection $j:\text{colim}(X_n\times\Bbb Q)\to X\times\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1255678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Help in evaluating limit of the given function.
Evaluate the limit $$\lim_{x\to \infty} \left(\cfrac{x^2+5x+3}{x^2+x+2}\right)^x $$
I'm not sure how to evaluate this limit. This is what I've done yet:
$$\begin{align}
\lim_{x\to \infty} \left\{1 + \left(\cfrac{x^2+5x+3}{x^2+x+2} - 1\right) \right\} ^x\\
=\ \lim_{x\... | $$\begin{align}\lim_{x\to\infty}\left(\frac{x^2+5x+3}{x^2+x+2}\right)^x&=\lim_{x\to\infty}\left(1+\frac{1}{\frac{x^2+x+2}{4x+1}}\right)^{\frac{x^2+x+2}{4x+1}\cdot\frac{x(4x+1)}{x^2+x+2}}\\&=\lim_{x\to\infty}\left(\left(1+\frac{1}{\frac{x^2+x+2}{4x+1}}\right)^{\frac{x^2+x+2}{4x+1}}\right)^{\frac{4+\frac 1x}{1+\frac 1x+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1255779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
To mark up in retail by $20$%, do I add $0.20$ times the original cost, or divide by $0.80$? Why is it that when I take a cost of say $\$15.60$ and want to mark the item up at retail 20% that I'm being told two different ways with two different answers?
The first way (my way) would be to multiply the original cost by ... | Simply because adding $0.2x$ to $x$, giving you $1.2x$, is not the same thing as $x/0.8$. This is because
$$1/0.8 = 1.25$$
You're not increasing by the same factor, the second option would be $1.25x$ instead of the very correct $1.2x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1255964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$1+\frac{1}{2} +\frac{1}{3} +...+\frac{1}{p-1} =\frac{a}{b}$ Let $p\gt 3$, be a prime number and $1+\frac{1}{2} +\frac{1}{3} +...+\frac{1}{p-1} =\frac{a}{b}$ when $a,b\in \mathbb N$ and $gcd(a,b)=1$.
prove that $p^2|a$.
I proved that $p|a$, but I cant prove $p^2|a$, and my idea is as follow:
By multiplying $(p-1)!$... | This is a consequence/version of Wolstenholme's theorem. A proof is available here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Why $\frac{x}{\sqrt{x+1}-1}$ can be written as $\sqrt{x+1}+1$? I've evaluated the first formula on $W|A$ and it says that $\sqrt{x+1}+1$ is an alternate form to the first expression. I just don't see how it's possible.
The first thing I imagined was to write:
$$\frac{x}{\sqrt{x+1}-1}=x^1(\sqrt{x+1}-1)^{-1}$$
But it do... | Just multiiply by 1, in a special form :
$$
\frac{x}{\sqrt{x+1}-1} \cdot 1 = \frac{x}{\sqrt{x+1}-1} \cdot \frac{\sqrt{x+1}+1}{\sqrt{x+1}+1} = \cdots
$$
Can you take it from here ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
solve the integral bu substitution I tried substituting $x=sin\ u$ but I didn't get nowhere, can someone just give me a hint how to solve this integral?
$$\int\frac{dx}{(x^2-1)^2}$$
| The integration can be accomplished using the substitution that you favor, but the partial fraction decomposition is a much simpler method.
Let $x=sin\ y$.
$\ \ \ \ dx = cos\ y\ dy$
$$\displaystyle\int\frac{dx}{(x^2-1)^2}=\displaystyle\int\frac{dx}{(1-x^2)^2}=\displaystyle\int\frac{cos\ y\ dy}{(1-sin^2\ y)^2}=\displ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How does $\inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2} \le \lVert \nabla u \rVert_{L^2}$ imply this inequality? Let $M$ be a compact Riemann manifold with boundary. I want to know, given the inequalities $$
\vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u \rVert_{L^2(M)},
$... | Since constant functions have zero $H^{1/2}$ seminorm, it follows that
$$\vert T u \vert_{H^{1/2} (\partial M)} = \vert (T u) -c \vert_{H^{1/2} (\partial M)} = \vert T (u -c) \vert_{H^{1/2} (\partial M)}\tag{1}$$
for every $c\in\mathbb{R}$. (Trace operator commutes with adding a constant, because the trace of a const... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Linear Logic, what is it used for? I read a lot about Linear Logic recently but I failed to find any real use to the logic.
I'd like to know how and where Linear Logic could be applied. Something like lambda calculus can be clearly used as a programming language (scheme, lisp). But I don't see how Linear Logic could be... | It matches Quantum Mechanics quite well, because the built-in conservation rules for propositions match the QM prohibitions on copying and deleting information:
"Physics, Topology, Logic and Computation: A Rosetta Stone"
John Baez, Mike Stay
http://math.ucr.edu/home/baez/rosetta.pdf
"Linear Logic for Generalized Quant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Show that this integral is finite $\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx$ Let $p > -1$ and $r \in \mathbb{N}$, show that
$$\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx = \int_0^\infty x^p (\ln x)^r e^{-x} dx$$
and that this integral is finite.
To solve that problem, I wan... | Oh I almost forgot about this question.
So here's my proof (maybe some parts are a bit messy, so if you want to edit some parts so that it is more rigorous/better, feel free to do so).
As mentioned in the question, we will use Lebesgue's dominated convergence theorem, thus we have two parts to verify:
*
*Measurabil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Are linear reductive algebraic groups closed under extensions? Say we have a ses of algebraic groups
$1 \to A \to B \to C \to 1$
where $A,C$ are linear reductive algebraic groups.
Does it follow that $B$ is also a linear reductive algebraic group? In other words, are linear reductive algebraic groups extension-closed... | Let $$1 \rightarrow A \rightarrow B \rightarrow C \rightarrow 1$$
be an exact sequence of algebraic groups where $A,C$ are linear algebraic groups.
Claim : $B$ is also a linear algebraic group.
If not, let $A' \subset B$ be the unique normal linear algebraic group contained in $B$ such that $B/A'$ is an Abelian variety... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Fitting a polynomial with arbitrary derivative values Given two points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_2 > x_1$ and $y_2 < y_1$, I can obviously fit a line (order $1$ polynomial) to them. But if I want to fit a quadratic function by specifying real, finite values of the derivatives at the two points, are there an... | If you want to fit a quadratic $y=ax^2+bx+c$ to the two points $(x_1,y_1)$ and $(x_2,y_2)$ then you have to satisfy the pair of equations
$$y_1=ax_1^2+bx_1+c$$
and
$$y_2=ax_2^2+bx_2+c$$
If, moreover, you want to force the derivative at $(x_1,y_1)$ to be $m_1$ and the derivative at $(x_2,y_2)$ to be $m_2$, you need to a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is this function defined in terms of elliptic $\mathrm{K}$ integrals even? Let $R,z > 0$ be positive real constants, and consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by
$$ f(v) = \frac{1}{\sqrt{(R+v)^2+z^2}}\ \mathrm{K}\!\left( \frac{4 R v}{(R+v)^2+z^2} \right)$$
where $\mathrm{K}(m)$ is the complete el... | Okay, cracked it. It's annoyingly simple in the end:
$$ K(m) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-m\sin^2{\theta}}} \\
= \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-m(1-\cos^2{\theta})}} \\
= \int_0^{\pi/2} \frac{d\theta}{\sqrt{(1-m)+m\cos^2{\theta}}} \\
= \frac{1}{\sqrt{1-m}}\int_0^{\pi/2} \frac{d\theta}{\sqrt{1-\frac{m}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why is $n^2+4$ never divisible by $3$? Can somebody please explain why $n^2+4$ is never divisible by $3$? I know there is an example with $n^2+1$, however a $4$ can be broken down to $3+1$, and factor out a three, which would be divisible by $4$.
| Consider cases. For example, what happens if $n=3k$, $n=3k+1$, and $n=3k+2$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1256915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 2
} |
How many elements are in the quotient ring $\frac{\mathbb Z_3[x]}{\langle 2x^3+ x+1\rangle} $ How many elements are in the quotient ring $\displaystyle \frac{\mathbb Z_3[x]}{\langle 2x^3+ x+1\rangle}$ ?
I guess I should be using the division algorithm but I'm stuck on how to figure it out.
| Using the division algorithm, you can show that each element in the quotient ring is represented by a unique polynomial in $\mathbb{Z}_3[x]$ of degree $\leq 2$. How many such polynomials are there?
It's exactly like counting the number of elements of $\mathbb{Z}/n\mathbb{Z}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1257000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Automorphism of $\Bbb Q[x]$
Question: Find a nonidentity automorphism $\varphi$ of $\Bbb Q[x]$ such that $\varphi^2$ is the identity automorphism of $\Bbb Q[x]$.
Is $\varphi(\Bbb Q[x]) = \Bbb Q[-x]$ a solution? I think it is correct by I have no confidence to write it down.
| If it's confidence you need:
Let $\varphi$ such that $\varphi^2=id$.
Since it's a ring-homomorphism, $\varphi|_\mathbb{Q}$ must be a field-automorphism. But $\text{Aut}(\mathbb{Q})=\{id\}$. Therefore, it holds $\varphi(q(x))=q(\varphi(x))\ \forall q(x)\in\mathbb{Q}[x]$.
Moreover, $\varphi(x)=p(x)$ for some $p(x)\in\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1257201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Formula of parabola from two points and the $y$ coordinate of the vertex The parabola has a vertical axis of symmetry. Given two points and the $y$ coordinate of the vertex, how to determine its formula?
For example:
| Hints: I will first give a straight-forward method, and then give a cleverer method.
Straightforward method:
You know the formula for a parabola is $$y=ax^2+bx+c.$$
The idea now is just to plug in your points and solve the resulting system of equations. The nonvertex points are easy to deal with - they give you the equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1257301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove there isn't a continuous surjection $f: [0, 1] \to \Bbb R$ (without compactness) Definitions:
Continuous: A map $f: X \to Y$, where $X$ and $Y$ are topological spaces, is continuous if the preimage in $X$ of any open set in $Y$ is open.
Subspace topology: If $(X, \mathcal{T})$ is a topological space, the subspac... | Let's cheat and only use that $[0,1]$ is closed and bounded:
For each $k\in\mathbb N$ pick $x_k\in[0,1]$ with $f(x_k)=k$.
Starting with $I_0=[0,1]$, which contains all $x_k$, we can repeatedly split $I_n=[a_n,b_n]$ into two subintervals $[a_n,\frac{a_n+b_n}2]$, $[\frac{a_n+b_n}2,b]$ and one of these contains infinitel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1257405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Finding a function into a closed form of the generating function I have the following equation:$$a_n = n((-1)^n(1-n) + 3^{n-1})$$
How do I convert this into a closed form of the generating function?
| Hint(s): This splits up into three terms: $(-1)^nn$, $-(-1)^nn^2$, and $3^{n-1}n$.
The first gives the series $\sum (-1)^n nx^n$; the easiest way to figure out the generating function is to write
\begin{equation*}
\sum_{n=1}^\infty (-1)^n nx^n
= \sum_{n=1}^\infty (-1)^n (n-1)x^n + \sum_{n=1}^\infty (-1)^n x^n.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1257493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many points do $f(x) = x^2$ and $g(x) = x \sin x +\cos x$ have in common? How many points do $f(x) = x^2$ and $g(x) = x \sin x +\cos x$ have in common?
Attempt:
Suppose $f(x) = g(x) \implies x^2 - x\sin x -\cos x = 0$.
Treating it as a quadratic equation in $x$, the discriminant $ = \sin^2x + 4 \cos x = 1-\cos^2x+4... | May be this would be an idea for you. We have $h(x) =x^2-x\sin x-\cos x$. Then $h'(x)=2x-\sin x -x\cos x +\sin x=x(2-\cos x) $. Hence $h'(x)=0 $ iff $x=0$, and for $$x<0 \implies h'(x) <0 \implies h \text{ decreases on } ] -\infty, 0[ $$ and $$x>0 \implies h'(x)>0 \implies h \text{ increases on } ] 0,\infty[ $$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1257596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
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