Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solve Integration area I am confused why the answer for $$\int_0^{\infty}\left(x-\frac{1}{\lambda}\right)^2\lambda e^{-\lambda x}\ dx$$
is $$\frac{1}{\lambda^2}$$
I get mine as $$\frac{5}{\lambda^2}$$
Official answer is as follows
but I do not get the last part when it is $$\frac{-2}{\lambda^2}$$
instead of $$\fra... | The last integral is
$$-\frac{2}{\lambda} \int_0^{\infty} e^{-\lambda x}\; dx = \left.\frac{-2}{\lambda} \frac{e^{-\lambda x}}{-\lambda} \right|_{0}^{\infty} = \frac{2}{\lambda^2} \left(e^{-\infty} - e^0 \right) = \frac{2}{\lambda^2}(0-1) = -\frac{2}{\lambda^2} .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2374963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
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Calculate root interval of $x^5+x^4+x^3+x^2+1$ So I have to find an interval (in the real numbers) such that it contains all roots of the following function:
$$f(x)=x^5+x^4+x^3+x^2+1$$
I've tried to work with the derivatives of the function but it doesn't give any information about the interval, only how many possible ... | The common factoring formula $x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \cdots + x^2+1)$ tells us that your $f(x)$ is equivalent to $g(x)=\dfrac {x^6-1}{x-1}$ as long as $x \not = 1$. Setting $g(x)=0$ yields $x^6-1=0$. Clearly the only real solutions to this are $\pm1$. But only $-1$ is a zero of $g$ and of $f$, so $f$ has only one... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2375023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 6
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How to arrive at the particular form of inequality as answer?
The stratosphere is the layer of earth's atmosphere that is more than $10$ km and less than $50$ km above the earth's surface. Write an inequality which describes all possible heights $x$, in km, above the earth's surface that are in stratosphere.
Answer: $... | More generally:
$$a \lt x \lt b \;\;\iff\;\; \left|x - \frac{a+b}{2}\right| \lt \frac{b-a}{2}$$
That's saying that $x$ is in the interval $\,(a,b)\,$ iff the distance between $\,x\,$ and the midpoint $\,\frac{a+b}{2}\,$ is smaller than half the length of the interval $\,\frac{b-a}{2}\,$.
The given problem follows from ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2375135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Can proofs be arbitrarily long? Suppose that I have a set of starting logical formulas (axioms) and some inference rules that produce new formulas from old ones. I am curious if there is an upper bound to the number of steps (number of times an inference rules has to be used) in the shortest proof of some decidable sta... | First of all, if $c$ is an integer then the function $c\cdot f$ is computable if $f$ is. Moreover, any constant at all is bounded by an integer, so regardless of what $c$ is, there is some computable function $h$ with $cf(x)\le h(x)$ for all $x$. So we might as well forget about $c$. (That is: to get a fast-growing fun... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2375356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Help with airthmetic progression query I am currently working on solving questions on Arithmetic Progression and need to understand where I am going wrong with below derivation.
Let us say there are two A.P.
*
*AP1 => First term a, common difference d
*AP2 => First term A, common difference D
If first 5 terms ... | You have defined $a$ and $A$ as the third, not first, terms of the progressions. The ratio of the sums of the first five terms is equal to the ratio of the third terms (assuming the denominator is not zero). The ratio $\frac {30}{65}$ of the sums is equal to the ratio $\frac {6}{13}$ of the third terms.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Different summation notation? In school, I learned that
$$\sum_{k=1}^{n}f(k)=f(1)+f(2)+f(3)+ ... +f(n)$$
But in some physics or mathematics book, I saw this kind of representation such like:
$$\lim_{\Delta x\to 0} \sum_{j} f(x_j) \Delta x$$
What is the difference between these kinds of summation notation?
| The notation
\begin{align*}
\lim_{\Delta x\to 0} \sum_{j} f(x_j) \Delta x\tag{1}
\end{align*}
is typically used for Riemann sums when introducing integration and adresses more concepts than only finite summation $$\sum_{k=1}^{n}f(k)=f(1)+f(2)+f(3)+ ... +f(n)$$
*
*Finite sum: One part of (1) is a finite sum g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Math subject GRE exam 9768 Q.30 (condition that a basis of a real vector space must satisfy)
I am sure that A,B and E are wrong, but I do not know which is right C or D, and why, could anyone help me please?
| Take nontrivial scalar multiples of elements of $B $. This gives a basis $B'$ which is disjoint from $B $. I.e. if $B=\{b_1, b_2,\dots,b_n\}$, let $B'=\{cb_1, cb_2,\dots, cb_n\}$, where $c\neq1$.Then $ B'$ and $B $ are disjoint. The answer is D.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2375656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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how to prove $\frac{b^2}{b_1^2}=\frac{ac}{a_{1c_1}}$? If the Ratio of the roots of $ax^2+bx+c=0$ be equal to the ratio of the roots of $a_1x^2+b_1x+c_1=0$, then how one prove that $\frac{b^2}{b^2_1}=\frac{ac}{a_1 c_1}$?
| Hint :
let $\alpha$ and $\beta$ be the roots of $ax^2+bx+c=0$ & let $\gamma$ and $\delta$ be the roots of $a_1 x^2+b_1 x+c_1 =0$.
The ratio of their roots are equal if
\begin{eqnarray*}
\frac{\alpha}{\beta} = \frac{\gamma}{\delta}.
\end{eqnarray*}
Further hint : $\color{red}{\alpha+\beta=-\frac{b}{a}}$ & $\alpha \beta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2375736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that the equation $x^3+y^3+z^3-(x^2z+y^2x+z^2y)=2$ has no solution in natural numbers I asked myself which primes $p$ can be written as $p=x^3+y^3+z^3-(x^2z+y^2x+z^2y)$ with $x,y,z \in \mathbb{N}$.
But for $p \neq 2$ we have the solution $x=y=\frac{p-1}{2}$ and $z=\frac{p+1}{2}$. So the only prime for which I ca... | EDIT: You're right stackExchangeUser; my proof doesn't work. With a similar tack, we can still salvage this:
\begin{align*}
&x^3 + y^3 + z^3 - (x^2 z + y^2 x + z^2 y) \\
= ~ &(x + y + z)^3 - 4(x^2 z + y^2 x + z^2 y) - 3(x^2y + y^2 z + z^2 x) - 6xyz
\end{align*}
So, we are solving,
$$(x + y + z)^3 = 2 + 4(x^2 z + y^2 x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2375828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Volume form induced by a metric I know that a Riemannian manifold of dimension $n$ is naturally endowed with a volume form induced by the Riemannian metric $\omega=\sqrt{|g|}dx_1\wedge\dots\wedge dx_n$.
Is the same thing true with only topological assumptions?
With this I mean: does a metric induce something similar t... | Let me first try to make sense of your question. First of all, the notion of a differential form is meaningless for general metric spaces $(X,d)$. However, one can still talk about Borel measures $\mu$ on the topological space $X$ (topologized using the metric $d$). The Borel condition still leaves too much freedom sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2375953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Finding the minimum value of $\cot^2A + \cot^2B+ \cot^2C$ where $A$, $B$ and $C$ are angles of a triangle. The question is:
If $A+B+C= \pi$, where $A>0$, $B>0$, $C>0$, then find the minimum value of $$\cot^2A+\cot^2B +\cot^2C.$$
My solution:
$(\cot A + \cot B + \cot C)^2\ge0$ // square of a real number
$\implies ... | it is equivalent to
$$\frac{1}{\sin(A)^2}+\frac{1}{\sin(B)^2}+\frac{1}{\sin(C)^2}\geq 4$$
with $$\sin(A)=\frac{a}{2R}$$ etc and $$S=\sqrt{s(s-a)(s-b)(s-c)}$$ and $$S=\frac{abc}{4R}$$ we get
$$b^2c^2+c^2a^2+a^2b^2-(-a+b+c)(a-b+c)(a+b-c)(a+b+c)\geq 0$$
and this is equivalent to
$$a^4+b^4+c^4\geq a^2b^2+b^2c^2+c^2a^2$$
wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2376094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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How many knights are there?
I cannot deduce the solution. I cannot deduce a solution.
Consider the case in which Anne is a Knave.
If Anne answers "Yes", then they are not both knights. Berne might be a knight, or they might be a knave. If Anne answers "No", then they are both knights. But Anne is a knave, so this lead... |
Consider the case in which Anne is a Knave. If Anne answers "Yes", then they are not both knights.
Anne is lying, so it could be that either Bernie is a knight and she is a knave or they are both knaves.
If Anne answers "No", then they are both knights. But Anne is a knave, so this leads to a contradiction.
Yup, s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2376194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Recurrence relation $a_n = 11a_{n-1} - 40a_{n-2} + 48a_{n-3} + n2^n$ I didn't do a lot of maths in my career, and we asked me to solve the following recurrence relation:
$$a_{n} = 11a_{n-1} - 40a_{n-2} + 48a_{n-3} + n2^n$$
with
$a_0 = 2$, $a_1 = 3$ and $a_2 = 1$
What is the procedure to solve such relation? So far, I ... | Just to offer another approach, generating functions can be used as well:
$\begin{align}
G(x) &= \sum_{n=0}^{\infty} a_n x^n \\
G(x) &= 2x^0 + 3x^1 + x^2 + \sum_{n=3}^{\infty}(11a_{n-1} - 40a_{n-2} + 48a_{n-3} + n2^n)x^n \\
G(x) &= 2 + 3x + x^2 + 11\sum_{n=3}^{\infty}a_{n-1}x^n - 40\sum_{n=3}^{\infty}a_{n-2}x^n + 48\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2376294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Has anyone come across a geometric interpretation for fractional exponents of pi? Once in a while I'll see pi, not squared, but to a fractional power. For instance when dealing with a bell curve with its integral to infinity, you obtain $$ \frac{\sqrt{\pi}}{2}$$
When you evaluate certain elliptic integrals or fractiona... | To generalize Professor Vector's comment, the $d$-dimensional hypersphere of radius 1 has hypervolume $A_d\pi^s$ for some easily computed rational constant $A_d$, where $s$ is the integer part of $d/2$ – see Wikipedia. Then the hypercube with the same hypervolume has side $A_d^{1/d}\pi^{s/d}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2376385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Find minimum value of the trigonometrical expression If $A+B+C=\pi$
then find the minimum value of
$\sin 3A+\sin 3B+\sin 3C$
where $0\le A\le \pi,0\le B \le \pi,0\le C\le \pi$
| The minimum value is $-2$.
Let be $A\leq B\leq C$. So we have $3A\leq \pi$ and so $\sin(3A)\geq 0$. So we get the minimum for $A=0, B=C=\frac{\pi}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2376450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Express a formula in terms of trigonometric expressions I am studying Kerr Black holes using Hobson's General relativity an introduction for physicists book.
In order to find circular radius for photons, two conditions need to be satisfied:
$$r_c=3\mu\frac{b-a}{b+a}$$ and
$$(b+a)^3=27\mu^2(b-a)$$
According to the book... | \begin{align}
r_c&=3\mu\frac{b-a}{b+a} \tag{1}\label{1}
\\
(b+a)^3&=27\mu^2(b-a) \tag{2}\label{2}
\end{align}
To get the expression for b from \eqref{1},
\begin{align}
b-a&=\frac{r_c}{3\mu}(b+a)
,
\end{align}
combined with \eqref{2},
\begin{align}
(b+a)^3&=27\mu^2\frac{r_c}{3\mu}(b+a)
,\\
(b+a)^2&=9\mu{r_c}
,\\
b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2376551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Need help integrating $\int e^x(\frac{x+2}{x+4})^2 dx $ I have simplified the problem a bit using integration by parts, with $u = e^x(x+2)^2$ and $v = 1/(x+4)^2$ but I'm then stuck with how to integrate this:
$$\int\frac{e^x(x^2+4x+8)}{x+4}dx. $$
I've considered substituting $t = e^x$, but this doesn't seem to make the... | \begin{align*}\int e^x\left(\frac{x+2}{x+4}\right)^2\,\mathrm dx&=\int\frac1{(x+4)^2}e^x(x+2)^2\,\mathrm dx\\&=-\frac{e^x(x+2)^2}{x+4}+\int e^x(x+2)\,\mathrm dx\\&=-\frac{e^x(x+2)^2}{x+4}+e^x(x+2)-\int e^x\,\mathrm dx\\&=-\frac{e^x(x+2)^2}{x+4}+e^x(x+1)\\&=\frac{xe^x}{x+4}.\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2376729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is my understanding of the definition of limit correct? The definition;
$(\forall \epsilon > 0)(\exists \delta > 0)(\forall x$ satisfying $0 <$
$|x-a| < \delta$ also satisfies $|f(x) - L| < \epsilon) \iff \lim_{x\to a} f(x) = L$
To explain my understanding, lets consider the following example;
$$\lim_{x\to 4} 3x^... | I'll try to answer your questions one at a time:
First of all, is there any flow or misunderstanding in my understanding of the definition of the limit?
No, you have stated the definition of the limit accurately. I would have dispensed with the logical symbols and simply said that $\lim_{x\to a}f(x) = L$ if correspon... | {
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Cauchy sequence in compact metric space converges; incorrect proof? I'm self-studying real analysis, and I've been trying to prove the following statement:
If $X$ is a compact metric space and if $\{p_n\}$ is a Cauchy sequence in $X$, then $\{p_n\}$ converges to some point of $X$.
My proof is as follows:
Fix $\epsilon>... | You did not prove that that it converges to $p_N$. Convergence to $p_N$ means that for every $\varepsilon>0$, you have $d(p_N,p_n)<\varepsilon$ for all $n$ large enough. But when you proved the inequality $d(p_N,p_n)<\varepsilon$, you proved it for a fixed $\varepsilon$, not for all of them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2376896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the maximum positive integer that divides $n^7+n^6-n^5-n^4$
Find the maximum positive integer that divides all the numbers of the form $$n^7+n^6-n^5-n^4 \ \ \ \mbox{with} \ n\in\mathbb{N}-\left\{0\right\}.$$
My attempt
I can factor the polynomial
$n^7+n^6-n^5-n^4=n^4(n-1)(n+1)^2\ \ \ \forall n\in\mathbb{N}.$
If... | Without using the link I gave you (which is overkilling the problem, by the way), you can see that $2^4$ divides $$f(n):=n^7+n^6-n^5-n^4=(n-1)\,n^4\,(n+1)^2$$
by considering the case $n$ is odd and the case $n$ is even. Since $n-1$, $n$, and $n+1$ are consecutive integers, $3$ must divide $f(n)$. That is, $2^4\cdot 3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2377112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Existence of a sequence of elements picked arbitrarly How to prove with the axiom of choice that : Given a family of non empty sets $(A_n)_{n\in\mathbb N}$, there exists a sequence $(x_n)_{n\in\mathbb N}$ such that for any $n\in\mathbb N$, $x_n \in A_n$.
I don't know how to proceed... fixing $n$ then apply the axiom of... | The definition here expresses the axiom of choice as
$$ \forall X \left[ \emptyset\not\in X \implies \exists f:X\to\bigcup X\quad
\forall A\in X\ (f(A) \in A) \right]. $$
By taking $X:=\{A_n \mid n\in\mathbb{N}\}$, since each $A_n$ is nonempty, we are given a function $f:X\to\bigcup X$ such that $f(A) \in A$ for eve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2377265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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What does "trivial solution" mean? What does "trivial solution" mean exactly?
Must the trivial solution always be equal to the zero-solution (where all unknowns/variables are zero)?
| Depending on the context, trivial solutions can be:
*
*the zero function $y = 0$
*singular solutions of the differential equation (e.g. where you divide by $0$ in the process of solving the differential equation)
*constant functions $y = c, c \in \mathbb{R}$
*or just solutions which one can see immediately
Howe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Inequalities in category theory I am trying to understand the definition of a relational $\beta$-module as described here: https://ncatlab.org/nlab/show/relational+beta-module.
The definition given in section $2$ under the title "Bridge to a Concrete Description" is as follows:
A relational $\beta$-module is a set $S$ ... | As mentioned in some of the comments, those diagrams take place in the category of sets and relations, $\operatorname{Rel}$. $\operatorname{Rel}$ isn't "just" a category: it's enriched over posets. That is, there isn't just a set of relations between sets $A$ and $B$, there's a poset of them (ordered by $\subseteq$).
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2377469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is the intersection of two $\sigma$-algebras the $\sigma$-algebra generated by the intersection of the generators? Let $\Omega$ be a set and $A, B \subseteq \mathcal{P}(\Omega)$ be two collections of subsets closed under intersection (sometimes called $\pi$-system).
I would like to know if
$$ \sigma(A\cap B) = \sigma(A... | Here's a counterexample: let $\Omega=\mathbb{R}$, $A=\{(-\infty,t]|t\in\mathbb{R}\}$ and $B=\{(a,b)|a,b\in\mathbb{R}\}\cup\{\emptyset\}$ - then $A$ and $B$ are closed under intersection and disjoint from each other but they generate the same $\sigma$-algebra (i.e. the standard Borel $\sigma$-algebra on $\mathbb{R}$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2377570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Maximizing the Integral
Find the interval $[a,b]$ for which the value of the integral $\int_{a}^{b} (2+x-x^2)dx$ is maximized.
To solve this problem, I believe I need to the largest interval over which the integrand is nonnegative. To that end, $2+x-x^2 \ge 0$ if and only if $(x+1)(x-2) \le 0$. This occurs if $x \ge ... | How about this:
Let $a$ be a fixed number. Consider the function
$$F(t)= \int_a^t (2+x-x^2) \ \mathrm dx$$
Then, $F'(t)= 2+t-t^2 = - (t+1)(t-2)$.
Observe that $F'$ is positive on $(-1,2)$ and negative on $(2,\infty)$.
This implies that $t=2$ is a local maximum for $F$.
Now similarly, you can consider $$G(t)= \int_t^2(2... | {
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"url": "https://math.stackexchange.com/questions/2377686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Isomorphism between normal subgroups I'm trying to work out this group theory proof:
Suppose that $G$ is a group have that has a normal subgroup $H$ such that $H$ is isomorphic to $D_3$. Prove that exists a subgroup $K$ of $G$ such that $G$ is isomorphic to $H\oplus K$.
Here is what I was thinking:
I was thinking $K... | Let $H \cong D_3$ and $K = C_G(H)$ (centralizer of $H$ in $G)$. Then $G/K \cong A \le Aut(D_3)$, $K \trianglelefteq G$.
Here you have to prove that $|Aut(D_3)| = |D_3|$ (this can be done "manually", by checking all the options where automorphism can send the generators of $D_3$).
Since $Z(D_3) = \{e\}$, every automor... | {
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Definite integral for a 4 degree function
The integral is:
$$\int_0^a \frac{x^4}{(x^2+a^2)^4}dx$$
I used an approach that involved substitution of x by $a\tan\theta$. No luck :\ . Help?
| $\displaystyle\int_0^a \frac{x^4}{(x^2+a^2)^4}dx$
Where do we get with the substitution you have suggested?
$x = a\tan\theta\\
dx = a\sec^2\theta\\
\displaystyle\int_0^{\frac \pi 4} \frac{(a^4\tan^4\theta)(a\sec^2\theta)}{(a^2\tan^2\theta+a^2)^4}d\theta\\
$
Looks promising:
Keep simplifying
$\displaystyle\int_0^{\frac ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$f^n = f^k \circ f^{n-k}$ $f : A \to A$ and $n \in n$, Let $f^n$ be defined by $f^1 = f$ and $$f^n = f \circ f^{n-1}$$ for $n \gt 1$.
Let $n$ and $k$ be natural numbers with $k \lt n$. Prove $$f^n = f^k \circ f^{n-k}$$
Induction: $n=2$
$f^2 = f \circ f^{2-1} \implies f^2 = f^1 \circ f^1$
Hence, the base case holds true... | Hint: fix $m$ and prove by induction on $n$ that $f^{n+m} = f^n \circ f^m$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Geometric interpretation regarding square of distances. Can anyone give an alternative solution or give a geometric "illustration/interpretation" to the constant relative to the distances(see picture). I could not do it without resorting to coordinates. The constant I found using coordinates is 2*side^2 or 6*radius^2.
... | Assuming a circle with radius $1$, then the side of the equilateral triangle = $\sqrt3$. By Ptolemy's theorem on the quadrilateral in a circle: $$DE*FC+DC*FE=EC*FD$$ Hence, substituting $DE$ for $DC$ and $EC$, we have$$DE(FC+FE)=DE*FD$$or$$FC+FE=FD$$Squaring, transposing, and factoring gives [1]$$FC^2+FE^2+FD^2=2(FD^2-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
An interesting divisibility about a Dirchlet convolution of binomial coefficients with Mobius function I have found some interesting divisibility properties which I don't know how to prove.
If we set $$T(n,k)=\sum_{l|n}\mu(l)(-1)^{\frac{n}{l}}\binom{k\frac{n}{l}}{\frac{n}{l}}$$ where $\mu(.)$ is the Mobius function, t... | I have found the following paper:
https://arxiv.org/abs/1504.06327
Proposition 1.2. gives the integrality we want since these invariants are integers! But I am still waiting for a direct proof. For exemple, the paper here
https://arxiv.org/abs/1703.00990
gives a partial proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2378273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Continuity implies the intermediate value property How can someone prove that continuity implies the intermediate value property?
P.S.: If $I$ is an interval, and $f:I\rightarrow\mathbb{R}$, we say that $f$ has the intermediate value property (IVP) iff whenever $a<b$ are points in $I$ and $f(a)\leq c\leq f(b)$, there i... | I guess it depends on the logical framework you're using.
In smooth infinitesimal analysis (which uses intuitionistic logic rather than classical logic), every function is continuous, and the intermediate value theorem fails!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2378360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Writing a simple expression in set notation I am trying to state that for every pair of even integers, the sum of the two integers is even.
$\forall m, n \in 2 \mathbb Z, \exists a = m + n \ni a \in 2 \mathbb Z$
Does this make sense?
| While your expression is logically true, according to your statement, it's better to say:
$$\forall m, n \in 2 \mathbb Z, m + n \in 2 \mathbb Z$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
Inequality: $\ln(\frac{a+y}{y})-\frac{a}{a+y}> 0$ I want to show that the function $g(y)=y\ln(1+\frac{a}{y})$ is increasing for $y>0, a>0$.
I've found the derivative and set up the inequality that I need to show:
$\ln(\frac{a+y}{y})-\frac{a}{a+y}> 0$
I'm not sure about how to show it. Would appreciate a suggestion or ... | Problem:
$\ln (\frac{a+y}{y}) - \frac{a}{a+y} \gt 0$, for $a, y, \gt 0$.
LHS:
$\ln( \frac{a+y}{y}) - 1 + \frac{y}{a+y}$.
Let $z: = \frac{a+y}{y}$ , then $z \gt 1$.
Problem reduces to:
$\star)$ $\ln (z) + \frac{1}{z} \gt 1$ for $z \gt 1$.
$f(z) := \ln(z) + \frac{1}{z}$ ;
$f(1) = 0+ 1 = 1$.
$f'(z) = \frac{1}{z} - \frac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Differentiation of logistic function The logistic function is $g(x) = \frac{1}{1+e^{-x}}$, and it's derivative is $g'(x) = (1-g(x))g(x)$.
Now if the argument of my logistic function is say $x+2x^2+ab$, with $a,b$ being constants, and I derive with respect to x: $(\frac{1}{1+e^{-x+2x^2+ab}})'$, is the derivative still $... | Suppose $g(x) = (1+\exp(-h(x))^{-1}$
then \begin{align}g'(x)&=-(1+\exp(-h(x))^{-2}\exp(-h(x))(-h'(x))\\
&=g(h(x))\frac{\exp(-h(x))}{1+\exp(-h(x))}h'(x)\\
&=g(h(x))\frac{1+\exp(-h(x))-1}{1+\exp(-h(x))}h'(x) \\
&=g(h(x))(1-g(h(x)))h'(x)\end{align}
In this case $h(x)=x+2x^2+ab$, then $h'(x)=4x+1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Should we think of lines as sets of points? I recently heard about a mathematician who denied that lines are sets of points, preferring to think of them as objects to which points may be incident.
What are the advantages of this point of view? These might be practical advantages or philosophical advantages. I have a fe... | The "advantage" of thinking of points and lines in geometry as primitive objects related by incidence (I classify this as synthetic geometry) is that it is very general and easy to get started. The board is relatively free of clutter compared to starting the other way with "a plane is a set of points, and there are spe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2378710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Basic probability question- multiplication rule I came across the following question in a textbook (bear in mind that this is the only information given)-
There is a $50$ percent chance of rain today. There is a $60$ percent chance of rain tomorrow. There is a $30$ percent chance that it will not rain either day. What... | Yes, the multiplication rule only applies when events are independent, and there is no reason to assume the events of rain on each day are independent.
For this type of problem, it helps to make a Venn diagram. There are two circles, $A$ and $B$, representing the events of rain today and tomorrow. This gives four regi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2378844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Prove this inequality $2(a+b+c)\ge\sqrt{a^2+3}+\sqrt{b^2+3}+\sqrt{c^2+3}$ For $a,b,c$ are positive real numbers satisfy $a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. Prove that $$2\left(a+b+c\right)\ge\sqrt{a^2+3}+\sqrt{b^2+3}+\sqrt{c^2+3}$$
We have:$a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{9}{a+b+c}\Leftrigh... | Since the function $f(t) := \sqrt{1+t}$ is concave in $[-1, +\infty)$, we have that
$$
f(t) \leq f(3) + f'(3) (t-3)
\qquad \forall t\geq -1,
$$
i.e.
$$
f(t) \leq 2 + \frac{1}{4}(t-3) = \frac{5}{4} + \frac{1}{4} t
\qquad \forall t\geq -1.
$$
Using this inequality we have that
$$
\sqrt{a^2+3} = a \sqrt{1+ 3/a^2} \leq
a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2378941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Show that a compact set of real numbers contains its greatest lower bound and its least upper bound. Show that a compact set of real numbers contains its greatest lower bound and its least upper bound. Can this occur for a set of real numbers that is not compact?
My attempt:
By Hein-borel theorem, compact set is closed... | Your reasoning is fine. However, this can also hold for non-compact sets; consider:
$$[-2,-1) \cup (0,1]$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2379025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inequality $\frac{x_1^2}{x_1^2+x_2x_3}+\frac{x_2^2}{x_2^2+x_3x_4}+\cdots+\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1}+\frac{x_n^2}{x_n^2+x_1x_2}\le n-1$
Show that for all $n\ge 2$
$$\frac{x_1^2}{x_1^2+x_2x_3}+\frac{x_2^2}{x_2^2+x_3x_4}+\cdots+\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1}+\frac{x_n^2}{x_n^2+x_1x_2}\le n-1$$
where $x_i$... | Let $\frac{x_2x_3}{x_1^2}=\frac{a_1}{a_2}$,... and similar, where $a_i>0$ and $a_{n+1}=a_1$.
Thus, we need to prove that:
$$\sum_{i=1}^n\frac{1}{1+\frac{a_i}{a_{i+1}}}\leq n-1$$ or
$$\sum_{i=1}^n\left(\frac{1}{1+\frac{a_i}{a_{i+1}}}-1\right)\leq-1$$ or
$$\sum_{i=1}^n\frac{a_i}{a_i+a_{i+1}}\geq1,$$
which is true because... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2379113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
what is the nth-derivative in 0 of $\frac{e^x}{1-x}$ Using $$f^n(0)=n! .a_n$$
I have$$\frac{e^x}{1-x}=\sum{\left(\frac{x^n}{n!}\right)}\sum{x^n}=\sum_{n \geq 0}\left({\sum_{k=0}^{n}{\frac{n!}{k!n!}}}\right)$$
How to get the combination?
| The General Leibniz rule tells us that for two smooth functions $f,g$, the $n$th derivative of $h(x) = f(x)g(x)$ is
$$h^{(n)}(x) = \sum_{k=0}^n {n \choose k} f^{(n-k)}(x)g^{(k)}(x)$$
The $n$ derivative of $e^x$ at $x=0$ is $1$, and the $n$th derivative of $\frac{1}{1-x}$ at $x=0$ is $n!$ by considering the power serie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2379207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to sum $\sum _{n=0}^{120}\:\frac{1}{\sqrt{n+1}+\sqrt{n}}\:$ without the use of a calculator? I'm learning about series and a textbook gives me the problem:
$\sum _{n=0}^{120}\:\frac{1}{\sqrt{n+1}+\sqrt{n}}\:$
But I can't figure out how to solve it, what process to follow or formula to use. I just know it diverges ... | HINT
Notice that
\begin{align*}
\frac{1}{\sqrt{n+1} + \sqrt{n}} = \frac{1}{\sqrt{n+1} + \sqrt{n}}\times\frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} = \sqrt{n+1} - \sqrt{n}
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2379369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Can I produce $1 - y$ with $y x_0 + x_1$ I want to know whether it is possible to calculate $1 - y$ using only a multiplication then an addition of two numbers which are not a function of $y$. I suspect it isn't possible but I would like to know how to prove this.
What I've tried so far:
I tried manipulating the equat... | Rewrite as
$$1 - y = 1 + (-1)\cdot y = 1 + y \cdot (-1) = y\cdot(-1) + 1$$
Now equate the coefficients to get
$$ x_0 = -1, x_1 = 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2379462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $ x \in \left(0,\frac{\pi}{2}\right)$. Then value of $x$ in $ \frac{3}{\sqrt{2}}\sec x-\sqrt{2}\csc x = 1$
If $\displaystyle x \in \left(0,\frac{\pi}{2}\right)$ then find a value of $x$ in $\displaystyle \frac{3}{\sqrt{2}}\sec x-\sqrt{2}\csc x = 1$
$\bf{Attempt:}$ From $$\frac{3}{\sqrt{2}\cos x}-\frac{\sqrt{2}}{\s... | I think it's better to make the following.
Let $x=\frac{\pi}{4}+t$, where $t\in\left(-\frac{\pi}{4},\frac{\pi}{4}\right)$.
Hence, we need to solve that
$$3\sin{x}-2\cos{x}=\sqrt2\sin{x}\cos{x}$$ or
$$3(\sin{t}+\cos{t})-2(\cos{t}-\sin{t})=\cos^2t-\sin^2t$$ or
$$\sin{t}(5+\sin{t})+(1-\cos{t})\cos{t}=0$$ or
$$\sin\frac{t}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2379567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
functions with the properties: $f(x) \rightarrow x$ when $x\rightarrow 0$ and $f(x) \rightarrow \frac{1}{x}$ when $x \rightarrow \infty$ I want to find some functions which have both the following asymptotic behaviors:
$f(x) \rightarrow x$ when $x\rightarrow 0$ and $f(x) \rightarrow \frac{1}{x}$ when $x\rightarrow \in... | Another pretty simple function is
$$
\frac{\tanh^2(x)}x
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2379674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Why can we use induction when studying metamathematics? In fact I don't understand the meaning of the word "metamathematics". I just want to know, for example, why can we use mathematical induction in the proof of logical theorems, like The Deduction Theorem, or even some more fundamental proposition like "every formul... | I am reminded of this remark at the beginning of Kleene's book Mathematical Logic:
"It will be very important as we proceed to keep in mind this distinction between the logic we are studying (the object logic) and our use of logic in studying it (the observer's logic). To any student who is not ready to do so, we sugge... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "59",
"answer_count": 4,
"answer_id": 2
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Solving a quadratic equation with perturbation on the exponent. This is for analysis of chemical rate equations. I have the equation $$ A(1-x)^2 = Bx^{2 + \epsilon} $$
which for the trivial case $\epsilon = 0 $, and ignoring the negative root, has the solution, $$ x = \frac{\sqrt A}{\sqrt A + \sqrt B}$$
I would like ... | Put $u = \sqrt{\dfrac{B}{A}}$, then we have:
$$1-x = u x^{1+\frac{\epsilon}{2}}= u x\exp\left[\frac{\epsilon}{2}\log(x)\right] = u x \left[1 + \frac{\epsilon}{2}\log(x) + \frac{\epsilon^2}{8}\log^2(x)+\cdots\right]$$
We then substitute the formal expansion:
$$x = x_0 + \epsilon x_1 + \epsilon^2 x_2 +\cdots$$
and expand... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2379863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Motivation for predicate logic. I am studying predicate logic,I came across this para-
Propositional logic, studied in Sections 1.1–1.3, cannot adequately express the meaning of all statements in mathematics and in natural language.
For example, suppose that we know that
“Every computer connected to the university... |
No rules of propositional logic allow us to conclude the truth of the statement “MATH3 is functioning properly",where MATH3 is one of the computers connected to the university network.
This is supposed to illustrate the motivation for predicate logic.
However,we can interpret the given statement as the conjunction o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2379913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Are smooth functions generically immersions? Let $T^2$ be the torus and let $\mathcal{C}^{\infty}(T^2, \mathbb{R}^3)$ be the space of smooth functions from $T^2$ to $\mathbb{R}^3$ endowed with the norm $\|f\| = \sup_x |f(x)| + \sup_x \|df_x\|$.
Is a generic function $f \in \mathcal{C}^{\infty}(T^2, \mathbb{R}^3)$ an im... | Consider the following function
$$ f: S^1 \times [-1,1] \to \mathbb R^3\,,\quad
f(\theta,z) = (z \cos \theta, z \sin \theta, z)\,,$$
that maps a cylinder into $\mathbb R^3$. If needed, we can extend it to a map from the torus into $\mathbb R^3$.
I would claim that it is not possible to approximate $f$ by an immersion i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2380035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is Fourier transform of unilateral sinc function? $$\int_{0}^{+\infty} \frac{\sin(x)}{x} e^{itx} dx = ?$$
I know the Laplace transform of sinc is $\arctan(1/t)$. However, what if $t$ is a complex number?
| In term of distributions, the Fourier transform of $\frac{\sin(\pi x)}{\pi x}$ is $1_{|\xi| < 1/2}$ and the FT of $1_{x > 0}$ is $\frac{1}{2i \pi}\frac{d}{d\xi} \log |\xi| + \frac{1}{2} \delta(\xi) $,
thus the FT of $\frac{\sin(\pi x)}{\pi x}1_{x > 0}$ is $$1_{|\xi| < 1/2} \ast (\frac{1}{2i \pi}\frac{d}{d\xi} \log |\xi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2380169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $0^\circ\leqslant x<360^\circ$, what is the maximum number of solutions to the equation $\sin x = a$ where a is a real number? I tried solving the question, but I kept getting $5$ solutions. My book only has $4$ choices: $0$, $1$, $2$, or $3$ solutions. My solutions were $0^\circ$, $90^\circ$, $150^\circ$, $180^\cir... | Clearly, we need $-1\le a\le1$ for at least one real solution
If $\sin x_1=\sin x_2$
Using Prosthaphaeresis Formulas, $$\sin x_1-\sin x_2=2\sin\dfrac{x_1-x_2}2\cos\dfrac{x_1+x_2}2.$$
If $\sin\dfrac{x_1-x_2}2=0\implies\dfrac{x_1-x_2}2=m180^\circ\iff x_1\equiv x_2\pmod{360^\circ}.$
If $\cos\dfrac{x_1+x_2}2=0\implies\dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2380284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find an interval such that the intersection has measure $\epsilon$ Suppose $E$ is measurable with finite lebesgue measure. Show that for each $0<\epsilon<m(E)$, there exists $x>0$ such that $m(E\cap (-x,x))=\epsilon$
I tried to use the $m(E)=\inf\{\sum_{k=1}^{\infty}l(I_k): \{I_k\}$ is a cover of open intervals of $E\}... | Hint If $0<x <y$ then $g(x) \leq g(y)$ and
$$g(y)-g(x) \leq 2(y-x)$$
Use this to show that $g$ is continuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2380527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Showing the Fourier sine series converges
The Fourier sine series for $f(x) = x$, $-2 < x < 2$ is
$$f(x) = \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\sin \frac{n\pi x}{2}$$
For each $x$ in the interval to what does the Fourier since series for $f(x)$ converge, can we prove pointwise convergence, converg... | *
*Your proof of convergence in $L^2$ is correct.
*Let us prove the pointwise convergenge in the open interval $(-2,2)$.
Consider, for $-2 < x < 2$,
$$ F(x) = \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}e^{ \frac{n\pi x}{2}i}$$
If $x\in (-2,2)$, then we have that $\sum_{n=1}^{\infty}(-1)^{n+1}e^{ \frac{n\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2380643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Element of certain order in special linear space. What would be the conditions (if any) on the trace of an element in $SL(2,p)$ in order for it to have order 5 ? (assuming $p= \pm1 mod 10$)
For example, any traceless element in $SL(2,p)$ has order 4 (straightforward proof).
Any suggestion or comment is tremendously va... | Suppose $A$ is an element of order $5$ in $\DeclareMathOperator{\SL}{SL} \SL_2(\mathbb{F}_p)$ and let $m(x)$ be its minimal polynomial. If $m$ has degree $1$, then $A$ is a scalar matrix hence must be of the form
$$
\begin{pmatrix}
c & 0\\
0 & c
\end{pmatrix}
$$
for some $c$. But then $c^2 = \det(A) = 1$, so $A$ has ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2380745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find the function $f$ such that $f(m^2 + f(n))=(f(m))^2 + n$
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$f(m^2 + f(n))=(f(m))^2 + n, \forall m,n \in \mathbb{Z} \tag 1$
If $m=0$ then $f(f(n))=f^2(0) + n \tag 2, \forall n$
From (2) $f \circ f$ is injective, therefore $f$ is injective.
Replac... | Well, if it's really easy to prove $f(1)=1$: $m=1$ gives $f(f(n)+1)=n+1,$ replacing $n$ by $f(n)$ and using $f(f(n))=n$ gives $f(n+1)=f(n)+1.$ The rest is obvious.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2381000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Pythagorean Triple: $\text{Area} = 2 \cdot \text{perimeter}$ Find the unique primitive Pythagorean triple whose area is equal to twice the perimeter.
So far I set the sides of the triangle to be $a, b,~\text{and}~c$ where $a$ and $b$ are the legs of the triangle and c is the hypotenuse.
I came up with 2 equations whic... | Rewrite the first equation as $c = \frac{ab}{4} - a - b$. Square it to get $$c^2 = a^2 + b^2 + \frac{a^2b^2}{16} - \frac{a^2b}{2} - \frac{ab^2}{2} + 2ab$$
Now using the other equation, we see that
$$\frac{a^2b^2}{16} - \frac{a^2b}{2} - \frac{ab^2}{2} + 2ab = 0$$
Since $a,b > 0$ divide by $ab$ and multiply by $16$ to ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2381494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Factor $x^4-7x^2+1$ Is there a general method of factoring fourth order polynomials into a product of two irreducible quadratics?
As I am reviewing on finding roots of polynomials in $\mathbb Z_n$ for abstract algebra, I am trying to factor the polynomial $x^4-7x^2+1$, and I was given the answer of $(x^2+3x+1)(x^2-3x+... | Hint:
Try with $$(x^2\pm1)^2-*x^2$$
The coefficient of $x^2$ has to be perfect square
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2381579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 2
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Bounding norm in $\ell_p$ by the norm in $\ell_{\infty}$ using multiplication by a vector Let $p \in [1, \infty)$. Is there a vector $y \in \mathbb{R}^{\mathbb{N}}$ such that for every $x \in \ell_p$ we have $\|x\|_p \leq \|xy\|_{\infty}$?
The multiplication is pointwise, and the norm on the right might be infinite.
Th... | Some observations:
*
*if $y$ works, then considering $x$ as the vector whose $n$-th coordinate is $1$ and all the others $0$, we get that $1\leqslant \left\lvert y_n\right\rvert$.
*Consider $x_n= \left\lvert y_n\right\rvert^{-1}$ for $0\leqslant n\leqslant N$, and zero for the others $n$. Then the $\ell^p$ norm of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2381699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Passing Shillings I am trying to solve one of Carroll's pillow problems, but even with the solution I can't really graps it.
The problem is as follow
Some men sat in a circle, so that each had 2 neighbours; and each had
a certain number of shillings. The first had I/ more than the second,
who had I/ more than the... | After $k$ circuits, i.e. $mk$ steps, the amount most recently passed was $mk$ shillings from the last man to the first man, and the last man has nothing left.
After a further $m-1$ steps the amount most recently passed was $mk+m-1$ shillings from the penultimate man to the last man, and so is the amount held by the l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2381791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Formula for consecutive residue of primitive modulo n. \begin{align*}
3^0 \equiv 1\mod 7\\
3^1 \equiv 3\mod 7\\
3^2 \equiv 2\mod 7\\
3^3 \equiv 6\mod 7\\
3^4 \equiv 4\mod 7\\
3^5 \equiv 5\mod 7\\
3^6 \equiv 1\mod 7\\
3^7 \equiv 3\mod 7\\
\end{align*}
Now just focusing on 1, 3, 2, 6, 4, 5, 1....
How to devise a formula ... | Fermat says $3^{6k+r}$mod$7=3^r$mod$7,0\le r \le 5$. Up to $r=5$ the calculation is very simple, no?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2381921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Determine if a line segment passes "through" a triangle ... Math gurus,
I apologize because in a previous post I asked how to determine if a line passes through a triangle; and an answer was given. Then, while testing, I came across the scenario where a vertical line “segment” was below the base of the triangle and the... | Two conditions must be met:
*
*the line of support crosses the triangle. This can be checked by counting the triangle vertices on either side of the line, i.e. 3 LeftOf tests (PQV0, PQV1, PQV2).
*the two endpoints may not both lie on the same side of the lines of support of the sides that are crossed. Takes 2 or 4 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2382016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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How do I prove my Fejer kernel definition is equivalent? In my notes, the definition of the Fejer kernel is
$$
F_{n} = \sum_{j=-N}^{N} \left(1 - \frac{|j|}{N+1}\right) e^{ijt}.
$$
But in most of the reference material I come across online, it is immediately defined as the average of the Dirichlet kernels
$$
F_{N} = \f... | It is sufficient to compare the coefficients of $e^{ijx}$ in both expressions.
What is the coefficient of $e^{ijx}$ in the second expression? Since
$$D_n=\sum_{k=-n}^n e^{ikx}$$
the number of Dirichlet Kernels in the average that contain $e^{ijx}$ is clearly
$N-j+1$, (because for $k<j$ the Dirichlet kernel $D_k$ does n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2382118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find minimum value that the trigonometric expression may take For $x\in\left(0, \frac{\pi}{2}\right)$ find a minimal value, which the expression
$$\sec x+\csc x+\sec^{2}x+\csc^{2}x$$
can take.
My attempt:
I followed the trigonometrical approach and obtained
$$\sec x+\csc x+\sec^{2}x+\csc^{2}x=\sqrt{\left(2\csc 2x+1\r... | Let $\sin{x}=a$ and $\cos{x}=b$.
Hence, $a^2+b^2=1$ and by AM-GM we obtain:
$$\sec x+\csc x+\sec^{2}x+\csc^{2}x=$$
$$=\frac{a+b}{ab}+\frac{1}{a^2b^2}\geq\frac{2\sqrt2}{\sqrt{a^2+b^2}}+\frac{4}{(a^2+b^2)^2}=4+2\sqrt2.$$
The equality occurs for $a=b=\frac{1}{\sqrt2}$, which says that we got a minimal value.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2382235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Geometry : Prove that $PE=PC$ Let $l$ be a line not intersecting circle $\omega$ that has center $O$. Draw $OP$ perpendicular to $l$ at point $P$ and draw $PA$ tangent to $\omega$ at point $A$. Extend $OA$ to cut $\omega$ again at point $B$ and cut $l$ at point $C$. $PB$ cuts $\omega$ at point $D$ and $AD$ cuts $l$ at ... | From the Menelaus Theorem on $\triangle CBP$ and the line $A-D-E$ we have:
$$\frac{CA}{AB} \times \frac{BD}{DP} \times \frac{PE}{CE} = 1$$
So from this it's enough to prove that $\frac{CA}{AB} \times \frac{BD}{DP} = 2$.
Now we have that $AB = 2R$, while from the power of point $P$ we get: $DP = \frac{PA^2}{PB}$. Using... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2382332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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evaluate series and express sum with harmonic numbers I've computed the series but have had trouble expressing the sum in harmonic numbers
For $M\geq1$ compute the sum of the series below at $x=\dfrac{1}{\sqrt{M}}$ and express the sum in harmonic numbers. e.g $3H + H $ where
$3H=3\sum_{k=1}^{n}\dfrac{1}{k}$
$$ \sum... | Substitute $x=\frac{1}{\sqrt{m}}$ into the sum & do partial fractions
\begin{eqnarray*}
\sum_{n=1}^{\infty} \frac{\frac{1}{\sqrt{M}}}{n(1+\frac{n}{M})}=\sqrt{M} \sum_{n=1}^{\infty} \frac{1}{n(n+M)} = \frac{1}{\sqrt{M}}\sum_{n=1}^{\infty} \left( \frac{1}{n} -\frac{1}{n+M} \right) =\color{red}{ \frac{H_M}{\sqrt{M}}}.
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2382408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Closure in the Discrete Topology
If $\tau$ is the discrete topology on the real numbers, find the closure of $(a,b)$
Here is the solution from the back of my book:
Since the discrete topology contains all subsets of $\Bbb{R}$, every subset of $\Bbb{R}$ is both open and closed. Therefore, the closure of $(a,b)$ is $[... | The above comments and answers are absolutely correct.
But we can prove that there is a mistake in your book,by contradiction.
Suppose that $cl(a,b)=[a,b]$ Then $a \in cl(a,b)=[a,b]$ and form definition of closure,we know that a point $x$ is in the closure of a set $A$ in a metric space $X$ if every open ball with cent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2382599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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$[F(a):F]<\infty \implies a$ algebraic over F Let $E/F$ be an extention field and $a\in E$. We want to show that
$$[F(a):F]<\infty \implies a\text{ algebraic over } F$$
without the theorem which tells us that every finite extension is algebraic.
Proof. Let $[F(a):F]<\infty$. If $a\in E$ was transcendental over $F$, t... | If you know that $\langle a \rangle$ spans $F(a)$ (as an $F$-vector space):
Observe that $F(a)$ is singly generated over $F$. In particular, $\{1, a, a^2, \dots\}$ is a spanning set of $F(a)$ (seen as an $F$-vector space). Since $[F(a):F] = n < \infty$, there are $f_0, \dots, f_n \in F$ such that $$\sum_{i=0}^n f_i a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2382697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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What are the projection maps? I've read that projection maps are an important type of maps whose domain is a product of n possibly different sets. My question is that why do they name them "projection" maps? What are we projecting exactly? Yes we are mapping or relating the product of the sets to an output (thats what ... | Generally, a projection map "projects" elements onto a lower dimensional subspace which is a product of some subset of those sets. For example, consider the map $(x,y) \mapsto x$ from $\mathbb{R}\times\mathbb{R} \to \mathbb{R}$, which projects onto the first coordinate. If you were to express this graphically, you woul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2382785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find all entire functions with $|f(z)|\geq e^{|z|}$ for all $z \in \mathbb{C}$. Find all entire functions with $|f(z)|\geq e^{|z|}$ for all $z \in \mathbb{C}$.
I don't think there is any such entire function, and here is my thought: since $\Re(z) \leq |z|$, we know $|e^z|\leq e^{|z|}\leq |f(z)|$ for all $z$. Consider... | It's a bit more complicated than necessary. $|f(z)| \ge e^{|z|} \ge 1$, so why not just use $f$ instead of $g$?
And, by the way, what are you contradicting?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2382897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof of $1+x\leq e^x$ for all x? Does anyone provide proof of $1+x\leq e^x$ for all $x$?
What is the minimum $a(>0)$ such that $1+x\leq a^x$ for all x?
| By using induction I have showded that $$1+n ≤ e^n$$
for $n \in \mathbb{N}$
Induction start $$P(1):1+1≤ e^1$$
$$2 ≤ e $$
Induction Step
$$P(n):1+n≤ e^n$$ Adding plus 1 to both sides
$$n+2 ≤ e^n+1$$
We know now that:$$e^n+1≤ e^{n+1}$$ Since we know that multiplying by a number will yield a higher result compared to a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 9,
"answer_id": 5
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What are the steps for deriving a complicated generalization of a partial sum of a taylor series? I looked at the Taylor series for $$-\frac{x}{x-2}$$ and found it to be $$ \sum_{k=1}^{\infty}\frac{x^k}{2^k}$$
but then I also found that this series' partial sum is a bit more complicated in the form of
$$\frac{x 2^{-k}(... | In this particular case, the partial sum is easy to find:
$$f(x)=\sum_{k=1}^\infty\frac{x^k}{2^k}\implies \frac{x^n}{2^n}f(x)=\frac{x^n}{2^n}\sum_{k=1}^\infty\frac{x^k}{2^k}=\sum_{k=1}^\infty\frac{x^{k+n}}{2^{k+n}}=\sum_{k=n+1}^\infty\frac{x^k}{2^k}.$$
Then by subtraction,
$$\sum_{k=1}^n\frac{x^k}{2^k}=f(x)\left(1-\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Finding the value of a given trigonometric series.
Find the value of $\tan^2\dfrac{\pi}{16}+\tan^2\dfrac{2\pi}{16}+\tan^2\dfrac{3\pi}{16}+\tan^2\dfrac{4\pi}{16}+\tan^2\dfrac{5\pi}{16}+\tan^2\dfrac{6\pi}{16}+\tan^2\dfrac{7\pi}{16}.$
My attempts:
I converted the given series to a simpler form:
$\tan^2\dfrac{\pi}{16}+... | $$\tan^2\dfrac{\pi}{16}+\tan^2\dfrac{2\pi}{16}+\tan^2\dfrac{3\pi}{16}+\tan^2\dfrac{4\pi}{16}+\tan^2\dfrac{5\pi}{16}+\tan^2\dfrac{6\pi}{16}+\tan^2\dfrac{7\pi}{16}=$$
$$=\tan^2\dfrac{\pi}{16}+\cot^2\dfrac{\pi}{16}+\tan^2\dfrac{3\pi}{16}+\cot^2\dfrac{3\pi}{16}+\tan^2\dfrac{\pi}{8}+\cot^2\dfrac{\pi}{8}+1=$$
$$=\left(\tan\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Show that $G = \{x \in \mathbb{R}: 0 < x < 1\}$ is open.
As i am reading up on introduction to point set topology, i saw this
example but they did not provide full details. Please help me take a
look and see if it is correct! Thanks!
We have to make a good gauge and we pick epsilon $\epsilon$ to be either $x$ or ... | G is the open ball with radius 1/2 centered at 1/2.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Computing a Galois group and well-definedness of isomorphism Thank you for watching this question. On the way of studying Galois theory I had one question about well-definedness of isomorphism.
For example, let $\alpha $ be $\sqrt{3}+\sqrt{5}$ and $F$ be $\mathbb{Q}(\alpha)$ . I can understand the minimal polynomial of... | As mentioned in the comments in this particular case an automorphism of the field $\Bbb{Q}[\alpha]$ is determined by its action on $\sqrt{3}$ and $\sqrt{5}$. To answer your question as to what is going on more generally one can look at the discriminant $\Delta = \prod_{i<j}(\alpha_i-\alpha_j)^2$. $\Delta$ is fixed by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Ordered pairs in NBG set theory I'm reading a set theory textbook by Pinter, and this book follows the spirit of NBG set theory and talks about sets and (proper) classes in a very early stage. This is quite satisfactory because the book can (and does) talk about 'functions' whose domains are classes, and 'partially ord... | There is a useful generalization of this: One can encode a class-valued function as an object by flattening it to the relation $R_f$ given by
$$ y \in f(x) \Longleftrightarrow (x,y) \in R_f $$
So if you encode an ordered pair as a function $\{ 0, 1 \} \to \mathbf{Cls}$, this transposition into a relation gives the enc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Why is this Cayley diagram valid? I'm reading A Book of Abstract Algebra by Charles C. Pinter to give me an introduction to the topic, but one of the exercises has me confused. The following Cayley diagram is shown:
But I can't understand why this isn't invalid - isn't each vertex connected to two vertices which repre... | Yes -- you're partially correct! Each edge represents multiplying by the same generator (element) but each vertex represents a different element of the group. So this particular diagram represents the group $C_6$ (some people write it as $(\mathbb{Z}/6\mathbb{Z})$ or $\mathbb{Z}_6$). This group consists of six elements... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why should we have $\sin^2(x) = \frac{1-\cos(2x)}{2}$ knowing that $\sin^2(x) = 1 - \cos^2(x)$?
Why should we have $\sin^2(x) = \frac{1-\cos(2x)}{2}$ knowing that $\sin^2(x) = 1 - \cos^2(x)$?
Logically, can you not subtract $\cos^2(x)$ to the other side from this Pythagorean identity $\sin^2(x)+\cos^2(x)=1?$
When I ... | Both formulas are true, however, both are useful in different contexts (applications).
*
*You use $\sin^2(x) = \frac{1-\cos(2x)}{2}$ for integrating $\sin^2(x)$.
*You use $\sin^2(x) = 1 - \cos^2(x)$, for example, when solving $\sin^2(x) = 2\cos(x)$.
Note that it is just in some way more "natural" to write $\sin^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Find $\cos2\theta+\cos2\phi$, given $\sin\theta + \sin\phi = a$ and $\cos\theta+\cos\phi = b$
If
$$\sin\theta + \sin\phi = a \quad\text{and}\quad \cos\theta+\cos\phi = b$$
then find the value of $$\cos2\theta+\cos2\phi$$
My attempt:
Squaring both sides of the second given equation:
$$\cos^2\theta+ \cos^2\phi + 2... | HINT: use that $$\cos(2\theta)+\cos(2\phi)=\cos(\theta-\phi)\cos(\theta+\phi)$$
and $$\sin(\theta)+\sin(\phi)=2\cos\left(\frac{\theta-\phi}{2}\right)\sin\left(\frac{\theta+\phi}{2}\right)$$
and
$$\cos(\theta)+\cos(\phi)=2\cos\left(\frac{\theta-\phi}{2}\right)\cos\left(\frac{\theta+\phi}{2}\right)$$
so another idea, and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Same convergent subsequence for two compact operators? I was reading a proof on how any linear combination of compact operators is compact.
Let $U,V: X \to Y$ be compact linear operators and let $\alpha,\beta \in \mathbb{C}$. Then each bounded sequence $(x_n)$ in $X$ contains a subsequence $(x_{n(k)})$ such that $(Ax_{... | Such manipulations are often left implicit in more advanced textbooks. You can choose a subsequence $x_{n_k}$ for the operator $A$ such that $Ax_{n_k}$ converges. Then you can choose a subsequence of $x_{n_k}$ (a bounded sequence, being a subsequence of a bounded sequence) $x_{n_{k_l}}$ such that $Bx_{n_{k_l}}$ converg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2383894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
calculating $\int_0^{\infty}\frac{1}{(x^2+y)^n}dx$ I would like to know if I solved this improper integral right:
$$\int_0^{\infty}\frac{1}{(x^2+y)^n}dx$$
for $y\gt 0$
My solution:
$$\int_0^{\infty}\frac{1}{(x^2+y)^n} \, dx=\lim_{M\rightarrow \infty}\int_0^M1\cdot\frac{1}{(x^2+y)^n} \, dx$$
now I used integration by pa... | By my essay, putting $a=y$ yields the result $$
\boxed{\int_{-\infty}^{\infty} \frac{d x}{\left(x^{2}+y\right)^{n+1}}=\frac{(2n-1)!!\pi}{2^{n} n !}y^{-\frac{2 n+1}{2}}}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solve $\log_7 3 = y$; $\log_7 2 = z$; $x = \log_3 2$ I am trying to figure out they logarithmic equation but it has 3 semicolons and I can not find out what they mean. How do I go about solving it?
I have to solve for $x$. The answer should be $z/y$.
$\log_7 3 = y$; $\log_7 2 = z$; $x = \log_3 2$
| Note that the 3 equations are equivalent to
$$7^y =3, 7^z=2, \mbox{ and } 3^x=2.$$
Replace the $3$ in the last equation with $7^y$, by dint of the first equation. And replace the $2$ in the last equation with $7^z$ per the second equation,
and you have
$$(7^y)^x = 7^z$$
or
$$7^{xy} = 7^z.$$
Therefore $xy = z$ and so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Polynomial $ax^2 + (b+c)x + (d+e)$ Let $a, b, c, d$ be real number such that polynomial $ax^2 + (b+c)x + (d+e)$ has real roots greater than $1$. Prove that polynomial $ax^4+bx^3+cx^2+dx+e$ has at least one real root.
Is my work correct ?
Let $r$ be real root of $ax^2+(c+b)x+(e+d)$, so $ar^2+cr+e=(br+d)(-1)$.
Let $P(x)... | Assume the roots are $r_1,r_2$. Then:
$$a(x-r_1)^2(x-r_2)^2=ax^2+(-ar_1-ar_2)x+ar_1r_2=0.$$
Hence the second equation:
$$f(x)=ax^4-ar_1x^3-ar_2x^2+(ar_1r_2-e)x+e=0.$$
Note:
$$f(r_1)=-er_1+e$$
$$f(0)=e$$
Now IVT is applicable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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How to find the general solution for this ODE? I'm really stuck on how to go about solving the following first order ODE; I've got little idea on how to approach it, and I'd really appreciate if someone could give me some hints and/or working for a solution so I can have a reference point on how to approach these sorts... | This kind of ODE should be solved as follows:
*
*Solve the corresponding homogeneous equation.
In your case it is $y'+y\cos(x)=0$ which has solution $y=c\cdot e^{-\sin(x)}$.
*Consider constant in previous solution as a function of variable $x$ and substitute it in original equation.
So, we have $y(x)=c(x)\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Polynomial equal to polynomial of lower degree I am studying Linear Algebra Done Right, chapter 2 problem 6 states:
Prove that the real vector space consisting of all continuous real valued functions on the interval $[0,1]$ is infinite dimensional.
My solution:
Consider the sequence of functions $x, x^2, x^3, \dots$ ... | Then the polynomial $\displaystyle x^n-\sum_{k=0}^{n-1}a_kx^k$ would have infinitely many roots, but it can have $n$, at most.
Another way of dealing with this problem is based upon defining polynomials (in one variable $x$) as expressions of the type $a_0+a_1x+a_2x^2+\cdots+a_nx^n$, where $n\in\{0,1,2,\dots\}$ and ea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Orthogonal Bases w.r.t a given Bilinear Form. Under what conditions there exists an orthogonal basis? Or even better, is there a characterization of the existence of an orthogonal basis in terms of a given bilinear form and/or the base field?
For instance, if the characteristic of the field is not 2 and the bilinear fo... | Let's take $B$ to be a bilinear form $B : V \times V \rightarrow K$ on a finite-dimensional $K$-vector space $V$; where $K$ is a field of characteristic $p$.
Take an orthogonal basis, i.e. a basis $e_1, \ldots, e_n$ such that $B(e_i, e_j) = 0$ if $i \neq j$.
*
*If $B$ is alternating, then $B(e_i, e_i) = 0$, so in fa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How fast can one move around an ellipse with bounded acceleration? Given a smooth closed planar curve $\Gamma$, I'm looking for its periodic parametrization $\phi : \mathbb{R}\to\Gamma$ such that
*
*the second derivative $\phi''$ is bounded by $1$ in the norm: $|\phi''|\le 1$
*the period $T$ of the parameterizatio... | I see the problem as follows. For an ellipse with semi-major/minor axes $(A,B)$, the trajectory is parameterized in the complex plane as
$$z=A\cos(t/\sqrt{a})+iB\sin(t/\sqrt{a})$$
Here, $a$ must have the dimensions [$\text{t}^2$], and, of course $(A,B)$ are [L].
The velocity and acceleration are given by
$$
v=\dot z=\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 1
} |
Partial fractions and linear vs quadratic factors I was watching some videos on partial fraction decompistion and I got confused on one of the examples:
Say for example you have $$\frac{x+4}{x^2(x^2 +3)^2}.$$
The partial fraction equation of this is apparently:
$$\frac{A}{x} + \frac{B}{x^2} + \frac{Cx+E}{x^2 +3} ... | Hint: $$\frac{ax+b}{x^2} = \frac{a}{x}+\frac{b}{x^2}.$$ Therefore, $$\frac{A}{x}+\frac{ax+b}{x^2}=\frac{A'}{x}+\frac{b}{x^2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Expectation of a die roll summed I know this problem involves conditional probability, but I'm confused as to how to tackle it.
Assume a die is rolled over and over, where the total is summed. If the die's roll is $\geq 3$ the game stops and the summed total is read out. What is the expectation of the total? What is th... | You have $p=1/3$ to roll a die and get only $1$ or $2$.
So you have $P(n)=p^n q=p^n (1-p)$ to roll the die $n$ times getting less than $3$, and then more or equal $3$ at the $n+1$-th roll.
We include $n=0$, meaning that you get $\ge 3$ at the first roll.
The sum P(n) over $0 \le n < \infty$ correctly gives $1$.
Now, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2384987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
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Brownian Motion Differential Equation The behavior of $S(t)$ through time is modeled by $() = \, + \,()$ for a $()$ standard Brownian motion and real value μ and σ > 0. Now, let $() =
\frac{1}{S(t)}$ Show that $U(t)$ satisfies the following stochastic differential equation.
$$() = (^2 − )\, − \,()$$
I solved the DE re... | It seems like there might be some typos in your question. Firstly, $S_t$ is not a standard Brownian motion since it has a non-zero "drift term" and non-unity "diffusion coefficient". Secondly, the equation:
$$ dS_t = \mu \,dt + \sigma\,dW_t $$
has solution $$ S_t=\mu t+\sigma W_t +S_0 $$
On the other hand, geometric Br... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Reduce a set of vectors of $\mathbb{R}^3$ to a basis $V_1 =(1,0,0); V_2=(0,1,-1); V_3= (0,4,-3); V_4=(0,2,0)$ reduce this to obtain a basis for $\mathbb{R}^3$ . I obtained a matrix as follows and then obtained its row echelon form, but then i cannot understand how I obtain basis from that matrix.
$$\begin{bmatr... | Since $v_1, v_2, v_3$ are linearly independent, and we know that $\mathbb{R}$ has the dimension 3, the vectors $v_1, v_2, v_3$ forms a basis for $\mathbb{R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Is a constant function periodic? Can we regard a constant function "$f(x)=\text{constant}$" to be a periodic function? If yes, what is its period?
| Nowhere in the definition of a period function is it stated that the function must have a least period.
If $f(x) = c$ then for any $p$ we have $f(x+p) = f(x)$. So $f$ is periodic and $p$ is a period. Obviously any other non-negative value will also be a period.
There is nothing in the definition of periodic function ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Stuck on proof of step 4 on Thm 7.32 on baby Rudin. According to baby Rudin's thm 7.32
Then the uniform closure $B$ of $A$ consists of 'all' real continuous functions of $K$.
and Step 4 in proof:
Given a real function $f$, continuous on $K$, and $\epsilon>0$ there exists a function $h\in B$ such that
$$|h(x)-f(x)|<\eps... | The statement of part 4 can be through of like this. Given any real continuous function $f$ on $K$, we can find a sequence of $h_{n}\in B$ such that $h_{n}\to f$ uniformly.
Uniformly closed means that for any uniformly convergent sequence $h_{n}\in B$, $\lim_{n\to\infty} h_{h}\in B$. Thus, we conclude via part 4 that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Sum of series of fractions I am trying to find the $f$ formula that returns the sum of the series created by fractions that have constant nominator and shifting by one denominator.
Here are some examples:
$$f(3) = \frac{3}{1} + \frac{3}{2} + \frac{3}{3} = 5.5$$
or
$$f(4) = \frac{4}{1} + \frac{4}{2} + \frac{4}{3} + \fr... | The numbers
$$1+\frac12+\frac13+\cdots+\frac1n$$
are called the harmonic numbers and often denoted $H_n$. There is
no simple closed formula for $H_n$, but $H_n$ is approximately $\ln n+\gamma$ for large $n$, where $\gamma$ is Euler's constant.
You are considering $nH_n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
To prove that $a^m.a^n=a^{m+n}$
In a group $G$, I want to prove this theorem:
$$\forall a\in G,\; a^m.a^n=a^{m+n},\;m,n\in \Bbb Z.$$
I am thinking that only associative law is sufficient to prove this.please give some suggestions or hint to prove this.thanks in advance
| It is clear that $a^{m+1}=a^m.a$. On the other hand, if $a^{m+n}=a^m.a^n$, then$$a^{m+n+1}=a^{m+n}.a=a^m.a^n.a=a^m.a^{n+1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Simplification of this integral I have an integral $\int_0^1 \sqrt{e^{2x} + e^{-2x} + 2}$ which the solution says simplifies to $ \int_0^1 e^{-x}(e^{2x} + 1)$. I understand the simplification but what happened to the constant $\sqrt{2}$? It's been awhile since I've done any computational math stuff so maybe it's just s... | Alternatively, take $e^{-2x}$ out:
$$\int_0^1 \sqrt{e^{-2x}}\sqrt{(e^{4x}+2e^{2x}+1)}dx=\int_0^1 e^{-x}\sqrt{(e^{2x}+1)^2}dx=\int_0^1 e^{-x}(e^{2x}+1)dx.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Does this collection form an algebra over $\mathbb{R}$? Does the collection
$$\mathcal{A}=\{V\cup K\mid V\subset\mathbb{R}\textrm{ is open and } K \subset \mathbb{R} \textrm{ is closed}\}$$
form an algebra over $\mathbb{R}$?
| No, it does not. Note that every open set $U$ is in $\mathcal{A}$ since $U = U \cup \emptyset$ and $\emptyset$ is closed; and similarly, every closed set $K$ is in $\mathcal{A}$. Now, if $\mathcal{A}$ were an algebra, that would imply every intersection of an open set and a closed set in $\mathbb{R}$ would also be a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2385852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
formula for $a_n$ where $a_{n+1}=4-a_n-\frac1{a_n}$ and $a_1=1$? There's a sequence $a_{n+1}=4-a_n-\frac{1}{a_n}$ staring with $a_1=1$.
Is it possible to find a general formula for $a_n$?
| Not a full answer, but a few steps in a possibly-fruitful direction: first of all, let's remove the inhomogeneous term. Set $a_n=2+b_n$; then we can write $(2+b_{n+1})=4-(2+b_n)-\frac1{2+b_n}$ $=2-b_n-\frac1{2+b_n}$, or in other words, $b_{n+1}=-b_n-\frac1{2+b_n}$.
Now, we write $b_n=\frac{x_n}{y_n}$ and equate numera... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2386068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Writing product and remainder Given some positive integers, an operation is to choose two integers $a\geq b$, delete them, and write $ab$ and $a\pmod b$ instead. Must the number $0$ eventually appear?
For example, when there are two numbers, the number $0$ must eventually appear. This is because one number keeps increa... | If you have $(a_1,a_2,...,a_n)$ and you choose $a_i$ and $a_j$ with $i<j$, and replace $a_i$ with the modulo and $a_j$ with the product, you may notice that the modulo is less than the minimum of $a_i$ and $a_j$.
So the new tuple is smaller than the initial one via lexicographical order.
Now use that $\mathbb N^n$ is w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2386184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prob. 15, Chap. 6, in Baby Rudin: If $f$ is a real, continuously differentiable function on $[a, b]$, . . . Here is Prob. 15, Chap. 6, in the book Principles of Mathematical Analysis, by Walter Rudin, 3rd edition:
Suppose $f$ is a real, continuously differentiable function on $[a, b]$, $f(a) = f(b) = 0$, and $$ \int_... | If the equality was not strict, it is not to hard to prove that $$xf(x) = \lambda \cdot f'(x) \tag{1}$$ for some $\lambda \in \mathbb{R}$, i.e, the Cauchy-Schwarz inequality is an equality when the terms are linearly dependent.
Case 1 ($\lambda = 0 $)
Then $f = 0$ which leads to a contradiction with the fact that $$\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2386295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the difference between the normal vector to a surface given by the traditional formula and the one given by the gradient?
What is the difference between the normal vector to a surface given by the traditional formula and the one given by the gradient?
In my class we learnt that the normal vector to a surface ... | Well, there are different way to describe a surface with a function
When you say "a surface described by a function $f(x,y)$", you probably mean a surface described by the equation $$
z=f(x,y) \tag{*}$$
and not $$f(x,y)=0$$ because in the second case, the normal vector would have zero $z$ component.
The gradient formul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2386409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find all polynomials $p(x)$ satisfying condition Find all polynomials $p(x)$ such that $\sin{p(x)}$ is periodic.
I think that the condition for the solution is $\deg{p(x)}\le 1$, but i can't find a formal way to prove it.
| You need to meet
$$p(x+T)=p(x)+2k\pi$$ or
$$p(x+T)-p(x)=2k\pi.$$
But for the LHS to be continuous, $k$ must be constant, hence $p(x)$ must indeed be linear.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2386540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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