Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Showing Linear Dependence My task is to show that the set of vectors: $\bf x_1, x_2, x_3, x_4$ where
$\bf x_1=[1,0,0]$
$\bf x_2=[t,1,1]$
$\bf x_3=[1,t,t^2]$ and
$\bf x_4=[t+2,t+1,t^2+1]$ are linearly dependent. (Note: $x_i$ can also be written in matrix format.)
To show that they are linearly dependent I form the equa... | Hint:
You can make an easy solution if you use the fact that if some vector in a list of vectors is a linear combination of other vectors in that same list, then the list is linearly dependent.
| {
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"url": "https://math.stackexchange.com/questions/1147152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show $z=0$ is a simple pole or removable singularity if $r\int_{0}^{2\pi}|f(re^{i\theta})|d\theta \lt M$ I am trying to show that if $f$ analytic in $0 \lt |z| \lt R$ and there exists $M \gt 0$ such that for every $0 \lt r \lt R$ we have:
$$r\int_{0}^{2\pi}|f(re^{i\theta})|d\theta \lt M$$
then $z=0$ is either a simple ... | I was able to find a solution using a different approach:
Using Laurent's theorem we have:
$$a_n = \frac{1}{2\pi i}\int_C \frac{f(\zeta)}{\zeta^{n+1}}d\zeta$$
where $C=re^{i\theta}$
and then using the given property of $f$:
$$
|a_n| = \left| \frac{1}{2\pi i}\int_C \frac{f(\zeta)}{\zeta^{n+1}}d\zeta \right| \le \frac{M}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1147224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What is the mathematical meaning of this statement made by Gödel (see details)? It appears as Proposition VI, in Gödel's 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I" :
"To every $\omega$-consistent recursive class $\kappa$ of formulae there correspond recursive clas... | See :
*
*Jean van Heijenoort, From Frege to Gödel : A Source Book in Mathematical Logic (1967), page 592-on.
$\kappa$ is a set of ("decidable") formulas and $Flg(\kappa)$ is the set of consequnces of $\kappa$, i.e. the set of formulas derivable from the formulas in $\kappa$ plus the (logical) axioms by rules of in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1147360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$ \lim_{x \to \infty} (\frac {x+1}{x+2})^x $ For the limit
$ \lim_{x \to \infty} (\frac {x+1}{x+2})^x $
could you split it up into the fraction $ \lim_{x \to \infty} (1 - \frac{1}{x+2})^x$
and apply the standard limit $ \lim_{x\to+\infty} \left(1+\frac{k}{x}\right)^x=e^k $
or would you have to apply L'Hopitals rule
edi... | $$\left( \frac{(x+2) - 1}{x+2} \right)^x = \left( 1 - \frac{1}{x+2} \right)^{x+2} \cdot \left( 1 - \frac{1}{x+2} \right)^{-2} \to e^{-1} \cdot 1 $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Creating a Hermitian function Say I have an operator $A$ such that $A^\dagger = B$.
I want to construct a Hermitian function, $f$, of these operators, $f(A,B)^\dagger = f(A,B)$. Is it possible to construct a function $f$ such that $f$ is not a function of $A+B$?: $f(A,B)\neq f(A+B)$ but $f(A,B)^\dagger = f(A,B)$?
| A trivial answer might be any product of $A$ and $A^{\dagger}=B$ is an hermitian function take $f_1(A,B)=AB=AA^{\dagger}$ and than since also every power of $AB$ is an hermitian function: ${(AB)^n}^{\dagger}=B^{\dagger}A^{\dagger}B^{\dagger}A^{\dagger}....B^{\dagger}A^{\dagger}=ABAB....AB = (AB)^n$ and since you can de... | {
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Can we prove that there is no limit at $x=0$ for $f(x)=1/x$ using epsilon-delta definition? The $\varepsilon-\delta$ definition of limit is:
$$\lim_{x \to a}f(x)=l \iff (\forall \varepsilon>0)(\exists \delta>0)(\forall x \in A)(0<|x-a|<\delta \implies |f(x)-l|<\varepsilon)$$
Consider there is $\lim\limits_{x \to 0}f(x)... | The task is for any given number $l$ and for any given tolerance challenge $\epsilon$ to find a positive $\delta$ that will for all $x \in (-\delta,0) \cup (0, \delta)$ keep
$$
\left\lvert \frac{1}{x} - l \right\rvert < \epsilon \quad (*)
$$
The problematic item is the $0$ end of the intervals where to choose the $x$ f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1147623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simplify a series of $\cos x$ or $\sin x$ Suppose we have this series
$$\sum\limits_{k=0}^n a^k \cos(kx) = 1 + a\cos(x) + a^2 \cos(2x) + a^3 \cos(3x) + \cdots + a^n\cos(nx)$$
What should we do to simplify this?
| You should give more information about what you have tried. In the meantime
Hint: $$\cos kx =\frac {e^{ikx}+e^{-ikx}}2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1147717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Cyclotomic fields of finite fields There is a fact that if $K=\mathbb Q$, then the Galois group of the $n$th cyclotomic field over $\mathbb Q$ is isomorphic to $\mathbb Z_n^{*}$.
In the case were $K$ is an arbitrary field, we have that the Galois group of the $n$th cyclotomic field is isomorphic to a subgroup of $\math... | If you only want to know what the Galois group is, in case the base is a finite field, the answer is easy, when you remember that the groups always are cyclic. In the case $\Bbb F_7$ and $X^5-1$, all you need to do is ask what the first power of $7$ is such that $5|(7^n-1)$, that’s the order of the multiplicative group... | {
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Count good sets with given conditions We need to find the numbers of good sets in a sequence. The good set is:
A good set is a sequence of $P$ positive integers which satisfies the following 2 condition :
*
*If an integer $L$ appears in a sequence then $L-1$ should also appear in the sequence
*The first occurrence ... | Here's one way to attack it:
Let $P(n)$ denote the number of good tuples of length $n$. Claim: $P(n) = n!$. Here is a proof sketch using induction on $n$.
$P(1) = 1$, OK. Suppose it's true up to and including $n-1$. You can map each of the $(n-1)!$ good tuples of length $n-1$ to a unique good $n$-tuple as follows:
Let... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can someone verify the reasoning of why these two congruences are equivalent? I've wondered why two congruences like $x\equiv81\pmod {53}$ and $x\equiv28\pmod{53}$ were equivalent.
I've come up with this proof. I am not sure if this is how you prove it though
I know that $x\equiv81\pmod{53}$ is saying that the differen... | Your argument is fine, but a bit longer than necessary.
Note that, in general $d\mid n \iff d\mid (n-d)$.
So $53\mid (x-28) \iff 53\mid ((x-28)-53)=x-81$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1148079",
"timestamp": "2023-03-29T00:00:00",
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Finite sum of the Leibniz formula for $\pi/4$ The Leibniz formula for $\pi/4$ is:
$$\sum_{i=0}^\infty \frac{(-1)^i}{2i+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ... $$
How to calculate the summation of the first $n$ terms in the above series?
| $$\sum_{i=0}^n\frac{(-1)^i}{2i+1}=\frac{(-1)^n}{2}[\psi_0(3/2+n)-\psi_0(1/2)],$$
where $\psi_0(x)$ is the Polygamma function (also known as the DiGamma function for this case). There are likely no further simplifications possible (if this even counts as one).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1148194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solving the Ornstein-Uhlenbeck Stochastic Differential Equation I am asked to solve the following SDE:
$$dX_t = (a-bX_t)dt + cdB_t,\ \text{ where }X(0) = x.$$
($(B_t)_{t\ge0}$ is a brownian motion.)
For constants $a$, $b$ and $c$ and $X$ is a random variable independent of brownian motion.
Also, find $m$ and $\sigma$ ... | Your solution to the ODE is incorrect. Indeed,
$$
\mathrm d\alpha_t = (a-b\alpha_t)\mathrm dt,\quad\alpha_0=1,
$$
has solution
$$
\alpha(t)={\mathrm e}^{-bt}+\frac ab\left(1-{\mathrm e}^{-bt}\right),\quad t\ge0.
$$
Additionally you made a mistake when equating (I) and (II), namely you should get
$$
aY_t\mathrm dt+\alph... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$\alpha = \frac{1}{2}(1+\sqrt{-19})$ I have asked a similar question here before, which was about ring theory, but it is slightly different today and very trivial.
$\alpha = \frac{1}{2}(1+\sqrt{-19})$
Here, $\alpha$ is a root of $\alpha^2 - \alpha + 5$ and $\alpha^2 = \alpha - 5$, but I can't seem to understand this.... | $\alpha^2-\alpha+5=\frac{1}{4}(1+\sqrt{-19})^2-\frac{1}{2}(1+\sqrt{-19})+5=\frac{1}{4}(1+2\sqrt{-19}+(-19))-\frac{1}{4}(2+2\sqrt{-19})+\frac{1}{4}\cdot 20=\frac{1}{4}(1+2\sqrt{-19}+(-19)-2-2\sqrt{-19}+20)=0$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving $\frac{n^n}{e^{n-1}} 2$. I am trying to prove
$$\frac{n^n}{e^{n-1}}<n!<\frac{(n+1)^{n+1}}{e^{n}} \text{ for all }n > 2.$$
Here is the original source (Problem 1B, on page 12 of PDF)
Can this be proved by induction?
The base step $n=3$ is proved: $\frac {27}{e^2} < 6 < \frac{256}{e^3}$ (since $e^2 > 5$ and $... | Proof: We will prove the inequality by induction. Since $e^2 > 5$ and $e^3 < 27$, we have
$$\frac {27}{e^2} < 6 < \frac{256}{e^3}.$$ Thus, the statement for $n=3$ is true. The base step is complete.
For the induction step, we assume the statement is true for $n=k$. That is, assume $$\frac{k^k}{e^{k-1}}<k!<\frac{(k+1)^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1148442",
"timestamp": "2023-03-29T00:00:00",
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Necessary condition for the convergence of an improper integral. My calculus professor mentioned the other day that whenever we separate an improper integral into smaller integrals, the improper integral is convergent iff the two parts of the integral are convergent. For example:
$$ \int_0^{+\infty} \frac{\log t}{t} \m... | $$
\int_{-\infty}^{+\infty} x\,dx = \text{what?}
$$
$$
\int_{-\infty}^{+\infty} x\,dx = \underbrace{\int_{-\infty}^0 x\,dx}_{\text{Diverges to }-\infty} + \underbrace{\int_0^\infty x\,dx}_{\text{Diverges to }+\infty}
$$
But
$$
\lim_{a\to\infty} \int_{-a}^a x\,dx = 0.
$$
Here's a more involved example:
$$
\lim_{a\to\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1148527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why must there be an infinite number of lines in an absolute geometry? Why must there be an infinite number of lines in an absolute geometry? I see that there must be an infinite number of points pretty trivially due to the protractor postulate, and there are an infinite number of real numbers. Hence, theres an infinit... | Pick one point $P$. Through $P$ and each other point $Q$ there's a line, $\ell_Q$, by axiom 1.
Case 1: There are infinitely many $\ell_Q$, and we're done.
Case 2: There are finitely many lines $\ell_X$. In that case, one of them -- say $\ell_Q$ -- must contain infinitely many points $R_1, R_2, \ldots$. (Because we kno... | {
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How are first digits of $\pi$ found? Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am talking about accurate digits by either multiplication or division or any other ope... | In the 18th century, Leonard Euler discovered an elegant formula:
$$\frac{π^4}{90}=\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\frac{1}{5^4}+\frac{1}{6^4}+\dots$$
The more terms you add, the more accurate the calculation of $π$ gets.
$$\frac{π^4}{90}=\frac{1}{1^4}=1.000$$
then $π=3.080$
$$\frac{π^4}{90}=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1148682",
"timestamp": "2023-03-29T00:00:00",
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Computing probabilities involving increasing event outcomes Suppose I have an unfair die with three outcomes: $A,B,C$
\begin{align*}
A &\sim 1/512 \\
B &\sim 211/512 \\
C &\sim 300/512 \\
\end{align*}
The case of one dice is trivial. However, if I roll two of these dice, what is the probability of at least one of the d... | The previous answers flesh out the standard take quite well. Here's another slant on these kinds of problems.
Treat the die faces as a polynomial:
$(\frac{a}{512}+\frac{211 b}{512}+\frac{300 c}{512})$
Squaring this represents rolling two (or twice), cubing it three, etc.
Squared, we get:
$(\frac{a^2}{262144}+\frac{211 ... | {
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Every $2$-coloring of $K_n$ contains a hamiltonian ...
Every $2$-coloring of $K_n$ contains a hamiltonian cycle that is made out of two monochromatic paths.
If there is a monochromatic hamiltonian cycle then we are done. Elseway I thought about using Dirac's theorem somehow (I know that there is in both monochromatic... | Choose any vertex $v \in K_n$. Then $K_n \setminus v$ is a 2-colored copy of $K_{n-1}$, and thus by induction contains a Hamiltonian cycle that is the union of a red path and a blue path. Let $u$ be a vertex where these two paths meet, and let $r$ be $u$'s neighbor along the red path, and $b$ be $u$'s neighbor along... | {
"language": "en",
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Identify quotient ring $\mathbb{R}[x]/(x^2-k), k>0$ I need to identify $\mathbb{R}[x]/(x^2-k)$, where $k>0$ (if $k<0$ I believe it's isomorphic to $\mathbb{C}$).
If we let $f(x) = x^2-k$, then according to Artin, since $\sqrt{k}$ satisfies $f(x)=0$ the set $(1, \sqrt{k})$ is a basis for $\mathbb{R}[\sqrt{k}]$ = $\mathb... | The mistake you made is in the assertion that $\{1,\surd k\}$ is a basis. This is true only in case $\surd k$ is not in the field you started with. But the field of real numbers has square roots (two) for every positive real number. So you get a ring that is a direct product of two copies of R. In particular it has zer... | {
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Generating $\pi$-system for $\mathbb{Z}_+$ As we know from the definition a $\pi$-system is a collection where the intersection of two sets is again in that collection. In fact we can use $\pi$-systems to generate $\sigma$-algebra's. We know that a $\pi$-system generates a $\sigma$-algebra if the minimum $\sigma$-algeb... | Notice that for $n\geqslant 1$, $\{ n\}=\{0,\dots,n\}\setminus\{0,\dots,n-1\}$, which is an element of the $\sigma$-algebra generated by $\mathcal I$. We thus deduce that $\{0\}\in\sigma(\mathcal I)$.
We have shown that $\{n\} \in\sigma(\mathcal I)$ for each $n\in\mathbb Z_+$. Since each element of the power set of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1149109",
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Why is the intersection of the empty set the universe? I haven't found an answer yet that I can make complete sense of, so I'm asking again to try and clear it up if I have any misconceptions.
If one defines $\cap_{i\in I}\alpha_i = \{x:\forall i\in I, x\in\alpha_i\}$ then apparently if $I = \emptyset$ this definition... | Any statement of the form $\forall i\in I, ...$ is true if $I$ is empty, so in particular the statement $\forall i\in I, x\in \alpha_i$ is true for all $x$. Perhaps the easiest way to see this, is that the negation of this statement is: $\exists i\in I$ such that $x\notin \alpha_i$, which is clearly false if are no $i$... | {
"language": "en",
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Probability that binomial random variable is even Suppose $X$ is binomial $B(n,p)$. How can I find the probability that $X$ is even ?
I know $$P(X = k ) = \frac{ n!}{(n-k)!k!} p^k(1-p)^{n-k} $$
where $X=1,....,n$. Are they just asking to find $P(X = 2m )$ for some $m > 0$ ?
| I claim that the answer is $\frac{1}{2}(1+(1-2p)^n)$.
To see why, first of we have to realise what the binomial distribution actually stands for. If an event happens with probability $p$, and we make $n$ trials, then $P(X=k)$ as you defined tells us the chance that we have $k$ "successes", i.e. the chance that the even... | {
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$\beta \mathbb {N}$ is compact I want to show the compactness of the set $\beta \mathbb N := \{ \mathcal U \mid \mathcal U \text{ is an ultrafilter on } \mathbb N \}$, with the topology induced by the basis $U_M = \{\mathcal U \in \beta \mathbb N \mid M \in \mathcal U \}$.
Since $\beta \mathbb N \subset \prod\limits_{... | Suppose that $x\in{^{\wp(\Bbb N)}\{0,1\}}\setminus\beta\Bbb N$, and let $\mathscr{X}=\{A\subseteq\Bbb N:x(A)=1\}$. Then $\mathscr{X}$ is not an ultrafilter on $\Bbb N$, so it must satisfy one of the following conditions:
*
*$\varnothing\in\mathscr{X}$;
*there are $A,B\subseteq\Bbb N$ such that $B\supseteq A\in\ma... | {
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Find The Equation This is the question:
Find the equations of the tangent lines to the curve $y = x − \frac 1x + 1$ that are parallel to the line $x − 2y = 3$.
There are two answers: 1) smaller y-intercept; 2) larger y-intercept
The work:
The slope of the line is (1/2).
$y' = ((x + 1) - (x - 1))/(x + 1)^2 = 2/(x + 1)^2... | there are no points on the graph of $y = x - \frac1x + 1$ has a tangent that is parallel to $x - 2y = 3.$
i will explain why.
you will get a better idea of the problem if you can draw the graph of $y = x - \frac1x + 1$ either on your calculator or by hand. the shape of the graph is called a hyperbola, similar looking t... | {
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A descending chain and choice axiom.
Let $R$ be a relation on a set $A$ such that for every $x\in A$ there exist a element $y\in A$ such that $x\mathrel{R}y$ the there exist a function $f\colon\omega\rightarrow A$ such that $f(n^+)Rf(n)$.
I would like some hint to this exercise using axiom of choice.
Thanks!
| HINT: Fix a choice function $G$ on non-empty subsets of $A$, and some $x_0$ in $A$, and by induction define $f(0)=x_0$, and $f(n+1)$ to be a chosen element from the relevant set of witnesses.
| {
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Is {∅} a subset of the power set of X, for every set X? I am reading the book "A Transition to Advanced Mathematics" 2014 edition, written by Smith, Eggen and St. Andre.
For every set X:
∅ ⊆ X
∅ ∈ power set of X
∅ ⊆ power set of X
{∅} ⊆ power set of X
Is {∅} ⊆ power set of X wrong?
I am an undergraduate student. Pl... | The empty set $\emptyset$ is a subset of every set $X$. Hence $\emptyset$ is a member of the power set of $X$, $P(X)$. Therefore $\{\emptyset\}$ is a subset of $P(X)$.
| {
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Proving that on a Lie group $G$ the space of left-invariant vector fields is isomorphic to $T_e G$ Let $G$ be a Lie Group. Given a vector $v\in T_e G$, we define the left-invariant vector field $L^v$ on $G$ by
$$L^v(g)=L^v|_g=(dL_g)_e v.$$
I want to show that $v\mapsto L^v$ is a linear isomorphism between $T_e G$ and ... | This is sufficient, though it's perhaps slightly awkward in that it defines $v$ to be the image of different vectors of different maps $(dL_{g^{-1}})_g$, all of which happen to coincide. One may as well take $v$ to be the image under just one of these maps, say, the one for $g = e$. Explicitly, this is just the map $\m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1149862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f$ is locally Lipschitz on $X$ and $X$ is compact, then $f$ is Lipschitz on $X$. If $f$ is locally Lipschitz on $X$ and $X$ is compact, then $f$ is Lipschitz on $X$.
My proof: Since $f$ is locally Lipschitz on $X$, for each $x ∈ X$ there exists an
open $B_x$ containing $x$ such that $f$ is Lipschitz on $B_x$. Consi... | First note that locally Lipschitz implies continuous. This said, the argument suggested can be completed as follows. For every $x\in X$ there is an open ball $B_x$ centered at $x$ of radius $\varepsilon_x$ and a constant $M_x$ such that $d(f(x),f(y))\le M_xd(x,y)$ for $x,y\in B_x$. Now consider the open convering consi... | {
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"source": "stackexchange",
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Solve the differential equation $y'-xy^2 = 2xy$ I get it to the form $\left | \dfrac{y}{y+2} \right |=e^{x^2}e^{2C}$ but I'm not sure how to get rid of the absolute value and then solve for y. I've heard the absolute value can be ignored in differential equations. Is this true?
| If $|A|=B$, then $A$ must be either $B$ or $-B$, so your equation is equivalent to
$$
\frac{y}{y+2}=\pm e^{2C} e^{x^2}
.
$$
Now let $D=\pm e^{2C}$ and solve for $y$. (Since $C$ is an arbitrary constant, $D$ will be an arbitrary nonzero constant.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1150055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find c for which tangent line meets $c/(x+1)$ Here is the problem:
Determine $c$ so that the straight line joining $(0, 3)$ and
$(5, -2)$ is tangent to the curve $y = c/(x + 1)$.
So, what I know: the line connecting the points is $y=-x+3$.
I need to find a value for c such that the derivative of $c/(x+1)$ can be $-x+... | Hint:
Here are two curves whose slope are supposed to be equal:
$$
y=-x + 3
$$
and
$$
\frac{c}{x+1}=y
$$
equating the slopes of both of curves,
we have
$$
-1=\frac{-c}{(x+1)^2}
$$
this gives you
$$
(x+1)^2=c\implies x=-1\pm \sqrt c
$$
note that from here we need $c$ should be non-negative. i.e. $c\geq 0$
Now at $x=-1\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1150171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding Radical of an Ideal
Given the ideal $J^\prime=\langle xy,xz-yz\rangle$, find it's radical.
I know that the ideal $\langle xy,yz,zx\rangle$ is radical ideal but that's not the case. How can I compute the radical here?
| The ideals $J^\prime=\langle xy,xz-yz\rangle\subset J= \langle xy,yz,zx \rangle$ have the same vanishing set in $\mathbb A^3 $, namely the union of the three coordinate axes, so that $\sqrt {J'}=\sqrt J$.
Since you know that $\sqrt J=J$, you can conclude that $$ \sqrt {J'}=J $$
[The same result can be reached ... | {
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Why $s(1-s)$ numbers which are squares in a field are written $\frac{u^2}{1+u^2}$? Trying to exercise in Math again after years of other activities, I need a little help on this :
Let $F$ be some arbitrary field with characteristic > 2.
Let $S \subseteq F$ be the set of numbers s such that $s(1-s)=a^2$ for some $a \in ... | Because $\frac{s}{1-s}$ is also a square.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\sqrt{4 -2 \sqrt{3}} = a + b\sqrt{3}$, where numbers $a$ and $b$ are rational If $a$ and $b$ are rational numbers such that $\sqrt{4 -2 \sqrt{3}} = a + b\sqrt{3}$
Then what is the value of $a$? The answer is $-1$.
$$\sqrt{4 - 2\sqrt{3}} = a + b\sqrt{3}$$
$$4 - 2\sqrt{3} = 2^2 - 2\sqrt{3}$$ Let $u =2$ hence,
$$\sqrt{u... | Using formula
$$\sqrt{a\pm\sqrt{b}}=\sqrt{\dfrac{a+\sqrt{a^2-b}}{2}}\pm\sqrt{\dfrac{a-\sqrt{a^2-b}}{2}}$$
you will get
$$\sqrt{4-2\sqrt3}=\sqrt{4-\sqrt{12}}=\sqrt{\dfrac{4+\sqrt{16-12}}{2}}-\sqrt{\dfrac{4-\sqrt{16-12}}{2}}=-1+\sqrt3$$
So, $a=-1$ and $b=1$.
| {
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How to show that if there's a fast matrix inversion algorithm, then there's a fast multiplication algorithm? Is there a way to show this and vice versa?
Suppose $F_n$ is the number of flops required by some algorithm to perform the inversion of an $n-by-n$ matrix. Assume that there exists a constant $c_1$ and a real nu... | To compute $XY$ using $*^{-1}$ function.
Since there is no upper limit on $c_2$, you can construct a matrix larger than $X$ and $Y$ by any constant factor (two times larger, three times larger, etc) and invert it.
So looking at http://en.wikipedia.org/wiki/Invertible_matrix#Blockwise_inversion we want to find a matrix ... | {
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"source": "stackexchange",
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Sniper probability question
There are 2 snipers and they are competing each other in game where winner is the one who hits the target first. First sniper's hit probability is 80%, the second's probability is 50%. In the same time second sniper shoots two times faster than the first.
The question is which sniper has t... | Answer:
Sniper1 hits the target in a shot $= 0.8$
Sniper1 does not hit the target in a shot$= 0.2$
Sniper 2 can fire two shots in the same time as Sniper 1. Let us look at the possibilities.
Case 1: Sniper 2 hits the target in the first shot(H)$ = 0.5$
Case 2: Sniper 2 hits the target in the second shot (NH)$ = 0.25$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A question concerning binary operations and isomorphisms Consider two nonempty sets $A$ and $B$, such that $A$ is isomorphic to $B$. Now, if $B$ is a group under some binary operation $*$, does it necessarily imply that there exists an operation $*'$, under which $A$ is a group? If this is true, what happens if I stren... | If there is a bijection $f\colon A\to B$ and $*$ is a group operation on $B$, you can define, for $x,y\in A$,
$$
x*'y=f^{-1}(f(x)*f(y))
$$
This defines an operation on $A$ and $f$ becomes an isomorphism between $(A,*')$ and $(B,*)$, since, by definition,
$$
f(x*'y)=f(x)*f(y)
$$
and $f$ is bijective. It's just routine t... | {
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The sum of the reciprocal of primeth primes A few days ago, a friend of mine taught me that the sum of the reciprocal of primeth primes $$\frac{1}{3}+\frac{1}{5}+\frac{1}{11}+\frac{1}{17}+\frac{1}{31}+\cdots$$
converges.
Does anyone know some papers which have a rigorous proof with the convergence value?
| It is a duplicate. We have $p_n\gg n\log n$ by Chebyshev's theorem (or the PNT) hence $$p_{p_n}\gg p_n \log n \gg n\log^2 n$$ and $$\sum_{n\geq 1}\frac{1}{n\log^2 n}$$
is convergent by Cauchy's condensation test.
| {
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"timestamp": "2023-03-29T00:00:00",
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prove that there are infinitely many k such that $\phi(n) = k$ has precisely two solutions I am asked to prove that there are infinitely many $k$ such that $\phi(n) = k$ has precisely two solutions.
I think for any prime number $p \mid n$, we have $(p-1) \mid \phi(n)$. But I am not sure how to proceed from here.
Coul... | HINT: Suppose that $p_1^{k_1}p_2^{k_2}\ldots p_m^{k_m}$ is the prime factorization of $n$. Then
$$\varphi(n)=\prod_{i=1}^m\left(p_i^{k_i}-p_i^{k_i-1}\right)=\prod_{i=1}^mp_i^{k_i-1}(p_i-1)\;.$$
Thus, you have to find numbers that can be expressed in the form
$$\prod_{i=1}^mp_i^{k_i-1}(p_i-1)\tag{1}$$
in exactly two dif... | {
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Discrete Math Boys and Girls Problem 4: Boys and Girls Consider a set of m boys and n girls. A group is called homogeneous if it consists of all boys or all girls. In the following questions, practice the multiplication and the addition rule, and state exactly where you have used them.
(a)How many teams of two can we m... | You’re a bit off track, I’m afraid. I’ll discuss the case of two-person teams in some detail; see if you can then apply the ideas to correct your answers for the case of three-person teams.
Your $n(n-1)$ is the number of ways to choose an ordered team of two girls. There are two problems with this: we don’t want to cou... | {
"language": "en",
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"source": "stackexchange",
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Covariance of Two Dependent Variables So I'm looking at the following:
$\operatorname{Cov}(X, X^2)$ where $X$ is the standard normal. So I know that $E[X] = 0$ and we have:
$$\operatorname{Cov}(X, X^2) = E(X \cdot X^2) - E[X] \cdot E[X^2] = E[X^3] - 0 \cdot E[X^2] = E[X^3]$$
From googling around, apparently this $= 0$... | The standard normal distribution is symmetric about $0$, so $E[X^n] = 0$ for any odd positive integer $n$.
| {
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"source": "stackexchange",
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proving an equality of two sets Suppose $x,y $ are real numbers, then
$$ (a,b) = \bigcup_{n=N}^{\infty} \left(a, b- \frac{1}{n} \right] \; \; $$
for large enough $N$.
TRY:
Say $x$ is in the union, then there is some $n_0 \geq N$ such that $ a < x \leq b - \frac{1}{n_0} $. I wanna show $x \in (a,b)$. Already I have $a <... | The other direction: if $x \in (a,b) \to a < x < b \to \exists N: N(b-x) \geq 1$ by Archemedian Principle. Thus $\dfrac{1}{N} \leq b-x \to x \leq b - \dfrac{1}{N} \to x \in \left(a, b-\frac{1}{N}\right] \to x \in \displaystyle \bigcup_{n=N}^\infty \left(a,b-\frac{1}{n}\right]$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Colored card probabilities A box contains 3 cards. One card is red on both sides, one card is green on both sides, and one card is red on one side and green on the other. One card is selected from the box at random , and the color on one side is observed. If this side is green, what is the probability that the other si... | Everything fine.
$$Pr (GG \mid G)=\frac{Pr GG}{Pr G}=Pr G=1/2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1151431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the tangent plane to the graph of $f:\mathbb R^2\to \mathbb R^2$, $f(x,y)=(\sin(x-y),\cos(x+y))$
Let $f:\mathbb R^2\to \mathbb R^2$, $f(x,y)=(\sin(x-y),\cos(x+y))$, find the tangent plane to the graph of $f$ in $\mathbb R^4$ at $({\pi\over 4},{\pi\over 4},0,0)$.
What I did:
The equation of the tangent plane is g... | Here is an alternative method that verifies your result.
First of all, your Jacobian matrix is evaluated correctly—
$$
\frac{\partial \vec F}{\partial(x,y)} =
\left[\begin{matrix}
\partial F_x/\partial x & \partial F_x/\partial y \\
\partial F_y/\partial x & \partial F_y/\partial y
\end{matrix}... | {
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how to compute this limits given these conditions.
if $f(1)=1$ and $f'(x)=\frac{1}{x^2+[f(x)]^2}$ then compute $\lim\limits_{x\to+\infty}f(x)$
i tried to write it was
$$\frac{dy}{dx}=\frac{1}{x^2+y^2}\\
(x^2+y^2)\frac{dy}{dx}=1\\
(x^2+y^2)dy=dx$$
by the help
$$\begin{align}
f(x)&\le1+\int_1^x\frac{dt}{1+t^2}\\
&\le1+... | Since $f'(x)>0 $ and $f(1)=1$ then we have
$$ \frac{1}{x^2+f(x)^2}\leq \frac{1}{x^2+1}. $$
Also we will have
$$ \int_{1}^{x}f'(t)dt = \int_{1}^{x}\frac{dt}{t^2+f(t)^2} \leq \int_{1}^{x}\frac{dt}{t^2+1} $$
$$ \implies f(x) \leq 1+ \int_{1}^{x}\frac{dt}{t^2+1} . $$
Try to finish the problem.
| {
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Non-linear analysis 1) I am looking for a book which would give the proof of the following theorem(see below). I didn't find any book who does it:
*
*in infinite dimension (in Rockafellar Convex analysis book we are in finite dimension)
*considering the linear continuous application (written A or L in the following... | The best result of the form you want are in "Convex Analysis in General Vector Spaces" by C Zalinescu
see also http://www.worldscientific.com/worldscibooks/10.1142/5021
| {
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prove $\lim_{n→∞}\left(\frac{∑^{n}_{k=1}k^{m}}{n^{m+1}}\right)=1/(m+1)$ I'm having trouble proving the following equation:
$$\lim_{n→∞}\left(\frac{∑^{n}_{k=1}k^{m}}{n^{m+1}}\right)=\frac{1}{(m+1)}$$
A link to a proof would suffice.
Thank you.
| Hint: Regard the expression as a limit of a Riemann sum, and evaluate the (easy) corresponding integral.
| {
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Explaining the non-application of the multiplication law of logarithms, when logs are in the denominators. I have an A' Levels student who had to solve the following problem:
$ log_2 x + log_4 x = 2$
This was to be solved using the Change of base rule, and then substitution, as follows:
$ \frac{1}{log_x 2} + \frac{1}{l... | You could try an example. Is it true that
$$\frac12+\frac12=\frac1{2+2}=\frac14\mathrm{?}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Every section of a measurable set is measurable? (in the product sigma-algebra) I'm studying measure theory, and I came across a theorem that says that, given two $\sigma$-finite measure spaces $\left(X,\mathcal M,\mu\right)$ and $\left(Y,\mathcal N,\nu\right)$, and a set $E\in \mathcal M\otimes\mathcal N$, the functio... | Since the map $u_{x}:Y\to X\times Y$ given by $u_x(y)=(x,y)$ for a given $x$ is $\mathcal{N}$-$(\mathcal{M}\otimes\mathcal{N})$-measurable and $E_x=u_x^{-1}(E)$ it follows directly that $E_x\in\mathcal{N}$ for any $x$.
To convince yourself that $u_x$ is indeed measurable, just note that
$$
u_x^{-1}(A\times B)=
\begin{... | {
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"url": "https://math.stackexchange.com/questions/1153161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question in relation to completing the square In description of "completing the square" at http://www.purplemath.com/modules/sqrquad.htm the following is given :
I'm having difficulty understanding the third part of the transformation.
Where is
$ -\frac{1}{4}$ derived from $-\frac{1}{2}$ ?
Why is $ -\frac {1}{4}$ squ... | Let's begin with the equation
$$x^2 - \frac{1}{2}x = \frac{5}{4}$$
What the author wants to do is to create a perfect square on the left hand side. That is, the author wants to transform the expression on the left hand side into the form $(a + b)^2 = a^2 + 2ab + b^2$. Assume that
$$a^2 + 2ab = x^2 - \frac{1}{2}x$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1153260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the range of the following function Find the range of the following function:
$$f(x)=\frac{x-1}{x^2-5x-6}$$
I know how to find the domain only
Please provide an explanation.
EDIT: Another question: find the range of
$$f(x)=\frac{1}{(1-x)(5-x)}$$
How would we do that?
| the range of $f(x) = \dfrac{x-1}{(x+1)(x-6)}$ is $(-\infty, \infty).$ here is the reason. show that $\lim_{x \to -1+}f(x) = \infty$ and $\lim_{x \to 6-} f(x)= -\infty$ and $f$ is continuous in $(-1, 6).$ these three conditions imply the conclusion that the range of $f$ is $(-\infty, \infty).$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1153466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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Does there exist a set satisfying the following conditions?
Does there exist an infinite set $S$ of positive integers satisfying the following three properties? (Prove your claim!)
$(i)$ any two numbers in $S$ have a common divisor greater than $1$,
$(ii)$ no positive integer $k>1$ divides all numbers in $S$,
$(iii)$ ... | I think that it's possible infact let's suppose:
$$A\ = \{ p*3*2\ |\ p>5\ prime\ number\} $$
$$B\ =\ \{2*5\}\ \cup\ \{3*5\}\ \cup\ A $$
B should be our set,infact:
i)
$$ \forall\ a,b\in A\ \Rightarrow\ GCD (a,b) = 6$$
$$ \forall a \in A\ \Rightarrow\ GCD(a,2*5)=2\ and\ GCD(a,3*5)=3 $$
$$ GCD(2*5,3*5)=5$$
Then it's alwa... | {
"language": "en",
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"question_score": "1",
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On the coequaliser of a kernel pair How does one prove the following statement about kernel pairs?
If a pair of parallel morphisms is a kernel pair and has a coequalizer, then it is the coequalizer of its kernel pair.
| I can't make any sense of this statement; in particular I don't understand what "it" refers to.
Here is the correct statement: suppose $f : X \to Y$ is a morphism with a kernel pair $g_1, g_2 : X \times_Y X \to X$ which is also the coequalizer of some other pair of maps $h_1, h_2 : Z \to X$. Then in fact $f$ must be th... | {
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"question_score": "1",
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Integrate $\frac{\sin^2(2t)}{4}$ from $0$ to $2\pi$ Im trying to integrate $$\int_{0}^{2\pi}\dfrac{\sin^2(2t)}{4} dt$$ and I'm stuck..
Im at the end of multivariable calculus so no good solutions for these "simpler" problems are shown in the book at this point.
| $$\int_{0}^{2\pi}\frac{1}{4}sin^2(2t)dt = \frac{2}{4}\int_{0}^{\pi}sin^2(2t) = \frac{t-\sin(t)\cos(t)}{4}|_{0}^{\pi} = \pi/4$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the inverse of a sum of matrices not the sum of their inverses? Suppose $A + B$ is invertible, then is it true that $(A + B)^{-1} = A^{-1} + B^{-1}$?
I know the answer is no, but don't get why.
| $$(A+B)(A^{-1}+B^{-1})=2I+AB^{-1}+BA^{-1}$$
so your statement is true if and only if $I+AB^{-1}+BA^{-1}=0$ (which is of course not always true, and even usually wrong).
| {
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"answer_id": 2
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Convergence in probability implies convergence of characteristic functions If $X_n$ converge in probability to $X$, then they also converge weakly, and we can conclude by the continuity theorem that the characteristic functions $\phi_n$ of $X_n$ converge pointwise to the characteristic function $\phi$ of $X$. How can w... | An alternative way of seeing this is as follows (solely for my own recording purposes). Let $Y_n(\theta)=\exp(e^{i \theta X_n})$. Note that, $|Y_n(\theta)|\leq 1$, and thus for every fixed $\theta$, the sequence $\{Y_n(\theta)\}_{n\geq 1}$ is uniformly integrable. This, together with $Y_n \to Y$ (using continuous mapp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1153892",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that group of symmetries is isomorphic to $S_n$ In my algebra book the first section has the following exercise:
Prove that group of all symmetries(isometric bijections under composition) of a regular tetrahedral is isomorphic to $S_4$.
I did it by explicitly writing out all the symmetries of a regular tetrahedra... | Imagine a plane through one edge and passing through the midpoint of the opposite edge. This opposite edge will be perpendicular to the new plane. Now reflect the tetrahedron through this plane. Clearly, the tetrahedron remains unchanged, but you have interchanged two vertices. Combining all possible interchanges of tw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1154010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Eigenvalues that are complex numbers Have a square matrix problem that involves complex numbers and am at a loss. $M$ is a square matrix with real entries. $\lambda = a + ib$ is a complex eigenvalue of $M$, show that the complex conjugate $\bar{\lambda} = a - ib$ of $\lambda$ is also an eigenvalue of $M$. Does solvi... | If $Mv=\lambda v$ where $v=(v_1,...,v_n)$ is a vector of complex numbers, then $$M\bar v=\overline{Mv}=\overline{\lambda v}=\bar\lambda \bar v$$
Here by writing a line over a matrix (or vector) I mean to take the complex conjugate of every entry; the first equality holds because $M$ is real, so $\bar M=M$, and because ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1154116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Computing the size of the stabilizers when $U(q)$ acts on $\Bbb Z_q$ Let $\mathbb{Z}_q$ be the additive group of integers modulo $q$ and $U(q):=\{g\in\mathbb{Z}_q:(g,q)=1\}$. If $a\in\mathbb{Z}_q$, then what is the cardinality of the set $\{g\in U(q):ga\equiv a(\mod q)\}$.
Is there a way to count the same? Thank you. ... | The argument Gerry mentions is the way to go. The problem is equivalent to computing the size of the kernel of the modulo map $U(n)\to U(d)$ when $d\mid n$. It's enough to prove this map is onto, since then the kernel's size is $\varphi(n)/\varphi(d)$. Observe the following diagram commutes:
$$\begin{array}{ccc} U(n) &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1154215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Series of Functions of integrals How do you solve this question? We had that in a test and I've been staring on it for around 30 minutes without any solution.
Given $g(x)$ a differentiable and bounded function over $\mathbb{R}$. Define,
\begin{align}
f_0(x) & =g(x) \\
f_n(x) &= \int _0^xdt_{n-1}\int _0^{t_{n-1}}dt_{n-... | Show by induction that for $n \ge 1$,
$$f_n(x) = \frac{1}{(n-1)!}\int_0^x (x - t)^{n-1} g(t)\, dt = \frac{1}{(n-1)!}\int_0^x t^{n-1}g(x - t)\, dt.$$
Let $a > 0$, and set $M = \max\{|g(x)|: 0\le x \le a\}$. For all $x\in [0,a]$, $|f_0(x)| \le M$ and for $n\ge 1$,
$$|f_n(x)| \le \frac{M}{(n-1)!}\int_0^a t^{n-1}\, dt = \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$f(x)=q$ if $x=p/q$, properly reduced is unbounded at every point. Define the function $f$ as follows:
$$f(x) =
\begin{cases}
q, & \text{if $x=p/q$,properly reduced} \\
0, & \text{if $x$ is irrational}
\end{cases}$$
Prove that for every real number $x_0$, $f$ fails to be bounded at $x_0$, i.e. there does not exist an... | "It's enough to consider only rational points." << If you want to be rigorous, you may start by developing that (it's true but useless nonetheless).
Then, in general if you doubt your own argument's rigour, I'd suggest to go "$\epsilon, \delta$" and go back to axioms/statements you're 100% sure about.
Here's a draft ex... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there a difference between transform and transformation? I was told that there is a difference between a transform and a transformation. Can anyone point out clearly?
For example, is Laplace transform not a transformation?
| To my mind (a mathematical physicist), a transform is a specific kind of transformation, namely, a transformation not of values, but of functions. So, a transformation $N$ maps a value $x$ onto its image value:
$N: x \to N(x)$
whereas the transform $M$ maps a function $f$ onto its image function:
$M: f(x) \to F(y)$
The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1154581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Prove that $a_1^2+a_2^2+\dots+a_n^2\ge\frac14\left(1+\frac12+\dots+\frac1n\right)$
Let $a_1,a_2,\dots,a_n$ be non-negative real numbers such that for any $k\in\mathbb{N},k\le n$
$$a_1+a_2+\dots+a_k\ge\sqrt k$$
Prove that
$$a_1^2+a_2^2+\dots+a_n^2\ge\frac14\left(1+\frac12+\dots+\frac1n\right)$$
I just want to ve... | let $b(k)=\sqrt{k}-\sqrt{k-1} ,k=1$ to $n \implies \sum_{i=1 }^k b(i)=\sqrt{k} $
$b(k)=\dfrac{1}{\sqrt{k}+\sqrt{k-1} }>\dfrac{1}{\sqrt{k}+\sqrt{k} }=\dfrac{1}{2\sqrt{k}} $
it is trivial $b(k)$ is mono decreasing function $\implies b(k-1)>b(k)$, ie,
$b(1 )>b(2)> b(3)>...>b(n)>0$
$a_k=b(k)+d_k,\sum a_i=\sum b(i)+\sum d_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1154661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Continuity of a piecewise function of two variable I'm given this equation:
$$
u(x,y) =
\begin{cases}
\dfrac{(x^3 - 3xy^2)}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\
0\quad& \text{if} \quad (x,y)=(0,0).
\end{cases}
$$
It seems like L'hopitals rule has been used but I'm confused because
*
*there is no limi... | u(0,0) can't possibly exist as it would be a division by 0. You haven't given us enough information. What is the point of this equation? What is the problem you have to solve?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the logic behind Jacobi iterative method? The book I follow and on net also, all that I can find is the algorithm to find the solution, but I don't quite understand the physical significance or logic behind the algorithm.
Can someone please help.
| You are trying to solve $A \textbf{x} = \textbf{b}$.
The matrix $A$ could be written
$A = D + N$ where $D$ is diagonal.
If the sequence $\textbf{x}^k$ generated by $D\textbf{x}^{k+1} = b - N\textbf{x}^k$ converges, then you have $D\textbf{x}^{\infty} = b - N\textbf{x}^{\infty}$ or $A \textbf{x}^{\infty} = \textbf{b}^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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On the intuition behind a conditional probability problem. This is a very similar question to this one. But notice the subtle difference that the event that I define $B$ is that I am dealt at least an ace.
Suppose I get dealt 2 random cards from a standard deck of poker (52 cards).
Let the event $A$ be that both cards... | Be careful about the meaning of the events $A$, $B$, and $C$. The event $B$ is "of the two cards dealt, at least one of them is an ace". And I think you intended the event $C$ to be "the first card is an ace". As you've calculated above, event $A$ is contained in both events $B$ and $C$, so the conditional probabilit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1154943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Divisibility property proof: If $e\mid ab$, $e\mid cd$ and $e\mid ac+bd$ then $e\mid ac$ and $e\mid bd$. $a,b,c,d,e\in \mathbb{Z}$. Prove that if $e\mid ab$, $e\mid cd$ and $e\mid ac+bd$ then $e\mid ac$ and $e\mid bd$.
I could use some hints on how to prove this property.
| Let $\,m = ac/e,\ n = bd/e.\,$ By hypothesis $\, \color{#c00}{m+n,\, mn \in \Bbb Z}\,$ and $\,m,n\,$ are roots of $\,(x-m)(x-n).\,$ This has $\color{#c00}{\rm integer}\,$ coeffs so, by the Rational Root Test, $\,m,n\in\Bbb Z,\,$ so $\ e\mid ac,bd.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Explicit formula for the sum of an infinite series with Chebyshev's $U_k$ polynomials $$\sum _{k=0}^{\infty } \frac{U_k(\cos (\text{k1}))}{k+1}=\frac{1}{2} i \csc (\text{k1}) \left(\log \left(1-e^{i \text{k1}}\right)-\log (i \sin (\text{k1})-\cos (\text{k1})+1)\right)$$ where $U_k$ is the Chebyshev $U$ polynomial
$$\su... | The first should come from
$$\sum_{k=1}^\infty \dfrac{\sin(kt)}{k} = -i/2 \left( -\ln \left( 1-{{\rm e}^{it}} \right) +\ln \left( 1-{
{\rm e}^{-it}} \right) \right)$$
which you get by expressing $\sin(kt)$ as a complex exponential and using the Maclaurin series for $\ln$ (with Dirichlet's test showing convergence a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Factoring Trinomials: Dealing with Variables I'm current working with Trinomials, doing things such as $2w^2 + 38w + 140$. I know how to solve this, however, I encountered a different type of problem, where the last term has a variable in it: $x^3 - 10x^2 + 21x$. I'm not sure how to solve it when the $x$ is in that $21... | The first step for any factoring problem is to factor out the greatest common factor, in this case $x$.
So $x^3 + 10x^2 + 21x$, we can factor out $x$ to get $x(x^2+10x+21)$. Can you then factor the remaining expression using the methods you already know?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I am trying to calculate percentage, and my results are not correct. Suppose a Student Scores 18.70 out of 30 in Subject1 & 29.75 out of 40 in Subject2.
I have calculated percentage of Subject1 as 62.33% & for Subject2 74.38%.
While calculating the final percentage If I average both percentages I get 68.35% but If I ca... | It is not clear from the original post just what is being asked,
There are two subjects. The assessment tool to evaluate each one has a different maximum number of possible marks. This decision does not have anything to do with the relative importance of the two subjects.
You have correctly converted these marks to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A real and normal matrix with all eigenvalues complex but some not purely imaginary? I'm trying to construct a normal matrix $A\in \mathbb R^{n\times n}$ such that all it's eigenvalues are complex but at least one of then also has a positive real part (well, at least two then, since the entries are real they'd come in ... | In order to get a real matrix with no real eigenvalue, $n$ must be even. Then you can construct any normal matrix with nonreal eigenvalues having positive real parts as follows:
*
*Take a block diagonal matrix $D:=D_1\oplus\cdots\oplus D_k$, $k=n/2$, with the diagonal blocks $D_i$ of the form
$$\tag{1}
D_i=\begin{bm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Changing the Modulo congruence base? This is a conversion someone on SE made:
$$77777\equiv1\pmod{4}\implies77777^{77777}\equiv77777^1\equiv7\pmod{10}$$
But I don't understand how this is done?
| What was actually used is Euler's theorem:
$$a^{\varphi(n)} \equiv 1 \pmod n$$
Where $\varphi$ is the totient function. $\varphi(10) = 4$ so
$$77777^{a + 4k} \equiv 77777^a \pmod{10}\\
\Rightarrow 77777^{77777} \equiv 77777^{77777 \bmod 4} = 77777^1 \pmod{10}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solving a separable equation Solve the following separable equations subject to the given boundary conditions
a) $\frac{dy}{dx}$ = $x\sqrt{1−y^2}$, $y=0$ at $x=0$,
b)$(ω^2+x^2)\frac{dy}{dx} = y$, $y=2$ at $x=0$ ($ω>0$ is a parameter).
Edit:
Sorry I wasn't clear, I am fine with everything up to and including integratio... | a) $$ \int \frac{dy}{\sqrt{1-y^{2}}}=\int x dx$$
b) $$\int\frac{dy}{y} =\int \frac{dx}{\omega^{2}+x^{2}}$$
If you're studying separation of variables, I think you'd be able to evaluate these integrals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1155879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Need help to understand the equation Here is a small equation, which says
$$E[min\{{D, Q}\}] = \int_{0}^{Q}{xf(x)dx} + \int_{Q}^{\infty}{Qf(x)dx}$$, where D is a continuous random variable denoting the demand on a specific day and f(x) is the representative density function of demand D. Q represents the Quantity to be... | We have
$$\mathbb E[\min(D,Q)] = \int_0^\infty \min(x,Q)f(x)\mathsf dx. $$
For $x<Q$, $\min(x,Q)=x$, and for $x>Q$, $\min(x,Q)=Q$. So the above is equal to
$$\int_0^Q xf(x)\mathsf dx + \int_Q^\infty Qf(x)\mathsf dx. $$
Recall that $(D-Q)^+=\max(D-Q,0)$, and $\max(x-Q,0)=x-Q$ for $x>Q$. So
$$
\begin{align*}
\mathbb E[D]... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $A$ is a square matrix that is linearly independent, is $AA$? I'm just not sure how to start this problem from Linear Algebra Done Wrong. The problem is to prove that if the columns of $A$, square matrix, are linearly independent, then the columns of $A^2$ = $AA$ are also linearly independent. I'm mostly just not su... | Since $A$ is linearly independent, no (non-zero) linear combination of the rows of $A$ result in 0. This means that for any non-zero vector $x$, $Ax\neq 0$. The rows of $A^2$ are linearly independent if and only if there is no (non-zero) linear combination of them that results in zero, i.e. no nonzero $x$ such that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1156043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to put this into set notation? if you could please help me put this sentence into set notation, examine my (flawed?)attempts, and tell me where I went wrong with them?
So we call a function $f\colon \mathbb{N} \to \mathbb{N}$ zero almost everywhere iff $f(n)=0$ for all except a finite number of arguments.
My quest... | Your $E$ doesn't ever consider the "for all except a finite number of arguments". I think the most accessible answer (although perhaps not the one you are fishing for) is simply:
$E=\{f:\mathbb{N} \to \mathbb{N} | \exists S \subset \mathbb{N}$ where $|S|<\infty$ with $f(n) \neq 0$ for $n \in S$ and $f(n)=0$ otherwise}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Exact values of the equation $\ln (x+1)=\frac{x}{4-x}$ I'm asking for a closed form (an exact value) of the equation solved for $x$
$$\ln (x+1)=\frac{x}{4-x}$$
$0$ is trivial but there is another solution (approximately 2,2...).
I've tried with Lambert's W Function or with Regularized Gamma Functions but I didn't get s... | Burniston and Siewert solved the equation:
$$ze^z=a(z+b)$$
Your equation can be reduced to the form:
$$5z-4= e^{z-1}z$$
which can be solved by Burniston Siewert integral
== References ==
[68] C. E. Siewert and E. E. Burniston, "Solutions of the Equation $ze^z=a(z+b)$," Journal of Mathematical Analysis and Applications,... | {
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"timestamp": "2023-03-29T00:00:00",
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Probability and dice rolls
Two fair six sided die are rolled fairly and the scores are noted. What is the probability that it takes 3 or more rolls to get a score of 7?
Here is what I have got so far:
The probability that it takes $3$ or more rolls to get a score of $7$ is equal to $$1-P(\text{you get a score of $7$ ... | The basic approach in the question is a good one,
except for an error in combining the
probability that the first roll is a $7$ and the
probability that the second roll is a $7$.
In general, for two events $A$ and $B$,
$$P(A \cup B) = P(A) + P(B) - P(A \cap B).$$
If $A$ is the event of rolling $7$ on the first roll,
a... | {
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"timestamp": "2023-03-29T00:00:00",
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How does one show that for $k \in \mathbb{Z_+},3\mid2^{2^k} +5$ and $7\mid2^{2^k} + 3, \forall \space k$ odd.
For $k \in \mathbb{Z_+},3\mid2^{2^k} +5$ and $7\mid2^{2^k} + 3, \forall \space k$ odd.
Firstly,
$k \geq 1$
I can see induction is the best idea:
Show for $k=1$:
$2^{2^1} + 5 = 9 , 2^{2^1} + 3 = 7$
Assume for... | Hint: What if you tried to prove that:
*
*$3\mid 2^\ell+5$ whenever $\ell$ is even, and
*$7\mid 2^\ell+3$ whenever $\ell\equiv2\pmod6$ (or whenever $\ell-2$ is divisible by six).
And then prove ...
If you follow this path the real question is, of course, how to know what you should try to prove? The answer is to... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Reduction of $\frac a{b +1}$ Can we convert $\frac a{b+1}$ into a fraction having only $b$ as a denominator? What are the steps in doing so?
| Lemma $ $ If $\ q = \dfrac{a}{b\!+\!1}\,$ can also be written with denominator $\,b,\,$ i.e. $\,q = \dfrac{n}b,\,$ then $\,q\,$ is an integer.
Proof $\ $ Since $\,(b\!+\!1)q\,$ and $\,bq\,$ are integers then so too is their difference $\,(b\!+\!1)q-bq\, =\, q.\ \ $
Remark $ $ Generally, a fraction can be written with... | {
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Antiderivative of $|\sin(x)|$ Because $$\int_0^{\pi}\sin(x)\,\mathrm{d}x=2,$$ then $$\int_0^{16\pi}|\sin(x)|\,\mathrm{d}x=32.$$
And Wolfram Alpha agrees to this, but when I ask for the indefinite integral $$\int|\sin(x)|\,\mathrm{d}x,$$ Wolfram gives me $$-\cos(x)\,\mathrm{sgn}(\sin(x))+c.$$ However, $$[-\cos(x)\,\math... | let $$f(x) = \int_0^x |\sin x| \, dx = (1-\cos x), 0\le x \le \pi.$$ since the integrand is $\pi$-periodic, we can extend the formula for $$f(x) = f(\pi) + f(x-\pi), \pi \le x \le 2\pi$$ and so on. you can verify that $$f(n\pi) = 2n \text{ for all integer } n.$$
in particular $f(16\pi) = 32.$
| {
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Square root of determinant equals determinant of square root? Is it true that for a real-valued positive definite matrix $X$, $\sqrt{\det(X)} = \det(X^{1/2})$?
I know that this is indeed true for the $2 \times 2$ case but I haven't been able to find the answer or prove it myself for larger $X$.
Thanks a lot for any hel... | Call $Y = x^{\frac{1}{2}}$. Then $\det X = \det (Y^2) = (\det Y)^2$, so $\sqrt{\det X} = |\det Y|$. If $Y$ is positive definite, then $|\det Y| = \det Y$, and so we are done.
| {
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} |
The set of functions that are zero almost everywhere is enumerable I have become somewhat overwhelmed with a problem I am working on I had a friend tell me that my proof was wrong. I would be grateful if someone could explain why I am wrong, and possibly offer a alternate solution.
Problem: A function $f\colon \mathbb... | Each $\Bbb N\to\Bbb N$ function is a natural sequence. Let's define a 'portrait' of an almost-everywhere zero function as a finite sequence of all arguments, for which the function has non-zero value, interleaved with those values. For example $f=(0, 0, 15, 0, 27, 0,...)$ has a portrait $P(f)=(3, 15, 5, 27)$. Of course... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1156818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
How to prove the convergence of the series $\sum\frac{(3n+2)^n}{3n^{2n}}$ I need to prove the convergence of this series:
$$\sum\limits_{n=1}^\infty\frac{(3n+2)^n}{3n^{2n}}$$
I've tried using Cauchy's criterion and ended up with a limit of $1$, but I already know it does converge.
How am I suppose to prove it? Note tha... | The root test applies nicely here. Taking $\displaystyle a_n=\left(\frac{3\,n+2}{n^2}\right)^n$ gives
$$
\lim_{n\to\infty}\lvert a_n\rvert^{1/n}
= \lim_{n\to\infty}\frac{3\,n+2}{n^2} =0<1
$$
This proves that $\sum a_n$ converges so your series $\frac{1}{3}\sum a_n$ converges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1156917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Trouble understanding Iterated expection problem I'm having trouble understanding iterated expectations. Whenever I try to search for solid examples online, I get proofs or theoretical solutions. I found an example, but it doesn't have a solution or steps I can follow to understand the method. Here's the problem: $E(M)... | By the corresponding definitions:
$$E[M]=\sum_{j=0}^{4}jP(M=j)=\sum_{j=0}^{4}j\sum_{i=0}^{2} P(M=j,Z=i)=$$ $$\sum_{j=0}^{4}j\sum_{i=0}^{2} \frac{P(M=j,Z=i)}{P(Z=i)}P(Z=i)=$$
$$\sum_{i=0}^{2}\sum_{j=0}^{4} jP(M=j|Z=i)P(Z=i),$$
where
$$\sum_{j=0}^{4} jP(M=j|Z=i)=E[M|Z=i].$$
So
$$E[M]=\sum_{i=0}^{2}E[M|Z=i]P(Z=i).$$
But w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Writing elements of a finitely generated field extension in terms of the generators I have a question that arose in the process of solving a different problem in Galois theory. That problem asked to show that if $\alpha_1, \dots, \alpha_n$ are the generators of an extension $K/F$, then any automorphism $\sigma$ of $K$ ... | The sophisticated way of doing this is the following:
Take two automorphisms $\sigma, \tau$ of $K$, which fix $F$ and agree on $\{\alpha_1, \dotsc, \alpha_n\}$. The set $M := \{x \in K | \sigma(x)=\tau(x)\} \subset K$ is a field, which contains $F$ and all $\alpha_i$. By the definition (minimality of $K$ as you mention... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
A set without the empty set By definition, the empty set is a subset of every set, right? Then how would you interpret this set: $A\setminus\{\}$? On one hand it looks like a set without the empty set, on the other hand, the empty set is in every set... Can you explain?
| The notation $A-\{\}$ roughly translates to "the set $A$ without the elements of $\{\}$." The difference is that the empty set is not an element of $A$ and this notation just means you're not removing any elements from your original set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Prove $\lim\limits_{n \to \infty} \sup \left ( \frac{(2n - 1)^{2n - 1}}{2^{2n} (2n)!)} \right ) ^ {\frac 1 n} = \frac {e^2} 4$ This is a problem in Heuer (2009) "Lerbuch der Analysis Teil 1" on page 366. I assume that the proof should use $e = \sum\limits_{k = 0}^{\infty} \frac 1 {k!}$, but I cannot come further.
| If $a_n=\frac{(2n - 1)^{2n - 1}}{2^{2n} (2n)!}$. Compute the limit
$$\begin{align}\lim \frac{a_{n+1}}{a_n}&=\lim \frac{\frac{(2n + 1)^{2n + 1}}{2^{2n+2} (2n+2)!}}{\frac{(2n - 1)^{2n - 1}}{2^{2n} (2n)!}}\\&=\lim\frac{(2n+1)(2n+1)}{4(2n+1)(2n+2)}\left(\frac{2n+1}{2n-1}\right)^{2n-1}\\&=\frac{1}{4}\lim\left[\left(1+\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Help to understand this property: $\int\limits_{ka}^{kb}s\left(\frac{x}{k}\right)dx = k\int\limits_a^bs(x)dx$ I'm reading the part explaining the properties of the integral of a step function in Apostol's Calculus I and he explains this property:
$$\int\limits_{ka}^{kb}s\left(\frac{x}{k}\right)dx = k\int\limits_a^bs(x)... | Hint: Make the substitution $u = \frac{x}{k}$ then $du = \frac{1}{k} dx \implies kdu = dx$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Survival probability up to time $n$ in a branching process. Let $\{Z_n : n=0,1,2,\ldots\}$ be a Galton-Watson branching process with time-homogeneous offspring distribution
$$\mathbb P(Z_{n,j} = 0) = 1-p = 1 - \mathbb P(Z_{n,j}=2), $$
where $0<p<1$. That is, $Z_0 = 1$ and
$$Z_n = \sum_{j=1}^{Z_{n-1}}Z_{n,j}$$
for $n\ge... | Consider the first generation in the branching process: either (1) we have $0$ offspring or (2) $2$ offsprings. Suppose $t_n$ is known. We would like to compute $t_{n+1}$, the probability that $T > n+1$. In order to have $T > n+1$, the first generation has to have $2$ offsprings, say $A$ and $B$. Now one of those two o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Integral $\int_{-\infty}^{\infty} \frac{\sin^2x}{x^2} e^{ix}dx$ How do I determine the value of this integral?
$$\int_{-\infty}^{\infty} \frac{\sin^2x}{x^2} e^{ix}dx$$
Plugging in Euler's identity gives
$$\int_{-\infty}^{\infty} \frac{i\sin^3x}{x^2}dx + \int_{-\infty}^{\infty} \frac{\sin^2x \cos x}{x^2}dx$$
and since $... | Define $\displaystyle I(a,b)=\int_{-\infty}^{\infty} \frac{\sin^2ax\cos bx}{x^2} {\rm d}x \displaystyle$, and take Laplace transform of $I(a,b)$ with respect to $a$ and $b$, with Laplace domain variables $s$ and $t$, respectively:
\begin{align}
\mathcal{L}_{a\rightarrow s,b\rightarrow t}\{I(a,b)\}&=\int_{-\infty}^{\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 0
} |
Trouble in a proof , functional analysis So, here is the point in the proof that I don't understand, he uses that it holds for f and g which are boundered(limited, e.g. there is some M such that $|f|<M$ and $|g|<M$) then for all p, $1\leq p < +\infty$ and $\alpha \in (0,1)$ it holds that:
$|f+g|^p \leq (1-\alpha)^{1-p}... | This is due to convexity: Because of $1 \leq p <\infty$, the map $[0,\infty) \to [0,\infty), x \mapsto x^p$ is convex (check that the first derivative is increasing).
Hence,
\begin{eqnarray*}
|f+g|^p &\leq& \bigg( (1-\alpha) \frac{|f|}{1-\alpha} + \alpha \frac{|g|}{\alpha}\bigg)^p
\\ & \leq & (1 - \alpha) \cdot (|f|/(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to interpret max(min(expression))?? I am reading this paper:
ai.stanford.edu/~ang/papers/icml04-apprentice.pdf
Step 2 of section 3 is to compute an expression of the form max(min(expr)). What does this mean?
I made a simple example of such an expression and am equally puzzled as to how to evaluate it:
$\max_{x \in... | Fix $x \in (-3,5)$. Then, solve the problem $f(x) = \min_{y \in (-1,1)} \frac{x-2}{y}$. Then, you have $f(x)$ defined on $(-3,5)$. Now solve $\max_{x \in (-3,5)} f(x)$.
There are various theorems on problems of the sort, known as minimax theorems (such as sion's minimax theorem, von neumann's minimax theorem, etc.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1157962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Integral of cosine multiplied by zeroth-order Bessel function I am looking for the result of the integral below,
$$ \int_0^1 \cos(ax)\ J_0(b\sqrt{1-x^2})\ \mathrm{d}x$$
where $J_0(x)$ is the zeroth order Bessel function of the first kind. Variables $a$ and $b$ are known and real.
| I found the answer from Gradshteyn and Ryzhik's book, 7th edition section 6.677 the 6th equation.
$$\int_0^1 \cos(ax)\ J_0\left(b\sqrt{1-x^2}\right)\ \mathrm{d}x = \frac{\sin\left(\sqrt{a^2+b^2}\right)}{\sqrt{a^2+b^2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proving that the symmetric difference of sets is countable at most Let $A$, $B$, $C$ be sets. How do you prove that if $|A\Delta B|\leq \aleph_{0}$ and $|B\Delta C|\leq \aleph_{0}$ then $|A\Delta C|\leq \aleph_{0}$?
| HINT: Remember that $A\triangle C=(A\triangle B)\triangle(B\triangle C)$, and that a subset of a countable set is countable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $n\ge2$, Prove $\binom{2n}{3}$ is even. Any help would be appreciated. I can see it's true from pascal's triangle, and I've tried messing around with pascal's identity and the binomial theorem to prove it, but I'm just not making any headway.
| In the spirit of various other answers, let $S=\{\color{red}1,\color{blue}1,\color{red}2,\color{blue}2\dots, \color{red}n,\color{blue}n\}$. If $T$ is a $3$-element subset of $S$, one color predominates among its elements, and the set $T'$, obtained by changing the color of every number in $T$, is a different $3$-elemen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 3
} |
Area of portion of circle inside a square. Consider a square grazing field with each side of length 8 metres. There is a pillar at the
centre of the field (i.e. at the intersection of the two diagonals). A cow is tied to the pillar
using a rope of length $8\over\sqrt3$ meters. Find the area of the part of the field tha... | First let's calculate the central angle of each segment in the circle which is cut off by the square.
Let $\angle BAC=\theta$. It is given that $AC=4$, and $AB={8 \over \sqrt3}$.
From the figure it is easy to deduce that in $\Delta ABC$, $\cos\theta={\sqrt3 \over 2}$,
which means that $\theta=30^\circ$ and hence the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to calculate derivative of $f(x) = \frac{1}{1-2\cos^2x}$? $$f(x) = \frac{1}{1-2\cos^2x}$$
The result of $f'(x)$ should be equals
$$f'(x) = \frac{-4\cos x\sin x}{(1-2\cos^2x)^2}$$
I'm trying to do it in this way but my result is wrong.
$$f'(x) = \frac {1'(1-2\cos x)-1(1-2\cos^2x)'}{(1-2\cos^2x)^2} =
\frac {1-2\cos... | The problem is in this step, from here
$$f'(x) = \frac {1'(1-2\cos x)-1(1-2\cos^2x)'}{(1-2\cos^2x)^2} $$
to here
$$\frac {1-2\cos^2x-(1-(2\cos^2x)')}{(1-2\cos^2x)^2}$$
because $$1'=0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
finding roots of polynomial in MAGMA or GAP Please hint me. I want to find the roots of $x^3+(1-2n)x^2-(1+4n)x+8n^2-10n-1$ with Magma or GAP. In Magma:
$P<x> := PolynomialRing(Rationals())$;
$P<n>:= PolynomialRing(Rationals())$;
$f:=x^3+(1-2n)x^2-(1+4n)x+8n^2-10n-1$;
Roots(f);
[]
but this polynomial has roots for $n\... | I don't think you would be able to do better than the generic formulas for roots of a degree polynomial (e.g. Wolfram $\alpha$ will give you these expressions), and these roots are not conveniently represented in the systems you refer to:
The Galois group of the polynomial in $Q(n)[x]$ is $S_3$. So the roots are not cy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let $ f(x)= ( \log_e x) ^2 $ and (Integration by parts. Comparing integrals of different limits ) Let $ f(x)=( \log_e x) ^2 $ and let
$ I_1= \int_{2}^{12} f(x) dx $ , $ I_2= \int_{5}^{15} f(x) dx $ and $ \int_{8}^{18} f(x) dx$
Then which of the following is true?
(A)$I_3 <I_1 < I_2 $
(B)$I_2 <I_3 < I_1 $
(C)$I_3 <I_... | You can see that each interval of integration is of the same length.
$I_2-I_1 = \int_2^5 f(x) dx - \int_{12}^{15} f(x) dx$
But f is increasing, so $\forall x \in [2,5] \forall y \in [12,15], f(x) < f(y)$
This imply that $\int_2^5 f(x) dx < \int_{12}^{15} f(x)$
Same idea to compare $I_2$ and $I_3$
Edit : You can, also... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1158903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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