Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Rolles theorem used for solving equation $ax^3+bx^2+cx+d=0$ If a,b,c,d are Real number such that
$\frac{3a+2b}{c+d}+\frac{3}{2}=0$. Then the equation $ax^3+bx^2+cx+d=0$ has
(1) at least one root in [-2,0]
(2) at least one root in [0,2]
(3) at least two root in [-2,2]
(4) no root in [-2,2]
I am doing hit and trial m... | Finding f'(x)=0 gives you a point of maxima or minima which need not be the root of the equation.
From Rolle's theorem we know that is the function in continuous and diffrentiable in [a,b] and if f(a)=f(b), then there exists a c $\epsilon$(a,b) such that f'(c)=0.
So integrate the given equation and find for which of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2939224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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What is the probability that Fra wins? Fra and Sam want to play a game. They have two classic coins Head-Tail.
They flip the coins at the same time.
If the result is $HH$, Fra wins. If the result is $HT$ (or $TH$), they flip again and result is again $HT$( or $TH$) Sam wins. In the other cases they continue.
So for ex... | ns=10000
For t=1 to ns
[alfa]
x=rnd(1)
if x<0.25 then f=f+1:goto[beta]
if x>0.75 then goto[alfa]
y=rnd(1)
if y<0.25 then f=f+1:goto[beta]
if y>0.75 then goto[alfa]
[beta]
next t
print f/ns
Simulation with Just Basic; ns is number of simulations, try other values.
| {
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"url": "https://math.stackexchange.com/questions/2939374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simultaneous diagonalization of two matrices if one does not has $n$ independent eigenvectors I have a small confusion. Suppose there are two $n \times n$ matrices $A$ and $B$ such that $A$ does not has $n$ independent eigenvectors. The $A$ is not diagonalizable. But $A$ and $B$ commute and I can find a matrix that dia... | In your example, matrices $A$ and $B$ are both diagonalizable (and both have $n$ independent eigenvectors), so it's not an instance of the thing you're describing:
*
*$A$ has eigenvector $(1,0,1)$ to the eigenvalue $2$, and eigenvectors $(0,1,0)$ and $(1,0,-1)$ to the eigenvalue $0$.
*$B$ has eigenvector $(1,-2,-1)... | {
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"timestamp": "2023-03-29T00:00:00",
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Relatively Prime Fibonacci numbers We can call the $x$th Fibonacci number Fib($x$). What's the best asymptotic lower bounds on the amount of relatively prime Fibonacci numbers between Fib($n$) and Fib($n+m$)?
In other words, if we take the $m$ Fibonacci numbers that lie between Fib($n$) (inclusive) and Fib($n+m$), wha... | Although this appears to be intrinsically a question about Fibonacci numbers, in fact the Fibonacci numbers are a guise. The key aspect to notice here is that
$$ \gcd(Fib(n), Fib(m)) = Fib(\gcd(n,m)).$$
(This is proved, for instance, in this other post on this site).
Thus $Fib(n)$ and $Fib(m)$ are relatively prime exac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2939783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is this set determined to be empty? From Eccles' Introduction to Mathematical Reasoning, problem 7.1 asks you to determine the set for:
$${\{n \in \mathbb{Z}^+ \mid \forall m \in \mathbb{Z}^+, m \leq n \}}$$
The answer provided in the back of the book is $\emptyset$. Why is $\{1\}$ not an answer? It satisfies $\mat... | Is $1\geq m$ for every positive integer $m$? In words, the set is the set of all positive integers greater than or equal to all positive integers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2939918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof of quotient rule $(\frac{f}{g})'(x_{0})=\frac{f'(x_0)g(x_0)-f(x_0)g'(x_0)}{g^2(x_0)}$ $$\left(\frac{f}{g}\right)'(x_{0})=\frac{f'(x_0)g(x_0)-f(x_0)g'(x_0)}{g^2(x_0)}$$
So, $\frac{1}{g}.f=\frac{f}{g}$, then $$\frac{f}{g}'(x_0)=\frac{f(x)\frac{1}{g(x)}-f(x_0)\frac{1}{g(x_0)}}{x-x_0}=f(x)\frac{\frac{1}{g(x)}-\frac{1... | $(1/g)'\neq 1/g'$ this is where you made the mistake.
| {
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"url": "https://math.stackexchange.com/questions/2940072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integral $\int\frac{\sqrt{4x^2-1}}{x^3}dx$ using trig identity substitution!
$$\int \frac{\sqrt{4x^2-1}}{x^3}\ dx$$
So, make the substitution
$ x = \sqrt{a \sec \theta}$, which simplifies to $a \tan \theta$.
$2x = \sqrt{1} \sec \theta$,
$ d\theta = \dfrac{\sqrt{1}\sec\theta\tan\theta}{2}$
$\int \dfrac{\sqrt{1}\tan\... | With the substitution $x=\frac {\sec \theta }{2}$ you get $dx = \frac {\sec \theta \tan \theta }{2} d\theta $ and the integral changes to $$ \int \frac {4\tan^2 \theta \sec \theta }{ \sec ^3 \theta } d\theta =4 \int \frac {\tan^2 \theta }{ \sec ^2 \theta } d\theta =4 \int \sin ^2 \theta d\theta$$
Now you can use the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2940220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the conjugate of a function always convex? The conjugate of a function $f$ is given (for some $y \in \operatorname{dom}(f)$) as:
$$f^*(y) = \sup_{x \in \operatorname{dom}(f)}\left(y^Tx - f(x)\right)$$
It is known that $f^*$ is convex even if $f$ is not. I would like to know how to prove this.
| As I wrote this question I recalled the following fact, and have attempted to prove this property of conjugate functions using it.
If $h(y,\,x)$ is convex in $y$ for each $x \in \mathcal{A}$, then $g(y) = \sup_{x \in \mathcal{A}}h(y,\,x)$ (the pointwise supremum) is convex.
Let us try to apply this fact to our proble... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prob. 4, Sec. 27, in Munkres' TOPOLOGY, 2nd ed: Any connected metric space having more than one point is uncountable Here is Prob. 4, Sec. 27, in the book Topology by James R. Munkres, 2nd edition:
Show that a connected metric space having more than one point is uncountable.
Here is a solution. Although I do unders... | You are just using $T_2$-ness of metric spaces to show finite ones are disconnected.
But there are $T_2$ spaces (even $T_3$) that are countable and connected, so
the last step cannot work based purely on case 1. You need to really use the metric (or normality) etc. to get the disconnectedness.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2940546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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should I search $f^{-1}(x)$ or is there an easier way to solve it? $\forall x\in\mathbb R : f(x) = x^3 +x -8$
solve : $2f(x) +3f^{-1}(x) =10 $
I actually tried to write it as : $f^{-1}(x) = \frac{10-2f(x)}{3}$
Hence : $x=f(\frac{10-2f(x)}{3})$
But it seems to be so hard to solve , do you have any suggestions for solv... | With $y:=f^{-1}(x)$, the equation becomes
$$\tag12f(f(y)) +3y=10$$which produces an awfully high degree equation:
$$ 2y^9 + 6y^7 - 48y^6 + 6y^5 - 96y^4 + 388y^3 - 48y^2 + 389y - 1066=0.$$
Solving such an equation exactly is beyond hope, in general. By sheer luck we may find a solution by trying a few small integer valu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2940689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find variance and expectation of $(n+1)X_{(1)}$ Consider a $X_{1} \dots X_{n}$ be a i.r.v with uniform distribution $[0,\theta)$. Now we may consider $Y = (n+1)X_{(1)} = (n+1)\min (\{ X_{i}\}^{n}_{i=1})$
Suppose we want to know variance and distribution of this r.v.
First of lets consider : $\mathrm{P}((n+1)^{2}X^{2}_... | You already know that
$$
f_{X_{(1)}}(x) = \frac n\theta \left(1 - \frac x\theta\right)^{n-1}.
$$
I'm going to ignore the scaling by $n+1$ as the hard part is the moments of $X_{(1)}$, not the scaling.
This means that
$$
E(X_{(1)}^p) = \frac n\theta\int_0^\theta x^p \left(1 - \frac x\theta\right)^{n-1}\,\text dx.
$$
Let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2940824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the set $T = \{1,\frac12,\frac13,\ldots,\frac1n,\ldots\}$ closed?
Exercise:Is the set $(T = \{1,\frac12,\frac13,\ldots,\frac1n,\ldots\})$ closed in $\mathbb{R}$?
I tried to answer this question by computing $\mathbb{R}\setminus T=\bigcup_\limits{2}^{\infty}(\frac{1}{n-1},\frac{1}{n})\cup(1,\infty)\cup(-\infty,1)$
... | Your answer does not work because $\mathbb{R}\setminus T$ does not contain the interval $(-\infty, 1)$. For example, $\frac12\in T$.
One way to solve your question: can you show that if $T$ is closed, you must have $0\in T$?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove/What laws to use for $(A\times B)∩(B\times A)$ $=$ $(A∩B)\times (A∩B)$. I'm stuck on where to start for my homework. I'm trying to re-write the left side using one of the sets laws. But I'm either blind or I just have no idea.
Is there any easier to start or is using the laws the best way to go about it?
| First, welcome to MSE. I hope you enjoy your stay.
With regards to your problem I would recommend using the definition of equality of sets. Specifically two sets $C$ and $D$ are said to be equal if $C\subseteq D$ and $D\subseteq C$. That is, if $x\in C$ then $x\in D$ and if $x\in D$ then $x\in C$.
You might try this fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2941051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are continuous functions dense in bounded measurable functions of a compact metric space? Let $X$ be a compact metric space equipped with the Borel $\sigma$-algebra.
Then we have $C(X)$, the set of all the real-valued continuous maps on $X$, equipped with the sup-norm.
We may also define $BM(X)$ as the set of all the r... | No, this is not true even for an interval in $\mathbb{R}$.
Recall uniform limit of continuous is continuous, so $C(X)$ is closed in $BM(X)$ (or its quotient $L^\infty(X)$). However, there are bounded discontinuous but measurable functions such as
$$
f(x)=\begin{cases}1 & x\geq\frac12\\
0 & x<\frac12
\end{cases}
$$
on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2941164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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L'Hôpital's rule only applicable if right-hand limit exist? In a source I have been reading, this statement was made regarding L'Hôpital's rule:
Why is it the case that L'Hôpital's rule is applicable only if the right-hand limit exists? Why not the left-hand? Why not both? I have read other sources on L'Hôpital's ... | My pocket example of such a limit is
$$ \lim_{x \to \infty} \frac{x + \sin x}{x + \cos x}.$$
This limit is very clearly $1$. But an application of l'Hopital's rule would lead to the consideration of
$$ \lim_{x \to \infty} \frac{1 + \cos x}{1 - \sin x},$$
which doesn't exist! Thus existence of the left hand limit does n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2941303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Finding value of $ \lim_{n\rightarrow \infty}\prod^{n}_{k=1}\frac{4k^2}{4k^2-1}$
Finding value of $\displaystyle \lim_{n\rightarrow \infty}\prod^{n}_{k=1}\frac{4k^2}{4k^2-1}$
Try: $$\lim_{n\rightarrow \infty}\prod^{n}_{k=1}\frac{2k}{2k-1}\cdot \frac{2k}{2k+1} = \lim_{n\rightarrow \infty}\prod^{n}_{k=1}\frac{2k}{2k-1... | You may find the following approach useful which avoids Stirling'approximation.
Let $$a_n=\int_{0}^{\pi/2}\sin^nx\,dx\tag{1}$$ and using integration by parts we have $$a_n=\left.-\sin^{n-1}x\cos x\right|_{x=0}^{x=\pi/2}+(n-1)\int_{0}^{\pi/2}\sin^{n-2}x\cos^2x\,dx$$ and the last integral can be written as $a_{n-2}-a_n$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simplify $(3\log x) - (2\log x)$ How to simplify $(3\log x) - (2\log x)$? Would this become $(\log x )^ {\frac{3}{2}}$ or would this be just $3\log x-2\log x =\log x$? If so how to get $\log x$?
I was given this question: solve for $x$ if $\log x + \log x^2 +...+ \log x^n =n(n+1)$. But, the answer to my main question w... | $3\log x - 2\log x = \log x$, just like $3y-2y=y$ no matter what $y$ is equal to.
Alternatively, you can get
$$3\log x - 2\log x = \log(x^3)-\log(x^2) = \log\left(\frac{x^3}{x^2}\right) = \log x$$
but you can never under any manipulation get $$3\log x - 2\log x = \log(x)^\frac{3}{2}$$ because that equality is simply no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2941892",
"timestamp": "2023-03-29T00:00:00",
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Solving an integer (boolean) constraint satisfaction problem I have a 0-1 integer constraint satisfaction problem of the following form: find binary vectors $x = (x_1,\dots,x_m) \in \{0,1\}^m$ and $y = (y_1, \dots,y_n) \in \{0,1\}^n$ that satisfy the constraints
*
*$x_i \le \sum_{j,k} a_{ijk} x_j y_k\ $ for $i = 1,\... | The problem you describe is a non-convex binary program. The non-convexity comes from the first and second set of constraints: In $x_i \le \sum_{j,k} a_{ijk} x_j y_k\ $ two decision variables $x_j$ and $y_k$ are multiplied by each other. This will make the problem very hard to solve for most solvers. However, there is ... | {
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"url": "https://math.stackexchange.com/questions/2942006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Recursive formulae in logic? Given domain $A$ and variables $x,y,z$, we could define the following "recursive formula":
$$\phi(x,y): \psi(x,y) \lor \exists z,[\phi(x,z)\land\phi (z,y)]\tag {*}$$
Where $\psi(x,y)$ is a first-order formula.
Clearly, this formula is not logically equivalent to any formula in first order l... | First-order logic on its own is completely neutral about axioms that might be interpreted as recursive definitions: as a simple example, $\forall x(f(x) = f(x))$ is trivially true in any first-order theory even though it will lead to a non-terminating function if you treat it as a definition in a functional programming... | {
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"timestamp": "2023-03-29T00:00:00",
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Use an appropriate change of variables to solve the differential equation. Use an appropriate change of variables to solve the differential equation.
$$t\frac{dy}{dt}-y=\sqrt{t^2+y^2}$$
My friend and I are trying to figure out how to solve this equation. Our professor has given us several methods but we aren't sure whi... | Divide by $t$ to obtain,
\begin{equation}
\frac{dy}{dt} - \frac{y}{t} = \sqrt{1+\left ( \frac{y}{t} \right )^2},
\end{equation}
then use a change of variables $u = \frac{y}{t}$. The transformed equation should be integrable using standard methods.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $2^q+q^2$ is divisible by 3 where $q$ is a prime and $q\geq5$. I'm looking to prove that $2^q+q^2$ is divisible by $3$ where $q$ is a prime such that $q\geq5$.
I know that primes greater than five will be congruent to either $1\ (\text{mod}\ 3)$ or $2\ (\text{mod}\ 3)$, which means that the $q^2$-term will a... | To show that congruence of $2^q\pmod 3$ is equal to 2 for a prime $q\geq 5$, it's sufficient to prove it for odd numbers, because any prime $q\geq 5$ is odd. So it reduces to verifying that $2^1\equiv 2\pmod 3$ and $2^2\equiv 1\pmod 3$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Selection of the numerical method
Given the transcendental equation:
$$ \frac{\tan x}{x} + c = 0, $$ where $c$ is any real number.
I tried Newton's method, but it is very bad fit. Which numerical method will be the smartest solution in this case?
| If you can find an interval which contains a single root with a sign change, and no singularities, then the secant method or a variant on it should work. This method avoids the problems associated with picking a bad starting point.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2942400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding an elementary function Can someone please help me find any elementary function that satisfies
*
*$f(0) = 6$
*$f(1) = f(-1) = 4$
*$f(2) = f(-2) = 1$?
I have been trying for nearly an hour, but I still can't figure it out. Only the points listed above matter. Nothing else matters (I don't care what $f(1.5)$... | Note that if given a function $h(x)$ where $h(-x)=x$, if we can find a function $g(x)$ such that $g(h(0))=6$, $g(h(1))=4$, $g(h(2))=1$, then we can take $f(x) = g(h(x))$. Rakibul Islam Prince seems to have taken $h(x) = x^2$ and $g(x)=\frac14 x^2-\frac94x+6$. lulu seems to have taken $h(x) = |x|$ and $g(x) = 6-\frac12... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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At a minimum or a maximum why does the first approximation make no difference with small variations? In an ordinary function like the temperature—one of the properties of the minimum is that if we go away from the minimum in the first order, the deviation of the function from its minimum value is only second order.
At ... | We say that a function $f: \mathbb{R} \rightarrowtail \mathbb{R}$ is differentiable at a point $a \in \text{Int}(\text{dom}(f))$ if there exists $A \in \mathbb{R}$ and $r: \text{dom}(f) \to \mathbb{R}$ function with $\lim\limits_{x \to a} \frac{r(x)}{x-a}$ so that
$$f(x)=f(a)+A(x-a)+r(x)$$
And of course $A=f'(a)$.
So i... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$p|\Phi_n(2)$ then $p|\Phi_{pn}(2)$
Prove that $p|\Phi_n(2)$ then $p|\Phi_{pn}(2)$
Here $\Phi_n(x)$ is nth cyclotomic polynomial.
I don't know what I should use. $$\Phi_n(x)=\prod_{{1\leq a\leq n } \& {(a,n)=1}}(x-\zeta_n^a) $$ or $$\Phi_n(x)=\prod_{d|n}(x^{n/d}-1)^{\mu(d)}$$
| Observe that the map $(-)^p:\mathbb F_p[x]\to \mathbb F_p[x]$ is ring morphism (with trivial kernel). For any positive integer $m$ indivisible by prime $p$ and nonnegative integer $k$ one has $$\begin{align*}\Phi_{p^km}(x)&=_{\mathbb F_p[x]}\prod_{d\mid p^km}\left(x^{p^km/d}-1\right)^{\mu(d)}\\ &=_{\mathbb F_p[x]}\prod... | {
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Evaluating the limit : $ \lim_{n \to \infty} \frac{ \sum_{k=1}^n n^k}{ \sum_{k=1}^n k^n}$ Here I'm given this limit.
$$\displaystyle \lim_{n \to \infty} \dfrac{\displaystyle \sum_{k=1}^n n^k}{\displaystyle \sum_{k=1}^n k^n}$$
$\displaystyle \sum_{k=1}^n n^k$ simplifies to $\dfrac{n(n^n-1)}{n-1}$ but I'm unable to tackl... | Note that
$$ \sum_{k=0}^n k^n = \sum_{j=0}^n (n-j)^n = n^n \sum_{j=0}^n (1-j/n)^n$$
and using dominated convergence,
$$ \sum_{j=0}^n (1-j/n)^n \to \sum_{j=0}^\infty e^{-j} = \frac{e}{e-1}$$
Thus
$$ \frac{\sum_{k=0}^n n^k}{\sum_{k=0}^n k^n} \to \frac{e-1}{e}$$
| {
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"url": "https://math.stackexchange.com/questions/2942989",
"timestamp": "2023-03-29T00:00:00",
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Solution to the equation of a polynomial raised to the power of a polynomial. The problem at hand is, find the solutions of $x$ in the following equation:
$$ (x^2−7x+11)^{x^2−7x+6}=1 $$
My friend who gave me this questions, told me that you can find $6$ solutions without needing to graph the equation.
My approach was ... | If this is just a casual riddle, then I can agree with the accepted answer. However, if we want to be mathematically strict, I claim that $3$ and $4$ are not solutions of the equation because they lie outside the domain.
Disclaimer: in this post I only consider real exponentiation. It is not my intention to dive into t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "52",
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How to come up with a greedy solution and prove it? Say we have a function $S(x)$, which gives the sum of the digits in the number $x$. So $S(452)$ would be $4 + 5 + 2 = 11$.
Given a number $x$, find two integers $a, b$ such that $0 <= a, b <= x$ and $a + b = x$. Objective is to maximize $S(a) + S(b)$. I came across t... | Show the following two statements (I guess they would be lemmas):
*
*When adding $a+b$ the way you learn in school, if you get no carries, then $S(a+b)=S(a)+S(b)$
*For each carry you get when adding $a+b$, the sum $S(a)+S(b)$ increases by $9$.
Together they mean that you want to have as many carries as you can. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2943253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
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Find intervals where $f$ is increasing/decreasing $f(x) = e^{-3x} -3e^{-2x} + 1$ I'm doing a problem where I'm asked to find the intervals where $f$ is decreasing or increasing of:
$$f(x)=e^{-3x}-3e^{-2x}+1.$$
I've found that the derivative is $f'(x)=-3e^{-3x}+6e^{-2x}$ and that $f'(x)=0$ when $x=-\ln(2)$.
As far as I... | Hint: Solve the inequality $$f'(x)\geq 0$$ this means $$-3e^{-3x}+6e^{-2x}\geq 0$$
This means $$e^x\geq \frac{1}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2943343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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( True/False) $f(x)=0$ has no positive solution if $f(0)=0$ and $f'(0)>0$
True/False:
Let $f$ be a twice differentiable function on $\mathbb{R}$ with $f''(x)>0$ for all $x\in \mathbb{R}$. If $f(0)=0$ and $f'(0)>0$, then $f(x)=0$ has no positive solution
Attempt [trying to show that this statement is TRUE]
$f''(x)>0$... | The result is true.
As $f^{\prime \prime}(x) > 0$ for all $x \in \mathbb R$, $f^\prime$ is an increasing map. As $f^\prime(0) > 0$, you have $f^\prime(x) > 0$ for all $x > 0$. Hence $f$ is a strictly increasing map.
As $f(0) =0$, you have $f(x) > 0$ for all $x>0$ proving the desired result.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Subgroup of order $4$ in $D_8$ Let me ask you a question on group theory which confuses me.
Consider the group $D_8$ the dihedral group of order $8$ generated by $\sigma$ and $\tau$ with $o(\sigma)=4,o(\tau)=2$ and $\tau\sigma=\sigma^{-1}\tau$.
Consider the following elements, namely $\sigma\tau$ and $\tau$ which have ... | I think your mistake is to assume that $\tau$ and $\tau \sigma$ commute. This is not the case
$$
\tau \cdot \tau \sigma = \sigma,
$$
while
$$
\tau \sigma \cdot \tau = \sigma^{-1} \tau \tau = \sigma^{-1} \ne \sigma.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2943559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove if matrix has right inverse then also has left inverse. I tried to prove that if $A$ and $B$ are both $n\times n$ matrices and if $AB = I_n$ then $BA = I_n$ (i.e. the matrix $A$ is invertible). So first I managed to conclude that if exists both $B$ and $C$ such that $AB = I_n$ and $CA = I_n$, then trivially $B=C$... | A matrix $A\in M_n(\mathbb{F})$ has a right inverse $B$ (which means $AB=I$) if and only if it has rank $n$. I assume you know that. So now you need to prove that $BA=I$. Well, let's multiply the equation $AB=I$ by $A$ from the right side. We get $A(BA)=A$ and hence $A(BA-I)=0$. Well, now we can split the matrix $BA-I$... | {
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"source": "stackexchange",
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Exercise 2, chapter 3 of Barry Simon. A comprehensive course in analysis part 1.
*For $x, y$ in $V$, an inner product space, and for $\lambda=re^{i\theta}\in\mathbb{C}$, use $|x+\lambda y|\geq 0$ for $\theta$ fixed to get a quadratic equation in $r$ whose roots must be
either equal or nonreal. Show that this implies ... | A straightforward proof of the CS inequality is obtained from the orthogonal (right triangle) decomposition where $x$ is the hypotenuse:
$$
x = \left(x-\frac{\langle x,y\rangle}{\langle y,y\rangle}y\right)+\frac{\langle x,y\rangle}{\langle y,y\rangle}y.
$$
From this it follows that
$$
\|x\|... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Am I allowed to write $(\mathbf e_1\cdot\frac{d}{dt}\mathbf v)_\varepsilon=(\frac{d}{dt}[\mathbf e_1\cdot\mathbf v])_\varepsilon$? Let $\varepsilon$ be the Euclidean space with basis $(\mathbf e_1,\mathbf e_2,\mathbf e_3)$.
For a rigid body in $\varepsilon$ suppose we have $$\mathbf a=\left(\frac{d}{dt}\mathbf v\right)... | Yes, the equivalence
$\mathbf e_i \cdot \dfrac{d\mathbf v}{dt} = \dfrac{d(\mathbf e_i \cdot \mathbf v)}{dt} \tag 1$
is valid, and the reason is that the $\mathbf e_i$ are constant with respect to $t$; then the ordinary Leibniz rule for product differentiation yields
$\dfrac{d(\mathbf e_i \cdot \mathbf v)}{dt} = \dfrac{... | {
"language": "en",
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Is $\Bbb Q$ a decomposable module over $\Bbb Z$ or not?
Is $\Bbb Q$ a decomposable module over $\Bbb Z$ or not?
My attempt:
let, $p_1,p_2,\dots,p_k,\dots$ be an enumeration of primes in $\Bbb N$. Then, can't we write $\Bbb Q = \Bbb Z \oplus Z(\frac{1}{p_1})\oplus \Bbb Z(\frac{1}{p_2})\oplus \dots \oplus \Bbb Z(\frac... | Suppose $\mathbb{Q}$ is decomposable as $\mathbb{Q}=X\oplus Y$. Then $X$ is
*
*divisible, because it is a homomorphic image of $\mathbb{Q}$;
*torsionfree, because it is a subgroup of $\mathbb{Q}$.
Similarly for $Y$. Therefore $X$ and $Y$ are vector spaces over $\mathbb{Q}$ and $\mathbb{Q}=X\oplus Y$ is a decompo... | {
"language": "en",
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Integrate $\int \frac {dx}{\sqrt {(x-a)(x-b)}}$ where $b>a$ Integrate: $\displaystyle\int \dfrac {dx}{\sqrt { (x-a)(x-b)}}$ where $b>a$
My Attempt:
$$\int \dfrac {dx}{\sqrt {(x-a)(x-b)}}$$
Put $x-a=t^2$
$$dx=2t\,dt$$
Now,
\begin{align}
&=\int \dfrac {2t\,dt}{\sqrt {t^2(a+t^2-b)}}\\
&=\int \dfrac {2\,dt}{\sqrt {a-b+t^2}... | Alternatively, you can use an Euler substitution to rationalize the integrand.
Option 1 Change variable to $t$, where $$\sqrt{(x - a) (x - b)} = x + t.$$
rearranging gives $$x = \frac{ab - t^2}{2 t + (a + b)},$$
and substituting gives $$\int \frac{dx}{\sqrt{(x - a) (x - b)}} = \int \frac{dt}{t + \frac{1}{2}(a + b)} .$... | {
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Find all the functions in $\mathbb{R}$, satisfying the given equation $(x+y)(f(x)-f(y)) = (x-y)f(x+y)$ for all $x,y$ in $\mathbb{R}$. Find all the functions in $\mathbb{R}$, satisfying the given equation
$(x+y)(f(x)-f(y)) = (x-y)f(x+y)$ for all $x,y$ in $\mathbb{R}$.
I tried to find something like a pattern that woul... | The equation from Cesaro requires "differentiable".
A class of solutions is f(x) = ax
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Equivalent Capital Pi Notation Expressions So I was working on a probability question and then this expression came up.
When I consulted the answers, I struggled to understand exactly how I would get from one expression to the other myself.
Substituting a constant such as
let $n=5$ makes it a bit clearer how they got ... | Perhaps a picture will help.
Think about the set of points $(j,k)$ in the plane at which you are evaluating the fraction $(40-j)/(52-j)$ (which happens not to depend on $k$). You want to find the product of all the values.
The points (with integer coordinates) will form a triangle. You can think of the product as find... | {
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"source": "stackexchange",
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"answer_id": 2
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How to sort out different cases in a proof related to a metric space Problem:
Let $M = (X, d)$ be a metric space.
Show that
$e(x, y) = min(1, d(x, y))$ is a metric.
To prove the triangle inequality for metric $e$,
three cases are considered.
Let $x, y, z \in X$.
(A) $d(x, y) \le 1$ and $d(y, z) \le 1$
(B) $d(x, y) > ... | You have a metric space $X$ with a metric $d$, and you are given three points $x,y,z$. You are told to consider $d(x,y)$ and $d(y,z)$.
Your statement simply asserts that there are only certain possibilities. There's no extra trickery here. You either have that
*
*Both $d(x, y) \le 1$ and $d(y, z) \le 1$,
*Only on... | {
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Does exponentiation of ideals factor through intersection? I am trying to prove that $(I \cap J)^2 = I^2 \cap J^2$. So far I have established the forward direction:
$(\subseteq)$ Let $a \in (I \cap J)^2$. Then $ a = a_1a_2 + a_3a_4 + a_5a_6 + \dots + a_{n-1}a_n$, where $a_i \in I \cap J.$ Then $a_i \in I, a_i \in J \i... | I am not sure that this holds in general, but I know that it is true for Dedekind Domains (so therefore also true for a PID) due to the unique factorisation of prime ideals. Here is a proof of that:
Write $I=\prod\limits_{i}\mathfrak{p_{i}}^{e_{i}}$ and $J=\prod\limits_{i}\mathfrak{p_{i}}^{f_{i}}$ where the $\mathfrak{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2944701",
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"source": "stackexchange",
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Is this theorem about girth and bipartite graph wrong? In this paper, the abstract mentions that the classical work of Andrásfai, Erdős, and Sós implies:
Every $n$-vertex graph with odd girth $2k+1$ and minimum degree bigger than $\dfrac{2}{2k+1}n$ must be bipartite.
I think the statement is wrong.
My idea is that if... | If you look about the middle of page 2, they define "a graph has odd girth at least $g$ if it contains no odd-length cycle of length less than $g$".
In other words they use a concept called "odd girth" which is not the same as saying the girth is odd.
I agree it looks quite confusing though.
| {
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Convert complex number to polar coordinates Problem
Compute when $x \in \mathbb{C}$:
$$ x^2-4ix-5-i=0 $$
and express output in polar coordinates
Attempt to solve
Solving this equation with quadratic formula:
$$ x=\frac{4i \pm \sqrt{(-4i)^2-4\cdot (-5-i)}}{2} $$
$$x= \frac{4i \pm \sqrt{4(i+1)}}{2} $$
$$ x = \frac{4i... | Let $a,b\in\mathbb{R}$ so that $$\sqrt{i+1} = a+bi$$
$$ i+1 = a^2 -b^2 +2abi $$
Equating real and imaginary parts, we have
$$2ab = 1$$
$$a^2 -b^2 = 1$$
Now we solve for $(a,b)$.
$$
\begin{align*}
b &= \frac{1}{2a}\\\\
\implies \,\,\, a^2 - \left(\frac{1}{2a}\right)^2 &= 1 \\\\
a^2 &= 1 + \frac{1}{4a^2}\\\\
4a^4 &= 4a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the determinant of an $n \times n$ square matrix $A$ whose entries are $a_{ij} = \max(i,j)$ I figured the matrix would look like this,
$$
A = \begin{bmatrix}
1 & 2 & 3 & \dots & n \\
2 & 2 & 3 & \dots & n \\
3 & 3 & 3 & \dots & n \\
\vdots & \vdots & \vdots & \ddots &\vdots \\
n-1 & n-1 & n-... | As in this answer: if you substract the $(i+1)$-th row to the $i$-th one, you will end up with a lower triangular matrix with all diagonal entries being $-1$ except $A_{nn} = n$, so $\det A = n(-1)^{n-1}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Limit of a sequence when the algebraic limit theorem 'breaks down' Background
There is the very well-known technique of computing the limit of a sequence by taking limit on both sides whenever recurrence relation arises. For example, $b_n = \frac{\alpha^n}{n!} $ for $0 < \alpha < 1$.
Question
But now there is this s... | $$a_n = \frac13a_{n-1} + \frac23a_{n-2}$$
The characteristic equation is $x^2-\frac13x-\frac23=0$.
$$3x^2-x-2=0$$
$$(3x+2)(x-1)=0$$
$$x=-\frac23,1$$
$$a_n = \alpha \left( -\frac23\right)^n+\beta$$
We have $a_1=0$ and $a_2=1$,
$$0=\alpha\left( -\frac23\right)+\beta$$
$$1=\alpha\left( -\frac23\right)^2+\beta$$
$$1=\alpha... | {
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Prove that $1/6 < \int_0^1 \frac{1-x^2}{3+\cos(x)}dx < 2/9$
Prove that $$\frac{1}{6}<\int_0^1 \frac{1-x^2}{3+\cos(x)}dx < \frac{2}{9}. $$
I tried using known integral inequalities (Cauchy-Schwarz, Chebyshev) but I did not arrive at anything. Then I also tried considering functions of the form $$f(x) = \int_0^x \frac{... | Hint: Since ${\pi\over 3}>1\geq x$ we have $\cos x > {1\over 2}$ so $$3+{1\over 2}<3+\cos x\leq 4$$ so $${1\over 4}\leq {1\over 3+\cos x}< {2\over 7}$$
$${1\over 6}\leq \int_0^1 {1-x^2\over 3+\cos x}dx<{4\over 21}<{2\over 9}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2945601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Limit Epsilon Delta everybody!
I am a new user here. Please correct me if I make any mistakes.
Show that for any $\epsilon$>0 there exists N such that for all n $\geqq$ N it is true that $|x^n - 0|$<$\epsilon$
x $\in$ (-1,1)
$x^n \to$ 0 as n$\to$ ∞
I tried to solve this problem
$lim_{x\to ∞}$ $x^n$ = 0
|$x^n$-0|... | For the second one you can do this.
$x_{n+k}=\frac{a^{n+k}-b^{n+k}}{a-b}$
So the given expression is
$\frac{x_{n+k}}{x_n}=\frac{a^{n+k}-b^{n+k}}{a^n-b^n}$
Taking $a^{n+k}$ common in numerator and $a^n$ in denominator we get
$\lim: \lim_{n\to \infty} \frac{a^k(1-\left({\frac{b}{a}}\right)^{n+k})}{1-\left({\frac{b}{a}}\r... | {
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Solving $2^x + 3^x = 12$ I need to solve $2^x + 3^x = 12$ for real $x$. I tried the following:
$$
2^x + 3^x = 3\times 2^2 \\
1+(3/2)^x= 3\times 2^{2-x}
$$
But from here on I don't know how to apply logarithms.
| Value of $x=1.917685944888545$
(15 digits round off)
| {
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"timestamp": "2023-03-29T00:00:00",
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A solution for $f(ax+b)=f(x)+1$ Let $a,b$ be two constant real numbers with $a\neq 0$. Can anyone give a special solution of the functional equation $f(ax+b)=f(x)+1$, where $f:\mathbb{R}\rightarrow \mathbb{R}$?
Note. It is a type of the Abel functional equations, and if $a=1$, then
$f(x)=[\frac{x}{b}]$ is an its solut... | For fixed $a\in\mathbb{R}\setminus\{1\}$ and $b\in\mathbb{R}$, let now consider a function $f:\mathbb{R}\to\mathbb{R}$ which satisfies the functional equation
$$f(ax+b)=f(x)+1\text{ for all }x\in\mathbb{R}\setminus\left\{\frac{b}{1-a}\right\}\,.\tag{#}$$
Firstly, we assume that $a=0$. Then, we see that $f(x)=f(b)-1$ f... | {
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Prove that the quantity is an integer I want to prove that $\frac{n^3}{3}-\frac{n^2}{2}+\frac{n}{6} \in \mathbb{Z}, \forall n \geq 1$.
I have thought to use induction.
Base Case: For $n=1$, $\frac{n^3}{3}-\frac{n^2}{2}+\frac{n}{6}=\frac{1}{3}-\frac{1}{2}+\frac{1}{6}=0 \in \mathbb{Z}$.
Induction hypothesis: We suppose t... | $$F=\frac{n^3}{3}-\frac{n^2}{2}+\frac{n}{6}=\frac{n(n-1)(2n-1)}{6}$$
You can see that for any n(odd or even) the numerator is always a multiple of 6, so the sum of fractions is an integer; in fact we may have:
*
*$n=6k ⇒ F=6m/6=m $
*$n=6k+1 ⇒ F=(6k+1)(6k+1-1)(12k+2-1)=6m/6=m$
*$n=6k+2 ⇒ F=(6k+2)(6k+2-1)(12k+4-1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\sup(S-T)=\sup S-\inf T$
Let $S$ and $T$ be nonempty sets of real numbers and define
$$S-T=\{s-t|s\in S,t\in T\}$$
Show that if S and T are bounded then
$$\sup(S-T)=\sup S-\inf T\\
\inf(S-T)=\inf S-\sup T.$$
My proof:
Since $S,T\subset\mathbb{R}$ are nonempty and bounded, then, by the Completeness A... | You fix $\varepsilon > 0$ and choose $x$ and $y$ such that $\alpha - \varepsilon < x \le \alpha$ and $-\beta - \varepsilon < -y \le -\beta$. Choosing such $x$ and $y$ represent a small concession: you know that $\alpha$ and $-\beta$ may not be achievable, but you know that you can get as close as you want to these boun... | {
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Fredholm Alternative for Singular ODE Consider the following inhomogeneous boundary value problem,
$$t^2 u'' + tpu' +qu = f(t), \ t \in [-1,1], \ \ u(1) = \alpha, \ u(-1) = \beta,$$
where $p$ and $q$ are constants.
I would like to determine a condition for the existence of a solution to this problem using the Fredholm... | As RHowe remarked,
the homogeneous equation $t^2 u'' + t p u' + q u = 0$ is Cauchy-Euler. Its indicial roots are $r_\pm = (1-p \pm \sqrt{(1-p)^2 - 4 q})/2$. Thus if those are distinct, the general solution of the homogeneous equation for $t > 0$ is $c_+ t^{r_+} + c_- t^{r_-}$. Now since you want a solution on an i... | {
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"answer_id": 0
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Distribution of $x^2+y^2$ vs. $x^2+y^2+z^2$ where $x, y$, and $z$ are each uniform random ~$ (0,1)$ I'm trying to figure out why $x^2+y^2$ is uniform distributed while $x^2+y^2+z^2$ appears to be distributed as $\sqrt(x)$. Both distributions drop off once $x^2+y^2 $ is bigger than 1 or $x^2+y2+z^2$ is bigger than 1 pre... | This result is part of the paper published by Ishay Weissman in Statistics and Probability Letters in Statistics and Probability Letters $129, (2017), 147–154$. The constant property was first noticed by Adi Ben-Israel.
It is shown that
$$f_2(s) =\begin{cases} \frac{\pi}4 & ,0 \le s\le 1 \\ \arcsin\left( \frac{1}{\sqr... | {
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Solving the system $a^2-c^2=x^2-z^2$, $ab=xy$, $ac=xz$, $bc=yz$ I've stumbled upon these equations, and am struggling to find a manual way to solve this in $\Re$:
$$\begin{align}
a^2-c^2&=x^2-z^2 \\
ab&=xy \\
ac&=xz \\
bc&=yz\end{align}$$
I've used Wolfram Alpha to compute this and I found that this is only possible ... | There is in fact another solution: $a=-x$, $b=-y$, $c=-z$.
Multiply the bottom three equations together:
$$(abc)^2=(xyz)^2$$
$$abc=\pm xyz$$
This leads to $c=\pm z$, $b=\pm y$, $a=\pm x$. The first equation is automatically satisfied after these relations. To see that all $\pm$ assignments must be the same, try letting... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2946743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Is it true that in a partial order the intersection of two upper cones is either disjoint or again an upper cone? Suppose $(P, \leq)$ is a partially ordered set.
For $x \in P$, define $U_x := \{ y \in P \ | \ y \geq x\}$.
Is it true that for any $x,y \in P$, either $U_x \cap U_y = \emptyset$ or $U_x \cap U_y = U_z$ fo... | What does it meant that $U_x\cap U_y$ is empty? It means that no element is larger than both of them.
What does it meant that $U_x\cap U_y=U_z$? It means that any element which is larger than both is also larger than $z$ (or $z$ itself).
So in order to find a counterexample, we need to engineer a partial order in whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2946986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $A(AA^T)^{-1}A^T$ a diagonal matrix? $A$ is a $n\times k$ matrix with rank $k<n$. I was wondering if $A(A^TA)^{-1}A^T$ a diagonal matrix where $k$ entries are one and other entries are zero.
I'm not sure if this is correct. If this is correct, how to prove it?
| My first instinct would be to try out an example or two, using (for instance) WolframAlpha. Then, once I had gotten the correct order of transopses and non-transopses so that all the dimensions line up and make sense, I would see that it is not the case:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2947084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Understanding a proof that $17\mid (2x+3y)$ iff $17\mid(9x +5y)$ I was studying the number theory and came across this question.
Example 1.1. Let $x$ and $y$ be integers. Prove that $2x + 3y$ is divisible by 17 if and only if $9x + 5y$ is divisible by 17.
Solution. $17 \mid (2x + 3y) \implies 17 | [13(2x + 3y)]$, or $... | I think the "if and only if" (abbreviated "iff" in the title of this question and in the textbook you're studying) is confusing you. It kind of suggests that one requires the other but the other does not necessarily require the one, when in fact the two conditions are mutually dependent: one requires the other and the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2947201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 3
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Find value of n such the function has local mimima at x=1. If $f$ is defined by
$$f(x)=(x^2-1)^n(x^2+x-1)$$
then $f$ has a local minimum at $x=1$, when
*
(i) $n=2$
(ii) $n=3$
(iii) $n=4$
(iv) $n=5$
Multiple options are correct.
The given answer is $n=2$ and $n=4$.
I tried putting derivative equal to zero and ... | $x=1$ is a local minimum if it is a double root for $f$. Since $1$ is not a root of $x^2+x-1$ it must be a even root for $(x^2-1)^n = (x+1)^n(x-1)^n$ and that is when $n$ is even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2947307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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isomorphism between category of sheaves and morphisms of abelian groups I am working on theory of category and I found this exercise. I tried a lot but I didn't know how I could do. Let $A$ a discrete valuation ring. Show that the category of sheaves of abelian groups on $Spec(A)$ is equivalent to the category which ob... | Let us write $X=\operatorname{Spec} A$. It seems that your main confusion is about what the topology on $X$ looks like in this case. If $A$ is a discrete valuation ring, it has two prime ideals $P=\{0\}$ and $Q$, the maximal ideal. So $X=\{P,Q\}$.
Now we need to determine the topology on $X$. By definition, a subse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2947412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Transitive group action implies conjugate stabilizers even when group is infinite? The same question has been asked. But answer doesn't address my concerns here.
This was an exercise in textbook. They didn't mention that the group order has to be finite.
It is immediately obvious to me that take two stabilizers $G_x$,... | $gx=y \Rightarrow g^{-1}y=x \Rightarrow g^{-1}G_yg \le G_x \Rightarrow G_y \le gG_xg^{-1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2947549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence of $\sum_{n=1}^{\infty}e^{-\sqrt{n}}$ using the integral test Given the series : $$\sum_{n=1}^{\infty}e^{-\sqrt{n}}$$ Determine if convergent or divergent.
The function is positive and monotonically decreasing function so I've used the "Integral Test"
$$\int_{1}^{\infty}e^{-\sqrt{x}}$$
then:$$ e^{-\sqrt{x}... | I usually think that Integral Test is the least elegant way to show the convergence of a sequence. I do have to admit that it is convenient, simple and powerful, though.
In order to show convergence of series involving negative power of $e$, using the Taylor Expansion of $e^x$ is a good way.
Notice that
$$e^\sqrt{n} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2947693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 1
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Is there an analytic solution for such problem? Given function $$f_n(x) = \cos x - (\cos \cos x) + (\cos \cos \cos x) - (\cos \cos \cos \cos x) + \dots + (-1)^{n-1} \underbrace{ \cos \cos \dots \cos }_n x,$$
where $n \in \mathbb{N}$ and $\underbrace{ \cos \cos \dots \cos }_n$ means cosine of cosine of cosine and so on ... | This is not a full answer, but should help you to derive bounds for the value in both directions.
Split the sum
$$ \sum_{n=1}^\infty (-1)^{n-1} \cos^n(x) $$
into a finite part of leading terms with odd length and the remaining higher terms
$$ \sum_{n=1}^\infty (-1)^{n-1} \cos^n(x)
= \sum_{n=1}^{2N +1} (-1)^{n-1} \cos^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2947779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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How to prove double negation elimination without using $\bot$? $\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}$
I am trying to derive some rules without the use of the $\bot$ symbol.
First I want to describe how I am defining certain inference rules:
Negation Introduction: $\{(a\to b), (a \to \lnot b) \}... | This prover is helpful to reply to this question. See example 3
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Multiplying by $10$'s place voodoo: Why is $30\times 50= 15\times 100$? This is a very trivial question, but I can't seem to reason out why
$$30\times 50= 15\times 100$$
As a kid, I never really thought about why it works, but now I can't figure it out and the idea is really troubling me. I understand that we can bre... | $$30\cdot50=(3\cdot10)(5\cdot10)=(3\cdot5)(10\cdot10)=15\cdot100$$
It's all just the property that $$(ab)(cd)=(ac)(bd)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2948005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Trying to find the error in my attempt at basic probability set complement problem I am working through a textbook on probability for actuaries and I am having trouble with this problem:
In a universe $U$ of $100$, let $A$ and $B$ be subsets of $U$ such that $|A \cup B| = 70$ and $|A \cup B'| = 90$. Then what is $|A|$... | Use the additivity for cardinality of the union of disjoint sets.
${\def\abs#1{{\lvert #1 \rvert}}\abs {S\cup T}=\abs S+\abs T}$ when $S,T$ are disjoint (finite) sets.
$${\def\abs#1{\lvert #1\rvert}\begin{split} \abs {A\cup B}+\abs{A\cup B'} &=\abs {A\cup(A^\complement\cap B)}+\abs {A\cup(A^\complement\cap B^\complem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2948168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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why using $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to factorize a polynomial degree 2 dose not always work i tried to factorize $5x^3-11x^2+2x$ so i took out $x$ and used $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to find the roots 2 and $\frac{1}{5}$ but to my surprise multiplying the roots like so $x(x-2)\cdot(x-\frac{1}{5})$ produces... | What you did would be right if your polynomial was monic, which means if its leading coefficient was $1$. But the truth is its leading coefficient is $5$. So what you have to do is take out $\frac{1}{5}x$ instead of just taking out $x$, and then find the roots.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2948303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence for series failed using "Ratio Test" $$\sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5\cdot ...\cdot (2n-1)}{1\cdot 4\cdot 7\cdot ...(3n-2)}$$
Using Ratio test: $$\lim_{n\rightarrow \infty}\frac{\frac{2(n+1)-1}{3(n+1)-2}}{\frac{2n-1}{3n-2}}$$
which equals to : $$\lim_{n \to \infty}\frac{6n^{2}-n-2}{6n^{2}-n-1}$$
t... | List out your $a_n$ clearly.$$a_n =\prod_{i=1}^n \left(\frac{2i-1}{3i-2}\right)$$
$$\lim_{n \to \infty}\frac{a_{n+1}}{a_n}=\lim_{n \to \infty}\frac{2(n+1)-1}{3(n+1)-2}=\frac{2}{3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2948533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluate $\lim_{x \to 0} x^2\left(1+2+3+...+\left[ \frac{1}{|x|} \right] \right)$ I was thinking about the squeeze theorem here. We can denote the $\left[\frac{1}{|x|}\right] =n$, and then try something like:
$$x^2(1+2+3+...+(n-1)+(n-1)) \leq x^2 \frac{n(n+1)}{2} \leq x^2(1+2+3+...+(n-1)+(n+1))$$
But I don't know what ... | Given that
$$\left[\frac1{|x|}\right]=n$$
we have
$$n\le\frac1{|x|}<n+1$$
Therefore
$$\frac1n\ge|x|>\frac1{n+1}$$
and thus,
$$\frac1{(n+1)^2}<x^2\le\frac1{n^2}$$
Because you seem worry only about an upper bound of the function, I feel that you expect that the limit is $0$. I think it isn't...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2948673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Do we have $R\simeq S$ for two submodules $R,S$ of $A^n$?
Let $A$ be a commutative ring with identity. Given two submodules $R,S$ of $A^n$ (where $n\in\Bbb N$), if there exists an isomorphism of $A$-modules $A^n/R\simeq A^n/S$, then do we have $R\simeq S$?
Note that this is definitely false for quotients of non-free ... | No.
Let $A=\Bbb Z^\Bbb N=\{\,f\colon \Bbb N\to\Bbb Z\,\}$, $n=1$, $R=\Bbb Z=\{\,f\in A\mid \forall n>0\colon f(n)=0\,\}$, and $S=0$. Then $A^1/R\cong A^1/S\cong A$, but of coure $R\not\cong S$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $R$ finitely generated? Let $A$ be a commutative ring with identity. Given two submodules $R,S$ of $A^n(n\in\Bbb N)$ and suppose $S$ is finitely generated, if there exists an isomorphism of $A$-modules $A^n/R\simeq A^n/S$, is $R$ finitely generated?
| My example (below) was wrong, I misinterpreted what OP was asking for. I'll leave it up so others don't get confused as I did.
No: Let $A = \mathbb{C}[x_1, x_2, \ldots, x_n, \ldots]$ be the polynomial ring in infinitely many variables over $\mathbb{C}$, considered as a module over itself (i.e. the $n$ in your question ... | {
"language": "en",
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"source": "stackexchange",
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What does this command mean? I am unable to interpret the following command in a MATLAB code
while (fa > 0) == (fb > 0)
I thought it says: if fa>0 , fb > 0 and both are equal to each other then do some commands.
However, while running in debug mode, at I found that even then fa and fb were < 0 and not equal to each ot... | fa > 0 returns a true/false vector for each entry, which is true iff the entry is positive.
Therefore, (fa > 0) == (fb > 0) is true iff all corresponding entries of fa and fb ahve the same signs.
In other words, $fa = [1,-1]$ and $fb = [-1,1]$ would cause it to be false, but $fa = [-1,-1]$ and $fb = [-2,-3]$ would make... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Maximum value on a circle I need to find the maximum value of a function on a circle: Let $C$ denote the circle of radius $6$ centered at the origin in the $xy$-plane. Find the maximum value of $x^2y$ on $C$. Where do I even start with this?
| Hint: For $(x,y)$ on the circle of radius $6$, we have
$$
x^2=36-y^2
$$
So you can find a single variable function to maximize.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do I differentiate $f(x) = 7 + 6/x + 6/x^2$? the problem is the following:
with the definition of the derivative, calculate
f(x) = $7+\frac 6x+ \frac6{x^2}$
I tried to solve it a bunch of times but I just don't get the correct answer
**edit: I must solve it with the def of the derivative
| $$f’(x) = \frac{-6}{x^2} + \frac{-12}{x^3}$$
Here we are basically using the formula :
$$(\frac{u}{v} )’ = \frac{u’v-v’u}{v^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Definition of totally bounded set
A set $A$ in a metric space $(M, d)$ is said to be totally bounded if,
given any $\epsilon>0$, there exist finitely many points
$x_1,\ldots,x_n\in M$ such that
$A\subset\bigcup_{i=1}^nB_\epsilon(x_i)$.
That is, each $x\in A$ is within $\epsilon$ of some $x_i$.
The author then... | What happens when $A\cap B(x_j)=\emptyset$ for some $j$? Then
$$A=A\backslash B(x_j)\subseteq\bigg(\bigcup B(x_i)\bigg)\backslash B(x_j)\subseteq\bigcup_{i\neq j} B(x_i)$$
In particular we can refine our covering $\{B(x_i)\}$ by removing $B(x_j)$ and still preserving all required properties.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Square root of two positive integers less than or equal to the sum of both integers direct proof Please help with this problem.
If x and y positive integers, show:
$$2\sqrt{xy} \le x + y $$
| Observe that $(\sqrt{x} - \sqrt{y})^2 = x - 2\sqrt{xy} + y \geq 0$.
Rationale: For any real numbers a,b, $(a - b)^2 = a^2 - 2ab + b^2 \geq 0 \implies a^2 + b^2 \geq 2ab$. Since $x, y > 0$, $\sqrt{x}, \sqrt{y}$ are real numbers. Thus, if we set $a = \sqrt{x}$ and $b =\sqrt{y}$, we obtain $$(\sqrt{x} - \sqrt{y})^2 =... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Use set equalities to prove $A−(B\cup C) = (A−B)∩(A−C)$ This is what needs to be proved: $$A−(B\cup C) = (A−B)∩(A−C)$$
I've tried working from both sides, but have gotten further from working with the left. Here's my attempt:$$A \cap (B \cup C)^{'}$$ $$A \cap (B^{'} \cap C^{'})$$ $$(A \cap B^{'}) \cap C^{'}$$ $$(A-B)-C... | You have the following, $A-(B\cup C) = A \cap (B\cup C)^c = A \cap (B^c\cap C^c) = (A \cap B^c) \cap (A\cap C^c) = (A-B)\cap (A-C)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2949615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Standard line element example
In Euclidean three-space, we can define paraboloidal coordinates $(u,v,\phi)$ via
\begin{align*}
x = uv\cos\phi,\quad y = uv\sin\phi,\quad z = \frac{1}{2}(u^2-v^2)
\end{align*}
Find $ds^2 = dx^2 + dy^2 + dz^2$
So I just want people to check my working and answer for this question. I ... | \begin{align*}
dz &= u\,du - v\,dv\\
\\
dz^2 &= u^2du^2 - 2uv\,du\,dv + v^2dv^2\\
\\
\Rightarrow ds^2 &= du^2v^2\cos^2\phi + 2du\,dv\,uv\cos^2\phi - 2du\,d\phi\,uv^2\cos^2\phi\\
&+ dv^2u^2\cos^2\phi - 2dv\,d\phi\,u^2v\sin\phi\cos\phi + d\phi^2u^2v^2\sin^2\phi\\
&+ du^2v^2\sin^2\phi + 2du\,dv\,uv\sin^2\phi + 2du\,d\phi\... | {
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"timestamp": "2023-03-29T00:00:00",
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proof that area of convergence is bounded by a circle I'm currently looking into the topic of holomorphic functions and their radii of convergence. While I do understand according to the Cauchy's integral formula why a Taylor series converges in a radius r, which is the distance to the nearest singularity from the cent... | This happens because the region $C$ of convergence of a power series about $a$ is always is always such that $D(a,r)\subset C\subset\overline{D(a,r)}$, for some $r>0$, with two exceptions: when $C=\{a\}$ and when $C=\mathbb C$. This is so because if a power series $\sum_{n=0}^\infty a_n(z-a)^n$ converges at some $z_0\n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\sup \{\frac{a}{b}; a \in \mathbb{N}, b \in \mathbb{N}, a < b\}=1$ Consider a subset of rational numbers $S = \{\frac{a}{b}; a \in \mathbb{N}, b \in \mathbb{N}, a < b\}$. I want to prove that $\sup S = 1$. By the definition of supremum, for $\epsilon > 0$, it suffices to show that there exists $\frac{a}{b}... | Let $s=sup(S)$
Suppose $s<1$
Between two real numbers $x<y$ there's always $q\in\mathbb{Q}$ (i.e. $q \in (x, y)$)
Let $q=\frac{q_1}{q_2}$ be a rational number in $(max(s, 0), 1)$
By definition $s<q$
then choose $a=q_1$ and $b=q_2$ $\Rightarrow s<q=\frac{a}{b}$
But $q\in S$.
So $s \geq 1$
But $S$ is bounded by $1$, then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2950154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Why can't set cover be reduced to min-cost max-flow? Okay, so I know obviously I'm making some kind of easy mistake here, since set cover is NP-complete and min-cost max-flow is in P, but I can't figure out what the mistake is.
So, given a universe $U$ and a set $S$ such that the union of all sets in $S$ is $U$, the se... | To put it simply: your algorithm only guarantees to fill every node in the last(right) layer but it is not restricted anywhere on the number of subsets it uses from the first(left) layer. For example in the wikipedia case you mention it could use $\{1,2,3\}$ for the $1$, $\{2,4\}$ for the $4$ and $\{4,5\}$ for the $5$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Find $\lim_{x \to 0^{+}} \frac{\pi^{x\ln x} - 1}{x}$ if it exists . Let $f(x) = \frac{\pi^{x\ln x} - 1}{x}$ . Find $\lim_{x \to 0^{+}}f(x)$ if it exists .
My try : $f(x) = \frac{\pi^{x\ln x} - 1}{x} = \frac{e^{x\ln x \ln \pi} - 1}{x}$ . Using $(\forall u\in\mathbb{R}):e^u=1+u+\frac{u^2}{2!}+\frac{u^3}{3!}+\cdots$ and ... | I think your answer is right.
Since $x\ln{x}\rightarrow0$, we obtain:$$\frac{\pi^{x\ln{x}}-1}{x}=\frac{e^{x\ln{x}\ln\pi}-1}{x\ln{x}\ln\pi}\cdot\ln\pi\ln{x}\rightarrow-\infty$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2950431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the probability of an event using set theory Given: $P(A \cup B) = 0.7$, $P(A \cup B') = 0.9$,
Find $P(A)$.
I feel like the answer has something to do with the property that
$P(A \cup B) = P(A) + P(B) - P(AB)$ and $P(A \cup B') = P(A) + P(B') - P(AB')$, but I don't know how to get rid of $B$ and $B'$ using th... | Hint: Try adding $P(A\cup B)$ and $P(A\cup B')$ and see what cancels.
Remember in particular the law of total probability: $Pr(X\cap Y)+Pr(X\cap Y') = Pr(X)$
$0.7 + 0.9 = Pr(A\cup B) + Pr(A\cup B') = Pr(A)+Pr(B)-Pr(A\cap B) + Pr(A)+Pr(B')-Pr(A\cap B')$
$ = 2Pr(A) + \left(Pr(B)+Pr(B')\right) - \left(Pr(A\cap B) + Pr(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2950684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If the equation $\alpha x^2+4\gamma xy+\beta y^2+4p(x+y+1)=0$ represents a pair of lines. Find the range of $p$ in terms of $\alpha,\beta$
For $\alpha,\beta,\gamma\in\mathbb{R}$ with $0<\alpha<\beta$, if
$$\alpha x^2+4\gamma xy+\beta y^2+4p(x+y+1)=0$$
represent a pair of lines. Then which one is right?
(a) $p... | This answer shows that there are no correct options and that the range of $p$ is $$p\in\bigg(-\infty,0\bigg]\cup \bigg[\beta,\infty\bigg)$$
For the condition "a pair of lines", we have two cases to consider :
*
*intersecting lines
*parallel lines (including "coincident lines")
Here, let
$$\begin{align}\Delta&:=\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2950937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determine a line such that all its points lie at equal distance to three non-parallel planes. I am supposed to determine the parametric equation of a line such that all it's points lie at equal distance to the three planes,
$$x+2y+2z+3=0$$
$$x-2y+2z-1=0$$
$$2x+y+2z+1=0$$
So far I've been able to determine the point wh... | You did right the first step: if the planes have a common point the line shall pass through it.
However the planes do not need in general to have a common point.
The concept to apply is that, given two planes, the points equi-distant from them lie on one of the two planes bisecting the angles between the given plane... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2951113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Why doesn't the Stone-Weierstrass theorem imply that every function has a power series expansion? I know that not every function has a power series expansion.
Yet what I don't understand is that for every $C^{\infty}$ functions there is a sequence of polynomial $(P_n)$ such that $P_n$ converges uniformly to $f$. That'... | $$\lim_{n\to\infty}\left(\lim_{k\to\infty} a_{n,k}\right)$$ is, in general, not the same as $$\lim_{k\to\infty}\left(\lim_{n\to\infty} a_{n,k}\right)$$
and in order to switch the order of your infinite sum (which is in its definition a limit) and your limit, you would need something like that.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2951249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 4,
"answer_id": 0
} |
Dividing $n^k+1$ by $n+1$ if and only if $k$ is odd For that question, I can use modular arithmetic to prove divisibility. Look at the following:
$$n \equiv-1\mod(n+1)$$
raising to $k^{th}$ power, if $k$ is odd, then
$$n^k \equiv(-1)^k \equiv-1\mod(n+1)$$ hence
$$n^k+1 \equiv0\mod(n+1)$$ as desired. On the other hand, ... | $((n+1)-1)^k +1=$
$(n+1)^k + k(-1)(n+1)^{k-1}+......$
$..+k(n+1)(-1)^{k-1}+(-1)^k +1.$
All terms, except the last term $(-1)^k$, in the binomial expansion have a factor $(n+1)$.
For odd $k$: $(-1)^k +1=0.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2951389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Finding X for Mod? If I have this:
*
*$x \pmod p = 1$
*$x \pmod q = 0$
Is there any way I can find a possible natural number for $x$ that satisfies both equations. I know it has something to do with the Chinese Reminder Theorem; however, I have been unable to solve it.
| No need for the Chinese Remainder Theorem.
I'll assume you intended to require $p,q$ to be relatively prime positive integers.
Then by Bezout's Theorem, there exist integers $a,b$ such that
$$ap+bq=1$$
If $p,q$ are given, qualifying values of $a,b$ can be found via the Extended Euclidean Algorithm.
Moreover, if $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2951496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Do I have to discharge an antecedent that I assume? For example, if I have the premise:
$P \rightarrow (Q \rightarrow R)$
Can I assume P to get:
$Q\rightarrow R$
And then assume Q to get R.
For reductio ad absurdum and arrow introduction I know that you have to discharge the assumptions that you use, I was just wonderi... | You could do so, but the proof would be unfinished. You could discharge each premise in turn to get:
$$P\implies [Q \implies R] \implies [P\implies [Q \implies R]]$$
If you introduced all of your premises at once, you could also prove:
$$[P\implies [Q \implies R] \land P \land Q] \implies R$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2951677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $f$ is holomorphic in a compact Riemann surface, it is constant. Why doesn't this work for compact subsets of $\mathbb{C}$?
Let $X$ be a compact Riemann surface. Suppose that $f$ is holomorphic over all of $X$. Then $f$ is constant
I proved this in the following way:
The function $f$ is continuous and hence $|f|$ ... | I would point out two reasons for why the proof does not work for compact subsets of $\mathbb{C}$.
1) In general a compact subspace $K \subset \mathbb{C}$ is not connected. Of course this is not the real problem, because there are non-constant functions on connected and compact subspaces, so you can ask the question "w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2951793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Cardinality of a set defined on the Cartesian product of a power set. $2^A$ is the power set of some finite set A.
Let $R:= \{(B, C) \in 2^A \times 2^A | B \subseteq C\}$. Show that $\lvert R\rvert = 3^{\lvert A\rvert}$.
It is the $B \subseteq C$ part in the definition of $R$ that I cannot understand nor its implicati... | First, we takt $A$ to be the empty set. In this case $\vert A \vert = 0$, and
$$ 2^A = \{ \ \emptyset \ \} $$
so that $$\left\vert 2^A \right\vert = 1. $$
And, in this case
$$ R = \big\{ \ ( \emptyset, \emptyset ) \ \big\} $$
so that
$$ \left\vert R \right\vert = 1 = 3^0 = 3^{\vert A \vert}. \tag{0} $$
Now let us su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2951923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Testing if an implicit relation is a solution to an implicit equation I am asked to show whether the relation
$$ f(x,y) = x^3 + y^3 - 3xy = 0, -\infty < x < \infty $$
is a solution to the equation
$$ F(x,y,y') = (y^2 - x)y' - y + x^2 = 0, -\infty < x < \infty $$
The difficulty I am having is not in implicitly differe... | $$
f(x,y)=x^3+y^3 - 3xy=0\\
3x^2 + 3 y^2 y' - 3xy' - 3y =0\\
(3y^2-3x)y'=3y-3x^2\\
y' = \frac{y-x^2}{y^2-x}
$$
So for all points $(x,y)$ on $f(x,y)=0$, we have $y'$.
$$
F=(y^2-x)y'-y+x^2\\
$$
Assume we are on $f(x,y)=0$, then we can substitute the expression for $y'$.
$$
F=(y^2-x)\frac{y-x^2}{y^2-x}-y+x^2 = 0
$$
so all... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2952051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Can identically distributed random variables $X$ and $Y$ have $P(X < Y) \geq p$? Question comes from Joe Blitzstein's "Introduction to Probability".
Let $X$ denote days of the week, encoded as $1, 2, ..., 7$ with equal probabilities. Set $Y = (X + 1)$ mod $7.$
It is easy to see that $Y$ and $X$ are identically distribu... | For Question 1, note that if $P(X<Y)=1$ and $X$, $Y$ take values in $\{1,\dots,n\}$ then $P(X=n)=0$.
For Question 2, if $X$, $Y$ are iid then $(X,Y)$ has the same distribution as $(Y,X)$ since there is only one joint distribution of $X$, $Y$ for which $X$ and $Y$ are independent. So $P(X<Y)=P(Y<X)$ and so $1=P(X=Y)+2P(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2952222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is there a relationship between the stationary points and the inflection point of a cubic polynomial function? Determine the stationary points, A and B, and point of inflection, G, for each of the following cubic polynomials.
(a) $y=x^3 -3x^2 -9x+7$
(b) $y=x^3 -12x^2 +21x-14$
(c) $y=x^3 +9x^2 -12$
Is there any common ... | The first derivative of a cubic polynomial is a quadratic polynomial $q$, and as such is even with respect to its stationary point $\xi$. It follows that the real zeros of $q$, if there are any, are at equal distance on both sides of $\xi$. This implies $G={A+B\over2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2952325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Residue of $\cot(z)/(z-\frac{\pi}{2})^2$ at $\frac{\pi}{2}$ I want to know what type of singularity has $f(z)=\cot(z)/(z-\frac{\pi}{2})^2$ at $\frac{\pi}{2}$ and what is the residue of $f(z)$ at $\frac{\pi}{2}$. I thought that $f$ has a pole of order $2$ at $\frac{\pi}{2}$, but the problem is that $\cot(\pi/2)=0$. Can ... | Write $\cos z$ as $(z-\frac {\pi} 2) g(z)$ near $\frac {\pi} 2$ and check (by looking at the derivative) that $g(\frac {\pi} 2)\neq 0$. Hence the function has a pole of order $1$ at $\pi /2$. Can you use $g$ to find the residue now? (The answer is $-1$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2952460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Dividing polynomial $f(x)$ by $x-3$ and $x+6$ leaves respective remainders $7$ and $22$. What's the remainder upon dividing by $(x-3)(x+6)$? If I have a polynomial $f(x)$ and is divided by $(x- 3)$ and $(x + 6)$ the respective remainders are $7$ and $22$, what is the remainder when $f(x)$ is divided by $(x-3)(x + 6)$?
... | $f(x)=(x-3)a(x)+7\Rightarrow f(3)=7$
$f(x)=(x+6)b(x)+22\Rightarrow f(-6)=22$
If you can write $f$ is of the form $f(x) =(x-3)(x+6)q(x) + ax+b$.
Solution is so easy:
$$f(3)=3a+b=7$$
$$f(-6)=-6a+b=22$$
Hence, $a=-\dfrac53$ and $b=12$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2952535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Help factorising this matrix series Let $x_i$ be a series of vectors of equal length, and let $\beta$ be a constant vector of equal length to $x_i$'s
I have the following sum
$$\sum_{i=1}^p (x_i^T \beta)^2 = \sum_{i=1}^p x_i^T \beta \beta^T x_i = x_1^T \beta \beta^T x_1 + x_2^T \beta \beta^T x_2 + \dots + x_p^T \beta \... | Let us give a small example for
$$I_p \otimes \beta$$
will be if p=3 and $\beta = [1,2,3]$:
$$\left[\begin{array}{ccccccccc}1&2&3&0&0&0&0&0&0\\0&0&0&1&2&3&0&0&0\\0&0&0&0&0&0&1&2&3\end{array}\right]$$
We see that if we stuff $[x_1,x_2,x_3]^T$ into column vector we can do
$$(I_p \otimes \beta)[x_1,x_2,x_3]^T$$
and then w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2952646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Classical intro to modern Number theory I'm self-studying Classical Intro to Modern Number Theory, by Kenneth Ireland and Michael Rosen, and I am stuck on a simple proof on page $34$:
Suppose $a_1, a_2 ...,a_t$ all divide $n$, and that $gcd(a_i, a_j) = 1$ for $i \neq j$. then $a_1\cdot\ a_2\cdot \ldots \cdot a_t$ all... | We have $n\cdot r\cdot a_t +n \cdot s\cdot a_1\cdots a_{t-1} = n$.
Since $a_t$ divides $n$ by hypothesis, $ a_1\cdots a_t$ divides $n\cdot s\cdot a_1\cdots a_{t-1}$. And $a_1\cdots a_{t-1}$ divides $n$ by induction hypothesis,thus $ a_1\cdots a_t $ divides $n\cdot r\cdot a_t$.
Therefore $ a_1\cdots a_t $ divides th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2953067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is the convolution of 2 dirac functions? I ran into a problem, which wants to find the convolution $\delta (3-t) * \delta (t-2)$ and I am stuck. How can I approach it?
| You could do it using the Laplace transform and the convolution theorem for Laplace transforms. The Laplace transform of a Dirac delta is
$$\mathcal{L}(\delta(t-a)) = e^{-as}$$
and the convolution theorem states that $\mathcal{L} ((f*g)(t)) = \mathcal{L}(f(t))\mathcal{L}(g(t))$, so you can multiply the Laplace transfor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2953358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
proof of the Jacobi Identity for certain poisson brackets I have to prove that these are effectively Poisson bracket. Specifically that the satisfy Jacobi Identity when $a_{ij}=-a_{ji}$. $$ \left\{ f,g\right\} =\stackrel{\scriptscriptstyle i,j=1..3}{\sum}\left(a_{ij}+\stackrel{\scriptscriptstyle k=1..3}{\sum}\epsilon_{... | Yes, it is much easier than doing pages of calculus:
In coordinates it always suffices to show the Jacobi identity for coordinate functions $(f,g,h)=(x_i, x_j, x_k)$ with $i<j<k$. Since we are on $\mathbb{R}^3$ we only have to show it for $(x_1,x_2,x_3)$: $$\{\{x_1,x_2\},x_3\} + \{\{x_2,x_3\},x_1\} + \{\{x_3,x_1\},x_2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2953533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to map interval $[0, 100]$ to the interval $[100, 350]$? I have an interval $[0; 100]$ and would like to map it to this new interval: $[100;350]$.
I thought about multiplying it by $3.5$, but that would give the interval $[0;350]$. And adding to each of these elements $100$ would give: $[100;450]$. Hence my questio... | Since $100\cdot t^0=100$ for any positive $t$, we find $t$ such that $100\cdot t^{100}=350\implies t=3.5^{0.01}$. $$\boxed{y=100\cdot3.5^{0.01x}}$$ In general an exponential mapping from $[a,b]$ to $[c,d]$ is $y=c\left(\frac dc\right)^{\frac{x-a}{b-a}}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2953600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Determine if $\sum_{n=0}^{\infty}(-1)^n\frac{n+1}{n^2+1}$ converges or diverges I'm having trouble figuring this one out.
$$\sum_{n=0}^{\infty} (-1)^n\frac{n+1}{n^2+1}$$
I think this is conditionally converging as it has $(-1)^n$ so we should take $\lvert(-1)^n\rvert$? I'm a little lost on this one.
Any help would be... | It is trivially convergent by Leibniz' test, an not absolutely convergent by asymptotic comparison with the harmonic series. Convergent to what? is a more interesting question.
We may notice that
$$ \frac{n+1}{n^2+1} = \int_{0}^{+\infty} e^{-nx}\left(\sin x+\cos x\right)\,dx $$
hence
$$ \sum_{n\geq 0}\frac{n+1}{n^2+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2953758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
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