Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Does there exist a theorem whose known proofs encapsulate all possible proofs of that theorem? I was just reading a book on number theory which gave ten different proofs of the infinitude of the primes. This caused me to wonder whether or not it would be possible in principle to find every proof of the infinitude of pr... | If there is one proof, then there are infinitely many, since you can always add trivial steps. You see this really clear in formal proofs: once I have a statement $P$ I can always derive $P \lor Q$ for any of an infinite number of statements $Q$, before getting to the theorem.
Of course, one can complain that adding no... | {
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Is the set $\left\{x \in Q\colon x^2 \le 2\right\}$ open or closed? Is the following set open or closed? I am almost certain it is open as the limits are not included in the rational set.
$$\left\{x \in \mathbb{Q}\colon x^2 \le 2\right\}$$
What I really don’t understand is the proper closure of the set. I think it woul... | Let $A$ be your set of rationals. You ask if $A$ is open or closed.
Open or closed within what space? The reals $\mathbb{R}$ or the rationals $\mathbb{Q}$?
Notice that $$A = [-r,r] \cap \mathbb{Q} = (-r,r) \cap \mathbb{Q}$$
where r = sqrt 2. Thus within $\mathbb{Q}$, $A$ is clopen.
However, within $\mathbb{R}$ it ... | {
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"url": "https://math.stackexchange.com/questions/2326833",
"timestamp": "2023-03-29T00:00:00",
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Equation of the line through the point $(\frac{1}{2},2)$ and tangent to the parabola $y=\frac{-x^2}{2}+2$ and secant to the curve $y=\sqrt{4-x^2}$ Find the equation of the line through the point $(\frac{1}{2},2)$ and tangent to the parabola $y=\frac{-x^2}{2}+2$ and secant to the curve $y=\sqrt{4-x^2}$
Let the required... | Let $\left(t,-\frac{t^2}{2}+2\right)$ be a tangent point.
Since $\left(-\frac{x^2}{2}+2\right)'=-x$, we get an equation of the tangent line:
$$y+\frac{t^2}{2}-2=-t(x-t).$$
Now, substitute $x=\frac{1}{2}$ and $y=2$, find a values of $t$ (I got $t=0$ or $t=1$) and choose a value, which you need.
| {
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"url": "https://math.stackexchange.com/questions/2326909",
"timestamp": "2023-03-29T00:00:00",
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Isolating $t$ on a parametric function I have to plot the graph of
$$\gamma(t)=(4t^2-4t,1-4t^2)$$
for $t\in\mathbb{R}$.
I tried to isolate $t$ in $x=4t^{2}-4t$ (or for $y=1-4t^2$), but in this case, I got something with square root. Is there a better way to do this?
| Hint. Eliminate $t$ and get that
$$
{x}^{2}-2\,xy+{y}^{2}+2\,x+2\,y-3=0.
$$
| {
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How do I solve this double definite integral? $$
\int_{0}^{\pi}\int_{0}^{x/8}\ln\left(\,\sin\left(\,x - 8y\,\right)\,\right)
\,\mathrm{d}y\,\mathrm{d}x
$$
I am pretty sure the solution
is $\displaystyle-\,\frac{\ln\left(\,2\,\right)\,\pi^{2}}{16}$. I just don't know how to get there.
I tried using the method for
solvi... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Let f be a differentiable function with f(0)=0 and f(1)=1, f'(0)=f'(1)=0. Prove that |f''(x)| > 4 for some x in (0,1). (Without invoking integrals) Let $f$ be a twice differentiable function such that $f(0)=0$ and $f(1)=1$.
Also, $f'(0)=f'(1)=0$.
Prove that $f''(x)>4$ for some $x \in (0,1)$.
Any help would be apprecia... | Note that
$$1 = |f(1) - f(0)| \leqslant |f(1/2) - f(0)| + |f(1/2) - f(1)|,$$
and by Taylor's theorem there exist $c_1 \in (0,1/2)$ and $c_2 \in (1/2,1)$ such that
$$f(1/2) = f(0) + \frac{1}{2} f''(c_1)\left(\frac{1}{2}\right)^2, \\ f(1/2) = f(1) + \frac{1}{2} f''(c_2)\left(\frac{1}{2}\right)^2$$
Hence,
$$1 \leqslant \f... | {
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Question regarding integration by substitution. The theorem on integration by substitution says that
$$\int_{\phi(a)}^{\phi(b)}f(x)dx=\int_{a}^{b}f(\phi(t))\phi'(t)dt$$
provided that $\phi$ has an integrable derivative. My question is, shouldn't $\phi$ be monotonic on $[a,b]$? I have this doubt as I am unable to prove... | You do not need that $\phi $ is bijective.
*
*$\phi $ continuously differentiable at $[a,b] $
*$f $ continuous at $\phi ([a,b]) $.
For the proof, consider
$$g (x)=\int_{\phi (a)}^{\phi (x)} f(t)dt-\int_a^x f (\phi (t))\phi'(t)dt $$
and you show by FTC , that $g'(x)=0$.
thus
$$g (b)=g (a)=0$$
| {
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Completion unique up to isomorphism I have a question concerning having a unique completion (up to isomorphism), which I managed to solve partially, but still need some help for finishing.
Namely,
Let $\mathbb P$ = $(P, ≤, . . .)$ be a partial order. Given $A ⊆ P$, we
say that
$a =$ sup $A$
in case
$a ∈ P$ is the le... | Every element $a\in B$ is the supremum of the elements in $P$ below it, and by completeness, there is also such an element in $C$, and this gives the isomorphism. Every element in the completion is the join of a subset (in fact, an antichain) of the partial order.
| {
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"question_score": "2",
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$L^p$ boundedness of a function In a Banach space $X$, let $T:X\rightarrow X$ be a bounded operator and $f:[0,a]\rightarrow X$ a measurable function. Assume that $Tof\in L^p([0,a],X)$, is $f\in L^p([0,a],X)$?
Thank you.
| Following gives you a category of examples where $T \neq 0$
Let $T : X \to X$ be a bounded linear map which map in which $Ker (T) \neq \{0\}$
Now consider any function $g:[0,\frac{1}{2}a]\rightarrow Ker(T)$ such that $g \notin L^p ([0,\frac{1}{2}a], Ker(T))$ (take $g$ so that $\int \|g\|_{X} = + \infty$.)
Now define ... | {
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Why the determinant of an invertible matrix $A$ must be equal to $\pm1$? I've been asked the following question: considering that $A$ is an invertible matrix / $A$ and $A^{-1}$ have integer coefficients, why both determinants must be $1$ or $-1$?
We know that, in linear algebra, an $n$-by-$n$ square matrix $A$ is calle... | If matrices $A$ and $A^{-1}$ have only integer coefficients, that means that both of them must have integer-valued determinant.
And by Cauchy–Binet formula we get:
$$ det(AA^{-1})=det(A)det(A^{-1})=1=det(I).$$
From here we directly get statement you want to prove.
| {
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About primes and cyclotomic extensions I have the following problem
Let $p\geq3$ a prime. Show that $\mathbb{Q}(\sqrt[p]{p})$ is not contained in any cyclotomic extension.
I don't know how to start the problem. Any hint or help will be appreciated !
Thanks in advance.
| *
*$Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \sim \mathbb{Z}_n^\times$ is an abelian (and Galois) extension. Thus for any field $F \subseteq \mathbb{Q}(\zeta_n)$, $Gal(F/\mathbb{Q})$ is a subgroup of $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ and $ F/\mathbb{Q}$ is an abelian extension.
*Let $K = \mathbb{Q}(\sqrt[p]{p},\zeta_... | {
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Is it the case that a proposition is equal to the product of all propositions implied by it? Is it the case that a proposition is equal to the product of all propositions implied by it?
More formally, it the case that
$$ \forall A. \left( A \leftrightarrow \underset{A \rightarrow B}{\bigwedge} B \right)$$
Also I think ... | Bear in mind that the infinite conjunction:
$$\bigwedge_{A \to B}B$$
is not valid object language syntax in ordinary (finitary) first-order logic. If you treat it as a metanotation, you can explain it semantically as in the other answers. The natural way to view it syntactically is in second-order (propositional) logic... | {
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Count the number of fish caught in total if we are given the number of fishermen who caught at least $n$ fish.
Some fishermen caught some fish. No one caught more than 20 fish. $a_i$ is the number of fishermen who caught at least $i$ fish. How many fish were caught?
So my guess is that the number of fish caught has t... | Yes, you are right. Here is a different (more concrete, I guess) way of seeing it:
Put out barrels numbered 1 through 20 and tell each fisherman to put their first fish in barrel 1, their second in barrel 2, and so on. Each fish gets put in a barrel, and the number of fish is therefore necessarily equal to the sum of f... | {
"language": "en",
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grasping uniform convergence for function series I would like some help with my grasping the idea of uniformly convergence for a series of function.
I know that if $\lim_{n\rightarrow\infty}\sup|f_n(x)-f(x)| = 0$ than the series $ {f_n(x)}$ converges uniformly, where $f(x)$ is the limit function.
If we look at the func... | Uniquesolution's comments answer the question, but in the hope a picture helps:
You're correct that if a sequence $(f_{n})$ of continuous functions converges to a discontinuous limit $f$, the convergence is not uniform.
Note carefully, however, that discontinuity of the limit is not necessary; a sequence of continuous... | {
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Why do hyperkaehler manifolds have complex structure valued in a 2-sphere? I understand that hyperkaehler manifolds have almost quaternionic structure, whereby there are three complex structures $I,J$ and $K$ which satisfy
$$
I^2=J^2=K^2=IJK=-1.
$$
It is also said that
$aI+bJ+cK$ is also a complex structure, so long ... | Note that
\begin{align*}
& (aI + bJ + cK)(aI + bJ + cK)\\
=&\ a^2I^2 + abIJ + acIK + abJI + b^2J^2 + bcJK + acKI + bcKJ + c^2K^2\\
=&\ -a^2\operatorname{id} + abIJ + acIK - abIJ -b^2\operatorname{id} + bcJK -acIK - bcJK - c^2\operatorname{id}\\
=&\ -(a^2 + b^2 + c^2)\operatorname{id}.
\end{align*}
So $aI + bJ + cK$ i... | {
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If $p(x)$ is a polynomial of degree 3 such that $p(i) = {1\over1+i}$ for all $a=\{1,2,3,4\}$. Then find $p(5)$. If $p(x)$ is a polynomial of degree $3$ such that $p(i)$ = $\frac{1}{1+i}$ for all $a=\{1,2,3,4\}$. Then find $p(5)$.
My attempt : (1) First obviously I thought of solving the four
equations which can be g... | I am writing this answer based on the hints provided by Daniel Fischer( in the comments section) so that others can also benefit from this answer.
Consider a polynomial function $h(x)=p(x)(x+1)-1$
$x=1,2,3,4$ are clearly the roots of this polynomial therefore $h(x)$ can also be written as
$h(x)=k(x-1)(x-2)(x-3)(x-4)$
... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "4",
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Simplification of Trigo expression
Simplify $$\frac{\tan^2 x+\cos^2 x}{\sin x+ \sec x}$$
My attempt,
$$=\frac{(\cos^2x+\frac{\sin^2x}{\cos^2x})}{\frac{\sin x \cos x+1}{\cos x}} $$
$$=\frac{\cos^4 x+\sin^2 x}{\cos^2 x}\cdot \frac{\cos x}{\sin x \cos x+1}$$
$$=\frac{\cos^4 x+\sin^2 x}{\cos x(\sin x \cos x+1)}$$
I'm st... | Hint
Use $\tan^2x=\sec^2x -1$
$$\frac{\tan^2 x+\cos^2 x}{\sin x+ \sec x}=\frac{(\sec^2 x-1)+\cos^2 x}{\sin x+ \sec x}=\frac{\sec^2 x-\sin^2 x}{\sin x+ \sec x}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Find the minimal value of $\sum\limits_{n=1}^8|x-n|$
Find the minimum value of $( |x-1|+|x-2|...+|x-8|) $
My attempt
Using triangle inequality i.e.
$|x-y|≤|x|+|y|$ $|x-1|=|x-1-0|≤|x-1|+0$ $|x-2|=|x-1-1|≤|x-1|+1$ $|x-3| =|x-1-2| ≤|x-1|+2$
...
...
...
$|x-8| =|x-1-7| ≤|x-1|+7$
Adding all these inequalities, $( |x-1|+... | Your answer is wrong, but there is nothing contradictory with your derivation. You used the inequality $(|x-1|+\cdots+|x-8|)\leq 8|x-1|+28$, which is loose at the optimal solution of $x$. In other words, $x=1$ minimizes $8|x-1|+28$ but not $(|x-1|+\cdots+|x-8|)$, because of the inequality.
A principled way to approach ... | {
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"question_score": "3",
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"answer_id": 1
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Convert a Pair of Integers to a Integer, Optimally?
What's the best algorithm that takes in two positive integers $a,b$
and returns a positive integer $c$, such that all $c$'s are unique and
$(a,b)$ is distinguishable from $(b,a)$; where the best means that the
length of $c$ in terms of digits is shortest possib... | The Cantor pairing function covers the non-negative integers nicely. As a bijection, it uses all the target values for $f(a,b)$, so you can't get more efficient than that. The value of the pairing function is roughly $\frac 12(a+b)^2$. To handle integers, you can just compose it with a bijection between the integers... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Relative error in solution when the matrix is perturbed Suppose that $A \in \mathbb{R}^{n \times n}$ is an invertible matrix and that $Ax=b$ for some $x,b \in \mathbb{R}^{n}$. If $\kappa(A)$ denotes the condition number of $A$, it is well known that if $b$ is perturbed by $\Delta b$, then $x$ gets perturbed by $\Delta ... | Yes, something like that is true. You can find the proof, e.g., in this book and can go around these lines.
We have $Ax=b$ and $(A+\Delta A)(x+\Delta x)=b+\Delta b$. This gives
$(A+\Delta A)\Delta x=\Delta b-\Delta Ax$.
If $A+\Delta A$ is invertible, we have $\Delta x=(A+\Delta A)^{-1}(\Delta b-\Delta Ax)$. Taking a no... | {
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What is the sup of the cardinalities of the chains in $\mathcal P(X)$?
What is the sup of the cardinalities of the chains in $\mathcal P(X)$, where $X$ is a set?
Here chain means totally ordered set, and $\mathcal P(X)$ is the power set of $X$ (ordered by inclusion).
We can assume that $X$ is infinite, because otherw... | This post of Eric Wofsey answers the question.
I'm posting this answer as a community wiki, and I'm panning to accept it as soon as possible in order that the question be considered as answered. If you think the question should be closed as a duplicate, please let me know, and kindly tell me what I should do.
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that the set of odd integers has the same cardinality as $\{2^n\mid n\in\mathbb N\}$ How do you show that the set of odd integers $(2k + 1)$ has the same cardinality as the set of positive powers of $2$ $(2^n)?$
| This is a bijection from the integers $\mathbb{Z}$ to the odd integers:
$$
f(k) = 2k + 1.
$$
This is a bijection from the nonnegative integers, $\mathbb{N}$, to the positive powers of two:
$$
g(n) = 2^n.
$$
So, you have to show these two functions, $f$ and $g$, are bijections.
Finally, you may already know that $\mathb... | {
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Computing $\lim\limits_{x \to \infty} \left( \cos\sqrt{ {2 \pi \over x}} \right)^x$ I have tried to solve the below limit using the exponential formula and then applying l'Hospital but the problem turns hard to solve. Does anyone knows an easier way for it?
$$\lim_{x\rightarrow\infty}\left(\cos\sqrt{\frac{2\pi}{x}}\rig... | $$\cos\sqrt{\frac{2\pi}x}=1-2\sin^2\sqrt{\frac{\pi}{2x}}$$
and
$$x=\frac{\pi\left(\dfrac{\sin\sqrt{\dfrac{\pi}{2x}}}{\sqrt{\dfrac\pi{2x}}}\right)^2}{2\sin^2\sqrt{\dfrac{\pi}{2x}}}.$$
The quotient inside the parenthesis tends to one, and the limit is that of
$$\left(\left(1-2\sin^2\sqrt{\frac{\pi}{2x}}\right)^{1/2\sin^2... | {
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Criteria for a prime degree field extension to be Galois Let $k$ be a field of characteristic zero, not algebraically closed,
and let $k \subset L$ be a field extension of prime degree $p \geq 3$.
I am looking for an additional condition which guarantees that $k \subset L$ is Galois.
An example for an answer: Here is... | Let $k(\alpha)/k$ a Galois extension of prime degree $[k(\alpha):k] = p$ and $char(k) \ne p$.
If $\zeta_p \not \in k$ then $k(\zeta_p)/k$ as well as $k(\alpha,\zeta_p)/k(\alpha)$ are Galois and (by looking at the possible automorphisms) $[k(\zeta_p):k]=[k(\alpha,\zeta_p):k(\alpha)] = n$ where $n \ |\ p-1$.
Thus $k(\... | {
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"timestamp": "2023-03-29T00:00:00",
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If $x^3 - 5x^2+ x=0$ then find the value of $\sqrt {x} + \dfrac {1}{\sqrt {x}}$ If $x^3 - 5x^2+ x=0$ then find the value of $\sqrt {x} + \dfrac {1}{\sqrt {x}}$
My Attempt:
$$x^3 - 5x^2 + x=0$$
$$x(x^2 - 5x + 1)=0$$
Either,
$x=0$
And,
$$x^2-5x+1=0$$
??
| $x^3-5x^2+x$ gives $x=0$ or $x^2+1=5x$.
For $x\leq0$ the needed value does not exist.
For $x>0$ we have $\sqrt{x}+\frac{1}{\sqrt{x}}>0$.
Thus,
$$x+\frac{1}{x}=5$$
or $$\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^2=7,$$
which gives $\sqrt{x}+\frac{1}{\sqrt{x}}=\sqrt7.$
| {
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Prove that $x_n=x$ eventually by the following condition.
Let $(X,d)$ be a metric space and $\{x_n\}$ be sequence in $X$ which converges to some $x \in X$. Let $S$ denote the range set of the sequence $\{x_n\}$. If $S$ is finite then show that $\exists$ $m \in \mathbb N$ such that $x_n=x$ for all $n \ge m$.
My attemp... | Let $S = \{x_n: n \in \mathbb{N}; x_n \neq x\}$ be the non-$x$ range of the sequence. (It's $x[\mathbb{N}] \setminus \{x\}$, where $x: \mathbb{N} \to X$ is the sequence, seen, as it should, as a function). Then $X \setminus S$ is an open neighbourhood of $x$ (as finite subsets of metric spaces are closed).
So the conve... | {
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Can a binary operation have an identity element when it is not associative and commutative? I tried getting the answers in similar questions, everyone says that it's not necessary, but if $e$ is the identity element for any binary operation $*$, which is not associative and commutative, how can
$$a*e=a=e*a$$
when it is... | An operation is commutative if for any $a$ and $b$, we have $ab=ba$. Finding one pair $a,b$ such that $ab=ba$ doesn't prove the operation is commutative; this has to hold for every pair.
Consider the set $\{a,b,c\}$ whose binary operation $\cdot$ is given by the following:
$$a\cdot a = a\,\,\,\,\,\,\,\,\,\,\, a\cdot b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2329528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
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Can I say that $\mu_{d}=(\mathbb{F}^*_q)^{(q-1)/d}$? Asume that $\mu_d=\{x\in\mathbb{F}^*_q:x^d=1\}$, I want to proof that if $d|(q-1)$ then $\mu_d=(\mathbb{F}^*_q)^{(q-1)/d}$.
I all ready do that proof if we have
$\zeta\in\mu_d\Leftrightarrow\zeta^d=1\Leftrightarrow\left(\zeta^{\frac{d}{q-1}}\right)^{q-1}=1\Leftright... | Your claim is correct, $\mu_d=(\Bbb{F}_q^*)^{(q-1)/d}$.
To show that every element of $\mu_d$ is of the form $x^{(q-1)/d}$ for some $x\in\Bbb{F}_q^*$ it is probably easiest to use the fact that $\Bbb{F}_q^*$ is cyclic. If $g$ is a generator (aka a primitive element), then it is not difficult to deduce that if $\zeta\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2329631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is the tensor product of irreducible representations of different groups irreducible? $\DeclareMathOperator{\Aut}{Aut}$
Let $G_1$ and $G_2$ be two groups, provided with two irreducible linear representations
$$R_1 : G_1 \to \Aut(V_1) \text{ and } R_2 : G_2 \to \Aut(V_2),$$
$V_1$ and $V_2$ being two finite-dimensional ... | $\DeclareMathOperator{\End}{End}$
So long as $V_1$ and $V_2$ are still finite dimensional this will still hold. Moreover it holds for finite dimensional simple modules over algebras not just groups, but I'll stick to the group case.
First note that the maps $f:\mathbb{C}G_1 \to \End_\mathbb{C}(V_1)$ and $g: \mathbb{C}G... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2329873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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proving f(n) is strictly less than 2.4 when $n\geq 1$ $f(1)= 2, f(n+1) = \sqrt{3+f(n)}$.
Prove $f(n) < 2.4$ for all $n ≥ 1$.
Would this be a proof by induction? If so, could somebody start me off?
| Hint: $\alpha=\frac{1}{2}\left(1+\sqrt{13}\right)$ is the only solution of $x=\sqrt{3+x}$. If you prove that
$$ \forall x\geq 2,\qquad \left|\sqrt{3+x}-\alpha\right|\leq \left|x-\alpha\right| \tag{1} $$
it follows that for every $n\geq 3$
$$ f(n)\in [2,\alpha]\tag{2} $$
since $\sqrt{3+x}\geq x$ on $[2,\alpha]$.
You ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2329940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Wavelets beside the Daubechies- and Meyer-wavelet?
*
*my first time asking a question on this forum.
I'm self studying the theory of wavelets.
I have one unanswered question regarding this; besides the Daubechies wavelets, Battle-Lemarié, and the Meyer wavelet, do we currently know of any other?
If yes, do any of t... | We know of lots of wavelets. Most do not have closed form expressions.
However see http://ieeexplore.ieee.org/document/705452/ for ones with a closed form expression.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2330032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
how to compute $E(X^4)$ when $X$ follow the normal law $N(0,1)$ Suppose $X$ follow the normal law $N(0,1)$
We have the density $f= \frac{1}{\sqrt{\sigma^2 2\pi}} e^{- \frac{(x-m)^2}{2\sigma^2}}$
We want to compute $E(X^4)$
We have by definition that $\displaystyle E(X^4) = \frac{1}{\sqrt{2\pi}} \int_\infty^\infty{e^{-... | Let $W = X^2$. Then $E[X^4] = E[W^2]$. From the formula for variance,
$$E[W^2] = Var(W) + E[W]^2 $$
$$E[W]^2 = E[X^2]^2 = (Var(X) + E[X]^2)^2 = (1 + 0^2)^2 = 1$$
Note that $W$ is a $\chi^2$ r.v. with one degree of freedom. The variance of a chi-squared is twice its degrees of freedom, thus $Var(W)=2$.
Then: $$E[X^4] = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Finding a function $\phi$ so that $\phi(X)$ is a random variable In my notes, Chebyshev's inequality is formulated as follows:
$$
\forall a \geq 0: P(X \geq a) \leq \dfrac{1}{\phi(a)}E[\phi(X)]
$$
Which is true when $X$ is a random variable and $\phi\geq 0$ and non-decreasing for $-\infty < x < \infty$.
$\phi(X)$ is s... | Edited answer after realizing my mistake:
The function $\phi$ only needs to be measurable for $\phi(X)$ to be a random variable. As @Did says, a non-decreasing function is trivially measurable.
More formally, if $X$ is a random variable defined on $(\Omega,\mathscr A)$, then for any Borel (measurable) function $\phi$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do you calculate the gradient of a scalar product? I am trying to follow the calculation that yields
$$\nabla\langle x, Ax \rangle=2Ax$$
For a symmetric, real matrix $A$.
Do I use the bilinearity of the scalar product? The product rule for the gradient? I do not know where to start and what is legitimate as I am un... | Simply without any coordinates just with Leibniz for any inner product:
$$d_p(\langle x,Ax\rangle)=\langle p,Ax\rangle+\langle x,Ap\rangle
=\langle Ax,p\rangle+\langle A^\star x,p\rangle=\langle(A+A^\star)x,p\rangle,$$
where $A^\star$ denotes the adjoint (see https://en.wikipedia.org/wiki/Hermitian_adjoint) of $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2330352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Cauchy sequence and convergence - $ \frac {1}{n}$ I have read that every convergent sequence is also a cauchy sequence and every cauchy sequence is convergent. I have found that the sequence given by $ \frac {1}{n}$ is Cauchy but $\sum_{i=1}^\infty \frac {1}{n}$ isn't obviously convergent because it has an infinite sum... | It is important to specify the space in which the points lie. The Cauchy criterion is equivalent to convergence to a limit if the underlying space is complete (the real numbers, for example, are complete). The sum $\sum_{n=1}^{\infty} \frac{1}{n}$ doesn't converge since the sequence $\{\sum_{n=1}^N \frac{1}{n}\}_{N=1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2330494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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$T_P V$ only depends on a neighbourhood of $P\in V$ up to isomorphism I'm having trouble understanding the proof of this statement, this is what my notes say
$T_P V$ only depends on a neighbourhood of $P\in V$ up to isomorphism. More precisely, if $P\in V_0\subset V$ and $Q\in W_0\subset W$ are open subsets of affine v... | *
*Proposition: Let $X$ be an affine variety. Let $p \in X$ and $U \subset X$ be an open set containing $p$. Then there is a neighborhood of $p$ contained in $U$ which is isomorphic to an affine variey. Proof: Suppose that $V(f_1, \ldots, f_n) = U^c$, so $U = \cup D(f_i)$. In particular, there is some $f_i$ so that $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2330575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Set of all sets those are equipotent to a given set?
Two sets are said to be 'equipotent' if there is a bijection between them. For a given set $A,$ consider the class $\Bbb{A}$ of all sets those are equipotent with $A.$ Is $\Bbb{A}$ form a set?
My answer is "No" unless $A=\emptyset.$ In order to prove this, my ide... | Your conclusion is right.
There are several ways to prove it, and I'm not entirely sure which one you have in mind based on your idea, but here's a hint that follows that idea: what is the cardinality of $\{x\}\times A$, for any set $x$?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question about my solution of this integral $\int_{0}^{1} \frac{a^x-a}{a^x+a} dx$ So i had to solve this integral:
$$\int_{0}^{1} \frac{a^x-a}{a^x+a} dx$$
$a\in\mathbb{R^+}$
So first i used substitution:
$t=a^x+a \implies a^x=t-a $
$ dx= \frac{dt}{\ln(a)(t-a)}$
Then with partial fractions i got this:
$$\frac{1}{\ln ... | using a slightly different route:
$$
\frac{a^x-a}{a^x+a} = \frac{a^{x-1}-1}{a^{x-1}+1} =\frac{e^{(x-1)\log a} -1}{e^{(x-1)\log a} +1}
$$
set $u=(x-1)\log a$, then
$$
I=\int_0^1 \frac{a^x-a}{a^x+a} dx = \frac1{\log a}\int_{-\log a}^0 \frac{e^u-1}{e^u+1}du = \frac1{\log a}\int_{-\log a}^0 \tanh \frac{u}{2} \quad du \\
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2330843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Solving equation that contains cdf and pdf of standard normal distribution I have the following equation: $ x = \frac{1-\Phi(x)}{a\phi(x)}$, where $\Phi$ is the cdf and $\phi$ is the pdf of the standard normal distribution.
How can one solve for $x$? Is there an analytical approach? Or can this only be done numerically... | Use the series expansions. You can see here how they can be derived. Let $a=1$.
The equation is $x\cdot \phi(x)=1-\Phi(x)$. The equation can be multipied by $\sqrt{2\cdot \pi}$. It becomes
$$x\cdot e^{-x^2/2}=\sqrt{2\cdot \pi}-\left(0.5\cdot \sqrt{2\cdot \pi}+\int_0^x e^{-t^2/2} \,dt \right)$$
Using the series expansi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Conformal mapping from unit disc onto a square I want to refer to question in this topic: Characterization of one-to-one conformal mapping from unit disc onto a square
I understand the solution and all, but is there a way to find explicite the value $|f'(0)|$?
| You can find it from the direct expression of the map (given what the value turns out to be, I think it's unlikely that there's another way). The expression can be computed using the Schwarz–Christoffel map as an elliptic integral:
$$ w = \frac{1}{a\omega K_e} F(\arcsin{(\omega z)},-1), $$
where $K_e$ is a constant giv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2331395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Linear algebra find a parallel to the same plane Find ''$a$'' for which all three vectors $(1,2,3)$,$(1,2+a,6)$ and $(1,10,a-1)$
are parallel to the same plane.
I am more or less sure about the process of doing this.
Iam thinking about doing dot the product of each vector and solve for $a$.
Any hints on this?
| You have to determine at which condition the three vectors are collinear. You can use the determinantal criterion:
$$0=\begin{vmatrix}1&1&1\\2&2+a&10\\3&6&a-1\end{vmatrix}=\begin{vmatrix}1&0&0\\2&a&8\\3&3&a-4\end{vmatrix}=a(a-4)-24=a^2-4a-24=(a-2)^2-28$$
so $\;a=\color{red}{2(1\pm\sqrt 7)}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2331489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Reference request: Crystals and quasicrystals I'm interested in the mathematics of crystals and quasicrystals--things like the enumeration of Bravais lattices and space groups, Bieberbach's results on higher dimensional lattices, and the cut and project construction for quasicrystals. Could someone suggest a reference... | The book On Quaternions and Octonions has a systematic enumeration of the 3 and 4 dimensional groups in Chapters 3 and 4.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2331576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Complex matrix decomposition into the sum of a diagonalizable and a nilpotent matrix. Let $A$ be any $n \times n$ complex matrix. Prove that $A$ can be written as $A = B + N$
where $B$ is diagonalizable, $N$ is nilpotent (some power is the zero matrix) and the
matrices $B$ and $N$ commute.
|
Every $n\times n$ matrix $A$ is (unitarily) similar to a triangular matrix $T$.
If the above statement is accepted, then the requested decomposition is easy to write down: the triangular matrix $T$ can be seen as $T_0+N_0$, where $T_0$ is diagonal with the same entries on the diagonal as $T$. Then $N_0=T-T_0$ is nilp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the area of the region inside the limacon with equation $r=3+2 \sin (\theta)$ that lies below the line $y=x$? What is the area of the region inside the limacon with equation $r=3+2 \sin (\theta)$ that lies below the line $y=x$?
I know I should use Riemann sum but how to write the equation? And what about the ... |
Notice that, as mentioned in the comments as well, the line is a combination of the rays corresponding to $\displaystyle \theta = \frac{π}{4}$ and $\displaystyle \theta = -\frac{3π}{4}$.
Just reminding you, polar curves are traced out by angle, and so are their integrals.
We have that the area enclosed by the two ray... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2331777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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continuity of a function series I need help finding for which $x$ the function:
$$\sum_{n=1}^{\infty}\frac{x+n(-1)^n}{x^2+n^2}$$
is continuous.
I first need to show that the series converges uniformly, the thing is I don't know how to deal with the $(-1)^n$. I tried comparing to the series $\sum_{n=1}^{\infty}\frac{x+... | as I mentioned in Doug answer, if we will prove the series coverges uniformly we can easily show it's contionous.
when $n\rightarrow \infty$ than the series acts as $\sum_{n=1}^{\infty}(-1)^n\frac{n}{x^2+n^2}$. using Leibniz test we know the series converges so from that we conclude that for all x's the series is cont... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2331849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve this integro-differential equation? I came across this integro-differential equation to solve
$$\frac{du(x;t)}{dt}=-\lambda\int_0^xu(\xi;t)\;d\xi\tag{1}$$
under the initial condition $u(x;0)=f(x)$ where $x$ is a parameter, $\lambda$ is a constant, and $0<t<\infty$.
My first thought is that I can just direc... | The solution detailed below is :
With $\quad F(s)=$ Laplace transform of $f(x)$.
$$\Phi(s,t)=e^{-\frac{\lambda \:t}{s}}F(s)$$
$$\boxed{u(x,t)= \text{ Inverse Laplace Transform of } \Phi(s,t)}$$
The result cannot be expressed more explicitly until the function $f(x)$ be explicitly given.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2331930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Is it true that $\lim_{n\to \infty}\int_0^1 f_n(x) dx=\int_0^1\lim_{n\to \infty} f_n(x) dx$? Let $f_n(x)=\dfrac{1}{1+n^2x^2};n\in \Bbb N;x\in \Bbb R$.
Is it true that $\lim_{n\to \infty}\int_0^1 f_n(x) dx=\int_0^1\lim_{n\to \infty} f_n(x) dx$?
$f(x)=\lim_{n\to \infty} f_n(x)=$\begin{cases} 1 & x=0 \\ 0 &x\neq 0\end{cas... | Yes, it is true. Note that
$$\lim_{n\to \infty}\int_0^1 \dfrac{1}{1+n^2x^2} dx=\lim_{n\to \infty}\left[\frac{\arctan(nx)}{n}\right]_0^1=\lim_{n\to \infty}\frac{\arctan(n)}{n}=0$$
where in the last step we used the fact that $\lim_{n\to \infty}\arctan(n)=\pi/2$.
Moreover
$$\int_0^1 f(x) dx=0$$
where $$\lim_{n\to \infty}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding probability of a region I have a question that says:
Given $f_{X,Y} (x, y) = 1/17(x\cdot y + y^2) , 0 < x < 3 , 0 < y < 2$.
Set up the integral that gives $P(X+Y>2)$.
They set up the integral like this:
$$1/17\cdot \int^2_0\int^3_{2-y}xy+y^2 \, dx\,dy$$
But I set up my integral differently... and I get a diffe... | Sketch a diagram of the domain. Observe that the region of interest is a trapezoid. In particular, if we slice the region vertically, then there are two types of "lower" boundaries: the diagonal at $y = 2 - x$ for $x \in (0, 2)$, and the horizontal at $y = 0$ for $x \in (2, 3)$. Thus, if we want to change the order of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Difference between the dimensions $years^{-1}$ and, for instance, $\dfrac{animals}{years}$ in mathematical modelling using differential equations? I am told that an animal population has a natural growth rate of $k \ years^{-1}$ and is harvested at a rate of $a \ \dfrac{animals}{years}$.
Note that the constants $k$ an... | I think it is the same reason that angular speed is sometimes measured in $\mathrm{time}^{-1}$ (such as $\mathrm{s}^{-1}$) rather than $\frac{\mathrm{angle}}{\mathrm{time}}$ (such as $\frac{\mathrm{rad}}{\mathrm{s}}$).
It is because you consider some variables (angles, animals...) as dimensionless variables. This is be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to calculate approximate changes in a nonlinear function first off I've not really studied math so I'm kind of in deep water.
Lets assume that $f(a,b)=y=1+a/b$ and that both variables can increase and decrease in a given scenario.
Then lets assume that the above function is used to calculate $y$ in two different ti... | You can write the variation of $y$ vs. $(a,b)$ as:
$$\Delta y=|\dfrac{\partial y}{\partial a}|\Delta a+|\dfrac{\partial y}{\partial b}|\Delta b $$ so you have:
$$\Delta y=|\dfrac{1}{b}|\Delta a+|\dfrac{a}{b^2}|\Delta b$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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bijective continuous map and the preimage of a convergent sequence At first my question
Let $X,Y$ are metric spaces with $X$ compact also let $T : X \to Y$ be a bijective continuous map and $(y_n)_{n=1}^{\infty}$ a sequence of $Y$. Is it true the following fact? $$(y_n) \text{ convergent } \implies (T^{-1} y_n) \text{... | The answer to both questions is "yes", and the assumption that $X$ is compact is necessary for both. Foobaz John has already provided an elegant proof for the first question, but here is another: since $X$ is compact, it suffices to show that $T^{-1}y$ is the only limit point of $\{T^{-1}y_n\}$ where $y=\lim_{n\to\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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find $M^{\perp}$ in Hilbert space Given $n \in \mathbb{Z}^+$ and $M = \{ (x_1,x_2,..,x_n,0,0,...) \mid x_1,x_2,..,x_n \in \mathbb{R} \}$. Find $M^{\perp}$ in $l^2$?
I can show $M$ is a closed subspace of $l^2$ and a Hilbert space.
Let $P_M : l^2 \to M$:
$$x= (x_1,x_2,..,x_n,..) \mapsto (x_1,x_2,..,x_n,0,0,..)$$
$P_M$... | Let $X$ be a Hilbert space and $P_H$ be the projection onto a subspace $H$ of $X$. We always have $X = P_HX\oplus (P_HX)^{\perp}$. Recall that the direct sum of two subspaces is the space of all sums of elements from the two spaces assuming they only have zero in common. Specifically, if $X = \ell^2$ and $H$ is a finit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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7 friends are going to the cinema. They will be sitting in a row with 7 seats. What is the probability that John and Mary don't sit together?
To watch a movie, John, Mary and 5 friends will sit randomly in a row with 7 seats. What is the probability John and Mary won't sit together?
$$(\mathbf A)\ \frac{2\times5!}{7!}... | See the total ways are $7! $ now let $jm $ be one guy (not biologically) just assume. So now we have total $1+5=6$ ways. We can now arrange these as $6! $ and these two persons can be arranged within themselves in $2! $ thus total ways where they sit together are $2!.6! $hence probability that they wont sit together$=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 15,
"answer_id": 3
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How to show this mapping is quasiconformal. And the integrability of the gradient. A complex map $f$ on the unit disk defined as $f(re^{i\theta})=r^ke^{i\theta}$ where $k>1$. I hope to know how to show this map is quasiconformal and what is the largest $p$ such that the gradient of the map is $L^p$ on the disk. Are th... | For the first part of your questions, I think that you can look at this maps as
$$
f(z)=\frac{z}{|z|}|z|^k\,,
$$
and they are classical examples of very importat class of mappings caled \emph{radial stretchings} (a complete discussion of the basics this class of maps can be found in section 2.6 of Astala, Iwaniec and M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that the equation has at least one solution in an interval, using Lagrange theorem Prove that the equation has at least one solution, in interval $x \in [ \frac{1}{ \sqrt[]{3} }, \sqrt[]{3}]$
$$ \frac{1-2x\arctan(x)}{(1+x^2)^2} = - \frac{\pi\sqrt3}{48}$$
I am stumped by this assignment.
I understand Lagrange the... | $$f(x)=\frac{\arctan (x)}{1+x^2}\implies f'(x)=\left(\frac{\arctan (x)}{1+x^2}\right)'= \frac{1-2x\arctan(x)}{(1+x^2)^2}$$
Now you need to apply Lagrange theorem to $f(x)$ in the given interval.
Since $f(x)$ is continuous and differentiable in the given interval,
\begin{align}
&\implies \frac{f(\sqrt 3)-f\left(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Given a line and a point, how do you calculate the point on the line that is exactly 45 degrees? Given a line connected by two points, (x1, y1) & (x2, y2).
And given a point not on the line, (x3, y3).
How do you calculate the point on the line that creates a line at a 45 degree angle. We'll call this point (x4, y4).
T... | One way to do it, could be a more elegant way:
First, we find the equation of a line through points $(x_{1},y_{1})$ and $(x_{2},y_{2})$. After that, we find the line through the point $(x_{3},y_{3})$ which is perpendicular to the line through points $(x_{1},y_{1})$ and $(x_{2},y_{2})$. After we do that, we find the int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2332898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Difficult Ordinary Differential Equation? I was wondering how to solve the following ODE:
Find $y(1)$ given $y(0)=1$ and
$\frac{dy}{dx}=2ye^{-5x^2}$
So far I got:
$\int \frac{1}{y} dy = 2 \int e^{-5x^2} dx$
On the left hand side I got $\ln|y|$ but did not know how to continue on the right hand integral. Any help would ... | Following Robert Israel's guidance the solution has the form
$$y(x) = c_{0} \, \exp\left[\frac{\pi}{\sqrt{5}} \, \operatorname{erf}(\sqrt{5} \, x)\right].$$
Since $y(0) = 1$ and $\operatorname{erf}(0) = 0$ then
$$y(x) = \exp\left[\frac{\pi}{\sqrt{5}} \, \operatorname{erf}(\sqrt{5} \, x)\right].$$
Now,
$$y(1) = e^{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2333000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve this question using the comparison test The problem is this
$$ \int_0^\infty \frac{x}{x^2+1}\,dx $$
You're supposed to find if this converges or diverges. Now you could figure that by solving the integral but if we were supposed to use the comparison test, how would this be solved?
My teacher taught me to ... | $$
\frac x {x^2+1} > \frac x {x^2+x^2} = \frac 1 {2x}.
$$
PS in response to comments:
$$
\int_0^\infty \frac x {x^2+1} \, dx = \int_0^1 \frac x {x^2+1} \, dx + \int_1^\infty \frac x {x^2+1} \, dx \ge \int_0^1 \frac x {x^2+1} \, dx + \int_1^\infty \frac{dx}{2x} = +\infty.
$$
If the integral from $0$ to $1$ were $-\infty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2333056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$\frac{\partial \mathbf{x}\mathbf{x}^{T}}{\partial \mathbf{x}}$ is this calculatable? I know $\frac{\partial \mathbf{x}^{T}\mathbf{x}}{\partial \mathbf{x}} = \mathbf{x}$, but what is the
$\frac{\partial \mathbf{x}\mathbf{x}^{T}}{\partial \mathbf{x}}$?
Is this calculatable? If it is, can anyone tell me how to derivative... | Well to me the operator
$$
\frac{\partial}{\partial \textbf{x}}=\sum_{j=1}^{N}\hat{e}_j\frac{\partial}{\partial x_j}
$$
$$
\textbf{x}\textbf{x}^{T}=\sum_{m,n}\hat{e}_m\hat{e}_n x_m x_n
$$
$$
\frac{\partial}{\partial\textbf{x}}\textbf{x}\textbf{x}^{T}=\sum_{j,m,n}\hat{e}_j\hat{e}_m\hat{e}_n(x_n\frac{\partial x_m}{\parti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2333205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is absolute value of an analytic function a harmonic function? It is well known that if $f(x+i y) = u(x, y) + i v(x, y)$ is an analytic function of variable $z = x + iy$ then both $u$ and $v$ are harmonic functions.
Does $|f| = \sqrt{u^2 + v^2}$ have any special properties, in particular is $|f|$ harmonic?
| It is generally not harmonic. For example let $f(z)=z$.
Then $\Delta |f|(z)=|z|^{-1},$ which is certainly not zero.
However, $\log |f|$ is harmonic if $f$ is analytic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2333417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Construction of Picard group and set theoretic issue. I am currently learning Algebraic geometry and I came over something that is bothering me.
I have an algebraic variety and the Picard group over it is defined as the set of classes of line bundles quotiented by the the relation of being isomorphic.
First of all it... | This is a common issue you will have to get used to. It's similar to the observation that zero-dimensional $k$-vector spaces are a proper class, because for any set $A$ we can give $\lbrace A\rbrace$ the structure of a $k$-vector space.
However up to isomorphisms you of course obtain an actual set.
Ususally this is res... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2333536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Euler's transformation to derive that $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}=\sum\limits_{n=1}^{\infty}\frac{3}{n^2\binom{2n}{n}}$ According to the accepted answer of this question, we can apply Euler's series transformation to derive that $$\displaystyle{\sum_{n=1}^{\infty}\frac{1}{n^2}=\sum_{n=1}^{\infty}\frac{3}{n... | A proof by pure creative telescoping has been linked in the comments above, but there also is an interesting proof that comes from manipulations of a logarithmic integral.
If we set
$$ I = -\int_{0}^{\pi/2}\log\left(1-\frac{1}{4}\sin^2 x\right)\frac{dx}{\sin x}$$
by expanding $-\log\left(1-\frac{1}{4}\sin^2 x\right)$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2333737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Poles and Laurent series of $\tan z$ I am very confused on how to obtain the principal part, and in general the Laurent series of functions of complex variable.
I will write an exercise and try to point out where are my doubts.
Consider the function $\tan(z)$ in the annulus $\lbrace3<|z|<4\rbrace$. Let $f(z)=f_0(z)+f_... | It is true that there are no poles inside the annulus $3 < \lvert z \rvert < 4$. But there are poles in the disk $\lvert z \rvert <3 $ (at $z=\pm \pi/2$), so we have to include these in $f_1$ so that $f_0$ can be analytic there. $f_1$ has no poles outside $\lvert z \rvert <3$, so we need to find out how to cancel out t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2333863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove $5^n +5 <5^{n+1}$ $∀n∈N$ using induction Prove $5^n +5 <5^{n+1}$ $∀n∈N$
Base Case: $n=1$
$\implies 5^1 +5 <25$
$\implies 10<25$ ; holds true
Induction hypothesis: Suppose $5^k +5 < 5^{k+1}$ is true for k∈N
Then;
$\implies 5^{k+1} +5 < 5^{k+2}$
$\implies 5\cdot 5^k +5 < 25*5^k$
I don't know how to proceed after t... | I often find it helpful in a problem like your own to simply rewrite "the left hand side" so that it is immediately set up so you can apply the inductive hypothesis. Your remaining task, usually, is to then make sure you did not change the value.
In this particular problem, for example, you need to somehow obtain the i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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Proof of the Jordan Holder theorem from Serge Lang
Why is there precisely one index $j$ such that $G_i/ G_{i+1} = G_{ij}/G_{i,j+1}$? How does the conclusion follow?
| This is my understanding of it.
To say $G_i/G_{i + 1}$ is simple means that there it has no normal subgroups other than $\{e\}$ and $G_i/G_{i + 1}$. This implies that there are no normal subgroups $G_i \unrhd H \unrhd G_{i + 1}$ other than $G_i$ and $G_{i+1}$ because $H/G_i \unlhd G_{i+1}/G_i$.
Therefore, if we take a ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why is (-3/77) mod 65 equal to 16? Getting an X for Chinese Remainder Theorem (CRT)
In the "Easy CRT" part of the answer to this problem, the author demonstrates that (-3/77) mod 65 is equal to 16. I don't understand - how is this accurate? I sort of understand the steps, but wouldn't the answer just be 62/77?
Thanks, ... | The definition of $\frac{1}{x}$ is that $\frac{1}{x}$ is the quantity such that $x \cdot \frac1x = 1$ (which may or may not exist). Therefore
$$ \frac{-3}{77} \equiv 16 \pmod {65} \text{ if and only if } -3 \equiv 77 \cdot 16 \pmod {65} $$
This happens if and only if
$$77 \cdot 16 + 3\equiv 0 \pmod {65}$$
which by defi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2334215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the inverse of $I+N$ where $N$ is nilpotent Suppose that $N \in M_{n,n}(\mathbb{C})$ is nilpotent (that is, $N^k = 0$ for some integer $k > 0$). Show that $I+N$ is invertible, and find its inverse as a polynomial in $N$.
I think I got the first part down "intuitively". Noticing that $N$ is nilpotent, so $N$ wil... | $(I+N)(I-N+N^2-N^3+...+(-1)^{k-1}N^{k-1})=I$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $\tan(\frac{1}{2}\arcsin(\frac{5}{13}))$ I wanted to solve it by using this formula: $\tan(\frac{x}{2})=\pm\sqrt{\frac{1-x^2}{1+x^2}}.$ I thought it wouldn't work (because there are $\pm$). Then used the right triangle method: $$\frac{1}{2}\arcsin(\frac{5}{13})=\alpha\Rightarrow\frac{5}{26}=\sin\alpha$$ $$a^2+b^2=... | I think you need to check you first formula. You can use $\arcsin (\frac {2x}{1+x^2})=2\arctan (x) $ and $\arctan (\frac {2x}{1-x^2})=2\arctan (x) $. Both of which can be proved using $x=\tan (t)$. I think you can continue from here.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all functions of the form $f(x)=\frac{b}{cx+1}$ where $f(f(f(x)))=x .$ This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with functions and polynomials, but other than that, the textbook gave no hints really and I'm really not sure about how to... | If $b=0$ or $c=0$ the function $f(x):={b\over cx+1}$ is constant, hence cannot satisfy $f^{\circ3}={\rm id}$. If $bc\ne0$ then $f$ is a Moebius transformation with matrix
$$\left[\matrix{0& b\cr c&1\cr}\right]\ .$$
The map $f^{\circ3}$ then has matrix
$$\left[\matrix{0& b\cr c&1\cr}\right]^3=\left[\matrix{bc& b(bc+1)\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2334497",
"timestamp": "2023-03-29T00:00:00",
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If $a \equiv r_i^2 \mod p_i$ for every prime divisor $p_i$ of $b$, then $a$ is a square modulo $b$. Let $a,b,r_i\in \mathbb{Z}$. How to show that if $b$ is squarefree and $a \equiv r_i^2 \mod p_i$ for every prime divisor $p_i$ of $b$, then $a$ is also a square modulo $b$.
Approach: I think I have to use CRT in some way... | $b$ square-free $\Longrightarrow b = \prod_{B\subset\mathbb{P}}b'$ for a finite set of prime numbers $B$.
When $a$ is a perfect square $\pmod{b'}$, for all $b' \in B$, $a$ is a perfect square $\pmod{b}$.
Because $b$ is square free, we know that the prime factors are unique and the exponents in the factorization are all... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2334587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Pick's theorem for a triangle I'm trying to show that Pick's theorem holds for any right triangle with vertices at the points $(0,0),(a,0),(0,b)$ with $a$ and $b$ both being positive integers.
I've managed to express the number of interior points as the sum
$$\sum_{k=1}^{a-1} \left( \left\lceil k \frac{b}{a} \right\rc... | We will prove both of the equations
$$f_1(a,b):=\sum_{k=1}^a \left\lceil k \frac{b}{a} \right\rceil=\frac{ab+a+b-\gcd(a,b)}{2} =:g_1(a,b),
\\
f_2(a,b):=\sum_{k=0}^{a-1} \left\lfloor k \frac{b}{a} \right\rfloor=\frac{ab-a-b+\gcd(a,b)}{2}:=g_2(a,b),$$
hold for positive integers $a$, and nonnegative integers $b$.
It is c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2334676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Using maxima / minima to find roots
Suppose $c$ is a given positive number. The equation $\ln x=cx^2$ must
have a solution if
A) $c<\frac{1}{2e}$
B) $c>\frac{1}{2e}$
C) $c<\frac{1}{e}$
D) $c>\frac{1}{e}$
I have no idea how to approach this, my professor used a method that used the maximum of $\ln x - cx^2$, but I c... | Define $f(x)=$In$(x)-cx^2$, $c>0$ on $x\in (0,\infty)$.
$f'(x)=0$ gives you $x=\frac{1}{\sqrt{2c}}$ as the stationary point in $(0,\infty)$. Since $f''(\frac{1}{\sqrt{2c}})<0$, so $x=\frac{1}{\sqrt{2c}}$ is point of maxima.
As $f(0)<0$ so you just need that $f_{max}>0$ in order to make the graph of $f$ to cross the X-a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Every compact manifold is geodesically complete An $n$-dimensional Riemannian manifold $(M,g)$ is said to be geodesically complete if every geodesic $\gamma:(-\varepsilon,\varepsilon) \to M$ can be extended to a geodesic $\widetilde{\gamma}:\mathbb{R}\to M$ defined on the whole real line. There is a theorem stating tha... | This is just a sketch of how you could prove this directly:
Let $\gamma:I\to M$ be a geodesic. Assume, for the sake of contradiction, that the maximal domain of existence is not all of $\mathbf{R}$. Say $I=(a,b)$ for some $a,b\in\mathbf{R}$. Now consider what happens to $\gamma(t)$ in the limit as $t\to b$. If $\gamma(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2334915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Value of an exponential function What is the value of the exponential function $y=e^{x^{x}-1}$ at $x=0$? I graphed the function on desmos and the value at $x=0$ is $1$. However, I do not know how to show this analytically.
| This function is not defined at $0$, but you can compute the limit $\lim_{x\to0}e^{x^x-1}=e^{\lim_{x\to0}x^x-1}$. It happens that$$\lim_{x\to0}x^x=\lim_{x\to0}e^{x\log x}=e^{\lim_{x\to0}x\log x}$$and that\begin{align}\lim_{x\to0}x\log x&=\lim_{x\to0}\frac{\log x}{\frac1x}\\&=\lim_{x\to0}\frac{\frac1x}{-\frac1{x^2}}\\&=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2335031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Simplifying $\sqrt{xy\mathstrut}\sqrt{x^3y}$ - two different paths, different results I'm trying to simplify $\sqrt{xy\mathstrut}\sqrt{x^3y}$, for which the book has the solution below:
$$\sqrt{xy\phantom{\big|}}\sqrt{x^3y} = \sqrt{(xy)(x^3y)} = \sqrt{x^4y^2} = x^2|y|$$
I understand and agree with the above solution. ... | At the stage where you simplify $\sqrt{(xy)^2}$ you should get $|xy|$, not $xy$. Therefore your final line should be $|xy||x|$, which is equal to $|x|^2 |y| = x^2 |y|$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Does Top category has both (epi, extremal mono) and (extremal epi, mono) factorization property? Let ${\bf{Top}}$ be a category of topological spaces with continuous functions. Does ${\bf{Top}}$ share extremal epi-mono factorization (and epi-extremal mono factorization) property? If yes, are these factorization the sam... | An extremal mono in $\mathbf{Top}$ is just an embedding and an extremal epi is just a quotient map. So both factorizations you as for exist: the epi-extremal mono factorization of $f:X\to Y$ is the factorization $X\to I\to Y$ where $I\to Y$ is the inclusion of the image of $f$ as a subspace of $Y$, and the extremal ep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2335237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why we can still use this function when it's dividing zero my teacher gave my class a question in calculus but the method he used to attempt the question was kind of odd.
Question: An aircraft at a constant 500 metres height is flying towards an observer at 80 m/s. How is the angle of elevation changing if the aircraft... | It's kind of like cancelling the $x$'s in the equation $y = \frac{x(x+3)(2x+5)}{x}$ when you try to find the limit as x approaches 0. The function is undefined at x = 0, but the limit still exists and still makes sense. The function $y = x(x+3)(2x+5)$ matches everywhere else except x = 0, similar to your problem with t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2335328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Meaning of the notation $L^2(\mathbb{R}^3)$ and its generalization I'm not sure whether this question really belongs to this website. In quantum physics texts, and physics stackechange website, I have often seen the notation $L^2(\mathbb{R}^3)$. My glossary of mathematical notations are limited. What does this symbol p... | The space $L^2(\mathbb{R}^3)$ is indeed the vector space of all square-integrable functions from $\mathbb{R}^3$ into $\mathbb R$. Here, integrable means Lebesgue-integrable. This space has a natural norm: $\|f\|_2=\sqrt{\int_{\mathbb{R}^3}f^2}$.
More generally, if $p\geqslant1$ you have the space $L^p(\mathbb{R}^n)$ of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2335450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving sum of infinite series My statistics textbook states that $\sum\limits_{n=0}^{\infty} \begin{pmatrix}2n \\ n \end{pmatrix}(pqs^2)^n=\frac{1}{\sqrt{1-4pqs^2}}$
where $|s|<1$, $p \in ]0,1[$ and $q=1-p$.
That thing is that prof of this is omitted since its "not relevant". How would one prove it?
Soo far i have n... | It is a consequence of the generalized binomial theorem, for $|x|<1$,
$$(1+x)^{-1/2}=\sum_{n=0}^{\infty}\binom{-1/2}{n}x^n,$$
where
$$\binom{-1/2}{n}=\frac{(-1/2)(-1/2-1)\cdots(-1/2-(n-1))}{n!}=\frac{(-1)^n}{4^n}\binom{2n}{n}.$$
Hence for $|4pqs^2|<1$,
$$\frac{1}{\sqrt{1-4pqs^2}}=(1+(-4pqs^2))^{-1/2}=\sum_{n=0}^{\infty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2335539",
"timestamp": "2023-03-29T00:00:00",
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Find the value of $\int_0^{\infty} \frac{x^{a-1}\,\mathrm{d}x}{1+x^{2}},\,$ where $\,0Using the Residue Theorem calculate
$$
\int_0^{\infty} \frac{x^{a-1}\,\mathrm{d}x}{1+x^{2}},\,\,\,\, \text{where}\,\,\,0<a<2.
$$
My solution comes out to be
$$ \frac{\pi}{2}[i^{a-1}-(-i)^{a-1}]$$
How to proceed from here since the sol... | In this answer, it is shown that for $m\gt0$ and $-1<n<m-1$
$$
\frac{\pi}{m}\csc\left(\pi\frac{n+1}{m}\right)=\int_0^\infty\frac{x^n}{1+x^m}\,\mathrm{d}x
$$
Plug in $n=a-1$ and $m=2$ to get that for $0\lt a\lt2$
$$
\int_0^\infty\frac{x^{a-1}}{1+x^2}\,\mathrm{d}x=\frac\pi2\csc\left(\frac{\pi a}2\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2335645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
No quantifier on bounded variable This is how subset is defined in set theory:
$A \subseteq B \iff \forall x \in A \implies x \in B$.
So, for how many elements this $x$ without quantifier in $x \in B$ actually stands for? All or some (at least one, maybe all)?
Isn't $\exists x \in B$ what actually is assumed?
| The quantifier is over the whole implication, so $A \subseteq B$ means:
$$\forall x: (x \in A \implies x \in B)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2335720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $p=a^2+4b^2$ for some $a,b \in \mathbb{Z}$, then $a$ is quadratic residu modulo $p$? If $p=a^2+4b^2$ for some $a,b \in \mathbb{Z}$ and $p$ prime, then $a$ is quadratic residu modulo $p$?
Approach: I thought it was true. (I could't find a counterexample). So I tried to prove it. I deduced that $a$ is a quadratic resi... | We easily see that $p\equiv1\pmod4$, so for any prime factor $q\mid a$ we have, by quadratic reciprocity
$$
\left(\frac pq\right)=\left(\frac qp\right).
$$
OTOH we have $a^2=p-4b^2$. Therefore
$$
p\equiv 4b^2=(2b)^2\pmod q
$$
and $\left(\dfrac pq\right)=1$ for all those primes $q$.
So all the prime factors of $a$ are Q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2335914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Prove that if $(f_n)$ converges to $f$ in measure then $(f_n^2)$ converges to $f^2$ in measure. Let $E$ be a measurable set with finite measure, $(f_n)$ be a sequence of real-valued measurable functions on $E$ and $f$ be a real valued measurable function on $E$. It is required to prove that if $(f_n)$ converges to $f$ ... | Convergence in measure means that $\mu(\{x: |f_n(x) - f(x)| \ge \epsilon\})$ tends to $0$ as $n \to \infty$. Fix $\epsilon$ and $\eta > 0$, and choose sufficiently large $M, N > 0$ such that $\mu(\{x: |f(x)|> M\}) < \frac{\eta}{3}$ and $\mu(\{x: |f_n(x)|> M\}) < \frac{\eta}{3}$ for $n \ge N$.
(We can do this because $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Calculate $\tan^2{\frac{\pi}{5}}+\tan^2{\frac{2\pi}{5}}$ without a calculator The question is to find the exact value of:
$$\tan^2{\left(\frac{\pi}{5}\right)}+\tan^2{\left(\frac{2\pi}{5}\right)}$$
without using a calculator.
I know that it is possible to find the exact values of $\tan{\left(\frac{\pi}{5}\right)}$ and $... | Let $\tan\left(\frac{\pi}{5}\right)=x$
$$\tan\left(\frac{\pi}{5}+\frac{\pi}{5}+\frac{\pi}{5}+\frac{\pi}{5}+\frac{\pi}{5}\right)=0$$
$$\implies S_1-S_3+S_5=0$$
($S_k$ represents sum of tangents taken $k$ at a time)
$$\implies 5x-10x^3+x^5=0$$
Now the roots of this equation are $\tan\left(\frac{\pi}{5}\right),\tan\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 3
} |
How do I solve the below system of congruence? I'm new here , i want to solve the below system which $(x, y)$ are a paire in $\mathbb{N²}$.
$2x+2y= 1\bmod 10,4x+y= 7\bmod 10$
Thank you for your help
| The first congruence $2x+2y=1\pmod{10}$ doesn't have any solution. If $(a,b)$ is a solution, $1=10k-2a-2b$. RHS is even, LHS is odd.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a mathematical description of three-part ratios? Rational numbers have many interpretations, but one of the simplest is as a ratio of one number to another. The fraction $1/2$ can be interpreted as the ratio 1:2 (i.e. one apple for every two oranges). Rational numbers are also considered an extension of the nu... | To represent a multi-part "ratio" $a_1:\cdots:a_n$, where each $a_i$ is an integer, I would suggest an element of the projective space $\mathrm{P}_\mathbb{Q}(\mathbb{Q}^n)$ (see Wikipedia) which is the set of equivalence classes of
$$\mathbb{Q}^n\setminus\{(0,\ldots,0)\}$$
under the equivalence relation $\sim$, where
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Why do computer programs (Wolfram Alpha, Symbolab) think $\sum_{n=0}^{\infty} \sin(\frac{\pi}{2}n!)$ is divergent? I've been recently looking at the series
$$\sum_{n=0}^{\infty} \sin(\frac{\pi}{2}n!)=1+1+0+0+0+0+0+\cdots,$$
which should equal $2$. However, programs such as Wolfram Alpha, Symbolab, etc., tell me that th... | Apparently, the software isn't clever enough to get the trick here. I suspect that when Wolfram Alpha says the series is divergent, what it really means is that it is unable to determine that the series is convergent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Limit involving cosine function
How to find
$$\lim_{n \rightarrow \infty}\; \frac{1}{n}\;\sum_{k=1}^{\Big\lfloor\frac{n}{2}\Big\rfloor} \cos\Big(\frac{k\pi}{n}\Big)$$?
I know the method when the upper limit is simply $n$, namely it converges to $\int_0^1 f(x)\;dx$ where $f$ is monotonically increasing on an interval ... | Define $C_n=\sum_{i=1}^{\lfloor n/2\rfloor}\cos(k\pi/n)$ and note that the sequence $(C_n)$ is non-decreasing, hence it is enough to check for even values of $n$, indeed:
$$
\frac{2n}{2n+1}\cdot \frac{C_{2n}}{2n} \le \frac{C_{2n+1}}{2n+1} \le \frac{2n+2}{2n+1}\cdot \frac{C_{2n+2}}{2n+2}.
$$
Therefore
$$
\lim_{n\to \inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The maximum volume of water. We are a group of math enthusiasts and we design and present our mathematical problems to societies. This week I designed this problem and I thought it might be interesting to share it with you here. If you think sharing such problems are not appropriate for this site, then I can remove it.... | Here is my solution to the maximum volume of water in a glass. The important step in this solution is to find the angle of rotation of the glass when it is resting on the table.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Monte Carlo For Integral Involving Dirac-Delta How does one do MC for integrand which has dirac-delta function like following:
$I=\int e^{-S(x)} \delta(f(x))dx$
where $x$ would be multi-dimensional, and hence this is a multi-dimensional integral over coordinates such as $x_1,x_2,x_3$ and so on.
I want to do importance... | Use the identity
\begin{align*}
\int_{\mathbb{R}^{n}}\exp(-S(\mathbf{x})) \delta(f(\mathbf{x})) \, \mathrm{d}\mathbf{x} = \int_{f^{-1}(0) \subset \mathbb{R}^{n-1}} \frac{\exp(-S(\mathbf{x})) }{|\nabla f(\mathbf{x})|} \, \mathrm{d}\mathbf{x}
\end{align*}
and then use Monte-Carlo on the surface integral. The difficulty ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Sketching a graph of $f(x)$ Given a function $f(x)$ is increasing in the intervals $(-\infty, -1 )$ and $(3, \infty)$, and decreasing in the interval $(-1,3)$
If I want to sketch a graph of $f(x)$,
I know that the maximum point $\implies x=-1$
Minimum Point $\implies x=3$
However, let's say given the graph
$-5<x<5... | Good question. I'll assume your function is of the form $f:\mathbf{R}\to \mathbf{R}$. When you're asked to sketch a function or curve like this, you should just seek to include the information specified. In this case, just draw the graph so that it is increasing on $(-\infty,-1)$ and $(3,\infty)$, while decreasing on $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2336954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
The following condition fails to define a function on any domain. State. why?" $\sin f(x) =x$
Quoting" The following condition fails to define a function on any domain. State. why?"
$$\sin f(x) =x$$
I understand that it yields a sinusoid oscillating vertically defined between $x=-1$ and $x=1$.
It shows that $\forall ... | If $|x|\leq 1$ the statement $\sin f(x)=x$ does not DEFINE $f(x)$ although the statement may be a property of a function $f.$ There are infinitely many $y$ for which $\sin y=x$ and the statement $\sin f(x)=x$ does not tell you which one of these $y$ is actually $f(x).$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2337043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find $\arctan(\frac{1}{3})+\arctan(\frac{1}{9})+\arctan(\frac{7}{19})$ Firstly used this formula $$ \begin{align} \arctan(\alpha)+\arctan(\beta)
& =\arctan(\frac{1-xy}{x+y}),\quad x\gt0,y\gt0 \\ &=\arctan(\frac{1-\frac{1}{3}\frac{1}{9}}{\frac{1}{3}+\frac{1}{9}}) \\ &=\arctan(2) \end{align}$$ So it is $\arctan(2)+\arct... | Just calculate:
$$\tan\left(\arctan\frac{1}{3}+\arctan\frac{1}{9}+\arctan\frac{7}{19}\right)=$$
$$=\frac{\frac{\frac{1}{3}+\frac{1}{9}}{1-\frac{1}{3}\cdot\frac{1}{9}}+\frac{7}{19}}{1-\frac{\frac{1}{3}+\frac{1}{9}}{1-\frac{1}{3}\cdot\frac{1}{9}}\cdot\frac{7}{19}}=1,$$
which gives the answer: $45^{\circ}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2337146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Prove: $x^4+mx^2+x$ have only two roots when $m>0$ I got the question:
Prove that $x^4+mx^2+x$ have only two roots when $m>0$.
I know that it is a continuous function.
I tried to use solve this question with two steps:
*
*Use intermediate value theorem to prove that there are at least two roots.
*Use Rolle's ... | Since the polynomial factors as $x(x^3+mx+1)$, you have one root at $x=0$. So now you just have to show that $x^3+mx+1$ has exactly one root, and your plan of
action above should do the trick.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2337213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Dimension of irreps of $C_3v$ The $C_{3v}$ point group (symmetry group of a regular triangle) has 6 elements: $A$,$B$,$C$,$E$,$D$,$F$ (3 reflections, identity, 2 rotations). It has 3 conjugate classes $\phi_1=\{E\}$, $\phi_2=\{A,B,C\}$ and $\phi_3=\{D,F\}$. A reducible representation $R(g)$ with $2\times 2$ real matric... | We can do this without really thinking about what the group is.
If $G$ is a non-abelian group of order $6$ then $G$ has $3$ irreducible representations and their degrees are $1$, $1$ and $2$.
To see this, we first note that not all the irreducible representations can have degree $1$ since then the group would be abelia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2337311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Concerning the closedness of a subset of $L^1[0,1]$ I was reading some stuff about that involved $L^p$ spaces and came up with this question: Given $X=L^1[0,1]$, I need to show a closed, convex subset of a Banach space with some properties, the thing is that apparently the the set
$$\mathcal{C}=\left\{f\in X:\int_0^1f... | Why should there be such a function $f$? The set $[-1,1]$ is a closed subset of $\mathbb R$. However, the sequence $\bigl((-1)^n\bigr)_{n\in\mathbb N}$ doesn't converge there.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2337417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Calculation with function For as long as I go around it I do not get to anything
If $\phi \left ( f(x)-1\right )=2x+5$
and
$\phi (x)=2f(x+1)+1$
find $f(4)$
| just an attempt
$$\phi (f (x)-1)=2x+5=2f(f (x))+1$$
thus
$$f (f (x))=x+2$$
$$f (4)=f (f (f (f (0)))) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2337559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solution to second order second degree equation $ x^2 (d^2y/dx^2)^2=(dy/dx)^2+1$. I am unawae of any method that is there to solve such an equation. IT is totaly different from the usual linear differential equation of degree 2. Any Hint as to how to solve.
| It certainly isn't linear. However, one thing we can do is reduce the order - let $u(x) = y' = \frac{dy}{dx}$, so that $u' = y''$. Then our equation becomes:
$$\begin{eqnarray}x^2u'^2 & = & u^2 + 1 \\
\frac{u'^2}{u^2 + 1} & = & \frac{1}{x^2} \\
\frac{u'}{\sqrt{u^2 + 1}} & = & \frac{1}{x} \end{eqnarray} $$
which is a se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2337681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
If $A$ and $ B$ be real invertible invertible matrices such that $AB=-BA$ then Trace Let $A$ and $B$ be real invertible matrices such that $AB = - BA$. Then
1.$Trace(A)=Trace(B)=0$
2.$Trace(A)=Trace(B)=1$
3.$Trace(A)=0,Trace(B)=1$
4.$Trace(A)=1,Trace(B)=0$
$A$ is invertible $\Rightarrow$ $ABA^{-1}= -B$$\Rightarrow$ $B... | You don't need to use eigen values because you don't know if they have real eigen values and using complex eigen values seems overkill.
For example with $$A = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}$$ and $$B = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}$$
You have $BA = -AB$ with $A$ and $B$ invertible but ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2337792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Find the magnitude of the vertex angle of an isosceles triangle of the given area $A$ Find the magnitude of the vertex angle $\alpha$ of an isosceles triangle with the given area $A$ such that the radius $r$ of the circle inscribed into the triangle is maximal.
My attempt:
| We know that every triangle has a unique incircle; we also have the following 2 theorems:
Theorem 1: Among all triangles of given perimeter, the equilateral one has the
largest area.
Theorem 2: The radius $r$ of the incircle for a triangle $\triangle ABC$ is
given by $ r = 2 \frac{Area(\triangle ABC) } {Perimeter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2337999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 2
} |
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