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Is there an easy enough way to show that for a polynomial $q(x)=\sum_{p\ge2}\gamma_{p}x^{p}$ with $\gamma_{p}\ge0$ and $\sum_{p\ge2}\gamma_{p}=1$ where there are finitely many non-zero $\gamma_{p}$, we have for $x>0$: $\frac{q\left(x\right)}{q^{\prime}\left(x\right)\cdot x}\ge\frac{1}{p_{\text{max}}}$?
I wrote expli... |
The following is a part of the theorem in Section 7.8. in Titchmarsh's book *The Theory of the Riemann Zeta-Function*:
[![enter image description here][1]][1]
My questions:
1- I need to study Parseval's formula but lets say that the formula $\frac{1}{2\pi}\int_{-\infty}^\infty \lvert F(it)\rvert^2\,dt = \int_... |
My teacher said that the following set E={x: range(phi_x) != {x} } (phi_x is the x-th Turning Machine) is productive. Why? I think it should be recursive enumerable. In fact, to decide wether x belongs to E, you can simulate the x-th Turing Machine with dovetail method and, if x in E, the machine will terminate for sur... |
Productive set? |
One of tool for determinate the behaviours of the series is to check if $$\lim \sup \sqrt(a_n)<1$$ or $$\lim \inf sqrt(a_n)>1$$
But when the sequences $a_n$ converges to some $l$, the limit is not equal to the $\lim \inf$ and $\lim \sup$? So it's wrong determinate directly the $\lim \sqrt(a_n)$?
I don't understand w... |
Let $X$ be a smooth surface and $C$ a smooth curve, over $\mathbb C$. Let $P\in X$, $c_0\in C$, and $C'$ a smooth curve in $X\times C$ such that $x=(P,c_0)\in C'$. Let
$$p:X\times C\longrightarrow C$$
be the projection.
Suppose that $C'$ is transverse to $p^{-1}(c_0)$ at $x$.
Question: does there exist neigbour... |
One can check that the coproduct of $G=\langle A \mid R \rangle$ and $G'=\langle A' \mid R' \rangle$ is $G * G'= \langle A \amalg A' \mid R \amalg R' \rangle$. I'm wondering if such an argument can be generalized for arbitrary categories $C$ with a right adjoint forgetful functor to $Sets$.
More concretely, given $... |
Suppose I have a degree 2 line bundle $L$ on an elliptic curve $E$. Then we may
write $L = \mathcal{O}(p+q)$ for some points $p,q \in E$. If we let $\oplus$ denote the addition map on the elliptic curve $E$. Then we can map $L$ to the degree 1 line bundle $L' = \mathcal{O}(p \oplus q)$.
Is this map well defined? ... |
One of tool for determinate the behaviours of the series is to check if $$\lim \sup \sqrt(a_n)<1$$ or $$\lim \inf \sqrt(a_n)>1$$
But when the sequences $a_n$ converges to some $l$, the limit is not equal to the $\lim \inf$ and $\lim \sup$? So it's wrong determinate directly the $\lim \sqrt(a_n)$?
I don't understand ... |
So the question is the following: Given that $S_n = 3n^2 - 2$ , Find the nth term of the sequence $a_n$.
$S_n$ is the sum of the sequence from 1 to n. I know that the answer should be an arithmetic sequence because all the choices are as such and anyway we know that from the form of $S_n$.
The weird thing is that... |
# Background
The famous [Evaluating Large Language Models Trained on Code](https://arxiv.org/abs/2107.03374) from OpenAI proposed the code LLM; GitHub uses it as the code assistant called GitHub copilot. It also proposed a metric that evaluates the performance of new code LLMs:
$$
\mathrm{Pass}@k=\mathbb{E} _ \mat... |
So the question is the following:
> Given that $S_n = 3n^2 - 2$ , Find the nth term of the sequence $a_n$.
>
> $S_n$ is the sum of the sequence from $1$ to $n$.
I know that the answer should be an arithmetic sequence, because all the choices are as such, and anyway we know that from the form of $S_n$.
The ... |
The correct thing to quotient in a general algebraic theory is a **congruence**. It is a subobject $R \hookrightarrow G \times G$, satisfying reflexivity, symmetry and transitivity. The quotient is simply the coequalizer of the two maps $R \rightrightarrows G$. *Exercise*: prove that congruences in the category of grou... |
In a physics related problem, I am trying to compute the Fourier transform
\begin{align}
\mathcal{F}\left[\frac{1}{\sinh^{n}\left[\pi T_R\left(x+ i\epsilon-t\right)\right]\sinh^{m}\left[\pi T_L\left(x+i\epsilon+t\right)\right]}\right]\left(\omega\right)=\int_{-\infty}^\infty\frac{dt e^{i\omega t}}{\sinh^{n}\left[\p... |
In the glossary for mathematical operators in Unicode<sup>[1]</sup>, the last one, u+22FF "⋿", is said to represent "Z Notation Bag Membership".
Looking up "z notation bag membership" gives me this info:
>The symbol "Z Notation Bag Membership" is included in the "Relations" subblock of the "Mathematics Operators" b... |
"Z Notation Bag Membership": What does this represent in math? |
I am wondering if anybody knows good exact bounds for the Median of a Hypergeometric distribution:
$$
X \sim \text{Hypergeometric}(N,K,n)
$$
One exact bound could be given by exploiting the [inequality][1] between Mean, Median and Standard Deviation.
Another direction might be the Binomial approximation of H... |
Which are the countable ordinals $\lambda$ such that, for every sequence of ordinals $\alpha_i\ (i\in\mathbb{N})$ such that
1) $\ $it is strictly increasing for all sufficiently large $i$
2) $\ $$\alpha_i<\lambda\ $ for all $i\in\mathbb{N}$
3) $\ $$\sup\{ \alpha_i:i\in\mathbb{N}\}=\lambda$,
there exists an ordin... |
Let $k = \mathbb{F}_q$ be a finite field of characteristic $p$ and $C$ a projective, smooth curve over $k$. Denote by $\bar C$ the base change of $C$ to a separable closure $\bar k$ of $k$. Let $\ell$ be a prime $\ell \neq p$. Then we can consider the first étale *homology* by which I mean
$$
H_1^{et}(\bar C, \mathbb... |
Which are the countable ordinals $\lambda$ such that, for every sequence of ordinals $\alpha_i\ (i\in\mathbb{N})$ such that
1) $\ $it is strictly increasing for all sufficiently large $i$
2) $\ $$\alpha_i<\lambda\ $ for all $i\in\mathbb{N}$
3) $\ $$\sup\{ \alpha_i:i\in\mathbb{N}\}=\lambda$,
there exists an ordin... |
First of all, I know that there is an extremely similar question from yesterday that has been closed due to Mathematics Stack Exchange guidelines, so I can't comment and find what is incorrect in my way.
The question : find the coefficient of $(x^3y^4z)$ in $ (x+y+z)^5 (1+x+y+z)^{5}$.
We know that: $(x+y+z)^5$ = ... |
One of tool for determinate the behaviours of the series is to check if $$\limsup_{n\to\infty} \sqrt{a_n}<1$$ or $$\liminf_{n\to\infty} \sqrt{a_n}>1$$
But when the sequences $a_n$ converges to some $l$, the limit is not equal to the $\liminf$ and $\limsup$? So it's wrong determinate directly the $\lim \sqrt{a_n}$?
I... |
Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_5} \times \mathbb{Z_7}$.
I know that this would be isomorphic to $\mathbb{Z_{70}}$, so that I would be looking for $8$ subgroups because $70$ has $8$ factors.
I tried finding an isomorphism between $\mathbb{Z_2} \times \mathbb{Z_5} \times \mathbb{Z_7}$ and $\ma... |
One of tool for determinate the behaviours of the series is to check if $$\limsup_{n\to\infty} \sqrt[n]{a_n}<1$$ or $$\liminf_{n\to\infty} \sqrt[n]{a_n}>1$$
But when the sequences $a_n$ converges to some $l$, the limit is not equal to the $\liminf$ and $\limsup$? So it's wrong determinate directly the $\lim \sqrt[n]{... |
Let $K\subset\mathbb R^d$ be compact. Then how is $C^1(K)$ defined? I've seen this at many places, but no one seems to care about the problems at the boundary of $K$. |
Short question: How is $C^1(K)$ defined for a compact set $K$? |
How to find the coefficient of $x^3y^4z$ in $ (x+y+z)^5 (1+x+y+z)^{5}$? |
My teacher said that the following set $E=\{x: range(\phi_x) \not= \{x\} \}$ ($\phi_x$ is the $x$-th Turning Machine) is productive. Why? I think it should be recursive enumerable. In fact, to decide wether $x$ belongs to $E$, you can simulate the $x$-th Turing Machine with dovetail method and, if $x$ in $E$, the machi... |
Let p be a prime number $>=$ 3.
We consider $2p$ black beads and $2p$ blue beads (both indistinguishable).
How many unique necklaces size $4p$, created from these beads, are there? (Considering only rotations.)
*I need help with the Burnside's lemma.*
(I have not found a similar question with a satisfactory... |
Multivariable function having partial derivatives almost everywhere are necessarily measurable? |
Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p,\ell$ be two prime numbers.
Consider the mod $\ell$ representation $$\rho:Gal(\mathbb{\overline{Q}}/\mathbb{Q})\to Aut(E[\ell])= GL(2,\ell).$$
I want to know the results [or references] about the ramification of $\rho$ at different primes $p$.
For example... |
Suppose that $(X_t)_{t\in [0, T]}$ is a stationary stochastic process with continuous sample paths such that $\mathbb E[X_t] = 0$ and $\mathbb E[|X_t|^2] < \infty$, and assume that
$$
S := \mathbb E \left[ \sup_{t \in [0, T]} |X_t|\right] < \infty.
$$
Take $J$ independent copies $(X^1_t), \dotsc, (X^J_t)$ of the pr... |
I am pondering a question in number theory that touches upon the representation of natural numbers as sums of consecutive elements from a subset S of $\mathbb{N}$. Specifically, the question is:
**Conjecture:** Does there exist a subset $S$ of $\mathbb{N}$ such that all but finitely many natural numbers are **no... |
Let $p \geq 3$ be a prime number.
We consider $2p$ black beads and $2p$ blue beads (both indistinguishable).
How many unique necklaces of size $4p$, created from these beads, are there? (Consider only rotations.)
*I need help with the Burnside's lemma.*
(I have not found a similar question with a satisfacto... |
On my last test, there was a question:
>Estimate, as well as possible from above, the probability that $X \geq 10\$, if $E(X^3) = 100$.
I used the fact that $$P(X \geq c) = P(f(X) \geq f(c)) \leq \frac{E[f(X)]}{f(c)},$$ where $f$ is non-negative and monotonically increasing.
If you substitute $f(x) = x^3$ and $c... |
On my last test, there was a question:
>Estimate, as well as possible from above, the probability that $X \geq 10$, if $E(X^3) = 100$.
I used the fact that $$P(X \geq c) = P(f(X) \geq f(c)) \leq \frac{E[f(X)]}{f(c)},$$ where $f$ is non-negative and monotonically increasing.
If you substitute $f(x) = x^3$ and $c=... |
I am pondering a question in number theory that touches upon the representation of natural numbers as sums of consecutive elements from a subset S of $\mathbb{N}$. Specifically, the question is:
**Conjecture:** Does there exist a subset $S$ of $\mathbb{N}$ such that all but finitely many natural numbers are the ... |
Consider the following formulation of the AM-GM Inequalities:
$$f_n(a_1, a_2, a_3,\cdots,a_n) := \left(\sum_{k=1}^n a_k^n\right) - n \prod_{k=1}^n a_k \geqslant 0$$
As required by the classical formulation, we must have $a_k \geqslant 0$
For $n=2,3$, the inequalities admit proof by factorisation
$f_2 \equiv... |
I've recently become a little obsessed with all the ways limits can be composed - and the preconditions for this to take place. I took $A,B_{1},B_{2},C \subseteq \mathbb{R}$ and $f:A\to B_{1},\ g:B_{2} \to C$ and further assumed $f[A] \subseteq B_{2}$. And then I listed out all the ways to compose $f$ and $g$ with limi... |
I will denote a pair of neighbour numbers from the sequence $(a_n)$ as $x,y$ going in that order. As you already stated, once we get a pair of coprime $x,y$, the sequence is done: it strictly increases after and hence is unbounded. Indeed, $\frac{x+y}{\gcd(x,y)}= x+y > y$ and new pair numbers $y, x+y$ are also coprime.... |
On my last test, there was a question:
>Find an upper bound on the probability that $X \geq 10$, if $E(X^3) = 100$.
I used the fact that $$P(X \geq c) = P(f(X) \geq f(c)) \leq \frac{E[f(X)]}{f(c)},$$ where $f$ is non-negative and monotonically increasing.
If you substitute $f(x) = x^3$ and $c=10$, it turns out t... |
Let ABCD be a convex quadrilateral, point E on BC, F and G different points on CD such that (GFC), and H is a point on AD. Prove that the lengths EG and FH intersect.
[![enter image description here][1]][1]
My attempt was to look at triangle GCE and line HF and to use Pasch axiom on them. To use that axiom first ... |
Let $R_1$, $R_2 \in \mathbb R^{N \times M}$ with $M > N$ be matrices of rank $N-1$. When can we find a symmetric matrix $P\in \mathbb R^{M\times M}$ such that the matrix equation
$$ R_1 P = R_2, $$
is satisfied? When does a solution exist? I was thinking that an approach could be rewriting the equation by vecot... |
Let me first set the scene. This is how we learned the variables and their names and symbols.
[![enter image description here][2]][2]
[![enter image description here][1]][1]
[![enter image description here][3]][3]
We have the sugar content of 77 different cereal brands. It firsts asks "estimate the true average... |
It is well-known that the Catalan numbers have an asymptotic approximation
$$C_n=\frac{4^n}{\sqrt{\pi}n^{3/2}}.$$
I am curious about combinatorial interpretations of this formula, rather than a formal derivation (for example using the Stirling approximation.) For example, why would I expect $4^n$ and $n^{3/2}$ to... |
I want to solve the following exercise:
Let $k$ be an algebraically closed field and consider a smooth projective irreducible curve $C$ over $k$. Set $S$ $:= C ×C$. Show that $p_{1}^* \oplus p^*_2 : \operatorname{Pic}(C)\oplus \operatorname{Pic}(C) \to \operatorname{Pic}(S)$ is injective, where $p_i : S \to C$ is th... |
Theorem 5.99 in Rotman's "An Introduction to Homological Algebra" is
>If $\mathcal{A}$ is a small abelian category, then there is a covariant full faithful exact functor $F: \mathcal{A} \to Ab$.
Is this correct? I thought one gets only a covariant full faithful exact functor $F: \mathcal{A} \to \text{R-Mod}$ for s... |
There are $n$ students. In how many ways can they sign up to courses $A, B, C$ if each of the students can choose either $0$, $1$, $2$ or $3$ courses and also (*) none of the parts of Venn Diagram visualizing the problem are empty.
**My solution:**
I assumed that (*) is equivalent to simply having to subtract the... |
You are told you have a container containing N balls. There is only 1 white ball in the container.
S = The number of draws needed to obtain a white ball for the first time.
Prove that P(S = i) = 1/N if 1 <= i <= N
I do not understand how the probability of drawing a white ball can be 1/N on such a large range ... |
I am pondering a question in number theory that touches upon the representation of natural numbers as sums of consecutive elements from a subset S of $\mathbb{N}$. Specifically, the question is:
**Conjecture:** Does there exist a subset $S$ of $\mathbb{N}$ such that all but finitely many natural numbers are the sum ... |
I am pondering a question in number theory that touches upon the representation of natural numbers as sums of consecutive elements from a subset S of $\mathbb{N}$. Specifically, the question is:
Does there exist a subset $S$ of $\mathbb{N}$ such that all but finitely many natural numbers are the sum of multiple con... |
The following computes that $\lim_{j \to \infty} \lim_{N \to \infty} S(N,j,h) = \int_0^{1/2} \frac{h}{\sqrt{h^2 + \cos^2(\pi x)}} \mathrm{d}x.$ This at least tells us the limiting value seen in the post, but in my opinion it is not satisfactory as an explanation of the observations because a) it doesn't explain the beh... |
So I am studying Catalan numbers, and I see there is no one single definition of Catalan numbers .
However, I was wondering how can you explain the Catalan number formula combinatorically ?
so the nth Catalan number can be found using the formula $$C_n = {2n \choose n} - {2n \choose n +1}$$
So ${2n \choose n}$... |
There are $n$ students. In how many ways can they sign up to courses $A, B, C$ if each of the students can choose either $0$, $1$, $2$ or $3$ courses and also (*) none of the parts of Venn Diagram visualizing the problem are empty.
**My solution:**
I assumed that (*) is equivalent to simply having to subtract the... |
Consider $h\in B(H)$, $h=h_1-h_2$, where $h_1=\frac{1}{2}(\|h\|+h)$ and $h_2=\frac{1}{2}(\|h\|-h)$.
In a proof of a theorem about positive elements of $C^*$-algebra, I see that $\|h_1\|\leq\|h\|$ and $\|h_2\|\leq\|h\|$. How?
I know that $h\leq h_1,h_2$, easy. But I tried proof $\|h_1\|\leq\|h\|$, my attempted (fa... |
Why $\|\frac{1}{2}(\|h\|+h)\|\leq\|h\|$? |
The following computes that $\lim_{j \to \infty} \lim_{N \to \infty} S(N,j,h) = \int_0^{1/2} \frac{h}{\sqrt{h^2 + \cos^2(\pi x)}} \mathrm{d}x.$ This at least tells us the limiting value, $a$, seen in the post, but in my opinion it is not satisfactory as an explanation of the observations because a) it doesn't explain t... |
I have a function $f(n)$ defined over $n \in \mathbb{N}$ which I would like to bound asymptotically. I have proven that $f(a) \leq g(a)$ for some subset of naturals $a \in A \subset \mathbb{N}$ and I know $g(n) \sim h(n)$. The function $g(n)$ is rather ugly, but $h(n)$ is simple, so I would prefer to relate the growth ... |
There are $n$ students. In how many ways can they sign up to courses $A, B, C$ if each of the students can choose either $0$, $1$, $2$ or $3$ courses and also (*) none of the parts of Venn Diagram visualizing the problem are empty.
**My solution:**
I assumed that (*) is equivalent to simply having to subtract the... |
If $\pi$ is a $*$-representation (i.e. a $*$-homomorphism) of a unital Banach$*$-algebra, then $\pi(\mathscr{A})$, is trivially self-adjoint, but why is $\pi(\mathscr{A})'$ a von Neumann algebra? This is used in the following proof that such a representation is irreducible iff $\pi(\mathscr{A})' = \mathbb{C}\mathbb{1}$... |
Why is the commutant of a unital $*$-representation a von Neumann algebra? |
$C_a(u) = au\, \text{mod} p \, mod \, \Delta , \; C_a(v) = au\, \text{mod} p \, mod \, \Delta$ where $a \in \{0,1,2 \ldots p-1 \}, p$ is a prime number and $\Delta $ is any number less than $p$. If $C_a(u) = C_a(v)$, then it is given that $a(u-v) \, mod \, p \in \{-1 \Delta, -2 \Delta, \ldots \ -\lceil\frac{p-1}{\Delt... |
In Klenke's book on probability he states Slutzky's theorem as:
> Let $X, X_1, X_2, \ldots$ and $Y_1, Y_2\ldots$ be random varaibles with values in $E$. Assume $X_n \xrightarrow{\mathcal{D}} X$ and $d(X_n, Y_n) \xrightarrow{n\rightarrow \infty} 0$ in probability. Then $Y_n \xrightarrow{\mathcal{D}} X$.
Here $E$ is ... |
Ok, what would an example have to look like? Say $x\in f^{-1}(\overline{B})\setminus\overline{f^{-1}(B)}$. We know every neighbourhood $U$ of $f(x)$ intersects $B$, pulling back to a neighbourhood $f^{-1}(U)$ of $x$ intersecting $f^{-1}(B)$. As $x$ fails to be in the closure, there must be a neighbourhood $V$ of $x$ wh... |
I'm studying the following theorem in (Schindler, 2014: Set Theory
Exploring Independence and Truth), p. 178-180:
**Theorem 9.16 (Mokobodzki) No rapid filter F $\subset$ ${}^\omega 2$ is Lebesgue measurable.**
Proof:\
Let $\epsilon_n = \frac{1}{2n+1}$ for n < $\omega$. By Lemma 9.14 *(A non-trivial filter o... |
I come from the applied math and statistics world, but I was talking to my friend who comes from the pure math and number theory world—in particular Galois representation theory.
We got together to watch [the Superbowl][1] and were talking about mathematical topics. Now I mentioned something about the confusin... |
I'm sorry if this is a noob question but I got this statement asked on a test:
$\forall x \exists y \,F(x,y) \rightarrow \exists y \forall x \, G(x,y)$
where
$M = \lbrace \text{Natural numbers > 0} \rbrace \\
F(x,y): \, x <y \\
G(x,y): \, x > y\\$
The statement is supposed to be false? I get that the left... |
It is well-known that the Catalan numbers have an asymptotic approximation
$$C_n\sim \frac{4^n}{\sqrt{\pi}n^{3/2}}.$$
I am curious about combinatorial interpretations of this formula, rather than a formal derivation (for example using the Stirling approximation.) For example, why would I expect $4^n$ and $n^{3/2}... |
There are $n$ students. In how many ways can they sign up to courses $A, B, C$ if each of the students can choose either $0$, $1$, $2$ or $3$ courses and also (*) none of the parts of Venn Diagram visualizing the problem are empty.
**My solution:**
I assumed that (*) is equivalent to simply having to subtract the... |
We let $R$ denote a reduced finitely generated $k$-algebra where $\dim_kR = n$ as a $k$ vector space and $n$ is a positive integer.
If we write $M$ to be the set of all proper maximal ideals of $R$, how might one show that $|M| = n$? Namely, we are looking to show that $R \cong k^n$ as $k$-algebras through this. |
You are almost there. From $a(ab-1)+b(1-ab)=0$, you get
$$(a-b)(ab-1)=0 $$
A product is $0$ if and only if one of the factors is $0$. So either $a=b$ (and we are done), or: $ab=1$. Now recall that $a,b\in V$, i.e., $\operatorname{Im} a>0$ and $\operatorname{Im} b>0$. Do you know how to find $b=\frac1a$ if $a=x+iy$? |
There are $n$ distinguishable students. In how many ways can they sign up to courses $A, B, C$ if each of the students can choose either $0$, $1$, $2$ or $3$ courses and also (*) none of the $7$ parts of Venn Diagram visualizing the problem are empty.
**My solution:**
I assumed that (*) is equivalent to simply ha... |
We let $k$ denote a fixed algebraically closed field of characteristic zero.
We let $R$ denote a reduced finitely generated $k$-algebra where $\dim_kR = n$ as a $k$ vector space and $n$ is a positive integer.
If we write $M$ to be the set of all proper maximal ideals of $R$, how might one show that $|M| = n$? Nam... |
Is this valid?
$ p\wedge(q\vee r)\equiv(p\wedge q)\vee r$
And if it is, would this equivalency be valid?
$ p\wedge(q\vee r) \equiv q\wedge ( p\vee r)$
|
About the logic associative and commutative laws, does this apply? |
Let $G$ be a group action on the set $X$, and let $f:(X,\Sigma_X)\to (Y,\Sigma_Y)$ be a measurable maximal invariant ($f$ is constant on each orbit and take different values on different orbits). Am asked to show that if $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ are standard Borel then
$$\sigma(f)=\{B\in \Sigma_X:g^{-1}(B)=... |
Problem :
Show by hand without any computer assistance :
>>$$e^{e^2}>1000\phi$$
I come across this limit showing :
$$\lim_{x\to 0}x!^{\frac{x!!^{\frac{2}{x!!-1}}}{x!-1}}=e^{e^2}$$
I cannot show it without knowing some decimal and if so using power series or continued fraction .
It seems challenging an... |
Scenario: we have reached River, i.e. there are 5 cards on the table, and three of them are hearts. I have no heart on my hand, and there are 6 remaining players other than me. What is the probability, that (at least) one of them will have (any) two hearts on hand?
Now if there was just one player other than me, the... |
Question:
>Find the smallest $n$ for which there exists a Markov chain with $n$ states such that it is periodic simultaneously with period $4$ and $6$? Put "does not exist" if such a Markov chain does not exist.
My answer is $n = 12$, because we can create a chain as in the picture.
[![Markov chain][1]][1]
... |
A cylindrical tank has a diameter of $60 cm$ and a height of $100 cm$. It is filled with water to a depth of $10 cm$, then tilted by an angle $\theta = 30^\circ$. What will be the new depth of water? i.e. how high above the ground is the water surface in the tilted tank?
**My Approach:**
By taking horizontal ... |
There is standart well known geometric way to add two points $P$ and $Q$ of elliptic curve, that is by drawing a straight through the two points $P$ and $Q$ and getting a third point $R$ and **using symmetry** of the curve along the x-axis we call $P+Q$ is our new point.
My **question** is why do we need to take sym... |
I am trying to linearize $\frac{1}{x}+\frac{1}{y} = \frac{1}{k}$, ie. getting it on the form $y = ax+b$ and identifying $a$ and $b$.
With a straight forward approach, I end up with the following:
\begin{align}
\frac{1}{x}+\frac{1}{y} = \frac{1}{k} \Rightarrow ky+kx=xy \Rightarrow y(k-x) = -kx \Rightarrow y = -\f... |
There is standart well known geometric way to add two points $P$ and $Q$ of elliptic curve, that is by drawing a straight through the two points $P$ and $Q$ and getting a third point $R$. Take **symmetry** of $R$ along the x-axis and we call $P+Q$ is our new point.
My **question** is why do we need to take symmetry?... |
Let $G$ be a group action on the set $X$, and let $f:(X,\Sigma_X)\to (Y,\Sigma_Y)$ be a measurable maximal invariant ($f$ is constant on each orbit and take different values on different orbits). Am asked to show that if $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ are standard Borel then
$$\sigma(f)=\{B\in \Sigma_X:g^{-1}(B)=... |
Since the Compactness theorem of FOL is equivalent to the ultrafilter Lemma, which implies Hahn-Banach, the implication is clear to me. I was more just wondering if there is a nice direct proof? I saw in another thread that Compactness very nicely proves the existence and uniqueness of Algebraic closures, so I thought ... |
Scenario: we have reached River, i.e. there are 5 cards on the table, and three of them are hearts. I have no heart on my hand, and there are 6 remaining players other than me. What is the probability, that (at least) one of them will have (any) two hearts on hand?
Now if there was just one player other than me, the... |
In Kadison's article he says that:
"Since the commutant of an irreducible representation consists of scalars, two such are either unitarily equivalent or disjoint."
Ref:”https://msp.org/pjm/1960/10-2/pjm-v10-n2-p12-s.pdf”
That the commutant of irreducible representations consist of $\mathbb{C}\mathbb{1}$ I got as ... |
If two representations have the scalars a its commutants, why does it imply that they are either unitarily equivalent or disjoint? |
There are $n$ distinguishable students. In how many ways can they sign up to courses $A, B, C$ if each of the students can choose either $0$, $1$, $2$ or $3$ courses and also (*) none of the $7$ parts of Venn Diagram visualizing the problem are empty.
**My solution:**
I assumed that (*) is equivalent to simply ha... |
From [1], we have the well known Lindeberg–Levy Central-Limit Theorem:
>[Lindeberg–Levy Central-Limit Theorem]
Suppose $X_1, X_2, \ldots, X_n$ is a sequence of independent and identically distributed random variables with $E[X_i] = \mu$ and $\operatorname{Var}[X_i] = \sigma^2 < \infty.$ Then, as $n$ approaches in... |
From the Central-Limit Theorem, what does a sum of IID random variables converge in distribution to? |
From the text:
If $A$ is an object in a category $\mathcal{C}$, show that the invertible elements of Mor($\mathcal{A}$, $\mathcal{A}$) form a group (called the automorphism group of $\mathcal{A}$, denoted Aut($\mathcal{A}$)). Show that two isomorphic objects have isomorphic automorphism groups.
(For readers with ... |
So, I got up to expanding $\frac{(Re(z))^2}{|z|}$ to $\frac{z^2+2z\overline z + \overline z^2}{4|z|}$ but I don't see other sequences that converge to $0$ to spark inspiration on how to continue.
This one has just really thrown me off. Any tips? |
From [1], we have the well known Lindeberg–Levy Central-Limit Theorem:
>[Lindeberg–Levy Central-Limit Theorem]
Suppose $X_1, X_2, \ldots, X_n$ is a sequence of independent and identically distributed random variables with $E[X_i] = \mu$ and $\operatorname{Var}[X_i] = \sigma^2 < \infty.$ Then, as $n$ approaches in... |
We take $X = V(xy-zw)$ for $V(xy-zw)$ the set of points in $\mathbb A^4_{x,y,z,w}$ in which $xy-zw = 0$.
Given this, is it always the case that the rational function $f = x/z \in k(X)$ cannot be expressed as $g/h$ for $g,h \in k[X]$ with $V_X(h) \subseteq V_X(y,z)$? I was thinking contradiction would perhaps be the ... |
[Proof of Hutton's 1776 formula][1]
[1]: https://i.stack.imgur.com/2ZMe3.png
Why does the figure establish this identity?
$\begin{equation}
\frac{\pi}{4} = \arctan\left(\frac{1}{3}\right) + \arctan\left(\frac{1}{7}\right),
\end{equation}$
|
I'll try to answer to your questions following the order of their appearance.
>Are there interesting approaches that focus on the minimization of the energy directly?
**Yes, there are several approaches to the solution of this problem: they are all known under the collective name of "*Variational Method*"**. Thi... |
I tried to find the indefinite integral
$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx)dx$$
by using euler formula and put $x=\frac{\ln y}{2i}$ I got
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(n+1)(2n+1)}{12}-1} \prod_{k=1}^n (y^p+1)^p dy$$
now lets define $a(n,k)$ as the coefficient of $x^k$ in the expression $\... |
how to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$? |
So, I got up to expanding $\frac{(Re(z))^2}{|z|}$ to $\frac{z^2+2z\overline z + \overline z^2}{4|z|}$ but I don't see other sequences that converge to $0$ to spark inspiration on how to continue.
This one has just really thrown me off. Any tips?
Edit: Actually, $\frac{(Re(z))^2}{|z|}$ = $\frac{r^2cos^2(n)}{|r|e^... |
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