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Does a kind of real plane algebraic curve always have a factorization corresponding to connected components? |
Let
$$
Q =
\begin{bmatrix}
Q_0 & Q_1 & Q_2 & \cdots\\
Q_{-1} & Q_{0} & Q_1 & \cdots\\
Q_{-2} & Q_{-1} & Q_0 & \cdots\\
\vdots & \vdots & \vdots & \ddots
\end{bmatrix}
$$
be a block Toeplitz matrix with $Q_n\in \mathbb{R}^{p\times p}$ and $Q_{-n} = Q_{n}^{\sf{T}}$, i.e., $Q$ is symmetric. Additionally, let $Q_... |
When is a Symmetric Block Toeplitz Matrix Invertible? |
alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even well-known) literature i haven’t found or by pointing out a fault in my reasoning.
joel david hamkins is a proponent of... |
I have some data that can be fit reasonably well with an exponential function. However, a colleague mentioned that it would be better to use the sum of two exponentials:
$$
f(x; a, b_1, b_2, \lambda_1, \lambda_2) = a + b_1 e^{\lambda_1 x} + b_1 e^{\lambda_2 x}
$$
There was no mathematical justification given, a... |
Calculate: $$I = \int_{\substack{x_1 \geq 0, \ldots, x_n \geq 0 \\ x_1+\cdots+x_n \leq 2}} \sqrt[n]{x_1 \cdots x_n\left(x_1+\cdots+x_n\right)}\, d x_1 \cdots d x_n$$
Attempt: Perform the substitution $ x_i = 2y_i $, the Jacobian will be $ 2^n $ and so $$I=2^{n+1+\frac{1}{n}} \cdot \int_{\substack{x_1 \geq 0, \ldo... |
Let f be a continuous function on $\mathbb{R}$ and B_t a brownian motion.
Is there any density result of the neuronal network class for function of the form $t \mapsto f(B_t)$ ?
|
All following notation and assumptions follow [Gödel's Theorems and Zermelo's Axioms][1] by Halbeisen and Krapf.
Exercise 11.4 c) states "Conclude that the Second Incompleteness Theorem is provable within PA."
As the Second Incompleteness Theorem is stated as $\text{PA} \not\vdash \neg \text{prv}(\lceil 0=1 \rce... |
Is Goedels second incompleteness theorem provable within peano arithmetic? |
$\int_0^{+\infty}\frac{(\sin x)^a}{x^b}dx=?,\,\text{suppose} \ a,b \in \mathbb{Z^+},a-b\geq 0,b\geq 2$ |
Consider the ODE $$y''+y'+f(y)=0,\quad (*)$$
with $f(y)=y+y^3$.
I'm interested in the asymptotic behavior of solutions $y(x)$ as $x\to\infty$. From Wolfram Alpha's example plots (and from what I've managed), I suspect that $\lim y=\lim y'=0.$
Since a priori, no term seems to be small compared to the others, my app... |
Is Gödels second incompleteness theorem provable within peano arithmetic? |
Consider we have $A,B,C$.
What is the formula in combinatorics to get the value of 15 we have here for this special example:
$\{A,B,C,AB,AC,BC,BA,CA,CB,ABC,BCA,CBA,BAC,ACB,CAB\}$
but in the general case for $n$ items?
|
What 1-Form Satisfies this Property? |
I am trying to understand the conditions for time series $\{y_t\}$ to be stationary, i.e.:
- $E(y_t)=\mu$ is constant for all $t >0,$
- $Var(y_t) = V$ is constant for all $t >0$ and
- $cov(y_t, y_s)$ depends only on $|t-s|$ and not $t$ itself.
I came across a [youtube video by ritvikmath](https://www.youtube.com/... |
All following notation and assumptions follow [Gödel's Theorems and Zermelo's Axioms][1] by Halbeisen and Krapf.
Exercise 11.4 c) states "Conclude that the Second Incompleteness Theorem is provable within PA."
As the Second Incompleteness Theorem is stated as $\text{PA} \not\vdash \neg \text{prv}(\lceil 0=1 \rce... |
*Long comment*.
Here is evidence that $0.457<P<0.459$.
Assume that the circle is a unit circle centred at the origin, and the vertices of the triangle are:
$A\space(\cos(-2Y),\sin(-2Y))$ where $0\le Y\le\pi$
$B\space(\cos(2X),\sin(2X))$ where $0\le X\le\pi$
$C\space(1,0)$
So:
$a=BC=2\sin X$
$b... |
The Facebook Birthday Problem:
This problem stems from the classic Birthday Paradox.
It asks:
> How many friends do you need for the probability of having at least one friend with a birthday each day to be greater than 50%?
> Answer: If there are 23 people in a class, the probability of two students shari... |
I am trying to comprehend the epsilon-delta proofs and additive limit laws proofs and I have reached the following conclusions (A,B). If anyone could verify that my understanding on either is correct it would be very appreciated.
**A**. Let functions $f(x), t(x)$ with the same domain. Let $lim_{x\rightarrow a}f(x) =... |
I'm trying to calculate this integral using residue theorem:
$$
lim_{\epsilon \rightarrow 0}\int_{-\infty}^{\infty}\text{d}\omega\frac{\omega^2-k^2v_F^2}{(\omega+kv+i\epsilon)(\omega-kv-i\epsilon)}\frac{1}{(\omega-kv_q+i\epsilon)(\omega-kv_q-i\epsilon)}
$$
It seems to me that the integrand is proportional to $\fr... |
What is the meaning behind the terminology *"point-set level (object or category)”* in context of stable homotopy theory? This apeared e.g. in following excerpt from Tom Bachmann's [Thesis][1]Invertible Objects in Motivic Homotopy Theory (p 2):
>Just as the homotopy category of spaces can be obtained from seve... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
After school hours, all students will take their bikes and pay 1 dollar. Supposing that there are m students with 1-dollar bills, k students with 2-dollar bills (every student has either 1-dollar or 2-dollar, there is no one with both) and the keeper has no cash at the start. How many ways can all students take their b... |
How many ways students can take their bikes without the keeper having to use his own money to pay? |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
I am trying to comprehend the epsilon-delta proofs and additive limit laws proofs and I have reached the following conclusions (A,B). If anyone could verify that my understanding on either is correct it would be very appreciated.
**A**. Let functions $f(x), t(x)$ with the same domain. Let $lim_{x\rightarrow a}f(x) =... |
[1]: https://doku.pub/documents/an-introduction-to-the-theory-of-groups-rotmanpdf-oq1nzoynzz02
At the end of the Existence Proof, Rotman says that Q char K follows from K=HQ.Can anyone explain why this is true? |
The Facebook Birthday Problem:
This problem stems from the classic Birthday Paradox.
It asks:
> How many friends do you need for the probability of having at least one friend with a birthday each day to be greater than 50%?
> Answer: If there are 23 people in a class, the probability of two students shari... |
> Theorem 4 The discretized signature kernel over $k$,
$$
\mathrm{k}^{+}: X^{+} \times X^{+} \rightarrow \mathbb{R}, \quad \mathrm{k}^{+}(x, y)=\left\langle\mathrm{S}^{+}\left(\mathrm{k}_x\right), \mathrm{S}^{+}\left(\mathrm{k}_x\right)\right\rangle
$$
> 1. is a positive definite kernel,
> 2. $$\mathrm{k}^{+}(x, y... |
I ran across an integral that is rather tough and I am wondering if anyone could give me a shove in the right direction.
$$\int_{0}^{1}\frac{\tanh^{-1}x\ln x}{x(1-x^{2})}{ d}x=-\frac{7}{16}\zeta(3)-\frac{{\pi}^{2}}{8}\ln(2)$$
This solution is almost exactly like the solution to $\int_{0}^{\frac{\pi}{2}}x\ln(\si... |
That's not quite true, since it can run into problems when $p = 2$:
$$
{\rm Gal}(\mathbf Q(\mu_{2^\infty})/\mathbf Q(\mu_2)) = {\rm Gal}(\mathbf Q(\mu_{2^\infty})/\mathbf Q) \cong \mathbf Z_2^\times,
$$
which is not isomorphic to $\mathbf Z_2$ since it has nontrivial torsion (namely the number $-1$). Maybe you mean... |
I wouldn't say this isomorphism is "really simple" when learning these things for the first time. It involves an isomorphism in infinite Galois theory and an isomorphism between a certain subgroup of $\mathbf Z_p^\times$ with $\mathbf Z_p$. In particular, the group
${\rm Gal}(F(\mu_{p^\infty})/F(\mu_{p}))$ is not r... |
Just have a look at this drawing, where the left green angle equals the right green angle (quite obvious):
[![enter image description here][1]][1]
The sum of the angles of the triangles $AEC$ and $DEB$ are equal to $180°$.
Hence, the sum of the angles $A+C$ equals $B+D$.
Why would something, so obvious, nee... |
I am trying to solve the following problem taken the book Markov Chains and Mixing Times 2nd edition (Exercise 12.2):
Let $P$ be irreducible transition matrix, and suppose that A is a matrix with $0 \le A(i,j) \le P(i,j)$ and $A \ne P$. Show that any eigenvalue $\lambda$ of $A$ satisfies $|\lambda| < 1$.
Now I kn... |
I wouldn't say this isomorphism is "really simple" when learning these things for the first time. It involves an isomorphism in infinite Galois theory and an isomorphism between a certain subgroup of $\mathbf Z_p^\times$ with $\mathbf Z_p$. In particular, the group
${\rm Gal}(F(\mu_{p^\infty})/F(\mu_{p}))$ is not r... |
Good morning everyone, I encountered a calculation procedure (structural dimensioning) that proceeds by trial and error in a recursively manner, which converges to a value. From various steps I arrived at the following nested infinite radical which unfortunately I cannot resolve further with my skills.
$$x_{1} = \fr... |
We need to pick a set of unique numbers (**with replacement**!) out of a group of n unique numbers. Each number has the same probability to be picked. Each iteration we pick randomly one number from this group. We end once we found a set of unique numbers where the amount of numbers in the set is exactly 10% of the gro... |
Good morning everyone, I encountered a calculation procedure (structural dimensioning) that proceeds by trial and error in a recursively manner, which converges to a value. From various steps I arrived at the following nested infinite radical which unfortunately I cannot resolve further with my skills.
$$x_{1} = \fr... |
Let $Y_i = \frac{1}{i} X_i$, for $i \in \mathbb{N}$, where $X_i$'s are i.i.d. r.v. such that $\mathbb{E}[X_i] = 0$ and $\mathbb{E}[e^{\lambda X_i}] < \infty$ for some $\lambda > 0$. I want to obtain a concentration inequality of the form
$$\mathbb{P}\left(\sum_{i=1}^{n}Y_i \geq \epsilon\ln(n)\right) \leq 1 - \frac{1... |
We need to pick a set of unique numbers (**with replacement**!) out of a group of `n` unique numbers. Each number has the same probability to be picked. Each iteration we pick randomly one number from this group. We end once we found a set of unique numbers where the amount of numbers in the set is exactly 10% of the... |
I have recently been thinking about the housing allocation problem where we have a set of players and a set of houses where players have strict preferences over the houses. I am aware of the Top Trading Cycle algorithm which can be used to assign houses to players in a Pareto Optimal, Strategy Proof, and Individual Rat... |
Given a set of coordinates on a euclidean plane, is it possible to sort the points by sweeping a line through them where the line might not be strictly horizontal or vertical?
I know that for a simply vertical or horizontal sweep line, you obviously only need to account for the x value (for a vertical sweep) or the ... |
Is there an algorithm for sorting points in a field based on sweeping a line through them at an arbitrary angle? |
We need to pick a set of unique numbers (**with replacement**!) out of a collection of `n` unique numbers. Each number has the same probability to be picked. Each iteration we pick randomly one number from this collection. We end once we have a set of unique numbers where the amount of numbers in the set is exactly 1... |
let $f$ be an endomorphism from some compact space S.<br>
let $b$ a measurable and bounded function.<br>
let $g(y)=sup(y \cdot f(x)+b(x)+1/2\lVert f(x) \rVert)^2$ for $y \in S \bigcup f(S)$<br>
I found the assertion that $g$ is convex, lipschitz continue and differentiable.
How can these properties be proven?
... |
$sup(y \cdot f(x)+b(x)+1/2\lVert f(x) \rVert)^2$ convex and regular with respect to $y$? |
We need to pick a set of unique numbers (**with replacement**!) out of a collection of `n` unique numbers. Each number has the same probability to be picked. Each iteration we pick randomly one number from this collection. We end once we have a set of unique numbers where the amount of numbers in the set is exactly 1... |
It is known that the derivative of an inverse function is given as
$$
g'(y)=\frac{1}{f'(x)} \implies \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}
$$
So if $\arcsin(y)$ is differentiated:
$$
\arcsin(y)' = \frac{1}{\cos(x)} = \frac{1}{\sqrt{1-y^2}}
$$
but then when I graph them, I get different results, and I was also... |
What is the different between these two forms of the derivative of arcsine? |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/QBSb6.png
Given the matrix above, what is the easiet way to calculate it's determinant? I've tried getting as many zeroes on the first column for example, but I still end up with 2 non-zero elements. And it repeats for every minor, and bec... |
Can you please tell me where you found the above formula?\
If you have poles on or outside the unit circle you may use this formula:
$$f(\tilde{f},T,\rho,n)\approx \frac{\rho ^T}{n}\cdot \sum _{k=0}^{n-1} \Re\left(\tilde{f}\left(\rho\cdot \exp \left(\frac{i ((2 \pi ) k)}{n}\right)\right)\cdot \exp \left(\frac{i ((2... |
let $f$ be a function from some compact space S, $f:S \mapsto S$.<br>
let $b$ a measurable and bounded function.<br>
let $g(y)=sup(y \cdot f(x)+b(x)+1/2\lVert f(x) \rVert)^2$ for $y \in S \bigcup f(S)$<br>
I found the assertion that $g$ is convex, lipschitz continue and differentiable.
How can these properties be... |
The Facebook Birthday Problem:
This problem stems from the classic Birthday Paradox.
It says:
> How many friends do you need for the probability of having at least one friend with a birthday each day to be greater than 50%?
> Answer: If there are 23 friends, the probability of two or more people sharing a... |
Everything that follows is from Rudin's _Functional Analysis_:
---
$\def\L{\Lambda} \def\DDD{\mathcal{D}} \def\sbe{\subseteq} \def\W{\Omega} \def\RR{\mathbb{R}} \def\CC{\mathbb{C}}$
Below $\DDD$ is the space of test (smooth, compactly supported) functions from an open set $\Omega\sbe\RR^n$ to $\CC$. Such space... |
I wouldn't say this isomorphism is "really simple" when learning these things for the first time. It involves an isomorphism in infinite Galois theory and an isomorphism between a certain subgroup of $\mathbf Z_p^\times$ with $\mathbf Z_p$. In particular, the group
${\rm Gal}(F(\mu_{p^\infty})/F(\mu_{p}))$ is not r... |
there is one thing I'm not sure about regarding cosine similarity. Does the magnitude of the vector matter (I think the answer is yes, especially if you look at the picture below where the word count is an important factor). However, when we take the angle from the beginning, the magnitude of the vector is not relevant... |
there is one thing I'm not sure about regarding cosine similarity. Does the magnitude of the vector matter (I think the answer is yes, especially if you look at the picture below where the word count is an important factor). However, when we take the angle from the beginning, the magnitude of the vector is not relevant... |
Say you have a list of $n$ random integers in the range $[1, m]$. The binomial PMF can find the probability of exactly $k$ occurrences of a number with $p = \frac{1}{m}$
$$ {n \choose k} p^k (1-p)^{n-k} $$
Now say the list is constructed in the following way: the $i$th element in the list is a random number in $[1,... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
Given $s,t \in \mathbb{R}^+$ and $i,j \in \mathbb{N}$, Let $X,Y,Z$ be random variables with Poisson distribution with rate $s,t$ and $s+t$, respectively. Is it true that
$$ P(Z = i+j) \geq P(X=i) \cdot P(Y=j).$$
I would need to have
$$ \frac{s^i}{i!}\cdot \frac{t^j}{j!} \leq \frac{(s+t)^{i+j}}{(i+j)!}.$$
By the You... |
Good morning everyone, I encountered a calculation procedure (structural dimensioning) that proceeds by trial and error in a recursive manner, which converges to a value. From various steps I arrived at the following nested infinite radical which unfortunately I cannot resolve further with my skills.
$$x_{1} = \frac... |
Given a set of coordinates on a euclidean plane, is it possible to sort the points by sweeping a line through them where the line might not be strictly horizontal or vertical? The inputs to the problem would be the set of unsorted 2d points, and a unit vector.
I know that for a simply vertical or horizontal sweep li... |
$$A=\begin{pmatrix}4x+24 & x-12 & x-12 & x-12 & x-12\\x-12 & 4x+24 & x-12 & x-12 & x-12\\x-12 &x-12 &4x+24 & x-12 & x-12\\x-12 & x-12 & x-12 & 4x+24 & x-12\\ x-12 & x-12 & x-12 & x-12 & 4x+24\end{pmatrix}$$
Given the matrix above, what is the easiet way to calculate it's determinant? I've tried getting as many zero... |
$$A=\begin{pmatrix}4x+24 & x-12 & x-12 & x-12 & x-12\\x-12 & 4x+24 & x-12 & x-12 & x-12\\x-12 &x-12 &4x+24 & x-12 & x-12\\x-12 & x-12 & x-12 & 4x+24 & x-12\\ x-12 & x-12 & x-12 & x-12 & 4x+24\end{pmatrix}$$
Given the matrix above, what is the easiet way to calculate its determinant? I've tried getting as many zeroe... |
there is one thing I'm not sure about regarding cosine similarity. Does the magnitude of the vector matter? I think the answer is yes, especially if you look at the picture below where the word count is an important factor. However, when we take the angle from the beginning, the magnitude of the vector is not relevant ... |
So I'm mainly looking for Algebra books for the purpose of Olympiads, but I also want to learn just a bit higher too for personal enjoyment. I currently have 2 in mind: the ones in the title. Since both are of the same name, and there wasn't enough info that I could find online regarding the latter, I just wanted to as... |
What's different between Hall and Knight "Higher Algebra" and Barnard and Child "Higher Algebra"? |
I think I have an idea using coupling. I define $X \sim \text{Pois}(s)$,$Y \sim \text{Pois}(t)$ independently in $P$. Then $Z = X+Y \sim \text{Pois}(t+s)$. Then
$$ P(X = i)\cdot P(Y=j) = P(X=i,Y=j) \leq P(X+Y = i+j) = P(Z = i+j).$$
Do you think is right? |
I'm studying the article "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de vecteurs" (polar factorization and monotone rearrangement of vector-valued functions).
In this work one function is build which is expected to have some properties:
let $f$ be a function from some compact sp... |
>The limit in question
$$\lim_{n\to \infty} \frac{n}{n!!^\frac{2}{n}}.$$
What I have done:
$$
\lim_{n\to \infty}\frac{(n^\frac{2}{n})^\frac{n}{2}}{n!!^\frac{2}{n}}.
$$
Then taking the logarithm and converting it into a summation series
$$\lim_{n\to \infty}\frac{2}{n}\sum_{r=0}^{n-1}\ln\frac{1}{1-\frac{2r}... |
In bayesian networks we often represent conditional independence assumptions within the network like the following:
$$
(eq.1)P(X|A, B, C) = P(X|A)
$$
This is assuming that B, C are conditionally independent to X given A. In a bayesian network, I'd imagine the simplest representation of this as being a Y-shaped... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
Can you please tell me where you found the above formula?\
If you have poles on or outside the unit circle you may use this formula:
$$f(\tilde{f},T,\rho,n)\approx \frac{\rho ^T}{n}\cdot \sum _{k=0}^{n-1} \Re\left(\tilde{f}\left(\rho\cdot \exp \left(\frac{i ((2 \pi ) k)}{n}\right)\right)\cdot \exp \left(\frac{i ((2... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satisfying... |
Let $f(x) \in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n$ with roots $\alpha_1, \alpha_2, \dots, \alpha_k$ in some splitting field $K$, none of which are $0$. Prove that $r = \displaystyle\sum_{i=1}^k \frac{1}{\alpha_i} \in \mathbb{Q}$.
---
This question is very similar in flavor to [this post](htt... |
If I know that $\sin(x)/x \leq 2/\pi$ for $x$ in $[0, \pi/2]$, how can I deduce that $\sum_1^\infty \sin(1/n)$ converges? I'm not seeing any connection between the two. By inspection, it looks like the series converges to 1, but I don't even need to know the limit. I just need to argue that it converges.
My pro... |
This is an extension (and more distilled version) of https://math.stackexchange.com/q/4752960/460999, with new information. I am fairly sure that my constructions are an alternate description of the [De Brujn constant method][1]. Specifically I think $\Lambda=0$ corresponds to $\phi_0.$ This is what my question will be... |
Is $\phi_0$ equivalent to the Riemann hypothesis? |
I'm studying the article "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de vecteurs" (polar factorization and monotone rearrangement of vector-valued functions).
In this work one function is build which is expected to have some properties:
let $f$ be a function from some compact sp... |
$sup(y \cdot f(x)+b(x)-1/2\lVert f(x) \rVert)^2$ convex and regular with respect to $y$? |
If I know that $\sin(x)/x \geq 2/\pi$ for $x$ in $[0, \pi/2]$, how can I deduce that $\sum_1^\infty \sin(1/n)$ converges? I'm not seeing any connection between the two. By inspection, it looks like the series converges to 1, but I don't even need to know the limit. I just need to argue that it converges.
My pro... |
If I know that $\sin(x)/x \geq 2/\pi$ for $x$ in $[0, \pi/2]$, how can I deduce that $\sum_1^\infty \sin(1/n)$ diverges? I'm not seeing any connection between the two. By inspection, it looks like the series converges to 1, but I don't even need to know the limit. I just need to argue that it converges.
My prof... |
Consider the Cauchy problem $(E)$: $x'(r)=f(r,x(r))$ with initial condition $x_0.$
If we proved that there exists an increasing sequence $(r_k)_k$ such that $r_0=0,$ for every $k,(E)$ has unique (local) solution over $[0,r_k[,$ then does is lead to a maximal solution?
|
I have $\frac{1}{z+z^2}$ defined on the annulus $|z+1|>1$.
I write:
$f(z) = \frac{1}{z} - \frac{1}{z+1} = -\frac{1}{z+1} + \frac{1}{(z+1)-1} = -\frac{1}{z+1} + \frac{1}{z+1} \cdot \frac{1}{1-\frac{1}{z+1}} = -\frac{1}{z+1} + \frac{1}{z+1} \sum_{n=0}^{\infty}(z+1)^{-n} = -(z+1)^{-1} + \sum_{n=0}^{\infty}(z+1)^{-n... |
This is not an answer but motivated by the question
**Can we use this method to find the integral: for $a<b$**
$$\int_0^{\infty} \frac{\cos(ax)- \cos(bx)}{x} dx$$
by **user008**.
I try to use Laplace method to prove it. Since we have
\begin{align}
\mathcal{L_s}\left[\frac{f(t)}{t}\right]_{s=0}=\int_0^{\infty}... |
I'm trying to determine the number of binary relations between sets $A$ and $B$ that are both left-unique and right-unique, which we call one-to-one. That is, relations $R$ such that every $a \in A$ and every $b \in B$ appear in at most one pair in $R$. Here's what I'm thinking:
We can have such a relation containin... |
Is this formula for number of binary relations correct? |
Let $C:F(x,y,z)=0$ be a projective plane curve, where $F$ is a homogenous polynomial of degree $d$. I want to show that there exists $p\in \mathcal{P}_2$ such that all lines through $p$ are tangent to $C$ at one point, at the most. I know, for example, that a general quartic plane curve has 28 bitangent lines. I'd like... |
I'm studying the article "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de vecteurs" (polar factorization and monotone rearrangement of vector-valued functions).
In this work one function is build which is expected to have some properties:
let $f$ be a function from some compact sp... |
This is not an answer but motivated by the question
**Can we use this method to find the integral: for $a<b$**
$$\int_0^{\infty} \frac{\cos(ax)- \cos(bx)}{x} dx$$
by **user008**.
I try to use Laplace method to prove it. Since we have
\begin{align}
\mathcal{L_s}\left[\frac{f(t)}{t}\right]_{s=0}=\int_0^{\infty}... |
In arithmetic without total exponentiation, the exponentiable numbers are closed under addition. It is thus common to think of natural numbers as binary strings $n : |n| \to 2$, whose (unary) lengths $|n|$ may be added but not necessarily multiplied.
Likewise, the *twice* exponentiable numbers form a cut. We could t... |
What is the meaning behind the terminology *"point-set level (object or category)”* in context of stable homotopy theory? This appears e.g. in following excerpt quoted from Tom Bachmann's [Thesis][1]Invertible Objects in Motivic Homotopy Theory (p 2):
> [...] Just as the homotopy category of spaces can be obta... |
What does a "$\bmod{2 \pi}$ function" mean? |
Let $f:(M, g) \rightarrow (N, h)$ be a pseudo-Riemannian immersion, $\overline D$ the linear connection on $f^*(TN)$ induced from the Levi-Civita connection of $h$, and $\nabla$ the connection induced on the normal bundle $NM$ (which is the orthogonal complement of $TM$ in $f^*(TN)$). Consider also the second fundament... |
I couldn't fit the whole question in the title so here it is re-typed.
Let $k$ and $w$ be digits and let $X$ be some positive integer with one or more digits. Using the two digits, $kw_7$ is a two-digit base 7 number and $wk_9$ is a two-digit base 9 number. Then $kw_7 = wk_9 = X_8$. Determine the value of the int... |
Complex function completely simplified and i cant figure out how? |
Given two 2D attractors, for example two Clifford Attractors, is it possible to combine them in an attractor in 4D? For example something of the form:
$x_{n+1} = \sin(a y_n) + c \cos(a x_n)$,
$y_{n+1} = \sin(b x_n) + d \cos(b y_n)$,
$w_{n+1} = \sin(a z_n) + c \cos(a w_n)$,
$z_{n+1} = \sin(b w_n) + d \cos(b z_n)$
i... |
I'm learning about intersection homology, and I'm trying to write a proof of the following statement:
>Take X,Y two filtered spaces with perversities $p$ and $q$ resp. , a continuous function $f:X\to Y$ such that for all stratum $S$ of Y, there is a strata $S^f$ of Y such that $f(S)\subset S^f$ and $codim_Y(S^f)\le... |
I'm studying the article "Brenier, Y. (1987) Décomposition polaire et réarrangement monotone des champs de vecteurs" (polar factorization and monotone rearrangement of vector-valued functions).
In this work one function is build which is expected to have some properties:
let $f$ be a function from some compact sp... |
Would be enough to demonstrate that:
$f$ is continuous (1)
$∇f = 0$ at that given point - $ x^* $ (2)
$H(f) = 0 $ at that given point - $ x^* $ (3)
in order to make the case that a particular function $ f $ has only one local minimizer $ x^* $
If yes, why is this the case?
What if we consider f a non-... |
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