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I am trying to solve a question from Leaving Cert. Ordinary Level Maths 2018 Paper 2.
A solid sphere is made of gold.
It has a volume of 0.113cm3 (cubed).
Each cm3 of pure gold weighs 19.3grams.
I am trying to find the number of grams of pure gold in the sphere.
The marking scheme says I am to multip... |
Finding the number of grams of gold in a solid sphere? |
In *Graphs and Digraphs* by Chartrand and Lesniak, a proof of a generalized Turan's Theorem uses the following fact: the inequalities $r \leq n \leq 2(r-1)$ are equivalent to the inequality
$$\binom{r-1}{2} + (n-r+1)(r-2) + \binom{n-r+1}{2} \leq \frac{r-2}{2r-2}\cdot n^2$$
Where $\binom{n}{k}$ is the binomial coe... |
A domain $\Omega \subset \mathbb{R}^n$ is called an *$(\varepsilon, \delta)$-domain* (or *locally uniform domain*) if there exists an $\varepsilon > 0$ such that for all $x, y \in \Omega$ with $\lvert{x - y}\rvert < \delta$ there is a rectifiable curve $\gamma : [0, l(\gamma)] \to \Omega$ joining $x$ to $y$ and satisfy... |
Does anyone have an idea if this double sum could be simplified?
$\sum_{p=0}^\infty\sum_{q=0}^\infty \frac{(-1)^p(-1)^q}{(2p+1)(2q+1) } e^{-\pi\sqrt{(2p+1)^2+(2q+1)^2} \bar y}\sqrt{(2p+1)^2+(2q+1)^2}$
(Context: I computed the electric potential, solution of Laplace's equation, in some geometry as a double cosine... |
In *Probability and Stochastics*, (Cinlar), isomorphic measurable spaces are defined as follows:
[![][1]][1]
[1]: https://i.stack.imgur.com/WdbAR.jpg
So we have a bijection $f$. But then it's emphasized that we have a *functional inverse* $\hat{f}$. Why is this distinction necessary? Is $\hat{f}$ not al... |
*Rem:* I have mostly doubt on my part "3-" but any general remark on my answer will be greatly appreciated.
### Question:
We have the sequence of r.v. $ (Y_n= X_{n+1} \cdot X_n ) , n \in \mathbb{N}$ with $X_n$ a i.i.d. r.v. defined as follow $\forall n \in \mathbb{N}, X_n \in \{0; n \} $ and $P(X_n=0)=1- \fra... |
Certain types of triangles and quadrilaterals have acquired special names, for example the *drafter* triangle (30°-60°-90°), the *golden* rectangle ($\phi$ $\times$ 1), the *lozenge* (rhombus in which the acute angles are 45°), and so on.
Has a special name been given to the 2 $\times$ 1 rectangle? |
Is there a name for a 2 $\times$ 1 rectangle? |
For localizationg of a ring in a prime number, I mean (that's how my book defines it) all the elements of the field of fractions of the ring in question of the form $\frac{a}{b}$ such that b can't be divided by p. In the case of $\mathbb{Z}$, we are talking of {$\frac{a}{b} \in \mathbb{Q}$ such that p doesn't divide b}... |
Are $\mathbb{Z}_{(2)}$ and $\mathbb{Z}_{(3)}$ isomophic? |
For localizationg of a ring in a prime number (noted p), I mean (that's how my book defines it) all the elements of the field of fractions of the ring in question of the form $\frac{a}{b}$ such that b can't be divided by p. In the case of $\mathbb{Z}$ and p, we are talking of {$\frac{a}{b} \in \mathbb{Q}$ such that p d... |
For localizationg of a ring in a prime number (noted p), I mean (that's how my book defines it) all the elements of the field of fractions of the ring in question of the form $\frac{a}{b}$ such that b can't be divided by p. In the case of $\mathbb{Z}$ and p, we are talking of {$\frac{a}{b} \in \mathbb{Q}$ such that p d... |
For localizationg of a ring in a prime number (noted p), I mean (that's how my book defines it) all the elements of the field of fractions of the ring in question of the form $\frac{a}{b}$ such that b can't be divided by p. In the case of $\mathbb{Z}$ and p, we are talking of {$\frac{a}{b} \in \mathbb{Q}$ such that p d... |
For localizationg of a ring in a prime number (noted p), I mean (that's how my book defines it) all the elements of the field of fractions of the ring in question of the form $\frac{a}{b}$ such that b can't be divided by p. In the case of $\mathbb{Z}$ and p, we are talking of {$\frac{a}{b} \in \mathbb{Q}$ such that p d... |
*Rem:* I have mostly doubt on my part "3-" but any general remark on my answer will be greatly appreciated.
### Question:
We have the sequence of r.v. $ (Y_n= X_{n+1} \cdot X_n ) , n \in \mathbb{N}$ with $X_n$ a i.i.d. r.v. defined as follow $\forall n \in \mathbb{N}, X_n \in \{0; n \} $ and $P(X_n=0)=1- \fra... |
Let $S$ be some subset of $\mathbb{R}^3$ and $\text{Conv}(S)$ be its convex hull. Say $S$ has property $\mathcal{P}$ if each point $x \notin S$ lies on some line contained in $\mathbb{R}^3 \setminus S$ such that there is a ruled surface in $\mathbb{R}^3 \setminus S$ giving a homotopy to a line in $\mathbb{R}^3 \setminu... |
I am trying to solve a question from Leaving Cert. Ordinary Level Maths 2018 Paper 2.
A solid sphere is made of gold.
It has a volume of $0.113$cm$^3$.
Each cm$^3$ of pure gold weighs $19.3$ grams.
I am trying to find the number of grams of pure gold in the sphere.
The marking scheme says I am to mul... |
I'm going through the paper of D. Zagier on Short Proof of Prime Number Theorem. There it says in V that
$\phi(s)=\int_1^\infty \frac{d\vartheta(x)}{x^s}$ . Can someone please explain why that is the case ?
And also how The Analytic Theorem is being applied here to the functions defined as f(t) and g(z) ?
Link:
... |
I'm going through the paper of D. Zagier on Short Proof of Prime Number Theorem. There it says in V that
$\phi(s)=\int_1^\infty \frac{d\vartheta(x)}{x^s}$ . Can someone please explain why that is the case ?
And also how The Analytic Theorem is being applied here to the functions defined as f(t) and g(z) ?
MSE qu... |
I'm going through the paper of D. Zagier on Short Proof of Prime Number Theorem. There it says in V that
$\phi(s)=\int_1^\infty \frac{d\vartheta(x)}{x^s}$ . Can someone please explain in details why that is the case ?
And also how The Analytic Theorem is being applied here to the functions defined as f(t) and g(z) ... |
*Remark:* I have mostly doubt on **Part 3** but any general remark on my answer will be greatly appreciated.
## Question:
Let $\alpha > 0$. For $n \in \mathbb{N}$, let the random variable $X_n$ on $\{0,n\}$ be defined by
$$
\begin{cases}
P(X_n=0)=1- \frac{1}{n^{\alpha}}\\
P(X_n=n)=\frac{1}{n^{\alpha}}.
\end... |
Certain types of triangles and quadrilaterals have acquired special names; for example, the *drafter* triangle ($30^\circ$-$60^\circ$-$90^\circ$), the *golden* rectangle ($\phi\times1$), the *lozenge* (rhombus in which the acute angles are $45^\circ$), and so on.
Has a special name been given to the $2\times1$ recta... |
Is there a name for a $2 \times 1$ rectangle? |
I am stuck at simplifying the following integral
$$
I =
\int_{-\pi/2}^{+\pi/2}
d\phi
\left[
1-M(\phi) \theta^*(\phi)
\right]
$$
where
$
M=
p \cos \phi + q \sin \phi
$.
Here $p,q$ are fixed.
The $\theta^\star$ is defined through
$\cos \theta^*=
\frac{M}{\sqrt{1+M^2}}
$
Can anybody help me to c... |
I'am asking myself the following: let $\lambda$ be a positive $\sigma$-finite measure on $\mathbb{R}$ and $(\mu_t)_{t \in \mathbb{R}}$ a family of probability measures on $\mathbb{R}$ with the proberty: $t\mapsto \mu_t(A)$ is measurable for all Borel subsets $A \subseteq \mathbb{R}$.
Furthermore, let $\mu_t=\mu_t^a... |
Let $\mathbb{H}=\{q=t+xi+yj+zk:t,x,y,z\in\mathbb{R}\}$ is quternion set. $f:\mathbb{H}\to\mathbb{H}$ is a function satisfying$$\overline{\partial_c}f=0,$$which
$$\overline{\partial_c}=\frac{1}{2}(\frac{\partial}{\partial t}+\frac{Imq}{r}\frac{\partial}{\partial r}), \ Imq=xi+yj+zk, \ r^2=x^2+y^2+z^2.$$
Then we have... |
Is this statement about quaternion function true? |
Let $\mathbb{H}=\{q=t+xi+yj+zk:t,x,y,z\in\mathbb{R}\}$ is quternion set. $f:\mathbb{H}\to\mathbb{H}$ is a function satisfying$$\overline{\partial_c}f=0,$$which
$$\overline{\partial_c}=\frac{1}{2}(\frac{\partial}{\partial t}+\frac{Imq}{r}\frac{\partial}{\partial r}), \ Imq=xi+yj+zk, \ r^2=x^2+y^2+z^2.$$
Then we have... |
I came across with the sobolev embedding $H_0^1(\Omega)\stackrel{c}{\hookrightarrow} L^2(\Omega)\hookrightarrow H^{-1}(\Omega)$, where $\Omega$ is an open bounded set in $\mathbb{R^n}$. My doubt is whether the embedding $H^1(\Omega) \stackrel{c}{\hookrightarrow} L^2(\Omega) \hookrightarrow (H^1(\Omega))^*$, where * rep... |
I have the following statement.
Suppose $(a_n)$ is a sequence of complex numbers satisfying the following properties.
i) $a_{nm} = a_na_m$ for all positive integers $n, m$.
ii)
\begin{align}
\sum_{n \geq 1} |a_n|^2 < \infty.
\end{align}
iii) Not all $a_n$ are zero.
Then $|a_n| \leq n^{-1/2}$ for al... |
I studied graph theory at the university but "flow networks" were outside the course topics. While reading material about flow networks it is not clear for me if the latency concept (beyond capacity) is there or not. Could you please suggest references on this topic?
I am looking to simulate a computer network with ... |
Does Flow Networks in Graph Theory includes latency or is it another model? |
When we have a discrete groupoid, we have the concept of equivalence of groupoids as categories. Given two equivalent discrete groupoids, we obtain a simplicial homotopy between their respective nerves. Now, I'm curious about the topological case. How is the equivalence of topological groupoids defined? How do we defin... |
I'm currently trying to understand Silvermans example for the valuation on curves discussed in the answer to this post: https://math.stackexchange.com/questions/2000409/definition-and-example-of-order-of-a-function-at-a-point-of-a-curve
Silverman has already shown that $M_P / M_P^2$ is generated by (y), but why does... |
Why is (y) a uniformizer for $y^2 = x^3 + x$ at (0,0) and why is its order equal to 1? |
[CMI May 23, 2022 BSc entrance exam](https://www.cmi.ac.in/admissions/sample-qp/ugmath2022-23may.pdf), Question A2(7):
>You are asked to take three *distinct* points $1, \omega_1, \omega_2$ in the complex plane such that $|\omega_1|=|\omega_2|=1$. Consider the triangle $T$ formed by the the complex numbers $1, \ome... |
In the J. Tate's paper "Relations Between K2 and Galois Cohomology" Lemma 5.2, it says that given $\alpha_1,\cdots,\alpha_r\in $Br$_lF$, i.e. $\alpha_i$ is killed by $l$, where $l$ is prime and $l\neq$ char $F$ and $F$ is a global field. If there exists for each place $v$ of Fan element $x_v\in F_v^{\times}$ such that... |
In I.M. James' *Topological and uniform spaces*, the proof for the second half of Proposition 11.5, on p.143, makes me confused. There, he proves a unformizable space is completely regular.
>The outline of his proof is, for a uniform space $X$, a closed subset $H$ and a point $x\in X\setminus H$, he wants to find a... |
According to some sources, i.e. Wiki, the application of [L'Hospital's Rule][1] requires four conditions to be satisfied. One of them is the existence of limit of the quotient of the derivatives.
We have the following limit:
$$
\lim_{x\to \pi}\frac{x-\pi}{1+\cos x}=\left[\frac{0}{0}\right]=-\lim_{x\to \pi}\frac{1}{\... |
Is L'Hospital applicable when the limit of derivative quotient does not exist? |
In an attempt to solve for stable solutions of certain non-autonomous differential equations, I arrived at the following quadratic equations
$$ 0= A_n x_n^2 + B_n x_n + C_n + \sum_{m=1 | m\neq n}^N D_{n,m} x_m $$
for the variables $x_n$ and real constants $A_n, B_n, C_n, D_{nm}$ for $n,m =
1,...,N $
Can we say u... |
I have boolean function of next variables: $x_{ij}$, $ 1 \leq i \leq n$ and $1 \leq j \leq m$. And the function is equal to 1 only when we have a row of ones. Can I determine the value of the function in less than $n*m$ questions? |
I think there might be an error in the official solution to the following question:
[CMI May 23, 2022 BSc entrance exam](https://www.cmi.ac.in/admissions/sample-qp/ugmath2022-23may.pdf), Question A2(7):
>You are asked to take three *distinct* points $1, \omega_1, \omega_2$ in the complex plane such that $|\omega... |
I have a question regarding the techniques of finding a uniform random variable that is independent of sequence of random variables.
**Context**: For some powerful statistical results (e.g., Theorem 5.1 of [E. Rio][1]) it is necessary to prove that there exists such a random variable $U \sim Uniform([0, 1])$, that i... |
In *Graphs and Digraphs* by Chartrand and Lesniak, a proof of a generalized Turan's Theorem uses the following fact: the inequalities $r \leq n \leq 2(r-1)$ are equivalent to the inequality
$$\binom{r-1}{2} + (n-r+1)(r-2) + \binom{n-r+1}{2} \leq \frac{r-2}{2r-2}\cdot n^2$$
Where $\binom{n}{k}$ is the binomial coe... |
I am attempting to prove Lobachevsky's integral formula, which states that for an integrable even function $f$ with a period of
π, the following integral equality must hold,$$\int_{0}^{\infty} \frac{\sin(x)}{x}f(x)dx = \int_{0}^{\pi/2}f(x)dx$$
I am aware of a classic proof,but I want to try proving this integral for... |
What do I do wrong when computing this commutator of differential operators? |
I have a question regarding the techniques of finding a uniform random variable that is independent of sequence of random variables.
**Context**: For some powerful statistical results (e.g., Theorem 5.1 of [E. Rio][1]) it is necessary to prove that there exists such a random variable $U \sim Uniform([0, 1])$, that i... |
You can use L'Hôpital's rule for one-sided limits. Here you can use it twice, once on each side, and it says the one-sided limits are the same in the first limit as in the second limit (which they are). See [Does L'hopital work for one sided limits?](https://math.stackexchange.com/q/1266111/139123) |
A domain $\Omega \subset \mathbb{R}^n$ is called an *$(\varepsilon, \delta)$-domain* (or *locally uniform domain*) if there exists an $\varepsilon > 0$ such that for all $x, y \in \Omega$ with $\lvert{x - y}\rvert < \delta$ there is a rectifiable curve $\gamma : [0, l(\gamma)] \to \Omega$ joining $x$ to $y$ and satisfy... |
I have a function $f(x, y) = \csc(x) \csc(x + y) \sin(y) + \cot(y)$, and I know from graphing it that the function is symmetric in $x$ and $y$ such that $f(x, y) = f(y, x)$. However, that fact isn't obvious just by looking at the equation, since the two variables don't seem to be treated the same.
**Can the expressi... |
Can this function be rewritten to make its symmetry more obvious? |
In exponential growth, what does it mean for the derivative to be proportional to the quantity iself? |
I think there might be an error in the official solution to the following question:
[CMI May 23, 2022 BSc entrance exam](https://www.cmi.ac.in/admissions/sample-qp/ugmath2022-23may.pdf), Question A2(7):
>You are asked to take three *distinct* points $1, \omega_1, \omega_2$ in the complex plane such that $|\omega... |
Brainstorm: If $a+c \le w \le b+d$ then $w$ is "some proportion" between $a+c$ and $b+d$. Pick $x,y$ so that the are the same proportion between $a$ and $b$ and $c$ and $d$ respectively. How?
Let $r = \frac {w- (a+c)}{(b+d)-(a+c)}$. Thats the "some proportion". Note: $0 \le r \le 1$.
Let $x = a + r\times (b-a... |
Can $\csc(x)\csc(x + y)\sin(y)+\cot(y)$ be rewritten to make its symmetry in $x$ and $y$ more obvious? |
Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a joint PDF, i.e., $\|f\|_{L^1}=1$, which satisfies $f(x,y)>0$ for all $(x,y)\in \mathbb{R^2}$.
What is a necessary and sufficient condition under which the marginal $$ f^y (y)\equiv\int\limits_{\mathbb{R}} f(x,y)\, dx $$
exists *for all* $y\in \mathbb{R}$?
*Known facts:* ... |
Brainstorm: If $a+c \le w \le b+d$ then $w$ is "some proportion" between $a+c$ and $b+d$. Pick $x,y$ so that the are the same proportion between $a$ and $b$ and $c$ and $d$ respectively. How?
Let $r = \frac {w- (a+c)}{(b+d)-(a+c)}$. Thats the "some proportion". Note: $0 \le r \le 1$.
Let $x = a + r\times (b-a... |
I have an operator defined by $\displaystyle{H(P)=P^{2}=PP}$. I want to formulate the superoperator of this operator, in this case, we let the superoperator be the operator derivative.
By the literature, the “operator derivative” is simply the Jacobian matrix of the function (of operators) where one simply treats t... |
In I.M. James' *Topological and uniform spaces*, the proof for the second half of Proposition 11.5, on p.143, makes me confused. There, he proves a unformizable space is completely regular.
>The outline of his proof is, for a uniform space $X$, a closed subset $H$ and a point $x\in X\setminus H$, he wants to find a... |
I encountered this integral
$$\int_{-2}^2 \frac{\sqrt{4-\epsilon ^2} }{1-\beta \epsilon ^2} e^{-i t \epsilon }\mathrm{d}\epsilon$$
where $\beta\in\mathbb{R}$ and $t\in \mathbb{R}^+$.
Any hint on how to perform this integral?
|
How do I perform this integral? |
I am stuck at simplifying the following integral:
$$
I =
\int_{-\pi/2}^{+\pi/2}
d\phi
\left[
1-M(\phi) \theta^*(\phi)
\right]
$$
where
$
M=
p \cos \phi + q \sin \phi
$.
Here $p,q$ are fixed.
The $\theta^\star$ is defined through
$\cos \theta^*=
\frac{M}{\sqrt{1+M^2}}
$
Can anybody help me to c... |
I'm studying the equation $(2m^2-1)^2=2n^2-1$ ($\ast$), where $m$, $n$ are positive integers. It is known that $m$ can only be 1 or 2 by using Wolfram Alpha. Now I want to prove that result.
That is my attempt: If a prime $p$ | gcd($m$, $n$), ($\ast$) is equivalent to $(2m^2-1)(m^2-1)=(n-m)(n+m)$, $p$ divides RHS bu... |
I'm going through the paper of D. Zagier on Short Proof of Prime Number Theorem. There it says in V that
$\Phi(s)=\int_1^\infty \frac{d\vartheta(x)}{x^s}$ . Can someone please explain in details why that is the case ?
And also how The Analytic Theorem is being applied here to the functions defined as f(t) and g(z) ... |
There is a $N \times N$ diagonal matrix *A* with an infinite cycle with a period of 3, and an infinite cyclic matrix *B*, as follows. Assume $\Phi$ is a matrix of eigenvectors of $A^{-1}B$, where $\Phi^{T}A\Phi=I$, *I* is the $N \times N$ identity matrix.
Currently, a $N \times N$ diagonal matrix *C* has been constr... |
I am currently working on ridge regression, which can be interpreted using Bayesian statistics (DOI: 10.1016/j.electacta.2015.03.123). In particular, I know that the maximum-a-posteriori (MAP) estimate can be computed with its uncertainty. However, is it possible to evaluate the MAP when the 1-norm is used instead of t... |
How to sample with the 1-norm? |
I have to prove a simple property but i have doubts about a property used in the demonstration.
I've to prove that
$\lVert\vec{u}$ x $\vec{v}\rVert^{2} =\lVert \vec{u}\rVert^{2}\lVert \vec{v}\rVert^{2} - (\mathrm{\vec{u}}\cdot\mathrm{\vec{v}})^{2}$
where x is the vectorial product.
I decide to use a formula give... |
$x^n = y^n \Leftrightarrow x = y$ for all real numbers? |
In I.M. James' *Topological and uniform spaces*, the proof for the second half of Proposition 11.5, on p.143, makes me confused. There, he proves a unformizable space is completely regular.
>The outline of his proof is, for a uniform space $X$, a closed subset $H$ and a point $x\in X\setminus H$, he wants to find a... |
I have an operator defined by $\displaystyle{H(P)=P^{2}=PP}$. I want to formulate the superoperator of this operator, in this case, we let the superoperator be the operator derivative.
By the literature, the “operator derivative” is simply the Jacobian matrix of the function (of operators) where one simply treats t... |
>Find the values of the constant $a$ for which the problem $y''(t)+ay(t)=y(t+\pi)$ has a solution with period $2\pi$ which is not identically zero. Also, determine all such solutions
We use a Fourier series solution anstaz, that is $y(t)=\sum_{n\in \mathbb{Z}}c_ne^{int}$. Plugging this into the differential equation... |
In I.M. James' *Topological and uniform spaces*, the proof for the second half of Proposition 11.5, on p.143, makes me confused. There, he proves a unformizable space is completely regular.
>The outline of his proof is, for a uniform space $X$, a closed subset $H$ and a point $x\in X\setminus H$, he wants to find a... |
I have to prove a simple property but i have doubts about a property used in the demonstration.
I've to prove that
$\lVert\vec{u}$ x $\vec{v}\rVert^{2} =\lVert \vec{u}\rVert^{2}\lVert \vec{v}\rVert^{2} - (\mathrm{\vec{u}}\cdot\mathrm{\vec{v}})^{2}$
where x is the vectorial product.
I decide to use a formula give... |
I am reading [a paper](https://www.sciencedirect.com/science/article/pii/0012365X76900352)
On page 223, they used the property that
$\displaystyle \sum_{j=0}^{+\infty}y^j \sum_{n=0}^{+\infty} \binom{n+j}{j} \frac{x^n}{n!n!}A_n(y) = \sum_{n=0}^{+\infty}(1-y)^{-n-1}\frac{x^n}{n!n!}A_n(y)$
But I don't know how to... |
In I.M. James' *Topological and uniform spaces*, the proof for the second half of Proposition 11.5, on p.143, makes me confused. There, he proves a unformizable space is completely regular.
>The outline of his proof is, for a uniform space $X$, a closed subset $H$ and a point $x\in X\setminus H$, he wants to find a... |
Are there infinitely many squares which cannot be written as the sum of a smaller non-zero square and twice a triangular number?
In other words, is the list given at https://oeis.org/A230312 infinite?
And if yes, can you explain/cite its proof please?
Thanks in advance! |
You prob want a subroutine that can take a function as an argument. You might also want a variable step size.
If step size is $\Delta x>0, dy/dx=f(x,y)$ then $y_{i+1}\approx y_i + \Delta x f(x,y), x_{n+1}=x_n + \Delta x$. Then iterate.
That's not the most stable or accurate way of performing an integration.... |
I have the following question before me:
q. Use epsilon − delta definition to show that f(x, y) = [(x^3) + (y^3)] is continuous at (0,0).
i started with:
epsilon > 0 and mod(x^3 + y^3) < epsilon
how do i proceed after this?
|
I'm trying to solve the following ODE with the Laplace transform $y'' + y = \cos \omega t$ with initial conditions $y(0)=y'(0)=0$.
I get $Y(s) = \frac{s}{(s^2 + 1)^2}$ but then, I don't know how to compute the inverse transformation. Decomposing this fraction into partial fractions does not help in this specific cas... |
I was solving the following question:
*Find the CDF of a Binomial Distribution.*
For some reason, I missed the fact that the CDF of a Binomial Distribution is simply
$$F(x) = \sum_{i=0}^x \binom{n}{i}p^x(1-p)^{n-i}$$ and simply did the following
\begin{align*}
P(X \leq x) &= p(0) + p(1) + p(2) + \ldots +... |
Let $f$ be a Schwartz function on $\mathbb{R}^2$ such that $f(x,x)=0$ for all $x \in \mathbb{R}$.
Then, I wonder if the integral
\begin{equation}
\int_{\mathbb{R}^2} \frac{f(x,y)}{\lvert x-y \rvert^{n}}dxdy
\end{equation}
is finite for each $n \in \mathbb{N}$.
I have tried to find a bound for the denominator ... |
For Schwartz $f$ on $\mathbb{R}^2$ with $f(x,x)=0$, is the integral $\int_{\mathbb{R}^2} \frac{f(x,y)}{\lvert x-y \rvert^{n}}dxdy$ finite? |
I assume you mean something like
\begin{equation}
\frac{dy}{dx} = ky(x),
\end{equation}
where the derivative of $y$ with respect to $x$ is proportional to $y$ itself (which here is a function of $x$). Your question is then (I think) why are exponential functions described in this way?
If so, the answer is b... |
I recently came up with a method of constructing the graph of allowed moves in two-player boardgames like tic-tac-toe or chess, and I would like to turn this into a research project with a goal of aligning LLMs to play these games. I solved the problem for tic-tac-toe, and I'm working on extending the solution to chess... |
What you are asking for is the [expected value][1]
of the number of balls in a crate. It's essentially an average of all the possibilities weighted by their probabilities.
The expected value is calculated by multiplying the outcomes by the probability of each outcome.
For example, consider finding the expected v... |
If $J=[0,T]\subset \mathbb{R}$, and $X$ is a Banach space? Then, is $C[J,X]$ a compact metric space with respect to supnorm defined as $\|x\|_{\infty}=\sup \{x(t): t\in[0,T]\}$? |
Compactness of $C[J,X]$? |
If i am given a function \[ f: [0,1] \rightarrow \mathbb{R} \] which is continuous, and for which \(\int_{0}^{1} f(x) \, dx = 1\) , how do i prove the existence of a<b from (0,1) that verify f(a)*f(b)=1? Also, how could i prove that there are an infinite amount of such pairs of values?
I have tried various things, s... |
If $J=[0,T]\subset \mathbb{R}$, and $X$ is a Banach space? Then, is $C[J,X]$ (Banach space of all continuous functions from $J$ to $X$) a compact metric space with respect to supnorm defined as $\|x\|_{\infty}=\sup \{x(t): t\in[0,T]\}$? |
I am seeking advice on methods for uniformly sampling from the space of increasing vectors within the interval [0, 1]. Specifically, I require an efficient algorithm that can handle high-dimensional vectors (e.g., dimensionality around 100), where each vector component satisfies $0 \leq x_1 \leq x_2 \leq \ldots \leq x_... |
If I am given a function $$ f: [0,1] \rightarrow \mathbb{R} $$ which is continuous, and for which $ \int_{0}^{1} f(x) \, dx = 1$ , how do I prove the existence of $a,b\in (0,1)$ with $a<b$ such that $f(a)f(b)=1$? Also, how could I prove that there are infinitely many such pairs?
I have tried various things, such as ... |
I am seeking advice on methods for uniformly sampling from the space of increasing vectors within the interval $[0, 1]$. Specifically, I require an efficient algorithm that can handle high-dimensional vectors (e.g., dimensionality around 100), where each vector component satisfies $0 \leq x_1 \leq x_2 \leq \ldots \leq ... |
>In the café, 4 people are having lunch, whose names are $v_1$, $v_2$, $v_3$, and $v_4$. Some of them know each other. The number of acquaintances of person $v_i$ (who are in the café) is denoted by $d(v_i)$. Prove that among the numbers $d(v_1)$, $d(v_1)$, $d(v_1)$, and $d(v_1)$, there are equal ones.
I guess I hav... |
I found a question in the very end of [this][1] video, which essentially is the following:
[1]: https://www.youtube.com/watch?v=Sv7VseMsOQc&t=5s
> Let $n$ be some natural number. Suppose there is a function $f : \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ such that it satisfies the following 2 properties:
>-... |
Let $J = (0,T)$, $T > 0$, $G = (a,b) \subset \mathbb{R}$ (finite interval), and $f \in C(J;L^2(G))$. I consider the heat equation with zero Dirichlet boundary conditions and with initial value $u_0 \in L^2(G)$,
\begin{align}
\begin{cases}
\frac{\partial u(t,x)}{\partial t} - \frac{\partial^2 u(t,x)}{\partial x^2} &=... |
The original title of my post was
<h3>Why is it so hard for mathematicians to say <i>exactly one</i>?</h3>
The proof under critique seems to be an elaborate way of saying: given a function $s$ let $a=s\left(x\right)$ and $b=s\left(y\right)$ then $x=y\implies a=b.$ Which is the definition of a function.
The w... |
Given a circle $C$ of radius $1$ around the origin, I want to determine the locus of the centers of all circles that touch $C$ and the $x$-axis. This is the red curve in the following [Desmos plot](https://www.desmos.com/calculator/e8upuzdtke), where the blue circe touches $C$ and the $x$-axis:
[![enter image desc... |
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