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Let
$$
f: \mathbb{R}^m \longrightarrow \mathbb{R}
$$
be a smooth, real-valued function. Then it holds that
$$
f(x)- f(0) = x\cdot \nabla f(0)+ x\ \mbox{Hess}f(0)\ x^T + \sum_{|\alpha|=3,\alpha\in\mathbb{N}^m} \frac{x^\alpha}{\alpha!} \int_{0}^{1}(1-t)^2\ {\partial^\alpha} f(tx)\ dt.
$$
Now if we assume that... |
A binary operation is a mapping from $S \times S$ to $S$.To each ordered pair from $S \times S$, there are $|S|$ possibilities of a paring from the codomain. That means if $|S|=n$, then n gets multiplied as many times as there are ordered pairs, so the total is $n^{n^2}$.
There are fewer ordered pairs to work with... |
I am doing this problem: A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
What I tried to do is to look at total permuta... |
I have been following Stanley's [Enumerative combinatorics][2] for the definition of an Eulerian poset. It is defined as follows:
**Definition:** A finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ is eulerian if $\mu_P(s,t)=(-1)^{l(s,t)}$ for all $s\leq t$ in $P$.
Also after checking [online][1], I found ano... |
How are the two definitions of Eulerian posets equivalent? |
For which values of $\alpha$ and $\beta$ is the set S a linear subspace? |
I have been following Stanley's [Enumerative combinatorics][2] for the definition of an Eulerian poset. It is defined as follows:
**Definition:** A finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ is Eulerian if $\mu_P(s,t)=(-1)^{l(s,t)}$ for all $s\leq t$ in $P$.
Also after checking [online][1], I found a... |
I've been looking at the inequality: for $x, y, z > 0$,
$$\left(1-\frac{3}{5}\right)\left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)+\left(1+3\cdot\frac{3}{5}\right)\frac{3\sqrt[3]{xyz}}{x+y+z}\geq4$$
Which proved to be surprisingly resistant to more 'elegant methods' or I'm just oblivious to something obvious.
Bu... |
I want to understand what's meant by "Lie type" in the paper "Character degrees and random walks in finite groups of Lie type" by Liebeck and Shalev, available [here][1]. Several theorems involve statements such as "fix a (possibly twisted) Lie type $L$'' (Thm 1.1).
What is a Lie type? Examples from the paper includ... |
What's a "Lie type"? |
To my knowledge, the Jacobian matrix `A` for the 1D shallow water equations is
```
|0 1 |
|-u^2 + g * h 2*u|
```
that is known to give the eigenvalues such as
```
Lamba1 = u - sqrt(9.81 * h)
Lamba2 = u + sqrt(9.81 * h)
```
Now, let's assume the values as follows:
```
h = 1
u = 2
Lamba1... |
It might be interesting to posit a more formal answer based on an application of the general chain rule. Consider this, if $A = \sum_{kl} A_{kl}E_{kl}$ then
$$ \frac{\partial A}{\partial A_{ij}} = \sum_{kl} \frac{\partial A_{kl}}{\partial A_{ij}}E_{kl} = \sum_{kl} \delta_{ik}\delta_{jl}E_{kl} = E_{ij}. $$
Thus, by th... |
I have a hunch that the following sum can be expressed as a single $G$-function:
$$
S(t,y) = \sum_{k=0}^\infty \frac{t^{2k}}{(2k)!} G_{p+2,q+2}^{m+2,n}\left(y\left|\begin{smallmatrix}\vec{a}-k,-k,-k-\frac{1}{2}\\ \newline
0,-\frac{1}{2},\vec{b}-k-\frac{1}{2}\end{smallmatrix}\right.\right)
$$
where $\vec{a}$ and... |
I'm not sure if the last part is ok. Can you check my proof, please? Any observation wi'll be very valuable.
Let $A=\{(x_1,x_2)\in \Bbb R^2:max(x_1,x_2)<5\}$, then show that $A$ is an open set in $\Bbb R^2$
So we want to show that $\forall y\in A (y\in \Bbb R^2), \exists \varepsilon_y\gt 0: B(y,\varepsilon_y)\su... |
Recently I am studying real analysis, when I read about the introduction part of stein's 《real analysis》, I get really confused about the first example he gave in page xvi. From where we can see Riemann integration has some limitations and need to be refined? Is the problem lies just at the incompleteness of Riemann in... |
I am trying to find the differential equation which implies a smooth path $q:[a,b] \rightarrow \mathbb{R}^n$ subject to $g(\dot{q}(t)) = 0$ is a stationary point of
$$
\int_a^b L(t,q(t), \dot{q}(t)) \ dt
$$
for a certain $L(t,q(t), \dot{q}(t))$. This question and answer [here][1] appears to be exactly what I need, ... |
Let's say, in $d$-dimensional space, we have two hyperplanes: $\mathbf{n_1}\cdot\mathbf{x}+b_1=0$ and $\mathbf{n_2}\cdot\mathbf{x}+b_2=0$ respectively, and their normal directions satisfy $\Vert\mathbf{n_1}\Vert=1$ and $\Vert\mathbf{n_2}\Vert=1$. We define the exterior as the intersection of two positive hyperplane reg... |
Consider the set $V$ consisting of all infinite sequences $a = (a_0, a_1, \ldots)$ where each $a_i \in \mathbb{R}$ and $a_0 \neq 0$. How can we demonstrate that the operation $(a + b) + c = a + (b + c)$ is associative for all $a, b, c \in V$?
$a + b = (a_0b_0, a_0b_1 + a_1b_0, \ldots)$ and it is equal to: $(a + b)_... |
The general linear group $\operatorname{GL}_n(\mathbb R)$ (as a Lie group) acts smoothly on the real projective space $\mathbb R \mathbb P^n$, via $A \cdot [x] := [Ax]$. By $[x]$ here, I mean the $1$-dimensional subspace of $\mathbb R^{n+1}$ spanned by $x \in \mathbb R^{n+1} \backslash\{0\}$. This action is transitive,... |
I am trying to find the differential equation which implies a smooth path $q:[a,b] \rightarrow \mathbb{R}^n$ subject to $|\dot{q}(t)| = 1$ (i.e. $q$ has unit speed) is a stationary point of
$$
\int_a^b L(t,q(t), \dot{q}(t)) \ dt
$$
for a certain $L(t,q(t), \dot{q}(t))$. This question and answer [here][1] appears to... |
Find the sum of the following sequence: $\frac{7}{10} *\sum_{n=1}^{\infty}{n*(3/10)^n}$.
The original question is:
You have a 30% chance to gain an extra "reroll" after every reroll. What is the expected number of reroll's after one reroll? For context, this is from the game TFT and is the augment "Golden Ticket".... |
The general linear group $\operatorname{GL}_{n+1}(\mathbb R)$ (as a Lie group) acts smoothly on the real projective space $\mathbb R \mathbb P^n$, via $A \cdot [x] := [Ax]$. By $[x]$ here, I mean the $1$-dimensional subspace of $\mathbb R^{n+1}$ spanned by $x \in \mathbb R^{n+1} \backslash\{0\}$. This action is transit... |
Let $\mathbb{H}_+\subseteq \mathbb{C}$ be the upper half plane,and $\bar{\mathbb{H}}_+=\mathbb{H}_+\cup \mathbb{R}$. Then consider the following equation:
$$z=-\frac{1}{m}-m+\frac{a^2}{ca^2m+1}.$$
Show that for each fixed $a,c\in \mathbb{R}_+$ and $z\in \mathbb{H}_+$, there is a unique $m=m(z)\in \mathbb{H}_+$ and ... |
I'm trying to solve this exercise but I'm stuck. The exercise is the following: \
Let $\mathcal{B}(\mathbb{R})$ and $\mathcal{B}(\mathbb{R}^2)$ be the borel $\sigma$ algebra on $\mathbb{R}$ and $\mathbb{R}^2$ respectively. We denote $\lambda$ the Lebesgue measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ and by $\lam... |
[enter image description here][1]
[1]: https://i.stack.imgur.com/CjCp9.jpg
This proof for the proposition stated is from Marsden's *Elementary Classical Analysis*. My question here is that in this proof, we assume **F** to be a complete ordered field (which also means that **F** follows Monotone Sequence Prop... |
For any positive real numbers $a,b,c$ then prove $$(a+b+c-3)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-3\right)+abc+\frac{1}{abc}\ge 2.$$
I've try to assume $a+b+c\ge 3.\quad(1)$
By AM-GM $$abc+\frac{1}{abc}\ge 2$$ and we need to prove $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-3\ge 0$$
By Cauchy-Schwarz $$\frac{1}{a}+\... |
I'm studying Lie derivatives. So i want to derive right Jacobian $\bf{J}_\it {r}$ on $SO(3)$ as below:
$$
\bf{J}_\it{r} \rm{=}\bf I+\frac{\rm1-\cos\rm\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymbol{\phi}\Vert^\rm2}[\boldsymbol\phi]_\times+\frac{\Vert\boldsymbol{\phi}\Vert-\sin\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymb... |
What does it mean to say that similar matrices represent the same linear map under (possibly) different bases? |
In the bisection method, we look at the first derivative and then depending on whether it is positive or negative replace the boundary point a or b. I tried to implement it in python and it worked (at least for the simple examples I replicated). But it seems that I can arbitrarily change a=c and b=c positions and the c... |
Why can I replace both sides of the interval in the bisection method? |
I'm studying Lie derivatives. So i want to derive right Jacobian $\bf{J}_\it {r}$ on $SO(3)$ as below:
$$
\bf{J}_\it{r} \rm{=}\bf I+\frac{\rm1-\cos\rm\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymbol{\phi}\Vert^\rm2}[\boldsymbol\phi]_\times+\frac{\Vert\boldsymbol{\phi}\Vert-\sin\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymb... |
Suppose $\Gamma\vdash\lnot (A\to B)$. How do I show that both $\Gamma\vdash A$ and $\Gamma\vdash \lnot B$ using sequent calculus inference rules?
Here is my attempt to obtain $\Gamma\vdash A$: From $\Gamma\vdash\lnot (A\to B)$, I can obtain $A\to B,\Gamma\vdash$ using $\lnot$L rule and the cut rule using process out... |
> **Proposition.** Complete ordered fields are Archimedean.
>
>**Proof.** Let $\mathbb{F}$ be a complete ordered field and consider $x\in\mathbb{F}$. We must show that there is an integer $N$ such that $x<N.$ If not, the monotone sequence $1,2,3,\ldots$ is bounded above by $x$ and converges by the monotone sequence p... |
In the bisection method for optimization, we look at the first derivative and then depending on whether it is positive or negative replace the boundary point a or b. I tried to implement it in python and it worked (at least for the simple examples I replicated). But it seems that I can arbitrarily change a=c and b=c po... |
Why can I replace both sides of the interval in the bisection method for optimization? |
In the bisection method for optimization, we look at the first derivative and then depending on whether it is positive or negative replace the boundary point a or b. I tried to implement it in python and it worked (at least for the simple examples I replicated). But it seems that I can arbitrarily change a=c and b=c po... |
Consider the 8-dimensional euclidean space $\mathbb R^8$. I denote a point in $\mathbb R^8$ by
$$(a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$$
I want to sample a point from a convex subset $S$ of $\mathbb R^8$, where $S$ is defined by five equations and one inequality (it is indeed convex)
$$
\begin{cases}
a_1+a_2+a_3... |
I'm studying Lie derivatives. So i want to derive right Jacobian $\bf{J}_\it {r}$ on $SO(3)$ as below:
$$
\bf{J}_\it{r} \rm{=}\bf I+\frac{\rm1-\cos\rm\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymbol{\phi}\Vert^\rm2}[\boldsymbol\phi]_\times+\frac{\Vert\boldsymbol{\phi}\Vert-\sin\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymb... |
i have this XOR equations:
(x1 ⊕ x2 =1)
(x2 ⊕ x3 =1)
(x3 ⊕ x4 =1)
(x4 ⊕ x5 =1)
(x5 ⊕ x1 =1)
i have a few questions about it.
1. what is the rank of this equations matrix and how to calculate it ? iam not sure how to handle this xor equations.
2. i know that thi... |
I'm studying Lie derivatives. So i want to derive right Jacobian $\bf{J}_\it {r}$ on $SO(3)$ as below:
$$
\bf{J}_\it{r} \rm{=}\bf I+\frac{\rm1-\cos\rm\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymbol{\phi}\Vert^\rm2}[\boldsymbol\phi]_\times+\frac{\Vert\boldsymbol{\phi}\Vert-\sin\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymb... |
I'm studying Lie derivatives. So i want to derive right Jacobian $\bf{J}_\it {r}$ on $SO(3)$ as below:
$$
\bf{J}_\it{r} \rm{=}\bf I+\frac{\rm1-\cos\rm\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymbol{\phi}\Vert^\rm2}[\boldsymbol\phi]_\times+\frac{\Vert\boldsymbol{\phi}\Vert-\sin\Vert\boldsymbol{\phi}\Vert}{\Vert\boldsymb... |
Your equation seems incorrect. The number of rolls is always one more than the number of rerolls. So it should be $(n+1)$ in the GP, not $(n)$, where n goes from $1$ to infinity.
The correct answer is $\textbf{$\frac{10}7$}$. Let's solve it using your method of GP and using an intuitive faster method as well.
> M... |
Simpler version of the Josephus problem. Circle of n people, every second is removed
let f(n) represent the last one from a circle of n people
I've proved f(1)=1; f(2n)= 2f(n) - 1; f(2n+1) = 2f(n) + 1
Hypothesis: f(2^m + l)=1+2l, 2^m > l
Can the proof be: 2^m + l is either 2n or 2n+1?
when its even:
f... |
Your equation seems incorrect. The number of rolls is always one more than the number of rerolls. So it should be $(n+1)$ in the summation of GP, not $(n)$, where n goes from $0$ to infinity.
The correct answer is $\textbf{$\frac{10}7$}$. Let's solve it using your method of GP and using an intuitive faster method as... |
Let
$$
f: \mathbb{R}^m \longrightarrow \mathbb{R}
$$
be a smooth, real-valued function. Then it holds that
$$
f(x)- f(0) = x\cdot \nabla f(0)+ x\ \mbox{Hess}f(0)\ x^T + \sum_{|\alpha|=3,\alpha\in\mathbb{N}^m} \frac{x^\alpha}{\alpha!} \int_{0}^{1}(1-t)^2\ {\partial^\alpha} f(tx)\ dt.
$$
Now if we assume that... |
I'm trying to study if the following function has continuous partial derivatives at the origin:
$$
f(x, y) =
\begin{cases}
0 & \text{if } (x, y) = (0, 0) \\
\frac{x^4y^3}{x^8 + y^4} & \text{if } (x, y) \neq (0, 0)
\end{cases}
$$
I proved $f$ is continuous at the origin, and I also proved its partial deri... |
Let $T$ be the right shift operator on $\ell^2$ defined by $T(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots)$.
The numerical radius of $T$ is defined by $w(T)=\sup\{|\langle Th,h\rangle:\, \|h\|=1\}$. It is well known that $r(T)\le w(T)\le \lVert T\rVert$ where $r(T)$ denotes the spectral radius of $T$. Since $T^n$ is isometry ... |
How to compute the numerical radius of the right shift operator? |
I am trying to find the differential equation which implies a smooth path $q:[a,b] \rightarrow \mathbb{R}^n$ subject to $|\dot{q}(t)| = 1$ (i.e. $q$ has unit speed) is a stationary point of
$$
\int_a^b L(t,q(t), \dot{q}(t)) \ dt
$$
for a certain $L(t,q(t), \dot{q}(t))$. This question and answer [here][1] appears to... |
Let $T$ be the right shift operator on $\ell^2$ defined by $T(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots)$.
The numerical radius of $T$ is defined by $w(T)=\sup\{|\langle Th,h\rangle|:\, \|h\|=1\}$. It is well known that $r(T)\le w(T)\le \lVert T\rVert$ where $r(T)$ denotes the spectral radius of $T$. Since $T^n$ is isometry... |
i have this XOR equations:
(x1 ⊕ x2 =1)
(x2 ⊕ x3 =1)
(x3 ⊕ x4 =1)
(x4 ⊕ x5 =1)
(x5 ⊕ x1 =1)
i want to show that i can transform this unsolvable matrix into antoher solveable matrix , for example if x1 ⊕ x2 =0 then the new matrix is solveable. but before i can do... |
Simpler version of the Josephus problem. Circle of $n$ people, every second is removed
let $f(n)$ represent the last one from a circle of $n$ people
I've proved $f(1)=1$; $f(2n)= 2f(n) - 1$; $f(2n+1) = 2f(n) + 1$
Hypothesis: $f(2^m + l)=1+2l$, $2^m > l$
Can the proof be: $2^m + l$ is either $2n$ or $2n+1$?
... |
Let $E_n$ be the elliptic curve defined by $y^2=x^3-n^2x$, and let $\mathbb{F}_{p^r}$ be a finite field of $p^r$ elements, where $p$ is prime and divides $n$. How can I determine how many points the curve has on such field? I found this problem on the book "Introduction to elliptic curves and modular forms" chapter II:... |
I'm attacking to solve the graph isomorphism problem using the adjacent matrices.
I would appreciate it if someone could show me whether following conjecture works or not. I expect this conjecture to work.
Let $A$ and $B$ be $n \times n$ symmetric matrices. Let there is no permutation matrix $P$ such that $A=P^t... |
Let $X$ be a random variable with mgf $M_X(t)=a+be^{2t}$.Given that mean of the random variable $X$ is $1.5$. Then find the expected value of the random variable $2^X.$
**My attempt:**
We know that $E(X^r)=\frac{d^r}{dt^r}M_X(t) \rvert_{t=0}$.
So, we have that $E(X) = 2b = 1.5$.
This means that $b=0.75$.
... |
Let $X$ be a random variable with mgf $M_X(t)=a+be^{2t}$.Given that mean of the random variable $X$ is $1.5$. Then find the expected value of the random variable $2^X.$
**My attempt:**
We know that $E(X^r)=\frac{d^r}{dt^r}M_X(t) \rvert_{t=0}$.
So, we have that $E(X) = 2b = 1.5$.
This means that $b=0.75$.
... |
> Consider $n\geq 4$ points arranged clockwise on a circle. Label them $0,\ldots, n-1$. Let $X_1,X_2,\ldots$ be iid Rademacher and define $S_0=0$, $S_p=(\sum_{k=1}^p X_k) \text{ mod } n$. What is the probability that $(S_p)_{p\geq 1}$ visits all of $\{0,2,3,\ldots, n-1\}$ before it ever visits $1$ ?
Simulations (con... |
While attempting to express the magnetic field induced by a single coil of current (at any point in space, not just on the coil's axis), I tried visualising the set of the infinitesimal contributions $\text{d}\mathbf{B}$ given by the [Biot-Savart law][1] at specific points. The goal is to study the shapes and curves pr... |
Does this family of curves appearing in the magnetic field of a coil have a name? |
One can conclude from the truth table below, with $y_{pq}:=x_p\oplus x_q$. Do you agree ?
$$\begin{array}&&x_1&x_2&x_3&x_4&x_5&y_{12}&y_{23}&y_{34}&y_{45}&y_{51}\\
&0&0&0&0&0&0&0&0&0&0\\
&0&0&0&0&1&0&0&0&1&1\\
&0&0&0&1&0&0&0&1&1&0\\
&0&0&0&1&1&0&0&1&0&1\\
&0&0&1&0&0&0&1&1&0&0\\
&0&0&1&0&1&0&1&1&1&1\\
&0&0&1&... |
I am trying to find the differential equation which implies a smooth path $q:[a,b] \rightarrow \mathbb{R}^n$ subject to $|\dot{q}(t)| = 1$ (i.e. $q$ has unit speed) is a stationary point of
$$
\int_a^b L(t,q(t), \dot{q}(t)) \ dt
$$
for a certain $L(t,q(t), \dot{q}(t))$. This question and answer [here][1] appears to... |
I have a question about a statement I came across which I'd be happy to understand more.
On $L^2(0,1)$, we can consider two self-adjoint operators. The first operator $H_0$ acts as $H_0f=-f''$, with domain consisting of all $H^1$ functions satisfying $f(0)=0$ and $f'(1)=f(1)$. The second operator $H_1$ acts as $H_1f... |
How is changing the boundary conditions a finite rank perturbation? |
Are there any prime solutions to p^q - q^p = 3 ?
(For the prime numbers p and q)
If so what are they?
This problem is meant to be solved with modular arithmetic.
I tried proving that firstly they must be of different parity (congruence mod 2) since two numbers must be of differnt parity to equal an odd differ... |
Let's consider the following deterministic constrained optimisation problem, where $c_t$ is the control, and $x_t$ and $y_t$ are the state variables:
\begin{align}
J_t = \inf_{c_t} \ &\int_t^\infty e^{-\delta (u - t)} \left( x_u^2 + \lambda y_u^2 \right) du \\
s.t. \ &c_t \geq - \xi
\end{align}
where:
\begin{alig... |
Are there any prime solutions to $p^q - q^p = 3$?
(For the prime numbers $p$ and $q$)
If so what are they?
This problem is meant to be solved with modular arithmetic.
I tried proving that firstly they must be of different parity (congruence $\bmod 2$) since two numbers must be of different parity to equal an ... |
Consider the 8-dimensional euclidean space $\mathbb R^8$. I denote a point in $\mathbb R^8$ by
$$(a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$$
I want to sample a point from a convex subset $S$ of $\mathbb R^8$, where $S$ is defined by five equations and one inequality (it is indeed convex)
$$
\begin{cases}
a_1+a_2+a_3... |
[Here][1], the Ramanujan Master Theorem is proven in the following way:
$$ \int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} \lambda(n)x^n = \int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} E^n\lambda(0)x^n = \int_0^\infty x^{s-1}e^{-Ex}dx \;\lambda(0) = \frac{\Gamma(s)}{E^s}\lambda(0) = \Gamma(s)\lambda(-s).$$
Th... |
Can the vertex of a polygon be a convex combination of two other points? |
i'm trying to solve the equation $x^3-2x^2-x+1 = 0 mod(7^3) $ . i tried first to find a solution $mod 7$ , i got that $3$ is a solution , denote $x =$ $3+7t $ . now find a solution $mod 7^2 $ , i got that for $t= 37 $ , there is a solution , thus $x =$ $3+7 *37 (mod 49 ) $ = $ 17 (mod... |
For example: how exactly do $$\begin{pmatrix} 0 & -1 \\\ 1 &0\end{pmatrix}$$ and $$\begin{pmatrix} 2 & -5/3 \\\ 3 & -2\end{pmatrix}$$ represent the same linear map?
I do understand how to express the same vector under different bases.
I believe my question is the same as https://math.stackexchange.com/questions/2... |
I am reading "Analysis on Manifolds" by James R. Munkres.
>In $\mathbb{R}^3$, a frame $(a_1,a_2,a_3)$ is right-handed if curling the fingers of one's right hand in the direction from $a_1$ to $a_2$ makes one's thumb point in the direction of $a_3$. See Figure 20.2.
One way to justify this statement is to note tha... |
Newton's 1676 formulas are
$$
{}_2F_1\left(\genfrac{}{}{0pt}{0}{{1}/{2}+a,{1}/{2}-a}{{1}/{2}};\sin^2x\right)\cos x=\cos(2ax),\tag{1}
$$
$$
{}_2F_1\left(\genfrac{}{}{0pt}{0}{a,-a}{{1}/{2}};\sin^2x\right)=\cos(2ax),\tag{2}
$$
$$
{}_2F_1\left(\genfrac{}{}{0pt}{0}{{1}/{2}+a,{1}/{2}-a}{{3}/{2}};\sin^2x\right)\sin x=... |
I recently stumbled across what looks like a very interesting [paper][1] concerning a simplex-based bisection algorithm for multidimensional optimization.
The authors provide results from their own MATLAB code, which is apparently not publicly available.
I have not heard about this method on SE - has it been veri... |
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... |
I'm trying to solve the equation $x^3-2x^2-x+1 = 0 \pmod {7^3} $ . I tried first to find a solution $\mod 7$ , I got that $3$ is a solution , denote $x =$ $3+7t $ . now find a solution $\mod 7^2 $ , I got that for $t= 37 $ , there is a solution , thus $x =$ $3+7\times37 \pmod {49} $ $... |
I'm trying to study if the following function has continuous partial derivatives at the origin:
$$f(x, y) = \frac{x^4y^3}{x^8 + y^4} \qquad \quad (x, y) \neq (0, 0)$$
(and $f(x, y) = 0$ for $(x, y) \neq (0, 0)$).
I proved $f$ is continuous at the origin, and I also proved its partial derivatives exist at the origi... |
The set $R$ of rational points on the curve $y^2=x^2+1$ can be viewed as a group under the following operation: if $X_1=(x_1,y_1),\ X_2=(x_2,y_2)\in R$, define $X_1*X_2:=(x_1y_2+x_2y_1,x_1x_2+y_1y_2)$. Does this group finitely generated, i.e. does the Mordell-Weil theorem holds for this case? I don't think so by the fo... |
I'm trying to study if the following function has continuous partial derivatives at the origin:
$$f(x, y) = \begin{cases}\frac{x^4y^3}{x^8 + y^4} & (x, y) \neq (0, 0) \\\\ \quad 0 & (x, y) = (0, 0)\end{cases}$$
I proved $f$ is continuous at the origin, and I also proved its partial derivatives exist at the origin.... |
The theater is configured so that in any row there is a wall, 6 seats, an aisle, 6 more seats, and then a wall. In total there are 2* 6! arrangements = 1400.
"How many ways can they sit if Jane and Mary refuse to sit next to each other?"
I have tried to approach this in 3 different ways. First that the number of a... |
> Let $m; n \in N$ with $m > n$. Then there does not exist an injection from $N_m$ into $N_n$.
We will do this by induction on $n$.
Let's formulate the inductive hypothesis $P(k)$ is if $f : N_m \to N_k$ and $m > k$ then $f$ is not an injection.
Base case: $k = 1$: then $f(1)=f(2)=...=f(m)=1$. So, $f$ is not an ... |
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... |
This is a question given in the book Mathematics in Machine Learning.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/KItAZ.png
I know how to solve the first part a. of the question, finding the kernel and image of the transformation matrix.
But I don't understand the second part of t... |
The correspondence between Grothendieck and Serre during the period 1955-1987 illustrates how Grothendieck's mathematical work shifted from functional analysis to algebraic geometry. The correspondence edited by the American Mathematical Society has the special flavour of being bilingual.
**Added**: Grothendieck's i... |
Do any whole number solutions exist for a^2 + b^3 = 2^2023
And if so, would it be possible to give an example of one?
This problem is meant to be solved with modular-arithmetic and i could not even get started.
Guessing that you must find the right modulus such that no squares or cubes can be added to reach the ... |
Let's consider the following deterministic constrained optimisation problem, where $c_t$ is the control, and $x_t$ and $y_t$ are the state variables:
\begin{align}
J_t = \inf_{c_t} \ &\int_0^\infty e^{-\delta u} \left( x_u^2 + \lambda y_u^2 \right) du \\
s.t. \ &c_t \geq - \xi
\end{align}
where:
\begin{align}
dx... |
I would try a logarithmic approach. What is $$L:=\lim_{x \to 0}\ln(x^x)=\lim_{x \to 0} x\ln(x)?$$
Take that value and evaluate $e^L$. |
Define a 3D axis in space, at (0, 0, 0) with an X, Y and Z axis visually, and a point at (1, 1, 1) is in one of eight octant sections the axis form. Let’s trade up 1 for +# as being any positive value, and –# as any negative value. Starting with (+#, +#, +#), if we clockwise rotate all values 45 degrees, we and up at (... |
Let's consider the following deterministic constrained optimisation problem, where $c(t)$ is the control, and $x(t)$ and $y(t)$ are the state variables:
\begin{align}
J(t) = \inf_{c(t)} \ &\int_0^\infty e^{-\delta u} \left( x(u)^2 + \lambda y(u)^2 \right) du \\
s.t. \ &c(t) \geq - \xi
\end{align}
where the dynamic... |
Define a 3D axis in space, at (0, 0, 0) with an X, Y and Z axis visually, and a point at (1, 1, 1) is in one of eight octant sections the axis form. Let’s trade up 1 for +# as being any positive value, and –# as any negative value. Starting with (+#, +#, +#), if we clockwise rotate all values 45 degrees, we and up at (... |
Do any whole number solutions exist for $a^2 + b^3 = 2^{2023}$?
And if so, would it be possible to give an example of one?
This problem is meant to be solved with modular-arithmetic and I could not even get started.
Guessing that you must find the right modulus such that no squares or cubes can be added to reach... |
>What is $(-1)^{2r-1}$?
$(-1)^{2r-1}=\dfrac{(-1)^{2r}}{(-1)}=\dfrac{[(-1)^2]^r}{(-1)}=\dfrac{1^r}{-1}=\dfrac{1}{-1}=-1$
Would this be correct? It's quite confusing. Also is it mathematically legal to have a negative sign in the denominator only? Is there a difference being in the denominator vs. the numerator? Th... |
What is $(-1)^{2r-1}$? |
I recently stumbled across what looks like a very interesting [paper][1] concerning a simplex-based bisection algorithm for multidimensional optimization.
The authors provide results from their own MATLAB code, which is apparently not publicly available.
I have not heard about this method on SE - has it been veri... |
Let $A_n$ be an $n\!\times\!n$ matrix whose inverse is known and let $A_{n+1}$ be a regular matrix obtained by adding an $(n\!+\!1)$'st row and an $(n\!+\!1)$'st column to $A$, so it is "augmented" both on the left and at the bottom. Can knowing $A_n^{-1}$ help to make inversion of $A_{n+1}$ more efficient than it woul... |
i have this XOR equations:
(x1 ⊕ x2 =1)
(x2 ⊕ x3 =1)
(x3 ⊕ x4 =1)
(x4 ⊕ x5 =1)
(x5 ⊕ x1 =1)
i want to show that i can transform this unsolvable matrix into antoher solveable matrix , for example if x1 ⊕ x2 =0 then the new matrix is solveable. but before i can do... |
Let $(X,\mathcal{A})$ be a measurable space and $(\mu_n)$ be a sequence of measures which satisfies $\mu_n(X)=1$. I want to show that the function defined as
$$
\nu(E)=\sum_{n=1}^{\infty}2^{-n}\mu_n(E)
$$
is a measure with $\nu(X)=1$.
In the proof of countable additivity part: Let $(E_j)$ be a sequence of pairw... |
Let $A_n$ be an $n\!\times\!n$ matrix whose inverse is known and let $A_{n+1}$ be a regular $(n\!+\!1)\!\times\!(n\!+\!1)$ matrix obtained by adding an $(n\!+\!1)$'st row and an $(n\!+\!1)$'st column to $A$, so it is "augmented" both on the left and at the bottom. Can knowing $A_n^{-1}$ help to make inversion of $A_{n+... |
Define a 3D axis in space, at (0, 0, 0) with an X, Y and Z axis visually, and a point at (1, 1, 1) is in one of eight octant sections the axis form. Let’s trade up 1 for +# as being any positive value, and –# as any negative value. Starting with (+#, +#, +#), if we clockwise rotate all values 45 degrees, we and up at (... |
Let $A_n$ be an $n\!\times\!n$ matrix whose inverse is known and let $A_{n+1}$ be a regular $(n\!+\!1)\!\times\!(n\!+\!1)$ matrix obtained by adding an $(n\!+\!1)$'st row and an $(n\!+\!1)$'st column to $A$, so it is "augmented" both on the right and at the bottom. Can knowing $A_n^{-1}$ help to make inversion of $A_{n... |
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