instruction stringlengths 12 30k |
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I attached the image below for reference. I am trying to color the figure using three colors and find the number of distinguishable colorings.
The long sides of the figure are one side and not two short sides. We do not count reflections as indistinguishable
I was trying to solve it for using casework: When the... |
Random variables on $\mathbb{R}$ with finite expectation form an $\mathbb{R}$-linear space with respect to addition of random variables and multiplication by elements of $\mathbb{R}$. Is there an understandable way to characterize all linear functionals on random variables, i.e. $\mathbb{R}$-linear functions that map r... |
> How many compositions of $n$ are there where each part is at least $m$?
**My attempt:**
So far I have found the generating function and have that the desired number of compositions is
$$
[x^{n}]\frac{1-x}{1-x-x^{m}}.
$$
But I am unsure how to proceed, I have made a few attempts to transform this (for exa... |
**What set of points on the $256 \times 256$ integer grid maximizes the number of vertices in its convex hull?**
For some context, I was given an assignment to test algorithms that find the vertices of the convex hull of a set of points in $2D$. One natural thing to test is when we set the input set such that all th... |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
I'm trying to study if the following function has continuous partial derivatives at the origin:
$$f(x, y) = \frac{x^4y^3}{x^8 + y^4} \qquad \quad (x, y) \neq (0, 0)$$
(and $f(x, y) = 0$ for $(x, y) \neq (0, 0)$).
I proved $f$ is continuous at the origin, and I also proved its partial derivatives exist at the origi... |
I'm currently working on a physics exercise where I'm told:
Given two functions $f$ and $g$ which are both continuously differentiable and depend on $x$ and $y$. Show that the following relation holds
$$
\left(
\frac{\partial f}{\partial x}
\right)_g =\left(
\frac{\partial f}{\partial x}
... |
This is **not an exercise** found in a textbook, just a question that arose while I was working on a larger project. The question may sound vague, so let me know If I have skipped on some information. \
So $f, g : \mathbb{R}\rightarrow\mathbb{R}$ are continuous in an interval that contains zero.
$$\lim\limits_{x\righ... |
I am curious what the object you obtain is when you cut the $d$-dimensional flat torus $\mathbb{T}^d :=\left(\mathbb{R} / \mathbb{Z}\right)^d$ along certain hyper-planes. Specifically, if you identify it with $[0,1)^d$ and consider a typical element $x=(x_1,\dots,x_d)$, I would like to cut it along the planes $x_i=x_j,... |
**Task:** Let p and q be prime numbers with $p \equiv 3 \ (mod \ 4)$ and $q=2p+1$. Prove that $2^{p} - 1$ is prime only when $p=3$.
This question has been asked on this forum for a long time, but I have some questions about the solution that was proposed. That's why I asked a new question, because I'm not sure if s... |
The version of Hall's Marriage Theorem I'd like to consider is the following:
---
**Theorem** Let $G$ be a finite bipartite graph with bipartition $\{X,Y\}$ and edge set $E$, so that we can view $E$ as a relation from $X$ to $Y$. Suppose that, for all $S \subseteq X$,
$$\lvert \{y \in Y : \exists x \in S (xEy)\}... |
**Task:** Let p and q be prime numbers with $p \equiv 3 \ \pmod 4$ and $q=2p+1$. Prove that $2^{p} - 1$ is prime only when $p=3$.
This question has been asked on this forum for a long time, but I have some questions about the solution that was proposed. That's why I asked a new question, because I'm not sure if som... |
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... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/NeREW.png
Hi, this is one of the questions from my Discrete Mathematics exam that I got wrong. I believe I answered 2 since I did not see the "not" in the question.
Which of the following statements is true?
1: The greatest lower bound... |
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... |
In problems like this I find it useful to use a homogenized form of polar coordinates: Let $x^4=r\cos\theta$ and $y^2=r\sin\theta$, with $0\le\theta\le\pi/2$. Then
$$\left|\frac{x^3y^3(y^4-x^8)}{(y^4+x^8)^2}\right| = \frac{r^{3/4}r^{3/2}r^2|\cos 2\theta||\cos\theta|^{3/4}||\sin\theta|^{3/2}}{r^4} \le r^{3/4+3/2-2} = r... |
Denote $(A_n,B_n,C_n)$ the number of sequences for a given $n$ that begins with $0$, $1$ or $2$. We observe that:
$$\begin{align}
&A_{n+1} = A_n + B_n\\
&B_{n+1} = A_n + B_n+C_n\tag{1}\\
&C_{n+1} = B_n +C_n\\
\end{align}$$
Write $(1)$ in matrix form:
$$\underbrace{\pmatrix{A_{n+1}\\B_{n+1}\\C_{n+1}}}_{:=X_{n+1}... |
Let $M$ be a smooth manifold and $\nabla$ the Levi-Civita connection. Now, I am a bit puzzled by a serious notational ambiguity, namely for the second covariant derivative.
To explain myself, let us consider a vector field $v\in\Gamma(TM)$. Then, the notation $\nabla_{a}\nabla_{b}$ can mean two things:
1. Some a... |
Denote $(A_n,B_n,C_n)$ the number of sequences for a given $n$ that begins with $0$, $1$ or $2$. We observe that:
$$\begin{align}
&A_{n+1} = A_n + B_n\\
&B_{n+1} = A_n + B_n+C_n\tag{1}\\
&C_{n+1} = B_n +C_n\\
\end{align}$$
Write $(1)$ in matrix form:
$$\underbrace{\pmatrix{A_{n+1}\\B_{n+1}\\C_{n+1}}}_{:=X_{n+1}... |
In problems like this I find it useful to use a homogenized form of polar coordinates: Let $x^4=r\cos\theta$ and $y^2=r\sin\theta$, with $0\le\theta\le\pi/2$. Then
$$\left|\frac{x^3y^3(y^4-x^8)}{(y^4+x^8)^2}\right| = \frac{r^{3/4}r^{3/2}r^2|\cos 2\theta||\cos\theta|^{3/4}|\sin\theta|^{3/2}}{r^4} \le r^{3/4+3/2-2} = r^... |
Let $M$ be a smooth manifold and $\nabla$ the Levi-Civita connection. Now, I am a bit puzzled by a serious notational ambiguity, namely for the second covariant derivative.
To explain myself, let us consider a vector field $v\in\Gamma(TM)$. Then, the notation $\nabla_{a}\nabla_{b}$ can mean two things:
1. Some a... |
$\def\calS{\mathcal{S}}$
$\def\calW{\mathcal{W}}$
$\def\qty#1{\left( #1 \right)}$
$\def\calX{\mathcal{X}}$
$\def\calY{\mathcal{Y}}$
$\def\abs#1{\lvert #1 \rvert}$
$\def\pdv#1#2{\frac{\partial #1}{\partial #2}}$
$\def\expect{\mathbb{E}}$
$\def\indic{\mathbb{1}}$
In their book "Understanding Machine Leaning" the... |
In definitions of linear independence that I've learned earlier in my class, if a matrix has a free variable it must be linearly dependent. All of the theorems in that section of the textbook state that.
Now that we've moved on to bases, it doesn't seem to be the case anymore and I'm very confused as to why.
If ... |
How can a basis have free variables if it also has to be linearly independent? |
Let $M$ be a smooth manifold and $\nabla$ the Levi-Civita connection. Now, I am a bit puzzled by a serious notational ambiguity, namely for the second covariant derivative.
To explain myself, let us consider a vector field $v\in\Gamma(TM)$. Then, the notation $\nabla_{a}\nabla_{b}$ can mean two things:
1. Some a... |
Denote $(A_n,B_n,C_n)$ the number of sequences for a given $n$ that begins with $0$, $1$ or $2$. We observe that:
$$\begin{align}
&A_{n+1} = A_n + B_n\\
&B_{n+1} = A_n + B_n+C_n\tag{1}\\
&C_{n+1} = B_n +C_n\\
\end{align}$$
Write $(1)$ in matrix form:
$$\underbrace{\pmatrix{A_{n+1}\\B_{n+1}\\C_{n+1}}}_{:=X_{n+1}... |
Let $K \subset \mathbb{R}^2$ is closed convex set. Is $f(x) = \rho(x, K) = \min\limits_{k \in K}\rho(x,k)$ is also convex function?
I know that without the convexity assumption, this is not true. I can only prove reverse statement: if function $f$ is convex then $K$ is convex: $$0≤f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y).$$ And... |
Let $M$ be a smooth manifold and $\nabla$ the Levi-Civita connection. Now, I am a bit puzzled by a serious notational ambiguity, namely for the second covariant derivative.
To explain myself, let us consider a vector field $v\in\Gamma(TM)$. Then, the notation $\nabla_{a}\nabla_{b}$ can mean two things:
1. Some a... |
In Complex analysis by Lars Ahlfors 1.1.5.4
>4. Show that there are complex numbers $z$ satisfying
$|z - a| + |z + a| = 2|c|$
if and only if $|a| \le |c|$- If this condition is fulfilled, what are the smallest
and largest values of |z|?
The first two questions are easy if $|c|>|a|$, $z=\frac{|c|a}{|a|}$ wi... |
The step $\boldsymbol{\beta}^{t+1}$ in solving an optimization problem
is given below.
$$\boldsymbol{\beta}^{t+1}=\mathrm{argmin}_{\boldsymbol{\beta}}\frac{1}{2}\left\Vert \mathbf{u}^{t}-\boldsymbol{\alpha}^{t+1}+\mathbf{r}-\mathbf{A}\boldsymbol{\beta}\right\Vert _{2}^{2}+\frac{1}{\rho}\left\Vert \boldsymbol{\beta}... |
This comes from Usa Harvard-MIT Mathematics Tournament (i don't know from witch year)
>a particle moves along the x-axis such that its velocity at the x-position is given by the formula v(x) = 2 + sin(x)
What is its acceleration at x = π/6 ?
(answer is 2,2)
how do i do this if I don't know velocity as a functio... |
This problem is from the 2018 admission test to Scuola Normale Superiore in Pisa, Italy. I am having trouble with the second question.
**Problem:** Let $V^1, V^2: \mathbb{R}^N \to \mathbb{R}^N$ be two Lipschitz-continuous vector fields. Given $\lambda \in ]0,1[$ consider, for $n \in \mathbb{N}$, the vector fields $... |
Random variables on $\mathbb{R}$ with finite expectation form an $\mathbb{R}$-linear space with respect to addition of random variables and multiplication by elements of $\mathbb{R}$. Is there an understandable way to characterize all linear functionals on random variables with finite expectation, i.e. $\mathbb{R}$-lin... |
A friend of mine told me about this problem.
Given a indexed sequence of 13 cards from Ace to King, there is a machine which shuffles them and gives a certain (indexed) output sequence. Now inserting this output again in the machine, it gives another indexed sequence in the order:
Jack, 9, Queen, Ace, King, 3, 5... |
This comes from USA Harvard-MIT Mathematics Tournament (I don't know from which year)
>A particle moves along the $x$-axis such that its velocity at the $x$-position is given by the formula $v(x) = 2 + \sin(x)$.
What is its acceleration at $x = \pi/6$?
(Answer is $2.2$)
How do I do this if I don't know velocity... |
What is $\sqrt{-1}$? and what is $i$? (How to define the square root of negatives ) |
The Question:
---
>For $g\in\operatorname{SL}(2,q)$ and $q=p^r$ for a prime $p$ with $r\in\Bbb N$, do we have $\operatorname{tr}(g)=\operatorname{tr}(g^{-1})?$
*Here $\operatorname{tr}(h)$ is the trace of the matrix $h$.*
Thoughts:
---
I think so. According to [a preprint](https://arxiv.org/abs/1808.01815... |
For $g\in\operatorname{SL}(2,q)$, do we have $\operatorname{tr}(g)=\operatorname{tr}(g^{-1})?$ |
**What set of points on the $256 \times 256$ integer grid maximizes the number of vertices in its convex hull?**
For some context, I was given an assignment to test algorithms that find the vertices of the convex hull of a set of points in $2D$. One natural thing to test is when we set the input set such that all th... |
Let $\mathscr{C}$ be an $\infty$-category (for example taking quasicategories as a model). Recall that the arrow category is $\mathsf{Ar}(\mathscr{C}) = \mathsf{Fun}([1], \mathscr{C})$. We denote by $\mathsf{Isom}(\mathscr{C})$ the full sub-$\infty$-category spanned by the equivalences. Then, there is the expected resu... |
Consider the integral
$$I(t) = \int_0^{1/2} \frac{t dx}{x^{-t} - x^t}$$
in the vicinity of $t=0$.
When $t$ is viewed as a real variable, $I(t)$ is very well-behaved. For example, it's infinitely differentiable.
On the other hand, when $t$ is on the imaginary axis, the integrand is very poorly behaved as a ... |
I read a proof from @chess64 's [AoPS blog][1]:
> If $T$ is a linear operator on $V$ and $n = \dim V$, $ V = \text{null } T^n \oplus \text{range } T^n$.
>
> Proof:\
>Any $v$ in $V$ can be written $ v = T^n(v) + (v - T^n(v))$, where $ T^n(v) \in \text{range }T^n$ and $ T^n(v-T^n(v)) = T^n(v) - T^{2n}(v) = 0$ (follow... |
Layman question here.
I have a square matrix, $Z_1$ as shown below. It is found by inverting a matrix named $Y_1$. The $x$ values in row 4 and col 4 may be zero or non-zero and the off-diagonals are equal (e.g. the 1,4 element equals the 4,1 element). The 4,4 and at least one of the off-diagonals will be non-zero.... |
I read a proof from @chess64 's [AoPS blog][1]:
> If $T$ is a linear operator on $V$ and $n = \dim V$, $ V = \text{null } T^n \oplus \text{range } T^n$.
>
> Proof:\
>Any $v$ in $V$ can be written $ v = T^n(v) + (v - T^n(v))$, where $ T^n(v) \in \text{range }T^n$ and $ T^n(v-T^n(v)) = T^n(v) - T^{2n}(v) = 0$ (follow... |
I'm trying to see if I understood the integration by parts correctly. By setting $u(t) = a(t)$ and $dv(t) = b(t)dt$, the integral by part gives $\int_0^k a(t) b(t) dt = u(k)v(k) - u(0)v(0) - \int_0^k v(t) du(t)$. Since $v(t) = \int_0^t b(t') dt'$, is this integral become
\begin{align}
\int_0^k a(t) b(t) dt &= a(k)\in... |
Q. Evaluate by Gauss divergence theorem ∫∫(xz²dy dz + (x²y – z³)dz dx + (2xy +
y2z)dx dy), where S is the surface bounded by z = 0 and z = √ (a2 a² – x2 – y2)
I was not able to understand this question properly and when I solved this question I got the answer (2/3)π(a^5). Is this answer correct?
In this question,... |
[enter image description here][1]
Q. Evaluate by Gauss divergence theorem ∫∫(xz²dy dz + (x²y – z³)dz dx + (2xy +
y2z)dx dy), where S is the surface bounded by z = 0 and z = √ (a2 a² – x2 – y2)
I was not able to understand this question properly and when I solved this question I got the answer (2/3)π(a^5). Is this... |
Can I use Sturm's comparison theorem to say something about the average?
Let $f(t)$ be a continuous function such that $\lim_{t \to \infty } f(t) = \infty$. Let us consider the following function
$$ \ddot{x}(t) + t^2 x(t) = 0. $$
Let $C$ be a constant. Then, we can find $T$ such that $C^2 < t^2$ for all $t > T... |
I read a proof from @chess64 's [AoPS blog][1]:
> If $T$ is a linear operator on $V$ and $n = \dim V$, $ V = \text{null } T^n \oplus \text{range } T^n$.
>
> Proof:\
>Any $v$ in $V$ can be written $ v = T^n(v) + (v - T^n(v))$, where $ T^n(v) \in \text{range }T^n$ and $ T^n(v-T^n(v)) = T^n(v) - T^{2n}(v) = 0$ (follow... |
> Q. Evaluate by Gauss divergence theorem
$$\iint_S xz^2\,dy\,dz + \left(x^2y-z^3\right)\,dz\,dx + \left(2xy+y^2z\right)\,dx\,dy$$
where $S$ is the surface bounded by $z=0$ and $z=\sqrt{a^2-x^2-y^2}$.
I was not able to understand this question properly and when I solved this question I got the answer (2/3)π(a^5).... |
In the book Introduction to Partial Differential Equations by Folland, he states the claim
> Let $ \Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary $ S$. The Green's function $G$ for $\Omega$ exists and for each $x \in \Omega , G(x, \cdot) \in C^\infty(\overline{\Omega} \backslash \{x\})$.
The... |
Since $24 = 3 + 5 + 7 +9$, the number $24$ can be written as the sum of at least two consecutive odd positive
integers.
(a) Can $2005$ be written as the sum of at least two
consecutive odd positive integers? If yes, give
an example of how it can be done. If no, provide
aproof why not.
(b) Can $2006$ be writte... |
All the literature on reflections in minkowski space, that I have found, have defined ways to reflect about an arbitrary planes or lines and they always add the disclaimer eventually that the plane or line one wishes to reflect with must be **NOT lightlike**.
For example, the formula that is utilized in general and ... |
What are All the Reflections in Minkowski Space $\mathbb{R}^1,n$? |
What are All the Reflections in Minkowski Space $\mathbb{R}^{1,n}$? |
**What set of points on the $256 \times 256$ integer grid maximizes the number of vertices in its convex hull?**
For some context, I was given an assignment to test algorithms that find the vertices of the convex hull of a set of points in $2D$. One natural thing to test is when we set the input set such that all th... |
How to compute
$$\int \limits _0 ^{2 \pi} \frac {(r \cos \phi +x) \cos n\phi} {r^2+2xr \cos \phi +x^2} d\phi ?$$
The answer I am provided with is $\dfrac {(-1)^n\pi r^n} {x^{n+1}}$ for $\ x>r$, but I have no idea whether this is actually correct and how to get this. |
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 $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function, and it is known that for any constant $u$ it holds that $\Psi(x,x^{1/u})$ tends to $x \cdot u^{-u}$ as $x$ tends to infinity. There is also a large literature on computin... |
Best explicit upper bound(s) for count of smooth numbers? |
In the book Introduction to Partial Differential Equations by Folland, he states the claim
> Let $ \Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary $ S$. The Green's function $G$ for $\Omega$ exists and for each $x \in \Omega , G(x, \cdot) \in C^\infty(\overline{\Omega} \backslash \{x\})$.
The... |
I am revising the lecture notes and i stumbled across this proposition:
> Given $(\Omega, \mathcal{A}, P)$ a probability space, the following holds:
>
> 1. If a process $(X_t)_{t\geq 0}$ has continuous trajectories (that is the map $t\to X_t$ is continuous), then it is measurable (that is $X\colon(\Omega\times[0,\... |
I am thinking about the consequences of adding prediction intervals and the consequence it has on the resulting interval. For example, I am considering when to expect the sum of two such intervals to be sub-additive or sup-additive, which I believe relates to the concavity/convexity of the quantile function or cumulati... |
When is the quantile function of a gamma distribution concave? |
Layman question here.
I have a square matrix, $Z_1$ as shown below. It is found by inverting a matrix named $Y_1$. The $x$ values in row 4 and col 4 may be zero or non-zero and the off-diagonals are equal (e.g. the 1,4 element equals the 4,1 element). The 4,4 and at least one of the off-diagonals will be non-zero.... |
I'm trying to solve this exercise but I'm stuck. The exercise is the following: \
Let $\mathcal{B}(\mathcal{R})$ and $\mathcal{B}(\mathcal{R^2})$ be the borel $\sigma$ algebra on $\mathcal{R}$ and $\mathcal{R^2}$ respectively. We denote $\lambda$ the Lebesgue measure on $(\mathcal{R}, \mathcal{B(R)})$ and by $\lambda_... |
I was thinking about the notation of function mappings, such as
$$f: \mathbb R^n \rightarrow \mathbb R^m,$$
and wondered if there would be any point to considering functions with variable output (or input) dimensions. In this case I would imagine we could define
\begin{gather}
f:\mathbb R^n\rightarrow \mathbb R^... |
Does it make sense to define a function with a dependent number of dimensions? |
I'm trying to understand the notation I've found in a paper which doesn't seem to be defined within the text. I can't quite recreate it in MathJax but it looks something like:
$$\propto 1(\overset{\Delta}{\pi}(f^a_t), u^a_t)$$
In the paper the 1 is blackboard bold style. I'm not sure what the significance of the **1*... |
Bold 1 without subscript - indicator function? |
Is it possible (using ZFC or any other axiom system which is known to be as consistent as ZF) to prove the existence of an ultrafilter $\mathcal F$ on the set of prime numbers s.t. there is an embedding $\mathbb R\hookrightarrow \tilde{\mathbb F}=\prod_p \mathbb F_p/\mathcal F$, but without this field $\tilde{\mathbb F... |
Can the reals be embedded in an ultraproduct of finite fields? |
Let $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function, and it is known that for any constant $u$ it holds that $\Psi(x,x^{1/u})$ tends to $x \cdot u^{-u}$ as $x$ tends to infinity. There is also a large literature on computin... |
Best explicit bound(s) for count of smooth numbers? |
First choose an encoding of the cards as integers $1, 2, \dots, 13$. The usual one is to assign $1$ to Ace, use the value of the cards for $2$ through $10$, then assign $11$ to Jack, $12$ to Queen, and $13$ to King. Now we have the a permutation $\sigma \in S_{13}$ such that
$$
\tau = \sigma^2 = \begin{pmatrix}
11 ... |
I'm trying to solve this exercise but I'm stuck. The exercise is the following: \
Let $\mathcal{B}(\mathcal{R})$ and $\mathcal{B}(\mathcal{R}^2)$ be the borel $\sigma$ algebra on $\mathcal{R}$ and $\mathcal{R}^2$ respectively. We denote $\lambda$ the Lebesgue measure on $(\mathcal{R}, \mathcal{B(R)})$ and by $\lambda_... |
I'm trying to solve this exercise but I'm stuck. The exercise is the following: \
Let $\mathcal{B}(\mathcal{R})$ and $\mathcal{B}(\mathcal{R}^2)$ be the borel $\sigma$ algebra on $\mathcal{R}$ and $\mathcal{R}^2$ respectively. We denote $\lambda$ the Lebesgue measure on $(\mathcal{R}, \mathcal{B(R)})$ and by $\lambda_... |
I'm trying to solve this exercise but I'm stuck. The exercise is the following: \
Let $\mathcal{B}(\mathcal{R})$ and $\mathcal{B}(\mathcal{R}^2)$ be the borel $\sigma$ algebra on $\mathcal{R}$ and $\mathcal{R}^2$ respectively. We denote $\lambda$ the Lebesgue measure on $(\mathcal{R}, \mathcal{B(R)})$ and by $\lambda_... |
Can we say that a linear constant coefficient difference equation can always represent a linear shift invariant system? Are there any conditions which need to be satisfied additionally by these kind of equations to be able to do that? |
Can One Represent Any LTI system via a Linear Constant Coefficient Difference Equation? |
Suppose $f: \mathbb{R} \rightarrow \mathbb{R^3}$
$T(t)$ - the unit tangent at $t$
$N(t)$ - the principal normal at $t$
$\kappa{t}$ - the curvature at $t$
Is it correct that:
* when $f'(a) = \overrightarrow{0}$
then $N(a)$ and $\kappa{a}$ are not defined
* the image of $f$ ... |
Suppose $f: \mathbb{R} \rightarrow \mathbb{R^3}$ is for example 3 times differentiable, not necessarily smooth function (smooth meaning $f'(t) \ne \overrightarrow{0}$ for each $t$.
$T(t)$ - the unit tangent at $t$
$N(t)$ - the principal normal at $t$
$\kappa{t}$ - the curvature at $t$
Is it ... |
$\def\calS{\mathcal{S}}$
$\def\calW{\mathcal{W}}$
$\def\qty#1{\left( #1 \right)}$
$\def\calX{\mathcal{X}}$
$\def\calY{\mathcal{Y}}$
$\def\abs#1{\lvert #1 \rvert}$
$\def\pdv#1#2{\frac{\partial #1}{\partial #2}}$
$\def\expect{\mathbb{E}}$
$\def\indic{\mathbb{1}}$
In their book "Understanding Machine Leaning" the... |
Suppose $f: \mathbb{R} \rightarrow \mathbb{R^3}$ is for example 3 times differentiable, not necessarily smooth function (smooth meaning $f'(t) \ne \overrightarrow{0}$ for each $t$.
$T(t)$ - the unit tangent at $t$
$N(t)$ - the principal normal at $t$
$\kappa{t}$ - the curvature at $t$
Is it ... |
$\def\calS{\mathcal{S}}$
$\def\calW{\mathcal{W}}$
$\def\qty#1{\left( #1 \right)}$
$\def\calX{\mathcal{X}}$
$\def\calY{\mathcal{Y}}$
$\def\abs#1{\lvert #1 \rvert}$
$\def\pdv#1#2{\frac{\partial #1}{\partial #2}}$
$\def\expect{\mathbb{E}}$
$\def\indic{\mathbb{1}}$
In their book "Understanding Machine Leaning" the... |
Suppose $f: \mathbb{R} \rightarrow \mathbb{R^3}$ is for example 3 times differentiable, not necessarily smooth function (smooth meaning $f'(t)$ exists and $f'(t) \ne \overrightarrow{0}$ for each $t$).
$T(t)$ - the unit tangent at $t$
$N(t)$ - the principal normal at $t$
$\kappa(t)$ - the curvature... |
What is the maximum number of vertices in a connected graph $G$ if we are given that every vertex has degree $\binom{n}{2}$?
For small cases (1,2,3,4), the answer seems to be $2^{n-1},$ but how would you go about proving this? Is there a way to find an isomorphism between subsets of edges from a vertex and vertices? |
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... |
Suppose $f: \mathbb{R} \rightarrow \mathbb{R^3}$ is for example 3 times differentiable, not necessarily smooth function (smooth meaning $f'(t)$ exists and $f'(t) \ne \overrightarrow{0}$ for each $t$).
$T(t)$ - the unit tangent at $t$
$N(t)$ - the principal normal at $t$
$\kappa(t)$ - the curvature... |
Let Y ∼ Gamma(a; λ) , a, λ > 0: that means random variable Y has continuous density:
$$f_Y(y) = \begin{cases}
\frac{\lambda^a y^{a-1} e^{-\lambda y}}{\Gamma(a)}, & y > 0, \\
0, & \text{elsewhere.}
\end{cases}$$
Let random variable X, conditional to Y have normal distribution with
expected value 0 and va... |
$\def\calS{\mathcal{S}}$
$\def\calW{\mathcal{W}}$
$\def\qty#1{\left( #1 \right)}$
$\def\calX{\mathcal{X}}$
$\def\calY{\mathcal{Y}}$
$\def\abs#1{\lvert #1 \rvert}$
$\def\pdv#1#2{\frac{\partial #1}{\partial #2}}$
$\def\expect{\mathbb{E}}$
$\def\indic{\mathbb{1}}$
In their book "Understanding Machine Leaning" the... |
Suppose $f: \mathbb{R} \rightarrow \mathbb{R^3}$ is for example 3 times differentiable, not necessarily smooth function (smooth meaning $f'(t)$ exists and $f'(t) \ne \overrightarrow{0}$ for each $t$).
$T(t)$ - the unit tangent at $t$
$N(t)$ - the principal normal at $t$
$\kappa(t)$ - the curvature... |
Let $\mathscr{C}$ be an $\infty$-category (for example taking quasicategories as a model). Recall that the arrow category is $\mathsf{Ar}(\mathscr{C}) = \mathsf{Fun}([1], \mathscr{C})$. We denote by $\mathsf{Isom}(\mathscr{C})$ the full sub-$\infty$-category spanned by the equivalences. Then, there is the expected resu... |
As I was surfing the Mathematics side of Instagram (as usual), I came across this integral:
$$I=\int\limits_{0}^{1}\frac{\ln x\sin^{-1}x\cos^{-1}x}{x}dx$$
It encouraged me to embark on a very satisfying journey to try and evaluate it. The result turned out very nice, of course. I've found that $$I=\frac{1}{24}\ln^{4}... |
Is there an easier way to evaluate the integral $I=\int\limits_{0}^{1}\frac{\ln x\sin^{-1}x\cos^{-1}x}{x}dx$? |
We are going to find the definite integral $${I=\int\limits_{0}^{1}\frac{\ln x\sin^{-1}x\cos^{-1}x}{x}dx}$$
Ah, this integral. The more I looked into it the more it looked like it might have a closed form. Arriving at it just feels so satisfying!
<hr>
Firstly, by the substitution $x=\sin \theta$, we obtain
$$\b... |
I recently stumbled across what looks like a very interesting [paper][1] concerning a simplex-based bisection algorithm for multidimensional optimization.
The authors provide results from their own MATLAB code, which is apparently not publicly available.
I have not heard about this method on SE - has it been veri... |
Suppose $X$ is a real-valued random variable with a CDF $F$, and density/mass function $f$. In elementary probability theory, we often define expectations in the form:
$$ \mathbb{E}[X] = \int_{\mathbb{R}} x f(x) \ dx;\qquad \mathbb{E}[X] = \sum_{x \in \text{supp}(X)} x f(x) $$
depending on if $X$ is continuous or di... |
Let $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function, and it is known that for any constant $u$ it holds that $\Psi(x,x^{1/u})$ tends to $x \cdot u^{-u}$ as $x$ tends to infinity. There is also a large literature on computin... |
To be proved: for all integers $a$ and $b$, if there exist integers $x$ and $y$ such that $ax + by = \gcd(a, b)$, then $\gcd(x,y)=1$.
Suppose that $ax+by = \gcd(a,b)$. **I first want to prove that every common divisor of $x$ and $y$ divides $\gcd(a,b)$.**
Let $n$ be such a common divisor. We have: $n$ divides $x$... |
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