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We have just started covering the limit of sequences and I've stumbled upon this limit in our uni's excercises:
$$\lim_{n\rightarrow \infty} {\sqrt[n]{\ln(n)}}$$
I've considered solving it using the fact that $\lim_{n\rightarrow \infty} {\sqrt[n]{a}}=1$ for $a>0$. And since we're dealing with natural numbers, wit... |
Is this a correct approach to calculating $\lim_{n\rightarrow \infty} {\sqrt[n]{\ln(n)}}$? |
Lets say we have a series $a_n(-1)^n$
- lets assume we are finding the remainder of the series
Looking at the series $\sum_1^{\infty}a_n(-1)^n$ I know that if the series converges to S that the nth partial sum is $S_n$
- Cool, but the nth partial sum does not give us the sum of the whole series
I underst... |
I want to know, whether the following calculations for a vector of variables $A_{i}$ of size N, drawn independently at random from a complex normal distribution, with mean zero and standard deviation one are correct:
$$\sum_{ij} \mathbb{E}(A_i \cdot A_j^*) = \sum_i \mathbb{E} (|A_i|^2) = N$$
$$\sum_{ijk} \mathbb{... |
Lets say we have a series $a_n(-1)^n$
- lets assume we are finding the remainder of the series
Looking at the series $\sum_1^{\infty}a_n(-1)^n$ I know that if the series converges to S that the nth partial sum is $S_n$
- Cool, but the nth partial sum does not give us the sum of the whole series
I underst... |
Lets say we have a series $a_n(-1)^n$
- lets assume we are finding the remainder of the series
Looking at the series $\sum_1^{\infty}a_n(-1)^n$ I know that if the series converges to S that the nth partial sum is $S_n$
- Cool, but the nth partial sum does not give us the sum of the whole series
I underst... |
Lets say we have a series $a_n(-1)^n$
- lets assume we are finding the remainder of the series
Looking at the series $\sum_1^{\infty}a_n(-1)^n$ I know that if the series converges to S that the nth partial sum is $S_n$
- Cool, but the nth partial sum does not give us the sum of the whole series
I underst... |
Lets say we have a series $a_n(-1)^n$
- lets assume we are finding the remainder of the series
Looking at the series $\sum_1^{\infty}a_n(-1)^n$ I know that if the series converges to S that the nth partial sum is $S_n$
- Cool, but the nth partial sum does not give us the sum of the whole series
I underst... |
**Context**
With some effort we can show that Ramanujan's fastest series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)... |
With $A$ = diseased, $B$ = tests positive
$P(A|B) = \Large\frac{P(A \cap B)}{P(B)} = \frac{P(B|A)\cdot P(A)}
{P(B|A)\cdot P(A) + P(B|A^c)\cdot P(A^c)}$
**For Z1**
$P(A|B) =\Large\frac{\frac34\frac15}{\frac34\frac15 + \frac14\frac45}$
Compute similarly for $Z_2$, and then apply the law of total probability... |
$A_{\mathfrak{p}}$-submodules of the residue field $k = A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$? |
Well I was doing some Laplace transform problems.I was really intrigued by it and I came across one problem which was interesting
The Laplace transform of f(t)^n
I attempted to solve it using definition of Laplace but it was really complex
Is there a way to find it using some other method?
|
What is the lapace transform of (f(t))^n? |
This question is from QuantGuide(Busted 6 II):\
Suppose you play a game where you continually roll a die until you obtain either a 5 or a
6. If you receive a
5, then you cash out the sum of all of your previous rolls (excluding the
5). If you receive a
6, then you receive no payout. You do not have the decisio... |
Find the number of solutions of $$2^x+3^x+4^x-5^x=0$$
----------
Answer is given $1$. I tried to take the derivative of the function $$f(x)=2^x+3^x+4^x-5^x\\f'(x)=\ln2 (2^x)+\ln3(3^x)+\ln4(4^x)-\ln5(5^x)$$ but I could not make the conclusion wheather it is $\gt0$ or, $\lt0$. Projecting $f'(x)$ in desmos it is sho... |
How to solve for the f(x) in the **integral equation**?
$$
f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz \\
\alpha \in (1,2), C_2\text{ is a positive constant.} \\
\text{What is the definition of the function } f(x) \text{ in the given integral equation?}
$$
I have tried ... |
What is the definition of the function f(x) in the given integral equation? Can iterative methods be usedοΌ |
Find the number of solutions of $$2^x+3^x+4^x-5^x=0$$
----------
Answer is given $1$. I tried to take the derivative of the function $$f(x)=2^x+3^x+4^x-5^x\\f'(x)=\ln2 (2^x)+\ln3(3^x)+\ln4(4^x)-\ln5(5^x)$$ but I could not make the conclusion wheather it is $\gt0$ or, $\lt0$. Projecting $f'(x)$ in desmos it is sho... |
I am trying to read G.E. Sacks's book on higher order recursion theory, and he has this result (where $O$ is Kleene's $O$).
Chapter 2, Theorem 1.3: Each of the following predicates is $\Pi_1^1$:
- (i) $x \in O$ and $y \in H_x$
- (ii) $x \in O$ and $y \not \in H_x$
The proof for Theorem 1.3 in his book howev... |
Did I understand this right ? - proof that $x \in O \wedge y \in H_x$ is $\Pi_1^1$. |
Name the first two axioms as follows:
$$K: a β b β a,\quad S: (a β b β c) β (a β b) β a β c,$$
where we adopt the prevailing convention of of associating to the right, i.e. $a β b β c = a β (b β c)$. Name the third axiom
$$Z: (Β¬a β Β¬b) β b β a,$$
and for modus ponens, applied to $f: a β b$ and $g: a$, write $fg: b$... |
A faster method would be assuming 6 is also cashing out, then dividing by 2 (because 6 and 5 are identical other than the fact that 6 multiplies your score by 0). you have a 1/3 chance of cashing out each turn, meaning each outcome is expected to be rolled "half a time". the calculation $\frac{\frac{1+2+3+4}{2}}{2} =2.... |
I am trying to read G.E. Sacks's book on higher order recursion theory, and he has this result (where $O$ is Kleene's $O$).
Chapter 2, Theorem 1.3: Each of the following predicates is $\Pi_1^1$:
- (i) $x \in O$ and $y \in H_x$
- (ii) $x \in O$ and $y \not \in H_x$
The proof for Theorem 1.3 in his book howev... |
Suppose $b\in L^\infty(\mathbb{R})$ and $u\colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, where $(t,x)\mapsto u(t,x)$ is a solution to $$
\partial_t^2 u = \partial_x^2 u +b(x)\partial_x u,\quad (x,t)\in \mathbb{R}\times \mathbb{R}
$$ with initial conditions $u(0,\cdot)=u_0$ and $\partial_t u(0,\cdot)=u_1$.<br>
Do... |
I am trying to read G.E. Sacks's book on higher order recursion theory, and he has this result (where $O$ is Kleene's $O$).
Chapter 2, Theorem 1.3: Each of the following predicates is $\Pi_1^1$:
- (i) $x \in O$ and $y \in H_x$
- (ii) $x \in O$ and $y \not \in H_x$
The proof for Theorem 1.3 in his book howev... |
Well, I was doing some Laplace transform problems. I was intrigued by it and I came across one interesting problem
The Laplace transform of $f(t)^n$
I attempted to solve it using the definition of Laplace but it was complex
Is there a way to find it using some other method? |
Consider the following questions, the whole exercise is dedicated to determining the square root of negative numbers, after introducing the complex numbers. Eg.
$$\sqrt{-144}$$
Solutions for the whole exercise only give one solution for each problem, the above being $12i$.
However, this is providing to be r... |
Here is my question which seems trivial:
> Let $X$ be a projective manifold with a vector bundle $\scr{E}$ on it. If $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(1)$ is generated by global sections, whether the $\scr{E}$ generated by global sections or not? If not, is there some easy counterexamples? In which special case... |
Globally-generated property of vector bundle and the tautological bundle on its projective bundle? |
I am trying to read G.E. Sacks's book on higher order recursion theory, and he has this result (where $O$ is Kleene's $O$ and $H_x$ is a hyperarithmetic set, i.e. where we have sets of the form $H_{2^a} = (H_e)'$ for some $e$ and $'$ is the jump).
Chapter 2, Theorem 1.3: Each of the following predicates is $\Pi_1^1$... |
Consider optimization problem
$$
\max_x f(x)
$$
$x$ could be of any dimension.
Question 1: There is only one constraint $g(x)\ge 0$.
Question 2: There are two constraints $g(x)\ge 0, h(x)\ge 0$.
Suppose I can show that $h(x)\ge 0$ is always binding in question 2. Is there any relationship between the bind... |
Let $g$ be given by $g(x,y) = -x^2+y^2$. Suppose I have a bilinear function $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ for which $f \geq 0$ whenever $|x|=|y|$.
* how do I demonstrate the bound $f \geq c |g|$?
* how do I generalize it from two-dimensional space to general dimensions?
----
My attempts:
1. ... |
I have read that a locally bounded family is not a bounded family, and that the following family shows that:
$\mathcal{F} = \{ f_n : n \in \mathbb{N} \}$, $f_n : \mathbb{E} \to \mathbb{C}$, $f_n(z) =n z^n$. Where $\mathbb{E} = \{z \in \mathbb{C} : \vert z \vert < 1\}$
(So that this is a locally bounded family but ... |
Let F be an ordered field. Let k be a positive element of F. Define a binary operator * on F:
x*y = x+y - xy/k
Then I claim ([0,k],*,0) is a commutative monoid with k as an absorbing element.
Moreover if you have an element a, such that 0 < a < k,
then
a < a*a < a*a*a < ....< a^n < a^{n+1} < ...... < k.
... |
Zeno's Monoid: Has anyone got a reference for this? |
At [this answer](https://math.stackexchange.com/a/4883740/567304) it is provided an interval of the form $[kn,(k+1)n]$ where each integer is divisible by at least one of the integers of the interval $[2,n]$.
We define a permutation of the integers of the interval $[2,n]$ dividing the integers of the interval $[kn,(k... |
Let F be an ordered field. Let k be a positive element of F. Define a binary operator * on F:
x*y = x+y - xy/k
Then I claim ([0,k],*,0) is a commutative monoid with k as an absorbing element.
Moreover if you have an element a, such that 0 < a < k,
then
a < a * a < a * a * a < ....< a^n < a^{n+1} < ...... <... |
I am trying to read G.E. Sacks's book on higher order recursion theory, and he has this result (where $O$ is Kleene's $O$ and $H_x$ is a hyperarithmetic set, i.e. where we have sets of the form $H_{2^a} = (H_e)'$ for some $e$ and $'$ is the jump).
Chapter 2, Theorem 1.3: Each of the following predicates is $\Pi_1^1$... |
This question is from my countryβs (Indonesia) university entrance exam, called [UTBK (also SNBT)](https://en.wikipedia.org/wiki/SNBT). I'm very confused about this data sufficiency problem.
Is the set $\{x\,|\; x < -1 \; \text{or} \; x > 5, \; x \; \text{is a real number}\}$ the solution set for $|m β 3x| > n$? Dec... |
In this snip taken below from Bishop's PRML book, I'm having trouble following the derivation that allows us to conclude that the minimizing hypothesis function, labeled y(t) here, can be shown to be equal to $\mathbb{E}[$t|x].
Perhaps my issues is 1) lack of knowledge of the calculus of variations that allows for r... |
Let's consider $2024$ variables $x_i$ that are arbitrary reals in $[-2;3]$ such that $x_1+x_2+...+x_{2024}=500$.
What is the greatest value of :
$$x_1^4+x_2^4+...+x_{2024}^4$$
Here is my approach but it is not rigorous. I wanted to choose the $x_i$ with extreme values. I wanted to take $167$ times a $3$ and $2$ ti... |
> Does he talk about Poincare's disk here?
No, with all those references to a βfundamental planeβ this has to be the half-space model.
> I cannot understand the meaning "Sphere cutting orthogonally the fundamental plane".
The wording is a bit hard to understand as it refers to three dimensions. Go for 2d only ... |
Is it known, for a general nontrivial ring $R$, whether the finitely generated free algebra $R \langle x_1, \cdots, x_n\rangle $ has the IBN property? What if $R$ is more restricted? Thanks. |
Does $R \langle x_1, \cdots, x_n\rangle $ have the IBN property? |
> Let $z=f(u,v), u=xy, v=\dfrac{x}{y}, f$ is second differentiable.
> Compute $\dfrac{{\partial}^2z}{\partial x\partial y}$ and
> $\dfrac{{\partial}^2z}{\partial x^2}.$
My attempt for $\dfrac{{\partial}^2z}{\partial x\partial y}$ is to use The chain rule:\
$\dfrac{\partial z}{\partial y}=\dfrac{\partial z}{\parti... |
B is a set where $x_1,x_2$ belong to $\mathbb{R}^2$ and $x_1^2 + x_2^2\leq 1\wedge x_1\geq 0.$
A function $f:B \rightarrow \mathbb{R}$ is given by
\begin{equation*}
f(x_1 ,x_2 )=x_1^2 x_2^2 +3x_1 x_2 +x_2 -4.
\end{equation*}
State the image of $f$.
From what I understand, this equation represents a unit... |
I am currently proving a theorem on the expressivity of functions which I have called "autoregressive". Take a finite set $T$ and two natural numbers $n,m\in\mathbb N$. For any function $a:T^n\to T$ we can define a function $\phi[a]:T^n\to T^m$ by applying $a$ $m$ times in an "autoregressive" way, by which I mean for $... |
I am reading a real analysis book which states that the function $f:\mathbb{R}^2\to\mathbb{R}$ defined by
$$f(x,y)=\begin{cases}
\frac{4(x^2-y)(2y-x^2)}{y^2} & \text{if } y>0\\
0 & \text{if }y\leq 0.
\end{cases}$$
is continuous on $\mathbb{R}^2\setminus\{(0,0)\}$, is not continuous at $(0,0)$ and if we fix... |
why is $f(x,y)=\begin{cases}\frac{4(x^2-y)(2y-x^2)}{y^2} & y>0\\ 0 & y\leq 0.\end{cases}$ continuous on $\mathbb{R}^2\setminus\{(0,0)\}$? |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
>Find all entire functions such that $|f(z+z')|\leq |f(z)| + |f(z')|$, for all $z,z'\in\mathbb{C}$
In particular, let $z=z'$ yields $|f(2z)|\leq2|f(z)|$. This gives that $\frac{f(2z)}{f(z)}=c, $ for some $c\in\mathbb{C}$. By considering continuity at 0, we have $f(0)=\lim_{n\to\infty}\frac{1}{c^n} f(1)$. Then $c\in\... |
<h4>Question Summary (to Make Easier to Reference in the Answer)</h4>
$$
\text{If }x^2-16\sqrt{x}=12 \text{ what is the value of }f=x-2\sqrt{x}
\tag{Eq. 1}$$
<h4>Checking for Consistency by Plugging in the $f=x-2\sqrt{x}=2$</h4>
Check $ x - 2\sqrt{x}-2=0$ implies $(\sqrt{x}-1)^2=2+1=3$ so $\sqrt{x}=1+\sqrt{3... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
I have the following problem:
> Let $B$ be the set $B=\{(x_1 , x_2) \in \mathbb{R}^2\mid x_1^2 + x_2^2 \leq 1\text{ and }x_1 \geq 0\}$.
>
> A function $f:B \rightarrow \mathbb{R}$ is given by
$$
f(x_1 ,x_2 )=x_1^2 x_2^2 +3x_1 x_2 +x_2 -4.$$
> State the image of $f$.
To find the image, I want to do an extrema... |
> Let $z=f(u,v)$ where $u=xy$ and $v=\frac{x}{y}$ such that $f$ is second differentiable.
>
> Compute $\tfrac{{\partial}^2z}{\partial x\partial y}$ and
> $\tfrac{{\partial}^2z}{\partial x^2}$.
My attempt for $\tfrac{{\partial}^2z}{\partial x\partial y}$ is to use the chain rule:
\begin{align*}
&\dfrac{\partial... |
>$\cos(y)\sin(t)\frac{dy}{dt}=\sin(y)\cos(t)$
(This ODE is taken from 'Differential Equations
and Their Applications' by Martin Braun)
The ODE above is separable if $\sin(y) \neq 0$ and so, we can verify that $y(t)=2k\pi; k \in \mathbb{Z}$ are solutions of the given ODE; now if $y$ isn't any of these solutions,... |
I am trying to solve an HJB equation in a very simple case that put me in trouble.
For a function $v(t,x)$ regular enough defined on $[0,T]$ with condition $v(T,x) = 0$ , I want to find a solution of
$$
\partial_{t}v(t,x) -\frac{1}{2}\left(\partial_{x}v(t,x)\right)^2 + 2x\partial_{x}v(t,x) + \frac{1}{2}x^2 = 0... |
I am currently working with linear differential equations, but I lack some very foundational understanding for why they are solved certain ways. Suppose we have a simple equation, such as:
$y'-2y=8x$
To find the general solution, we need to find the homogenous solution to the equation and one particular solution.... |
Why does finding only one particular solution yield the general solution? |
**Let P0 be an equilateral triangle of area 10. Each side of P0 is trisected into three segments
of equal length, and the corners of P0 are snipped off, creating a new polygon (in fact, a
hexagon) P1. What is the area of P1? Now repeat the process to P1 β i.e. trisect each side
and snip off the corners β to obtain a... |
$AGL(1, \mathbb F_p)$ is the group formed by transformations from $\gamma_{A,b}: \mathbb{F_p}$ to $\mathbb F_p$ with $\gamma_{a,b}(u) = au+b$.
I know that the following group is a subgroup
$$ T = \{ \gamma_{1,b} | b \in \mathbb{F_p} \} $$
and that
$$ AGL(1, \mathbb{F_p}) / T \simeq \mathbb{F^{\times}_p}$$
alth... |
Let $(X_n : n\in \mathbb{N})$ be a sequence of i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Let $S_0 = 0$ and $S_n = \sum _{i=1}^{n}X_i$ for $n \geq 1$. Let $N$ be a non-negative integer-valued random variable.
The question I am working on asks to assume further that $N$ is independent of the ran... |
Source : Ayres, *Outline of Modern Algebra*
There is a stage in Ayre's construction of the set $I$ of integers that I do not understand. More precisely, I recognize the truth of the claim he makes, but I do not see the point of the claim regarding the final goal.
See image below $Β§4.5$
Ayres has defined
-... |
**Theorem 70.1 (Seifert-van Kampen theorem).** Let $X=U \cup V$, where $U$ and $V$ are open in $X$; assume $U, V$, and $U \cap V$ are path connected; let $x_0 \in U \cap V$. Let $H$ be a group, and let $$ \phi_1: \pi_1\left(U, x_0\right) \longrightarrow H \quad \text { and } \quad \phi_2: \pi_1\left(V, x_0\right) \long... |
Source : Ayres, *Outline of Modern Algebra*
There is a stage in Ayre's construction of the set $I$ of integers that I do not understand. More precisely, I recognize the truth of the claim he makes, but I do not see the ***function or necessity*** of the claim regarding the final goal.
See image below $Β§4.5$
A... |
I have come across an integral that involves a hypergeometric function, which can be expressed as follows:
$$I = \int_0^1 x^{1/2}(1-x)^{\epsilon-1} {_{2}F_1}(\frac{1}{2}+\epsilon,1+\epsilon;\frac{3}{2};x) dx.$$
Here, $\epsilon$ is a small complex quantity where $|\epsilon|\ll1$.
I found an integral formula i... |
> **Theorem 70.1 (Seifert-van Kampen theorem).** Let $X=U \cup V$, where $U$ and $V$ are open in $X$; assume $U, V$, and $U \cap V$ are path
> connected; let $x_0 \in U \cap V$. Let $H$ be a group, and let $ \phi_1: \pi_1\left(U, x_0\right) \longrightarrow H \ \text { and } \ \phi_2: \pi_1\left(V, x_0\right) \longri... |
I know how to prove when $R$ is a commutative ring, $M\otimes_R N$ is a module, but how to prove when $R$ is non-commutative ring, we cannot give any module structure on $M\otimes_R N$? I see many answers say that we cannot define $r(m\otimes n)$= $m\otimes rn$ because it will broke the bilinearity. However, it is not ... |
One task asks to find the spectrum of the matrix
$$
\left( \begin{array}{*{20}{c}}
-21&6&2&-20&-6\\
-21&12&-7&-14&-7\\
-2 &1&-4&-1&-1\\
19&-6&-2&18&0\\
0&7&-7&7&-2\\
\end{array} \right).
$$
I can find this spectrum using a computer, however, the task involves searching with a pen and paper. How is this possib... |
In a round robin tournament with $n\geq 10$ teams, each match ends in either a win or loss. What is the minimum number of losses to ensure that a team is out of contention of top 2 spots?
**Clarification:** *If two teams score the same, it's good enough because they aren't eliminated.*
**My attempt:** assign the... |
A total order $I$ is said to be weakly self-similar if there exists a proper subset $J \subsetneq I$ together with a bijective, order-preserving function $f:I \to J$ (that is, $J$ is isomorphic to $I$). Analogously, $I$ is said to be strongly self-similar if there exists an element $i \in I$ such that one of the total ... |
In Conway's *Functions of one complex variable I* the following definition of *total boundedness* is given:
Let $(X, d)$ be a metric space;
For every $\epsilon>0$ there are a finite number of points $x_1, ..., x_n$ in $X$ s.t.
$X = \cup_{k=1}^{n}B(x_k;\epsilon).$
Then we have the following theorem:
$X$ ... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
Source : Ayres, *Outline of Modern Algebra*
There is a stage in Ayres' construction of the set $I$ of integers that I do not understand. More precisely, I recognize the truth of the claim he makes, but I do not see the ***function or necessity*** of the claim regarding the final goal.
See image below $Β§4.5$
A... |
I know how to prove when $R$ is a commutative ring, $M\otimes_R N$ is a module, but how to prove when $R$ is non-commutative ring, we cannot give any module structure on $M\otimes_R N$? I see many answers say that we cannot define $r(m\otimes n)$= $m\otimes rn$ because it will broke the bilinearity. However, it is not ... |
Proposition: If $A\subset\mathbb{N},\ A$ has natural density $d> \frac{1}{2},$ then $\exists\ N\in\mathbb{N}\ $ such that $\ n>N \implies \exists\ a,b\in A\ $ such that $\ a+b=n.$
Proof sketch: Since $d> \frac{1}{2},\ \exists\ N\ $ such that $\ \left\lvert A\cup \{1,2,\ldots, n\} \right\rvert > \frac{1}{2}\ \forall\... |
$A\subset\mathbb{N},$ natural density $1/2.$ Half the members of $A$ are even, half are odd. Is $A$ an (eventual) additive basis of $\mathbb{N}?$ |
I am stuck trying to solve this equation using Laplace transform:
> $ yβ²β²(t)+2y(t)=3β«(tβu)y(u)du $
(the intergal is from 0 to t)
with the initial conditions: y(0)=2 , y'(0)=3.
I got stuck after performing the Laplace transformation and can't get the equation to a form I can do the inverse Laplace tranf... |
I am stuck trying to solve this equation using Laplace transform:
> $ yβ²β²(t)+2y(t)=\displaystyle 3\int_0^3 (tβu)y(u)du $
with the initial conditions: $y(0)=2 , y'(0)=3$.
I got stuck after performing the Laplace transformation and can't get the equation to a form I can do the inverse Laplace transforma... |
>$\cos(y)\frac{dy}{dt}=\frac{-t\sin(y)}{1+t^2} \\
y(1)=\frac{\pi}{2}$
As $y(1)=\frac{\pi}{2}; \ \sin(y(1))=1$; we have by continuity that there exists $h>0$:
$$\sin(y(t)) \neq 0; t \in J=(1-h,1+h)$$
And in such an interval $J$:
\begin{align*}
& -\frac{\cos(y)}{\sin(y)}\frac{dy}{dt}=\frac{t}{1+t^2} \\
\iff ... |
**PROBLEM**
Suppose that
$$
\sum_{m=-l}^{l} c_m Y_m^l(\theta, \phi) Y_m^l(\theta', \phi')^*
$$
is rotationally invariant, then how can we show that the $c_m$'s must be all equal?
**ATTEMPT AT A SOLUTION**
I reformulated the rotational invariance in Dirac Notation:
$$
\sum_{m=-l}^{l} c_m \hat D^{\dagge... |
> **Theorem 70.1 (Seifert-van Kampen theorem).** Let $X=U \cup V$, where $U$ and $V$ are open in $X$; assume $U, V$, and $U \cap V$ are path
> connected; let $x_0 \in U \cap V$. Let $H$ be a group, and let $ \phi_1: \pi_1\left(U, x_0\right) \longrightarrow H \ \text { and } \ \phi_2: \pi_1\left(V, x_0\right) \longri... |
In the book [1] the covariance of an SDE is derived. I am not sure about a particular step. Let me describe it in a TLDR version, then in a longer version.
We have an SDE
$$dx = f(x,t) dt + L(x,t) d\beta $$
where $x$ is a vector, $\beta$ is a Brownian motion and $L$ is a matrix.
In the derivation of the va... |
Just as the title says:
Can there be an infinite set S whose elements all contain S?
If so, how is it called?
An example of a finite set could be the set of all Humans, where each human h knows about all other humans (where 'knowing' means being part of a set).
If there are infinite things in reality, and a... |
can there be an infinite set S whose elements all contain S? |
I have read that a locally bounded family is not a bounded family, and that the following family shows that:
$\mathcal{F} = \{ f_n : n \in \mathbb{N} \}$, $f_n : \mathbb{E} \to \mathbb{C}$, $f_n(z) =n z^n$. Where $\mathbb{E} = \{z \in \mathbb{C} : \vert z \vert < 1\}$
(So that this is a locally bounded family but ... |
Let $\Theta$ be an $\mathbb R^d$-valued Gaussian-mixture random variable with $k$ Gaussian components $\Phi_1,\Phi_2,\ldots,\Phi_k$, where $\Phi_i$ has mean vector $\mu_i$ and covariance matrix $\Sigma_i$, and mixing probabilities $p_1,p_2,\ldots,p_k$. Let $C$ be the conditioning categorical random variable. Then, $Y_x... |
Is there a Gaussian-mixture process? |
>$\cos(y)\frac{dy}{dt}=\frac{-t\sin(y)}{1+t^2} \\
y(1)=\frac{\pi}{2}$
As $y(1)=\frac{\pi}{2}; \ \sin(y(1))=1$; we have by continuity that there exists $h>0$:
$$\sin(y(t)) \neq 0; t \in J=(1-h,1+h)$$
And in such an interval $J$:
\begin{align*}
& -\frac{\cos(y)}{\sin(y)}\frac{dy}{dt}=\frac{t}{1+t^2} \\
\iff ... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
Using the type of natural numbers $\mathrm{N}$ and the type of propositions $\mathrm{Prop}$, we can make Peano arithmetic in the style of type theory.
https://ncatlab.org/nlab/show/Peano+arithmetic
Since Peano arithmetic is a first-order theory, the range of quantification is limited to $\mathrm{N}$.
If I all... |
I have a doubt about this task, have I done it correctly?
The task is: Construct a minimal DFA that accepts the following language:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/CIKC1.jpg |
I'm considering quasi-polynomials of the form
$$ P(z) = z^n + a_1z^{n-1}+...+a_n + K_1e^{-z\tau}(z^m + b_1z^{m-1}+...+b_m) + K_2e^{-2z\tau}(z^j + c_1z^{j-1}+...+c_j), $$
where all the constants are real. For the case where $K_2=0$, there is this article: https://community.ams.org/journals/bull/1964-70-02/S0002-... |
Your approach to applying the Stone-Weierstrass Theorem through the lens of neural networklike functions is indeed intriguing. One aspect that might further illuminate your proof is considering the richness of the space generated by your function $G(x)$ under different choices of the activation function $\varphi$. Spec... |
A total order $I$ is said to be weakly self-similar if there exists a proper subset $J \subsetneq I$ together with a bijective, order-preserving function $f:I \to J$ (that is, $J$ is isomorphic to $I$). Analogously, $I$ is said to be strongly self-similar if there exists an element $i \in I$ such that one of the total ... |
Choose positive integers $D$ and $N$. Roll a fair $D$-sided die $N$ times, recording the number of times each of the $D$ outcomes are rolled, say $r_1, r_2, \ldots, r_D.$ What are the asymptotics of $\min(r_1, r_2, \ldots, r_D)$?
If it makes it easier you can assume $N \gg D.$ Of course the leading term is $N/D$ but... |
I have the following Definition with I struggle with. Namely it is the definition of a open set of class $C^1$.
We say an open set $O$ in $\mathbb{R}^d$ is of class $C^1$ iff for sets
1) $S_1:=\lbrace (x_1,\ldots,x_{d})\in \mathbb{R}^d : (x_1,\ldots,x_{d-1}) \text{ is an element of the open unit ball in }\mathbb{... |
> What is the smallest prime factor of $$\sum_{j=1}^{10} j!^{j!}$$ ?
- Trial :
This number has $23\ 804\ 069$ digits , so if it were prime it would be a record prime. I do not think however that this number is prime. According to my calculations , it has no prime factor below $10^{10}$. To establish its composit... |
<h4>Question Summary (to Make Easier to Reference in the Answer)</h4>
$$
\text{If }x^2-16\sqrt{x}=12 \text{ what is the value of }f=x-2\sqrt{x}
\tag{Eq. 1}$$
<h4>Checking for Consistency by Plugging in the $f=x-2\sqrt{x}=2$</h4>
Check $ x - 2\sqrt{x}-2=0$ implies $(\sqrt{x}-1)^2=2+1=3$ so $\sqrt{x}=1+\sqrt{3... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
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