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Consider the equation
$$t\frac{\partial^2}{\partial t^2} \sum_{n=1}^\infty \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^\infty \Phi_n(x,t)+\sum_{n=2}^{\infty}\Lambda(n)\Phi_n(x,t)$$
where $\Lambda(n)$ represents an arbitrary number theoretic function, that is a function $\Lambda:\Bbb N \to \Bbb C.$ So far ... |
Let $A$ be a non-symmetric matrix of real numbers, $A\in\mathbb{R}^{n\times n}$.
If the quadratic form $x^\intercal Ax$ is positive definite in the sense that $x^\intercal Ax>0 \\; \forall x\neq 0\\,\in\mathbb{R}^n$, then the real part of all eigenvalues of $A$ is positive, $\Re(\lambda_i)>0$ for $i=1,\dots,n$.
H... |
Let $X$ be a normed vector space. We define the weak-$^*$ topology on $X^*$ to be the initial topology generated by all linear functionals $ev_x: X^*\rightarrow F$ defined by
$$
f\mapsto f(x).
$$
Here is my professor's description about the subbasic open sets and basic open sets with respect to the weak-$^*$ topol... |
$A$ and $B$ play a game where they take turns throwing darts at a dart board. The winner is the person to hit first. $A$ hits with probability $a$ and $B$ with probability $b$.
I showed (by computing a sum) that the expected number of darts thrown in one game is
$$\frac{\alpha}{a}+\frac{\beta}{b}$$
where $\alpha$... |
Consider the family $$C_{\alpha}:\big({2(\alpha-\cos{t})\,\cos{t},\,2(\alpha-\cos{t})\,\sin{t}}\big)\,,\;t\in[0,2\pi]\,,\;\alpha\in\mathbb{R},$$ of closed parametric curves. Let $X_{\alpha}$ is the image set of $C_{\alpha}$ equiped with the induced topology of $\mathbb{R}^2$.
1. For which values of $\alpha$ these ... |
I've done a bit of digging but I haven't found any sources to confirm that this is true.
Take $\limsup{\{x_n\}}=L$ and $f(x): D \to \mathbb{R}$ to be a monotonically increasing function such that $\{x_n\} \in D$. Does $\limsup{f(x_n)} = f(L)$?
I take it to be true because $f$ preserves order, and as there are onl... |
Is the limsup perserved under monotonically increasing functions? |
Specifically, From reading a some texts, it states that the Riemann condition for showing that the function is integratable is by showing that there is an
$\epsilon > 0$ and condition $s \leq f \leq t$ (s and t are step functions below and above f respectively). To prove integrability it is said to then show that
$... |
How to prove that a function is integratable using the Riemann condition? |
In *algebra*, you can combine functions and then set $x$ equal to a value or you can set $x$ equal to a value and then combine the values in the same way, and you get the same result, e.g., $(f+g)(3) = f(3) + g(3)$. When $h(3) = f(3)$, we have $(h + g)(3) = h(3) + g(3) = f(3) + g(3) = (f+g)(3)$. So algebraic operatio... |
When $X_t$ is conditionally normal distributed and has density $p_t$, how can we compute $\text E\left[\left\|\nabla\ln p_t(X_t)\right\|^2\right]$? |
I am a high school student and we firstly learn something about abstract algebra this week.I want to learn the idea behing the difference between $a*b$ and $b*a$
It is given that $a*b=a^3 -b^2+4a+3b$ where $*$ is an algebraic operation. It is asked to find $b*a$.
I thought that I should put "a" in place of $b$ i... |
In the following equation
$$10^{10^z} = 10^{10^x}+10^{10^y}$$
I want to find an algorithm that computes $z$ in a floating point accurate manner given any values of $x$ and $y$ (e.g. $x=y=2000$). The algorithm should thus avoid evaluating $10^x$ or $10^y$ directly otherwise a memory usage error happens. What would be ... |
I'm unable to prove if the following series converge uniformly on $[0,+\infty]$ for the values of the parameter $\beta$ between $2$ and $3$.
$$
\sum_{n\geq 1}\frac{n^\beta x}{x^4+n^4}
$$
The maximum of the summed functions is realized (for fixed $n$) in $\frac{n}{\sqrt[4]{3}}$ and it is $\frac{\sqrt[4]{27}}{4}n^{... |
Consider the following population model: $x'(t) = r x(t-d) - h x(t)$, with growth rate $r > 0$ and death rate $h > 0$, where only adults that have reached the age $d > 0$ can reproduce.
I have to draw a bifurcation diagram. For $d = 0$ it looks easy but I don't know how to do it for $d > 0$ since it's my first enco... |
I will call an _ordered semicategory with action_ (OSA) an ordered semicategory that is also a semicategory with actions acting on ordered elements, where the action preserves the order of both elements and morphisms. I denote the action of a morphism $f$ as $\langle f\rangle$.
I call two morphisms $\mu$ and $\nu$ o... |
One can write down the following power series expansion near $x = 0$ (i.e., Maclaurin series):
\begin{align*}
\ln(1 + x) & = \sum_{n = 1}^\infty \frac{(-1)^{n - 1}x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \\
e^{-x^4} & = \sum_{n = 0}^\infty \frac{(-1)^nx^{4n}}{n!} = 1 - x^4 + \frac{x^8}{... |
From an old notebook, I found a question mark (not known existing how many years) about a contradition between the integral form and the original exponential decaying function $f(t)$.
$$
f(t)=\begin{cases}
e^{-bt},& t\ge 0\\
0 & t<0
\end{cases}
$$
where $b>0$ and $f(0)=1$. The Fourier transfor... |
Where is the $1/2$ lost in Fourier transform? |
I will call an _ordered semicategory with action_ (OSA) an ordered semicategory that is also a semicategory with actions acting on ordered elements, where the action preserves the order of both elements and morphisms. I denote the action of a morphism $f$ as $\langle f\rangle$.
I call two morphisms $\mu$ and $\nu$ o... |
I've done a bit of digging but I haven't found any sources to confirm that this is true.
Take $\limsup{\{x_n\}}=L$ and $f(x): D \to \mathbb{R}$ to be a continuous and monotonically increasing function such that $\{x_n\} \in D$. Does $\limsup{f(x_n)} = f(L)$?
I take it to be true because $f$ preserves order, and a... |
I am a bit confused. We have two different random variables $X,Y$ and another two random variables $G,F$, such that $G$ has the same distribution as $X$ and $F$ has the same distribution as $Y$.\
Does
\begin{align*}
E(XY)=E(GF)
\end{align*}
hold?\
Or do we need some independence assumptions?\
Sorry for this sill... |
Recall that $e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$. Then,
$$\frac{2x\ln(1+x) + x^3}{\sqrt{1 - e^{-x^4}}} = \frac{2x\ln(1+x) + x^3}{\sqrt{1 - \sum_{n=0}^{\infty}(-1)^n\frac{x^{4n}}{n!}}} = \frac{2x\ln(1+x) + x^3}{\sqrt{ \sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{4n}}{n!}}} = \frac{2x\ln(1+x) + x^3}{x^2\sqrt{ \sum_{n=0}^{... |
Can someone explain to me with the help of a diagram/illustration the subject of dual mapping
I'm getting confused.
I know that W maps using a linear function f, vectors to scalars.
Can I write that R is the w*?
and why
S*(f) = fS
After all, S* is a transition from W* to V*. If I run a function f on it, how do I ... |
Say I am solving the following differential equation:
$$y'' - y = x^2$$
It is perfectly permissible for me to proceed as follows:
$$D^2y- y = x^2$$
$$y(D^2-1) = x^2$$
$$y = (D^2 - 1)^{-1} \cdot x^2$$
$$y=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} D^n\cdot x^2$$
$$y = (-1 - D^2 - D^4...) x^2$$
$$y_p = -x^... |
What are the properties of the differential operator that justify this? |
Suppose that we have a sequence of random variable $X(n)$ and we know that each $X(n)$ has moments of all orders. The well-known `method of moments' tells us that if, for each $k \ge 1$, $\mathbb{E}(X(n)^k) \to \mathbb{E}(X^k)$ as $n\to\infty$ where $X$ is a normal distribution then $X(n)$ converges in distribution to ... |
Say I am solving the following differential equation:
$$y'' - y = x^2$$
It is perfectly permissible for me to proceed as follows:
$$D^2y- y = x^2$$
$$y(D^2-1) = x^2$$
$$y = (D^2 - 1)^{-1} \cdot x^2$$
$$y=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} D^n\cdot x^2$$
$$y = (-1 - D^2 - D^4...) x^2$$
$$y_p = -x^... |
The question, briefly
---------------------
How does the calculation $P(H|E) = \frac{h (c +a \overline c)}{hc+a\overline c}$ work?
Some background
---------------------
I'm trying to work through the proof of the following theorem in Heinzelmann and Hartmann (2022):
*Theorem 1.* $P(H | E, \neg O ) > P(H|E... |
**Note:** This question takes a pluralistic / ["multiverse"][1] view of sets (including "classes" and "collections") and set theor***ies***.
My understanding (from reading many nLab articles) is that there are at least two category-theoretic notions$^\dagger$ of "set":
1. ("intramural" / "within-category" / "int... |
Are the "intramural" and "extramural" definitions of "set" from category theory equivalent? |
Neznayka drew a rectangle, divided it into 64 smaller rectangles by drawing 7 straight lines parallel to each of the original rectangle's sides. After that, Znayka points to $n$ rectangles of the division at the same time, and Neznayka names the area of each of these rectangles. What is the smallest value of $n$ for Zn... |
I am studying Jacobson rings, using [this file](https://www.math.uchicago.edu/~emerton/pdffiles/jacobson.pdf) by Matthew Emerton and [this](https://www2.math.upenn.edu/~chai/602f06/hw602f06/jacobson_ring.pdf) source.
I am trying to understand the proof given for the fact that
> if $A$ is a Jacobson ring (that is, ... |
I am new to measure theory. I was asked to prove the following result:
> **Proposition**$\quad$ *The $\sigma$-algebra $\mathcal{B}(\mathbb{R}^d)$ of Borel subsets of $\mathbb{R}^d$ is generated by each of the following collections of sets:*
> 1. *the collection of all closed subsets of $\mathbb{R}^d$;*
> 2. *t... |
Can someone explain to me with the help of a diagram/illustration the subject of dual mapping
I'm getting confused.
I know that $W$ maps using a linear function $f$, vectors to scalars.
Can I write that $R$ is the $W*$?
and why
$S^*(f) = fS$
After all, $S^*$ is a transition from $W^*$ to $V^*$. If I run a functio... |
I am new to measure theory. I was asked to prove the following result:
> **Proposition**$\quad$ *The $\sigma$-algebra $\mathcal{B}(\mathbb{R}^d)$ of Borel subsets of $\mathbb{R}^d$ is generated by each of the following collections of sets:*
> 1. *the collection of all closed subsets of $\mathbb{R}^d$;*
> 2. *t... |
>Let $\boldsymbol{\sigma}(u,v)=\big(f(v)\cos u, f(v) \sin u, g(v)\big)$, where
$U=[(u,v):0<u<2\pi,~v\in I]$ ($I$ open interval of $\mathbb{R}$) be the parametric surface of revolution by rotating the one-to-one curve $\boldsymbol{\gamma}(v)=(f(v),0,g(v)),~v\in I$ around the $z-$axis (we let $f(v)>0$ for all $v$).
P... |
I am new to measure theory. I was asked to prove the following result:
> **Proposition**$\quad$ *The $\sigma$-algebra $\mathcal{B}(\mathbb{R}^d)$ of Borel subsets of $\mathbb{R}^d$ is generated by each of the following collections of sets:*
> 1. *the collection of all closed subsets of $\mathbb{R}^d$;*
> 2. *t... |
Find all values of $x$ such that
$$\left(x^2-\frac{1}{x^2}\right)^2\:+\:2\,\left(x+\frac{1}{x}\right)^2 \:=\: 2024\,.$$
I arrived at the solutions $\,x=\pm \sqrt{22 \pm \sqrt{483}}$, $\,\pm \sqrt{-23 \pm 4 \sqrt{33}}\,$, by simplifying to a point where there is no more linear term, and then substituting $y=x^2$. It t... |
How to elegantly solve this Quadratic/Quartic Equation? |
**tl;dr** If you could integrate 0-forms on a smooth manifold, you'd automatically get a natural definition of *volume* of all (say, open) subsets $U\subset M$ by $\textrm{vol}(U):=\int_U 1$. But as you know, angles, lengths, volumes require an additional structure - a metric, so this should be impossible. The differen... |
For any module with vector set $V$ and scalar set $C$, must there exist a set $X$ such that $V$ and $(X → C)$ are isomorphic? |
I am new to measure theory. I was asked to prove the following result:
> **Proposition**$\quad$ *The $\sigma$-algebra $\mathcal{B}(\mathbb{R}^d)$ of Borel subsets of $\mathbb{R}^d$ is generated by each of the following collections of sets:*
> 1. *the collection of all closed subsets of $\mathbb{R}^d$;*
> 2. *t... |
Consider the limit $\lim_{n\to \infty}\sqrt[n]{x\^n}=x$ for any real positive x, but I will proceed in a different way:<br>
$\lim_{n\to \infty}\sqrt[n]{e^{ln(x\^n)}}=x$<br>
$\lim_{n\to \infty}\sqrt[n]{(1+nln(x)/n)^n}=x$<br>
$1+ln(x)=x$<br>
Which is obviously wrong, but why? What is the restriction I ignored when ev... |
Consider the limit $\lim_{n\to \infty}\sqrt[n]{x^n}=x$ for any real positive x, but I will proceed in a different way:<br>
$\lim_{n\to \infty}\sqrt[n]{e^{ln(x^n)}}=x$<br>
$\lim_{n\to \infty}\sqrt[n]{(1+nln(x)/n)^n}=x$<br>
$1+ln(x)=x$<br>
Which is obviously wrong, but why? What is the restriction I ignored when eval... |
I am studying Jacobson rings, using [this file](https://www.math.uchicago.edu/~emerton/pdffiles/jacobson.pdf) by Matthew Emerton and [this](https://www2.math.upenn.edu/~chai/602f06/hw602f06/jacobson_ring.pdf) source.
I am trying to understand the proof given for the fact that
> if $A$ is a Jacobson ring (that is, ... |
> Is there a prime number $p$ with $$p \mid \sum_{j=1}^{p-1} j!^2$$ ?
I checked the primes upto $600\ 000$ without finding a solution.
Heuristic : If we can assume that the probability that $p$ is a solution is $\frac{1}{p}$ , then there should be infinite many solutions and therefore the desired prime should exi... |
Is there a prime number $p$ dividing $1+2!^2+3!^2+\cdots (p-1)!^2$? |
Neznayka draws a rectangle, divides it into 64 smaller rectangles by drawing $7$ straight lines parallel to each of the original rectangle's sides.
After that, Znayka points to $n$ rectangles of the division at the same time, and Neznayka names/gives/reveals the areas of each of these rectangles.
What is the ... |
If (X,Y) are jointly normal $(\mu,\Sigma)$,
How to calculate $E(X|Y>0)$? |
Is this a valid proof for continuity using epsilon delta definition? |
Following this question: https://math.stackexchange.com/questions/242779/limit-of-lp-norm?noredirect=1&lq=1
>Given a finite measure space $(X, \mathcal{M}, \mu)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$
Can we find a counterexamp... |
Given the following differential equation:
$$y^2+(2xy+1)\frac{dy}{dx}=0$$
I have found that the solution is given implictly by:
$$ (\ast) \quad xy^2(x)+y(x)=k \: ; k \in \mathbb{R}$$
If I was interested in a domain that doesn't contain $0$, then I'd write that:
$$y(x)=\frac{-1 \pm \sqrt{1+4kx}}{2x}$$ and $... |
Consider the limit
$$
\lim_{n\to \infty}\sqrt[n]{x^n}=x
$$ for any real positive $x$, but I will proceed in a different way:
$$
\begin{split}
&\lim_{n\to \infty}\sqrt[n]{e^{\ln(x^n)}}=x\\
&\lim_{n\to \infty}\sqrt[n]{(1+n\ln(x)/n)^n}=x\\
&1+\ln(x)=x
\end{split}
$$
which is obviously wrong, but why? What is t... |
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... |
If (X,Y) are jointly normal $(\mu,\Sigma)$,
How to calculate $E(X|Y>0)$?
I have a solution using $E(X|Y>0)=E{E(X|Y)|Y>0}$. It is $\mu_x+\rho \sigma_x \phi(m)/\Phi(m)$ where $m=\mu_y/\sigma_y$. Do you think it's correct? |
If (X,Y) are jointly normal $(\mu,\Sigma)$,
How to calculate $E(X|Y>0)$?
I have a solution using $E(X|Y>0)=E\{E(X|Y)|Y>0\}$, since we know $E(X|Y)=\mu_x+\rho\sigma_x/\sigma_y(Y-\mu_y)$. It is $\mu_x+\rho \sigma_x \phi(m)/\Phi(m)$ where $m=\mu_y/\sigma_y$. Do you think it's correct? |
If (X,Y) are jointly normal $(\mu,\Sigma)$,
How to calculate $E(X|Y>0)$?
I have a solution using $E(X|Y>0)=E\{E(X|Y)|Y>0\}$, since we know $E(X|Y)=\mu_x+\rho\sigma_x/\sigma_y(Y-\mu_y)$. After a very long algebra I get the final solution is $\mu_x+\rho \sigma_x \phi(m)/\Phi(m)$ where $m=\mu_y/\sigma_y$. Do you think... |
We know that the unit circle can be a boundary of complete convergence or complete nonconvergence or of conditional convergence for power series in one complex variable...If the subset of the boundary where there is conditional convergence is P, what can we say about the nature of P for power series in many, but finite... |
We know that the unit circle can be a boundary of complete convergence or complete nonconvergence or of conditional convergence for power series in one complex variable...If the subset of the boundary where there is conditional convergence is P, what can we say about the nature of P for power series in many, but finite... |
This is the proof I have obtained to show that any ideal of a Banach algebra is closed:
If $\cal B$ is a Banach algebra, then let $I$ be an ideal (bilateral). Let $\{y_n\}_{n\in{\mathbb N}}$ a sequence of elements of $I$ converging to $y$ in $\cal B$, then $y$ must be in $I$. Indeed, if it is not in $I$, then there ... |
Is any ideal of a Banach algebra closed? |
I will call an _ordered semicategory with action_ (OSA) an ordered semicategory that is also a semicategory with actions acting on ordered elements, where the action preserves the order of both elements and morphisms. I denote the action of a morphism $f$ as $\langle f\rangle$.
I call two morphisms $\mu$ and $\nu$ o... |
Let $G$ and $H$ be Lie groups. Is there an easy way to see why a surjective Lie group homomorphism $\varphi: G \to H\:$ with finite kernel is a covering map?
The only thing I am not sure about is how to prove that $\varphi$ is open, the rest is clear.
For context, I am interested in proving that the spin group is... |
**First step:**
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators... |
In $\mathbb{C}[x,y]$: When $\langle u,v \rangle$ is maximal and when $\langle u,v,w \rangle$ is maximal? |
A system consists of components 1, 2, 3, and 4 that work with probability p1, p2, p3, and p4 respectively. The signal can only pass through a component if it is working. The components are independent of each other. The system works if a signal can pass from S to E. What is the probability that the system works? [Image... |
I've read about the estimation lemma, also known as the ML-inequality, on Wikipedia and in the complex analysis textbook by Saff and Snider. According to both these references, what the lemma says is the following:
If a function f is continuous on a given contour gamma, and if the absolute value of f is bounded abov... |
Why is continuity assumed in the formulation of the estimation lemma? |
Let $G$ and $H$ be Lie groups. Is there an easy way to see why a surjective Lie group homomorphism $\varphi: G \to H\:$ with finite kernel is a covering map?
When assuming $G$ is a Hausdorff space, the only thing I am not sure about is how to prove that $\varphi$ is open, the rest is clear. Do we need such an assump... |
**First step:**
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators... |
> Find all values of $x$ such that
$$\left(x^2-\frac{1}{x^2}\right)^2\:+\:2\,\left(x+\frac{1}{x}\right)^2 \:=\: 2024$$
I arrived at the solutions $\,x=\pm \sqrt{22 \pm \sqrt{483}}$, $\,\pm \sqrt{-23 \pm 4 \sqrt{33}}\,$, by simplifying to a point where there is no more linear term, and then substituting $y=x^2$. It ... |
How to elegantly solve $\left(x^2-\frac1{x^2}\right)^2+2\left(x+\frac1{x}\right)^2 = 2024$? |
**First step:**
Let $I=\langle x-y,xy \rangle \subset \mathbb{C}[x,y]$ be the ideal generated by $u=x-y$ and $v=xy$.
Notice that $x^2 \in I$, since $I \ni xu+v=x(x-y)+xy=x^2-xy+xy=x^2$.
$I$ is not a radical ideal, sicne $x^2 \in I$ but $x \notin I$, hence $I$ is not a maximal ideal.
Notice that the generators... |
Given two infinite cardinals $\kappa_1,\kappa_2$, what's the number $\tau$ of functions $f:\kappa_1\to\kappa_2$ that are eventually constant? I think that, if $\tau_0$ is the number of eventually zero functions, then $\tau=\kappa_2\cdot\tau_0=\max\{\kappa_2,\tau_0\}$ but I don't exactly know how to compute $\tau_0$. We... |
How can I count the number of eventually constant functions $\kappa_1\to\kappa_2$? |
Quoting your question:
"Take$ \limsup{\{x_n\}}=L$ and $f(x): D \to \mathbb{R}$ to be a continuous and monotonically increasing function such that $\{x_n\} \in D$. Does $\limsup{f(x_n)} = f(L)$?"
i.) Given that $f(x)$ is continuous and monotonically increasing has no logical connection to $\{x_n\} \in D$.
ii.) I am a... |
Quoting your question:
"Take $\limsup{\{x_n\}}=L$ and $f(x): D \to \mathbb{R}$ to be a continuous and monotonically increasing function such that $\{x_n\} \in D$. Does $\limsup{f(x_n)} = f(L)$?"
i.) Given that $f(x)$ is continuous and monotonically increasing has no logical connection to $\{x_n\} \in D$.
ii.) ... |
Quoting your question:
"Take $\limsup{\{x_n\}}=L$ and $f(x): D \to \mathbb{R}$ to be a continuous and monotonically increasing function such that $\{x_n\} \in D$. Does $\limsup{f(x_n)} = f(L)$?"
i.) Given that $f(x)$ is continuous and monotonically increasing has no logical connection to $\{x_n\} \in D$.
ii.) ... |
> Is there a continuous function that describes the Hilbert transform of (a periodic) x?
Yes,
we can start from the Fourier series expansion of a saw tooth wave:
$$
\begin{align*}
&s(x) = \frac{x}{\pi}, for -\pi < x < \pi \\[5pt]
&A_n = 0, n \geq 0 \\[5pt]
&B_n = \frac{2}{\pi} \frac{(-1)^{n+1}}{n}, n \g... |
Given two infinite cardinals $\kappa_1,\kappa_2$, what's the number $\tau$ of functions $f:\kappa_1\to\kappa_2$ that are eventually constant? I think that, if $\tau_0$ is the number of eventually zero functions, then $\tau=\kappa_2\cdot\tau_0=\max\{\kappa_2,\tau_0\}$ but I don't exactly know how to compute $\tau_0$. We... |
In "Algebraic Number Theory and Fermat's Last Theorem" by Stewart & Tall the volume is, as usually, defined as an integral. Now, exercise 6.1 (page 138) asks to show that the volume of a fundamental domain $T$ of a lattice $L \subseteq \mathbb{Z}^2$ is equal to the number of integer points lying in $T$.
I understand... |
Let $A=\frac{\mathbb C[X,Y]}{(X^3-Y^2)}$.I am asked to show that $\mathbf m=(\overline X-1,\overline Y-1)$ is a maximal ideal of $A$ which I have shown successfully. Now I am asked to compute the $\mu(\mathbf mA_m)$ which is the unique cardinality of a minimal generating set of $\mathbf mA_m$ which I am unable to do. I... |
For $\mu \in \mathcal P(\mathbb R)$, let $m(\mu)$ be the median of $\mu$.
Assume that $\mathcal P(\mathbb R)$ is endowed with the weak convergence topology. Is $m$ continuous?
If no, assume that $\mathcal P(\mathbb R)$ is endowed with the Borel sigma algebra associated to the above topology. Is $m$ measurable? |
Is the median a continuous function of the probability distribution? |
In how many ways can the numbers from the set $ \{1, 2, ..., 3n \}$ be arranged in a sequence such that the sum of every three consecutive numbers is divisible by $3$?
Solution:
All the numbers from the set $ \{1, 2, ..., 3n \}$ can be divided in 3 subsets:
1. numbers that are equal to $0$ mod $3$
2. number... |
In how many ways can numbers $ \in \{1, 2, ..., 3n \}$ be arranged in such way that the sum of every $3$ consecutive numbers is divisible by $3$? |
Let $n \in \mathbb{N}$, $0<r<1$ and $\omega_j = \exp\left(\frac{2 \pi i}{n}\right)$ for $j = 0, 1, \ldots, n-1$ be the roots of unity.
Define $a_j = r \omega^j$, these are the vertices of a regular $n$-gon inside the circle of radious $r$.
Now for which $z \in \mathbb{C}$ on the unit circle, that is $|z|=1$, does ... |
Let $n \in \mathbb{N}$, $0<r<1$ and $\omega_j = \exp\left(\frac{2 \pi i}{n}\right)$ for $j = 0, 1, \ldots, n-1$ be the roots of unity.
Define $a_j = r \omega^j$, these are the vertices of a regular $n$-gon inside the circle of radious $r$.
Now for which $z \in \mathbb{C}$ on the unit circle, that is $|z|=1$, does ... |
I've read about the estimation lemma, also known as the ML-inequality, on Wikipedia and in the complex analysis textbook by Saff and Snider. According to both these references, what the lemma says is the following:
If a function $f$ is continuous on a given contour $\gamma$, and if $|f|$ is bounded above there by a ... |
Let $n \in \mathbb{N}$, $0<r<1$ and $\omega_j = \exp\left(\frac{2 \pi i}{n}\right)$ for $j = 0, 1, \ldots, n-1$ be the roots of unity.
Define $a_j = r \omega^j$, these are the vertices of a regular $n$-gon inside the circle of radious $r$.
Now for which $z \in \mathbb{C}$ on the unit circle, that is $|z|=1$, does ... |
I've read about the estimation lemma, also known as the ML-inequality, on Wikipedia and in the complex analysis textbook by Saff and Snider. According to both these references, what the lemma says is the following:
If a function $f$ is continuous on a given contour $\gamma$, and if $|f|$ is bounded above there by a ... |
Is the Gram-Schmidt Orthogonalization process for functions the same as it is for vectors? |
The answer is $n=15$.
To convince yourself it is true try to prove using induction the more general property: For a partition of the rectangle into $m\times m$ smaller rectangles (using $m-1$ straight lines parallel to each of the original rectangle's sides) the minimal number is $n=2m-1$ (In your case $m=8$).
Af... |
Is this a valid proof of the continuity of $f(x)=\sqrt{x}$ at $x=2$, using epsilon-delta definition? |
>In a topological space $(X,\mathcal{T})$, consider any descending sequence of nonempty open sets: $A_{1} \supset A_{2} \supset A_{3} \supset \dots$, when does there exist a a descending sequence of nonempty closed sets $C_{1} \supset C_{2} \supset C_{3} \supset\dots$ such that for all $i\in \mathbb{N}$, $C_{i} \subset... |
Select a subset of the set $ \{1, 2, . . . , 200 \}$ in such a way that it contains the same number of even and odd elements. In how many ways can this be done?
My solution:
To ensure that a subset has an equal number of even and odd numbers we need to choose $k$ even numbers (out of $100$) and choose $k$ odd num... |
Let $A=\frac{\mathbb C[X,Y]}{(X^3-Y^2)}$. I am asked to show that $\mathbf m=(\overline X-1,\overline Y-1)$ is a maximal ideal of $A$ which I have shown successfully. Now I am asked to compute the $\mu(\mathbf mA_m)$ which is the unique cardinality of a minimal generating set of $\mathbf mA_m$ which I am unable to do. ... |
How to compute a minimal spanning set and the minimal spanning number of $\Bbb C[x,y]_{(x-1,y-1)}/(x^3-y^2)$? |
I've read about the estimation lemma, also known as the ML-inequality, on Wikipedia and in the complex analysis textbook by Saff and Snider. According to both these references, what the lemma says is the following:
If a function $f$ is continuous on a given contour $\gamma$, and if $|f|$ is bounded above there by a ... |
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