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Mathematics Stack Exchange
18h ago
We are still in the saga of solving the 2002 qualifier. This question 6b has stumped me and I am mostly clueless about it:
Say $f:\mathbb{R}\rightarrow \mathbb{R}$ is bounded with a finite constant $B$ such that: $$\frac{|f(x+y)+f(x-y)-2f(x)|}{|y|}\leq B$$ Prove there exists $M(\lVert f \rVert_\infty, B)$ such that for all $x\not=y$: $$|f(x)-f(y)|\leq M |x-y|\left(1+\ln_+(\frac{1}{|x-y|})\right)$$ Where $\ln_+(x)=\max \{0,\ln(x)\}$
Intuitively this means that away from $y=x$, $f(x+y)\rightarrow f(x)$ linearly. Close to $x$, it is still true $f(x+y)\rightarrow f(x)$ but it is slightly perturb ..read more
Mathematics Stack Exchange
2d ago
The radius of the quarter circle is $6\sqrt 5$ and we assume that $OA= 5$ and $OC=10$. What is the maximum area of the blue triangle?
Interpreting the problem statement, I believe that points $A$ and $C$ are fixed and point $B$ can move on the arc. To solve this problem, I assumed that the coordinate of $O$ is $(0,0)$ and then assigned coordinates for each vertex of the triangle: $A(5,0), C(0,10), B(x,\sqrt{180-x^2})$ where $x \in [0, 6\sqrt5]$. Then I applied the formula for the area of the triangle given its vertices, and the problem is reduced to maximizing
$$A(x)= \left|25-(\frac52\sqrt ..read more
Mathematics Stack Exchange
2d ago
Well of course $0^\sharp$ cannot be added by forcing. What I mean is the following. Suppose $0^\sharp$ exists. Let $\mathbb{P}=\mathrm{Add}(\omega,1)$ be the Cohen poset, and $L^\mathbb{P}$ be the class of $\mathbb{P}$-names in $L$. Let $W=\{\sigma[0^\sharp]:\sigma\in L^\mathbb{P}\}$ where $\sigma[0^\sharp]$ is the interpretation of the name $\sigma$ using (the filter corresponding to) $0^\sharp$. Is it true that $W=L[0^\sharp]$? Clearly $W\subseteq L[0^\sharp]$. For the reverse direction, it's enough to show $W\models\mathsf{ZF}$, for which it's enough to show $W$ is closed under Gödel operat ..read more
Mathematics Stack Exchange
2d ago
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\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{19},\textbf{23},\textbf{-1}\}$$ because every prime except $2,3$ and $5$ does belong to one of theses classes of numbers. There is a lot of things to say about $\mathbb Z/30\mathbb Z$, but those are the types of objects that interest me : $$\boxed{J\overset{def1}=\{p\in (\mathbb ..read more
Mathematics Stack Exchange
2d ago
How to determine the value of $\displaystyle f(x) = \sum_{n=1}^\infty\frac{\sqrt n}{n!}x^n$? No context, this is just a curiosity o'mine.
Yes, I am aware there is no reason to believe a random power series will have a closed form in terms of well established functions, but also I have no way to know if that is the case here, so that is why I'm asking. Do you know this power series or any method I could use to determine its value?
In my research I've found out about the polylogarithm, which is defined as $$\mathrm{Li}_s(x) = \sum_{n=1}^\infty\frac{x^n}{n^s} = \frac1{\Gamma(s)}\int_0^\infty\frac ..read more
Mathematics Stack Exchange
2d ago
We are still in the saga of solving the 2002 qualifier. This question 6b has stumped me and I am mostly clueless about it:
Say $f:\mathbb{R}\rightarrow \mathbb{R}$ is bounded with a finite constant $B$ such that: $$\frac{|f(x+y)+f(x-y)-2f(x)|}{|y|}\leq B$$ Prove there exists $M(\lVert f \rVert_\infty, B)$ such that for all $x\not=y$: $$|f(x)-f(y)|\leq M |x-y|\left(1+\ln_+(\frac{1}{|x-y|})\right)$$ Where $\ln_+(x)=\max \{0,\ln(x)\}$
Intuitively this means that away from $y=x$, $f(x+y)\rightarrow f(x)$ linearly. Close to $x$, it is still true $f(x+y)\rightarrow f(x)$ but it is slightly perturb ..read more
Mathematics Stack Exchange
2d ago
Let two smooth $v_1$ and $v_2$ both satisfy the system
$$\partial_t{v}-\Delta v=f \quad \text{in} \quad U \times (0,\infty), $$ $$v = g \quad \text{on} \quad \partial U \times (0,\infty),$$
for some fixed given smooth $f: \bar{U}\times (0,\infty) \rightarrow \mathbb{R}$ and $g: \partial U \times (0,\infty).$ $U$ is open, bounded and $U \subset \mathbb{R}^n.$ Show that $$\sup_{x \in U} |v_1(t, x) − v_2(t, x)| \rightarrow 0,$$ as $t \rightarrow \infty.$
This is my work:
Let $ u =v_1 -v_2,$ it is sufficient to prove $\sup_{x \in U} |u(x,t)| \rightarrow 0,$ as $t \rightarrow \infty. (1)$
$u$ obeys ..read more
Mathematics Stack Exchange
2d ago
This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-dimensional Hilbert space. First, some context:
We note that, by concentration of measures for spheres, we have the following: Let $S^{n-1} = \{x \in \mathbb{R}^n: \|x\|_2 = 1\}$ denote the unit sphere of $\mathbb{R}^n$. Let $v_n \in S^{n-1}$ be randomly chosen according to the canonical probability measure on $S^{n-1}$. For any fixed $\epsilon > 0$, as $n \to \infty$, the probability that $v_n$ lies $\epsilon$-close to any given equator goes to $1$. See, for exampl ..read more
Mathematics Stack Exchange
2d ago
I was watching a video on "system identification" in control theory, in which the creator says that we don't have solutions to nonlinear systems. And I have heard this many times in many contexts, related to control problems or nonlinear odes, etc. I think I am reacting to these kinds of blanket statements, and I would like to understand more precisely what is meant.
But I wanted to understand precisely what is meant that we can't solve nonlinear systems? Indeed, there are probably hundreds of questions on Math SE regarding numerical solutions to nonlinear systems. There are many algorithms fo ..read more
Mathematics Stack Exchange
2d ago
Let $(W_t)_{t\geq 0}$ be a standard one-dimensional Brownian motion and let
$$ Y_t := e^{-t}W(e^{2t}), \qquad X_t := \int^t_0 Y_s ds $$
Show that the joint process $(X_t,Y_t)$ is Markovian and find the generator of the process.
This is an exercise that I came across while reading a book. I can show the first part where the joint process is indeed Markovian. However, I don't know how to find the generator
$$ L[f](0,x) := \lim_{t \rightarrow 0}\frac{E_{(0,x)}[f(X_t,Y_t)] -f(0,x)}{t}. $$
for $f$ sufficiently regular.
My first thought is to use Ito, but I only know the form of $f(t,X_t)=...$ not t ..read more