Dynamics of an inequality

The dynamics D∋(ri,ri+1)↦(ri+1,ri+2)∈D on the set D:={(x,y)∈R2:x>0,y>x2/2} is given by the recurrence ri+2=r2i+12+1r3i+1(ri+1−r2i2) for i=0,1,…. Questions: Is it true that the only periodic sequence (ri) here is the constant one with ri=1 for all i? Take any natural n. Suppose that the sequence (ri) is periodic with period n and r0⋯rn−1=1. Does it then always … Read more

Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function? For example lets say I have the function log(z2+a2) for a>0 and I choose my branch-cuts to be starting at ±ia and moving up and down the y−axis respectively. Now I am trying to integrate around a small circle around such … Read more

Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem has more complicated operator other then just $\Delta$ below. And it works in $BV$ space. But here I give … Read more

Well-definedness on C∞0(Rn)C_{0}^{\infty}(\mathbb{R}^{n})

Let T be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel K∈CZKα of order α>0 and b∈BMO(Rn). Then for f∈C∞0(Rn) define[b,T](f)=b⋅Tf−T(bf). Question. Is [b,T] is well-defined on C∞0(Rn) for all b∈BMO(Rn)? Answer AttributionSource : Link , Question Author : Timothy , Answer Author : Community

Classifying countable sets of weighted dots on a real line

Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two(or more) dots located at the same place whose weights sum is equal to that of the original dot. The countable sets of such dots have some property, let call it “class”. … Read more

Level sets of function of inner products of vectors on hypercube

Let $H = \{ 0, 1\}^d$ be the $d$-th Cartesian product of $\{0, 1\}$ in $\mathbb{R}^d$. Suppose $v_1, \ldots, v_k$ are $k$ vectors in $H$ in general position. We define function $F \colon H^{k}\rightarrow \mathbb{R}$ as \begin{align} F (u_1, u_2, \ldots, u_k) = \sum_{i,j = 1}^k f( | \langle v_i, u_j \rangle |), \end{align} where … Read more

The asymptotic behavior of the ratio between the largest two of nn i.i.d. chi-square random variables

My question is about the asymptotic behavior of the ratio between the largest and second largest values of n independent chi-square random variables. Let X1,…,Xn be n independent and identically distributed random variables with distribution χ21. Let X(n) be the largest and X(n−1) be the second largest of these n random variables. I was wondering … Read more

Approximation of monotone Sobolev functions

Let f∈W1,2loc(R2) be a continuous monotone (real valued) function (monotone in the sense that the maximum and minimum of f in a precompact open set are attained at the boundary). Is it true that there exists a sequence of smooth monotone functions fn converging to f in W1,2loc(R2)? We could ask the same question for … Read more

Operator topologies

Let L(H) be the space of bounded operators on some Hilbert space. We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT). It is immediate that the norm topology is stronger than the SOT which is again stronger than the WOT. My question is … Read more

Bessel in matrix?

Let Mn be the matrix M_n=\begin{pmatrix} 1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\ 1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \qquad\qquad\dots & 0\\ 1&\binom{3}{1}\binom{3-1}{1-1} &\binom{3}{2}\binom{3-1}{2-1} &\binom{3}{3}\binom{3-1}{3-1}\qquad\dots &0\\ \qquad \qquad \dots\\ \qquad \qquad \dots\\ 1&\binom{n}{1}\binom{n-1}{1-1} &\binom{n}{2}\binom{n-1}{2-1} &\binom{n}{3}\binom{n-1}{3-1}\qquad\dots &\binom{n}{n-1}\binom{n-1}{n-2}\end{pmatrix}. Question. It appears that the modified Bessel function of the first kind, of order 0, has the expansion I_0(x)=\exp\left(\sum_{j=1}^{\infty}\frac{(-1)^{j-1}\det(M_j)}{j!^2}\left(\frac{x}2\right)^{2j}\right). Is this true? Answer … Read more