Osculating ellipsoids

Let K be a given smooth, origin-symmetric, strictly convex body in n dimensional euclidean space. At every point x on the boundary of K there exists an origin-symmetric ellipsoid Ex that touches x of second-order, the osculating ellipsoid at x. Denote the family of osculating ellipsoids by F:={Ex:x∈boundary of K}. Moreover, set G:={TE:E∈F & T∈SL(n)}. Is it true … Read more

Steklov averages in PDE: what to do when we have time-dependent elliptic operator

One may have an equation (with boundary conditions omitted below) ut−Au=f u(0)=u0 which has a weak solution u∈L2(0,T;V)∩C([0,T];H) in the sense that −∫T0∫Ωu(t)ϕt(t)+∫T0∫ΩA12u(t)A12ϕ(t)=∫T0∫Ωf(t)ϕ(t)+∫Ωu0ϕ(0) for all test functions ϕ vanishing at t=T. Since we do not know if ut exists or is a function, the concept of Steklov averages vh(t)=1h∫t+htv(s)ds is useful, since the weak formulation … 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

Weakenings of the Bounded Approximation Property

Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$. Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that $S_i\to\mathrm{id}_X$ pointwise, that is, $\|S_i(x)-x\|\to 0$ for every $x\in X$. What is the connection to the following variants of this property: (Explanation of … Read more

The weak-star closure of closed left ideals corresponding to pure states

I asked this question at math.stackexchange and received no comment. Let A be a C*-algebra and ϕ be a positive linear functional on A. Let ˜ϕ be its unique w∗-continuous extension on A∗∗. Let us put Nϕ:={a∈A:ϕ(a∗a)=0}   ,   N˜ϕ:={x∈A∗∗:ϕ(x∗x)=0} Nϕ forms a closed left ideal in A and N˜ϕ forms a w∗-closed left ideal in A∗∗. It … Read more

Compactly supported distributions as a projective G-module

For a Lie group G and a locally convex space V let E(G,V) be the locally convex space of smooth functions from G to V, and accordingly E′c(G,V) the space of compactly supported distributions. A G-module V is called differentiable if v→(g→gv) defines a continuous map from V to E(G,V) for all v∈V. In particular … Read more

Functional equations about Conway’s box function

Conway’s box function is the inverse of Minkowski’s question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). The question is are there functional equations known about the function, which would allow recursive computation? Answer AttributionSource : Link , Question Author : Dimiter P … Read more

A strongly open set which is not measurable in the weak operator topology

Let H be a non-separable Hilbert space and {ei}i∈I be an orthonormal basis for H. Let J be a uncountable proper subset in I. Let us put E={x∈B(H):‖ One may check that E is an open set in the strong operator topology but not in the weak operator topology. Question1: I feel E is not … Read more