Consider a Riemannian manifold with bounded Ricci curvature $|\mathrm{Ric}|\leq n-1$ and the noncollapsing lower volume bound $\mathrm{Vol}(B_1(p))>\mathrm{v}>0$. The first main result of this paper is to prove that we have $L^2$ $⨏_{B_1(p)}|\mathrm{Rm}|^2(x)\, dx \lt C(n,\mathrm{v})$,which proves conjecture. In order this, will need show following structural for limits. Namely, if $(M^n_j,d_j,...

In this paper, we consider Hankel operators on domains with bounded intrinsic geometry. For these characterize the $$L^2$$ -symbols where associated operator is compact (respectively bounded) space of square integrable holomorphic functions.

In Part I of this paper, we introduced a method of making two isomorphic intervals of a bounded lattice congruence equivalent. In this paper, we make one interval dominate another one. Let L be a bounded lattice, let [a, b] and [c, d] be intervals of L, and let φ be a homomorphism of [a, b] onto [c, d]. We construct a bounded (convex) extension K of L such that a congruence Θ of L has an extens...

We consider the problem of simultaneous nite gain L p-stabilization and internal stabilization of linear systems subject to input saturation via linear static state feedback. We show that bounded input nite-gain L p-stabilization and local asymptotic stabilization can always be achieved simultaneously no matter where the poles of the open loop system are, and the locations of these poles play a...

Solution: Since {xn} ∞ n=1 is a bounded sequence in R, the Bolzano-Weierstrass Theorem ensures that there is at least one convergent subsequence: let L be its limit. If the entire sequence does not converge to L, then for some > 0 there must be infinitely many terms of the sequence outside the ball B = (L− , L+ ). This infinite but bounded collection of terms likewise by Bolzano-Weierstrass has...