Let ? be an open connected subset of the complex plane C and let T be a bounded linear operator on a Hilbert space H. For ? in ? let e the orthogonal projection onto the null-space of T-?I . We discuss the necessary and sufficient conditions for the map ?? to b e continuous on ?. A generalized Gram- Schmidt process is also given.

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in Hausdorff based on locally compactable neighborhood systems without considering strictly (metrized) Borel variety. null extension gives rise to quasi sigma-semiring sigma-neighborhoods, which are rectifiable view Dieudonné n-space. concepts symmetric signed measure, un...

Consider the semidefinite relaxation (SDR) of the quadratic integer program (QIP): γ := maxx∈{−1,1}n xT Qx ≤ minD−Qo0 trace(D) =: γ̄ where Q is a given symmetric matrix and D is diagonal. In this paper we consider the relaxation gap γ̄ − γ. We establish the uniqueness of the solution of the semidefinite relaxation problem and prove that γ = γ̄ if and only if γr := 1 n maxx∈{−1,1}n xT V V T x = 1 w...

This paper deals with the stability of null-space velocity control algorithms in extended operational space for redundant robots. We compare the performance of the control algorithm based on the minimal null-space projection and generalized-inverse-based projection into the Jacobian nullspace. We show how the null-space projection affects the performance of the null-space tracking algorithm. Th...

A. The filling length of an edge-circuit η in the Cayley 2-complex of a finitely presented group is the least integer L such that there is a combinatorial null-homotopy of η down to a basepoint through loops of length at most L. We introduce similar notions in which the null-homotopy is not required to fix a basepoint, and in which the contracting loop is allowed to bifurcate. We exhibit...

We seek the solution of banded, Toeplitz, Hankel, Vandermonde, Cauchy and other structured linear systems of equations with integer coefficients. By combining Hensel’s symbolic lifting with either divide-and-conquer algorithms or numerical iterative refinement, we unify the solution for all these structures. We yield the solution in nearly optimal randomized Boolean time, which covers both solu...