This paper is devoted to the multivariable $$H^\infty $$ functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if $$(A_1,\ldots , A_d)$$ such family, $$A_k$$ R-sectorial R-type $$\omega _k\in (0,\pi )$$ $$k=1,\ldots ,d$$ and admits bounded (\Sigma _{\theta _1}\times \cdots \times \Sigma _d})$$ joint for some $$\theta (\omega _k...

For every k > 3, we give a construction of planar point sets with many collinear k-tuples and no collinear (k + 1)-tuples. We show that there are n0 = n0(k) and c = c(k) such that if n ≥ n0, then there exists a set of n points in the plane that does not contain k + 1 points on a line, but it contains at least n 2− c √ logn collinear k-tuples of points. Thus, we significantly improve the previou...

Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are...

We study the variant of the k-local hamiltonian problem which is a natural generalization of k-CSPs, in which the hamiltonian terms all commute. More specifically, we consider a hamiltonian H = ∑i Hi over n qubits, where each Hi acts non-trivially on O(log n) qubits and all the terms Hi commute, and show the following 1. We show that a specific case of O(log n) local commuting hamiltonians over...

Following the spirit of a recent work of one of the authors (J. Phys. A: Math. Theor. 44 (2011) 045301), the essential structure of the generalized Pauli group of a qubit-qudit, where d = 2 and an integer k ≥ 2, is recast in the language of a finite geometry. A point of such geometry is represented by the maximum set of mutually commuting elements of the group and two distinct points are regard...