While any fault-tolerant asynchronous consensus algorithm requires two communication steps even in failure-free executions, it is known that we can construct an algorithm terminating in one step for some good inputs (e.g. all processes propose a same value). In this paper, we present the necessary and sufficient constraint for the set of inputs for which we can construct an asynchronous consens...

The solvability and Fredholm properties of refinement equations in spaces of square-integrable functions are studied. Necessary and jointly necessary and sufficient conditions for the solvability of homogeneous and nonhomogeneous refinement equations are established. It is shown that in the space L2(R) the kernel space of any homogeneous equation with a non-trivial solution is infinite dimensio...

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted L classes. We establish: (1) Mapping properties for the double and single layer potentials, as well as the Newton potential; (2) Extrapolation-type solvability results: the fact that solv...

This paper is concerned with a mean-field linear quadratic (LQ, for short) optimal control problem with deterministic coefficients. It is shown that convexity of the cost functional is necessary for the finiteness of the mean-field LQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex ...

We have earlier shown by construction that a proposition can have a well-defined nonzero probability, even if it is justified by an infinite probabilistic regress. We thought this to be an adequate rebuttal of foundationalist claims that probabilistic regresses must lead either to an indeterminate, or to a determinate but zero probability. In a comment, Frederik Herzberg has argued that our cou...

where 2* = 2 N / ( N 2) is the critical Sobolev exponent for the embedding H’(52) C L2* (0). Criticality and subcriticality play important rBles concerning the solvability of equation (1.1). Using (by now classical) variational methods one sees that equation (1.1) is solvable i f f has subcritical growth (and satisfies some additional hypotheses), while for f(u) = IuIP-~u the well-known Pohoiae...

A nonassociative algebra is nilpotent if there is some n such that the product of any n elements, no matter how they are associated, is zero. Several related, but more general, notions are left nilpotency, solvability, local nilpotency, and nillity. First the complexity of several decision problems for these properties is examined. In finite-dimensional algebras over a finite field it is shown ...