A general construction of transmutation operators is developed for selfadjoint operators in Gelfand triples. Theorems regarding analyticity of generalized eigenfunctions and Paley-Wiener properties are proved.
We study properties of quantum strategies, which are complete specifications of a given partys actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantum strategies that generalizes the Choi-Jamio{l}kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting only on the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simple proof of Kitaevs lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games.
We recall some of the fundamental achievements of formal deformation quantization to argue that one of the most important remaining problems is the question of convergence. Here we discuss different approaches found in the literature so far. The recent developments of finding convergence conditions are then outlined in three basic examples: the Weyl star product for constant Poisson structures, the Gutt star product for linear Poisson structures, and the Wick type star product on the Poincare disc.
We study products of general topological spaces with Mengers covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming CH{}, every productively Lindelof space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.
The collapse of large social systems, often referred to as civilizations or empires, is a well known historical phenomenon, but its origins are the object of an unresolved debate. In this paper, we present a simple biophysical model which we link to the concept that societies collapse because of the diminishing returns of complexity proposed by Joseph Tainter. Our model is based on the description of a socioeconomic system as a trophic chain of energy stocks which dissipate the energy potential of the available resources. The model produces various trajectories of decline, in some cases rapid enough that they can be defined as collapses. At the same time, we observe that the exploitation of the resource stock (production) has a strongly nonlinear relationship with the complexity of the system, assumed to be proportional to the size of the stock termed bureaucracy. These results provide support for Tainter s hypothesis.
Dilation theory is a paradigm for studying operators by way of exhibiting an operator as a compression of another operator which is in some sense well behaved. For example, every contraction can be dilated to (i.e., is a compression of) a unitary operator, and on this simple fact a penetrating theory of non-normal operators has been developed. In the first part of this survey, I will leisurely review key classical results on dilation theory for a single operator or for several commuting operators, and sample applications of dilation theory in operator theory and in function theory. Then, in the second part, I will give a rapid account of a plethora of variants of dilation theory and their applications. In particular, I will discuss dilation theory of completely positive maps and semigroups, as well as the operator algebraic approach to dilation theory. In the last part, I will present relatively new dilation problems in the noncommutative setting which are related to the study of matrix convex sets and operator systems, and are motivated by applications in control theory. These problems include dilating tuples of noncommuting operators to tuples of commuting normal operators with a specified joint spectrum. I will also describe the recently studied problem of determining the optimal constant $c = c_{theta,theta}$, such that every pair of unitaries $U,V$ satisfying $VU = e^{itheta} UV$ can be dilated to a pair of $cU, cV$, where $U,V$ are unitaries that satisfy the commutation relation $VU = e^{itheta} UV$. The solution of this problem gives rise to a new and surprising application of dilation theory to the continuity of the spectrum of the almost Mathieu operator from mathematical physics.