Collapse models with non-white noises II: particle-density coupled noises

We continue the analysis of models of spontaneous wave function collapse with stochastic dynamics driven by non-white Gaussian noise. We specialize to a model in which a classical "noise" field, with specified autocorrelator, is coupled to a local nonrelativistic particle density. We derive general results in this model for the rates of density matrix diagonalization and of state vector reduction, and show that (in the absence of decoherence) both processes are governed by essentially the same rate parameters. As an alternative route to our reduction results, we also derive the Fokker-Planck equations that correspond to the initial stochastic Schr\"odinger equation. For specific models of the noise autocorrelator, including ones motivated by the structure of thermal Green's functions, we discuss the qualitative and qantitative dependence on model parameters, with particular emphasis on possible cosmological sources of the noise field.Comment: Latex, 43 pages; versions 2&3 have minor editorial revision

Exponential mixing for the Teichmuller flow

We study the dynamics of the Teichmuller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Holder observables. A geometric consequence is that the \SL(2,\R) action in the moduli space has a spectral gap.Comment: 49 page

Intuitionistic S4 is decidable

In this paper we demonstrate decidability for the intuitionistic modal logic S4 first formulated by Fischer Servi. This solves a problem that has been open for almost thirty years since it had been posed in Simpson's PhD thesis in 1994. We obtain this result by performing proof search in a labelled deductive system that, instead of using only one binary relation on the labels, employs two: one corresponding to the accessibility relation of modal logic and the other corresponding to the order relation of intuitionistic Kripke frames. Our search algorithm outputs either a proof or a finite counter-model, thus, additionally establishing the finite model property for intuitionistic S4, which has been another long-standing open problem in the area.Comment: 13 pages conference paper + 26 pages appendix with examples and proof

Marshall vs. Walras on Equilibrium and Disequilibrium

In all introductory and intermediate textbooks in microeconomics, price theory is at first developed in a partial equilibrium framework which, though invariably Marshallian from the point of view of the graphical apparatus, is almost always Walrasian from the point of view of the analysis. In discussing the well-known Marshallian graphical treatment of the equilibrium problem of a single, isolated, assumedly "competitive" market for a producible consumers' good, a few diligent textbook writers 2 occasionally point out that placing the price and quantity variables on the ordinate and abscissa axes, respectively, is somewhat incongruous with the role played by the same two variables in the analytical development of the theory: for, from a Walrasian perspective, price is the independent variable, while quantity is the dependent one (whatever this may mean). Such incongruousness is sometimes historically justified by recalling that price had indeed been regarded as the independent variable by Marshall, to whom the demand-and-supply graphical apparatus can be traced back; but this suggestion, never elaborated upon, hovers about as a mysterious reference to a by now forgotten past. As a matter of fact, in the partial equilibrium framework of an isolated market, to which microeconomic primers confine most of their discussions of price theory, almost all methodological, epistemological, as well as analytical distinctions between Marshall's and Walras's approaches are skipped over. Moreover, when intermediate and advanced microeconomic textbooks eventually deal with the issue of price formation in a multi-market framework, general equilibrium theory is invariably presented as the natural extension of partial equilibrium analysis, as if Walrasian and Marshallian approaches to price theory only differed in scope and intended applications, being otherwise essentially similar in their foundations and results 3 . In this paper we want to oppose the received view on the basic equivalence of the two traditional approaches to price theory. Specifically, we want to show that Marshall's analysis of the equilibration process and his related interpretation of the equilibrium concept are essentially different from, and irreducible to, Walras's analysis and interpretation. Further, we want to show that the patent difference in scope of their respective theories (that is, partial vs. general analysis), far from being an accidental outcome of history or the innocuous consequence of the idiosyncratic preferences of the two economists, is in effect the unavoidable and irremediable by-product of their different analytical assumptions and explanatory aims. To this end, we shall first identify a common ground for our discussion, that is, a model economy that, being explicitly examined by both authors in their respective writings, will allow us to contrast their analyses and to precisely single out what distinguishes them from one another. Such model economy is represented by the two-commodity, pure-exchange economy which is dealt with by both Walras and Marshall right at the beginning of their respective expositions of price theory, under the assumption that there exists a fixed finite number of traders in the economy. Yet, while Walras's analysis is entirely developed under the assumption of an arbitrary finite number of traders (greater than or equal to two), in Marshall's case one finds a distinction between two sorts of economies: the former, called by Marshall a "barter" economy and more recently referred to as an "Edgeworth Box" economy, is an economy where the number 1 I would like to thank Michel De Vroey, Antonio Guccione, and Enrico Minelli for their comments and suggestions. The usual disclaimer applies

Superconducting qubits for quantum annealing applications

Over the last two decades, Quantum Annealing (QA) has grown to be a commercial technology with machines reaching the scale of 5000 interconnected qubits. Two reasons for this progress are the relative ease of implementing adiabatic Hamiltonian control and QA’s partial robustness against errors caused by decoherence. Despite the success of this approach to quantum computation, proving a scaling advantage over classical computation remains an elusive goal to this date. Different strategies are therefore being considered to boost the performance of quantum annealing. These include using more coherent qubit architectures and error-suppression to limit the effect of environmental noise, implementing non-stoquastic driver terms and tailored annealing schedules to enhance the success probability of the algorithm, and using many-body couplers to embed higher-order binary optimisation problems with less resource overhead. This thesis contributes to these efforts in two different ways. The first part provides a detailed numerical analysis and a physical layout for a threebody coupler for flux qubits based on ancillary spins. The application of the coupler in a coherence-signature QA Hamiltonian is also considered and the results of the simulated quantum evolution are compared to the outcomes of classical optimisation on the problem Hamiltonian showing that the classical algorithms cannot correctly reproduce the state distribution at the end of QA. In the second part of the thesis, we develop a numerical method for mapping the Hamiltonian of a composite superconducting circuit to an effective many-qubit Hamiltonian. By overcoming drawbacks of standard reduction methods, this protocol can be used to guide the design of non-stoquastic and many-body Hamiltonian terms, as well as to get a more precise evaluation of the QA schedule parameters, which can greatly improve the outcomes of the optimisation. This numerical work is accompanied by a proposal for an experimental verification of the predictions of the reduction protocol and by some preliminary experimental results

Frontiers of Conditional Logic

Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional obligation). Despite the close connections between conditional and modal logic, both the technical development and philosophical exploitation of the latter has outstripped that of the former, with the result that noticeable lacunae exist in the literature on conditional logic. My dissertation addresses a number of these underdeveloped frontiers, producing new technical insights and philosophical applications. I contribute to the solution of a problem posed by Priest of finding sound and complete labeled tableaux for systems of conditional logic from Lewis\u27 V-family. To develop these tableaux, I draw on previous work on labeled tableaux for modal and conditional logic; errors and shortcomings in recent work on this problem are identified and corrected. While modal logic has by now been thoroughly studied in non-classical contexts, e.g. intuitionistic and relevant logic, the literature on conditional logic is still overwhelmingly classical. Another contribution of my dissertation is a thorough analysis of intuitionistic conditional logic, in which I utilize both algebraic and worlds semantics, and investigate how several novel embedding results might shed light on the philosophical interpretation of both intuitionistic logic and conditional logic extensions thereof. My dissertation examines deontic and connexive conditional logic as well as the underappreciated history of connexive notions in the analysis of conditional obligation. The possibility of interpreting deontic modal logics in such systems (via embedding results) serves as an important theoretical guide. A philosophically motivated proscription on impossible obligations is shown to correspond to, and justify, certain (weak) connexive theses. Finally, I contribute to the intensifying debate over counterpossibles, counterfactuals with impossible antecedents, and take—in contrast to Lewis and Williamson—a non-vacuous line. Thus, in my view, a counterpossible like If there had been a counterexample to the law of the excluded middle, Brouwer would not have been vindicated is false, not (vacuously) true, although it has an impossible antecedent. I exploit impossible (non-normal) worlds—originally developed to model non-normal modal logics—to provide non-vacuous semantics for counterpossibles. I buttress the case for non-vacuous semantics by making recourse to both novel technical results and theoretical considerations