On the relative strengths of fragments of collection

Let $\mathbf{M}$ be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, $\Delta_0$-separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to $\mathbf{M}$. We focus on two common parameterisations of collection: $\Pi_n$-collection, which is the usual collection scheme restricted to $\Pi_n$-formulae, and strong $\Pi_n$-collection, which is equivalent to $\Pi_n$-collection plus $\Sigma_{n+1}$-separation. The main result of this paper shows that for all $n \geq 1$, (1) $\mathbf{M}+\Pi_{n+1}\textrm{-collection}+\Sigma_{n+2}\textrm{-induction on } \omega$ proves the consistency of Zermelo Set Theory plus $\Pi_{n}$-collection, (2) the theory $\mathbf{M}+\Pi_{n+1}\textrm{-collection}$ is $\Pi_{n+3}$-conservative over the theory $\mathbf{M}+\textrm{strong }\Pi_n \textrm{-collection}$. It is also shown that (2) holds for $n=0$ when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity and $V=L$) that does not include the powerset axiom.Comment: 22 page

Definable orthogonality classes in accessible categories are small

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Levy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from the existence of a proper class of supercompact cardinals if S is \Sigma_2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a new hierarchy, and we show that Vopenka's principle is equivalent to the existence of C(n)-extendible cardinals for all n. As a consequence, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This result follows from the fact that cohomology equivalences are \Sigma_2. In contrast with this fact, homology equivalences are \Sigma_1, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.Comment: 38 pages; some results have been improved and former inaccuracies have been correcte

Pressure buildup during CO2 injection in brine aquifers using the Forchheimer equation

If geo-sequestration of CO2 is to be employed as a key emissions reduction method in the global effort to mitigate climate change, simple yet robust screening of the risks of disposal in brine aquifers will be needed. There has been significant development of simple analytical and semi-analytical techniques to support screening analysis and performance assessment for potential carbon sequestration sites. These techniques have generally been used to estimate the size of CO2 plumes for the purpose of leakage rate estimation. A common assumption has been that both the fluids and the geological formation are incompressible. Consequently, calculation of pressure distribution requires the specification of an arbitrary radius of influence. In this talk, a new similarity solution is derived using the method of matched asymptotic expansions. By allowing for slight compressibility in the fluids and formation, the solution improves on previous work by not requiring the specification of an arbitrary radius of influence. A large-time approximation of the solution is then extended to account for non-Darcy inertial effects using the Forchheimer equation. Both solutions are verified by comparison with finite difference solutions. The results show that inertial losses will often be comparable, and sometimes greater than, the viscous Darcy-like losses associated with the brine displacement, although this is strongly dependent on formation porosity and permeability

Locating regions in a sequence under density constraints

Several biological problems require the identification of regions in a sequence where some feature occurs within a target density range: examples including the location of GC-rich regions, identification of CpG islands, and sequence matching. Mathematically, this corresponds to searching a string of 0s and 1s for a substring whose relative proportion of 1s lies between given lower and upper bounds. We consider the algorithmic problem of locating the longest such substring, as well as other related problems (such as finding the shortest substring or a maximal set of disjoint substrings). For locating the longest such substring, we develop an algorithm that runs in O(n) time, improving upon the previous best-known O(n log n) result. For the related problems we develop O(n log log n) algorithms, again improving upon the best-known O(n log n) results. Practical testing verifies that our new algorithms enjoy significantly smaller time and memory footprints, and can process sequences that are orders of magnitude longer as a result.Comment: 17 pages, 8 figures; v2: minor revisions, additional explanations; to appear in SIAM Journal on Computin

Noise and thermal stability of vibrating micro-gyrometers preamplifiers

The preamplifier is a critical component of gyrometer's electronics. Indeed the resolution of the sensor is limited by its signal to noise ratio, and the gyrometer's thermal stability is limited by its gain drift. In this paper, five different kinds of preamplifiers are presented and compared. Finally, the design of an integrated preamplifier is shown in order to increase the gain stability while reducing its noise and size.Comment: Submitted on behalf of EDA Publishing Association (http://irevues.inist.fr/EDA-Publishing

Exploiting the causal tensor network structure of quantum processes to efficiently simulate non-Markovian path integrals

In the path integral formulation of the evolution of an open quantum system coupled to a Gaussian, non-interacting environment, the dynamical contribution of the latter is encoded in an object called the influence functional. Here, we relate the influence functional to the process tensor -- a more general representation of a quantum stochastic process -- describing the evolution. We then use this connection to motivate a tensor network algorithm for the simulation of multi-time correlations in open systems, building on recent work where the influence functional is represented in terms of time evolving matrix product operators. By exploiting the symmetries of the influence functional, we are able to use our algorithm to achieve orders-of-magnitude improvement in the efficiency of the resulting numerical simulation. Our improved algorithm is then applied to compute exact phonon emission spectra for the spin-boson model with strong coupling, demonstrating a significant divergence from spectra derived under commonly used assumptions of memorylessness.Comment: 6+5 pages, 4 figure

A discrete memory-kernel for multi-time correlations in non-Markovian quantum processes

Efficient simulations of the dynamics of open systems is of wide importance for quantum science and tech-nology. Here, we introduce a generalization of the transfer-tensor, or discrete-time memory kernel, formalism to multi-time measurement scenarios. The transfer-tensor method sets out to compute the state of an open few-body quantum system at long times, given that only short-time system trajectories are available. Here, we showthat the transfer-tensor method can be extended to processes which include multiple interrogations (e.g. measurements) of the open system dynamics as it evolves, allowing us to propagate high order short-time correlation functions to later times, without further recourse to the underlying system-environment evolution. Our approach exploits the process-tensor description of open quantum processes to represent and propagate the dynamics in terms of an object from which any multi-time correlation can be extracted. As an illustration of the utility of the method, we study the build-up of system-environment correlations in the paradigmatic spin-boson model, and compute steady-state emission spectra, taking fully into account system-environment correlations present in the steady state.Comment: 9 pages, 2 figure

Protocol for the Reconstructing Consciousness and Cognition (ReCCognition) Study

Important scientific and clinical questions persist about general anesthesia despite the ubiquitous clinical use of anesthetic drugs in humans since their discovery. For example, it is not known how the brain reconstitutes consciousness and cognition after the profound functional perturbation of the anesthetized state, nor has a specific pattern of functional recovery been characterized. To date, there has been a lack of detailed investigation into rates of recovery and the potential orderly return of attention, sensorimotor function, memory, reasoning and logic, abstract thinking, and processing speed. Moreover, whether such neurobehavioral functions display an invariant sequence of return across individuals is similarly unknown. To address these questions, we designed a study of healthy volunteers undergoing general anesthesia with electroencephalography and serial testing of cognitive functions (NCT01911195). The aims of this study are to characterize the temporal patterns of neurobehavioral recovery over the first several hours following termination of a deep inhaled isoflurane general anesthetic and to identify common patterns of cognitive function recovery. Additionally, we will conduct spectral analysis and reconstruct functional networks from electroencephalographic data to identify any neural correlates (e.g., connectivity patterns, graph-theoretical variables) of cognitive recovery after the perturbation of general anesthesia. To accomplish these objectives, we will enroll a total of 60 consenting adults aged 20–40 across the three participating sites. Half of the study subjects will receive general anesthesia slowly titrated to loss of consciousness (LOC) with an intravenous infusion of propofol and thereafter be maintained for 3 h with 1.3 age adjusted minimum alveolar concentration of isoflurane, while the other half of subjects serves as awake controls to gauge effects of repeated neurobehavioral testing, spontaneous fatigue and endogenous rest-activity patterns

Dynamical Surface Gravity in Spherically Symmetric Black Hole Formation

We study dynamical surface gravity in a general spherically symmetric setting using Painlev\'{e}-Gullstrand (PG) coordinates. Our analysis includes several definitions that have been proposed in the past as well as two new definitions adapted to PG coordinates. Various properties are considered, including general covariance, value at extremality, locality and static limit. We illustrate with specific examples of "dirty" black holes that even for spacetimes possessing a global timelike Killing vector, local definitions of surface gravity can differ substantially from "non-local" ones that require an asymptotic normalization condition. Finally, we present numerical calculations of dynamical surface gravity for black hole formation via spherically symmetric scalar field collapse. Our results highlight the differences between the various definitions in a dynamical setting and provide further insight into the distinction between local and non-local definitions of surface gravity.Comment: Final version to appear in Phys. Rev. D. Slight name change, further improvements to numerics and presentation, 25 pages, 7 figure

Selfdual 2-form formulation of gravity and classification of energy-momentum tensors

It is shown how the different irreducibility classes of the energy-momentum tensor allow for a Lagrangian formulation of the gravity-matter system using a selfdual 2-form as a basic variable. It is pointed out what kind of difficulties arise when attempting to construct a pure spin-connection formulation of the gravity-matter system. Ambiguities in the formulation especially concerning the need for constraints are clarified.Comment: title changed, extended versio