Stashing And Parallelization Pentagons

Parallelization is an algebraic operation that lifts problems to sequences in a natural way. Given a sequence as an instance of the parallelized problem, another sequence is a solution of this problem if every component is instance-wise a solution of the original problem. In the Weihrauch lattice parallelization is a closure operator. Here we introduce a dual operation that we call stashing and that also lifts problems to sequences, but such that only some component has to be an instance-wise solution. In this case the solution is stashed away in the sequence. This operation, if properly defined, induces an interior operator in the Weihrauch lattice. We also study the action of the monoid induced by stashing and parallelization on the Weihrauch lattice, and we prove that it leads to at most five distinct degrees, which (in the maximal case) are always organized in pentagons. We also introduce another closely related interior operator in the Weihrauch lattice that replaces solutions of problems by upper Turing cones that are strong enough to compute solutions. It turns out that on parallelizable degrees this interior operator corresponds to stashing. This implies that, somewhat surprisingly, all problems which are simultaneously parallelizable and stashable have computability-theoretic characterizations. Finally, we apply all these results in order to study the recently introduced discontinuity problem, which appears as the bottom of a number of natural stashing-parallelization pentagons. The discontinuity problem is not only the stashing of several variants of the lesser limited principle of omniscience, but it also parallelizes to the non-computability problem. This supports the slogan that "non-computability is the parallelization of discontinuity".

Non-Naturalism and Reasons-Firstism: How to Solve the Discontinuity Problem by Reducing Two Queerness Worries to One

A core tenet of metanormative non-naturalism is that genuine or robust normativity—i.e., the kind of normativity that is characteristic of moral requirements, and perhaps also of prudential, epistemic and even aesthetic requirements—is metaphysically special in a way that rules out naturalist analyses or reductions; on the non-naturalist view, the normative is sui generis and metaphysically discontinuous with the natural (or descriptive or non-normative). Non-naturalists agree, however, that the normative is modally as well as explanatorily dependent on the natural. These two commitments—discontinuity and dependence—at least initially pull in opposite directions, and one of the central challenges to non-naturalism is how to reconcile them. In this paper I spell out the most pressing version of this discontinuity problem, as I propose to call it, and I go on to offer a novel solution. Drawing on the ideology of reasons-firstism, I formulate an account of normative explanation which reconciles the two commitments, and I argue that competing accounts either do not solve the problem or are implausible on independent grounds.

Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem

In this work, goal-oriented adjoint-based a posteriori error estimates are derived for a nonlinear phase-field discontinuity problem in which a scalar-valued displacement field interacts with a scalar-valued smoothed indicator function. The latter is subject to an irreversibility constraint, which is regularized using a simple penalization strategy. The main advancements in the current work are error identities, resulting estimators, and two-sided estimates employing the dual-weighted residual method, which address the influence of the phase-field regularization, penalization, and spatial discretization parameters. Some numerical tests accompany our derived estimates.

Numerical dissipation of flux splitting schemes for contact discontinuities

The contact discontinuity is simulated by three kinds of flux splitting schemes to evaluate and analyse the influence of numerical dissipation in this paper. The numerical results of one-dimensional contact discontinuity problem show that if the flow velocity on both sides of the contact discontinuity is not simultaneously supersonic, the non-physical pressure and velocity waves may occur when the initial theoretically contact discontinuity is smeared into a transition zone spanning several grid-cells caused by numerical dissipations. Since these non-physical waves have no effect on the corresponding density dissipation, this paper considers these fluctuations as only numerical errors and are not part of the numerical dissipation. In addition, for two-dimensional flow field, the characteristics of high-order accuracy difference schemes, i.e. low dissipation and high resolution, may induce the multi-dimensional non-physical waves that interfere with each other to produce more complex non-physical flow structures, so the fluctuations in the calculated results should be treated with caution.

Saccades along spatial neural circuit discontinuities

Saccades are realized by six extraocular muscles that define the final reference frame for eyeball rotations. However, upstream of the nuclei innervating the eye muscles, eye movement commands are represented in two-dimensional retinocentric coordinates, as is the case in the superior colliculus (SC). In such spatial coordinates, the horizontal and vertical visual field meridians, relative to the line of sight, are associated with neural tissue discontinuities due to routing of binocular retinal outputs when forming retinotopic sensory-motor maps. At the level of the SC, a functional discontinuity along the horizontal meridian was additionally discovered, beyond the structural vertical discontinuity associated with hemifield lateralization. How do such neural circuit discontinuities influence purely cardinal saccades? Using thousands of saccades from 3 rhesus macaque monkeys and 14 human subjects, we show how the likelihood of purely horizontal or vertical saccades is infinitesimally small, nulling a discontinuity problem. This does not mean that saccades are sloppy. On the contrary, saccades exhibit remarkable direction and amplitude corrections to account for small initial eye position deviations due to fixational variability: “purely” cardinal saccades can deviate, with an orthogonal component of as little as 0.03 deg, to correct for tiny target position deviations from initial eye position. In humans, probing perceptual target localization additionally revealed that saccades show different biases from perception when targets deviate slightly from purely cardinal directions. These results demonstrate a new functional role for fixational eye movements in visually-guided behavior, and they motivate further neurophysiological investigations of saccade trajectory control in the brainstem.New and NoteworthyPurely cardinal saccades are often characterized as being straight. We show how a small amount of curvature is inevitable, alleviating an implementational problem of dealing with neural circuit discontinuities in the representations of the visual meridians. The small curvature functionally corrects for minute variability in initial eye position due to fixational eye movements. Saccades are far from sloppy; they deviate by as little as <1% of the total vector size to adjust their landing position.

DEMONSTRATION OF A LINEAR PROLONGATION CMFD METHOD ON MOC

Coarse Mesh Finite Difference (CMFD) method is a very effective method to accelerate the iterations for neutron transport calculation. But it can degrade and even fail when the optical thickness of the mesh becomes large. Therefore several methods, including partial current-based CMFD (pCMFD) and optimally diffusive CMFD (odCMFD), have been proposed to stabilize the conventional CMFD method. Recently, a category of “higherorder” prolongation CMFD (hpCMFD) methods was proposed to use both the local and neighboring coarse mesh fluxes to update the fine cell flux, which can solve the fine cell scalar flux discontinuity problem between the fine cells at the bounary of the coarse mesh. One of the hpCMFD methods, refered as lpCMFD, was proposed to use a linear prolongation to update the fine cell scalar fluxes. Method of Characteristics (MOC) is a very popular method to solve neutron transport equations. In this paper, lpCMFD is applied on the MOC code MPACT for a variety of fine meshes. A track-based centroids calculation method is introduced to find the centroids coordinates for random shapes of fine cells. And the numerical results of a 2D C5G7 problem are provided to demonstrate the stability and efficiency of lpCMFD method on MOC. It shows that lpCMFD can stabilize the CMFD iterations in MOC method effectively and lpCMFD method performs better than odCMFD on reducing the outer MOC iterations.

Regularization of free discontinuity problem and its application in the simulation of brittle fracture of concrete

As a free discontnuity problem, the predicton of the path that a crack chooses while it propagates through a britle material has been a long standing problem in fracture mechanics. To circumvent the mathematcal difcultes in the modeling of fracture behaviors, this paper takes advantage of the regularizaton approach to approximate the Munford-Shah functonal with Γ-convergence. The governing equatons of physical felds are constructed based on the concept of energy minimizaton. This method is capable of capturing the crack initaton and propagaton without a specifc tracking algorithm. Additonally, a pure elastc phase can be considered by introducing an energy threshold. By the use of the concept of equivalent stress, a stress-based fracture governing criterion was established. The numerical implementaton is based on a standard fnite element discretzaton and on the staggered algorithmic. Two classic concrete fracturing tests were investgated to demonstrate the performance of the model by comparing the numerical results with experimental results.

The Effects of Incidence and Deviation on the CFD-Based Throughflow Analysis

A major concern in the Computational Fluid Dynamics (CFD)-based throughflow calculation is the treatment of the incidence and deviation. This paper investigates the effects of the incidence and deviation on the CFD-based throughflow analysis by a time-marching throughflow model. The model is realized by solving complete Navier-Stokes equations with a single grid in the g-wise direction. The inviscid blade force is determined by calculating a pressure difference between the pressure and suction surfaces. The losses are introduced by imposing a blade surface skin friction factor converted from the pressure loss coefficient. And the flow discontinuity problem at the leading and trailing edges is resolved by modifying the mean blade surface to accommodate the incidence and deviation. The sensitivity of the throughflow results to the modification strategy of the mean blade surface is studied through response surface method. And the Kriging-assisted genetic algorithm (GA) is applied to determine the optimal distributions of incidence and deviation in the streamwise direction for NASA Rotor 37. Finally, four examples are provided to validate the throughflow model and to demonstrate the effects of incidence and deviation on CFD-based throughflow analysis at off-design conditions.