Automatic linearity detection

Given a function, or more generally an operator, the question "Is it linear?" seems simple to answer. In many applications of scientific computing it might be worth determining the answer to this question in an automated way; some functionality, such as operator exponentiation, is only defined for linear operators, and in other problems, time saving is available if it is known that the problem being solved is linear. Linearity detection is closely connected to sparsity detection of Hessians, so for large-scale applications, memory savings can be made if linearity information is known. However, implementing such an automated detection is not as straightforward as one might expect. This paper describes how automatic linearity detection can be implemented in combination with automatic differentiation, both for standard scientific computing software, and within the Chebfun software system. The key ingredients for the method are the observation that linear operators have constant derivatives, and the propagation of two logical vectors, $\ell$ and $c$, as computations are carried out. The values of $\ell$ and $c$ are determined by whether output variables have constant derivatives and constant values with respect to each input variable. The propagation of their values through an evaluation trace of an operator yields the desired information about the linearity of that operator

Automatic Frechet differentiation for the numerical solution of boundary-value problems

A new solver for nonlinear boundary-value problems (BVPs) in Matlab is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton's method in function space, where instead of the usual Jacobian matrices, the derivatives involved are Frechet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Frechet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable to enable a recursive, delayed evaluation of derivatives with forward mode AD. The AD techniques are applied within a new Chebfun class called chebop which allows users to set up and solve nonlinear BVPs in a few lines of code, using the "nonlinear backslash" operator (\). This framework enables one to study the behaviour of Newton's method in function space

The chebop system for automatic solution of differential equations

In MATLAB, it would be good to be able to solve a linear differential equation by typing u = L\f, where f, u, and L are representations of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described in which such calculations are indeed possible, based on the previously developed chebfun system in object-oriented MATLAB. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution

Cultural threads in primary schools

This paper explores headteachers’ and teachers’ perceptions of foreign languages (FL) and cultural learning in three primary schools in areas of disadvantage in England. Drawing upon a new theoretical frame for primary languages – Critical Cosmopolitanism (Delanty, 2006; Beck and Sznaider, 2006) and The Grammar of Culture (Holliday, 2018), we argue that the grand narrative of a target language inhabited by a target culture is outdated and approaches to cultural learning in primary schools could lead the way. There is substantial evidence that most learners find language lessons fun, particularly activities such as songs, stories and intercultural events (Driscoll et al.,2004, 2014; Cable et al. 2010). The discourse on conditions for inclusive practice is less commonplace and little is known about FL learning in areas of high deprivation (Nikolov & Mihaljević Djigunović, 2011). Teachers and headteachers in this study were committed to cultural learning and staff adopted creative approaches to teaching. The findings, however, also indicate that traditional notions of a mono or homogenised national culture with associated stereotypes linger in teachers’ framing of FL. There exists a need for a more personalised approach to cultural learning drawing upon children’s own cultural experiences. Data was analysed thematically following strict ethical guidelines and all names were anonymised to ensure confidentiality

The high-frequency backscattering angular response of gassy sediments: Model/data comparison from the Eel River Margin, California

A model for the high-frequency backscatter angular response of gassy sediments is proposed. For the interface backscatter contribution we adopted the model developed by Jackson et al. @J. Acoust. Soc. Am. 79, 1410–1422 ~1986!#, but added modifications to accommodate gas bubbles. The model parameters that are affected by gas content are the density ratio, the sound speed ratio, and the loss parameter. For the volume backscatter contribution we developed a model based on the presence and distribution of gas in the sediment. We treat the bubbles as individual discrete scatterers that sum to the total bubble contribution. This total bubble contribution is then added to the volume contribution of other scatters. The presence of gas affects both the interface and the volume contribution of the backscatter angular response in a complex way that is dependent on both grain size and water depth. The backscatter response of fine-grained gassy sediments is dominated by the volume contribution while that of coarser-grained gassy sediments is affected by both volume and interface contributions. In deep water the interface backscatter is only slightly affected by the presence of gas while the volume scattering is strongly affected. In shallow water the interface backscatter is severely reduced in the presence of gas while the volume backscatter is only slightly increased. Multibeam data acquired offshore northern California at 95 kHz provides raw measurements for the backscatter as a function of grazing angle. These raw backscatter measurements are then reduced to scattering strength for comparison with the results of the proposed model. The analysis of core samples at various locations provides local measurements of physical properties and gas content in the sediments that, when compared to the model, show general agreement

Water conservation and nutrient, sediment, and herbicide movement in furrow-irrigated tillage systems

2013 Summer.Includes bibliographical references.Due to an increase of population in the state of Colorado, as well as several years of receiving less than average precipitation, water allocation has become a state-wide concern. Agricultural, municipal, and recreational sectors demand ever-increasing volumes of water, which has caused the state to re-assess the amount and location of where water will be best economically and environmentally utilized. From an agricultural standpoint, furrow irrigation is a less effective method of irrigation than sprinkler or drip irrigation, however land suitability and socio-economic factors keep furrow irrigated acres high throughout Colorado. Therefore, there is a need to develop cropping systems that increase the irrigating efficiency of furrow irrigation in the state while decreasing sediment and nutrient contamination of water sources. Adoption of conservation tillage in furrow-irrigated cropland is limited compared to rain-fed and sprinkler irrigated systems. Residue on the soil surface impeding furrow irrigation flow and establishing a quality seed bed are the primary concerns. A two year field-based study was conducted in Fort Collins, Colorado during 2011 and 2012 to compare (a) soil moisture and irrigation requirements, (b) water outflow, infiltration, and advance in furrows and (c) sediment and nutrients in runoff for minimum till (MT) and strip till (ST) systems to a conventional till, plow-based system (CT). The MT and ST systems included a modified row-cleaning operation to move residue from irrigated furrows to adjacent non-irrigated furrows. Crop residue was greater on the soil surface in MT and ST than in CT, which resulted in higher soil moisture content at planting yet still allowed for successful irrigation. Average advance of water through furrows in 2011 was faster in MT (79 min) than CT (101 min) and ST (108 min), and in the order of ST (109 min) > MT (99 min) > CT (88 min) during 2012. Penetration resistance measurements showed that CT (567 kPa) and ST (275 kPa) created good seedbeds, but hard soil on MT (848 kPa) beds caused poor seed placement. Within individual irrigation events, tillage practice had little effect on the concentrations of sediment or nutrients in runoff, except MT had higher concentration and load of nitrate (NO3-) than CT and ST. Sediment concentrations and loads were similar for all tillage systems and average annual sediment loss in irrigation runoff was 4.9 Mg ha-1. Conservation tillage systems can be successfully modified for application to furrow irrigation systems and can decrease the dependence on irrigation at planting by maintaining crop residue at the soil surface, although high amounts of residue on the seedbed can be of concern for effective planting in MT ST was a better approach than MT because it created better seedbed conditions. An additional concern for growers in regards to conservation tillage is decreased efficacy of herbicides due to interception of herbicides by residue on the soil surface. To address this concern, an herbicide dissipation study was included in this tillage study to compare the fate and movement of atrazine, s-metolachlor, and pyroxasulfone in the three tillage systems. Pyroxasulfone is a newly released herbicide that is applied pre- or post-emergence to corn and has a similar weed control spectrum to atrazine and s-metolachlor, but with significantly lower application rates. The molecule has low water solubility and the potential for longer persistence in the soil than atrazine and s-metolachlor. More information is needed about the behavior of pyroxasulfone in the environment and about interaction with varying management systems. This field study was performed at the same site as the tillage study during 2011 and 2012 to 1) compare sorption of pyroxasulfone to that of atrazine and s-metolachlor for an alkaline, loam soil, and 2) to evaluate and compare the persistence and movement of pyroxasulfone, atrazine, and s-metolachlor in conventional (CT), minimum (MT), and strip (ST) tillage systems under furrow irrigated grain corn. In each year, labeled rates of 0.28 kg ai ha-1 for pyroxasulfone, 0.74 kg ai ha-1 for atrazine, and 1.71 kg ai ha-1 for s-metolachlor were applied pre-emergence to corn. Four depth increments of soil samples were taken over the top 30 cm in each tillage system at five time intervals over 60 days. Herbicides were extracted and analyzed by GC/MS to determine the dissipation and movement in soil. All three herbicides had low to moderate sorption and the rank order of sorption coefficients (Kd) was s-metolachlor (0.96 L kg-1) > pyroxasulfone (0.56 L kg-1) > atrazine (0.45 L kg-1). Pyroxasulfone had a much longer half life in all tillage systems when compared to atrazine and s-metolachlor. For pyroxasulfone, DT50 was longest in ST both years, and were not quantifiable because its persistence was longer than the 60 day sample period. Tillage practice affected DT50 of all herbicides, mainly due to residue coverage differences, with herbicides persisting longer in the conservation tillage systems than in conventional tillage

Multiple layer local oxidation for fabricating semiconductor nanostructures

Coupled semiconductor nanostructures with a high degree of tunability are fabricated using local oxidation with a scanning force microscope. Direct oxidation of the GaAs surface of a Ga[Al]As heterostructure containing a shallow two-dimensional electron gas is combined with the local oxidation of a thin titanium film evaporated on top. A four-terminal quantum dot and a double quantum dot system with integrated charge readout are realized. The structures are tunable via in-plane gates formed by isolated regions in the electron gas and by mutually isolated regions of the Ti film acting as top gates. Coulomb blockade experiments demonstrate the high quality of this fabrication process.Comment: 3 pages, 3 figure