Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations I: Method Formulation

In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier-Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear self-adjoint second-order elliptic partial differential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton-GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher-order polynomials are presented.

A scale variational principle of Herglotz

The Herglotz problem is a generalization of the fundamental problem of the calculus of variations. In this paper, we consider a class of non-differentiable functions, where the dynamics is described by a scale derivative. Necessary conditions are derived to determine the optimal solution for the problem. Some other problems are considered, like transversality conditions, the multi-dimensional case, higher-order derivatives and for several independent variables.

What can be learned from failed proofs of non-theorems?

This paper reports an investigation into the link between failed proofs and non-theorems. It seeks to answer the question of whether anything more can be learned from a failed proof attempt than can be discovered from a counter-example. We suggest that the branch of the proof in which failure occurs can be mapped back to the segments of code that are the culprit, helping to locate the error. This process of tracing provides finer grained isolation of the offending code fragments than is possible from the inspection of counter-examples. We also discuss ideas for how such a process could be automated.

A Comparison of two Proof Critics: Power vs. Robustness

Proof critics are a technology from the proof planning paradigm. They examine failed proof attempts in order to extract information which can be used to generate a patch which will allow the proof to go through. We consider the proof of the $quot;whisky problem$quot;, a challenge problem from the domain of temporal logic. The proof requires a generalisation of the original conjecture and we examine two proof critics which can be used to create this generalisation. Using these critics we believe we have produced the first automatic proofs of this challenge problem. We use this example to motivate a comparison of the two critics and propose that there is a place for specialist critics as well as powerful general critics. In particular we advocate the development of critics that do not use meta-variables.

Integrating SVC and HOL with the PROSPER Toolkit

We describe an integration of the SVC decision procedure with the HOL theorem prover. This integration was achieved using the PROSPER toolkit. The SVC decision procedure operates on rational numbers, an axiomatic theory for which was provided in HOL. The decision procedure also returns counterexamples and a framework has been devised for handling counterexamples in a HOL setting.

Ketamine alters oscillatory coupling in the hoppocampus

Recent studies show that higher order oscillatory interactions such as cross-frequency coupling are important for brain functions that are impaired in schizophrenia, including perception, attention and memory. Here we investigated the dynamics of oscillatory coupling in the hippocampus of awake rats upon NMDA receptor blockade by ketamine, a pharmacological model of schizophrenia. Ketamine (25, 50 and 75 mg/kg i.p.) increased gamma and high-frequency oscillations (HFO) in all depths of the CA1-dentate axis, while theta power changes depended on anatomical location and were independent of a transient increase of delta oscillations. Phase coherence of gamma and HFO increased across hippocampal layers. Phase-amplitude coupling between theta and fast oscillations was markedly altered in a dose-dependent manner: ketamine increased hippocampal theta-HFO coupling at all doses, while theta-gamma coupling increased at the lowest dose and was disrupted at the highest dose. Our results demonstrate that ketamine alters network interactions that underlie cognitively relevant theta-gamma coupling.

Comparison and validation of physical wave parameterizations in spectral wave models

Recent developments in the physical parameterizations available in spectral wave models have already been validated, but there is little information on their relative performance especially with focus on the higher order spectral moments and wave partitions. This study concentrates on documenting their strengths and limitations using satellite measurements, buoy spectra, and a comparison between the different models. It is confirmed that all models perform well in terms of significant wave heights; however higher-order moments have larger errors. The partition wave quantities perform well in terms of direction and frequency but the magnitude and directional spread typically have larger discrepancies. The high-frequency tail is examined through the mean square slope using satellites and buoys. From this analysis it is clear that some models behave better than the others, suggesting their parameterizations match the physical processes reasonably well. However none of the models are entirely satisfactory, pointing to poorly constrained parameterizations or missing physical processes. The major space-time differences between the models are related to the swell field stressing the importance of describing its evolution. An example swell field confirms the wave heights can be notably different between model configurations while the directional distributions remain similar. It is clear that all models have difficulty in describing the directional spread. Therefore, knowledge of the source term directional distributions is paramount in improving the wave model physics in the future.

SEEKING CULTURAL FAIRNESS IN A MEASURE OF RELATIONAL REASONING

Relational reasoning, or the ability to identify meaningful patterns within any stream of information, is a fundamental cognitive ability associated with academic success across a variety of domains of learning and levels of schooling. However, the measurement of this construct has been historically problematic. For example, while the construct is typically described as multidimensional—including the identification of multiple types of higher-order patterns—it is most often measured in terms of a single type of pattern: analogy. For that reason, the Test of Relational Reasoning (TORR) was conceived and developed to include three other types of patterns that appear to be meaningful in the educational context: anomaly, antinomy, and antithesis. Moreover, as a way to focus on fluid relational reasoning ability, the TORR was developed to include, except for the directions, entirely visuo-spatial stimuli, which were designed to be as novel as possible for the participant. By focusing on fluid intellectual processing, the TORR was also developed to be fairly administered to undergraduate students—regardless of the particular gender, language, and ethnic groups they belong to. However, although some psychometric investigations of the TORR have been conducted, its actual fairness across those demographic groups has yet to be empirically demonstrated. Therefore, a systematic investigation of differential-item-functioning (DIF) across demographic groups on TORR items was conducted. A large (N = 1,379) sample, representative of the University of Maryland on key demographic variables, was collected, and the resulting data was analyzed using a multi-group, multidimensional item-response theory model comparison procedure. Using this procedure, no significant DIF was found on any of the TORR items across any of the demographic groups of interest. This null finding is interpreted as evidence of the cultural-fairness of the TORR, and potential test-development choices that may have contributed to that cultural-fairness are discussed. For example, the choice to make the TORR an untimed measure, to use novel stimuli, and to avoid stereotype threat in test administration, may have contributed to its cultural-fairness. Future steps for psychometric research on the TORR, and substantive research utilizing the TORR, are also presented and discussed.

Why are some companies more successful than others? An exploration into organizational culture and dynamic capabilities

Why are some companies more successful than others? This thesis approaches the question by enlisting theoretical frameworks that explain the performance with internal factors, deriving from the resource-based view, namely the dynamic capabilities approach. To deepen the understanding of the drivers and barriers towards developing these higher order routines aiming at improving the operational level routines, this thesis explores the organisational culture and identity research for the microfoundational antecedents that might shed light on the formation of the dynamic capabilities. The dynamic capabilities framework in this thesis strives to take the theoretical concept closer to practical applicability. This is achieved through creation of a dynamic capabilities matrix, consisting of four dimensions often encountered in dynamic capabilities literature. The quadrants are formed along internal-external and resources-abilities axes, and consist of Sensing, Learning, Reconfiguration and Partnering facets. A key element of this thesis is the reality continuum, which illustrates the different levels of reality inherent in any entity of human individuals. The theoretical framework constructed in the thesis suggests a link between the collective but constructivist understanding of the organisation and both the operational and higher level routines, evident in the more positivist realm. The findings from three different case organisations suggest that the constructivist assumptions inherent to an organisation function as a generative base for both drivers and barriers towards developing dynamic capabilities. From each organisation one core assumption is scrutinized to identify its connections to the four dimensions of the dynamic capabilities. These connections take the form of drivers or barriers – or have the possibility to develop into one or the other. The main contribution of this thesis is to show that one key for an organisation to perform well in a turbulent setting, is to understand the different levels of realities inherent in any group of people. Recognising the intangible levels gives an advantage in the tangible ones.

Radiation from a D-dimensional collision of shock waves: two-dimensional reduction and Carter–Penrose diagram

We analyze the causal structure of the two-dimensional (2D) reduced background used in the perturbative treatment of a head-on collision of two D-dimensional Aichelburg–Sexl gravitational shock waves. After defining all causal boundaries, namely the future light-cone of the collision and the past light-cone of a future observer, we obtain characteristic coordinates using two independent methods. The first is a geometrical construction of the null rays which define the various light cones, using a parametric representation. The second is a transformation of the 2D reduced wave operator for the problem into a hyperbolic form. The characteristic coordinates are then compactified allowing us to represent all causal light rays in a conformal Carter–Penrose diagram. Our construction holds to all orders in perturbation theory. In particular, we can easily identify the singularities of the source functions and of the Green’s functions appearing in the perturbative expansion, at each order, which is crucial for a successful numerical evaluation of any higher order corrections using this method.

Discrete breathers in a two-dimensional hexagonal Fermi-Pasta-Ulam lattice

We consider a two-dimensional Fermi-Pasta-Ulam (FPU) lattice with hexagonal symmetry. Using asymptotic methods based on small amplitude ansatz, at third order we obtain a eduction to a cubic nonlinear Schr{\"o}dinger equation (NLS) for the breather envelope. However, this does not support stable soliton solutions, so we pursue a higher-order analysis yielding a generalised NLS, which includes known stabilising terms. We present numerical results which suggest that long-lived stationary and moving breathers are supported by the lattice. We find breather solutions which move in an arbitrary direction, an ellipticity criterion for the wavenumbers of the carrier wave, symptotic estimates for the breather energy, and a minimum threshold energy below which breathers cannot be found. This energy threshold is maximised for stationary breathers, and becomes vanishingly small near the boundary of the elliptic domain where breathers attain a maximum speed. Several of the results obtained are similar to those obtained for the square FPU lattice (Butt \& Wattis, {\em J Phys A}, {\bf 39}, 4955, (2006)), though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics, and the isotropy of the generalised NLS equation.

Discrete breathers in a two-dimensional Fermi-Pasta-Ulam lattice

Using asymptotic methods, we investigate whether discrete breathers are supported by a two-dimensional Fermi-Pasta-Ulam lattice. A scalar (one-component) two-dimensional Fermi-Pasta-Ulam lattice is shown to model the charge stored within an electrical transmission lattice. A third-order multiple-scale analysis in the semi-discrete limit fails, since at this order, the lattice equations reduce to the (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation which does not support stable soliton solutions for the breather envelope. We therefore extend the analysis to higher order and find a generalised $(2+1)$-dimensional NLS equation which incorporates higher order dispersive and nonlinear terms as perturbations. We find an ellipticity criterion for the wave numbers of the carrier wave. Numerical simulations suggest that both stationary and moving breathers are supported by the system. Calculations of the energy show the expected threshold behaviour whereby the energy of breathers does {\em not} go to zero with the amplitude; we find that the energy threshold is maximised by stationary breathers, and becomes arbitrarily small as the boundary of the domain of ellipticity is approached.

Generalizations of Poisson Structures Related to Rational Gaudin Model

International audience

Essays on Supply Chain Finance

I study how a larger party within a supply chain could use its superior knowledge about its partner, who is considered to be financially constrained, to help its partner gain access to cheap finance. In particular, I consider two scenarios: (i) Retailer intermediation in supplier finance and (ii) The Effectiveness of Supplier Buy Back Finance. In the fist chapter, I study how a large buyer could help small suppliers obtain financing for their operations. Especially in developing economies, traditional financing methods can be very costly or unavailable to such suppliers. In order to reduce channel costs, in recent years large buyers started to implement their own financing methods that intermediate between suppliers and financing institutions. In this paper, I analyze the role and efficiency of buyer intermediation in supplier financing. Building a game-theoretical model, I show that buyer intermediated financing can significantly improve supply chain performance. Using data from a large Chinese online retailer and through structural regression estimation based on the theoretical analysis, I demonstrate that buyer intermediation induces lower interest rates and wholesale prices, increases order quantities, and boosts supplier borrowing. The analysis also shows that the retailer systematically overestimates the consumer demand. Based on counterfactual analysis, I predict that the implementation of buyer intermediated financing for the online retailer in 2013 improved channel profits by 18.3%, yielding more than $68M projected savings. In the second chapter, I study a novel buy-back financing scheme employed by large manufacturers in some emerging markets. A large manufacturer can secure financing for its budget-constrained downstream partners by assuming a part of the risk for their inventory by committing to buy back some unsold units. Buy back commitment could help a small downstream party secure a bank loan and further induce a higher order quantity through better allocation of risk in the supply chain. However, such a commitment may undermine the supply chain performance as it imposes extra costs on the supplier incurred by the return of large or costly-to-handle items. I first theoretically analyze the buy-back financing contract employed by a leading Chinese automative manufacturer and some variants of this contracting scheme. In order to measure the effectiveness of buy-back financing contracts, I utilize contract and sales data from the company and structurally estimate the theoretical model. Through counterfactual analysis, I study the efficiency of various buy-back financing schemes and compare them to traditional financing methods. I find that buy-back contract agreements can improve channel efficiency significantly compared to simple contracts with no buy-back, whether the downstream retailer can secure financing on its own or not.