The performance of approximating ordinary differential equations by neural nets

The dynamics of many systems are described by ordinary differential equations (ODE). Solving ODEs with standard methods (i.e. numerical integration) needs a high amount of computing time but only a small amount of storage memory. For some applications, e.g. short time weather forecast or real time robot control, long computation times are prohibitive. Is there a method which uses less computing time (but has drawbacks in other aspects, e.g. memory), so that the computation of ODEs gets faster? We will try to discuss this question for the assumption that the alternative computation method is a neural network which was trained on ODE dynamics and compare both methods using the same approximation error. This comparison is done with two different errors. First, we use the standard error that measures the difference between the approximation and the solution of the ODE which is hard to characterize. But in many cases, as for physics engines used in computer games, the shape of the approximation curve is important and not the exact values of the approximation. Therefore, we introduce a subjective error based on the Total Least Square Error (TLSE) which gives more consistent results. For the final performance comparison, we calculate the optimal resource usage for the neural network and evaluate it depending on the resolution of the interpolation points and the inter-point distance. Our conclusion gives a method to evaluate where neural nets are advantageous over numerical ODE integration and where this is not the case. Index Terms—ODE, neural nets, Euler method, approximation complexity, storage optimization

TEACHING CONSUMER ECONOMICS USING TASTE PANELS

Consumer/Household Economics,

Contrarian Investment, Extrapolation, and Risk

For many years, stock market analysts have argued that value strategies outperform the market. These value strategies call for buying stocks that have low prices relative to earnings, dividends, book assets, or other measures of fundamental value. While there is some agreement that value strategies produce higher returns, the interpretation of why they do so is more controversial. This paper provides evidence that value strategies yield higher returns because these strategies exploit the mistakes of the typical investor and not because these strategies are fundamentally riskier.

Ring opening polymerization of lactides and lactones by multimetallic alkyl zinc complexes derived from the acids Ph₂C(X)CO₂2H (X = OH, NH₂ )

The reaction of the dialkylzinc reagents R₂Zn with the acids 2,2-Ph₂C(X)(CO₂H), where X = NH₂, OH, i.e. 2,2′-diphenylglycine (dpgH) or benzilic acid (benzH2), in toluene at reflux temperature afforded the tetra-nuclear ring complexes [RZn(dpg)]₄, where R = Me (1), Et (2), 2-CF₃C₆H₄ (3), and 2,4,6-F₃C₆H₂ (4); complex 2 has been previously reported. The crystal structures of 1·(2MeCN), 3 and 4·(4(C₇H₈)·1.59(H₂O)) are reported, along with that of the intermediate compound (2-CF₃C₆H₄)3B·MeCN and the known compound [ZnCl₂(NCMe)₂]. Complexes 1–4, together with the known complex [(ZnEt)₃(ZnL)₃(benz)₃] (5; L = MeCN), have been screened, in the absence of benzyl alcohol, for their potential to act as catalysts for the ring opening polymerization (ROP) of ε-caprolactone (ε-CL), δ-valerolactone (δ-VL) and rac-lactide (rac-LA); the co-polymerization of ε-CL with rac-LA was also studied. Complexes 3 and 4 bearing fluorinated aryls at zinc were found to afford the highest activities

International Rent Sharing in Multinational Firms

We use a unique firm-level panel data set of multinational parents and their foreign affiliates to analyze whether profits are shared across borders within multinational firms. Using both fixed-effects and generalized method-of-moments estimators, affiliate wage levels are estimated to respond to both affiliate and parent profitability. The elasticity of affiliate wages to parent profits per worker is approximately 0.03, which can explain over 20 percent of the observed variation in affiliate wages. These results reveal a previously ignored aspect of labor-market rent sharing. They also reveal an important micro-level linkage with potential macro-level implications. International rent sharing can transmit economic conditions across national borders, and can thereby provide an implicit cross-country risk-sharing mechanism.

Social Behaviors of Modern and Indigenous Peoples Impacting the Ecology of the Amazon Rain Forest in Brazil

Human induced disruption of the environment is prevalent in every culture. In Brazil, the effects of massive deforestation have become apparent since the nineteen eighties. However, along with deforestation, and a coinciding loss in an economic resource for the country, is a significant loss of natural habitat and species extinction. The Amazon in Brazil contains a large proportion of the world’s species diversity that is threatened by the socio-economic activities of modern Brazilian culture. Historically and presently, indigenous groups have contributed to insignificant levels of ecological disruption and are themselves threatened by the activities of modern Brazilians. The effects of ecological disruption range from shifting climate patterns, degradation of soils, loss of floral and faunal species, displacement of indigenous cultures, and the loss of many unknown resources. The preservation of this unique environment is vital to the continued viability of the region

New routes to highly functionalised heterocycles

This study focuses mainly on the synthesis of tetrahydrofuran and pyrrolidine derivatives. The heterocyclic skeletons have a core structure of four carbons and one heteroatom and occur naturally and synthetically. [Illustration omitted.] The five membered monocyclic nitrogen and oxygen containing ring systems are constructed by reacting activated vinylcyclopropane intermediates with a range of carbon heteroatom double bond electrophiles. Such reactions are known as palladium (0)-mediated cycloadditions. [Illustration omitted.] Variables such as functional groups, solvent changes and catalysis, as well as the effect of Lewis acids on the yield and stereochemical outcome of the product was investigated. The final part of this study details the application of these cycloadditions towards the synthesis of the pyrrolizidine and indolizidine alkaloids. [Illustration omitted.