We combine exponential sums, character sums and Fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids.
We provide a comparative overview of the process of implementation, harmonization and stabilization of public oversight systems for statutory auditors across the European Union (EU) after Directive 2006/43/EC. We build on institutional change theory to identify potential determinants as to why some countries still lag in this harmonization process.Oversight systems are a key institutional factor to guarantee the quality of financial information, essential to maintain investors' confidence and deep and stable capital markets.Thus, the harmonization of these systems has long been an objective of the EU. Our analyses serve to identify, analyse and compare how EU countries have incorporated European-wide requirements into their national legal systems. Particularly, we study: (1) basic characteristics of the system and bodies for public oversight, (2) organizational structure, (3) financing (4) transparency, (5) supervisory, and (6) disciplinary mechanisms. We show that significant diversity still exists across systems and that both the incentives for institutional change and the distance between pre-existing systems and the Directive are important explanatory factors of the achieved level of harmonization.
We improve the upper bound for the lattice point discrepancy of large spheres under conjectural properties of the real L-functions. In connection with this we give some new unconditional estimates for exponential and character sums of independent interest.
This study aims to introduce the robust optimum (RO) method as an alternative to the classical weighted averaging (WA) method for estimating the ecological optimum as well as the optimum and tolerance ranges of a taxon with respect to an environmental variable in limnological studies. The RO method is based on robust location and scale estimates rather than on the mean and the standard deviation used by the WA method. The results of our study support the well-known fact that the presence of outliers and the asymmetry of the distribution of the environmental variable might cause a significant effect on the WA-calculated ecological optimum as well as on tolerance ranges. We compared both methods through the identification of potential biological indicators of global warming. Biological data included several benthic, oligostenotherm macroinvertebrate families inhabiting the Ju´car River Basin (JRB, eastern Spain). The results of this comparison suggest that the RO method is more appropriate for estimating the distribution of taxa and, consequently, that it provides more realistic information for identifying potential sentinels of global warming in running aquatic systems. Currently, the identification of such sentinels is a goal for several environmental protection laws, such as the European Union Water Framework Directive.
The number of visible (primitive) lattice points in the sphere of radius R is well approximated by 4π 3ζ(3) R 3 . We consider an integral expression involving the error term E * (R), which leads to E * (R) = Ω(R(log R) 1/2 ). This is comparable to what is known in the sphere problem. We can avoid the use of the second power moment (which is in this case unknown) by employing an auxiliary trigonometric series correlated to E * (R). This approach to prove Ω-results seems to be new and could be useful in other problems.
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