Rate of convergence of a particle method to the solution of the Mc Kean-Vlasov's equation

This paper studies the rate of convergence of an appropriatediscretization scheme of the solution of the Mc Kean-Vlasovequation introduced by Bossy and Talay. More specifically,we consider approximations of the distribution and of thedensity of the solution of the stochastic differentialequation associated to the Mc Kean - Vlasov equation. Thescheme adopted here is a mixed one: Euler/weakly interactingparticle system. If $n$ is the number of weakly interactingparticles and $h$ is the uniform step in the timediscretization, we prove that the rate of convergence of thedistribution functions of the approximating sequence in the $L^1(\Omega\times \Bbb R)$ norm and in the sup norm is of theorder of $\frac 1{\sqrt n} + h $, while for the densities is ofthe order $ h +\frac 1 {\sqrt {nh}}$. This result is obtainedby carefully employing techniques of Malliavin Calculus.

The gain-loss asymmetry and single-self preferences

Kahneman and Tversky asserted a fundamental asymmetry between gains and losses, namely a reflection effect which occurs when an individual prefers a sure gain of $ pz to anuncertain gain of $ z with probability p, while preferring an uncertain loss of $z with probability p to a certain loss of $ pz.We focus on this class of choices (actuarially fair), and explore the extent to which thereflection effect, understood as occurring at a range of wealth levels, is compatible with single-self preferences.We decompose the reflection effect into two components, a probability switch effect,which is compatible with single-self preferences, and a translation effect, which is not. To argue the first point, we analyze two classes of single-self, nonexpected utility preferences, which we label homothetic and weakly homothetic. In both cases, we characterize the switch effect as well as the dependence of risk attitudes on wealth.We also discuss two types of utility functions of a form reminiscent of expected utility but with distorted probabilities. Type I always distorts the probability of the worst outcome downwards, yielding attraction to small risks for all probabilities. Type II distorts low probabilities upwards, and high probabilities downwards, implying risk aversion when the probability of the worst outcome is low. By combining homothetic or weak homothetic preferences with Type I or Type II distortion functions, we present four explicit examples: All four display a switch effect and, hence, a form of reflection effect consistent a single self preferences.

Riesz-Nágy singular functions revisited

In 1952 F. Riesz and Sz.Nágy published an example of a monotonic continuous function whose derivative is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-N´agy family of functions and we relate it to a system for real numberrepresentation which we call (t, t-1) expansions. With the help of these real number expansions we generalize the family. The singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.

Exploiting fitness distance correlation of set covering problems

The set covering problem is an NP-hard combinatorial optimization problemthat arises in applications ranging from crew scheduling in airlines todriver scheduling in public mass transport. In this paper we analyze searchspace characteristics of a widely used set of benchmark instances throughan analysis of the fitness-distance correlation. This analysis shows thatthere exist several classes of set covering instances that have a largelydifferent behavior. For instances with high fitness distance correlation,we propose new ways of generating core problems and analyze the performanceof algorithms exploiting these core problems.

On the Chern classes and the Euler characteristic for nonsingular complete intersections

It is shown that Hirzebruch's result on the Chern classes of a complete intersection of nonsingular hypersurfaces in general position in PN(C), is also valid for any nonsingular complete intersection. Then rela- tions between Euler characteristic, class and Milnor number are pointed out.

Neuroprotective role of PrPC against kainate-induced epileptic seizures and cell death depends on the modulation of JNK3 activation by GluR6/7-PSD-95 binding

Cellular prion protein (PrPC) is a glycosyl-phosphatidylinositol¿anchored glycoprotein. When mutated or misfolded, the pathogenic form (PrPSC) induces transmissible spongiform encephalopathies. In contrast, PrPC has a number of physiological functions in several neural processes. Several lines of evidence implicate PrPC in synaptic transmission and neuroprotection since its absence results in an increase in neuronal excitability and enhanced excitotoxicity in vitro and in vivo. Furthermore, PrPC has been implicated in the inhibition of N-methyl-D-aspartic acid (NMDA)¿mediated neurotransmission, and prion protein gene (Prnp) knockout mice show enhanced neuronal death in response to NMDA and kainate (KA). In this study, we demonstrate that neurotoxicity induced by KA in Prnp knockout mice depends on the c-Jun N-terminal kinase 3 (JNK3) pathway since Prnpo/oJnk3o/o mice were not affected by KA. Pharmacological blockage of JNK3 activity impaired PrPC-dependent neurotoxicity. Furthermore, our results indicate that JNK3 activation depends on the interaction of PrPC with postsynaptic density 95 protein (PSD-95) and glutamate receptor 6/7 (GluR6/7). Indeed, GluR6¿PSD-95 interaction after KA injections was favored by the absence of PrPC. Finally, neurotoxicity in Prnp knockout mice was reversed by an AMPA/KA inhibitor (6,7-dinitroquinoxaline-2,3-dione) and the GluR6 antagonist NS-102. We conclude that the protection afforded by PrPC against KA is due to its ability to modulate GluR6/7-mediated neurotransmission and hence JNK3 activation.

Neuroprotective role of PrPC against kainate-induced epileptic seizures and cell death depends on the modulation of JNK3 activation by GluR6/7-PSD-95 binding

Cellular prion protein (PrPC) is a glycosyl-phosphatidylinositol¿anchored glycoprotein. When mutated or misfolded, the pathogenic form (PrPSC) induces transmissible spongiform encephalopathies. In contrast, PrPC has a number of physiological functions in several neural processes. Several lines of evidence implicate PrPC in synaptic transmission and neuroprotection since its absence results in an increase in neuronal excitability and enhanced excitotoxicity in vitro and in vivo. Furthermore, PrPC has been implicated in the inhibition of N-methyl-D-aspartic acid (NMDA)¿mediated neurotransmission, and prion protein gene (Prnp) knockout mice show enhanced neuronal death in response to NMDA and kainate (KA). In this study, we demonstrate that neurotoxicity induced by KA in Prnp knockout mice depends on the c-Jun N-terminal kinase 3 (JNK3) pathway since Prnpo/oJnk3o/o mice were not affected by KA. Pharmacological blockage of JNK3 activity impaired PrPC-dependent neurotoxicity. Furthermore, our results indicate that JNK3 activation depends on the interaction of PrPC with postsynaptic density 95 protein (PSD-95) and glutamate receptor 6/7 (GluR6/7). Indeed, GluR6¿PSD-95 interaction after KA injections was favored by the absence of PrPC. Finally, neurotoxicity in Prnp knockout mice was reversed by an AMPA/KA inhibitor (6,7-dinitroquinoxaline-2,3-dione) and the GluR6 antagonist NS-102. We conclude that the protection afforded by PrPC against KA is due to its ability to modulate GluR6/7-mediated neurotransmission and hence JNK3 activation.

On the dimension of the core of the assignment game

The set of optimal matchings in the assignment matrix allows to define a reflexive and symmetric binary relation on each side of the market, the equal-partner binary relation. The number of equivalence classes of the transitive closure of the equal-partner binary relation determines the dimension of the core of the assignment game. This result provides an easy procedure to determine the dimension of the core directly from the entries of the assignment matrix and shows that the dimension of the core is not as much determined by the number of optimal matchings as by their relative position in the assignment matrix.

Energy and structure of dilute hard- and soft-sphere gases

The energy and structure of dilute hard- and soft-sphere Bose gases are systematically studied in the framework of several many-body approaches, such as the variational correlated theory, the Bogoliubov model, and the uniform limit approximation, valid in the weak-interaction regime. When possible, the results are compared with the exact diffusion Monte Carlo ones. Jastrow-type correlation provides a good description of the systems, both hard- and soft-spheres, if the hypernetted chain energy functional is freely minimized and the resulting Euler equation is solved. The study of the soft-sphere potentials confirms the appearance of a dependence of the energy on the shape of the potential at gas paremeter values of x~0.001. For quantities other than the energy, such as the radial distribution functions and the momentum distributions, the dependence appears at any value of x. The occurrence of a maximum in the radial distribution function, in the momentum distribution, and in the excitation spectrum is a natural effect of the correlations when x increases. The asymptotic behaviors of the functions characterizing the structure of the systems are also investigated. The uniform limit approach is very easy to implement and provides a good description of the soft-sphere gas. Its reliability improves when the interaction weakens.

On the dimension of the core of the assignment game

The set of optimal matchings in the assignment matrix allows to define a reflexive and symmetric binary relation on each side of the market, the equal-partner binary relation. The number of equivalence classes of the transitive closure of the equal-partner binary relation determines the dimension of the core of the assignment game. This result provides an easy procedure to determine the dimension of the core directly from the entries of the assignment matrix and shows that the dimension of the core is not as much determined by the number of optimal matchings as by their relative position in the assignment matrix.

Nonempirical cluster-model study of the chemisorption of atomic hydrogen on the (111) surface of diamondlike crystals

The interaction of atomic hydrogen with C4H9, Si4H9, and Ge4H9 model clusters has been studied using all-electron and pseudopotential ab initio Hartree-Fock computations with basis sets of increasing flexibility. The results show that the effect of polarization functions is important in order to reproduce the experimental findings, but their inclusion only for the atoms directly involved in the chemisorption bond is usually sufficient. For the systems H-C4H9 and H-Si4H9 all-electron and pseudopotential results are in excellent agreement when basis sets of comparable quality are used. Besides, semiempirical modified-neglect-of-differential-overlap computations provide quite reliable results both for diamond and silicon and have been used to investigate larger model clusters. The results confirm the local nature of chemisorption and further justify the use of minimal X4H9 model clusters.

Ab initio cluster-model study of the on-top chemisorption of F and Cl on Si(111) and Ge(111) surfaces

The interaction of atomic F and Cl with Si4H9 and Ge4H9 cluster models has been studied by using ab initio pseudopotentials and basis sets of increasing complexity. The results show that the effect of d orbitals is important in order to reproduce the experimental findings. However, the use of polarization functions in the atoms which are directly involved in the chemisorption bond leads to results which are very close to those obtained using extended basis sets. The local nature of the chemisorption bond is also interpreted by means of a Mulliken population analysis. For F-Si4H9 and Cl-Si4H9 the present results are in good agreement with previous ab initio all-electron calculations, and for the chemisorption of Cl on Si(111) and Ge(111) surfaces, good agreement is found with respect to the available experimental results as well as with previous slab calculations based on the local-density-functional formalism.

Chemical composition of the small coastal lagoons of the Mediterranean Spanish littoral

About sixty small water bodies (coastal lagoons, marshes, salt pans, channels, springs, etc.) of the Spanish Mediterranean coast were sampled seasonally for one year (1979-1980), in order to study different aspects of their chemical composition. The concentrations of major ions (alkalinity, Cl-, Ca2+, Mg2+, Na+, and K+), nutrients (N.NO-3, N.NO2-, TRP and Si), oxygen and pH were determined for this purpose. The salt concentrations measured range between 0.4 and 361.3 g l-1. The samples have been divided into four classes of salinity (in g l-1): Cl, S < 5; C2, 5 40. Within these classes, the pattern of ionic dominance recorded is remarkably constant and similar to that found in most coastal lagoons (Cl- > So42- > Alk., for the anions, and Na+ > Mg2+ > Ca2+ > K+, for the cations), although other models occur especially in the first class. The dominance of Na+ and Cl-, as well as the molar ratios Mg2+/Ca2+ and Cl- / SO42- ,clearly increase from class Cl to class C4. The hyperhaline waters include different subtypes of the major brine type"c",, of EUGSTER & HARDIE (1978), the Na+ - (Mg2+) - Cl- - (SO42-) being the most frequent. Nutrient concentrations fall within a wide range (N.NO3 from 0.1 to 1100 mg-at 1-1; PRT from 0.01 to 23.56 mg-at l-1 and Si from 1.0 to 502.0 mg-at l-1). The oxygen values are very variable too, ranging between 0 and 14.4 ml l-1. Four different patterns of nutrient distribution have been distinguished based on the mean concentrations of N.NO3-, and TRP (mean values in mg-at l-1): A, N.NO3- < 10, TRP > l ; B, N.NO3- > 100, TRP < 1; C, 10 < N.NO3- < 100, TRP < 1; C, D, N.NO3- < 10, TRP < 1. As a rule, lagoons of low salinity (C1 and C2 classes) display the nutrient pattern C, and lagoons of high salinity (C3 and C4) show the nutrient pattern D. Model A only appears in waters of very low salinity, whereas model B does not seem to be related to salinity.

Effects of total fishing prohibition on the rocky fish assemblages of Medes Islands marine reserve (NW Mediterranean)

Visual scuba diving censuses were used to assess the effects of fishing prohibition on abundance and size structure of littoral fish populations by comparing the same benthic communities inside and outside the protected area of Medes Islands (NE Catalonia, Spain). The total number of species found was 43 in the reserve and 44 outside, but the mean value of species richness per sampling station was significantly higher in the protected area. However, diversity, heavily affected by the presence or absence of large schools of pelagic species, showed no significant differences between sites. The prohibition of fishing for 6 years is the first factor affecting the qualitative and quantitative structure of fish populations ('reserve effect'), and depth is the second factor. Thus, except in the cases of Serranus cabrilla and Mullus surmuletus, all other vulnerable species are highly sensitive to the protection measures. The size structure of all vulnerable species was found to be absolutely different at the reserve sites than in the unprotected zones, and the modal size classes of size frequency distributions were always higher in the reserve than outside. The reserve effect was significantly responsible of the differences observed in this change on size structure. Some highly vulnerable species, such as Epinephelus guaza and Sciaena umbra, have only been found in the protected area. Others, such as Sparus aurata, Diplodus cervinus and Dicentrarchus labrax, were much more frequent inside the reserve.