Homogenization and concentrated capacity for the heat equation with non-linear variational data in reticular almost disconnected structures and applications to visual transduction

Out of a right, circular cylinder of height H and cross-section a disc of radius R+ one removes a stack of nH/ parallel, equi-spaced cylinders C_j,j=1,2,...,n, each of radius R and height . Here , are fixed positive numbers and is a positive parameter to be allowed to go to zero. The union of the C_j almost fills in the sense that any two contiguous cylinders C_j are at a mutual distance of the order of and that the outer shell, i.e., the gap S=-_o has thickness of the order of (_o is obtained from by formally setting =0). The cylinder from which the C_j are removed, is an almost disconnected structure, it is denoted by , and it arises in the mathematical theory of phototransduction.For each >0 we consider the heat equation in the almost disconnected structure , for the unknown function u, with variational boundary data on the faces of the removed cylinders C_j. The limit of this family of problems as 0 is computed by concentrating heat capacity and diffusivity on the outer shell, and by homogenizing the u within the limiting cylinder _o.It is shown that the limiting problem consists of an interior diffusion in _o and a boundary diffusion on the lateral boundary S of _o. The interior diffusion is governed by the 2-dimensional heat equation in _o, for an interior limiting function u. The boundary diffusion is governed by the Laplace–Beltrami heat equation on S, for a boundary limiting function u_S. Moreover the exterior flux of the interior limit u provides the source term for the boundary diffusion on S. Finally the interior limit u, computed on S in the sense of the traces, coincides with the boundary limit u_S. As a consequence of the geometry of , local arguments do not suffice to prove convergence in _o, and also we have to take into account the behavior of the solution in S. A key, novel idea consists in extending equi-bounded and equi-Hölder continuous functions in -dependent domains, into equi-bounded and equi-Hölder continuous functions in the whole ^N, by means of the Kirzbraun–Pucci extension technique.The biological origin of this problem is traced, and its application to signal transduction in the retina rod cells of vertebrates is discussed. Mathematics Subject Classification (2000) 35B27, 35K50, 92C37     

TDDFT studies of electronic structure and first hyperpolarizability of terta-nuclear cubane-like transition metal clusters

The TDDFT method is first applied in a series of tetra-nuclear transition metal clusters studies for nonlinear optical properties. The results indicate that the charge transfer inside the metal core [MCu3X4] (M=W, Mo; X=S, O, Cl, Se, Br) makes contribution to the optical nonlinearity. It is possible to enhance the hyperpolarizability by substituting the ligands of the clusters.     

On the Intermittency in Nuclear Fragmentations

A simple Monte Carlo simulation and even a partition method can be used to reproduce quite well the power law behavior between the factorial moment and the charge interval of fragment charge distributions from the multifragmentation of ¹⁹⁷Au nucleus bombarding emulsion at～1 GeV/nucleon. This indicates that the above power law seems not to be a unique precursor of the intermittent behavior in nuclear multifragmentations.     

NUCLEAR CHARGE DISPERSION OF FRAGMENTS IN HIGH ENERGY PROTON-NUCLEUS COLLISION

The charge dispersion of fragments in high energy p+Cu,Kr and Xe reactions are calculated by statistical model and Monte Carlo technique.The corresponding data are reproduced quite well.It is shown that the charge dispersions are all nearly gaussian.The mass dependence of the most probable fragment charge reflects the rest target memory effect.     

EQUATION OF STATE OF HOT NUCLEI BEFORE BREAK-UP

Following the method developed by the authors,recently,the equation of state of hot nuclei(²³⁸U in concrete) before break-up was investigated numerically.The isotherms are drawn in the plan of the general pressure P versus volume V_RT. They are similar to those of Van der Waals gas.The critical temperature of phase transition should correspond to the isotherm with one turning point only.It turns out that the data of mass yield distribution can be reproduced by many pairs of parameters T and V_RT (freeze-out temperature and freeze-out volume) varying in certain range.For each isotherm (each T),the data are always best reproduced by the value of V_RT located at the maximum general pressure within two phases coexistence region.     

Λ Production and Λ/p Ratio in Relativistic Heavy Ion Collisions

The Λ multiplicity and Λ/p ratio are studied by hadron transportation-string fragmentation model in relativistic heavy ion collisions. Firstly, the dependence of Λ multiplicity and Λ/p ratio on the system size and the collision centrality is studied. It shows that the Λ and p multiplicities go up as the increase of system size and the increase of collision centrality. However, their ratio keeps almost a constant. The effect of Λ annihilation cross section to Λ multiplicity and Λ/p ratio is also studied. It is found that this effect is weak: Λ multiplicity and Λ/p ratio have a little amount of increase by the decrease of Λ annihilation cross section. Even the cross section is down to zero, Λ/p ratio is only 1.2 in 200A GeV AuAu head on collision. The Λ/p ratio is obtained to be 0.28 in pp collision, lying in the range of experimental data:0.2—0.3. It is also obtained that the ratio in AA collisions is 3—5 times of that in pp collision.     

相对论性核与乳胶(Ag)碰撞中灰粒子的系统研究 *

用描写相对论性核 核碰撞的LUCIAE模型和相应的事件产生器系统地研究了 1 4.6 ,6 0和 2 0 0AGeV的O核、2 0 0AGeV的Si和S核以及 1 1 .6AGeV的Au核与乳胶 (Ag)碰撞中灰粒子产生的平均多重数、多重数分布以及角分布 3个物理量同入射能量、射弹质量及碰撞中心度间的关系 ,还研究了再散射在灰粒子产生机制中的作用 .LUCIAE的这些研究结果与相应的EMU0 1乳胶实验结果都相一致 .     

Energy Dependence of String Fragmentation Function and φ Meson Production

The φ meson productions in Au+Au and/or Pb+Pb collisions at AGS, SPS, RHIC, and LHC energies have been studied systematically with a hadron and string cascade model LUCIAE. After considering the energy dependence of the model parameter α in string fragmentation function and adjusting it to the experimental data of charged multiplicity to a certain extent, the model predictions for φ meson yield, rapidity, and transverse mass distributions are compatible with the experimental data at AGS, SPS and RHIC energies. A calculation for Pb+Pb collisions at LHC energy is given as well. The obtained fractional variable in string fragmentation function shows a saturation in energy dependence. It is discussed that the saturation of fractional variable in string fragmentation function might be a qualitative representation of the energy dependence of nuclear transparency.     

Charged Multiplicity Density and Number of Participant Nucleons in Relativistic Nuclear Collisions

The energy and centrality dependencies of charged particle pseudorapidity density in relativistic nuclearcollisions were studied using a hadron and string cascade model, JPCIAE. Both the relativistic p+p experimental dataand the PHOBOS and PHENIX Au+Au data at RHIC energy could be fairly reproduced within the framework ofJPCIAE model and without retuning the model parameters. The predictions for Pb + Pb collisions at the LHC energywere also given. We computed the participant nucleon distributions using different methods. It was found that thenumber of participant nucleons is not a well defined variable both experimentally and theoretically. Thus it may beinappropriate to use the charged particle pseudorapidity density per participant pair .as a function of the number ofparticipant nucleons for distinguishing various theoretical models.     

相对论重离子碰撞中Λ粒子产生和Λ/p比率

利用强子输运–弦碎裂模型研究了200A GeV重离子碰撞中Λ多重产生和Λ/p比率.研究了它们随反应系统大小和碰撞中心度的变化.研究表明Λ和p多重数都随反应系统增大和随碰撞中心度提高而增大,但它们的比率基本保持不变.Λ的湮没截面大小变化对Λ的多重数产生有一定影响,但对Λ/p比率影响不大.结果很好地再现了pp反应中Λ/p比率.并获得这一比率在AA反应中是pp反应中的3—5倍.