Quantitative property A, Poincaré inequalities, Lp-compression and Lp-distortion for metric measure spaces

We introduce a quantitative version of Property A in order to estimate the L ^p -compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to be useful to yield upper bounds on the L ^p -distortion of finite metric spaces. Namely, we obtain new optimal results for finite subsets of homogeneous Riemannian manifolds. We also introduce a general form of Poincaré inequalities that provide constraints on compressions, and lower bounds on distortion. These inequalities are used to prove the optimality of some of our results.      

A high sensitive assay platform based on surface-enhanced Raman scattering for quantification of protease activity

In this study, a new, sensitive, and rapid assay was developed to quantitatively measure the proteolytic enzyme activity using the surface-enhanced Raman scattering (SERS) probe. Two different shapes of gold nanoparticles, gold nanosphere and nanorod particles were produced. SERS label, comprising self-assembled monolayers (SAMs) of Raman reporter molecule (5,5-Dithiobis (2-Nitrobenzoic acid), DTNB), was coated on the surface of the nanoparticles. Two different SERS-based analysis platforms were designed using gold-coated glass slide and polystyrene microtiter plate. The calibration curves were obtained by plotting the intensity of the SERS signal of symmetric NO₂ stretching of DTNB at 1326 cm⁻¹vs. the protease concentration. The effects of nanoparticle geometry and assay platform on the protease assay were investigated and the best working combination of the parameters was selected as rod shaped SERS probe and gold-coated glass slide. The correlation between the protease activity and SERS signal was found to be linear within the range of 0.1-2 mU/mL (R² = 0.979). The limit of detection (LOD) and limit of quantification (LOQ) values of the validated method were found as 0.43 and 1.30 mU/mL, respectively. The intra-day and inter-day precisions of the method, as relative standard deviation (RSD), were determined as 2.5% and 3.6%, respectively. The developed method was successfully applied for quantitative analysis of the commercial enzyme preparate that is used in cheese making process. It was also used for investigation of substrate specificity of protease enzyme towards the casein and bovine serum albumin. The proposed method has a flexibility to try different substrates for the detection of various enzyme activities.     

Particle dynamics on Snyder space

We examine Hamiltonian formalism on Euclidean Snyder space. The latter corresponds to a lattice in the quantum theory. For any given dynamical system, it may not be possible to identify time with a real number parametrizing the evolution in the quantum theory. The alternative requires the introduction of a dynamical time operator. We obtain the dynamical time operator for the relativistic (nonrelativistic) particle, and use it to construct the generators of Poincaré (Galilei) group on Snyder space.     

动力学变换法探测连续系统不稳定周期轨道

本文利用动力学变换方法和庞加莱截面方法对两种连续混沌动力学系统进行不稳定周期轨道探测研究, 并对Lorenz系统进行了替代数据法检验.结果表明:基于庞加莱截面的动力学变换改进算法 可有效探测连续混沌动力学系统中的不稳定周期轨道.     

On the existence and uniqueness of minima and maxima on spheres of the integral functional of the calculus of variations

Given a bounded domain Ω⊂Rn, we prove that if is a C¹ function whose gradient is Lipschitzian in Rn+1 and non-zero at 0, then, for each r>0 small enough, the restriction of the integral functional to the sphere has a unique global minimum and a unique global maximum.     

Refinement of a formula for decay after weak coherent excitation of a sphere

A new formula has been given recently by A.A. Svidzinsky and M.O. Scully to describe the temporal evolution of the excitation function in a large sphere satisfying the Markov condition after excitation by a single photon. This formula is based on a physically reasonable Ansatz from which differential equations are inferred for the undetermined radial functions in the Ansatz. The solution to these differential equations leads to the formula for β. Numerical calculations from this formula yield a value ∼10% for the maximum probability of occupancy of secondary excited states.In this paper, we refine the formula of Svidzinsky and Scully by allowing the radial functions in the Ansatz to depend on the angular index l of the spherical Bessel functions. By using the Debye formula for the asymptotic behavior of j_l(u) for large l as well as u, we obtain differential equations in each angular sector, similar to theirs but with a dependence on l. The solution to these equations yields our improved formula, from which we calculate 17.1% for the maximum probability of secondary excited states.     

A combinatorial Yamabe flow in three dimensions

A combinatorial version of Yamabe flow is presented based on Euclidean triangulations coming from sphere packings. The evolution of curvature is then derived and shown to satisfy a heat equation. The Laplacian in the heat equation is shown to be a geometric analogue of the Laplacian of Riemannian geometry, although the maximum principle need not hold. It is then shown that if the flow is nonsingular, the flow converges to a constant curvature metric.     

Almost Everywhere Convergence of Orthogonal Expansions of Several Variables

For a weighted L¹ space on the unit sphere of R^d+1, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in h-harmonics. The result applies to various methods of summability, including the de La Vallée Poussin means and the Cesàro means. Similar results are also established for weighted orthogonal expansions on the unit ball and on the simplex of R^d.