Optical design for enhanced prism type fingerprint scanner image contrast with asymmetrical aspheric lens

The design of optical prism fingerprint scanner with aspheric lens is researched. We propose a modified structure for optical prism fingerprint scanner, which has an asymmetrical aspheric lens on the output plane. It is numerically shown under elliptic valley similar fingerprint model that the optical image qualities of such a scanner are better than the scanner with symmetrical aspheric lens. When the fingerprint range is 15 mm, the results show that the minimum image contrast is 0.49 for an optical prism with asymmetrical aspheric lens, much better than a symmetrical aspheric lens with 0.20. The relative illumination variation is from 0.62 to 1.00 for optical prism with asymmetrical aspheric lens, and for the symmetrical aspheric lens it is 0.35 to 1.00. We conclude that an optical prism with asymmetrical aspheric lens is better than an optical prism with symmetrical aspheric lens.     

Complex geometrical optics of Kerr type nonlinear media

The paper generalizes paraxial complex geometrical optics (PCGO) for Gaussian beam (GB) propagation in nonlinear media of Kerr type. Ordinary differential equations for the beam amplitude and for complex curvature of the wave front are derived, which describe the evolution of axially symmetric GB in a Kerr type nonlinear medium. It is shown that PCGO readily provides the solutions of NLS equation obtained earlier from diffraction theory on the basis of the aberration-free approach. Besides reproducing classical results of self-focusing PCGO readily describes an influence of the initial curvature of the wave front on the beam evolution in a medium of Kerr type including a nonlinear graded-index fiber. The range of applicability of the PCGO theory is discussed as well which is helpful for avoiding nonphysical solutions.     

Exact solution of a coagulation equation with a product kernel in the multicomponent case

In this paper, we obtain the general solution for the continuous Smoluchowski equation in the multicomponent case with a product kernel as a series expansion. The solution of the problem involves the Laplace transform in several dimensions. We obtain a nonlinear partial differential equation (PDE) of the advective kind generalizing the one previously given by other authors for the mono-component case.As in its relative mono-component case, gelation is produced at some point, the conditions for its occurrence being the same as those for the mono-component case, though substituting a sum of derivatives by a derivative in the Laplace transform field. We demonstrate that for a multicomponent particle size distribution (PSD) of multiplicative form, it is sufficient for one of the marginal PSDs to generate instantaneous gelation for the occurrence of instantaneous gelation in the multicomponent PSD.The general solution is applied to several specific cases, a discrete case that recovers a previously known solution, and another two continuous cases which can be used to check numerical methods designed to directly solve the Smoluchowski equation in more general cases.We have compared the solutions for the multicomponent PSD for constant, additive and product kernels and we conjecture about the relation existing between the functional forms for the solutions both in the mono-component and the multicomponent case.Finally, we have analysed the shape of the solutions for multicomponent PSD for constant, additive and product kernels for very small masses of components, obtaining a qualitatively different behaviour for the product kernel. This has effects in the mixing state of the sol phase as time passes.     

New reproducing kernel Hilbert spaces on semi-infinite domains with existence and uniqueness results for the nonhomogeneous telegraph equation

We introduce new reproducing kernel Hilbert spaces on a semi-infinite domain and demonstrate existence and uniqueness of solutions to the nonhomogeneous telegraph equation in these spaces if the driver is square-integrable and sufficiently smooth.     

A 2D kernel determination problem in a visco-elastic porous medium with a weakly horizontally inhomogeneity

We consider a system of hyperbolic integro-differential equations of SH waves in a visco-elastic porous medium. In this work, it is assumed that the visco-elastic porous medium has weakly horizontally inhomogeneity. The direct problem is the initial-boundary problem: the initial data is equal to zero, and the Neumann-type boundary condition is specified at the half-plane boundary and is an impulse function. As additional information, the oscillation mode of the half-plane line is given. It is assumed that the unknown kernel has the form K(x,t)=K₀(t)+ϵxK₁(t)+…, where ϵ is a small parameter. In this work, we construct a method for finding K₀,K₁ up to a correction of the order of O(ϵ²).     

Several logarithmic Caffarelli–Kohn–Nirenberg inequalities and applications

In this paper, based on the Caffarelli–Kohn–Nirenberg inequalities on the Euclidean space and the weighted Hölder inequality, we establish the logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities, and give applications for the weighted ultracontractivity of positive strong solutions to a kind of evolution equations. We also prove corresponding logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities on the Heisenberg group and related to generalized Baouendi–Grushin vector fields. Some applications are provided.     

On semi‐isogenous mixed surfaces

Let C be a smooth projective curve and G be a finite subgroup of whose action is mixed, i.e. there are elements in G exchanging the two isotrivial fibrations of . Let be the index two subgroup . If G⁰ acts freely, then is smooth and we call it semi‐isogenous mixed surface. In this paper we give an algorithm to determine semi‐isogenous mixed surfaces with given geometric genus, irregularity and self‐intersection of the canonical class. As an application we classify irregular semi‐isogenous mixed surfaces with and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with . We provide an example of a minimal surface of general type with and .     

Gradient estimates of Dirichlet heat kernels for unimodal Lévy processes

Under some mild assumptions on the Lévy measure and the symbol we obtain gradient estimates of Dirichlet heat kernels for pure‐jump isotropic unimodal Lévy processes in .     

Lie symmetries of a system arising in plasma physics

Lie group classification for a diffusion‐type system that has applications in plasma physics is derived. The classification depends on the values of 5 parameters that appear in the system. Similarity reductions are presented. Certain initial value problems are reduced to problems with the governing equations being ordinary differential equations. Examples of potential symmetries are also presented.