A nonstandard finite difference method for solving a mathematical model of HIV-TB co-infection

We design and analyze an unconditionally convergent nonstandard finite-difference method to study transmission dynamics of a mathematical model of HIV-TB co-infection. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves positivity of the solution which is one of the essential requirements when modelling epidemic diseases. Furthermore, we show that the numerical method is unconditionally stable. Competitive numerical results confirming theoretical investigations are provided. Comparisons are also made with other conventional approaches that are routinely used to solve these types of problems.     

The graph structure of a deterministic automaton chosen at random

An n‐state deterministic finite automaton over a k‐letter alphabet can be seen as a digraph with n vertices which all have k labeled out‐arcs. Grusho (Publ Math Inst Hungarian Acad Sci 5 (1960), 17–61). proved that whp in a random k‐out digraph there is a strongly connected component of linear size, i.e., a giant, and derived a central limit theorem. We show that whp the part outside the giant contains at most a few short cycles and mostly consists of tree‐like structures, and present a new proof of Grusho's theorem. Among other things, we pinpoint the phase transition for strong connectivity. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 428–458, 2017     

Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3

The main results of this paper are in two directions. First, the family of finite local Frobenius non-chain rings of length 4 (hence of nilpotency index 3) is determined. As a by-product all finite local Frobenius non-chain rings with $p^4$ elements, (p a prime) are given. Second, the number and structure of γ-constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, of length relatively prime to the characteristic of the residue field of the ring, are determined.     

Precision Matrix Estimation With ROPE

It is known that the accuracy of the maximum likelihood-based covariance and precision matrix estimates can be improved by penalized log-likelihood estimation. In this article, we propose a ridge-type operator for the precision matrix estimation, ROPE for short, to maximize a penalized likelihood function where the Frobenius norm is used as the penalty function. We show that there is an explicit closed form representation of a shrinkage estimator for the precision matrix when using a penalized log-likelihood, which is analogous to ridge regression in a regression context. The performance of the proposed method is illustrated by a simulation study and real data applications. Computer code used in the example analyses as well as other supplementary materials for this article are available online.     

Norm equalities and inequalities for three circulant operator matrices

In this paper, three new circulant operator matrices, scaled circulant operator matrices, diag-circulant operator matrices and retrocirculant operator matrices, are given respectively. Several norm equalities and inequalities for these operator matrices are proved. We show the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norm. Pinching type inequality is also given for weakly unitarily invariant norms. These results are closely related to the nice structure of these special operator matrices. Furthermore, some special cases and specific examples are also considered.     

Rees–Shishikura’s theorem for geometrically finite rational maps

In this paper, we generalize Rees–Shishikura’s theorem to the class of geometrically finite rational maps.     

Plasmonic Pollen Grain Nanostructures: A Three-Dimensional Surface-Enhanced Raman Scattering (SERS)-Active Substrate

A new route has been developed to design plasmonic pollen grain-like nanostructures (PGNSs) as surface-enhanced Raman scattering (SERS)-active substrate. The nanostructures consisting of silver (Ag) and gold (Au) nanoparticles along with zinc oxide (ZnO) nanoclusters as spacers were found highly SERS-active. The morphology of PGNSs and those obtained in the intermediate stage along with each elemental evolution has been investigated by a high-resolution field emission scanning electron microscopy. The optical band gaps and crystal structure have been identified by UV-vis absorption and X-ray powder diffraction (XRD) measurements, respectively. For PGNSs specimen, three distinct absorption bands related to constituent elements Ag, Au, and ZnO were observed, whereas XRD peaks confirmed the existence of Ag, Au, and ZnO within the composition of PGNSs. SERS-activity of PGNSs was confirmed using Rhodamine 6G (R6G) as Raman-active dyes. Air-cooled solid-state laser kits of 532 nm were used as excitation sources in SERS measurements. SERS enhancement factor was estimated for PGNSs specimen and was found as high as 3.5×10⁶. Finite difference time domain analysis was carried out to correlate the electromagnetic (EM) near-field distributions with the experiment results achieved under this investigation. EM near-field distributions at different planes were extracted for s-, p- and 45° of incident polarizations. EM near-field distributions for such nanostructures as well as current density distributions under different circumstances were demonstrated and plausible scenarios were elucidated given SERS enhancements. Such generic fabrication route as well as correlated investigation is not only indispensable to realize the potential of SERS applications but also unveil the underneath plasmonic characteristics of complex SERS-active nanostructures.     

A two-grid finite element method for the Allen-Cahn equation with the logarithmic potential

In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H¹ norm is achieved when the mesh sizes satisfy h = O(H²). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate.     

Weighted isoperimetric inequalities in warped product manifolds

We prove some sharp isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We also relate them to inequalities involving the higher order mean-curvature integrals. Applications include some sharp eigenvalue estimates, Pólya-Szegö inequality, Faber-Krahn inequality, Sobolev inequality and some sharp geometric inequalities in some warped product spaces.