Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.     

Diameter of a direct power of a finite group

We present two conjectures concerning the diameter of a direct power of a finite group. The first conjecture states that the diameter of Gⁿ with respect to each generating set is at most n(|G|?rank(G)); and the second one states that there exists a generating set 𝒜, of minimum size, for Gⁿ such that the diameter of Gⁿ with respect to 𝒜 is at most n(|G|?rank(G)). We will establish evidence for each of the above mentioned conjectures.     

A modified pseudospectral method for solving trajectory optimization problems with singular arc

This paper presents a direct method based on Legendre–Radau pseudospectral method for efficient and accurate solution of a class of singular optimal control problems. In this scheme, based on a priori knowledge of control, the problem is transformed to a multidomain formulation, in which the switching points appear as unknown parameters. Then, by utilizing Legendre‐Radau pseudospectral method, a nonlinear programming problem is derived which can be solved by the well‐developed parameter optimization algorithms. The main advantages of the present method are its superior accuracy and ability to capture the switching times. Accuracy and performance of the proposed method are examined by means of some numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.     

Effects of awareness program and delay in the epidemic outbreak

In the present paper, an epidemic model has been proposed and analyzed to investigate the impact of awareness program and reporting delay in the epidemic outbreak. Awareness programs induce behavioral changes within the population, and divide the susceptible class into two subclasses, aware susceptible and unaware susceptible. The existence and the stability criteria of the equilibrium points are obtained in terms of the basic reproduction number. Considering time delay as the bifurcating parameter, the Hopf bifurcation analysis has been performed around the endemic equilibrium. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and central manifold theorem. To verify the analytical results, comprehensive numerical simulations are carried out. Copyright © 2016 John Wiley & Sons, Ltd.     

A free boundary problem arising from DCIS mathematical model

Ductal carcinoma in situ – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma in situ model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.     

Crystal structure of 5,6‐bis(9H‐carbazol‐9‐yl)benzo[c][1,2,5]thiadiazole: distortion from a hypothetical higher‐symmetry structure

Nucleophilic substitution of F atoms in 5,6‐difluorobenzo[c ][1,2,5]thiadiazole (DFBT) for carbazole could be potentially interesting as a novel way of synthesizing building blocks for new conjugated materials for applications in organic chemistry. The crystal structures of 5,6‐bis(9H‐carbazol‐9‐yl)benzo[c ][1,2,5]thiadiazole (DCBT), C₃₀H₁₈N₄S, and its hydrate, C₃₀H₁₈N₄S·0.125H₂O, were investigated using single‐crystal X‐ray analysis. The hydrate contains two symmetry‐independent DCBT molecules. The dihedral angles between the plane of the central benzothiadiazole fragment and that of the carbazole units vary between 50.8 and 69.9°, indicating conformational flexibility of the DCBT molecule in the crystals, which is consistent with quantum chemical calculations. The analysis of the crystal packing of DCBT revealed that the experimental triclinic structure could be described as a distortion from a hypothetical higher‐symmetry monoclinic structure. The quantum chemical calculations of two possible monoclinic structures, which are related to the experimental structure by a shifting of molecular layers, showed that the proposed structures are higher in energy by 5.4 and 10.1 kcal mol⁻¹. This energy increase is caused by less dense crystal packings of the symmetric structures, which results in a decrease of the number of intermolecular interactions.     

Virtual screening of B-Raf kinase inhibitors: A combination of pharmacophore modelling,molecular docking, 3D-QSAR model and binding free energy calculation studies

B-Raf kinase has been identified as an important target in recent cancer treatment. In order to discover structurally diverse and novel B-Raf inhibitors (BRIs), a virtual screening of BRIs against ZINC database was performed by using a combination of pharmacophore modelling, molecular docking, 3D-QSAR model and binding free energy (ΔG_bind) calculation studies in this work. After the virtual screening, six promising hit compounds were obtained, which were then tested for inhibitory activities of A375 cell lines. In the result, five hit compounds show good biological activities (IC₅₀ < 50 μM). The present method of virtual screening can be applied to find structurally diverse inhibitors, and the obtained five structurally diverse compounds are expected to develop novel BRIs.     

3D Laser Micro‐ and Nanoprinting: Challenges for Chemistry

3D printing is a powerful emerging technology for the tailored fabrication of advanced functional materials. This Review summarizes the state‐of‐the art with regard to 3D laser micro‐ and nanoprinting and explores the chemical challenges limiting its full exploitation: from the development of advanced functional materials for applications in cell biology and electronics to the chemical barriers that need to be overcome to enable fast writing velocities with resolution below the diffraction limit. We further explore chemical means to enable direct laser writing of multiple materials in one resist by highly wavelength selective (λ‐orthogonal) photochemical processes. Finally, chemical processes to construct adaptive 3D written structures that are able to respond to external stimuli, such as light, heat, pH value, or specific molecules, are highlighted, and advanced concepts for degradable scaffolds are explored.     

g-C3N4/SnS2 Heterostructure: a Promising Water Splitting Photocatalyst

Graphite-like carbon nitride (g-C₃N₄) based heterostrutures has attracted intensive attention due to their prominent photocatalytic performance. Here, we explore the g-C₃N₄/SnS₂ coupling effect on the electronic structures and optical absorption of the proposed g-C₃N₄/SnS₂ heterostructure through performing extensive hybrid functional calculations. The obtained geometric structure, band structures, band edge positions and optical absorptions clearly reveal that the g-C₃N₄ monolayer weakly couples to SnS₂ sheet, and forms a typical van der Waals heterojunction. The g-C₃N₄/SnS₂ heterostructure can effectively harvest visible light, and its valence band maximum and conduction band minimum locate in energetically favorable positions for both water oxidation and reduction reactions. Remarkably, the charge transfer from the g-C₃N₄ monolayer to SnS₂ sheet leads to the built-in interface polarized electric field, which is desirable for the photogenerated carrier separation. The built-in interface polarized electric field as well as the nice band edge alignment implys that the g-C₃N₄/SnS₂ heterostructure is a promising g-C₃N₄ based water splitting photocatalyst with good performance.     

Study on convergence and error of a numerical method for solving systems of nonlinear Fredholm–Volterra integral equations of Hammerstein type

The aim of this paper is to present an efficient analytical and numerical procedure for solving systems of nonlinear Fredholm–Volterra integral equations of the Hammerstein type with the aid of fixed point techniques and the usual Schauder basis in an adequate Banach space.