Approximation by complex szász-durrmeyer operators in compact disks

In the present paper, we deal with the complex Szász-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. Also, the exact order of approximation is found.     

The Ablowitz–Ladik hierarchy integrability analysis revisited: the vertex operator solution representation structure

A regular gradient-holonomic approach to studying the Lax type integrability of the Ablowitz–Ladik hierarchy of nonlinear Lax type integrable discrete dynamical systems in the vertex operator representation is presented. The relationship to the Lie-algebraic integrability scheme is analyzed and the connection with the τ-function representation is discussed.     

A simplified adaptive Cartesian grid system for solving the 2D shallow water equations

This paper presents a new simplified grid system that provides local refinement and dynamic adaptation for solving the 2D shallow water equations (SWEs). Local refinement is realized by simply specifying different subdivision levels to the cells on a background uniform coarse grid that covers the computational domain. On such a non‐uniform grid, the structured property of a regular Cartesian mesh is maintained and neighbor information is determined by simple algebraic relationships, i.e. data structure becomes unnecessary. Dynamic grid adaptation is achieved by changing the subdivision level of a background cell. Therefore, grid generation and adaptation is greatly simplified and straightforward to implement. The new adaptive grid‐based SWE solver is tested by applying it to simulate three idealized test cases and promising results are obtained. The new grid system offers a simplified alternative to the existing approaches for providing adaptive mesh refinement in computational fluid dynamics. Copyright © 2011 John Wiley & Sons, Ltd.     

An Endoplasmic Reticulum Targeting Type I Photosensitizer for Effective Photodynamic Therapy against Hypoxic Tumor Cells

Organelle-targeted type I photodynamic therapy (PDT) shows great potential to overcome the hypoxic microenvironment in solid tumors. The endoplasmic reticulum (ER) is an indispensable organelle in cells with important biological functions. When the ER is damaged due to the production of reactive oxygen species (ROS), the accumulation of misfolded proteins will interfere with ER homeostasis, resulting in ER stress. Here, an ER-targeted benzophenothiazine-based photosensitizer NBS-ER was presented. ER targeting modification significantly reduced the dark toxicity and improved phototoxicity index (PI). NBS-ER could effectively produce O₂⁻⋅ with near-infrared irradiation, making its phototoxicity under hypoxia close to that under normoxia. Meanwhile, the photoinduced ROS triggered ER stress and induced apoptosis. In addition, NBS-ER possessed excellent photodynamic therapeutic effect in 4T1-tumor-bearing mice.     

Split‐type octonion matrix

The split and hyperbolic (countercomplex) octonions are eight‐dimensional nonassociative algebras over the real numbers, which are in the form , where e_m's have different properties for them. The main purpose of this paper is to define the split‐type octonion and its matrix whose inputs are split‐type octonions and give some properties for them by using the real quaternions, split, and hyperbolic (countercomplex) octonions. On the other hand, to make some definitions, we present some operations on the split‐type octonions. Also, we show that every split‐type octonions can be represented by 2 × 2 real quaternion matrix and 4 × 4 complex number matrix. The information about the determinants of these matrix representations is also given. Besides, the main features of split‐type octonion matrix concept are given by using properties of  real quaternion matrices. Then, 8n × 8nreal matrix representations of split‐type octonion matrices are shown, and some algebraic structures are examined. Additionally, we introduce real quaternion adjoint matrices of split‐type octonion matrices. Moreover, necessary and sufficient conditions and definitions are given for split‐type octonion matrices to be special split‐type octonion matrices. We describe some special split‐type octonion matrices. Finally, oct‐determinant of split‐type octonion matrices is defined. Definitive and understandable examples of all definitions, theorems, and conclusions were given for a better understanding of all these concepts.     

Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Let be the class of all sense‐preserving homeomorphic self‐mappings of . The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω) = g in unit disk with associated boundary value conditions and . Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when and are small enough.     

PDα‐type distributed learning control for nonlinear fractional‐order multiagent systems

This paper considers the consensus tacking problem for nonlinear fractional‐order multiagent systems by presenting a PD^α‐type iterative learning control update law with initial learning mechanisms. The asymptotical convergence of the proposed distributed learning algorithm is strictly proved by using the properties of fractional calculus. A sufficient condition is derived to guarantee the whole multiagent system achieving an asymptotic output consensus. An illustrative example is given to verify the theoretical results.     

The maximum number of columns in supersaturated designs with

Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows () and a maximum in absolute value correlation between any two columns (). In particular, they proved that for (mod ) and . However, the only known SSD satisfying this upper bound is when . By utilizing a computer search, we prove that for , and . These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the lower bound for SSDs with rows and columns, for , and . Finally, we show that a skew‐type Hadamard matrix of order can be used to construct an SSD with rows and columns that proves . Hence, we establish for and for all (mod ) such that . Our result also implies that when is a prime power and (mod ). We conjecture that for all and (mod ), where is the maximum number of equiangular lines in with pairwise angle .