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 .     

New findings for two new type sine hyperbolic potentials

We propose two new type sine hyperbolic potentials $V(x)=a^2{\sinh }^2?(x)?k{\tanh }^2?(x)$ and $V(x)=c^2{\sinh }^4?(x)?k{\tanh }^2?(x)$. They may become single- or double-well potentials depending on the potential parameters $a,c$ and k. We find that its exact solutions can be written as the confluent Heun functions $H_c(\alpha ,\beta ,\gamma ,\delta ,\eta ;z)$, in which the energy level E is involved inside the parameter η. The properties of the wave functions, which is strongly relevant for the potential parameters $a,c$ and k, are illustrated.     

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.     

Well‐defined non‐linear polyethylene‐based macromolecular architectures

Polyethylene (PE)‐based 3‐ and 4‐miktoarm star [PE(PCL)₂, PE(PCL)₃] and H‐type [(PCL)₂PE(PCL)₂] block copolymers [polycaprolactone (PCL)] were synthesized by a combination of polyhomologation, chlorosilane chemistry, and ring opening polymerization (ROP). The following steps were used for the synthesis of the miktoarm stars: (a) reaction of a hydroxy‐terminated polyethylene (PE‐OH), prepared by polyhomologation of dimethylsulfoxonium methylide with a monofunctional boron initiator followed by oxidation/hydrolysis, with chloromethyl(methyl)dimethoxysilane or chloromethyltrimethoxysilane; (b) hydrolysis of the produced ω‐di(tri)methoxysilyl‐polyethylenes to afford ω‐dihydroxy‐polyethylene (difunctional initiator) and ω‐trihydroxy‐polyethylene (trifunctional initiator); and (c) ROP of ɛ‐caprolactone with the difunctional (3‐miktoarm star) or trifunctional macroinitiator (4‐miktoarm star), in the presence of 1‐tert‐butyl‐2,2,4,4,4‐pentakis(dimethylamino)‐2λ⁵,4λ⁵‐catenadi(phosphazene) (t‐BuP₂). The H‐type block copolymers were synthesized using the same strategy, but with a difunctional polyhomologation initiator. All intermediates and final products were characterized by HT‐GPC, ¹H NMR and FTIR analyses. Thermal properties of the PE precursors and all final products were investigated by DSC and TGA. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2018 , 56, 2129–2136     

Some Regularity Criteria for Affine Semilocal Rings

The purpose of this article is to study an existence of p-bases and mainly to give some regularity criteria for semilocal rings essentially of finite type over fields.     

Structural investigations of anthranilimide derivatives by CoMFA and CoMSIA 3D-QSAR studies reveal novel insight into their structures toward glycogen phosphorylase inhibition

In the present work, three-dimensional quantitative structure–activity relationship (3-D QSAR) studies on a set of 70 anthranilimide compounds has been performed using docking-based as well as substructure-based molecular alignments. This resulted in the selection of more statistically relevant substructure-based alignment for further studies. Further, molecular models with good predictive power were derived using CoMFA (r ^2?=?0.997; Q ^2?=?0.578) and CoMSIA (r ^2?=?0.976; Q ^2?=?0.506), for predicting the biological activity of new compounds. The so-developed contour plots identified several key features of the compounds explaining wide activity ranges. Based on the information derived from the CoMFA contour maps, novel leads were proposed which showed better predicted activity with respect to the already reported systems. Thus, the present study not only offers a highly significant predictive QSAR model for anthranilimide derivatives as glycogen phosphorylase (GP) inhibitors which can eventually assist and complement the rational drug-design attempts, but also proposes a highly predictive pharmacophore model as a guide for further development of selective and more potent GP inhibitors as anti-diabetic agents.