Syntheses, Crystal Structures and Properties of Two Supramolecules Assembled from Weak Interactions

Two complexes, [Cu2(Htdb)4(H2O)2] (1) and [Cd(bipy)2(Hmcmbc)2] (2) (H2tdb = 2,2-thiodibenzoic acid, H2mcmbc = m-(carboxyl-methyloxy)-benzenecarboxylic acid, bipy = 4,4'-bipyridine), have been prepared, and were characterized by elemental analysis, FT-IR and single-crystal X-ray diffraction. Structures indicate that complex 1 is a single molecule, and 2 is a one-dimensional chain. Their two-and three-dimensional frameworks are constructed through hydrogen bonding, π…π or C–H…π stacking, and such other weak interactions. The cyclic voltametric behavior of complex 1 and luminescence property of complex 2 were investigated. 1 belongs to the triclinic system, space group P with a = 7.8607(6), b = 11.7619(9), c = 15.3481(12) , α = 109.3670(10), β = 92.4420(10), γ = 92.0450(10)°, V = 1335.65(18) 3, Mr = 1256.26, Dc = 1.5619(2) g/cm3, F(000) = 642, μ = 1.029 mm–1, Z = 1, the final R = 0.0289 and wR = 0.0763 for 5199 observed reflections with I > 2σ(I). Complex 2 crystallizes in triclinic, space group P with a = 9.6378(10), b = 9.7508(10), c = 19.055(2) , α = 88.7020(10), β = 80.5260(10), γ = 69.2000(10)o, V = 1649.9(3) 3, Mr = 815.07, Dc = 1.641 g/cm3, μ = 0.732 mm-1, F(000) = 828, Z = 2, the final R = 0.0511 and wR = 0.1149 for 4729 observed reflections (I > 2σ(I)).     

On constant elasticity of substitution – Constant elasticity of transformation Directional Distance Functions

This paper aims to present non-linear CES (Constant Elasticity of Substitution)–CET (Constant Elasticity of Transformation) Directional Distance Functions. These measures inherit the structure of the standard Directional Distance Functions and that of the CES–CET technology. These functions allow non-parametric estimation of efficiency scores through linear programming method. Besides, the CES –CET technology gives the opportunity to explore α-returns to scale assumption for the new distance functions. The duality theory is investigated through pseudo profit, cost and revenue functions. The dual standpoint provides non-linear adjusted prices that can occur into non-linear pricing practices. An application is proposed to give an illustrative example of the primal CES–CET Directional Distance Functions.     

Weighted core–EP inverse of an operator between Hilbert spaces

We introduce and study the weighted core–EP inverse of an operator between two Hilbert spaces as a generalization of the weighted core–EP inverse for a rectangular matrix. Several new properties of weighted core–EP inverses are given and some known results are extended. Using a weighted operator, the core–EP pre-order and the minus partial order of corresponding operators, we define new pre-orders on the set of all Wg–Drazin invertible operators between two Hilbert spaces. As consequences of our results, we present a new characterization and new representations of the core–EP inverse, new characterizations of the core–EP pre-order and extend the core–EP pre-order to a partial order.     

On explicit order 1.5 approximations with varying coefficients: The case of super-linear diffusion coefficients

A conjecture appears in Kumar and Sabanis (2016), in the form of a remark, where it is stated that it is possible to construct, in a specified way, any high order explicit numerical schemes to approximate the solutions of SDEs with superlinear coefficients. We answer this conjecture to the positive for the case of order 1.5 approximations and show that the suggested methodology works. Moreover, we explore the case of having Hölder continuous derivatives for the diffusion coefficients.     

Blow-up and global solutions for a class of time fractional nonlinear reaction–diffusion equation with weakly spatial source

This paper investigates the blow-up of solutions for a time fractional nonlinear reaction–diffusion equation with weakly spatial source. We first derive two sufficient conditions under which the solutions may blow up in finite time. Then, we prove the existence of global solution when the initial data are small enough. Moreover, the long time behavior of bounded solutions will be analyzed.     

Breather and rogue wave solutions for the (2+1)-dimensional nonlinear Schrödinger–Maxwell–Bloch equation

In this paper, we mainly study the (2+1)-dimensional Schrödinger–Maxwell–Blochequation (SMBE). We have constructed the generalized N-fold Darboux transformations (DT), and based on the plane wave solutions, the breather and rogue wave solutions are systematically generated, the dynamical features of those solutions are graphically represented.     

An improvement of the Kolmogorov–Riesz compactness theorem

The purpose of this short note is to provide a new and very short proof of a result by Sudakov (1957), offering an important improvement of the classical result by Kolmogorov–Riesz on compact subsets of Lebesgue spaces.     

Remark on a lower diameter bound for compact shrinking Ricci solitons

In this paper, inspired by Fernández-López and García-Río [11], we shall give a new lower diameter bound for compact non-trivial shrinking Ricci solitons depending on the range of the potential function, as well as on the range of the scalar curvature. Moreover, by using a universal lower diameter bound for compact non-trivial shrinking Ricci solitons by Chu and Hu [7] and by Futaki, Li, and Li [13], we shall provide a new sufficient condition for four-dimensional compact non-trivial shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Furthermore, we shall give a new lower diameter bound for compact self–shrinkers of the mean curvature flow depending on the norm of the mean curvature. We shall also prove a new gap theorem for compact self–shrinkers by showing a necessary and sufficient condition to have constant norm of the mean curvature.     

Superconvergence of finite element method for singularly perturbed convection–diffusion equations in 1D

In this article, a singularly perturbed convection–diffusion equation is solved by a linear finite element method on a Shishkin mesh. By means of an analysis exploiting symmetries in the convective term of the bilinear form, a new superconvergence rate, which improves the existing result, is obtained.     

Nonautonomous soliton solutions for a nonintegrable Korteweg–de Vries equation with variable coefficients by the variational approach

A nonintegrable Korteweg–de Vries equation with variable coefficients is investigated in this paper. Due to the existence of variable coefficients, the equation becomes nonintegrable, which leads to the invalidity of the traditional analytical methods to obtain soliton solutions. In order to overcome this difficulty, the variational approach is employed in this paper. The variational principle corresponding to this nonintegrable equation is established. Based on that, the first- and second-order nonautonomous soliton solutions are derived. We note that the obtained solutions can be degenerated to the integrable cases. Properties of the nonautonomous solitons and influence of the variable coefficients are discussed.