A numerical method based on hybrid orthonormal Bernstein and improved block-pulse functions for solving Volterra–Fredholm integral equations

This paper deals with the numerical solution of the integral equations of linear second kind Volterra–Fredholm. These integral equations are commonly used in engineering and mathematical physics to solve many of the problems. A hybrid of Bernstein and improved block-pulse functions method is introduced and used where the key point is to transform linear second-type Volterra–Fredholm integral equations into an algebraic equation structure that can be solved using classical methods. Numeric examples are given which demonstrate the related features of the process.     

Dynamical instability of shear-free collapsing star in extended teleparallel gravity

We study the spherically symmetric collapsing star in terms of dynamical instability. We take the framework of extended teleparallel gravity with a non-diagonal tetrad, a power-law form of the model presenting torsion and a matter distribution as a non-dissipative anisotropic fluid. The vanishing shear scalar condition is adopted to gain insight in a collapsing star. We apply a first order linear perturbation scheme to the metric, the matter, and f(T) functions. The dynamical equations are formulated under this perturbation scheme to develop collapsing equation for finding dynamical instability limits in two regimes, such as the Newtonian and the post-Newtonian regime. We obtain a constraint-free solution of a perturbed time dependent part with the help of a vanishing shear scalar. The adiabatic index exhibits the instability ranges through the second dynamical equation which depend on physical quantities such as the density, the pressure components, the perturbed parts of the symmetry of the star, etc. We also develop some constraints on the positivity of these quantities and obtain instability ranges to satisfy the dynamical instability condition.     

Soliton solutions of Burgers equations and perturbed Burgers equation

This paper carries out the integration of Burgers equation by the aid of tanh method. This leads to the complex solutions for the Burgers equation, KdV-Burgers equation, coupled Burgers equation and the generalized time-delayed Burgers equation. Finally, the perturbed Burgers equation in (1+1) dimensions is integrated by the ansatz method.     

Modified simple equation method for nonlinear evolution equations

This paper reflects the implementation of a reliable technique which is called modified simple equation method (MSEM) for solving evolution equations. The proposed algorithm has been successfully tested on two very important evolution equations namely Fitzhugh-Nagumo equation and Sharma-Tasso-Olver equation. Numerical results are very encouraging.     

Study on the normalized Laplacian of a penta-graphene with applications

Let L _n denote a linear pentagonal chain with 2n pentagons. The penta-graphene (penta-C), denoted by R _n is the graph obtained from L _n by identifying the opposite lateral edges in an ordered way, whereas the pentagonal Möbius ring is the graph obtained from the L _n by identifying the opposite lateral edges in a reversed way. In this paper, through the decomposition theorem of the normalized Laplacian characteristic polynomial and the relationship between its roots and the coefficients, an explicit closed-form formula of the multiplicative degree-Kirchhoff index (resp. Kemeny's constant, the number of spanning trees) of R _n is obtained. Furthermore, it is interesting to see that the multiplicative degree-Kirchhoff index of R _n is approximately of its Gutman index. Based on our obtained results, all the corresponding results are obtained for .     

Oligosaccharide Analogues of Polysaccharides. Part 4. Synthesis of a monosaccharide-derived octamer

NaSMe in toluene leads to regioselective de-C-silylation of the bis[(trimethylsilyl)ethynyl]saccharide 2 , but to decomposition of butadiynes such as 1 or 12 . We have, therefore, combined the known reagent-controlled, regioselective desilylation of 2 and of 12 (AgNO₂/KCN) with a substrate-controlled regioselective de-C-silylation, based on C-silyl groups of different size. This combination was studied with the fully protected 3 which was mono-desilylated to 4 or to 5 (Scheme 1). Triethylsilylation of 5 (→ 6 ) was followed by removal of the Me₃Si group (→ 7 ), introduction of a (t-Bu)Me₂Si group (→ 8 ) and removal of the Et₃Si group yielded 9 ; these high-yielding transformations proceed with a high degree of selectivity. Iodination of 4 gave 10 . The latter was coupled with 5 to the homodimer 11 and the heterodimer 12 , which was desilylated to 13 . The second building block for the tetramer was obtained by coupling 14 (from 7 ) with 5 , leading to 15 and 16 . Removal of the Me₃Si group (→ 17 ) and iodination led to 18 which was coupled with 13 to the homotetramer 20 and the heterotetramer 19 (Scheme 2). Deprotection of 19 gave 21 , which was, on the one hand, iodinated to 22 , and, on the other hand, protected by the (t-Bu)Me₂Si group (→ 23 ). Removal of the Et₃Si group (→ 24 ) and coupling afforded the homooctamer 26 and the heterooctamer 25 . Yields of iodination, silylation, and desilylation were consistently high, while heterocoupling proceeded in only 50–55%. Cleavage of the (i-Pr)₃SiC and MeOCH₂O groups of 11 (→ 27 ), 15 (→ 28 ), 20 (→ 29 ) and 26 (→ 30 ) proceeded in high yields (Scheme 3). Complete deprotection in two steps of the heterocoupling products 16 (→ 31 → 32 ), 19 (→ 33 → 34 ), and 25 (→ 35 → 36 ) gave the unprotected dimer 32 , tetramer 34 , and octamer 36 in high yields (Scheme 4). Only the dimer 32 is soluble in H₂O; the ¹H-NMR spectra of 32 , 34 , and 36 in (D₆)DMSO (relatively low concentration) show no signs of association.     

Self-assembly, characterization, and chemical stability of isocyanide-bound molecular wire monolayers on gold and palladium surfaces

Self-assembled monolayers (SAMs) of the isocyano derivative of 4,4'-di(phenylene-ethynylene)benzene (1), a member of the "OPE" family of "molecular wires" of current interest in molecular electronics, have been prepared on smooth, {111} textured films of Au and Pd. For assembly in oxygen-free environments with freshly deposited metal surfaces, infrared reflection spectroscopy (IRS) indicates the molecules assume a tilted structure with average tilt angles of 18-24 degrees from the surface normal. The combination of IRS, X-ray photoelectron spectroscopy, and density functional theory calculations all support a single sigma-type bond of the -NC group to the Au surface and a sigma/pi-type of bond to the Pd surface. Both SAMs show significant chemical instability when exposed to typical ambient conditions. In the case of the Au SAM, even a few hours storage in air results in significant oxidation of the -NC moieties to -NCO (isocyanate) with an accompanying decrease in surface chemical bonding, as evidenced by a significant increase in instability toward dissolution in solvent. In the case of the Pd SAM, similar air exposure does not result in incorporation of oxygen or loss of solvent resistance but rather results in a chemically altered interface which is attributed to polymerization of the -NC moieties to quasi-2D poly(imine) structures. Conductance probe atomic force microscope measurements show the conductance of the degraded Pd SAMs can diminish by approximately 2 orders of magnitude, an indication that the SAM-Pd electrical contact has severely degraded. These results underscore the importance of careful control of the assembly procedures for aromatic isocyanide SAMs, particularly for applications in molecular electronics where the molecule-electrode junction is critical to the operational characteristics of the device.     

Supramolecular structure of tetrakis-μ-(4-chloro-3-nitrobenzoato)-bis(methanol)dicopper(II)

The title compound, [Cu₂(II)(4-Cl-3-NO₂–C₆H₃CO₂)₄(CH₃OH)₂] or [Cu₂(4-chloro-3-nitrobenzoate)₄(MeOH)₂] has been prepared and its structure determined using X-ray crystallography. The complex crystallizes in the triclinic space group P-1 with a = 9.6887(9)Å, b = 10.6448(9)Å, c = 11.4194(10)Å, = 108.094(2), = 110.682(2), = 105.055(2), V = 952.691(15)ų, and Z = 1. The structure consists of centrosymmetric dimers in which the Cu(II) atoms display a square pyramidal CuO₅ coordination, with four carboxylate O atoms in the basal plane [CuO 1.951(2)–1.968(2)Å] and the methanol O atoms in the apical position [CuO 2.170(2)Å]. The Cu atoms are 2.614(1)Å apart and are bridged by four benzoate groups. The discrete dimers are extended into 1D chains that result from hydrogen bonding between the coordinated methanol on one Cu(II) dimer and the nitro substituent on an adjacent Cu(II) dimer. The chains are interdigited and held by – stacking interactions forming 3D supramolecular array.     

Insights on the Inhibitory Power of Flavonoids on Tyrosinase Activity: A Survey from 2016 to 2021

Tyrosinase is a multifunctional copper-containing oxidase enzyme that initiates melanin synthesis in humans. Excessive accumulation of melanin pigments or the overexpression of tyrosinase may result in skin-related disorders such as aging spots, wrinkles, melasma, freckles, lentigo, ephelides, nevus, browning and melanoma. Nature expresses itself through the plants as a source of phytochemicals with diverse biological properties. Among these bioactive compounds, flavonoids represent a huge natural class with different categories such as flavones, flavonols, isoflavones, flavan-3-ols, flavanones and chalcones that display antioxidant and tyrosinase inhibitor activities with a diversity of mechanistic approaches. In this review, we explore the role of novel or known flavonoids isolated from different plant species and their participation as tyrosinase inhibitors reported in the last five years from 2016 to 2021. We also discuss the mechanistic approaches through the different studies carried out on these compounds, including in vitro, in vivo and in silico computational research. Information was obtained from Google Scholar, PubMed, and Science Direct. We hope that the updated comprehensive data presented in this review will help researchers to develop new safe, efficacious, and effective drug or skin care products for the prevention of and/or protection against skin-aging disorders.     

An authentication protocol based on chaos and zero knowledge proof

Nonlinear Dynamics - Port Knocking is a method for authenticating clients through a closed stance firewall, and authorising their requested actions, enabling severs to offer services to...