SIMULTANEOUSE APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVES BY LAGRANGE INTERPOLATING POLYNOMIALS

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).

A NEW METHOD FOR THE PREPARATION OF PHENOLS AND ITS DERIVATIVES

a-Oxo ketene dithioacetals 2 via 1,2-nucleophilie addition by methallyl magnesius chloride afforded corresponding alcohols (3). Treated with water or methanol and catalyzed by Lewis acid, the alcohols 3 were converted regiospecifical ly to substituted phenols 5' or related phenol methyl ethers 5 respectively. This reaction is a novel approach to the synthesis of phenols and their derivatives starting from non-aromatic precursors.

SPECTROPHOTOMETRIC STUDY OF DDQ CHARGE-TRANSFER COMPLEXES WITH 1,10-PHENANTHROLINE AND ITS DERIVATIVES

1,10-phenanthroline and its derivatives can form charge-transfer complexes with 2,3-dichloro-5,6-dicyano-1,4-benzoquinone(DDQ).The absorption spectra and conditions of complex formation,such as the reaction time,the quantity of DDQ, and the solvents, have been studied.After the HMO calculation of phenanthrolines,the molar absorptivities were found to depend on the substituents on the phenanthroline rings.

Investigation of Plant Systemic Organoselena Compounds (Ⅰ) Synthesis of Selena-Morpholind and its Derivatives

The synthesis of selena-morpholine and its derivatives is described.

ON APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES BY A POLYNOMIAL AND ITS DERIVATIVES

Let g∈C^q[-1, 1] be such that g^((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n^((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n^((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi- mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x^2)^(1/2)~q‖ This result is easily extended to an arbitrary f∈C^q[-1, 1], by subtracting from f the polynomial of minnimal degree which interpolates f^((0))…,f^((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a given Polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation in C^q[-1, 1] and trigonometric polynomial approximation in the space of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation proedures for small values of q.

Reactivity of C-10 Function in Sinenxan A and Its Derivatives

The allylic nature of C-10 in sinenxan A (1) leads to facile SN1 and eliminion reactions

Copper-Catalyzed Asymmetric Hydroamination:A Unified Strategy for the Synthesis of Chiralβ-Amino Acid and Its Derivatives

Catalytic asymmetric aza-Michael represents one of the most convenient and atom-economical approaches for the rapid construction of biologically active chiralβ-amino acid frameworks.However,the direct enantioselective addition of nitrogen-based nucleophiles to intrinsically low reactivity ofα,β-unsaturated carboxylic acid,ester,and amide,as well as simpleα,β-unsaturated nitrile,remains a long-standing challenge.

Enantiomeric separation of α-phenylethylamine and its derivatives by series coupled open tubular columns

A series coupled fused-silica open tubular columns of a chiral and an achiral stationary phase were used to achieve the enantiomeric separation of a-phenylethylamine, o,m,p-methyl-and o,m,p-methoxy-a-phenylethylamine. A simplified equation describing the retention values of series coupled columns was derived, which relates the capacity factors of series coupled columns to the capacity factor of individual columns, their lengths, inlet and outlet pressure. Very good agreement was obtained both between calculated values by this equation and those by Krupcik's equation and between values measured experimentally and calculated theoretically. By means of this simplified equation a computer program was developed in locating the operating range of optimum amin, enantiomeric separation of all a-phenylethylamine and its derivatives were achieved in a single run.

CHARGE TRANSFER POLYMERIZATION——THE POLYMERIZATION OF ACRYLONITRILE INITIATED BY CARBAZOLE AND ITS DERIVATIVES

The photopolymerization of acrylonitrile (AN) initited by aromatic tertiary amine, phenothiazine, diphenylsulfide triphenylphosphine, etc. are known to proceed by a charge transfer process with the formation of an exciplex. Recently we have briefly reported that the N-vinylcarbazole (VCZ) is an excellent photoinitiator for AN polymeriza-

FLAVONOIDS AND ITS DERIVATIVES FROM CALLISTEPHUS CHINENSIS FLOWERS AND SIMULTANEOUS DERTERMINATION BY HPLC

The aim of the study was to study the chemical constituents of Callistephus chinensis flowers(CCF)(Chinese name is Cui Ju)and establish a method for determination of seven flavonoids in CCF by using HPLC simultaneously.The compounds were isolated by repeated column chromatography with silica gel,and HPLC.

Boundary Conditions for Sturm-Liouville Equation with Transition Regions and Barriers or Wells

By means of expansions of rapidly in infinity decreasing functions in delta functions and their derivatives, we derive generalized boundary conditions of the Sturm-Liouville equation for transitions and barriers or wells between two asymptotic potentials for which the solutions are supposed as known. We call such expansions “moment series” because the coefficients are determined by moments of the function. An infinite system of boundary conditions is obtained and it is shown how by truncation it can be reduced to approximations of a different order (explicitly made up to third order). Reflection and refraction problems are considered with such approximations and also discrete bound states possible in nonsymmetric and symmetric potential wells are dealt with. This is applicable for large wavelengths compared with characteristic lengths of potential changes. In Appendices we represent the corresponding foundations of Generalized functions and apply them to barriers and wells and to transition functions. The Sturm-Liouville equation is not only interesting because some important second-order differential equations can be reduced to it but also because it is easier to demonstrates some details of the derivations for this one-dimensional equation than for the full three-dimensional vectorial equations of electrodynamics of media. The article continues a paper that was made long ago.

Studies on the Electronic Structures of the C_60, C_60H_60 and C_60F_60

The present paper represents the electronic structures of the cluster C60 and its derivatives C60H60 and C60F60 with FOOT-type of structure. On the basis of the total energies and molecular orbital energies from CNDO/2 calculations the clusters' stabilities, electronic configurations and charge distributions are discussed. The calculation results and discussion are very beneficial to understanding physical and chemical properties of the cluster C60 and its derivatives.

The utilization of carbon-based nanomaterials in bone tissue regeneration and engineering:Respective featured applications and future prospects

In recent years,with advancements in bone tissue regeneration and engineering technologies,carbon-based nanomaterials(CNMs)have progressively demonstrated advantages in the therapies of critical bone defects and related diseases that conventional substances fail to develop,such as excellent mechanical properties,large specific surface,tunable surface characteristics,and superior biocompatibility.More importantly,carbon-based nanomaterials with efficient cell proliferation and osteogenic differentiation could have a significant impact on bone tissue regeneration.In this paper,we have reviewed the characteristic applications of extraordinary types of carbon-based nanomaterials(fullerene,carbon dots,carbon nanotubes,as well as graphene and its derivatives)in bone tissue regeneration and engineering based on their structural properties,with a view to presenting their unique advantages,applications,and current directions of development as specifically and comprehensively as possible.