Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives

In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.

On the Variational Problems of the Functionals with Derivatives of Higher Orders and Undetermined Boundary

According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.

The Normal Meromorphic Functions Family Concerning Higher Order Derivative and Shared Set by One-Way

Let F be a meromorphic functions family on the unit disc Δ, If for every (the zeros of f is a multiplicity of at least k) and if then and ( ), then F is normal on Δ.

First-order Derivative Spectrophotometry for the Determination of Vitamin C in Medicament

A novel method for the determination of vitamin C（Vc） is proposed in this article. After the reaction with Folin-Ciocalteau reagent at ambient temperature, Vc solution was scanned at 750--1100 nm, and its first-order derivative spectrum were obtained from the original spectrum. The values of derivative selected at 995 nm were used for determination. It was proved that Vc could quickly react with Folin-Ciocalteau reagent within 5 min and the product was quite stable for a long time. The conditions required for this method is not very complicated, its precision and accuracy are similar to those of the iodometric titration described in Chinese Pharmacopoeia, and the limit of detection is 0.312 μg/mL. The determination of the results of vitamin C tablet, pill, and injection demonstrates that this method has wide pharmaceutical applications.

Extremum of second-order directional derivatives

In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other.

Catalytic Kinetic Determination of Micro Amounts of Oxalic Acid by Second-Order Derivative Oscillopolarography

It was found that micro amounts of oxalate showed a very strong catalytic effect on the slow reaction between K 2Cr 2O 7 and Orange Ⅳ in a diluted sulfuric acid medium in a water bath at 70 ℃ . Orange Ⅳ exhibited a sensitive second order derivative polarographic wave at -0 50 V( vs . SCE). This provides the basis for a sensitive and selective catalytic kinetic method for oxalate determination with second order derivative oscillopolarography. The effects of sulphuric acid, K 2Cr 2O 7, and orange Ⅳ concentrations, reaction temperature and reaction time were investigated. A calibration curve of oxalate in the range of 0 1-2 0 μg/mL was obtained by the fixed time procedure. The detection limit was 0 03 μg/ mL. The possible interference from co existing substances or ions was examined. The new method has a high sensitivity and a good selectivity compared to other existing methods for oxalic acid determination. It has been applied to the determination of micro amounts of oxalate in real urine samples with satisfactory results.

Peano Derivatives and Singular Integrals of Arbitrary Order

The Peano derivatives are introduced for functions along an arc in the complex plane. Singular integrals of arbitrary order with singularities at its end-points are defined so that a unified theory for such integrals and Cauchy principal value integrals is established.

Pfaff-Birkhoff Variational Problem and Noether Symmetry in Terms of RiemannLiouville Fractional Derivatives of Variable Order

The Pfaff-Birkhoff variational problem and its Noether symmetry are studied in terms of Riemann-Liouville fractional derivatives of variable order. Based on the combination of variational principle and fractional calculus of variable order,the Pfaff-Birkhoff variational principle with Riemann-Liouville fractional derivatives of variable order is proposed, and the fractional Birkhoff's equations of variable order are derived. Then,the Noether 's theorem for the fractional Birkhoffian system of variable order is given. At last,an example is expressed to illustrate the application of the results.

Fractional Order Proportional Integral Derivative Controller Design and Simulation for Bioengineering Systems

This paper deals with the study of fractional order system tuning method based on Factional Order Proportional Integral Derivative( FOPID) controller in allusion to the nonlinear characteristics and fractional order mathematical model of bioengineering systems. The main contents include the design of FOPID controller and the simulation for bioengineering systems. The simulation results show that the tuning method of fractional order system based on the FOPID controller outperforms the fractional order system based on Fractional Order Proportional Integral( FOPI) controller. As it can enhance control character and improve the robustness of the system.

Fractal Dimension of Graph of Weierstrass Function and Its Derivative of the Fractional Order

In this paper,we obtain the fractal dimension of the graph of the Weierstrass function, its derivative of the fractional order and the relation between the dimension and the order of the fractional derivative.

A PRELIMINARY STUDY ON THE THEORY OF THE SECOND AND THIRD ORDER DERIVATIVE ADSORPTION CHRONOPOTENTIOMETRY FOR A REVERSIBLE REACTION

The equations of the second and third order derivative curves of time with respect to potential for a reversible process in adsorption chronopotentiometry are derived and experimentally verified.

Synthesis of Imidazole Derivatives for Their Second-order Nonlinear Optics

The design and the synthesis of two conjugated donor acceptor imidazole derivatives(1, 2) were carried out for second order nonlinear optics. The thermal properties, the transparency and second order nonlinear optical properties of the molecules were investigated. The experimental results indicate that a good nonlinearity transparency thermal stability trade off is achieved for them.

Study on State of Terbium Ion in Liposome by One order Derivative Fluorescence Quenching Method

The state of Tb3+ is investigated in liposome. When the concentration of PC is below CMC (critical micell concentration), most of Tb3+ is associated with PC, the binding constant is about 3.35×103 L/mol. When the concentration of PC is beyond CMC, most of Tb3+ is dimerized, the dimerization constant is about 3.92×104L/mol. In PC?CH?H2O system, the binding constant of Tb3+?CH complex 2.93×104L/mol is obtained.

Optimality conditions in set optimization employing higher-order radial derivatives

There are two approaches of defining the solutions of a set-valued optimization problem： vector criterion and set criterion. This note is devoted to higher-order optimality conditions using both criteria of solutions for a constrained set-valued optimization problem in terms of higher-order radial derivatives. In the case of vector criterion, some optimality conditions are derived for isolated （weak） minimizers. With set criterion, necessary and sufficient optimality conditions are established for minimal solutions relative to lower set-order relation.

Effect of Substituting Group on the Calculated Nonlinear Second－Order Optical Susceptibilities of Substituted Thiophene Derivatives with Quinoidlike Conformation

alculations of the nonlinear second-order optical susceptlbilities(β_(ijk))for sub- stituted tl1iophene derivative;with quinoidlike conformation are reported.These systetems possess small dipole moment;and large differences between dipole mo- ments of ground and first-excited states.Geometry optimizations of the molecules investigated were carried out using AM 1 method.The calculations were performed using INDO/CI method comboned with a sum-over-states expression for β_(jik). The calculated results sbw that the second-order susceptibility is a function of the na- ture and location of substituents and is larger for disubstituted molecules than monosubstituted molecules. Bipolymeric thiophenemetmne with NH_2/NO_2 groups was calctilated to have a β_μof 79. 920 × 10￣(-30) esu. It was found that the NH_2 and NO_2 groups in above disubstituted molecules are pull-pull groups in ground states,but are usual push-pull groups in the first excited states.

Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

This paper stresses the theoretical nature of constructing the optimal derivative-free iterations. We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence. We also establish the connection of derivative-free and derivative presence three-point iterations. The use of the sufficient convergence conditions allows us to design wide class of optimal derivative-free iterations. The proposed family of iterations includes not only existing methods but also new methods with a higher order of convergence.

A Hybrid ESA-CCD Method for Variable-Order Time-Fractional Diffusion Equations

In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.

Estimation of the galanthamine using derivative spectrophotometry in bulk drug and formulation

Two simple, rapid, accurate, precise, reliable and economical spectrophotometric methods have been proposed for the determination of galanthamine hydrobromide (GH) in bulk and pharmaceutical formulation. First method is zero order UV spectrophotometry and second is 1st derivative zero crossing spectrophotometry. The developed methods have shown best results in terms of linearity, accuracy, precision, LOD and LOQ for bulk drugs and marketed formulations. Absorbance was measured at 287 nm for zero order and 277.4 nm for first derivative. It obeyed Lambert-Beer’s law in the range of 30-80 μg mL-1. Both methods have good linearity (r2 = 0.9997) and accuracy found to be 100.5% and 101.2% for both methods respectively.

A NEW WAY FOR DERIVING HIGH ORDER STOKES WAVES

Variational principle for non-vortex, non-linear wave theories is established in this paper. By using this variational principle and related functional minimum condition, the fifth and sixth order Stokes Vaves are given as an example and the results are compared with those in Reference (Skjel-breia, 1961).

Financial Derivatives Market and Storage Management

It’s known to all that under ideal condition the s to rage cost is kept in lower level when storage management be arranged by Economic Order Quantity(EOQ).Does this mean that any companies should set up their own storing system in proportion to the scale of the commodities’ producing or sell ing Furthermore, even if they manage storage in EOQ, because of different oper ation scale, geographical condition or ability borrowing money from financial ma rket, different companies pay unequal cost in storing the same commodity.In thi s paper, except for supplying commodities from our own storage system, the autho rs have analyzed other two supplying ways without whole storage system, they are forward contracts and futures contracts.The authors have discussed variable su pply cost for above different supply measures.According to the cost of each sup ply way, the managers can choose the most economical way in supplying the commod ity and predict the price of futures from storage management arranged by EOQ.Th e summary content is as follow: 1. The comparing of supply cost between forward contracts and storing system a rranged by EOQ. (1) The supply cost from forward contracts (2) The supply cost from storage system arranged by Economic Order Quantity (3) The application example for comparing cost in different supply way 2.The comparing of supply cost between futures going physical and storing syst em arranged by Economic Order Quantity. (1) The supply cost from futures going physical (2) The correlation between futures contracts and storage management arranged b y EOQ (3) The application example for comparing cost in different supply way 3.How does storing system of scale economic affect the price of forward and fu tures contracts (1) How does the price of forward and futures contracts fluctuate (2) How do we calculate the price of a commodity at future point from the cost of scale economic storing (3) How do we operate efficiently in derivatives market by using the cost of sc ale economic storing (4) The application example for analyzing the price of futures 4.The correlation among storage managementforward contracts and futures mark et.