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In this article the aim is to propose a new form to densify parallelepipeds of R^N by sequences of α‐dense curves with accumulated densities.

This will be done by using a basic α‐densification technique and adding the new concept of sequence of α‐dense curves with accumulated density to improve the resolution of some global optimization problems.

It is found that the new technique based on sequences of α‐dense curves with accumulated densities allows to simplify considerably the process consisting on the exploration of the set of optimizer points of an objective function with feasible set a parallelepiped K of R^N. Indeed, since the sequence of the images of the curves of a sequence of α‐dense curves with accumulated density is expansive, in each new step of the algorithm it is only necessary to explore a residual subset. On the other hand, since the sequence of their densities is decreasing and tends to zero, the convergence of the algorithm is assured.

The results of this new technique of densification by sequences of α‐dense curves with accumulated densities will be applied to densify the feasible set of an objective function which minimizes the quadratic error produced by the adjustment of a model based on a beta probability density function which is largely used in studies on the transition‐time of forest vegetation.

A sequence of α‐dense curves with accumulated density represents an original concept to be added to the set of techniques to optimize a multivariable function by the reduction to only one variable as a new application of α‐dense curves theory to the global optimization.

A recursive scheme for the ALIENOR method is proposed as a remedy for the difficulties induced by the method. A progressive focusing on the most promising region, in combination with a variation of the density of the alpha-dense curve, is proposed.

ALIENOR method is aimed at reducing the space dimensions of an optimization problem by spanning it by using a single alpha-dense curve: the curvilinear abscissa along the curve becomes the only design parameter for any design space. As a counterpart, the transformation of the objective function in the projected space is much more difficult to tackle.

A fine tuning of the procedure has been performed in order to identity the correct balance between the different elements of the procedure. The proposed approach has been tested by using a set of algebraic functions with up to 1,024 design variables, demonstrating the ability of the method in solving large scale optimization problem. Also an industrial application is presented.

In the knowledge of the author there is not a similar paper in the current literature.

The purpose of this paper is to use α‐dense curves for solving some Diophantine equations, such as Pythagorean triples, Linear Diophantine equations, the Pell Fermat equation, the Mordell equation for positive values.

The paper's aim is to present the applications in Number Theory of a new method based on α‐dense curves first developed at the beginning of the 1980s by Yves Cherruault and Arthur Guillez. The α‐dense curves generalize the space filling curves (Peanocurves,…) and fractal curves. This technique can be used for solving all problems of operational research in a simple way. The main idea consists in expressing n variables by means of a single one.

Apply the method to Number Theory. One of the most important applications is related to global optimization. Multivariable optimization problems coming from operational research or from industry can be quickly and easily solved.

The paper presents a new method based on α‐dense curves for solving Diophantine equations.

The purpose of this paper is to describe a general method for solving all problems arising in industrial processes and more generally in operational research.

The paper's aim is to present a new method based on α‐dense curves first developed at the beginning of the 1980s by Yves Cherruault and Arthur Guillez. This technique allows to solve all problems of operational research in a simple way. For instance, industrial problems leading to optimization or optimal control problems can be easily and precisely solved by this very general technique. The main idea consists in expressing n variables by means of a single one.

This new method, based on “alpha‐dense curves” allows to express n variables in function of a single variable. One of the most important applications is related to global optimization. Multivariable optimization problems can be quickly and easily solved, even for great numbers of variables and for integer or boolean variables. Every problem (linear or nonlinear) coming from operational research or from industry becomes simple to solve in a very short time on micro‐calculators.

This method is deduced from the original works of Yves Cherruault et al. of MEDIMAT laboratory. The reducing transformations were initiated at the beginning of the 1980s by Yves Cherruault and Arthur Guillez. Then they were generalized by the notion of α‐dense curves. A lot of applications were derived covering entirely the operational research and a part of functional analysis.

The purpose of this paper is to use α‐dense curves for solving Boolean equations, 0‐1 integer programming problems such as the shortest path problem or the knapsack problem.

The paper's aim is to present the applications in Boolean algebra and 0‐1 integer programming of a new method based on α‐dense curves first developed at the beginning of the 1980s by Yves Cherruault and Arthur Guillez. The α‐dense curves generalize the space filling curves (Peanocurves,…) and fractal curves. The main idea consists in expressing n variables by means of a single one.

Apply the method to Boolean algebra and 0‐1 integer programming.

The paper presents a new method based on α‐dense curves for solving Boolean equations and 0‐1 integer programming problems.

The purpose of this paper is to show that a combination of Adomian and Alienor methods can be used to solve the problem of parameters identification of HIV/AIDS model. This model involves a system of three ordinary differential equations.

Parameters identification leads to the minimization of an error functional given by the sum of variations between measured variables and calculated variables obtained by solving the system of differential equations. We assume that the quantity of healthy cells CD4 + T and the viral load contents in the blood are measured.

The identification was realized by applying the combined Adomian/Alienor method allowing to reduce the problem to a minimization problem in dimention one.

Simulation results are given for illustration.

Application to parameter identification in an HIV model‐compatible, results to other methods.

This paper deals with the use of the combined Adomian/Alienor methods for solving the problem of optimal cancer chemotherapy model.

The combination of the Adomian decomposition method and the Alienor reducing transformation method allows us to solve the control problem as if it were a classical one dimensional minimization problem. The mathematical model used here describes specific model based on cell‐cycle kinetics.

It was found that the goal is to maintain the number of the cancerous cells around a desired value while keeping the toxicity to the normal tissues acceptable.

Simulation results are given for illustration.

New combined approach to optimal control of cancer chemotherapy using Adomian/Alienor Methods.

This paper seeks to present an original method for transforming multiple integrals into simple integrals.

This can be done by using α‐dense curves invented by Y. Cherruault and A. Guillez at the beginning of the 1980s.

These curves allow one to approximate the space Rⁿ (or a compact of Rⁿ) with the accuracy α. They generalize fractal curves of Mandelbrobdt. They can be applied to global optimization where the multivariables functional is transformed into a functional depending on a single variable.

Applied to a multiple integral, the α‐dense curves using Chebyshev's kernels permit one to obtain a simple integral approximating the multiple integral. The accuracy depends on the choice of α.

The paper presents an original method for transforming integrals into simple integrals.