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This paper aims to introduce a new class of entire functions whose zeros (z_k)_k≥1 satisfy ∑_k=1^∞Im z_k=O(1).

This is done by means of a Ritt's formula which is used to prove that every partial sum of the Riemann Zeta function, ζ_n(z):=∑_k=1ⁿ1/k^z, n≥2, has zeros (s_nk)_k≥1 verifying ∑_k=1^∞Re s_nk=O(1) and extending this property to a large class of entire functions denoted by A_O.

It is found that this new class A_O has a part in common with the class A introduced by Levin but is distinct from it. It is shown that, in particular, A_O contains every partial sum of the Riemann Zeta function ζ_n(iz) and every finite truncation of the alternating Dirichlet series expansion of the Riemann zeta function, T_n(iz):=∑_k=1ⁿ(−1)^k−1/k^iz, for all n≥2.

With the exception of the n=2 case, numerical experiences show that all zeros of ζ_n(z) and T_n(z) are not symmetrically distributed around the imaginary axis. However, the fact consisting of every function ζ_n(iz) and T_n(iz) to be in the class A_O implies the existence of a very precise physical equilibrium between the zeros situated on the left half‐plane and the zeros situated on the right half‐plane of each function. This is a relevant fact and it points out that there is certain internal rule that distributes the zeros of ζ_n(z) and T_n(z) in such a way that few zeros on the left of the imaginary axis and far away from it, must be compensated with a lot of zeros on the right of the imaginary axis and close to it, and vice versa.

The paper presents an original class of entire functions that provides a new point of view to study the approximants and the alternating Dirichlet truncations of the Riemann zeta function.

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.

To use α‐dense curves for solving an optimization problem with constraints involving integer variables.

α‐dense curves are curves in Rⁿ depending on a single variable able to approximate a compact K⊂Rⁿ with precision α. It is proposed α‐dense curves allowing to obtain all integer points of a compact domain in Rⁿ. This transformation allows to transform the functional into a new function depending on a single variable. Then we can calculate the global optimum of the functional.

Alienor method invented by Y. Cherruault allows to find global minimum of n‐continuous variables functions. Here, α‐dense curves are extended to problems involving integer variables. The curves pass through all points having integer coordinates and belonging to the compact domain. By this method integer programming (nonlinear) problems arising in operational research have been easily and exactly solved.

It is the first time the technique based on α‐dense curves to optimization problems with integer variables are extended. This approach is totally original and allows to solve very easily and fastly nonlinear optimization problems.

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.

This chapter explores the perceptions that custodians of African cultures such as parents, grand-parents, village chiefs, and other community leaders hold regarding what higher education institutions ought to achieve for their children, and the extent to which these final ends are being, or can be, achieved through teaching and learning in cyberspace. Using the African context of Botswana as a reference point, the chapter situates learning and teaching practices online in the context of culture. After exploring the literature on cultural worldviews, African expectations of higher education, and online pedagogy and technology, the chapter documents the understandings and perceptions of the final ends that higher education institutions ought to pursue held by 39 village chiefs, community leaders, and parents from different cultural groups and communities. Evidence from phenomenological interviews revealed that Africans held that Afro-communal purposes for higher education in which promoting virtue, supporting culture, facilitating cooperation, and contributing to socio-economic development were seen as the paramount final ends for higher education institutions. The findings support Metz’s (2019) theoretical model of the ends of African higher education institutions. The chapter argues that, for online pedagogy in African universities and colleges to be successful, it should be grounded in collectivism where emphasis is given to the use of synchronous communication systems for the delivery of learning experiences, pedagogical practices are framed within community of practice, students are exposed to education for morality to foster a deeper sense of being real humans, and a socio-economic development agenda is emphasised.

Codes of ethics are a moral reflection centred on the duties and rights of a given profession that establishes the minimum moral standards required. These codes imply self-regulation and therefore an individual application on the conduct of professionals. In this chapter we reflect on the main values that guide PR practice based on Schwartz's theory of basic human values, which measures universal values that are recognised throughout all major cultures. A qualitative and quantitative content analysis was carried out of the codes of ethics of six national PR and communication associations and of the Global Alliance's code. The ethics codes were analysed to study the priorities of values stressed by PR professional associations and to highlight the motivational values that may be present in them. Findings show that values contained in the codes of ethics are based on a system of 32 human values; three of the values – the common good, integrity and truthfulness – are identified in all the codes; motivational values relating to universalism, benevolence and conformity are also covered to varying degrees in all the texts. It has been confirmed that the Global Alliance code is the only text that deals with the values of all the motivations described by Schwartz. The PR codes of ethics are based on a list of common ethical values of a collective nature, which are mostly contemplated by the Global Alliance; the main difference at the national level is that Latin countries include in their texts more principles of ethical universalism.

In this paper, some kinds of operators, defined on a class of real functions depending on a single real variable, preserving some fundamental properties for solving global optimization problems are introduced. These operators allow the transfer of the search of the global extremum of a determined objective function, f into those of its transformed function, T_f, for which a global extremum is easy to obtain.

To introduce an algorithm to solve inequalities defined by real functions on a certain compact set D of a general metric space (E, D). The device is based on α‐dense curves.

The solution of inequalities using α‐dense curves and also an approach to a global optimization technique, similarly obtained to that of the inequalities.

A new method is presented. The algorithm for solving inequalities is described which is based on a proven result. Inequalities of n‐variable dependence are reduced by transformation using α‐dense curves.

The research is a continuation of studies that resulted in a new method called Alienor for solving global optimization problems associated with multi‐variable functions.

Based on earlier research by the authors α‐dense curves have been used which allow the transformation of a n‐variables global optimization problem into a one‐variable global one. This paper gives a new method for quickly solving the one‐variable problem.

In this paper, we consider problems of numerical integration of fast oscillatory functions of one variable, obtained by using α‐dense curves and approximating multiple integrals. Using first, periodic and regular α‐dense curves we propose a trapezoidal formula for calculating the periodic integrand obtained. Then, we consider the simple integrals as integrals with weight. We propose a method to evaluate the moments of the weight function. This allows us to build a recurrent formula for the orthogonal polynomials family and to use a Gaussian rule to estimate the simple integral. Finally, we adapt the Filon's method, consisting in evaluating the Fourier coefficients of a function, to the oscillatory integrand obtained by using reducing transformations.

To use a new method based on α‐dense curved for solving problems of operational research.

The method of global optimization (called Alienor) is used for solving problems involving integer or mixed variables. A reducing transformation using α‐dense curves allows to transforms a n‐variables problem into a problem of a single variable.

Extends the basic method valid for continuous variables to problems involving integer, Boolean or mixed variables. All problems of operational research, linear or nonlinear, may be easily solved by or technique based on α‐dense curves (filling a n‐dimensional space). Industrial problems can be quickly solved by this technique obtaining the best solutions.

This method is deduced from the original works of Y. Cherruault and colleagues about global optimization and α‐dense curves. It proposes new techniques for solving operational research problems.