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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.

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.

The Alienor method offers a powerful approximation technique for the optimization of continuous multivariable functions defined on a compact set H of Rⁿ. Its computational efficiency is completed by the fact that it gave rise to the theory of space‐densifying curves. Presents a survey of these curves, analysing their most important properties and characteristics. Finally, the concept of theoretic calculation time (t.c.t.) associated with each curve suggests an interesting geometric problem on the existence of a curve with minimal t.c.t.

Gives a new method for defining and calculating multiple integrals. More precisely proposes that it is possible to define a multiple integral by means of a simple integral. This can be performed by using α‐dense curves in Rⁿ, already introduced for global optimization using the ALIENOR method.

This paper deals with the existence of a curve of minimal length, expressed in parametric coordinates, which densifies the square J²=≤ft [ −1,1\right ] × ≤ft [ −1,1\right ] \ with a given degree of density α. Nevertheless, the same problem has no solution if we consider the family of curves defined by means of the graphics of continuous and rectifiable functions f: J→ J. Their consequences on the approximation method to the global optimization are also derived.

The purpose of this paper is to solve a global optimization problem raised from industry, named the M‐problem, posed in the space L₁ of Lebesgue measurable functions of integrable absolute value.

The paper introduces the new concept of V‐dense curve (VDC), a generalization of that of α‐dense curve, to densify subsets of topological vector spaces not necessarily metrisable. It is proved that the feasible set of the M‐problem, namely the subset D of L₁ of all probability functions, is densifiable by VDC provided that L₁ to be endowed with the weak topology.

It is proved that the M‐problem, consisting of finding a probability function f of D associated to the mean life of an electronic devise that minimizes the expectation defined by a certain functional on L₁, is not a well‐posed problem in D. Nevertheless, by virtue of the compactness, the M‐problem has solution on each weak VDC in D for arbitrary weak 0‐neighbourhood V, which allows to find an approximate probability function with arbitrary precision.

The paper has designed, by means of the VDC‐method, a convergent algorithm to find approximate solutions in ill‐posed global optimization problems when the feasible set is contained in a non‐metrisable topological vector space.

This paper aims to present a new method for obtaining points of the set determined by the closure of the real projections of the zeros of each partial sum 1+2^s+ċ+n^s, n≥2, s=σ+it, of the Riemann zeta function and to show several applications of this result.

The authors utilize an auxiliary function related to a known result of Avellar that characterizes the set of points of interest. Several figures and numerical experiences are presented to illustrate the various properties which are studied.

It is first shown that each point of the image of the auxiliary function can be approximated by points of the image of the function formed by the approximants. Secondly, conditions are given on the auxiliary function to obtain points satisfying the property of density which is studied. Finally, by using these conditions, several useful applications are presented to the case n=4 and σ₀=0 which a more specific criterion is also given.

This research is applicable for finding accumulation points of the set of the real projection of the zeros of the approximants on its critical interval. An exact interval included in this set is given for the case n=4. Also, it is demonstrated that the point 0 is included for a large set of values of n.

The method employed is original and it contributes to the study on the properties of the density of the real parts of the zeros of a particular class of entire functions.

The purpose of this paper is to obtain an axiomatic characterization of Lowen's fuzzy compactness.

The paper shows a characterization of Lowen's fuzzy compactness.

The paper shows new results on fuzzy compactness.

Clearly, this paper is devoted to fuzzy topological spaces.

The main applications are in the mathematical field.

The paper presents original results on fuzzy topology.

The theoretic calculation time associated to every α‐dense curve into a fixed H of Rⁿ is inversely proportional to the discretization step depending on the length of the curve and, more directly, of the derivatives of its coordinate functions. For a given degree of density α, it is interesting to seek curves into H which may minimize the theoretic calculation time and then to solve the practical problem of computing approximations for global optimization of a given continuous function defined in H, by means of its restriction over a family of curves with the same degree of density into the cube H.