Axioms doi: 10.3390/axioms8010036

Authors: Valery Y. Glizer

A singularly perturbed linear time-dependent controlled system with multiple point-wise delays and distributed delays in the state and control variables is considered. The delays are small, of order of a small positive multiplier for a part of the derivatives in the system. This multiplier is a parameter of the singular perturbation. Two types of the considered singularly perturbed system, standard and nonstandard, are analyzed. For each type, two much simpler parameter-free subsystems (the slow and fast ones) are associated with the original system. It is established in the paper that proper kinds of controllability of the slow and fast subsystems yield the complete Euclidean space controllability of the original system for all sufficiently small values of the parameter of singular perturbation. Illustrative examples are presented.

]]>Axioms doi: 10.3390/axioms8010035

Authors: Nicolas Behr Giuseppe Dattoli Ambra Lattanzi

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.

]]>Axioms doi: 10.3390/axioms8010034

Authors: Hamid Faraji Dragana Savić Stojan Radenović

In this paper, some new results are given on fixed and common fixed points of Geraghty type contractive mappings defined in b-complete b-metric spaces. Moreover, two examples are represented to show the compatibility of our results. Some applications for nonlinear integral equations are also given.

]]>Axioms doi: 10.3390/axioms8010033

Authors: Taras Banakh Igor Protasov

We survey some results and pose some open problems related to boundedness of real-valued functions on balleans and coarse spaces. Special attention is paid to balleans on groups. The boundedness of functions that respect the coarse structure of a ballean could be considered as a coarse counterpart of pseudo-compactness.

]]>Axioms doi: 10.3390/axioms8010032

Authors: Benjamín Barán Marcos Villagra

In this work we show how to use a quantum adiabatic algorithm to solve multiobjective optimization problems. For the first time, we demonstrate a theorem proving that the quantum adiabatic algorithm can find Pareto-optimal solutions in finite-time, provided some restrictions to the problem are met. A numerical example illustrates an application of the theorem to a well-known problem in multiobjective optimization. This result opens the door to solve multiobjective optimization problems using current technology based on quantum annealing.

]]>Axioms doi: 10.3390/axioms8010031

Authors: Juan Pablo Ramírez

We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf&rsquo;s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

]]>Axioms doi: 10.3390/axioms8010030

Authors: Vasile Drăgan

In this paper a stochastic optimal control problem described by a quadratic performance criterion and a linear controlled system modeled by a system of singularly perturbed It&ocirc; differential equations with two fast time scales is considered. The asymptotic structure of the stabilizing solution (satisfying a prescribed sign condition) to the corresponding stochastic algebraic Riccati equation is derived. Furthermore, a near optimal control whose gain matrices do not depend upon small parameters is discussed.

]]>Axioms doi: 10.3390/axioms8010029

Authors: Micheal Pawliuk Michael Alexander Waddell

Given a dataset, we quantify the size of patterns that must always exist in the dataset. This is done formally through the lens of Ramsey theory of graphs, and a quantitative bound known as Goodman&rsquo;s theorem. By combining statistical tools with Ramsey theory of graphs, we give a nuanced understanding of how far away a dataset is from correlated, and what qualifies as a meaningful pattern. This method is applicable to a wide range of datasets. As examples, we analyze two very different datasets. The first is a dataset of repeated voters ( n = 435 ) in the 1984 US congress, and we quantify how homogeneous a subset of congressional voters is. We also measure how transitive a subset of voters is. Statistical Ramsey theory is also used with global economic trading data ( n = 214 ) to provide evidence that global markets are quite transitive. While these datasets are small relative to Big Data, they illustrate the new applications we are proposing. We end with specific calls to strengthen the connections between Ramsey theory and statistical methods.

]]>Axioms doi: 10.3390/axioms8010028

Authors: Metod Saniga Edyta Bartnicka

In this paper, it is shown that there exists a particular associative ring with unity of order 16 such that the relations between non-unimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.

]]>Axioms doi: 10.3390/axioms8010027

Authors: Abduhafiz Bobodzhanov Valeriy Safonov Vasiliy Kachalov

We consider a singularly perturbed integral equation with weakly and rapidly varying kernels. The work is a continuation of the studies carried out previously, but these were focused solely on rapidly changing kernels. A generalization for the case of two kernels, one of which is weakly, and the other rapidly varying, has not previously been carried out. The aim of this study is to investigate the effects introduced into the asymptotics of the solution of the problem by a weakly varying integral kernel. In the second part of the work, the problem of constructing exact (more precise, pseudo-analytic) solutions of singularly perturbed problems is considered on the basis of the method of holomorphic regularization developed by one of the authors of this paper. The power series obtained with the help of this method for the solutions of singularly perturbed problems (in contrast to the asymptotic series constructed in the first part of this paper) converge in the usual sense.

]]>Axioms doi: 10.3390/axioms8010026

Authors: Emer Bernal Oscar Castillo José Soria Fevrier Valdez

Galactic swarm optimization (GSO) is a recently created metaheuristic which is inspired by the motion of galaxies and stars in the universe. This algorithm gives us the possibility of finding the global optimum with greater precision since it uses multiple exploration and exploitation cycles. In this paper we present a modification to galactic swarm optimization using type-1 (T1) and interval type-2 (IT2) fuzzy systems for the dynamic adjustment of the c3 and c4 parameters in the algorithm. In addition, the modification is used for the optimization of the fuzzy controller of an autonomous mobile robot. First, the galactic swarm optimization is tested for fuzzy controller optimization. Second, the GSO algorithm with the dynamic adjustment of parameters using T1 fuzzy systems is used for the optimization of the fuzzy controller of an autonomous mobile robot. Finally, the GSO algorithm with the dynamic adjustment of parameters using the IT2 fuzzy systems is applied to the optimization of the fuzzy controller. In the proposed approaches, perturbation (noise) was added to the plant in order to find out if our approach behaves well under perturbation to the autonomous mobile robot plant; additionally, we consider our ability to compare the results obtained with the approaches when no perturbation is considered.

]]>Axioms doi: 10.3390/axioms8010025

Authors: Andrea Masini Margherita Zorzi

We know that quantum logics are the most prominent logical systems associated to the lattices of closed Hilbert subspaces. However, what happen if, following a quantum computing perspective, we want to associate a logic to the process of quantum registers measurements? This paper gives an answer to this question, and, quite surprisingly, shows that such a logic is nothing else that the standard propositional intuitionistic logic.

]]>Axioms doi: 10.3390/axioms8010024

Authors: Marta Dudek Janusz Garecki

In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein&ndash;Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein&ndash;Palatini (EP) action, and then we present how Einstein fields equations can be derived from it. In the next section, we study Einstein&ndash;Palatini action integral for general relativity with a positive cosmological constant &Lambda; in terms of the corrected curvature &Omega; c o r. We see that in terms of &Omega; c o r this action takes the form typical for a gauge field. Finally, we give a geometrical interpretation of the corrected curvature &Omega; c o r.

]]>Axioms doi: 10.3390/axioms8010023

Authors: João Fialho Feliz Minhós

The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arz&egrave;la Ascoli theorem and Schauder&rsquo;s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions.

]]>Axioms doi: 10.3390/axioms8010022

Authors: Feliz Minhós

In this paper, we consider the second order discontinuous differential equation in the real line, a t , u ϕ u &prime; &prime; = f t , u , u &prime; , a . e . t &isin; R , u ( &minus; &infin; ) = &nu; &minus; , u ( + &infin; ) = &nu; + , with ϕ an increasing homeomorphism such that ϕ ( 0 ) = 0 and ϕ ( R ) = R , a &isin; C ( R 2 , R ) with a ( t , x ) &gt; 0 for ( t , x ) &isin; R 2 , f : R 3 &rarr; R a L 1 -Carath&eacute;odory function and &nu; &minus; , &nu; + &isin; R such that &nu; &minus; &lt; &nu; + . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities ϕ and f . To the best of our knowledge, this result is even new when ϕ ( y ) = y , that is for equation a t , u ( t ) u &prime; ( t ) &prime; = f t , u ( t ) , u &prime; ( t ) , a . e . t &isin; R . Moreover, these results can be applied to classical and singular ϕ -Laplacian equations and to the mean curvature operator.

]]>Axioms doi: 10.3390/axioms8010021

Authors: Viktor Abramov

We show that given a Lie superalgebra and an element of its dual space, one can construct the 3-Lie superalgebra. We apply this approach to Lie superalgebra of ( m , n ) -block matrices taking a supertrace of a matrix as the element of dual space. Then we also apply this approach to commutative superalgebra and the Lie superalgebra of its derivations to construct 3-Lie superalgebra. The graded Lie brackets are constructed by means of a derivation and involution of commutative superalgebra, and we use them to construct 3-Lie superalgebras.

]]>Axioms doi: 10.3390/axioms8010020

Authors: Michael Gil’

The paper is devoted to the discrete Lyapunov equation X &minus; A * X A = C , where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.

]]>Axioms doi: 10.3390/axioms8010019

Authors: Jan Andres Denis Pennequin

As a nontrivial application of the abstract theorem developed in our recent paper titled &ldquo;Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions&rdquo;, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied.

]]>Axioms doi: 10.3390/axioms8010018

Authors: Haci Mehmet Baskonus

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson&ndash;Pickering model is applied. A set of new complex soliton solutions to the Gilson&ndash;Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given.

]]>Axioms doi: 10.3390/axioms8010017

Authors: Nizar Souayah Hassen Aydi Thabet Abdeljawad Nabil Mlaiki

In this paper, we ensure the existence and uniqueness of a best proximity point in rectangular metric spaces endowed with a graph structure.

]]>Axioms doi: 10.3390/axioms8010016

Authors: Luigi Brugnano Felice Iavernaro

The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.

]]>Axioms doi: 10.3390/axioms8010015

Authors: Ivan G. Ivanov Tonya Mateva

In this paper, based on Kou&rsquo;s classical iterative methods with fifth-order of convergence, we propose new interval iterative methods for computing a real root of the nonlinear scalar equations. Some numerical experiments have executed with the program INTLAB in order to confirm the theoretical results. The computational results have described and compared with Newton&rsquo;s interval method, Ostrowski&rsquo;s interval method and Ostrowski&rsquo;s modified interval method. We conclude that the proposed interval schemes are effective and they are comparable to the classical interval methods.

]]>Axioms doi: 10.3390/axioms8010014

Authors: Fernando Gaxiola Patricia Melin Fevrier Valdez Juan R. Castro Alain Manzo-Martínez

A dynamic adjustment of parameters for the particle swarm optimization (PSO) utilizing an interval type-2 fuzzy inference system is proposed in this work. A fuzzy neural network with interval type-2 fuzzy number weights using S-norm and T-norm is optimized with the proposed method. A dynamic adjustment of the PSO allows the algorithm to behave better in the search for optimal results because the dynamic adjustment provides good synchrony between the exploration and exploitation of the algorithm. Results of experiments and a comparison between traditional neural networks and the fuzzy neural networks with interval type-2 fuzzy numbers weights using T-norms and S-norms are given to prove the performance of the proposed approach. For testing the performance of the proposed approach, some cases of time series prediction are applied, including the stock exchanges of Germany, Mexican, Dow-Jones, London, Nasdaq, Shanghai, and Taiwan.

]]>Axioms doi: 10.3390/axioms8010013

Authors: Mohammad Asim A. Rauf Khan Mohammad Imdad

In this paper, we first introduce the class of partial symmetric spaces and then prove some fixed point theorems in such spaces. We use one of the our main results to examine the existence and uniqueness of a solution for a system of Fredholm integral equations. Furthermore, we introduce an analogue of the Hausdorff metric in the context of partial symmetric spaces and utilize the same to prove an analogue of the Nadler contraction principle in such spaces. Our results extend and improve many results in the existing literature. We also give some examples exhibiting the utility of our newly established results.

]]>Axioms doi: 10.3390/axioms8010012

Authors: Mohammad Masjed-Jamei Wolfram Koepf

We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38).

]]>Axioms doi: 10.3390/axioms8010011

Authors: Alexey A. Petrov

In this paper, we study relations between shadowing and inverse shadowing for homeomorphisms of a compact space. We present an example of a smooth diffeomorphism of a compact three-dimensional manifold that has the shadowing property and does not have the inverse shadowing property. For some classes of inverse shadowing, we construct examples of homeomorphisms that have the inverse shadowing property but do not have the shadowing property.

]]>Axioms doi: 10.3390/axioms8010010

Authors: Rutwig Campoamor-Stursberg Francisco Oviaño García

The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum spec ( t ) = 1 , k , k + 1 , ⋯ , n + k &minus; 3 , n + 2 k &minus; 3 for k &ge; 2 are analyzed, with special emphasis on the resulting Lie algebras for which the second Chevalley cohomology space vanishes. From the detailed inspection of the values k &le; 5 , some series of cohomologically rigid algebras for arbitrary values of k are determined.

]]>Axioms doi: 10.3390/axioms8010009

Authors: Vasily E. Tarasov Valentina V. Tarasova

In this paper, we propose a macroeconomic growth model, in which we take into account memory with power-law fading and gamma distributed lag. This model is a generalization of the standard Harrod&ndash;Domar growth model. Fractional differential equations of this generalized model with memory and lag are suggested. For these equations, we obtain solutions, which describe the macroeconomic growth of national income with fading memory and distributed time-delay. The asymptotic behavior of these solutions is described.

]]>Axioms doi: 10.3390/axioms8010008

Authors: Juan Carlos Guzmán Ivette Miramontes Patricia Melin German Prado-Arechiga

The use of artificial intelligence techniques such as fuzzy logic, neural networks and evolutionary computation is currently very important in medicine to be able to provide an effective and timely diagnosis. The use of fuzzy logic allows to design fuzzy classifiers, which have fuzzy rules and membership functions, which are designed based on the experience of an expert. In this particular case a fuzzy classifier of Mamdani type was built, with 21 rules, with two inputs and one output and the objective of this classifier is to perform blood pressure level classification based on knowledge of an expert which is represented in the fuzzy rules. Subsequently different architectures were made in type-1 and type-2 fuzzy systems for classification, where the parameters of the membership functions used in the design of each architecture were adjusted, which can be triangular, trapezoidal and Gaussian, as well as how the fuzzy rules are optimized based on the ranges established by an expert. The main contribution of this work is the design of the optimized interval type-2 fuzzy system with triangular membership functions. The final type-2 system has a better classification rate of 99.408% than the type-1 classifier developed previously in &ldquo;Design of an optimized fuzzy classifier for the diagnosis of blood pressure with a new computational method for expert rule optimization&rdquo; with 98%. In addition, we also obtained a better classification rate than the other architectures proposed in this work.

]]>Axioms doi: 10.3390/axioms8010007

Authors: Marisa Fernández Victor Manero Jonatan Sánchez

We consider the Laplacian flow of locally conformal calibrated G 2 -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G 2 -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G 2 manifolds and, in both cases, we obtain a flow of locally conformal calibrated G 2 -structures, which are ancient solutions, that is they are defined on a time interval of the form ( &minus; &infin; , T ) , where T &gt; 0 is a real number. Moreover, for each of these examples, we prove that the underlying metrics g ( t ) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to &minus; &infin; , and they blow-up at a finite-time singularity.

]]>Axioms doi: 10.3390/axioms8010006

Authors: Domenico Perrone

There is one-to-one correspondence between contact semi-Riemannian structures ( &eta; , &xi; , &phi; , g ) and non-degenerate almost CR structures ( H , &thetasym; , J ) . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H 1 , 0 : = X &minus; i J X , X &isin; H is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case.

]]>Axioms doi: 10.3390/axioms8010005

Authors: Axioms Editorial Office

Rigorous peer-review is the corner-stone of high-quality academic publishing [...]

]]>Axioms doi: 10.3390/axioms8010004

Authors: Ravi Agarwal Snezhana Hristova Donal O’Regan

In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We consider the case of impulses that start abruptly at some points and their actions continue on given finite intervals. The study of Lipschitz stability by Lyapunov functions requires appropriate derivatives among fractional differential equations. A brief overview of different types of derivative known in the literature is given. Some sufficient conditions for uniform Lipschitz stability and uniform global Lipschitz stability are obtained by an application of several types of derivatives of Lyapunov functions. Examples are given to illustrate the results.

]]>Axioms doi: 10.3390/axioms8010003

Authors: Arkady Leiderman Sidney Morris

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

]]>Axioms doi: 10.3390/axioms8010002

Authors: Zhuang-Dan Daniel Guan Pilar Orellana Anthony Van

This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new K&auml;hler&ndash;Einstein manifolds as well as Fano manifolds without K&auml;hler&ndash;Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds N n , m when n &gt; 15 and N n , m &prime; when n &gt; 4 . We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian&ndash;Einstein metrics. These also provide open Calabi&ndash;Yau manifolds.

]]>Axioms doi: 10.3390/axioms8010001

Authors: John R. Martin

An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum S + T of two (L)-semigroups S and T having boundary B is the quotient space obtained from the union of S &times; { 0 } and T &times; { 1 } by identifying the point ( x , 0 ) in S &times; { 0 } with ( x , 1 ) in T &times; { 1 } for each x in B. It is shown that no (L)-semigroup sum of dimension less than or equal to five admits an H-space structure, nor does any (L)-semigroup sum obtained from (L)-semigroups having an Abelian boundary. In particular, such sums cannot be a retract of a topological group.

]]>Axioms doi: 10.3390/axioms7040094

Authors: Ray-Ming Chen

In this article, we show how to define a metric on the finite power multisets of positive real numbers. The metric, based on the minimal matching, consists of two parts: the matched part and the mismatched part. We also give some concrete applications and examples to demonstrate the validity of this metric.

]]>Axioms doi: 10.3390/axioms7040093

Authors: Kazumine Moriyasu Kazuhiro Sakai Naoya Sumi

In this paper, the notion of shadowable measures is introduced as a generalization of the shadowing property from the measure theoretical view point, and the set of diffeomorphisms satisfying the notion is considered. The dynamics of the C 1 -interior of the set of diffeomorphisms possessing the shadowable measures is characterized as the uniform hyperbolicity.

]]>Axioms doi: 10.3390/axioms7040092

Authors: Hayat Zouiten Ali Boutoulout Delfim F. M. Torres

We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann&ndash;Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge of the state.

]]>Axioms doi: 10.3390/axioms7040091

Authors: Angelamaria Cardone Dajana Conte Raffaele D’Ambrosio Beatrice Paternoster

We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation of the long-term character of stochastic oscillators over discretized dynamics and the analysis of mean-square and asymptotic stability properties of &thetasym; -methods for Volterra integral equations.

]]>Axioms doi: 10.3390/axioms7040090

Authors: Giovanni Bazzoni Alberto Raffero

Motivated by known results in locally conformal symplectic geometry, we study different classes of G 2 -structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G 2 -structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G 2 -structures.

]]>Axioms doi: 10.3390/axioms7040089

Authors: Manuel D. Echeverry Carlos E. Mejía

We consider a two-dimensional time fractional diffusion equation and address the important inverse problem consisting of the identification of an ingredient in the source term. The fractional derivative is in the sense of Caputo. The necessary regularization procedure is provided by a two-dimensional discrete mollification operator. Convergence results and illustrative numerical examples are included.

]]>Axioms doi: 10.3390/axioms7040088

Authors: Sorin Dragomir Francesco Esposito

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere S m &sub; R m + 1 . Given a codimension two totally geodesic submanifold &Sigma; &sub; S m , we show that every nonconstant exponentially harmonic map ϕ : M &rarr; S m either meets or links &Sigma; . If H 1 ( M , Z ) = 0 then ϕ ( M ) &cap; &Sigma; &ne; &empty; .

]]>Axioms doi: 10.3390/axioms7040087

Authors: Aiman Mukheimer

In this paper, we introduce the concept of extended partial S b -metric spaces, which is a generalization of the extended S b -metric spaces. Basically, in the triangle inequality, we add a control function with some very interesting properties. These new metric spaces generalize many results in the literature. Moreover, we prove some fixed point theorems under some different contractions, and some examples are given to illustrate our results.

]]>Axioms doi: 10.3390/axioms7040086

Authors: Dmitri Shakhmatov Víctor Yañez

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence ( U n ) of non-empty open subsets of G, one can choose a point x n ∈ U n for all n ∈ N in such a way that the resulting sequence ( x n ) has a p-limit in G; that is, { n ∈ N : x n ∈ V } ∈ p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo- ω -bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ⨁ i ∈ I X i , where each space X i is either maximal or discrete, contains no infinite separable pseudocompact subsets.

]]>Axioms doi: 10.3390/axioms7040085

Authors: Florin F. Nichita

We consider several unification problems in mathematics. We refer to transcendental numbers. Furthermore, we present some ways to unify the main non-associative algebras (Lie algebras and Jordan algebras) and associative algebras.

]]>Axioms doi: 10.3390/axioms7040084

Authors: Keonhee Lee C. A. Morales

We prove that a linear operator of a complex Banach space has a shadowable point if and only if it has the shadowing property. In addition, every equicontinuous linear operator does not have the shadowing property and its spectrum is contained in the unit circle. Finally, we prove that if a linear operator is expansive and has the shadowing property, then the origin is the only nonwandering point.

]]>Axioms doi: 10.3390/axioms7040083

Authors: Muhammad Akram Hina Gulzar Florentin Smarandache Said Broumi

Neutrosophic sets and soft sets are two different mathematical tools for representing vagueness and uncertainty. We apply these models in combination to study vagueness and uncertainty in K-algebras. We introduce the notion of single-valued neutrosophic soft (SNS) K-algebras and investigate some of their properties. We establish the notion of ( &isin; , &isin; &or; q ) -single-valued neutrosophic soft K-algebras and describe some of their related properties. We also illustrate the concepts with numerical examples.

]]>Axioms doi: 10.3390/axioms7040082

Authors: Hari M. Srivastava

Website: http://www.math.uvic.ca/faculty/harimsri/ [...]

]]>Axioms doi: 10.3390/axioms7040081

Authors: Teresa Laudadio Nicola Mastronardi Paul Van Dooren

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the Q R and L U factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur algorithm. We show that it helps to prove some theoretical properties of the R factor of the Q R factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based algorithm for computing the null-space of polynomial matrices.

]]>Axioms doi: 10.3390/axioms7040080

Authors: Nabil Mlaiki Nihal Taş Nihal Yılmaz Özgür

In this paper, we consider the fixed-circle problem on metric spaces and give new results on this problem. To do this, we present three types of F C -Khan type contractions. Furthermore, we obtain some solutions to an open problem related to the common fixed-circle problem.

]]>Axioms doi: 10.3390/axioms7040079

Authors: Stefan Wagner

A dynamical system is a triple ( A , G , &alpha; ) consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism &alpha; : G &rarr; Aut ( A ) that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units A &times; is open in A and the inversion map &iota; : A &times; &rarr; A &times; , a ↦ a &minus; 1 is continuous at 1 A . Given a dynamical system ( A , G , &alpha; ) with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A.

]]>Axioms doi: 10.3390/axioms7040078

Authors: Susana Díaz José Carlos R. Alcantud Susana Montes

We focus on the relationships among some consistency axioms in the framework of fuzzy choice functions. In order to help disclose the role of the t-norm in existing analyses, we start to study the situation that arises when we replace the standard t-norm with other t-norms. Our results allow us to conclude that unless we impose further structure on the domain of application for the choices, the use of the Łukasiewicz t-norm as a replacement for the standard t-norm does not guarantee a better performance.

]]>Axioms doi: 10.3390/axioms7040077

Authors: Michael Megrelishvili

A well-known result of Ferri and Galindo asserts that the topological group c 0 is not reflexively representable and the algebra WAP ( c 0 ) of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame ( c 0 ) of tame functions. Respectively, it is an open question if c 0 is representable on a Rosenthal Banach space. In the present work we show that Tame ( c 0 ) is small in a sense that the unit sphere S and 2 S cannot be separated by a tame function f &isin; Tame ( c 0 ) . As an application we show that the Gromov&rsquo;s compactification of c 0 is not a semigroup compactification. We discuss some questions.

]]>Axioms doi: 10.3390/axioms7040076

Authors: José Rodrigo González Granada Joachim Gwinner Victor A. Kovtunenko

This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour&ndash;Zol&eacute;sio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet&ndash;Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method.

]]>Axioms doi: 10.3390/axioms7040075

Authors: Igor V. Protasov

We survey different topologizations of the set S ( G ) of closed subgroups of a topological group G and demonstrate some applications using Topological Groups, Model Theory, Geometric Group Theory, and Topological Dynamics.

]]>Axioms doi: 10.3390/axioms7040074

Authors: Haitham Qawaqneh Mohd Noorani Wasfi Shatanawi Habes Alsamir

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty ( &alpha; , &psi; , ϕ ) -quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show the superiority of our results. The obtained results progress some well-known fixed (common fixed) point results in the literature. Our main results cannot be specifically attained from the corresponding metric space versions. This paper is scientifically novel because we take Geraghty contraction self-mapping in partially ordered metric-like spaces via &alpha; &minus; admissible mapping. This opens the door to other possible fixed (common fixed) point results for non-self-mapping and in other generalizing metric spaces.

]]>Axioms doi: 10.3390/axioms7040073

Authors: Muhammad Imran Muhammad Kamran Siddiqui Ali Ahmad Usman Ali Nazia Hanif

Tickysim is a clock tick-based simulator for the inter-chip interconnection network of the SpiNNaker architecture. Network devices such as arbiters, routers, and packet generators store, read, and write forward data through fixed-length FIFO buffers. At each clock tick, every component executes a &ldquo;read&rdquo; phase followed by a &ldquo;write&rdquo; phase. The structures of any finite graph which represents numerical quantities are known as topological indices. In this paper, we compute degree-based topological indices of the Tickysim SpiNNaker Model ( T S M ) sheet.

]]>Axioms doi: 10.3390/axioms7040072

Authors: Maurizio Parton Paolo Piccinni

Starting from the 2001 Thomas Friedrich’s work on Spin ( 9 ) , we review some interactions between Spin ( 9 ) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin ( 9 ) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin ( 9 ) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin ( 9 ) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians.

]]>Axioms doi: 10.3390/axioms7040071

Authors: Pierpaolo Natalini Paolo Emilio Ricci

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell&ndash;Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear.

]]>Axioms doi: 10.3390/axioms7040070

Authors: Mina Torabi Mohammad-Mehdi Hosseini

This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order accurate in time. A comparative study between the proposed method and the one-step wavelet collocation method is provided. In order to verify the stability of these methods, asymptotic stability analysis is employed. Numerical illustrations are investigated to show the reliability and efficiency of the proposed method. An important property of the presented method is that unlike the one-step wavelet collocation method, it is not necessary to choose a small time step to achieve stability.

]]>Axioms doi: 10.3390/axioms7040069

Authors: Makrina Agaoglou Michal Fečkan Michal Pospíšil Vassilis M. Rothos Alexander F. Vakakis

In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system.

]]>Axioms doi: 10.3390/axioms7030068

Authors: Dae Ho Jin Jae Won Lee

We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces.

]]>Axioms doi: 10.3390/axioms7030067

Authors: Memet Şahin Abdullah Kargın

The notion of Neutrosophic triplet (NT) is a new theory in Neutrosophy. Also, the v-generalized metric is a specific form of the classical metrics. In this study, we introduced the notion of neutrosophic triplet v-generalized metric space (NTVGM), and we obtained properties of NTVGM. Also, we showed that NTVGM is different from the classical metric and neutrosophic triplet metric (NTM). Furthermore, we introduced completeness of NTVGM.

]]>Axioms doi: 10.3390/axioms7030066

Authors: Chenkuan Li Thomas Humphries Hunter Plowman

The goal of this paper is to study fractional calculus of distributions, the generalized Abel&rsquo;s integral equations, as well as fractional differential equations in the distributional space D &prime; ( R + ) based on inverse convolutional operators and Babenko&rsquo;s approach. Furthermore, we provide interesting applications of Abel&rsquo;s integral equations in viscoelastic systems, as well as solving other integral equations, such as &int; &theta; &pi; / 2 y ( &phi; ) cos &beta; &phi; ( cos &theta; &minus; cos &phi; ) &alpha; d &phi; = f ( &theta; ) , and &int; 0 &infin; x 1 / 2 g ( x ) y ( x + t ) d x = f ( t ) .

]]>Axioms doi: 10.3390/axioms7030065

Authors: Jean-Daniel Djida Arran Fernandez

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative.

]]>Axioms doi: 10.3390/axioms7030064

Authors: Ashraf Al-Quran Shawkat Alkhazaleh

The basic aim of soft computing is to trade precision for a tractableness and reduction in solution cost by pushing the limits of tolerance for imprecision and uncertainty. This paper introduces a novel soft computing technique called complex neutrosophic relation (CNR) to evaluate the degree of interaction between two complex neutrosophic sets (CNSs). CNSs are used to represent two-dimensional information that are imprecise, uncertain, incomplete and indeterminate. The Cartesian product of CNSs and subsequently the complex neutrosophic relation is formally defined. This relation is generalised from a conventional single valued neutrosophic relation (SVNR), based on CNSs, where the ranges of values of CNR are extended to the unit circle in complex plane for its membership functions instead of [0, 1] as in the conventional SVNR. A new algorithm is created using a comparison matrix of the SVNR after mapping the complex membership functions from complex space to the real space. This algorithm is then applied to scrutinise the impact of some teaching strategies on the student performance and the time frame(phase) of the interaction between these two variables. The notion of inverse, complement and composition of CNRs along with some related theorems and properties are introduced. The performance and utility of the composition concept in real-life situations is also demonstrated. Then, we define the concepts of projection and cylindric extension for CNRs along with illustrative examples. Some interesting properties are also obtained. Finally, a comparison between different existing relations and CNR to show the ascendancy of our proposed CNR is provided.

]]>Axioms doi: 10.3390/axioms7030063

Authors: Michael Dumbser Francesco Fambri Maurizio Tavelli Michael Bader Tobias Weinzierl

In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication-avoiding and cache-blocking, and are furthermore very well-suited for vectorization, and so they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing levels of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes.

]]>Axioms doi: 10.3390/axioms7030062

Authors: Giuseppe Dattoli Bruna Germano Silvia Licciardi Maria Renata Martinelli

The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.

]]>Axioms doi: 10.3390/axioms7030061

Authors: Nonthamon Chaikham Wannika Sawangtong

A dynamical model of Zika virus (ZIKV) epidemic with direct transmission, sexual transmission, and vertical transmission is developed. A sub-optimal control problem to counter against the disease is proposed including three controls: vector elimination, vector-to-human contact reduction, and sexual contact reduction. Each control variable is discretized into piece-wise constant intervals. The problem is solved by Differential Evolution (DE), which is one of the evolutionary algorithm developed for optimization. Two scenarios, namely four time horizons and eight time horizons, are compared and discussed. The simulations show that models with controls lead to decreasing the number of patients as well as epidemic period length. From the optimal solution, vector elimination is the prioritized strategy for disease control.

]]>Axioms doi: 10.3390/axioms7030060

Authors: Alexander Šostak

In this note, we present an alternative approach to the concept of a fuzzy metric, calling it a revised fuzzy metric. In contrast to the traditional approach to the theory of fuzzy metric spaces which is based on the use of a t-norm, we proceed from a t-conorm in the definition of a revised fuzzy metric. Here, we restrict our study to the case of fuzzy metrics as they are defined by George-Veeramani, however, similar revision can be done also for some other approaches to the concept of a fuzzy metric.

]]>Axioms doi: 10.3390/axioms7030059

Authors: Igor Líška Beloslav Riečan Anna Tirpáková

In the present paper we consider one of the basic theorems of probability theory on real numbers. We prove that it is equivalent with the supremum axiom of real numbers.

]]>Axioms doi: 10.3390/axioms7030058

Authors: Francesca Mazzia Alessandra Sestini

The class of A-stable symmetric one-step Hermite&ndash;Obreshkov (HO) methods introduced by F. Loscalzo in 1968 for dealing with initial value problems is analyzed. Such schemes have the peculiarity of admitting a multiple knot spline extension collocating the differential equation at the mesh points. As a new result, it is shown that these maximal order schemes are conjugate symplectic, which is a benefit when the methods have to be applied to Hamiltonian problems. Furthermore, a new efficient approach for the computation of the spline extension is introduced, adopting the same strategy developed for the BS linear multistep methods. The performances of the schemes are tested in particular on some Hamiltonian benchmarks and compared with those of the Gauss&ndash;Runge&ndash;Kutta schemes and Euler&ndash;Maclaurin formulas of the same order.

]]>Axioms doi: 10.3390/axioms7030057

Authors: Qiaoyan Li Yingcang Ma Florentin Smarandache Shuangwu Zhu

Data clustering is an important field in pattern recognition and machine learning. Fuzzy c-means is considered as a useful tool in data clustering. The neutrosophic set, which is an extension of the fuzzy set, has received extensive attention in solving many real-life problems of inaccuracy, incompleteness, inconsistency and uncertainty. In this paper, we propose a new clustering algorithm, the single-valued neutrosophic clustering algorithm, which is inspired by fuzzy c-means, picture fuzzy clustering and the single-valued neutrosophic set. A novel suitable objective function, which is depicted as a constrained minimization problem based on a single-valued neutrosophic set, is built, and the Lagrange multiplier method is used to solve the objective function. We do several experiments with some benchmark datasets, and we also apply the method to image segmentation using the Lena image. The experimental results show that the given algorithm can be considered as a promising tool for data clustering and image processing.

]]>Axioms doi: 10.3390/axioms7030056

Authors: Taekyun Kim Cheon Seoung Ryoo

In this paper, we study some special polynomials which are related to Euler and Bernoulli polynomials. In addition, we give some identities for these polynomials. Finally, we investigate the zeros of these polynomials by using the computer.

]]>Axioms doi: 10.3390/axioms7030055

Authors: Fernando S. Silva Davidson M. Moreira Marcelo A. Moret

In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, the analytical solution for a class of fractional models associated with the logistic model, the von Foerster model and the Bertalanffy model is presented graphically for various fractional orders. The solution of the corresponding classical model is recovered as a particular case.

]]>Axioms doi: 10.3390/axioms7030054

Authors: Hafize Gümüş Nihal Demir

In our paper, by using the concept of W&minus;asymptotically J&minus; statistical equivalence of order &alpha; which has been previously defined, we present the definitions of W&minus;asymptotically J&lambda;&minus;statistical equivalence of order &alpha;, W&minus;strongly asymptotically J&lambda;&minus;statistical equivalence of order &alpha;, and W&minus;strongly Ces&aacute;ro asymptotically J&minus;statistical equivalence of order &alpha; where 0&lt;&alpha;&le;1. We also extend these notions with a sequence of positive real numbers, p=(pk), and we investigate how our results change if p is constant.

]]>Axioms doi: 10.3390/axioms7030053

Authors: Kelvin C. K. Chan Raymond H. Chan Mila Nikolova

The goal of edge-histogram specification is to find an image whose edge image has a histogram that matches a given edge-histogram as much as possible. Mignotte has proposed a non-convex model for the problem in 2012. In his work, edge magnitudes of an input image are first modified by histogram specification to match the given edge-histogram. Then, a non-convex model is minimized to find an output image whose edge-histogram matches the modified edge-histogram. The non-convexity of the model hinders the computations and the inclusion of useful constraints such as the dynamic range constraint. In this paper, instead of considering edge magnitudes, we directly consider the image gradients and propose a convex model based on them. Furthermore, we include additional constraints in our model based on different applications. The convexity of our model allows us to compute the output image efficiently using either Alternating Direction Method of Multipliers or Fast Iterative Shrinkage-Thresholding Algorithm. We consider several applications in edge-preserving smoothing including image abstraction, edge extraction, details exaggeration, and documents scan-through removal. Numerical results are given to illustrate that our method successfully produces decent results efficiently.

]]>Axioms doi: 10.3390/axioms7030052

Authors: John C. Butcher

The traditional derivation of Runge&ndash;Kutta methods is based on the use of the scalar test problem y&prime;(x)=f(x,y(x)). However, above order 4, this gives less restrictive order conditions than those obtained from a vector test problem using a tree-based theory. In this paper, stumps, or incomplete trees, are introduced to explain the discrepancy between the two alternative theories. Atomic stumps can be combined multiplicatively to generate all trees. For the scalar test problem, these quantities commute, and certain sets of trees form isomeric classes. There is a single order condition for each class, whereas for the general vector-based problem, for which commutation of atomic stumps does not occur, there is exactly one order condition for each tree. In the case of order 5, the only nontrivial isomeric class contains two trees, and the number of order conditions reduces from 17 to 16 for scalar problems. A method is derived that satisfies the 16 conditions for scalar problems but not the complete set based on 17 trees. Hence, as a practical numerical method, it has order 4 for a general initial value problem, but this increases to order 5 for a scalar problem.

]]>Axioms doi: 10.3390/axioms7030051

Authors: Carmela Scalone Nicola Guglielmi

In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for the closest rank-r matrix X to M, where the distance is measured in the Frobenius norm. A key element in the solution of this matrix nearness problem consists of the use of a constrained gradient system of matrix differential equations. The obtained results, compared to those obtained by different approaches show that the method has a correct behaviour and is competitive with the ones available in the literature.

]]>Axioms doi: 10.3390/axioms7030050

Authors: Sumera Naz Muhammad Akram Florentin Smarandache

A single-valued neutrosophic set is an instance of a neutrosophic set, which provides us an additional possibility to represent uncertainty, imprecise, incomplete and inconsistent information existing in real situations. In this research study, we present concepts of energy, Laplacian energy and signless Laplacian energy in single-valued neutrosophic graphs (SVNGs), describe some of their properties and develop relationship among them. We also consider practical examples to illustrate the applicability of the our proposed concepts.

]]>Axioms doi: 10.3390/axioms7030049

Authors: Carlo Garoni Mariarosa Mazza Stefano Serra-Capizzano

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}n, which often turns out to be a GLT sequence or one of its &ldquo;relatives&rdquo;, i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations.

]]>Axioms doi: 10.3390/axioms7030048

Authors: Konstantin Zhukovsky Dmitrii Oskolkov Nadezhda Gubina

One-dimensional equations of telegrapher&rsquo;s-type (TE) and Guyer&ndash;Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated&mdash;fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed.

]]>Axioms doi: 10.3390/axioms7030047

Authors: Muhammad Akram Sidra Sayed Florentin Smarandache

In this research study, we introduce the notion of single-valued neutrosophic incidence graphs. We describe certain concepts, including bridges, cut vertex and blocks in single-valued neutrosophic incidence graphs. We present some properties of single-valued neutrosophic incidence graphs. We discuss the edge-connectivity, vertex-connectivity and pair-connectivity in neutrosophic incidence graphs. We also deal with a mathematical model of the situation of illegal migration from Pakistan to Europe.

]]>Axioms doi: 10.3390/axioms7030046

Authors: Francesca Pitolli

Efficient numerical methods to solve fractional differential problems are particularly required in order to approximate accurately the nonlocal behavior of the fractional derivative. The aim of the paper is to show how optimal B-spline bases allow us to construct accurate numerical methods that have a low computational cost. First of all, we describe in detail how to construct optimal B-spline bases on bounded intervals and recall their main properties. Then, we give the analytical expression of their derivatives of fractional order and use these bases in the numerical solution of fractional differential problems. Some numerical tests showing the good performances of the bases in solving a time-fractional diffusion problem by a collocation&ndash;Galerkin method are also displayed.

]]>Axioms doi: 10.3390/axioms7030045

Authors: Angelamaria Cardone Dajana Conte Raffaele D’Ambrosio Beatrice Paternoster

We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step and multistep collocation methods, and illustrate main convergence results, making some comparisons in terms of accuracy and efficiency. Some numerical experiments complete the paper.

]]>Axioms doi: 10.3390/axioms7030044

Authors: Benedetta Morini

This work studies limited memory preconditioners for linear symmetric positive definite systems of equations. Connections are established between a partial Cholesky factorization from the literature and a variant of Quasi-Newton type preconditioners. Then, a strategy for enhancing the Quasi-Newton preconditioner via available information is proposed. Numerical experiments show the behaviour of the resulting preconditioner.

]]>Axioms doi: 10.3390/axioms7030043

Authors: Cesare Bracco Carlotta Giannelli Rafael Vázquez

The construction of suitable mesh configurations for spline models that provide local refinement capabilities is one of the fundamental components for the analysis and development of adaptive isogeometric methods. We investigate the design and implementation of refinement algorithms for hierarchical B-spline spaces that enable the construction of locally graded meshes. The refinement rules properly control the interaction of basis functions at different refinement levels. This guarantees a bounded number of nonvanishing (truncated) hierarchical B-splines on any mesh element. The performances of the algorithms are validated with standard benchmark problems.

]]>Axioms doi: 10.3390/axioms7020042

Authors: Sergio Manzetti

Anomalous waves and rogue events are closely associated with irregularities and unexpected events occurring at various levels of physics, such as in optics, in oceans and in the atmosphere. Mathematical modeling of rogue waves is a highly active field of research, which has evolved over the last few decades into a specialized part of mathematical physics. The applications of the mathematical models for rogue events is directly relevant to technology development for the prediction of rogue ocean waves and for signal processing in quantum units. In this survey, a comprehensive perspective of the most recent developments of methods for representing rogue waves is given, along with discussion of the devised forms and solutions. The standard nonlinear Schrödinger equation, the Hirota equation, the MMT equation and other models are discussed and their properties highlighted. This survey shows that the most recent advancement in modeling rogue waves give models that can be used to establish methods for the prediction of rogue waves in open seas, which is important for the safety and activity of marine vessels and installations. The study further puts emphasis on the difference between the methods and how the resulting models form the basis for representing rogue waves in various forms, solitary or with a wave background. This review has also a pedagogic component directed towards students and interested non-experts and forms a complete survey of the most conventional and emerging methods published until recently.

]]>Axioms doi: 10.3390/axioms7020041

Authors: Young Bae Jun Seok-Zun Song Florentin Smarandache Hashem Bordbar

The notion of a neutrosophic quadruple BCK/BCI-number is considered, and a neutrosophic quadruple BCK/BCI-algebra, which consists of neutrosophic quadruple BCK/BCI-numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple BCK-algebra and a closed ideal in a neutrosophic quadruple BCI-algebra are studied. Given subsets A and B of a BCK/BCI-algebra, the set NQ(A,B), which consists of neutrosophic quadruple BCK/BCI-numbers with a condition, is established. Conditions for the set NQ(A,B) to be a (positive implicative) ideal of a neutrosophic quadruple BCK-algebra are provided, and conditions for the set NQ(A,B) to be a (closed) ideal of a neutrosophic quadruple BCI-algebra are given. An example to show that the set {0&tilde;} is not a positive implicative ideal in a neutrosophic quadruple BCK-algebra is provided, and conditions for the set {0&tilde;} to be a positive implicative ideal in a neutrosophic quadruple BCK-algebra are then discussed.

]]>Axioms doi: 10.3390/axioms7020040

Authors: Alessandra Aimi Lorenzo Diazzi Chiara Guardasoni

This paper aims to illustrate how SABO (Semi-Analytical method for Barrier Option pricing) is easily applicable for pricing floating strike Asian barrier options with a continuous geometric average. Recently, this method has been applied in the Black–Scholes framework to European vanilla barrier options with constant and time-dependent parameters or barriers and to geometric Asian barrier options with a fixed strike price. The greater efficiency of SABO with respect to classical finite difference methods is clearly evident in numerical simulations. For the first time, a user-friendly MATLAB® code is made available here.

]]>Axioms doi: 10.3390/axioms7020039

Authors: Raffaele Chiappinelli

A nonlinear eigenvalue problem is generally described by an equation of the form F(&lambda;,x)=0, where F(&lambda;,0)=0 for all &lambda;, and contains by definition two unknowns: the eigenvalue parameter &lambda; and the &ldquo;nontrivial&rdquo; vector(s) x corresponding to it. The nonlinear dependence of F can be in either of them (and of course in both), and also the research in this area seems to follow two quite different directions. In this review paper, we try to collect some points of possible common interest for both fields.

]]>Axioms doi: 10.3390/axioms7020038

Authors: Mohammad Masjed-Jamei Wolfram Koepf

Essentially, whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature. In this paper, we apply two identities of generalized hypergeometric series in order to extend some classical summation theorems of hypergeometric functions such as Gauss, Kummer, Dixon, Watson, Whipple, Pfaff&ndash;Saalsch&uuml;tz and Dougall formulas and also obtain some new summation theorems in the sequel.

]]>Axioms doi: 10.3390/axioms7020037

Authors: Humberto Bustince Javier Fernandez Esteban Induráin

We focus on the articles recently published in the special issue of Axioms devoted to &ldquo;New Trends in Fuzzy Set Theory and Related Items&rdquo;.

]]>Axioms doi: 10.3390/axioms7020036

Authors: Luigi Brugnano Felice Iavernaro

In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references.

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Authors: Marta Cardin

In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex spaces to formalize the notion of quantiles. We also put in evidence that many important models of decision problems can be viewed as convex spaces-based models. Several properties of aggregation operators are translated into this general setting, and independence and invariance are used to provide axiomatic characterizations of quantiles.

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Authors: Hsien-Chung Wu

The T 1 -spaces induced by the pre-metric spaces along with many forms of triangle inequalities are investigated in this paper. The limits in pre-metric spaces are also studied to demonstrate the consistency of limit concept in the induced topologies.

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Authors: Muhammad Akram Shumaiza Florentin Smarandache

Technique for the order of preference by similarity to ideal solution (TOPSIS) and elimination and choice translating reality (ELECTRE) are widely used methods to solve multi-criteria decision making problems. In this research article, we present bipolar neutrosophic TOPSIS method and bipolar neutrosophic ELECTRE-I method to solve such problems. We use the revised closeness degree to rank the alternatives in our bipolar neutrosophic TOPSIS method. We describe bipolar neutrosophic TOPSIS method and bipolar neutrosophic ELECTRE-I method by flow charts. We solve numerical examples by proposed methods. We also give a comparison of these methods.

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Authors: Igor Protasov

A class M of coarse spaces is called a variety if M is closed under the formation of subspaces, coarse images, and products. We classify the varieties of coarse spaces and, in particular, show that if a variety M contains an unbounded metric space then M is the variety of all coarse spaces.

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Authors: Ann-Eva Christensen Jon Johnsen

This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax&ndash;Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.

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