Axioms — Open Access Journal

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. Full article

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. Full article

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. Full article

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. Full article

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, $Cof\left(\mathbb{N}\right)$, is isomorphic to the natural numbers. Then, we prove the power set of integers, $2^{\mathbb{Z}}$, 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, $2^{\mathbb{N}}$, 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’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. Full article

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 Itô 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. Full article

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’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. Full article

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. Full article

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. Full article

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. Full article

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. Full article

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

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 Arzèla Ascoli theorem and Schauder’s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions. Full article

In this paper, we consider the second order discontinuous differential equation in the real line, ${\left(a\left(t,u\right)\varphi \left(u^{\prime }\right)\right)}^{\prime }=f\left(t,u,u^{\prime }\right),a.e.t\in \mathbb{R},u(-\infty )={\nu }^{-},u(+\infty )={\nu }^{+},$ with $\varphi$ an increasing homeomorphism such that $\varphi \left(0\right)=0$ and $\varphi \left(\mathbb{R}\right)=\mathbb{R}$, $a\in C({\mathbb{R}}^2,\mathbb{R})$ with $a(t,x)>0$ for $(t,x)\in {\mathbb{R}}^2$, $f:{\mathbb{R}}^3\to \mathbb{R}$ a $L^1$-Carathéodory function and ${\nu }^{-},{\nu }^{+}\in \mathbb{R}$ such that ${\nu }^{-}<{\nu }^{+}$. 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 $\varphi$ and $f$. To the best of our knowledge, this result is even new when $\varphi \left(y\right)=y$, that is for equation ${\left(a\left(t,u\left(t\right)\right)u^{\prime }\left(t\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right),a.e.t\in \mathbb{R}$. Moreover, these results can be applied to classical and singular $\varphi$-Laplacian equations and to the mean curvature operator. Full article

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. Full article

The paper is devoted to the discrete Lyapunov equation $X-A^{*}XA=C$, where A and C are given operators in a Hilbert space $\mathcal{H}$ 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 $\mathcal{H}$ is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in $\mathcal{H}$. Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators. Full article

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, 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. Full article

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–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. Full article