Axioms — Open Access Journal

The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum $\mathrm{spec}\left(\mathfrak{t}\right)=\left(1,k,k+1,\cdots ,n+k-3,n+2k-3\right)$ 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. Full article

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–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. Full article

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 “Design of an optimized fuzzy classifier for the diagnosis of blood pressure with a new computational method for expert rule optimization” with 98%. In addition, we also obtained a better classification rate than the other architectures proposed in this work. Full article

We consider the Laplacian flow of locally conformal calibrated ${\mathrm{G}}_2$-structures as a natural extension to these structures of the well-known Laplacian flow of calibrated ${\mathrm{G}}_2$-structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated ${\mathrm{G}}_2$ manifolds and, in both cases, we obtain a flow of locally conformal calibrated ${\mathrm{G}}_2$-structures, which are ancient solutions, that is they are defined on a time interval of the form $(-\infty ,T)$, where $T>0$ is a real number. Moreover, for each of these examples, we prove that the underlying metrics $g\left(t\right)$ of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to $-\infty$, and they blow-up at a finite-time singularity. Full article

There is one-to-one correspondence between contact semi-Riemannian structures $(\eta ,\xi ,\phi ,g)$ and non-degenerate almost CR structures $(\mathcal{H},\vartheta ,J)$. In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for ${\mathcal{H}}^{1,0}:=\left\{X-iJX,X\in \mathcal{H}\right\}$ 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. Full article

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. Full article

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 $C\left(K\right)$ for metrizable compact spaces K; and ${\ell }_p$, for $p\ge 1$. 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. Full article

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 Kähler–Einstein manifolds as well as Fano manifolds without Kähler–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 $n>15$ and $N_{n,m}^{\prime }$ when $n>4$. 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–Einstein metrics. These also provide open Calabi–Yau manifolds. Full article

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 $S\times \left\{0\right\}$ and $T\times \left\{1\right\}$ by identifying the point $(x,0)$ in $S\times \left\{0\right\}$ with $(x,1)$ in $T\times \left\{1\right\}$ 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. Full article

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. Full article

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. Full article

We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann–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. Full article

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 $\vartheta$-methods for Volterra integral equations. Full article

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. Full article

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. Full article

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere $S^m\subset {\mathbb{R}}^{m+1}$. Given a codimension two totally geodesic submanifold $\Sigma \subset S^m$, we show that every nonconstant exponentially harmonic map $\varphi :M\to S^m$ either meets or links $\Sigma$. If $H^1(M,\mathbb{Z})=0$ then $\varphi \left(M\right)\cap \Sigma \ne \varnothing$. Full article

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. Full article

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 $\mathbb{N}$ of natural numbers and every sequence $\left(U_n\right)$ of non-empty open subsets of G, one can choose a point $x_n\in U_n$ for all $n\in \mathbb{N}$ in such a way that the resulting sequence $\left(x_n\right)$ has a p-limit in G; that is, $\{n\in \mathbb{N}:x_n\in V\}\in p$ 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-$\omega$-bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ${⨁}_{i\in I}{X}_i$, where each space $X_i$ is either maximal or discrete, contains no infinite separable pseudocompact subsets. Full article