Axioms, Volume 8, Issue 3 (September 2019) – 29 articles

In convex optimization, it is often inevitable to work with projectors onto convex sets composed with a linear operator. Such a need arises from both the theory and applications, with signal processing being a prominent and broad field where convex optimization has been used recently. In this article, a novel projector is presented, which generalizes previous results in that it admits to work with a broader family of linear transforms when compared with the state of the art but, on the other hand, it is limited to box-type convex sets in the transformed domain. The new projector is described by an explicit formula, which makes it simple to implement and requires a low computational cost. The projector is interpreted within the framework of the so-called proximal splitting theory. The convenience of the new projector is demonstrated on an example from signal processing, where it was possible to speed up the convergence of a signal declipping algorithm by a factor of more than two. Full article

In this paper, we give a mathematical model of the logic of determination of objects (LDO) based on preordered sets, and a mathematical model of the logic of typical and atypical instances (LTA). We prove that LTA is an extension of LDO. It can manipulate several types of “exceptions”. Finally, we show that the structural part of LTA can be modeled by a quasi topology structure (QTS). Full article

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two different ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier. Full article

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes four restrictions imposed on the parameters: two of the singularities should have non-zero integer characteristic exponents and the accessory parameters should obey polynomial equations. Full article

This study considers the mathematical analysis framework aimed at the adequate description of the modes of lasers on the threshold of non-attenuated in time light emission. The lasers are viewed as open dielectric resonators equipped with active regions, filled in with gain material. We introduce a generalized complex-frequency eigenvalue problem for such cavities and prove important properties of the spectrum of its eigensolutions. This involves reduction of the problem to the set of the Muller boundary integral equations and their discretization with the Nystrom technique. Embedded into this general framework is the application-oriented lasing eigenvalue problem, where the real emission frequencies and the threshold gain values together form two-component eigenvalues. As an example of on-threshold mode study, we present numerical results related to the two-dimensional laser shaped as an active equilateral triangle with a round piercing hole. It is demonstrated that the threshold of lasing and the directivity of light emission, for each mode, can be efficiently manipulated with the aid of the size and, especially, the placement of the piercing hole, while the frequency of emission remains largely intact. Full article

Using the defined notion of the inference with multiply-conclusion rules, we show that in the logics enjoying the disjunction property, any derivable rule can be inferred from the single-conclusion rules and a single multiple-conclusion rule, which represents the disjunction property. Also, the conversion algorithm of single- and multiple-conclusion deductive systems into each other is studied. Full article

In this paper, we establish some results on coincidence point and common fixed point theorems for a hybrid pair of single valued and multivalued mappings in complete $C^{*}$ -algebra valued fuzzy soft metric spaces. In addition, we provided some coupled fixed point theorems. Finally, we have given examples which support our main results. Full article

This paper introduces a variant of entropy measures based on vertex eccentricity and applies it to all graphs representing the isomers of octane. Taking into account the vertex degree as well (degree-ecc-entropy), we find a good correlation with the acentric factor of octane isomers. In particular, we compute the degree-ecc-entropy for three classes of dendrimer graphs. Full article

In this paper, a new class of $(C,G_f)$ -invex functions introduce and give nontrivial numerical examples which justify exist such type of functions. Also, we construct generalized convexity definitions (such as, $(F,G_f)$ -invexity, C-convex etc.). We consider Mond–Weir type fractional symmetric dual programs and derive duality results under $(C,G_f)$ -invexity assumptions. Our results generalize several known results in the literature. Full article

Our main goal of this research is to present the theory of points for relatively cyclic and relatively relatively noncyclic p-contractions in complete locally $\mathbb{K}$ -convex spaces by providing basic conditions to ensure the existence and uniqueness of fixed points and best proximity points of the relatively cyclic and relatively noncyclic p-contractions map in locally $\mathbb{K}$ -convex spaces. The result of this paper is the extension and generalization of the main results of Kirk and A. Abkar. Full article

The main ideas of F-transform came from representing expert rules. It would be therefore reasonable to expect that the more accurately the membership functions describe human reasoning, the more successful will be the corresponding F-transform formulas. We know that an adequate description of our reasoning corresponds to complicated membership functions—however, somewhat surprisingly, many successful applications of F-transform use the simplest possible triangular membership functions. There exist some explanations for this phenomenon, which are based on local behavior of the signal. In this paper, we supplement these local explanations by a global one: namely, we prove that triangular membership functions are the only one that provide the exact reconstruction of the appropriate global characteristic of the signal. Full article

In many application problems, F-transform algorithms are very efficient. In F-transform techniques, we replace the original signal or image with a finite number of weighted averages. The use of a weighted average can be naturally explained, e.g., by the fact that this is what we get anyway when we measure the signal. However, most successful applications of F-transform have an additional not-so-easy-to-explain feature: the fuzzy partition requirement that the sum of all the related weighting functions is a constant. In this paper, we show that this seemingly difficult-to-explain requirement can also be naturally explained in signal-measurement terms: namely, this requirement can be derived from the natural desire to have all the signal values at different moments of time estimated with the same accuracy. This explanation is the main contribution of this paper. Full article

A Structure Theorem for Protori is derived for the category of finite-dimensional protori (compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are parameterized in the structure theorem by the dual category of finite rank torsion-free abelian groups. A consequence is a universal resolution for a finite-dimensional protorus, independent of a choice of a particular subgroup. A resolution is also given strictly in terms of the path component of the identity and the union of all zero-dimensional subgroups. The structure theorem is applied to show that a morphism of finite-dimensional protori lifts to a product morphism between products of periodic locally compact groups and real vector spaces. Full article

This document introduces a method to solve linear optimization problems. The method’s strategy is based on the bounding condition that each constraint exerts over the dimensions of the problem. The solution of a linear optimization problem is at the intersection of the constraints defining the extreme vertex. The method decomposes the n-dimensional linear problem into n-1 two-dimensional problems. After studying the role of constraints in these two-dimensional problems, we identify the constraints intersecting at the extreme vertex. We then formulate a linear equation system that directly leads to the solution of the optimization problem. The algorithm is remarkably different from previously existing linear programming algorithms in the sense that it does not iterate; it is deterministic. A fully c-sharp-coded algorithm is made available. We believe this algorithm and the methods applied for classifying constraints according to their role open up a useful framework for studying complex linear problems through feasible-space and constraint analysis. Full article

The sampling reconstruction theory is one of the great areas of the analysis in which Paul Leo Butzer earned longstanding and excellent theoretical results. Thus, we are forced either by earlier exhaustive presentations of his research activity and/or the highly voluminous material to restrict ourselves to a more narrow and precise sub-area in consideration; we discuss here, giving deeper insight, Paul Butzer’s sampling theoretical work with special attention concerning sampling stochastic signals. Full article

In this paper, we establish a topological version of the notion of an Eilenberg–Mac Lane space. If X is a pointed topological space, ${\pi }_1(X)$ has a natural topology coming from the compact-open topology on the space of maps $S^1\to X$ . In general, the construction does not produce a topological group because it is possible to create examples where the group multiplication ${\pi }_1(X)\times {\pi }_1(X)\to {\pi }_1(X)$ is discontinuous. This discontinuity has been noticed by others, for example Fabel. However, if we work in the category of compactly generated, weakly Hausdorff spaces, we may retopologise both the space of maps $S^1\to X$ and the product ${\pi }_1(X)\times {\pi }_1(X)$ with compactly generated topologies to see that ${\pi }_1(X)$ is a group object in this category. Such group objects are known as k-groups. Next we construct the Eilenberg–Mac Lane space $K(G,1)$ for any totally path-disconnected k-group G. The main point of this paper is to show that, for such a G, ${\pi }_1(K(G,1))$ is isomorphic to G in the category of k-groups. All totally disconnected locally compact groups are k-groups and so our results apply in particular to profinite groups, answering a question of Sauer’s. We also show that analogues of the Mayer–Vietoris sequence and Seifert–van Kampen theorem hold in this context. The theory requires a careful analysis using model structures and other homotopical structures on cartesian closed categories as we shall see that no theory can be comfortably developed in the classical world. Full article

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space $\mathcal{H}$ and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space; instead, they are functionals on the dual space, ${\mathsf{\Phi }}^{\times }$ , of a rigged Hilbert space, $\mathsf{\Phi }\subset \mathcal{H}\subset {\mathsf{\Phi }}^{\times }$ . In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors $\mathsf{\Phi }$ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on $\mathsf{\Phi }$ with its own topology, so that they admit continuous extensions to the dual ${\mathsf{\Phi }}^{\times }$ and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider $SO\left(2\right)$ and functions on the unit circle, $SU\left(2\right)$ and associated Laguerre functions, Weyl–Heisenberg group and Hermite functions, $SO(3,2)$ and spherical harmonics, $su(1,1)$ and Laguerre functions, $su(2,2)$ and algebraic Jacobi functions and, finally, $su(1,1)\oplus su(1,1)$ and Zernike functions on a circle. Full article

In this paper, for a given direction $\mathbf{b}\in {\mathbb{C}}^n\backslash \left\{\mathbf{0}\right\}$ we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line $\{z^0+t\mathbf{b}:t\in \mathbb{C}\}$ for any $z^0\in {\mathbb{C}}^n$ . Unlike to quaternionic analysis, we fix the direction $\mathbf{b}$ . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable $z_1$ and continuous in variable $z_2.$ For this class of functions there is introduced a concept of boundedness of L-index in the direction $\mathbf{b}$ where $\mathbf{L}:{\mathbb{C}}^n\to {\mathbb{R}}_{+}$ is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function $L:{\mathbb{C}}^n\to {\mathbb{R}}_{+}.$ Full article

In the theory of generalized hypergeometric functions, classical summation theorems for the series ${}_2{F}_1$ , ${}_3{F}_2$ , ${}_4{F}_3$ , ${}_5{F}_4$ and ${}_7{F}_6$ play a key role. Very recently, Masjed-Jamei and Koepf established generalizations of the above-mentioned summation theorems. Inspired by their work, the main objective of the paper is to provide a new class of Laplace-type integrals involving generalized hypergeometric functions ${}_p{F}_p$ for $p=2,3,4,5$ and 7 in the most general forms. Several new and known cases have also been obtained as special cases of our main findings. Full article

We study factorization properties of continuous homomorphisms defined on submonoids of products of topologized monoids. We prove that if S is an ω-retractable submonoid of a product $D={\prod }_{i\in I}{D}_i$ of topologized monoids and $f:S\to H$ is a continuous homomorphism to a topologized semigroup H with $\psi \left(H\right)\le \omega$ , then one can find a countable subset E of I and a continuous homomorphism $g:p_E\left(S\right)\to H$ satisfying $f=g\circ p_E\upharpoonright S$ , where $p_E$ is the projection of D to ${\prod }_{i\in E}{D}_i$ . The same conclusion is valid if S contains the $\mathsf{\Sigma }$ -product $\mathsf{\Sigma }D\subset D$ . Furthermore, we show that in both cases, there exists the smallest by inclusion subset $E\subset I$ with the aforementioned properties. Full article

Let $\mathbf{A}$ be an epireflective subcategory of the category of all semitopological groups that consists only of abelian groups. We describe maximal hereditary coreflective subcategories of $\mathbf{A}$ that are not bicoreflective in $\mathbf{A}$ in the case that the $\mathbf{A}$ -reflection of the discrete group of integers is a finite cyclic group, the group of integers with a topology that is not $T_0$ , or the group of integers with the topology generated by its subgroups of the form $⟨p^n⟩$ , where $n\in \mathbb{N}$ , $p\in P$ and P is a given set of prime numbers. Full article

In [Fixed Point Theory Appl., 2015 (2015):185], the authors introduced a new concept of modified contractive mappings, generalizing Ćirić, Chatterjea, Kannan, and Reich type contractions. They applied the condition $\left({\theta }_4\right)$ (see page 3, ^{Section 2} of the above paper). Later, in [Fixed Point Theory Appl., 2016 (2016):62], Jiang et al. claimed that the results in [Fixed Point Theory Appl., 2015 (2015):185] are not real generalizations. In this paper, by restricting the conditions of the control functions, we obtain a real generalization of the Banach contraction principle (BCP). At the end, we introduce a weakly JS-contractive condition generalizing the JS-contractive condition. Full article

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers–Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald. Full article

We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over ${\widehat{\mathfrak{sl}}}_2$ . In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights. Full article

In this study, we establish the existence and uniqueness theorems of the best proximity points for Geraghty type Ƶ-proximal contractions defined on a complete metric space. The presented results improve and generalize some recent results in the literature. An example, as well as an application to a variational inequality problem are also given in order to illustrate the effectiveness of our generalizations. Full article

Two types of singularly-perturbed nonlinear time delay controlled systems are considered. For these systems, sufficient conditions of the functional null controllability are derived. These conditions, being independent of the parameter of singular perturbation, provide the controllability of the systems for all sufficiently small values of the parameter. Illustrative examples are presented. Full article

To date, the usage of electromyography (EMG) signals in myoelectric prosthetics allows patients to recover functional rehabilitation of their upper limbs. However, the increment in the number of EMG features has been shown to have a great impact on performance degradation. Therefore, feature selection is an essential step to enhance classification performance and reduce the complexity of the classifier. In this paper, a hybrid method, namely, binary particle swarm optimization differential evolution (BPSODE) was proposed to tackle feature selection problems in EMG signals classification. The performance of BPSODE was validated using the EMG signals of 10 healthy subjects acquired from a publicly accessible EMG database. First, discrete wavelet transform was applied to decompose the signals into wavelet coefficients. The features were then extracted from each coefficient and formed into the feature vector. Afterward, BPSODE was used to evaluate the most informative feature subset. To examine the effectiveness of the proposed method, four state-of-the-art feature selection methods were used for comparison. The parameters, including accuracy, feature selection ratio, precision, F-measure, and computation time were used for performance measurement. Our results showed that BPSODE was superior, in not only offering a high classification performance, but also in having the smallest feature size. From the empirical results, it can be inferred that BPSODE-based feature selection is useful for EMG signals classification. Full article

Nonassociative algebras with metagroup relations and their modules are studied. Their cohomology theory is scrutinized. Extensions and cleftings of these algebras are studied. Broad families of such algebras and their acyclic complexes are described. For this purpose, different types of products of metagroups are investigated. Necessary structural properties of metagroups are studied. Examples are given. It is shown that a class of nonassociative algebras with metagroup relations contains a subclass of generalized Cayley–Dickson algebras. Full article

Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus. Full article