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Axioms, Volume 11, Issue 1 (January 2022) – 34 articles

Cover Story (view full-size image): We consider the Banach space consisting of real functions defined on a metric space which is locally compact and countable at infinity. Additionally, we assume that functions belonging to the space in question have increments tempered by a given modulus of continuity. The mentioned space is normed by a suitable norm connected with the given modulus of continuity. We investigate as an important particular case the Euclidean space Rk as the locally compact and countable at infinity metric space. Moreover, we also provide a few particular examples of spaces of such a type with moduli of continuity generated by the Lipschitz or Holder conditions. The main result of the paper presents a sufficient condition for relative compactness in the discussed Banach space consisting of real functions defined on the space Rk with increments tempered by an arbitrary modulus of continuity. View this paper.
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Article
Brain Tumor Classification Using Dense Efficient-Net
Axioms 2022, 11(1), 34; https://doi.org/10.3390/axioms11010034 - 17 Jan 2022
Cited by 1 | Viewed by 1040
Abstract
Brain tumors are most common in children and the elderly. It is a serious form of cancer caused by uncontrollable brain cell growth inside the skull. Tumor cells are notoriously difficult to classify due to their heterogeneity. Convolutional neural networks (CNNs) are the [...] Read more.
Brain tumors are most common in children and the elderly. It is a serious form of cancer caused by uncontrollable brain cell growth inside the skull. Tumor cells are notoriously difficult to classify due to their heterogeneity. Convolutional neural networks (CNNs) are the most widely used machine learning algorithm for visual learning and brain tumor recognition. This study proposed a CNN-based dense EfficientNet using min-max normalization to classify 3260 T1-weighted contrast-enhanced brain magnetic resonance images into four categories (glioma, meningioma, pituitary, and no tumor). The developed network is a variant of EfficientNet with dense and drop-out layers added. Similarly, the authors combined data augmentation with min-max normalization to increase the contrast of tumor cells. The benefit of the dense CNN model is that it can accurately categorize a limited database of pictures. As a result, the proposed approach provides exceptional overall performance. The experimental results indicate that the proposed model was 99.97% accurate during training and 98.78% accurate during testing. With high accuracy and a favorable F1 score, the newly designed EfficientNet CNN architecture can be a useful decision-making tool in the study of brain tumor diagnostic tests. Full article
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Article
A New Proof for a Result on the Inclusion Chromatic Index of Subcubic Graphs
Axioms 2022, 11(1), 33; https://doi.org/10.3390/axioms11010033 - 17 Jan 2022
Viewed by 562
Abstract
Let G be a graph with a minimum degree δ of at least two. The inclusion chromatic index of G, denoted by χ(G), is the minimum number of colors needed to properly color the edges of [...] Read more.
Let G be a graph with a minimum degree δ of at least two. The inclusion chromatic index of G, denoted by χ(G), is the minimum number of colors needed to properly color the edges of G so that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. We prove that every connected subcubic graph G with δ(G)2 either has an inclusion chromatic index of at most six, or G is isomorphic to K^2,3, where its inclusion chromatic index is seven. Full article
(This article belongs to the Special Issue Graph Theory with Applications)
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Article
A Model of Directed Graph Cofiber
Axioms 2022, 11(1), 32; https://doi.org/10.3390/axioms11010032 - 16 Jan 2022
Viewed by 529
Abstract
In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give [...] Read more.
In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give a construction of a cofiber of a directed graph map whose image is contractible in the cofiber. Our work reveals that a category-theoretically correct construction in continuous setup is no longer correct when it is discretized and hence leads to look at canonical constructions in category theory in a different perspective. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application)
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Article
Vector-Valued Entire Functions of Several Variables: Some Local Properties
Axioms 2022, 11(1), 31; https://doi.org/10.3390/axioms11010031 - 15 Jan 2022
Cited by 1 | Viewed by 519
Abstract
The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in join variables, where L:CnR+n is a positive continuous function. For vector-valued functions from this class we prove some propositions [...] Read more.
The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in join variables, where L:CnR+n is a positive continuous function. For vector-valued functions from this class we prove some propositions describing their local properties. In particular, these functions possess the property that maximum of norm for some partial derivative at a skeleton of polydisc does not exceed norm of the derivative at the center of polydisc multiplied by some constant. The converse proposition is also true if the described inequality is satisfied for derivative in each variable. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces
Axioms 2022, 11(1), 30; https://doi.org/10.3390/axioms11010030 - 14 Jan 2022
Viewed by 559
Abstract
We establish some properties of the bilateral Riemann–Liouville fractional derivative Ds. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by Ws,1(a,b), and the fractional [...] Read more.
We establish some properties of the bilateral Riemann–Liouville fractional derivative Ds. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by Ws,1(a,b), and the fractional bounded variation spaces of fractional order s, denoted by BVs(a,b). Examples, embeddings and compactness properties related to these spaces are addressed, aiming to set a functional framework suitable for fractional variational models for image analysis. Full article
Article
A Family of Generalized Legendre-Based Apostol-Type Polynomials
Axioms 2022, 11(1), 29; https://doi.org/10.3390/axioms11010029 - 14 Jan 2022
Viewed by 424
Abstract
Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate [...] Read more.
Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
Article
Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations
Axioms 2022, 11(1), 28; https://doi.org/10.3390/axioms11010028 - 10 Jan 2022
Viewed by 591
Abstract
A numerical scheme for nonlinear hyperbolic evolution equations is made based on the implicit Runge-Kutta method and the Fourier spectral method. The detailed discretization processes are discussed in the case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with third-order accuracy is [...] Read more.
A numerical scheme for nonlinear hyperbolic evolution equations is made based on the implicit Runge-Kutta method and the Fourier spectral method. The detailed discretization processes are discussed in the case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with third-order accuracy is presented. The order of total calculation cost is O(Nlog2N). As a benchmark, the relations between numerical accuracy and discretization unit size and that between the stability of calculation and discretization unit size are demonstrated for both linear and nonlinear cases. Full article
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Article
Multi-Layered Blockchain Governance Game
Axioms 2022, 11(1), 27; https://doi.org/10.3390/axioms11010027 - 09 Jan 2022
Viewed by 706
Abstract
The research designs a new integrated system for the security enhancement of a decentralized network by preventing damages from attackers, particularly for the 51 percent attack. The concept of multiple layered design based on Blockchain Governance Games frameworks could handle multiple number of [...] Read more.
The research designs a new integrated system for the security enhancement of a decentralized network by preventing damages from attackers, particularly for the 51 percent attack. The concept of multiple layered design based on Blockchain Governance Games frameworks could handle multiple number of networks analytically. The Multi-Layered Blockchain Governance Game is an innovative analytical model to find the best strategies for executing a safety operation to protect whole multiple layered network systems from attackers. This research fully analyzes a complex network with the compact mathematical forms and theoretically tractable results for predicting the moment of a safety operation execution are fully obtained. Additionally, simulation results are demonstrated to obtain the optimal values of configuring parameters of a blockchain-based security network. The Matlab codes for the simulations are publicly available to help those whom are constructing an enhanced decentralized security network architecture through this proposed integrated theoretical framework. Full article
(This article belongs to the Special Issue Strategic Decision Models and Applications)
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Article
(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space
Axioms 2022, 11(1), 26; https://doi.org/10.3390/axioms11010026 - 09 Jan 2022
Viewed by 305
Abstract
We introduce the real minimal surfaces family by using the Weierstrass data (ζm,ζm) for ζC, mZ2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional [...] Read more.
We introduce the real minimal surfaces family by using the Weierstrass data (ζm,ζm) for ζC, mZ2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space E3. In addition, we propose that family has a degree number (resp., class number) 2m(m+1) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c). Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
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Article
Semigroup Structures and Commutative Ideals of BCK-Algebras Based on Crossing Cubic Set Structures
Axioms 2022, 11(1), 25; https://doi.org/10.3390/axioms11010025 - 09 Jan 2022
Viewed by 316
Abstract
First, semigroup structure is constructed by providing binary operations for the crossing cubic set structure. The concept of commutative crossing cubic ideal is introduced by applying crossing cubic set structure to commutative ideal in BCK-algebra, and several properties are investigated. The relationship between [...] Read more.
First, semigroup structure is constructed by providing binary operations for the crossing cubic set structure. The concept of commutative crossing cubic ideal is introduced by applying crossing cubic set structure to commutative ideal in BCK-algebra, and several properties are investigated. The relationship between crossing cubic ideal and commutative crossing cubic ideal is discussed. An example to show that crossing cubic ideal is not commutative crossing cubic ideal is given, and then the conditions in which crossing cubic ideal can be commutative crossing cubic ideal are explored. Characterizations of commutative crossing cubic ideal are discussed, and the relationship between commutative crossing cubic ideal and crossing cubic level set is considered. An extension property of commutative crossing cubic ideal is established, and the translation of commutative crossing cubic ideal is studied. Conditions for the translation of crossing cubic set structure to be commutative crossing cubic ideal are provided, and its characterization is processed. Full article
(This article belongs to the Special Issue Cubic Set Structure and Its Applications)
Article
Reconstruction of Differential Operators with Frozen Argument
Axioms 2022, 11(1), 24; https://doi.org/10.3390/axioms11010024 - 09 Jan 2022
Cited by 2 | Viewed by 322
Abstract
We study spectral properties of a wide class of differential operators with frozen arguments by putting them into a general framework of rank-one perturbation theory. In particular, we give a complete characterization of possible eigenvalues for these operators and solve the inverse spectral [...] Read more.
We study spectral properties of a wide class of differential operators with frozen arguments by putting them into a general framework of rank-one perturbation theory. In particular, we give a complete characterization of possible eigenvalues for these operators and solve the inverse spectral problem of reconstructing the perturbation from the resulting spectrum. This approach provides a unified treatment of several recent studies and gives a clear explanation and interpretation of the obtained results. Full article
(This article belongs to the Special Issue Analytic Functions and Nonlinear Functional Analysis)
Tutorial
From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods
Axioms 2022, 11(1), 23; https://doi.org/10.3390/axioms11010023 - 07 Jan 2022
Viewed by 302
Abstract
The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. [...] Read more.
The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date. Full article
(This article belongs to the Special Issue Advances in Finite-Difference Time-Domain Methods and Applications)
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Article
On Complex Numbers in Higher Dimensions
Axioms 2022, 11(1), 22; https://doi.org/10.3390/axioms11010022 - 07 Jan 2022
Viewed by 300
Abstract
The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this [...] Read more.
The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented. Full article
(This article belongs to the Special Issue Complex Analysis)
Article
Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces
Axioms 2022, 11(1), 21; https://doi.org/10.3390/axioms11010021 - 05 Jan 2022
Viewed by 297
Abstract
The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To [...] Read more.
The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To obtain the result, we assume the Mosco convergence of the sets of minimizers of these functions. Full article
(This article belongs to the Special Issue Theory and Application of Fixed Point)
Article
On the Truncated Multidimensional Moment Problems in Cn
Axioms 2022, 11(1), 20; https://doi.org/10.3390/axioms11010020 - 05 Jan 2022
Viewed by 232
Abstract
We consider the problem of finding a (non-negative) measure μ on B(Cn) such that Cnzkdμ(z)=sk, kK. Here, K is an arbitrary finite [...] Read more.
We consider the problem of finding a (non-negative) measure μ on B(Cn) such that Cnzkdμ(z)=sk, kK. Here, K is an arbitrary finite subset of Z+n, which contains (0,,0), and sk are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. Firstly, one may consider this problem as an extension of the truncated multidimensional moment problem on Rn, where the support of the measure μ is allowed to lie in Cn. Secondly, the moment problem is a particular case of the truncated moment problem in Cn, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
Article
New Fixed Point Theorem on Triple Controlled Metric Type Spaces with Applications to Volterra–Fredholm Integro-Dynamic Equations
Axioms 2022, 11(1), 19; https://doi.org/10.3390/axioms11010019 - 02 Jan 2022
Viewed by 324
Abstract
The objective of the research article is two-fold. Firstly, we present a fixed point result in the context of triple controlled metric type spaces with a distinctive contractive condition involving the controlled functions. Secondly, we consider an initial value problem associated with a [...] Read more.
The objective of the research article is two-fold. Firstly, we present a fixed point result in the context of triple controlled metric type spaces with a distinctive contractive condition involving the controlled functions. Secondly, we consider an initial value problem associated with a nonlinear Volterra–Fredholm integro-dynamic equation and examine the existence and uniqueness of solutions via fixed point theorem in the setting of complete triple controlled metric type spaces. Furthermore, the theorem is applied to illustrate the existence of a unique solution to an integro-dynamic equation. Full article
(This article belongs to the Special Issue Theory and Application of Fixed Point)
Article
The Influence Factor Analysis of Symmetrical Half-Bridge Power Converter through Regression, Rough Set and GM(1,N) Model
Axioms 2022, 11(1), 18; https://doi.org/10.3390/axioms11010018 - 02 Jan 2022
Viewed by 284
Abstract
Analysis of power converter performance has tended to be engineering-oriented, focusing mainly on voltage stability, output power and efficiency improvement. However, there has been little discussion about the weight relations between these factors. In view of the previous inadequacy, this study employs regression, [...] Read more.
Analysis of power converter performance has tended to be engineering-oriented, focusing mainly on voltage stability, output power and efficiency improvement. However, there has been little discussion about the weight relations between these factors. In view of the previous inadequacy, this study employs regression, rough set and GM(1,N) to analyze the relations among the factors that affect the converter, with a symmetrical half-bridge power converter serving as an example. The four related affecting factors, including the current conversion ratio, voltage conversion ratio, power conversion ratio and output efficiency, are firstly analyzed and calculated. The respective relative relations between output efficiency and the other three factors are obtained. This research can be referred to by engineers in their design of symmetrical half-bridge power converters. Full article
(This article belongs to the Special Issue Grey System Theory and Applications in Mathematics)
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Article
An Improved Evaluation Methodology for Mining Association Rules
Axioms 2022, 11(1), 17; https://doi.org/10.3390/axioms11010017 - 31 Dec 2021
Cited by 4 | Viewed by 378
Abstract
At present, association rules have been widely used in prediction, personalized recommendation, risk analysis and other fields. However, it has been pointed out that the traditional framework to evaluate association rules, based on Support and Confidence as measures of importance and accuracy, has [...] Read more.
At present, association rules have been widely used in prediction, personalized recommendation, risk analysis and other fields. However, it has been pointed out that the traditional framework to evaluate association rules, based on Support and Confidence as measures of importance and accuracy, has several drawbacks. Some papers presented several new evaluation methods; the most typical methods are Lift, Improvement, Validity, Conviction, Chi-square analysis, etc. Here, this paper first analyzes the advantages and disadvantages of common measurement indicators of association rules and then puts forward four new measure indicators (i.e., Bi-support, Bi-lift, Bi-improvement, and Bi-confidence) based on the analysis. At last, this paper proposes a novel Bi-directional interestingness measure framework to improve the traditional one. In conclusion, the bi-directional interestingness measure framework (Bi-support and Bi-confidence framework) is superior to the traditional ones in the aspects of the objective criterion, comprehensive definition, and practical application. Full article
(This article belongs to the Section Mathematical Analysis)
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Article
A Note on Hermite–Hadamard–Fejer Type Inequalities for Functions Whose n-th Derivatives Are m-Convex or (α,m)-Convex Functions
Axioms 2022, 11(1), 16; https://doi.org/10.3390/axioms11010016 - 31 Dec 2021
Viewed by 282
Abstract
In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of n-th derivatives are (α,m)-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. [...] Read more.
In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of n-th derivatives are (α,m)-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. As a special case, the generalization of the remainder term of the midpoint and trapezoidal quadrature formulas are obtained. Full article
(This article belongs to the Special Issue Current Research on Mathematical Inequalities)
Article
Automated Detection and Classification of Meningioma Tumor from MR Images Using Sea Lion Optimization and Deep Learning Models
Axioms 2022, 11(1), 15; https://doi.org/10.3390/axioms11010015 - 30 Dec 2021
Viewed by 385
Abstract
Meningiomas are the most prevalent benign intracranial life-threatening brain tumors, with a life expectancy of a few months in the later stages, so this type of tumor in the brain image should be recognized and detected efficiently. The source of meningiomas is unknown. [...] Read more.
Meningiomas are the most prevalent benign intracranial life-threatening brain tumors, with a life expectancy of a few months in the later stages, so this type of tumor in the brain image should be recognized and detected efficiently. The source of meningiomas is unknown. Radiation exposure, particularly during childhood, is the sole recognized environmental risk factor for meningiomas. The imaging technique of magnetic resonance imaging (MRI) is commonly used to detect most tumor forms as it is a non-invasive and painless method. This study introduces a CNN-HHO integrated automated identification model, which makes use of SeaLion optimization methods for improving overall network optimization. In addition to these techniques, various CNN models such as Resnet, VGG, and DenseNet have been utilized to give an overall influence of CNN with SeaLion in each methodology. Each model is tested on our benchmark dataset for accuracy, specificity, dice coefficient, MCC, and sensitivity, with DenseNet outperforming the other models with a precision of 98%. The proposed methods outperform existing alternatives in the detection of brain tumors, according to the existing experimental findings. Full article
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Article
Approximation of Fixed Points for Enriched Suzuki Nonexpansive Operators with an Application in Hilbert Spaces
Axioms 2022, 11(1), 14; https://doi.org/10.3390/axioms11010014 - 29 Dec 2021
Viewed by 275
Abstract
In this article, we introduce the class of enriched Suzuki nonexpansive (ESN) mappings. We show that this new class of mappings properly contains the class of Suzuki nonexpansive as well as the class of enriched nonexpansive mappings. We establish existence of fixed point [...] Read more.
In this article, we introduce the class of enriched Suzuki nonexpansive (ESN) mappings. We show that this new class of mappings properly contains the class of Suzuki nonexpansive as well as the class of enriched nonexpansive mappings. We establish existence of fixed point and convergence of fixed point in a Hilbert space setting under the Krasnoselskii iteration process. One of the our main results is applied to solve a split feasibility problem (SFP) in this new setting of mappings. Our main results are a significant improvement of the corresponding results of the literature. Full article
(This article belongs to the Special Issue Theory and Application of Fixed Point)
Article
Banach Actions Preserving Unconditional Convergence
Axioms 2022, 11(1), 13; https://doi.org/10.3390/axioms11010013 - 27 Dec 2021
Viewed by 622
Abstract
Let A,X,Y be Banach spaces and A×XY, (a,x)ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if [...] Read more.
Let A,X,Y be Banach spaces and A×XY, (a,x)ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)nω in A and unconditionally convergent series nωxn in X, the series nωanxn is unconditionally convergent in Y. We prove that a Banach action A×XY preserves unconditional convergence if and only if for any linear functional y*Y* the operator Dy*:XA*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from 1 to 2, we prove that a Banach action A×XY preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)nω such that for every xX, the series nωenx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r[1,] with 1r1p+1q, the coordinatewise multiplication p×qr preserves unconditional convergence if and only if one of the following conditions holds: (i) p2 and qr, (ii) 2<p<qr, (iii) 2<p=q<r, (iv) r=, (v) 2q<pr, (vi) q<2<p and 1p+1q1r+12. Full article
(This article belongs to the Special Issue Analytic Functions and Nonlinear Functional Analysis)
Article
On the Asymptotics and Distribution of Values of the Jacobi Theta Functions and the Estimate of the Type of the Weierstrass Sigma Functions
Axioms 2022, 11(1), 12; https://doi.org/10.3390/axioms11010012 - 25 Dec 2021
Viewed by 604
Abstract
A refined asymptotics of the Jacobi theta functions and their logarithmic derivatives have been received. The asymptotics of the Nevanlinna characteristics of the indicated functions and the arbitrary elliptic function have been found. The estimation of the type of the Weierstrass sigma functions [...] Read more.
A refined asymptotics of the Jacobi theta functions and their logarithmic derivatives have been received. The asymptotics of the Nevanlinna characteristics of the indicated functions and the arbitrary elliptic function have been found. The estimation of the type of the Weierstrass sigma functions has been given. Full article
(This article belongs to the Special Issue Analytic Functions and Nonlinear Functional Analysis)
Article
The Space of Functions with Tempered Increments on a Locally Compact and Countable at Infinity Metric Space
Axioms 2022, 11(1), 11; https://doi.org/10.3390/axioms11010011 - 25 Dec 2021
Viewed by 986
Abstract
The aim of the paper is to introduce the Banach space consisting of real functions defined on a locally compact and countable at infinity metric space and having increments tempered by a modulus of continuity. We are going to provide a condition that [...] Read more.
The aim of the paper is to introduce the Banach space consisting of real functions defined on a locally compact and countable at infinity metric space and having increments tempered by a modulus of continuity. We are going to provide a condition that is sufficient for the relative compactness in the Banach space in question. A few particular cases of that Banach space will be discussed. Full article
(This article belongs to the Special Issue Operator Theory and Its Applications)
Article
Design, Analysis and Comparison of a Nonstandard Computational Method for the Solution of a General Stochastic Fractional Epidemic Model
Axioms 2022, 11(1), 10; https://doi.org/10.3390/axioms11010010 - 24 Dec 2021
Cited by 1 | Viewed by 696
Abstract
Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the [...] Read more.
Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R0<1. A similar result is obtained for the endemic equilibrium when R0>1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge–Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)
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Article
Some Identities on the Twisted q-Analogues of Catalan-Daehee Numbers and Polynomials
Axioms 2022, 11(1), 9; https://doi.org/10.3390/axioms11010009 - 23 Dec 2021
Viewed by 643
Abstract
In this paper, the author considers twisted q-analogues of Catalan-Daehee numbers and polynomials by using p-adic q-integral on Zp. We derive some explicit identities for those twisted numbers and polynomials related to various special numbers and polynomials. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
Article
A Hypergraph Model for Communication Patterns
Axioms 2022, 11(1), 8; https://doi.org/10.3390/axioms11010008 - 23 Dec 2021
Viewed by 644
Abstract
The article deals with interaction in concurrent systems. A calculus able to express specific communication patterns is defined, together with its abstract control structures. A hypergraph model for these structures is presented. The hypergraphs are able to properly express the communication patterns, providing [...] Read more.
The article deals with interaction in concurrent systems. A calculus able to express specific communication patterns is defined, together with its abstract control structures. A hypergraph model for these structures is presented. The hypergraphs are able to properly express the communication patterns, providing a fully abstract model for the pattern calculus. It is also proved that the hypergraph model preserves the operational reductions of processes from pattern calculus and of the actions from the control structures. Full article
(This article belongs to the Special Issue In Memoriam, Solomon Marcus)
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Article
Multiplicity of Positive Solutions to Nonlocal Boundary Value Problems with Strong Singularity
Axioms 2022, 11(1), 7; https://doi.org/10.3390/axioms11010007 - 23 Dec 2021
Viewed by 634
Abstract
In this paper, we consider generalized Laplacian problems with nonlocal boundary conditions and a singular weight, which may not be integrable. The existence of two positive solutions to the given problem for parameter λ belonging to some open interval is shown. Our approach [...] Read more.
In this paper, we consider generalized Laplacian problems with nonlocal boundary conditions and a singular weight, which may not be integrable. The existence of two positive solutions to the given problem for parameter λ belonging to some open interval is shown. Our approach is based on the fixed point index theory. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Related Fixed Point Problems)
Article
Baire-Type Properties in Metrizable c0(Ω, X)
Axioms 2022, 11(1), 6; https://doi.org/10.3390/axioms11010006 - 23 Dec 2021
Viewed by 720
Abstract
Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered [...] Read more.
Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms .nN defining its topology, then c0(Ω,X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f:ΩX such that for each ε>0 and nN the set ωΩ:f(ω)n>ε is finite or empty, with the topology defined by the semi-norms fn=supf(ω)n:ωΩ, nN. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space c0(Ω,X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c0(Ω,X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented. Full article
(This article belongs to the Collection Mathematical Analysis and Applications)
Article
Zero-Aware Low-Precision RNS Scaling Scheme
Axioms 2022, 11(1), 5; https://doi.org/10.3390/axioms11010005 - 23 Dec 2021
Viewed by 701
Abstract
Scaling is one of the complex operations in the Residue Number System (RNS). This operation is necessary for RNS-based implementations of deep neural networks (DNNs) to prevent overflow. However, the state-of-the-art RNS scalers for special moduli sets consider the 2k modulo as [...] Read more.
Scaling is one of the complex operations in the Residue Number System (RNS). This operation is necessary for RNS-based implementations of deep neural networks (DNNs) to prevent overflow. However, the state-of-the-art RNS scalers for special moduli sets consider the 2k modulo as the scaling factor, which results in a high-precision output with a high area and delay. Therefore, low-precision scaling based on multi-moduli scaling factors should be used to improve performance. However, low-precision scaling for numbers less than the scale factor results in zero output, which makes the subsequent operation result faulty. This paper first presents the formulation and hardware architecture of low-precision RNS scaling for four-moduli sets using new Chinese remainder theorem 2 (New CRT-II) based on a two-moduli scaling factor. Next, the low-precision scaler circuits are reused to achieve a high-precision scaler with the minimum overhead. Therefore, the proposed scaler can detect the zero output after low-precision scaling and then transform low-precision scaled residues to high precision to prevent zero output when the input number is not zero. Full article
(This article belongs to the Special Issue Computing Methods in Mathematics and Engineering)
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