Axioms

9 pages, 265 KiB

Open AccessArticle

Balances in the Set of Arithmetic Progressions

by Chan-Liang Chung, Chunmei Zhong and Kanglun Zhou

Axioms 2021, 10(4), 350; https://doi.org/10.3390/axioms10040350 - 20 Dec 2021

Viewed by 2377

This article focuses on searching and classifying balancing numbers in a set of arithmetic progressions. The sufficient and necessary conditions for the existence of balancing numbers are presented. Moreover, explicit formulae of balancing numbers and various relations are included. Full article

(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)

16 pages, 490 KiB

Open AccessFeature PaperArticle

A Probabilistic Approach for Solutions of Deterministic PDE’s as Well as Their Finite Element Approximations

by Joël Chaskalovic

Axioms 2021, 10(4), 349; https://doi.org/10.3390/axioms10040349 - 20 Dec 2021

Cited by 5 | Viewed by 2233

Abstract

A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation

u_{h}^{(k)}

performed by

P_{k}

Lagrange finite element. Two limitations motivated our approach: On the one [...] Read more.

A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation

u_{h}^{(k)}

performed by

P_{k}

Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution u relative to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation

u_{h}^{(k)}

. We, thus, fill this knowledge gap by considering the exact solution u together with its corresponding approximation

u_{h}^{(k)}

as random variables. By a method of consequence, any function where u and

u_{h}^{(k)}

are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on

W^{m, p}

Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation

u_{h}^{(k)}

in order to consider their respective

W^{m, p}

-norm as a random variable, as well as the

W^{m, p}

approximation error with regards to

P_{k}

finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements

P_{k_{1}}

and

P_{k_{2}}, (k_{1} < k_{2})

. Full article

(This article belongs to the Special Issue Differential Equations: Theories, Methods and Modern Applications)

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12 pages, 301 KiB

Open AccessArticle

Wiman’s Type Inequality in Multiple-Circular Domain

by Andriy Kuryliak and Oleh Skaskiv

Axioms 2021, 10(4), 348; https://doi.org/10.3390/axioms10040348 - 17 Dec 2021

Cited by 4 | Viewed by 2007

Abstract

In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions

f \in A_{0}^{p} (G)

in an arbitrary complete Reinhard domain

G \subset C^{p}

,

p \in N

[...] Read more.

In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions

f \in A_{0}^{p} (G)

in an arbitrary complete Reinhard domain

G \subset C^{p}

,

p \in N

represented by the power series of the form

f (z) = f (z_{1}, \dots, z_{p}) = \sum_{‖ n ‖ = 0}^{+ \infty} a_{n} z^{n}

with the domain of convergence

G .

We have proven the following statement: If

f \in A^{p} (G)

and

h \in H^{p}

, then for a given

ε = (ε_{1}, \dots, ε_{p}) \in R_{+}^{p}

and arbitrary

δ > 0

there exists a set

E \subset | G |

such that

\int_{E \cap Δ_{ε}} \frac{h (r) d r_{1} \dots d r_{p}}{r_{1} \dots r_{p}} < + \infty

and for all

r \in Δ_{ε} ∖ E

we have

M_{f} (r) \leq μ_{f} (r) {(h (r))}^{\frac{p + 1}{2}} {ln}^{\frac{p}{2} + δ} h (r) {ln}^{\frac{p}{2} + δ} {μ_{f} (r) h (r)} \prod_{j = 1}^{p} (ln \frac{e r_{j}}{ε_{j}})^{\frac{p - 1}{2} + δ} .

Note, that this assertion at

p = 1,

G = C,

h (r) \equiv const

implies the classical Wiman–Valiron theorem for entire functions and at

p = 1,

the

G = D : = {z \in C : | z | < 1},

h (r) \equiv 1 / (1 - r)

theorem about the Kővari-type inequality for analytic functions in the unit disc

D

;

p > 1

implies some Wiman’s type inequalities for analytic functions of several variables in

C^{n} \times D^{k}

,

n, k \in Z_{+}, n + k \in N .

Full article

(This article belongs to the Special Issue Complex Analysis)

14 pages, 5823 KiB

Open AccessArticle

Comparative Analysis of the Simple WISP and Some Prominent MCDM Methods: A Python Approach

by Dragiša Stanujkić, Darjan Karabašević, Gabrijela Popović, Edmundas Kazimieras Zavadskas, Muzafer Saračević, Predrag S. Stanimirović, Alptekin Ulutaş, Vasilios N. Katsikis and Ieva Meidute-Kavaliauskiene

Axioms 2021, 10(4), 347; https://doi.org/10.3390/axioms10040347 - 17 Dec 2021

Cited by 10 | Viewed by 3279

Abstract

This article presents a comparison of the results obtained using the newly proposed Simple Weighted Sum Product method and some prominent multiple criteria decision-making methods. For comparison, several analyses were performed using the Python programming language and its NumPy library. The comparison was [...] Read more.

This article presents a comparison of the results obtained using the newly proposed Simple Weighted Sum Product method and some prominent multiple criteria decision-making methods. For comparison, several analyses were performed using the Python programming language and its NumPy library. The comparison was also made on a real decision-making problem taken from the literature. The obtained results confirm the high correlation of the results obtained using the Simple Weighted Sum Product method and selected multiple criteria decision-making methods such as TOPSIS, SAW, ARAS, WASPAS, and CoCoSo, which confirms the usability of the Simple Weighted Sum Product method for solving multiple criteria decision-making problems. Full article

(This article belongs to the Special Issue Multiple-Criteria Decision-Making and Computational Intelligence: Recent Applications)

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17 pages, 335 KiB

Open AccessArticle

Oscillatory Behavior of Third-Order Quasi-Linear Neutral Differential Equations

by Belgees Qaraad, Osama Moaaz, Shyam Sundar Santra, Samad Noeiaghdam, Denis Sidorov and Elmetwally M. Elabbasy

Axioms 2021, 10(4), 346; https://doi.org/10.3390/axioms10040346 - 17 Dec 2021

Cited by 3 | Viewed by 2177

Abstract

In this paper, we consider a class of quasilinear third-order differential equations with a delay argument. We establish some conditions of such certain third-order quasi-linear neutral differential equation as oscillatory or almost oscillatory. Those criteria improve, complement and simplify a number of existing [...] Read more.

In this paper, we consider a class of quasilinear third-order differential equations with a delay argument. We establish some conditions of such certain third-order quasi-linear neutral differential equation as oscillatory or almost oscillatory. Those criteria improve, complement and simplify a number of existing results in the literature. Some examples are given to illustrate the importance of our results. Full article

(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)

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15 pages, 267 KiB

Open AccessArticle

Axiomatic and Dynamic Results for Power Indexes under Symmetry

by Ruey-Rong Huang and Yu-Hsien Liao

Axioms 2021, 10(4), 345; https://doi.org/10.3390/axioms10040345 - 16 Dec 2021

Viewed by 1852

Abstract

Symmetry exists in a multitude of phenomena in varying forms. The main aim of this article is to analyze the plausibility of the equal allocation non-separable costs, the efficient Banzhaf–Owen index and the efficient Banzhaf–Coleman index from the perspective of symmetry. First, based [...] Read more.

Symmetry exists in a multitude of phenomena in varying forms. The main aim of this article is to analyze the plausibility of the equal allocation non-separable costs, the efficient Banzhaf–Owen index and the efficient Banzhaf–Coleman index from the perspective of symmetry. First, based on the difference between “participation processes” and “allocating results”, different forms of symmetry are proposed. Next, building on these forms of symmetry, axiomatic results are put forth for the three power indexes, whereby the plausibility of the three power indexes is analyzed. Finally, on the basis of these different forms of symmetry and related axiomatic results, this article introduces different dynamic processes to analyze how an initial allocation result approaches the results derived from the three power indexes through dynamically modification. Full article

25 pages, 3396 KiB

Open AccessArticle

GPU Based Modelling and Analysis for Parallel Fractional Order Derivative Model of the Spiral-Plate Heat Exchanger

by Guanqiang Dong and Mingcong Deng

Axioms 2021, 10(4), 344; https://doi.org/10.3390/axioms10040344 - 16 Dec 2021

Cited by 7 | Viewed by 2458

Abstract

Heat exchangers are commonly used in various industries. A spiral-plate heat exchanger with two fluids is a compact plant that only requires a small space and is excellent in high heat transfer efficiency. However, the spiral-plate heat exchanger is a nonlinear plant with [...] Read more.

Heat exchangers are commonly used in various industries. A spiral-plate heat exchanger with two fluids is a compact plant that only requires a small space and is excellent in high heat transfer efficiency. However, the spiral-plate heat exchanger is a nonlinear plant with uncertainties, considering the difference between the heat fluid, the heated fluid, and other complex factors. The fractional order derivation model is more accurate than the traditional integer order model. In this paper, a parallel fractional order derivation model is proposed by considering the merit of the graphics processing unit (GPU). Then, the parallel fractional order derivation model for the spiral-plate heat exchanger is constructed. Simulations show the relationships between the output temperature of heated fluid and the orders of fractional order derivatives with two directional fluids impacted by complex factors, namely, the volume flow rate in hot fluid, and the volume flow rate in cold fluid, respectively. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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17 pages, 320 KiB

Open AccessArticle

Hankel Transform of the Type 2 (p,q)-Analogue of r-Dowling Numbers

by Roberto Corcino, Mary Ann Ritzell Vega and Amerah Dibagulun

Axioms 2021, 10(4), 343; https://doi.org/10.3390/axioms10040343 - 16 Dec 2021

Viewed by 2316

Abstract

In this paper, type 2

(p, q)

-analogues of the r-Whitney numbers of the second kind is defined and a combinatorial interpretation in the context of the A-tableaux is given. Moreover, some convolution-type identities, which are useful in [...] Read more.

In this paper, type 2

(p, q)

-analogues of the r-Whitney numbers of the second kind is defined and a combinatorial interpretation in the context of the A-tableaux is given. Moreover, some convolution-type identities, which are useful in deriving the Hankel transform of the type 2

(p, q)

-analogue of the r-Whitney numbers of the second kind are obtained. Finally, the Hankel transform of the type 2

(p, q)

-analogue of the r-Dowling numbers are established. Full article

(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)

12 pages, 854 KiB

Open AccessArticle

Generalized Quantum Integro-Differential Fractional Operator with Application of 2D-Shallow Water Equation in a Complex Domain

by Rabha W. Ibrahim and Dumitru Baleanu

Axioms 2021, 10(4), 342; https://doi.org/10.3390/axioms10040342 - 12 Dec 2021

Cited by 1 | Viewed by 2214

Abstract

In this paper, we aim to generalize a fractional integro-differential operator in the open unit disk utilizing Jackson calculus (quantum calculus or q-calculus). Next, by consuming the generalized operator to define a formula of normalized analytic functions, we present a set of integral [...] Read more.

In this paper, we aim to generalize a fractional integro-differential operator in the open unit disk utilizing Jackson calculus (quantum calculus or q-calculus). Next, by consuming the generalized operator to define a formula of normalized analytic functions, we present a set of integral inequalities using the concepts of subordination and superordination. In addition, as an application, we determine the maximum and minimum solutions of the extended fractional 2D-shallow water equation in a complex domain. Full article

(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)

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15 pages, 1006 KiB

Open AccessArticle

Applying Grey Relational Analysis in the Evaluation of the Balance of Children with Intellectual Disability

by Chih-Sheng Chang

Axioms 2021, 10(4), 341; https://doi.org/10.3390/axioms10040341 - 11 Dec 2021

Viewed by 2356

Abstract

In addition to intellectual performance, children with intellectual disability also seem to have lower performance than children without intellectual disability in terms of balance. Therefore, they often experience walking instability or fall due to imbalance, causing injuries. With regard to balance training courses [...] Read more.

In addition to intellectual performance, children with intellectual disability also seem to have lower performance than children without intellectual disability in terms of balance. Therefore, they often experience walking instability or fall due to imbalance, causing injuries. With regard to balance training courses provided by medical or special education personnel for children with intellectual disability, although there are subjective observation scales that describe their balance in a qualitative way, there are still few direct measurement methods that can provide personnel with the ability to evaluate the training results of an intervention program. The purpose of this study was to provide a method for evaluating the balance of children with intellectual disability to facilitate a general inspection or evaluation of balance before and after the implementation of various intervention programs that help movement development. In recent years, the force platform system has been widely used in the research of the elderly balance, yet the research on balance assessment tools applied to children is rare. This study used the objective, fast, and accurate characteristics of the force platform system to analyze the key points of the sit-to-stand movement and the movement balance parameters of children with intellectual disability and children without intellectual disability. Using the grey relational analysis (GRA) method, the time factors and weight factors from the average performance of children without intellectual disabilities was used as the analysis data. After analyzing the relevance between each participant and the target, a norm for evaluating the balance of children with intellectual disability was established. Hence, this valuable result can provide researchers, special education teachers, and related professionals with an effective and time-saving evaluation of the balance of children with intellectual disability. Full article

(This article belongs to the Special Issue Grey System Theory and Applications in Mathematics)

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17 pages, 5490 KiB

Open AccessEditor’s ChoiceArticle

Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals

by Wolf-Dieter Richter

Axioms 2021, 10(4), 340; https://doi.org/10.3390/axioms10040340 - 10 Dec 2021

Cited by 5 | Viewed by 3029

Abstract

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal [...] Read more.

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities. Full article

(This article belongs to the Special Issue Complex Analysis)

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11 pages, 281 KiB

Open AccessArticle

Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces

by Samer Al Ghour

Axioms 2021, 10(4), 339; https://doi.org/10.3390/axioms10040339 - 10 Dec 2021

Cited by 5 | Viewed by 2172

Abstract

As a weaker form of

ω

-paracompactness, the notion of

σ

-

ω

-paracompactness is introduced. Furthermore, as a weaker form of

σ

-

ω

-paracompactness, the notion of feebly

ω

-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are [...] Read more.

As a weaker form of

ω

-paracompactness, the notion of

σ

-

ω

-paracompactness is introduced. Furthermore, as a weaker form of

σ

-

ω

-paracompactness, the notion of feebly

ω

-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly

ω

-paracompact. Furthermore, it is proven hereinthat countably

ω

-paracompact

σ

-

ω

-paracompact topological spaces are

ω

-paracompact. Furthermore, it is proven hereinthat

σ

-

ω

-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat

ω

-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if

A

is a locally finite closed covering of a topological space

(X, τ)

with each

A \in A

being

ω

-paracompact and normal, then

(X, τ)

is

ω

-paracompact and normal, and as a corollary, a sum theorem for

ω

-paracompact normal topological spaces follows. Moreover, three open questions are raised. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

14 pages, 330 KiB

Open AccessArticle

Two Extensions of Cover Automata

by Cezar Câmpeanu

Axioms 2021, 10(4), 338; https://doi.org/10.3390/axioms10040338 - 10 Dec 2021

Viewed by 2152

Abstract

Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get [...] Read more.

Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get even more compact representations of finite languages. DFCAs are extended by accepting either “do not care” symbols or considering multiple entry DFCAs. We study for each of the two models the existence of the minimization or simplification algorithms and their computational complexity, the state complexity of these representations compared with other representations of the same language, and the bounds for state complexity in case we perform a representation transformation. Minimization for both models proves to be NP-hard. A method is presented to transform minimization algorithms for deterministic automata into simplification algorithms applicable to these extended models. DFCAs with “do not care” symbols prove to have comparable state complexity as Nondeterministic Finite Cover Automata. Furthermore, for multiple entry DFCAs, we can have a tight estimate of the state complexity of the transformation into equivalent DFCA. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

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16 pages, 1083 KiB

Open AccessArticle

Bäcklund Transformations for Liouville Equations with Exponential Nonlinearity

by Tatyana V. Redkina, Robert G. Zakinyan, Arthur R. Zakinyan and Olga V. Novikova

Axioms 2021, 10(4), 337; https://doi.org/10.3390/axioms10040337 - 9 Dec 2021

Cited by 2 | Viewed by 2058

Abstract

This work aims to obtain new transformations and auto-Bäcklund transformations for generalized Liouville equations with exponential nonlinearity having a factor depending on the first derivatives. This paper discusses the construction of Bäcklund transformations for nonlinear partial second-order derivatives of the soliton type with [...] Read more.

This work aims to obtain new transformations and auto-Bäcklund transformations for generalized Liouville equations with exponential nonlinearity having a factor depending on the first derivatives. This paper discusses the construction of Bäcklund transformations for nonlinear partial second-order derivatives of the soliton type with logarithmic nonlinearity and hyperbolic linear parts. The construction of transformations is based on the method proposed by Clairin for second-order equations of the Monge–Ampere type. For the equations studied in the article, using the Bäcklund transformations, new equations are found, which make it possible to find solutions to the original nonlinear equations and reveal the internal connections between various integrable equations. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)

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19 pages, 300 KiB

Open AccessArticle

On q-Horn Hypergeometric Functions H₆ and H₇

by Ayman Shehata

Axioms 2021, 10(4), 336; https://doi.org/10.3390/axioms10040336 - 8 Dec 2021

Cited by 3 | Viewed by 2581

Abstract

This work aims to construct various properties for basic Horn functions

H_{6}

and

H_{7}

under conditions on the numerator and denominator parameters, such as several q-contiguous function relations, q-differential relations, and q-differential equations. Special cases of our main [...] Read more.

This work aims to construct various properties for basic Horn functions

H_{6}

and

H_{7}

under conditions on the numerator and denominator parameters, such as several q-contiguous function relations, q-differential relations, and q-differential equations. Special cases of our main results are also demonstrated. Full article

(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)

15 pages, 391 KiB

Open AccessArticle

A New Entropy Measurement for the Analysis of Uncertain Data in MCDA Problems Using Intuitionistic Fuzzy Sets and COPRAS Method

by Parul Thakur, Bartłomiej Kizielewicz, Neeraj Gandotra, Andrii Shekhovtsov, Namita Saini, Arsham Borumand Saeid and Wojciech Sałabun

Axioms 2021, 10(4), 335; https://doi.org/10.3390/axioms10040335 - 7 Dec 2021

Cited by 21 | Viewed by 3221

Abstract

In this paper, we propose a new intuitionistic entropy measurement for multi-criteria decision-making (MCDM) problems. The entropy of an intuitionistic fuzzy set (IFS) measures uncertainty related to the data modelling as IFS. The entropy of fuzzy sets is widely used in decision support [...] Read more.

In this paper, we propose a new intuitionistic entropy measurement for multi-criteria decision-making (MCDM) problems. The entropy of an intuitionistic fuzzy set (IFS) measures uncertainty related to the data modelling as IFS. The entropy of fuzzy sets is widely used in decision support methods, where dealing with uncertain data grows in importance. The Complex Proportional Assessment (COPRAS) method identifies the preferences and ranking of decisional variants. It also allows for a more comprehensive analysis of complex decision-making problems, where many opposite criteria are observed. This approach allows us to minimize cost and maximize profit in the finally chosen decision (alternative). This paper presents a new entropy measurement for fuzzy intuitionistic sets and an application example using the IFS COPRAS method. The new entropy method was used in the decision-making process to calculate the objective weights. In addition, other entropy methods determining objective weights were also compared with the proposed approach. The presented results allow us to conclude that the new entropy measure can be applied to decision problems in uncertain data environments since the proposed entropy measure is stable and unambiguous. Full article

(This article belongs to the Special Issue Multiple-Criteria Decision Making II)

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11 pages, 273 KiB

Open AccessArticle

List Approximation for Increasing Kolmogorov Complexity

by Marius Zimand

Axioms 2021, 10(4), 334; https://doi.org/10.3390/axioms10040334 - 7 Dec 2021

Viewed by 2104

Abstract

It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. However, is it possible to construct a few strings, no longer than the input string, so that most of them have larger complexity? We show that the answer [...] Read more.

It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. However, is it possible to construct a few strings, no longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that takes as input a string x of length n and returns a list with

O (n^{2})

strings, all of length n, such that 99% of them are more complex than x, provided the complexity of x is less than

n - log log n - O (1)

. We also present an algorithm that obtains a list of quasi-polynomial size in which each element can be produced in polynomial time. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

10 pages, 286 KiB

Open AccessArticle

A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings

by Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

Axioms 2021, 10(4), 333; https://doi.org/10.3390/axioms10040333 - 5 Dec 2021

Cited by 2 | Viewed by 2561

Abstract

This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented [...] Read more.

This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application)

21 pages, 356 KiB

Open AccessArticle

A New Approach on Transforms: Formable Integral Transform and Its Applications

by Rania Zohair Saadeh and Bayan fu’ad Ghazal

Axioms 2021, 10(4), 332; https://doi.org/10.3390/axioms10040332 - 1 Dec 2021

Cited by 46 | Viewed by 4443

Abstract

In this paper, we introduce a new integral transform called the Formable integral transform, which is a new efficient technique for solving ordinary and partial differential equations. We introduce the definition of the new transform and give the sufficient conditions for its existence. [...] Read more.

In this paper, we introduce a new integral transform called the Formable integral transform, which is a new efficient technique for solving ordinary and partial differential equations. We introduce the definition of the new transform and give the sufficient conditions for its existence. Some essential properties and examples are introduced to show the efficiency and applicability of the new transform, and we prove the duality between the new transform and other transforms such as the Laplace transform, Sumudu transform, Elzaki transform, ARA transform, Natural transform and Shehu transform. Finally, we use the Formable transform to solve some ordinary and partial differential equations by presenting five applications, and we evaluate the Formable transform for some functions and present them in a table. A comparison between the new transform and some well-known transforms is made and illustrated in a table. Full article

18 pages, 1812 KiB

Open AccessArticle

Change Point Detection Using Penalized Multidegree Splines

by Eun-Ji Lee and Jae-Hwan Jhong

Axioms 2021, 10(4), 331; https://doi.org/10.3390/axioms10040331 - 1 Dec 2021

Viewed by 2583

Abstract

We consider a function estimation method with change point detection using truncated power spline basis and elastic-net-type

L^{1}

-norm penalty. The

L^{1}

-norm penalty controls the jump detection and smoothness depending on the value of the parameter. In terms of the [...] Read more.

We consider a function estimation method with change point detection using truncated power spline basis and elastic-net-type

L^{1}

-norm penalty. The

L^{1}

-norm penalty controls the jump detection and smoothness depending on the value of the parameter. In terms of the proposed estimators, we introduce two computational algorithms for the Lagrangian dual problem (coordinate descent algorithm) and constrained convex optimization problem (an algorithm based on quadratic programming). Subsequently, we investigate the relationship between the two algorithms and compare them. Using both simulation and real data analysis, numerical studies are conducted to validate the performance of the proposed method. Full article

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8 pages, 237 KiB

Open AccessArticle

New Expressions for Sums of Products of the Catalan Numbers

by Conghui Xie and Yuan He

Axioms 2021, 10(4), 330; https://doi.org/10.3390/axioms10040330 - 1 Dec 2021

Cited by 1 | Viewed by 2441

Abstract

In this paper, we perform a further investigation for the Catalan numbers. By making use of the method of derivatives and some properties of the Bell polynomials, we establish two new expressions for sums of products of arbitrary number of the Catalan numbers. [...] Read more.

In this paper, we perform a further investigation for the Catalan numbers. By making use of the method of derivatives and some properties of the Bell polynomials, we establish two new expressions for sums of products of arbitrary number of the Catalan numbers. The results presented here can be regarded as the development of some known formulas. Full article

(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)

7 pages, 278 KiB

Open AccessArticle

Applying Set Theory

by Saharon Shelah

Axioms 2021, 10(4), 329; https://doi.org/10.3390/axioms10040329 - 30 Nov 2021

Viewed by 2465

Abstract

We prove some results in set theory as applied to general topology and model theory. In particular, we study

ℵ_{1}

-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets. Full article

(This article belongs to the Collection Mathematical Analysis and Applications)

28 pages, 370 KiB

Open AccessArticle

A Comprehensive Analysis of Hermite–Hadamard Type Inequalities via Generalized Preinvex Functions

by Muhammad Tariq, Hijaz Ahmad, Hüseyin Budak, Soubhagya Kumar Sahoo, Thanin Sitthiwirattham and Jiraporn Reunsumrit

Axioms 2021, 10(4), 328; https://doi.org/10.3390/axioms10040328 - 30 Nov 2021

Cited by 5 | Viewed by 2249

Abstract

The principal objective of this article is to introduce the idea of a new class of n-polynomial convex functions which we call n-polynomial s-type m-preinvex function. We establish a new variant of the well-known Hermite–Hadamard inequality in the mode [...] Read more.

The principal objective of this article is to introduce the idea of a new class of n-polynomial convex functions which we call n-polynomial s-type m-preinvex function. We establish a new variant of the well-known Hermite–Hadamard inequality in the mode of the newly introduced concept. To add more insight into the newly introduced concept, we have discussed some algebraic properties and examples as well. Besides, we discuss a few new exceptional cases for the derived results, which make us realize that the results of this paper are the speculations and expansions of some recently known outcomes. The immeasurable concepts and chasmic tools of this paper may invigorate and revitalize additional research in this mesmerizing and absorbing field. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

12 pages, 295 KiB

Open AccessArticle

P Systems with Evolutional Communication and Division Rules

by David Orellana-Martín, Luis Valencia-Cabrera and Mario J. Pérez-Jiménez

Axioms 2021, 10(4), 327; https://doi.org/10.3390/axioms10040327 - 30 Nov 2021

Cited by 3 | Viewed by 2145

Abstract

A widely studied field in the framework of membrane computing is computational complexity theory. While some types of P systems are only capable of efficiently solving problems from the class P, adding one or more syntactic or semantic ingredients to these membrane [...] Read more.

A widely studied field in the framework of membrane computing is computational complexity theory. While some types of P systems are only capable of efficiently solving problems from the class P, adding one or more syntactic or semantic ingredients to these membrane systems can give them the ability to efficiently solve presumably intractable problems. These ingredients are called to form a frontier of efficiency, in the sense that passing from the first type of P systems to the second type leads to passing from non-efficiency to the presumed efficiency. In this work, a solution to the SAT problem, a well-known NP-complete problem, is obtained by means of a family of recognizer P systems with evolutional symport/antiport rules of length at most (2,1) and division rules where the environment plays a passive role; that is, P systems from

\hat{CDEC} (2, 1)

. This result is comparable to the one obtained in the tissue-like counterpart, and gives a glance of a parallelism and the non-evolutionary membrane systems with symport/antiport rules. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

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16 pages, 3840 KiB

Open AccessArticle

Advanced Proportional-Integral-Derivative Control Compensation Based on a Grey Estimated Model in Dynamic Balance of Single-Wheeled Robot

by Mao-Lin Chen, Chun-Yen Chen, Chien-Hung Wen, Pin-Hao Liao and Kai-Jung Chen

Axioms 2021, 10(4), 326; https://doi.org/10.3390/axioms10040326 - 30 Nov 2021

Cited by 1 | Viewed by 2444

Abstract

This paper aims to design a one-wheeled robot as regards its pitch freedom and balance control on the one hand and to assess the application feasibility of the GM (1,1) swing estimation controller on the other. System control focuses mainly on one-wheeled robot [...] Read more.

This paper aims to design a one-wheeled robot as regards its pitch freedom and balance control on the one hand and to assess the application feasibility of the GM (1,1) swing estimation controller on the other. System control focuses mainly on one-wheeled robot stability, body swings in position, and speed control. Mathematical modeling and GM (1,1) prediction control are under investigation. The mathematical modeling is firstly conducted through referencing to the Newtonian mechanics and the Lagrange equation, from which the robot transfer function and state-space differential equation are derived. Next, the linear quadratic regulator is applied as the control rule at the balance point. Applying GM (1,1) to assess the robot gyro signal at a dynamic state is a discussion. Next, model reference estimation control is processed, and a mathematical model of the balance control method is completed. Finally, a simulation is conducted to verify the feasibility of the GM (1,1) estimation reference model. The linear quadratic regulator, which is credited with tenacity, can provide pitch swing and balance control of the one-wheeled robot. Full article

(This article belongs to the Special Issue Grey System Theory and Applications in Mathematics)

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22 pages, 388 KiB

Open AccessArticle

Asymptotic Analysis of Spectrum and Stability for One Class of Singularly Perturbed Neutral-Type Time-Delay Systems

by Valery Y. Glizer

Axioms 2021, 10(4), 325; https://doi.org/10.3390/axioms10040325 - 30 Nov 2021

Cited by 5 | Viewed by 2047

Abstract

In this study, a singularly perturbed linear time-delay system of neutral type is considered. It is assumed that the delay is small of order of a small positive parameter multiplying a part of the derivatives in the system. This system is decomposed asymptotically [...] Read more.

In this study, a singularly perturbed linear time-delay system of neutral type is considered. It is assumed that the delay is small of order of a small positive parameter multiplying a part of the derivatives in the system. This system is decomposed asymptotically into two much simpler parameter-free subsystems, the slow and fast ones. Using this decomposition, an asymptotic analysis of the spectrum of the considered system is carried out. Based on this spectrum analysis, parameter-free conditions guaranteeing the exponential stability of the original system for all sufficiently small values of the parameter are derived. Illustrative examples are presented. Full article

(This article belongs to the Special Issue Non-local Mathematical Models and Applications: A Theme Issue in Honor of Prof. Carlo Cattani)

6 pages, 235 KiB

Open AccessArticle

Quadruple Integral Involving the Logarithm and Product of Bessel Functions Expressed in Terms of the Lerch Function

by Robert Reynolds and Allan Stauffer

Axioms 2021, 10(4), 324; https://doi.org/10.3390/axioms10040324 - 30 Nov 2021

Cited by 3 | Viewed by 2329

Abstract

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s

ζ (3)

constants are produced. Some [...] Read more.

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s

ζ (3)

constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new. Full article

(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)

21 pages, 1634 KiB

Open AccessArticle

Dynamics of an Impulsive Stochastic Predator–Prey System with the Beddington–DeAngelis Functional Response

by Yuanfu Shao

Axioms 2021, 10(4), 323; https://doi.org/10.3390/axioms10040323 - 28 Nov 2021

Cited by 6 | Viewed by 2403

Abstract

Taking impulsive effects into account, an impulsive stochastic predator–prey system with the Beddington–DeAngelis functional response is proposed in this paper. First, the impulsive system is transformed into an equivalent system without pulses. Then, by constructing suitable functionals and applying the extreme-value theory of [...] Read more.

Taking impulsive effects into account, an impulsive stochastic predator–prey system with the Beddington–DeAngelis functional response is proposed in this paper. First, the impulsive system is transformed into an equivalent system without pulses. Then, by constructing suitable functionals and applying the extreme-value theory of quadratic functions, sufficient conditions on the existence of periodic Markovian processes are provided. The uniform continuity and global attractivity of solutions are also investigated. Additionally, we investigate the extinction and permanence in the mean of all species with the help of comparison methods and inequality techniques. Sufficient conditions on the existence and ergodicity of the stationary distribution of solutions for the autonomous and non-impulsive case are given. Finally, numerical simulations are performed to illustrate the main results. Full article

(This article belongs to the Special Issue Recent Advances in Stochastic Differential Equations)

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14 pages, 455 KiB

Open AccessArticle

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

by Ricardo Almeida, Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan

Axioms 2021, 10(4), 322; https://doi.org/10.3390/axioms10040322 - 27 Nov 2021

Cited by 17 | Viewed by 2751

Abstract

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, [...] Read more.

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

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20 pages, 2557 KiB

Open AccessArticle

Roots of Elliptic Scator Numbers

by Manuel Fernandez-Guasti

Axioms 2021, 10(4), 321; https://doi.org/10.3390/axioms10040321 - 27 Nov 2021

Cited by 3 | Viewed by 2150

Abstract

The Victoria equation, a generalization of De Moivre’s formula in

1 + n

dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in

S^{1 + n}

of a real or a scator number, there [...] Read more.

The Victoria equation, a generalization of De Moivre’s formula in

1 + n

dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in

S^{1 + n}

of a real or a scator number, there are

q^{n}

possible roots. For

n = 1

, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For

n \geq 2

, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in

S^{1 + 2}

. Full article

(This article belongs to the Section Algebra and Number Theory)

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Axioms, Volume 10, Issue 4 (December 2021) – 114 articles

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