Axioms

17 pages, 366 KiB

Open AccessArticle

by Gao Zhang and Shaoqun Zhang

Axioms 2023, 12(6), 610; https://doi.org/10.3390/axioms12060610 - 20 Jun 2023

Viewed by 656

For a quantale

I

, which is a unit interval endowed with a continuous triangular norm and the Barr extension

{\bar{β}}_{I}

of the ultrafilter monad to

I - Rel,

a characterization of the discrete presheaf monad associated to [...] Read more.

For a quantale

I

, which is a unit interval endowed with a continuous triangular norm and the Barr extension

{\bar{β}}_{I}

of the ultrafilter monad to

I - Rel,

a characterization of the discrete presheaf monad associated to

{\bar{β}}_{I}

is given. It is also proved that, when & is the Łucasiewicz triangular norm, the discrete presheaf monad is isomorphic to the saturated prefilter monad, and when & is the product triangular norm, the prime functional ideal monad is isomorphic to a submonad of the discrete presheaf monad. Full article

18 pages, 595 KiB

Open AccessArticle

A Fractional Analysis of Zakharov–Kuznetsov Equations with the Liouville–Caputo Operator

by Abdul Hamid Ganie, Fatemah Mofarreh and Adnan Khan

Axioms 2023, 12(6), 609; https://doi.org/10.3390/axioms12060609 - 19 Jun 2023

Cited by 9 | Viewed by 954

Abstract

In this study, we used two unique approaches, namely the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM), to derive approximate analytical solutions for nonlinear time-fractional Zakharov–Kuznetsov equations (ZKEs). This framework demonstrated the behavior of weakly nonlinear ion-acoustic waves [...] Read more.

In this study, we used two unique approaches, namely the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM), to derive approximate analytical solutions for nonlinear time-fractional Zakharov–Kuznetsov equations (ZKEs). This framework demonstrated the behavior of weakly nonlinear ion-acoustic waves in plasma containing cold ions and hot isothermal electrons in the presence of a uniform magnetic flux. The density fraction and obliqueness of two compressive and rarefactive potentials are depicted. In the Liouville–Caputo sense, the fractional derivative is described. In these procedures, we first used the Yang transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. The results of these methods also made clear the connections between the precise solutions to the issues under study. Illustrations of the reliability of the proposed techniques are provided. The results are clarified through graphs and tables. The reliability of the proposed procedures is demonstrated by illustrative examples. The proposed approaches are attractive, though these easy approaches may be time-consuming for solving diverse nonlinear fractional-order partial differential equations. Full article

(This article belongs to the Special Issue New Trends in Fractional Operators with Applications in Mathematical Physics)

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18 pages, 587 KiB

Open AccessArticle

Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment

by Fahad Al Basir, Biru Rajak, Bootan Rahman and Khalid Hattaf

Axioms 2023, 12(6), 608; https://doi.org/10.3390/axioms12060608 - 19 Jun 2023

Cited by 1 | Viewed by 1131

Abstract

Infectious diseases continue to be a significant threat to human health and civilization, and finding effective methods to combat them is crucial. In this paper, we investigate the impact of awareness campaigns and optimal control techniques on infectious diseases without proper vaccines. Specifically, [...] Read more.

Infectious diseases continue to be a significant threat to human health and civilization, and finding effective methods to combat them is crucial. In this paper, we investigate the impact of awareness campaigns and optimal control techniques on infectious diseases without proper vaccines. Specifically, we develop an SIRS-type mathematical model that incorporates awareness campaigns through media and treatment for disease transmission dynamics and control. The model displays two equilibria, a disease-free equilibrium and an endemic equilibrium, and exhibits Hopf bifurcation when the bifurcation parameter exceeds its critical value, causing a switch in the stability of the system. We also propose an optimal control problem that minimizes the cost of control measures while achieving a desired level of disease control. By applying the minimum principle to the optimal control problem, we obtain analytical and numerical results that show how the infection rate of the disease affects the stability of the system and how awareness campaigns and treatment can maintain the stability of the system. This study highlights the importance of awareness campaigns in controlling infectious diseases and demonstrates the effectiveness of optimal control theory in achieving disease control with minimal cost. Full article

(This article belongs to the Special Issue Control Theory and Its Application in Mathematical Biology)

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21 pages, 23038 KiB

Open AccessArticle

Noninteger Dimension of Seasonal Land Surface Temperature (LST)

by Sepideh Azizi and Tahmineh Azizi

Axioms 2023, 12(6), 607; https://doi.org/10.3390/axioms12060607 - 19 Jun 2023

Cited by 2 | Viewed by 997

Abstract

During the few last years, climate change, including global warming, which is attributed to human activities, and its long-term adverse effects on the planet’s functions have been identified as the most challenging discussion topics and have provoked significant concern and effort to find [...] Read more.

During the few last years, climate change, including global warming, which is attributed to human activities, and its long-term adverse effects on the planet’s functions have been identified as the most challenging discussion topics and have provoked significant concern and effort to find possible solutions. Since the warmth arising from the Earth’s landscapes affects the world’s weather and climate patterns, we decided to study the changes in Land Surface Temperature (LST) patterns in different seasons through nonlinear methods. Here, we particularly wanted to estimate the noninteger dimension and fractal structure of the Land Surface Temperature. For this study, the LST data were obtained during the daytime by a Moderate Resolution Imaging Spectroradiometer (MODIS) on NASA’s Terra satellite. Depending on the time of the year data were collected, temperatures changed in different ranges. Since equatorial regions remain warm, and Antarctica and Greenland remain cold, and also because altitude affects temperature, we selected Riley County in the US state of Kansas, which does not belong to any of these location types, and we observed the seasonal changes in temperature in this county. According to our fractal analysis, the fractal dimension may provide a complexity index to characterize different LST datasets. The multifractal analysis confirmed that the LST data may define a self-organizing system that produces fractal patterns in the structure of data. Thus, the LST data may not only have a wide range of fractal dimensions, but also they are fractal. The results of the present study show that the Land Surface Temperature (LST) belongs to the class of fractal processes with a noninteger dimension. Moreover, self-organized behavior governing the structure of LST data may provide an underlying principle that might be a general outcome of human activities and may shape the Earth’s surface temperature. We explicitly acknowledge the important role of fractal geometry when analyzing and tracing settlement patterns and urbanization dynamics at various scales toward purposeful planning in the development of human settlement patterns. Full article

(This article belongs to the Section Geometry and Topology)

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11 pages, 275 KiB

Open AccessArticle

Applications of Fuzzy Semiprimary Ideals under Group Action

by Asma Ali, Amal S. Alali and Arshad Zishan

Axioms 2023, 12(6), 606; https://doi.org/10.3390/axioms12060606 - 19 Jun 2023

Cited by 1 | Viewed by 839

Abstract

Group actions are a valuable tool for investigating the symmetry and automorphism features of rings. The concept of fuzzy ideals in rings has been expanded with the introduction of fuzzy primary, weak primary, and semiprimary ideals. This paper explores the existence of fuzzy [...] Read more.

Group actions are a valuable tool for investigating the symmetry and automorphism features of rings. The concept of fuzzy ideals in rings has been expanded with the introduction of fuzzy primary, weak primary, and semiprimary ideals. This paper explores the existence of fuzzy ideals that are semiprimary but neither weak primary nor primary. Furthermore, it defines a group action on a fuzzy ideal and examines the properties of fuzzy ideals and their level cuts under this group action. In fact, it aims to investigate the relationship between fuzzy semiprimary ideals and the radical of fuzzy ideals under group action. Additionally, it includes the results related to the radical of fuzzy ideals and fuzzy

G

-semiprimary ideals. Moreover, the preservation of the image and inverse image of a fuzzy

G

-semiprimary ideal of a ring

R

under certain conditions is also studied. It delves into the algebraic nature of fuzzy ideals and the radical under

G

-homomorphism of fuzzy ideals. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

13 pages, 304 KiB

Open AccessArticle

A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators

by Sofia Giuffrè

Axioms 2023, 12(6), 605; https://doi.org/10.3390/axioms12060605 - 18 Jun 2023

Viewed by 794

Abstract

The purpose of the research is the study of a nonconstant gradient constrained problem for nonlinear monotone operators. In particular, we study a stationary variational inequality, defined by a strongly monotone operator, in a convex set of gradient-type constraints. We investigate the relationship [...] Read more.

The purpose of the research is the study of a nonconstant gradient constrained problem for nonlinear monotone operators. In particular, we study a stationary variational inequality, defined by a strongly monotone operator, in a convex set of gradient-type constraints. We investigate the relationship between the nonconstant gradient constrained problem and a suitable double obstacle problem, where the obstacles are the viscosity solutions to a Hamilton–Jacobi equation, and we show the equivalence between the two variational problems. To obtain the equivalence, we prove that a suitable constraint qualification condition, Assumption S, is fulfilled at the solution of the double obstacle problem. It allows us to apply a strong duality theory, holding under Assumption S. Then, we also provide the proof of existence of Lagrange multipliers. The elements in question can be not only functions in

L^{2}

, but also measures. Full article

(This article belongs to the Special Issue Optimization Models and Applications)

16 pages, 960 KiB

Open AccessArticle

Aristotelian Fragments and Subdiagrams for the Boolean Algebra

B

₅

by Koen Roelandt and Hans Smessaert

Axioms 2023, 12(6), 604; https://doi.org/10.3390/axioms12060604 - 18 Jun 2023

Viewed by 700

Abstract

On a descriptive level, this paper presents a number of logical fragments which require the Boolean algebra

B_{5}

, i.e., bitstrings of length five, for their semantic analysis. Two categories from the realm of natural language quantification are considered, namely, proportional quantification [...] Read more.

On a descriptive level, this paper presents a number of logical fragments which require the Boolean algebra

B_{5}

, i.e., bitstrings of length five, for their semantic analysis. Two categories from the realm of natural language quantification are considered, namely, proportional quantification with fractions and percentages—as in two thirds/66 percent of the children are asleep—and normative quantification—as in not enough/too many children are asleep. On a more theoretical level, we study two distinct Aristotelian subdiagrams in

B_{5}

, which are the result of moving from

B_{5}

to

B_{4}

either by collapsing bit positions or by deleting bit positions. These two operations are also argued to shed a new light on earlier results from Logical Geometry, in which the collapsing or deletion of bit positions triggers a shift from

B_{4}

to

B_{3}

. Full article

(This article belongs to the Special Issue Modal Logic and Logical Geometry)

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18 pages, 905 KiB

Open AccessArticle

RPCGB Method for Large-Scale Global Optimization Problems

by Abderrahmane Ettahiri and Abdelkrim El Mouatasim

Axioms 2023, 12(6), 603; https://doi.org/10.3390/axioms12060603 - 18 Jun 2023

Viewed by 791

Abstract

In this paper, we propose a new approach for optimizing a large-scale non-convex differentiable function subject to linear equality constraints. The proposed method, RPCGB (random perturbation of the conditional gradient method with bisection algorithm), computes a search direction by the conditional gradient, and [...] Read more.

In this paper, we propose a new approach for optimizing a large-scale non-convex differentiable function subject to linear equality constraints. The proposed method, RPCGB (random perturbation of the conditional gradient method with bisection algorithm), computes a search direction by the conditional gradient, and an optimal line search is found by a bisection algorithm, which results in a decrease of the cost function. The RPCGB method is designed to guarantee global convergence of the algorithm. An implementation and testing of the method are given, with numerical results of large-scale problems that demonstrate its efficiency. Full article

(This article belongs to the Special Issue Advances in Combinatorial Optimization and Discrete Mathematics with Applications)

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26 pages, 1126 KiB

Open AccessArticle

Local Refinement and Adaptive Strategy for a System of Free Boundary Power Options with High Order Compact Differencing

by Chinonso Nwankwo and Weizhong Dai

Axioms 2023, 12(6), 602; https://doi.org/10.3390/axioms12060602 - 17 Jun 2023

Viewed by 753

Abstract

In this research, we propose fourth-order non-uniform Hermitian differencing with a fifth-order adaptive time integration method for pricing system of free boundary exotic power put options consisting of the option value, delta sensitivity, and gamma. The main objective for implementing the above scheme [...] Read more.

In this research, we propose fourth-order non-uniform Hermitian differencing with a fifth-order adaptive time integration method for pricing system of free boundary exotic power put options consisting of the option value, delta sensitivity, and gamma. The main objective for implementing the above scheme is to carefully account for the irregularity in the locality of the left corner point after fixing the free boundary. Specifically and mainly, we stretch the performance of our proposed method threefold. First, we exploit the non-uniform fourth-order Hermitian scheme to locally concentrate space grid points arbitrarily close to the left boundary. Secondly, we further leverage the adaptive nature of the embedded time integration method, which allows optimal selection of a time step based on the space grid point distribution and regional variation. Thirdly, we introduce a fourth-order combined Hermitian scheme, which requires fewer grid points for computing the near boundary point of the delta sensitivity and gamma. Another novelty is how we approximate the optimal exercise boundary and its derivative using a fifth-order Robin boundary scheme and fourth-order combined Hermitian scheme. Our proposed method consistently achieves reasonable accuracy with very coarse grids and little runtime across the numerical experiments. We further compare the results with existing methods and the ones we obtained from the uniform space grid. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics)

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26 pages, 13370 KiB

Open AccessArticle

Statistical Analysis of Type-II Generalized Progressively Hybrid Alpha-PIE Censored Data and Applications in Electronic Tubes and Vinyl Chloride

by Ahmed Elshahhat, Osama E. Abo-Kasem and Heba S. Mohammed

Axioms 2023, 12(6), 601; https://doi.org/10.3390/axioms12060601 - 16 Jun 2023

Cited by 2 | Viewed by 819

Abstract

A new Type-II generalized progressively hybrid censoring strategy, in which the experiment is ensured to stop at a specified time, is explored when the lifetime model of the test subjects follows a two-parameter alpha-power inverted exponential (Alpha-PIE) distribution. Alpha-PIE’s parameters and reliability indices, [...] Read more.

A new Type-II generalized progressively hybrid censoring strategy, in which the experiment is ensured to stop at a specified time, is explored when the lifetime model of the test subjects follows a two-parameter alpha-power inverted exponential (Alpha-PIE) distribution. Alpha-PIE’s parameters and reliability indices, such as reliability and hazard rate functions, are estimated via maximum likelihood and Bayes estimation methodologies in the presence of the proposed censored data. The estimated confidence intervals of the unknown quantities are created using the normal approximation of the acquired classical estimators. The Bayesian estimators are also produced using independent gamma density priors under symmetrical (squared-error) loss. The Bayes’ estimators and their associated highest posterior density intervals cannot be calculated theoretically since the joint likelihood function is derived in a complicated form, but they can potentially be assessed using Monte Carlo Markov-chain algorithms. We next go through four optimality criteria for identifying the best progressive design. The effectiveness of the suggested estimation procedures is assessed using Monte Carlo comparisons, and certain recommendations are offered. Ultimately, two different applications, one focused on the failure times of electronic tubes and the other on vinyl chloride, are analyzed to illustrate the effectiveness of the proposed techniques that may be employed in real-world scenarios. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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18 pages, 339 KiB

Open AccessArticle

New Applications of Faber Polynomials and q-Fractional Calculus for a New Subclass of m-Fold Symmetric bi-Close-to-Convex Functions

by Mohammad Faisal Khan, Suha B. Al-Shaikh, Ahmad A. Abubaker and Khaled Matarneh

Axioms 2023, 12(6), 600; https://doi.org/10.3390/axioms12060600 - 16 Jun 2023

Cited by 1 | Viewed by 697

Abstract

Using the concepts of q-fractional calculus operator theory, we first define a

(λ, q)

-differintegral operator, and we then use m-fold symmetric functions to discover a new family of bi-close-to-convex functions. First, we estimate the general Taylor–Maclaurin coefficient [...] Read more.

Using the concepts of q-fractional calculus operator theory, we first define a

(λ, q)

-differintegral operator, and we then use m-fold symmetric functions to discover a new family of bi-close-to-convex functions. First, we estimate the general Taylor–Maclaurin coefficient bounds for a newly established class using the Faber polynomial expansion method. In addition, the Faber polynomial method is used to examine the Fekete–Szegö problem and the unpredictable behavior of the initial coefficient bounds of the functions that belong to the newly established class of m-fold symmetric bi-close-to-convex functions. Our key results are both novel and consistent with prior research, so we highlight a few of their important corollaries for a comparison. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

23 pages, 17240 KiB

Open AccessArticle

The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

by Xiaoming Wang, Rimsha Ansar, Muhammad Abbas, Farah Aini Abdullah and Khadijah M. Abualnaja

Axioms 2023, 12(6), 599; https://doi.org/10.3390/axioms12060599 - 16 Jun 2023

Cited by 3 | Viewed by 830

Abstract

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the [...] Read more.

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the

β

-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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18 pages, 1109 KiB

Open AccessArticle

The Dynamics of a General Model of the Nonlinear Difference Equation and Its Applications

by Osama Moaaz and Aseel A. Altuwaijri

Axioms 2023, 12(6), 598; https://doi.org/10.3390/axioms12060598 - 16 Jun 2023

Cited by 1 | Viewed by 838

Abstract

This article investigates the qualitative properties of solutions to a general difference equation. Studying the properties of solutions to general difference equations greatly contributes to the development of theoretical methods and provides many pieces of information that may help to understand the behavior [...] Read more.

This article investigates the qualitative properties of solutions to a general difference equation. Studying the properties of solutions to general difference equations greatly contributes to the development of theoretical methods and provides many pieces of information that may help to understand the behavior of solutions of some special models. We present the sufficient and necessary conditions for the existence of prime period-two and -three solutions. We also obtain a complete perception of the local stability of the studied equation. Then, we investigate the boundedness and global stability of the solutions. Finally, we support the validity of the results by applying them to some special cases, as well as numerically simulating the solutions. Full article

(This article belongs to the Special Issue Advances in Difference Equations)

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10 pages, 294 KiB

Open AccessArticle

The Recursive Structures of Manin Symbols over

Q

, Cusps and Elliptic Points on X₀ (N)

by Sanmin Wang

Axioms 2023, 12(6), 597; https://doi.org/10.3390/axioms12060597 - 16 Jun 2023

Viewed by 680

Abstract

Firstly, we present a more explicit formulation of the complete system

D (N)

of representatives of Manin’s symbols over

Q

, which was initially given by Shimura. Then, we establish a bijection between [...] Read more.

Firstly, we present a more explicit formulation of the complete system

D (N)

of representatives of Manin’s symbols over

Q

, which was initially given by Shimura. Then, we establish a bijection between

D (M) \times D (N)

and

D (M N)

for

(M, N) = 1

, which reveals a recursive structure between Manin’s symbols of different levels. Based on Manin’s complete system

Π (N)

of representatives of cusps on

X_{0} (N)

and Cremona’s characterization of the equivalence between cusps, we establish a bijection between a subset

C (N)

of

D (N)

and

Π (N)

, and then establish a bijection between

C (M) \times C (N)

and

C (M N)

for

(M, N) = 1

. We also provide a recursive structure for elliptical points on

X_{0} (N)

. Based on these recursive structures, we obtain recursive algorithms for constructing Manin symbols over

Q

, cusps, and elliptical points on

X_{0} (N)

. This may give rise to more efficient algorithms for modular elliptic curves. As direct corollaries of these recursive structures, we present a recursive version of the genus formula and prove constructively formulas of the numbers of

D (N)

, cusps, and elliptic points on

X_{0} (N)

. Full article

(This article belongs to the Special Issue Discrete Curvatures and Laplacians)

10 pages, 291 KiB

Open AccessArticle

Estimates for Generalized Parabolic Marcinkiewicz Integrals with Rough Kernels on Product Domains

by Mohammed Ali and Hussain Al-Qassem

Axioms 2023, 12(6), 596; https://doi.org/10.3390/axioms12060596 - 15 Jun 2023

Viewed by 630

Abstract

We prove

L^{p}

estimates of a class of generalized Marcinkiewicz integral operators with mixed homogeneity on product domains. By using these estimates along with an extrapolation argument, we obtain the boundedness of our operators under very weak conditions on the kernel functions. [...] Read more.

We prove

L^{p}

estimates of a class of generalized Marcinkiewicz integral operators with mixed homogeneity on product domains. By using these estimates along with an extrapolation argument, we obtain the boundedness of our operators under very weak conditions on the kernel functions. Our results in this paper improve and extend several known results on both generalized Marcinkiewicz integrals and parabolic Marcinkiewicz integrals on product domains. Full article

19 pages, 648 KiB

Open AccessArticle

On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions

by Sergey Sekatskii

Axioms 2023, 12(6), 595; https://doi.org/10.3390/axioms12060595 - 15 Jun 2023

Cited by 1 | Viewed by 715

Abstract

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties [...] Read more.

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions sn(z, k) and others understood as functions of the index k. The consideration of the case of the derivative of the Weierstrass rho-function,

℘_{z} (z,_{} τ)

, leads to quite easy and transparent proof of numerous equalities between the sums over inverse powers of the lattice points

m + n τ

and “demi-lattice” points

m + 1 / 2 + n τ

,

m + (n + 1 / 2) τ

,

m + 1 / 2 + (n + 1 / 2) τ

. We also prove theorems showing that, in most cases, fundamental parallelograms contain exactly one simple zero for the first derivative

θ_{1}' (z | τ)

of the elliptic theta-function and the Weierstrass

ζ

-function, and that far from the origin of coordinates such zeroes of the

ζ

-function tend to the positions of the simple poles of this function. Full article

(This article belongs to the Section Mathematical Analysis)

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18 pages, 16907 KiB

Open AccessArticle

Settlement of a Foundation on an Unsaturated Sandy Base Taking Vibrocreep into Account

by Armen Z. Ter-Martirosyan, Alexander N. Shebunyaev and Evgeny S. Sobolev

Axioms 2023, 12(6), 594; https://doi.org/10.3390/axioms12060594 - 15 Jun 2023

Cited by 1 | Viewed by 1255

Abstract

Dynamic loading causes (1) a substantial change in the strength and deformation parameters of sandy soil and (2) excessive viscoplastic deformation. The goal of this study is to create a new analytical solution to the problem of the settlement of (1) the foundation [...] Read more.

Dynamic loading causes (1) a substantial change in the strength and deformation parameters of sandy soil and (2) excessive viscoplastic deformation. The goal of this study is to create a new analytical solution to the problem of the settlement of (1) the foundation that is the source of dynamic loading, and (2) a nearby foundation, taking into account the rheological properties of sandy soil subjected to vibration, given that these rheological properties depend on shear stresses. The proposed solution allows the progress of deformation over time to be described. The present paper states and provides an analytical solution for the problem of evaluating the settlement of a single foundation that transmits static and dynamic harmonic pressure to the base. The authors also analyze the settlement of another foundation located at some distance from the transmitting foundation. The second foundation transmits static pressure to the base. The dependence of the viscosity coefficient on the shear stress intensity and vibration intensity, as well as the vibrocreep decay over time, are based on the exponential and homographic dependencies previously identified by two of the authors (A.Z. Ter-Martirosyan and E.S. Sobolev). The solution to the problem is obtained by numerical integration in the Mathcad program of an analytical expression for nonlinear viscoplastic deformations. As a result of the research, the authors have found that the dynamic viscoplastic component makes the greatest contribution to foundation settlement. The settlement of the transmitting foundation increases along with increasing static and dynamic pressure transmitted to the base. The settlement of the nearby foundation increases when the pressure increases under the foundation, but it reduces when static pressure from the transmitting foundation, the depth of the foundation, and the distance between the foundations increase. General analytical dependencies obtained by the authors comply with the results of laboratory and field experiments performed by other researchers. These dependencies can be used to predict the settlement of foundations in whose unsaturated sandy bases mechanical vibrations propagate. Full article

(This article belongs to the Special Issue Applied Numerical Analysis in Civil Engineering)

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16 pages, 348 KiB

Open AccessArticle

An Efficient Non-Standard Numerical Scheme Coupled with a Compact Finite Difference Method to Solve the One-Dimensional Burgers’ Equation

by Komalpreet Kaur and Gurjinder Singh

Axioms 2023, 12(6), 593; https://doi.org/10.3390/axioms12060593 - 15 Jun 2023

Cited by 1 | Viewed by 940

Abstract

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of [...] Read more.

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of initial-boundary value partial differential equations (PDEs). Further, a method of this family is considered as a special case and coupled with a fourth-order compact finite difference resulting in a combined numerical scheme to solve initial-boundary value PDEs. The combined scheme has first-order accuracy in time and fourth-order accuracy in space. Some basic characteristics of the scheme are analysed and a section concerning the numerical experiments is presented demonstrating the good performance of the combined numerical scheme. Full article

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13 pages, 5485 KiB

Open AccessArticle

Study on the Nonlinear Dynamics of the (3+1)-Dimensional Jimbo-Miwa Equation in Plasma Physics

by Peng Xu, Bing-Qi Zhang, Huan Huang and Kang-Jia Wang

Axioms 2023, 12(6), 592; https://doi.org/10.3390/axioms12060592 - 15 Jun 2023

Viewed by 840

Abstract

The Jimbo-Miwa equation (JME) that describes certain interesting (3+1)-dimensional waves in plasma physics is studied in this work. The Hirota bilinear equation is developed via the Cole-Hopf transform. Then, the symbolic computation, together with the ansatz function schemes, are utilized to seek exact [...] Read more.

The Jimbo-Miwa equation (JME) that describes certain interesting (3+1)-dimensional waves in plasma physics is studied in this work. The Hirota bilinear equation is developed via the Cole-Hopf transform. Then, the symbolic computation, together with the ansatz function schemes, are utilized to seek exact solutions. Some new solutions, such as the multi-wave complexiton solution (MWCS), multi-wave solution (MWS) and periodic lump solution (PLS), are successfully constructed. Additionally, different types of travelling wave solutions (TWS), including the dark, bright-dark and singular periodic wave solutions, are disclosed by employing the sub-equation method. Finally, the physical characteristics and interaction behaviors of the extracted solutions are depicted graphically by assigning appropriate parameters. The obtained outcomes in this paper are more general and newer. Additionally, they reveal that the used methods are concise, direct, and can be employed to study other partial differential equations (PDEs) in physics. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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18 pages, 1038 KiB

Open AccessArticle

Dynamics of a Fractional-Order COVID-19 Epidemic Model with Quarantine and Standard Incidence Rate

by Trisilowati, Isnani Darti, Raqqasyi Rahmatullah Musafir, Maya Rayungsari and Agus Suryanto

Axioms 2023, 12(6), 591; https://doi.org/10.3390/axioms12060591 - 15 Jun 2023

Cited by 3 | Viewed by 1157

Abstract

In this paper, we propose a fractional-order COVID-19 epidemic model with a quarantine and standard incidence rate using the Caputo fractional-order derivative. The model consists of six classes: susceptible (S), exposed (E), infected (I), quarantined (Q [...] Read more.

In this paper, we propose a fractional-order COVID-19 epidemic model with a quarantine and standard incidence rate using the Caputo fractional-order derivative. The model consists of six classes: susceptible (S), exposed (E), infected (I), quarantined (Q), recovered (R), and deceased (M). In our proposed model, we simultaneously consider the recovery rate and quarantine rate of infected individuals, which has not been considered in other fractional-order COVID-19 epidemic models. Furthermore, we consider the standard incidence rate in the model. For our proposed model, we prove the existence, uniqueness, non-negativity, and boundedness of the solution. The model has two equilibrium points: disease-free equilibrium and endemic equilibrium. Implementing the spectral radius of the next-generation matrix, we obtain the basic reproduction number (

R_{0}

). The disease-free equilibrium always exists and is locally and globally asymptotically stable only if

R_{0} < 1

. On the other hand, endemic equilibrium exists and is globally asymptotically stable if

R_{0} > 1

. Our numerical simulation confirms the stability properties of the equilibrium. The smaller the order of the derivative, the slower the convergence of the solution of the model. Both the recovery rate and quarantine rate of the infected class are important parameters determining the stability of the equilibrium point. Based on parameter estimation from COVID-19 data in Indonesia, the fractional-order model has better performance than the first-order model for both the calibration and 20-day forecasting of confirmed daily active cases of COVID-19. Full article

(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)

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30 pages, 4187 KiB

Open AccessArticle

Knowable Moments in Stochastics: Knowing Their Advantages

by Demetris Koutsoyiannis

Axioms 2023, 12(6), 590; https://doi.org/10.3390/axioms12060590 - 14 Jun 2023

Cited by 2 | Viewed by 866

Abstract

Knowable moments, abbreviated as K-moments, are redefined as expectations of maxima or minima of a number of stochastic variables that are a sample of the variable of interest. The new definition enables applicability of the concept to any type of variable, continuous or [...] Read more.

Knowable moments, abbreviated as K-moments, are redefined as expectations of maxima or minima of a number of stochastic variables that are a sample of the variable of interest. The new definition enables applicability of the concept to any type of variable, continuous or discrete, and generalization for transformations thereof. While K-moments share some characteristics with classical and other moments, as well as with order statistics, they also have some unique features, which make them useful in relevant applications. These include the fact that they are knowable, i.e., reliably estimated from a sample for high orders. Moreover, unlike other moment types, K-moment values can be assigned values of distribution function by making optimal use of the entire dataset. In addition, K-moments offer the unique advantage of considering the estimation bias when the data are not an independent sample but a time series from a process with dependence. Both for samples and time series, the K-moment concept offers a strategy of model fitting, including its visualization, that is not shared with other methods. This enables utilization of the highest possible moment orders, which are particularly useful in modelling extremes that are closely associated with high-order moments. Full article

(This article belongs to the Section Mathematical Analysis)

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18 pages, 330 KiB

Open AccessArticle

Some Construction Methods for Pseudo-Overlaps and Pseudo-Groupings and Their Application in Group Decision Making

by Diego García-Zamora, Rui Paiva, Anderson Cruz, Javier Fernandez and Humberto Bustince

Axioms 2023, 12(6), 589; https://doi.org/10.3390/axioms12060589 - 14 Jun 2023

Viewed by 695

Abstract

In many real-world scenarios, the importance of different factors may vary, making commutativity an unreasonable assumption for aggregation functions, such as overlaps or groupings. To address this issue, researchers have introduced pseudo-overlaps and pseudo-groupings as their corresponding non-commutative generalizations. In this paper, we [...] Read more.

In many real-world scenarios, the importance of different factors may vary, making commutativity an unreasonable assumption for aggregation functions, such as overlaps or groupings. To address this issue, researchers have introduced pseudo-overlaps and pseudo-groupings as their corresponding non-commutative generalizations. In this paper, we explore various construction methods for obtaining pseudo-overlaps and pseudo-groupings using overlaps, groupings, fuzzy negations, convex sums, and Riemannian integration. We then show the applicability of these construction methods in a multi-criteria group decision-making problem, where the importance of both the considered criteria and the experts vary. Our results highlight the usefulness of pseudo-overlaps and pseudo-groupings as a non-commutative alternative to overlaps and groupings. Full article

(This article belongs to the Special Issue Fuzzy Systems and Decision Making Theory)

17 pages, 338 KiB

Open AccessArticle

Mittag-Leffler-Type Stability of BAM Neural Networks Modeled by the Generalized Proportional Riemann–Liouville Fractional Derivative

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

Axioms 2023, 12(6), 588; https://doi.org/10.3390/axioms12060588 - 14 Jun 2023

Cited by 6 | Viewed by 728

Abstract

The main goal of the paper is to use a generalized proportional Riemann–Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved, such as the fractional derivative [...] Read more.

The main goal of the paper is to use a generalized proportional Riemann–Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved, such as the fractional derivative of a squared function. Additionally, some comparison results for GPRLFD are provided. Two types of equilibrium of the BAM model with GPRLFD are defined. In connection with the applied fractional derivative and its singularity at the initial time, the Mittag-Leffler exponential stability in time of the equilibrium is introduced and studied. An example is given, illustrating the meaning of the equilibrium as well as its stability properties. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

26 pages, 5158 KiB

Open AccessArticle

Steam Gasification in a Fluidized Bed with Various Methods of In-Core Coal Treatment

by Nikolay Abaimov, Alexander Ryzhkov, Vladimir Tuponogov, Leonid Simbiriatin, Alexey Dubinin, Lu Ding and Sergey Alekseenko

Axioms 2023, 12(6), 587; https://doi.org/10.3390/axioms12060587 - 13 Jun 2023

Cited by 2 | Viewed by 1266

Abstract

The aim of this work is to study coal steam gasification with various methods of coal in-core treatment in FB using a newly developed thermodynamic calculation method. A calculational study of subbituminous coal steam non-catalytic gasification was carried out using four different methods [...] Read more.

The aim of this work is to study coal steam gasification with various methods of coal in-core treatment in FB using a newly developed thermodynamic calculation method. A calculational study of subbituminous coal steam non-catalytic gasification was carried out using four different methods of coal in-core treatment in single-vessel multisectional fluidized-bed gasifiers. A semi-empirical model based on the entropy maximization thermodynamic method and “restricted equilibria” based on previously obtained experimental data has been developed. Based on thermodynamic calculations, the effect of the leading thermochemical processes and operating parameters of the fluidized bed (temperature, fluidization number, steam/coal ratio feed rate) was revealed. New information was obtained regarding the composition of char and syngas at the gasifier outlet, the syngas heating value, and the cold gas efficiency of the steam gasification of Borodinskiy subbituminous coal char. The results indicate the possibility of significantly accelerating and improving non-catalytic steam gasification in fluidized bed gasifiers through the appropriate organization of in-core coal treatment. Based on the results obtained, the following recommendation is made—when designing multi-section and multi-vessel steam-blown gasifiers, the ratio of residence times should be set in favor of increasing the coal residence time in the steam-blown carbonization zone. Structurally, this can be achieved by increasing the volume and/or area of the steam-blown carbonization section (vessel). Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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30 pages, 8777 KiB

Open AccessArticle

Modified Flower Pollination Optimization Based Design of Interval Type-2 Fuzzy PID Controller for Rotary Inverted Pendulum System

by Mukhtar Fatihu Hamza

Axioms 2023, 12(6), 586; https://doi.org/10.3390/axioms12060586 - 13 Jun 2023

Cited by 5 | Viewed by 965

Abstract

The Type 2 Fuzzy Logic System (T2FLS) is an enhanced form of the classical Fuzzy Logic System (FLS). The T2FLS based control technics demonstrated a lot of improvements for the past few decades. This is based on the advantage of its membership function [...] Read more.

The Type 2 Fuzzy Logic System (T2FLS) is an enhanced form of the classical Fuzzy Logic System (FLS). The T2FLS based control technics demonstrated a lot of improvements for the past few decades. This is based on the advantage of its membership function (MF). Many experimental studies indicated the superiority of Type 2 Fuzzy Logic Controller (T2FLC) over the ordinary Type 1 Fuzzy Logic Controller (T1FLC), particularly in the event of non-linearities and complex uncertainties. However, the organized design method of T2FLCs is still an interesting problem in the control engineering community. This is due to the difficulties in computing the parameters associated it. A novel application of the Modified Flower Pollination (MFP) optimization algorithm in the design of T2FL is presented. The optimized Cascade Interval Type 2 Fuzzy PID Controller (IT2FPIDC) structure is proposed in this study. The best values of the parameters of the antecedent MFs and the PID gains of IT2FPIDC are found using the MFP algorithm. The MFP optimization technique was used because of its lower computational effort and high convergence speed, in view of the higher number of variables to be optimized in cascaded IT2FPIDC. The MFP-based Type-1 Fuzzy Proportional Integral Derivative Controller (T1FPIDC) is compared with the proposed MFP-based cascade-optimized IT2FPIDC. The rotary inverted pendulum (RIP) which is a non-minimum phase, non-linear, and unstable system is employed as a benchmark for testing the proposed controller. Balance and trajectory-tracking controls of the RIP are considered. Furthermore, the disturbance rejection ability of the proposed controller is analysed. The presented control methos is evaluated on the RIP manufactured by Quanser over many simulations and real-world experiments. The performance characteristics considered are steady state error

{(E}_{s s})

, settling time

(t_{s})

, maximum overshoot

(M_{p})

and rise time

(t_{r})

. The improvement of the effectiveness and robustness proposed controller in the presence of load disturbance, noise effects and parameter variation is shown. Full article

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19 pages, 352 KiB

Open AccessArticle

Faber Polynomial Coefficient Estimates for Bi-Close-to-Convex Functions Defined by the q-Fractional Derivative

by Hari Mohan Srivastava, Isra Al-Shbeil, Qin Xin, Fairouz Tchier, Shahid Khan and Sarfraz Nawaz Malik

Axioms 2023, 12(6), 585; https://doi.org/10.3390/axioms12060585 - 13 Jun 2023

Cited by 6 | Viewed by 978

Abstract

By utilizing the concept of the q-fractional derivative operator and bi-close-to-convex functions, we define a new subclass of

A

, where the class

A

contains normalized analytic functions in the open unit disk

E

and is invariant or symmetric under rotation. First, [...] Read more.

By utilizing the concept of the q-fractional derivative operator and bi-close-to-convex functions, we define a new subclass of

A

, where the class

A

contains normalized analytic functions in the open unit disk

E

and is invariant or symmetric under rotation. First, using the Faber polynomial expansion (FPE) technique, we determine the lth coefficient bound for the functions contained within this class. We provide a further explanation for the first few coefficients of bi-close-to-convex functions defined by the q-fractional derivative. We also emphasize upon a few well-known outcomes of the major findings in this article. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

27 pages, 455 KiB

Open AccessArticle

Overlapping of Lévai’s and Milson’s e-Tangent-Polynomial Potentials along Symmetric Curves

by Gregory Natanson

Axioms 2023, 12(6), 584; https://doi.org/10.3390/axioms12060584 - 12 Jun 2023

Cited by 1 | Viewed by 709

Abstract

The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm–Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the [...] Read more.

The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm–Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators of the given rational density function as ‘tangent polynomial’ (TP) and thereby term the aforementioned potentials as ‘e-TP’. Special attention is given to the overlap between the two potentials along symmetric curves which represent two different rational forms of the Ginocchio potential exactly quantized via Gegenbauer and Masjed-Jamei polynomials, respectively. Our analysis reveals that the actual interconnection between Lévai’s parameters for these two rational realizations of the Ginocchio potential is much more complicated than one could expect based on the striking resemblance between two quartic equations derived by Lévai for ‘averaged’ Jacobi indexes. Full article

(This article belongs to the Section Mathematical Physics)

15 pages, 1777 KiB

Open AccessArticle

Fault Detection and Identification with Kernel Principal Component Analysis and Long Short-Term Memory Artificial Neural Network Combined Method

by Nahid Jafari and António M. Lopes

Axioms 2023, 12(6), 583; https://doi.org/10.3390/axioms12060583 - 12 Jun 2023

Viewed by 1247

Abstract

A new fault detection and identification approach is proposed. The kernel principal component analysis (KPCA) is first applied to the data for reducing dimensionality, and the occurrence of faults is determined by means of two statistical indices, T² and Q. The [...] Read more.

A new fault detection and identification approach is proposed. The kernel principal component analysis (KPCA) is first applied to the data for reducing dimensionality, and the occurrence of faults is determined by means of two statistical indices, T² and Q. The K-means clustering algorithm is then adopted to analyze the data and perform clustering, according to the type of fault. Finally, the type of fault is determined using a long short-term memory (LSTM) neural network. The performance of the proposed technique is compared with the principal component analysis (PCA) method in early detecting malfunctions on a continuous stirred tank reactor (CSTR) system. Up to 10 sensor faults and other system degradation conditions are considered. The performance of the LSTM neural network is compared with three other machine learning techniques, namely the support vector machine (SVM), K-nearest neighbors (KNN) algorithm, and decision trees, in determining the type of fault. The results indicate the superior performance of the suggested methodology in both early fault detection and fault identification. Full article

(This article belongs to the Special Issue Control Theory and Control Systems: Algorithms and Methods)

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17 pages, 741 KiB

Open AccessArticle

Fuzzy vs. Traditional Reliability Model for Inverse Weibull Distribution

by Eslam Hussam, Mohamed A. Sabry, M. M. Abd El-Raouf and Ehab M. Almetwally

Axioms 2023, 12(6), 582; https://doi.org/10.3390/axioms12060582 - 12 Jun 2023

Cited by 2 | Viewed by 846

Abstract

In this paper, fuzzy stress strengths

R_{F} = P (Y ≺ X)

and traditional stress strengths

R = P (Y < X)

are considered and compared when X and Y are independently inverse Weibull random variables. When axiomatic [...] Read more.

In this paper, fuzzy stress strengths

R_{F} = P (Y ≺ X)

and traditional stress strengths

R = P (Y < X)

are considered and compared when X and Y are independently inverse Weibull random variables. When axiomatic fuzzy set theory is taken into account in the stress–strength inference, it enables the generation of more precise studies on the underlying systems. We discuss estimating both conventional and fuzzy models of stress strength utilizing a maximum product of spacing, maximum likelihood, and Bayesian approaches. Simulations based on the Markov Chain Monte Carlo method are used to produce various estimators of conventional and fuzzy dependability of stress strength for the inverse Weibull model. To generate both conventional and fuzzy models of dependability, we use the Metropolis–Hastings method while performing Bayesian estimation. In conclusion, we will examine a scenario taken from actual life and apply a real-world data application to validate the accuracy of the provided estimators. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications)

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14 pages, 518 KiB

Open AccessArticle

Some Stability Results and Existence of Solutions for a Backward Differential Equation with Time Advance via ζ—Caputo Fractional Derivative

by Abdellatif Ben Makhlouf, Lassaad Mchiri and Mohamed Rhaima

Axioms 2023, 12(6), 581; https://doi.org/10.3390/axioms12060581 - 12 Jun 2023

Viewed by 729

Abstract

In this paper, using a fixed point method, we proved the existence and uniqueness of solutions for a backward differential equation with time advance via

ζ -

Caputo fractional derivative. Furthermore, the Ulam–Hyers–Rassias and the Ulam–Hyers stabilities of the backward differential equation with [...] Read more.

In this paper, using a fixed point method, we proved the existence and uniqueness of solutions for a backward differential equation with time advance via

ζ -

Caputo fractional derivative. Furthermore, the Ulam–Hyers–Rassias and the Ulam–Hyers stabilities of the backward differential equation with time advance via

ζ -

Caputo fractional derivative are investigated. Finally, some experiments are given to illustrate the theoretical results. Full article

(This article belongs to the Section Mathematical Analysis)

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Axioms, Volume 12, Issue 6 (June 2023) – 106 articles

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