https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/issue/feedThai Journal of Mathematics2024-03-31T12:23:09+00:00Prof.Dr. Suthep Suantaithaijmath@cmu.ac.thOpen Journal Systems<p><strong>Thai Journal of Mathematics</strong><br />Thai Journal of Mathematics (TJM) is a peer-reviewed, open access international journal publishing original research works of high standard in all areas of pure and applied mathematics.</p> <p><br /><strong>Publication Frequency</strong><br />From 2020 onwards, TJM publishes one volume per year which consists of four regular issues (March, June, September, and December). All manuscripts are refereed under the same standards as those used by the finest-quality printed mathematical journals. Accepted papers will be published online in their final forms using the TJM template and will be published in four issues annually.</p>https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1595Novel Exact Traveling Wave Solutions for the (2 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation with Atangana's Space and Time Beta-Derivatives via the Sardar Subequation Method2024-03-30T14:59:03+00:00Chanidaporn Pleumpreedapornchanidaporn.p@rbru.ac.thSongkran Pleumpreedapornsongkran.p@rbru.ac.thElvin J. Mooreelvin.j@sci.kmutnb.ac.thSekson Sirisubtaweesekson.s@sci.kmutnb.ac.thSurattana Sungnulsurattana.s@sci.kmutnb.ac.th<p>The (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation usually describes the interaction of a Riemann wave propagating along the y-axis with a long wave propagating along the x-axis. This equation can also be regarded as a generalization of a Korteweg–de Vries (KdV) equation. In this paper, we generalize the BLMP equation by using Atangana's space and time beta-derivatives. We then use the Sardar subequation method and an appropriate traveling wave transformation to derive exact traveling wave solutions for the (2 + 1)-dimensional BLMP equation with fractional derivatives. The exact solutions of the equation are expressed in terms of generalized trigonometric and hyperbolic functions. These functions, which include both real- and complex-valued functions, are defined in this paper for the first time. Exact solutions are derived for a range of values of fractional orders and 2D, 3D and contour plots of the solutions are shown. Solutions are obtained for a range of parameter values to show some of the types of solution that can occur. As examples, we show solutions with physical behaviors such as a singular bell-shaped solitary wave solution, a solitary wave soliton of kink type and a periodic wave solution. We demonstrate that the proposed technique gives a straightforward and efficient method for deriving new exact traveling wave solutions for nonlinear partial differential equations such as the BLMP equation.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1596A Novel Technology-Based Stochastic Epidemic Model2024-03-30T15:26:46+00:00Chananun Onjanchananun_o@cmu.ac.thParkpoom Phetpradapparkpoom.phetpradap@cmu.ac.th<p>In this article, we propose a new discrete-time stochastic epidemic compartment model to study and analyze the spread of disease. The SUQIHR model consists of six compartments; Susceptible (S), Unsafe (U), Quarantined (Q), Infected (I), Hospitalized (H) and Recovered (R). The Unsafe class (U) comprises individuals who are at higher risk of infection compared to the general susceptible population, such as those with close contact to infected individuals. Transitions between compartments are assumed to follow certain probability distributions that capture the movement of individuals. The advancement of tracking technologies enables the differentiation of unsafe individuals from susceptible ones through the use of tracking equipment or mobile applications. Therefore, this model finds relevance in technology-ready societies. In this study, we utilize the SUQIHR model to forecast the future spread of diseases. The model incorporates both the transmission dynamics of epidemics and measures to control their spread. We examine the mathematical analysis of the model such as long-term behavior, the basic reproduction number and sensitivity analysis. Moreover, the Monte Carlo simulation can be employed to study the survival distribution of the outbreak, the final size of infected individuals, and the expected duration of the epidemic. By this comprehensive approach, our model provides valuable insights for understanding and managing disease outbreaks in various scenarios.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1597Mathematical Models for Brown Planthopper Infestation of Rice under Habitat Complexity and Monsoon Effects2024-03-30T16:25:52+00:00Nattawut Khansainattawut.khansai@gmail.comSekson Sirisubtaweesekson.s@sci.kmutnb.ac.thSanoe Koonprasertsanoe.k@sci.kmutnb.ac.thWatchareewan Jamboonsriwatchareewan@biotec.or.thWisut Kitchainukoonwisut.kit@lru.ac.th<p>In this research, we study mathematical predator-prey models for brown planthopper (BPH) infestation of rice under the effects of habitat complexity and monsoon for two time scales. Using a fast time scale, we obtain a complete model which is a system of first-order differential equations including logistic growth terms, modified Holling type II functional responses and migration terms due to the monsoon. The positivity and boundedness of the fast time-scale model are proved. Using a slow time scale and the aggregated method, we obtain an aggregated model which is less complicated than the former model in terms of the number of variables and parameters. We investigate the existence of equilibrium points and their local asymptotic stability for this aggregrated model. Hopf bifurcation of the aggregrated model is also shown to occur as the maximum carrying capacity is varied. Numerical simulations are performed to illustrate the theoretical results. Finally, some ecological discussion of methods for reducing the BPH dispersion of the model is given.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1598Some Generalized Derivations on n-ary Multiplicative Semilattices2024-03-30T16:47:08+00:00Pongsatorn Chotchayapongsatorn.cho@ku.thUtsanee Leerawatfsciutl@ku.ac.th<p>In this paper, we introduce a generalization of derivation on n-ary multiplicative semilattices; namely permuting n-(f, g)-derivation, and investigate some related properties. Moreover, we study the notion of trace of permuting n-(f, g)-derivation in multiplicative semilattices and we also obtain some results concerning identities with traces and permuting n-(f, g)-derivation in multiplicative semilattices.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1599Regularity in Ternary Semihypergroups Induced by Subsets of Ternary Semigroups2024-03-30T17:52:48+00:00Anak Nongmaneeanak_nongmanee@cmu.ac.thSorasak Leeratanavaleesorasak.l@cmu.ac.th<p>Ternary semihypergroups, which can be considered as a generalization of arbitrary ternary semigroups, are algebraic hyperstructures together with one ternary associative hyperoperation. In this article, we construct a new algebraic hyperstructure which can be formed as a ternary semihypergroup. In particularly, such a ternary semihypergroup is induced by a nonempty subset of the base set of the ternary semigroup. We investigate the regularity of the ternary semihypergroups. Furthermore, we present some of its characterizations via significant conditions.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1600Jordan f-Derivations on Prime and Semiprime Rings2024-03-30T17:58:51+00:00Utsanee Leerawatfsciutl@ku.ac.thPatipat Tokapatipat.to@ku.th<p>Let $R$ be a ring and $f : R\rightarrow R$ be a mapping. A mapping $D : R\rightarrow R$ is said to be a $Jordan\ f-derivation$ on $R$ if $D(a+b) = D(a) + D(b)$ for all $a, b\in R$, and $D(a^2) = D(a)f(a) + f(a)D(a)$ for all $a \in R.$ In this paper, we present some conditions on prime ring or semiprime ring $R$ that force the zero map to be the only Jordan $f$-derivation on $R$.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1601Self-Conjugate-Reciprocal Polynomials over Finite Fields and Self-Conjugate-Reciprocal Transformation2024-03-30T18:05:51+00:00Hataiwit Palasakhataiwit.p@gmail.comOuamporn Phuksuwanouamporn.p@chula.ac.thTuangrat Chaichanatuangrat.c@chula.ac.th<p>An interesting class of polynomials over finite fields, namely self-conjugate-reciprocal polynomials, has been studied here. Some elementary properties on their roots and a way to find all self-conjugate-reciprocal irreducible monic polynomials of a given degree are provided. Moreover, in the last part, we define a map taking a polynomial over a finite field with some conditions to a self-conjugate-reciprocal polynomial. Certain properties of the polynomial obtained from this map are investigated.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1602On the Relative Rank of Orientation-preserving or Orientation-reversing Transformation Semigroups with Restricted Range2024-03-30T18:30:00+00:00Krittapon Chaikanprabchaikan@gmail.comAniruth Phon-Onaniruth.p@psu.ac.thKittisak Tinpunkittisak.ti@psu.ac.th<p>Let $G$ and $U$ be subsets of a semigroup $S$. The rank of a semigroup $S$ is the minimal size of a generating set of $S$. By the definition of rank, it gives a new idea of definition of rank which is called the relative rank of $S$ modulo $U$ which is the minimal size of a subset $G$ such that $G\cup U$ is a generating set of $S$. A set $G$ is called a generating set of $S$ modulo $U$. Let $X$ be a finite chain and let $Y$ be a subchain of $X$. The semigroup $\mathcal{T}(X,Y)$ is so-called the full transformation semigroup on $X$ with restricted range $Y$ which is a subsemigroup of the semigroup $\mathcal{T}(X)$. In this work, we determine the relative rank of the semigroup $\mathcal{OPR}(X,Y)$ of all orientation-preserving or orientation-reversing transformations with restricted range modulo the semigroup $\mathcal{OD}(X,Y)$ of all order-preserving or order-reversing transformations with restricted range.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1618Powers of Some Special Upper Triangular Matrices2024-03-31T12:23:09+00:00Charn Khetchatturatfscichk@ku.ac.thUtsanee Leerawatfsciutl@ku.ac.thPimchana Siricharuanunfscispns@ku.ac.th<p><img src="https://thaijmath2.in.cmu.ac.th/public/site/images/pjailoka/screen-shot-2567-03-31-at-7.13.04-pm.png" alt="" width="1279" height="475"></p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1604Two-parameter Taxicab Trigonometric Identities2024-03-30T19:01:59+00:00Siravit Boonleangsiravit.eiei@g.swu.ac.thChanoknan Changklangchanoknan.cha@g.swu.ac.thPhatarapol Pakongphatarapol.pakong@g.swu.ac.thThunwa Theerakarnthunwa@g.swu.ac.th<p>The metric on $\mathbb{R}^2$ defined by $d((x_1, x_2), (y_1, y_2)) = |x_1 - y_1| + |x_2 - y_2|$ is known as the $\ell^1$ or the taxicab metric. Delp and Filipski define and provide explicit formulas for sine and cosine functions for the taxicab space. Their version agrees with the right-triangle definition of the standard trigonometric functions. In particular, the sine (cosine) of an acute angle in a right triangle is equal to the ratio of the length of its opposite (adjacent) side and the length of the hypotenuse. These functions must have two parameters because a general rotation is not an isometry in the taxicab metric. We derive new identities for the taxicab sine and cosine functions. Specifically, we derive the Pythagorean, angle sum, double-angle, half-angle, and negative-angle identities. Additionally, we derive derivative identities for the taxicab tangent, secant, cotangent, and cosecant functions.<br />We find that the derivatives of these functions behave similarly to their Euclidean counterparts.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1605On (f, g)-Semi-Derivations of Hyperrings2024-03-30T19:10:50+00:00Utsanee Leerawatfsciutl@ku.ac.thJirat Sanguansatjirat.sang@ku.th<p>In this paper, we introduce a generalization of semi-derivation on the Krasner hyperrings R. Specifically, we propose a new type of semi-derivation called (f, g)-semi-derivation, where f and g are mappings from R into itself. Our aim is to explore the properties of this new type of semi-derivation. Additionally, we conduct investigations some results on (f, g)-semi-derivations either on 2-torsion free prime hyperrings or on prime hyperrings.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1606New Exact Traveling Wave Solutions of the (3+1)-Dimensional Chiral Nonlinear Schrodinger Equation Using Two Reliable Techniques2024-03-30T19:27:29+00:00Montri Torvattanabunmontri_kai@windowslive.comNattawut Khansainattawut.khansai@gmail.comSekson Sirisubtaweesekson.s@sci.kmutnb.ac.thSanoe Koonprasertsanoe.k@sci.kmutnb.ac.thNguyen Minh Tuannmtuanbm2006@gmail.com<p>In this research, we study a (3+1)-dimensional chiral nonlinear Schrodinger equation (CNLSE) and find its exact traveling wave solutions via the extended simplest equation method (ESEM) and the improved generalized tanh-coth method (IGTCM). The exact solutions of the CNSLE are complex-valued functions that can be expressed in terms of exponential, hyperbolic, trigonometric, and rational functions. The magnitudes of some representative solutions are plotted as 3D and contour plots to illustrate the physical interpretations of the solutions. The findings establish that the used methods are simple, powerful, and reliable tools for obtaining new exact traveling wave solutions for complex nonlinear partial differential equations.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1607Completeness of Low-Dimensional Leibniz Algebras2024-03-31T09:32:18+00:00Yannawat Kongsomprachyannawat.f@gmail.comSuchada Pongprasertsuchadapo@g.swu.ac.thThitarie Rungratgasamethitarie@g.swu.ac.thSatrirat Tiansa-ardsatrirat.tiansaard@gmail.com<p>Leibniz algebras are generalizations of Lie algebras. By using the classification results of low-dimensional non-Lie nilpotent and non-nilpotent solvable Leibniz algebras obtained earlier, we define a basis of the derivation algebra Der(A) of each Leibniz algebra A and study their properties. It is known that for a Leibniz algebra A if the Lie algebra A / Leib(A) is complete, then A is a complete Leibniz algebra. We show that the converse holds when A is a complete solvable Leibniz algebra with dim(A) \leq 3. It is also known that for the derivation algebra of a complete Lie algebra is complete. However, our results show that this is not true for Leibniz algebras.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1608A Fibonacci Galerkin Method for Solving Certain Types of Boundary Value Problems2024-03-31T09:52:55+00:00Sanoe Koonprasertsanoe.k@sci.kmutnb.ac.thSekson Sirisubtaweesekson.s@sci.kmutnb.ac.thPruchayaporn Srisutatppruchayaa@gmail.comPattarawadee Petkongkaewrose.petkongkaew2000@gmail.comElvin J. Mooreelvin.j@sci.kmutnb.ac.th<p>In the Fibonacci sequence, the first two numbers are 0 and 1, and the next numbers are equal to the sum of the previous two numbers. In this paper, the concept of the Fibonacci sequence is extended to polynomial functions which can be effectively applied in various function approximations. Based on these Fibonacci-based polynomials and the Galerkin method, we develop a Fibonacci Galerkin method (FGM) to solve some types of boundary value problems (BVPs) such as a linear singular two-point BVP and a nonlinear multi-point BVP. The FGM process constructs a residual function for a BVP by utilizing an approximate solution formed by the method and then evaluating the integral of the product between residual functions and weight functions over a domain. Equating the value of the integral close to zero, one obtains an analytical solution of the BVP. As examples, we apply the method to certain types of BVPs including a linear singular two-point BVP, a nonlinear multi-point BVP and a regular two-point BVP whose exact solutions are given. Their semi-analytical solutions are obtained. By comparing the solutions of the proposed boundary value problems obtained by this technique with their exact solutions, we believe that the technique is highly accurate and effective.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1609More on Continuity in Rough Set Theory2024-03-31T10:06:59+00:00Gun Sunyeekhanfscigsy@ku.ac.thPuntitra Noonoipuntitra.no@ku.th<p>It is generally known that in rough set theory, we can define the topology imposed by the lower approximation operator in the approximation space (U, R). As a result, any approximation can be viewed as a topological space. In this article, we present the sufficient condition for a map between two approximation spaces to be a continuous map. Some elementary properties of the maps are stated.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1610Some Properties of a Trinomial Random Walk Conditioned on End Points2024-03-31T10:20:16+00:00Poramin Gaengaew6234324023@student.chula.ac.thWasamon JantaiWasamon.J@chula.ac.thMonchai KooakachaiMonchai.K@chula.ac.th<p>Given a sequence of trinomial random variables $\displaystyle\{ X_i \}^\infty_{i=1}$ and define $S_n =\sum_{i=1}^n X_i$ and $S_0 = 0$, we study some properties of $X_i $ conditioned on $S_n = 0.$ The mathematical expressions of expectation, variance and covariance were investigated. We found that the a finite sequence $(X_1, X_2, \ldots, X_n)$ conditioned on $S_n = 0$ is exchangeable. Moreover, the expectation of $X_i$ is zero and the covariance of $X_i$ and $X_j$ where $i \neq j$ is nonpositive. Furthermore, we extend the previous setting to a rescaled trinomial random walk. Some properties on the extension were derived.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1611Arithmetic Functions Associated with Exponentially Odd and Exponentially Even Integers2024-03-31T10:32:00+00:00Vichian Laohakosolfscivil@ku.ac.thPussadee Yangklanpussadee.yang@kmutt.ac.th<p>An exponentially even integer is a positive integer whose prime factorization contains only even prime powers, while an exponentially odd integer is a positive integer whose prime factorization contains only odd prime powers. We investigate here the problem of counting the number of positive integers that are semi-prime to an exponentially even integer, and to an exponentially odd integer. Basic properties of the functions involved are established.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1612On Cubic Exponential Diophantine Equations $\pm 3^x \pm a^y = z^3$2024-03-31T10:40:19+00:00Poonyanoot Khanompoonyanoot.kh@mail.wu.ac.thSaeree Wananiyakuls.wananiyakul@hotmail.comJanyarak Tongsomporntjanyarak@gmail.com<p>We investigate the integer solutions of the cubic exponential Diophantine equations in the form $\pm 3^x \pm a^y = z^3$ using elementary techniques. In particular, we also prove that there are infinitely many cubic exponential Diophantine equations in that form that have no integer solutions.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1613The (s, t)−Jacobsthal Hybrid Numbers and (s, t)−Jacobsthal-Lucas Hybrid Numbers2024-03-31T11:09:10+00:00Tanupat Petpanwongtinapat008@gmail.comNarawadee Phudolsitthiphatnarawadee_n@hotmail.co.th<p>The (s, t)−Jacobsthal and (s, t)−Jacobsthal-Lucas hybrid numbers are introduced. Several properties of these numbers are derived, including the Binet formulas, generating functions, exponential generating functions, summation formulas and identities such as those due to Catalan, Cassini and d'Ocagne. In addition, a matrix generator for these numbers is presented. The obtained results extend and generalize well-known theorems.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1614Properties of the Graph Arising from Certain Map over a Finite Field2024-03-31T11:33:56+00:00Pratchayaporn Doemlimpratchayaporn2539@gmail.comVichian Laohakosolfscivil@ku.ac.thTuangrat Chaichanatuangrat.c@chula.ac.th<p>For primes $p$ and $q$, the graph obtained from iterating the map $x \mapsto x^p$ over the finite field of $q^2$ elements is considered. Asymptotic formulas for the sum, over bounded primes $q$, of the total number of elements in all cycles and that of all tail lengths are derived.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1615Derivation of Some Identities and Applications2024-03-31T11:48:09+00:00Tuangrat Chaichanatuangrat.c@chula.ac.thVichian Laohakosolfscivil@ku.ac.thRattiya Meesarattiya3328@gmail.com<p>Certain identities based on recurrently published results about explicit solutions of general second-order linear recurrences are proved and are used to derive explicit solutions of four well-known recurrence relations with polynomial coefficients.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1616A Comparison of Capture-recapture Modeling to Estimate the Number of Patients with Psoriasis in Trang Province2024-03-31T11:54:13+00:00Orasa Nunkawaorasa@tsu.ac.thPreedaporn Kanjanasamranwongpreedaporn@tsu.ac.th<p>This study aims to estimate the number of patients with psoriasis in Trang Province between 2015 and 2018. The population size estimator based on Poisson distribution called the MLEPoi was used as a basic model for homogeneity population. However, the heterogeneity often occurs in capture-recapture experiment. The population size estimators based on the Poisson mixture model; the MLEGeo, LCMP, TG, Chao, Zelterman and LB estimators were selected for heterogeneous population. By using the ratio plot of the Poisson model for investigating a suitable model for psoriasis data, the results suggested that the Poisson mixture model performs better than the Poisson model. Therefore, the LCMP and MLEGeo estimators provided the best accuracy with excellent goodness-of-fit over competitors. The TG estimator showed promise as an alternative choices to estimate population size in all cases, wile the LB estimator provided the worst result with violated assumption parameters. Using the most appropriate population size estimator determined the average number of hidden patients with psoriasis in Trang Province at 536 persons per year. Combination of reported and unreported cases to the true number of patients, an estimate of patients with psoriasis was 1,104 (95%: 804–1,045) people per year.</p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematicshttps://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1617Interpolation Theorems for the Arboricity and the Vertex Arboricity of Graphs2024-03-31T12:08:14+00:00Teerasak Khoployklangteerasak.k@chandra.ac.thAraya Chaemchanaraya@mathstat.sci.tu.ac.th<p><img src="https://thaijmath2.in.cmu.ac.th/public/site/images/pjailoka/screen-shot-2567-03-31-at-7.05.47-pm.png" alt="" width="1279" height="409"></p>2024-03-31T00:00:00+00:00Copyright (c) 2024 Thai Journal of Mathematics