دانشگاه اصفهان نشریه ریاضی و جامعه 2345-6493 8 3 2023 10 10 Some Pascal’s like triangles چند مثلث‌ عددیِ مشابه مثلث خیام-پاسکال 1 30 27796 10.22108/msci.2023.137812.1578 FA سید محمد امین خاتمی گروه علوم کامپیوتر، دانشگاه صنعتی بیرجند، بیرجند، ایران محمدهادی مصلحی گروه ریاضی، دانشگاه پیام نور، تهران، ایران Journal Article 2023 05 24 Abstract. In this article, we delve into the fascinating world of numerical triangles similar to Khayyam-Pascal triangle. Our focus is on triangles that are comprised of natural numbers. Along with a combi-natorial interpretation, we also propose an algebraic interpretation for the elements in most cases. We explore in details the triangle of coefficients of Chebyshev polynomials (Chebyshev triangle). Through our analysis, we derive a recursive relation for its elements. This finding sheds new light on the properties of this intriguing numerical construction. To further enhance our understanding of these triangles, we also present new images related to the Catalan, Bell, and Chebyshev triangles. These images provide a clearer visualization of the numerical triangle construction. Overall, this article offers a comprehensive exploration of numerical triangles similar to Khayyam-Pascal triangle and examine some of their properties and relationships for better understanding of these constructions. در این مقاله نگاهی به دنیای شگفت‌انگیز مثلث‌های عددیِ مشابه مثلث خیام-پاسکال می‌اندازیم. تمرکز ما البته به مثلث‌هایی است که از اعداد طبیعی تشکیل شده‌اند. سعی کرده‌ایم علاوه بر تعابیر ترکیبیاتی و جبری اعداد مثلث‌ها، به پاره‌ای از خواص مهمِ آنها نیز اشاره کنیم ولی کمتر به جزئیات پرداخته‌ایم. البته در این بین تجزیه و تحلیلی که از روابط عناصر در مثلث چبیشف داشته‌ایم با تفصیل بیشتر بوده است که ویژگی‌های بیشتری از ساختار عددی این مثلث را روشن می‌کند. برای تقویت درک خود از این مثلث‌ها، بعضی از اثبات‌ها و تصاویر را تغییر داده‌ایم و بخصوص تصاویر جدیدی برای مثلث‌های کاتالان، بل، و چبیشف رسم کرده‌ایم.

https://math-sci.ui.ac.ir/article_27796_7d38619bfc49198ab99ad4d2fab5ff72.pdf

دانشگاه اصفهان نشریه ریاضی و جامعه 2345-6493 8 3 2023 10 16 Star compressed zero divisors graph and partitions of vector spaces گراف ستاره مقسوم علیه صفر فشرده و افراز فضاهای برداری 31 39 27820 10.22108/msci.2023.138202.1587 FA حمید رضا دربیدی گروه ریاضی، دانشکده علوم پایه، دانشگاه جیرفت Journal Article 2023 06 26 Let $R$ be a commutaive ring and $Zd(R)$ be the set of zero divisors of $R$. Define an equivalence relation $\sim$ on $Zd(R)$ as follows: $x\sim y$ if and only if $ann(x)=ann(y)$. The graph $\Gamma_E(R)$ is a graph associated to R whose vertices are the classes of elements in $Zd(R)^*=Zd(R)\backslash\{0\}$, and two distinct classes $[x]\neq[y]$ are joined by an edge if and only if $xy=0$. We show that if $R$ is a local ring and $\Gamma_E(R)$ is a star graph with at least four elements then $m/Soc(R)$ has a partition of vector spaces where $m$ is the maximal ideal of $R$. Also, We construct from a special partition of vector spaces, a ring whose associated graph is a star graph. 1. Introduction The compressed zero divisor graph or the graph of equivalence classes of zero divisors of a ring $R$ is denoted by $\Gamma_E(R)$, and is defined in [20]. Let $Zd(R)$ denotes the set of zero divisors of a ring $R$ and $Zd(R)^*=Zd(R)\backslash\{0\}$. Define a relation $\sim$ on $Zd(R)$ as follows [14]: $x\sim y$ if and only if $ann(x)=ann(y)$. It is easily seen that $\sim$ is an equivalence relation. The graph $\Gamma_E(R)$ is a graph associated to R whose vertices are the classes of elements in $Zd(R)^*$, and two distinct classes $[x]\neq[y]$ are joined by an edge if and only if $xy=0$. Another interpretation of $\Gamma_E(R)$ is as follows: The vertices are the elements of $\mathcal{J}=\{ann(a):a\in Zd(R)^*\}$ and two distinct elements $ann(x)$ and $ann(y)$ are adjacent if and only if $xy=0$. In [20], some necessary conditions are obtained for a ring $R$ such that $\Gamma_E(R)$ is a star graph. For example, If $\Gamma_E(R)$ is a star graph with at least four vertices then $|Ass(R)|=1$ and $Char(R)=2,4,8$. In [9] a method for constructing star compressed zero divisor graph is obtained. They used a quotient of a symmetric algebra of a vector space whose relations come from a special partition of that vector space. But it seems that the authors were not aware of partition of vector spaces. By analyzing the proofs in [20] and [9], we see that the star graphs give a partition of a vector space and conversely some partitions give a star graph. An interesting problem about star compressed zero divisor graph is the size of them. For example is there any ring whose compressed zero divisor graph be a star graph with $36$ vertices? 2. Main Results Theorem 2.1. Let $R$ be a ring such that $\Gamma_E(R)$ is a star graph with at least four vertices. Let $[y]$ be the unique vertex with maximal degree and $K=[y]\bigcup \{0\}$. Then $R$ satisfies the following properties: (1) $Ass(R)=\{P\}$ where $ann(y)=P$. Also $ann(P)=K$. In particular $K$ is an ideal of $R$ and $Zd(R)=P$. (2) $P^3=0$. (3) If $ann(x_0)=K$ then $x_0^3=0$ and $[x_0+y]=[x_0]$. (4) If $J=\{x\in R:K\subsetneqq ann(x)\}$ then $J$ is an ideal of $R$ and $ann(x)=[x]\bigcup K$ for each $x\in J\backslash K$. Also $[x+y]=[x]$ for each $x\in J\backslash K$. (5) $Char(R)=2,4,8$. Corollary 2.2. Let $(R,m)$ be a local Artinian ring such that $\Gamma_E(R)$ is a star graph with at least four vertices. Let $[y]$ be the unique vertex with maximal degree and $K=[y]\bigcup \{0\}$. If $J=\{x\in R:K\subsetneqq ann(x)\}$ then $\{ann(x)/K:x\in J\backslash (K)\}$ is a partition of $R/m-$vector space $J/K$. Also $Soc(R)=ann(m)=K$. Theorem 2.3. Let $V=V_1\bigoplus\cdots\bigoplus V_t$ be an $n-$dimensional vector space over $\mathbb{F}_2$ and $dim(V_i)=n_i$. Assume $\{X_{i,k}:1\leq k\leq n_i\}$ is a basis of $V_i$. Let $S=\mathbb{F}_2[X_{i,k}:1\leq i\leq t,1\leq k\leq n_i]$ be a polynomial ring over $n$ indeterminates. Let $I=\langle V_i^2,V^3\rangle$ and $R=\frac{S}{I}$. Then $\Gamma_E(R)$ is a star graph with $2^n-(2^{n_1}+\cdots+2^{n_t})+2t$ vertices. 3. Summary of Proofs/Conclusions In this article we show that every star compressed zero divisor graph correspond to a partition of vector spaces. Conversely, we construct from a special vector space partition a star compressed zero divisor graph. فرض کنیم $R$یک حلقه جابجایی باشد و $Zd(R)$ مجموعه مقسوم علیه‌های صفر آن باشد. رابطه هم‌ارزی $\sim$ را روی $Zd(R)$ به‌صورت زیر در نظر می‌گیریم: $x\sim y$ اگر و تنها اگر $ann(x)=ann(y)$. گراف $\Gamma_E(R)$ گرافی است که رئوس آن رده‌های هم‌ارزی اعضای $Zd(R)^*$ است و دو رأس متمایز $[x]\neq[y]$ به هم متصل هستنند اگر و تنها اگر $xy=0$. ما نشان می‌دهیم که اگر حلقه $R$ یک حلقه موضعی با ایده‌آل بیشین $m$ باشد و گراف $\Gamma_E(R)$ گراف ستاره با حداقل 4 رأس باشد آنگاه $m/Soc(R)$، به‌عنوان فضایی برداری، یک افراز دارد. همچنین با استفاده ازیک افراز خاص فضاهای برداری، حلقه‌ای می‌سازیم که گراف وابسته آن گراف ستاره است.

https://math-sci.ui.ac.ir/article_27820_e9e8fb6b35148a2424f83a24471ac379.pdf

دانشگاه اصفهان نشریه ریاضی و جامعه 2345-6493 8 3 2023 10 28 Resolving sets of vertices with the minimum size in graphs مجموعه‌های تفکیک‌کننده رأس‌ها در گراف‌ها با کوچکترین اندازه 41 54 27889 10.22108/msci.2023.138631.1599 FA علی ظفری گروه ریاضی، دانشکده علوم‌پایه، دانشگاه پیام‌نور، تهران، ایران نادر حبیبی گروه ریاضی، دانشکده علوم‌پایه، دانشگاه آیت الله العظمی بروجردی (ره)، بروجرد، ایران سعید علیخانی دانشکده علوم ریاضی، دانشگاه یزد، یزد، ایران Journal Article 2023 08 06 Suppose that $G$ is a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. A subset $S=\{s_1, s_2,\ldots , s_l \}$ of vertices of graph $G$ is called a doubly resolving set of $G$, if for any distinct vertices $u$ and $v$ in $G$ there are elements $x$ and $y$ in the set $S$ such that $d(u, x)-d(u, y)\ne d(v, x)-d(v, y)$. The minimum size of a doubly resolving set of the vertices of graph $G$ is denoted by ${\psi} (G)$. In this paper, we calculate the resolving sets of vertices with the minimum size for the line graph $L(C_n\circ{\overline{K}}_m)$ and graph $\left((C_n\circ{\overline{K}}_m)\square P_k\right)$, in which the symbols $\circ$ and $\square$ denote the Corona product and Cartesian product between two graphs, respectively. In particular, we show that if $n\geq 3$ and $m, k\geq 2$ are integers, then ${\psi}((C_n\circ{\overline{K}}_m)\square P_k )={\psi}(C_n\circ{\overline{K}}_m)+{\psi}(P_k )-1$, which gives a partial answer to the problem of characterizing graphs $G$ and $H$ satisfying the equality ${\psi}(G\square H)={\psi}(G)+{\psi}(H)-1$, which is recently posed in [K. Nie and K. Xu, The doubly metric dimension of cylinder graphs and torus graphs, Bull. Malays. Math. Sci. Soc., 46 (2023) 19 pp]. 1. Introduction Let $G$ be a simple and connected graph with the set of vertices $V(G)$ and the set of edges $E(G)$. We denote the length of the shortest path between two vertices $u$ and $v$ in the graph $G$ by $d(u, v)$. We use $C_n$, $\overline{K_m}$ and $P_k$ to denote the cycle graph of order $n$, complement of the complete graph on $m$ vertices and the path graph of order $k$, respectively. Also, the line graph $G$ is denoted by $L(G)$, that the set of vertices of $L(G)$ are the same as the edges of the graph $G$, and two vertices are adjacent in the graph $L(G)$, if their corresponding edges in graph $G$ have a common vertex [6]. Our goal is to calculate some resolving sets depending on the line graph of the Corona product $C_{n}\circ \overline{K_{m}} $ and the Cartesian product $(C_{n}\circ \overline{K_{m}} )\square P_{k}$, so we give first some explanations about the Corona product and Cartesian product of graphs. Suppose $G$ and $H$ are two graphs with $n$ and $m$ vertices, respectively. If we consider $n$ copies of $H$ and for $i=1, 2,\cdots ,n$, all the vertices of the $i^{th}$ copy of $H$ are adjacent to the vertex $i$ of $G$, then the desired graph is called the Corona product of two graphs $G$ and $H$ and we denote it by $G\circ H$. Also, if $G$ and $H$ are two graphs, we denote the Cartesian product of these graphs by $G \square H$ or $G \times H$ and define in this way $V\left(G\square H\right)=V(G)\times V(H)$ and two vertices $(g, h)$ and $(g',h')$ in $G\square H$ are adjacent if and only if $g=g'$ and $hh'\in E(H)$ or $h=h'$ and $gg'\in E(G)$. For any ordered subset $S=\{ s_{1}, s_{2},\cdots,s_{k}\}$ of the vertices of graph $G$ and the vertex $v$ of $G$, representation the vertex $v$ with respect to the ordered set $S$ denoted by $r(v{|S)}$ and so it is $ r(v{|S)}=\left({d}\left(v, s_1\right),{d}\left(v, s_2\right),\ldots,{d}\left(v, s_k\right)\right)$. If all the vertices of the graph $G$ have distinct metric representations with respect to the ordered set $S$, then $S$ is called a resolving set of vertices of $G$. A resolving set of vertices of $G$ with the minimum size is called the metric dimension of the graph $G$ and it is represented by ${\beta }\left(G\right)$. The study of resolving set of vertices in graph theory dates back to the 1970s, and such concepts were first introduced in the articles [7,18]. The metric dimension of complete graphs, trees, paths, and the Cartesian product have been taken into consideration, also in the article [5] all graphs of order $n$ that have the metric dimension greater than or equal to $n-2$ are fully characterised. The subset $ S=\{s_{1},s_{2},\cdots,s_{l}\}$ of the vertices of graph $G$ is called a doubly resolving set for $G$, if for any distinct pair vertices $u$ and $v$ of $G$, there are the elements $x$ and $y$ of $S$, in which $d(u, x)-d(u, y)\ne d(v, x)-d(v, y)$. We denote the size of the minimum doubly resolving set in graph $G$ by ${\psi}\left(G\right) $. Concepts related to resolving sets and doubly resolving sets of graphs have been studied in the articles [4,5]. The graph $(C_5\circ \overline{K}_{3})$ and graph $L(C_{5}\circ \overline{K}_{3})$ are drawn in Figure 1. Also, the graph $(C_3\circ \overline{K}_{3})$ and graph $(C_{3}\circ \overline{K}_{3})\square P_{2}$ are drawn in Figure 2. 2. Main Results Theorem 2.1. If $n$ and $m$ are fixed positive integers, in which $n\geq 3$, $m\geq 2$, then the cardinality of minimum doubly resolving set of the line graph of graph $C_n\circ{\overline{K}}_m$ is $nm-n$. Theorem 2.2. If $n$, $m$ and $k$ are fixed positive integers, in which $n\geq 3$ and $m, k\geq 2$, then ${\beta }((C_n\circ{\overline{K}}_m)\square P_k )=nm-n+1.$ Theorem 2.3. If $n$, $m$ and $k$ are fixed positive integers, in which $n\geq 3$ and $m, k\geq 2$, then ${\psi }\left((C_n\circ \overline{K}_{m})\square P_{k}\right)={\psi } (C_n\circ \overline{K}_{m})+{\psi }(P_{k})-1.$ 3. Conclusions Considering the importance of the minimum resolving sets in graphs, in this paper, we first calculated some resolving sets of vertices with the minimum size for the line graph of graph $C_n\circ{\overline{K}}_m$ and graph $\left((C_n\circ{\overline{K}}_m)\square P_k\right)$. In particular, we show that if $n\geq 3$ and $m, k\geq 2$ are integers, then ${\psi}((C_n\circ{\overline{K}}_m)\square P_k )={\psi}(C_n\circ{\overline{K}}_m)+{\psi}(P_k )-1$, which gives a partial answer to the problem of characterizing graphs $G$ and $H$ satisfying the equality ${\psi}(G\square H)={\psi}(G)+{\psi}(H)-1$, which is recently posed in [15]. فرض کنیم $G$ یک گراف ساده همبند با مجموعه رأس‌های $V(G)$ و مجموعه یال‌های $E(G)\ $ باشد. زیرمجموعه $ S=\{s_1, s_2,\ldots,s_l \}$ از رأس‌های گراف $G$ یک مجموعه تفکیک‌کننده دوگانه برای گراف $G$ نامیده می‌شود، هرگاه برای هر دو رأس متمایز $u$ و $v$ از گراف $G$، عضوهای $x$ و $y$ از $S$ موجود باشند که $.d\left(u,\ x\right)-d\left(u,\ y\right)\neq \ d\left(v,\ x\right)\mathrm{-}d\left(v,\ y\right)$ اندازه کوچک‌ترین مجموعه تفکیک‌کننده دوگانه در گراف $G$ را با ${\psi} (G)$ نشان می‌دهند. در این مقاله، ضمن آشنایی با مفهوم و خواص ${\psi} (G)$, برخی مجموعه‌های تفکیک‌کننده رأس‌ها با کوچکترین اندازه را برای گراف یالی $L(C_n\circ{\overline{K}}_m)$ و گراف $(C_n\circ{\overline{K}}_m)\square P_k$ محاسبه می‌کنیم، که در آن نمادهای $\circ$ و $\square$ به‌ترتیب حاصل‌ضرب کرونا و حاصل‌ضرب دکارتی بین دو گراف را مشخص می‌کنند. به‌ویژه، در پاسخ به مسأله مشخص نمودن گراف‌های $G$ و $H$، که برای آن‌ها تساوی ${\psi}(G\square H)={\psi}(G)+{\psi}(H)-1$ برقرار است \cite{15}، ما نشان می‌دهیم که اگر $ n\ge 3$ و $m,k\ge 2$ عددهای صحیح باشند، آن‌گاه ${\psi} \left((C_n\circ{\overline{K}}_m)\square P_k\right)$ برابر است با $.{\psi} \left(C_n\circ{\overline{K}}_m)+{\psi} (P_k\right)-1$

https://math-sci.ui.ac.ir/article_27889_5b79157dd0ab0effff7459beabfc0c47.pdf

دانشگاه اصفهان نشریه ریاضی و جامعه 2345-6493 8 3 2023 11 04 On 4-dimensional Einsteinian manifolds with parallel null distribution مطالعه خمینه‌های اینشتینی 4-بعدی با توزیع پوچ موازی 55 79 27904 10.22108/msci.2023.138258.1589 FA مهدی جعفری دانشکده ریاضی- دانشگاه پیام نور-تهران-ایران Journal Article 2023 07 01 In this paper, we investigate the Einsteinian manifolds with parallel null distribution. For this purpose, we first obtain the equations, which are known as Einstein's equations, that lead to finding the mentioned manifolds and then, we reduce Einstein's equations by using Lie symmetry method. In this method, we first obtain the generators of the symmetry algebra and then calculate the differential Invariants for each of the generators and calculate the group invariant solutions of this equation. We also obtain the optimal system of the one-dimensional sub-algebras of these equations which helps us to have a classification on group invariant solutions by using conjugate mapping. 1. Introduction In the theory of general relativity, the volume of space-time matter can be described by the energy tensor. The fact that the mass of matter in the universe can be considered as a complete fluid in the standard cosmological model led to the hypothesis that it is possible to model the tendency field with appropriate 4-dimensional Lorentzian metrics. This fact made the role of such metrics and the manifolds resulting from it colorful in physics and especially in the theory of relativity [1]. In the meantime, 4-dimensional manifolds that have an null and parallel distribution play an essential role. These manifolds are called Walker manifolds. According to the conditions that must be established on the metric of 4-dimensional Walker manifolds, a system of equations is created, which is called Einstein's equations [20]. So, solving the Einstein's equations leads to finding the general metric form of Walker manifolds. These manifolds play a significant role in modeling physical problems and have been studied recently by many mathematicians and physicists [8]. It is worth mentioning that the Walkers' manifolds were first introduced by Walker and was investigated around the years 1950 to 1953 (he studied the manifolds that have a null parallel distribution). In honor of this great mathematician and due to his findings on this field, these manifolds are called Walker manifolds [18,19]. Einstein's equations play an essential role in making cosmological models. These models express the fact that matter determines the geometry of space-time and, opposite to the movement of matter, it is defined by the space meter tensor [17]. Accordingly, the importance four-dimensional Einstein Walker manifolds as background space for physical models and geometrical as well as solving Einstein's equations to obtain such manifolds are specified. One of the most important and practical methods for analyzing and solving equations is the Lie symmetry method which was first introduced by Marius Sophos Lie in the late 19th century [13]. This method is based on the integration of differential equations and it caused Lie to devote his efforts to the development of the theory of differential equations based on the concept of Lie groups. The Lie groups have many applications in algebraic topology, differential geometry, special functions, numerical analysis, control theory, mechanics classical and quantum relativity. It should be noted that differential equations are the main source of the application of Lie groups in this strings. If we want to briefly talk about the symmetry group of a system of differential equations, we can say that these groups include transformations that transform the solutions of the system. From Lie's point of view, symmetries are geometric transformations that act on the space containing the independent and dependent variables of the system and also act on the solutions of the system with the changes applied to their graph. One of Lie's most important discoveries in the field of differential equations is that he was able to show that the complex nonlinear conditions governing a dynamic system can be locally converted into linear conditions by the infinitesimal derivatives of that system under the generators of its symmetry group, which is important in physics. This led to the use of computers and many software in this field because working with symmetric groups has calculations that follow an algorithmic process [4]. Having the symmetry group of a system of differential equations has many advantages, for example, we can mention the classification of the solutions of the system of differential equations. The basis of this classification is that the solutions that are placed in a category can be converted to each other by some generators of the symmetry group [16]. If we are involved with an ordinary differential equation system, the symmetry group helps us to get the solution by one time integration by reducing the order of the equation to one, and in the case that the desired equation is of the first order, the general solution is also obtained. But unfortunately, this is not the case for partial differential equations, that is, the general solution of such equations cannot necessarily be obtained by having a group of symmetries. (Except in the case where the system can be converted into a linear system.) In this situation, only some solutions are obtained, which fall under some subgroups of the symmetry group. These solutions are known as group invariant solutions, which are obtained from the solution of a system that contains a smaller number of independent variables than the original system [5]. In recent years, several equations have been investigated using this method and their group invariant solutions have been obtained. For example, the KdV equation and its different modes, which is one of the important models in the theory of long waves in shallow waters and other physical systems, have been studied in several sources, including [2,11]. Also, exact solutions of the BBM equation, which is one of the important equations in the modeling of gravity waves with small amplitude, have also been obtained with this method [10,12]. Our goal in this article is to analyze the symmetric group and obtain group-invariant solutions for Einstein's equations. In addition, we obtain a classification on these solutions based on the optimal system of one-dimensional subalgebras of symmetric algebra. The division of this article is as follows: In the second section, the prerequisites and preliminary definitions and some important results about Riemannian pseudo-geometry and Einsteinian tunnels are stated. In the third section, Lie's symmetry method is stated and the symmetry group of Einstein's equations is obtained. Also, a complete analysis of this group and Lie algebras like it will be presented. In the fourth section, the optimal system of the one-parameter subalgebras of the symmetric algebra of Einstein's equations is presented, and then the reduced equations are obtained using differential derivations. In the end, by solving the reduced equations, the solutions of Einstein's equations are obtained. 2. Main Results We first prove that the infinitesimal generators of the symmetric Lie algebra of the Einstein's equations and its general form, the second order extension and the characteristic of vector field $V$ are as follows: (2.1) \begin{equation} \begin{array}{lclclclclc}\label{22.2} V_1=\partial_x,&& && V_5=b\partial_c+2c\partial_a+t\partial_x,\\ V_2=\partial_t,&& && V_6=c\partial_c+2b\partial_b+t\partial_t,\\ V_3=-c\partial_c+-2b\partial_b+x\partial_x,&& && V_7=c\partial_c+b\partial_b+a\partial_a.\\ V_4=a\partial_c+2c\partial_b+x\partial_t,&& \end{array} \end{equation} (2.2) \begin{equation}\label{16.2} \begin{array}{lclclc} V=\xi^1(x,t,a,b,c)\partial_x+\xi^2(x,t,a,b,c)\partial_t+\\ \phi_1(x,t,a,b,c)\partial_a+\phi_2(x,t,a,b,c)\partial_b+\phi_3(x,t,a,b,c)\partial_c. \end{array} \end{equation} \begin{eqnarray*} V^{(2)} &=&V+\phi_1^x\partial_{a_x}+\phi_1^t\partial_{a_t}+\phi_2^x\partial_{b_x}+\phi_2^t\partial_{b_t}+ \phi_3^x\partial_{c_x}+ \phi_3^t\partial_{c_t}\\&& +\phi_1^{xx}\partial_{a_{xx}}+\phi_2^{xx}\partial_{b_{xx}}+\phi_3^{xx}\partial_{c_{xx}}+\phi_1^{ xt}\partial_{a_{xt}}+\phi_2^{xt}\partial_{b_{xt}}\\&& +\phi_3^{xt}\partial_{c_{xt}}+\phi_1^{tt}\partial_{a_{tt}}+\phi_2^{tt}\partial_{b_{tt}}+\phi_3^{ tt}\partial_{c_{tt}}. \end{eqnarray*} \begin{eqnarray*}\label{18.2} \begin{array}{lclclc} Q_1=\phi_1-\xi^1 a_x-\xi^2a_t,&&Q_2=\phi_2-\xi^1b_x-\xi^2b_t,&& Q_3=\phi_3-\xi^1c_x-\xi^2c_t. \end{array} \end{eqnarray*} Then, by using the invariant theorem, we obtain the following irreducibility conditions: (2.3) \begin{equation}\label{19.2} \begin{array}{lclclclclc} V^{(2)}[a_{xx}-b_{tt}]=0,&& && a_{xx}-b_{tt}=0,\\ V^{(2)}[c_{xx}+b_{xy}]=0,&& &&c_{xx}+b_{xy}=0,\\ \vdots &&\\ V^{(2)}[-b_{}c_{xt}+2c_{}b_{xt}+a_{}b_{xx}-{c_{x}}^{2} +b_{t}c_{x}-b_{x}c_{t}+a_{x}b_{x}]=0,\\ -b_{}c_{xt}+2c_{}b_{xt}+a_{}b_{xx}-{c_{x}}^{2}+b_{t}c_{ x}-b_{x}c_{t}+a_{x}b_{x}=0. \end{array} \end{equation} Now, by replacing the extension coefficients in (2.3), we get the following determining system: (2.4) \begin{equation}\label{20.2} \begin{array}{lclcl} a^3\xi^{1}_{a}=0, b^2\xi^{1}_{b}=0, a\xi^{2 }_{b}=0,&& b^2\xi^{2}_{b}=0,&& ac\phi_{{2}_{a}}=0,&& cb\xi^{1} _{b}=0,\ldots\\ abc(-3\xi^{1}_t-\phi_{3_b})+2c^2a(-\xi^{2}_t+\xi^{1}_x)+ba^2(\phi_{2_b}- \phi_{3_c}-\xi^{1}_x+\xi^{2}_t)+\\ ca^2(-2\xi^{2}_x+\phi_{{2}_c})+\phi_1(ab-2c^2)-a^2\phi_2+2ca\phi_3+4c^ 3\xi^{1}_t=0. \end{array} \end{equation} In the following theorem, by solving the above system of PDEs, the vector field coefficients (2.2) is obtained. Theorem 2.1. The Lie algebra corresponding to the symmetry group of the Einstein's system of equations is produced by the vector field (2.2), whose coefficients are the following functions: (2.5) \begin{equation}\label{21.2} \begin{array}{lclclclcl} \phi_1=k_7a+2k_1c ,&&\xi_{{1}}={k_2}+{k_1}{ t}+{k_3}{x},\\ \phi_{{2}}=2{k_6}c_{{}}+({k_7}-2{k_3}+2{k_4}) b_{{}},&&\xi_{{2}}={k_5 }+{k_4}{t}+{k_6}{x},\\ \phi_{{3}} ={k_1}b_{{}}+{k_6}a_{{}}+( -{k_3}+{k_4}+{k_7} ) c_{{}}, \end{array} \end{equation} where $k_i$s, $i=1,\ldots,7$, are arbitrary constants. The one-parameter group related to the vector fields (2.1), which is denoted by $g_k(\varepsilon)$ for $V_k$, $k=1,\ldots,6$, is defined as follows: (2.6) \begin{equation}\label{23.2} \begin{array}{lclcl} g_1({\varepsilon} ):(x,t,a,b,c)\longmapsto(x+{\varepsilon} ,t,a,b,c), & g_2({\varepsilon } ):(x,t,a,b,c)\longmapsto(x,t+{\varepsilon} ,a,b,c),\\ g_3({\varepsilon} )(x,t,a,b ,c)\longmapsto(e^{\varepsilon}x ,t,a,e^{-2{\varepsilon}}b ,e^{-{\varepsilon} }c),\\ g_4({\varepsilon} ):(x,t,a,b,c)\longmapsto(x,t+{\varepsilon} x,a,b+2{\varepsilon}c+{\varepsilon} ^2a,c+{ \varepsilon}a),\\ g_5({\varepsilon} ):(x,t,a,b,c)\longmapsto(x+{\varepsilon}t ,t,a+2{\varepsilon}c +{\varepsilon} ^2b,b,c+{\varepsilon}b), \\ g_6({\varepsilon} ):(x,t,a,b,c)\longmapsto(x,e^{\varepsilon}t ,a,e^{2{\varepsilon} }b,e^{ {\varepsilon} }c),& g_7({\varepsilon} ):(x,t,a,b,c)\longmapsto(x,t,e^{\varepsilon}a , e^{\varepsilon}b ,e^{\varepsilon}c ). \end{array} \end{equation} 3. Conclusions As one of the most important equations in mathematics and physics, we analyzed the Einstein's equations, by using the Lie symmetry method. We first obtained the symmetric group of these equations and the optimal system of one-parameter subalgebras. Then, using differential invariants, we reduced the equations and obtained the group invariants solutions, which lead to the new solutions of the Einstein's equations and to the new Einsteinian Walker manifolds. در این مقاله به بررسی خمینه‌های اینشتینی با توزیع پوچ موازی می‌پردازیم. نخست معادلاتی که منجر به یافتن خمینه‌های مذکور می‌شود را به‌دست می‌آوریم، که به معادلات اینشتین معروف هستند. سپس با استفاده از روش تقارنی‌لی این معادلات را کاهش می‌دهیم. در این روش ابتدا مولدهای جبر تقارن را به‌دست می‌آوریم و سپس ناورداهای دیفرانسیلی را برای هر کدام از مولدها محاسبه کرده و جواب‌های ناوردای گروهی این معادله را محاسبه می‌کنیم. همچنین دستگاه بهینه زیرجبرهای یک بعدی این معادلات را نیز به‌دست می‌آوریم، این دستگاه بهینه به ما کمک می‌کند که یک طبقه‌بندی روی جواب‌های ناوردای گروهی با استفاده از نگاشت مزدوجی داشته باشیم.

https://math-sci.ui.ac.ir/article_27904_1ff834516cf3bf8f08bf78e66f44c51e.pdf

دانشگاه اصفهان نشریه ریاضی و جامعه 2345-6493 8 3 2023 12 03 Investigating the skills of junior high school students in posing problems in field of proportional reasoning بررسی مهارت‌های طرح مسئله دانش‌آموزان دوره اول متوسطه در زمینه استدلال تناسبی 81 117 27914 10.22108/msci.2023.137726.1575 FA حمیدرضا برخورداری دانشکده علوم پایه، دانشگاه تربیت دبیر شهید رجایی تهران، ایران ابراهیم ریحانی دانشکده علوم پایه ، دانشگاه تربیت دبیر شهید رجایی ، تهران، تهران، ایران سعید حق جو دانشکده علوم پایه ، دانشگاه تربیت دبیر شهید رجایی ، تهران، تهران، ایران Journal Article 2023 05 18 This study investigates the skills of junior high school students in posing problems in the field of proportional reasoning. This study, by considering purpose and implementation consecutively, is applied and descriptive (survey type) in nature. The sample of this study was 442 of Qazvin City’s junior high school students, who were chosen based on randomized cluster sampling. The measurement tool was a questionnaire with five problem-posing tasks related to proportional reasoning witch its content and face validity were examined by some of the mathematics professors and mathematics education professors. The Cronbach's alpha coefficient for the questionnaire was $0.83$. Data analysis was done using SPSS26 software and descriptive and inferential statistics methods. The analysis of the results showed that the student’s problem-posing skill was generally evaluated significantly at the level of ``replacement" from the theoretical framework of the study. Also, the analysis of data showed that there is a significant difference between the performance of students in 7th, 8th, and 9th grades in problem-posing in the field of proportional reasoning, and with the increasing educational grade, their problem-posing skills will be increased. On the other hand, by studying the effect of students' gender on problem-posing performance, it was found that there was no significant difference between boy and girl students in problem-posing. Also, it was observed that the school type is not an effective factor in problem-posing in proportional reasoning problems and there is no significant difference between the performance of ordinary and gifted school students. The results of this study can be used in teacher Training and textbook authoring. 1- Introduction Proportional reasoning is the cornerstone of high school mathematics and is known as the high goal of elementary mathematics [1,2]. Proportional reasoning including ratio, proportion, rate, and fraction are among the important concepts of school mathematics that learning is necessary for students but it is difficult for teachers to teach them [4,5,2]. Proportional reasoning is a special mathematical topic in mathematics education research because many subjects need knowledge and understanding in mathematics curriculum (e.g. scale, probability, percentage, rate, calculus, algebra, geometry) [16] and science (density, molarity, speed, force) [17,13]. Problem posing is one of the most important aspects of pure and applied mathematics and can be a part of the modeling cycle that is required in modeling real-world phenomena. Based on Leung [43], problem posing is the new organization of the given problem. In this research, according to the part of the literature review (Vistro-Yu model and Leung model) and evaluation and modification of them, we observed that in the Vistro-Yu model, the incorrect problems were not been considered and in the Leung model, a suitable classifier was proposed for incorrect problem posing in this study, we have proposed a suitable classifier for incorrect classification. therefore, by choosing the three levels of the five levels of the Leung model that were related to incorrect responses and applying some small changes in it and also levels of Vistro-Yu levels, a combination framework composed of 9 levels was formed and the ability of the junior high school students was studied. In this framework, after reviewing and analyzing project issues, first, the performance of the students is divided into two categories correct problem posing and incorrect problem posing. Incorrect problem posing is classified in three levels and the correct problem posing is classified in six levels. The reason for choosing this combination is that the frameworks used in it were complementary and supported each other, and by combining them, a complete classification was achieved that covers the wide range of students ' problem-posing. 2- Main Results In this research, five questions related to the concepts need to be provided to the students with different formats and include a specific goal. These five questions were selected as selection criteria and were selected for use in the study to cover all objectives of the study and the extracted data from them were adapted and adapted with the research framework. Therefore, the students' responses to test questions after careful examination and extraction of the results from each of the levels of the proportional problem-posing framework and then the data obtained, led the researchers to the secondary goals of this research, the effect of educational grades, gender, and school type on the performance of students in proportional reasoning. For this purpose, the test questions the target, and the answers provided by students to test questions were analyzed. In general, according to the results of the analysis of students' responses to test questions, the performance of the students in posing proportional problems is fairly good. However, most of the students responded to test questions and posed a new problem by changing the variables, numbers, and numerical relationships of the problem and their performance was evaluated mainly at the level of substitution. in addition, a considerable portion of the answers were related to the questions that were posed incorrectly and different factors can have caused this matter such as lack of accuracy to problem assumptions, problem posing with rely on intuition and without reasoning, insufficient understanding of rate concept, incorrect comparison of fractions, misunderstanding in the concept of percent and how to use it and so on. On the other hand, a major part of the incorrect responses is related to the lack of understanding of the nature of the problem, because understanding the nature of a problem is the first step in the solution of the problem [37]. As it was observed a considerable portion of the students with collective strategy and vice versa. After careful examination and analysis, the responses of students to test questions were categorized according to their collective or multiplicative understanding and their proportional reasoning and their proportional reasoning and achieve their goal in each of the nine levels of the theoretical framework of the study. Students’ Correct responses were classified in levels of 4 to 9, (replacement level to reformatting level) and similarly incorrect responses at levels of 1 to 3, (proposition level to the impossible problem level). The results showed that most of the answers were in level 4 (replacement). In conclusion, according to the distribution of students' responses to levels of the research framework, it can be concluded that the performance of students in posing problems related to proportional reasoning is placed at the replacement level. the outcome of the study is that the percent of the answers to the research questions at the second level (irrelevant problem) can be attributed to the inability of students to distinguish between proportional and non-proportional problems and inappropriate use of proportional reasoning in collective and multiplicative situations, which according to [55,5,7] is one of the most important weaknesses of the junior high school students in dealing with proportional reasoning. The root of this issue can be considered the lack of understanding of the existing multiplicative or multiplicative relations between the variables and insufficient understanding of concepts such as ratios, fractions, rates, and proportions. According to the results obtained from the study of the frequency of correct and false responses to research questions, it can be found that $86.1\%$ were correct and the rest of the answers, $13.9\%$, were incorrect indicating that most of the students were able to pose problems related to proportional reasoning. the results of the chi-square test and Spearman correlation coefficient at a $95\%$ confidence level showed that there is a direct and significant relationship between the performance of junior high school students in posing problems related to proportional reasoning and their educational grade with increasing the educational grade of students their problem posing skills in proportional reasoning problems increases too. in general, it makes the development of understanding proportional reasoning abilities. It was also observed that there is no significant relationship between students gender and their performance in posing problems related to proportional reasoning. Despite observing the general differences in the performance of students in gifted and general schools, this difference is not significant in terms of inferential statistics and chi-square correlation coefficients. 3- Conclusions According to the results of the analysis of students' responses to test questions, it is observed that students in response to test and problem posing displayed a variety of functional levels. Some of them did not understand the concept of the problem. Some posed a new problem by only changing the variables, names, numbers, etc. in the problem. Some added a new variable. Also, it can be noted that the fundamental changes in the problem, Contextualizing the problem and making the problem near to the real world, reversing and shifting of demand and data exchanged with each other. Among the causes of low success and low performance in problem posing in proportional reasoning situations, we can mention the lack of accuracy to the problem assumptions, insufficient understanding of rate concept, incorrect comparison of fractions, misunderstanding of the concept of percent and how to use it, etc. The results showed that more than half of the posed problems were in level five (replacement). Although a considerable portion of the responses was in level two (irrelevant problem). Also, educational grade and school type influence the performance of students' problem-posing problems with proportional reasoning, and at the $95\%$ confidence level, there was a significant difference between students' problem-posing ability and their educational grades. But there was no significant difference between boy and girl students and also students in general and gifted schools. پژوهش حاضر مهارت‌های طرح مسئلة دانش‌آموزان دورة اول متوسطه در زمینه استدلال تناسبی را مورد بررسی قرار می‌دهد. این پژوهش از نظر هدف، کاربردی و از نظر ماهیت و نوع مطالعه از نوع توصیفی و پیمایشی است. در این پژوهش، 442 نفر از دانش‌آموزان متوسطه اول شهرستان قزوین پس از انتخاب به روش نمونه‌گیری خوشه‌ای تصادفی، مورد مطالعه قرار گرفتند. ابزار اندازه‌گیری، آزمونی شامل 5 پرسش طرح مسئله مرتبط با استدلال تناسبی است که روایی صوری و محتوایی پرسشنامه توسط تعدادی از اساتید ریاضی و آموزش ریاضی تایید شد و ضریب آلفای کرونباخ 0.83 به دست آمد. تجزیه و تحلیل داده‌ها، با نرم افزارSPSS26 و روش‌های آمار توصیفی و استنباطی انجام شد. تحلیل نتایج حاکی از آن بود مهارت طرح مسئله دانش آموزان به طور کلی در سطح«جایگزینی» از چارچوب نظری پژوهش، معنادار ارزیابی می‌شود. همچنین تجزیه و تحلیل داده‌ها نشان داد که بین عملکرد دانش‌آموزان پایه‌های هفتم، هشتم و نهم در طرح مسائل مرتبط با استدلال تناسبی تفاوت معناداری وجود دارد و با افزایش پایه تحصیلی دانش آموزان شاهد افزایش مهارت طرح مسئله هستیم. از طرفی با بررسی تاثیر جنسیت دانش‌آموزان بر عملکرد طرح مسئله مشخص شد تفاوت عملکرد دانش‌آموزان دختر و پسر در طرح مسائل مرتبط با استدلال تناسبی معنادار نبود.همچنین مشاهده گردید که نوع مدرسه عامل تاثیر‌گذاری در طرح مسئله دانش‌آموزان در مسائل مرتبط با استدلال تناسبی محسوب نمی‌شود و بین عملکرد دانش‌آموزان مدارس عادی و تیزهوشان تفاوت معناداری وجود ندارد. نتایج این پژوهش می‌تواند در امر آموزش معلمان و تالیف کتاب‌های درسی مورد استفاده قرار گیرد.

https://math-sci.ui.ac.ir/article_27914_681301b1fca36f10da8f006a5d5157b3.pdf

دانشگاه اصفهان نشریه ریاضی و جامعه 2345-6493 8 3 2023 12 16 Prediction of gold price pattern by fractal interpolation پیش‌بینی الگوی قیمتی طلا با درونیابی فراکتال 119 144 27900 10.22108/msci.2023.137920.1583 FA حمیدرضا یوسف زاده گروه ریاضی، دانشگاه پیام نور، تهران، ایران اعظم فتوت گروه ریاضی، دانشگاه پیام نور، تهران، ایران Journal Article 2023 06 07 Analyzing and examining the price trend of an asset is a fundamental step in managing investment risk on that asset. Therefore, in markets, predicting the price trend of an asset is of special interest to traders and even plays a crucial role in a country's monetary policies. Based on this, in this paper, we will try to use the concept of fractal interpolation to predict the price trend of gold, given its price fluctuations and greater importance compared to other metals in markets. By analyzing the gold’s price trend using time series data with a fractal structure, we aim to determine the pattern of price trend to predict the price trend of gold ounces. Such an approach can provide the necessary tool to help investment decision-making in different time periods (short-term, medium-term, and possibly long-term). To achieve this, we first identify the presence of long-term memory in gold's price trend using the Hurst exponent. After confirming stability, we generate fractal data by calling the fractal interpolation algorithm and then predict the behavior of the corresponding time series data using a neural network algorithm based on fractal data. Finally, we compare the results obtained from calling the algorithms present in the literature on gold data. 1- Introduction The financial market data is unstable and irregular, and it sometimes contains missing data. To address these issues, researchers have developed different approaches, such as the fractal interpolation method. The aim of this study is to investigate the time series related to the gold market and determine whether it exhibits fractal characteristics. To generate fractal data, we can use the improved fractal interpolation algorithm (IFI). Then, we can use the support vector regression (SVR) algorithm, which is a type of machine learning method (SVM), to predict the price trend of gold in a specific time period. The price of an asset is directly proportional to its risk or fluctuation. Therefore, in the first phase, business owners and investors can determine the appropriate fee rate by analyzing the time series of data with fractal structure, and in the second phase, in the asset management phase, they can control losses caused by big fluctuations in returns and investment. The fluctuation of asset prices is a significant topic that has been studied by many researchers. They use linear or non-linear methods to predict and make appropriate use of these price fluctuations. In economic and social fields, fractal interpolation is often used to fit missing data and predict short-term trends due to the abundance of unstable and irregular data. In time series, data are collected at regular intervals according to a certain rule, and by analyzing the obtained data and with the help of different methods, the behavior of the series in the future can be approximately predicted. There are methods to determine whether a system is fractal or not, and thus, to calculate its fractal dimension. The method of calculating the dimension of the system depends on the type of its fractal structure. By determining the fractal dimension, the stability of the system can be investigated. Stability is a key factor in time series analysis, and the Hurst exponent is one of the criteria used to assess stability. Therefore, the fractal interpolation method can be implemented when the time series is stable. Hurst's exponent is a measure that identifies the long-term memory in time series. The R/S analysis criterion is one of the methods used to calculate the Hurst exponent. This criterion was first proposed by Hurst in his studies of natural phenomena such as the hydrological characteristics of the Nile Basin in 1951. In financial markets, the R/S analysis criterion is used to distinguish fractal from non-fractal systems, to identify the stability of trends, and to determine the length of life-time cycles. The range value of R in this index is equal to the difference between the lowest and highest deviation values from the cumulative average of the time series. Hurst normalized the value of the R range by using the standard deviation of the time series, relative to the fluctuations of the inputs of different time series, and defined the analysis criterion in a certain period of time. In this study, we aim to evaluate the performance of the fractal interpolation algorithm in predicting the trend of gold price based on time series data. The global ounce of gold is of great importance in world markets and experiences fluctuations, making it an ideal candidate for this analysis. To achieve this, we first calculate the Hurst exponent to determine the long-term memory of the gold price trend. We then generate fractal data using the fractal interpolation algorithm and apply the support vector regression (SVR) algorithm to predict the gold price trend. We compare the performance of Wang's algorithm and the Fracsion algorithm to determine the best method for predicting the gold price trend. The primary objective of this research is to examine the predictability of the price trend of gold and determine its price pattern. We analyze and evaluate this process by comparing it with past-oriented methods such as Wang's method. 2- Main Results We have analyzed the results of the Wang algorithm and the Fracsion algorithm separately for their ability to predict the final price of gold in 2020, 2021, and 2022 using well-known evaluation criteria.Two algorithms, the Wong and Fracsion methods, are presented below for the purpose of numerical analysis. Wang, et al., in 2018, using the fractal property of the Shanghai stock market and employing the contribution of algorithm of fractal interpolation and Support Vector Machin (SVM), have focused on predicting price patterns [21]. Algorithm 1 Wang Algorithm: Require: Gold closing price. Ensure: Prediction of gold price pattern in a short-term time interval. Start 1: Examine the stability and fractal structure of gold price data using the Hurst exponent. 2: Predict the data using the SVM algorithm. 3: Adjust the points obtained from Step 2 using the fractal interpolation algorithm. 4: Predict a short-term period based on the corresponding fractal interpolation function of the points from Step 3. End Algorithm 2 Fracsion Algorithm: Require: Gold closing price in a priod. Ensure: Prediction of gold price pattern in a short-term time interval. Start 1: Examine the stability and confirm the fractal structure of the data using the Hurst exponent. 2: Generate a set of fractal points for data with a fractal structure. 3: Call and train the SVR algorithm based on the obtained fractal data. 4: Predict the trend of gold price in a time interval based on the regression function obtained in Step 3. End The results for 2022 are presented in the following figure, for instance. The comparison of the results indicates that although both algorithms exhibit errors in price prediction, the adaptive Fracsion algorithm outperforms the Wang algorithm in predicting the price trend of gold in a short term memory. 3- Summary of Proofs/Conclusions In this paper, we analyzed the gold price time series in two phases. Firstly, by applying Hurst's method for each year, we investigated the stability of the gold price pattern (fractal structure of the system). Secondly, we utilized existing algorithms on the time series corresponding to the price of gold to predict its price trend, which provides important information to investors who seek to predict the gold market. The fundamental analysis of the gold market with the fractal structure presents a new approach to the analysis of the gold market and a non-linear perspective on this issue. The characteristic of long-term memory as well as the fractal structure of a time series corresponding to gold price data can play an effective role in predicting gold fluctuations and hence, the return based on gold fluctuations. However, predicting the price trend of the gold market in the long term is difficult due to many different factors that affect the global markets. Nevertheless, using algorithms for predicting the behavior of a time series, as a technical analysis tool, in stable economic and political conditions can have satisfactory results in predicting the price trend of gold in the short term. However, when there are conditions that strongly affect the price trend, such as war, global inflation, and in recent years, the global pandemic of Covid-19, algorithms cannot perform well in predicting the price pattern trend, because the structural order of the price pattern in such a situation, it is faced to distribution. تحلیل و بررسی روند قیمت یک دارایی، گام اساسی در مدیریت ریسک سرمایه‌گذاری بر روی آن دارایی به شمار می‌رود. بنابراین در بازارهای جهانی، پیش‌بینی روند قیمتی یک دارایی مورد توجه ویژه معامله‌گران می‌باشد و حتی در سیاست‌های پولی یک کشور نقش اساسی را ایفا می‌کند. براین اساس، در این مقاله سعی خواهیم کرد با توجه به نوسانات قیمتی و اهمیت بیشتر اونس جهانی طلا نسبت به سایر فلزات در بازارهای جهانی، با استفاده از مفهوم درونیابی فراکتال در پیش‌بینی روند قیمتی داده‌های با ساختار سری‌های زمانی، روند قیمتی این فلز گرانبها را مورد تجزیه و تحلیل قرار دهیم تا به کمک آن، الگوی روند قیمتی طلا را به‌منظور پیش‌بینی روند قیمتی اونس جهانی طلا تعیین کنیم. چنین رویکردی، ابزار لازم در جهت کمک به‌نحوه انجام سرمایه‌گذاری در دوره‌های زمانی مختلف (کوتاه‌مدت، میان‌مدت و احتمالاً بلند‌مدت) را می‌تواند فراهم نماید. برای رسیدن به این مهم در ابتدا به تشخیص وجود حافظه بلندمدت در روند قیمتی طلا، با استفاده از نمای هرست می‌پردازیم. پس از تأیید پایداری، با فراخوانی الگوریتم درونیابی فراکتال به تولید داده‌های فراکتالی می‌پردازیم و در پایان با فراخوانی الگوریتم مبتنی بر شبکه‌های عصبی بر روی داده‌های فراکتالی، به پیش‌بینی رفتار سری زمانی متناظر با داده‌های قیمتی طلا می‌پردازیم. در پایان به مقایسه نتایج حاصل از فراخوانی دو الگوریتم موجود در ادبیات موضوع بر روی داده‌های طلا می‌پردازیم.

https://math-sci.ui.ac.ir/article_27900_39e010229fb8aa5e61999c7b08fd3f90.pdf