The graph AG(R) of a commutative ring R with identity has an edge linking two unique vertices whe... more The graph AG(R) of a commutative ring R with identity has an edge linking two unique vertices when the product of the vertices equals the zero ideal and its vertices are the nonzero annihilating ideals of R. The annihilatingideal graph with respect to an ideal (I), which is denoted by AGI (R), has distinct vertices K and J that are adjacent if and only if KJ ⊆ I. Its vertices are {K | KJ ⊆ I for some ideal J and K, J I, K is a ideal of R}. The study of the two graphs AGI (R) and AG(R/I) and extending certain prior findings are two main objectives of this research. This studys among other things, the findings of this study reveal that AGI (R) is bipartite if and only if AGI (R) is triangle-free.
In emergency situations, accurate demand forecasting for relief materials such as food, water, an... more In emergency situations, accurate demand forecasting for relief materials such as food, water, and medicine is crucial for effective disaster response. This research is presented a novel algorithm to demand forecasting for relief materials using extended Case-Based Reasoning (CBR) with the best-worst method (BWM) and Hidden Markov Models (HMMs). The proposed algorithm involves training an HMM on historical data to obtain a set of state sequences representing the temporal fluctuations in demand for different relief materials. When a new disaster occurs, the algorithm first determines the current state sequence using the available data and searches the case library for past disasters with similar state sequences. The effectiveness of the proposed algorithm is demonstrated through experiments on real-world disaster data of Iran. Based on the results, the forecasting error index for four relief materials is less than 10%; therefore, the proposed CBR-BWM-HMM is a strong and robust algorithm.
From the viewpoint of "extra dimension detecting," the phenomenon of the transition of the free p... more From the viewpoint of "extra dimension detecting," the phenomenon of the transition of the free point particle into 3d space is investigated. In this way, we formulate the problem using the second-class constrained system. To investigate it using a gauge theoretical approach, we use two methods to convert its two second-class constraints to first-class ones. In symplectic embedding, we construct a pair of scaler and vector gauge potentials, which can be interpreted as interactions for detecting extra dimensions. A Wess-Zumino variable appears as a new coordinate in potentials, and the particle's mass plays the role of a globally conserved charge related to the constructed gauge theory for extra dimensions.
Ricci bi-conformal vector fields have find their place in geometry as well as in physical applica... more Ricci bi-conformal vector fields have find their place in geometry as well as in physical applications. In this paper, we consider the Siklos spacetimes and we determine all the Ricci bi-conformal vector fields on these spaces.
The representation theory of groups is one of the most interesting examples of the interaction be... more The representation theory of groups is one of the most interesting examples of the interaction between physics and pure mathematics, where group rings play the main role. The group ring RΓ is actually an associative ring that inherits the properties of the group Γ and the ring of coefficients R. In addition to the fact that the theory of group rings is clearly the meeting point of group theory and ring theory, it also has applications in algebraic topology, homological algebra, algebraic K-theory and algebraic coding theory. In this article, we provide a complete description of Gorenstein flat-cotorsion modules over the group ring RΓ, where Γ is a group and R is a commutative ring. It will be shown that if Γ Γ is a finite-index subgroup, then the restriction of scalars along the ring homomorphism RΓ → RΓ as well as its right adjoint RΓ ⊗ RΓ −, preserve the class of Gorenstein flat-cotorsion modules. Then, as a result, Serre's Theorem is proved for the invariant GhdRΓ, which refines the Gorenstein homological dimension of Γ over R, GhdRΓ, and is defined using flat-cotorsion modules. Moreover, we show that the inequality GFCD(RΓ) GFCD(R) + cdRΓ holds for the group ring RΓ, where GFCD(R) denotes the supremum of Gorenstein flat-cotorsion dimensions of all R-modules and cdRΓ is the cohomological dimension of Γ over R.
We present an enhanced approach to solving the combined non-linear time-dependent Burgers-Fisher ... more We present an enhanced approach to solving the combined non-linear time-dependent Burgers-Fisher equation, which is widely used in mathematical biology and has a broad range of applications. Our proposed method employs a modified version of the finite element method, specifically the virtual element method, which is a robust numerical approach. We introduce a virtual process and an Euler-backward scheme for discretization in the spatial and time directions, respectively. Our numerical scheme achieves optimal error rates based on the degree of our virtual space, ensuring high accuracy. We evaluate the efficiency and flexibility of our approach by providing numerical results on both convex and non-convex polygonal meshes. Our findings indicate that the proposed method is a promising tool for solving non-linear time-dependent equations in mathematical biology.
Graph coloring is the assignment of one color to each vertex of a graph so that two adjacent vert... more Graph coloring is the assignment of one color to each vertex of a graph so that two adjacent vertices are not of the same color. The graph coloring problem (GCP) is a matter of combinatorial optimization, and the goal of GCP is determining the chromatic number χ(G). Since GCP is an NP-hard problem, then in this paper, we propose a new approximated algorithm for finding the coloring number (it is an approximation of chromatic number) by using a graph adjacency matrix to colorize or separate a graph. To prove the correctness of the proposed algorithm, we implement it in MATLAB software, and for analysis in terms of solution and execution time, we compare our algorithm with some of the best existing algorithms that are already implemented in MATLAB software, and we present the results in tables of various graphs. Several available algorithms used the largest degree selection strategy, while our proposed algorithm uses the graph adjacency matrix to select the vertex that has the smallest degree for coloring. We provide some examples to compare the performance of our algorithm to other available methods. We make use of the Dolan-Moré performance profiles to assess the performance of the numerical algorithms, and demonstrate the efficiency of our proposed approach in comparison with some existing methods.
This article investigates and studies the dynamics of infectious disease transmission using a fra... more This article investigates and studies the dynamics of infectious disease transmission using a fractional mathematical model based on Caputo fractional derivatives. Consequently, the population studied has been divided into four categories: susceptible, exposed, infected, and recovered. The basic reproduction rate, existence, and uniqueness of disease-free as well as infected steady-state equilibrium points of the mathematical model have been investigated in this study. The local and global stability of both equilibrium points has been investigated and proven by Lyapunov functions. Vaccination and drug therapy are two controllers that may be used to control the spread of diseases in society, and the conditions for the optimal use of these two controllers have been prescribed by the principle of Pontryagin's maximum. The stated theoretical results have been investigated using numerical simulation. The numerical simulation of the fractional optimal control problem indicates that vaccination of the susceptible subjects in the community reduces horizontal transmission while applying drug control to the infected subjects reduces vertical transmission. Furthermore, the simultaneous use of both controllers is much more effective and leads to a rapid increase in the cured population and it prevents the disease from spreading and turning into an epidemic in the community.
Consider a simple, undirected graph G = (V, E), where A represents the adjacency matrix and Q rep... more Consider a simple, undirected graph G = (V, E), where A represents the adjacency matrix and Q represents the Laplacian matrix of G. The second smallest eigenvalue of Laplacian matrix of G is called the algebraic connectivity of G. In this article, we present a Python program for studying the Laplacian eigenvalues of a graph. Then, we determine the unique graph of minimum algebraic connectivity in the set of all tricyclic graphs.
Our examination of quadratic curvature functionals in Generalized Symmetric Spaces has resulted i... more Our examination of quadratic curvature functionals in Generalized Symmetric Spaces has resulted in the comprehensive classification of critical metric sets within diverse categories of these spaces.
The utilization of model checking has been suggested as a formal verification technique for analy... more The utilization of model checking has been suggested as a formal verification technique for analyzing critical systems. However, the primary challenge in applying to complex systems is the state space explosion problem. To address this issue, bisimulation minimization has emerged as a prominent method for reducing the number of states in a system, aiming to overcome the difficulties associated with the state space explosion problem. For systems with stochastic behaviors, probabilistic bisimulation is employed to minimize a given model, obtaining its equivalent form with fewer states. In this paper, we propose a novel technique to partition the state space of a given probabilistic model to its bisimulation classes. This technique uses the PRISM program of a given model and constructs some small versions of the model to train a classifier. It then applies supervised machine learning techniques to approximately classify the related partition. The resulting partition is then used to accelerate the standard bisimulation technique, significantly reducing the running time of the method. The experimental results show that the approach can decrease significantly the running time compared to state-of-the-art tools.
The Hub Location Problem (HLP) is a significant problem in combinatorial optimization consisting ... more The Hub Location Problem (HLP) is a significant problem in combinatorial optimization consisting of two main components: location and network design. The HLP aims to develop an optimal strategy for various applications, such as product distribution, urban management, sensor network design, computer network, and communication network design. Additionally, the upgrading location problem arises when modifying specific components at a cost is possible. This paper focuses on upgrading the uncapacitated multiple allocation p-hub median problem (u-UMApHMP), where a predetermined budget and bound of changes are given. The aim is to modify certain network parameters to identify the p-hub median that improves the objective function value concerning the modified parameters. We propose a non-linear mathematical formulation for u-UMApHMP to achieve this goal. Then, we employ the McCormick technique to linearize the model. Subsequently, we solve the linearized model using the CPLEX solver and the Benders decomposition method. Finally, we present experimental results to demonstrate the effectiveness of the proposed approach.
A significant development in the field of gyrogroups was the introduction of the space of all rel... more A significant development in the field of gyrogroups was the introduction of the space of all relativistically admissible velocities, which brought gyrogroups into the mainstream. A group has various generalizations, one of which is the notion of gyrogroups. Moreover, for any pair (a, b) in this structure, there exists an automorphism gyr[a, b] that fulfills left associativity and left loop property. The motivation behind this study is to generalize gyrogroups and semigroups, which has led to the introduction of gyrosemigroups. Accordingly, in this paper, some classes of gyrosemigroups are presented. Also, all gyrosemigroups of order 2 are characterized. Furthermore, the gyrosemigroups with an identity or a zero are studied.
This paper introduces a novel approach to enhance the performance of the stochastic gradient des... more This paper introduces a novel approach to enhance the performance of the stochastic gradient descent (SGD) algorithm by incorporating a modified decay step size based on $\frac{1}{\sqrt{t}}$. The proposed step size integrates a logarithmic term, leading to the selection of smaller values in the final iterations. Our analysis establishes a convergence rate of $O(\frac{\ln T}{\sqrt{T}})$ for smooth non-convex functions without the Polyak-Łojasiewicz condition. To evaluate the effectiveness of our approach, we conducted numerical experiments on image classification tasks using the Fashion-MNIST and CIFAR10 datasets, and the results demonstrate significant improvements in accuracy, with enhancements of $0.5\%$ and $1.4\%$ observed, respectively, compared to the traditional $\frac{1}{\sqrt{t}}$ step size. The source code can be found at https://github.com/Shamaeem/LNSQRTStepSize.
The graph AG(R) of a commutative ring R with identity has an edge linking two unique vertices whe... more The graph AG(R) of a commutative ring R with identity has an edge linking two unique vertices when the product of the vertices equals the zero ideal and its vertices are the nonzero annihilating ideals of R. The annihilatingideal graph with respect to an ideal (I), which is denoted by AGI (R), has distinct vertices K and J that are adjacent if and only if KJ ⊆ I. Its vertices are {K | KJ ⊆ I for some ideal J and K, J I, K is a ideal of R}. The study of the two graphs AGI (R) and AG(R/I) and extending certain prior findings are two main objectives of this research. This studys among other things, the findings of this study reveal that AGI (R) is bipartite if and only if AGI (R) is triangle-free.
In emergency situations, accurate demand forecasting for relief materials such as food, water, an... more In emergency situations, accurate demand forecasting for relief materials such as food, water, and medicine is crucial for effective disaster response. This research is presented a novel algorithm to demand forecasting for relief materials using extended Case-Based Reasoning (CBR) with the best-worst method (BWM) and Hidden Markov Models (HMMs). The proposed algorithm involves training an HMM on historical data to obtain a set of state sequences representing the temporal fluctuations in demand for different relief materials. When a new disaster occurs, the algorithm first determines the current state sequence using the available data and searches the case library for past disasters with similar state sequences. The effectiveness of the proposed algorithm is demonstrated through experiments on real-world disaster data of Iran. Based on the results, the forecasting error index for four relief materials is less than 10%; therefore, the proposed CBR-BWM-HMM is a strong and robust algorithm.
From the viewpoint of "extra dimension detecting," the phenomenon of the transition of the free p... more From the viewpoint of "extra dimension detecting," the phenomenon of the transition of the free point particle into 3d space is investigated. In this way, we formulate the problem using the second-class constrained system. To investigate it using a gauge theoretical approach, we use two methods to convert its two second-class constraints to first-class ones. In symplectic embedding, we construct a pair of scaler and vector gauge potentials, which can be interpreted as interactions for detecting extra dimensions. A Wess-Zumino variable appears as a new coordinate in potentials, and the particle's mass plays the role of a globally conserved charge related to the constructed gauge theory for extra dimensions.
Ricci bi-conformal vector fields have find their place in geometry as well as in physical applica... more Ricci bi-conformal vector fields have find their place in geometry as well as in physical applications. In this paper, we consider the Siklos spacetimes and we determine all the Ricci bi-conformal vector fields on these spaces.
The representation theory of groups is one of the most interesting examples of the interaction be... more The representation theory of groups is one of the most interesting examples of the interaction between physics and pure mathematics, where group rings play the main role. The group ring RΓ is actually an associative ring that inherits the properties of the group Γ and the ring of coefficients R. In addition to the fact that the theory of group rings is clearly the meeting point of group theory and ring theory, it also has applications in algebraic topology, homological algebra, algebraic K-theory and algebraic coding theory. In this article, we provide a complete description of Gorenstein flat-cotorsion modules over the group ring RΓ, where Γ is a group and R is a commutative ring. It will be shown that if Γ Γ is a finite-index subgroup, then the restriction of scalars along the ring homomorphism RΓ → RΓ as well as its right adjoint RΓ ⊗ RΓ −, preserve the class of Gorenstein flat-cotorsion modules. Then, as a result, Serre's Theorem is proved for the invariant GhdRΓ, which refines the Gorenstein homological dimension of Γ over R, GhdRΓ, and is defined using flat-cotorsion modules. Moreover, we show that the inequality GFCD(RΓ) GFCD(R) + cdRΓ holds for the group ring RΓ, where GFCD(R) denotes the supremum of Gorenstein flat-cotorsion dimensions of all R-modules and cdRΓ is the cohomological dimension of Γ over R.
We present an enhanced approach to solving the combined non-linear time-dependent Burgers-Fisher ... more We present an enhanced approach to solving the combined non-linear time-dependent Burgers-Fisher equation, which is widely used in mathematical biology and has a broad range of applications. Our proposed method employs a modified version of the finite element method, specifically the virtual element method, which is a robust numerical approach. We introduce a virtual process and an Euler-backward scheme for discretization in the spatial and time directions, respectively. Our numerical scheme achieves optimal error rates based on the degree of our virtual space, ensuring high accuracy. We evaluate the efficiency and flexibility of our approach by providing numerical results on both convex and non-convex polygonal meshes. Our findings indicate that the proposed method is a promising tool for solving non-linear time-dependent equations in mathematical biology.
Graph coloring is the assignment of one color to each vertex of a graph so that two adjacent vert... more Graph coloring is the assignment of one color to each vertex of a graph so that two adjacent vertices are not of the same color. The graph coloring problem (GCP) is a matter of combinatorial optimization, and the goal of GCP is determining the chromatic number χ(G). Since GCP is an NP-hard problem, then in this paper, we propose a new approximated algorithm for finding the coloring number (it is an approximation of chromatic number) by using a graph adjacency matrix to colorize or separate a graph. To prove the correctness of the proposed algorithm, we implement it in MATLAB software, and for analysis in terms of solution and execution time, we compare our algorithm with some of the best existing algorithms that are already implemented in MATLAB software, and we present the results in tables of various graphs. Several available algorithms used the largest degree selection strategy, while our proposed algorithm uses the graph adjacency matrix to select the vertex that has the smallest degree for coloring. We provide some examples to compare the performance of our algorithm to other available methods. We make use of the Dolan-Moré performance profiles to assess the performance of the numerical algorithms, and demonstrate the efficiency of our proposed approach in comparison with some existing methods.
This article investigates and studies the dynamics of infectious disease transmission using a fra... more This article investigates and studies the dynamics of infectious disease transmission using a fractional mathematical model based on Caputo fractional derivatives. Consequently, the population studied has been divided into four categories: susceptible, exposed, infected, and recovered. The basic reproduction rate, existence, and uniqueness of disease-free as well as infected steady-state equilibrium points of the mathematical model have been investigated in this study. The local and global stability of both equilibrium points has been investigated and proven by Lyapunov functions. Vaccination and drug therapy are two controllers that may be used to control the spread of diseases in society, and the conditions for the optimal use of these two controllers have been prescribed by the principle of Pontryagin's maximum. The stated theoretical results have been investigated using numerical simulation. The numerical simulation of the fractional optimal control problem indicates that vaccination of the susceptible subjects in the community reduces horizontal transmission while applying drug control to the infected subjects reduces vertical transmission. Furthermore, the simultaneous use of both controllers is much more effective and leads to a rapid increase in the cured population and it prevents the disease from spreading and turning into an epidemic in the community.
Consider a simple, undirected graph G = (V, E), where A represents the adjacency matrix and Q rep... more Consider a simple, undirected graph G = (V, E), where A represents the adjacency matrix and Q represents the Laplacian matrix of G. The second smallest eigenvalue of Laplacian matrix of G is called the algebraic connectivity of G. In this article, we present a Python program for studying the Laplacian eigenvalues of a graph. Then, we determine the unique graph of minimum algebraic connectivity in the set of all tricyclic graphs.
Our examination of quadratic curvature functionals in Generalized Symmetric Spaces has resulted i... more Our examination of quadratic curvature functionals in Generalized Symmetric Spaces has resulted in the comprehensive classification of critical metric sets within diverse categories of these spaces.
The utilization of model checking has been suggested as a formal verification technique for analy... more The utilization of model checking has been suggested as a formal verification technique for analyzing critical systems. However, the primary challenge in applying to complex systems is the state space explosion problem. To address this issue, bisimulation minimization has emerged as a prominent method for reducing the number of states in a system, aiming to overcome the difficulties associated with the state space explosion problem. For systems with stochastic behaviors, probabilistic bisimulation is employed to minimize a given model, obtaining its equivalent form with fewer states. In this paper, we propose a novel technique to partition the state space of a given probabilistic model to its bisimulation classes. This technique uses the PRISM program of a given model and constructs some small versions of the model to train a classifier. It then applies supervised machine learning techniques to approximately classify the related partition. The resulting partition is then used to accelerate the standard bisimulation technique, significantly reducing the running time of the method. The experimental results show that the approach can decrease significantly the running time compared to state-of-the-art tools.
The Hub Location Problem (HLP) is a significant problem in combinatorial optimization consisting ... more The Hub Location Problem (HLP) is a significant problem in combinatorial optimization consisting of two main components: location and network design. The HLP aims to develop an optimal strategy for various applications, such as product distribution, urban management, sensor network design, computer network, and communication network design. Additionally, the upgrading location problem arises when modifying specific components at a cost is possible. This paper focuses on upgrading the uncapacitated multiple allocation p-hub median problem (u-UMApHMP), where a predetermined budget and bound of changes are given. The aim is to modify certain network parameters to identify the p-hub median that improves the objective function value concerning the modified parameters. We propose a non-linear mathematical formulation for u-UMApHMP to achieve this goal. Then, we employ the McCormick technique to linearize the model. Subsequently, we solve the linearized model using the CPLEX solver and the Benders decomposition method. Finally, we present experimental results to demonstrate the effectiveness of the proposed approach.
A significant development in the field of gyrogroups was the introduction of the space of all rel... more A significant development in the field of gyrogroups was the introduction of the space of all relativistically admissible velocities, which brought gyrogroups into the mainstream. A group has various generalizations, one of which is the notion of gyrogroups. Moreover, for any pair (a, b) in this structure, there exists an automorphism gyr[a, b] that fulfills left associativity and left loop property. The motivation behind this study is to generalize gyrogroups and semigroups, which has led to the introduction of gyrosemigroups. Accordingly, in this paper, some classes of gyrosemigroups are presented. Also, all gyrosemigroups of order 2 are characterized. Furthermore, the gyrosemigroups with an identity or a zero are studied.
This paper introduces a novel approach to enhance the performance of the stochastic gradient des... more This paper introduces a novel approach to enhance the performance of the stochastic gradient descent (SGD) algorithm by incorporating a modified decay step size based on $\frac{1}{\sqrt{t}}$. The proposed step size integrates a logarithmic term, leading to the selection of smaller values in the final iterations. Our analysis establishes a convergence rate of $O(\frac{\ln T}{\sqrt{T}})$ for smooth non-convex functions without the Polyak-Łojasiewicz condition. To evaluate the effectiveness of our approach, we conducted numerical experiments on image classification tasks using the Fashion-MNIST and CIFAR10 datasets, and the results demonstrate significant improvements in accuracy, with enhancements of $0.5\%$ and $1.4\%$ observed, respectively, compared to the traditional $\frac{1}{\sqrt{t}}$ step size. The source code can be found at https://github.com/Shamaeem/LNSQRTStepSize.
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