https://mmr1.khu.ac.ir/ Mathematical Research en daily 1 Mon, 08 Jun 2026 00:00:00 +0330 Mon, 08 Jun 2026 00:00:00 +0330 https://mmr1.khu.ac.ir/article_8929.html In this paper, assuming that G is a discrete topological group and H is a subgroup of G, we introduce the notion of a (G, H)-operator system and define the Hamana boundary for pairs of groups (G, H). In fact, we show that the Furstenberg boundary for pairs of groups is isomorphic to the Hamana boundary for pairs of groups. In other words, the natural extension of the classical theory of Furstenberg boundaries remains valid for the Furstenberg boundary associated with pairs of groups (G, H). https://mmr1.khu.ac.ir/article_8930.html In statistics, the existence of independence between statistics is of particular importance. Without calculating the joint distribution of two statistics, the existence of this independence is proven by meeting the necessary conditions, including the sufficient completeness of one statistic and the ancillarity of the other. In this paper, generalizations of this issue, as well as the cases of countably completeness and Bayesian framework , will be examined. Finally, under the action of topological groups, this fact is generalized and the independence of a sufficiently equivariant and invariant function is proven. https://mmr1.khu.ac.ir/article_8931.html The classical functional linear regression model with a scalar response has been widely used. However, the local effects of the predictor function on the response cannot be evaluated with this model, as it considers only a weighted average of the entire predictor trajectory in relation to the response. To ad-dress this issue, we propose a generalization of the classical functional linear regression model by adding an unknown number of points of impact, where the predictor values at these points have a significant effect on the response. The points of impact and their coefficients, the slope function, the eigenvalues, and the eigenfunctions of the covariance operator of the predictor are unknown and need to be estimated. In this paper, we derive the convergence rate of a quantity constructed based on these unknown parameters. This result can be used in future statistical inferences for this model, including the calculation of confidence in-tervals and bootstrap confidence intervals. Additionally, a simulation study has been conducted to evaluate the obtained results. https://mmr1.khu.ac.ir/article_8932.html The Quicksort algorithm sorts the data stored in an array. The Partial Quicksort algorithm sorts the l smallest numbers out of numbers stored in an array of length n. The Quicksort on the fly provides online first the smallest, then the second smallest and so on. If we stop at the l-th smallest, we obtain Partial Quicksort. In this paper, we analyze Ynln, a correctly normalized version of the number of comparisons needed to sort the l smallest numbers out of numbers stored in an array of length n, by the Median Quicksort algorithm. We show that whenever l≔nt، t∈0,1 and n→∞, then the process Ynt≔Ynntn converges to Yt in the space of cadlag functions. https://mmr1.khu.ac.ir/article_8933.html Let $G$ be a finite group. A subset $X subseteq G$ is called a pairwise non-commuting set if $xy neq yx$ for all distinct elements $x, y in X$. If $|X| geq |Y|$ for every other pairwise non-commuting set $Y$ in $G$, then $X$ is called a maximal pairwise non-commuting set. A $p$-group $G$ is defined as an $A_2$-group if it contains a nonabelian subgroup of index $p$ such that all subgroups of index $p^2$ are abelian. In this paper, we determine the exact maximum size of such sets in finite $A_2$-groups. https://mmr1.khu.ac.ir/article_8934.html Background Cancer is one of the great human challenges in all countries, both advanced and developing. Cancer treatment management can include surgery, chemotherapy, or radiation therapy [1]. Radiation therapy is done in two ways: Teletherapy and Brachytherapy. Brachytherapy involves the use of radiation sources to treat cancer by irradiating cancerous tissue from within the patient’s body [2]. But the dose and how to use this method has always been questionable for researchers. The purpose of this paper is to present a two-objective optimization model that directly summarizes the multidimensional nature of the problem of brachytherapy treatment planning in two objectives. Therefore, it simplifies the decision-making process of treatment planners when creating a clinically acceptable plan. Methods In the present study, the dose prescribed for an organ was evaluated by dosimetric indices listed in Table 1. For the present study, data from patients in the age range of 50 to 74 and mean age 62 years with a wide range of prostate volume between 23 and 103 cubic centimeters, and for the treatment of prostate cancer by brachytherapy from the Academic Medical Center (AMC, Amsterdam, the Netherlands) had participated. To compare brachytherapy programs with high interstitial dose, the dose rate was calculated with 192Ir beam with a radiation dose of 13 Gy, according to the standard protocol TG-43. To begin with, computed tomography (CT) scans or magnetic resonance imaging (MRI) were taken from the patients pelvis, and entered into the treatment planning software for use in treatment planning sessions. BT treatment planners and specialists then determined the input catheters, target volumes, and OARs obtained from the medical images. Depending on the size and exact location of the target volumes, between 14 and 20 catheters entered the patient’s body, reaching the target volumes. After designing and approving an acceptable treatment plan, the catheters inserted into the patient’s body were connected to a retractor that controls the movement of the radiation source. After the treatment program, the source was returned to the retractor. Results According to this study, four multi-objective evolutionary algorithms, NSGA-II, MOEA / D, SPEA-II algorithms and G-NSGA-II (gray multi-objective evolutionary algorithm) have been used. Instead of providing only one optimal answer, these algorithms create a set of Pareto optimal answers, none of which is superior to the other. But they have better results compared to other methods. The results show that the G-NSGA-II (gray multi-objective evolutionary algorithm) is the best multi-objective evolutionary algorithm for both the quality of the answers and the diversity of the answers, as well as the gray structure and the gray operators used in it. Dose optimization is a problem in brachytherapy that uses the interdependence between decision variables to solve it efficiently. These results indicate the effectiveness of this algorithm in shortening the treatment period and increasing the accuracy of the brachytherapy program. Conclusion According to the obtained results, it can be stated that whether the main goal is the maximum coverage or the goal is the shortest possible time to reach the coverage above 95%, the best algorithm that can get a good answer for each patient is the G-NSGA-II (gray multi-objective evolutionary algorithm). Ethical Considerations and Compliance with ethical guidelines All ethical principles were considered in this article. In order to observe the ethical points in the truth, the researcher undertook to keep all the information in the questionnaire confidential. It also provides the results of the research to the respondents. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for profit sectors. Conflicts of interest The authors declared no conflict of interest https://mmr1.khu.ac.ir/article_8935.html Delay systems are important and it is difficult to obtain their analytical solution. Therefore, numerical methods are used to solve optimal control problems with delay systems. Solving optimal control problems using orthogonal functions has attracted much attention in recent years. Specific orthogonal functions that have been used so far in this field are the Walsh, block-pulse, and Laguerre functions. In this paper, the optimal control piecewise constant delay systems using shifted chebyshev functions are transformed into a nonlinear programming problem, so that by solving it, the approximate solution of the main problem is obtained. Numerical examples are given to evaluate the efficiency of the method. https://mmr1.khu.ac.ir/article_8936.html The space of trace class or nuclear operators was noticed by Grothendieck. In recent years, Neufang defined a new product on the space of nuclear operators making it into a Banach algebra. Many scientists investigated the properties of this Banach algebra. In this article, we investigate the existence of (weakly) compact left and right multipliers on the closed ideal of the convolution algebra of nuclear operators. This characterize is employed to find a necessary and sufficient condition for to be an ideal in its second dual space, where the second dual is equipped with an Arens product. https://mmr1.khu.ac.ir/article_8937.html The primary objective of this study is to develop a hybrid numerical scheme for approximating the solutions of the one-dimensional fractional Burgers’ equation governed by a variable-order Riemann–Liouville derivative. In the proposed framework, spatial discretization is accomplished via a spectral element method based on the collocation approach, whereas temporal discretization is handled using a Crank–Nicolson Leap-Frog scheme. To enhance the stability of the derivative matrix resulting from the spectral element formulation, an adaptive penalty enforcement strategy is employed, wherein the penalty coefficient is dynamically adjusted in response to the local variations in the fractional derivative order. The accuracy, stability, and computational efficiency of the proposed method are validated through a series of numerical experiments, demonstrating its robustness and effectiveness across diverse test cases. https://mmr1.khu.ac.ir/article_8938.html The present study presents a new and applicable method to analyze and decompose the weights set in the weighted sum method. To do this, a linear programming (LP) problem is first designed for each extreme point of the expanded outcome space. Then, it is shown that the extreme optimal solutions of the presented LPs are the gradient vectors of the defining (weak) non-dominated hyperplanes of the expanded outcome space. Then, the set of eligible weights in the weighted sum method is comprehensively studied. The polyhedral and structural soft engineering of the set of all weights yielding the same non-dominated solution is meticulously checked. As a consequence, some structural insights related to the properties of the set of (weak) non-dominated solutions are discussed. Finally, the advantage and validity of the method is presented by a numerical example. https://mmr1.khu.ac.ir/article_8939.html Let R be a commutative noetherian ring and I be an ideal of R. The aim of this paper is to establish the local-global principle for the generalized local homology modules a_I(M,N), where a_I(M,N) is the smallest integer such that the generalized local homology module of M,N is not artinian. For a finitely generated R- module M and a linearly compact R-module N with the set CoassR(HI_aI (M;N)(M;N)) finite, we show that aI (M;N) = inffaIRp (Mp;p N)jp 2 Spec(R)g: https://mmr1.khu.ac.ir/article_8940.html Introduction As the Forex market becomes increasingly complex, accurate trend forecasting has gained critical importance for traders and researchers. Unlike most studies that focus on price prediction, this paper introduces a novel bi-timeframe framework (1-hour and 4-hour) that integrates the Ichimoku Kinko Hyo strategy with deep learning models to predict directional movements in currency pairs. Materials and Methods The approach employs convolutional neural networks (CNNs) and hybrid architectures (CNN-LSTM, CNN-GRU), with hyperparameters optimized using the Particle Swarm Optimization Algorithm (PSO). Models are trained on historical EURUSD data (2019--2024) from MetaTrader5 and evaluated on eight highly correlated ($pm$80%) currency pairs. Due to the limitations of regression metrics (MAE, MSE, MAPE) in trading contexts, regression outputs are used solely for 4-hour trend classification, with Accuracy and F1-score as primary performance measures. Results and Discussion Results show that PSO-optimized models, particularly Ichimoku-CNN-GRU-PSO (ICGP), consistently outperform standard variants, achieving the highest Accuracy (up to 80.23% on USDSGD) and F1-score across most pairs. Conclusion The findings confirm that Ichimoku-based features, combined with hybrid deep learning and metaheuristic optimization, significantly enhances trend forecasting reliability and generalization in volatile financial markets.