Decomposition Problem Research Articles

Optimization of multi-user fairness in RIS-enhanced UAV secure transmission systems

This paper proposes a reconfigurable intelligent surface (RIS)-enhanced unmanned aerial vehicle (UAV) secure communicating scheme with multiple mutually-untrusted ground users (GUs). To resist eavesdropping, a pre-defined artificial noise (AN) is added to the intended signal at UAV for scheduled GUs in each time slot. The GU scheduling schemes, UAV trajectories and transmit power allocations in different time slots are jointly designed to ensure communication security for all GUs. To boost secrecy rate of the system while ensuring secure fairness among GUs, we conduct a joint optimization process, involving optimizing UAV trajectories, RIS phase shifts, GU scheduling schemes, and UAV transmit power splitting factors, adhering to the constraints of UAV mobility, RIS phase shifts and other related constraints. To solve this non-convex optimization, we utilize the block coordinate descent (BCD) method for problem decomposition, coupled with an iterative algorithm to optimize the resulting sub-problems. To further solve the non-convex sub-problems, we employ the variable substitution to convexify the transmit power allocation, the semi-deterministic relaxation (SDR) to convexify the RIS passive beamforming and successive convex approximation (SCA) to convexify UAV mobility constraints. Simulation results show the convergence and effectiveness of the proposed scheme, compared to the benchmark schemes, the average worst-case secrecy rate increases by 32.33%, 60.38% and 80.09‘% respectively.

The Weighted p-Norm Weight Set Decomposition for Multiobjective Discrete Optimization Problems

Many solution algorithms for multiobjective optimization problems are based on scalarization methods that transform the multiobjective problem into a scalar-valued optimization problem. In this article, we study the theory of weighted p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-norm scalarizations. These methods minimize the distance induced by a weighted p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-norm between the image of a feasible solution and a given reference point. We provide a comprehensive theory of the set of eligible weights and, in particular, analyze the topological structure of the normalized weight set. This set is composed of connected subsets, called weight set components which are in a one-to-one relation with the set of optimal images of the corresponding weighted p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-norm scalarization. Our work generalizes and complements existing results for the weighted sum and the weighted Tchebycheff scalarization and provides new insights into the structure of the set of all Pareto optimal solutions.

On Students' Computational Thinking Skills for Solving SRAC and its Theoretical Framework on Multi-Step Time Series Forecasting on River Erosion using GNN under RBL-STEM Learning Stages

Computational thinking involves the use of computer science principles to solve complex problems, extending beyond simple programming to various life applications. In today’s educational landscape, the promotion of these skills in the classroom is critical, yet students’ computational thinking skills remain underdeveloped due to inadequate learning models. Key indicators of computational thinking include problem decomposition, algorithmic thinking, pattern recognition, abstraction, and generalization. This study presents RBL-STEM learning activities aimed at enhancing students’ computational thinking through solving the Strong Rainbow Antimagic Coloring or SRAC problem and applying it to multi-step time series forecasting on river erosion using Graph Neural Networks or GNN. The research adopts a qualitative narrative method, beginning with the development of a prototype for multi-step time series forecasting on river erosion using SRAC and GNN, and progressing to the formulation of RBL-STEM learning steps. The results include a comprehensive RBL-STEM learning framework ready for implementation in future research. Learning framework offers student and educator a structured approach to integrating STEM on real life issues. By employing RBL-STEM, students are encouraging to solve river erosion problem systematically based on RBL stages. These finding suggest that the implementation of RBL-STEM with innovative mathematical problems such as SRAC can enhance students’ combinatorial skills, leading to practical solutions for everyday issues through education.

Packing spheres with quasi-containment conditions

A novel sphere packing problem is introduced. A maximum number of spheres of different radii should be placed such that the spheres do not overlap and their centers fulfill a quasi-containment condition. The latter allows the spheres to lie partially outside the given cuboidal container. Moreover, specified ratios between the placed spheres of different radii must be satisfied. A corresponding mixed-integer nonlinear programming model is formulated. It enables the exact solution of small instances. For larger instances, a heuristic strategy is proposed, which relies on techniques for the generation of feasible points and the decomposition of open dimension problems. Numerical results are presented to demonstrate the viability of the approach.

Signal Decomposition for Monitoring Systems of Processes

This article is devoted to the problem of signal decomposition into periodic and aperiodic components. According to the proposed approach, there is no need to evaluate the aperiodic component as a difference between the total signal of its periodic components. This research aims to create a general analytical approach that combines the Fourier and Maclaurin series methodologies into a single comprehensive series. As a result, analytical expressions for determining deposition coefficients were established for an aperiodic signal with a monoharmonic overlay. Recurrence relations were established to determine the coefficients of this series. These relations allow direct integrations of the obtained values of integrals to be avoided. The evaluated numerical values of the coefficients are also presented graphically and tabulated. It was proven that the values of these coefficients are universal numbers since they do not depend on the period/frequency of oscillations. The reliability of the proposed approach was confirmed by the fact that the evaluated coefficients are equal to the Fourier series coefficients in the case of a periodic signal. Also, for an aperiodic signal, these coefficients were reduced to the coefficients of the Maclaurin series. The usability of the proposed generalized analytical approach for signal decomposition is for control and monitoring systems of processes.

A Novel Privacy Preservation Scheme by Matrix Factorized Deep Autoencoder

Data transport entails substantial security to avoid unauthorized snooping as data mining yields important and quite often sensitive information that must be and can be secured using one of the myriad Data Privacy Preservation methods. This study aspires to provide new knowledge to the study of protecting personal information. The key contributions of the work are an imputation method for filling in missing data before learning item profiles and the optimization of the Deep Auto-encoded NMF with a customizable learning rate. We used Bayesian inference to assess imputation for data with 13%, 26%, and 52% missing at random. By correcting any inherent biases, the results of decomposition problems may be enhanced. As the statistical analysis tool, MAPE is used. The proposed approach is evaluated on the Wiki dataset and the traffic dataset, against state-of-the-art techniques including BATF, BGCP, BCPF, and modified PARAFAC, all of which use a Bayesian Gaussian tensor factorization. Using this approach, the MAPE index is decreased for data which avails privacy safeguards than its corresponding original forms.

Apportionable matrices and gracefully labelled graphs

To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There are connections between apportionment and classical graph decomposition problems, including graceful labellings of graphs, Hadamard matrices, and equiangular lines, and potential applications to instantaneous uniform mixing in quantum walks. The connection between apportionment and graceful labellings allows the construction of apportionable matrices from trees. A generalization of the well-known Eigenvalue Interlacing Inequalities using graceful labellings is also presented. It is shown that every rank one matrix can be apportioned by a unitary similarity, but there are 2×2 matrices that cannot be apportioned. A necessary condition for a matrix to be apportioned by unitary matrix is established. This condition is used to construct a set of matrices with nonzero Lebesgue measure that are not apportionable by a unitary matrix.

THE METHOD OF RANKING EFFECTIVE PROJECT SOLUTIONS IN CONDITIONS OF INCOMPLETE CERTAINTY

The subject of research in the article is the process of ranking options in support systems for project decision-making under conditions of incomplete certainty. The goal of the work is to increase the efficiency of technologies for automated design of complex systems due to the development of a combined method of ranking effective options for building objects in conditions of incomplete certainty of input data. The following tasks are solved in the article: analysis of the current state of the problem of ranking options in support systems for project decision-making; decomposition of problems of system optimization of complex design objects and support of project decision-making; development of a variant ranking method that combines the procedures of lexicographic optimization and cardinal ordering in conditions of incomplete certainty of input data. The following methods are used: systems theory, utility theory, optimization, operations research, interval and fuzzy mathematics. Results. According to the results of the analysis of the problem of supporting project decision-making, the existence of the problem of correctly reducing subsets of effective options for ranking, taking into account factors that are difficult to formalize and the experience of the decision-maker (DM), was established. Decomposition of the problems of system optimization of complex design objects and support for project decision-making was carried out. For the case of ordinalistic presentation of preferences between local criteria, an estimate of the size of the rational reduction of subsets of optimal and suboptimal options for each of the indicators is proposed. Its use allows for one approach to obtain a subset of effective variants of a given capacity for analysis and final selection of the DM. A method of transforming the ordinalistic presentation of preferences between local criteria to their quantitative presentation in the form of weighting coefficients is proposed. Conclusions. The developed methods expand the methodological foundations of the automation of processes supporting the adoption of multi-criteria project decisions. They make it possible to correctly reduce the set of effective alternatives in conditions of incomplete certainty of the input data for the final choice, taking into account factors that are difficult to formalize, knowledge and experience of ODA. The practical use of the obtained results will allow to reduce the time and capacity complexity of the procedures for supporting project decision-making, and due to the use of the technology of selection of subsets of effective options with intervally specified characteristics - to guarantee the quality of project decisions and to provide a more complete assessment of them.

In Situ Hybrid Crosslinking Polymerization of Nanoparticles for Composite Polymer Electrolytes to Achieve Highly‐Stable Solid Lithium–Metal Batteries

AbstractThe composite solid electrolyte, which combines the advantages of inorganic conductors and organic polymer electrolytes, has become a crucial strategy for the construction of solid‐state batteries. However, the physical deposition and agglomeration of traditional composite fillers seriously affect their structural uniformity and ion transport performance, and the construction of uniform and stable composite electrolytes is still an insurmountable challenge. Herein, a strategy of in situ hybrid crosslinking polymerization of TiO2 nanoparticles is proposed for highly stable polymer composite electrolytes (NHCPE) with an ultrahigh ionic conductivity of 1.74 × 10−3 S cm−1 at 25 °C, and a high lithium‐ion transference number of 0.725. These properties enable the composed lithium symmetric battery to be stably deposited/plating off at 0.5 mA cm−2 for more than 1000 h. Moreover, the assembled LFP|PDOL@nanoTiO2|Li battery exhibits a superior specific discharge capacity of 142.6 mAh g−1 at 1 C and 25 °C, and an ultrahigh capacity retention rate of 90% after 1000 cycles. The proposed PDOL@nanoTiO2 NHCPE greatly inhibits the defects of easy agglomeration of composite electrolytes, solves the problems of easy decomposition, low thermal stability, and poor safety of polyether electrolytes, and opens up a new way for the design and industrial application of high‐stability composite polymer electrolytes.

Using Rotations to Control Observable Relativistic Effects

This paper examines the possibility of controlling the outcome of measured (flat space-time) relativistic effects, such as time dilation or length contractions, using pure rotations and their nontrivial interactions with Lorentz boosts in the isometry group SO+(3,1). In particular, boost contributions may annihilate leaving only a geometric phase (Wigner rotation), which we see in the complex solutions of the generalized Euler decomposition problem in R3. We consider numerical examples involving specific matrix factorizations, along with possible applications in special relativity, electrodynamics and quantum scattering. For clearer interpretation and simplified calculations we use a convenient projective biquaternion parametrization which emphasizes the geometric phases and for a large class of problems allows for closed-form solutions in terms of only rational functions.

A key exchange protocol and a cryptosystem based on the generalized decomposition problem

The most crucial factor in the internet-based environment is the encryption and security of personal data. In this paper, a key exchange protocol based on Generalized Decomposition Problem (GDP) over semiring is proposed. The security and complexity analysis of the proposed protocol is examined. Also, based on the proposed key exchange protocol, an ElGamal cryptosystem for securing Sensitive Health Information (SHI) in Healthcare systems is presented. Abbreviations: CSP: conjugacy search problem; DHP: diffie hellman problem; DLP: discrete logarithm problem; DP: decomposition problem; FDLP: factorization discrete logarithm problem; GDP: generalized decomposition problem; RSA: rivest shamir adleman; SHI: sensitive health information.

Enhancing computational thinking and spatial reasoning skills in gamification programming learning: A comparative study of tangible, block and paper‐and‐pencil tools

AbstractTangible programming tools have become a mainstream teaching aid in gamification programming learning (GPL) due to their interactivity and ability to enhance novice learners' computational thinking and spatial reasoning skills. However, comparing the relative efficacy of different programming tools that simultaneously support these skills was not adequately explored. This study designed and evaluated three programming tools: the tangible programming tool (TPG), which uses real touchable objects; the block programming tool (BPG), which employs virtual programming blocks and 3D game scenarios; and the paper‐and‐pencil programming tool (PPG), which uses paper and pen to draw. The study involved 112 seventh‐grade students from three natural classes: Class A (TPG, n1=37), Class B (BPG, n2=38), and Class C (PPG, n3=37). These students completed four gamification programming tasks and CT skills, spatial reasoning skills, enjoyment, cognitive load and GPL task list measurements. The results indicated that the tangible programming tool led to lower cognitive load, significant improvement in spatial reasoning skills and better abstraction and problem decomposition skills. The block programming tool provided a more enjoyable experience and facilitated students' algorithm design and efficiency. The paper‐and‐pencil programming tool was found to be less effective in improving spatial reasoning skills. This study's findings can help programming educators cultivate students' thinking skills and improve their learning experience by effectively selecting the most appropriate programming tools.

The Hadamard decomposition problem

AbstractWe introduce the Hadamard decomposition problem in the context of data analysis. The problem is to represent exactly or approximately a given matrix as the Hadamard (or element-wise) product of two or more low-rank matrices. The motivation for this problem comes from situations where the input matrix has a multiplicative structure. The Hadamard decomposition has potential for giving more succint but equally accurate representations of matrices when compared with the gold-standard of singular value decomposition (svd). Namely, the Hadamard product of two rank-$$h$$ h matrices can have rank as high as $${h}^2$$ h 2 . We study the computational properties of the Hadamard decomposition problem and give gradient-based algorithms for solving it approximately. We also introduce a mixed model that combines svd and Hadamard decomposition. We present extensive empirical results comparing the approximation accuracy of the Hadamard decomposition with that of the svd using the same number of basis vectors. The results demonstrate that the Hadamard decomposition is competitive with the svd and, for some datasets, it yields a clearly higher approximation accuracy, indicating the presence of multiplicative structure in the data.

Classical orthogonal decomposition of a modular [formula omitted]

An orthogonal decomposition problem of Lie algebras over the complex numbers has been studied since the 1980s. It has many applications and relations to other areas of mathematics and sciences. In this paper, we consider this decomposition problem over a field of prime characteristic. We define a classical orthogonal decomposition of a modular Lie algebra and construct it for sln under certain sufficient conditions. Additionally, we provide more detailed analysis of the problem when n=2 and 3.

Performance degradation assessment of rolling bearings based on the comprehensive characteristic index and improved SVDD

Rolling bearing is one of the most critical parts of mechanical equipment, so the performance degradation assessment of rolling bearing is vital to ensure the normal operation of the whole mechanical equipment. Aiming at the problems that a single degradation characteristic can only contain limited performance degradation information of rolling bearings, a large number of redundant characteristics exist in the high-dimension characteristic set resulting in the inability to effectively mine the characteristic information of rolling bearings, and that traditional degradation assessment models are not suitable for the shortage of fault data during the actual operation of rolling bearings, a performance degradation assessment method of rolling bearings based on the comprehensive characteristic index and improved support vector data description (SVDD) is proposed in this paper. Firstly, to solve the parameter selection problem of variational mode decomposition (VMD), a parameter-adaptive VMD method based on salp swarm algorithm based on mixed strategy (MSSSA) is proposed. Secondly, to extract the performance degradation information of rolling bearings more comprehensively and fully, the comprehensive characteristic index is proposed. Then, a kernel locality preserving projection orthogonal kernel principal component analysis (KLPPOKPCA) method is proposed to reduce the dimensionality of the extracted multi-domain characteristic set of the rolling bearing. Finally, a support vector data description with negative samples (NSSVDD) is proposed and optimized by MSSSA to solve the problem that traditional degradation assessment models are not suitable for the shortage of fault data during the actual operation of rolling bearings and improve the detection performance of abnormal data. The experimental results show that the proposed method can accurately divide the performance degradation process of the rolling bearing. Moreover, the comparison with other methods further highlights the superiority of the proposed method in determining the point in time of early fault of the rolling bearing.

An enhanced hybrid scheme for ship roll prediction using support vector regression and TVF-EMD

In order to represent the complex characteristics of ship roll motion such as nonlinearity, uncertainty, and time-varying dynamics thus improving the ship roll prediction accuracy, an enhanced hybrid prediction scheme is proposed by using improved swarm intelligent optimization, time series decomposition, and machine learning. Firstly, the novel time-varying filtering-based empirical mode decomposition (TVF-EMD) is employed to decompose the ship roll motion data into multiple mode components with dynamic time-varying characteristics, which immunes of the mode mixing and intermittency problems of traditional empirical mode decomposition (EMD). Then, the support vector regression is utilized to train and predict mode components. Finally, the predicted results of individual components are reconstructed to achieve the final ship roll prediction angles. To avoid the unfavorable influence of manual parameter selection for TVF-EMD, an improved black widow optimization algorithm is employed to optimize the parameter configuration. The feasibility and effectiveness of the enhanced hybrid model are validated by ship roll prediction simulation based on the measured data of M.V. YuKun at sea. The experimental results show that the enhanced hybrid scheme can improve the prediction accuracy of ship roll motion and outrank those by using methods of EMD, ensemble empirical mode decomposition, and variational mode decomposition.

Secure Key Exchange in Tropical Cryptography: Leveraging Efficiency with Advanced Block Matrix Protocols

In the quest for robust and efficient digital communication, this paper introduces cutting-edge key exchange protocols leveraging the computational prowess of tropical semirings and the structural resilience of block matrices. Moving away from the conventional use of finite fields, these protocols deliver markedly faster processing speeds and heightened security. We present two implementations of our concept, each utilizing a different platform for the set of commuting matrices: one employing tropical polynomials of matrices and the other employing Linde–de la Puente matrices. The inherent simplicity of tropical semirings leads to a decrease in operational complexity, while using block matrices enhances our protocols’ security profile. The security of these protocols relies on the Matrix Decomposition Problem. In addition, we provide a comparative analysis of our protocols against existing matrix block-based protocols in finite fields. This research marks a significant shift in cryptographic protocol design, is specifically tailored for demanding engineering applications, and sets a new standard in secure and efficient digital communication.

Unpacking the challenges and predictors of elementary–middle school students’ use of the distributive property

The distributive property plays a pivotal role in advancing students’ understanding of multiplication, enabling the decomposition of problems and the acquisition of new facts. However, this property of multiplication is difficult for students to understand. We used two unique data sets to explore middle school students’ use of the distributive property. Study 1 involved data from 1:1 structured interviews of students (N = 24) discussing worked examples and solving associated practice problems. We examined whether or not students used the distributive property to solve the problems and whether or not interviewers followed the recommended distributive property prompts or defaulted to more conventional methods. Despite exposure to worked examples using the distributive property and a protocol calling for attention to it, students and interviewers favored methods like PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction) or long multiplication. Study 2 used a data set with middle school students’ (N = 131) item-level responses on Kirkland’s (2022; doctoral dissertation, University of Notre Dame) Brief Assessment of Mature Number Sense along with several related measures of domain-general and domain-specific skills. We extracted problems involving the distributive property for analysis. Surprisingly, there was no evidence that students’ use of the distributive property improved from sixth grade to eighth grade. However, both grade-level mathematics achievement and cognitive reflection uniquely predicted the correct use of the distributive property. Results suggest that middle school students who exhibit stronger reflective thinking tend to perform better on distributive property problems. Findings highlight cognitive reflection as a potentially important construct involved in the understanding and use of the distributive property.

Learning Obstacles of Junior High School Students in Computational Thinking on Number Pattern Lessons

This qualitative research employed a phenomenological approach with the aim of describing students’ learning obstacles in computational thinking related to the lesson on number patterns. This study took place at one of the MTsN schools in Makassar, involving 74 students of grade 8th , out of which 12 students were selected for interviews. The research employed tests, questionnaires, and interview guidelines as research instruments. The results revealed that students faced various learning obstacles, including (a) ontogenical obstacles such as instrumental ontogenical obstacle, psychological ontogenical obstacle, and conceptual ontogenical obstacle; (b) epistemological obstacles, encompassing difficulties in pattern recognition, abstraction, and generalization due to limitations in students’ contextual abilities to solve problems related to number patterns; and (c) didactical obstacles, including limitations in the teaching of number pattern lesson. The learning process fails to involve students in the process of abstraction and , resulting in incomplete material presentation. Moreover, it lacks emphasis on students’ thinking process during abstraction and generalization process, and places less emphasis on problem decomposition and algoritmic thinking. Keywords: computational thinking, junior high school, learning obstacles, number pattern lessons

Studi Literatur: Pentingnya Berpikir Komputasional dalam Meningkatkan Kemampuan Pemecahan Masalah Matematis Peserta Didik

Computational Thinking is a way of thinking that refers to the ability to solve problems using concepts used in computing. Computational thinking is a way of thinking in solving problems that students must develop in the digital era. This research aims to examine the importance of computational thinking in improving students' mathematical problem solving abilities. The type of research used is qualitative research using a literature study approach in the form of books, notes, journals, reports of relevant research results, as well as the results of observations. The results of the analysis show that it is very important for students to have computational thinking in order to improve their mathematical problem solving abilities. Computational thinking involves solving problems using a logical and systematic mindset that involves selecting and using algorithms, data representation, problem decomposition, abstraction, pattern recognition and hypothesis testing. Computational thinking has become an important and essential skill for humans in the 21st century. Educators must advocate the importance of integrating learning with computational thinking concepts into the educational curriculum. Computational thinking has an important role in learning. This can be useful for improving mathematical problem solving abilities. Apart from that, computational thinking can also develop critical, creative and analytical thinking skills in solving complex problems, both in the context of computing and everyday life.