Complex Mathematical Problems Research Articles

Comparative Analysis of Bitcoin Mining Machines and Their Global Environmental Impact

The amount of power required to mine one Bitcoin (BTC) can vary significantly depending on several factors, including the type of mining hardware being used, its efficiency, the cost of electricity, and the overall network difficulty at any given time. Mining BTC involves solving complex mathematical problems to validate transactions on the blockchain network, which requires significant computational power. This research paper focuses on dedicated mining machines, combining essential data and information into a singular comparison evaluation of these machines.

ОСОБЛИВОСТІ ВИКОРИСТАННЯ СУЧАСНИХ ІНФОРМАЦІЙНОКОМУНІКАЦІЙНИХ ТЕХНОЛОГІЙ У ВИЩІЙ МАТЕМАТИЧНІЙ ОСВІТІ

The article discusses the relevance of using information and communication technologies (ICT) in higher mathematical education within the context of the contemporary educational environment characterized by rapid technological changes. It is noted that effective utilization of ICT, such as electronic journals, multimedia technologies, testing systems, and computer mathematics systems, contributes to the improvement of the teaching and learning process. Emphasis is placed on the necessity of reviewing and modifying teaching methodologies for mathematical disciplines to integrate innovative solutions. Furthermore, the use of information and communication technologies in higher mathematical education is deemed highly relevant due to the rapid development of the modern technological environment and the needs of the contemporary job market. The implementation of electronic journals, multimedia technologies, testing systems, and computer mathematics not only enhances the efficiency of teaching mathematical disciplines but also helps create a solid foundation for the future professional success of students. Electronic journals serve not only as convenient tools for administrative processes but also as platforms for active interaction among students, teachers, and administration, fostering the development of academic community and stimulating discussions and collaboration. Multimedia technologies, such as video lessons, interactive exercises, and simulations, provide students with opportunities to better grasp material and understand complex concepts, opening up new possibilities for personalized learning and adaptation to various learning styles. Testing systems, particularly those based on interactive platforms, enable instructors to effectively monitor students' knowledge and track their progress, as well as provide opportunities for personalized learning through automatic adaptation of task complexity Computer mathematics systems, including software packages for numerical and symbolic computations, become an integral part of modern mathematical education, empowering students to efficiently solve complex mathematical problems and explore new concepts, thus stimulating their creative potential and innovative approach to problem-solving. Together with educational platforms and distance learning systems, these tools contribute to the modernization and optimization of the educational process, leading to the improvement of the quality of higher mathematical education and the preparation of qualified professionals for the contemporary world.

Angle Trisection with Growth Rate of The Golden Ratio

In this paper, we explore the triangulation of angles using the golden ratio and geometric compass principles. By dividing angles into three equal parts based on the golden ratio, we demonstrate a method for solving this geometric problem. Furthermore, we extend our exploration into the third dimension, where a geometric compass can effectively divide a two-dimensional angle into three equal parts. Additionally, our investigation delves into the realm of six-dimensional space-time, where we observe the simultaneous movement of two arms of the compasses based on the golden ratio to achieve the division of every angle into three equal parts. We emphasize the significance of the growth rate associated with the golden ratio in resolving complex mathematical and physical problems. This study provides valuable insights into the application of geometric principles and the golden ratio in solving challenging problems within mathematics and physics.

Solving linear systems of fractional integro-differential equations by Haar and Legendre wavelets techniques

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

Angle Trisection with Growth Rate of The Golden Ratio

In this paper, we explore the triangulation of angles using the golden ratio and geometric compass principles. By dividing angles into three equal parts based on the golden ratio, we demonstrate a method for solving this geometric problem. Furthermore, we extend our exploration into the third dimension, where a geometric compass can effectively divide a two-dimensional angle into three equal parts. Additionally, our investigation delves into the realm of six-dimensional space-time, where we observe the simultaneous movement of two arms of the compasses based on the golden ratio to achieve the division of every angle into three equal parts. We emphasize the significance of the growth rate associated with the golden ratio in resolving complex mathematical and physical problems. This study provides valuable insights into the application of geometric principles and the golden ratio in solving challenging problems within mathematics and physics.

Spectral methods on the hydrogen atom

This study investigates the computational implementation of spectral theory and its applications in analyzing complex mathematical problems. It explores the use of modern programming languages and scientific libraries for implementing and visualizing complex mathematical concepts, particularly focusing on spectral theory within quantum physics. The research employs computational methods to address the challenges in interpreting spectral properties, utilizing the Lanczos spectral method for eigenvalue calculation in large, sparse matrices. The results illustrate the effectiveness of these computational techniques in visualizing quantum states, demonstrating the potential of advanced programming in understanding and solving intricate problems in quantum physics and spectral graph theory. The study's findings are significant in bridging computational methods with theoretical spectral analysis, offering a new perspective on the application of computational techniques in scientific research.

Evolution and role of virtual power plants: Market strategy with integration of renewable based microgrids

The use of distributed energy resources (DER) programs, which include separate renewable energy (RE) programs, has expanded in residential as well as commercial structures as an outcome of recent technological developments. The virtual power plant (VPP) may improve the security and reliability of an electricity grid's operations through including energy storage, changeable loads, and distributed energy resources (DER), among other characteristics. Consequently, a growing number of scholars tend to focus on VPP and providing recommendations for its improvement. In order to facilitate the incorporation of distributed generation into electricity distribution systems, cutting-edge management ideas for facilities have emerged in the last few years. As a result, it is now more crucial than ever to improve management abilities for the combining of dispersed energy output and consumption using various virtual power plants (VPPs). To increase operational profits, it's crucial to make use of their capacity to play a role in power markets. This work provides a classification scheme for, and an in-depth analysis of, recent studies that provide VPP models featuring interactions with various forms of electricity markets. The objective of this article is to find out which VPP strategy will yield the greatest possible return in each legislative scenario. These consist of the model's design, strategies for resolving complex mathematical problems, involvement in various marketplaces, and the application of the suggested models to real-world case studies. The research has been evaluated, and it has been concluded that contemporary models are more comprehensive and realistic. This article additionally elaborates on and emphasizes the VPP concept from other researchers' findings. Moreover, a number of prevalent VPP projects from throughout the globe are included. The VPP research also addresses some potential challenges and recommendations for prospective growth.

A novel approach to optimize an integrated network design and pricing of a healthcare supply chain

Adopting various pricing policies has been highly regarded in recent years for setting prices and increasing firms' profits. One of the most common steps in pricing is to identify costs. Since a significant part of costs is related to the corresponding supply chain, many researchers in different fields have used decision-making for simultaneous pricing and network design. However, there is no such approach in the field of healthcare. This paper tries to fill this gap by formulating a mixed-integer nonlinear bi-level programming model examining the interaction of hospitals and their medicines suppliers. At the upper level, there is a competitive market where a new firm (entrant) intends to enter the market and faces the challenge of pricing medicines and making network design decisions. At the lower level, there is a private hospital competing with a public hospital, and it also struggles with healthcare services pricing and supplier selection. A comprehensive utility function that considers healthcare services prices, quality, waiting time, health insurance, readmission rate, and referral rate is extended at this level. Three novel meta-heuristic algorithms are recommended, including bi-level, improved fruit fly, jellyfish optimization, and forensic-based investigation optimization algorithms to solve the presented complex mathematical problem.

Angle Trisection with Growth Rate of The Golden Ratio

In this paper, we explore the triangulation of angles using the golden ratio and geometric compass principles. By dividing angles into three equal parts based on the golden ratio, we demonstrate a method for solving this geometric problem. Furthermore, we extend our exploration into the third dimension, where a geometric compass can effectively divide a two-dimensional angle into three equal parts. Additionally, our investigation delves into the realm of six-dimensional space-time, where we observe the simultaneous movement of two arms of the compasses based on the golden ratio to achieve the division of every angle into three equal parts. We emphasize the significance of the growth rate associated with the golden ratio in resolving complex mathematical and physical problems. This study provides valuable insights into the application of geometric principles and the golden ratio in solving challenging problems within mathematics and physics.

Turning Dust into Gold: Distilling Complex Reasoning Capabilities from LLMs by Leveraging Negative Data

Large Language Models (LLMs) have performed well on various reasoning tasks, but their inaccessibility and numerous parameters hinder wide application in practice. One promising way is distilling the reasoning ability from LLMs to small models by the generated chain-of-thought reasoning paths. In some cases, however, LLMs may produce incorrect reasoning chains, especially when facing complex mathematical problems. Previous studies only transfer knowledge from positive samples and drop the synthesized data with wrong answers. In this work, we illustrate the merit of negative data and propose a model specialization framework to distill LLMs with negative samples besides positive ones. The framework consists of three progressive steps, covering from training to inference stages, to absorb knowledge from negative data. We conduct extensive experiments across arithmetic reasoning tasks to demonstrate the role of negative data in distillation from LLM.

Modified Viscosity Iterative Algorithm for Solving Variational Inclusion and Fixed Point Problems in Real Hilbert Space

This paper introduces a new iterative algorithm, called the Modified Viscosity Iterative algorithm, designed to solve problems related to Variational Inclusion and Fixed point in real Hilbert spaces. The algorithm is specifically tailored to handle Multivalued Quasi-Nonexpansive and Demicontractive operators. The convergence properties of the algorithm are analyzed and established, ensuring its effectiveness in finding solutions for complex mathematical problems in the field of optimization and equilibrium.

Innovative Solutions for the Kadomtsev–Petviashvili Equation via the New Iterative Method

This research paper presents a new iterative method (NIM) for obtaining the solution to the potential Kadomtsev–Petviashvili (PKP) equation. NIM is a promising approach to solving complex mathematical problems, and its effectiveness and efficiency are highlighted through its application to the PKP equation. The results obtained through the use of NIM are compared to the exact solutions of the PKP equation, and it is found that the NIM approach provides results that are in close agreement with the exact solutions. This demonstrates the utility and accuracy of NIM and makes it a valuable tool for solving similar mathematical problems in the future. Furthermore, the lack of discretization in the NIM approach makes it a more convenient method for solving the PKP equation compared to traditional approaches that require discretization. Overall, the findings of this research paper suggest that NIM is a highly effective and convenient method for obtaining approximate analytical solutions to complex mathematical problems, such as the 2+1-dimensional PKP equation.

The Development of RBL-STEM Learning Materials to Improve Students' Information Literacy in Solving Rainbow Antimagic Coloring Problem for ETLE Technology

Students often struggle to solve complex mathematical problems in real-world contexts due to low information literacy skills. To improve information literacy skills, an effective learning approach can be RBL-STEM, which provides research-based learning and can be practically applied in the real world. This study aims to investigate RBL-STEM activities, describe the process and results of developing RBL-STEM materials, and analyze data from the results of developing these materials to improve students’ information literacy skills. The method of research used is research and development (R&D). The purpose of this research is to develop RBL-STEM materials and produce learning material products in the form of semester learning plans, student assignment designs, student worksheets, and learning outcomes tests. The results of the materials development showed validity with a validity criterion of 92.75%. The trial was conducted with 40 students, and the implementation using the RBL-STEM materials was found to be practical with a practical criterion of 98.75% and effective with an effectiveness criterion of 94%. In addition, the students were highly engaged and responded very positively to the learning experience. Pretest and posttest analysis showed an increase in students’ information literacy in solving the rainbow antimagic coloring problem. The study also identified three levels of information literacy skills: high, medium, and low. Statistical analysis, phase portrait, NVivo, and word cloud confirmed the research findings and showed an increase in students’ information literacy skills. Thus, RBL-STEM has the potential to improve students’ information literacy in real-world contexts, such as the application of ETLE using graph neural network techniques.

A comprehensive analysis of Fokas–Lenells equation using Lie symmetry method

This article extracts analytical solutions of the Fokas–Lenells equation describing pulse propagation in optical fibers and presents a comprehensive analysis of the analytical solutions. This paper focuses on deriving analytical, non‐perturbative solutions for the Fokas–Lenells equation by reducing it to a system of ordinary differential equations using the Lie symmetry method. Three generators for the Lie algebra are obtained, and three reductions are explored by combining vector generators. These algebraic structures provide a powerful framework for simplifying complex mathematical and physical problems by revealing underlying symmetries and patterns. The graphical representations of obtained solutions are rendered. Besides, the influence of parameters within the equation on the behavior of the solutions has been demonstrated in the figures. These results contribute to a deeper understanding of the Fokas–Lenells equation and have implications in various fields of physics and engineering.

A Geometric Flow Approach for Segmentation of Images with Inhomongeneous Intensity and Missing Boundaries

Image segmentation is a complex mathematical problem, especially for images that contain intensity inhomogeneity and tightly packed objects with missing boundaries in between. For instance, Magnetic Resonance (MR) muscle images often contain both issues, making muscle segmentation especially difficult. In this paper we propose a novel intensity correction and a semi-automatic active contour-based segmentation approach. The approach uses a geometric flow that incorporates a Reproducing Kernel Hilbert Space (RKHS) edge detector and a geodesic distance penalty term from a set of markers and anti-markers. We test the proposed scheme on MR muscle segmentation and compare with some state-of-the-art methods. To help deal with the intensity inhomogeneity in this kind of image, a new approach to estimate the bias field using a fat fraction image, called Prior Bias-Corrected Fuzzy C-means (PBCFCM), is introduced. Numerical experiments show that the proposed scheme leads to significantly better results than compared ones. The average dice values of the proposed method are 92.5%, 85.3%, 85.3% for quadriceps, hamstrings and other muscle groups while other approaches are at least 10% worse.

(Invited) Integration of Functional Nanostructures and Nanoparticles into Micro- and Nanoelectronic Components and Systems

In recent decades, there has been a significant evolution of microelectronic and semiconductor technologies towards the nanoscale. Both bottom-up and top-down integration concepts of functional nanostructures and nanoparticles into micro- and nanofabricated electronic components have become critical issues that decisively affect the performance of smart electronic systems in various application areas such as environmental sensing and energy harvesting.Functional nanostructures and nanoparticles of different shapes, sizes, morphologies, and materials are now used in various technical and industrial applications, medicine, pharmacy, biology, electronics, power engineering, ecology, and many other fields. However, the complex engineering challenge here is the integration of functional nanostructures, such as molecular components (e.g., functional self-assembled molecular monolayers or entities of desoxyribonucleic acid, DNA), organic or inorganic nanoparticles (metal, semiconductor, colloidal materials), or similar building blocks into nanolithographically fabricated environments.Manufacturing paradigms for this integration go beyond current CMOS- and MEMS-fabrication schemes. Scale-bridging and hybrid manufacturing schemes that combine established top-down manufacturing techniques for microelectronic devices, including aspects of nanotechnology, with bottom-up nanoassembly of molecules on unstructured and pre-structured substrates are required. Microsystems technology based on the combination of conventional semiconductor processing with biotechnological approaches has thus become an emerging field with manifold applications, especially in the field of sensor technology.This contribution describes two illustrative and innovative research questions in nanopatterning and system integration at the micro-nano-interface to motivate applications with new and unprecedented functionalities. The first research question focuses on network-based biocomputing (NBC) using biological agents driven by biomolecular motor proteins in a lithographically fabricated nanofluidic channel system. This approach opens up opportunities for energy-efficient solutions to complex mathematical problems as they make their way through the NBC network. Methods are required to create switchable and rewritable nano-networks to address different mathematical problems using a single NBC-chip. Therefore, stimuli-responsive materials must be selectively and locally deposited into the nano-network to control the transport of the filaments through the network.The second research question focuses on nanophotonic devices based on DNA origami. The high-precision arrangement of nanoscopic substructures using specific binding sites (capture ends) of the DNA origami enables the precise arrangement of DNA origami functionalized with individualized metal nanoparticles. This arrangement could be used to exploit individual spectral response mediated by plasmonic effects.From a technological perspective, top-down lithographic nanostructuring technologies and bottom-up nanostructuring technologies are the basis for new device concepts. The adaptability of nanostructures to the world of biological molecules and other nanoscopic building blocks such as quantum dots and nanoparticles enables novel, even personalized, devices and engineering solutions.

A New Iterative Method for Investigating Modified Camassa–Holm and Modified Degasperis–Procesi Equations within Caputo Operator

In this paper, we employ the new iterative method to investigate two prominent nonlinear partial differential equations, namely the modified Camassa–Holm (mCH) equation and the modified Degasperis–Procesi (mDP) equation, both within the framework of the Caputo operator. The mCH and mDP equations are fundamental in studying wave propagation and soliton dynamics, exhibiting complex behavior and intriguing mathematical structures. The new iterative method (NIM), a powerful numerical technique, is utilized to obtain analytical and numerical solutions for these equations, offering insights into their dynamic properties and behavior. Through systematic analysis and computation, we unveil the unique features of the mCH and the mDP equations, shedding light on their applicability in various scientific and engineering domains. This research contributes to the ongoing exploration of nonlinear wave equations and their solutions, emphasizing the versatility of the new iterative method in tackling complex mathematical problems. Numerical results and comparative analyses are presented to validate the effectiveness of the new iterative method in solving these equations, highlighting its potential for broader applications in mathematical modeling and analysis.

Ensuring Data Integrity And Security In Diverse Cloud Environments To Prevent Duplicacy.

Data deduplication is a valuable technique for compressing and minimizing data duplication during data transfers, especially in cloud environments. By eliminating redundant data, it optimizes transmission capacity and reduces memory usage. To ensure the integrity of sensitive data, encryption is applied throughout the deduplication process. The SHA algorithm is commonly used for storing text data during deduplication. It generates security bits by padding the text and computes a hash consisting of hexadecimal, string, and integer data. Hash-based deduplication involves hashing the entire file and treating the hash values of text data as unique identifiers. This allows clients to identify duplicate data within the cloud. The Proof-of-Work (PoW) algorithm is widely utilized in blockchain networks like Bitcoin and Ethereum. Its primary function is to verify transactions through a process called mining. Miners engage in competition to solve complex mathematical problems, and the first one to find a solution is granted the right to add a new block to the blockchain. PoW relies on cryptographic hashing, such as the SHA-256 hashing function, to validate and secure transactions. On the other hand, the Proof of Retrievability (PoR) algorithm finds application in cloud computing systems. It serves as a consensus mechanism, ensuring that cloud storage providers store and retrieve data accurately. PoR enables cloud providers to demonstrate to consumers that their files can be fully recovered. This algorithm incorporates cryptographic proofs that validate the integrity and availability of the stored data. In cloud storage, deduplication is often implemented using the Memory mapping technique (MPT). This allows multiple data owners to store the same data in a single copy, enhancing storage efficiency. To maintain data security, encryption is applied both before and during the deduplication process, ensuring that sensitive information remains protected. In summary, data deduplication is a powerful method for compressing data, minimizing duplication, and optimizing transmission capacity. With encryption techniques and hash-based identification, it provides secure and efficient deduplication in cloud environments, while memory deduplication and MPT support further enhance performance and storage efficiency.

Quantum Computing Algorithms for Solving Complex Mathematical Problems

The power of quantum mechanics, that is too complex for conventional computers, can be solved by an innovative model of computing known as quantum computing. Quantum algorithms can provide exponential speedups for some types of problems, such as many difficult mathematical ones. In this paper, we review some of the most important quantum algorithms for hard mathematical problems. When factoring large numbers, Shor's algorithm is orders of magnitude faster than any other known classical algorithm. The Grover's algorithm, which searches unsorted databases much more quickly than conventional algorithms, is then discussed.

Chatbots Put to the Test in Math and Logic Problems: A Comparison and Assessment of ChatGPT-3.5, ChatGPT-4, and Google Bard

In an age where artificial intelligence is reshaping the landscape of education and problem solving, our study unveils the secrets behind three digital wizards, ChatGPT-3.5, ChatGPT-4, and Google Bard, as they engage in a thrilling showdown of mathematical and logical prowess. We assess the ability of the chatbots to understand the given problem, employ appropriate algorithms or methods to solve it, and generate coherent responses with correct answers. We conducted our study using a set of 30 questions. These questions were carefully crafted to be clear, unambiguous, and fully described using plain text only. Each question has a unique and well-defined correct answer. The questions were divided into two sets of 15: Set A consists of “Original” problems that cannot be found online, while Set B includes “Published” problems that are readily available online, often with their solutions. Each question was presented to each chatbot three times in May 2023. We recorded and analyzed their responses, highlighting their strengths and weaknesses. Our findings indicate that chatbots can provide accurate solutions for straightforward arithmetic, algebraic expressions, and basic logic puzzles, although they may not be consistently accurate in every attempt. However, for more complex mathematical problems or advanced logic tasks, the chatbots’ answers, although they appear convincing, may not be reliable. Furthermore, consistency is a concern as chatbots often provide conflicting answers when presented with the same question multiple times. To evaluate and compare the performance of the three chatbots, we conducted a quantitative analysis by scoring their final answers based on correctness. Our results show that ChatGPT-4 performs better than ChatGPT-3.5 in both sets of questions. Bard ranks third in the original questions of Set A, trailing behind the other two chatbots. However, Bard achieves the best performance, taking first place in the published questions of Set B. This is likely due to Bard’s direct access to the internet, unlike the ChatGPT chatbots, which, due to their designs, do not have external communication capabilities.