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"Mathematical functions"
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Improved pelican optimization algorithm with chaotic interference factor and elementary mathematical function
Pelican optimization algorithm (POA) is a new heuristic algorithm that simulates the pelican’s natural behavior in the hunting process. In order to improve the convergence speed and accuracy of the original algorithm and to solve the problem that the original algorithm is easy to fall into local optimization, an improved POA based on chaotic interference factor and elementary mathematical function is proposed. In this paper, ten different chaotic interference factors are introduced in the exploration stage of POA. After selecting an improved POA with the best performance, six different elementary mathematical functions are introduced in the exploitation stage of POA to improve its optimization performance. Then 30 benchmark functions in CEC-BC-2017 were used to test the performance of different improved algorithms. The experimental results showed that the performance of the improved algorithms have been improved effectively compared with the original POA, and the accuracy and optimization ability to balance exploration and exploitation were significantly improved. Compared with seven different algorithms, the feasibility of the improved POA proposed in this paper is proved. Finally, four engineering design problems are optimized, and the simulation results show that among four different engineering design problems, the improved POA proposed in this paper is obviously superior to the original POA, which proves that the improved POA based on chaotic interference factor and elementary function is competitive in optimization performance on function optimization and practical engineering applications.
Journal Article
De Bello Homomorphico: Investigation of the extensibility of the OpenFHE library with basic mathematical functions by means of common approaches using the example of the CKKS cryptosystem
Cloud computing has become increasingly popular due to its scalability, cost-effectiveness, and ability to handle large volumes of data. However, entrusting (sensitive) data to a third party raises concerns about data security and privacy. Homomorphic encryption is one solution that allows users to store and process data in a public cloud without the cloud provider having access to it. Currently, homomorphic encryption libraries only support addition and multiplication; other mathematical functions must be implemented by the user. To this end, we discuss and implement the division, exponential, square root, logarithm, minimum, and maximum function, using the CKKS cryptosystem of the OpenFHE library. To demonstrate that complex applications can be realized with this extended function set, we have used it to homomorphically realize the Box–Cox transform, which is used in many real-world applications, e.g., time-series forecasts. Our results show how the number of iterations required to achieve a given accuracy varies depending on the function. In addition, the execution time for each function is independent of the input and is in the range of ten seconds on a reference machine. With this work, we provide users with insights on how to extend the original restricted function set of the CKKS cryptosystem of the OpenFHE library with basic mathematical functions.
Journal Article
Gamma : exploring Euler's constant
\"Among the many constants that appear in mathematics, [pi], e, and i are the most familiar. Following closely behind is [gamma] or gamma, a constant that arises in many mathematical areas yet remains profoundly mysterious. Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + ... up to 1/n , minus the natural logarithm of n -- and the numerical value is 0.5772156 ... But unlike its more celebrated colleagues [pi] and e, the exact nature of gamma remains a mystery. In fact, we don't even know if gamma is a fraction. In this tantalizing blend of history and mathematics, Julian Havil takes readers on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics. Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.\"--Back cover.
Book
Fuzzy Logic in Dynamic Parameter Adaptation of Harmony Search Optimization for Benchmark Functions and Fuzzy Controllers
Nowadays the use of fuzzy logic has been increasing in popularity, and this is mainly due to the inference mechanism that allows simulating human reasoning in knowledge-based systems. The main contribution of this work is using the concepts of fuzzy logic in a method for dynamically adapting the main parameters of the harmony search algorithm during execution. Dynamical adaptation of parameters in metaheuristics has been shown to improve performance and accuracy in a wide range of applications. For this reason, we propose and approach for fuzzy adaptation of parameters in harmony search. Two case studies are considered for testing the proposed approach, the optimization of mathematical functions, which are unimodal, multimodal, hybrid, and composite functions and a control problem without noise and when noise is considered. A statistical comparison between the harmony search algorithm and the fuzzy harmony search algorithm is presented to verify the advantages of the proposed approach.
Journal Article
Interval type-2 fuzzy logic for dynamic parameter adaptation in the bat algorithm
We describe in this paper a proposed enhancement of the bat algorithm (BA) using interval type-2 fuzzy logic for dynamically adapting the BA parameters. The BA is a metaheuristic algorithm inspired by the behavior of micro bats that use the echolocation feature for hunting their prey, and this algorithm has been recently applied to different optimization problems obtaining good results. We propose a new method for dynamic parameter adaptation in the BA using interval type-2 fuzzy logic, where an especially design fuzzy system is responsible for determining the optimal values for the parameters of the algorithm. Simulations results on a set of benchmark mathematical functions with the interval type-2 fuzzy bat algorithm outperform the traditional bat algorithm and a type-1 fuzzy variant of BA. The proposed integration of the type-2 fuzzy system into the BA has the goal of improving the performance of BA for the future applicability of the algorithm in more complex optimization problems where higher levels of uncertainty need to be handled, like in the optimization of fuzzy controllers.
Journal Article
Optimization of Mathematical Function-Shaped Fracture Distribution Patterns for Multi-Stage Fractured Horizontal Wells
A conventional oil and gas well does not have a natural production capacity, which necessitates a hydraulic fracturing operation. The effectiveness of the fracturing directly impacts the economic benefit of a single well. Among the various parameters, including fracture spacing, fracture width, and conductivity, fracture half-length is one of the main influencing factors on the productivity of horizontal wells. For conventional homogeneous reservoirs, research mainly focuses on fracture patterns with equal fracture lengths. However, in actual production processes, due to mutual interference and the superimposition of drainage areas between fractures, the production distribution of each fracture is non-uniform. Typical fracture distribution patterns mainly include uniform, staggered, dumbbell, and spindle. While many believe that the dumbbell-shaped fracture distribution pattern has the best effect, there has been no quantitative study on the length of each fracture under the dumbbell-shaped pattern. Based on this, this paper proposes a modeling approach for function-shaped fracture distribution that takes advantage of the high production of edge fractures and the low output of middle fractures in horizontal wells. The influence of this approach on production capacity is studied. Constant, linear, and parabolic functions are used to establish the relationship between fracture position and fracture half-length, optimizing the fracture distribution function to achieve the best production effect. This method can guide the horizontal well fracture distribution in the block to maximize productivity. The results show that the parabolic function-shaped model is better than the linear function-shaped model and the constant function-shaped model is the least effective. The research presented in this paper offers a new idea for optimizing on-site fracturing plans. It utilizes mathematical expressions to describe the parameters that affect productivity, which provides valuable guidance for designing multi-stage fractured horizontal wells in the field. In the future, this research will be extended by exploring the optimal fracture distribution function under different formation conditions.
Journal Article
Cake Moisture Estimation Based on Image Analysis and Regression Model for Controlling the Compression Time of Filter Press in Sludge Dewatering
This study proposes practical methods for estimating the moisture content of sludge, represented by the cake moisture, in the filter press dewatering process. Because the cake moisture and filtrate volume are difficult to measure directly, the proposed approaches utilize indirectly measurable data, including drain outlet images and the differential pressure during the compression phase. By analyzing the correlations between these parameters and the cake moisture, estimation models were developed using mathematical approximations. In the image-based approach, image processing techniques were applied to isolate the dewatered region, and the relationship between the pixel count and actual filtrate volume was analyzed to estimate the cake moisture based on the calculated filtrate volume per minute. In the pressure-based approach, two models were proposed: one that directly estimates the cake moisture from the differential pressure, and another that models the relationship among the differential pressure, filtrate volume, and cake moisture. Unlike complex machine learning techniques, the proposed methods employ simple and interpretable mathematical functions, offering both practicality and reliability. Validation using real-world operational data confirmed the accuracy and effectiveness of the proposed approaches.
Journal Article