Fraction Decomposition Research Articles

A New Method of Fault Localization for 500 kV Transmission Lines Based on FRFT-SVD and Curve Fitting

The paper presents the Fractional Fourier Transform-Singular Value Decomposition (FRFT-SVD) method for the localization of various power system faults in a 200 km long, 500 kV Egyptian transmission line using sent end-line current signals. Transient simulations are carried out using Alternating Transient Program/Electromagnetic Transient Program (ATP-EMTP), and the outcomes are then examined in MATLAB to carry out a sensitivity analysis against measurement noises, sampling frequency, and fault characteristics. The proposed work employs current fault signals of five distinct kinds at nineteen intermediate points throughout the length of the line. The approach utilized to construct the localizer model is FRFT-SVD. It is much more effortless, precise, and effective. The FRFT-SVD is utilized in this technique to calculate 19 sets of indices of the greatest S value throughout the length of the line. The FRFT-SVD localizer model is also designed to be realistic, with power system noise corrupting fault signals. To generate fault curves, the curve fitting technique is applied to these 19 sets of indices. Reduced chi-squared and modified R-squared criteria are used to choose the best-suited curve. The proposed work results in a very precise localization, with only a 0.0016% average percentage error for fault localization and a maximum percentage error of 0.002% for the 200 km Egyptian transmission line. Finally, this work can be employed as a proper link between the nuclear power plant and the grid. The proposed method is an efficient fault distance estimation method that might contribute to creating a dependable transient-based approach to power system protection.

Determining the Inverse of a Vandermonde Matrix Using Partial Fraction Decomposition Method

One method that can be used to calculate the inverse of a matrix is ​​the adjoin method. In this method, the process begins by calculating the value of the determinant and adjoin of a matrix. This study discusses a method for calculating the inverse, especially on the Vandermonde matrix using partial fraction decomposition. The advantage of this method is that it can calculate the inverse of a matrix, without the need to calculate the value of the determinant and adjoin of a matrix. The steps taken are to define a rational function and then write it in the form of a partial fraction, then by using a formula to calculate the coefficient of a partial fraction, a formula is derived to calculate the inverse of the Vandermonde matrix. After obtaining the formula for calculating the inverse, then comparing the results of the inverse calculation of the Vandermonde matrix using the partial fraction decomposition method with the adjoin method. The results obtained a formula to calculate the inverse of the Vandermonde matrix, V-1=WxA. Based on the case examples given, it can be concluded that the results of the inverse calculations performed using the partial fraction decomposition method are the same as the results of the calculations performed using the adjoin method. However, the calculations performed using the partial fraction decomposition method are more effective and efficient than using the adjoin method

Partial-fraction decomposition of a rational function and its application

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function frac{x^{M}}{(x+1)^{lambda}_{n}} (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu’ results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind.

Convergence analysis for the fractional decomposition method applied to class of nonlinear fractional Fredholm integro-differential equation

For scientists conducting research, fractional integral differential equation analysis is crucial. Therefore, in this study, we investigate analysis utilizing a novel method called the fractional decomposition method, which is applicable to fractional nonlinear fractional Fredholm integro-differential equations. Then, we apply the approach to five test problems for a general fractional derivative [Formula: see text] involving fractional Fredholm integro-differential equations. To the best of our knowledge, we are the first to ever do so because of the very complicated calculations involved when dealing with the general case [Formula: see text]. For fractional Fredholm integral-differential equations, we provide both exact and approximate solutions. Throughout this work, the fractional Caputo derivative is discussed. This technique leads us to say that the method is precise, accurate, and efficient, according to the theoretical analysis.

SHORTER PROOF OF THE RIESZ INTERPOLATION FORMULA FOR TRIGONOMETRIC POLYNOMIALS

By means of partial fraction decompositions, a shorter proof is presented for the important interpolation formula of trigonometric polynomials discovered by Riesz (1914).

Conformable Fractional Elzaki Decomposition Method of Conformable Fractional Space-Time Fractional Telegraph Equations

Conformable space-time fractional linear telegraph equations are examined using a new method known as conformable fractional Elzaki decomposition method. The suggested method combines the Adomian decomposition method with the conformable fractional Elzaki transform. It is found that numerical simulations confirm the effectiveness and reliability of the proposed method.

Fractional Decomposition Tree Algorithm: A tool for studying the integrality gap of Integer Programs

We present a new algorithm, Fractional Decomposition Tree (FDT), for finding a feasible solution for an integer program (IP) where all variables are binary. FDT runs in polynomial time and is guaranteed to find a feasible integer solution provided the integrality gap of an instance’s polyhedron, independent of objective function, is bounded. The algorithm gives a construction for Carr and Vempala’s theorem that any feasible solution to the IP’s linear-programming relaxation, when scaled by the instance integrality gap, dominates a convex combination of feasible solutions. FDT is also a tool for studying the integrality gap of IP formulations. The upper bound on the integrality gap of an FDT solution can be exponentially large. However our experiments demonstrate that FDT can be effective in practice. We study the integrality gap of two problems: optimally augmenting a tree to a 2-edge-connected graph and finding a minimum-cost 2-edge-connected multi-subgraph (2EC). We also give a simplified algorithm, DomToIP, that finds a feasible solution to an IP instance, or concludes that it has unbounded integrality gap. We show that FDT’s speed and approximation quality compare well to that of the original feasibility pump heuristic on moderate-sized instances of the vertex cover problem. For a particular set of hard-to-decompose fractional 2EC solutions, FDT always gave a better integer solution than the Best-of-Many Christofides Algorithm (BOMC).

The carbon-quality temperature hypothesis: Fact or artefact?

Climate warming can reduce global soil carbon stocks by enhancing microbial decomposition. However, the magnitude of this loss remains uncertain because the temperature sensitivity of the decomposition of the major fraction of soil carbon, namely resistant carbon, is not fully known. It is now believed that the resistance of soil carbon mostly depends on microbial accessibility of soil carbon with physical protection being the primary control of the decomposition of protected carbon, which is insensitive to temperature changes. However, it is still unclear whether the temperature sensitivity of the decomposition of unprotected carbon, for example, carbon that is not protected by the soil mineral matrix, may depend on the chemical recalcitrance of carbon compounds. In particular, the carbon-quality temperature (CQT) hypothesis asserts that recalcitrant low-quality carbon is more temperature-sensitive to decomposition than labile high-quality carbon. If the hypothesis is correct, climate warming could amplify the loss of unprotected, but chemically recalcitrant, carbon and the resultant CO2 release from soils to the atmosphere. Previous research has supported this hypothesis based on reported negative relationships between temperature sensitivity and carbon quality, defined as the decomposition rate at a reference temperature. Here we show that negative relationships can arise simply from the arbitrary choice of reference temperature, inherently invalidating those tests. To avoid this artefact, we defined the carbon quality of different compounds as their uncatalysed reaction rates in the absence of enzymes. Taking the uncatalysed rate as the carbon quality index, we found that the CQT hypothesis is not supported for enzyme-catalysed reactions, which showed no relationship between carbon quality and temperature sensitivity. The lack of correlation in enzyme-catalysed reactions implies similar temperature sensitivity for microbial decomposition of soil carbon, regardless of its quality, thereby allaying concerns of acceleration of warming-induced decomposition of recalcitrant carbon.

Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition

Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions.

Magnesium Oxide Production by Plasma Chemical Conversion from Fluorine-Containing Industrial Waste

This work discusses the possibility of decomposing magnesium fluoride by ionized water vapor to form solid magnesium oxide and hydrogen gas in the reaction: MgF2 + H2O → MgO + 2HF. The technology and individual apparatuses of the plasma-chemical installation are described, and the influence of the fractional composition of magnesium fluoride powder on the productivity of the plasma conversion process is considered. To improve the efficiency of the plasma pyrolysis process, a method for making magnesium fluoride briquettes was developed. The completeness of the conversion process of magnesium fluoride to an oxide was evaluated by energy dispersive X-ray spectroscopy in the study of objects in scanning electron microscopy (SEM) and by X-ray diffractometry. It was found that the conversion process of magnesium fluoride to magnesium oxide has a relatively high degree of decomposition of magnesium fluoride fraction ≤75 µm. The use of the proposed processing method makes it possible to obtain pure magnesium oxide as a commercial product and to utilize fluorine-containing industrial waste.

Particulate and mineral-associated organic carbon fractions reveal the roles of soil aggregates under different land-use types in a karst faulted basin of China

Soil aggregation has been recognized as the main mechanism of organic carbon (C; OC) stabilization, and OC dynamics within aggregates are closely linked to soil OC (SOC) sequestration. To unravel the roles of different-sized aggregates and the associated OC fractions in SOC accumulation, SOC stocks (SOCS), aggregate distributions, and OC fractions within aggregates were measured under five typical land-use types (cropland, grassland, shrubland, plantation forest, and natural forest) in a karst faulted basin, China. Land-use type had distinct effects on SOCS in the topsoil (0–10 cm) and along the soil profile (0–60 cm), and the primary factors influencing SOCS varied with the soil horizons. Plantation forest had the highest SOCS (51.2 Mg C ha−1) in the topsoil, whereas the SOCS in shrubland (182.7 Mg C ha−1) was distinctively higher than those of other land-use types across the soil profile. Soil microbial biomass C and mean weight diameter were the major factors driving SOCS variation in the topsoil layer and the entire soil profile, respectively. OC content within macroaggregates (>0.25 mm) was significantly positively correlated with the SOC content of bulk soil, whereas only the mass proportion of large macroaggregates (>2 mm) showed a positive relationship with SOC. Additionally, mineral-associated OC (MAOC) was the dominant fraction of SOC and was abundant in macroaggregates. However, particulate OC (POC) within aggregates was relatively enriched in the topsoil and microaggregates. Considering the decomposition and distribution characteristics of OC fractions in different-sized aggregates, we suggest that the POC/MAOC ratios of macro- and microaggregates could respectively reflect aggregate capacity to sequestrate MAOC and POC across different land-use types. Overall, shrubland is the optimal land-use type for rocky desertification control in the karst fault basin, due to the protection of POC and MAOC within different-sized aggregates.

Thermal stability of limestone calcined clay cement (LC3) at moderate temperatures 100–400 °C

The mechanical properties, phase evolution, microstructure development and (re)hydration properties of limestone calcined clay cement (LC3) paste and mortar were investigated to understand their thermal stability under moderate temperatures (100–400 °C). The residual compressive strengths of LC3 and Portland cement (PC) mortars kept constant or slightly increased from ambient to 300 °C, and sharply dropped at 400 °C. At 100 °C, monocarboaluminate (Mc) and ettringite (AFt) converted into hemicarboaluminate (Hc) and monosulfate (Ms), and caused partial dehydroxylation of C-(A)-S-H. As the temperature increased, at 200–300 °C, due to the decomposition of large fraction of hydration products, macroscopic cracks and microstructure damage occurred. At 400 °C, the mean chain length (MCL) of C-(A)-S-H gel decreased, accompanied by the shift of Si and Al-coordination number and the formation of active sites. Under water saturation conditions, the microstructure showed self-repairing performance.

Tapir: A tool for topologies, amplitudes, partial fraction decomposition and input for reductions

The demand for precision predictions in the field of high energy physics has dramatically increased over recent years. Experiments conducted at the LHC, as well as precision measurements at the intensity frontier such as Belle II require equally precise theoretical predictions to make full use of the acquired data. To match the experimental precision, second-, third- and, for certain quantities, even higher-order calculations in perturbative quantum field theory are required.To facilitate such calculations, computer software automating as many steps as possible is required. Yet, each calculation poses different challenges and thus, a high level of configurability is required. In this context we present tapir: a tool for identification, manipulation and minimization of Feynman integral families. It is designed to integrate in toolchains based on the computer algebra system FORM, the use of which is common practice in the field. tapir can be used to reduce the complexity of multi-loop problems with cut-filters, topology mapping, partial fraction decomposition and alike. Program summaryProgram Title:tapirCPC Library link to program files:https://doi.org/10.17632/ptc9t46xyn.1Developer's repository link:https://gitlab.com/tapir-devs/tapirLicensing provisions: GPLv3Programming language:python 3, C++Nature of problem: Multi-loop computations require the automatization of a large number of different tasks related to Feynman integral topologies. Among them are the identification and minimization of integral topologies, partial fraction decomposition of topologies in the case of linearly dependent propagators as well as mapping scalar products of loop momenta to scalar functions.Solution method: The minimization of topologies is performed by comparison of their respective Nickel indices [1], even further minimization utilizes Pak's algorithm [2]. To efficiently map scalar products of loop momenta to scalar functions FORM [3] code is generated.Additional comments including restrictions and unusual features: Minimization based on Pak's algorithm slows down for many lines and scales. A coarser minimization using the Nickel indices, however, is still possible.

Convergence analysis of the fractional decomposition method with applications to time‐fractional biological population models

AbstractIn this study, we present convergence analysis along with an error estimate for time‐fractional biological population equation in terms of the Caputo derivative using a new technique called the fractional decomposition method (FDM). Further, we present exact solutions to four test problems of nonlinear time‐fractional biological population models to show the accuracy and efficiency of the FDM. This method based on constructing series solutions in a form of rapidly convergent series with easily computable components and without the need of linearization, discretization and perturbations. The results prove that the FDM is very effective and simple for solving fractional partial differential equations in multi‐dimensional spaces, special cases of which we have described in this paper.

Method of data-driven mode decomposition for cavitating flow in a Venturi nozzle

Cavitation is a common phenomenon that occurs in ocean engineering. In the present work, experimental measurement of cavitating flow in a Venturi nozzle is carried out at cavitation number σ = 1.5, and then the simulation based on the Zwart cavitation model is conducted with validation of experimental data. Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) are introduced to investigate the three-dimensional coherent structures of cavitation and flow fields. For decomposition of vapor volume fraction, both POD and DMD results can characterize cavitation structures and associated frequencies, while different dominant structures can be obtained. For decomposition of streamwise velocity, POD results extract the coherent structures with converse polarities induced by the vortices related to cavity evolution, and DMD results shed light on the structures related to the overall periodic dynamics of cloud cavitation under shedding frequency. Based on the mode decomposition of vapor volume fraction and streamwise velocity, the cavitation-velocity interaction is revealed. Under the studied cavitation condition, cavitation and velocity mainly interact in the range of 0-3Lref downstream the Venturi throat, where the modes of vapor volume fraction and streamwise velocity show similar coherent structures and mode value fluctuations.

Computational Analysis of Local Fractional LWR Model Occurring in a Fractal Vehicular Traffic Flow

In this paper, we implement computational methods, namely the local fractional natural homotopy analysis method (LFNHAM) and local fractional natural decomposition method (LFNDM), to examine the solution for the local fractional Lighthill–Whitham–Richards (LFLWR) model occurring in a fractal vehicular traffic flow. The LWR approach preferably models the traffic flow and represents the traffic patterns via the supposition of speed–density equilibrium relationship and continuity equation. This model is mostly preferred for modeling of traffic flow because of its simple approach and interpretive ability to examine the qualitative patterns of traffic flow. The methods applied here incorporate the local fractional natural transform (LFNT) and derive the solutions for the LFLWR model in a closed form. Two examples are provided to demonstrate the accuracy and efficiency of the suggested methods. Furthermore, the numerical simulations have also been presented for each of the examples in the fractal domain. Additionally, the explored solutions for both examples have also been compared and are in good match with already existing solutions in literature. The methods applied in this work make the computational process easier as compared to other iterative methods and still provide precise solutions.

Ericoid shrubs shape fungal communities and suppress organic matter decomposition in boreal forests.

Mycorrhizal fungi associated with boreal trees and ericaceous shrubs are central actors in organic matter (OM) accumulation through their belowground carbon allocation, their potential capacity to mine organic matter for nitrogen (N) and their ability to suppress saprotrophs. Yet, interactions between co-occurring ectomycorrhizal fungi (EMF), ericoid mycorrhizal fungi (ERI), and saprotrophs are poorly understood. We used a long-term (19 yr) plant functional group manipulation experiment with removals of tree roots, ericaceous shrubs and mosses and analysed the responses of different fungal guilds (assessed by metabarcoding) and their interactions in relation to OM quality (assessed by mid-infrared spectroscopy and nuclear magnetic resonance) and decomposition (litter mesh-bags) across a 5000-yr post-fire boreal forest chronosequence. We found that the removal of ericaceous shrubs and associated ERI changed the composition of EMF communities, with larger effects occurring at earlier stages of the chronosequence. Removal of shrubs was associated with enhanced N availability, litter decomposition and enrichment of the recalcitrant OM fraction. We conclude that increasing abundance of slow-growing ericaceous shrubs and the associated fungi contributes to increasing nutrient limitation, impaired decomposition and progressive OM accumulation in boreal forests, particularly towards later successional stages. These results are indicative of the contrasting roles of EMF and ERI in regulating belowground OM storage.

Optimal Power Allocation for Achieving Secure Green Cognitive Radio Networks

In cognitive radio networks, wireless nodes adapt to the surrounding radio environment and utilize the spectrum of licensed users. The cognitive radio environment is dynamic, and wireless channels are accessible by both legitimate and illegitimate users. Therefore, maintaining the security of cognitive radio networks is a challenging task, which must be addressed thoroughly. Further, with the recent exponential surge in wireless nodes and associated high data rate requirements, energy consumption is also growing at an unprecedented rate. Hence, energy efficiency becomes an important metric that must be considered in the design of future wireless networks. Accordingly, by considering the great ecological and economic benefits of green wireless networks, this work focus on energy-efficient resource allocation in secure cognitive radio networks. Since physical-layer security is an emerging technique that improves the security of communication devices, in this paper, an ergodic secure energy efficiency problem for a cognitive radio network is formulated with a primary user, a secondary user, and an eavesdropper. As the formulated problem is non-convex, a concave lower bound is applied to transform the original non-convex problem into a convex one. Further, by adopting the fractional programming and dual decomposition techniques, optimal power allocation strategies are obtained with the aim of maximizing the ergodic secure energy efficiency of the secondary user with constraints on the average interference power and average transmit power. Numerical examples are used to demonstrate the effectiveness of the proposed algorithm.

Decomposition of the weight fractions for modelling ridge filters in a proton wobbling nozzle

In proton therapy, the width of a pristine Bragg peak is too narrow to cover the entire tumor volume, and should be spread out using the Spread out Bragg peak (SOBP) techniques such as energy stacking, range modulator wheel and ridge filter. The main purpose of this study was to demonstrate feasibility of using a simple least square technique to decompose the simulated weight fractions of elemental Bragg peaks of ridge filters used in a clinical wobbling beam nozzle. First, initial simulated beam parameters including mean energy and energy spread at the nozzle entrance for all available clinical energies were commissioned against measurements for the open middle wobbling field at the Proton and Radiation Center of the Linkou Chang Gung Memorial Hospital (LCGMH) using the Particle Therapy Simulation Framework (PTSim) Monte Carlo package. Material composition and geometry of nozzle components were provided by the vendor. Depth dose distribution of beam energies from 70 to 230 MeV at 20-MeV increment were simulated and validated with measurement. Curve fitting to those beam parameters was performed to extend simulation energy range to all available clinical energies. For the reconstructed ridge filter simulation, elemental Bragg peaks (BPs) with layer filters corresponding to ridge filter step thicknesses were obtained as bases of the weight factor decomposition. The least square technique was implemented to decompose weight factors for individual ridge filter at a selected beam energy within its applicable energy range. Finally, the reconstructed ridge filters using decomposed weight factors were tested at one other energy to validate the study design. Four different ridge filters (5 cm-LE, 5 cm-HE, 10 cm and 11 cm SOBP) were examined in this study. Commission results for tuned beam parameter settings showed maximum disagreement of BP characteristics, represented by the 80% dose depth (R80d) and width of distal fall-off (W80-20), occurred at 230 MeV and was smaller than 0.2 and 0.5 mm respectively. Furthermore, analyzed outcome for SOBPs from the four ridge filters revealed general improvement of agreement to measurement with decomposition and satisfactory agreement between measurements and those with least square technique. Deviation from measurement of SOBP characteristics evaluated for R90d, W80-20 and SOBP flatness after weight decomposition showed values smaller than 1mm/1%. Similar evaluation of SOBP width generated larger difference with maximum value of 2.7 mm for the 11-cm ridge filter. Results of the validation testes were similar with maximum deviation of 1.4 mm, 0.6 mm, 0.2% and 3.8 mm for R90d, W80-20, flatness and width respectively.

New families of mock theta functions and partial fraction decomposition

In 2000, Andrews challenged mathematicians in the 21st century to elucidate the overlap between classes of q-series and modular forms. Motivated by Andrews' challenge, seeking q-hypergeometric series which are mock theta functions in the modern sense and finding their alternative representations have attracted the attention of many mathematicians in recent years. In this paper, we use partial fraction decomposition to give rise to several families of two-parameter mock theta functions and express them in the form of Appell-Lerch sums. In particular, we establish some identities involving several families of mock theta functions and Appell-Lerch sums which imply three identities proved by Cui and Gu.