Fraction Decomposition Research Articles

An Effective Partial Fraction Decomposition Method for Undergraduate Students

An Effective Partial Fraction Decomposition Method for Undergraduate Students

High biorenewable content acrylate photocurable resins for DLP 3D printing

AbstractGreen chemistry and green engineering concepts have been combined to develop novel sustainable polymeric materials. Solvent free photocurable acrylate resins with biorenewable carbon content of 75%–82% suitable for application in DLP 3D printing technology were composed by commercially available bio‐based materials, acrylated epoxidized soybean oil (AESO), isobornyl methacrylate (IBOMA), methacrylic ester (ME), tetrahydrofurfuryl acrylate (THFA), and tetrahydrofurfuryl methacrylate (THFMA). They demonstrated high printing accuracy and good adhesion between layers. The monitoring of photocross‐linking kinetics of high biorenewable content acrylate photoresins by the real‐time photorheometry and analysis of their rheological parameters was carried out. Synthesized polymers exhibited high yield of insoluble fraction and thermal decomposition temperature at the weight loss of 10% above 300°C. Polymers AESO/IBOMA and AESO/THFMA showed the highest values of tensile modulus and tensile strength. Biodegradability of the synthesized polymers AESO/ME, AESO/THFA, and AESO/THFMA was investigated by measuring oxygen consumption in a closed respirometer. Such AESO‐based polymers can be a competitive solution to replace petroleum‐derived polymeric materials in additive manufacturing and reduce the environmental impact.

Progress towards Nash-Williams' conjecture on triangle decompositions

Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on n vertices with minimum degree at least 0.75n admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely 1.We show that for any graph on n vertices with minimum degree at least (7+2114)n⪅0.82733n admits a fractional triangle decomposition. Combined with results of Barber, Kühn, Lo, and Osthus, this implies that for all ε>0, every sufficiently large triangle divisible graph on n vertices with minimum degree at least (7+2114+ε)n admits a triangle decomposition.

Image decomposition and denoising using fractional‐order partial differential equations

In this study, the authors propose a fractional derivative-based image decomposition and denoising model which decomposes the image into the cartoon component (the component formed by homogeneous regions and with sharp boundaries) and the texture (or noise) component. The cartoon component is modelled by a function of the fractional-order total bounded variation, while the texture component is modelled by an oscillatory function, bounded in the negative Sobolev space norm. The authors give the corresponding minimisation functional, after some transformations, and then the resulting fractional-order partial differential equation can be solved using the Fourier transform. By symmetry and asymmetry of the fractional-order derivative, some generalisations and variants of the proposed model are also introduced. Finally, the authors implement the algorithm by the fractional-order finite difference in the frequency-domain. The experimental results demonstrate that the proposed models make objective and visual improvements compared with other standard approaches in the task of decomposition and denoising.

Hydrothermal Conversion of Thorium Oxalate into ThO2·nH2O Oxide.

Hydrothermal conversion of thorium oxalate, Th(C2O4)2·nH2O, into thorium dioxide was explored through a multiparametric study, leading to some guidelines for the preparation of crystallized samples with the minimum amount of impurities. As the formation of the oxide appeared to be operated through the hydrolysis of Th4+ after decomposition of oxalate fractions, pH values typically above 1 must be considered to recover a solid phase. Also, because of the high stability of the thorium oxalate precursor, hydrothermal treatments of more than 5 h at a temperature above 220 °C were required. All the ThO2·nH2O samples prepared presented amounts of residual carbon and water in the range 0.2-0.3 wt % and n ≈ 0.5, respectively. A combined FTIR, PXRD, and EXAFS study showed that these impurities mainly consisted of carbonates trapped between elementary nanosized crystallites, rather than substituted directly in the lattice, which generated a tensile effect over the crystal lattice. The presence of carbonates at the surface of the elementary crystallites could also explain their tendency to self-assembly, leading to the formation of spherical aggregates. Hydrothermal conversion of oxalates could then find its place in different processes of the nuclear fuel cycle, where it will provide an interesting opportunity to set up dustless routes leading from ions in solution to dioxide powders in a limited number of steps.

Predictor-corrector schemes for nonlinear space-fractional parabolic PDEs with time-dependent boundary conditions

We propose novel second-order accurate methods for nonlinear space-fractional PDEs with time-dependent boundary conditions. The matrix transfer technique is used for spatial discretization of the parabolic PDEs to obtain systems of ordinary differential equations. The treatment of the boundary conditions gave rise to a source term which depends on the temporal variable only. Methods based on rational approximations to the exponential function and which uses Gaussian quadrature points are developed for integrating the systems of differential equations. We develop new schemes based on the (1,1)-, (0,2)-Padé approximations and the real distinct pole approximation to the exponential function with Gaussian quadrature points. Error estimates are provided for the convergence analysis. Stability and reliability of the resulting schemes are discussed. A partial fraction decomposition technique is used for the efficient implementation of the schemes. Finally, numerical results in one and two spatial dimensions are presented to corroborate our theoretical observations.

A high-order implicit–explicit Runge–Kutta type scheme for the numerical solution of the Kuramoto–Sivashinsky equation

This manuscript is concerned with the development and the implementation of a numerical scheme to study the spatio-temporal solution profile of the well-known Kuramoto–Sivashinsky equation with appropriate initial and boundary conditions. A fourth-order Runge–Kutta based implicit–explicit scheme in time along with compact higher-order finite difference scheme in space is introduced. The proposed scheme takes full advantage of the method of line (MOL) and partial fraction decomposition techniques, therefore, it just needs to solve two backward Euler-type linear systems at each time step to get the solution. Performance of the scheme is investigated by testing it on some test examples and by comparing numerical results with relevant known schemes. The numerical results showed that the proposed scheme is more accurate and reliable than existing schemes to solve Kuramoto–Sivashinsky equation.

Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions

We consider the problems of calculating the dynamical order parameter two-point function at finite temperatures and the one-point function after a quantum quench in the transverse field Ising chain. Both of these can be expressed in terms of form factor sums in the basis of physical excitations of the model. We develop a general framework for carrying out these sums based on a decomposition of form factors into partial fractions, which leads to a factorization of the multiple sums and permits them to be evaluated asymptotically. This naturally leads to systematic low density expansions. At late times these expansions can be summed to all orders by means of a determinant representation. Our method has a natural generalization to semi-local operators in interacting integrable models.

Decomposição da serapilheira foliar em plantios de bambu, nim indiano e eucalipto

O processo de decomposição da serapilheira regula o acúmulo de matéria orgânica no solo e a ciclagem de nutrientes, desempenhando importante papel na manutenção de ecossistemas florestais. Objetivou-se avaliar a decomposição da fração foliar da serapilheira de plantios homogêneos de Bambusa vulgaris, Azadirachta indica e de Eucalyptus urophylla, utilizando como referência uma área de Floresta Estacional Semidecidual. As áreas de estudo estão localizadas no município de Vitória da Conquista, Bahia, Brasil. Para a avaliação da decomposição utilizou-se o método dos litterbags, sendo as coletas realizadas aos 30, 75, 120, 180, 240 e 350 dias. A partir dos resultados de massa remanescente, estimou-se a constante de decomposição (k) e o tempo de meia-vida. A serapilheira foliar foi submetida à análises químicas para determinação dos teores de carbono, nitrogênio, polifenóis, celulose e lignina. Ao final de um ano, bambu e nim indiano tiveram mais de 65% da sua serapilheira decomposta, com constantes k iguais a 0,0033 e 0,0069 g g-1 dia, respectivamente. O plantio de eucalipto e a floresta nativa apresentaram perdas de massa inferiores a 50% e valores de k menores que 0,0020 g g-1 dia. A serapilheira foliar do bambu e do nim indiano mostra-se mais propensa à decomposição, com taxas superiores ao eucalipto e à floresta nativa, o que proporciona a estas duas espécies uma maior capacidade de reciclar matéria orgânica e nutrientes.

New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques

AbstractThe pivotal aim of the present work is to find the numerical solution for fractional Benney–Lin equation by using two efficient methods, called q‐homotopy analysis transform method and fractional natural decomposition method. The considered equation exemplifies the long waves on the liquid films. Projected methods are distinct with solution procedure and they are modified with different transform algorithms. To illustrate the reliability and applicability of the considered solution procedures we consider eight special cases with different initial conditions. The fractional operator is considered in Caputo sense. The achieved results are drowned through two and three‐dimensional plots for different Brownian motions and classical order. The numerical simulations are presented to ensure the efficiency of considered techniques. The behavior of the obtained results for distinct fractional order is captured in the present framework. The outcomes of the present investigation show that, the considered schemes are efficient and powerful to solve nonlinear differential equations arise in science and technology.

Thermal Characteristics, Kinetic Models, and Volatile Constituents during the Energy Conversion of Bituminous SARA Fractions in Air.

To understand the thermal characteristics, nonisothermal kinetic models, and volatile constituents during the energy conversion of bituminous materials at the fraction level, differential scanning calorimetry–mass spectrometry tests were performed on bituminous four fractions, including saturates, aromatics, resins, and asphaltenes (SARA). Then, three-dimensional (3D) nonisothermal kinetic models of SARA fractions were established and volatile constituents of SARA fractions were discussed. Results indicate that when the heating rate is increased, the decomposition temperature ranges in each stage increase and the initial decomposition, peak, and burn-out temperatures of each SARA fraction all shift to high temperatures. Also, the whole energy conversion processes of SARA fractions are mainly exothermic reactions. Additionally, the energy conversion mechanism in each stage of saturates and aromatics accords with different nonisothermal kinetic models. However, the energy conversion mechanisms of resins and asphaltenes are similar and both accord with the 3D diffusion models. Further, the established nonisothermal kinetic models in each decomposition stage of SARA fractions are feasible to describe the energy conversion processes of SARA fractions. The released small molecular volatiles from saturates and aromatics increase when the heating rate is increased, but the macromolecular volatiles are decreased. The opposite is true for resins, but all volatiles emitted from asphaltenes are increased. Finally, the heating rate has little influence on the constituents of emitted gaseous products from SARA fractions but shows an effect on the release amount of volatiles from SARA fractions. The main common volatiles of SARA fractions are CO2, H2O, methanol, hydrazine, propyne, acetaldehyde, and propane. This study contributes to further reveal the energy conversion mechanisms of bituminous materials.

Non-commutative double-sided continued fractions

We study double-sided continued fractions whose coefficients are non-commuting symbols. We work within the formal approach of the Mal’cev–Neumann series and free division rings. We start with presenting the analogs of the standard results from the theory of continued fractions, including their (right and left) simple fractions decomposition, the Euler–Minding summation formulas, and the relations between nominators and denominators of the simple fraction decompositions. We also transfer to the non-commutative double-sided setting the standard description of the continued fractions in terms of 2 × 2 matrices presenting also a weak version of the Serret theorem. The equivalence transformations between the double continued fractions are described, including also the transformation from generic such fractions to their simplest form. Then we give the description of the double-sided continued fractions within the theory of quasideterminants and we present the corresponding version of the LR and qd-algorithms. We study also (strictly and ultimately) periodic double-sided non-commutative continued fractions and we give the corresponding version of the Euler theorem. Finally we present a weak version of the Galois theorem and we give its relation to the non-commutative KP map, recently studied in the theory of discrete integrable systems.

Thermal effects of asphalt SARA fractions, kinetic parameter calculation using isoconversional method and distribution models

To discuss the thermal decomposition mechanism of asphalt materials, the kinetic parameters were calculated and their distribution models were established based on differential scanning calorimetry (DSC) test results of saturates, aromatics, resins and asphaltenes (SARA) fractions in asphalt. Results indicate that the thermal decomposition processes of SARA fractions are accompanied by complicated endothermic and exothermic reactions, and the exothermic reactions are predominant. The DSC curve is shifted to higher temperature when the heating rate is elevated, and the thermal effects, reaction progress and decomposition rate are increased simultaneously. Also, the Friedman–Reich–Levi (FRL) isoconversional method is suitable to calculate the kinetic parameters during the thermal decomposition of SARA fractions. The conversion ratio (α) of each SARA fraction shows obvious effects on the calculated activation energy (E). As the α of each SARA fraction is increased, the calculated E of saturates is more accurate at a lower α at the decomposition stages 1 and 3, but is always accurate at stage 2. Additionally, the calculated E is more accurate at the lower and higher α than that at the middle α at the two decomposition stages of aromatics. The calculated E of resins is fluctuated between 99 and 112 kJ mol−1, and its accuracy is satisfactory. The calculated E of asphaltenes is changed between 8 and 80 kJ mol−1, and is more accurate at the lower and higher α than that at the middle α. Finally, the distributed activation energy models of each SARA fraction are established to provide an insight into thermal decomposition mechanism of asphalt materials.

On Mordell–Tornheim double Eisenstein series

Inspired by the Mordell–Tornheim double zeta function, in this paper, we introduce the Eisenstein series of Mordell–Tornheim type. Based on the theory of Cohen series and partial fraction decomposition, we explicitly compute these series. We end by discussing several identities about products of Eisenstein series and their Fourier expansions.

Adaptive fractional multi-scale edge-preserving decomposition and saliency detection fusion algorithm

Image fusion expands the space–time scope of the detection target by making full use of the information complementarity between images, which is useful for automatic target recognition. An adaptive fractional multi-scale edge-preserving decomposition based on the weighted least square framework is proposed to fuse infrared (IR) and visible (VIS) images. It decomposes images into the base layer and the detail layer. The saliency map is obtained based on adaptive fractional saliency detection of the IR image. This map is beneficial to fuse the base layer and highlights the saliency information significantly. Then, the detail layer is fused by classical choose-max. The experiments indicate that the proposed method outperforms state-of-the-art fusion methods in extracting the target and preserving background information of the IR and VIS images.

ECG signal denoising by fractional wavelet transform thresholding

The analysis of electrocardiogram (ECG) signals allows experts to diagnose several cardiac disorders. However, the accuracy of such diagnosis depends heavily on the signal quality. In this paper, an efficient method based on fractional wavelet decomposition coupled with thresholding techniques is proposed for noise removal. The usual low-pass and high-pass filters of the wavelet transform are replaced by fractional-order ones. Thus, fractional wavelets are proposed, simulated, and compared to other wavelets for ECG denoising. The denoising process was made operational by the means of an appropriate choice of the wavelet transform coefficient thresholding and the wavelet decomposition level of the signal. Considering the relative error metrics, the best wavelet function for efficient denoising is the fractional one. In our study, we have used eight real ECG signals from the Physionet MITBIH. In order to prove the effectiveness of our method, we investigated the filtering of two types of noises, namely Gaussian white noise and power-line interference (PLI) noise. The proposed method removed the Gaussian white noise completely and had better performance on the PLI noise. Considering classical metrics of assessment, results show the advantage of the proposed method compared to other types of wavelets. The proposed method is the most suitable one for removing PLI and Gaussian white noise from ECG signals with superior performance than other wavelets. Also, it can be applied for high-frequency denoising even without a priori frequency knowledge.

Refuse Derived Fuel Production through Biodrying Process (Case study: Solid Waste from Canteens)

Due to its calorific value, wastes could be treated into Refuse Derived Fuel (RDF) through several processes. In order to get higher calorific value, moisture content in the wastes could be removed by utilizing the heat generated from decomposition of organic fraction by microorganism (biodrying process). The study aims to treat solid wastes generated from canteens in Ganesha Campus of Institut Teknologi Bandung into RDF through biodrying process. Through standard sampling procedure, total waste generated from 59 canteens was 228 kg/day and organic fraction became the dominant (74%). There were 3 biodrying piles prepared, namely aeration, windrow, and control pile. Temperature in all piles increased in first and second weeks, then it gradually decreased and the average temperature were between 23-48 oC. The heat generated during the process could be remove water content and the optimum time 17-22 days could reach 20-30% of moisture content. The highest calorific value could be obtained from aerated pile (14.98 MJ/kg). By considering several parameters, the best RDF were produced from aerated pile. The parameters which still did not comply with the international standard of RDF were ash content, fixed carbon, and organic carbon. The quality of RDF was affected significantly by the composition of the feed. Though it could not meet with all parameters as an international standard of RDF, the product could be used as co-fuel to substitute coal or other fossil fuels for industrial activities. By knowing that the wastes could be converted into valuable product, the local municipality may shift the conventional paradigm of the waste management which is only collect-haul-dispose into a new paradigm by prioritizing waste recycle.

Oxidation as an Early Stage in the Multistep Thermal Decomposition of Uranium(IV) Oxalate into U3O8.

The thermal decomposition of actinide oxalates is greatly dependent on the oxidation state of the cation, the gas involved, and the physical characteristics of the precursor. In the actinides series, uranium(IV) oxalate U(C2O4)2·6H2O can be viewed as a peculiar case, as its sensibility toward oxidation leads to a specific series of reactions when heating under an oxygen atmosphere. In order to clarify the disagreements existing in the literature, particularly concerning potential carbonate intermediates and the possible transitory existence of UO3, we show here an extended characterization of the different intermediates through a combination of X-ray diffraction, vibrational spectroscopies and X-ray absorption near-edge spectroscopy. In this frame, uranium oxidation was found to occur at low temperature (200 °C) concomitantly to the onset of oxalate groups decomposition, leading to an amorphous oxo-oxalato compound. Pursuing the thermal conversion up to 350 °C led to complete oxidation of U(IV) into U(VI), then to the formation of amorphous UO3 still bearing adsorbed carbonates. The first pure oxide formed during the thermal conversion was further identified to substoichiometric UO3-δ after heating at 550 °C. Finally, U3O8 was obtained as the final stable phase after heating above 660 °C. The mechanism of thermal conversion of uranium(IV) oxalate into oxide under oxygen is then driven by a complex interplay between redox reactions and decomposition of the organic fractions. Such chemical reactions were also found to significantly modify the morphology of the powder through high-temperature environmental scanning electron microscopy observations: decomposition led to a 20% reduction in the size of the aggregates, while uranium oxidation clearly promoted growth within the agglomerates.

Novel Dynamic Structures of 2019-nCoV with Nonlocal Operator via Powerful Computational Technique.

In this study, we investigate the infection system of the novel coronavirus (2019-nCoV) with a nonlocal operator defined in the Caputo sense. With the help of the fractional natural decomposition method (FNDM), which is based on the Adomian decomposition and natural transform methods, numerical results were obtained to better understand the dynamical structures of the physical behavior of 2019-nCoV. Such behaviors observe the general properties of the mathematical model of 2019-nCoV. This mathematical model is composed of data reported from the city of Wuhan, China.

Estoque de nutrientes e decomposição de serapilheira em matas ciliares do bioma Cerrado em Coxim - MS, Brasil

A serapilheira é considerada como um dos principais componentes-chave no processo de funcionamento dos ecossistemas florestais. Com base nisso, o objetivo deste estudo foi avaliar se a dinâmica de decomposição de serapilheira foliar se correlaciona com variáveis ambientais e pode ser empregada como indicadora ambiental da ciclagem de nutrientes em duas áreas de matas ciliares (Área Córrego Criminoso e Área do Rio Taquari), localizadas no bioma Cerrado, no município de Coxim – MS, Brasil. Para a avaliação das taxas de decomposição e teores de nutrientes na serapilheira, foram coletadas folhas recém-caídas, posteriormente submetidas à secagem, pesagem e acondicionamento em 30 litterbags (25 cm x 25 cm), distribuídos aleatoriamente na superfície do solo em cada área de estudo. Da serapilheira acumulada em cada área, realizou-se a análise química dos nutrientes Cu, Ca, Fe, Mg, Mn, K. A massa remanescente foliar foi equivalente em ambas às áreas, ao final dos 150 dias de estudo, detendo de um percentual de 74,4%. A decomposição da fração foliar teve maior correlação com a temperatura do ar do que com a precipitação pluviométrica. Na serapilheira na Área Rio Taquari houve menores concentrações de nutrientes Cu, K e Fe quando comparada à Área Córrego Criminoso. De modo geral, a serapilheira se correlacionou com variável ambiental (temperatura do ar) e pode ser considerada uma indicadora de ciclagem de nutrientes em áreas de matas ciliares no Cerrado.