Search results for: Fukui function

Authors: Subhojyoti Chatterjee, Peter J. Mahon, Feng Wang

Abstract:

Resveratrol (RvL), a phenolic compound, is a key ingredient in wine and tomatoes that has been studied over the years because of its important bioactivities such as anti-oxidant, anti-aging and antimicrobial properties. Out of the two isomeric forms of resveratrol i.e. trans and cis, the health benefit is primarily associated with the trans form. Thus, studying the structural properties of the isomers will not only provide an insight into understanding the RvL isomers, but will also help in designing parameters for differentiation in order to achieve 99.9% purity of trans-RvL. In the present study, density function theory (DFT) study is conducted, using the B3LYP/6-311++G** model to explore the through bond and through space intramolecular interactions. Properties such as vibrational spectroscopy (IR and Raman), nuclear magnetic resonance (NMR) spectra, excess orbital energy spectrum (EOES), energy based decomposition analyses (EDA) and Fukui function are calculated. It is discovered that the structure of trans-RvL, although it is C1 non-planar, the backbone non-H atoms are nearly in the same plane; whereas the cis-RvL consists of two major planes of R1 and R2 that are not in the same plane. The absence of planarity gives rise to a H-bond of 2.67Å in cis-RvL. Rotation of the C(5)-C(8) single bond in trans-RvL produces higher energy barriers since it may break the (planar) entire conjugated structure; while such rotation in cis-RvL produces multiple minima and maxima depending on the positions of the rings. The calculated FT-IR spectrum shows very different spectral features for trans and cis-RvL in the region 900 – 1500 cm-1, where the spectral peaks at 1138-1158 cm-1 are split in cis-RvL compared to a single peak at 1165 cm-1 in trans-RvL. In the Raman spectra, there is significant enhancement of cis-RvL in the region above 3000cm-1. Further, the carbon chemical environment (13C NMR) of the RvL molecule exhibit a larger chemical shift for cis-RvL compared to trans-RvL (Δδ = 8.18 ppm) for the carbon atom C(11), indicating that the chemical environment of the C group in cis-RvL is more diverse than its other isomer. The energy gap between highest occupied molecular orbital (HOMO) and the lowest occupied molecular orbital (LUMO) is 3.95 eV for trans and 4.35 eV for cis-RvL. A more detailed inspection using the recently developed EOES revealed that most of the large energy differences i.e. Δεcis-trans > ±0.30 eV, in their orbitals are contributed from the outer valence shell. They are MO60 (HOMO), MO52-55 and MO46. The active sites that has been captured by Fukui function (f + > 0.08) are associated with the stilbene C=C bond of RvL and cis-RvL is more active at these sites than in trans-RvL, as cis orientation breaks the large conjugation of trans-RvL so that the hydroxyl oxygen’s are more active in cis-RvL. Finally, EDA highlights the interaction energy (ΔEInt) of the phenolic compound, where trans is preferred over the cis-RvL (ΔΔEi = -4.35 kcal.mol-1) isomer. Thus, these quantum mechanics results could help in unwinding the diversified beneficial activities associated with resveratrol.

Keywords: resveratrol, FT-IR, Raman, NMR, excess orbital energy spectrum, energy decomposition analysis, Fukui function

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Authors: Imad Eddine Charif, Wafaa Benchouk, Sidi Mohamed Mekelleche

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The regioselectivity of a series of 16 1,3-dipolar cycloaddition reactions of nitrile oxides with 2(5H)-furanones has been analysed by means of global and local electrophilic and nucleophilic reactivity indices using density functional theory at the B3LYP level together with the 6-31G(d) basis set. The local electrophilicity and nucleophilicity indices, based on Fukui and Parr functions, have been calculated for the terminal sites, namely the C1 and O3 atoms of the 1,3-dipole and the C4 and C5 atoms of the dipolarophile. These local indices were calculated using both Mulliken and natural charges and spin densities. The results obtained show that the C5 atom of the 2(5H)-furanones is the most electrophilic site whereas the O3 atom of the nitrile oxides is the most nucleophilic centre. It turns out that the experimental regioselectivity is correctly reproduced, indicating that both Fukui- and Parr-based indices are efficient tools for the prediction of the regiochemistry of the studied reactions and could be used for the prediction of newly designed reactions of the same kind.

Keywords: 1, 3-dipolar cycloaddition, density functional theory, nitrile oxides, regioselectivity, reactivity indices

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Authors: Samir Hmaimou, Marouane Ait Lahcen, Mohamed Adardour, Mohamed Maatallah, Abdesselam Baouid

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Due to (3 + 2) cycloaddition and condensation reaction, a wide range of synthetic routes can be used to obtain biologically active heterocyclic compounds. Condensation and (3+2) cycloaddition reactions in heterocyclic syntheses are versatile due to the wide variety of possible combinations of several atoms of the reactants. In this article, we first outline the synthesis of benzodiazepine 4 with two dipolarophilic centers (C=C and C=N) by condensation reaction. Then, we use it for cycloaddition reactions (3+2) with nitrile oxides to prepare oxadiazole-benzodiazepines and pyrazole-benzodiazepine compounds. ¹H and ¹³C NMR are used to establish all the structures of the synthesized products. These condensation and cycloaddition reactions were then analyzed using density functional theory (DFT) calculations at the B3LYP/6-311G(d,p) theoretical level. In this study, the mechanism of the one-step cycloaddition reaction was investigated. Molecular electrostatic potential (MEP) was used to identify the electrophilic and nucleophilic attack sites of the molecules studied. Additionally, Fukui investigations (electrophilic f- and nucleophilic f+) in the various reaction centers of the reactants demonstrate that, whether in the condensation reaction or cycloaddition, the reaction proceeds through the atomic centers with the most important Fukui functions, which is in full agreement with experimental observations. In the condensation reaction, thermodynamic control of regio, chemo, and stereoselectivity is observed, while those of cycloaddition are subject to kinetic control.

Keywords: cycloaddition reaction, regioselectivity, mechanism reaction, NMR analysis

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Authors: Karim Hrratha, Khaled Essalahb, Christophe Morellc, Henry Chermettec, Salima Boughdiria

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Among the carbon-carbon bond formation types, allylation of active methylene compounds with cyclic Baylis-Hillman (BH) alcohols is a reliable and widely used method. This reaction is a very attractive tool in organic synthesis of biological and biodiesel compounds. Thus, in view of an insistent and peremptory request for an efficient and straightly method for synthesizing the desired product, a thorough analysis of various aspects of the reaction processes is an important task. The product afforded by the reaction of active methylene with BH alcohols depends largely on the experimental conditions, notably on the catalyst properties. All experiments reported that catalysis is needed for this reaction type because of the poor ability of alcohol hydroxyl group to be as a suitable leaving group. Within the catalysts, several transition- metal based have been used such as palladium in the presence of acid or base and have been considered as reliable methods. Furthemore, acid catalysts such as BF3.OEt2, BiX3 (X= Cl, Br, I, (OTf)3), InCl3, Yb(OTf)3, FeCl3, p-TsOH and H-montmorillonite have been employed to activate the C-C bond formation through the alkylation of active methylene compounds. Interestingly a report of a smoothly process for the ability of 4-imethyaminopyridine(DMAP) to catalyze the allylation reaction of active methylene compounds with cyclic Baylis-Hillman (BH) alcohol appeared recently. However, the reaction mechanism remains ambiguous, since the C- allylation process leads to an unexpected product (noted P1), corresponding to a direct allylation instead of conjugate allylation, which involves the most electrophilic center according to the electron withdrawing group CO effect. The main objective of the present theoretical study is to better understand the role of the DMAP catalytic activity as well as the process leading to the end- product (P1) for the catalytic reaction of a cyclic BH alcohol with active methylene compounds. For that purpose, we have carried out computations of a set of active methylene compounds varying by R1 and R2 toward the same alcohol, and we have attempted to rationalize the mechanisms thanks to the acid–base approach, and conceptual DFT tools such as chemical potential, hardness, Fukui functions, electrophilicity index and dual descriptor, as these approaches have shown a good prediction of reactions products.The present work is then organized as follows: In a first part some computational details will be given, introducing the reactivity indexes used in the present work, then Section 3 is dedicated to the discussion of the prediction of the selectivity and regioselectivity. The paper ends with some concluding remarks. In this work, we have shown, through DFT method at the B3LYP/6-311++G(d,p) level of theory that: The allylation of active methylene compounds with cyclic BH alcohol is governed by orbital control character. Hence the end- product denoted P1 is generated by direct allylation.

Keywords: DFT calculation, gas phase pKa, theoretical mechanism, orbital control, charge control, Fukui function, transition state

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Authors: Nur Rokhman

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A mathematical function gives relationship between the variables composing the function. Interpolation can be viewed as a process of finding mathematical function which goes through some specified points. There are many interpolation methods, namely: Lagrange method, Newton method, Spline method etc. For some specific condition, such as, big amount of interpolation points, the interpolation function can not be written explicitly. This such function consist of computational steps. The solution of equations involving the interpolation function is a problem of solution of non linear equation. Newton method will not work on the interpolation function, for the derivative of the interpolation function cannot be written explicitly. This paper shows the use of Secton method to determine the numerical solution of the function involving the interpolation function. The experiment shows the fact that Secton method works better than Newton method in finding the root of Lagrange interpolation function.

Keywords: Secton method, interpolation, non linear function, numerical solution

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Authors: Faisel Eltuhami Alzaalik, Omar Imhemed Alramli, Ahmed Mohamed Elaieb

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The IEEE 802.11 defines two modes of MAC, distributed coordination function (DCF) and point coordination function (PCF) mode. The first sub-layer of the MAC is the distributed coordination function (DCF). A contention algorithm is used via DCF to provide access to all traffic. The point coordination function (PCF) is the second sub-layer used to provide contention-free service. PCF is upper DCF and it uses features of DCF to establish guarantee access of its users. Some papers and researches that have been published in this technology were reviewed in this paper, as well as talking briefly about the distributed coordination function (DCF) technology. The simulation of the PCF function have been applied by using a simulation program called network simulator (NS2) and have been found out the throughput of a transmitter system by using this function.

Keywords: DCF, PCF, throughput, NS2

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Authors: Shatha S. Alhily

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The purpose of this research paper is to show all the possible values of the pth power of the integrable function which make the integral means of the derivative of univalent function existing and finite.

Keywords: derivative, integral means, self conformal maps, univalent function

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Authors: F. U. Rahman, R. Q. Zhang

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This work aims to develop a systematic numerical technique which can be easily extended to many-body problem. The Lippmann Schwinger equation (integral form of the Schrodinger wave equation) is solved for the nonrelativistic radial wave of hydrogen atom using iterative integration scheme. As the unknown wave function appears on both sides of the Lippmann Schwinger equation, therefore an approximate wave function is used in order to solve the equation. The Green’s function is obtained by the method of Laplace transform for the radial wave equation with excluded potential term. Using the Lippmann Schwinger equation, the product of approximate wave function, the Green’s function and the potential term is integrated iteratively. Finally, the wave function is normalized and plotted against the standard radial wave for comparison. The outcome wave function converges to the standard wave function with the increasing number of iteration. Results are verified for the first fifteen states of hydrogen atom. The method is efficient and consistent and can be applied to complex systems in future.

Keywords: Green’s function, hydrogen atom, Lippmann Schwinger equation, radial wave

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Authors: Zhongzhi Hu, Jie Shen, Jiqiang Wang

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A high precision aeroengine model is needed when developing the engine control system. Compared with other main components, the axial compressor is the most challenging component to simulate. In this paper, a compressor map optimizing tool based on the introduction of a modifiable β function is developed for FWorks (FADEC Works). Three parameters (d density, f fitting coefficient, k₀ slope of the line β=0) are introduced to the β function to make it modifiable. The comparison of the traditional β function and the modifiable β function is carried out for a certain type of compressor. The interpolation errors show that both methods meet the modeling requirements, while the modifiable β function can predict compressor performance more accurately for some areas of the compressor map where the users are interested in.

Keywords: beta function, compressor map, interpolation error, map optimization tool

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Authors: Slobodan B. Tričković

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We present the results concerned with trigonometric series that include sine and cosine functions with a parameter appearing in the denominator. We derive two types of closed-form formulas for trigonometric series. At first, for some integer values, as we know that Riemann’s ζ function and Dirichlet η, λ equal zero at negative even integers, whereas Dirichlet’s β function equals zero at negative odd integers, after a certain number of members, the rest of the series vanishes. Thus, a trigonometric series becomes a polynomial with coefficients involving Riemann’s ζ function and Dirichlet η, λ, β functions. On the other hand, in some cases, one cannot immediately replace the parameter with any positive integer because we shall encounter singularities. So it is necessary to take a limit, so in the process, we apply L’Hospital’s rule and, after a series of rearrangements, we bring a trigonometric series to a form suitable for the application of Choi-Srivastava’s theorem dealing with Hurwitz’s zeta function and Harmonic numbers. In this way, we express a trigonometric series as a polynomial over Hurwitz’s zeta function derivative.

Keywords: Dirichlet eta lambda beta functions, Riemann's zeta function, Hurwitz zeta function, Harmonic numbers

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Authors: Sasiporn Rattanasupha, Settapat Chinviriyasit

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The treatment function adopts a continuous and differentiable function which can describe the effect of delayed treatment when the number of infected individuals increases and the medical condition is limited. In this paper, the SEIR epidemic model with treatment function is studied to investigate the dynamics of the model due to the effect of treatment. It is assumed that the treatment rate is proportional to the number of infective patients. The stability of the model is analyzed. The model is simulated to illustrate the analytical results and to investigate the effects of treatment on the spread of infection.

Keywords: basic reproduction number, local stability, SEIR epidemic model, treatment function

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Authors: Naga Velamakuri, Jyothi K. Reddy

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Quality must be designed into a product and not inspected has become the main motto of all the companies globally. Due to the rapidly increasing technology in the past few decades, the nature of demands from the consumers has become more sophisticated. To sustain this global revolution of innovation in production systems, companies have to take steps to accommodate this technology growth. In this process of understanding the customers' expectations, all the firms globally take steps to deliver a perfect output. Most of these techniques also concentrate on the consistent development and optimization of the product to exceed the expectations. Quality Function Deployment(QFD) and Modular Function Deployment(MFD) are such techniques which rely on the voice of the customer and help deliver the needs. In this paper, Quality Function Deployment and Modular Function Deployment techniques which help in converting the quantitative descriptions to qualitative outcomes are discussed. The area of interest would be to understand the scope of each of the techniques and the application range in product development when these are applied together to any problem. The research question would be mainly aimed at comprehending the limitations using modularity in product development.

Keywords: quality function deployment, modular function deployment, house of quality, methodology

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Authors: Myunggon Yoon, Jung-Ho Moon

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In this paper, we present a transfer function representation of a general one-dimensional combustor. The input of the transfer function is a heat rate perturbation of a burner and the output is a flow velocity perturbation at the burner. This paper considers a general combustor model composed of multiple cans with different cross sectional areas, along with a non-zero flow rate.

Keywords: combustor, dynamics, thermoacoustics, transfer function

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Authors: İbrahim Aktaş, Árpád Baricz

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In this paper, the radii of star likeness of the Jackson and Hahn-Exton q-Bessel functions are considered, and for each of them three different normalizations is applied. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower, and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-Pólya class of real entire functions plays an important role in this study. In particular, we obtain some new bounds for the first positive zero of the derivative of the classical Bessel function of the first kind.

Keywords: bessel function, lommel function, radius of starlikeness and convexity, Struve function

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Authors: Christian Lavault

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In this paper, the generalized fractional integral operators of two generalized Mittag-Leffler type functions are investigated. The special cases of interest involve the generalized M-series and K-function, both introduced by Sharma. The two pairs of theorems established herein generalize recent results about left- and right-sided generalized fractional integration operators applied here to the M-series and the K-function. The note also results in important applications in physics and mathematical engineering.

Keywords: Fox–Wright Psi function, generalized hypergeometric function, generalized Riemann– Liouville and Erdélyi–Kober fractional integral operators, Saigo's generalized fractional calculus, Sharma's M-series and K-function

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Authors: Diogo Silva, Fadul Rodor, Carlos Moraes

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This work compares the results of multidimensional function approximation using two algorithms: the classical Particle Swarm Optimization (PSO) and the Quantum Particle Swarm Optimization (QPSO). These algorithms were both tested on three functions - The Rosenbrock, the Rastrigin, and the sphere functions - with different characteristics by increasing their number of dimensions. As a result, this study shows that the higher the function space, i.e. the larger the function dimension, the more evident the advantages of using the QPSO method compared to the PSO method in terms of performance and number of necessary iterations to reach the stop criterion.

Keywords: PSO, QPSO, function approximation, AI, optimization, multidimensional functions

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Authors: Vladimir S. Timofeev

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In this article the problem of distributional moments estimation is considered. The new approach of moments estimation based on usage of the characteristic function is proposed. By statistical simulation technique, author shows that new approach has some robust properties. For calculation of the derivatives of characteristic function there is used numerical differentiation. Obtained results confirmed that author’s idea has a certain working efficiency and it can be recommended for any statistical applications.

Keywords: characteristic function, distributional moments, robustness, outlier, statistical estimation problem, statistical simulation

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Authors: Rogelio Luck, Yucheng Liu

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This paper proposes the continuous-time singular value decomposition (SVD) for the impulse response function, a special kind of Green’s functions e⁻⁽ᵗ⁻ ᵀ⁾, in order to find a set of singular functions and singular values so that the convolutions of such function with the set of singular functions on a specified domain are the solutions to the inhomogeneous differential equations for those singular functions. A numerical example was illustrated to verify the proposed method. Besides the continuous-time SVD, a discrete-time SVD is also presented for the impulse response function, which is modeled using a Toeplitz matrix in the discrete system. The proposed method has broad applications in signal processing, dynamic system analysis, acoustic analysis, thermal analysis, as well as macroeconomic modeling.

Keywords: singular value decomposition, impulse response function, Green’s function , Toeplitz matrix , Hankel matrix

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Authors: Rashidah Omar, Suzeini Abdul Halim, Aini Janteng

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The coefficients estimate problem for Taylor-Maclaurin series is still an open problem especially for a function in the subclass of bi-univalent functions. A function f ϵ A is said to be bi-univalent in the open unit disk D if both f and f^-1 are univalent in D. The symbol A denotes the class of all analytic functions f in D and it is normalized by the conditions f(0) = f’(0) – 1=0. The class of bi-univalent is denoted by  The subordination concept is used in determining second and third Taylor-Maclaurin coefficients. The upper bound for second and third coefficients is estimated for functions in the subclasses of bi-univalent functions which are subordinated to the function φ. An analytic function f is subordinate to an analytic function g if there is an analytic function w defined on D with w(0) = 0 and |w(z)| < 1 satisfying f(z) = g[w(z)]. In this paper, two subclasses of bi-univalent functions associated with Hohlov operator are introduced. The bound for second and third coefficients of functions in these subclasses is determined using subordination. The findings would generalize the previous related works of several earlier authors.

Keywords: analytic functions, bi-univalent functions, Hohlov operator, subordination

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Authors: Samir Hmaimou, Marouane Ait Lahcen, Mohamed Adardour, Mohamed Maatallah, Abdesselam Baouid

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The stereoisomers (E)-2,2-dimethyl-4-(4-subsitutedstyryl)-2,3-dihydro-1H-[1,5]-benzodiazepine 3(a-d) were synthesized via the condensation reaction of 2,2,3 4-trimethyl-2,3-dihydro-1H-1,5-benzodiazepine (BZD) 1 with the benzaldehyde derivatives 2(a-d) in polar protic solvent as ethanol. The chemical structure of the prepared products was confirmed by NMR (¹H and ¹³C), HRMS, and X-ray analysis of the crystal structure 3d. The condensation reaction was examined using DFT calculations at the theoretical level of B3LYP/6-311G(d,p). Frontier molecular orbital analysis shows that the most favorable interaction is between the HOMO of BZD 1 and the LUMO of 2(a-d). On the other hand, the calculation of the global reactivity indices (softness, hardness, and chemical potential) confirmed that benzodiazepine BDZ 1 act as a nucleophile, whereas the aldehyde derivatives 2(a-d) play the role of electrophile. Furthermore, we identified each reagent's reactive sites by the measurement of the reactivity indices to explain the experimentally observed regioselectivity, using Fukui local reactivity descriptors. A one-step mechanism reaction and order 2 water elimination were investigated. We also looked at how the electron-withdrawing groups (EWG) of various aldehydes affected the reaction's mechanism and the stability of products 3(a-d).

Keywords: benzodiazepine, DFT calculations, crystal structure, regioselective, condensation Reaction

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Authors: Lanyan Peng, Ling Chen

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Leishan Miao New-Year is one of the grandest festivals in the southeastern of Guizhou Province in China. It was officially listed in the National Intangible Cultural Heritage List in 2008, as a traditional folk cultural activity organized by the local Miao people. With the rise of cultural tourism, after 19 years of exploration, the local government has successfully built Miao New-Year into a cultural card that is well-known at home and abroad. During the Miao New-Year period, it has attracted 3.8 million tourists and achieves a win-win situation in the economy and culture. However, tourism development has changed the living environment and living state of the local people. And it is accompanied by changes in the form of the festival, the content of the festival, and the local people’s needs and attitudes to the festival. This paper uses the field investigation method to achieve 410 questionnaires and 35 interviews, exploring the process and the reasons for changes of Leishan Miao New-Year’s cultural function. Among all the functions, the economic function, identity function, and entertainment function have been enhanced, and the marriage and love function has been extended. In the meanwhile, sacrificial function has been weakened. There are some trends in functions. The function of commemorating ancestor and self-entertainment has been changed to entertaining people and economic pursuit.

Keywords: Miao New-Year, Miao nationality, festival function, changes

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Authors: Irina Peterburgsky

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Let T be a unit circumference of a complex plane, E be a Banach space, E* and E** be its conjugate and second conjugate, respectively. In general, a Hardy space Hp(E), p ≥1, where functions act from the open unit disk to E, could contain a function for which even weak nontangential (angular) boundary value in the space E** does not exist at any point of the unit circumference T (C. Grossetete.) The situation is "better" when certain restrictions to the Banach space of values are applied (more or less resembling a classical case of scalar-valued functions depending on constrains, as shown by R. Ryan.) This paper shows that, nevertheless, in the case of a Banach space of a general type, the following positive statement is true: Proposition. For any function f(z) from Hp(E), p ≥ 1, there exists a function F(eiθ) on the unit circumference T to E** whose Poisson (in the Pettis sense) is integral regains the function f(z) on the open unit disk. Some characteristics of the function F(eiθ) are demonstrated.

Keywords: hardy spaces, Banach space-valued function, boundary values, Pettis integral

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Authors: Zuhier Altawallbeh

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In this paper, an explicit homotopic function is constructed to compute the Hochschild homology of a finite dimensional free k-module V. Because the polynomial algebra is of course fundamental in the computation of the Hochschild homology HH and the cyclic homology CH of commutative algebras, we concentrate our work to compute HH of the polynomial algebra.by providing certain homotopic function.

Keywords: hochschild homology, homotopic function, free and projective modules, free resolution, exterior algebra, symmetric algebra

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Authors: Bina Kumari, Subir K. Sarkar, Pradipta Bandyopadhyay

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The autocorrelation function of the density fluctuation is studied in each of the two phases in a Gibbs Ensemble Monte Carlo (GEMC) simulation of the problem of phase separation for a square well potential with various values of its range. We find that the normalized autocorrelation function is described very well as a linear combination of an exponential function with a time scale τ₂ and a stretched exponential function with a time scale τ₁ and an exponent α. Dependence of (α, τ₁, τ₂) on the parameters of the GEMC algorithm and the range of the square well potential is investigated and interpreted. We also analyse the issue of how to choose the parameters of the GEMC simulation optimally.

Keywords: autocorrelation function, density fluctuation, GEMC, simulation

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Authors: Yuan Zheng

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3D model-based vehicle matching provides a new way for vehicle recognition, localization and tracking. Its key is to construct an evaluation function, also called fitness function, to measure the degree of vehicle matching. The existing fitness functions often poorly perform when the clutter and occlusion exist in traffic scenarios. In this paper, we present a practical and efficient fitness function. Unlike the existing evaluation functions, the proposed fitness function is to study the vehicle matching problem from both local and global perspectives, which exploits the pixel gradient information as well as the silhouette information. In view of the discrepancy between 3D vehicle model and real vehicle, a weighting strategy is introduced to differently treat the fitting of the model’s wireframes. Additionally, a normalization operation for the model’s projection is performed to improve the accuracy of the matching. Experimental results on real traffic videos reveal that the proposed fitness function is efficient and robust to the cluttered background and partial occlusion.

Keywords: 3D-2D matching, fitness function, 3D vehicle model, local image gradient, silhouette information

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Authors: Mohammedi Ferhate

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This paper deals with the generalized the notion of the function of wave a micro particle moving free, the concept of the linear operator in the representation function delta of Dirac which is a generalization of the symbol of Kronecker to the case of a continuous variation of the sizes concerned with the condition of orthonormation of the Eigen functions the use of linear operators and their Eigen functions in connection with the solution of given differential equations, it is of interest to study the properties of the operators themselves and determine which of them follow purely from the nature of the operators, without reference to specific forms of Eigen functions. The models simulation examples are also presented.

Keywords: function, operator, simulation, wave

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Authors: Md. Rashidul Hasan, Atikur Rahman Baizid

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The Bayesian estimation approach is a non-classical estimation technique in statistical inference and is very useful in real world situation. The aim of this paper is to study the Bayes estimators of the parameter of exponential distribution under different loss functions and then compared among them as well as with the classical estimator named maximum likelihood estimator (MLE). In our real life, we always try to minimize the loss and we also want to gather some prior information (distribution) about the problem to solve it accurately. Here the gamma prior is used as the prior distribution of exponential distribution for finding the Bayes estimator. In our study, we also used different symmetric and asymmetric loss functions such as squared error loss function, quadratic loss function, modified linear exponential (MLINEX) loss function and non-linear exponential (NLINEX) loss function. Finally, mean square error (MSE) of the estimators are obtained and then presented graphically.

Keywords: Bayes estimator, maximum likelihood estimator (MLE), modified linear exponential (MLINEX) loss function, Squared Error (SE) loss function, non-linear exponential (NLINEX) loss function

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Authors: Jihad H. Asad

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An expression for the Green’s function (GF) of anisotropic face cantered cubic (IFCC) lattice is evaluated analytically and numerically for a single impurity problem. The density of states (DOS), phase shift and scattering cross section are expressed in terms of complete elliptic integrals of the first kind.

Keywords: lattice Green's function, elliptic integral, physics, cubic lattice

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Authors: X. Wang, T. J. Craft, H. Iacovides

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When handling the near-wall regions of turbulent flows, it is necessary to account for the viscous effects which are important over the thin near-wall layers. Low-Reynolds- number turbulence models do this by including explicit viscous and also damping terms which become active in the near-wall regions, and using very fine near-wall grids to properly resolve the steep gradients present. In order to overcome the cost associated with the low-Re turbulence models, a more advanced wall function approach has been implemented within OpenFoam and tested together with a standard log-law based wall function in the prediction of flows which involve 2-D shock wave/turbulent boundary layer interactions (SWTBLIs). On the whole, from the calculation of the impinging shock interaction, the three turbulence modelling strategies, the Lauder-Sharma k-ε model with Yap correction (LS), the high-Re k-ε model with standard wall function (SWF) and analytical wall function (AWF), display good predictions of wall-pressure. However, the SWF approach tends to underestimate the tendency of the flow to separate as a result of the SWTBLI. The analytical wall function, on the other hand, is able to reproduce the shock-induced flow separation and returns predictions similar to those of the low-Re model, using a much coarser mesh.

Keywords: SWTBLIs, skin-friction, turbulence modeling, wall function

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Authors: Takashi Mitsuishi, Koji Saigusa

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This paper presents circular polar coordinates transformation of periodic fuzzy membership function. The purpose is identification of domain of periodic membership functions in consequent part of IF-THEN rules. The proposed methods are applied to the simple color construct system.

Keywords: periodic membership function, polar coordinates transformation, defuzzification, circular coordinates

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