Search results for: shifted Jacobi polynomials

Authors: Falek Kamel

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Various approaches are usually used in the dynamic analysis of beams vibrating transversally. For this, numerical methods allowing the solving of the general eigenvalue problem are utilized. The equilibrium equations describe the movement resulting from the solution of a fourth-order differential equation. Our investigation is based on the finite element method. The findings of these investigations are the vibration frequencies obtained by the Jacobi method. Two types of the elementary mass matrix are considered, representing a uniform distribution of the mass along with the element and concentrated ones located at fixed points whose number is increased progressively separated by equal distances at each evaluation stage. The studied beams have different boundary constraints representing several classical situations. Comparisons are made for beams where the distributed mass is replaced by n concentrated masses. As expected, the first calculus stage is to obtain the lowest number of beam parts that gives a frequency comparable to that issued from the Rayleigh formula. The obtained values are then compared to theoretical results based on the assumptions of the Bernoulli-Euler theory. These steps are used for the second type of mass representation in the same manner.

Keywords: structural elements, beams vibrating, dynamic analysis, finite element method, Jacobi method

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Authors: Natalia D. Vaysfel’d, Zinaida Y. Zhuravlova

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The loaded plane elastic semistrip, the lateral boundaries of which are fixed, is considered. The integral transformations are applied directly to Lame’s equations. It leads to one dimensional boundary value problem in the transformations’ domain which is formulated as a vector one. With the help of the matrix differential calculation’s apparatus and apparatus of Green matrix function the exact solution of a vector problem is constructed. After the satisfying the boundary condition at the semi strip’s edge the problem is reduced to the solving of the integral singular equation with regard of the unknown stress at the semis trip’s edge. The equation is solved with the orthogonal polynomials method that takes into consideration the real singularities of the solution at the ends of integration interval. The normal stress at the edge of the semis trip were calculated and analyzed.

Keywords: semi strip, Green's Matrix, fourier transformation, orthogonal polynomials method

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Authors: M. Fakharian, M. I. Khodakarami

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In this paper, a new trend for improvement in semi-analytical method based on scale boundaries in order to solve the 2D elastodynamic problems is provided. In this regard, only the boundaries of the problem domain discretization are by specific sub-parametric elements. Mapping functions are uses as a class of higher-order Lagrange polynomials, special shape functions, Gauss-Lobatto -Legendre numerical integration, and the integral form of the weighted residual method, the matrix is diagonal coefficients in the equations of elastodynamic issues. Differences between study conducted and prior research in this paper is in geometry production procedure of the interpolation function and integration of the different is selected. Validity and accuracy of the present method are fully demonstrated through two benchmark problems which are successfully modeled using a few numbers of DOFs. The numerical results agree very well with the analytical solutions and the results from other numerical methods.

Keywords: 2D elastodynamic problems, lagrange polynomials, G-L-Lquadrature, decoupled SBFEM

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Authors: Yasaswi Palagummi, Sareh Rowlands

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Generalised Zero-Shot Learning, often known as GZSL, is an advanced variant of zero-shot learning in which the samples in the unseen category may be either seen or unseen. GZSL methods typically have a bias towards the seen classes because they learn a model to perform recognition for both the seen and unseen classes using data samples from the seen classes. This frequently leads to the misclassification of data from the unseen classes into the seen classes, making the task of GZSL more challenging. In this work of ours, to solve the GZSL problem, we propose an approach leveraging the Shifted Window based Self-Attention in the Swin Transformer (Swin-GZSL) to work in the inductive GSZL problem setting. We run experiments on three popular benchmark datasets: CUB, SUN, and AWA2, which are specifically used for ZSL and its other variants. The results show that our model based on Swin Transformer has achieved state-of-the-art harmonic mean for two datasets -AWA2 and SUN and near-state-of-the-art for the other dataset - CUB. More importantly, this technique has a linear computational complexity, which reduces training time significantly. We have also observed less bias than most of the existing GZSL models.

Keywords: generalised, zero-shot learning, inductive learning, shifted-window attention, Swin transformer, vision transformer

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Authors: Min Wang, Sergey Utev

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The lessons from the 2007-09 global financial crisis have driven scientific research, which considers the design of new methodologies and financial models in the global market. The quantum mechanics approach was introduced in the unpredictable stock market modeling. One famous quantum tool is Feynman path integral method, which was used to model insurance risk by Tamturk and Utev and adapted to formalize the path-dependent option pricing by Hao and Utev. The research is based on the path-dependent calculation method, which is motivated by the Feynman path integral method. The path calculation can be studied in two ways, one way is to label, and the other is computational. Labeling is a part of the representation of objects, and generating functions can provide many different ways of representing share price paths. In this paper, the recent works on graphical theoretical construction of individual share price path via matroid is presented. Firstly, a study is done on the knowledge of matroid, relationship between lattice path matroid and Tutte polynomials and ways to connect points in the lattice path matroid and Tutte polynomials is suggested. Secondly, It is found that a general binary tree can be validly constructed from a connected lattice path matroid rather than general lattice path matroid. Lastly, it is suggested that there is a way to represent share price paths via a general binary tree, and an algorithm is developed to construct share price paths from general binary trees. A relationship is also provided between lattice integer points and Tutte polynomials of a transversal matroid. Use this way of connection together with the algorithm, a share price path can be constructed from a given connected lattice path matroid.

Keywords: combinatorial construction, graphical representation, matroid, path calculation, share price, Tutte polynomial

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Authors: K. Khelil, H. Ammar, K. Saouchi

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Fiber Bragg Grating (FBG) structure is an periodically modulated optical fiber. It acts as a selective filter of wavelength whose reflected peak is called Bragg wavelength and it depends on the period of the fiber and the refractive index. The simulation of FBG is based on solving the Coupled Mode Theory equation by using the Transfer Matrix Method which is carried out using MATLAB. It is found that spectral reflectivity is shifted when the change of temperature and strain is uniform. Under non-uniform temperature or strain perturbation, the spectrum is both shifted and destroyed. In case of transverse loading, reflectivity spectrum is split into two peaks, the first is specific to X axis, and the second belongs to Y axis. FBGs are used in civil engineering to detect perturbations applied to buildings.

Keywords: Bragg wavelength, coupled mode theory, optical fiber, temperature measurement

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Authors: Prashant Pandey

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In the present article, an effective Laguerre collocation method is used to obtain the approximate solution of a system of coupled fractional-order non-linear reaction-advection-diffusion equation with prescribed initial and boundary conditions. In the proposed scheme, Laguerre polynomials are used together with an operational matrix and collocation method to obtain approximate solutions of the coupled system, so that our proposed model is converted into a system of algebraic equations which can be solved employing the Newton method. The solution profiles of the coupled system are presented graphically for different particular cases. The salient feature of the present article is finding the stability analysis of the proposed method and also the demonstration of the lower variation of solute concentrations with respect to the column length in the fractional-order system compared to the integer-order system. To show the higher efficiency, reliability, and accuracy of the proposed scheme, a comparison between the numerical results of Burger’s coupled system and its existing analytical result is reported. There are high compatibility and consistency between the approximate solution and its exact solution to a higher order of accuracy. The exhibition of error analysis for each case through tables and graphs confirms the super-linearly convergence rate of the proposed method.

Keywords: fractional coupled PDE, stability and convergence analysis, diffusion equation, Laguerre polynomials, spectral method

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Authors: Walaa Obaidallah Alqarafi, Wafaa Mohammed Fakieh, Alaa Abdallah Altassan

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Algebraic graph theory is defined as a bridge between algebraic structures and graphs. It has several uses in many fields, including chemistry, physics, and computer science. The prime graph is a type of graph associated with a ring R, where the vertex set is the whole ring R, and two vertices x and y are adjacent if either xRy=0 or yRx=0. However, the investigation of the prime graph over rings remains relatively limited. The behavior of this graph in extended rings, like R[x] and R[[x]], where R is a non-commutative ring, deserves more attention because of the wider applicability in algebra and other mathematical fields. To study the prime graphs over polynomials and power series rings, we used a combination of ring-theoretic and graph-theoretic techniques. This paper focuses on two invariants: the diameter and the girth of these graphs. Furthermore, the work discusses how the graph structures change when passing from R to R[x] and R[[x]]. In our study, we found that the set of strong zero-divisors of ring R represents the set of vertices in prime graphs. Based on this discovery, we redefined the vertices of prime graphs using the definition of strong zero divisors. Additionally, our results show that although the prime graphs of R[x] and R[[x]] are comparable to the graph of R, they have different combinatorial characteristics since these extensions contain new strong zero-divisors. In particular, we find conditions in which the diameter and girth of the graphs, as they expand from R to R[x] and R[[x]], do not change or do change. In conclusion, this study shows how extending a non-commutative ring R to R[x] and R[[x]] affects the structure of their prime graphs, particularly in terms of diameter and girth. These findings enhance the understanding of the relationship between ring extensions and graph properties.

Keywords: prime graph, diameter, girth, polynomial ring, power series ring

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Authors: Abdon Choque-Rivero

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The Jacobi system with the Dirichlet boundary condition is considered on a half-line lattice when the coefficients are real valued. The inverse problem of recovery of the coefficients from various data sets containing the so-called transmission eigenvalues is analyzed. The Marchenko method is utilized to solve the corresponding inverse problem.

Keywords: inverse scattering, discrete system, transmission eigenvalues, Marchenko method

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Authors: Navdeep Goel, Salvador Gabarda

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Digital images are widely used in computer applications. To store or transmit the uncompressed images requires considerable storage capacity and transmission bandwidth. Image compression is a means to perform transmission or storage of visual data in the most economical way. This paper explains about how images can be encoded to be transmitted in a multiplexing time-frequency domain channel. Multiplexing involves packing signals together whose representations are compact in the working domain. In order to optimize transmission resources each 4x4 pixel block of the image is transformed by a suitable polynomial approximation, into a minimal number of coefficients. Less than 4*4 coefficients in one block spares a significant amount of transmitted information, but some information is lost. Different approximations for image transformation have been evaluated as polynomial representation (Vandermonde matrix), least squares + gradient descent, 1-D Chebyshev polynomials, 2-D Chebyshev polynomials or singular value decomposition (SVD). Results have been compared in terms of nominal compression rate (NCR), compression ratio (CR) and peak signal-to-noise ratio (PSNR) in order to minimize the error function defined as the difference between the original pixel gray levels and the approximated polynomial output. Polynomial coefficients have been later encoded and handled for generating chirps in a target rate of about two chirps per 4*4 pixel block and then submitted to a transmission multiplexing operation in the time-frequency domain.

Keywords: chirp signals, image multiplexing, image transformation, linear canonical transform, polynomial approximation

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Authors: Zongyao Sha

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Green vegetation plays a vital role in energy flows and matter cycles in terrestrial ecosystems, and vegetation phenology may not only be influenced by but also impose active feedback on climate changes. The phenological events of vegetation, such as the start of the season (SOS), end of the season (EOS), and length of the season (LOS), can respond to climate changes and affect gross primary productivity (GPP). Here we coupled satellite remote sensing imagery with FLUXNET observations to systematically map the shift of SOS, EOS, and LOS in global vegetated areas and explored their response to climate fluctuations and feedback on GPP during the last two decades. Results indicated that SOS advanced significantly, at an average rate of 0.19 days/year at a global scale, particularly in the northern hemisphere above the middle latitude (≥30°N) and that EOS was slightly delayed during the past two decades, resulting in prolonged LOS in 72.5% of the vegetated area. The climate factors, including seasonal temperature and precipitation, are attributed to the shifts in vegetation phenology but with a high spatial and temporal difference. The study revealed interactions between vegetation phenology and climate changes. Both temperature and precipitation affect vegetation phenology. Higher temperature as a direct consequence of global warming advanced vegetation green-up date. On the other hand, 75.9% and 20.2% of the vegetated area showed a positive correlation and significant positive correlation between annual GPP and length of vegetation growing season (LOS), likely indicating an enhancing effect on vegetation productivity and thus increased carbon uptake from the shifted vegetation phenology. Our study highlights a comprehensive view of the vegetation phenology changes of the global terrestrial ecosystems during the last two decades. The interactions between the shifted vegetation phenology and climate changes may provide useful information for better understanding the future trajectory of global climate changes. The feedback on GPP from the shifted vegetation phenology may serve as an adaptation mechanism for terrestrial ecosystems to mitigate global warming through improved carbon uptake from the atmosphere.

Keywords: vegetation phenology, growing season, NPP, correlation analysis

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Authors: Imad Eddine Charif, Sidi Mohamed Mekelleche, Didier Villemin

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The solvent effects on the keto-enol tautomeric equilibriums of acetylacetone and dimedone are theoretically investigated at the correlated Becke-3-parameter-Lee-Yang-Parr (B3LYP) and second-order Møller-Plesset (MP2) computational levels. The present study shows that the most stable keto tautomer of acetylacetone corresponds to the trans-diketo, E,Z form; while the most stable enol tautomer corresponds to the closed cis-enol,Z,Z form. The keto tautomer of dimedone prefers the trans diketo, E, E form; while the most stable enol tautomer corresponds to trans-enol form. The calculated free Gibbs enthalpies indicate that, in polar solvents, the keto-enol equilibrium of acetylacetone is shifted toward the keto tautomer; whereas the keto-enol equilibrium of dimedone is shifted towards the enol tautomer. The experimental trends of the change of equilibrium constants with respect to the change of solvent polarity are well reproduced by both B3LYP and MP2 calculations.

Keywords: acetylacetone, dimedone, solvent effects, keto-enol equilibrium, theoretical calculations

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Authors: K. Godazandeh, K. Sadeghy

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In the present study, the effect of magnetic field on the hydrodynamic instability of Taylor-Couette flow between two concentric rotating cylinders has been numerically investigated. At the beginning the basic flow has been solved using continuity, Cauchy equations (with regards to Lorentz force) and the constitutive equations of a viscoelastic model called "Giesekus" model. Small perturbations, considered to be normal mode, have been superimposed to the basic flow and the unsteady perturbation equations have been derived consequently. Neglecting non-linear terms, the general eigenvalue problem obtained has been solved using pseudo spectral method (combination of Chebyshev polynomials). The objective of the calculations is to study the effect of magnetic fields on the onset of first mode of instability (axisymmetric mode) for different dimensionless parameters of the flow. The results show that the stability picture is highly influenced by the magnetic field. When magnetic field increases, it first has a destabilization effect which changes to stabilization effect due to more increase of magnetic fields. Therefor there is a critical magnetic number (Hartmann number) for instability of Taylor-Couette flow. Also, the effect of magnetic field is more dominant in large gaps. Also based on the results obtained, magnetic field shows a more considerable effect on the stability at higher Weissenberg numbers (at higher elasticity), while the "mobility factor" changes show no dominant role on the intense of suction and injection effect on the flow's instability.

Keywords: magnetic field, Taylor-Couette flow, Giesekus model, pseudo spectral method, Chebyshev polynomials, Hartmann number, Weissenberg number, mobility factor

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Authors: Nidal Ali

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Consider an algebraic number field K of degree n, A0 K is its ring of integers and a prime number p inert in K. Let F(u1, . . . , un, x) be the generic polynomial of integers of K. We will study in advance the stability of this polynomial and then, we will apply it in order to obtain all the monic irreducible polynomials in Fp[x] of degree d dividing n.

Keywords: generic polynomial, irreducibility, iteration, stability, inert prime, totally ramified

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Authors: Hossein Kheiri, Bashir Naderi, Mohamad Reza Niknam

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In this paper we study the optimal synchronization of chaotic T-system with complete uncertain parameter. Optimal control laws and parameter estimation rules are obtained by using Hamilton-Jacobi-Bellman (HJB) technique and Lyapunov stability theorem. The derived control laws are optimal adaptive control and make the states of drive and response systems asymptotically synchronized. Numerical simulation shows the effectiveness and feasibility of the proposed method.

Keywords: Lyapunov stability, synchronization, chaos, optimal control, adaptive control

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Authors: Chiou-Yng Lee, Wen-Yo Lee, Chieh-Tsai Wu, Cheng-Chen Yang

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Shifted polynomial basis (SPB) is a variation of polynomial basis representation. SPB has potential for efficient bit-level and digit-level implementations of multiplication over binary extension fields with subquadratic space complexity. For efficient implementation of pairing computation with large finite fields, this paper presents a new SPB multiplication algorithm based on Karatsuba schemes, and used that to derive a novel scalable multiplier architecture. Analytical results show that the proposed multiplier provides a trade-off between space and time complexities. Our proposed multiplier is modular, regular, and suitable for very-large-scale integration (VLSI) implementations. It involves less area complexity compared to the multipliers based on traditional decomposition methods. It is therefore, more suitable for efficient hardware implementation of pairing based cryptography and elliptic curve cryptography (ECC) in constraint driven applications.

Keywords: digit-serial systolic multiplier, elliptic curve cryptography (ECC), Karatsuba algorithm (KA), shifted polynomial basis (SPB), pairing computation

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Authors: K. S. Ghosh, Sagnik Bose, Kapil Tripati

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Potentiodynamic polarization studies carried out on AA2024 and AA2014 Al-Cu-Mg alloys of various tempers in 3.5 wt. % NaCl and in 3.5 wt. % NaCl + 1.0 % H2O2 solution characteristic E-i curves. Corrosion potential (Ecorr) value has shifted towards more negative potential with the increase of artificial aging time. The Ecorr value for the alloy tempers has also shifted anodically in presence of H2O2 in 3.5 % NaCl solution. Further, passivity phenomenon has been observed in all the alloy tempers when tested in 3.5 wt. % NaCl solution at pH 12. Stress corrosion cracking (SCC) behaviour of friction stir weld (FSW) joint of AA2014 alloy has been studied bu slow strain rate test (SSRT) in 3.5 wt. % NaCl solution. Optical micrographs of the corroded surfaces of polarised samples showed general corrosion, extensive pitting and intergranular corrosion as well. Further, potentiodynamic cyclic polarization curves displayed wide hysteresis loop indicating that the alloy tempers are susceptible to pit growth damage. Attempts have been made to explain the variation of observed electrochemical and SCC behaviour of the alloy tempers and the electrolyte conditions with the help of microstructural features.

Keywords: AA 2014 and AA 2024 Al-C-Mg alloy, artificial ageing, potentiodynamic polarization, TEM micrographs, stress corrosion cracking (SCC)

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Authors: Dimitris Varsamis, Nicholas Karampetakis

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It is well known that any interpolating polynomial P(x,y) on the vector space Pn,m of two-variable polynomials with degree less than n in terms of x and less than m in terms of y has various representations that depends on the basis of Pn,m that we select i.e. monomial, Newton and Lagrange basis etc. The aim of this paper is twofold: a) to present transformations between the coordinates of the polynomial P(x,y) in the aforementioned basis and b) to present transformations between these bases.

Keywords: bivariate interpolation polynomial, polynomial basis, transformations, interpolating polynomial

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Authors: R. Vidyasagar, G. L. S. Vilela, B. M. Guiraldelli, A. B. Henriques, J. Moodera

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Nano-scaled magnetic systems that can have both magnetic and optical transitions controlled and manipulated by external means have received enormous research attention for their potential applications in magneto-optics and spintronic devices. Among several ferrimagnetic insulators, the Tm₃Fe₅O₁₂ (TmIG) has become a prototype material displaying huge perpendicular magnetic anisotropy. Nevertheless, the optical properties of nano-scale TnIG films have not yet been investigated. We report the observation of giant splitting in the optical transitions of high-quality thin films of Tm₃Fe₅O₁₂ (TmIG) grown by rf sputtering on gadolinium gallium garnet substrates (GGG-111) substrate. The optical absorbance profiles measured with optical absorption spectroscopy show a dual optical transition in visible frequency regimes attributed to the transitions of electrons from the O-2p valence band to the Fe-3d conduction band and from the O-2p valence band to the Fe-2p⁵3d⁶ excitonic states at the Γ-symmetric point of the TmIG Brillouin zone. When the thickness of the film is reduced from 120 nm to 7.5 nm, the 1st optical transition energy shifted from 2.98 to 3.11 eV ( ~130 meV), and the 2nd transition energy shifted from 2.62 to 2.56 eV (~ 60 meV). The giant band splitting of both transitions can be attributed to the population of excited states associated with the atomic modification pertaining to the compressive or tensile strains.

Keywords: optical transitions, thin films, ferrimagnetic insulator, strains

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Authors: Ahmad Zahran, Ahmed Herzallah, Ahmad Ahmad, Mahran Quraan

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Generation of high DC voltages is necessary for testing the insulation material of high voltage AC transmission lines with long lengths. The harmonic and ripple contents of the output DC voltage supplied by high voltage DC circuits require the use of costly capacitors to smooth the output voltage after rectification. This paper proposes a new modular multiplier high voltage DC generator with embedded Cockcroft-Walton circuits that achieve a negligible harmonic and ripple contents of the output DC voltage without the need for costly filters to produce a nearly constant output voltage. In this new topology, Cockcroft-Walton modules are connected in series to produce a high DC output voltage. The modules are supplied by low input AC voltage sources that have the same magnitude and frequency and shifted from each other by a certain angle to eliminate the harmonics from the output voltage. The small ripple factor is provided by the smoothing column capacitors and the phase shifted input voltages of the cascaded modules. The constituent harmonics within each module are determined using Fourier analysis. The viability of the proposed DC generator for testing purposes and the effectiveness of the cascaded connection are confirmed by numerical simulations using MATLAB/Simulink.

Keywords: Cockcroft-Walton circuit, harmonics, ripple factor, HVDC generator

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Authors: Helmi Temimi

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In this paper, we study the super-convergence properties of the discontinuous Galerkin (DG) method applied to one-dimensional mth-order ordinary differential equations without introducing auxiliary variables. We found that nth−derivative of the DG solution exhibits an optimal O (hp+1−n) convergence rates in the L2-norm when p-degree piecewise polynomials with p≥1 are used. We further found that the odd-derivatives and the even derivatives are super convergent, respectively, at the upwind and downwind endpoints.

Keywords: discontinuous, galerkin, superconvergence, higherorder, error, estimates

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Authors: Panpan Feng

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The concept of green infrastructure originated in Europe and the United States. It aims to ensure smart growth of urban and rural ecosystems and achieve sustainable urban and rural ecological, social, and economic development by combining it with gray infrastructure in traditional planning. Based on the literature review of the theoretical origin, value connotation, and measurement methods of green infrastructure, this study summarizes the research content of green infrastructure at different scales from the three spatial levels of region, city, and block and divides it into functional dimensions, spatial dimension, and strategic dimension. The results show that in the functional dimension, from region-city-block, the research on green infrastructure gradually shifts from ecological function to social function. In the spatial dimension, from region-city-block, the research on the spatial form of green infrastructure has shifted from two-dimensional to three-dimensional, and the spatial structure of green infrastructure has shifted from single ecological elements to multiple composite elements. From a strategic perspective, green infrastructure research is more of a spatial planning tool based on land management, environmental livability and ecological psychology, providing certain decision-making support.

Keywords: green infrastructure, multi-scale, social and ecological functions, spatial strategic decision-making tools

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Authors: Said Algarni

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The purpose of this study was to investigate selected numerical methods that demonstrate good performance in solving PDEs. We adapted alternative method that involve rational polynomials. Padé time stepping (PTS) method, which is highly stable for the purposes of the present application and is associated with lower computational costs, was applied. Furthermore, PTS was modified for our study which focused on diffusion equations. Numerical runs were conducted to obtain the optimal local error control threshold.

Keywords: Padé time stepping, finite difference, reaction diffusion equation, PDEs

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Authors: Abdul Jalil M. Khalaf, Esraa M. Kadhim

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The Hosoya polynomial of a graph G is a graphical invariant polynomial that its first derivative at x= 1 is equal to the Wiener index and second derivative at x=1 is equal to the Hyper-Wiener index. In this paper we study the Hosoya polynomial of zero-divisor graphs.

Keywords: Hosoya polynomial, wiener index, Hyper-Wiener index, zero-divisor graphs

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Authors: Min Dai, Hong Liu, Shuaijie Qian

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We study a portfolio selection problem of an investor who faces constraints on rebalancing frequency, which is common in pension fund investment. We formulate it as a multiple optimal stopping problem and utilize the dynamic programming principle. By numerically solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation, we find a series of free boundaries characterizing optimal strategy, and the constraints significantly impact the optimal strategy. Even in the absence of transaction costs, there is a no-trading region, depending on the number of the remaining trading chances. We also find that the equivalent wealth loss caused by the constraints is large. In conclusion, our model clarifies the impact of the constraints on transaction frequency on the optimal strategy.

Keywords: portfolio selection, rebalancing frequency, optimal strategy, free boundary, optimal stopping

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Authors: Ling Zhang, Feng Liu

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A matrix is called a ray pattern matrix if its entries are either 0 or a ray in complex plane which originates from 0. A ray pattern A of order n is called spectrally arbitrary if the complex matrices in the ray pattern class of A give rise to all possible nth degree complex polynomial. Otherwise, it is said to be spectrally non-arbitrary ray pattern. We call that a spectrally arbitrary ray pattern A of order n is minimally spectrally arbitrary if any nonzero entry of A is replaced, then A is not spectrally arbitrary. In this paper, we find that is not spectrally arbitrary when n equals to 4 for any θ which is greater than or equal to 0 and less than or equal to n. In this article, we give several ray patterns A(θ) of order n that are not spectrally arbitrary for some θ which is greater than or equal to 0 and less than or equal to n. by using the nilpotent-Jacobi method. One example is given in our paper.

Keywords: spectrally arbitrary, nilpotent matrix , ray patterns, sign patterns

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Authors: Ceren Karaman, Onur Karaman

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The adsorption of Remazol Black B (RBB), an anionic dye, onto dried orange pulp (DOP) adsorbent prepared by only drying and by treating with cetyltrimetylammonium bromide (CTAB), a cationic surfactant, surface-modified orange pulp (SMOP) was studied in a stirred batch experiments system at 25°C. The adsorption of RBB on each adsorbent as a function of surfactant dosage, initial pH of the solution and initial dye concentration was investigated. The optimum amount of CTAB was found to be 25g/l. For RBB adsorption studies, while working pH value for the DOP adsorbent system was determined as 2.0, it was observed that this value shifted to 8.0 when the 25 g/l CTAB treated-orange pulp (SMOP) adsorbent was used. It was obtained that the adsorption rate and capacity increased to a certain value, and the adsorption efficiency decreased with increasing initial RBB concentration for both DOP and SMOP adsorbents at pH 2.0 and pH 8.0. While the highest adsorption capacity for DOP was determined as 62.4 mg/g at pH 2.0, and as 325.0 mg/g for SMOP at pH 8.0. As a result, it can be said that permanent cationic coating of the adsorbent surface by CTAB surfactant shifted the working pH from 2.0 to 8.0 and it increased the dye adsorption rate and capacity of orange pulp much more significantly at pH 8.0. The equilibrium RBB adsorption data on each adsorbent were best described by the Langmuir isotherm model. The adsorption kinetics of RBB on each adsorbent followed a pseudo-second-order model. Moreover, the intraparticle diffusion model was used to describe the kinetic data. It was found that diffusion is not the only rate controlling step. The adsorbent was characterized by the Brunauer–Emmett–Teller (BET) analysis, Fourier-transform-infrared (FTIR) spectroscopy, and scanning-electron-microscopy (SEM). The mechanism for the adsorption of RBB on the SMOP may include hydrophobic interaction, van der Waals interaction, stacking and electrostatic interaction.

Keywords: adsorption, Cetyltrimethylammonium Bromide (CTAB), orange pulp, Remazol Black B (RBB), surface modification

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Authors: Ih-Ching Hsu

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Inspired by the so called Jacobi Identity (x y) z + (y z) x + (z x) y = 0, the following class of functional equations EQ I: F [F (x, y), z] + F [F (y, z), x] + F [F (z, x), y] = 0 is proposed, researched and generalized. Research methodologies begin with classical methods for functional equations, then evolve into discovering of any implicit algebraic structures. One of this paper’s major findings is that EQ I, under two additional conditions F (x, x) = 0 and F (x, y) + F (y, x) = 0, proves to be a fundamental functional equation for Lie Algebras. Existence of non-trivial solutions for EQ I can be proven by defining F (p, q) = [p q] = pq –qp, where p and q are quaternions, and pq is the quaternion product of p and q. EQ I can be generalized to the following class of functional equations EQ II: F [G (x, y), z] + F [G (y, z), x] + F [G (z, x), y] = 0. Concluding Statement: With a major finding proven, and non-trivial solutions derived, this research paper illustrates and provides a new functional equation scheme for studies in two major areas: (1) What underlying algebraic structures can be defined and/or derived from EQ I or EQ II? (2) What conditions can be imposed so that conditional general solutions to EQ I and EQ II can be found, investigated and applied?

Keywords: fundamental functional equation, generalized functional equations, Lie algebras, quaternions

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Authors: Aloke Bapli, Debabrata Seth

Abstract:

Graphene oxide (GO) is the two-dimensional (2D) nanoscale allotrope of carbon having several physiochemical properties such as high mechanical strength, high surface area, strong thermal and electrical conductivity makes it an important candidate in various modern applications such as drug delivery, supercapacitors, sensors etc. GO has been used in the photothermal treatment of cancers and Alzheimer’s disease etc. The main idea to choose GO in our work is that it is a surface active molecule, it has a large number of hydrophilic functional groups such as carboxylic acid, hydroxyl, epoxide on its surface and in basal plane. So it can easily interact with organic fluorophores through hydrogen bonding or any other kind of interaction and easily modulate the photophysics of the probe molecules. We have used different spectroscopic techniques for our work. The Ground-state absorption spectra and steady-state fluorescence emission spectra were measured by using UV-Vis spectrophotometer from Shimadzu (model-UV-2550) and spectrofluorometer from Horiba Jobin Yvon (model-Fluoromax 4P) respectively. All the fluorescence lifetime and anisotropy decays were collected by using time-correlated single photon counting (TCSPC) setup from Edinburgh instrument (model: LifeSpec-II, U.K.). Herein, we described the photophysics of a hydrophilic molecule 7-(n,n׀-diethylamino) coumarin-3-carboxylic acid (7-DCCA) in the reverse micelles containing GO. It was observed that photophysics of dye is modulated in the presence of GO compared to photophysics of dye in the absence of GO inside the reverse micelles. Here we have reported the solvent relaxation and rotational relaxation time in GO containing reverse micelle and compare our work with normal reverse micelle system by using 7-DCCA molecule. Normal reverse micelle means reverse micelle in the absence of GO. The absorption maxima of 7-DCCA were blue shifted and emission maxima were red shifted in GO containing reverse micelle compared to normal reverse micelle. The rotational relaxation time in GO containing reverse micelle is always faster compare to normal reverse micelle. Solvent relaxation time, at lower w₀ values, is always slower in GO containing reverse micelle compare to normal reverse micelle and at higher w₀ solvent relaxation time of GO containing reverse micelle becomes almost equal to normal reverse micelle. Here emission maximum of 7-DCCA exhibit bathochromic shift in GO containing reverse micelles compared to that in normal reverse micelles because in presence of GO the polarity of the system increases, as polarity increases the emission maxima was red shifted an average decay time of GO containing reverse micelle is less than that of the normal reverse micelle. In GO containing reverse micelle quantum yield, decay time, rotational relaxation time, solvent relaxation time at λₑₓ=375 nm is always higher than λₑₓ=405 nm, shows the excitation wavelength dependent photophysics of 7-DCCA in GO containing reverse micelles.

Keywords: photophysics, reverse micelle, rotational relaxation, solvent relaxation

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Authors: Melusi Khumalo, Anastacia Dlamini

Abstract:

In this paper, we consider a numerical solution for nonlinear Fredholm integral equations of the second kind. We work with uniform mesh and use the Lagrange polynomials together with the Galerkin finite element method, where the weight function is chosen in such a way that it takes the form of the approximate solution but with arbitrary coefficients. We implement the finite element method to the nonlinear Fredholm integral equations of the second kind. We consider the error analysis of the method. Furthermore, we look at a specific example to illustrate the implementation of the finite element method.

Keywords: finite element method, Galerkin approach, Fredholm integral equations, nonlinear integral equations

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