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The infrared (IR) spectra of different white copy paper types tend to appear indifferent. Discrimination between white copy papers could lead to the solution of forgery cases. In this preliminary study, three varieties of white paper were purchased from local stationery shops in Kuala Lumpur, Malaysia. The papers were classified according to their manufacturers using Partial Least Squares-Discriminant Analysis (PLS-DA) and sparse PLS-DA (sPLS-DA) models. The error rate for the two models on the training and the test data sets were estimated and compared. Results show that the performance of the two models are comparable. By including the first six latent variables in both models, classification accuracy as high as 100% can be achieved.

Selection of the most significant variables, i.e. the wavenumber, from an infrared (IR) spectrum is always difficult to be achieved. In this preliminary paper, the feasibility of genetic algorithms (GA) in identifying most informative wavenumbers from 150 IR spectra of papers was investigated. The list of selected wavenumbers was then employed in Linear Discriminant Analysis (LDA). GA procedure was repeated 30 times to get different lists of variables. Then the performances of LDA models were estimated via leave-one-out cross-validation. A total of six to eight wavenumbers were identified to be valuable variables in the GA procedures. All the 30 LDA models achieve correct classification rates between 97.3% to 100.0%. Therefore the GA-LDA model could be a suitable tool for differentiating white papers that appeared to be highly similar in their IR fingerprints.

Infrared (IR) spectral data are always influenced by undesired random and systematic variations. As such, pre-processing of spectral data is normally required before chemometric modeling. Two most widely used pre-processing techniques, i.e. scatter-correction methods and spectral derivatives, were used to pre-process 150 IR spectral data of paper. The algorithms investigated in this preliminary study are Standard Normal Variate (SNV), Multiplicative Scatter Correction (MSC), Savitzky-Golay (SG) and Gap-Segment (GS). The visual examination of the clustering among three studied varieties of paper, i.e. IK Yellow, One Paper and Save Pack, is accomplished via Principal Component Analysis (PCA). Overall, separation of the three varieties of paper is greatly enhanced after pre-processing. The most significant improvement is obtained with pre-processing via 1st derivative using SG algorithms.

Vateria Indica is a multipurpose tree that is much used locally in India. It provides food, medicine and a range of other commodities. It is often planted along avenues in India. The use of vegetable oil as feedstock for biodiesel production is divisive as an outcome of the challenges of a food-fuel catastrophe linked with the use of fit to be eaten oils for biodiesel making. The present study is focused on the extraction of oil from vateria Indica seed using a solvent extraction method, evaluation of optimal conditions for oil extraction. A Box-Behnken design of response surface methodology (RSM) with 15 experimental runs is used to study the optimum environment for the withdrawal, and the variables of interest are effective solvent/seed ratio 1 ml/gm, 1.25 ml/gm and 1.5 ml/gm. extraction temperature 60°C, 65°C, 70°C, and extraction time 3 hr, 4 hr, 5 hr. From the current study it is observed that the optimized oil yield using the solvent extraction is found to be 22.85% at temperature of 66.6°C, extraction time of 4.41 hour keeping the solvent to seed ratio of 1.353 ml/g after adopting response optimization technique.

Pineapples are available all year round since they are unseasonal fruits. Due to the continuous harvesting of the fruit, the retailers and farmers had to find a solution such as the processing of pineapple into jam, to treat the unsuccessfully sold pineapples. The direct heating of pineapple puree during the production of pineapple jam can cause over degradation of quality of the fresh pineapple. Thus, this study aims to optimise the microwave-assisted processing conditions for producing pineapple jam which could reduce water activity and meets minimum requirement for pH and total soluble solids contents of fruit jam. The power and time of the microwave processing were chosen as the factors, while the water activity, pH and total soluble solids (TSS) content of the pineapple jam were determined as responses to be optimised. The microwave treatment on the pineapple jam was able to give significant effect on the water activity and TSS content of the pineapple jam. The optimum power and time for the microwav...

Color masterbatches are widely used in WPC production. As long as no product changes in terms of color will occur, the rheological behavior of color master batches as well as their influence on actual production conditions seems negligible. However, at short production runs or frequent color changes on the same product (within one shift), the rheological influence of color masterbatches to the WPC formulation is significant. Change of recorded production parameters are often observed and can also lead to production problems up to production stops. This presentation shows rheological analysis on color masterbatches, their influence to WPC formulations as well as their influence on WPC production in direct extrusion. Furthermore, possible strategies are discussed at the end.

Friction is a phenomenon which occurs commonly in the nature and in mechanical constructions. One can find numerous mathematical models describing friction as a one-dimensional process. On the other hand, the number of models which take into account second dimension is significantly smaller. The paper in hand introduce two-dimensional, dynamical model for orthotropic dry friction. Proposed model obeys maximum dissipation power principle by means of non-linear two-dimensional bristle stiffness. Numerical studies show influence of orthotropic friction on planar oscillator and 2D stick-slip system trajectories. Model is also verified against experimental results. Frictional pair with orthotropic properties have been prepared for laboratory rig which is a spherical pendulum with frictional contact. Comparison between experimental and simulation results shows good similarity, although further validation is required.Friction is a phenomenon which occurs commonly in the nature and in mechanical constructions. One can find numerous mathematical models describing friction as a one-dimensional process. On the other hand, the number of models which take into account second dimension is significantly smaller. The paper in hand introduce two-dimensional, dynamical model for orthotropic dry friction. Proposed model obeys maximum dissipation power principle by means of non-linear two-dimensional bristle stiffness. Numerical studies show influence of orthotropic friction on planar oscillator and 2D stick-slip system trajectories. Model is also verified against experimental results. Frictional pair with orthotropic properties have been prepared for laboratory rig which is a spherical pendulum with frictional contact. Comparison between experimental and simulation results shows good similarity, although further validation is required.

In this paper, we define soft D− metric spaces and give some fundamental definitions. In addition to this, we define a soft Δ− distance on a complete soft D− metric.In this paper, we define soft D− metric spaces and give some fundamental definitions. In addition to this, we define a soft Δ− distance on a complete soft D− metric.

The most systems in our environment contain components that interact through competition or cooperation, which can lead to system adaptation. Recently, it is especially important to study the behavior of such systems, and to develop and apply new appropriate mathematical methods for studying the processes in these systems. Such approaches have many applications in economy and sociology and are successfully used in mathematics, physics, ecology, biology and technical sciences. In the last decades non–linear models are intensively used to model economic and social systems. In many cases the main features of such complex systems can be explained by a relatively small number of non–linear differential equations. Examples of such systems are some economic organizations. In this paper we model the behavior of a socio-economic system by partial differential equations. The model describes dynamics of populations competing for limited resources. In the model, migration is treated as a advection–diffusion process influenced by changing of the growth rates and the interactions among population individuals. The model describes several novel features of the interacting populations compared to the well-known classic models in population dynamics. Using the modified method of simplest equation and one of its extended versions, we obtain new wave solutions of the model system.The most systems in our environment contain components that interact through competition or cooperation, which can lead to system adaptation. Recently, it is especially important to study the behavior of such systems, and to develop and apply new appropriate mathematical methods for studying the processes in these systems. Such approaches have many applications in economy and sociology and are successfully used in mathematics, physics, ecology, biology and technical sciences. In the last decades non–linear models are intensively used to model economic and social systems. In many cases the main features of such complex systems can be explained by a relatively small number of non–linear differential equations. Examples of such systems are some economic organizations. In this paper we model the behavior of a socio-economic system by partial differential equations. The model describes dynamics of populations competing for limited resources. In the model, migration is treated as a advection–diffusion process i...

This paper presents a method for quantitatively calculating the responsibility of harmonic energy loss. First, we establish a Norton equivalent circuit at the PCC to obtain a harmonic loss calculation formula for the user and the background under the condition that the circuit equivalent parameters are known. For the background side of the PCC, the background fluctuation harmonic method is used to calculate the background equivalent harmonic impedance. For the user side of the PCC, the user equivalent harmonic impedance is estimated by calculating the fundamental impedance. Under the premise of accurately calculating the equivalent circuit parameters, by measuring the voltage and current data of the PCC during the analysis period, the user’s responsibility for harmonic energy loss can be quantitatively calculated. Through the IEEE14 node system simulation test, the calculation results show that our error control is within a small range, and the calculation method is accurate and easy to implement.This paper presents a method for quantitatively calculating the responsibility of harmonic energy loss. First, we establish a Norton equivalent circuit at the PCC to obtain a harmonic loss calculation formula for the user and the background under the condition that the circuit equivalent parameters are known. For the background side of the PCC, the background fluctuation harmonic method is used to calculate the background equivalent harmonic impedance. For the user side of the PCC, the user equivalent harmonic impedance is estimated by calculating the fundamental impedance. Under the premise of accurately calculating the equivalent circuit parameters, by measuring the voltage and current data of the PCC during the analysis period, the user’s responsibility for harmonic energy loss can be quantitatively calculated. Through the IEEE14 node system simulation test, the calculation results show that our error control is within a small range, and the calculation method is accurate and easy to implement.

Blades on the wind turbine are moving due to the lift and drag force. During the conversion process of wind energy into electrical energy arise energy loss naturally. One of the optimization techniques of power harvesting in wind turbine is pitch angle control on the turbine blades hence, the wind turbine power output lies on optimum condition. The wind turbine control system in this paper is composed of turbine blades, servo motor as actuator, rotational connector and rotary encoder as transmitter. The best result of wind turbine prototype test shows that Power Coefficient (CP) up to 0.57510. To obtain adaptive control conform to the existing wind speed, pitch angle adjustment is controlled using PID control, with each parameters value of Kp, Ki, and Kd are equal to 0,0239069, 0.0001382, 0,053518, respectively. The parameters value are achieved by implementing PSO optimization method. According to the result of the test, the set points are 40, 60, 80 and 100 rpm, which is the system is able to reach the set point with maximum overshoot attains 26.25% and the rise time in 181 seconds. Therefore, the PID-PSO produces a better response than uncontrolled and fuzzy-based controller.Blades on the wind turbine are moving due to the lift and drag force. During the conversion process of wind energy into electrical energy arise energy loss naturally. One of the optimization techniques of power harvesting in wind turbine is pitch angle control on the turbine blades hence, the wind turbine power output lies on optimum condition. The wind turbine control system in this paper is composed of turbine blades, servo motor as actuator, rotational connector and rotary encoder as transmitter. The best result of wind turbine prototype test shows that Power Coefficient (CP) up to 0.57510. To obtain adaptive control conform to the existing wind speed, pitch angle adjustment is controlled using PID control, with each parameters value of Kp, Ki, and Kd are equal to 0,0239069, 0.0001382, 0,053518, respectively. The parameters value are achieved by implementing PSO optimization method. According to the result of the test, the set points are 40, 60, 80 and 100 rpm, which is the system is able to reach the ...

In this paper, new results of domination in graphs called bi-domination are determined. Some properties for bi-dominating set are presented. The bi-domination number of G is the minimum cardinality of a bi-dominating set of G. In this paper, we investigate the bi-domination number for web graph, tadpole graph, lollipop graph, daisy graph, Dutch windmill graph, windmill graph, dumbbell graph, barbell graph, grid graph, product of two path, and the corona H ⊙ K2 and H + K2.

Motivated by the recent physicists’ interest in contact geometry, this review paper is devoted to some modern geometric insights upon the contact structures. In view of this, we start from the initial perspective on contact manifolds, namely that of an odd-dimensional orientable manifold whose volume form is generated by a 1-form θ and its differential dθ. This naturally arises from a coorientable maximally non-integrable hyperplane distribution. In this picture, we establish a one-to-one correspondence between the transitive Jacobi pairs over odd-dimensional manifolds and the coorientable contact structures over the same manifolds. Then, we introduce the geometric perspective on contact manifolds by omitting the coorientability of the maximally non-integrable hyperplane distribution, and we define the contact structure via an L-valued 1-form [with (L, π, M) a line bundle over an odd-dimensional manifold] with non-degenerate curvature. In this realm, it is shown that there exists a one-to-one correspondence between the transitive Jacobi line bundles over odd-dimensional manifolds and the contact structures over the same manifolds. This faithful ‘representation’ of contact structures brings them nearer to symplectic-like ones through the canonical [bracket] structures inherited from the corresponding Jacobi structures.Motivated by the recent physicists’ interest in contact geometry, this review paper is devoted to some modern geometric insights upon the contact structures. In view of this, we start from the initial perspective on contact manifolds, namely that of an odd-dimensional orientable manifold whose volume form is generated by a 1-form θ and its differential dθ. This naturally arises from a coorientable maximally non-integrable hyperplane distribution. In this picture, we establish a one-to-one correspondence between the transitive Jacobi pairs over odd-dimensional manifolds and the coorientable contact structures over the same manifolds. Then, we introduce the geometric perspective on contact manifolds by omitting the coorientability of the maximally non-integrable hyperplane distribution, and we define the contact structure via an L-valued 1-form [with (L, π, M) a line bundle over an odd-dimensional manifold] with non-degenerate curvature. In this realm, it is shown that there exists a one-to-one corresponden...

The recent paper (DOI: 10.1007/s10898-017-0537-6) suggests various practical tests (sufficient conditions) for checking pseudoconvexity of a twice differentiable function on an interval domain. The tests were implemented using interval extensions of the gradient and the Hessian of the function considered. In this paper, we present an alternative approach which is based on the use of linear interval parametric enclosures of the gradient and the Hessian. It is shown that the new approach results in more efficient tests for checking pseudoconvexity.

In this paper the objectivity in science, the Riemann-Liouville and the Caputo fractional order derivatives are presented shortly. This is followed by the presentation of some recent papers which propose the use of these fractional order derivatives, instead of the integer order derivatives, in the description of some physical phenomena. The objectivity of the new mathematical concepts, constitutive equations, evolution equations in these papers is not considered. In the present paper it is shown that in classical mechanics when Riemann-Liouville or Caputo fractional derivatives are used, the objectivity of the new concepts, constitutive relations, evolution equations, is lost. With this aim this study was undertaken.

Since the real structure is not perfect, the geometric imperfections are present. In this case, the bifurcation buckling load overestimates the buckling load. This problem is particularly interesting in experimental studies, because the amplitude of imperfection and its mode can differ for each sample. In this paper, a methodology for determination of the lowest buckling load of a thin-walled plate structure with imperfection using a load–axial shortening plot was presented. The proposed approach can be applied, when the post-buckling path is stable, only. It was shown that the load corresponding to an alternation in rigidity of the real structure on the load-axial shortening plot determines the buckling load with high accuracy. The P-w2 method and the inflection point method were applied, as well, to verify the obtained results. First, numerical calculations were performed by the finite element method and Koiter's method using Byskov-Hutchinson's formulation. The imperfections were defined explicitly. Other parameters of the system, such as boundary conditions, loads, geometrical dimensions were free from inaccuracies. A plate model of the thin-walled structure was applied. An eigenvalue buckling problem of perfect structures was solved to determine bifurcation loads and their eigenmodes. Then nonlinear problem of buckling was solved by Koiter's perturbation method for one mode approach or the finite element method employing Newton-Rawson's method. The load - axial shortening plots were made to analyse an influence of the imperfection amplitude on an approximate value of the lowest buckling load. The uncoupled local buckling was considered. Detailed computations were conducted for short Z-column made of general carbon-epoxy laminate under uniform compression. The Z-column was simply supported on both ends. Finally, the numerical results were verified in experimental tests using Aramis system. A static compression test was performed on a universal testing machine. Tests were performed at a constant velocity of the cross-bar equal to 2 mm/min. The compressive load was less than 150% of the bifurcation load. A very good agreement between the results attained with both the methods for solving the nonlinear problem was obtained.Since the real structure is not perfect, the geometric imperfections are present. In this case, the bifurcation buckling load overestimates the buckling load. This problem is particularly interesting in experimental studies, because the amplitude of imperfection and its mode can differ for each sample. In this paper, a methodology for determination of the lowest buckling load of a thin-walled plate structure with imperfection using a load–axial shortening plot was presented. The proposed approach can be applied, when the post-buckling path is stable, only. It was shown that the load corresponding to an alternation in rigidity of the real structure on the load-axial shortening plot determines the buckling load with high accuracy. The P-w2 method and the inflection point method were applied, as well, to verify the obtained results. First, numerical calculations were performed by the finite element method and Koiter's method using Byskov-Hutchinson's formulation. The imperfections were defined explicitly. Ot...

The aim of this paper deals with Caputo’s solution of fractional two-point boundary value problems by using second-order central difference discretization scheme and Caputo’s fractional operator to construct a Caputo’s finite differences approximation equation. Then this approximation equation was be used to generate a linear system. In this paper, the Successive Over-Relaxation (SOR) method has been considered as linear solver. To do this matter, this method is derive based on the Caputo’s approximation equation. Based on numerical results, solutions in this problem will show SOR method is requires less amount of number of iterations and computational time as compared with GS method. In term of number of iterations, performance analysis of SOR methods has drastically decreased between 95.00% and 99.99% with the execution time decline between 87.00% and 99.99% respectively as compared with GS method. The numerical result showed that the SOR method is more efficient as compared with GS method.

In this paper we introduce and examine the concepts of Δfm−lacunary statistical convergence and lacunary strongly Δfm−convergence and give some relations between Δfm−lacunary statistical convergence and lacunary strongly Δfm−convergence.In this paper we introduce and examine the concepts of Δfm−lacunary statistical convergence and lacunary strongly Δfm−convergence and give some relations between Δfm−lacunary statistical convergence and lacunary strongly Δfm−convergence.

In this paper, we extend the Montgomery identities for the generalized fractional integrals. These results are connected with the celebrated Ostrowski type integral inequality for generalized fractional integral operators by definition of Sarikaya et al.[6]. The results presented here would provide extensions of those given in earlier works.In this paper, we extend the Montgomery identities for the generalized fractional integrals. These results are connected with the celebrated Ostrowski type integral inequality for generalized fractional integral operators by definition of Sarikaya et al.[6]. The results presented here would provide extensions of those given in earlier works.

The paper considers the influence of the assumed error indicator on the final adapted mesh. Provided that threshold values of an error are increased by applying the adaptive procedure, it turns out that final mesh depends on the assumed error indicator. In the paper, there were used the standard error estimates and the error indicator proposed by the author. The proposed error indicator is based on applying hierarchically generalized finite difference method (FDM). In the case of the proposed error indicator, the final adapted mesh is the most optimal for the exact solution.

In this paper, we obtain sufficient conditions for compactness of the commutator for the Riesz potential [b, Iα] in global Morrey-type spaces GMpθw .In this paper, we obtain sufficient conditions for compactness of the commutator for the Riesz potential [b, Iα] in global Morrey-type spaces GMpθw .

This paper presents the calculation of the gas flow through the piston ring pack of the combustion engine. The paper compares the solution of ordinary differential equations using the standard numerical methods (ode solvers present in MATLAB) with a new method based on the Taylor series – Modern Taylor Series Method.This paper presents the calculation of the gas flow through the piston ring pack of the combustion engine. The paper compares the solution of ordinary differential equations using the standard numerical methods (ode solvers present in MATLAB) with a new method based on the Taylor series – Modern Taylor Series Method.

This paper illustrates that coefficients involved in iterative scheme play major role in determining the speed of its convergence. Here in this paper, we analyze the speed of convergence of Jungck-Mann &Jungck-Ishikawa iteration plans by the methodology of interchanging the coefficients involved in these iterative procedures.

Many mathematical models developed to predict the stress-strain curve of soil. Most of them are complicated because of the complexity nature of soil material. Moreover, soil behavior is apparently unpredictable due to the fact that every type of soil has its own inherent properties that cannot be represented by a single model. Hence, the objectives of this paper is to review the mathematical models available in the literature and to assess the applicability, limitation and the basic requirement in order to apply the models. Also, this paper will look on the basic principles of the models and the basic characteristic that should have on the models for the best prediction capability. This paper will review the basic models such as th elastic models (Hooke's law and hyperbolic models), simple elastic plastic models (Drucker-Prager, Mohr Coulomb, DiMaggio-Sandler, PLAXIS Soft Soil and Lade and Duncan), Critical State Soil Models (Modified Cam Clay, elasto-viscoplastic, Structured Cam Clay, Anisotropic Cam Clay) and Bounding surface models (Dafalias and MIT-E3). Some weaknesses will also be highlighted to shed some light for further improvements.Many mathematical models developed to predict the stress-strain curve of soil. Most of them are complicated because of the complexity nature of soil material. Moreover, soil behavior is apparently unpredictable due to the fact that every type of soil has its own inherent properties that cannot be represented by a single model. Hence, the objectives of this paper is to review the mathematical models available in the literature and to assess the applicability, limitation and the basic requirement in order to apply the models. Also, this paper will look on the basic principles of the models and the basic characteristic that should have on the models for the best prediction capability. This paper will review the basic models such as th elastic models (Hooke's law and hyperbolic models), simple elastic plastic models (Drucker-Prager, Mohr Coulomb, DiMaggio-Sandler, PLAXIS Soft Soil and Lade and Duncan), Critical State Soil Models (Modified Cam Clay, elasto-viscoplastic, Structured Cam Clay, Anisotropic Cam Cla...

Identifiability of any parametric model should be analyzed before econometric estimation of parameters. If a model’s structure is not identified, parameters cannot be estimated. This result is well known for the case of simultaneous equations models. The problem of identifiability does not, however, relates only to simultaneous equations models. The issue of local and global identifiability of an underlying theoretical structure is discussed in this paper in a very general class of parametric models. An important operative criterion for assessing local identifiability is the rank condition of Fisher information matrix. The paper analyzes this rank condition using Kullback information integral and comments on some results relating to global identifiability.

The paper deals with the numerical solution of partial differential equations. The one-dimensional wave equation was chosen for experiments; it is solved using Method of Lines which transforms the partial differential equation into the system of ordinary differential equations. The solution in time remains continuous, and the Modern Taylor Series Method is used for solving the system of initial value problems. On the other hand, the spatial discretization is performed using higher order finite difference formulas, which can be unstable. The necessity of the variable precision arithmetic to stabilize the solution is discussed in this paper. The seven point difference formula is analysed as an example of higher order finite difference formulas.

The biomass production is very essential in microalgae farming such that its growth rate is very important to be determined. This paper proposes the dynamics model of it that restricted by its nutrition. The model is developed by considers some related processes that are photosynthesis, respiration, nutrition absorption, stabilization, lipid synthesis and CO2 mobilization. The stability of the dynamical system that represents the processes is analyzed using the Jacobian matrix of the linearized system in the neighborhood of its critical point. There is a lipid formation threshold needed to require its existence. In such case, the absorption rate of respiration process has to be inversely proportional to the absorption rate of CO2 due to photosynthesis process. The Pontryagin minimal principal also shows that there are some requirements needed to have a stable critical point, such as the rate of CO2 released rate, due to the stabilization process that is restricted by 50%, and the threshold of its shifted critical point. In case of the rate of CO2 released rate due to the photosynthesis process is restricted in such interval; the stability of the model at the critical point could not be satisfied anymore. The simulation shows that the external nutrition plays a role in glucose formation such that sufficient for the biomass growth and the lipid production.The biomass production is very essential in microalgae farming such that its growth rate is very important to be determined. This paper proposes the dynamics model of it that restricted by its nutrition. The model is developed by considers some related processes that are photosynthesis, respiration, nutrition absorption, stabilization, lipid synthesis and CO2 mobilization. The stability of the dynamical system that represents the processes is analyzed using the Jacobian matrix of the linearized system in the neighborhood of its critical point. There is a lipid formation threshold needed to require its existence. In such case, the absorption rate of respiration process has to be inversely proportional to the absorption rate of CO2 due to photosynthesis process. The Pontryagin minimal principal also shows that there are some requirements needed to have a stable critical point, such as the rate of CO2 released rate, due to the stabilization process that is restricted by 50%, and the threshold of its shifted ...

In this paper, we propose for the first time a method based on Bernstein polynomials for solving directly a class of third-order ordinary differential equations (ODEs). This method gives a numerical solution by converting the equation into a system of algebraic equations which is solved directly. Some numerical examples are given to show the applicability of the method.In this paper, we propose for the first time a method based on Bernstein polynomials for solving directly a class of third-order ordinary differential equations (ODEs). This method gives a numerical solution by converting the equation into a system of algebraic equations which is solved directly. Some numerical examples are given to show the applicability of the method.

Spoon is a form of free-form surfaces. In this paper, a method for designing a spoon using partial differential equations (PDE) is discussed. Boundary curves are defined in term of Fourier series. By selecting the appropriate boundary curves, a design of spoon is generated using fourth order elliptic PDE. Then by solving the elliptic PDE, a smooth design of spoon can be generated. In addition, this paper also examined the impact of the choice of number of boundary curves in generating different spoon shapes.

The paper deals with linear-quadratic optimal control problems with a weak control and the fixed left point in a critical case, where a matrix standing in front of a state variable in the state equation is singular for any argument value if the small parameter is equal to zero.Using the direct scheme method, consisting in immediate substituting a postulated asymptotic solution into the problem condition and obtaining problems for finding asymptotics terms, the zero order asymptotic solution is constructed under some conditions.In contrast to the paper: N. T. Hoai, J. Optim. Theory Appl., vol. 175, 324–340 (2017), where the considered problem was studied, the projector approach is applied here. This approach allows us to make the algorithm of constructing asymptotic solution clearer and to correct some inaccuracies in the paper mentioned.

Graph theory was first introduced by Leonhard Euler in 1736 and still being one of the mathematic’s topics which is rapidly developing and can be used to simplify mathematic problems. There are many interesting topics in graph theory; one of them is graph labeling. There are many ways of labeling a graph, and one of them is graceful labeling. Let G(V, E is a graph. The injective mapping f: V → {0,1, …, |E|} is called graceful if the weights of edge w(uv) = |f(u) – f(v)| are all different for every edge uv. There is a famous conjecture in graceful labeling. It is said that all trees are graceful. To prove this conjecture, then we must show that every tree is graceful. There are many research papers dealing with the special cases of trees. Many classes of trees have been proven graceful, and one of them is supercaterpillar. Previous research had proved that supercaterpillars with certain conditions are also graceful. In this paper, we generalize the concept of supercaterpillar, and show that the subclass of supercaterpillar graphs that has not been discussed earlier is also graceful, using an adjacency matrix for the construction.

The aim of this paper is to give interpolation function for the families of numbers which are associated with not only the Apostol-Bernoulli numbers, but also the Apostol-Euler numbers and the Apostol-Genocchi numbers. We investigate some properties of these functions and these numbers. Moreover, we give some formulas including these functions and these numbers. Finally, we also give some observations related to well-known families of zeta-type functions.The aim of this paper is to give interpolation function for the families of numbers which are associated with not only the Apostol-Bernoulli numbers, but also the Apostol-Euler numbers and the Apostol-Genocchi numbers. We investigate some properties of these functions and these numbers. Moreover, we give some formulas including these functions and these numbers. Finally, we also give some observations related to well-known families of zeta-type functions.

This paper presents an introduction to ERKN integrators for multi-frequency highly oscillatory systems with applications to nonlinear Klein–Gordon equations. The result of error analysis stated in this paper is a nature extension of that for Gautschi-type methods of order two. In particular, the error bound of ERKN integrators, when applied to nonlinear Klein–Gordon equations, is shown to be independent of the refinement of spatial discretisation based on the finite energy condition which appeared in the literature.

In this paper, we investigate solutions of the system of difference equations xn+1=xn−1ynxn−1, yn+1=yn−1xnyn−1−1, zn+1=xnynzn−1, where x0,x−1,y0,y−1,z0,z−1 real numbers such that y0 x−1 ≠1 and x0y−1 ≠ 1In this paper, we investigate solutions of the system of difference equations xn+1=xn−1ynxn−1, yn+1=yn−1xnyn−1−1, zn+1=xnynzn−1, where x0,x−1,y0,y−1,z0,z−1 real numbers such that y0 x−1 ≠1 and x0y−1 ≠ 1

In this paper, the basic principle and application of extended finite element method (XFEM) are expounded based on the model of plate structure crack. Simulating crack location by level set method; the discontinuity of crack can be supplemented through introducing Heaviside functions and crack tip reinforcement function. On this basis, the damage model of edge crack and center crack under the same boundary condition are established. The displacements of each node in the two Situations are given and explained the application of damage model in two aspects.In this paper, the basic principle and application of extended finite element method (XFEM) are expounded based on the model of plate structure crack. Simulating crack location by level set method; the discontinuity of crack can be supplemented through introducing Heaviside functions and crack tip reinforcement function. On this basis, the damage model of edge crack and center crack under the same boundary condition are established. The displacements of each node in the two Situations are given and explained the application of damage model in two aspects.

This paper considers the problem of reachability and observability gramian computation for linear time-varying systems. Both gramians are obtained by using system trajectories. The reachability gramian is computed from the state impulse response and the observability gramian is obtained from the output response. The relation with the linear time invariant case is discussed and three different approaches for gramian computation are presented: the dual system approach, the empirical gramian approach and integral operator approach. The general theory of linear time-varying systems is shortly presented and the role of the transient matrix for solving the linear state equation is shown. An algorithm for computation of the transient matrix is proposed, which is based on the integration of the state equation by the method of Runge Kutta. The role of the adjoint system is also discussed and the two basic impulse response characteristics, namely the normal and the adjoint state impulse responses are presented.This paper considers the problem of reachability and observability gramian computation for linear time-varying systems. Both gramians are obtained by using system trajectories. The reachability gramian is computed from the state impulse response and the observability gramian is obtained from the output response. The relation with the linear time invariant case is discussed and three different approaches for gramian computation are presented: the dual system approach, the empirical gramian approach and integral operator approach. The general theory of linear time-varying systems is shortly presented and the role of the transient matrix for solving the linear state equation is shown. An algorithm for computation of the transient matrix is proposed, which is based on the integration of the state equation by the method of Runge Kutta. The role of the adjoint system is also discussed and the two basic impulse response characteristics, namely the normal and the adjoint state impulse responses are presented.

In this paper the reliability assessment of buckling resistance of steel beam is presented. A number of parameters such as: the boundary conditions, the section height to width ratio, the thickness and the span are considered. The examples are solved using FEM procedures and formulas proposed in the literature and standards. In the case of the numerical models the following parameters are investigated: support conditions, mesh size, load conditions, steel grade. The numerical results are compared with approximate solutions calculated according to the standard formulas. It was observed that for high slenderness section the deformation of the cross-section had to be described by the following modes: longitudinal and transverse displacement, warping, rotation and distortion of the cross section shape. In this case we face interactive buckling problem. Unfortunately, neither the EN Standards nor the subject literature give close-form formulas to solve these problems. For this reason the reliability of the critical bending moment calculations is discussed.In this paper the reliability assessment of buckling resistance of steel beam is presented. A number of parameters such as: the boundary conditions, the section height to width ratio, the thickness and the span are considered. The examples are solved using FEM procedures and formulas proposed in the literature and standards. In the case of the numerical models the following parameters are investigated: support conditions, mesh size, load conditions, steel grade. The numerical results are compared with approximate solutions calculated according to the standard formulas. It was observed that for high slenderness section the deformation of the cross-section had to be described by the following modes: longitudinal and transverse displacement, warping, rotation and distortion of the cross section shape. In this case we face interactive buckling problem. Unfortunately, neither the EN Standards nor the subject literature give close-form formulas to solve these problems. For this reason the reliability of the cri...

The aim of this paper is to give some combinatorial sums, identities and relations related to a family of combinatorial numbers and the Bernstein type basis functions. With the help of generating functions for the combinatorial numbers from this aforementioned family, we give some functional equations. By using these functional equations, we obtain several relationships including these combinatorial numbers and the Bernstein type basis functions. Moreover, we provide not only some identities, but also applications including combinatorial sums, integrals, the combinatorial numbers, the extended Bersntein basis functions, and the beta function.The aim of this paper is to give some combinatorial sums, identities and relations related to a family of combinatorial numbers and the Bernstein type basis functions. With the help of generating functions for the combinatorial numbers from this aforementioned family, we give some functional equations. By using these functional equations, we obtain several relationships including these combinatorial numbers and the Bernstein type basis functions. Moreover, we provide not only some identities, but also applications including combinatorial sums, integrals, the combinatorial numbers, the extended Bersntein basis functions, and the beta function.

In this paper, the concept of an Sθ-δ2-quasi-Cauchy sequence is investigated. In this investigation, we proved interesting theorems related to Sθ-δ2-ward continuity, and some other kinds of continuities. A real valued function f defined on a subset A of R, the set of real numbers, is called Sθ-δ2-ward continuous on A if it preserves Sθ-δ2-quasi-Cauchy sequences of points in A, i.e. (f(αk)) is an Sθ-δ2-quasi-Cauchy sequence whenever (αk) is an Sθ-δ2-quasi-Cauchy sequence of points in A, where a sequence (αk) is called Sθ-δ2-quasi-Cauchy if (Δ2αk) is an Sθ-quasi-Cauchy sequence. It turns out that the set of Sθ-δ2-ward continuous functions is a closed subset of the set of continuous functions.In this paper, the concept of an Sθ-δ2-quasi-Cauchy sequence is investigated. In this investigation, we proved interesting theorems related to Sθ-δ2-ward continuity, and some other kinds of continuities. A real valued function f defined on a subset A of R, the set of real numbers, is called Sθ-δ2-ward continuous on A if it preserves Sθ-δ2-quasi-Cauchy sequences of points in A, i.e. (f(αk)) is an Sθ-δ2-quasi-Cauchy sequence whenever (αk) is an Sθ-δ2-quasi-Cauchy sequence of points in A, where a sequence (αk) is called Sθ-δ2-quasi-Cauchy if (Δ2αk) is an Sθ-quasi-Cauchy sequence. It turns out that the set of Sθ-δ2-ward continuous functions is a closed subset of the set of continuous functions.

In this paper, a general theorem dealing with boundedness and |A, pn|k summability factors of infinite series have been established, where A is a positive normal matrix. This new theorem also reveals some results.In this paper, a general theorem dealing with boundedness and |A, pn|k summability factors of infinite series have been established, where A is a positive normal matrix. This new theorem also reveals some results.

In the present paper, a two dimensional elliptic-parabolic equation with Dirichlet-Neumann boundary condition is studied. The first and second order of accuracy difference schemes for the numerical solution of this problem are presented. Illustrative numerical results of these difference schemes are provided by using a procedure of modified Gauss elimination method.In the present paper, a two dimensional elliptic-parabolic equation with Dirichlet-Neumann boundary condition is studied. The first and second order of accuracy difference schemes for the numerical solution of this problem are presented. Illustrative numerical results of these difference schemes are provided by using a procedure of modified Gauss elimination method.

This paper is addressed to a comparative study of the vector control of a three phase induction motor based on two mathematical models. The first one is the conventional model based on the assumptions that the saturation and the iron losses are neglected; the second model fully accounts for both the fundamental iron loss and main flux saturation with and without compensation. A rotor resistance identifier is developed, so the compensation of its variation is achieved. The induction motor should be fed through a three levels inverter. The simulation results show the performances of the vector control based on the both models.This paper is addressed to a comparative study of the vector control of a three phase induction motor based on two mathematical models. The first one is the conventional model based on the assumptions that the saturation and the iron losses are neglected; the second model fully accounts for both the fundamental iron loss and main flux saturation with and without compensation. A rotor resistance identifier is developed, so the compensation of its variation is achieved. The induction motor should be fed through a three levels inverter. The simulation results show the performances of the vector control based on the both models.

The aim of this paper is to give a new type Bernstein polynomials which are different from the (p, q)-Bernstein polynomials of Mursaleen et.al.[6] based on (p, q)-integers. Also, we derived some properties and results such as generating function, symmetry property and recurrence relations.The aim of this paper is to give a new type Bernstein polynomials which are different from the (p, q)-Bernstein polynomials of Mursaleen et.al.[6] based on (p, q)-integers. Also, we derived some properties and results such as generating function, symmetry property and recurrence relations.

Hepatitis C is a liver disease which is caused by the hepatitis C virus. The virus can cause both acute and chronic hepatitis. As per the fact sheet of WHO updated on October 2017 for hepatitis C, 71 million people have been estimated for chronic hepatitis C infection in the whole world. Approximately 399,000 people die each year from hepatitis C. No vaccine for hepatitis C is available currently and research is going on in all possible directions. In order to control the hepatitis C virus and to better understand its mechanism, researchers have been proposing nonlinear integer as well as fractional mathematical models. But in general, no analytical method is available to solve these. In this paper, a Haar wavelet based numerical method has been implemented to find an approximate solution of a nonlinear fractional model for hepatitis C proposed by Ahmed and El-Saka [1, 2]. The error bounds have also been calculated.Hepatitis C is a liver disease which is caused by the hepatitis C virus. The virus can cause both acute and chronic hepatitis. As per the fact sheet of WHO updated on October 2017 for hepatitis C, 71 million people have been estimated for chronic hepatitis C infection in the whole world. Approximately 399,000 people die each year from hepatitis C. No vaccine for hepatitis C is available currently and research is going on in all possible directions. In order to control the hepatitis C virus and to better understand its mechanism, researchers have been proposing nonlinear integer as well as fractional mathematical models. But in general, no analytical method is available to solve these. In this paper, a Haar wavelet based numerical method has been implemented to find an approximate solution of a nonlinear fractional model for hepatitis C proposed by Ahmed and El-Saka [1, 2]. The error bounds have also been calculated.

In this paper, a multiple un-replicated linear functional relationship model is derived where its maximum likelihood estimators are obtained from a single p – 1 dimensional fitted plane. Its properties of unbiasedness and consistency were investigated using Taylor approximation and Fisher information matrix respectively. Simulations were conducted to investigate the effect of different sizes of error variances and sample sizes. The developed model is applied to real estate with housing data from Petaling Jaya, Selangor state. The results obtained show that the fitting and predictive abilities of the proposed model are stronger as compared to multiple regression model when applied to the training and testing samples respectively.In this paper, a multiple un-replicated linear functional relationship model is derived where its maximum likelihood estimators are obtained from a single p – 1 dimensional fitted plane. Its properties of unbiasedness and consistency were investigated using Taylor approximation and Fisher information matrix respectively. Simulations were conducted to investigate the effect of different sizes of error variances and sample sizes. The developed model is applied to real estate with housing data from Petaling Jaya, Selangor state. The results obtained show that the fitting and predictive abilities of the proposed model are stronger as compared to multiple regression model when applied to the training and testing samples respectively.

This paper presents a practical application of a new, simplified method of Lyapunov exponents estimation. The method has been applied to optimization of a real, nonlinear inverted pendulum system. Authors presented how the algorithm of the Largest Lyapunov Exponent (LLE) estimation can be applied to evaluate control systems performance. The new LLE-based control performance index has been proposed. Equations of the inverted pendulum system of the fourth order have been found. The nonlinear friction of the regulation object has been identified by means of the nonlinear least squares method. Three different friction models have been tested: linear, cubic and Coulomb model. The Differential Evolution (DE) algorithm has been used to search for the best set of parameters of the general linear regulator. This work proves that proposed method is efficient and results in faster perturbation rejection, especially when disturbances are significant.This paper presents a practical application of a new, simplified method of Lyapunov exponents estimation. The method has been applied to optimization of a real, nonlinear inverted pendulum system. Authors presented how the algorithm of the Largest Lyapunov Exponent (LLE) estimation can be applied to evaluate control systems performance. The new LLE-based control performance index has been proposed. Equations of the inverted pendulum system of the fourth order have been found. The nonlinear friction of the regulation object has been identified by means of the nonlinear least squares method. Three different friction models have been tested: linear, cubic and Coulomb model. The Differential Evolution (DE) algorithm has been used to search for the best set of parameters of the general linear regulator. This work proves that proposed method is efficient and results in faster perturbation rejection, especially when disturbances are significant.

The embedded predictor–predictor–predictor–corrector finite difference pairs with three–stages of prediction and with eliminated phase-lag and its derivatives are described in this paper. We use the symbol (EP3CM) because the new proposed methods are of three stages of prediction. It is noted that the first stage of the predictor of the new proposed finite difference pairs is based on the linear ten–step symmetric method of Quinlan–Tremaine [1]. The new finite difference pair is called non–linear or hybrid or Runke–Kutta type because has four–stages. The methods proposed in this paper can be used for the numerical solution of: (1) initial–value problems (IVPs) with oscillatory and/or periodical solutions, (2) boundary–value problems (IVPs) with oscillatory and/or periodical solutions, (3) orbital problems and (4) the Schrodinger equation and related problems.The methods proposed in this paper belong to the embedded finite difference pairs. The numerical and theoretical results show the efficiency of the n...

In the present paper we consider European style options with an exponential payoff function. The problem is transformed to evaluation of multidimensional integrals of the exponential function over the unit cube where the values of the parameters involved in the formula depend the values of the European options. We compare the performance of quasi-Monte Carlo methods based on lattice rules for multiple integrals up to 30 dimensions. The performance of a lattice rule depends on the choice of the generator vectors. When the integrand is sufficiently regular the lattice rules outperform not only the standard Monte Carlo methods, but also other types of methods using low discrepancy sequences. We consider “rank 1” rules whose lattices have a a single generator vector. The advantages and disadvantages of the different quasi-Monte Carlo methods for multidimensional integrals related to evaluation of European options are studied in the paper.

Over the years, many researchers have been examined the problems involving scattering phenomenon of water waves with floating rigid dock used as a breakwater in the constant depth of water. However, in nature, the seabed is often found to be an undulating type. One such undulation shape is a rectangular shelf. In this paper, the problem involving wave interaction with a floating dock having finite length and thickness over a rectangular shelf is examined for its solution. Employing the matching eigenfunction expansion method, the mixed boundary value problem is converted to a system of linear algebraic equations. The system of linear algebraic equations is solved to find out the quantities involving the reflection and transmission coefficients associated with reflected waves and transmission waves respectively. The numerical values of these quantities are evaluated for various values of system parameters and plotted through graphs. The energy balance relation is also verified.Over the years, many researchers have been examined the problems involving scattering phenomenon of water waves with floating rigid dock used as a breakwater in the constant depth of water. However, in nature, the seabed is often found to be an undulating type. One such undulation shape is a rectangular shelf. In this paper, the problem involving wave interaction with a floating dock having finite length and thickness over a rectangular shelf is examined for its solution. Employing the matching eigenfunction expansion method, the mixed boundary value problem is converted to a system of linear algebraic equations. The system of linear algebraic equations is solved to find out the quantities involving the reflection and transmission coefficients associated with reflected waves and transmission waves respectively. The numerical values of these quantities are evaluated for various values of system parameters and plotted through graphs. The energy balance relation is also verified.

A graph is a set of vertices and edges where each edge connects two vertices in the graph. There are several ways to represent a graph, e.g., it can be represented as a matrix: its adjacency matrix, anti-adjacency matrix, Laplacian matrix, or distance matrix. Two matrices which will be explored in this paper are the adjacency matrix and distance matrix. The adjacency matrix represents the presence or absence of an edge connecting two vertices, while the distance matrix represents the shortest path between two vertices. A graph with diameter two is a graph such that the longest distance between any two vertices is equal to two. Examples of two-diameter graphs include bipartite graphs, wheel graphs, and fan graphs. The relation between the distance and adjacency matrices is already known. The purpose of this paper is to find the relation between the square of the distance matrix and the square of the adjacency matrix of a two diameter graph and to find the characteristic polynomial of the distance matrix of a complete bipartite graph Kn,n (which is a special type of two-diameter graph). We prove that where D2=4(J−I)A−+A2 where D is the distance matrix, J is the matrix all of whose entries are 1, I is the identity matrix, A is the adjacency matrix and Ā is the complement of A. Moreover, we also find that the characteristic polynomial of the distance matrix of a complete bipartite graph is P(λ)=(λ+2)2n−2(λ−(n−2))(λ−(3n−2)).