Search results for: Coefficient inequalities

Authors: Lianglin Xiong, Xiuyong Ding, Shouming Zhong

Abstract:

In this paper, the problem of asymptotical stability of neutral systems with nonlinear perturbations is investigated. Based on a class of novel augment Lyapunov functionals which contain freeweighting matrices, some new delay-dependent asymptotical stability criteria are formulated in terms of linear matrix inequalities (LMIs) by using new inequality analysis technique. Numerical examples are given to demonstrate the derived condition are much less conservative than those given in the literature.

Keywords: Asymptotical stability, neutral system, nonlinear perturbation, delay-dependent, linear matrix inequality (LMI).

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Authors: Abimbola M. Enitan, Josiah Adeyemo

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Anaerobic modeling is a useful tool to describe and simulate the condition and behaviour of anaerobic treatment units for better effluent quality and biogas generation. The present investigation deals with the anaerobic treatment of brewery wastewater with varying organic loads. The chemical oxygen demand (COD) and total suspended solids (TSS) of the influent and effluent of the bioreactor were determined at various retention times to generate data for kinetic coefficients. The bio-kinetic coefficients in the modified Stover–Kincannon kinetic and methane generation models were determined to study the performance of anaerobic digestion process. At steady-state, the determination of the kinetic coefficient (K), the endogenous decay coefficient (Kd), the maximum growth rate of microorganisms (μmax), the growth yield coefficient (Y), ultimate methane yield (Bo), maximum utilization rate constant Umax and the saturation constant (KB) in the model were calculated to be 0.046 g/g COD, 0.083 (d¯¹), 0.117 (d-¹), 0.357 g/g, 0.516 (L CH4/gCODadded), 18.51 (g/L/day) and 13.64 (g/L/day) respectively. The outcome of this study will help in simulation of anaerobic model to predict usable methane and good effluent quality during the treatment of industrial wastewater. Thus, this will protect the environment, conserve natural resources, saves time and reduce cost incur by the industries for the discharge of untreated or partially treated wastewater. It will also contribute to a sustainable long-term clean development mechanism for the optimization of the methane produced from anaerobic degradation of waste in a close system.

Keywords: Brewery wastewater, methane generation model, environment, anaerobic modeling.

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Authors: Dana Kristalova, Jan Mazal

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This article deals with geographical conditions in terrain and their effect on the movement of vehicles, their effect on speed and safety of movement of people and vehicles. Finding of the optimal routes outside the communication is studied in the Army environment, but it occur in civilian as well, primarily in crisis situation, or by the provision of assistance when natural disasters such as floods, fires, storms etc., have happened. These movements require the optimization of routes when effects of geographical factors should be included. The most important factor is the surface of a terrain. It is based on several geographical factors as are slopes, soil conditions, micro-relief, a type of surface and meteorological conditions. Their mutual impact has been given by coefficient of deceleration. This coefficient can be used for the commander`s decision. New approaches and methods of terrain testing, mathematical computing, mathematical statistics or cartometric investigation are necessary parts of this evaluation.

Keywords: Movement in a terrain, geographical factors, surface of a field, mathematical evaluation, optimization and searching paths.

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Authors: Linghui Zhu, Junjie Gu

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Thermal-driven refrigeration systems have attracted increasing research and development interest in recent years. These systems do not cause ozone depletion and can reduce demand on electricity. The main objective of this work is to perform theoretical analyses of a thermal-driven refrigeration system using a new sorbent-sorptive pair as the working pair. The active component of sorbent is sodium thiocyanate (NaSCN). Ammonia (NH3) is chosen as sorptive. Based on the thermodynamic properties of the working solution, a mathematical model is introduced to analyze the system characteristics and performance. The results are used to compare with other thermal-driven refrigeration systems. It is shown that the advantages provided by this system over other absorption units include lower generator and evaporator temperatures, a higher coefficient of performance (COP). The COP is about 10 percent higher than the ones for the NH3-H2O system working at the same conditions.

Keywords: Absorption; Ammonia-Sodium thiocyanate, Exergy, coefficient of performance (COP)

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Authors: Sarka Velcovska

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The paper deals with quality labels used in the food products market, especially with labels of quality, labels of origin, and labels of organic farming. The aim of the paper is to identify perception of these labels by consumers in the Czech Republic. The first part refers to the definition and specification of food quality labels that are relevant in the Czech Republic. The second part includes the discussion of marketing research results. Data were collected with personal questioning method. Empirical findings on 150 respondents are related to consumer awareness and perception of national and European food quality labels used in the Czech Republic, attitudes to purchases of labelled products, and interest in information regarding the labels. Statistical methods, in the concrete Pearson´s chi-square test of independence, coefficient of contingency, and coefficient of association are used to determinate if significant differences do exist among selected demographic categories of Czech consumers.

Keywords: Food quality labels, quality labels awareness, quality labels perception, marketing research.

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Authors: Bhupendra C. S. Chauhan, P. S. Bisht, O. P. S. Negi

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According to celebrated Hurwitz theorem, there exists four division algebras consisting of R (real numbers), C (complex numbers), H (quaternions) and O (octonions). Keeping in view the utility of octonion variable we have tried to extend the three dimensional vector analysis to seven dimensional one. Starting with the scalar and vector product in seven dimensions, we have redefined the gradient, divergence and curl in seven dimension. It is shown that the identity n(n - 1)(n - 3)(n - 7) = 0 is satisfied only for 0, 1, 3 and 7 dimensional vectors. We have tried to write all the vector inequalities and formulas in terms of seven dimensions and it is shown that same formulas loose their meaning in seven dimensions due to non-associativity of octonions. The vector formulas are retained only if we put certain restrictions on octonions and split octonions.

Keywords: Octonions, Vector Space and seven dimensions

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Authors: Barenten Suciu

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In order to explain the damping mechanism, related to the hunting motion of the wheel axle of a high-speed railway vehicle, a generalized dynamic model is proposed. Based on such model, analytic expressions for the damping coefficient and damped natural frequency are derived, without imposing restrictions on the ratio between the lateral and vertical creep coefficients. Influence of the travelling speed, wheel conicity, dimensionless mass of the wheel axle, ratio of the creep coefficients, ratio of the track span to the yawing diameter, etc. on the damping coefficient and damped natural frequency, is clarified.

Keywords: High-speed railway vehicle, hunting motion, wheel axle, damping, creep, vibration model, analysis.

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Authors: N. Thongjub, B. Puangkird, V. Ngamaramvaranggul

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The numerical simulation of the slip effect via vicoelastic fluid for 4:1 contraction problem is investigated with regard to kinematic behaviors of streamlines and stress tensor by models of the Navier-Stokes and Oldroyd-B equations. Twodimensional spatial reference system of incompressible creeping flow with and without slip velocity is determined and the finite element method of a semi-implicit Taylor-Galerkin pressure-correction is applied to compute the problem of this Cartesian coordinate system including the schemes of velocity gradient recovery method and the streamline-Upwind / Petrov-Galerkin procedure. The slip effect at channel wall is added to calculate after each time step in order to intend the alteration of flow path. The result of stress values and the vortices are reduced by the optimum slip coefficient of 0.1 with near the outcome of analytical solution.

Keywords: Slip effect, Oldroyd-B fluid, slip coefficient, time stepping method.

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Authors: Renata Carneiro, Jody Russon, Allena Moncrief, Erica Wilkins

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Views on therapists- attraction have influenced the ethical and professional development of the mental health fields. Because the majority of therapist attraction literature (63.6%) has been conducted from a psychoanalytic standpoint, approaches to attraction from feminist perspectives have not been adequately developed. Considering the lack of a feminist voice regarding attraction, this article attempts to offer a feminist perspective on this issue. The purpose of this article is to offer a feminist perspective on the phenomenon of attraction in order to raise awareness about the importance of power inequalities, intersectionalities, contextual variables and the need for action in the field.

Keywords: attraction, feminism, power inequality, silence

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Authors: Safae El Alaoui, Mohamed Ouzahra

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In this work, we investigate the exponential stability of a linear system described by x˙ (t) = Ax(t) − ρBx(t). Here, A generates a semigroup S(t) on a Hilbert space, the operator B is supposed to be of Desch-Schappacher type, which makes the investigation more interesting in many applications. The case of Miyadera-Voigt perturbations is also considered. Sufficient conditions are formulated in terms of admissibility and observability inequalities and the approach is based on some energy estimates. Finally, the obtained results are applied to prove the uniform exponential stabilization of bilinear partial differential equations.

Keywords: Exponential stabilization, unbounded operator, Desch-Schappacher, Miyadera-Voigt operator.

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Authors: Jaipong Kasemsuwan

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A numerical solution of the initial boundary value problem of the suspended string vibrating equation with the particular nonlinear damping term based on the finite difference scheme is presented in this paper. The investigation of how the second and third power terms of the nonlinear term affect the vibration characteristic. We compare the vibration amplitude as a result of the third power nonlinear damping with the second power obtained from previous report provided that the same initial shape and initial velocities are assumed. The comparison results show that the vibration amplitude is inversely proportional to the coefficient of the damping term for the third power nonlinear damping case, while the vibration amplitude is proportional to the coefficient of the damping term in the second power nonlinear damping case.

Keywords: Finite-difference method, the nonlinear damped equation, the numerical simulation, the suspended string equation

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Authors: J. Sakano, Y. Yajima, W. Ono, Y. Yamazaki, S. Sasahara, Y. Tomotsune, Y. Ohi, S. Suzuki, A. Seki, I. Matsuzaki

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The study aimed to verify a hypothesis that a sense of fulfillment in student life and perceived stress in training in the facilities could affect vocational identity among social welfare university students, in order to acquire implications for enhancing the vocational consciousness. A questionnaire survey was conducted with 388 third- and fourth-year students of training course for certified social workers in three universities in A prefecture in Japan. The questionnaire was returned by 338 students, and 288 responses (85.2%) were valid and used for the analysis. As a SEM result, the hypothesized model proved to be fit to the data. Path coefficient of sense of fulfillment of student life to vocational identity was statistically positive. Path coefficient of training stress to vocational identity was statistically negative.

Keywords: Training stress, Physical health, Sense of fulfillment of student life, structural equation modeling (SEM)

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Authors: Li Long, Thomas Ortlepp

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The transport properties of carriers in polycrystalline silicon film affect the performance of polycrystalline silicon-based devices. They depend strongly on the grain structure, grain boundary trap properties and doping concentration, which in turn are determined by the film deposition and processing conditions. Based on the properties of charge carriers, phonons, grain boundaries and their interactions, the thermoelectric properties of polycrystalline silicon are analyzed with the relaxation time approximation of the Boltzmann transport equation. With this approach, thermal conductivity, electrical conductivity and Seebeck coefficient as a function of grain size, trap properties and doping concentration can be determined. Experiment on heavily doped polycrystalline silicon is carried out and measurement results are compared with the model.

Keywords: Conductivity, polycrystalline silicon, relaxation time approximation, Seebeck coefficient, thermoelectric property.

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Authors: Tomoaki Hashimoto

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Recently, optimal control problems subject to probabilistic constraints have attracted much attention in many research field. Although probabilistic constraints are generally intractable in optimization problems, several methods haven been proposed to deal with probabilistic constraints. In most methods, probabilistic constraints are transformed to deterministic constraints that are tractable in optimization problems. This paper examines a method for transforming probabilistic constraints into deterministic constraints for a class of probabilistic constrained optimal control problems.

Keywords: Optimal control, stochastic systems, discrete-time systems, probabilistic constraints.

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Authors: Matthias Dehmer1 Jürgen Kilian

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Problems on algebraical polynomials appear in many fields of mathematics and computer science. Especially the task of determining the roots of polynomials has been frequently investigated.Nonetheless, the task of locating the zeros of complex polynomials is still challenging. In this paper we deal with the location of zeros of univariate complex polynomials. We prove some novel upper bounds for the moduli of the zeros of complex polynomials. That means, we provide disks in the complex plane where all zeros of a complex polynomial are situated. Such bounds are extremely useful for obtaining a priori assertations regarding the location of zeros of polynomials. Based on the proven bounds and a test set of polynomials, we present an experimental study to examine which bound is optimal.

Keywords: complex polynomials, zeros, inequalities

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Authors: Saijal K. K., K. Prabhakaran Nair

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This study aims to determine change in optimal locations of dual trailing-edge flaps for various thrust coefficient to solidity (Ct /σ) ratios of helicopter to achieve minimum hub vibration levels, with low penalty in terms of required trailing-edge flap control power. Polynomial response functions are used to approximate hub vibration and flap power objective functions. Single objective and multiobjective optimization is carried with the objective of minimizing hub vibration and flap power. The optimization result shows that the inboard flap location at low Ct /σ ratio move farther from the baseline value and at high Ct /σ ratio move towards the root of the blade for minimizing hub vibration.

Keywords: Helicopter rotor, Trailing-edge flap, Thrust coefficient to solidity (Ct /σ) ratio, Optimization.

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Authors: H.Mohammadi Majd, M.Jalali Azizpour , Foad Saadi

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In this paper back-propagation artificial neural network (BPANN) is employed to predict the deformation of the upsetting process. To prepare a training set for BPANN, some finite element simulations were carried out. The input data for the artificial neural network are a set of parameters generated randomly (aspect ratio d/h, material properties, temperature and coefficient of friction). The output data are the coefficient of polynomial that fitted on barreling curves. Neural network was trained using barreling curves generated by finite element simulations of the upsetting and the corresponding material parameters. This technique was tested for three different specimens and can be successfully employed to predict the deformation of the upsetting process

Keywords: Back-propagation artificial neural network(BPANN), prediction, upsetting

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Authors: M.J. Park, K.H. Kim, O.M. Kwon

Abstract:

This paper proposes a delay-dependent leader-following consensus condition of multi-agent systems with both communication delay and probabilistic self-delay. The proposed methods employ a suitable piecewise Lyapunov-Krasovskii functional and the average dwell time approach. New consensus criterion for the systems are established in terms of linear matrix inequalities (LMIs) which can be easily solved by various effective optimization algorithms. Numerical example showed that the proposed method is effective.

Keywords: Multi-agent systems, probabilistic self-delay, consensus, Lyapunov method, LMI.

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Authors: H. F. A. Abdul Ghani, R. R. Porle

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Speech to text in Malay language is a system that converts Malay speech into text. The Malay language recognition system is still limited, thus, this paper aims to investigate the performance of ten Malay words obtained from the online Malay news. The methodology consists of three stages, which are preprocessing, feature extraction, and speech classification. In preprocessing stage, the speech samples are filtered using pre emphasis. After that, feature extraction method is applied to the samples using Mel Frequency Cepstrum Coefficient (MFCC). Lastly, speech classification is performed using Feedforward Neural Network (FFNN). The accuracy of the classification is further investigated based on the hidden layer size. From experimentation, the classifier with 40 hidden neurons shows the highest classification rate which is 94%.  

Keywords: Feed-Forward Neural Network, FFNN, Malay speech recognition, Mel Frequency Cepstrum Coefficient, MFCC, speech-to-text.

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Authors: A. Manivannan, R. T. Narendhirakannan

Abstract:

The objective of this research was to investigate biodegradation of water hyacinth (Eichhornia crassipes) to produce bioethanol using dilute-acid pretreatment (1% sulfuric acid) results in high hemicellulose decomposition and using yeast (Pachysolen tannophilus) as bioethanol producing strain. A maximum ethanol yield of 1.14g/L with coefficient, 0.24g g^-1; productivity, 0.015g l^-1h^-1 was comparable to predicted value 32.05g/L obtained by Central Composite Design (CCD). Maximum ethanol yield coefficient was comparable to those obtained through enzymatic saccharification and fermentation of acid hydrolysate using fully equipped fermentor. Although maximum ethanol concentration was low in lab scale, the improvement of lignocellulosic ethanol yield is necessary for large scale production.

Keywords: Acid hydrolysis, Biodegradation, Hemicellulose, Pachysolen tannophilus, Water hyacinth.

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Authors: H.Mohammadi Majd, M.Jalali Azizpour

Abstract:

In this paper back-propagation artificial neural network (BPANN )with Levenberg–Marquardt algorithm is employed to predict the deformation of the upsetting process. To prepare a training set for BPANN, some finite element simulations were carried out. The input data for the artificial neural network are a set of parameters generated randomly (aspect ratio d/h, material properties, temperature and coefficient of friction). The output data are the coefficient of polynomial that fitted on barreling curves. Neural network was trained using barreling curves generated by finite element simulations of the upsetting and the corresponding material parameters. This technique was tested for three different specimens and can be successfully employed to predict the deformation of the upsetting process

Keywords: Back-propagation artificial neural network(BPANN), prediction, upsetting

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Authors: Kanungnit Chawong, Panarat Rattanaphanee

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Extraction of lactic acid from aqueous solution using n-butanol as an extractant was studied. Effect of mixing time, pH of the aqueous solution, initial lactic acid concentration, and volume ratio between the organic and the aqueous phase were investigated. Distribution coefficient and degree of lactic acid extraction was found to increase when the pH of aqueous solution was decreased. The pH Effect was substantially pronounced at pH of the aqueous solution less than 1. Initial lactic acid concentration and organic-toaqueous volume ratio appeared to have positive effect on the distribution coefficient and the degree of extraction. Due to the nature of n-butanol that is partially miscible in water, incorporation of aqueous solution into organic phase was observed in the extraction with large organic-to-aqueous volume ratio.

Keywords: Lactic acid, liquid-liquid extraction, n-Butanol, Solvating extractant.

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Authors: Michal Mardiak, Jaroslav Polec

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In this paper we proposed comparison of four content based objective metrics with results of subjective tests from 80 video sequences. We also include two objective metrics VQM and SSIM to our comparison to serve as “reference” objective metrics because their pros and cons have already been published. Each of the video sequence was preprocessed by the region recognition algorithm and then the particular objective video quality metric were calculated i.e. mutual information, angular distance, moment of angle and normalized cross-correlation measure. The Pearson coefficient was calculated to express metrics relationship to accuracy of the model and the Spearman rank order correlation coefficient to represent the metrics relationship to monotonicity. The results show that model with the mutual information as objective metric provides best result and it is suitable for evaluating quality of video sequences.

Keywords: Objective quality metrics, mutual information, region recognition, content based metrics

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Authors: Vladyslav Povoroznyuk, Galina Dubetska, Roksolana Povoroznyuk

Abstract:

In recent years, the problem of hyperuricemia is getting a particular importance due to its increased incidence in the world population. The aim of this study was to determine uriс acid level in blood serum, incidence of hyperuricemia among women in postmenopausal period and their association with body mass index and some components of metabolic syndrome (triglyceride, cholesterol, systolic and diastolic pressure). We examined 412 women in postmenopausal period. They were divided in to the following groups: I group (BMI = 18,5-24,9), II group (BMI = 25,0-29,9), III group (BMI = 30,0-34,9), IV group (BMI > 35). We determined uric acid level among women during postmenopausal period depending on their body mass index. The higher level of uric acid was found in patients with the maximal body mass index (BMI > 35). In the I group it was 277,52 ± 8,40; in the II group – 286,81 ± 7,79; in the III group – 291,81 ± 7,56; in the IV group – 327,17 ± 12,17. Incidence of hyperuricemia among women in the I group was 10,2%, in the II group – 15,9%; in the III group – 21,2%, in the IV group – 34,2%. We found an interdependence between an uric acid level and BMI in the examined women (r = 0,21, p < 0,05). We determined that the highest level of triglyceride (F = 18,62, p < 0,05), cholesterol (F = 3,64, p < 0,05), atherogenic coefficient (F = 22,64, p < 0,05), systolic (F = 10,5, p < 0,05) and diastolic pressure (F = 4,30, p < 0,05) was among women with hyperuricemia. It was an interdependence between an uric acid level and triglyceride (r = 0,26, p < 0,05), atherogenic coefficient (r = 0,24, p < 0,05) among women in postmenopausal period.

Keywords: Hyperuricemia, uric acid, body mass index, metabolic syndrome, triglyceride, cholesterol, atherogenic coefficient, systolic and diastolic pressure, women.

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Authors: Taweechai Nuntawisuttiwong, Natasha Dejdumrong

Abstract:

Geometric modeling plays an important role in the constructions and manufacturing of curve, surface and solid modeling. Their algorithms are critically important not only in the automobile, ship and aircraft manufacturing business, but are also absolutely necessary in a wide variety of modern applications, e.g., robotics, optimization, computer vision, data analytics and visualization. The calculation and display of geometric objects can be accomplished by these six techniques: Polynomial basis, Recursive, Iterative, Coefficient matrix, Polar form approach and Pyramidal algorithms. In this research, the coefficient matrix (simply called monomial form approach) will be used to model polynomial rectangular patches, i.e., Said-Ball, Wang-Ball, DP, Dejdumrong and NB1 surfaces. Some examples of the monomial forms for these surface modeling are illustrated in many aspects, e.g., construction, derivatives, model transformation, degree elevation and degress reduction.

Keywords: Monomial form, rectangular surfaces, CAGD curves, monomial matrix applications.

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Authors: K. Benhabib, X. Pierens, V-D Nguyen, G. Mimanne

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The main hypothesis of the dynamics of solid phase microextraction (SPME) is that steady-state mass transfer is respected throughout the SPME extraction process. It considers steady-state diffusion is established in the two phases and fast exchange of the analyte at the solid phase film/water interface. An improved model is proposed in this paper to handle with the situation when the analyte (atrazine) is in contact with colloid suspensions (carboxylate latex in aqueous solution). A mathematical solution is obtained by substituting the diffusion coefficient by the mean of diffusion coefficient between analyte and carboxylate latex, and also thickness layer by the mean thickness in aqueous solution. This solution provides an equation relating the extracted amount of the analyte to the extraction a little more complicated than previous models. It also gives a better description of experimental observations. Moreover, the rate constant of analyte obtained is in satisfactory agreement with that obtained from the initial curve fitting.

Keywords: Pesticide, SPME methods, polyacrylate, steady state.

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Authors: Wei Zha, Jingzeng Zhang, Chen Zhao, Ran Cai, Xueyuan Nie

Abstract:

Cathodic plasma electrolysis (CPE) is used to create surface textures on cast iron samples for improving the tribological properties. Micro craters with confined size distribution were successfully formed by CPE process. These craters can generate extra hydrodynamic pressure that separates two sliding surfaces, increase the oil film thickness and accelerate the transition from boundary to mixed lubrication. It was found that the optimal crater size was 1.7 μm, at which the maximum lubrication efficiency was achieved. The Taguchi method was used to optimize the process parameters (voltage and roughness) for CPE surface texturing. The orthogonal array and the signal-to-noise ratio were employed to study the effect of each process parameter on the coefficient of friction. The results showed that with higher voltage and lower roughness, the lower friction coefficient can be obtained, and thus the lubrication can be more efficiently used for friction reduction.

Keywords: Cathodic plasma electrolysis, friction, lubrication, plasma surface texturing.

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Authors: Aritras Roy, Rinku Mukherjee

Abstract:

The current study couples a quasi-steady Vortex Lattice Method and a camber correcting technique, ‘Decambering’ for unsteady post-stall flow prediction. The wake is force-free and discrete such that the wake lattices move with the free-stream once shed from the wing. It is observed that the time-averaged unsteady coefficient of lift sees a relative drop at post-stall angles of attack in comparison to its steady counterpart for some angles of attack. Multiple solutions occur at post-stall and three different algorithms to choose solutions in these regimes show both unsteadiness and non-convergence of the iterations. The distribution of coefficient of lift on the wing span also shows sawtooth. Distribution of vorticity changes both along span and in the direction of the free-stream as the wake develops over time with distinct roll-up, which increases with time.

Keywords: Post-stall, unsteady, wing, aerodynamics.

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Authors: Z. Pavlík, J. Žumár, M. Pavlíková, R. Černý

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Interaction of inorganic water-soluble salts and building stones is studied in the paper. Two types of sandstone and one type of spongillite as representatives of materials used in historical masonry are subjected to experimental testing. Within the performed experiments, measurement of moisture and chloride concentration profiles is done in order to get input data for computational inverse analysis. Using the inverse analysis, moisture diffusivity and chloride diffusion coefficient of investigated materials are accessed. Additionally, the effect of salt presence on water vapor storage is investigated using dynamic vapor sorption device. The obtained data represents valuable information for restoration of historical masonry and give evidence on the performance of studied stones in contact with water soluble salts.

Keywords: Moisture and chloride transport, sandstone, spongillite, moisture diffusivity, chloride diffusion coefficient.

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Authors: Liyuan Wang, Shouming Zhong

Abstract:

In this paper, the design problem of state estimator for neural networks with the mixed time-varying delays are investigated by constructing appropriate Lyapunov-Krasovskii functionals and using some effective mathematical techniques. In order to derive several conditions to guarantee the estimation error systems to be globally exponential stable, we transform the considered systems into the neural-type time-delay systems. Then with a set of linear inequalities(LMIs), we can obtain the stable criteria. Finally, three numerical examples are given to show the effectiveness and less conservatism of the proposed criterion.

Keywords: State estimator, Neural networks, Globally exponential stability.

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