Search results for: Strong and weak convergence

Authors: Raju Kumar

Abstract:

Information and communication technology (ICT) has become, within a very short time, one of the basic building blocks of modern society. Many countries now understanding the importance of ICT and mastering the basic skills and concepts of it as part of the core of education. Organizations, experts and practitioners in the education sector increasingly recognizing the importance of ICT in supporting educational improvement and reform. This paper addresses the convergence of ICT and education. When two technologies are converging to each other, together they will generate some great opportunities and challenges. This paper focuses on these issues. In introduction section, it explains the ICT, education, and ICT-enhanced education. In next section it describes need of ICT in education, relationship between ICT skills and education, and stages of teaching learning process. The next two sections describe opportunities and challenges in integrating ICT in education. Finally the concluding section summaries the idea and its usefulness.

Keywords: Education, Information and CommunicationTechnology, Learning, Teaching.

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Authors: LI qiu-min, TIAN yi-xiang, ZHANG gao-xun

Abstract:

GMDH algorithm can well describe the internal structure of objects. In the process of modeling, automatic screening of model structure and variables ensure the convergence rate.This paper studied a new GMDH model based on polynomial spline  stimation. The polynomial spline function was used to instead of the transfer function of GMDH to characterize the relationship between the input variables and output variables. It has proved that the algorithm has the optimal convergence rate under some conditions. The empirical results show that the algorithm can well forecast Consumer Price Index (CPI).

Keywords: spline, GMDH, nonparametric, bias, forecast.

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Authors: Thomas Shermer, Godfried T. Toussaint

Abstract:

A chord of a simple polygon P is a line segment [xy] that intersects the boundary of P only at both endpoints x and y. A chord of P is called an interior chord provided the interior of [xy] lies in the interior of P. P is weakly visible from [xy] if for every point v in P there exists a point w in [xy] such that [vw] lies in P. In this paper star-shaped, L-convex, and convex polygons are characterized in terms of weak visibility properties from internal chords and starshaped subsets of P. A new Krasnoselskii-type characterization of isothetic star-shaped polygons is also presented.

Keywords: Convex polygons, L-convex polygons, star-shaped polygons, chords, weak visibility, discrete and computational geometry

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Authors: Irina Maria Artinescu, Costin Radu Boldea, Eduard-Ionut Matei

Abstract:

The paper is a comparative study of two classical vari-ants of parallel projection methods for solving the convex feasibility problem with their equivalents that involve variable weights in the construction of the solutions. We used a graphical representation of these methods for inpainting a convex area of an image in order to investigate their effectiveness in image reconstruction applications. We also presented a numerical analysis of the convergence of these four algorithms in terms of the average number of steps and execution time, in classical CPU and, alternativaly, in parallel GPU implementation.

Keywords: convex feasibility problem, convergence analysis, ınpainting, parallel projection methods

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Authors: Wai-Sum Lee

Abstract:

This study investigates C-V and V-C co-articulation in Cantonese monosyllables of the CV, VC or CVC structure, with C = one of the three stop consonants [p, t, k] and V = one of the three corner vowels [i, a, u]. Five repetitions of each test syllable on a randomized list were elicited from Cantonese young adult speakers in their early-20s. A research tool, EMA AG500, was used to record the synchronized audio signals and articulatory data at three different locations of the tongue – tongue tip, tongue middle, and tongue back – and the positions of the upper and lower lips during the test syllables. The main findings based on the articulatory data collected from two male Cantonese speakers are as follows: (i) For the syllable-initial [p-], strong co-articulation is observed when [p-] preceding the high vowel [i] or [u], but not the low vowel [a]. As for the syllable-final [-p], it is strongly co-articulated with the preceding vowel, even when the vowel is [a]. (ii) The co-articulation between the initial [t-] and the following vowel of any type is weak. In the syllable-final position, the degree of co-articulatory resistance of [-t] is also large when following the vowel [u], but [-t] is largely co-articulated with the preceding vowel when the vowel is [i] or [a]. (iii) The strength of co-articulation differs when the initial [k-] precedes the different types of vowel. A stronger co-articulation between [k-] and [i] than between [k-] and [u], and the strength of co-articulation is much reduced between [k-] and [a]. However, in the syllable-final position, there is strong co-articulation between [-k] and the preceding vowel [a]. (iv) Among the three types of stop consonants in the syllable-initial position, the decreasing degree of co-articulatory resistance (CR) is [t-] > [k-] > [p-], and the degree of CR is reduced during all three types of stop in the syllable-final position. In general, the data on co-articulation between consonant and vowel in the Cantonese monosyllables are similar to those in other languages reported in previous studies.

Keywords: Cantonese, co-articulation, consonant, vowel.

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Authors: Yilun Shang

Abstract:

In this paper, we investigate the appearance of the giant component in random subgraphs G(p) of a given large finite graph family Gn = (Vn, En) in which each edge is present independently with probability p. We show that if the graph Gn satisfies a weak isoperimetric inequality and has bounded degree, then the probability p under which G(p) has a giant component of linear order with some constant probability is bounded away from zero and one. In addition, we prove the probability of abnormally large order of the giant component decays exponentially. When a contact graph is modeled as Gn, our result is of special interest in the study of the spread of infectious diseases or the identification of community in various social networks.

Keywords: subgraph, expander, random graph, giant component, percolation.

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Authors: Thomas Kerstein, Martin Laurowski, Philipp Klein, Michael Weyrich, Hubert Roth, Jürgen Wahrburg

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Optical 3D measurement of objects is meaningful in numerous industrial applications. In various cases shape acquisition of weak textured objects is essential. Examples are repetition parts made of plastic or ceramic such as housing parts or ceramic bottles as well as agricultural products like tubers. These parts are often conveyed in a wobbling way during the automated optical inspection. Thus, conventional 3D shape acquisition methods like laser scanning might fail. In this paper, a novel approach for acquiring 3D shape of weak textured and moving objects is presented. To facilitate such measurements an active stereo vision system with structured light is proposed. The system consists of multiple camera pairs and auxiliary laser pattern generators. It performs the shape acquisition within one shot and is beneficial for rapid inspection tasks. An experimental setup including hardware and software has been developed and implemented.

Keywords: automated optical inspection, depth from structured light, stereo vision, surface reconstruction

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Authors: S. Anna Durai, E. Anna Saro

Abstract:

Image Compression using Artificial Neural Networks is a topic where research is being carried out in various directions towards achieving a generalized and economical network. Feedforward Networks using Back propagation Algorithm adopting the method of steepest descent for error minimization is popular and widely adopted and is directly applied to image compression. Various research works are directed towards achieving quick convergence of the network without loss of quality of the restored image. In general the images used for compression are of different types like dark image, high intensity image etc. When these images are compressed using Back-propagation Network, it takes longer time to converge. The reason for this is, the given image may contain a number of distinct gray levels with narrow difference with their neighborhood pixels. If the gray levels of the pixels in an image and their neighbors are mapped in such a way that the difference in the gray levels of the neighbors with the pixel is minimum, then compression ratio as well as the convergence of the network can be improved. To achieve this, a Cumulative distribution function is estimated for the image and it is used to map the image pixels. When the mapped image pixels are used, the Back-propagation Neural Network yields high compression ratio as well as it converges quickly.

Keywords: Back-propagation Neural Network, Cumulative Distribution Function, Correlation, Convergence.

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Authors: Pongthep Bunrueng

Abstract:

This action research accentuates the outcome of a development in English pronunciation, using principles of phonetics for English major students at Loei Rajabhat University. The research is split into 5 separate modules: 1) Organs of Speech and How to Produce Sounds, 2) Monopthongs, 3) Diphthongs, 4) Consonant sounds, and 5) Suprasegmental Features. Each module followed a 4 step action research process, 1) Planning, 2) Acting, 3) Observing, and 4) Reflecting. The research targeted 2nd year students who were majoring in English Education at Loei Rajabhat University during the academic year of 2011. A mixed methodology employing both quantitative and qualitative research was used, which put theory into action, taking segmental features up to suprasegmental features. Multiple tools were employed which included the following documents: pre-test and post-test papers, evaluation and assessment papers, group work assessment forms, a presentation grading form, an observation of participants form and a participant self-reflection form.

All 5 modules for the target group showed that results from the post-tests were higher than those of the pre-tests, with 0.01 statistical significance. All target groups attained results ranging from low to moderate and from moderate to high performance. The participants who attained low to moderate results had to re-sit the second round. During the first development stage, participants attended classes with group participation, in which they addressed planning through mutual co-operation and sharing of responsibility. Analytic induction of strong points for this operation illustrated that learner cognition, comprehension, application, and group practices were all present whereas the participants with weak results could be attributed to biological differences, differences in life and learning, or individual differences in responsiveness and self-discipline.

Participants who were required to be re-treated in Spiral 2 received the same treatment again. Results of tests from the 5 modules after the 2^nd treatment were that the participants attained higher scores than those attained in the pre-test. Their assessment and development stages also showed improved results. They showed greater confidence at participating in activities, produced higher quality work, and correctly followed instructions for each activity. Analytic induction of strong and weak points for this operation remains the same as for Spiral 1, though there were improvements to problems which existed prior to undertaking the second treatment.

Keywords: Action research, English pronunciation, phonetics, segmental features, suprasegmental features.

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Authors: H. Wasag

Abstract:

The removal of hydrogen sulphide is required for reasons of health, odour problems, safety and corrosivity problems. The means of removing hydrogen sulphide mainly depend on its concentration and kind of medium to be purified. The paper deals with a method of hydrogen sulphide removal from the air by its catalytic oxidation to elemental sulphur with the use of Fe-EDTA complex. The possibility of obtaining fibrous filtering materials able to remove small concentrations of H2S from the air were described. The base of these materials is fibrous ion exchanger with Fe(III)- EDTA complex immobilized on their functional groups. The complex of trivalent iron converts hydrogen sulphide to elemental sulphur. Bivalent iron formed in the reaction is oxidized by the atmospheric oxygen, so complex of trivalent iron is continuously regenerated and the overall process can be accounted as pseudocatalytic. In the present paper properties of several fibrous catalysts based on ion exchangers with different chemical nature (weak acid,weak base and strong base) were described. It was shown that the main parameters affecting the process of catalytic oxidation are:concentration of hydrogen sulphide in the air, relative humidity of the purified air, the process time and the content of Fe-EDTA complex in the fibres. The data presented show that the filtering layers with anion exchange package are much more active in the catalytic processes of hydrogen sulphide removal than cation exchanger and inert materials. In the addition to the nature of the fibres relative air humidity is a critical factor determining efficiency of the material in the air purification from H2S. It was proved that the most promising carrier of the Fe-EDTA catalyst for hydrogen sulphide oxidation are Fiban A-6 and Fiban AK-22 fibres.

Keywords: hydrogen sulphide, catalytic oxidation, odour control, ion exchange, fibrous ion exchangers, air deodorization

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Authors: H. Abderrezek, M. N. Harmas

Abstract:

DC-DC converters are widely used as reliable power source for many industrial and military applications, computers and electronic devices. Several control methods were developed for DC-DC converters control mostly with asymptotic convergence. Synergetic control (SC) is a proven robust control approach and will be used here in a so called terminal scheme to achieve finite time convergence. Lyapounov synthesis is adopted to assure controlled system stability. Furthermore particle swarm optimization (PSO) algorithm, based on an integral time absolute of error (ITAE) criterion will be used to optimize controller parameters. Simulation of terminal synergetic control of a DC-DC converter is carried out for different operating conditions and results are compared to classic synergetic control performance, that which demonstrate the effectiveness and feasibility of the proposed control method.

Keywords: DC-DC converter, PSO, finite time, terminal, synergetic control.

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Authors: Zhenhuan Zhu, David Mulvaney, Vassilios Chouliaras

Abstract:

This paper presents a novel genetic algorithm, termed the Optimum Individual Monogenetic Algorithm (OIMGA) and describes its hardware implementation. As the monogenetic strategy retains only the optimum individual, the memory requirement is dramatically reduced and no crossover circuitry is needed, thereby ensuring the requisite silicon area is kept to a minimum. Consequently, depending on application requirements, OIMGA allows the investigation of solutions that warrant either larger GA populations or individuals of greater length. The results given in this paper demonstrate that both the performance of OIMGA and its convergence time are superior to those of existing hardware GA implementations. Local convergence is achieved in OIMGA by retaining elite individuals, while population diversity is ensured by continually searching for the best individuals in fresh regions of the search space.

Keywords: Genetic algorithms, hardware-based machinelearning.

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Authors: Kavita Burse, Manish Manoria, Vishnu P. S. Kirar

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The back propagation algorithm calculates the weight changes of artificial neural networks, and a common approach is to use a training algorithm consisting of a learning rate and a momentum factor. The major drawbacks of above learning algorithm are the problems of local minima and slow convergence speeds. The addition of an extra term, called a proportional factor reduces the convergence of the back propagation algorithm. We have applied the three term back propagation to multiplicative neural network learning. The algorithm is tested on XOR and parity problem and compared with the standard back propagation training algorithm.

Keywords: Three term back propagation, multiplicative neuralnetwork, proportional factor, local minima.

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Authors: M. Junaid , Muid Mufti, M. Umar Ilyas

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IEEE has recently incorporated CCMP protocol to provide robust security to IEEE 802.11 wireless LANs. It is found that CCMP has been designed with a weak nonce construction and transmission mechanism, which leads to the exposure of initial counter value. This weak construction of nonce renders the protocol vulnerable to attacks by intruders. This paper presents how the initial counter can be pre-computed by the intruder. This vulnerability of counter block value leads to pre-computation attack on the counter mode encryption of CCMP. The failure of the counter mode will result in the collapse of the whole security mechanism of 802.11 WLAN.

Keywords: Information Security, Cryptography, IEEE 802.11i, Computer security, Wireless LAN

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Authors: Z. Zainuddin, N. Mahat, Y. Abu Hassan

Abstract:

Since the presentation of the backpropagation algorithm, a vast variety of improvements of the technique for training a feed forward neural networks have been proposed. This article focuses on two classes of acceleration techniques, one is known as Local Adaptive Techniques that are based on weightspecific only, such as the temporal behavior of the partial derivative of the current weight. The other, known as Dynamic Adaptation Methods, which dynamically adapts the momentum factors, α, and learning rate, η, with respect to the iteration number or gradient. Some of most popular learning algorithms are described. These techniques have been implemented and tested on several problems and measured in terms of gradient and error function evaluation, and percentage of success. Numerical evidence shows that these techniques improve the convergence of the Backpropagation algorithm.

Keywords: Backpropagation, Dynamic Adaptation Methods, Local Adaptive Techniques, Neural networks.

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Authors: U. Idachaba, F. Z. Wang, A. Qi, N. Helian

Abstract:

Artificial Bee Colony (ABC) algorithm is a relatively new swarm intelligence technique for clustering. It produces higher quality clusters compared to other population-based algorithms but with poor energy efficiency, cluster quality consistency and typically slower in convergence speed. Inspired by energy saving foraging behavior of natural honey bees this paper presents a Quality and Quantity Aware Artificial Bee Colony (Q2ABC) algorithm to improve quality of cluster identification, energy efficiency and convergence speed of the original ABC. To evaluate the performance of Q2ABC algorithm, experiments were conducted on a suite of ten benchmark UCI datasets. The results demonstrate Q2ABC outperformed ABC and K-means algorithm in the quality of clusters delivered.

Keywords: Artificial bee colony algorithm, clustering.

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Authors: Chidera G. Eze, Kennedy K. Abrokwa, Chimaobi V. Okolo

Abstract:

This paper empirically examines the dynamic relationship between financial deepening and economic growth in a monetary union. We find positive but weak evidence of impacts of financial deepening on growth for Gambia, Gabon and Sierra Leone. There is no evidence of any positive significant impact for Ghana and Nigeria. We argue that, the weak evidence between financial deepening and economic growth can be a consequence of the inability of assessing credit (long-term loans), credit worthiness, lack of information and low level of bank deposits by the private sector despite the improvement in the financial sector.

Keywords: Financial deepening, economic growth, dynamics, innovation accounting.

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Authors: Wondwosen Teshome B.

Abstract:

The major aim of this paper is to investigate the opposition politics in Africa. The paper also examines the status and the role, the contributions and the weaknesses of opposition1 political parties in Africa, particularly in transitional democracies that emerged in the 1990s. In Africa, many of the opposition parties appear or become active only during an election, and disappear when the election is over. It is found out that most of the opposition parties in Africa are established around the personalities of individuals, lack internal democracy, suffer from inter-party and intra-party conflicts, have severe shortage of finance, and lack strong base and experience. Their weaknesses also include bad organization and weak connection with the popular constituencies. The paper concludes that most of the weaknesses of the African opposition parties emanate from the incumbents- hostile policies, which are mostly aimed at fragmenting and weakening the opposition groups.

Keywords: Africa, Hybrid regime, Incumbent party, Neopatrimonialsim, Opposition party, Political party, Pseudo-democracy.

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Authors: Xin Luo, Jin Huang, Chuan-Long Wang

Abstract:

The mechanical quadrature methods for solving the boundary integral equations of the anisotropic Darcy-s equations with Dirichlet conditions in smooth domains are presented. By applying the collectively compact theory, we prove the convergence and stability of approximate solutions. The asymptotic expansions for the error show that the methods converge with the order O (h3), where h is the mesh size. Based on these analysis, extrapolation methods can be introduced to achieve a higher convergence rate O (h5). An a posterior asymptotic error representation is derived in order to construct self-adaptive algorithms. Finally, the numerical experiments show the efficiency of our methods.

Keywords: Darcy's equation, anisotropic, mechanical quadrature methods, extrapolation methods, a posteriori error estimate.

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Authors: Amal Ezzidani, Abdelghani Ben Tahar, Mohamed Hanini

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A sequence of finite tandem queue is considered for this study. Each one has a single server, which operates under the egalitarian processor sharing discipline. External customers arrive at each queue according to a renewal input process and having a general service times distribution. Upon completing service, customers leave the current queue and enter to the next. Under mild assumptions, including critical data, we prove the existence and the uniqueness of the fluid solution. For asymptotic behavior, we provide necessary and sufficient conditions for the invariant state and the convergence to this invariant state. In the end, we establish the convergence of a correctly normalized state process to a fluid limit characterized by a system of algebraic and integral equations.

Keywords: Fluid Limit, fluid model, measure valued process, processor sharing, tandem queue.

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Authors: Sadeghi H., Mani M., Ardakani M. A.

Abstract:

The unsteady wake of an EPPLER 361 airfoil in pitching motion has been investigated in a subsonic wind tunnel by hot-wire anemometry. The airfoil was given the pitching motion about the one-quarter chord axis at reduced frequency of 0182. Streamwise mean velocity profiles (wake profiles) were investigated at several vertically aligned points behind the airfoil at one-quarter chord downstream distance from trailing edge. Oscillation amplitude and mean angle of attack were varied to determine the effects on wake profiles. When the maximum dynamic angle of attack was below the static stall angle of attack, weak effects on wake were found by increasing oscillation amplitude and mean angle of attack. But, for higher angles of attack strong unsteady effects were appeared on the wake.

Keywords: Unsteady wake, amplitude, mean angle, EPPLER 361 airfoil.

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Authors: Ikmal Hisyam Mohamad Paris, Lilly Suriani Affendey, Norwati Mustapha

Abstract:

In this paper we compare the accuracy of data mining methods to classifying students in order to predicting student-s class grade. These predictions are more useful for identifying weak students and assisting management to take remedial measures at early stages to produce excellent graduate that will graduate at least with second class upper. Firstly we examine single classifiers accuracy on our data set and choose the best one and then ensembles it with a weak classifier to produce simple voting method. We present results show that combining different classifiers outperformed other single classifiers for predicting student performance.

Keywords: Classification, Data Mining, Prediction, Combination of Multiple Classifiers.

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Authors: Pyone Lai Swe, Wanna Swe, Kyaw Myo Lin

Abstract:

Voltage stability has become an important issue to many power systems around the world due to the weak systems and long line on power system networks. In this paper, MATLAB load flow program is applied to obtain the weak points in the system combined with finding the voltage stability limit. The maximum permissible loading of a system, within the voltage stability limit, is usually determined. The methods for varying tap ratio (using tap changing transformer) and applying different values of shunt capacitor injection to improve the voltage stability within the limit are also provided.

Keywords: Load flow, Voltage stability, Tap changingtransformer, Shunt capacitor injection, Voltage stability limit

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Authors: Changjin Xu, Yusen Wu

Abstract:

In this paper, a asymptotically periodic predator-prey model with Modified Leslie-Gower and Holling-Type II schemes is investigated. Some sufficient conditions for the uniformly strong persistence of the system are established. Our result is an important complementarity to the earlier results.

Keywords: Predator-prey model, uniformly strong persistence, asymptotically periodic, Holling-type II.

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Authors: Jiming Yang

Abstract:

A kind of singularly perturbed boundary value problems is under consideration. In order to obtain its approximation, simple upwind difference discretization is applied. We use a moving mesh iterative algorithm based on equi-distributing of the arc-length function of the current computed piecewise linear solution. First, a maximum norm a posteriori error estimate on an arbitrary mesh is derived using a different method from the one carried out by Chen [Advances in Computational Mathematics, 24(1-4) (2006), 197-212.]. Then, basing on the properties of discrete Green-s function and the presented posteriori error estimate, we theoretically prove that the discrete solutions computed by the algorithm are first-order uniformly convergent with respect to the perturbation parameter ε.

Keywords: Convergence analysis, green's function, singularly perturbed, equi-distribution, moving mesh.

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Authors: R. Sabre

Abstract:

This work focuses on the symmetric alpha stable process with continuous time frequently used in modeling the signal with indefinitely growing variance, often observed with an unknown additive error. The objective of this paper is to estimate this error from discrete observations of the signal. For that, we propose a method based on the smoothing of the observations via Jackson polynomial kernel and taking into account the width of the interval where the spectral density is non-zero. This technique allows avoiding the “Aliasing phenomenon” encountered when the estimation is made from the discrete observations of a process with continuous time. We have studied the convergence rate of the estimator and have shown that the convergence rate improves in the case where the spectral density is zero at the origin. Thus, we set up an estimator of the additive error that can be subtracted for approaching the original signal without error.

Keywords: Spectral density, stable processes, aliasing, p-adic.

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Authors: M. Kalbac, T. Verhagen, K. Drogowska, J. Vejpravova

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One- and two-dimensional carbon nanostructures with sp2 hybridization of carbon atoms (single walled carbon nanotubes and graphene) are promising materials in future electronic and spintronics devices due to specific character of their electronic structure. In this paper we present a comparative study of graphene and single-wall carbon nanotubes by Raman spectro-microscopy in strong magnetic field. This unique method allows to study changes in electronic band structure of the two types of carbon nanostructures induced by a strong magnetic field.

Keywords: Carbon nanostructures, magnetic field, Raman spectroscopy.

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Authors: Jing-Gang Xie

Abstract:

As the continuation to the previous studies of gravitational frequency shift, gravitational time dilation, gravitational light bending, gravitational waves, dark matter, and dark energy are explained in the context of Newtonian mechanics. The photon is treated as the particle with mass of hν/C² under the gravitational field of much larger mass of M. Hence the quantum mechanics theory could be applied to gravitational field on cosmology scale. The obtained results are the same as those obtained by general relativity considering weak gravitational field approximation; however, the results are different when the gravitational field is substantially strong.

Keywords: Gravitational time dilation, gravitational light bending, gravitational waves, dark matter, dark energy, General Relativity, gravitational frequency shift.

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Authors: Aina Xu, Hongfang Ma, Haitao Zhang, Weiyong Ying, Dingye Fang

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Hydrothermally synthesized high silica borosilicates with the MFI structure was subjected to several characterization techniques. The effect of boron on the structure and acidity of HZSM-5 catalyst were studied by XRD, SEM, N2 adsorption, solid state NMR, NH3-TPD. It was confirmed that boron had entered the framework in the boron samples. The results also revealed that strong acidity was weakened and weak acidity was strengthened by the boron added zeolite framework compared with parent catalyst. The catalytic performance was carried out in a fixed bed at 460°C for methanol to propylene (MTP) reaction. The results of MTP reaction showed a great increment of the propylene selectivity and excellent stability for the B-HZSM-5. The catalyst exhibited about 81% selectivity to C2 = - C4 = olefins with 40% selectivity of propylene as major component at near 100% methanol conversion, and the stable performance in the studied period was 100h.

Keywords: Methanol to propylene, HZSM-5, boron.

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Authors: Rabah Haoui

Abstract:

The aim of this work is to analyze a viscous flow in the axisymmetric nozzle taken into account the mesh size both in the free stream and into the boundary layer. The resolution of the Navier- Stokes equations is realized by using the finite volume method to determine the supersonic flow parameters at the exit of convergingdiverging nozzle. The numerical technique uses the Flux Vector Splitting method of Van Leer. Here, adequate time stepping parameter, along with CFL coefficient and mesh size level is selected to ensure numerical convergence. The effect of the boundary layer thickness is significant at the exit of the nozzle. The best solution is obtained with using a very fine grid, especially near the wall, where we have a strong variation of velocity, temperature and shear stress. This study enabled us to confirm that the determination of boundary layer thickness can be obtained only if the size of the mesh is lower than a certain value limits given by our calculations.

Keywords: Supersonic flow, viscous flow, finite volume, nozzle

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