Search results for: convergence and smoothness

Authors: Muhammad Younis

Abstract:

A ternary 4-point approximating non-stationary subdivision scheme has been introduced that generates the family of $C^2$ limiting curves. The theory of asymptotic equivalence is being used to analyze the convergence and smoothness of the scheme. The comparison of the proposed scheme has been demonstrated using different examples with the existing 4-point ternary approximating schemes, which shows that the limit curves of the proposed scheme behave more pleasantly and can generate conic sections as well.

Keywords: ternary, non-stationary, approximation subdivision scheme, convergence and smoothness

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Authors: Tarul Garg, P. N. Agrawal

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We propose to define the q-analogue of the α-Bernstein Kantorovich operators and then introduce the q-bivariate generalization of these operators to study the approximation of functions of two variables. We obtain the rate of convergence of these bivariate operators by means of the total modulus of continuity, partial modulus of continuity and the Peetre’s K-functional for continuous functions. Further, in order to study the approximation of functions of two variables in a space bigger than the space of continuous functions, i.e. Bögel space; the GBS (Generalized Boolean Sum) of the q-bivariate operators is considered and degree of approximation is discussed for the Bögel continuous and Bögel differentiable functions with the aid of the Lipschitz class and the mixed modulus of smoothness.

Keywords: Bögel continuous, Bögel differentiable, generalized Boolean sum, K-functional, mixed modulus of smoothness

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Authors: Kwounghee Choi

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Technological convergence between the various sectors is expected to have a very large impact on future industrial and economy. This study attempts to do empirical approach between specific technologies’ classification. For technological convergence classification, it is necessary to set the target technology to be analyzed. This study selected target technology from national research and development plan. At first we found a source technology for analysis. Depending on the weight of source technology, NT-based, BT-based, IT-based, ET-based, CS-based convergence types were classified. This study aims to empirically show the concept of convergence technology and convergence types. If we use the source technology to classify convergence type, it will be useful to make practical strategies of convergence technology.

Keywords: technology convergence, source technology, convergence type, R&D strategy, technology classification

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Authors: Ruchi, A. M. Acu, Purshottam N. Agrawal

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The purpose of this paper to introduce the Durrmeyer type modification of q-generalized-Bernstein operators which include the Bernstein polynomials in the particular α = 0. We investigate the rate of convergence by means of the Lipschitz class and the Peetre’s K-functional. Also, we define the bivariate case of Durrmeyer type modification of q-generalized-Bernstein operators and study the degree of approximation with the aid of the partial modulus of continuity and the Peetre’s K-functional. Finally, we introduce the GBS (Generalized Boolean Sum) of the Durrmeyer type modification of q- generalized-Bernstein operators and investigate the approximation of the Bögel continuous and Bögel differentiable functions with the aid of the Lipschitz class and the mixed modulus of smoothness.

Keywords: Bögel continuous, Bögel differentiable, generalized Boolean sum, Peetre’s K-functional, Lipschitz class, mixed modulus of smoothness

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Authors: Sandeep Kaur Chouhan, Jatinderdeep Kaur, S. S. Bhatia

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The existence of sine and cosine series as a Fourier series, their L1-convergence seems to be one of the difficult question in theory of convergence of trigonometric series in L1-metric norm. In the literature so far available, various authors have studied the L1-convergence of cosine and sine trigonometric series with special coefficients. In this paper, we present a modified cosine and sine sums and criterion for L1-convergence of these modified sums is obtained. Also, a necessary and sufficient condition for the L1-convergence of the cosine and sine series is deduced as corollaries.

Keywords: conjugate Dirichlet kernel, Dirichlet kernel, L1-convergence, modified sums

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Authors: Rishikesh Yadav, Ramakanta Meher, Vishnu Narayan Mishra

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The main aim of this article is to investigate the statistical convergence of the summation of integral type operators and to obtain the weighted statistical convergence. The rate of statistical convergence by means of modulus of continuity and function belonging to the Lipschitz class are also studied. We discuss the convergence of the defined operators by graphical representation and put a better rate of convergence than the Szasz-Mirakjan-Kantorovich operators. In the last section, we extend said operators into bivariate operators to study about the rate of convergence in sense of modulus of continuity and by means of Lipschitz class by using function of two variables.

Keywords: The Szasz-Mirakjan-Kantorovich operators, statistical convergence, modulus of continuity, Peeters K-functional, weighted modulus of continuity

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Authors: Amjad Naveed

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This study analysed the role of structural change in the process of labour productivity convergence at country and regional levels. Many forms of structural changes occurred within the European Union (EU) countries i.e. variation in sectoral employment share, changes in demand for products, variations in trade patterns and advancement in technology which may have an influence on the process of convergence. Earlier studies on convergence have neglected the role of structural changes which can have resulted in different conclusion on the nature of convergence. The contribution of this study is to examine the role of structural change in testing labour productivity convergence at various levels. For the empirical purpose, the data of 19 EU countries, 259 regions and 6 industries is used for the period of 1991-2009. The results indicate that convergence varies across regional and country levels for different industries when considered the role of structural change.

Keywords: labor produvitivty, convergence, structural change, panel unit root

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Authors: F. Berna Benli, Özgür Keskin

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Lately, many mathematicians have been studied the statistical convergence of a sequence of fuzzy numbers. We know that Lambda-statistically convergence is a kind of convergence between ordinary convergence and statistical convergence. In this paper, we will introduce the new kind of convergence such as λ-levelwise statistical convergence. Then, we will define the concept of the λ-levelwise statistical cluster and limit points of a sequence of fuzzy numbers. Also, we will discuss the relations between the sets of λ-levelwise statistical cluster points and λ-levelwise statistical limit points of sequences of fuzzy numbers. This work has been extended in this paper, where some relations have been considered such that when lambda-statistical limit inferior and lambda-statistical limit superior for lambda-statistically convergent sequences of fuzzy numbers are equal. Furthermore, lambda-statistical boundedness condition for different sequences of fuzzy numbers has been studied.

Keywords: fuzzy number, λ-levelwise statistical cluster points, λ-levelwise statistical convergence, λ-levelwise statistical limit points, λ-statistical cluster points, λ-statistical convergence, λ-statistical limit points

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Authors: Neha Bhardwaj

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In this paper, we consider positive linear operators and study the Voronovskaya type result of the operator then obtain an error estimate in terms of the higher order modulus of continuity of the function being approximated and its A-statistical convergence. Also, we compute the corresponding rate of A-statistical convergence for the linear positive operators.

Keywords: Poisson distribution, Voronovskaya, modulus of continuity, a-statistical convergence

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Authors: Bastien Batardière, Joon Kwon

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For finite-sum optimization, variance-reduced gradient methods (VR) compute at each iteration the gradient of a single function (or of a mini-batch), and yet achieve faster convergence than SGD thanks to a carefully crafted lower-variance stochastic gradient estimator that reuses past gradients. Another important line of research of the past decade in continuous optimization is the adaptive algorithms such as AdaGrad, that dynamically adjust the (possibly coordinate-wise) learning rate to past gradients and thereby adapt to the geometry of the objective function. Variants such as RMSprop and Adam demonstrate outstanding practical performance that have contributed to the success of deep learning. In this work, we present AdaLVR, which combines the AdaGrad algorithm with loopless variance-reduced gradient estimators such as SAGA or L-SVRG that benefits from a straightforward construction and a streamlined analysis. We assess that AdaLVR inherits both good convergence properties from VR methods and the adaptive nature of AdaGrad: in the case of L-smooth convex functions we establish a gradient complexity of O(n + (L + √ nL)/ε) without prior knowledge of L. Numerical experiments demonstrate the superiority of AdaLVR over state-of-the-art methods. Moreover, we empirically show that the RMSprop and Adam algorithm combined with variance-reduced gradients estimators achieve even faster convergence.

Keywords: convex optimization, variance reduction, adaptive algorithms, loopless

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Authors: Berkay Buluş, Aytekin İşman, Kübra Yüzüncüyıl

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Developments in media and communication technologies have changed the way we use media. The importance of convergence culture has been increasing day by day within the framework of these developments. With new media, it is possible to say that social networks are the most powerful platforms that are integrated to this digitalization process. Although social networks seem like the place that people can socialize, they can also be utilized as places of production. On the other hand, audience has become users within the framework of transformation from national to global broadcasting. User generated contents make conventional media and new media collide. In this study, these communication platforms will be examined not as platforms that replace one another but mediums that unify each other. In the light of this information, information that is produced by users regarding new media platforms and all new media use practices are called convergence culture. In other words, convergence culture means intersections of conventional and new media. In this study, examples of convergence culture will be analyzed in detail.

Keywords: new media, convergence culture, convergence, use of new media, user generated content

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Authors: Sonia Sonia

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This paper presents a family of iterative scheme for solving nonlinear systems of equations which have wide application in sciences and engineering. The proposed iterative family is based upon some parameters which generates many different iterative schemes. This family is completely derivative free and uses first of divided difference operator. Moreover some numerical experiments are performed and compared with existing methods. Analysis of convergence shows that the presented family has fourth-order of convergence. The dynamical behaviour of proposed family and local convergence have also been discussed. The numerical performance and convergence region comparison demonstrates that proposed family is efficient.

Keywords: convergence, divided difference operator, nonlinear system, Newton's method

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Authors: Zahid Ullah, Atlas Khan

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The presented research is dedicated to an extensive examination of the regularity properties associated with a specific class of equations, namely extremely degenerate elliptic equations. This study holds significance in unraveling the complexities inherent in these equations and understanding the smoothness of their solutions. The focus is on analyzing the regularity of results, aiming to contribute to the broader field of mathematical theory. By delving into the intricacies of extremely degenerate elliptic equations, the research seeks to advance our understanding beyond conventional analyses, addressing challenges posed by degeneracy and pushing the boundaries of classical analytical methods. The motivation for this exploration lies in the practical applicability of mathematical models, particularly in real-world scenarios where physical phenomena exhibit characteristics that challenge traditional mathematical modeling. The research aspires to fill gaps in the current understanding of regularity properties within solutions to extremely degenerate elliptic equations, ultimately contributing to both theoretical foundations and practical applications in diverse scientific fields.

Keywords: investigating smoothness, extremely degenerate elliptic equations, regularity properties, mathematical analysis, complexity solutions

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Authors: Ahmad Alhawarat, Mustafa Mamat, Mohd Rivaie, Ismail Mohd

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Conjugate gradient method has been enormously used to solve large scale unconstrained optimization problems due to the number of iteration, memory, CPU time, and convergence property, in this paper we find a new class of nonlinear conjugate gradient coefficient with global convergence properties proved by exact line search. The numerical results for our new βK give a good result when it compared with well-known formulas.

Keywords: conjugate gradient method, conjugate gradient coefficient, global convergence

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Authors: César Garza

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In supervised binary classification and regression problems, it is well-known that learnability is equivalent to a uniform convergence of the hypothesis class, and if a problem is learnable, it is learnable by empirical risk minimization. For the general learning setting of unsupervised learning tasks, there are non-trivial learning problems where uniform convergence does not hold. We present here the task of learning centers of mass with an extra feature that “activates” some of the coordinates over the unit ball in a Hilbert space. We show that the learning problem is learnable under a stable RLM rule. We introduce a family of distributions over the domain space with some mild restrictions for which the sample complexity of uniform convergence for these problems must grow logarithmically with the dimension of the Hilbert space. If we take this dimension to infinity, we obtain a learnable problem for which the uniform convergence property fails for a vast family of distributions.

Keywords: statistical learning theory, learnability, uniform convergence, stability, regularized loss minimization

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Authors: Ramandeep Behl, Prashanth Maroju, S. S. Motsa

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In this paper, we study the semilocal convergence of a fifth order iterative method using recurrence relation under the assumption that first order Fréchet derivative satisfies the Hölder condition. Also, we calculate the R-order of convergence and provide some a priori error bounds. Based on this, we give existence and uniqueness region of the solution for a nonlinear Hammerstein integral equation of the second kind.

Keywords: Holder continuity condition, Frechet derivative, fifth order convergence, recurrence relations

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Authors: Firano Zakaria, Filali Adib Fatine

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The issue f convergence in theoretical models (classical or Keynesian) has been widely discussed. The results of the work affirm that most countries are seeking to get as close as possible to a steady state in order to catch up with developed countries. In this paper, we have retested this question whether it is absolute or conditional. The results affirm that the degree of convergence of countries like Morocco is very low and income is still far from its equilibrium state. Moreover, the analysis of financial convergence, of the countries in our panel, states that the pace in this sector is more intense: countries are converging more rapidly in financial terms. The question arises as to why, with a fairly convergent financial system, growth does not respond, yet the financial system should facilitate this economic convergence. Our results confirm that the degree of information exchange between the financial system and the economic system did not change significantly between 1985 and 2017. This leads to the hypothesis that the financial system is failing to serve its role as a creator of information in developing countries despite all the reforms undertaken, thus making the existence of an economic demon in the Maxwell prevail.

Keywords: economic convergence, financial convergence, financial system, entropy

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Authors: Apirak Sombat, Teerapol Saleewong, Poom Kumam, Parin Chaipunya, Wiyada Kumam, Anantachai Padcharoen, Yeol Je Cho, Thana Sutthibutpong

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This research is aimed to study a two-step iteration process defined over a finite family of σ-asymptotically quasi-nonexpansive nonself-mappings. The strong convergence is guaranteed under the framework of Banach spaces with some additional structural properties including strict and uniform convexity, reflexivity, and smoothness assumptions. With similar projection technique for nonself-mapping in Hilbert spaces, we hereby use the generalized projection to construct a point within the corresponding domain. Moreover, we have to introduce the use of duality mapping and its inverse to overcome the unavailability of duality representation that is exploit by Hilbert space theorists. We then apply our results for σ-asymptotically quasi-nonexpansive nonself-mappings to solve for ideal efficiency of vector optimization problems composed of finitely many objective functions. We also showed that the obtained solution from our process is the closest to the origin. Moreover, we also give an illustrative numerical example to support our results.

Keywords: asymptotically quasi-nonexpansive nonself-mapping, strong convergence, fixed point, uniformly convex and uniformly smooth Banach space

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Authors: Andreas Aceranti, Guido Bighiani, Francesca Crotto, Marco Colorato, Stefania Zaghi, Marino Zanetti, Simonetta Vernocchi

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The management of eye parameters such as convergence, accommodation, and miosis is very complex; in fact, both the neurovegetative system and the complex Oculocephalgiria system come into play. We have found the effectiveness of the "highvelocity low amplitude" technique directed on C7-T1 (where the cilio-spinal nucleus of the budge is located) in improving the convergence parameter through the measurement of the point of maximum convergence. With this research, we set out to investigate whether the improvement obtained through the High Velocity Low Amplitude maneuver lasts over time, carrying out a pre-manipulation measurement, one immediately after manipulation and one month after manipulation. We took a population of 30 subjects with both refractive and non-refractive problems. Of the 30 patients tested, 27 gave a positive result after the High Velocity Low Amplitude maneuver, giving an improvement in the point of maximum convergence. After a month, we retested all 27 subjects: some further improved the result, others kept, and three subjects slightly lost the gain obtained. None of the re-tested patients returned to the point of maximum convergence starting pre-manipulation. This result opens the door to a multidisciplinary approach between ophthalmologists and osteopaths with the aim of addressing oculomotricity and convergence deficits that increasingly afflict our society due to the massive use of devices and for the conduct of life in closed and restricted environments.

Keywords: point of maximum convergence, HVLA, improvement in PPC, convergence

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Authors: Sudhakar Kadiyala, Sripad R. Naik

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Large cavern in Bhutan Himalayas is being monitored since the construction period. The behavior of the cavern is being monitored for last 16 years. Instrumentation includes measurement of convergence of high walls by geodetic monitoring, load on the support systems with load cells and instrumented bolts. Analysis of the results of instrumentation showed that during the construction period of the cavern, the convergence of the cavern varied from 181 - 233 mm in the unit bay area with maximum convergence rate of 2.80mm/day. Whereas during the operational period the total convergence observed was in the range of 21 to 45 mm during a period of 11.30 years with convergence rate of 0.005 to 0.011 mm/day. During the last five years, there were no instances of high tensile stress recorded by the instrumented bolts. Load on the rock bolts have shown stabilization trend at most of the locations. This paper discusses in detail the results of long-term monitoring using the geotechnical instruments and how the data is being used in 3D numerical model to confirm the stability of the cavern.

Keywords: convergence, displacements, geodetic monitoring, long-term stability

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Authors: T. Martini, J. M. Martínez

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An algorithmic acceleration strategy based on quasi-Newton (or secant) methods is displayed for address the practical problem of accelerating the convergence of the Newton-Lagrange method in the case of convergence to critical multipliers. Since the Newton-Lagrange iteration converges locally at a linear rate, it is natural to conjecture that quasi-Newton methods based on the so called secant equation and some minimal variation principle, could converge superlinearly, thus restoring the convergence properties of Newton's method. This strategy can also be applied to accelerate the convergence of algorithms applied to fixed-points problems. Computational experience is reported illustrating the efficiency of this strategy to solve fixed-point problems with linear convergence rate.

Keywords: algorithmic acceleration, fixed-point problems, nonlinear programming, quasi-newton method

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Authors: Guillaume Leduc

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In this paper, we describe a broad setting under which the error of the approximation can be quantified, controlled, and for which convergence occurs at a speed of n⁻¹ for European and American options. We describe how knowledge of the error allows for arbitrarily fast acceleration of the convergence.

Keywords: random walk approximation, European and American options, rate of convergence, option pricing

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Authors: Pouria Ghidi

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Creating convergence in trade is very important, but in practice, this seems out of reach due to the conflict of interests and views of countries. The most important mission of UNCITRAL is to standardize and modernize international trade law through legislative and non-legislative tools on various issues of international trade law between governments. Unfortunately, the performance of governments has shown that, except in some cases, unity is not welcomed. Therefore, although unification is envisaged as a goal, it is more practical to create convergence between countries. In a variety of ways, UNCITRAL seeks to create a kind of common ground between influential actors in the international trade law system that approaches a degree of convergence of views. Accordingly, this realization seeks to find these mechanisms and their impact on creating convergence among actors in the field of international trade. In other words, this study seeks to address the question of what tools the UN Commission on International Trade Law uses to develop the convergence of rules and regulations in this area, which groups it targets, and at what levels they work.

Keywords: UNCITRAL, harmonization, unification in interpretation, international trade law, model laws

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Authors: Zhifeng Kong

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Over-parameterized neural networks have attracted a great deal of attention in recent deep learning theory research, as they challenge the classic perspective of over-fitting when the model has excessive parameters and have gained empirical success in various settings. While a number of theoretical works have been presented to demystify properties of such models, the convergence properties of such models are still far from being thoroughly understood. In this work, we study the convergence properties of training two-hidden-layer partially over-parameterized fully connected networks with the Rectified Linear Unit activation via gradient descent. To our knowledge, this is the first theoretical work to understand convergence properties of deep over-parameterized networks without the equally-wide-hidden-layer assumption and other unrealistic assumptions. We provide a probabilistic lower bound of the widths of hidden layers and proved linear convergence rate of gradient descent. We also conducted experiments on synthetic and real-world datasets to validate our theory.

Keywords: over-parameterization, rectified linear units ReLU, convergence, gradient descent, neural networks

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Authors: Sharfi Dirar, Shihab Elhaj, Ahmed El Fatih

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Computational fluid dynamics were used to simulate and study the heated water boiler tube for both normal and rifled tube with a refinement of the mesh to check the convergence. The operation condition was taken from GARRI power station and used in a boundary condition accordingly. The result indicates the rifled tube has higher heat transfer efficiency than the normal tube.

Keywords: boiler tube, convergence check, normal tube, rifled tube

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Authors: Ka-Yan Yim, Chi-Wai Kan

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This paper conducts a comparison study using KES-FB and PhabrOmeter to measure 58 selected warp knitted fabric hand properties. Fabric samples were selected and measured by both KES-FB and PhabrOmeter. Results show differences between these two measurement methods. Smoothness and stiffness values obtained by KES-FB were found significant correlated (p value = 0.003 and 0.022) to the PhabrOmeter results while softness values between two measurement methods did not show significant correlation (p value = 0.828). Disagreements among these two measurement methods imply limitations on different mechanism principles when facing warp knitted fabrics. Subjective measurement methods and further studies are suggested in order to ascertain deeper investigation on the mechanisms of fabric hand perceptions.

Keywords: fabric hand, fabric objective measurement, KES-FB, PhabrOmeter

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Authors: Vishnu Pratap Singh Kirar

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Artificial Neural Network (ANN) can be trained using backpropagation (BP). It is the most widely used algorithm for supervised learning with multi-layered feed-forward networks. Efficient learning by the BP algorithm is required for many practical applications. The BP algorithm calculates the weight changes of artificial neural networks, and a common approach is to use a two-term algorithm consisting of a learning rate (LR) and a momentum factor (MF). The major drawbacks of the two-term BP learning algorithm are the problems of local minima and slow convergence speeds, which limit the scope for real-time applications. Recently the addition of an extra term, called a proportional factor (PF), to the two-term BP algorithm was proposed. The third increases the speed of the BP algorithm. However, the PF term also reduces the convergence of the BP algorithm, and criteria for evaluating convergence are required to facilitate the application of the three terms BP algorithm. Although these two seem to be closely related, as described later, we summarize various improvements to overcome the drawbacks. Here we compare the different methods of convergence of the new three-term BP algorithm.

Keywords: neural network, backpropagation, local minima, fast convergence rate

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Authors: Vishal Jaunky, Jonas Grafström

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The development of the European Union’s (EU) single economic market and rapid technological change has resulted in major structural changes in EU’s member states economies. The general liberalization process that the countries has undergone together has convinced the governments of the member states of need to upgrade their economic and training systems in order to be able to face the economic globalization. Several signs of economic convergence have been found but less is known about the knowledge production. This paper addresses the convergence pattern of technological innovation in 13 European Union (EU) states over the time period 1990-2011 by means of parametric and non-parametric techniques. Parametric approaches revolve around the neoclassical convergence theories. This paper reveals divergence of both the β and σ types. Further, we found evidence of stochastic divergence and non-parametric convergence approach such as distribution dynamics shows a tendency towards divergence. This result is supported with the occurrence of γ-divergence. The policies of the EU to reduce technological gap among its member states seem to be missing its target, something that can have negative long run consequences for the market.

Keywords: convergence, patents, panel data, European union

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Authors: Manideepa Saha, Jahnavi Chakrabarty

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Generalized Jacobi (GJ) and Generalized Gauss-Seidel (GGS) methods are most effective than conventional Jacobi and Gauss-Seidel methods for solving linear system of equations. It is known that GJ and GGS methods converge for strictly diagonally dominant (SDD) and for M-matrices. In this paper, we study the convergence of GJ and GGS converge for symmetric positive definite (SPD) matrices, L-matrices and H-matrices. We introduce a generalization of successive overrelaxation (SOR) method for solving linear systems and discuss its convergence for the classes of SDD matrices, SPD matrices, M-matrices, L-matrices and for H-matrices. Advantages of generalized SOR method are established through numerical experiments over GJ, GGS, and SOR methods.

Keywords: convergence, Gauss-Seidel, iterative method, Jacobi, SOR

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Authors: Salah Al-Din I. Badran, Samad Ahmadi, Ismail Shahin

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A blind source separation method is proposed; in this method we use a non-uniform filter bank and a novel normalisation. This method provides a reduced computational complexity and increased convergence speed comparing to the full-band algorithm. Recently, adaptive sub-band scheme has been recommended to solve two problems: reduction of computational complexity and increase the convergence speed of the adaptive algorithm for correlated input signals. In this work the reduction in computational complexity is achieved with the use of adaptive filters of orders less than the full-band adaptive filters, which operate at a sampling rate lower than the sampling rate of the input signal. The decomposed signals by analysis bank filter are less correlated in each sub-band than the input signal at full bandwidth, and can promote better rates of convergence.

Keywords: blind source separation, computational complexity, subband, convergence speed, mixture

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