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Summarizes the construction of paper folding. Method for approximating rational subdivisions or arbitrary angles and line segments; Angle-folding algorithm; Approximating angles, regular polygons and star polygons; Dissection of angles into equal parts.

This Forum Note concerns the question of necessary optimality conditions in optimal control problems subject to state constraints. Some critical remarks about a recently published paper are made. [ABSTRACT FROM AUTHOR]

The simple question of how much paper is left on my toilet roll is studied from a mathematical modelling perspective. As is typical with applied mathematics, models of increasing complexity are introduced and solved. Solutions produced at each step are compared with the solution from the previous step. This process exposes students to the typical stages of mathematical modelling via an example from everyday life. Two activities are suggested for students to complete, as well as several extensions to stimulate class discussion. [ABSTRACT FROM PUBLISHER]

The concept of "approximate reasoning" is central to Zadeh's contributions in logic. Standard fuzzy logic as we use today is only one potential interpretation of Zadeh's concept. I discuss various meanings for the syntagme "approximate reasoning" as intuitively presented in the paper Zadeh dedicated to the memory of Grigore C. Moisil in 1975. [ABSTRACT FROM AUTHOR]

Stochastic cellular automata for rock-paper-scissors games are related to Lotka-Volterra model. Simulations are usually performed by two methods local and global interactions. It is well known that the population dynamics with local interaction is stable, where all species coexist. In contrast, global interaction leads to extinction. So far, theories such as mean-field theory and pair approximation have been presented, but they never explained the stable dynamics in local simulation. In the present article, we apply effective medium approximation (EMA) which has been developed in Physics. The effective medium is determined in a self-consistent way. The EMA theory well predicts the stability of population dynamics. Moreover, it fairly explains the aggregation of each species observed in the stationary state. [ABSTRACT FROM AUTHOR]

In the Russian school, the total normalized gradient method belongs to the most wide‐spread of direct interpretation methods for potential field data. This method was also used and partly developed by many experts from abroad. The main advantage of the total normalized gradient method is its relative independence of parameters such as the expected differential density of interpreted structures. The method is built from a construction of a specially transformed field (total normalized gradient) on a section crossing the potential field sources. The special properties of this transformed field allow it to be used to detect the source positions. From the 1960s, the mathematical basis of the method underwent enormous development and several modifications of the method have been elaborated. The total normalized gradient operator itself represents a relatively complicated, non‐linear band‐pass filter in the spectral domain. The properties of this operator can be handled by means of several parameters that act to separate the information about field sources at different depth levels. In this contribution, we describe the development of the method from its very beginning (based mostly on qualitative interpretation of simple total normalized gradient sections) through to more recent numerical improvements to the method. These improvements include the quasi‐singular points method, which refines the filter properties of the total normalized gradient operator and defines an objective criteria (so called criterion 'α' and 'Г') for the definition of source depths in the section. We end by describing possibilities for further development of the method in the future. [ABSTRACT FROM AUTHOR]

Although flexible neural networks (FNNs) have been used more successfully than classical neural networks (CNNs) in many industrial applications, nothing is rigorously known about their properties. In fact they are not even well known to the systems and control community. In the first part of this paper, existing structures of and results on FNNs are surveyed. In the second part FNNs are examined in a theoretical framework. As a result, theoretical evidence is given for the superiority of FNNs over CNNs and further properties of the former are developed. More precisely, several fundamental properties of feedforward and recurrent FNNs are established. This includes the universal approximation capability, minimality, controllability, observability, and identifiability. In the broad sense, the results of this paper help that general use of FNNs in systems and control theory and applications be based on firm theoretical foundations. Theoretical analysis and synthesis of FNN-based systems thus become possible. The paper is concluded by a collection of topics for future work. [ABSTRACT FROM PUBLISHER]

Though it is said that the paperless age arrived with the spread of personal computers, the amount of office paper used shows no tendency to decrease. In general, used paper is recovered and recycled in order to preserve the environment. But labor and expense are required for the recovery of used paper and the recycled paper is lower in quality. If we could reuse used paper repeatedly without these disadvantages, for example, by using copying machines that could print again by removing the toner on used paper, it would be very convenient and ecological. We therefore studied the removal of toner from used paper by the laser ablation technique. As an optical source, an SHG-YAG laser was chosen. The energy density of irradiation with an SHG-YAG laser for the removal of toner from used paper was examined. At an energy density of approximately 12 mJ/mm2, which was the average value, the toner was removed sufficiently to allow reuse of the paper. The conditions of the laser irradiation, specifically the laser scanning speed and the number of irradiations, were studied. © 2012 Wiley Periodicals, Inc. Electr Eng Jpn, 179(1): 8-14, 2012; Published online in Wiley Online Library (). DOI 10.1002/eej.21214 [ABSTRACT FROM AUTHOR]

Abstract: A new two-dimensional network model is proposed as a micromechanics model to simulate paper’s failure process due to sequentially breakages of fibers and/or bonds. Paper is approximated as a network composed of fibers any two of which link to each other by their intersecting point, namely so-called bond. Fibers distribute along three particular directions, leading to network’s macro-level isotropy. In the framework of finite element method, nodes correspond to fiber-to-fiber bonds, while elements are fiber segments between every two neighboring nodes and described by Timoshenko beam theory. Element breaks when its equivalent internal tensile stress reaches the tensile strength of fiber. Strength of nodes, i.e. fiber-to-fiber bonds is assumed to be dependant on shearing interaction between fibers, considering the dominant interaction is shearing in a plane problem. Numerical examples show the model’s capacity of reflecting basic failure characteristic in paper. Influences of fiber length and the ratio of fiber strength to bond strength are analyzed in detail. [ABSTRACT FROM AUTHOR]

In the framework of a 1D cyclic competition model, the Rock-Paper-Scissors model, where three kinds of generic agents are allowed to mutate, to interact and to move in space, we study the formation of stochastic patterns, where all the agents do coexist. We modelled the problem using an individual-based setting and we used the system size van Kampen expansion to deal with the Master Equation. We have hence been able to characterise the spatio-temporal patterns using the power spectrum of the fluctuations. We proved that such patterns are robust against the intrinsic noise and they can be found for parameter values beyond the ones fixed by the deterministic approach (mean field approximation). We complement such analytical results with numerical simulations based on the Gillespie algorithm. [ABSTRACT FROM AUTHOR]

Abstract: We show that the main results in the recent paper [4] by Sadiq Basha (2012) are not real generalizations but particular cases of existing fixed point theorems in the literature. [Copyright &y& Elsevier]

Abstract: This paper proposes a multi-attribute comprehensive evaluation method of individual research output (IRO). It highlights the fact that a single index can never give more than a rough approximation to IRO, and the evaluation of IRO is a multi-attribute complex problem. Firstly, an evaluation index system is established by determining evaluation attributes and choosing the appropriate bibliometric indicators. To address the multiple authorship problem, this paper develops an improved number-of-papers-published indicator. Following this, TOPSIS method is used to conduct a comprehensive IRO evaluation. Then this paper uses a case study to test the feasibility of the methodology. Finally, this paper discusses the effectiveness of the proposed method. Compared with traditional single-indicator evaluation approaches, the proposed multi-attribute evaluation takes more aspects into consideration, therefore it is able to effectively overcome the one-sidedness of a single indicator. The proposed method also has significant advantages compared with other comprehensive IRO evaluation methods. [Copyright &y& Elsevier]

Abstract: We report on the fabrication and characterization of densified and transferred carbon nanotube forests for electronic interconnect application. A simple, low cost and quality method is developed for densifying vertically-aligned carbon nanotube (VA-CNTs) forests at room temperature. Commercially available paper is utilized in this work to serve as a solvent carrier. Highly densified CNT bundles are formed by the sorption of evaporative liquid from the paper into carbon nanotube forests. An average Young’s modulus increase from approximately 15.8 to 111.9MPa is extrapolated from the measured load–displacement curves in the compression tests of the as-densified VA-CNTs. Subsequent low temperature transfer method is used to transfer the VA-CNT bundles onto the target substrate. Four-probe measurement of the transferred VA-CNTs shows resistance of 3.70±0.04Ω of each CNT bundle. [Copyright &y& Elsevier]

Abstract: In the present paper we studied the Voronovskaya type theorem for modified Gamma operators presented by A. İzgi in the paper “Voronovskaya type asymptotic approximation by modified Gamma operators” [Applied Mathematics and Computation 217 (2011) 8061–8067]. We show that this theorem is not true. We present a correct form of this theorem and its proof. Furthermore, we obtain an error estimate for modified Gamma operators. [Copyright &y& Elsevier]

Abstract: presented a bias adjustment to the jackknife variance estimator of in the presence of non-response. In their paper, they obtained a second-order approximation of the bias of the Rao-Sitter variance estimator and then proposed a bias-adjusted estimator based on this approximation. To compare their proposed variance estimator to various other variance estimators, they performed a simulation study and showed that their variance estimator is superior to the Rao-Sitter variance estimator. In fact they showed that the Rao-Sitter variance estimator suffers from severe underestimation. These results contradict those in the literature, which indicate that the Rao-Sitter variance estimator suffers from a positive bias if the sampling fractions are not negligible; see , and . Because of this contradiction, we felt that a further investigation was warranted. In this paper, we attempt to recreate the results of and, in fact, show that their second order approximation to the bias of the Rao-Sitter variance estimator is incorrect and that their simulation results are also questionable. [Copyright &y& Elsevier]

In his famous paper [15] Ibragim Ibishievich Ibragimov has given asymptotic valucs of the best uniform approximation of functions of the form (a - χ)s lnm(a - χ), (a ≥ 1). These results have led to the developmcnt of a series of new directions in approximation theory, including the following ones, to which we devote this paper. • Constructive characterization of approximation of functions on a closed interval. • Babenko spaces. • Ditzian-Totik moduli of smoothness. • Constructive characterization of approximation of functions on the sets of complex plane. • Shape preserving approximation. In particular, we will show how we have used the results by I. I. Ibragimov in our recent paper in Constructive Approximation. [ABSTRACT FROM AUTHOR]

Abstract: The paper presents the results of experimental investigations of paper–oil insulation samples from the point of view of the assessment of the influence of temperature on the parameters of the approximation method of the time characteristics analysis of depolarization current. Three types of equations: Debye’s function with a single relaxation time, Curie–von Schweidler’s function with triple Debye’s function (taking into account short, average and long relaxation times of paper–oil insulation), and Jonscher’s function with Debye’s function with a single relaxation time were selected as approximation functions, as recommended by literature. The investigated insulation samples were diversified with respect to the degree of dampness and aging. Summing up, the paper focuses on verification of the determined parameters of approximation equations in relation to the influence of insulation temperature and presents recommendations for their application in diagnostics carried out by the PDC (Polarization and Depolarization Current) Method. [Copyright &y& Elsevier]

Abstract: A physical model of a continuous paper pulp digester is simplified and two subprocesses selected from the digester are modelled by coupled linear partial differential equations. This study focuses on the parameter identification of the simplified linear models. Finite-dimensional approximation of the model is made and a software package developed for identification of distributed parameter processes is applied. This identification system is developed for flexibility to allow identification for different choices of subprocesses and process variables. Unknown parameters of the subprocess models are estimated and the results are illustrated by process simulation and model validation. [Copyright &y& Elsevier]

We show abstract versions for Banach couples of several limiting compact interpolation theorems established by Edmunds and Opic for couples of [ABSTRACT FROM AUTHOR]

In this note, we take care of a gap in Remark 2.1 of Mishra and Laha ('On approximately star-shaped functions and approximate vector variational inequalities', Mishra and Laha in J. Optim. Theory Appl. 156:278-293, ) and modify the corresponding results. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036–3047] and then correct it. [Copyright &y& Elsevier]

Abstract: Ahn, Lee, Park and Yoo proved that the best constrained degree reduction of a polynomial f in -norm equals the best weighted Euclidean approximation of the Bernstein–Bézier coefficients of f in a paper, published in the journal, Computer Aided Geometric Design 21 (2) (2004) 181–191. In this note, we point out an error in their paper and give the correct result. [Copyright &y& Elsevier]

In this paper, a special system of non-linear Abel integral equations (SNAIEs) is studied, which arises in astrophysics. Here, the well-known collocation method is extended to obtain an approximate solution of the SNAIEs. The existence and uniqueness conditions of the solution are investigated. Finally, some examples are solved to illustrate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we explore the numerical analysis of the microscale heat equation. We present the characteristics of numerical solutions obtained through both semi- and fully-discrete linear finite element methods. We establish a priori estimates and error bounds for both semi-discrete and fully-discrete finite element approximations. Additionally, the existence and uniqueness of the semi-discrete and fully-discrete finite element approximations have been confirmed. The study explores error bounds in various spaces, comparing the semi-discrete to the exact solutions, the semidiscrete against the fully-discrete solutions, and the fully-discrete solutions with the exact ones. A practical algorithm is introduced to address the system emerging from the fully-discrete finite element approximation at every time step. Additionally, the paper presents numerical error calculations to further demonstrate and validate the results. [ABSTRACT FROM AUTHOR]

We shall give an elementary proof for Lemma 2.4 and correct some errors in Table 1 of the author's paper of the title. Moreover, we shall extend this table up to integers below 232. [ABSTRACT FROM AUTHOR]

The aim of the paper is to develop some representation formulae for strongly continuous operator semigroups on Banach spaces, in terms of limits of integrated means with respect to some given family of probability Borel measures and other parameters. The cases where these limits hold true pointwise or uniformly on compact subintervals are discussed separately. In order to face them different methods have been required: the former case has been studied by using purely functional-analytic methods, the latter one by involving methods arising from Approximation Theory. The paper also contains some estimates of the rate of convergence in terms of the rectified modulus of continuity and the second modulus of continuity. In a final section some illustrative examples and applications are provided. [ABSTRACT FROM AUTHOR]

In this paper, the authors look for patterns among the symbols, including complete symbols. These symbols arise from a paper-folding algorithm which produces approsimations to regular polygons, to any desired level of accuracy. In order to quickly generate these symbols, they use a Maple program, which is included in Section 4 of this paper. [ABSTRACT FROM AUTHOR]

While working with usual software tools users have to change their many valued logical thinking (approximate reasoning) into the two-valued computer logic. Although the Structured Query Language (SQL) is a very powerful tool, it is unable to satisfy needs for data selection based on linguistic expressions and degrees of truth. In this paper, we are interested in flexible querying that is based on fuzzy set theory. Medina et al. have developed a server named fuzzy SQL, supporting flexible queries and based on a theoretic model called GEFRED. To model the flexible queries and the concept of fuzzy attributes, an extension of the SQL language named fuzzy SQL has been defined. The FRDB supposed has already been defined by the user. In this paper, we extend the work of medina et al. to implement a new architecture of fuzzy DBMS based on the GEFRED model. This architecture is based on the concept of weak coupling with the DBMS SQL Server. [ABSTRACT FROM AUTHOR]

Oil spill models are used to forecast the transport and fate of oil after it has been released. CranSLIK is a model that predicts the movement and spread of a surface oil spill at sea via a stochastic approach. The aim of this work is to identify parameters that can further improve the forecasting algorithms and expand the functionality of CranSLIK, while maintaining the run time efficiency of the method. The results from multiple simulations performed using the operational, validated oil spill model, MEDSLIK-II, were analysed using multiple regression in order to identify improvements which could be incorporated into CranSLIK. This has led to a revised model, namely CranSLIK v2.0, which was validated against MEDSLIK-II forecasts for real oil spill cases. The new version of CranSLIK demonstrated significant forecasting improvements by capturing the oil spill accurately in real validation cases and also proved capable of simulating a broader range of oil spill scenarios. [ABSTRACT FROM AUTHOR]

What role does our knowledge about the ideal society play in guiding policymaking in the real world? One intuitive answer is to approximate. Namely, we have a duty to approximate the ideal within the relevant constraints of feasibility. However, political philosophers seem to have what might be called 'approximatophobia'. Many philosophers, including idealists such as David Estlund, warn against approximation. Their criticism is chiefly motivated by 'the problem of second best', which points out that your second-best option may not be closest to your best option. This paper aims to dispel 'approximatophobia'. The difficulty posed by the problem of second best is often overstated. More positively, I present a novel defence of approximation, arguing that approximation of an ideal can be a reasonably reliable default strategy of action guidance in the real world. Difficulties that may afflict the project of approximation can be mitigated by sophistication of the project of approximation. After showing that critics of approximation overstep their mark in issuing the strong or moderate warning against approximation, I propose an account of sophisticated approximation. It seeks a series of reforms that make existing social institutions closer to the ideal based on careful selection of frame of analysis. Three virtuous correlations are identified as indicators for successful project of approximation, linking descriptive similarity to desirability, feasibility and knowledge of the ideal. I also explain a two-staged strategy of sophisticating the way you approximate the ideal, with an expected positive feedback effect. [ABSTRACT FROM AUTHOR]

This paper presents a novel deep-learning network designed to detect intervals of jump discontinuities in single-variable piecewise smooth functions from their noisy samples. Enhancing the accuracy of jump discontinuity estimations can be used to find a more precise overall approximation of the function, as traditional approximation methods often produce significant errors near discontinuities. Detecting intervals of discontinuities is relatively straightforward when working with exact function data, as finite differences in the data can serve as indicators of smoothness. However, these smoothness indicators become unreliable when dealing with highly noisy data. In this paper, we propose a deep-learning network to pinpoint the location of a jump discontinuity even in the presence of substantial noise. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative scheme to approximate the fixed point of generalized α-nonexpansive mappings. we first prove that the proposed iteration process is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas and Thakur processes for contractive mappings. We also obtain some weak and strong convergence theorems for generalized α-nonexpansive mappings. Using the example presented in [R. Pant and R. Shukla, Approximating fixed point of generalized α-nonexpansive mappings in Banach spaces, J. Numer. Funct. Anal. Optim. 38(2017) 248-266.], we compare the convergence behavior of the new iterative process with other iterative processes. [ABSTRACT FROM AUTHOR]

In this paper, a high order accuracy method is developed for finding the approximate solution of two-point boundary value problems. The present approach is based on a special algorithm, taken from Pascal’s triangle, for obtaining a generalized form of the parametric splines of degree (2k + 1), k = 1, 2, . . ., which has a lower computational cost and gives the better approximation. Some appropriate band matrices are used to obtain a matrix form for this algorithm. The approximate solution converges to the exact solution of order O(h4k ), where k is a quantity related to the degree of parametric splines and the number of matrix bands that are applied in this paper. Some examples are given to illustrate the applicability of the method, and we compare the computed results with other existing known methods. It is observed that our approach produced better results. [ABSTRACT FROM AUTHOR]

The article focuses on three approximate constructions of an angle of 1 radians. Topics covered include a generalised Lindemann theorem to demonstrate the construction of an angle of 1 radian, the approximate construction achieved by paper-folding, starting with the A4-sized sheet of paper, and the right-angled triangle OBG which is the heart of the construction.

To increase the efficiency of photovoltaic (PV) systems, maximum power point (MPP) tracking of the solar arrays is needed. Under partially shaded conditions (PSCs), the solar arrays power–current (P–I) characteristic has multiple MPP. This paper presents various methods and approaches of tracking the MPP from a PV generator operating under PSC. Some comparisons, advantages, drawbacks and critical analysis of each method are discussed. It was found that, indirect methods use empirical data or mathematical expressions of numerical approximations to estimate the MPP from the PV generator’s voltage, current and irradiance. Direct methods offer the advantage of obtaining the actual maximum power from the PV generator’s voltage and current. Artificial intelligence methods do not need exact mathematical models. They can perform under parameter variation, load and supply voltage disturbances. Finally, novel methods require less number of iterations to converge, independent to the initial conditions. All these algorithms can be included in some of the DC/DC converters and MPP trackers for stand-alone or grid-connected systems. [ABSTRACT FROM AUTHOR]