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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.

In this paper, we consider the problem of estimating the scale parameter of the inverse Rayleigh distribution based on general progressively Type-II censored samples and progressively Type-II censored samples. The pivotal quantity method is used to derive the estimator of the scale parameter. Besides, considering that the maximum likelihood estimator is tough to obtain for this distribution, we derive an explicit estimator of the scale parameter by approximating the likelihood equation with Taylor expansion. The interval estimation is also studied based on pivotal inference. Then we conduct Monte Carlo simulations and compare the performance of different estimators. We demonstrate that the pivotal inference is simpler and more effective. The further application of the pivotal quantity method is also discussed theoretically. Finally, two real data sets are analyzed using our methods. [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the approximation properties of Durrmeyer-Stancu type variant of Jain operators with the modified forms of the Beta basis functions. We establish some direct results, which include the asymptotic formula, the error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct a King modification of these operators which preserves the test functions e0 and e1. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniform approximation of functions by Meyer-König and Zeller operators of max-product type in some weighted spaces. We estimate the rate of approximation in terms of a suitable modulus of continuity. [ABSTRACT FROM AUTHOR]

In this paper, an optimal error estimate for system of parabolic quasi-variational inequalities related to stochastic control problems is studied. Existence and uniqueness of the solution is provided by the introduction of a constructive algorithm. An optimally L∞-asymptotic behavior in maximum norm is proved using the semiimplicit time scheme combined with the finite element spatial approximation. The approach is based on the concept of subsolution and discrete regularity. [ABSTRACT FROM AUTHOR]

In this work, we generated more topologies based on similarity relation for an information system and we found lower and upper approximations. This paper discussed two approaches for determining accuracy with Yao's method and Pawlak's method of qualitative data. From both ideas, it is seen that due to the uncertainty and vagueness of qualitative data, we get many topologies on one or two attributes. We determined the accuracies by the new method; this method showed the difference between one or two attributes. This method is clarified by application. [ABSTRACT FROM AUTHOR]

There is no general exact analysis for the class of generalized mixed models, and asymptotic procedures are widely used. Importantly, under small sample conditions equivalent asymptotic procedures can yield conflicting inference when applied to the same data set [Aubin, E. C. Q., Cordeiro, G. M. (2000). Bartlett-corrected tests for normal linear models when the covariance matrix is nonscalar. Commun. Statist.—Theory Methods 29:2405–2426]. For the classical likelihood ratio test (LRT), Bartlett’s [Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proc. R. Soc. London. Ser. A, Math. Phys. Sci. 160(901):268–282] correction may be used to yield improved small sample performance. Zucker et al. [Zucker, D. M., Lieberman, O., Manor, O. (2000). Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood. J. R. Statist. Soc., Ser. B 62:827–838] proposed and investigated methods for improved small sample inference in the mixed linear model using refined LRTs. The refinements included the use of a Bartlett correction and the Cox–Reid adjusted likelihood [Cox, D. R., Reid, N. (1987). Approximations to noncentral distributions. Can. J. Statist. 15(2):105–114], which using simulation studies (under a random-line model, and a two-period, four-treatment crossover design) were shown to yield Type I error rates very close to the nominal level. An alternative approach which has also been shown [Kowalchuk, R., Keselman, H. (2001). The analysis of repeated measurements with mixed-model Kenward Roger's adjusted F-tests. Paper presented at the Annual Meeting of the American Educational Research Association, Seattle, Washington] to have improved performance characteristics is a procedure involving t and F statistics for tests of fixed effects with modified degrees of freedom calculations detailed by Kenward and Roger [Kenward, M. G., Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53:983–997]. However, to date there has been no direct comparison of the Bartlett modified likelihood ratio and Kenward Roger procedures. This paper provides the results from a Monte Carlo simulation study examining the Type I error control and power profiles of modified likelihood ratio procedures including those described in Zucker et al. and proposed by Kenward and Roger for tests of fixed effect parameters in mixed linear models with data simulated from small sample unbalanced repeated measures designs, followed by results from a real data example. [ABSTRACT FROM AUTHOR]

In this paper, we give a review on rigorous and numerical results for amplitude equations. We focus on the Swift-Hohenberg equation and the Burgers' equation in order to determine the quality of the approximation and the impact of degenerate noise on the approximating equation. [ABSTRACT FROM AUTHOR]

Subdivision schemes are special multi-resolution analysis (MRA) methods that have become prevalent in computer-aided geometric design. This paper draws useful analogies between the mathematics of subdivision schemes and the hierarchical structures of music compositions. Based on these analogies, we propose new methods for music synthesis and analysis through MRA, which provide a different perspective on music composition, representation and analysis. We demonstrate that the structure and recursive nature of the recently proposed subdivision models [S. Hed and D. Levin, Subdivision models for varying-resolution and generalized perturbations, Int. J. Comput. Math. 88(17) (2011), pp. 3709–3749; S. Hed and D. Levin, A ‘subdivision regression’ model for data analysis, 2012, in preparation] are well suited to the synthesis and analysis of monophonic and polyphonic musical patterns, doubtless due in large part to the strongly hierarchical nature of traditional musical structures. The analysis methods demonstrated enable the compression and decompression (reconstruction) of selected musical pieces and derive useful features of the pieces, laying groundwork for music classification. [ABSTRACT FROM PUBLISHER]

Chemists often come across triangle domains - usually with the basic simplex in 2. A smooth surface is needed for approximating the chemical properties between the measured data for solving some model (differential) equations numerically. Our research group has been working on approximating ternary chemical surfaces of two special fields by smooth functions (vapour - Liquid equilibrium data and explosion-limit surfaces of ternary gas systems). A mathematical solution was given in both fields by special spline surfaces, and for visualization, our own software (TRIGON) was used. In this paper the summarized chemical background of each problem is provided, the mathematical solutions, the newest theoretical developments and their results are discussed. [ABSTRACT FROM AUTHOR]

This paper presents a framework for the study of generalizing the standard notion of rough set approximation space. We propose new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from a particular property on the universe and are called general lower and upper approximations. Properties of general approximations are investigated, and their connections are examined. [ABSTRACT FROM AUTHOR]

In this paper, we study two types of approximate solutions for a vector optimization problem in Banach space setting. Our main concern is to define two new concepts of approximate saddle points and relate them to the above solution concepts. As a result, a dual is formulated, and duality results are established. [ABSTRACT FROM AUTHOR]

This paper investigates the performance of quasi maximum likelihood (QML) and non-linear least squares (NLS) estimation applied to temporally aggregated GARCH models. As these are known to be only weak GARCH, the conditional variance of the aggregated process is in general not known. Thus, one major condition, often used in proving the consistency of QML, the correct specification of the first two moments, is absent. Indeed, our results suggest that QML is not consistent, with a substantial bias if both the initial degree of persistence and the aggregation level are high. In other cases, QML might be taken as an approximation with only a small bias. On the basis of the results for univariate GARCH models, NLS is likely to be consistent, although inefficient, for weak GARCH models. Our simulation study reveals that NLS does not reduce the bias of QML in considerably large samples. As the variation of NLS estimates is much higher than that of QML, one would obviously prefer QML in most practical situations. An empirical example illustrates some of the results. [ABSTRACT FROM AUTHOR]

This paper considers the problem of approximating an integrable function by piecewise linear functions that keep the integral and positivity of the original function. [ABSTRACT FROM AUTHOR]

Synchronous generators have a natural different time scale dynamics. That is why, for modelling and control design in such systems, the methods of singular perturbations are widely used. In this paper the possibilities of sliding mode control design for synchronous generators are analysed. With this aim the concept of singular perturbation is revised in order to use it for relay control systems. The obtained results are used for sliding mode control of synchronous generator. [ABSTRACT FROM AUTHOR]

The well-known Simpson rule is an optimal two-step fourth-order method which is unconditionally unstable. In the present paper we describe a new L-stable version of the method. A suitable combination of the arithmetic average approximation with the explicit backward Euler formula provides a third-order approximation at the midpoint which, when plugged into the Simpson rule, gives a third-order L-stable scheme. The L-stable Simpson-type rule (LSIMP3) obtained is then employed to derive a third-order time integration scheme for the diffusion equation. Numerical illustrations are provided to compare the performance of the new LSIMP3 scheme with the Crank–Nicolson scheme. While the Crank–Nicolson scheme can produce unacceptable oscillations in the computed solution, the present LSIMP3 scheme can provide both stable and accurate approximations. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism s ? Aut( R ), we use the notion of a s-compatible module M R to obtain results on the couniform dimension of the polynomial modules M [ x ], M [ x -1 ], and M [ x , x -1 ] over suitable skew extension rings. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of whether the best approximation to any x in a Hilbert space X from the set where C is a closed convex subset of X , S is a closed convex cone that does not necessarily have nonempty interior, Y is a Banach space, and g : X ? Y is a continuous S -convex function, can be characterized by the best approximation to a perturbation x - l of x from the set C for some l ? X . We provide a global approach to this problem by presenting a dual global constraint qualification, which is less restrictive than the Slater type (or interior-point) conditions, guaranteeing the strong conical hull intersection property (CHIP). We then show that the strong CHIP characterizes the perturbation property under a mild closure condition. The closure condition, for instance, holds whenever the explicit constraint set is described by finitely many linear inequality constraints. We also establish easily verifiable dual conditions that are equivalent to the best approximation from the set K . We finally demonstrate that our results recapture the corresponding known results in the particular case where Y is a finite dimensional space. [ABSTRACT FROM AUTHOR]

This paper extends Pawlak's rough set onto the basis of a fuzzy partition of the universe of discourse. Some basic properties of partition-based fuzzy approximation operators are examined. To measure uncertainty in generalized fuzzy rough sets, a new notion of entropy of a fuzzy set is introduced. The notion is demonstrated to be adequate for measuring the fuzziness of a fuzzy event. The entropy of a fuzzy partition and conditional entropy are also proposed. These kinds of entropy satisfy some basic properties similar to those of Shannon's entropy. It is proved that the measure of fuzziness of a partition-based fuzzy rough set, FR( A ), is equal to zero if and only if the set A is crisp and definable. [ABSTRACT FROM AUTHOR]

In the analysis of steel structures, several modern codes such as LRFD and Eurocode 3 provide for second-order approximate analysis. This paper presents a method of optimization for use in design that is directed toward improving the overall stability and strength of moment-resistant building frames. The method makes use of codified expressions for approximate second-order analysis. The objective function used in the optimization is the dominant eigenvalue of the linearized buckling problem. Only a first-order analysis is required, along with the calculation of the first eigenvalue of the linearized buckling problem of the structure. This new method provides a story by story procedure that is easily visualized. Several examples are provided to illustrate applications of the procedure. [ABSTRACT FROM AUTHOR]

In this paper we consider so called strong Taylor approximations for processes, which are solutions of Itô-type SDEs with jumps driven by the homogeneous Poisson process. We give the order of convergence of such approximations in L²-norm, that depends on an accepted difference scheme, and show the explicit formula of the order 1 scheme. [ABSTRACT FROM AUTHOR]

In this paper our propose is to find a common term which is included in the assumptions of theorems proving existence of zeros, implicit functions, fixed points or coincidence points. This new point of view allows us to weaken the assumptions which guarantee the solvability of nonlinear equations and to recommend a possible unified treatment of several classes of operators which appear in the theory of nonlinear equations. [ABSTRACT FROM AUTHOR]

In this paper the first two moments of the Statistic (treatment sum of squares)/(treatment +error sums of squares) over all possible random assignments of treatments to the experimental plots are obtained for two associate PBIB designs. They are compared with the corresponding moments of a central beta distribution to study the extent to which the normal theory test may serve as an approximation to the randomization test. It is found that the approximation is reasonable for some classes of PBIB designs. [ABSTRACT FROM AUTHOR]