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We investigate the characteristics of the error surface shape (ESS) induced by spatial attitude variations (SAVs) of the tested surface in the phase measuring deflectometry (PMD) system. In a PMD system, the spatial attitude of the target mirror might deviate from its initial status due to the instability and the stress release of the assembly structure. According to the principle of the PMD, there are such three SAVs as displacement, pitch and yaw angle of the target mirror that will bring about the ESS. In this paper, the principle of the ESS induced by the three SAVs are measured and analyzed based on an established PMD system. Based on the ESS characteristics, a two-step reset method (TSRM) is proposed to reset the spatial attitude of the target mirror in high precision. Experimental results verify that the PMD system could be used to monitor the SAVs of the target mirror and accurately reset it by using the TSRM; the measurement and alignment performance of the PMD system would definitely be improved accordingly.

We found in previous work that the error surfaces of recurrent networks have spurious valleys that can cause significant difficulties in training these networks. Our earlier work focused on single-layer networks. In this paper, we extend the previous results to general layered digital dynamic networks. We describe two types of spurious valleys that appear in the error surfaces of these networks. These valleys are not affected by the desired network output (or by the problem that the network is trying to solve). They depend only on the input sequence and the architecture of the network. The insights gained from this analysis suggest procedures for improving the training of recurrent neural networks.

Storage function method has been used for flood forecasting in the major rivers in Korea, however, the researches on the relationship between the parameters and runoff characteristics was not sufficient. In addition, there has been a controversy about the optimized parameters without the consideration of the physical characteristics of the basin. Therefore, in this study, the SCE-UA method is used to optimize the parameters and the proposed method was applied with two stage optimization in the Jeongseon and Yeongwol watersheds located in the most upstream in the South Han river. The contour map was developed to investigate parameters and the error surface calculated from the runoff. The proposed parameters is to provide a range of the possible parameter set in a watershed, rather than a specific value. However, the applicability is examined using the average value of the proposed ranged parameters. In this study, the criticism about the optimization technique to find an optimal value having no physical meaning on a watershed is tried to avoid. The objective of this study is to provide a range of parameters for the flood forecasting model and the intuition about the behavior of the parameters, so the efficiency of flood forecasting is increased.

[EN] The goal of this research is to analise and to develop a system that allows the determination of the most general movement dynamics of a tall building, as well as to quantify its evolution over time by means of Gaussian algorithms revision and by applying GNSS techniques and methodologies. In this thesis, the oscillation of the control quadrilateral located on the roof of the ``Torre Espacio'' bulding and determined by a high precision survey network is assessed rigorously with the method of sequential solution with addition of variables or parameters using VRS-RTK techniques. It is the data processing, together with the gaussian adjustment methodology based on a conditional tax own physical reality and analysis of partial and final results allowing us to achieve a high level of confidence that translates into effective management real-time risk. In parallel, the instantaneous and simultaneous precision of each antenna in every moment is determined. That is the error surface and the individual, simultaneous reliability of each GNSS receiver. Prior to assessing the structure dynamics, the sensitivity threshold is computed, under which nothing can be affirmed or denied with respect to the displacement produced in the control structure. It involves testing the accuracy of the instrumentation GNSS and gaussian initial adjustment mathematical model. The project concludes with the development of a warning system that is activated at the time when the movement of the building reaches a preset threshold., [ES] El objeto de la investigación es analizar y desarrollar un sistema que permita determinar la dinámica más general del movimiento de un edificio de gran altura, así como cuantificar su evolución en el tiempo. Dicho sistema se plantea mediante la revisión de los algoritmos gaussianos y la aplicación de metodologías y técnicas GNSS. En este trabajo se determina rigurosamente, mediante el método general de Ajustes Coordinados con adición de funciones de variables o parámetros, la situación de oscilación del cuadrilátero de control conformado por una red microgeodésica local y observado con técnicas VRS-RTK ubicado en la planta de coronación del edificio Torre Espacio de Madrid. Es precisamente el tratamiento de los datos, la metodología gaussiana de ajuste en función de un condicionado propio impuesto por la realidad física y el análisis de los resultados parciales y finales lo que nos permite alcanzar un alto nivel de fiabilidad que se traduce en una gestión eficaz del riesgo en tiempo real. Paralelamente se determina la precisión instantánea y simultánea de cada antena y en cada momento, esto es la superficie de error y la fiabilidad individual y simultánea de la posición de cada receptor GNSS. Previo a la evaluación de la dinámica de la estructura, se calcula el umbral de precisión o ``sensibilidad'', por debajo del cual nada puede afirmarse o negarse con respecto al desplazamiento producido en la estructura a controlar. Supone contrastar la precisión de la instrumentación GNSS y del modelo matemático inicial de ajuste gaussiano. El proyecto concluye con el desarrollo de un sistema de alerta que se activa en el momento en que el movimiento del edificio alcanza un umbral preestablecido., [CA] L'objecte de la investigació és analitzar i desenvolupar un sistema que permeta determinar la dinàmica més general del moviment d'un edifici de gran alçada, així com quantificar la seua evoluvió en el temps. Aquest sistema es plantaja mitjançant la revisió dels algorismes Gaussians i l'aplicació de metodologies y tècniques GNSS. En aquest treball es determina rigorosament, mitjançant el mètode general d'Ajusts Coordinats amb adició de de funcions de variables o paràmetres, la situació d'oscilació del quadrilàter de control conformat per una xarxa microgeodèsica local i observant amb tècniques VRS-RTK localitzat a la planta de coronació de l'edifici TorreEspacio de Madrid. És precisament el tractament de les dades, la metodologia Gaussiana d'ajust en funció d'un condicionat propi imposat per la realitat física y l'anàlisi dels resultats parcials i finals el que ens permet arribar a un alt nivell de fiabilitat, que es tradieix en una gestió eficaç del risc en temps real. Paral·lelament es determina la precisió instantània i simultània de cada antena i en cada moment, es a dir la superfície d'error i la fiabilitat individual i simultània de la posició de cada receptor GNSS. Previament a l'evaluació de la dinàmica de l'estructura, es calcula l'umbral de precisió o ``sensibilitat'', per davall de la qual no es por afirmar o negar res respecte al desplaçament produït en l'estructura a controlar. Això suposa contrastar la precisió de la instrumentació GNSS i del model matemàtic inicial d'ajust Gaussià. El projecte conclou amb el desenvolupament d'un sistema d'alerta que s'activa en el moment en que el moviment de l'edifici arriba a un umbral preestablert.

Back-propagation neural network and genetic-neural network were developed to predict mould properties of sodium silicate-bonded, carbon dioxide gas hardened moulding sand system from the input process parameters. The performance of back-propagation neural network was found to be comparable with that of the best statistical regression model in predicting the mould properties, whereas genetic-neural network showed large deviations from the target values. Both the said neural networks had been developed as the reverse mapping tools and the performance of genetic-neural network was found to be marginally better than the other. The performances of above neural networks were seen to be dependent on the nature of error surface.

Genetic Algorithms (GA's) are based on the principles of natural selection and natural genetics that originate in biology. The Genetic Algorithm (GA) has been used for IIR adaptive system identification to deal with its multimodal error surface. The Genetic Algorithm (GA) can perform well in many different IIR adaptive filter structures, while the Gradient Algorithm often does not perform well in these situations. This paper focuses on the performances of three different structures of IIR filters based on the conventional Genetic Algorithm (GA) and the more specialized Multi-Parents Genetic Algorithm (MPGA).

This paper addresses the cooperative consensus problem for nonlinear strict-feedback systems under directed graph topologies. For the case of multi-agent systems with unmatched nonlinear, we adapt a dynamic surface control which is a recursive and Lyapunov-based control design procedure yet avoid the problem of "explosion of terms". A new error surface based on graph theory is introduced to reduce the number of first-order filters which can simplify the recursive procedure. As an application of our algorithm, we study the problem of leaderless coordination of multiple quad rotors. The effectiveness of the theoretical result is demonstrated via simulations.

The computation of a scalar correspondence error is the fundamental step in most stereo correspondence algorithms. The quality of the results obtained by the reconstruction algorithm directly depends on the characteristics of such error. We have developed a procedure to evaluate different methods proposed for the computation of the correspondence error. The evaluation is based on exploring the shape of the error surface generated and testing it for uniqueness, isolation and compatibility. The scheme presented makes it possible to recognise the known characteristics of the tested methods for the computation of a correspondence error from the results of the evaluations. Our results show that, for the tested scenes, the evaluation scheme allows us to identify the most appropriate method to compute the correspondence error.

Results are presented of a study of the inverse problem of ellipsometry. In the lack of inverse equations, iterative methods (search) are needed to find the optimal parameters (global minimum) of the adaptive optical model. Different kinds of optical models and search strategies are demonstrated and compared for evaluating real ellipsometric measurements. It is also demonstrated that the well-known gradient search alone is useless if the search space is large (few a-priori information) or the error surface is hilly. The applied algorithms were built in a software, which is freely available at polgaro@mfa.kfki.hu.

Error surfaces are calculated for the fitting of concentration–depth profiles to angle-resolved x-ray photoelectron spectroscopy (ARXPS) data. The shapes of the error surfaces indicate that model parameters related to composition (especially at the very surface of the sample) are well constrained by the data, whereas parameters related to depth have a less significant impact on the fit. It is then shown that certain parameters in the different depth profile models employed are highly correlated and that the different models convey essentially the same information in different ways. Finally, a compromise profile definition is proposed for the fitting of constrained but flexible depth profiles to ARXPS data. Copyright © 2003 John Wiley & Sons, Ltd.

Adaptive infinite-impulse-response (IIR) filtering provides a powerful approach for solving a variety of practical problems. Because the error surface of IIR filters is generally multi-modal, global optimisation techniques are required in order to avoid local minima. We apply a global optimisation method, called the adaptive simulated annealing (ASA), to digital IIR filter design. An important advantage of the ASA is the simplicity in software programming. Simulation study involving system identification application shows that the proposed approach is accurate and has a fast convergence rate, and the results obtained demonstrate that the ASA offers a viable tool to digital IIR filter design.

The influence of the difference in initial tooth flank form of hypoid gears on the progress of tooth flank wear and its influence on gear performance is discussed. The tooth flank forms of two new gears from the same production line as that of wear tested gears were employed as the initial state. The wear simulation developed by this investigation has revealed that the tooth of pinion is prone to considerable wear, and when the tooth flank wear develops itself enough, the composite error surface of worn gears tends to have a resemble 3D figure. The change of the gear performance with progress of tooth flank wear is although somewhat different as function of the difference in the initial composite error form of individual gears. The difference in the change of vibrational excitation with progress of tooth flank wear is especially large. The maximum value of static stresses generally decrease in some extent with progress of tooth flank wear so long as wear is not severe enough.

A novel adaptive regularization parameter selection (ARPS) method is proposed in this paper to enhance the performance of the regularization method. The proposed ARPS method enables a gradient descent type training to tunnel through some of the undesired sub-optimal solutions on the composite error surface by means of changing the value of the regularization parameter. Undesired sub-optimal solutions are introduced inherently from regularized objective functions. Hence, the proposed ARPS method is capable of enhancing the regularization method without getting stuck at these sub-optimal solutions.

All local minima of the error surface of the 2d2d1 XOR network are described. A local minimum is defined as a point such that all points in a neighbourhood have an error value greater than or equal to the error value in that point. It is proved that the error surface of the twodlayer XOR network with two hidden units has a number of regions with local minima. These regions of local minima occur for combinations of the weights from the inputs to the hidden nodes such that one or both hidden nodes are saturated for at least two patterns. However, boundary points of these regions of local minima are saddle points. It will be concluded that from each finite point in weight space a strictly decreasing path exists to a point with error zero. This also explains why experiments using higher numerical precision find less “local minima”.

Addresses the convergence of IIR output error adaptive algorithms. We argue that convergence to global or local optimum is simultaneously determined by the shape of the transient error surface and the convergence speed of the algorithm. This argument is confirmed by simulation examples comparing the convergence of LMS-based and LS-based algorithms.

The position and shape of contact pattern on tooth flanks of hypoid gears have strong correlation with the gear performance. An experienced gear engineer can qualitatively estimate the performance of the gearing by that pattern. It is although difficult to evaluate the performance of the gearing, e.g. stress state of hypoid gears, quantitatively. Considering that the gear performance is decided by the composite error surface of mating gear pairs, an algorithm to generate the composite error surface as a function of tooth contact pattern is proposed in this paper. When the simulation program for analyzing the performance of hypoid gears incorporates this reconstructed composite error surface, it becomes possible to predict the gear performance as function of observed tooth contact pattern.

Contact pattern and transmission error were compared for two different methods of simulating tooth generation and power transmission characteristics of hypoid gears. One method incorporates pinion and gear tooth flanks that are generated individually as a function of the parameters of the tooth cutting machine, and the other method uses the same gear tooth flank, but a different pinion tooth flank. The reference pinion tooth flank is a virtual one generated mathematically by a computer and it is conjugate to the gear tooth flank. The latter method uses the composite error surface of mating tooth flanks in the analysis. The calculated results were also compared with measured results obtained for four hypoid gear with four different kinds of tooth contact patterns. These investigations show the validity of the latter method for simulating the performance of hypoid gears. With this method, a gear engineer can apply knowledge of cylindrical involute gears to hypoid or bevel gears in predicting their performance.

This paper demonstrates the relationship between detection errors and resolutions (ρ- and θ- directions) when the Hough Transform (HT) is employed to detect straight segments in images. The inflexion of the error-resolution curve was uncovered. To comprehensively study the location of the inflexion, the effects of several factors are considered, such as the positions (ρ and θ), widths and lengths of straight segments, noise level, and the ratio of resolutions. An error surface according to ρ- and θ- resolutions is obtained in the paper to guide seeking of the best resolutions. The area containing “good” resolution settings is uncovered and modelled.

This section contains 5 chapters presenting easy to implement tricks which modify either the architecture and/or the learning algorithm so as to enhance the network’s modeling ability. Better modeling means better solutions in less time.

Stochastic (“Monte Carlo”) inversion methods, which do not involve the calculation of misfit function gradients, are now gaining interest. One such approach is via the Neighbourhood Algorithm (Sambridge, 1999a) allows to replace forward modeling for some of the examined models by the construction of an approximate error surface. This is accomplished with the use of Voronoi cells, whose dimension equals the number of parameters describing a given forward model. A brief description of the theory and its underlying assumptions is then illustrated with two examples. The first is a 2-layer gas reservoir overlain by water, while the second is based on a single depth profile from the SEG SEAM model, which intersects a reservoir partially filled with oil.

The key to high performance of video compression lies in an efficient reduction of the temporal redundancy. For this purpose, the block-based motion estimation (BBME) technique has been successfully applied in the video compression standards from H.261 to H.264. The most straightforward BBME method must be full search algorithm (FSA) that searches every candidate position within the search range. Since FS consumes extremely high computational cost, development and refinement on ME algorithms have been fueled to archive better tradeoff between the computational cost and the ME speed. In this paper, we study the low complexity ME algorithms and classify them into three categories, namely modeling the matching error surface, fast full search and reduction of searching candidate points. In the following sections, we will have a brief review across all these categories. The aim of this review is to provide the succeeding researchers with some constructive information in design of the fast ME algorithms.

This paper presents a new linear neuron, the GeTLS EXIN neuron for the on-line estimation of the electrical parameters of the induction motor. This estimation problem is a non generic one, particularly for the flatness of the surface error along the K 2 direction, and current OLS neurons are not capable of facing it. Only the TLS EXIN neuron and the GeTLS EXIN neuron can cope with this issue for on-line applications. However, unlike the TLS EXIN neuron, the GeTLS EXIN neuron with its variation of the parameter ς, varies the landscape of the error surface, which then varies at each batch of data. This causes the surface to crumple up and reduce the flatness of the critical directions (K 2 in this case) with resulting improved accuracy.

A technique to predict the optimal sub-pel motion vector (MV) position based on integer-pel MVs is investigated in this work. Although it is possible to find the optimal MV position by fitting a local error surface using integer-pel MVs, the minimum of the error surface may fall outside of the region of interest. By analyzing the behavior of the error surface, we propose a classification scheme that divides blocks into regular and irregular types. For a regular block, we adopt the traditional error surface model and solve its minimum as the predicted sub-pel MV position. For an irregular block, we developed a novel prediction scheme to predict the optimal sub-pel MV position. Experimental results are given to show the low complexity and good RD performance of the proposed sub-pel MV prediction scheme.

In this paper, a new global method of error compensation is developed for the measuring system based on multi-linear structured light. Due to the complexity of the error which is related to the position of measured point, the mathematical model of the measuring system is established and analyzed in detail firstly. Then, according to the error's characters, an error surface which comprises all of the error data for compensation for each position in a light plane is conceived. And the error surface is produced by integrating and fitting the error data obtained by repeatedly measuring an elaborate standard plane at different heights. At last, a test is given to prove that the proposed method is effective.

The correlation between the transmission error of gears and some factors which are usually considered to express characteristics of a tooth contact pattern, such as areal percentage and total length of simultaneous contact lines, is examined, but very poor correlation is found. A new method of estimating the transmission error from a tooth contact pattern is introduced. This is a method of supposing a triangular shape for the composite error on the actual contact line, which results in the ridge figure of the composite error surface as a function of the tooth contact pattern. The transmission error is derived as a function of this ridge figure. The results estimated by this method showed fairly good correlation with the actual transmission error, especially for a low transmitting load. Based on this method, an index value to estimate the amplitude of transmission error of a gear pair by the observation of tooth contact pattern is also introduced.

This paper presents a different view on the issue of local minima and introduces a new search method for Backpropagation learning algorithm of Multi-Layer Perceptrons (MLP). As in conventional point of view, Back-propagation may be trapped at local minima instead of finding the global minimum. This concept often leads to less confidence that people may have on neural networks. However, one could argue that most of local minima may be caused by the limitation of search methods. Therefore a new search method to address this situation is proposed in this paper. This new method, "retreat and turn", has been applied to several different types of data alone or combined with other techniques. The encouraging results are included in this paper.

The problem of local minima in backpropagation [Rumelhart 86] based networks is addressed. Networks that should have fast learning behavior fall into large number of epochs learning. This is usually produced by a combination of initial conditions (network size, training set relative to the global problem size, learning factors, etc...) that forces the network to fall into local minima, large valleys or huge plateaus of the error surface. Several procedures have been proposed to solve this problem, some of them affecting the training set, others taking advantage of problem preprocessing to find the initial position of the network. We will show a fully local method to avoid the problem: the detection (by the neuron) of the local minimum in which it may be involved. Once this step is covered, several methods to pick the network out of the minimum are proposed. The first method is to increase the network size by producing new neurons (Meiosis[Hanson 90b]), other compatible methods are presented too.

An analysis of basin cells for an error surface of an multilayer perceptron is presented. Utilizing the topological structure of the basin cells, an escaping strategy is proposed to solve difficult local minima problems. A numerical example is given to illustrate the proposed method and is shown to have a potential to locate better local minima efficiently.

A method is proposed for designing three layer neural networks that assures global minimization of errors. The commonly used gradient-based learning algorithm suffers form the local minima problem, however, it can be solved if the error surface becomes convex. In the paper a number of possible network structures are provided together with their gradient-based learning algorithms. For a given set of training data, an appropriate network structure, i.e. the number of hidden nodes, the types of activation function, and the connections between them, is determined. All of the proposed structures give convex error surfaces and thus solve the local minima problem. The difference between them is in the level of locality and generalization ability. A numerical example is provided that supports the present approach.

This paper shows a quantitative relation between the regularization techniques, the generalization ability, and the sensitivity of the Multilayer Perceptron (MLP) to input noise. Although many studies about these topics have been presented, in most cases only one of the problems is addressed, and only experimentally obtained evidence is provided to illustrate some kind of correlation between generalization, noise immunity and the use of regularization techniques to obtain a set of weights after training that provides the corresponding MLP with generalization ability and noise immunity. Here, a new measurement of noise immunity for a MLP is presented. This measurement, which is termed Mean Squared Sensitivity (MSS), explicitly evaluates the Mean Squared Error (MSE) degradation of a MLP when it is perturbed by input noise, and can be computed from the statistical sensitivities (previously proposed) of the output neurons. The MSS provides an accurate evaluation of the MLP performance loss when its inputs are perturbed by noise and can also be considered a measurement of the smoothness of the error surface with respect to the inputs. Thus, as the MSS can be used to evaluate the noise immunity or the generalization ability, it gives a criterion to select among different weight configurations that present a similar MSE after training.

The genetic-algorithm-based method described in this paper can be used to identify a parameter set whose value defines the gene regulation circuit. To demonstrate the effectiveness of the approach we choose Drosophila segmentation processes. In the processes, we search the parameter set of diffusion constant and transcription ratio of each gene. The characteristics of convergence were also investigated in order to find out how to improve the method. The results suggest that (1) when the gene regulatory network is hierarchically structured, genetic algorithm optimize the upstream parameters earlier than that of downstream in the hierarchy structure, (2) some gene network has smooth concave error surface with no local minima, and (3) the method caji be used to test appropriateness of the basic model assumed.

This is an introductory overview of neural networks. The subject is now divided into two tracks, artificial neural networks (ANNs) and computational neuroscience (CNS). Artificial neural networks are being applied in finance, industry, and commerce. Computational neuroscience attempts to model the brain, from the analysis of a single living nerve cell to the study of consciousness and self-consciousness of the whole brain. There is no doubt that neural networks will enter into a new regime of application and explanatory power in the next millenium.

It is not the aim of this chapter to investigate or discuss all the possible digital forms of sensor calibration techniques, but rather to demonstrate the feasibility of the progressive polynomial calibration method proposed in chapter 3.

A connectionist model made up of a combination of RBF networks is proposed; the model decomposes multivalued dependencies into local single valued functions; theory and applications are presented.

When using cubic B-splines, the quality of approximation depends on the placement of the knots. This paper describes the practical application of a new method for the selection of knot densities. Using a filtering and merging algorithm to split the input space into distinct regions, the number of equidistant knots in each subdivision of the space can be calculated in order to keep the approximation error below a predefined limit. In addition to the smoothing of the error surface, the technique also has the advantage of reducing the computational cost of calculating the spline approximation parameters.

CMA fractionally spaced equalizers (CMA-FSEs) have been shown, under certain conditions, to be globally asymptotically convergent to a setting which provides perfect equalization. Such a result relies heavily on the assumptions of a white source and no channel noise (as is the case in much of the literature's analysis of CMA). Herein, we relax the white source assumption and examine the effect of source correlation on CMA. Analytic results are meshed with examples showing CMA-FSE source correlation effects. Techniques for finding all stationary and saddle points on the CMA-FSE error surface are presented using recent developments in the algebraic-geometry community. Upprättat; 1995; 20070109 (ysko)

Presents a technique, named ARON (adaptive region of nonlinearity), that accelerates learning processes through a dynamic adaptation of the error surface. The procedure implements a generalization of the basic McCulloch-Pitts type of neuron which gives to each unit the ability to automatically adapt its operational region according to the requirements of the problem. >

It is shown theoretically that for an arbitrary T-element training set with t(t=/T) different inputs, the backpropagation error surface does not have suboptimal local minima if the network is capable of exactly implementing an arbitrary training set consisting of t different patterns. As a special case, the error surface of a backpropagation network with one hidden layer and t-1 hidden units has no local minima, if the network is trained by an arbitrary T-element set with t different inputs.

in the above paper Yu (IEEE Trans. Neural Networks, vol.3, no.6, p.1019-21 (1992)) claims to prove that local minima do not exist in the error surface of backpropagation networks being trained on data with t distinct input patterns when the network is capable of exactly representing arbitrary mappings on t input patterns. The commenter points out that the proof presented is flawed, so that the resulting claims remain unproved. In reply, Yu points out that the undesired phenomenon that was sited can be avoided by simply imposing the arbitrary mapping capacity of the network on lemma 1 in the article. >

A method is presented for inverting sound‐speed perturbations in a stratified waveguide environment over several kilometers range. An adjoint Green’s function formalism is used to backpropagate the difference between a measured and modeled received field through the modeled waveguide environment, in a manner reminiscent of acoustic time‐reversal methods. Using a set of linearized equations similar in form to the Born approximation, the backpropagated error field is converted into an estimate of an error surface gradient, or Frechet derivative [S. J. Norton, J. Acoust. Soc. Am. 106, 2653–2660]. The gradient can then be used to locate the error surface minimum via a series of iterations. The use of a normal‐mode waveguide Green’s function enables a considerable analytic simplification of the equations. The similarities and differences between the adjoint approach, matched‐field processing, and diffraction tomography are discussed. [Work sponsored by ONR.]

A calculating method of tooth contact marking of involute helical gears is shown with the following results: 1) When the shape of composite gear error surface of a pair of involute helical gears is mathematically similar and the ratio between amplitude and the transmitting load is kept constant, the pattern of tooth contact marking does not change and the state of tooth loading distribution is similar. 2) When a pattern of tooth contact marking is the same, the ratio of stress increase owing to gear error is almost the same. Some restricting conditions for these two theorems are indicated. Measured and calculated tooth contact markings and tooth fillet stresses of gears which have concave tooth form error and simultaneous longitudinal crowning of tooth trace correction verify the propriety of these two theorems.