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In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the range of the L p -space under the Riemann-Liouville fractional integral. The connection between these new spaces and classical fractional-order Sobolev spaces is then elucidated. Under this framework, the optimal decaying rate of Jacobi expansion coefficients is obtained, based on which the projection errors under different norms are given. This work is expected to introduce fractional calculus into traditional fields in approximation theory and to explore the possibility in solving classical problems by this ‘new’ tool. [ABSTRACT FROM AUTHOR]

Almost all systems are inevitably subject to various uncertainties or disturbances from the external environment in practical applications. Taking these factors into consideration, in this paper a distributed algorithm with state noise and gradient disturbance is proposed for solving distributed optimization problem with closed convex set constraint based on multi-agent system under weight-balanced graph. Moreover, based on the gradient tracking and projection methods, the proposed distributed algorithm with gradient tracking can improve the convergence rate by introducing a projection error term and an auxiliary parameter. In contrast to some existing constrained distributed gradient algorithms, the proposed one can make the convergence faster and enhance the performance of convergence. The proposed algorithm is illustrated with two simulation examples to show its effectiveness and robustness. [ABSTRACT FROM AUTHOR]

Abstract Background Inaccurate projection on standard pelvic radiographs leads to the underestimation of femoral offset—a critical determinant of postoperative hip function—during total hip arthroplasty (THA) templating. We noted that the posteromedial facet of the greater trochanter and piriformis fossa form a double contour on radiographs, which may be valuable in determining the risk of underestimating femoral offset. We evaluate whether projection errors can be predicted based on the double contour width. Methods Plain anteroposterior (AP) pelvic radiographs and magnetic resonance images (MRIs) of 64 adult hips were evaluated retrospectively. Apparent femoral offset, apparent femoral head diameter and double contour widths were evaluated from the radiographs. X-ray projection errors were estimated by comparison to the true neck length measured on MRIs after calibration to the femoral heads. Multivariate analysis with backward elimination was used to detect associations between the double contour width and radiographic projection errors. Femoral offset underestimation below 10% was considered acceptable for templating. Results The narrowest width of the double line between the femoral neck and piriformis fossa is significantly associated with projection error. When double line widths exceed 5 mm, the risk of projection error greater than 10% is significantly increased compared to narrower double lines, and the acceptability rate for templating drops below 80% (p = 0.02). Conclusion The double contour width is a potential landmark for excluding pelvic AP radiographs unsuitable for THA templating due to inaccurate femoral rotation.

The extraction of facial features is the key to micro-expression recognition. This paper puts forward a micro-expression recognition algorithm through multi-feature fusion. In this algorithm, the change of local binary pattern (LBP) feature distribution is correlated with projection error. For fast and accurate detection, the research data were all extracted from professional facial expression databases, the images had basically the same positions in each expression library, and the pure face image was acquired through manual segmentation from the selected expression library. Through comparison and its application in an intelligent classroom environment, it is proved that the proposed algorithm clearly outperforms the original LBP algorithm. The proposed method can be extended to other image recognition and classification problems.

Background: Inaccurate projection on standard pelvic radiographs leads to the underestimation of femoral offset-a critical determinant of postoperative hip function-during total hip arthroplasty (THA) templating. We noted that the posteromedial facet of the greater trochanter and piriformis fossa form a double contour on radiographs, which may be valuable in determining the risk of underestimating femoral offset. We evaluate whether projection errors can be predicted based on the double contour width.Methods: Plain anteroposterior (AP) pelvic radiographs and magnetic resonance images (MRIs) of 64 adult hips were evaluated retrospectively. Apparent femoral offset, apparent femoral head diameter and double contour widths were evaluated from the radiographs. X-ray projection errors were estimated by comparison to the true neck length measured on MRIs after calibration to the femoral heads. Multivariate analysis with backward elimination was used to detect associations between the double contour width and radiographic projection errors. Femoral offset underestimation below 10% was considered acceptable for templating.Results: The narrowest width of the double line between the femoral neck and piriformis fossa is significantly associated with projection error. When double line widths exceed 5 mm, the risk of projection error greater than 10% is significantly increased compared to narrower double lines, and the acceptability rate for templating drops below 80% (p = 0.02).Conclusion: The double contour width is a potential landmark for excluding pelvic AP radiographs unsuitable for THA templating due to inaccurate femoral rotation. [ABSTRACT FROM AUTHOR]

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Recovering a 3D human-pose in the form of an abstracted skeleton from a 2D image suffers from loss of depth information. Assuming the projected human-pose is represented by a set of 2D landmarks capturing the human-pose limbs, recovering back the original 3D locations is an ill posed problem. To recover a 3D configuration, camera localization in 3D space plays a major role, an inaccurate camera localization might mislead the recovery process. In this paper, we propose a 3D camera localization model using only human-pose appearance in a 2D image (i.e., the set of 2D landmarks). We apply a supervised multi-class logistic regression to assign the camera location in 3D space. In the learning process, we assume a set of predefined labeled camera locations. The features we train consist of relative length of limbs and 2D shape context. The goal is to build a relation between these projected landmarks and the camera location in 3D space. This kind of analysis allows us to reconstruct 3D human-poses based on the 2D projection only without any predefined camera parameters. Also, makes real-time multimedia exchange more reliable specially for human-pose related tasks. We test our model on a set of real images showing a variety of camera locations. [ABSTRACT FROM AUTHOR]

Abstract Failures of countries to set and achieve renewable energy targets are prevalent, producing uncertainty as to the possibility of renewable energy contributing to a reduction in global emissions. Lack of policy and incorrect modelling analyses are among the sources of these failures and understanding these two sources is crucial for improving confidence in renewables. We assess errors in projections pertaining to the capacity and production of renewable energy in the United States and those countries of the European Union that have strong commitments to green energy supply. Our results show that solar energy has the lowest level of uncertainty as it has the most achievable capacity projections. However, other renewables entail more attractive policies and further research is needed for the advancement of reliable technology and accurate weather predictions. Our findings also provide ranges for the projection uncertainties for six renewable energy technologies, drawing attentions to ways that the dominant errors in these renewable energy projections may be rectified. Highlights • Projection errors of capacity and production of six renewable energy technologies in the US and EU countries are estimated. • Solar energy target is the most achievable. • Achieving capacity target does not guarantee to achieve production target. [ABSTRACT FROM AUTHOR]

This research deals with projection error evaluation for large data sets using only a personal computer without any particular technologies for high performance computing. A shortcoming of basic projection error calculation ways is such that they require a large amount of computer memory or computation time is not acceptable when large data sets are analyzed. This paper proposes two ways for projection error evaluation: the first one is based on calculating the projection error for not full data set, but only for representative data sample, the second one obtains the projection error by dividing a data set into the smaller data sets. The experiments have been carried out with twelve real and artificial data sets. The computational efficiency of the projection error evaluation ways is confirmed by a comprehensive set of comparisons. We demonstrate that dividing data set into the smaller data sets allows us to calculate the projection error for large data sets.

Background: The patient's head can be slightly rotated sagitally vertically or transversely within the head holding device. Because of such improper positions due to the head rotation, an error can occur in cephalometric measurements. The purpose of this study was to identify the potential projection errors of lateral cephalometric radiograph due to head rotation toward X-ray film in the vertical Z-axis. Materials and Methods: Totally, 10 human dry skulls with permanent dentition were collected from the Department of Anatomy, J.J.M.C. Medical College, Davanagere. Each dry skull was rotated from 0° to -20° at 5° intervals. A vertical axis, the Z-axis, was used as a rotational axis to have 100 lateral cephalometric radiographs exposed. Four linear (S-N, Go-Me, N-Me, S-Go) and six angular measurements (SNA, SNB, N-S-Ar, S-Ar-Go, Ar-Go- Me, AB-Mandibular plane angle) were calculated manually. Results: The findings were that: (1) Angular measurements have fewer projection errors than linear measurements; (2) the greater the number of landmarks on the midsagittal plane that are included in angular measurements, the fewer the projection errors occurring; (3) the horizontal linear measurements have more projection errors than vertical linear measurements according to head rotation. Conclusion: In summary the angular measurements of lateral cephalometric radiographs are more useful than linear measurements in minimizing the projection errors associated with head rotation on a vertical axis. [ABSTRACT FROM AUTHOR]

The classic algorithm for Sammon's projection and two new its modifications are examined in details. All the algorithms are oriented to minimize the projection error. The discovered new ways for minimization of the projection error makes a background for the further research in this field.

Basis functions play important roles in computational electromagnetics (GEM). It is interesting to investigate the errors in the projection of the equivalent current of a plane wave using various basis functions. In this work, the projection error of various basis functions is studied. The basis functions involved are the pulse basis function, the triangular basis function, higher-order versions of these basis functions, and the divergence-conforming basis function on rectangular and triangular elements. The projection errors are derived in closed form. The asymptotic expression of the closed form is given. The analytical results are verified by numerical results. The projection error of the pth order one-dimensional (ID) basis is asymptotically inversely proportional to the (p+1)th power of the density of unknowns. Based on the closed-form projection errors in the one- dimensional case, it is found that when the expansion basis is fixed, the application of different testing functions only affects the coefficient of the projection error, rather than the order. Generally, the error of the divergence-conforming basis in the projection of curl-free vectors is less than that of divergence-free vectors. [ABSTRACT FROM AUTHOR]

In this paper, we perform an a posteriori error analysis of a multiscale operator decomposition finite element method for the solution of a system of coupled elliptic problems. The goal is to compute accurate error estimates that account for the effects arising from multiscale discretization via operator decomposition. Our approach to error estimation is based on a well-known a posteriori analysis involving variational analysis, residuals, and the generalized Green's function. Our method utilizes adjoint problems to deal with several new features arising from the multiscale operator decomposition. In part I of this paper, we focus on the propagation of errors arising from the solution of one component to another and the transfer of information between different representations of solution components. We also devise an adaptive discretization strategy based on the error estimates that specifically controls the effects arising from operator decomposition. In part II of this paper, we address issues related to the iterative solution of a fully coupled nonlinear system. [ABSTRACT FROM AUTHOR]

A sensitivity analysis of plausible errors in population, employment, fuel price, and income projections is conducted using the travel demand and emissions models of the Sacramento, CA, USA, region for their transportation plan. The results of the analyses indicate that plausible error ranges for household income and fuel prices are not a significant source of uncertainty with respect to the region''s travel demand and emissions projections. However, plausible errors in population and employment projections (within approximately one standard deviation) may result in the region''s transportation plan not meeting the conformity test for nitrogens of oxides (NOx) in the year 2005 (i.e., an approximately 16% probability). This outcome is also possible in the year 2015 but less likely (within approximately two standard deviations or a 2.5% probability). Errors in socioeconomic projections are only one of many sources of error in travel demand and emissions models. These results have several policy implications. First, regions like Sacramento that meet their conformity tests by a very small margin should rethink new highway investment and consider contingency transportation plans that incorporate more aggressive emissions reduction policies. Second, regional transportation planning agencies should conduct sensitivity analyses as part of their conformity analysis to make explicit significant uncertainties in the methods and to identify the probability of their transportation plan not conforming. Third, the US Environmental Protection Agency (EPA) should clarify the interpretation of “demonstrate” conformity of transportation plans; that is, specify the level of certainty that it considers a sufficient demonstration of conformity. [Copyright &y& Elsevier]

The purpose of this study was to identify the potential projection errors of posteroanterior cephalometric radiographs due to head rotation in the vertical Z-axis. For this investigation, 20 human dry skull samples with permanent dentition were collected from the Department of Anatomy in the College of Medicine, Chosun University, Korea. They had no gross asymmetries and were well preserved. Each dry skull was rotated from 0° to ± 10° at 1° intervals. A vertical axis, the Z-axis, was used as a rotational axis for the exposure of 420 posteroanterior cephalometric radiographs. Most of the abscissa values of each landmark showed statistically significant differences in the head rotation from each rotational angle (P < .05), whereas the ordinate values were almost the same in all rotational angles regardless of the head rotation. The abscissa values of each landmark anterior to the vertical rotational axis displaced in the same direction as the head rotation, whereas those of other landmarks posterior to the vertical rotational axis displaced in the opposite direction. The mean differences of the abscissa values, per 1° of head rotation, were larger as the landmark was located further anteroposteriorly from the vertical rotational axis and smaller as the landmark was located nearer the vertical rotational axis. In view of the projection error, a posteroanterior cephalometric radiograph is a more valuable diagnostic tool when it is exposed with no head rotation about the vertical Z-axis. [ABSTRACT FROM AUTHOR]

The purpose of this study was to identify the potential projection errors of lateral cephalometric radiographs due to head rotation in the vertical Z-axis. For this investigation, 17 human dry skull samples with permanent dentition were collected from the Department of Anatomy in the College of Medicine, Chosun University. They had no gross asymmetry and were well preserved. Each dry skull was rotated from 0° to ±15° at 1° intervals. A vertical axis, the Z-axis, was used as a rotational axis to have 527 lateral cephalometric radiographs exposed. The findings were that: (1) angular measurements have fewer projection errors than linear measurements; (2) the greater the number of landmarks on the midsagittal plane that are included in angular measurements, the fewer the projection errors occurring; (3) horizontal linear measurements decrease gradually in length as the rotational angle toward the film increases, whereas a small increase and then decrease of the length occurs as the rotational angle toward the focal spot increases; (4) horizontal linear measurements have more projection errors than vertical linear measurements according to head rotation; and (5) projection errors of vertical linear measurements increase as the distance from the rotational axis increases. In summary, angular measurements of lateral cephalometric radiographs are more useful than linear measurements in minimizing the projection errors associated with head rotation on a vertical axis. [ABSTRACT FROM AUTHOR]

Objectives: The aim for the present study was to determine whether projection error of measuring Pauwels' angle in young femur neck fracture could be eliminated by CT plane manipulation., Methods: Clinical data of displaced femur neck fractures in young adults aged 20 to 64years old (13 females and 17 males) were retrospectively analyzed. Their average age was 47.9years (range: 22-64years; SD: 11.3). Using modified measurement method for Pauwels' angle using central line of the shaft as a guideline, the angle of a conventional coronal CT image was measured. CT images were imported into Mimics® software. The scanning plane was then reformatted parallel to the neck axis to eliminate projection error of injured limb. Measured angles were classified into three types (I<30°; II, 30-50°; and III>50°) and differences were analyzed., Results: Average Pauwels' angle was 52.9° (range: 28.6-68.3°; SD: 9.9; type II, 17 cases; type III, 13 cases) for conventional CT images and 68.7° (range: 29.8-91.2°; SD: 13.4; type II, 1 cases; type III, 29 cases) for reformatted CT images. Difference between these two measurements on average was 15.7° (range: 1.2-34.9°; SD: 7.3)., Conclusions: Reformatting CT scanning plane by manipulating the proximal fragment to be parallel with the neck axis of the distal neck-shaft fragment is a convenient and reliable technique for eliminating the projection error of measuring Pauwels' angle in the femur neck fractures., Level of Evidence: IV, cohort study., (Copyright © 2019 Elsevier Masson SAS. All rights reserved.)