Linear Widths Research Articles

Fault Diagnosis and Optimized Operation and Maintenance of Drop-out Fuses in Alpine Regions Using MKL-SVM and GRU

Extreme climate in alpine regions leads to frequent failures of drop-out fuses, but existing methods have low detection accuracy and response speed. This article combines MKL-SVM (Multiple Kernel Learning-Support Vector Machine) and GRU (Gated Recurrent Unit) model to achieve precise fault diagnosis of drop-out fuses, leveraging their efficient processing of high-dimensional time series data. Sensors are used to collect data such as voltage, current, temperature, and wind speed. Based on GRU, global trends, temporal hidden states, and dynamic nonlinear features of the data are extracted as inputs for subsequent MKL-SVM. In MKL-SVM, a linear kernel function is used to simplify the linear relationship between static features; targeting temporal features, RBF (Radial Basis Function) kernel effectively captures dynamic trends, fluctuations, and periodic changes in temporal features. To further enhance the classification ability of the model, the kernel function parameters, including linear kernel weights and RBF kernel widths, are adjusted through cross-validation. Finally, based on the model diagnosis results and predicted data, an optimized operation and maintenance plan is formulated to reduce maintenance costs and improve equipment operating efficiency. The results show that the fusion model can significantly improve the reliability of fault detection and adapt to the unique environmental challenges of alpine regions. The accuracy of fault diagnosis for drop-out fuses is 95%, with an MSE (mean square error) value of about 0.15 and a maintenance cost reduction of about 21.89%. This method can achieve accurate fault diagnosis and efficient optimization of operation and maintenance under extreme weather conditions.

Design of an adjustable low-temperature linear microwave plasma source for atmospheric pressure applications

Linear plasmas, compared to small-area low-temperature plasma jets, offer a larger single-treatment area with a brush-like pattern, making them highly promising for various applications. This paper introduces the design of an adjustable low-temperature linear plasma source that operates under atmospheric pressure at 2.45 GHz. The design integrated microwave theory with dielectric barrier discharge principles, utilizing a resonant structure based on a microstrip power divider with one end open and the other shorted. The ground plane of the microstrip structure was replaced by a metal plate featuring a large groove. Argon gas was introduced from the short-circuited end and exited through the groove at the open end, where plasma excitation occurred. Experimental results demonstrated that the device, operating at atmospheric pressure, can achieve adjustable linear plasma widths ranging from 10 mm to 50 mm by varying the incident power between 30 and 100 W. Optical emission measurements confirmed the uniformity of the linear plasma, and the gas temperature 5 mm away from the discharge area remained at only 65 °C, even with a microwave incident power of 100 W. This study offers a novel approach to designing linear atmospheric pressure microwave plasma sources, with significant potential for diverse material treatment applications.

Exploring Users Pointing Performance on Large Displays with Different Curvatures in Virtual Reality.

Large curved displays inside Virtual Reality environments are becoming popular for visualizing high-resolution content during analytical tasks, gaming or entertainment. Prior research showed that such displays provide a wide field of view and offer users a high level of immersion. However, little is known about users' performance (e.g., pointing speed and accuracy) on them. We explore users' pointing performance on large virtual curved displays. We investigate standard pointing factors (e.g., target width and amplitude) in combination with relevant curve-related factors, namely display curvature and both linear and angular measures. Our results show that the less curved the display, the higher the performance, i.e., faster movement time. This result holds for pointing tasks controlled via their visual properties (linear widths and amplitudes) or their motor properties (angular widths and amplitudes). Additionally, display curvatures significantly affect the error rate for both linear and angular conditions. Furthermore, we observe that curved displays perform better or similar to flat displays based on throughput analysis. Finally, we discuss our results and provide suggestions regarding pointing tasks on large curved displays in VR.

Accuracy of lingual straight-wire orthodontic treatment with passive self-ligating brackets and square slot: a retrospective study

ObjectivesTo investigate the accuracy of torque, tip and rotation and linear intra-arch movements yielded by passive self-ligating lingual straight-wire appliances with brackets featuring square slots.Materials and methodsTwenty-five adult Caucasian patients (16 females and 9 males; mean age 26.5 ± 4.3 years) with Class I or mild Class II head-to-head malocclusion were orthodontically treated via passive lingual self-ligating straight-wire appliances (ALIAS, Ormco, Orange, CA) with no extraction. Records were retrospectively analysed, and digital models of pre-treatment (T0), planned (T1) and achieved (T2) phase were acquired for both arches in each patient via an intraoral scanner (Medit I500 (iScan Medit, Seoul, Korea). VAM software (Vectra, Canfield Scientific, Fairfield, NJ, USA) was used to measure both angular values (torque, tip and rotation) and linear intra-arch widths (between canines, first and second premolars and first and second molars). Measurements were obtained for all the movements investigated for each tooth group (incisors, canines, premolars and molars), by arch (maxillary and mandibular) and for both arches at T0, T1 and T2. The accuracy of angular values was compared using Student's t-test against a hypothetical 100%, and among the various tooth groups by post-hoc tests. Transverse linear measurements were investigated by means of the non-parametric Friedman test. The significance threshold was set at 0.05.ResultsThe mean accuracy of angular values was 77.25 ± 7.71% for torque, 78.41 ± 6.17% for tip and 77.99 ± 6.58% for rotation. In all cases, however, there was a significant difference between planned and achieved movements, and accuracy was significantly lower than the hypothetical 100% for all tooth groups, individual arches and dentition (p < 0.001). For intra-arch diameters, the greatest accuracy values were found for the anterior sectors (83.54 ± 5.19% for the maxillary inter-canine distance) and the lowest for the posterior sectors (67.28% for the maxillary inter-second molar distance).ConclusionStraight-wire lingual treatment with passive self-ligating appliances featuring with square slot displayed excellent clinical accuracy, albeit with statistical accuracy decreasing antero-posteriorly.

Characteristics of linear and nonlinear approximation of isotropic classes of periodic multivariate functions

Exact order estimates for some characteristics of linear and nonlinear approximation of the isotropic Nikol'skii-Besov classes $\mathbf{B}^r_{p,\theta}$ of periodic multivariate functions in the spaces $B_{q,1}$, $1\leq q \leq \infty$, are obtained. Among them are the best orthogonal trigonometric approximations, best $m$-term trigonometric approximations, Kolmogorov, linear and trigonometric widths.&#x0D; For all considered characteristics, their estimates coincide in order with the corresponding estimates in the spaces $L_{q}$. Moreover, the obtained exact in order estimates (except the case $1&lt;p&lt;2\leq q &lt; \frac{p}{p-1}$) are realized by the approximation of functions from the classes ${\mathbf{B}}^r_{p,\theta}$ by trigonometric polynomials with the spectrum in cubic regions. In any case, they do not depend on the smoothness parameter $\theta$.

Mandibular Ramus: An Indicator for Gender Determination.

Background The goal of this study is to use digital orthopantomographs (OPGs) to find out the role the mandibular ramus plays in figuring out a person's gender. Methodology Six hundred digital OPGsof patients, aged 21 to 50 years of either gender, fulfilling the exclusion and inclusion criteria, were randomly selected from the department archives exclusively for this digital retrospective study. All the scans were anonymized before the analysis. A total of seven measurements (in mm) were performed on OPGs, namely, minimum and maximum widths of the ramus,minimum and maximum condylar heights,maximum height of the ramus and coronoid,gonial angle bilaterally, and bigonial width. The data obtained was statistically analyzed using IBM SPSS Statistics for Windows, Version 21.0. (IBM Corp., Armonk, NY, USA), by a stepwise discriminant functional analysis for gender determination. Results Linear measurements, including maximum and minimum widths of the ramus, maximum height of the condyle, height of the ramus, and coronoid and bigonial width, revealed more values in males than in females. However, the gonial angle showed higher average values in females than in males. Moreover, all seven parameters showed statistically insignificant age-related changes. Conclusions The mandibular ramus showed high sexual dimorphism, and its analysis on OPG can be a valuable aid in the determination of gender in the fields of forensic odontology and anthropology.

On the approximation of solutions of one singular differential equation on the axis

In this paper we study the problem of the best approximation by linear methods of solutions to one Triebel-type equation. This problem was solved by using estimates of the linear widths of the unit ball in corresponding spaces of differentiable functions. According to the definition, linear widths give the best estimates for the approximation of compact sets in a given normed space by linear methods which are implemented through finite-dimensional operators. The problem includes answers to the questions about the solvability of the studied equation, the construction of the corresponding weighted space of differentiable functions, the development of a method for estimating linear widths of compact sets in weighted polynomial Sobolev space. In this work, conditions are obtained under which the considered operator has a bounded inverse. The weighted Sobolev space corresponding to the posed problem is determined. Upper estimates are obtained for the counting function for a sequence of linear widths, which correspond to the posed problem. One example is constructed in which two-sided estimates of linear widths are given. The method for solving this problem can be applied to the numerical solution of non-standard ordinary differential equations on an infinite axis.

Post-Expansion and end of treatment outcomes of semi-rapid maxillary expansion with a modified removable appliance.

Maxillary expansion is the mainstay therapy for maxillary transverse deficiency. There has been a constant search for the most effective yet biologically friendly method of maxillary expansion, alternatives being, slow, rapid and semi rapid. The purpose of this study was to explore the outcome of palatal expansion achieved using a removable plate and low continuous forces brought about by a semi rapid screw activation protocol. Retrospective study. Plaster models of 56 consecutive patients treated for maxillary expansion were obtained pre-treatment (T0), post-expansion (T1), and post fixed appliance treatment (T2). The radiographic images of the models were traced using Image J software. Linear and angular measurements were evaluated to measure transverse change. Interclass Correlation Coefficient [ICC] and Dahlberg's formula were used for reliability test. The differences in the mean values between the three duration groups [T0, T1 and T2] were analysed using Analysis of Variance (ANOVA). For multiple comparisons, a post hoc Tukey honestly significant difference (HSD) test was performed. Significant increase in inter-molar, alveolar and palatal linear widths were observed from T0 to T1 with significant relapses from T1 to T2, with an overall net gain remaining at T2. Similarly, significant increases in all angular measurements were observed from T0 to T1 with significant relapses from T1 to T2 and an overall insignificant change at T2 as compared to T0. The appliance and protocol were effective in producing transverse expansion with minimal molar and alveolar tipping.

Jackson-type inequalities and widths of functional classes in the Musielak–Orlicz type spaces

In the Musielak–Orlicz-type spaces 𝒮M, exact Jackson-type inequalities are obtained in terms of best approximations of functions and the averaged values of their generalized moduli of smoothness. The values of Kolmogorov, Bernstein, linear, and projective widths in 𝒮M are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.

Function values are enough for L2-approximation: Part II

In the first part we have shown that, for L2-approximation of functions from a separable Hilbert space in the worst-case setting, linear algorithms based on function values are almost as powerful as arbitrary linear algorithms if the linear widths are square-summable. That is, they achieve the same polynomial rate of convergence. In this sequel, we prove a similar result for separable Banach spaces and other classes of functions.

Erratum to: Linear Widths of Weighted Sobolev Classes with Conditions on the Highest Order and Zero Derivatives

Erratum to: Linear Widths of Weighted Sobolev Classes with Conditions on the Highest Order and Zero Derivatives

Linear Widths of Weighted Sobolev Classes with Conditions on the Highest Order and Zero Derivatives

In this paper order estimates for the linear widths of some function classes are obtained; these classes are defined by restrictions on the weighted $L_{p_1}$-norm of the r-th derivative and the weighted $L_{p_0}$-norm of zero derivative.

Linear and Kolmogorov Widths of the Classes $$ {B}_{p,\uptheta}^{\Omega} $$ of Periodic Functions of One and Several Variables

Abstract. Some estimates exact in order for linear widths of the classes $$ {B}_{p,\uptheta}^{\Omega} $$ of periodic multivariable functions in the space Lq with certain relations between the parameters p, q; and 𝜃 are obtained. In the univariate case, the estimates exact in order for Kolmogorov and linear widths of the classes $$ {B}_{\infty, \uptheta}^{\omega } $$ in the space Lq, 1 ≤ q ≤ ∞; are established.

Soft Tissue Radiological Knee (SToRK) Index: An observational cohort study to produce an index that quantifies the magnitude of soft tissue around the knee using standard radiographs

IntroductionThe soft tissue envelope around the knee can have an impact on the ease of performing surgery such as total knee replacement (TKR). BMI is often used in planning theatre time but may be a poor indicator of the soft tissue around the knee due to varying distribution of adipose tissue. Radiological images directly show the soft tissue. We therefore aim to develop a method of quantifying the soft tissue envelope around the knee using radiographs. MethodPlain weight-bearing radiographs were used to measure the total knee (soft tissue and bony) width at the level of the epicondyles of the knee and the bony epicondylar width of the femur. The ratio of the two widths was defined as the Soft Tissue Radiological Knee (SToRK) Index. The validity of the index as a true measure of soft tissue envelope was assessed using cross sectional areas on axial MRI cuts at the level of the epicondyles. The inter-observer reliability was assessed using the intra-class correlation coefficient. SToRK Index values were correlated with patients’ BMI, gender and operative time. ResultsThe results show there is a close correlation between the ratio of cross sectional area of MRI axial cuts at the level of epicondyles and the ratio of linear widths measured on plain radiographs, validating the SToRK Index as a measure of soft tissue envelope. There was also good to excellent inter-rater reliability of measurements of these widths. There was a close correlation between BMI and SToRK Index with differences between men and women. DiscussionWe believe the SToRK Index is a validated method of quantifying soft tissue distribution around the knee and gives surgeons a better descriptor of the knee envelope than BMI. It is easy to use, needs simple investigations and is reproducible.

Linear and Kolmogorov widths of the classes B^{\Omega}_{p, \theta} of periodic functions of one and several variables

Some estimates exact in order for linear widths of the classes $B^{\Omega}_{p, \theta}$ of periodic multivariable functions in the space $L_q$ with certain relations between the parameters $p$, $q,$ and $\theta$ are obtained. In the univariate case, the estimates exact in order for Kolmogorov and linear widths of the classes $B^{\omega}_{\infty, \theta}$ in the space $L_q$, $1 \leq q \leq \infty,$ are established.

On the estimates of the values of various widths of classes of functions of two variables in the weight space L2,γ (ℝ2), γ = exp(− x2− y2)

For the classes of functions of two variables W2 (Ωm,γ, Ψ) = {f ∈ L2,γ (ℝ2) : Ωm,γ (f, t) ⩽ Ψ(t) ∀t ∈ (0, 1)}, m ∈ ℕ, where Ωm,γ is a generalized modulus of continuity of the m-th order, and Ψ is a majorant, the upper and lower bounds for the ortho-, Kolmogorov, Bernstein, projective, Gel’fand, and linear widths in the metric of the space L2,γ (ℝ2) are found. The condition for a majorant under which it is possible to calculate the exact values of the listed extreme characteristics of the optimization content is indicated. We consider the similar problem for the classes $$ {W}_2^{r,0} $$ (Ωm,γ, Ψ) = $$ {L}_{2,\upgamma}^{r,0} $$ (D, ℝ2) ∩ $$ {W}_2^r $$ (Ωm,γ, Ψ), r,m ∈ ℕ, ( $$ D=\frac{\partial^2}{\partial {x}^2}+\frac{\partial^2}{\partial {y}^2}-2x\frac{\partial }{\partial x}-2y\frac{\partial }{\partial y} $$ being the differential operator). Those classes consist of functions f ∈ $$ {L}_{2,\upgamma}^{r,0} $$ (ℝ2) whose Fourier–Hermite coefficients are ci0 (f) = c0j (f) = c00 (f) = 0 ∀i, j ∈ ℕ. The r-th iterations Drf = D (Dr−1f) (D0f ≡ f) belong to the space L2,γ (ℝ2) and satisfy the inequality Ωm,γ (Drf, t) ⩽ Ψ(t) ∀t ∈ (0, 1). On the indicated classes, we have determined the upper bounds (including the exact ones) for the Fourier–Hermite coefficients. The exact results obtained are specified, and a number of comments regarding them are given.

Theory of edge localized mode suppression by static resonant magnetic perturbations in the DIII-D tokamak

According to a recent paper [Hu et al., Phys. Plasmas 26, 120702 (2019)], mode penetration at the top of the pedestal is a necessary and sufficient condition for the suppression of edge localized modes (ELMs) by a resonant magnetic perturbation (RMPs) in an H-mode tokamak discharge. This paper employs asymptotic matching theory to model a particular DIII-D discharge in which ELMs were suppressed by an externally generated, static, n = 2, RMP whose amplitude was modulated at a frequency of 1 Hz. It is demonstrated that the response of the plasma to the applied RMP, in the immediate vicinities of the rational (i.e., resonant) surfaces, is governed by nonlinear, rather than by linear, physics. This is the case because the magnetic island widths associated with driven reconnection exceed the linear layer widths, even in cases where driven reconnection is strongly suppressed by plasma rotation. The natural frequency at a given rational surface (i.e., the helical frequency at which the locally resonant component of the RMP would need to propagate in order to maximize driven reconnection) is found to be offset from the local E×B frame in the ion diamagnetic direction. The size of the offset is mostly determined by neoclassical poloidal rotation. Finally, the predictions of a fully nonlinear plasma response model are found to be broadly consistent with the DIII-D experimental data.

Exact asymptotic orders of various randomized widths on Besov classes

We study the efficiency of the approximation of the functions from the Besov space \begin{document}$ B_{p\theta}^\Omega(\mathbf{T}^d) $\end{document} in the norm of \begin{document}$ L_q(\mathbf{T}^d) $\end{document} by various random methods. We determine the exact asymptotic orders of Kolmogorov widths, linear widths, and Gel'fand widths of the unit ball of \begin{document}$ B_{p\theta}^\Omega(\mathbf{T}^d) $\end{document} in \begin{document}$ L_q(\mathbf{T}^d) $\end{document} . Our results show that the convergence rates of the randomized linear and Gel'fand methods are faster than the deterministic counterparts in some cases. The maximal improvement can reach a factor \begin{document}$ n^{-1/2} $\end{document} roughly.

Arecibo 4.5/1.4/0.33-GHz polarimetric single-pulse emission survey

ABSTRACT We report on an Arecibo 4.5-GHz polarimetric single-pulse survey of the brightest pulsars at high frequency within its sky. The high-frequency profiles are accompanied by a collection of both previously published and unpublished high quality 1.4- and 0.33-GHz observations. Here our analyses and discussion primarily involve the average and statistical properties of the 46 pulsars polarimetric pulse sequences, profile classification and frequency evolution, and polarimetric profiles and peak-occurrence histograms. In most cases both the fractional linear polarization and profile widths decrease with frequency as expected, but there are some exceptions. Similarly, we were able to review and/or extend the profile classifications for this population of pulsars and work out their beaming characteristics quantitatively showing that almost all show properties compatible with the core/double-cone emission beam model. The entirety of these observations’ average profiles are accessible for download.

On the Estimates of Widths of the Classes of Functions Defined by the Generalized Moduli of Continuity and Majorants in the Weighted Space L2,x(0, 1)

The upper and lower estimates for the Kolmogorov, linear, Bernstein, Gelfand, projective, and Fourier widths are obtained in the space L2,x(0, 1) for the classes of functions $$ {W}_2^r\left({\Omega}_{m,x}^{(v)};\Psi \right), $$ where r ∈ ℤ+, m ∈ ℕ, v ≥ 0, and $$ {\Omega}_{m,x}^{(v)} $$ and Ψ are the m th order generalized modulus of continuity and the majorant, respectively. The upper and lower estimates for the suprema of Fourier–Bessel coefficients are also established for these classes. We also present the conditions for majorants under which it is possible to find the exact values of the indicated widths and the upper bounds of the Fourier–Bessel coefficients.