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We prove new theorems which describe a necessary and sufficient condition for linear (strong and non-strong) separability and inseparability of the sets in a finite-dimensional Euclidean space. We propose a universal measure for the thickness of the geometric margin (both the strong separation margin (separator) and the margin of unseparated points (pseudo-separator)) formed between the parallel generalized supporting hyperplanes of the two sets which are separated. The introduced measure allows comparing results of linear separation obtained by different techniques for both linearly separable and inseparable sets. An optimization program whose formulation provides a maximum thickness of the separator for the separable sets is considered. When the sets are inseparable, the same solver is guaranteed to construct a pseudo-separator with a minimum thickness. We estimate the distance between the convex and closed sets. We construct a cone of generalized support vectors for hyperplanes, each one of which linearly separates the considered sets. The interconnection of the problem of different types of linear separation of sets with some related problems is studied.

A variant of the embedding technique proposed earlier by the second author is suggested in which the sets to be embedded are support cones. Replacing the cones by simplices gives a modification with a polynomial convergence rate.

It has recently been established that the tube formula and the Euler characteristic method give an identical and valid asymptotic expansion of the tail probability of the maximum of a Gaussian random field when the random field has finite Karhunen-Loève expansion and the index set has positive critical radius. We show that the positiveness of the critical radius is an essential condition. When the critical radius is zero, we prove that only the main term is valid and that other higher-order terms are generally not valid in the formal asymptotic expansion based on the tube formula. This is done by first establishing an exact tube formula and comparing the formal tube formula with the exact formula. Furthermore, we show that the equivalence of the formal tube formula and the Euler characteristic method no longer holds when the critical radius is zero. We conclude by applying our results to some specific examples.

The classification of soils into categories with a similar range of properties is a fundamental geotechnical engineering procedure. At present, this classification is based on various types of cost- and time-intensive laboratory and/or in situ tests. These soil investigations are essential for each individual construction site and have to be performed prior to the design of a project. Since Machine Learning could play a key role in reducing the costs and time needed for a suitable site investigation program, the basic ability of Machine Learning models to classify soils from Cone Penetration Tests (CPT) is evaluated. To find an appropriate classification model, 24 different Machine Learning models, based on three different algorithms, are built and trained on a dataset consisting of 1339 CPT. The applied algorithms are a Support Vector Machine, an Artificial Neural Network and a Random Forest. As input features, different combinations of direct cone penetration test data (tip resistance qc, sleeve friction fs, friction ratio Rf, depth d), combined with “defined”, thus, not directly measured data (total vertical stresses σv, effective vertical stresses σ’v and hydrostatic pore pressure u0), are used. Standard soil classes based on grain size distributions and soil classes based on soil behavior types according to Robertson are applied as targets. The different models are compared with respect to their prediction performance and the required learning time. The best results for all targets were obtained with models using a Random Forest classifier. For the soil classes based on grain size distribution, an accuracy of about 75%, and for soil classes according to Robertson, an accuracy of about 97–99%, was reached.

In this article, we attempt to characterize optimum solutions for optimization problems with interval-valued functions of interval variables. As the constraint set or the underlying variable spaces of such optimization problems are a set of intervals, we introduce, analyze, and interrelate the notions of normal cone and tangent cone of a set of intervals. In the sequel, their various kinds of properties are defined, such as closedness, weak intersection rule, some algebraic preserving properties, etc. The dual correspondence between the tangent and normal cones is also analyzed. For constrained optimization problems, the normal cone is used to characterize efficient solutions. Furthermore, we derive a necessary condition for efficient solutions for interval optimization problems with the help of Lagrange multipliers, tangent cones, and normal cones. Lastly, an application of the proposed normal cone in solving an interval-valued support vector machines type problem is discussed.

Background Artificial intelligence has been proven to improve the identification of various maxillofacial lesions. The aim of the current study is two-fold: to assess the performance of four deep learning models (DLM) in external root resorption (ERR) identification and to assess the effect of combining feature selection technique (FST) with DLM on their ability in ERR identification. Methods External root resorption was simulated on 88 extracted premolar teeth using tungsten bur in different depths (0.5 mm, 1 mm, and 2 mm). All teeth were scanned using a Cone beam CT (Carestream Dental, Atlanta, GA). Afterward, a training (70%), validation (10%), and test (20%) dataset were established. The performance of four DLMs including Random Forest (RF) + Visual Geometry Group 16 (VGG), RF + EfficienNetB4 (EFNET), Support Vector Machine (SVM) + VGG, and SVM + EFNET) and four hybrid models (DLM + FST: (i) FS + RF + VGG, (ii) FS + RF + EFNET, (iii) FS + SVM + VGG and (iv) FS + SVM + EFNET) was compared. Five performance parameters were assessed: classification accuracy, F1-score, precision, specificity, and error rate. FST algorithms (Boruta and Recursive Feature Selection) were combined with the DLMs to assess their performance. Results RF + VGG exhibited the highest performance in identifying ERR, followed by the other tested models. Similarly, FST combined with RF + VGG outperformed other models with classification accuracy, F1-score, precision, and specificity of 81.9%, weighted accuracy of 83%, and area under the curve (AUC) of 96%. Kruskal Wallis test revealed a significant difference ( p  = 0.008) in the prediction accuracy among the eight DLMs. Conclusion In general, all DLMs have similar performance on ERR identification. However, the performance can be improved by combining FST with DLMs.

Visual performance varies around the visual field. It is best near the fovea compared to the periphery, and at iso-eccentric locations it is best on the horizontal, intermediate on the lower, and poorest on the upper meridian. The fovea-to-periphery performance decline is linked to the decreases in cone density, retinal ganglion cell (RGC) density, and V1 cortical magnification factor (CMF) as eccentricity increases. The origins of polar angle asymmetries are not well understood. Optical quality and cone density vary across the retina, but recent computational modeling has shown that these factors can only account for a small percentage of behavior. Here, we investigate how visual processing beyond the cone photon absorptions contributes to polar angle asymmetries in performance. First, we quantify the extent of asymmetries in cone density, midget RGC density, and V1 CMF. We find that both polar angle asymmetries and eccentricity gradients increase from cones to mRGCs, and from mRGCs to cortex. Second, we extend our previously published computational observer model to quantify the contribution of phototransduction by the cones and spatial filtering by mRGCs to behavioral asymmetries. Starting with photons emitted by a visual display, the model simulates the effect of human optics, cone isomerizations, phototransduction, and mRGC spatial filtering. The model performs a forced choice orientation discrimination task on mRGC responses using a linear support vector machine classifier. The model shows that asymmetries in a decision maker’s performance across polar angle are greater when assessing the photocurrents than when assessing isomerizations and are greater still when assessing mRGC signals. Nonetheless, the polar angle asymmetries of the mRGC outputs are still considerably smaller than those observed from human performance. We conclude that cone isomerizations, phototransduction, and the spatial filtering properties of mRGCs contribute to polar angle performance differences, but that a full account of these differences will entail additional contribution from cortical representations.

Visual performance depends on polar angle, even when eccentricity is held constant; on many psychophysical tasks observers perform best when stimuli are presented on the horizontal meridian, worst on the upper vertical, and intermediate on the lower vertical meridian. This variation in performance 'around' the visual field can be as pronounced as that of doubling the stimulus eccentricity. The causes of these asymmetries in performance are largely unknown. Some factors in the eye, e.g. cone density, are positively correlated with the reported variations in visual performance with polar angle. However, the question remains whether these correlations can quantitatively explain the perceptual differences observed 'around' the visual field. To investigate the extent to which the earliest stages of vision-optical quality and cone density-contribute to performance differences with polar angle, we created a computational observer model. The model uses the open-source software package ISETBIO to simulate an orientation discrimination task for which visual performance differs with polar angle. The model starts from the photons emitted by a display, which pass through simulated human optics with fixational eye movements, followed by cone isomerizations in the retina. Finally, we classify stimulus orientation using a support vector machine to learn a linear classifier on the photon absorptions. To account for the 30% increase in contrast thresholds for upper vertical compared to horizontal meridian, as observed psychophysically on the same task, our computational observer model would require either an increase of ~7 diopters of defocus or a reduction of 500% in cone density. These values far exceed the actual variations as a function of polar angle observed in human eyes. Therefore, we conclude that these factors in the eye only account for a small fraction of differences in visual performance with polar angle. Substantial additional asymmetries must arise in later retinal and/or cortical processing.

Accurate midpalatal suture maturation assessment is critical for orthodontic treatment planning, yet current manual staging methods exhibit substantial inter-examiner variability (kappa values 0.3-0.8), compromising treatment decision reliability. This study developed and validated DeepMSM, an automated multimodal deep learning framework integrating cone-beam computed tomography with clinical indicators for standardized midpalatal suture staging. We retrospectively analyzed cone-beam computed tomography and lateral cephalometric radiographs from 200 orthodontic patients aged 7-36 years. The DeepMSM framework integrated multimodal images with clinical variables including age, gender, cervical vertebral maturation stage, and mandibular third molar stage using attention-based fusion strategies. DeepMSM achieved 93.75% accuracy and 93.81% F1-score, substantially outperforming single-modality approaches (47.50%-71.25% accuracy) and dual-modality models (73.75–81.25% accuracy). The system demonstrated excellent performance in distinguishing critical stages C and D with F1-scores of 92%-93%, representing the decision point between conventional expansion and surgical intervention. All clinical parameters showed significant correlations with midpalatal suture maturation (p<0.05). DeepMSM, a novel multimodal midpalatal suture maturation assessment system, achieved a high accuracy of 93.75%, demonstrating the potential to reduce diagnostic variability and improve treatment reliability. This automated framework particularly benefits less experienced clinicians in making critical treatment decisions for maxillary expansion therapy.

The prediction of the potential of soil liquefaction induced by the earthquake is a vital task in construction engineering and geotechnical engineering. To provide a possible solution to such problems, this paper proposes two support vector machine (SVM) models which are optimized by genetic algorithm (GA) and grey wolf optimizer (GWO) to predict the potential of soil liquefaction. Field observation data based on cone penetration test (CPT), standard penetration test (SPT) and shear wave velocity (VS) test (SWVT) are employed to verify the reliability of the GA–SVM model and the GWO–SVM model, the numbers of input variables of these three field testing data sets are 6, 12 and 8, respectively, and the output result is the potential of soil liquefaction. To verify whether the two optimization algorithms GA and GWO have significantly improved the performance of SVM model, an unoptimized SVM model is served as a reference in this study. And five performance metrics, including classification accuracy rate (ACC), precision rate (PRE), recall rate (REC), F1 score (F1) and AUC are used to evaluate the classification performance of the three models. Results of the study confirm that when CPT-based, SPT-based and SWVT-based test sets are input into three classification models, the highest classification accuracy of 0.9825, 0.9032 and 0.9231, respectively, is achieved with GWO–SVM. And based on these three data sets, the values of AUC obtained by GWO–SVM are all higher than those obtained by GA–SVM. Further, by comparing the other metrics of the three classification models, it is found that the classification performance of the two hybrid models is very similar and significantly better than the SVM, which indicates that GWO–SVM, like GA–SVM, can also be used as a reliable model for predicting soil liquefaction potential.