Academic literature on the topic 'L1 Norms'

By utilizing kernel functions, support vector machines (SVMs) successfully solve the linearly inseparable problems. Subsequently, its applicable areas have been greatly extended. Using multiple kernels (MKs) to improve the SVM classification accuracy has been a hot topic in the SVM research society for several years. However, most MK learning (MKL) methods employ L1-norm constraint on the kernel combination weights, which forms a sparse yet nonsmooth solution for the kernel weights. Alternatively, the Lp-norm constraint on the kernel weights keeps all information in the base kernels. Nonetheless, the solution of Lp-norm constraint MKL is nonsparse and sensitive to the noise. Recently, some scholars presented an efficient sparse generalized MKL (L1- and L2-norms based GMKL) method, in which L1 L2 established an elastic constraint on the kernel weights. In this paper, we further extend the GMKL to a more generalized MKL method based on the p-norm, by joining L1- and Lp-norms. Consequently, the L1- and L2-norms based GMKL is a special case in our method when p=2. Experiments demonstrated that our L1- and Lp-norms based MKL offers a higher accuracy than the L1- and L2-norms based GMKL in the classification, while keeping the properties of the L1- and L2-norms based on GMKL.

AbstractWe investigate the geometry of manifolds with bounded Ricci curvature in L1-sense. In particular, we generalize the classical volume comparison theorem to our situation and obtain a generalized sphere theorem.

Ethical group might have differential skeletal forms, thus cephalometric analysis and orthodontic material have been developed for specific ethnic groups which might be different for Kurdish Ethics.Objectives: This study is to determine the cephalometric norms of an Erbil Kurdish population according to Steiner analysis, who attend to Tishk Hospital/Diagnosis Department Materials and Methods: In this cross-sectional study, 48 lateral cephalograms of adult orthodontic subjects between 20 and 29 years with normal occlusion, and with no previous orthodontic treatment were evaluated. Lateral cephalograms were traced and analyzed based on Steiner’s cephalometric parameters. Data were analyzed using SPSS. Differences between Erbil Kurdish and Steiner’s norms were analyzed using one-sample t-test (P < 0.05). Results: The SNA, ML-NSL (Mandibular Line to NS line), U1-NA (both angular and linear measurements), interincisal angle, L1-NB (angular only), Pog-NB, L1-Pog//NB, Ls-SL, and Li-SL values were significantly different between the Erbil Kurdish population and Steiner’s norms (P < 0.05). No significant differences existed in Occl-SN (occlusal plane to SN) and L1-Nb (linear only), SNB, and ANB between the Erbil Kurdish population and the Steiner’s norms (P > 0.05). Conclusions: Cephalometric norms for Erbil Kurdish adults are different from those of Steiner’s norms. The norms obtained in our study can be used for orthodontic treatments and orthognathic surgeries in Erbil Kurdish population.

The objective of this study was to investigate the upper incisors (U1), lower incisors (L1), and soft tissue profiles of bimaxillary protrusion (BM) adult patients among skeletal Class I (BM1), II (BM2) and III (BM3). Understanding these characteristics would be useful for incisor and lip diagnostics in different skeletal classifications. Fourteen linear and twelve angular variables of the incisors and lips were evaluated in 214 lateral cephalograms (BM1 = 91, BM2 = 84, BM3 = 39). ANOVA and Bonferroni tests compared the measurements. BM1 and BM3 exhibited a greater U1 position and U1 inclination than the norms, while BM2 presented only a greater U1 position than the norms but normal U1 inclination. BM1 and BM3 had a significantly greater U1 position than BM2. BM1 and BM2 demonstrated a greater L1 position and L1 inclination than the norms, whereas BM3 displayed only a greater L1 position than the norms but normal L1 inclination. BM2 had the most anterior L1 position, whereas BM3 had the least anterior position. Only BM2 had a longer anterior dental height (ADH) than the norms, while BM1 and BM3 had a normal ADH and the significantly shortest ADH, respectively. Only BM1 had a normal upper incisor display at rest (U1R), while BM2 and BM3 displayed an increased and decreased U1R, respectively, with significant differences among the three groups. The most significantly protruded upper and lower lips were presented in BM2, but these were exhibited the least in BM3. The most significant acute nasolabial angle (NLA) was found in BM3, whereas BM2 presented the least acute NLA. A normal lip–chin–throat angle (LCTA) was observed in BM1 and BM3, while only BM2 had a greater LCTA than the norms. The most significant obtuse LCTA was found in BM2, while BM3 had the least obtuse LCTA. Therefore, both U1 and L1 in all groups presented protrusion and proclination, except for U1 in BM2, while L1 in BM3 exhibited normal inclination. The ADH and U1R were increased in BM2 but decreased in BM3. The most acute NLA was found in BM3, whereas the least acute was found in BM2. The most obtuse LCTA was in BM2, while the least was in BM3.

AbstractIn this paper we determine the L1 ⟶ L1 and L∞ ⟶ L∞ norms of an integral operator N related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.