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We classify non-trivial, non-negative, positively homogeneous solutions of the equation Δu= γuγ1 in the plane. The problem is motivated by the analysis of the classical Alt–Phillips free boundary problem, but considered here with negative exponents γ . The proof relies on several bespoke results for ordinary differential equations [ABSTRACT FROM AUTHOR]

We study strictly positive solutions to the critical Laplace equation - Δ u = n (n - 2) u n + 2 n - 2 , decaying at most like d (o , x) - (n - 2) / 2 , on complete noncompact manifolds (M, g) with nonnegative Ricci curvature, of dimension n ≥ 3 . We prove that, under an additional mild assumption on the volume growth, such a solution does not exist, unless (M, g) is isometric to R n and u is a Talenti function. The method employs an elementary analysis of a suitable function defined along the level sets of u. [ABSTRACT FROM AUTHOR]

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In this paper, we prove Liouville type theorems for stable solutions to the weighted fractional Lane-Emden system (−∆)su = h(x)vp, (−∆)sv = h(x)uq, u, v > 0 in ℝN, where 1 < q ≤ p and h is a positive continuous function in ℝN satisfying lim inf / |x|→∞ h(x) / |x|l > 0 with l > 0. Our results generalize the results established in [23] for the Laplacian case (correspond to s = 1) and improve the previous work [12]. As a consequence, we prove classification result for stable solutions to the weighted fractional Lane-Emden equation (−∆)su = h(x)up in ℝN. [ABSTRACT FROM AUTHOR]

We consider a fractional version of Gauß capillarity energy. A suitable extension problem is introduced to derive a boundary monotonicity formula for local minimizers of this fractional capillarity energy. As a consequence, blow-up limits of local minimizers are shown to subsequentially converge to minimizing cones. Finally, we show that in the planar case there is only one possible fractional min- imizing cone, the one determined by the fractional version of Young’s law. [ABSTRACT FROM AUTHOR]

In this article, we study various analytic aspects of interpolating sesqui-harmonic maps between Riemannian manifolds where we mostly focus on the case of a spherical target. The latter are critical points of an energy functional that interpolates between the functionals for harmonic and biharmonic maps. In the case of a spherical target, we will derive a conservation law and use it to show the smoothness of weak solutions. Moreover, we will obtain several classification results for interpolating sesqui-harmonic maps. [ABSTRACT FROM AUTHOR]

Most gender classifications methods from near‐infrared (NIR) images have used iris information. Recent work has explored the use of the whole periocular iris region which has surprisingly achieved better results. This suggests the most relevant information for gender classification is not located in the iris as expected. In this work, the authors analyse and demonstrate the location of the most relevant features that describe gender in periocular NIR images and evaluate their influence in classification. Experiments show that the periocular region contains more gender information than the iris region. They extracted several features (intensity, texture, and shape) and classified them according to their relevance using the XgBoost algorithm. Support vector machine and nine ensemble classifiers were used for testing gender accuracy when using the most relevant features. The best classification results were obtained when 4000 features located on the periocular region were used (89.22%). Additional experiments with the full periocular iris images versus the iris‐occluded images were performed. The gender classification rates obtained were 84.35 and 85.75%, respectively. From results, they suggest focusing only on the surrounding area of the iris. This allows us to realise a faster classification of gender from NIR periocular images. [ABSTRACT FROM AUTHOR]

This study presents a comprehensive review of the automated classification in partial discharge (PD) source identification and probabilistic interpretation of the classification results based on the relationship between the variation of the phase-resolved PD (PRPD) patterns and the source of the PD. The proposed automated classification system consists of modern, high-performance statistical feature extraction methods and classifier algorithms. Their application in online monitoring and recognition of the PD patterns is investigated based on their low-processing time and high-performance evaluation. The application of modern statistical algorithms and pre-processing methods configured in this automated classification system improves the pattern recognition accuracy of the different PD sources that are suitable to be employed in different high-voltage (HV) insulation media. To evaluate the performance of the different combinations of the feature extraction/classier pairs, laboratory setups are designed and built that simulate various types of PDs. The test cells include three sources of PD in [inline-formula], two sources of PD in transformer oil, and corona in the air. Data samples for different classes of PD sources are captured under two levels of voltage and two different levels of noise. The results of this study evaluate the suitability of the proposed classification systems for probabilistic source identification in various insulation media. Furthermore, of importance to the problem of the PD source identification is to assign a ‘degree of membership’ to each PRPD pattern, besides assigning a class label to it. Some of the classifier algorithms studied in this study, such as fuzzy classifiers, are not only able to show high classification accuracy rate, but they also calculate the ‘degree of membership’ of a sample to a class of data. This enables probabilistic interpretation of a new PRPD pattern that is being classified. The determination of the degree of membership for future PRPD samples allows safer decision making based on the risk associated with the different sources of PD in HV apparatus.

Let P be a Poisson algebra, E a vector space and π : E → P an epimorphism of vector spaces with V = Ker ( π ) . The global extension problem asks for the classification of all Poisson algebra structures that can be defined on E such that π : E → P becomes a morphism of Poisson algebras. From a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component. All such Poisson algebra structures on E are classified by an explicitly constructed classifying set G P H 2 ( P , V ) which is the coproduct of all non-abelian cohomological objects P H 2 ( P , ( V , ⋅ V , [ − , − ] V ) ) which are the classifying sets for all extensions of P by ( V , ⋅ V , [ − , − ] V ) . The second classical Poisson cohomology group H 2 ( P , V ) appears as the most elementary piece among all components of G P H 2 ( P , V ) . Several examples are provided in the case of metabelian Poisson algebras or co-flag Poisson algebras over P : the latter being Poisson algebras Q which admit a finite chain of epimorphisms of Poisson algebras P n : = Q ⟶ π n P n − 1 ⋯ P 1 ⟶ π 1 P 0 : = P such that dim ⁡ ( Ker ( π i ) ) = 1 , for all i = 1 , ⋯ , n . [ABSTRACT FROM AUTHOR]

For the Dirichlet problem [image omitted] with Ω ⊂ N, N ≥ 2, a bounded domain and p > 1, blow-up phenomena necessarily arise as λ → + ∞. In the present paper, we address the asymptotic description for pointwise blow-up, as it occurs when either the 'energy' or the Morse index is uniformly bounded. A posteriori, we obtain an equivalence between the two quantities in the form of a double-side bound with essentially optimal constants, a sort of improved Rozenblyum-Lieb-Cwikel inequality for the equation under exam. Moreover, we prove the nondegeneracy of any 'low energy' or Morse index 1 solution under a suitable condition on the potential. [ABSTRACT FROM AUTHOR]

As is well known, harmonic functions satisfy the mean value property, i.e. the average of such a function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a feature characterizing only balls, namely, is a set, for which all harmonic functions satisfy the mean value property, necessarily a ball? This question was investigated by several authors, including Bernard Epstein (1962), Bernard Epstein and Schiffer (1965), Myron Goldstein and Wellington (1971), who obtained a positive answer to this question under suitable additional assumptions. The problem was finally elegantly, completely and positively settled by Ülkü Kuran (1972), with an artful use of elementary techniques. This classical problem has been recently fleshed out by Giovanni Cupini, et al. (in press) who proved a quantitative stability result for the mean value formula, showing that a suitable "mean value gap" (measuring the normalized difference between the average of harmonic functions on a given set and their pointwise value) is bounded from below by the Lebesgue measure of the "gap" between the set and the ball (and, consequently, by the Fraenkel asymmetry of the set). That is, if a domain "almost" satisfies the mean value property for all harmonic functions, then that domain is "almost" a ball. The goal of this note is to investigate some nonlocal counterparts of these results. Some of our arguments rely on fractional potential theory, others on purely nonlocal properties, with no classical counterpart, such as the fact that "all functions are locally fractional harmonic up to a small error". [ABSTRACT FROM AUTHOR]