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In this paper, the concepts of$L$ -concave structures, concave$L$ -interior operators and concave$L$ -neighborhood systems are introduced. It is shown that the category of$L$ -concave spaces and the category of concave$L$ -interior spaces are isomorphic, and they are both isomorphic to the category of concave$L$ -neighborhood systems whenever$L$is a completely distributive lattice. Also, it is proved that these categories are all isomorphic to the category of$L$ -convex spaces whenever$L$is a completely distributive lattice with an order-reversing involution operator.

Convergence structure and relation are useful tools in interpreting many mathematical structures such as topological spaces and convex spaces. The aim of this paper is to study convergence structures in the framework of L-concave spaces by using relations. Specifically, the notion of L-down—directed relations is introduced and some simple examples are presented. Based on this, notions of L-down—directed convergence relation spaces and L-concave down—directed convergence relations are introduced. It is proved that the category of L-concave internal relation spaces can be embedded into the category of L-down—directed convergence relation spaces as a reflective subcategory. In addition, the category of Lconcave down—directed convergence relation spaces is isomorphic to the category of L-concave internal relation spaces. In order to characterize L-down—directed convergence relation space and L-concave down—directed convergence relation space, notions of L-concave filters, L-filter convergence spaces and L-concave filter convergence spaces are introduced. It is showed that the category of L-down—directed convergence relation spaces is isomorphic to the category of L-filter convergence spaces. It also showed that the category of L-concave down—directed convergence relation spaces is isomorphic to the category of L-concave filter convergence spaces and the category of L-concave spaces.

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and s-concave densities on ℝ. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than n-2/5 when —1 < s < ∞ where s = 0 corresponds to the log-concave case. We also show that the MLE does not exist for the classes of s-concave densities with s <—1.

In the article, we prove that the function x → ( 1 − x ) p K ( x ) is logarithmically concave on (0, 1) if and only if p ≥ 7/32, the function x → K ( x ) / log ( 1 + 4 / 1 − x ) is convex on (0, 1) and the function x → d 2 d x 2 [ K ( x ) − log ( 1 + 4 1 − x ) ] is absolutely monotonic on (0, 1), where K ( x ) = ∫ 0 π / 2 ( 1 − x 2 sin 2 t ) − 1 / 2 d t is the complete elliptic integral of the first kind.

The present study uses OpenFOAM to simulate jet evolution during a cavitation bubble collapse above a concave wall. The generation mechanism and velocity characteristics of the needle jet are principally explored. Concurrently, the shock wave diffusion in the field and pressure fluctuation at the feature point are comprehensively analyzed. As a comparison, the regular jet cases near the concavity are also investigated, and the parameter range for producing needle jets is explored. The major findings are as follows: 1) When the non-dimensional distance between the bubble centroid and wall is less than 1.2, an extremely thin and high-speed needle jet is generated near the concavity. 2) The needle jet velocity is 10 times that of a regular jet. Combined with the effect of the shock wave and the needle jet, the pressure peak at the feature point can reach 40 times that of a regular jet. 3) The neck formed by the evolution of large curvature bubble walls under the influence of vortices is the key factor in the formation of the needle jet.

This work is devoted to the study of rates of convergence of the empirical measures \\mu_{n} = \\frac {1}{n} \\sum_{k=1}^n \\delta_{X_k}, n \\geq 1, over a sample (X_{k})_{k \\geq 1} of independent identically distributed real-valued random variables towards the common distribution \\mu in Kantorovich transport distances W_p. The focus is on finite range bounds on the expected Kantorovich distances \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) or \\big [ \\mathbb{E}(W_{p}^p(\\mu_{n},\\mu )) \\big ]^1/p in terms of moments and analytic conditions on the measure \\mu and its distribution function. The study describes a variety of rates, from the standard one \\frac {1}{\\sqrt n} to slower rates, and both lower and upper-bounds on \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

It is difficult to accurately identify and extract bodies of water and underwater vegetation from satellite images using conventional vegetation indices, as the strong absorption of water weakens the spectral feature of high near-infrared (NIR) reflected by underwater vegetation in shallow lakes. This study used the shallow Lake Ulansuhai in the semi-arid region of China as a research site, and proposes a new concave–convex decision function to detect submerged aquatic vegetation (SAV) and identify bodies of water using Gao Fen 1 (GF-1) multi-spectral satellite images with a resolution of 16 meters acquired in July and August 2015. At the same time, emergent vegetation, “Huangtai algae bloom”, and SAV were classified simultaneously by a decision tree method. Through investigation and verification by field samples, classification accuracy in July and August was 92.17% and 91.79%, respectively, demonstrating that GF-1 data with four-day short revisit period and high spatial resolution can meet the standards of accuracy required by aquatic vegetation extraction. The results indicated that the concave–convex decision function is superior to traditional classification methods in distinguishing water and SAV, thus significantly improving SAV classification accuracy. The concave–convex decision function can be applied to waters with SAV coverage greater than 40% above 0.3 m and SAV coverage 40% above 0.1 m under 1.5 m transparency, which can provide new methods for the accurate extraction of SAV in other regions.

Based on the traditional concave hexagonal honeycomb structure, three kinds of concave hexagonal honeycomb structures were compared. The relative densities of traditional concave hexagonal honeycomb structures and three other classes of concave hexagonal honeycomb structures were derived using the geometric structure. The impact critical velocity of the structures was derived by using the 1-D impact theory. The in-plane impact characteristics and deformation modes of three kinds of similar concave hexagonal honeycomb structures in the concave direction at low, medium, and high velocity were analyzed using the finite element software ABAQUS. The results showed that the honeycomb structure of the cells of the three types undergoes two stages: concave hexagons and parallel quadrilaterals, at low velocity. For this reason, there are two stress platforms in the process of strain. With the increase in the velocity, the joints and middle of some cells form a glue-linked structure due to inertia. No excessive parallelogram structure appears, resulting in the blurring or even disappearance of the second stress platform. Finally, effects of different structural parameters on the plateau stress and energy absorption of structures similar to concave hexagons were obtained during low impact. The results provide a powerful reference for the negative Poisson’s ratio honeycomb structure under multi-directional impact.

The purpose of this work is to present the quantum Hermite–Hadamard inequality through the Green function approach. While doing this, we deduce some novel quantum identities. Using these identities, we establish some new inequalities in this direction. We contemplate the possibility of expanding the method, outlined herein, to recast the proofs of some known inequalities in the literature.

In this study, scale-based runoff plots of concave grasslands were designed and simulated rainfall experiments were conducted to investigate their retention effectiveness for runoff volume and pollutant loads, and to analyze the influences of concave depths on runoff and pollution retention of grasslands. Results showed that mean time to runoff of concave grasslands was 88.5 minutes, which was 5.3 times than that of flat grassland. Average peak flow rate of concave grasslands was reduced by 36.2% compared with flat grassland. Concaved grasslands averagely retained 58.2% of stormwater runoff. Deeper concave depths significantly increased runoff detention and retention performance of grasslands. Total suspended solids (TSS) load reduction rates of concave grasslands were ranged from 50.8% to 97.3%. Total nitrogen (TN) load reduction rate was 49.8% for concave depth of 10 cm. Total phosphorus (TP) load reduction rates were 45.0% and 93.9% for grasslands with 5 cm and 10 cm concave depths, respectively. Pollution load reduction rates of TSS, TN and TP enhanced along with the increase in concave depths. The estimated minimum area ratios of upslope impervious surface to grasslands of 5 cm and 10 cm concave depths were approximately 1:1 under 20 mm rainfall events, and 38:1 under 5 mm rainfalls, respectively.