Inspired by a result of Carathéodory Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911) 193–217, the Carathéodory ...number of a convexity space is defined as the smallest integer k such that for every subset U of the ground set V and every element u in the convex hull of U, there is a subset F of U with at most k elements such that u in the convex hull of F. We study the Carathéodory number for generalized interval convexities and for convexity spaces derived from finite graphs. We establish structural properties, bounds, and hardness results.
Abstract
The local convexity of a
PN
space is discussed using the idea of probability metric, and the constraint condition of convexity preservation of probabilistic normed space is given. Based on ...the probability background, it is proved that the probability norm and its linear combination are convex functions. When
α
≥ −1, the lower horizontal set is convex. When
α
≥ − ½, the lower level set can form a probabilistic normed space.
In the present paper we solve a problem posed by Tomasz Szostok who asked about the solutions f and F to the system of inequalitiesf(x+y2)≤F(y)−F(x)y−x≤f(x)+f(y)2. We show that f and F are the ...solutions to the above system of inequalities if and only if f is a continuous convex function and F is primitive function of f. This result can be interpreted as a regularity phenomenon-the solutions to the system of functional inequalities turn out to be regular without any additional assumptions.
Developing a suitable and monotonous health index (HI) that can be used to represent a whole degradation process is a key step for continuous machine health monitoring during its life cycle. It is ...expected that the potential HI is able to inform incipient fault moment and then track machine degradation trajectories effectively and monotonically. Previously, nearest neighbor convex hull classification (NNCHC) has been widely applied for fault classification. In this paper, a HI construction methodology for machine life cycle health monitoring based on NNCHC is proposed. Firstly, a normal convex hull is modeled based on normal vibration data to fully characterize machine health conditions. Afterward, two HIs are constructed based on
ℓ
1
norm and
ℓ
2
norm distances between the normal convex hull and test points. The superiority of the developed approach in this study lies in the flexible and efficient development of a HI for fault progress tracking. Moreover, the only usage of a normal dataset in the proposed methodology is closer to real application.
In this paper, the Extended Heinz type function is introduced, and it is proved that the function FE(a, b;v) is a mean. Further investigated the various properties of mean such as symmetricity, ...homogeneity, monotonicity and convexity. As an application, some inequalities have been established.
Abstract
In this article, we investigate the concepts of monotonicity, Schur-geometric convexity, Schur-harmonic convexity, and Schur-power convexity for the lower and upper limits of the integral ...mean, focusing on convex functions on coordinate axes. Furthermore, we introduce novel and fascinating inequalities for binary means as a practical application.
We connect high-dimensional subset selection and submodular maximization. Our results extend the work of Das and Kempe In ICML (2011) 1057–1064 from the setting of linear regression to arbitrary ...objective functions. For greedy feature selection, this connection allows us to obtain strong multiplicative performance bounds on several methods without statistical modeling assumptions. We also derive recovery guarantees of this form under standard assumptions. Our work shows that greedy algorithms perform within a constant factor from the best possible subset-selection solution for a broad class of general objective functions. Our methods allow a direct control over the number of obtained features as opposed to regularization parameters that only implicitly control sparsity. Our proof technique uses the concept of weak submodularity initially defined by Das and Kempe. We draw a connection between convex analysis and submodular set function theory which may be of independent interest for other statistical learning applications that have combinatorial structure.
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