Topic
Concave function
About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.
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TL;DR: Li and Lv as mentioned in this paper considered self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting selfsimilarly are necessarily spheres.
Abstract: A recent article (Li and Lv, J Geom Anal 30:417–447, 2020) considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we consider self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting self-similarly are necessarily spheres.
3 citations
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01 Jan 1978TL;DR: In this article, a modification of Tui's algorithm is presented, which is slightly different from that of Zwart but which, of course, incorporates his basic observation, and it is indicated, for the modification, why the algorithm is convergent for E = 0, and (ii) degeneracy of the polyhedron presents virtually no difficulty.
Abstract: The Tui "cone-splitting" algomithm [2 ] fop minimlzin E a concave function on a convex polyhedron has been shown by Zwart [3] to be nonconvergent. Subsequently, Zwart [4] produced a modification of Tui's important idea and demonstrated, by a clever geometric amEument, that his modification would produce, in a finite number of steps, a point which is at least as good as some point in an E-neighborhood of an optimal point. The parameter ~ > 0 is cmucial to that demonstration and The parametem essentially represents a lower bound for how lap solution points of certain subproblems are f-corn the various hype~lanes that are generated during the course of the procedure. Zwamt has not proven convergence of his modification when the "geometric tolerance" pamameter g = 0 although he does point out [4] that his algorithm never failed to work in the case ~ = 0. In this paper we present a modification of Tui's algorithm which is slightly different from that of Zwart but which, of course, incorporates his basic observation. In particular, it is indicated, for the modification, why (i) the algorithm is convergent for E = 0, and (ii) degeneracy of the polyhedron presents virtually no difficulty. In Section 2 we state the basic ideas and present the alEorithm. Section 3 discusses the algorithm and its relationships and dlffe~ences with ~he modification of Zwart. Section 4 deals with convergence.
3 citations
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TL;DR: In this article, the authors show that the market impact of volume-weighted average price (VWAP) orders is a convex function of a trading rate, but most empirical estimates of transaction costs are concave functions.
Abstract: The market impact (MI) of volume-weighted average price (VWAP) orders is a convex function of a trading rate, but most empirical estimates of transaction costs are concave functions. How is this possible? We show that an isochronic (constant trading time) MI is slightly convex, and an isochoric (constant trading volume) MI is concave. We suggest a model that fits all trading regimes and guarantees no dynamic arbitrage.
3 citations
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TL;DR: In this article, a stability version of the Prekopa-Leindler inequality for log-concave functions on R n is established, and a stable version of this inequality is shown to hold for R n log n.
3 citations
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TL;DR: The work developed here takes the set membership approach as a starting point in order to obtain a container for ultimately pseudo-periodic functions representative of Discrete Event Dynamic Systems, and by approximating the exact system ensures to entirely include it in a guaranteed way.
Abstract: Based on the (min,+)-linear system theory, the work developed here takes the set membership approach as a starting point in order to obtain a container for ultimately pseudo-periodic functions representative of Discrete Event Dynamic Systems. Such a container, by approximating the exact system, ensures to entirely include it in a guaranteed way. To reach that point, the container introduced in this paper is given as an interval, the bounds of which are a convex function for the upper approximation and a concave function for the lower approximation. Thanks to the characteristics of the bounds, the aim is both to reduce data storage (that can be very high when exact functions are handled) and to reduce the algorithm complexity of the operations of sum, inf-convolution and subadditive closure. These operations are integrated into inclusion functions, the algorithms of which are of linear or quasi-linear complexity.
3 citations