Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation

We consider the problem of certifying an inequality of the form $f(x)\geq 0$, $\forall x\in K$, where $f$ is a multivariate transcendental function, and $K$ is a compact semialgebraic set. We introduce a certification method, combining semialgebraic optimization and max-plus approximation. We assume that $f$ is given by a syntaxic tree, the constituents of which involve semialgebraic operations as well as some transcendental functions like $\cos$, $\sin$, $\exp$, etc. We bound some of these constituents by suprema or infima of quadratic forms (max-plus approximation method, initially introduced in optimal control), leading to semialgebraic optimization problems which we solve by semidefinite relaxations. The max-plus approximation is iteratively refined and combined with branch and bound techniques to reduce the relaxation gap. Illustrative examples of application of this algorithm are provided, explaining how we solved tight inequalities issued from the Flyspeck project (one of the main purposes of which is to certify numerical inequalities used in the proof of the Kepler conjecture by Thomas Hales).Comment: 7 pages, 3 figures, 3 tables, Appears in the Proceedings of the European Control Conference ECC'13, July 17-19, 2013, Zurich, pp. 2244--2250, copyright EUCA 201

A Comparative Study of Different Blood Vessel Detection on Retinal Images

Detection of blood vessel plays an important stage in different medical areas, such as ophthalmology, oncology, neurosurgery, and laryngology. The significance of the vessel analysis was helped by the continuous overview in clinical studies of new medical technologies intended for improving the visualization of vessels. In this paper, several local segmentation techniques which include such as Vascular Tree Extraction, Tyler L. Coye and Line tracking, Kirsch’s Template and Fuzzy C Mean methods were studied. The main objective is to determine the best approaches in order to detect the blood vessel on the degraded retinal input image (DRIVE dataset). A few Image Quality Assessment (IQA) was obtained to prove the effectiveness of each detection methods. Overall, the result of sensitivity highest came from Kirsch Templates (96.928), while specificity from Fuzzy C means (77.573). However, in term of accuracy average, the Line Tracking method is more successful compared to the other methods

Versatile Preparation of Mesoporous Single-Layered Transition-Metal Sulfide/Carbon Composites for Enhanced Sodium Storage

Transition metal sulfides are promise electrochemical energy storage materials due to their abundant active sites, large inter-layer space and high theoretical capacities. Especially for sodium storage. However, the low conductivity and poor cycling stability at high current densities hampered their applications. Herein, we report a versatile dual-templates method to elaborate ordered mesoporous single layered MoS2 /carbon composite with high specific area, uniform pore size and large pore volume. The single layered MoS2 is confined in the carbon matrix. The mesopores between the composite nanorods provide fast electrolyte diffusion. The obtained nanocomposite shows a high sodium storage capability, excellent rate capacity, and very good cycling performance. A 310 mAh g-1 capacity can remains at 5.0 A g-1 after 2500 cycles. Furthermore, a SIB full cell composed the MoS2 /carbon composite anode and a Na3 V2 (PO4 )3 (NVP) cathode maintains a specific capacity of 330 mA h g-1 at 1.0 A g-1 during 100 cycles. The mechanism is investigated by in situ and ex situ characterizations as well as density functional theory (DFT) calculations. This article is protected by copyright. All rights reserved