Satisfiability Modulo Transcendental Functions via Incremental Linearization

In this paper we present an abstraction-refinement approach to Satisfiability Modulo the theory of transcendental functions, such as exponentiation and trigonometric functions. The transcendental functions are represented as uninterpreted in the abstract space, which is described in terms of the combined theory of linear arithmetic on the rationals with uninterpreted functions, and are incrementally axiomatized by means of upper- and lower-bounding piecewise-linear functions. Suitable numerical techniques are used to ensure that the abstractions of the transcendental functions are sound even in presence of irrationals. Our experimental evaluation on benchmarks from verification and mathematics demonstrates the potential of our approach, showing that it compares favorably with delta-satisfiability /interval propagation and methods based on theorem proving

A formal proof of the Kepler conjecture

This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project

Beta decay of the Tz=-2 nucleus 64Se and its descendants

International audience; The beta decay of the Tz=-2 nucleus 64Se has been studied in a fragmentation reaction at RIKEN-Nishina Center. 64Se is the heavies Tz=-2 nucleus that decays to bound states in the daughter nucleus and the heaviest case where the mirror reaction 64Zn(3He,t)64Ga on the Tz=+2 64Zn stable target exists and can be compared. Beta-delayed gamma and proton radiation is reported for the 64Se and 64As cases. New levels have been observed in 64As, 64Ge (N=Z), 63Ge and 63Ga. The associated T1/2 values have been obtained

The ß-decay of 71Kr: Precise measurement of the half-life

4 pags., 6 figs. --European Nuclear Physics Conference (EuNPC 2022), Section: P2 Nuclear Structure, Spectroscopy and DynamicsThe very proton-rich 71Kr isotope was produced through the in-flight fragmentation of 78Kr on a beryllium target at RIKEN ¿ Nishina Center in order to study its ß-decay properties. A stack of double-sided silicon strip detectors, called WAS3ABi, was used as the decay station, where the detection of ion implants, ß-decays and ß-delayed protons took place. Beta-delayed ¿-rays were measured using a system of 84 HPGe detectors, called EURICA, surrounding the decay station. The main goal of the present study was the precise measurement of the half-life of 71Kr, as in the literature there is an almost 10 ¿ difference between the most precise independent results. Implant¿ß time correlations, implant¿proton time correlations and implant¿ß¿¿ time correlations were all used to derive the half-life value, followed by a thorough investigation of systematic uncertainties for each method. As these values were found to be consistent, the weighted average t1/2 = 94.40+19ms is reported as a new half-life value in this work. Furthermore a total of 26 previously unreported ¿ following the ß-decay of 71Kr were also identified in the analysis.This work was carried out at the RIBF operated by RIKEN Nishina Center and CNS, University of Tokyo. We acknowledge the EUROBALL Owners Committee for the loan of germanium detectors and the PreSpec Collaboration for the readout electronics of the cluster detectors. This work was supported by the Spanish MICINN grants FPA2014-52823-C2-1-P, FPA2017-83946-C2-1-P (MCIU/AEI/FEDER); Ministerio de Ciencia e Innovacion grant PID2019-104714GB-C21; Centro de Excelencia Severo Ochoa del IFIC SEV - 2014 - 0398; Junta para la Ampliacion de E studios ´ Programme (CSIC JAEDoc contract) co-financed by FSE, by NKFIH (NN128072), the National Research, Development, and Innovation Fund of Hungary Project No. K 128729 the STFC (UK) through Grant No. ST/P005314/, the PROMETEO/2019/007 project and by the ÚNKP-20-5-DE-02 New National Excellence Program of the Ministry of Human Capacities of Hungary and by the JSPS KAKENHI of Japan (Grant No. 25247045). G. G. Kiss acknowledges support from the János Bolyai research fellowship of the Hungarian Academy of Sciences. A. A. acknowledges partial support of the JSPS Invitational Fellowships for Research in Japan (ID: L1955) P. S. acknowledges support from MCI/AEI/FEDER,UE (Spain) under grant PGC2018-093636-BI00. F. M. acknowledges support from ANID FONDECYT Regular Project 1221364 and ANID - Millennium Science Initiative Program - ICN2019-044. Supported by the ÚNKP-22-3 New National Excellence Program of the Ministry for Culture and Innovation from the source of the National Research, Development and Innovation Fund (Project No. ÚNKP-22-3-II-DE-132)

A FORMAL PROOF OF THE KEPLER CONJECTURE

Contains fulltext : 176365.pdf (publisher's version ) (Open Access

EXACT OPTIMIZATION VIA SUMS OF NONNEGATIVE CIRCUITS AND SUMS OF AM/GM EXPONENTIALS

International audienceWe provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm for polynomials lying in the interior of the SAGE cone. Each framework , inspired by previous contributions of Parrilo and Peyrl, is a rounding-projection procedure. For a polynomial lying in the interior of the SAGE cone, we prove that the decision algorithm terminates within a number of arithmetic operations, which is polynomial in the degree and number of terms of the input, and singly exponential in the number of variables. We also provide experimental comparisons regarding the implementation of the two optimization algorithms

A sparse version of Reznick's Positivstellensatz

19 pages, 2 tablesIf $f$ is a positive definite form, Reznick's Positivstellensatz [Mathematische Zeitschrift. 220 (1995), pp. 75--97] states that there exists $k\in\mathbf{N}$ such that ${\| x \|^{2k}_2}f$ is a sum of squares of polynomials. Assuming that $f$ can be written as a sum of forms $\sum_{l=1}^p f_l$, where each $f_l$ depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called {\em running intersection property}, we provide a sparse version of Reznick's Positivstellensatz. Namely, there exists $k \in \mathbf{N}$ such that $f=\sum_{l = 1}^p {{\sigma_l}/{H_l^{k}}}$, where $\sigma_l$ is a sum of squares of polynomials, $H_l$ is a uniform polynomial denominator, and both polynomials $\sigma_l,H_l$ involve the same variables as $f_l$, for each $l=1,\dots,p$. In other words, the sparsity pattern of $f$ is also reflected in this sparse version of Reznick's certificate of positivity. We next use this result to also obtain positivity certificates for (i) polynomials nonnegative on the whole space and (ii) polynomials nonnegative on a (possibly non-compact) basic semialgebraic set, assuming that the input data satisfy the running intersection property. Both are sparse versions of a positivity certificate due to Putinar and Vasilescu