Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.Comment: 15 page

Extrapolating SMBH correlations down the mass scale: the case for IMBHs in globular clusters

Empirical evidence for both stellar mass black holes M_bh<10^2 M_sun) and supermassive black holes (SMBHs, M_bh>10^5 M_sun) is well established. Moreover, every galaxy with a bulge appears to host a SMBH, whose mass is correlated with the bulge mass, and even more strongly with the central stellar velocity dispersion sigma_c, the `M-sigma' relation. On the other hand, evidence for "intermediate-mass" black holes (IMBHs, with masses in the range 1^2 - 10^5 M_sun) is relatively sparse, with only a few mass measurements reported in globular clusters (GCs), dwarf galaxies and low-mass AGNs. We explore the question of whether globular clusters extend the M-sigma relationship for galaxies to lower black hole masses and find that available data for globular clusters are consistent with the extrapolation of this relationship. We use this extrapolated M-sigma relationship to predict the putative black hole masses of those globular clusters where existence of central IMBH was proposed. We discuss how globular clusters can be used as a constraint on theories making specific predictions for the low-mass end of the M-sigma relation.Comment: 14 pages, 3 figures, accepted for publication in Astrophysics and Space Science; fixed typos and a quote in Sec.

Photon interferometry and size of the hot zone in relativistic heavy ion collisions

The parameters obtained from the theoretical analysis of the single photon spectra observed by the WA98 collaboration at SPS energies have been used to evaluate the two photon correlation functions. The single photon spectra and the two photon correlations at RHIC energies have also been evaluated, taking into account the effects of the possible spectral change of hadrons in a thermal bath. We find that the ratio $R_{side}/R_{out} \sim 1$ for SPS and $R_{side}/R_{out} <1$ for RHIC energy.Comment: text changed, figures adde

Combined Decision Techniques for the Existential Theory of the Reals

Methods for deciding quantifier-free non-linear arithmetical conjectures over *** are crucial in the formal verification of many real-world systems and in formalised mathematics. While non-linear (rational function) arithmetic over *** is decidable, it is fundamentally infeasible: any general decision method for this problem is worst-case exponential in the dimension (number of variables) of the formula being analysed. This is unfortunate, as many practical applications of real algebraic decision methods require reasoning about high-dimensional conjectures. Despite their inherent infeasibility, a number of different decision methods have been developed, most of which have &quot;sweet spots&quot; --- e.g., types of problems for which they perform much better than they do in general. Such &quot;sweet spots&quot; can in many cases be heuristically combined to solve problems that are out of reach of the individual decision methods when used in isolation. RAHD (&quot;Real Algebra in High Dimensions&quot;) is a theorem prover that works to combine a collection of real algebraic decision methods in ways that exploit their respective &quot;sweet-spots.&quot; We discuss high-level mathematical and design aspects of RAHD and illustrate its use on a number of examples

X-Ray and Radio Observations of Bright GeV Sources

We present X-ray and radio studies of sources which are brightabove 1 GeV (F_{>1GeV} > 4e-8 ph/cm^2/s. Only 11 out of ~30 of these gamma-ray sources have been identified with lower energy counterparts: 5 blazars and 6 pulsars. Three of these pulsars are surrounded by radio pulsar wind nebulae (PWN), two of which are also seen as bright, extended X-ray synchrotron nebulae. The ASCA X-ray telescope has observed 28 of the bright GeV sources, revealing an excess of F_{2-10keV} > 10e-12 ergs/cm^2/s sources within the {\it EGRET} error contours of the unidentified sources. Although several supernova remnants are positionally coincident with these sources, we find no X-ray evidence of high energy particle production in SNR shell shocks consistent with the GeV positions. We also present initial results from follow on radio imaging studies of several fields containing unidentified sources. We have discovered new X-ray/radio nebulae in three of these fields which are strong candidates for PWN. These sources, along with a similar nebula in CTA 1 and the PWN around PSR B1853+01 in W44, are all positionally coincident with variable EGRET sources. This suggests a class of variable gamma-ray sources associated with synchrotron emitting regions powered by the winds of young pulsars.Comment: 18 pages, 26 figures, To appear in the proceedings of the workshop: "The Nature of the Unidentified Galactic Gamma-Ray Sources" held at INAOE, Mexico, October 2000, (A.Carraminana, O. Reiner and D. Thompson,

Imaging Observations of Quasi-Periodic Pulsatory Non-Thermal Emission in Ribbon Solar Flares

Using RHESSI and some auxiliary observations we examine possible connections between spatial and temporal morphology of the sources of non-thermal hard X-ray (HXR) emission which revealed minute quasi-periodic pulsations (QPPs) during the two-ribbon flares on 2003 May 29 and 2005 January 19. Microwave emission also reveals the same quasi-periodicity. The sources of non-thermal HXR emission are situated mainly inside the footpoints of the flare arcade loops observed by the TRACE and SOHO instruments. At least one of the sources moves systematically both during the QPP-phase and after it in each flare that allows to examine the sources velocities and the energy release rate via the process of magnetic reconnection. The sources move predominantly parallel to the magnetic inversion line or the appropriate flare ribbon during the QPP-phase whereas the movement slightly changes to more perpendicular regime after the QPPs. Each QPP is emitted from its own position. It is also seen that the velocity and the energy release rate don't correlate well with the flux of the HXR emission calculated from the sources. The sources of microwaves and thermal HXRs are situated near the apex of the loop arcade and are not stationary either. Almost all QPPs and some spikes of HXR emission during the post-QPP-phase reveal the soft-hard-soft spectral behavior indicating separate acts of electrons acceleration and injection, rather than modulation of emission flux by some kinds of magnetohydrodynamic (MHD) oscillations of coronal loops. In all likelihood, the flare scenarios based on the successively firing arcade loops are more preferable to interpret the observations, although we can not conclude exactly what mechanism forces these loops to flare up.Comment: 22 pages, 10 figure

Globular cluster luminosity function as distance indicator

Globular clusters are among the first objects used to establish the distance scale of the Universe. In the 1970-ies it has been recognized that the differential magnitude distribution of old globular clusters is very similar in different galaxies presenting a peak at M_V ~ -7.5. This peak magnitude of the so-called Globular Cluster Luminosity Function has been then established as a secondary distance indicator. The intrinsic accuracy of the method has been estimated to be of the order of ~0.2 mag, competitive with other distance determination methods. Lately the study of the Globular Cluster Systems has been used more as a tool for galaxy formation and evolution, and less so for distance determinations. Nevertheless, the collection of homogeneous and large datasets with the ACS on board HST presented new insights on the usefulness of the Globular Cluster Luminosity Function as distance indicator. I discuss here recent results based on observational and theoretical studies, which show that this distance indicator depends on complex physics of the cluster formation and dynamical evolution, and thus can have dependencies on Hubble type, environment and dynamical history of the host galaxy. While the corrections are often relatively small, they can amount to important systematic differences that make the Globular Cluster Luminosity Function a less accurate distance indicator with respect to some other standard candles.Comment: Accepted for publication in Astrophysics and Space Science. Review paper based on the invited talk at the conference "The Fundamental Cosmic Distance Scale: State of the Art and Gaia Perspective", Naples, May 2011. (13 pages, 8 figures

A workshop on ‘Dietary Sweetness—Is It an Issue?’

This report summarises a workshop convened by ILSI Europe on 3 and 4 April 2017 to discuss the issue of dietary sweetness. The objectives were to understand the roles of sweetness in the diet, establish whether exposure to sweetness affects diet quality and energy intake, and consider whether sweetness per se affects health. Although there may be evidence for tracking of intake of some sweet components of the diet through childhood, evidence for tracking of whole diet sweetness, or through other stages of maturity are lacking. The evidence to date does not support adverse effects of sweetness on diet quality or energy intake, except where sweet food choices increase intake of free sugars. There is some evidence for improvements in diet quality and reduced energy intake where sweetness without calories replaces sweetness with calories. There is a need to understand the physiological and metabolic relevance of sweet taste receptors on the tongue, in the gut and elsewhere in the body, as well as possible differentiation in the effects of sustained consumption of individual sweeteners. Despite a plethora of studies, there is no consistent evidence for an association of sweetness sensitivity/preference with obesity or type 2 diabetes. A multifaceted integrated approach, characterising nutritive and sensory aspects of the whole diet or dietary patterns, may be more valuable in providing contextual insight. The outcomes of the workshop could be used as a scientific basis to inform the expert community and create more useful dialogue among health care professionals