Finding largest small polygons with GloptiPoly

A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices $n$. Many instances are already solved in the literature, namely for all odd $n$, and for $n=4, 6$ and 8. Thus, for even $n\geq 10$, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for $n=10$ and $n=12$. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic

Large Magellanic Cloud Microlensing Optical Depth with Imperfect Event Selection

I present a new analysis of the MACHO Project 5.7 year Large Magellanic Cloud (LMC) microlensing data set that incorporates the effects of contamination of the microlensing event sample by variable stars. Photometric monitoring of MACHO LMC microlensing event candidates by the EROS and OGLE groups has revealed that one of these events is likely to be a variable star, while additional data has confirmed that many of the other events are very likely to be microlensing. This additional data on the nature of the MACHO microlensing candidates is incorporated into a simple likelihood analysis to derive a probability distribution for the number of MACHO microlens candidates that are true microlensing events. This analysis shows that 10-12 of the 13 events that passed the MACHO selection criteria are likely to be microlensing events, with the other 1-3 being variable stars. This likelihood analysis is also used to show that the main conclusions of the MACHO LMC analysis are unchanged by the variable star contamination. The microlensing optical depth toward the LMC is = 1.0 +/- 0.3 * 10^{-7}. If this is due to microlensing by known stellar populations, plus an additional population of lens objects in the Galactic halo, then the new halo population would account for 16% of the mass of a standard Galactic halo. The MACHO detection exceeds the expected background of 2 events expected from ordinary stars in standard models of the Milky Way and LMC at the 99.98% confidence level. The background prediction is increased to 3 events if maximal disk models are assumed for both the MilkyWay and LMC, but this model fails to account for the full signal seen by MACHO at the 99.8% confidence level.Comment: 20 pages, 2 postscript figues, accepted by Ap

A Proper Motion Survey for White Dwarfs with the Wide Field Planetary Camera 2

We have performed a search for halo white dwarfs as high proper motion objects in a second epoch WFPC2 image of the Groth-Westphal strip. We identify 24 high proper motion objects with mu > 0.014 ''/yr. Five of these high proper motion objects are identified as strong white dwarf candidates on the basis of their position in a reduced proper motion diagram. We create a model of the Milky Way thin disk, thick disk and stellar halo and find that this sample of white dwarfs is clearly an excess above the < 2 detections expected from these known stellar populations. The origin of the excess signal is less clear. Possibly, the excess cannot be explained without invoking a fourth galactic component: a white dwarf dark halo. We present a statistical separation of our sample into the four components and estimate the corresponding local white dwarf densities using only the directly observable variables, V, V-I, and mu. For all Galactic models explored, our sample separates into about 3 disk white dwarfs and 2 halo white dwarfs. However, the further subdivision into the thin and thick disk and the stellar and dark halo, and the subsequent calculation of the local densities are sensitive to the input parameters of our model for each Galactic component. Using the lowest mean mass model for the dark halo we find a 7% white dwarf halo and six times the canonical value for the thin disk white dwarf density (at marginal statistical significance), but possible systematic errors due to uncertainty in the model parameters likely dominate these statistical error bars. The white dwarf halo can be reduced to around 1.5% of the halo dark matter by changing the initial mass function slightly. The local thin disk white dwarf density in our solution can be made consistent with the canonical value by assuming a larger thin disk scaleheight of 500 pc.Comment: revised version, accepted by ApJ, results unchanged, discussion expande

New approximations for the cone of copositive matrices and its dual

We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual), thus complementing previous inner (resp. outer) approximations for C (for the dual). In particular, both inner and outer approximations have a very simple interpretation. Finally, extension to K-copositivity and K-complete positivity for a closed convex cone K, is straightforward.Comment: 8

Photometric Confirmation of MACHO Large Magellanic Cloud Microlensing Events

We present previously unpublished photometry of three Large Magellanic Cloud (LMC) microlensing events and show that the new photometry confirms the microlensing interpretation of these events. These events were discovered by the MACHO Project alert system and were also recovered by the analysis of the 5.7 year MACHO data set. This new photometry provides a substantial increase in the signal-to-noise ratio over the previously published photometry and in all three cases, the gravitational microlensing interpretation of these events is strengthened. The new data consist of MACHO-Global Microlensing Alert Network (GMAN) follow-up images from the CTIO 0.9 telescope plus difference imaging photometry of the original MACHO data from the 1.3m "Great Melbourne" telescope at Mt. Stromlo. We also combine microlensing light curve fitting with photometry from high resolution HST images of the source stars to provide further confirmation of these events and to show that the microlensing interpretation of event MACHO-LMC-23 is questionable. Finally, we compare our results with the analysis of Belokurov, Evans & Le Du who have attempted to classify candidate microlensing events with a neural network method, and we find that their results are contradicted by the new data and more powerful light curve fitting analysis for each of the four events considered in this paper. The failure of the Belokurov, Evans & Le Du method is likely to be due to their use of a set of insensitive statistics to feed their neural networks.Comment: 29 pages with 8 included postscript figures, accepted by the Astrophysical Journa

Are there MACHOs in the Milky Way halo?

Microlensing searches aim to detect compact halo dark matter via its gravitational lensing effect on stars within the Large Magellanic Cloud. The most recent results have led to the claim that roughly one fifth of the galactic halo dark matter may be in the form of compact, solar-mass objects. We analyze this hypothesis by considering the goodness-of-fit of the best-fit halo dark matter solutions to the observational data. We show that the distribution of the durations of the observed microlensing events is narrower than that expected to result from a standard halo lens population at 90 to 95% confidence. Only when the fraction of expected background (non-halo) events is significantly increased does the discrepancy between the observed and theoretical event duration distributions disappear. This indicates that it is possible that most of the lenses responsible for the observed microlensing events are not located within the Milky Way halo.Comment: 5 pages, 2 figures, minor changes to discussion and additional references, version to appear in Astronomy and Astrophysic

Reionization Constraints on the Contribution of Primordial Compact Objects to Dark Matter

Many lines of evidence suggest that nonbaryonic dark matter constitutes roughly 30% of the critical closure density, but the composition of this dark matter is unknown. One class of candidates for the dark matter is compact objects formed in the early universe, with typical masses M between 0.1 and 1 solar masses to correspond to the mass scale of objects found with microlensing observing projects. Specific candidates of this type include black holes formed at the epoch of the QCD phase transition, quark stars, and boson stars. Here we show that accretion onto these objects produces substantial ionization in the early universe, with an optical depth to Thomson scattering out to z=1100 of approximately tau=2-4 [f_CO\epsilon_{-1}(M/Msun)]^{1/2} (H_0/65)^{-1}, where \epsilon_{-1} is the accretion efficiency \epsilon\equiv L/{\dot M}c^2 divided by 0.1 and f_CO is the fraction of matter in the compact objects. The current upper limit to the scattering optical depth, based on the anisotropy of the microwave background, is approximately 0.4. Therefore, if accretion onto these objects is relatively efficient, they cannot be the main component of nonbaryonic dark matter.Comment: 12 pages including one figure, uses aaspp4, submitted to Ap

The risk relationships between alcohol consumption, alcohol use disorder and alcohol use disorder mortality: A systematic review and meta-analysis.

Increasing levels of alcohol use are associated with a risk of developing an alcohol use disorder (AUD), which, in turn, is associated with considerable burden. Our aim was to estimate the risk relationships between alcohol consumption and AUD incidence and mortality. A systematic literature search was conducted, using Medline, Embase, PsycINFO and Web of Science for case-control or cohort studies published between 1 January 2000 and 8 July 2022. These were required to report alcohol consumption, AUD incidence and/or AUD mortality (including 100% alcohol-attributable deaths). The protocol was registered with PROSPERO (CRD42022343201). Dose-response and random-effects meta-analyses were used to determine the risk relationships between alcohol consumption and AUD incidence and mortality and mortality rates in AUD patients, respectively. Of the 5904 reports identified, seven and three studies from high-income countries and Brazil met the inclusion criteria for quantitative and qualitative syntheses, respectively. In addition, two primary US data sources were analyzed. Higher levels of alcohol consumption increased the risk of developing or dying from an AUD exponentially. At an average consumption of four standard drinks (assuming 10 g of pure alcohol/standard drink) per day, the risk of developing an AUD was increased sevenfold [relative risk (RR) = 7.14, 95% confidence interval (CI) = 5.13-9.93] and the risk of dying fourfold (RR = 3.94, 95% CI = 3.53-4.40) compared with current non-drinkers. The mortality rate in AUD patients was 3.13 (95% CI = 1.07-9.13) per 1000 person-years. There are exponential positive risk relationships between alcohol use and both alcohol use disorder incidence and mortality. Even at an average consumption of 20 g/day (about one large beer), the risk of developing an alcohol use disorder (AUD) is nearly threefold that of current non-drinkers and the risk of dying from an AUD is approximately double that of current non-drinkers

Existing opportunities to adapt the Rio Grande/Bravo Basin Water Resources Allocation Framework

The study of the Rio Grande/Bravo (RGB) Basin water allocation demonstrates how the United States (U.S.) and Mexico have consolidated a transboundary framework based on water sharing. However, the water supply no longer meets the ever-increasing demand for water or the expectations of different stakeholders. This paper explores opportunities for an enhanced management regime that will address past problems and better examine how to balance demands for a precious resource and environmental needs. Based on an overview of the RGB Basin context and the water allocation framework, as well as a discussion on stakeholders’ ability to achieve solutions, this paper explores three key questions: (1) Does the current binational water allocation framework meet current and future human and environmental needs? (2) How can the U.S.-Mexico water allocation framework be adapted to balance social and environmental water demands so it can support and preserve the RGB Basin ecosystem? (3) What are the main opportunities to be explored for expanding the U.S.-Mexico water resources allocation framework? The U.S.-Mexico water resources framework is subject to broad interpretation and may be adapted to the circumstances taking the fullest advantage of its flexibility. Policy recommendations highlight the existing flexibility of the binational framework, the potential to move forward with an ad hoc institutional arrangement, and the creation of political will to achieve change through stakeholders recommendations

The matricial relaxation of a linear matrix inequality

Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask: (Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X) PsD? (Q2) when do they have the same solution set? Such questions can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables x_j. With this relaxation, the domination questions (Q1) and (Q2) have elegant answers, indeed reduce to constructible semidefinite programs. Assume there is an X such that L_1(X) and L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^* L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are unitarily equivalent. Such algebraic certificates are typically called Positivstellensaetze and the above are examples of such for linear polynomials. The paper goes on to derive a cleaner and more powerful Putinar-type Positivstellensatz for polynomials positive on a bounded set of the form {X | L(X) PsD}. An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map from a subspace of matrices to a matrix algebra is "completely positive".Comment: v1: 34 pages, v2: 41 pages; supplementary material is available in the source file, or see http://srag.fmf.uni-lj.si