Constructing and exploring wells of energy landscapes

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features ``wells of duration $T$ and altitude $y$''. We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) which have been developed during the last two decades; we therefore propose a minimal appendix exposing them at the end of this paper to bridge the possible gap.Comment: 28 pages, submitted to J. Math. Phys -

Calculating the hadronic vacuum polarization and leading hadronic contribution to the muon anomalous magnetic moment with improved staggered quarks

We present a lattice calculation of the hadronic vacuum polarization and the lowest-order hadronic contribution to the muon anomalous magnetic moment, a_\mu = (g-2)/2, using 2+1 flavors of improved staggered fermions. A precise fit to the low-q^2 region of the vacuum polarization is necessary to accurately extract the muon g-2. To obtain this fit, we use staggered chiral perturbation theory, including the vector particles as resonances, and compare these to polynomial fits to the lattice data. We discuss the fit results and associated systematic uncertainties, paying particular attention to the relative contributions of the pions and vector mesons. Using a single lattice spacing ensemble (a=0.086 fm), light quark masses as small as roughly one-tenth the strange quark mass, and volumes as large as (3.4 fm)^3, we find a_\mu^{HLO} = (713 \pm 15) \times 10^{-10} and (748 \pm 21) \times 10^{-10} where the error is statistical only and the two values correspond to linear and quadratic extrapolations in the light quark mass, respectively. Considering systematic uncertainties not eliminated in this study, we view this as agreement with the current best calculations using the experimental cross section for e^+e^- annihilation to hadrons, 692.4 (5.9) (2.4)\times 10^{-10}, and including the experimental decay rate of the tau lepton to hadrons, 711.0 (5.0) (0.8)(2.8)\times 10^{-10}. We discuss several ways to improve the current lattice calculation.Comment: 44 pages, 4 tables, 17 figures, more discussion on matching the chpt calculation to lattice calculation, typos corrected, refs added, version to appear in PR

2+1 flavor simulations of QCD with improved staggered quarks

The MILC collaboration has been performing realistic simulations of full QCD with 2+1 flavors of improved staggered quarks. Our simulations allow for controlled continuum and chiral extrapolations. I present results for the light pseudoscalar sector: masses and decay constants, quark masses and Gasser-Leutwyler low-energy constants. In addition I will present some results for heavy-light mesons, decay constants and semileptonic form factors, obtained in collaboration with the HPQCD and Fermilab lattice collaborations. Such calculations will help in the extraction of CKM matrix elements from experimental measurements.Comment: To appear in the proceedings of QNP06, IVth International Conference on Quarks and Nuclear Physics, Madrid, June 200

Leptonic decay constants f_Ds and f_D in three flavor lattice QCD

We determine the leptonic decay constants in three flavor unquenched lattice QCD. We use O(a^2)-improved staggered light quarks and O(a)-improved charm quarks in the Fermilab heavy quark formalism. Our preliminary results, based upon an analysis at a single lattice spacing, are f_Ds = 263(+5-9)(+/-24) MeV and f_D = 225(+11-13)(+/-21) MeV. In each case, the first reported error is statistical while the is the combined systematic uncertainty.Comment: Talk presented at Lattice2004(heavy), Fermilab, June 21-26, 2004. 3 pages, 2 figure

Fixed points of dynamic processes of set-valued F-contractions and application to functional equations

The article is a continuation of the investigations concerning F-contractions which have been recently introduced in [Wardowski in Fixed Point Theory Appl. 2012:94,2012]. The authors extend the concept of F-contractive mappings to the case of nonlinear F-contractions and prove a fixed point theorem via the dynamic processes. The paper includes a non-trivial example which shows the motivation for such investigations. The work is summarized by the application of the introduced nonlinear F-contractions to functional equations

K to pi and K to 0 in 2+1 Flavor Partially Quenched Chiral Perturbation Theory

We calculate results for K to pi and K to 0 matrix elements to next-to-leading order in 2+1 flavor partially quenched chiral perturbation theory. Results are presented for both the Delta I=1/2 and 3/2 channels, for chiral operators corresponding to current-current, gluonic penguin, and electroweak penguin 4-quark operators. These formulas are useful for studying the chiral behavior of currently available 2+1 flavor lattice QCD results, from which the low energy constants of the chiral effective theory can be determined. The low energy constants of these matrix elements are necessary for an understanding of the Delta I=1/2 rule, and for calculations of epsilon'/epsilon using current lattice QCD simulations.Comment: 43 pages, 2 figures, uses RevTeX, added and updated reference

Order of the Chiral and Continuum Limits in Staggered Chiral Perturbation Theory

Durr and Hoelbling recently observed that the continuum and chiral limits do not commute in the two dimensional, one flavor, Schwinger model with staggered fermions. I point out that such lack of commutativity can also be seen in four-dimensional staggered chiral perturbation theory (SChPT) in quenched or partially quenched quantities constructed to be particularly sensitive to the chiral limit. Although the physics involved in the SChPT examples is quite different from that in the Schwinger model, neither singularity seems to be connected to the trick of taking the nth root of the fermion determinant to remove unwanted degrees of freedom ("tastes"). Further, I argue that the singularities in SChPT are absent in most commonly-computed quantities in the unquenched (full) QCD case and do not imply any unexpected systematic errors in recent MILC calculations with staggered fermions.Comment: 14 pages, 1 figure. v3: Spurious symbol, introduced by conflicting tex macros, removed. Clarification of discussion in several place

Research of metal solidification in zero-g state

An experiment test apparatus that allows metal melting and resolidification in the three seconds available during free fall in a drop tower was built and tested in the tower. Droplets (approximately 0.05 cm) of pure nickel and 1090 steel were prepared in this fashion. The apparatus, including instrumentation, is described. As part of the instrumentation, a method for measuring temperature-time histories of the free floating metal droplets was developed. Finally, a metallurgical analysis of the specimens prepared in the apparatus is presented