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

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 -

Light hadrons with improved staggered quarks: approaching the continuum limit

We have extended our program of QCD simulations with an improved Kogut-Susskind quark action to a smaller lattice spacing, approximately 0.09 fm. Also, the simulations with a approximately 0.12 fm have been extended to smaller quark masses. In this paper we describe the new simulations and computations of the static quark potential and light hadron spectrum. These results give information about the remaining dependences on the lattice spacing. We examine the dependence of computed quantities on the spatial size of the lattice, on the numerical precision in the computations, and on the step size used in the numerical integrations. We examine the effects of autocorrelations in "simulation time" on the potential and spectrum. We see effects of decays, or coupling to two-meson states, in the 0++, 1+, and 0- meson propagators, and we make a preliminary mass computation for a radially excited 0- meson.Comment: 43 pages, 16 figure

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

The Kaon B-parameter in Mixed Action Chiral Perturbation Theory

We calculate the kaon B-parameter, B_K, in chiral perturbation theory for a partially quenched, mixed action theory with Ginsparg-Wilson valence quarks and staggered sea quarks. We find that the resulting expression is similar to that in the continuum, and in fact has only two additional unknown parameters. At one-loop order, taste-symmetry violations in the staggered sea sector only contribute to flavor-disconnected diagrams by generating an O(a^2) shift to the masses of taste-singlet sea-sea mesons. Lattice discretization errors also give rise to an analytic term which shifts the tree-level value of B_K by an amount of O(a^2). This term, however, is not strictly due to taste-breaking, and is therefore also present in the expression for B_K for pure G-W lattice fermions. We also present a numerical study of the mixed B_K expression in order to demonstrate that both discretization errors and finite volume effects are small and under control on the MILC improved staggered lattices.Comment: 29 pages, 4 figures; Expanded spurion discussion, other discussions clarified, version to appear in PR

Pion and kaon physics with improved staggered quarks

We compute pseudoscalar meson masses and decay constants using staggered quarks on lattices with three flavors of sea quarks and lattice spacings $\approx 0.12$ fm and $\approx 0.09$ fm. We fit partially quenched results to ``staggered chiral perturbation theory'' formulae, thereby taking into account the effects of taste-symmetry violations. Chiral logarithms are observed. From the fits we calculate $f_\pi$ and $f_K$, extract Gasser-Leutwyler parameters of the chiral Lagrangian, and (modulo rather large perturbative errors) find the light and strange quark masses.Comment: Lattice2003(spectrum); 3 pages, 1 eps figur

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

Lattice Gauge Fixing as Quenching and the Violation of Spectral Positivity

Lattice Landau gauge and other related lattice gauge fixing schemes are known to violate spectral positivity. The most direct sign of the violation is the rise of the effective mass as a function of distance. The origin of this phenomenon lies in the quenched character of the auxiliary field $g$ used to implement lattice gauge fixing, and is similar to quenched QCD in this respect. This is best studied using the PJLZ formalism, leading to a class of covariant gauges similar to the one-parameter class of covariant gauges commonly used in continuum gauge theories. Soluble models are used to illustrate the origin of the violation of spectral positivity. The phase diagram of the lattice theory, as a function of the gauge coupling $\beta$ and the gauge-fixing parameter $\alpha$, is similar to that of the unquenched theory, a Higgs model of a type first studied by Fradkin and Shenker. The gluon propagator is interpreted as yielding bound states in the confined phase, and a mixture of fundamental particles in the Higgs phase, but lattice simulation shows the two phases are connected. Gauge field propagators from the simulation of an SU(2) lattice gauge theory on a $20^4$ lattice are well described by a quenched mass-mixing model. The mass of the lightest state, which we interpret as the gluon mass, appears to be independent of $\alpha$ for sufficiently large $\alpha$.Comment: 28 pages, 14 figures, RevTeX