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

Theory and experiment of the ESR of Co2+^{2+} in Zn2_2 % (OH)PO4_4 and Mg2_2(OH)AsO4_4

Experiments of Electron Spin Resonance (ESR) were performed on Co$$ substituting Zn$^{2+}$ or Mg$^{2+}$ in powder samples of Zn$_2$(OH)PO$_4$ and Mg$_2$(OH)AsO$_4$. The observed resonances are described with a theoretical model that considers the departures from the two perfect structures. It is shown that the resonance in the penta-coordinated complex is allowed, and the crystal fields that would describe the resonance of the Co$^{2+}$ in the two environments are calculated. The small intensity of the resonance in the penta-coordinated complex is explained assuming that this site is much less populated than the octahedral one; this assumption was verified by a molecular calculation of the energies of the two environments, with both Co and Zn as central ions in Zn$_2$(OH)PO$_4$.Comment: 43 pages, LaTex file, 6 figures, EPS. submitted to Journal of Physics Condens

Why are rhizobial symbiosis genes mobile?

Rhizobia are one of the most important and best studied groups of bacterial symbionts. They are defined by their ability to establish nitrogen-fixing intracellular infections within plant hosts. One surprising feature of this symbiosis is that the bacterial genes required for this complex trait are not fixed within the chromosome, but are encoded on mobile genetic elements (MGEs), namely plasmids or integrative and conjugative elements. Evidence suggests that many of these elements are actively mobilizing within rhizobial populations, suggesting that regular symbiosis gene transfer is part of the ecology of rhizobial symbionts. At first glance, this is counterintuitive. The symbiosis trait is highly complex, multipartite and tightly coevolved with the legume hosts, while transfer of genes can be costly and disrupt coadaptation between the chromosome and the symbiosis genes. However, horizontal gene transfer is a process driven not only by the interests of the host bacterium, but also, and perhaps predominantly, by the interests of the MGEs that facilitate it. Thus understanding the role of horizontal gene transfer in the rhizobium–legume symbiosis requires a ‘mobile genetic element's-eye view' on the ecology and evolution of this important symbiosis

A quantum mechanical relation connecting time, temperature, and cosmological constant of the universe: Gamow's relation revisited as a special case

Considering our expanding universe as made up of gravitationally interacting particles which describe particles of luminous matter and dark matter and dark energy which is described by a repulsive harmonic potential among the points in the flat 3-space, we derive a quantum mechanical relation connecting, temperature of the cosmic microwave background radiation, age, and cosmological constant of the universe. When the cosmological constant is zero, we get back the Gamow's relation with a much better coefficient. Otherwise, our theory predicts a value of the cosmological constant $2.0 10^{-56} {\rm {cm^{-2}}}$ when the present values of cosmic microwave background temperature of 2.728 K and age of the universe 14 billion years are taken as input.Comment: 4 pages, 1 figure, Study of the Universe from a condensed matter point of view, section III corrected with a single body potentia

A perspective on the landscape problem

I discuss the historical roots of the landscape problem and propose criteria for its successful resolution. This provides a perspective to evaluate the possibility to solve it in several of the speculative cosmological scenarios under study including eternal inflation, cosmological natural selection and cyclic cosmologies.Comment: Invited contribution for a special issue of Foundations of Physics titled: Forty Years Of String Theory: Reflecting On the Foundations. 31 pages, no figure

Towards a formal description of the collapse approach to the inflationary origin of the seeds of cosmic structure

Inflation plays a central role in our current understanding of the universe. According to the standard viewpoint, the homogeneous and isotropic mode of the inflaton field drove an early phase of nearly exponential expansion of the universe, while the quantum fluctuations (uncertainties) of the other modes gave rise to the seeds of cosmic structure. However, if we accept that the accelerated expansion led the universe into an essentially homogeneous and isotropic space-time, with the state of all the matter fields in their vacuum (except for the zero mode of the inflaton field), we can not escape the conclusion that the state of the universe as a whole would remain always homogeneous and isotropic. It was recently proposed in [A. Perez, H. Sahlmann and D. Sudarsky, "On the quantum origin of the seeds of cosmic structure," Class. Quant. Grav. 23, 2317-2354 (2006)] that a collapse (representing physics beyond the established paradigm, and presumably associated with a quantum-gravity effect a la Penrose) of the state function of the inflaton field might be the missing element, and thus would be responsible for the emergence of the primordial inhomogeneities. Here we will discuss a formalism that relies strongly on quantum field theory on curved space-times, and within which we can implement a detailed description of such a process. The picture that emerges clarifies many aspects of the problem, and is conceptually quite transparent. Nonetheless, we will find that the results lead us to argue that the resulting picture is not fully compatible with a purely geometric description of space-time.Comment: 53 pages, no figures. Revision to match the published versio

Daily carbohydrate accumulation in eight tall fescue cultivars

Eight cultivars of tall fescue (Loliumarundinaceum Schreb., S.J. Darbyshire = Festuca arundinacea Schreb.), Barcel, Kenhy, Kentucky-31, Missouri-96, Mozark, Stargrazer, C-1 (an experimental selection), and HiMag, were sampled at 2-h intervals during daylight on four cutting dates. Cultivars varied in concentrations of carbohydrate fractions but accumulation rates were not different. Daily mean total non-structural carbohydrate (TNC) concentrations for cutting dates in May, July, August and September declined from 239 to 231, 143 and 120 g TNC kg?1 adjusted dry weight (ADW) respectively. Concentrations of fructans were highest in July but sucrose, glucose and starch concentrations were highest in May. Sucrose was the largest contributor proportionately to TNC daily means across accessions in May (0·33), August (0·30) and September (0·38). Glucose composed an equivalent proportion of TNC in the August harvest. Starch concentration was highest in May at 53 g kg?1 ADW and lowest in August at 23 g kg?1 ADW. The TNC concentration increased by 22·4 (May), 16·8 (July), 21·0 (August) and 30·8 g kg?1 ADW (September) from dawn to dusk. Forage samples taken to estimate preference by ruminants or for TNC analyses should be cut and preserved within 1 h to control the diurnal variation of TNC proportionately within 0·05. Tall fescue should generally be cut between noon and sunset for TNC concentrations to be greater than the daily mean

Magnetothemopower study of quasi two-dimensional organic conductor α\alpha-(BEDT-TTF)2_2KHg(SCN)4_4

We have used a low-frequency magneto-thermopower (MTEP) method to probe the high magnetic field ground state behavior of $\alpha$-(BEDT-TTF)$_2$KHg(SCN)$_4$ along all three principal crystallographic axes at low temperatures. The thermopower tensor coefficients ($S_{xx}, S_{yx}$ and $S_{zz}$) have been measured to 30 T, beyond the anomalous low temperature, field-induced transition at 22.5 T. We find a significant anisotropy in the MTEP signal, and also observe large quantum oscillations associated with the de Haas - van Alphen effect. The anisotropy indicates that the ground state properties are clearly driven by mechanisms that occur along specific directions for the in-plane electronic structure. Both transverse and longitudinal magnetothermopower show asymptotic behavior in field, which can be explained in terms of magnetic breakdown of compensated closed orbits.Comment: 9 pages, 10 figure

Multispecies virial expansions

We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange–Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs

Quasiparticle spectra in the vicinity of a d-wave vortex

We discuss the evolution of the local quasiparticle spectral density and the related tunneling conductance measurable by the scanning tunneling microscope, as a function of distance r and angle \theta from the vortex core in a d_{x^2-y^2} superconductor. We consider the effects of electronic disorder and of a strongly anisotropic tunneling matrix element, and show that in real materials they will likely obscure the ~1/r asymptotic tail in the zero-bias tunneling conductance expected from the straightforward semiclassical analysis. We also give a prediction for the tunneling conductance anisotropy around the vortex core and establish a connection to the structure of the tunneling matrix element.Comment: 9 pages REVTeX + 5 PostScript figures. For related work and info visit http://www.pha.jhu.edu/~fran