Hitting spheres on hyperbolic spaces

For a hyperbolic Brownian motion on the Poincar\'e half-plane $\mathbb{H}^2$, starting from a point of hyperbolic coordinates $z=(\eta, \alpha)$ inside a hyperbolic disc $U$ of radius $\bar{\eta}$, we obtain the probability of hitting the boundary $\partial U$ at the point $(\bar \eta,\bar \alpha)$. For $\bar{\eta} \to \infty$ we derive the asymptotic Cauchy hitting distribution on $\partial \mathbb{H}^2$ and for small values of $\eta$ and $\bar \eta$ we obtain the classical Euclidean Poisson kernel. The exit probabilities $\mathbb{P}_z\{T_{\eta_1}<T_{\eta_2}\}$ from a hyperbolic annulus in $\mathbb{H}^2$ of radii $\eta_1$ and $\eta_2$ are derived and the transient behaviour of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the three dimensional sphere. For the hyperbolic half-space $\mathbb{H}^n$ we obtain the Poisson kernel of a ball in terms of a series involving Gegenbauer polynomials and hypergeometric functions. For small domains in $\mathbb{H}^n$ we obtain the $n$-dimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the $n$-dimensional case

A reduction principle for the critical values of random spherical harmonics

We study here the random fluctuations in the number of critical points with values in an interval I⊂R for Gaussian spherical eigenfunctions fℓ, in the high energy regime where ℓ→∞. We show that these fluctuations are asymptotically equivalent to the centred L2-norm of fℓ times the integral of a (simple and fully explicit) function over the interval under consideration. We discuss also the relationships between these results and the asymptotic behaviour of other geometric functionals on the excursion sets of random spherical harmonics

On the correlation between critical points and critical values for random spherical harmonics

We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval I ⊂ ℝ. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random L2-norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Wigman (2012) and Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics

Complexity of energy barriers in mean-field glassy systems

We analyze the energy barriers that allow escapes from a given local minimum in a complex high-dimensional landscape. We perform this study by using the Kac-Rice method and computing the typical number of critical points of the energy function at a given distance from the minimum. We analyze their Hessian in terms of random matrix theory and show that for a certain regime of energies and distances critical points are index-one saddles, or transition states, and are associated to barriers. We find that the transition state of lowest energy, important for the activated dynamics at low temperature, is strictly below the "threshold" level above which saddles proliferate. We characterize how the quenched complexity of transition states, important for the activated processes at finite temperature, depends on the energy of the state, the energy of the initial minimum, and the distance between them. The overall picture gained from this study is expected to hold generically for mean-field models of the glass transition

Dynamical Instantons and Activated Processes in Mean-Field Glass Models

We focus on the energy landscape of a simple mean-field model of glasses and analyze activated barrier-crossing by combining the Kac-Rice method for high-dimensional Gaussian landscapes with dynamical field theory. In particular, we consider Langevin dynamics at low temperature in the energy landscape of the pure spherical $p$-spin model. We select as initial condition for the dynamics one of the many unstable index-1 saddles in the vicinity of a reference local minimum. We show that the associated dynamical mean-field equations admit two solutions: one corresponds to falling back to the original reference minimum, and the other to reaching a new minimum past the barrier. By varying the saddle we scan and characterize the properties of such minima reachable by activated barrier-crossing. Finally, using time-reversal transformations, we construct the two-point function dynamical instanton of the corresponding activated process.Comment: v3, conclusions extended and minor revision

Explosion Behavior of Ethanol-Ethyl Acetate/Air Mixtures

Alcohol-ester mixtures and, among them, ethanol-ethyl acetate mixtures are widely used as solvents in the packaging industry. For the safe use of such mixtures, it is essential to characterize their explosion behavior. Specifically, knowledge is required about maximum pressure and the maximum rate of pressure rise (i.e., the deflagration index), which are among the most important parameters for the assessment of process hazards and the safe design of process equipment. To this aim, in this work, closed-vessel explosion tests were carried out for an ethanol-ethyl acetate composition (mole fraction of ethanol in ethanol + ethyl acetate equal to 0.62) of interest to the packaging industry, varying the fuel/air equivalence ratio from 1.0 to 1.7. Tests were also extended to ethanol/air and ethyl acetate/air to quantify the effects of the possible interaction between the two fuels in the mixture. All tests started from 25°C and 1 bar. Experimental results show that, as the fuel equivalence ratio is increased, a transition occurs from a regime in which synergistic effects arise making the explosion behavior of ethanol-ethyl acetate more severe (i.e., making the rate of explosion pressure rise of ethanol-ethyl acetate higher) than both ethanol and ethyl acetate, to a regime in which, as a result of a completely different interaction between ethanol and ethyl acetate, the explosion behavior of their mixture is less severe than both the individual components. The maximum rate of pressure rise falls within an intermediate regime in which non-linear interaction effects substantially disappear and, thus, the value of deflagration index for the mixture can be obtained by averaging the values of the two fuels according to their molar proportions

Sparse Inpainting and Isotropy

Sparse inpainting techniques are gaining in popularity as a tool for cosmological data analysis, in particular for handling data which present masked regions and missing observations. We investigate here the relationship between sparse inpainting techniques using the spherical harmonic basis as a dictionary and the isotropy properties of cosmological maps, as for instance those arising from cosmic microwave background (CMB) experiments. In particular, we investigate the possibility that inpainted maps may exhibit anisotropies in the behaviour of higher-order angular polyspectra. We provide analytic computations and simulations of inpainted maps for a Gaussian isotropic model of CMB data, suggesting that the resulting angular trispectrum may exhibit small but non-negligible deviations from isotropy.Comment: 18 pages, 6 figures. v3: matches version published in JCAP; formatting changes and single typo correction only. Code available from http://zuserver2.star.ucl.ac.uk/~smf/code.htm

PENGARUH PENGOLAHAN TANAH DAN PUPUK N SERTA PUPUK KANDANG TERHADAP SERAPAN Ca, S DAN KUALITAS HASIL KACANG TANAH (Arachis hypogaea L.) PADA ALFISOLS

The purpose of this research is to know influence of tillage,N and organic fertilizer application to Ca, S absorption and groundnut (Arachis hypogaea L.) quality on Alfisols. The research was done in field laboratory of Agriculture Faculty of Sebelas Maret University, Sukosari, Jumantono, Karanganyar and soil laboratory from December 2006 until June 2007. This research is the factorial experiment with Completely Randomized Design (CRD) as environmental design. First factor was tillage, i.e. no tillage (T0) and tillage (T1). Second factor was goat organic fertilizer , i.e 10 ton/ha (D1) and 20 ton/ha (D2). Third factor was two levels of N fertilizer,i.e urea 25 kg/ha + ZA 54,76 kg/ha (N1) and urea 50 kg/ha (N2). F-test, Kruskal Wallis test, DMRT 5 %, and Stepwisse Regression test was used in the experiment this research. The result of this research indicate that no interaction tillage, the goat organic and N fertilizer to Ca, S absorption between yield of groundnut and it’s effect of protein grade is significant. Tillage, goat organic fertilizer 20 ton/ha and urea fertilizer 25 kg/ha + ZA 54,76 kg/ha give the highest of Ca and S absorption, they were 0,032 % and 0,0047 % respectively. Tillage, goat organic fertilizer 20 ton/ha and urea fertilizer 50 kg/ha give the highest of protein grade groundnut, it was 27,06 %. Keywords : Tillage, goat organic fertilizer, N fertilizer, Ca absorption, S absorption, Arachis hypogaea L

Travelling Randomly on the Poincar\'e Half-Plane with a Pythagorean Compass

A random motion on the Poincar\'e half-plane is studied. A particle runs on the geodesic lines changing direction at Poisson-paced times. The hyperbolic distance is analyzed, also in the case where returns to the starting point are admitted. The main results concern the mean hyperbolic distance (and also the conditional mean distance) in all versions of the motion envisaged. Also an analogous motion on orthogonal circles of the sphere is examined and the evolution of the mean distance from the starting point is investigated

Cascades of Particles Moving at Finite Velocity in Hyperbolic Spaces

A branching process of particles moving at finite velocity over the geodesic lines of the hyperbolic space (Poincar\'e half-plane and Poincar\'e disk) is examined. Each particle can split into two particles only once at Poisson paced times and deviates orthogonally when splitted. At time $t$, after $N(t)$ Poisson events, there are $N(t)+1$ particles moving along different geodesic lines. We are able to obtain the exact expression of the mean hyperbolic distance of the center of mass of the cloud of particles. We derive such mean hyperbolic distance from two different and independent ways and we study the behavior of the relevant expression as $t$ increases and for different values of the parameters $c$ (hyperbolic velocity of motion) and $\lambda$ (rate of reproduction). The mean hyperbolic distance of each moving particle is also examined and a useful representation, as the distance of a randomly stopped particle moving over the main geodesic line, is presented