Steady shocks around black holes produced by sub-keplerian flows with negative energy

We discuss a special case of formation of axisymmetric shocks in the accretion flow of ideal gas onto a Schwarzschild black hole: when the total energy of the flow is negative. The result of our analysis enlarges the parameter space for which these steady shocks are exhibited in the accretion of gas rotating around relativistic stellar objects. Since keplerian disks have negative total energy, we guess that, in this energy range, the production of the shock phenomenon might be easier than in the case of positive energy. So our outcome reinforces the view that sub-keplerian flows of matter may significantly affect the physics of the high energy radiation emission from black hole candidates. We give a simple procedure to obtain analytically the position of the shocks. The comparison of the analytical results with the data of 1D and 2D axisymmetric numerical simulations confirms that the shocks form and are stable.Comment: 5 pages, 5 figures, accepted by MNRAS on 10 November 200

Primes and prime ideals in short intervals

We prove the analog of Cram\'er's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals.Comment: minor change to Proposition

Simulation of Thick Accretion Disks with Standing Shocks by Smoothed Particle Hydrodynamics

We present results of numerical simulation of inviscid thick accretion disks and wind flows around black holes. We use Smoothed Particle Hydrodynamics (SPH) technique for this purpose. Formation of thick disks are found to be preceded by shock waves travelling away from the centrifugal barrier. For a large range of the parameter space, the travelling shock settles at a distance close to the location obtained by a one-and-a-half dimensional model of inviscid accretion disks. Occasionally, it is observed that accretion processes are aided by the formation of oblique shock waves, particularly in the initial transient phase. The post-shock region (where infall velocity suddenly becomes very small) resembles that of the usual model of thick accretion disk discussed in the literature, though they have considerable turbulence. The flow subsequently becomes supersonic before falling into the black hole. In a large number of cases which we simulate, we find the formation of strong winds which are hot and subsonic when originated from the disk surface very close to the black hole but become supersonic within a few tens of the Schwarzschild radius of the blackhole. In the case of accretion of high angular momentum flow, very little amount of matter is accreted directly onto the black hole. Most of the matter is, however, first squeezed to a small volume close to the black hole, and subsequently expands and is expelled as a strong wind. It is quite possible that this expulsion of matter and the formation of cosmic radio jets is aided by the shock heating in the inner parts of the accretion disks.Comment: LaTeX, 16 pages, Astrophysical Journal (in press

In-plane structure and ordering at liquid sodium surfaces and interfaces from ab initio molecular dynamics

Atoms at liquid metal surfaces are known to form layers parallel to the surface. We analyze the two-dimensional arrangement of atoms within such layers at the surface of liquid sodium, using ab initio molecular dynamics (MD) simulations based on density functional theory. Nearest neighbor distributions at the surface indicate mostly 5-fold coordination, though there are noticeable fractions of 4-fold and 6-fold coordinated atoms. Bond angle distributions suggest a movement toward the angles corresponding to a six-fold coordinated hexagonal arrangement of the atoms as the temperature is decreased towards the solidification point. We rationalize these results with a distorted hexagonal order at the surface, showing a mixture of regions of five and six-fold coordination. The liquid surface results are compared with classical MD simulations of the liquid surface, with similar effects appearing, and with ab initio MD simulations for a model solid-liquid interface, where a pronounced shift towards hexagonal ordering is observed as the temperature is lowered

Explicit bounds for generators of the class group

Assuming Generalized Riemann's Hypothesis, Bach proved that the class group SICK of a number field K may be generated using prime ideals whose norm is bounded by 121og(2)delta(K), and by (4 + o(l)) log(2) delta(K) asymptotically, where delta(K) is the absolute value of the discriminant of K. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates SICK and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that SICK is generated by prime ideals whose norm is bounded by the minimum of 4.01 log(2) delta(K), 4(l + (2 pi e(gamma))N-_(K))(2) log(2) delta(k) and 4( log delta(k) + log log delta(K) - (gamma + log 2 pi)N-K + 1 + (N-K + 1) log(7log delta(K)/log delta(K))(2). Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedma's algorithms, confirming that it has size SIC (log delta(K) log log delta(K))(2). In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log(2) delta(K) except for 7 out of the 31292 fields we tested

Stable nitrile oxide dipolar cycloadditions in pure water

A systematic study on the behaviour of stable 2,4,6-trimethyl-3,5-dichlorobenzonitrile oxide versus a number of mono-, bi- and trisubstituted dipolarophiles in water was pursued obtaining simple as well as annulated isoxazolines. Reaction conditions changed with the dipolarophilic species, according to their solubility in water and the degree of substitution of the reactive carbon\u2013carbon multiple bond. The presence of sodium dodecylsulfate was also tested as potential micellar catalyst

Small values of signed harmonic sums

For every \u3c4 08R and every integer N, let mN(\u3c4) be the minimum of the distance of \u3c4 from the sums 11n=1Nsn/n, where s1,\u2026,sn 08 121,+1. We prove that mN(\u3c4

3D SPH Simulations of Shocks in Accretion Flows around black holes

We present the simulation of 3D time dependent flow of rotating ideal gas falling into a Schwarzschild black hole. It is shown that also in the 3D case steady shocks are formed in a wide range of parameters (initial angular momentum and thermal energy). We therefore highlight the stability of the phenomenon of shock formation in sub keplerian flows onto black holes, and reenforce the role of the shocks in the high luminosity emission from black hole candidates. The simulations have been performed using a parallelized code based on the Smoothed Particles Hydrodynamics method (SPH). We also discuss some properties of the shock problem that allow its use as a quantitative test of the accuracy of the used numerical method. This shows that the accuracy of SPH is acceptable although not excellent.Comment: 9 pages, 22 figure