To prove Theorem 13 we say that a satisfying assignment σ is α-coreless if its coarsening ﬁxed point w(σ) has at least αn ∗-variables.
On the Solution-Space Geometry of Random Constraint Satisfaction Problems
Let X be the random variable equal to the number of α-coreless satisfying assignments in a random k-CNF formula Fk (n, rn).
On the Solution-Space Geometry of Random Constraint Satisfaction Problems
Pr[0 is α-coreless | 0 is satisfying] . 0 solution, and amounts to Observe that conditioning on “ 0 is satisfying ” is exactly the same as “planting ” the selecting the m = rn random clauses in our formula, uniformly and independently from amongst all clauses having at least one negative literal.
On the Solution-Space Geometry of Random Constraint Satisfaction Problems
To estimate Pr[0 is α-coreless | 0 is satisfying] we consider a random k-CNF formula with rn clauses chosen uniformly among those satisfying 0.
On the Solution-Space Geometry of Random Constraint Satisfaction Problems
Lauer et al. (1995) called the coreless systems “power-law” galaxies, as they exhibited central light distribution that resembled steep power-laws as the HST resolution limit was approached.
Cores and the Kinematics of Early-Type Galaxies
After the encounter, planets with cores are more likely to be retained by their host stars in contrast with previous studies which suggested that coreless planets are often ejected.
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
In GRL the code was adapted to simulate the effects of strong tides on giant, coreless planets after both single and multiple close-in passages.
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
We ﬁnd that while the addition of a core produces qualitatively different results than coreless models, there are no qualitative differences when the core mass is varied for the values investigated here.
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
The non-monotonic relationship between rp and the change in orbital energy is considerably more complex than the results presented in FRW or GRL, where a coreless giant planet was assumed.
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
Although the results of encounters with rp/rt (cid:38) 1.75 are in general agreement with the coreless models, the discrepancy is apparent for closer encounters.
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
Open squares show the data for coreless planets as presented in FRW, whereas the dashed line shows the results of GRL.
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
Core Mass and Survivability Planets with cores not only lose less mass, but also maintain their internal structures more effectively than their coreless counterparts after the disruption has occurred.
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
The n = 1 single-layered polytrope, which corresponds to the coreless gas giant planets, does not change its radius when losing mass adiabatically, resulting in a decrease of the average density.
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
As a result, GRL found that coreless planets are always destroyed after several passages even if the initial periastron is fairly distant (the lower limit is 2.7 rt ).
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
We compared our results to the adiabatic response of composite polytropes to mass loss, and ﬁnd that while coreless planets always expand in response to mass loss, planets with cores contract, allowing them to retain a fraction of their initial envelopes.
On the Survivability and Metamorphism of Tidally Disrupted Giant Planets: the Role of Dense Cores
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