433 research outputs found
Universal quadratic forms, small norms and traces in families of number fields
We obtain good estimates on the ranks of universal quadratic forms over
Shanks' family of the simplest cubic fields and several other families of
totally real number fields. As the main tool we characterize all the
indecomposable integers in these fields and the elements of the codifferent of
small trace. We also determine the asymptotics of the number of principal
ideals of norm less than the square root of the discriminant.Comment: 20 page
Simulating Hard Rigid Bodies
Several physical systems in condensed matter have been modeled approximating
their constituent particles as hard objects. The hard spheres model has been
indeed one of the cornerstones of the computational and theoretical description
in condensed matter. The next level of description is to consider particles as
rigid objects of generic shape, which would enrich the possible phenomenology
enormously. This kind of modeling will prove to be interesting in all those
situations in which steric effects play a relevant role. These include biology,
soft matter, granular materials and molecular systems. With a view to
developing a general recipe for event-driven Molecular Dynamics simulations of
hard rigid bodies, two algorithms for calculating the distance between two
convex hard rigid bodies and the contact time of two colliding hard rigid
bodies solving a non-linear set of equations will be described. Building on
these two methods, an event-driven molecular dynamics algorithm for simulating
systems of convex hard rigid bodies will be developed and illustrated in
details. In order to optimize the collision detection between very elongated
hard rigid bodies, a novel nearest-neighbor list method based on an oriented
bounding box will be introduced and fully explained. Efficiency and performance
of the new algorithm proposed will be extensively tested for uniaxial hard
ellipsoids and superquadrics. Finally applications in various scientific fields
will be reported and discussed.Comment: 36 pages, 17 figure
Surface tension in the dilute Ising model. The Wulff construction
We study the surface tension and the phenomenon of phase coexistence for the
Ising model on \mathbbm{Z}^d () with ferromagnetic but random
couplings. We prove the convergence in probability (with respect to random
couplings) of surface tension and analyze its large deviations : upper
deviations occur at volume order while lower deviations occur at surface order.
We study the asymptotics of surface tension at low temperatures and relate the
quenched value of surface tension to maximal flows (first passage
times if ). For a broad class of distributions of the couplings we show
that the inequality -- where is the surface
tension under the averaged Gibbs measure -- is strict at low temperatures. We
also describe the phenomenon of phase coexistence in the dilute Ising model and
discuss some of the consequences of the media randomness. All of our results
hold as well for the dilute Potts and random cluster models
The distribution of polynomials over finite fields, with applications to the Gowers norms
n this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F^n -> F is poorly-distributed only if P is determined by the values of a few polynomials of lower degree, in which case we say that P has small rank.
We give several applications of this result, paying particular attention to consequences for the theory of the so-called Gowers norms. We establish an inverse result for the Gowers U^{d+1}-norm of functions of the form f(x)= e_F(P(x)), where P : F^n -> F is a polynomial of degree less than F, showing that this norm can only be large if f correlates with e_F(Q(x)) for some polynomial Q : F^n -> F of degree at most d.
The requirement deg(P) < |F| cannot be dropped entirely. Indeed, we show the above claim fails in characteristic 2 when d = 3 and deg(P)=4, showing that the quartic symmetric polynomial S_4 in F_2^n has large Gowers U^4-norm but does not correlate strongly with any cubic polynomial. This shows that the theory of Gowers norms in low characteristic is not as simple as previously supposed. This counterexample has also been discovered independently by Lovett, Meshulam, and Samorodnitsky.
We conclude with sundry other applications of our main result, including a recurrence result and a certain type of nullstellensatz
Ground state at high density
Weak limits as the density tends to infinity of classical ground states of
integrable pair potentials are shown to minimize the mean-field energy
functional. By studying the latter we derive global properties of high-density
ground state configurations in bounded domains and in infinite space. Our main
result is a theorem stating that for interactions having a strictly positive
Fourier transform the distribution of particles tends to be uniform as the
density increases, while high-density ground states show some pattern if the
Fourier transform is partially negative. The latter confirms the conclusion of
earlier studies by Vlasov (1945), Kirzhnits and Nepomnyashchii (1971), and
Likos et al. (2007). Other results include the proof that there is no Bravais
lattice among high-density ground states of interactions whose Fourier
transform has a negative part and the potential diverges or has a cusp at zero.
We also show that in the ground state configurations of the penetrable sphere
model particles are superposed on the sites of a close-packed lattice.Comment: Note adde
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