2,192 research outputs found
Combinatorial problems in finite geometry and lacunary polynomials
We describe some combinatorial problems in finite projective planes and
indicate how R\'edei's theory of lacunary polynomials can be applied to them
On the spectrum of fluctuations of a liquid surface: From the molecular scale to the macroscopic scale
We show that to account for the full spectrum of surface fluctuations from
low scattering vector qd << 1 (classical capillary wave theory) to high qd > 1
(bulk-like fluctuations), one must take account of the interface's bending
rigidity at intermediate scattering vector qd = 1, where d is the molecular
diameter. A molecular model is presented to describe the bending correction to
the capillary wave model for short-ranged and long-ranged interactions between
molecules. We find that the bending rigidity is negative when the Gibbs
equimolar surface is used to define the location of the fluctuating interface
and that on approach to the critical point it vanishes proportionally to the
interfacial tension. Both features are in agreement with Monte Carlo
simulations of a phase-separated colloid-polymer system.Comment: 18 pages, 11 figures, accepted for publication in The Journal of
Chemical Physic
The existence of a bending rigidity for a hard sphere liquid near a curved hard wall: Helfrich or Hadwiger?
In the context of Rosenfeld's Fundamental Measure Theory, we show that the
bending rigidity is not equal to zero for a hard-sphere fluid in contact with a
curved hard wall. The implication is that the Hadwiger Theorem does not hold in
this case and the surface free energy is given by the Helfrich expansion
instead. The value obtained for the bending rigidity is (1) an order of
magnitude smaller than the bending constant associated with Gaussian curvature,
(2) changes sign as a function of the fluid volume fraction, (3) is independent
of the choice for the location of the hard wall.Comment: 19 pages, 5 figures, to appear in Physical Review
On the spectrum of fluctuations of a liquid surface: From the molecular scale to the macroscopic scale
We show that to account for the full spectrum of surface fluctuations from
low scattering vector qd 1
(bulk-like fluctuations), one must take account of the interface's bending
rigidity at intermediate scattering vector qd = 1, where d is the molecular
diameter. A molecular model is presented to describe the bending correction to
the capillary wave model for short-ranged and long-ranged interactions between
molecules. We find that the bending rigidity is negative when the Gibbs
equimolar surface is used to define the location of the fluctuating interface
and that on approach to the critical point it vanishes proportionally to the
interfacial tension. Both features are in agreement with Monte Carlo
simulations of a phase-separated colloid-polymer system.Comment: 18 pages, 11 figures, accepted for publication in The Journal of
Chemical Physic
On the number of k-dominating independent sets
We study the existence and the number of -dominating independent sets in
certain graph families. While the case namely the case of maximal
independent sets - which is originated from Erd\H{o}s and Moser - is widely
investigated, much less is known in general. In this paper we settle the
question for trees and prove that the maximum number of -dominating
independent sets in -vertex graphs is between and
if , moreover the maximum number of
-dominating independent sets in -vertex graphs is between
and . Graph constructions containing a large number of
-dominating independent sets are coming from product graphs, complete
bipartite graphs and with finite geometries. The product graph construction is
associated with the number of certain MDS codes.Comment: 13 page
Flat-containing and shift-blocking sets in
For non-negative integers , how small can a subset
be, given that for any there is a -flat passing through and
contained in ? Equivalently, how large can a subset be, given that for any there is a linear -subspace not
blocked non-trivially by the translate ? A number of lower and upper
bounds are obtained
Reaction kinetics in open reactors and serial transfers between closed reactors
Kinetic theory and thermodynamics of reaction networks are extended to the
out-of-equilibrium dynamics of continuous-flow stirred tank reactors (CSTR) and
serial transfers. On the basis of their stoichiometry matrix, the conservation
laws and the cycles of the network are determined for both dynamics. It is
shown that the CSTR and serial transfer dynamics are equivalent in the limit
where the time interval between the transfers tends to zero proportionally to
the ratio of the fractions of fresh to transferred solutions. These results are
illustrated with finite cross-catalytic reaction network and an infinite
reaction network describing mass exchange between polymers. Serial transfer
dynamics is typically used in molecular evolution experiments in the context of
research on the origins of life. The present study is shedding a new light on
the role played by serial transfer parameters in these experiments.Comment: 11 pages, 7 figure
A finite version of the Kakeya problem
Let be a set of lines of an affine space over a field and let be a
set of points with the property that every line of is incident with at
least points of . Let be the set of directions of the lines of
considered as points of the projective space at infinity. We give a geometric
construction of a set of lines , where contains an grid and
where has size , given a starting configuration in the plane.
We provide examples of such starting configurations for the reals and for
finite fields. Following Dvir's proof of the finite field Kakeya conjecture and
the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, we prove a
lower bound on the size of dependent on the ideal generated by the
homogeneous polynomials vanishing on . This bound is maximised as
plus smaller order terms, for , when contains
the points of a grid.Comment: A few minor changes to previous versio
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