45 research outputs found
Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary
We derive the boundary condition for a subdiffusive particle interacting with
a reactive boundary with finite reaction rate. Molecular crowding conditions,
that are found to cause subdiffusion of larger molecules in biological cells,
are shown to effect long-tailed distributions with identical exponent for both
the unbinding times from the boundary to the bulk and the rebinding times from
the bulk. This causes a weak ergodicity breaking: typically, an individual
particle either stays bound or remains in the bulk for very long times. We
discuss why this may be beneficial for in vivo gene regulation by DNA-binding
proteins, whose typical concentrations are nanomolarComment: 4 pages, 1 figure, REVTeX4, accepted to Phys Rev Lett, some typos
correcte
The Fermat-Torricelli problem in normed planes and spaces
We investigate the Fermat-Torricelli problem in d-dimensional real normed
spaces or Minkowski spaces, mainly for d=2. Our approach is to study the
Fermat-Torricelli locus in a geometric way. We present many new results, as
well as give an exposition of known results that are scattered in various
sources, with proofs for some of them. Together, these results can be
considered to be a minitheory of the Fermat-Torricelli problem in Minkowski
spaces and especially in Minkowski planes. This demonstrates that substantial
results about locational problems valid for all norms can be found using a
geometric approach
Covering convex bodies by cylinders and lattice points by flats
In connection with an unsolved problem of Bang (1951) we give a lower bound
for the sum of the base volumes of cylinders covering a d-dimensional convex
body in terms of the relevant basic measures of the given convex body. As an
application we establish lower bounds on the number of k-dimensional flats
(i.e. translates of k-dimensional linear subspaces) needed to cover all the
integer points of a given convex body in d-dimensional Euclidean space for
0<k<d