6,806 research outputs found
Knot points of typical continuous functions
It is well known that most continuous functions are nowhere differentiable.
Furthermore, in terms of Dini derivatives, most continuous functions are
nondifferentiable in the strongest possible sense except in a small set of
points. In this paper, we completely characterise families S of sets of points
for which most continuous functions have the property that such small set of
points belongs to S. The proof uses a topological zero-one law and the
Banach-Mazur game.Comment: 24 page
Differentiability of Lipschitz Functions in Lebesgue Null Sets
We show that if n>1 then there exists a Lebesgue null set in R^n containing a
point of differentiability of each Lipschitz function mapping from R^n to
R^(n-1); in combination with the work of others, this completes the
investigation of when the classical Rademacher theorem admits a converse.
Avoidance of sigma-porous sets, arising as irregular points of Lipschitz
functions, plays a key role in the proof.Comment: 33 pages. Corrected minor misprints and added more detail to the
proofs of Lemma 3.2 and Lemma 8.
Diabetic microvascular complications as simple indicators of risk for cardiovascular outcomes and heart failure
No abstract available
Cyclicity in families of circle maps
In this paper we will study families of circle maps of the form x↦x+2πr+af(x)(mod2π) and investigate how many periodic trajectories maps from this family can have for a ‘typical’ function f provided the parameter a is small
Mean field dynamics of superfluid-insulator phase transition in a gas of ultra cold atoms
A large scale dynamical simulation of the superfluid to Mott insulator
transition in the gas of ultra cold atoms placed in an optical lattice is
performed using the time dependent Gutzwiller mean field approach. This
approximate treatment allows us to take into account most of the details of the
recent experiment [Nature 415, 39 (2002)] where by changing the depth of the
lattice potential an adiabatic transition from a superfluid to a Mott insulator
state has been reported. Our simulations reveal a significant excitation of the
system with a transition to insulator in restricted regions of the trap.Comment: final version, correct Fig.7 (the published version contains wrong
fig.7 by mistake
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