6,282 research outputs found
Fractal Dimension and Lower Bounds for Geometric Problems
We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension. In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension.
More specifically, we show that for any set of n points in d-dimensional Euclidean space, of fractal dimension delta in (1,d), for any epsilon>0 and c >= 1, any c-spanner must have treewidth at least Omega(n^{1-1/(delta - epsilon)} / c^{d-1}), matching the previous upper bound. The construction used to prove this lower bound on the treewidth of spanners, can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis. We provide two prototypical results of this type:
- For any delta in (1,d) and any epsilon >0, d-dimensional Euclidean TSP on n points with fractal dimension at most delta cannot be solved in time 2^{O(n^{1-1/(delta - epsilon)})}. The best-known upper bound is 2^{O(n^{1-1/delta} log n)}.
- For any delta in (1,d) and any epsilon >0, the problem of finding k-pairwise non-intersecting d-dimensional unit balls/axis parallel unit cubes with centers having fractal dimension at most delta cannot be solved in time f(k)n^{O (k^{1-1/(delta - epsilon)})} for any computable function f. The best-known upper bound is n^{O(k^{1-1/delta} log n)}. The above results nearly match previously known upper bounds from [Sidiropoulos & Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx & Sidiropoulos, SoCG 2014]
Fractal Weyl Law for Open Chaotic Maps
This contribution summarizes our work with M.Zworski on open quantum open
chaoticmaps (math-ph/0505034). For a simple chaotic scattering system (the open
quantum baker's map), we compute the "long-living resonances" in the
semiclassical r\'{e}gime, and show that they satisfy a fractal Weyl law. We can
prove this fractal law in the case of a modified model.Comment: Contribution to the Proceedings of the conference QMath9,
Mathematical Physics of Quantum Mechanics, September 12th-16th 2004, Giens,
Franc
Spatially independent martingales, intersections, and applications
We define a class of random measures, spatially independent martingales,
which we view as a natural generalisation of the canonical random discrete set,
and which includes as special cases many variants of fractal percolation and
Poissonian cut-outs. We pair the random measures with deterministic families of
parametrised measures , and show that under some natural
checkable conditions, a.s. the total measure of the intersections is H\"older
continuous as a function of . This continuity phenomenon turns out to
underpin a large amount of geometric information about these measures, allowing
us to unify and substantially generalize a large number of existing results on
the geometry of random Cantor sets and measures, as well as obtaining many new
ones. Among other things, for large classes of random fractals we establish (a)
very strong versions of the Marstrand-Mattila projection and slicing results,
as well as dimension conservation, (b) slicing results with respect to
algebraic curves and self-similar sets, (c) smoothness of convolutions of
measures, including self-convolutions, and nonempty interior for sumsets, (d)
rapid Fourier decay. Among other applications, we obtain an answer to a
question of I. {\L}aba in connection to the restriction problem for fractal
measures.Comment: 96 pages, 5 figures. v4: The definition of the metric changed in
Section 8. Polishing notation and other small changes. All main results
unchange
Standard Model in multiscale theories and observational constraints
We construct and analyze the Standard Model of electroweak and strong
interactions in multiscale spacetimes with (i) weighted derivatives and (ii)
-derivatives. Both theories can be formulated in two different frames,
called fractional and integer picture. By definition, the fractional picture is
where physical predictions should be made. (i) In the theory with weighted
derivatives, it is shown that gauge invariance and the requirement of having
constant masses in all reference frames make the Standard Model in the integer
picture indistinguishable from the ordinary one. Experiments involving only
weak and strong forces are insensitive to a change of spacetime dimensionality
also in the fractional picture, and only the electromagnetic and gravitational
sectors can break the degeneracy. For the simplest multiscale measures with
only one characteristic time, length and energy scale , and
, we compute the Lamb shift in the hydrogen atom and constrain the
multiscale correction to the ordinary result, getting the absolute upper bound
. For the natural choice of the
fractional exponent in the measure, this bound is strengthened to
, corresponding to and
. Stronger bounds are obtained from the measurement of the
fine-structure constant. (ii) In the theory with -derivatives, considering
the muon decay rate and the Lamb shift in light atoms, we obtain the
independent absolute upper bounds and
. For , the Lamb shift alone yields
.Comment: 25 pages. v2: authors' metadata corrected; v3: references added, new
material added including a comparison with varying-couplings and effective
field theories, a section on predictivity and falsifiability of multiscale
theories, a discussion on classical CPT, expanded conclusions, and new QED
constraints from the fine-structure constant; v3: minor typos corrected to
match the published versio
Fractal homogenization of multiscale interface problems
Inspired by continuum mechanical contact problems with geological fault
networks, we consider elliptic second order differential equations with jump
conditions on a sequence of multiscale networks of interfaces with a finite
number of non-separating scales. Our aim is to derive and analyze a description
of the asymptotic limit of infinitely many scales in order to quantify the
effect of resolving the network only up to some finite number of interfaces and
to consider all further effects as homogeneous. As classical homogenization
techniques are not suited for this kind of geometrical setting, we suggest a
new concept, called fractal homogenization, to derive and analyze an asymptotic
limit problem from a corresponding sequence of finite-scale interface problems.
We provide an intuitive characterization of the corresponding fractal solution
space in terms of generalized jumps and gradients together with continuous
embeddings into L2 and Hs, s<1/2. We show existence and uniqueness of the
solution of the asymptotic limit problem and exponential convergence of the
approximating finite-scale solutions. Computational experiments involving a
related numerical homogenization technique illustrate our theoretical findings
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