3,874 research outputs found
Hierarchies of hyper-AFLs
For a full semi-AFL K, B(K) is defined as the family of languages generated by all K-extended basic macro grammars, while H(K) B(K) is the smallest full hyper-AFL containing K; a full basic-AFL is a full AFL K such that B(K) = K (hence every full basic-AFL is a full hyper-AFL). For any full semi-AFL K, K is a full basic-AFL if and only if B(K) is substitution closed if and only if H(K) is a full basic-AFL. If K is not a full basic-AFL, then the smallest full basic-AFL containing K is the union of an infinite hierarchy of full hyper-AFLs. If K is a full principal basic-AFL (such as INDEX, the family of indexed languages), then the largest full AFL properly contained in K is a full basic-AFL. There is a full basic-AFL lying properly in between the smallest full basic-AFL and the largest full basic-AFL in INDEX
Numerics and Fractals
Local iterated function systems are an important generalisation of the
standard (global) iterated function systems (IFSs). For a particular class of
mappings, their fixed points are the graphs of local fractal functions and
these functions themselves are known to be the fixed points of an associated
Read-Bajactarevi\'c operator. This paper establishes existence and properties
of local fractal functions and discusses how they are computed. In particular,
it is shown that piecewise polynomials are a special case of local fractal
functions. Finally, we develop a method to compute the components of a local
IFS from data or (partial differential) equations.Comment: version 2: minor updates and section 6.1 rewritten, arXiv admin note:
substantial text overlap with arXiv:1309.0243. text overlap with
arXiv:1309.024
What can one learn about Self-Organized Criticality from Dynamical Systems theory ?
We develop a dynamical system approach for the Zhang's model of
Self-Organized Criticality, for which the dynamics can be described either in
terms of Iterated Function Systems, or as a piecewise hyperbolic dynamical
system of skew-product type. In this setting we describe the SOC attractor, and
discuss its fractal structure. We show how the Lyapunov exponents, the
Hausdorff dimensions, and the system size are related to the probability
distribution of the avalanche size, via the Ledrappier-Young formula.Comment: 23 pages, 8 figures. to appear in Jour. of Stat. Phy
Blackwell-Optimal Strategies in Priority Mean-Payoff Games
We examine perfect information stochastic mean-payoff games - a class of
games containing as special sub-classes the usual mean-payoff games and parity
games. We show that deterministic memoryless strategies that are optimal for
discounted games with state-dependent discount factors close to 1 are optimal
for priority mean-payoff games establishing a strong link between these two
classes
The geometry of variations in Batalin-Vilkovisky formalism
This is a paper about geometry of (iterated) variations. We explain why no
sources of divergence are built into the Batalin-Vilkovisky (BV) Laplacian,
whence there is no need to postulate any ad hoc conventions such as
"" and "" within BV-approach to quantisation of
gauge systems. Remarkably, the geometry of iterated variations does not refer
at all to the construction of Dirac's -function as a limit of smooth
kernels. We illustrate the reasoning by re-deriving - but not just "formally
postulating" - the standard properties of BV-Laplacian and Schouten bracket and
by verifying their basic inter-relations (e.g., cohomology preservation by
gauge symmetries of the quantum master-equation).Comment: XXI International Conference on Integrable Systems and Quantum
Symmetries (ISQS21) 11-16 June 2013 at CVUT Prague, Czech Republic; 51 pages
(9 figures). - Main Example 2.4 on pp.34-36 retained from arXiv:1302.4388v1,
standard proofs in Appendix A amended and quoted from arXiv:1302.4388v1
(joint with S.Ringers). - Solution to Exercise 11.6 from IHES/M/12/13 by the
same autho
Probability and Fourier duality for affine iterated function systems
Let be a positive integer, and let be a finite measure on \br^d.
In this paper we ask when it is possible to find a subset in \br^d
such that the corresponding complex exponential functions indexed
by are orthogonal and total in . If this happens, we say
that is a spectral pair. This is a Fourier duality, and the
-variable for the -functions is one side in the duality, while the
points in is the other. Stated this way, the framework is too wide,
and we shall restrict attention to measures which come with an intrinsic
scaling symmetry built in and specified by a finite and prescribed system of
contractive affine mappings in \br^d; an affine iterated function system
(IFS). This setting allows us to generate candidates for spectral pairs in such
a way that the sets on both sides of the Fourier duality are generated by
suitably chosen affine IFSs. For a given affine setup, we spell out the
appropriate duality conditions that the two dual IFS-systems must have. Our
condition is stated in terms of certain complex Hadamard matrices. Our main
results give two ways of building higher dimensional spectral pairs from
combinatorial algebra and spectral theory applied to lower dimensional systems
Sixty Years of Fractal Projections
Sixty years ago, John Marstrand published a paper which, among other things,
relates the Hausdorff dimension of a plane set to the dimensions of its
orthogonal projections onto lines. For many years, the paper attracted very
little attention. However, over the past 30 years, Marstrand's projection
theorems have become the prototype for many results in fractal geometry with
numerous variants and applications and they continue to motivate leading
research.Comment: Submitted to proceedings of Fractals and Stochastics
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