182,774 research outputs found
Randomness and Effective Dimension of Continued Fractions
Recently, Scheerer [Adrian-Maria Scheerer, 2017] and Vandehey [Vandehey, 2016] showed that normality for continued fraction expansions and base-b expansions are incomparable notions. This shows that at some level, randomness for continued fractions and binary expansion are different statistical concepts. In contrast, we show that the continued fraction expansion of a real is computably random if and only if its binary expansion is computably random.
To quantify the degree to which a continued fraction fails to be effectively random, we define the effective Hausdorff dimension of individual continued fractions, explicitly constructing continued fractions with dimension 0 and 1
Effective Continued Fraction Dimension versus Effective Hausdorff Dimension of Reals
We establish that constructive continued fraction dimension originally
defined using -gales is robust, but surprisingly, that the effective
continued fraction dimension and effective (base-) Hausdorff dimension of
the same real can be unequal in general.
We initially provide an equivalent characterization of continued fraction
dimension using Kolmogorov complexity. In the process, we construct an optimal
lower semi-computable -gale for continued fractions. We also prove new
bounds on the Lebesgue measure of continued fraction cylinders, which may be of
independent interest.
We apply these bounds to reveal an unexpected behavior of continued fraction
dimension. It is known that feasible dimension is invariant with respect to
base conversion. We also know that Martin-L\"of randomness and computable
randomness are invariant not only with respect to base conversion, but also
with respect to the continued fraction representation. In contrast, for any , we prove the existence of a real whose effective
Hausdorff dimension is less than , but whose effective continued
fraction dimension is greater than or equal to . This phenomenon is
related to the ``non-faithfulness'' of certain families of covers, investigated
by Peres and Torbin and by Albeverio, Ivanenko, Lebid and Torbin.
We also establish that for any real, the constructive Hausdorff dimension is
at most its effective continued fraction dimension
Badly approximable numbers over imaginary quadratic fields
We recall the notion of nearest integer continued fractions over the
Euclidean imaginary quadratic fields and characterize the "badly
approximable" numbers, ( such that there is a with for all ), by boundedness of the partial quotients in the
continued fraction expansion of . Applying this algorithm to "tagged"
indefinite integral binary Hermitian forms demonstrates the existence of entire
circles in whose points are badly approximable over , with
effective constants.
By other methods (the Dani correspondence), we prove the existence of circles
of badly approximable numbers over any imaginary quadratic field, with loss of
effectivity. Among these badly approximable numbers are algebraic numbers of
every even degree over , which we characterize. All of the examples
we consider are associated with cocompact Fuchsian subgroups of the Bianchi
groups , where is the ring of integers in an
imaginary quadratic field.Comment: v3: Improved exposition (hopefully), especially in the second half of
the pape
Explicit irrationality measures for continued fractions
AbstractLet τ=[a0;a1,a2,…], a0∈N, an∈Z+, n∈Z+, be a simple continued fraction determined by an infinite integer sequence (an). We are interested in finding an effective irrationality measure as explicit as possible for the irrational number τ. In particular, our interest is focused on sequences (an) with an upper bound at most (ank), where a>1 and k>0. In addition to our main target, arithmetic of continued fractions, we shall pay special attention to studying the nature of the inverse function z(y) of y(z)=zlogz
Potential Antimicrobial Activity of Marine Sponge Neopetrosia exigua
Neopetrosia exigua has received great attention in natural product chemistry. The diversity of N. exigua constituents has been demonstrated by the continued discovery of novel bioactive metabolites such as antimicrobial metabolites. In this study, in order to localise the active component of N. exigua biomass according to the polarity, a sequential gradient partition with different solvents (n-hexane, carbon tetrachloride, dichloromethane, n-butanol, and water) was performed to obtain fractions containing metabolites distributed according to their polarity. The antimicrobial activities of N. exigua fractions were then evaluated using disc diffusion and microdilution methods (influence on the growth curve, Minimum Inhibitory Concentration (MIC) and Minimum Bactericidal Concentration (MBC)). The results showed that the active metabolites were present in n-hexane, CH2Cl2, n-BuOH, and water fractions. n-hexane, CH2Cl2, and n-BuOH fractions were the most effective fractions. Among microbes tested, Staphylococcus aureus was the most susceptible microbe evaluated. The obtained results are considered sufficient for further study to isolate the compounds represent the antimicrobial activity
Non-Schwarzschild black-hole metric in four dimensional higher derivative gravity: analytical approximation
Higher derivative extensions of Einstein gravity are important within the
string theory approach to gravity and as alternative and effective theories of
gravity. H. L\"u, A. Perkins, C. Pope, K. Stelle [Phys.Rev.Lett. 114 (2015),
171601] found a numerical solution describing a spherically symmetric
non-Schwarzschild asymptotically flat black hole in the Einstein gravity with
added higher derivative terms. Using the general and quickly convergent
parametrization in terms of the continued fractions, we represent this
numerical solution in the analytical form, which is accurate not only near the
event horizon or far from black hole, but in the whole space. Thereby, the
obtained analytical form of the metric allows one to study easily all the
further properties of the black hole, such as thermodynamics, Hawking
radiation, particle motion, accretion, perturbations, stability, quasinormal
spectrum, etc. Thus, the found analytical approximate representation can serve
in the same way as an exact solution.Comment: 9 pages, 4 figures, 1 ancillary Mathematica(R) noteboo
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