3,404 research outputs found
Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals
No full textWe establish that constructive continued fraction dimension originally defined using s-gales [Nandakumar and Vishnoi, 2022] is robust, but surprisingly, that the effective continued fraction dimension and effective (base-b) 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 s-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 [Hitchcock and Mayordomo, 2013]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [Nandakumar and Vishnoi, 2022]. In contrast, for any 0 < ε < 0.5, we prove the existence of a real whose effective Hausdorff dimension is less than ε, but whose effective continued fraction dimension is greater than or equal to 0.5. This phenomenon is related to the "non-faithfulness" of certain families of covers, investigated by Peres and Torbin [Peres and Torbin] and by Albeverio, Ivanenko, Lebid and Torbin [Albeverio et al., 2020].
We also establish that for any real, the constructive Hausdorff dimension is at most its effective continued fraction dimension
Point-To-Set Principle and Constructive Dimension Faithfulness
No full textHausdorff Φ-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce a constructive analogue of Φ-dimension using the notion of constructive Φ-s-supergales. We prove a Point-to-Set Principle for Φ-dimension, through which we get Point-to-Set Principles for Hausdorff dimension, continued-fraction dimension and dimension of Cantor coverings as special cases. We also provide a Kolmogorov complexity characterization of constructive Φ-dimension.
A class of covering sets Φ is said to be "faithful" to Hausdorff dimension if the Φ-dimension and Hausdorff dimension coincide for every set. Similarly, Φ is said to be "faithful" to constructive dimension if the constructive Φ-dimension and constructive dimension coincide for every set. Using the Point-to-Set Principle for Cantor coverings and a new technique for the construction of sequences satisfying a certain Kolmogorov complexity condition, we show that the notions of "faithfulness" of Cantor coverings at the Hausdorff and constructive levels are equivalent.
We adapt the result by Albeverio, Ivanenko, Lebid, and Torbin [Albeverio et al., 2020] to derive the necessary and sufficient conditions for the constructive dimension faithfulness of the coverings generated by the Cantor series expansion, based on the terms of the expansion
Real Numbers Equally Compressible in Every Base
No full textThis work solves an open question in finite-state compressibility posed by Lutz and Mayordomo [Lutz and Mayordomo, 2021] about compressibility of real numbers in different bases.
Finite-state compressibility, or equivalently, finite-state dimension, quantifies the asymptotic lower density of information in an infinite sequence.
Absolutely normal numbers, being finite-state incompressible in every base of expansion, are precisely those numbers which have finite-state dimension equal to 1 in every base. At the other extreme, for example, every rational number has finite-state dimension equal to 0 in every base.
Generalizing this, Lutz and Mayordomo in [Lutz and Mayordomo, 2021] (see also Lutz [Lutz, 2012]) posed the question: are there numbers which have absolute positive finite-state dimension strictly between 0 and 1 - equivalently, is there a real number ξ and a compressibility ratio s ∈ (0,1) such that for every base b, the compressibility ratio of the base-b expansion of ξ is precisely s? It is conceivable that there is no such number. Indeed, some works explore "zero-one" laws for other feasible dimensions [Fortnow et al., 2011] - i.e. sequences with certain properties either have feasible dimension 0 or 1, taking no value strictly in between.
However, we answer the question of Lutz and Mayordomo affirmatively by proving a more general result. We show that given any sequence of rational numbers ⟨q_b⟩_{b=2}^∞, we can explicitly construct a single number ξ such that for any base b, the finite-state dimension/compression ratio of ξ in base-b is q_b. As a special case, this result implies the existence of absolutely dimensioned numbers for any given rational dimension between 0 and 1, as posed by Lutz and Mayordomo.
In our construction, we combine ideas from Wolfgang Schmidt’s construction of absolutely normal numbers from [Schmidt, 1961], results regarding low discrepancy sequences and several new estimates related to exponential sums
1/f noise in titanium doped aluminum thin film deposited by electron beam evaporation method and its dependence on structural variation with temperature
No full textDeepsea prawns
No full textFrom the beginning of this century, large varieties of prawns belonging to the families
Penaeidae, Pasiphaeidae, Oplophoridae and Pandalidae have been reported from
the west and east coast of India, particularly from southern regions beyond the
continental shelf (Alcock 1901, 1906; de Man 1911, 1920; Calman, 1939; John and
Kurian, 1959; Kurian 1964; George, 1966; George and Rao 1966; Suseelan and
Mohamed, 1968 and Suseelan 1974, 1989). However, existence of some of these
species in commercially exploitable concentrations in these deeper waters has been
brought to light only recently. Silas (1969) and Mohamed and Suseelan (1973)
gave general accounts on the distribution and relative abundance of common species
of prawns of the shelf-edge and upper continental slope of the southwest coast.
Detailed review on deep-sea fisheries of India has been given recently by Rao
(2009). First account on the deep-sea crustacean fishery carried out by commercial
prawn trawlers along Kerala coast were given by Rajan et al. (2001) and Nandakumar
et al. (2001)
Observations of Bºs→ψ(2S)η and Bº(s)→ψ(2S)π+π- decays
No full textFirst observations of the B0s
→ψ(2S)η, B0 →ψ(2S)π
+
π
− and B0s
→ψ(2S)π
+
π
− decays are made
using a dataset corresponding to an integrated luminosity of 1.0 fb−1 collected by the LHCb experiment in
proton–proton collisions at a centre-of-mass energy of
√
s = 7 TeV. The ratios of the branching fractions
of each of the ψ(2S) modes with respect to the corresponding J/ψ decays are
B(B0s
→ψ(2S)η)
÷
B(B0s
→J/ψη)
= 0.83± 0.14 (stat)±0.12 (syst) ±0.02 (B),
;
B(B0→ψ(2S)π
+
π
−
)
÷
B(B0→J/ψπ
+
π
−
)
= 0.56± 0.07 (stat)±0.05 (syst)± 0.01 (B),
;
B(B0s
→ψ(2S)π
+
π
−
)
÷
B(B0s
→J/ψπ
+
π
−
)
= 0.34± 0.04 (stat)±0.03 (syst)± 0.01 (B),
where the third uncertainty corresponds to the uncertainties of the dilepton branching fractions of the J/ψ
and ψ(2S) meson decays
Prompt charm production in pp collisions at √<span style="text-decoration:overline">s</span>=7 TeV
No full textCharm production at the LHC in pp collisions at s√=7 TeV is studied with the LHCb detector. The decays D0→K−π+, D+→K−π+π+, D⁎+→D0(K−π+)π+, D+s→ϕ(K−K+)π+, Λ+c→pK−π+, and their charge conjugates are analysed in a data set corresponding to an integrated luminosity of 15 nb−1. Differential cross-sections dσ/dpT are measured for prompt production of the five charmed hadron species in bins of transverse momentum and rapidity in the region 0<pT<8 GeV/c and 2.0<y<4.5. Theoretical predictions are compared to the measured differential cross-sections. The integrated cross-sections of the charm hadrons are computed in the above pT-y range, and their ratios are reported. A combination of the five integrated cross-section measurements gives
σ(cc¯)pT<8 GeV/c,2.0<y<4.5=1419±12(stat)±116(syst)±65(frag) μb,
where the uncertainties are statistical, systematic, and due to the fragmentation functions
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