5,204 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
Highly accurate solutions of the bifurcation structure of mixed-convection heat transfer using spectral method
No full textThis paper is concerned with producing highly accurate solution and bifurcation structure using the pseudo-spectral method for the two-dimensional pressure-driven flow through a horizontal duct of a square cross-section that is heated by a uniform flux in the axial direction with a uniform temperature on the periphery. Two approaches are presented. In one approach, the streamwise vorticity, streamwise momentum and energy equations are solved for the stream function, axial velocity, and temperature. In the second approach, the streamwise vorticity and a combination of the energy and momentum equations are solved for stream function and temperature only. While the second approach solves less number of equations than the first approach, a grid sensitivity analysis has shown no distinct advantage of one method over the other. The overall solution structure composed of two symmetric and four asymmetric branches in the range of Grashof number (Gr) of 0�2 � 106 for a Prandtl number (Pr) of 0.73 has been computed using the first approach. The computed structure is comparable to that found by Nandakumar and Weinitschke (1991) using a finite difference scheme for Grashof numbers in the range of 0�1�106. The stability properties of some solution branches; however, are different. In particular, the two-cell structure of the isolated symmetric branch that has been found to be unstable by the study of Nandakumar and Weinitschke is found to be stable by the current study.Wiley Online Librar
FIGURE 2. Rotala dhaneshiana A. Habit. B. A in Rotala dhaneshiana, a new species of Lythraceae from India
No full textFIGURE 2. Rotala dhaneshiana A. Habit. B. A portion of twig. C. Single flower. D. A portion of twig with mature fruits.Published as part of Ratheesh Narayanan, M. K., Sunil, C. N., Shaju, T., Nandakumar, M. K., Sivadasan, M. & Alfarhan, A. H., 2014, Rotala dhaneshiana, a new species of Lythraceae from India, pp. 227-232 in Phytotaxa 188 (4) on page 229, DOI: 10.11646/phytotaxa.188.4.5, http://zenodo.org/record/514745
Ergodic Theorems and Converses for PSPACE Functions
No full textWe initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x
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
Randomness and Effective Dimension of Continued Fractions
No full textRecently, 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
A Weyl Criterion for Finite-State Dimension and Applications
No full textFinite-state dimension, introduced early in this century as a finite-state
version of classical Hausdorff dimension, is a quantitative measure of the
lower asymptotic density of information in an infinite sequence over a finite
alphabet, as perceived by finite automata. Finite-state dimension is a robust
concept that now has equivalent formulations in terms of finite-state gambling,
lossless finite-state data compression, finite-state prediction, entropy rates,
and automatic Kolmogorov complexity. The Schnorr-Stimm dichotomy theorem gave
the first automata-theoretic characterization of normal sequences, which had
been studied in analytic number theory since Borel defined them. This theorem
implies that a sequence (or a real number having this sequence as its base-b
expansion) is normal if and only if it has finite-state dimension 1. One of the
most powerful classical tools for investigating normal numbers is the Weyl
criterion, which characterizes normality in terms of exponential sums. Such
sums are well studied objects with many connections to other aspects of
analytic number theory, and this has made use of Weyl criterion especially
fruitful. This raises the question whether Weyl criterion can be generalized
from finite-state dimension 1 to arbitrary finite-state dimensions, thereby
making it a quantitative tool for studying data compression, prediction, etc.
This paper does exactly this. We extend the Weyl criterion from a
characterization of sequences with finite-state dimension 1 to a criterion that
characterizes every finite-state dimension. This turns out not to be a routine
generalization of the original Weyl criterion. Even though exponential sums may
diverge for non-normal numbers, finite-state dimension can be characterized in
terms of the dimensions of the subsequence limits of the exponential sums. We
demonstrate the utility of our criterion though examples
FIGURE 1. Rotala dhaneshiana A. Habit. B. A in Rotala dhaneshiana, a new species of Lythraceae from India
No full textFIGURE 1. Rotala dhaneshiana A. Habit. B. A portion of flowering twig. C. Bract. D. Single flower. E. Floral tube cut-opened showing pistil, stamens, petals and epicalyx-lobes. F. Stamens—abaxial and adaxial views. G. Mature fruit enclosed in floral tube. H. Capsule. I. Seeds. Drawings by T. Shaju from live specimens.Published as part of Ratheesh Narayanan, M. K., Sunil, C. N., Shaju, T., Nandakumar, M. K., Sivadasan, M. & Alfarhan, A. H., 2014, Rotala dhaneshiana, a new species of Lythraceae from India, pp. 227-232 in Phytotaxa 188 (4) on page 228, DOI: 10.11646/phytotaxa.188.4.5, http://zenodo.org/record/514745
Nandakumar K., A finite element technique for multifluid incompressible flow using Eulerian grids
No full textAbstract The paper presents a finite element method for 3D incompressible fluid flows with capillary free boundaries. It uses a fixed Eulerian grid of 10 nodes (P 2 À P 1 ) tetrahedra and tracks the free boundary using a six nodes (P 2 ) triangular surface grid. In order to improve the mass conservation properties of the method, a local enrichment of the finite element basis in the elements intersected by the free boundaries is employed. In addition to the surface tracking, it also advects a smooth indicator function for an easy identification of the fluid properties in the different parts of the domain. The advective part of the Navier-Stokes equations is split and integrated with a characteristic method. The remaining generalized Stokes problem is resolved by means of an inexact outer-inner (Uzawa) iteration with a properly chosen preconditioner. The performance of this technique is evaluated on several problems involving droplets in viscous liquids
sj-docx-1-jhh-10.1177_15381927211052654 – Supplemental material for When It “Feels Like a Giant Living Room”: Implementing Peer Education at an Urban, Research-1 Hispanic Serving Institution
No full textSupplemental material, sj-docx-1-jhh-10.1177_15381927211052654 for When It “Feels Like a Giant Living Room”: Implementing Peer Education at an Urban, Research-1 Hispanic Serving Institution by Carla Amaro-Jiménez, Vandana Nandakumar, Holly Hungerford-Kresser, Oliver Patterson, Maria Martinez-Cosio and Jennifer Luken-Sutton in Journal of Hispanic Higher Education</p
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