Fraction Representation Research Articles

Volume preservation improvement for interface reconstruction hexahedral methods

We propose a new post-processing procedure for automatically adjusting node locations of an all-hex mesh to better match the volume of a reference geometry. Hexahedral meshes generated via an overlay grid procedure, where a precise reference geometry representation is unknown or is impractical to use, do not provide for precise volumetric preservation. A discrete volume fraction representation of the reference geometry MI on an overlay grid is compared with a volume fraction representation of a 3D finite element mesh MO. This work proposes a procedure that uses the localized discrepancy between MI and MO to drive node relocation operations to more accurately match a reference geometry. We demonstrate this procedure on a wide range of hexahedral meshes generated with the Sculpt code and show improved volumetric preservation while still maintaining acceptable mesh quality.

Dynamics of the transverse Ising model with next-nearest-neighbor interactions.

We study the effects of next-nearest-neighbor (NNN) interactions on the dynamics of the one-dimensional spin-1/2 transverse Ising model in the high-temperature limit. We use exact diagonalization to obtain the time-dependent transverse correlation function and the corresponding spectral density for a tagged spin. Our results for chains of 13 spins with periodic boundary conditions produce results which are valid in the infinite-size limit. In general we find that the NNN coupling produces slower dynamics accompanied by an enhancement of the central mode behavior. Even in the case of a strong transverse field, if the NNN coupling is sufficiently large, then there is a crossover from collective mode to central mode behavior. We also obtain several recurrants for the continued fraction representation of the relaxation function.

A simple approach for bubble modelling from multiphase fluid simulation

This article presents a novel and flexible bubble modelling technique for multi-fluid simulations using a volume fraction representation. By combining the volume fraction data obtained from a primary multi-fluid simulation with simple and efficient secondary bubble simulation, a range of real-world bubble phenomena are captured with a high degree of physical realism, including large bubble deformation, sub-cell bubble motion, bubble stacking over the liquid surface, bubble volume change, dissolving of bubbles, etc. Without any change in the primary multi-fluid simulator, our bubble modelling approach is applicable to any multi-fluid simulator based on the volume fraction representation.

Representações e Processos de Raciocínio na Comparação e Ordenação de Números Racionais numa Abordagem Exploratória

O presente estudo tem por base uma unidade de ensino de cunho exploratório, em que alunos do 5.º ano de escolaridade (de 10-11 anos) trabalham com várias representações dos números racionais, em diferentes significados e tipos de grandeza. Procuramos saber em que medida esta unidade leva os alunos a desenvolver a sua capacidade de comparação e ordenação de números racionais, levando-os a usar diferentes representações e processos de raciocínio informais e formais. A metodologia de investigação é qualitativa e interpretativa, com observação participante das aulas e o estudo de caso de uma aluna. Os resultados mostram que esta aluna desenvolveu a sua compreensão da ordenação e comparação de números racionais, mostrando-se mais proficiente nas representações decimal e em fração. O estudo sugere que o trabalho com diferentes representações, numa abordagem exploratória, permite que os alunos aprendam a comparar e ordenar números racionais combinando adequadamente processos de raciocínio formais e informais.

Multiple-Fluid SPH Simulation Using a Mixture Model

This article presents a versatile and robust SPH simulation approach for multiple-fluid flows. The spatial distribution of different phases or components is modeled using the volume fraction representation, the dynamics of multiple-fluid flows is captured by using an improved mixture model, and a stable and accurate SPH formulation is rigorously derived to resolve the complex transport and transformation processes encountered in multiple-fluid flows. The new approach can capture a wide range of real-world multiple-fluid phenomena, including mixing/unmixing of miscible and immiscible fluids, diffusion effect and chemical reaction, etc. Moreover, the new multiple-fluid SPH scheme can be readily integrated into existing state-of-the-art SPH simulators, and the multiple-fluid simulation is easy to set up. Various examples are presented to demonstrate the effectiveness of our approach.

Estimation for General Birth-Death Processes

Birth-death processes (BDPs) are continuous-time Markov chains that track the number of “particles” in a system over time. While widely used in population biology, genetics, and ecology, statistical inference of the instantaneous particle birth and death rates remains largely limited to restrictive linear BDPs in which per-particle birth and death rates are constant. Researchers often observe the number of particles at discrete times, necessitating data augmentation procedures such as expectation-maximization (EM) to find maximum likelihood estimates (MLEs). For BDPs on finite state-spaces, there are powerful matrix methods for computing the conditional expectations needed for the E-step of the EM algorithm. For BDPs on infinite state-spaces, closed-form solutions for the E-step are available for some linear models, but most previous work has resorted to time-consuming simulation. Remarkably, we show that the E-step conditional expectations can be expressed as convolutions of computable transition probabilities for any general BDP with arbitrary rates. This important observation, along with a convenient continued fraction representation of the Laplace transforms of the transition probabilities, allows for novel and efficient computation of the conditional expectations for all BDPs, eliminating the need for truncation of the state-space or costly simulation. We use this insight to derive EM algorithms that yield maximum likelihood estimation for general BDPs characterized by various rate models, including generalized linear models (GLM). We show that our Laplace convolution technique outperforms competing methods when they are available and demonstrate a technique to accelerate EM algorithm convergence. We validate our approach using synthetic data and then apply our methods to cancer cell growth and estimation of mutation parameters in microsatellite evolution.

A Simple Ladder Realization of Maximally Flat Allpass Fractional Delay Filters

This brief proposes a new ladder structure for the Thiran fractional delay filter (i.e., maximally flat allpass fractional delay filter given by the Thiran approximation). The proposed ladder structure is based on a continued fraction representation. Although there exists a similar approach that was proposed by Tassart and Depalle, their structure is not realizable because of delay-free loops. On the other hand, we show that the proposed method avoids generating delay-free loops and thus successfully yields a realizable ladder structure for the Thiran fractional delay filter in a very simple form.

Expansion in Lorentzian functions of spectra of quantum autocorrelations

We show that in a quantum mechanical many-body system of Boltzmann particles having space inversion symmetry the spectrum of the autocorrelation function of a local observable can always be given, similarly to the classical case [Phys. Rev. E 85, 022102 (2012)], in terms of a series of Lorentzian functions multiplied by the proper quantum detailed balance factor. This is done by transforming the continued fraction representation, which is derived via recurrent relations and without the use of the generalized Langevin equation hierarchy, into a series expansion. In this way characteristic frequencies can be defined, also in quantum mechanics, which refer to the particular autocorrelation. These are the frequencies of the eigenmodes of the relaxation function connected to the observable. We also show that in practical cases of interest in experimental spectroscopy, and particularly in inelastic neutron and x-ray scattering, the use of a finite number of Lorentzian shapes for an approximate description of the data is related to a reduction of the number of the relevant dynamical variables taken into account, equivalent to the lowering of the dimensionality of the orthogonalized space onto which the dynamic of the system is projected. Examples of application are given for the spectrum of the velocity autocorrelation function of liquid parahydrogen, calculated with a quantum simulation algorithm (path-integral centroid molecular dynamics), and for the molecular center-of mass dynamic structure factor in liquid carbon dioxide as computed by means of classical molecular dynamics simulation.

CONTINUED FRACTIONS WITH ODD PARTIAL QUOTIENTS

Every Euclidean algorithm is associated with a kind of continued fraction representation of a number. The representation associated with "odd" Euclidean algorithm we will call "odd" continued fraction. We consider the limit distribution function F(x) for sequences of rationals with bounded sum of partial quotients for "odd" continued fractions. In this paper we prove certain properties of the function F(x). Particularly this function is singular and satisfies a number of functional equations. We also show that the value F(x) can be expressed in terms of partial quotients of the "odd" continued fraction representation of a number x.

Application of a Continued-Fraction-Based Theory to Line-Profile in Mn-Doped GaN Film

Starting with the Kubo formalism and using the projection operator technique (POT) introduced by Kawabata, the optical quantum transition line-profiles (QTLPs) formula for a Mn-doped wurtzite GaN film was derived as a function of temperature at a frequency of 9.49 GHz (X-band), on the basis of continued fraction representation (CFR) which is a counterpart of the conventional series expansion (CSE). Utilizing this formula we obtained the fine-structure parameter a - F = 9.4 ×10-4 cm-1 and fitting parameter ζ= 4.1. The optical quantum transition half-widths (QTHWs) obtained with the use of these parameters agrees quite well with the existing experimental result in the temperature region T > 20 K. The QTHWs increase with increasing temperature due to the interaction of electrons with optical phonons. Thus, the present technique is considered to be more convenient to explain the resonant system as in the case of other optical transition problems.

Application of a Continued-Fraction-Based Theory to Line-Profile in Mn-Doped GaN Film

Starting with the Kubo formalism and using the projection operator technique (POT) introduced by Kawabata, the optical quantum transition line-profiles (QTLPs) formula for a Mn-doped wurtzite GaN film was derived as a function of temperature at a frequency of 9.49 GHz (X-band), on the basis of continued fraction representation (CFR) which is a counterpart of the conventional series expansion (CSE). Utilizing this formula we obtained the fine-structure parameter a - F = 9.4 ×10-4 cm-1 and fitting parameter ζ= 4.1. The optical quantum transition half-widths (QTHWs) obtained with the use of these parameters agrees quite well with the existing experimental result in the temperature region T > 20 K. The QTHWs increase with increasing temperature due to the interaction of electrons with optical phonons. Thus, the present technique is considered to be more convenient to explain the resonant system as in the case of other optical transition problems.

Two theorems concerning rational approximations

The two theorems of the title constitute the mathematical results underlying well-formed scale theory. This paper includes the purely mathematical portion of a manuscript from 1988, which the authors cited the following year in N. Carey and D. Clampitt [Aspects of well-formed scales, Music Theory Spectrum 11 (1989), pp. 187–206]. Both theorems concern finite sets of the fractional parts of multiples of a positive real number θ. Such sets may be taken to represent musical scales generated by a single interval, and they have nice properties when their cardinalities are the denominators of convergents or semi-convergents (intermediate convergents) in the continued fraction representation of θ. Here, the results and their proofs are brought together in a notationally consistent framework.

Modular categories, integrality and Egyptian fractions

It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank n. This follows from a double-exponential bound on the maximal denominator in an Egyptian fraction representation of 1. A na¨ove computer search approach to the classification of rank n integral modular categories using this bound quickly overwhelms the computer's memory (for n � 7). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. The first such condition is that the order of the twist matrix is 2,3,4 or 6 and we obtain a fairly complete description of these classes of modular categories. The second condition is that the unit object is the only simple non-self-dual object, which is equivalent to odd-dimensionality. In this case we obtain a (linear) improvement on the bounds and employ number-theoretic techniques to obtain a classification for rank at most 11 for odd-dimensional modular categories.

A Schur–Padé Algorithm for Fractional Powers of a Matrix

A new algorithm is developed for computing arbitrary real powers $A^p$ of a matrix $A\in\mathbb{C}^{n\times n}$. The algorithm starts with a Schur decomposition, takes $k$ square roots of the triangular factor $T$, evaluates an $[m/m]$ Pad\'e approximant of $(1-x)^p$ at $I - T^{1/2^k}$, and squares the result $k$ times. The parameters $k$ and $m$ are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for $T^{p/2^j}$, yielding increased accuracy. Since the basic algorithm is designed for $p\in(-1,1)$, a criterion for reducing an arbitrary real $p$ to this range is developed, making use of bounds for the condition number of the $A^p$ problem. How best to compute $A^k$ for a negative integer $k$ is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives, including the use of an eigendecomposition and approaches based on the formula $A^p = \exp(p\log(A))$.

The Scattering Factor Formula for Electron Spin Resonance in Fine Metallic Surfaces by Using a Continued Fraction Representation

The Scattering Factor Formula for Electron Spin Resonance in Fine Metallic Surfaces by Using a Continued Fraction Representation

Rational approximation to the Fermi–Dirac function with applications in density functional theory

We are interested in computing the Fermi---Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from density functional theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We describe Lanczos and sparse direct methods to address these systems. Each approach has advantages and disadvantages that are illustrated with experiments.

Noble magnetic barriers in the ASDEX UG tokamak

The second-order perturbation method of creating invariant tori inside chaos in Hamiltonian systems (Ali, H.; Punjabi, A. Plasma Phys. Contr. F. 2007, 49, 1565–1582) is applied to the axially symmetric divertor experiment upgrade (ASDEX UG) tokamak to build noble irrational magnetic barriers inside chaos created by resonant magnetic perturbations (m, n)=(3, 2)+(4, 3), with m and n the poloidal and toroidal mode numbers of the Fourier expansion of the magnetic perturbation. The radial dependence of the Fourier modes is ignored. The modes are considered to be locked and have the same amplitude δ. A symplectic mathematical mapping in magnetic coordinates is used to integrate magnetic field line trajectories in the ASDEX UG. Tori with noble irrational rotational transform are the last ones to be destroyed by perturbation in Hamiltonian systems. For this reason, noble irrational magnetic barriers are built inside chaos, and the strongest noble irrational barrier is identified. Three candidate locations for the strongest noble barrier in ASDEX UG are selected. All three candidate locations are chosen to be roughly midway between the resonant rational surfaces ψ32 and ψ43. ψ is the magnetic coordinate of the flux surface. The three candidate surfaces are the noble irrational surfaces close to the surface with q value that is a mediant of q=3/2 and 4/3, q value of the physical midpoint of the two resonant surfaces, and the q value of the surface where the islands of the two perturbing modes just overlap. These q values of the candidate surfaces are denoted by q MED, q MID, and q OVERLAP. The strongest noble barrier close to q MED has the continued fraction representation (CFR) [1;2,2,1∞] and exists for δ≤2.6599×10−4; the strongest noble barrier close to q MID has CFR [1;2,2,2,1∞] and exists for δ≤4.6311×10−4; and the strongest noble barrier close to q OVERLAP has CFR [1;2,2,6,2,1∞] and exists for δ≤1.367770×10−4. From these results, the strongest noble barrier is found to be close to the surface that is located physically exactly in the middle of the two resonant surfaces.

Efficient implementation of the nonequilibrium Green function method for electronic transport calculations

An efficient implementation of the nonequilibrium Green function (NEGF) method combined with the density functional theory (DFT) using localized pseudo-atomic orbitals (PAOs) is presented for electronic transport calculations of a system connected with two leads under a finite bias voltage. In the implementation, accurate and efficient methods are developed especially for evaluation of the density matrix and treatment of boundaries between the scattering region and the leads. Equilibrium and nonequilibrium contributions in the density matrix are evaluated with very high precision by a contour integration with a continued fraction representation of the Fermi-Dirac function and by a simple quadratureon the real axis with a small imaginary part, respectively. The Hartree potential is computed efficiently by a combination of the two dimensional fast Fourier transform (FFT) and a finite difference method, and the charge density near the boundaries is constructed with a careful treatment to avoid the spurious scattering at the boundaries. The efficiency of the implementation is demonstrated by rapid convergence properties of the density matrix. In addition, as an illustration, our method is applied for zigzag graphene nanoribbons, a Fe/MgO/Fe tunneling junction, and a LaMnO$_3/$SrMnO$_3$ superlattice, demonstrating its applicability to a wide variety of systems.

Vibrational properties of grain boundaries in nanocrystalline Ni using second moment potentials

The high frequency phonon properties of a computer-generated nanocrystalline (nc) Ni are investigated by directly calculating the onsite phonon Green function using a recursion technique based on a continued fraction representation. Past work found that a high frequency enhancement of the vibrational density of states exists and arises from localised vibrational modes forming within the nc grain boundary regions at atoms of reduced coordination [Phys. Rev. Lett. 92 (2004) p.035505]. In the present work, the positron wave function for such nanocrystalline grain boundaries was calculated and used to identify regions of localised free volume that, in turn, were found to correlate well with the localised vibrational modes. The work also demonstrates that, irrespective of the form of the central-force inter-atomic potential used, the observed effect can be more generally understood with respect to local variations in hydrostatic pressure within the grain boundary.

A restarted Lanczos approximation to functions of a symmetric matrix

In this paper the error term in the 'first' method of Eiermann & Ernst (2006, SIAM J. Numer. Anal., 44, 2481-2504) for restarting the Lanczos approximation of the matrix-vector product f (A)b, where A ∈ ℝ n×n is symmetric, is re-derived and expressed as an explicit partial fraction expansion of the divided differences. The partial fraction representation makes the new variant slightly more stable (albeit still unstable) than the former method because it requires fewer finite-difference evaluations. We then present an error bound for the restarted Lanczos approximation of f (A)b for symmetric positive-definite A when f is in a particular class of completely monotone functions and illustrate for some important matrix function applications the usefulness of these bounds for terminating the restart process once the desired accuracy in the matrix function approximation has been achieved. Finally, in an attempt to overcome the inherent instability of our restart procedure, we propose a simple heuristic that identifies when to halt the iterations.