Logarithmic Expansions Research Articles

Analytic moduli for parabolic Dulac germs

This paper gives moduli of analytic classification for parabolic Dulac germs (that is, almost regular germs). Dulac germs appear as first return maps of hyperbolic polycycles. Their moduli are given by a sequence of Écalle–Voronin-type germs of analytic diffeomorphisms. The result is stated in a broader class of parabolic generalized Dulac germs having power- logarithmic asymptotic expansions. Bibliography: 23 titles.

Differentiation via Logarithmic Expansions

In this note, we introduce a new finite difference approximation called the Black-Box Logarithmic Expansion Numerical Derivative (BLEND) algorithm, which is based on a formal logarithmic expansion of the differentiation operator. BLEND capitalizes on parallelization and provides derivative approximations of arbitrary precision, i.e., our analysis can be used to determine the number of terms in the series expansion to guarantee a specified number of decimal places of accuracy. Furthermore, in the vector setting, the complexity of the resulting directional derivative is independent of the dimension of the parameter.

Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction–Diffusion Systems in $${\mathbb {R}}^2$$ R 2

The linear stability of steady-state periodic patterns of localized spots in $\R^2$ for the two-component Gierer-Meinhardt (GM) and Schnakenburg reaction-diffusion models is analyzed in the semi-strong interaction limit corresponding to an asymptotically small diffusion coefficient $\eps^2$ of the activator concentration. In the limit $\eps\to 0$, localized spots in the activator are centered at the lattice points of a Bravais lattice with constant area $|\Omega|$. To leading order in $\nu={-1/\log\eps}$, the linearization of the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor diffusivity satisfies $D={D_0/\nu}$, for some $D_0$ independent of the lattice and the Bloch wavevector $\kb$. From a combination of the method of matched asymptotic expansions, Floquet-Bloch theory, and the rigorous study of certain nonlocal eigenvalue problems, an explicit analytical formula for the continuous band of spectrum that lies within an ${\mathcal O}(\nu)$ neighborhood of the origin in the spectral plane is derived when $D={D_0/\nu} + D_1$, where $D_1={\mathcal O}(1)$ is a de-tuning parameter. The periodic pattern is linearly stable when $D_1$ is chosen small enough so that this continuous band is in the stable left-half plane $\mbox{Re}(\lambda)<0$ for all $\kb$. Moreover, for both the Schnakenburg and GM models, our analysis identifies a model-dependent objective function, involving the regular part of the Bloch Green's function, that must be maximized in order to determine the specific periodic arrangement of localized spots that constitutes a linearly stable steady-state pattern for the largest value of $D$. From a numerical computation, it is shown within the class of oblique Bravais lattices that a regular hexagonal lattice arrangement of spots is optimal for maximizing the stability threshold in $D$.

Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations

We consider four approaches to relative Gromov-Witten theory and Gromov-Witten theory of degenerations: Jun Li's original approach, Bumsig Kim's logarithmic expansions, Abramovich-Fantechi's orbifold expansions, and a logarithmic theory without expansions due to Gross-Siebert and Abramovich-Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov-Witten invariants associated to all four of these theories are identical.

Lattice fusion rules and logarithmic operator product expansions

The interest in Logarithmic Conformal Field Theories (LCFTs) has been growing over the last few years thanks to recent developments coming from various approaches. A particularly fruitful point of view consists in considering lattice models as regularizations for such quantum field theories. The indecomposability then encountered in the representation theory of the corresponding finite-dimensional associative algebras exactly mimics the Virasoro indecomposable modules expected to arise in the continuum limit. In this paper, we study in detail the so-called Temperley–Lieb (TL) fusion functor introduced in physics by Read and Saleur [N. Read and H. Saleur, Associative-algebraic approach to logarithmic conformal field theories, Nucl. Phys. B 777 (2007) 316]. Using quantum group results, we provide rigorous calculations of the fusion of various TL modules at roots of unity cases. Our results are illustrated by many explicit examples relevant for physics. We discuss how indecomposability arises in the “lattice” fusion and compare the mechanisms involved with similar observations in the corresponding field theory. We also discuss the physical meaning of our lattice fusion rules in terms of indecomposable operator product expansions of quantum fields.

Hybrid HDMR method with an optimized hybridity parameter in multivariate function representation

High Dimensional Model Representation (HDMR) based methods are used to generate an approximation for a given multivariate function in terms of less variate functions. This paper focuses on Hybrid HDMR which is composed of Plain HDMR and Logarithmic HDMR. The Plain HDMR method works well for representing multivariate functions having additive nature. If the function under consideration has a multiplicative nature, then the Logarithmic HDMR method produces better approximation. Hybrid HDMR method aims to successfully represent a multivariate function having neither purely additive nor purely multiplicative nature under a hybridity parameter. The performance of the Hybrid HDMR method strongly depends on the value of this hybridity parameter because this parameter manages the contribution level of Plain and Logarithmic HDMR expansions. The main purpose of this work is to optimize the hybridity parameter to get the best approximations. Fluctuationlessness Approximation Theorem is used in this optimization process and in evaluating the multiple integrals appearing in HDMR based methods. A number of numerical implementations are given at the end of the paper to show the performance of our proposed method.

Psi-series method for equality of random trees and quadratic convolution recurrences

An unusual and surprising expansion of the form as , is derived for the probability pn that two randomly chosen binary search trees are identical (in shape, hence in labels of all corresponding nodes). A quantity arising in the analysis of phylogenetic trees is also proved to have a similar asymptotic expansion. Our method of proof is new in the literature of discrete probability and the analysis of algorithms, and it is based on the logarithmic psi-series expansions for nonlinear differential equations. Such an approach is very general and applicable to many other problems involving nonlinear differential equations; many examples are discussed in this article and several attractive phenomena are discovered.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 67–108, 2014

Universality of the Ising and theS=1model on Archimedean lattices: A Monte Carlo determination

The Ising models S=1/2 and S=1 are studied by efficient Monte Carlo schemes on the (3,4,6,4) and the (3,3,3,3,6) Archimedean lattices. The algorithms used, a hybrid Metropolis-Wolff algorithm and a parallel tempering protocol, are briefly described and compared with the simple Metropolis algorithm. Accurate Monte Carlo data are produced at the exact critical temperatures of the Ising model for these lattices. Their finite-size analysis provide, with high accuracy, all critical exponents which, as expected, are the same with the well-known 2D Ising model exact values. A detailed finite-size scaling analysis of our Monte Carlo data for the S=1 model on the same lattices provides very clear evidence that this model obeys, also very well, the 2D Ising model critical exponents. As a result, we find that recent Monte Carlo simulations and attempts to define effective dimensionality for the S=1 model on these lattices are misleading. Accurate estimates are obtained for the critical amplitudes of the logarithmic expansions of the specific heat for both models on the two Archimedean lattices.

Logarithmic Expansions for Reynolds Shear Stress and Reynolds Heat Flux in a Turbulent Channel Flow

The heat and fluid flow in a fully developed turbulent channel flow have been investigated. The closure model of Reynolds shear stress and Reynolds heat flux as a function of a series of logarithmic functions in the mesolayer variable have been adopted. The interaction between inner and outer layers in the mesolayer (intermediate layer) arising from the balance of viscous effect, pressure gradient and Reynolds shear stress (containing the maxima of Reynolds shear stress) was first proposed by Afzal (1982, “Fully Developed Turbulent Flow in a Pipe: An Intermediate Layer,” Arch. Appl. Mech., 53, 355–377). The unknown constants in the closure models for Reynolds shear stress and Reynolds heat flux have been estimated from the prescribed boundary conditions near the axis and surface of channel. The predictions are compared with the DNS data Iwamoto et al. and Abe et al. for Reynolds shear stress and velocity profile and Abe et al. data of Reynolds heat flux and temperature profile. The limitations of the closure models are presented.

Precision studies of the NNLO DGLAP evolution at the LHC with Candia

We summarize the theoretical approach to the solution of the NNLO DGLAP equations using methods based on the logarithmic expansions in x-space and their implementation into the C program Candia 1.0. 1 1 http://www.le.infn.it/candia. We present the various options implemented in the program and discuss the different solutions. The user can choose the order of the evolution, the type of the solution, which can be either exact or truncated, and the evolution either with a fixed or a varying flavor number, implemented in the varying-flavor-number scheme (VFNS). The renormalization and factorization scale dependencies are treated separately. In the non-singlet sector the program implements an exact NNLO solution. Program summary Program title: CANDIA Catalogue identifier: AEBK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBK_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 101 376 No. of bytes in distributed program, including test data, etc.: 5 865 234 Distribution format: tar.gz Programming language: C and Fortran Computer: All Operating system: Linux RAM: In the given examples, it ranges from 4 to 490 MB Classification: 11.1, 11.5 Nature of problem: The program provided here solves the DGLAP evolution equations for the parton distribution functions up to NNLO. Solution method: The algorithm implemented is based on the theory of the logarithmic expansions in Bjorken x-space. Additional comments: To be sure of getting the latest version of the program, the authors suggest downloading the code from their official CANDIA website ( http://www.le.infn.it/candia). Running time: In the given examples, it ranges from 1 to 40 minutes. The jobs have been executed on an Intel Core 2 Duo T7250 CPU at 2 GHz with a 64 bit Linux kernel. The test run script included in the package contains 5 sample runs and may take a number of hours to process, depending on the speed of the processor used and the size of the available RAM.

Incremental parameter evaluation from incomplete data with application to the population pharmacology of anticoagulants

We develop a method for parameter evaluation from incomplete data. Improved estimates of the desired parameters are evaluated step by step, from experiment to experiment by using both Bayesian and informational methods. We make dynamical, improved predictions while the experiments are still going on and keep and interpret information about local fluctuations, which is lost on applying global techniques. The input of information in small packets leads to semi-analytic methods for data processing. An evolution criterion for parameter evaluation, similar to Fisher's theorem of population selection, is derived. We develop direct processing methods, which can be applied to low dimensional systems, semi-analytic methods based on direct or double logarithmic phase expansions, steepest descent approaches, variation and perturbation methods. The techniques are illustrated by developing a method of long-term planning of treatments with oral anticoagulants based on limited clinical data. The efficiency of treatment by oral anticoagulants depends strongly on various anthropometric and genotypic factors, which lead to large variations of the clinical response. We use the clinical data, which accumulates from medical consultations, for extracting improved, incremental information about the statistical properties of the kinetic and anthropometric parameters for a given patient, which in turn is used for making repeated, improved clinical predictions as the treatment proceeds.

The Role of Logarithmic Expansions for Nonsinglet QCD Analysis of [formula omitted

In this paper the analysis of certain logarithmic expansions, which developed for precision studies of the evolution of the QCD parton distributions (pdf) at the Large Hadron Collider, is used. The x F 3 data of the CCFR collaboration is used to parameterize the non-singlet parton distributions in this order. In the fitting procedure, Bernstein polynomial method is employed. The results of valence quark distributions in the NLO are compered with exact and truncated solutions for non-singlet case.

NNLO logarithmic expansions and precise determinations of the neutral currents near the Z resonance at the LHC: the Drell-Yan case

We present a comparative study of the invariant mass and rapidity distributions in Drell-Yan lepton pair production, with particular emphasis on the role played by the QCD evolution. We focus our study around the Z resonance ($50 <Q < 200$ GeV) and perform a general analysis of the factorization/renormalization scale dependence of the cross sections, with the two scales included both in the evolution and in the hard scatterings. We also present the variations of the cross sections due to the errors on the parton distributions (pdf's) and an analysis of the corresponding $K$-factors. Predictions from several sets of pdf's, evolved by MRST and Alekhin are compared with those generated using \textsc{Candia}, a NNLO evolution program that implements the theory of the logarithmic expansions, developed in a previous work. These expansions allow to select truncated solutions of varying accuracy using the method of the $x$-space iterates. The evolved parton distributions are in good agreement with other approaches. The study can be generalized for high precision searches of extra neutral gauge interactions at the LHC.

Study of pressure–volume relationships and higher derivatives of bulk modulus based on generalized equations of state

We have generalized the pressure–volume (P–V) relationships using simple polynomial and logarithmic expansions so as to make them consistent with the infinite pressure extrapolation according to the model of Stacey. The formulations are used to evaluate P–V relationships and pressure derivatives of bulk modulus upto third order ( K′, K″ and K‴) for the earth core material taking input parameters based on the seismological data. The results based on the equations of state (EOS) generalized in the present study are found to yield good agreement with the Stacey EOS. The generalized logarithmic EOS due to Poirier and Tarantola deviates substantially from the seismic values for P, K and K′. The generalized Rydberg EOS gives almost identical results with the Birch—Murnaghan third-order EOS. Both of them yield deviations from the seismic data, which are in opposite direction as compared to those found from the generalized Poirier–Tarantola logarithmic EOS.

NNLO logarithmic expansions and exact solutions of the DGLAP equations from x-space: New algorithms for precision studies at the LHC

A NNLO analysis of certain logarithmic expansions, developed for precision studies of the evolution of the QCD parton distributions (pdf) at the large hadron collider, is presented. We elaborate on their relations to all the solutions of the DGLAP equations that have been hitherto obtained from Mellin space, to which are equivalent. Exact expansions, equivalent to exact solutions of the equations, are constructed in the non-singlet sector. The algorithmic features of our approach are also emphasized, since this method allows to obtain numerical solutions of the evolution equations with the same accuracy of other methods, based on Mellin space, and of brute force methods, which solve the equations by finite differences. The implementation of our analysis allows to compare with existing benchmarks for the evolution of the pdf's, useful for applications at the LHC, and to extend them significantly in a systematic fashion, especially when solutions that retain logarithmic corrections only of a certain accuracy are searched for.

Finite-Reynolds-number effects in turbulence using logarithmic expansions

Experimental or numerical data in turbulence are invariably obtained at finite Reynolds numbers whereas theories of turbulence correspond to infinitely large Reynolds numbers. A proper merger of the two approaches is possible only if corrections for finite Reynolds numbers can be quantified. This paper heuristically considers examples in two classes of finite-Reynolds-number effects. Expansions in terms of logarithms of appropriate variables are shown to yield results in agreement with experimental and numerical data in the following instances: the third-order structure function in isotropic turbulence, the mixed-order structure function for the passive scalar and the Reynolds shear stress around its maximum point. Results suggestive of expansions in terms of the inverse logarithm of the Reynolds number, also motivated by experimental data, concern the tendency for turbulent structures to cluster along a line of observation and (more speculatively) for the longitudinal velocity derivative to become singular at some finite Reynolds number. We suggest an elementary hydrodynamical process that may provide a physical basis for the expansions considered here, but note that the formal justification remains tantalizingly unclear.

The Logarithmic Asymptotic Expansions for the Norms of Evaluation Functionals

Let μ be a compactly supported finite Borel measure in ℂ, and let Πn be the space of holomorphic polynomials of degree at most n furnished with the norm of L 2(μ). We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials p ∈ Πn their values at a point z ∈ ℂ. The main results demonstrate how the asymptotic behavior depends on regularity of the complement of the support of μ and the Stahl-Totik regularity of the measure. In particular, we study the cases of pointwise and μ-a.e. convergence as n → ∞.

Reliability of a high energy one-loop expansion of e+e−→ W+W− in the SM and in the MSSM

We compare the logarithmic Sudakov expansions of the process e + e −→ W + W − in the one-loop approximation and in the resummed version, to subleading order accuracy, in the SM case and in a light SUSY scenario for the MSSM. We show that the two expansions are essentially identical below 1 TeV, but differ drastically at higher (2, 3 TeV) center of mass energies. Starting from these conclusions, we argue that a complete one-loop calculation in the energy region below 1 TeV does not seem to need extra two-loop corrections, in spite of the relatively large size of the one-loop effects.

Iterated logarithmic expansions of the pathwise code lengths for exponential families

Rissanen's (1978, 1986, 1989, 1996) minimum description length (MDL) principle is a statistical modeling principle motivated by coding theory. For exponential families we obtain pathwise expansions, to the constant order, of the predictive and mixture codelengths used in MDL. The results are useful for understanding different MDL forms.

A Hybrid Asymptotic-Numerical Method for Low Reynolds Number Flows Past a Cylindrical Body

The classical problem of slow, steady, two-dimensional flow of a viscous incompressible fluid around an infinitely long straight cylinder is considered. The cylinder cross section is symmetric about the direction of the oncoming stream but otherwise is arbitrary. For low Reynolds number, the well-known singular perturbation analysis for this problem shows that the asymptotic expansions of the drag coefficient and of the flow field start with infinite logarithmic series. We show that the entire infinite logarithmic expansions of the flow field and of the drag coefficient are contained in the solution to a certain related problem that does not involve the cross-sectional shape of the cylinder. The solution to this related problem is computed numerically using a straightforward finite-difference scheme. The drag coefficient for a cylinder of a specific cross-sectional shape, which is asymptotically correct to within all logarithmic terms, is given in terms of a single shape-dependent constant that is determi...