Series Analysis

Guttmann, A J; Jensen, Iwan

doi:10.1007/978-1-4020-9927-4_8

Cited by 19 publications

(23 citation statements)

References 20 publications

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“…We wrote a C program using message passing to run on a distributed system, and ran it in the Spartan [20] cluster at the High Performance Computing Centre at the University of Melbourne on 168 cores with 20GB per core. The computation was performed five times, each with computations performed modulo a number close to 2 16 , so that only 16 bits of storage were needed for each state. Each run took several hours.…”

Section: Runningmentioning

confidence: 99%

“…The most successful numerical method for extracting the asymptotics from the first few terms of the OGF of a function with a power-law singularity is the method of differential approximants, due to Guttmann and Joyce [17], with subsequent refinements due to Baker and Hunter [18] and Fisher and Au-Yang [10]. Details are given in [14,16,13]. In brief, the method fits available coefficients to a judiciously chosen family of D-finite ordinary differential equations (ODEs), and the singularity structure of the ODEs is extracted by standard methods [19,11].…”

Section: Differential Approximant Analysismentioning

confidence: 99%

“…The entries give an average value for the position and exponent of the singularity of the fitted ODEs. Full details of the method are given in [13,16]. It is seen that the 2nd-order DAs give estimates of the radius of convergence that can be summarised as 1/µ = 0.086205 ± 0.00003, or µ = 11.600 ± 0.004.…”

Section: Using Differential Approximantsmentioning

confidence: 99%

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1324-avoiding permutations revisited

Conway

Guttmann

Zinn-Justin

2018

Advances in Applied Mathematics

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We give an improved algorithm for counting the number of 1324-avoiding permutations, resulting in 14 further terms of the generating function, which is now known for all lengths ≤ 50. We re-analyse the generating function and find additional evidence for our earlier conclusion that unlike other classical length-4 pattern-avoiding permutations, the generating function does not have a simple power-law singularity, but rather, the number of 1324-avoiding permutations of length n behaves as B · µ n · µ √ n 1 · n g . We estimate µ = 11.600 ± 0.003, µ 1 = 0.0400 ± 0.0005, g = −1.1 ± 0.1 while the estimate of B depends sensitively on the precise value of µ, µ 1 and g. This reanalysis provides substantially more compelling arguments for the presence of the stretched exponential term µ √ n 1 . *

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Section: Runningmentioning

confidence: 99%

Section: Differential Approximant Analysismentioning

confidence: 99%

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1324-avoiding permutations revisited

Conway

Guttmann

Zinn-Justin

2018

Advances in Applied Mathematics

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“…The resulting estimates of the free-energy λ 0 (y) = − 1 2 log x c are plotted in Figure 12. The series were analysed using second and third order differential approximants [12]. At y = 1 the series is well behaved and has critical point 1/µ 2 with exponent α = 3/2, the exponent for self-avoiding polygons, which is unchanged if we consider SAPs attached to a surface (unlike the SAW case).…”

Section: Resultsmentioning

confidence: 99%

“…The calculation was performed in parallel using up to 32 processors, a maximum of some 70GB of memory and using a total of just under 1000 CPU hours. Details of the implementation and parallelization of our algorithm can be found in [35,34,12].…”

Section: Exact Enumerationsmentioning

confidence: 99%

Polygons pulled from an adsorbing surface

Guttmann¹,

Rensburg²,

Jensen³

et al. 2018

J. Phys. A: Math. Theor.

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Abstract. We consider self-avoiding lattice polygons, in the hypercubic lattice, as a model of a ring polymer adsorbed at a surface and either being desorbed by the action of a force, or pushed towards the surface. We show that, when there is no interaction with the surface, then the response of the polygon to the applied force is identical (in the thermodynamic limit) for two ways in which we apply the force. When the polygon is attracted to the surface then, when the dimension is at least 3, we have a complete characterization of the critical force-temperature curve in terms of the behaviour, (a) when there is no force, and, (b) when there is no surface interaction. For the 2-dimensional case we have upper and lower bounds on the free energy. We use both Monte Carlo and exact enumeration and series analysis methods to investigate the form of the phase diagram in two dimensions. We find evidence for the existence of a mixed phase where the free energy depends on the strength of the interaction with the adsorbing line and on the applied force.

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Critical isotherms from virial series using asymptotically consistent approximants

et al. 2014

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The low-density equation of state of a fluid along its critical isotherm is considered. An asymptotically consistent approximant is formed having the correct leading-order scaling behavior near the vapor-liquid critical point, while retaining the correct low-density behavior as expressed by the virial equation of state. The formulation is demonstrated for the Lennard-Jones fluid, and models for helium, water, and n-alkanes. The ability of the approximant to augment virial series predictions of critical properties is explored, both in conjunction with and in the absence of criticalproperty data obtained by other means. Given estimates of the critical point from molecular simulation or experiment, the approximant can refine the critical pressure or density by ensuring that the critical isotherm remains well-behaved from low density to the critical region. Alternatively, when applied in the absence of other data, the approximant remedies a consistent underestimation of the critical density when computed from the virial series alone. V C 2014 American Institute of Chemical Engineers AIChE J, 60: 3336-3349, 2014 Keywords: virial equation of state, critical phenomena, crossover, approximant, Lennard-Jones P c . When solved using a sufficiently high-order virial series, these predictions for T c and P c are often close to simulation data and/or experiments, but those for q c are consistently about 10% too low in such a comparison. 4,8 Underestimation of the critical density by the virial series is connected to the nonclassical behavior of real fluids at the critical point, for which critical scaling laws assert that q c is a branch-point singularity of the function P(q) along the critical isotherm. In the context of critical phenomena, the molar In this section, critical isotherms are constructed using given values of the critical density q c , critical pressure P c , Values are given in Lennard-Jones units. Virial coefficients are from Schultz et al. 5 q c,AJ is the corrected critical density given by the smallest positive root of (7), which takes the following as inputs: d 5 4.789; P c , q c , and T c as predicted by the virial series at each order J (given above); and the corresponding virial coefficients (taken from an interpolation). Numbers in parentheses specify the 68% uncertainty on the last digit, propagated from uncertainty in the virial coefficients (with the exception of the simulation values). a Predicted using (7) with P c,sim , T c,sim , and the first seven virial coefficients taken at T c,sim (from an interpolation) as inputs.

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Series Analysis

Cited by 19 publications

References 20 publications

1324-avoiding permutations revisited

1324-avoiding permutations revisited

Polygons pulled from an adsorbing surface

Critical isotherms from virial series using asymptotically consistent approximants

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