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Journal articles on the topic 'Exact partition function'

Author: Grafiati

Published: 14 March 2023

Last updated: 31 July 2025

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1

PUTROV, PAVEL, and MASAHITO YAMAZAKI. "EXACT ABJM PARTITION FUNCTION FROM TBA." Modern Physics Letters A 27, no. 34 (2012): 1250200. http://dx.doi.org/10.1142/s0217732312502008.

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We report on the exact computation of the S3 partition function of U (N)k × U (N)-k ABJM theory for k = 1, N = 1, …, 19. The result is a polynomial in π-1 with rational coefficients. As an application of our results, we numerically determine the coefficient of the membrane 1-instanton correction to the partition function.

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2

HATZINIKITAS, AGAPITOS, and IOANNIS SMYRNAKIS. "CLOSED BOSONIC STRING PARTITION FUNCTION IN TIME INDEPENDENT EXACT pp-WAVE BACKGROUND." International Journal of Modern Physics A 21, no. 05 (2006): 995–1013. http://dx.doi.org/10.1142/s0217751x06025493.

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The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the presence of certain homogeneous exact pp -wave backgrounds is studied. In the absence of an axion field, the partition function is found to be modular invariant and equal to the free field partition function. The partition function remains unchanged also in the presence of a fixed axion field. However, in this case, the covariant form of the action suggests summation over all possible twists generated by the axion field. This is shown to modify the partition function. In the light-c

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3

Hatsuda, Yasuyuki, Sanefumi Moriyama, and Kazumi Okuyama. "Exact instanton expansion of the ABJM partition function." Progress of Theoretical and Experimental Physics 2015, no. 11 (2015): 11B104. http://dx.doi.org/10.1093/ptep/ptv145.

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4

Izmailov, Alexander F., and Alexander R. Kessel. "Exact quantum partition function of the BCS model." International Journal of Theoretical Physics 29, no. 10 (1990): 1073–90. http://dx.doi.org/10.1007/bf00672086.

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5

Bouttier, J., P. Di Francesco, and E. Guitter. "Random trees between two walls: exact partition function." Journal of Physics A: Mathematical and General 36, no. 50 (2003): 12349–66. http://dx.doi.org/10.1088/0305-4470/36/50/001.

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6

Zhang, Degang. "Exact Solution for Three-Dimensional Ising Model." Symmetry 13, no. 10 (2021): 1837. http://dx.doi.org/10.3390/sym13101837.

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The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical

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7

Julian Lee. "Exact Partition Function Zeros of Two-Dimensional Lattice Polymers." Journal of the Korean Physical Society 44, no. 3 (2004): 617. http://dx.doi.org/10.3938/jkps.44.617.

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8

Basu-Mallick, B., and Nilanjan Bondyopadhaya. "Exact partition function of supersymmetric Haldane–Shastry spin chain." Nuclear Physics B 757, no. 3 (2006): 280–302. http://dx.doi.org/10.1016/j.nuclphysb.2006.09.009.

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9

González, Gabriel. "Exact Partition Function for the Random Walk of an Electrostatic Field." Advances in Mathematical Physics 2017 (2017): 1–5. http://dx.doi.org/10.1155/2017/6970870.

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The partition function for the random walk of an electrostatic field produced by several static parallel infinite charged planes in which the charge distribution could be either ±σ is obtained. We find the electrostatic energy of the system and show that it can be analyzed through generalized Dyck paths. The relation between the electrostatic field and generalized Dyck paths allows us to sum overall possible electrostatic field configurations and is used for obtaining the partition function of the system. We illustrate our results with one example.

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10

Kogan, Yaakov. "Exact analysis for a class of simple, circuit-switched networks with blocking." Advances in Applied Probability 21, no. 4 (1989): 952–55. http://dx.doi.org/10.2307/1427782.

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We consider the same circuit switching problem as in Mitra [1]. The calculation of the blocking probabilities is reduced to finding the partition function for a closed exponential pseudo-network with L−1 customers. This pseudo-network differs from that in [1] in one respect only: service rates at nodes 1, 2, …, p depend on the queue length. The asymptotic expansion developed in [1] follows from our exact expression for the partition function.

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11

Kogan, Yaakov. "Exact analysis for a class of simple, circuit-switched networks with blocking." Advances in Applied Probability 21, no. 04 (1989): 952–55. http://dx.doi.org/10.1017/s0001867800019200.

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We consider the same circuit switching problem as in Mitra [1]. The calculation of the blocking probabilities is reduced to finding the partition function for a closed exponential pseudo-network with L−1 customers. This pseudo-network differs from that in [1] in one respect only: service rates at nodes 1, 2, …, p depend on the queue length. The asymptotic expansion developed in [1] follows from our exact expression for the partition function.

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12

Iqbal, Amer, Babar A. Qureshi, and Khurram Shabbir. "(q, t) identities and vertex operators." Modern Physics Letters A 31, no. 11 (2016): 1650065. http://dx.doi.org/10.1142/s0217732316500656.

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Using vertex operators acting on fermionic Fock space we prove certain identities, which depend on a number of parameters, generalizing and refining the Nekrasov–Okounkov identity. These identities provide exact product representation for the instanton partition function of certain five-dimensional quiver gauge theories. This product representation also clearly displays the modular transformation properties of the gauge theory partition function.

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13

Andriushchenko, Petr Dmitrievich, and Konstantin V. Nefedev. "Partition Function and Density of States in Models of a Finite Number of Ising Spins with Direct Exchange between the Minimum and Maximum Number of Nearest Neighbors." Solid State Phenomena 247 (March 2016): 142–47. http://dx.doi.org/10.4028/www.scientific.net/ssp.247.142.

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The results of studies of 1D Ising models and Curie-Weiss models partition functions structure are presented in this work. Exact calculation of the partition function using the authors combinatorial approach for such system is discussed. The distribution of the energy levels degeneracy was calculated. Analytical solution for density of states of 1D Ising chain were obtained. Generating functions for these models were obtained. It was suggested that in Curie-Weiss model the transition to a low-energy state occurs without the formation of separation boundaries

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14

IZMAILOV, ALEXANDER F., and ALEXANDER R. KESSEL. "SOLUTION OF THE BCS MODEL." International Journal of Modern Physics A 04, no. 18 (1989): 4991–5002. http://dx.doi.org/10.1142/s0217751x89002120.

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The exact calculation of the reduced BCS model quantum partition function in the region of temperatures T > Tc was carried out by the path integration method. The partition function demonstrates the critical behavior at some temperature Tc. It turns out that this temperature is larger than the critical temperature T'c obtained in the traditional theories which are valid in the temperature region T < T'c.

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15

Skorik, Sergei. "Exact nonequilibrium current from the partition function for impurity-transport problems." Physical Review B 57, no. 20 (1998): 12772–80. http://dx.doi.org/10.1103/physrevb.57.12772.

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16

Chang, Shu-Chiuan, and Robert Shrock. "Exact Potts model partition function on strips of the triangular lattice." Physica A: Statistical Mechanics and its Applications 286, no. 1-2 (2000): 189–238. http://dx.doi.org/10.1016/s0378-4371(00)00225-9.

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17

Alvarez, P. D. "Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice." Journal of Statistical Mechanics: Theory and Experiment 2024, no. 8 (2024): 083101. http://dx.doi.org/10.1088/1742-5468/ad64bc.

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Abstract We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The

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18

Archibald, Margaret, Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, and Toufik Mansour. "Two by two squares in set partitions." Mathematica Slovaca 70, no. 1 (2020): 29–40. http://dx.doi.org/10.1515/ms-2017-0328.

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AbstractA partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 < min B2 < ⋯ < min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and $\begin{array}{} \displaystyle \bigcup _{i=1}^n \{\pi_i\}=[k], \end{array}$ and πj indicates that j ∈ Bπj. The canonical representations of all set par

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19

DAMGAARD, P. H., and J. LACKI. "PARTITION FUNCTION ZEROS OF AN ISING SPIN GLASS." International Journal of Modern Physics C 06, no. 06 (1996): 819–43. http://dx.doi.org/10.1142/s012918319500068x.

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We study the pattern of zeros emerging from exact partition function evaluations of Ising spin glasses on conventional finite lattices of varying sizes. A large number of random bond configurations are probed in the framework of quenched averages. This study is motivated by the relationship between hierarchical lattice models whose partition function zeros fall on Julia sets and chaotic renormalization group flows in such models with frustration, and by the possible connection of the latter with spin glass behavior. In any finite volume, the simultaneous distribution of the zeros of all partit

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20

Honchar, Yulian, Mariana Krasnytska, Bertrand Berche, Yurij Holovatch, and Ralph Kenna. "Partition function zeros for the Blume–Capel model on a complete graph." Low Temperature Physics 51, no. 5 (2025): 567–77. https://doi.org/10.1063/10.0036500.

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In this paper, we study finite-size effects in the Blume–Capel model through the analysis of the zeros of the partition function. We consider a complete graph and use the behavior of the partition function zeros to elucidate the crossover from effective to asymptotic properties. While in the thermodynamic limit the exact solution yields the asymptotic mean-field behavior, for finite size systems an effective critical behavior is observed. We show that even for large systems the criticality is not asymptotic. We also present insights into how partition function zeros in different complex fields

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21

CHANG, SHU-CHIUAN, and ROBERT SHROCK. "EXACT PARTITION FUNCTION FOR THE POTTS MODEL WITH NEXT-NEAREST NEIGHBOR COUPLINGS ON ARBITRARY-LENGTH LADDERS." International Journal of Modern Physics B 15, no. 05 (2001): 443–78. http://dx.doi.org/10.1142/s0217979201004630.

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We present exact calculations of partition function Z of the q-state Potts model with next-nearest-neighbor spin–spin couplings, both for the ferromagnetic and antiferromagnetic case, for arbitrary temperature, on n-vertex ladders with free, cyclic, and Möbius longitudinal boundary conditions. The free energy is calculated exactly for the infinite-length limit of these strip graphs and the thermodynamics is discussed. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus ℬ in the corresponding [Formula: see text] space, arising as the accum

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22

Guerin, H. "Exact classical vibrational-rotational partition function for Lennard-Jones and Morse potentials." Journal of Physics B: Atomic, Molecular and Optical Physics 25, no. 8 (1992): 1697–703. http://dx.doi.org/10.1088/0953-4075/25/8/006.

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23

Basu-Mallick, Bireswar, Hideaki Ujino, and Miki Wadati. "Exact Spectrum and Partition Function of SU(m|n) Supersymmetric Polychronakos Model." Journal of the Physical Society of Japan 68, no. 10 (1999): 3219–26. http://dx.doi.org/10.1143/jpsj.68.3219.

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24

Jian-wei, Pan, Zhang Yong-de, and G. G. Siu. "Exact Expressions of Energy Spectrum and Partition Function for Quantum Quadratic Systems." Chinese Physics Letters 14, no. 4 (1997): 241–44. http://dx.doi.org/10.1088/0256-307x/14/4/001.

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25

Zhelifonov, M. P., and B. S. Nikitin. "Exact solution for the partition function of a system of interacting fermions." Theoretical and Mathematical Physics 68, no. 1 (1986): 707–14. http://dx.doi.org/10.1007/bf01017800.

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26

Lee, Julian. "Exact Partition Function Zeros of the Wako-Saitô-Muñoz-Eaton Protein Model." Biophysical Journal 106, no. 2 (2014): 438a. http://dx.doi.org/10.1016/j.bpj.2013.11.2467.

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27

Izmailov, A. F., and A. R. Kessel. "Partition function of the BCS model in the nonregular phase: Exact results." Physica C: Superconductivity 168, no. 3-4 (1990): 450–56. http://dx.doi.org/10.1016/0921-4534(90)90539-q.

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28

Tsuzuki, Toshio. "Dynamic compensation theorem and exact partition function of a spin-boson system." Solid State Communications 74, no. 8 (1990): 743–46. http://dx.doi.org/10.1016/0038-1098(90)90927-4.

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29

Aguilar, A., and E. Braun. "Exact solution of a general two-dimensional Ising model: The partition function." Physica A: Statistical Mechanics and its Applications 170, no. 3 (1991): 643–62. http://dx.doi.org/10.1016/0378-4371(91)90011-z.

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30

LI, Y. M., Y. YU, N. D'AMBRUMENIL, L. YU, and Z. B. SU. "EXPRESSION OF THE PARTITION FUNCTION AND RENORMALIZATION EQUATION OF SPIN-1/2 TOMONAGA–LUTTINGER CHAINS." Modern Physics Letters B 08, no. 12 (1994): 749–57. http://dx.doi.org/10.1142/s0217984994000753.

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We derive an exact expression for the partition function of two spin-1/2 Tomonaga–Luttinger chains, and obtain the renormalization group equation for the normal state. Anderson's confinement is then discussed. We find that spin–charge separation is irrelevant to the confinement.

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31

Frasca, Marco, and Stefan Groote. "Some Exact Green Function Solutions for Non-Linear Classical Field Theories." Symmetry 16, no. 11 (2024): 1504. http://dx.doi.org/10.3390/sym16111504.

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We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be a

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32

Prat Pou, Arnau, Enrique Romero, Jordi Martí, and Ferran Mazzanti. "Mean Field Initialization of the Annealed Importance Sampling Algorithm for an Efficient Evaluation of the Partition Function Using Restricted Boltzmann Machines." Entropy 27, no. 2 (2025): 171. https://doi.org/10.3390/e27020171.

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Probabilistic models in physics often require the evaluation of normalized Boltzmann factors, which in turn implies the computation of the partition function Z. Obtaining the exact value of Z, though, becomes a forbiddingly expensive task as the system size increases. A possible way to tackle this problem is to use the Annealed Importance Sampling (AIS) algorithm, which provides a tool to stochastically estimate the partition function of the system. The nature of AIS allows for an efficient and parallel implementation in Restricted Boltzmann Machines (RBMs). In this work, we evaluate the parti

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33

LU, HUAI-XIN, ZENG-BING CHEN, LEI MA, and YONG-DE ZHANG. "EXACT SOLUTION FOR A GENERAL SUPERSYMMETRIC QUADRATIC SYSTEM." Modern Physics Letters B 16, no. 07 (2002): 241–50. http://dx.doi.org/10.1142/s0217984902003671.

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We construct a general supersymmetric quantum transformation for studying the supersymmetric quadratic Hamiltonian which contains the interaction of multi-mode fermions with multi-mode bosons. The Hamiltonian can be derived from that of multi-mode radiation field interacting with m two-level atoms. By using the supersymmetric quantum transformation, the diagonolized Hamiltonian is given. We also concisely derive the analytic expression of the super-partition function for the supersymmetric quadratic Hamiltonian without knowing the energy spectrum.

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34

LIN, K. Y. "EXACT RESULTS FOR THE ISING MODEL ON A 3–12 LATTICE." International Journal of Modern Physics B 03, no. 08 (1989): 1237–45. http://dx.doi.org/10.1142/s021797928900083x.

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We consider the Ising model on a 3–12 lattice with magnetic field. An exact functional relation is established for the partition function and our result is a generalization of Giacomini’s work on the Kagomé lattice. We calculate the zero-field magnetic susceptibility when an appropriate relation among the interaction parameters is satisfied.

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35

Saryal, Sushant, and Deepak Dhar. "Exact results for interacting hard rigid rotors on a d-dimensional lattice." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 4 (2022): 043204. http://dx.doi.org/10.1088/1742-5468/ac6038.

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Abstract We study the entropy of a set of identical hard objects, of general shape, with each object pivoted on the vertices of a d-dimensional regular lattice of lattice spacing a, but can have arbitrary orientations. When the pivoting point is situated asymmetrically on the object, we show that there is a range of lattice spacings a, where in any orientation, a particle can overlap with at most one of its neighbors. In this range, the entropy of the system of particles can be expressed exactly in terms of the grand partition function of coverings of the base lattice by dimers at a finite neg

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36

Verbaarschot, J. J. M., and T. Wettig. "Random Matrix Theory and Chiral Symmetry in QCD." Annual Review of Nuclear and Particle Science 50, no. 1 (2000): 343–410. http://dx.doi.org/10.1146/annurev.nucl.50.1.343.

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▪ Abstract Random matrix theory is a powerful way to describe universal correlations of eigenvalues of complex systems. It also may serve as a schematic model for disorder in quantum systems. In this review, we discuss both types of applications of chiral random matrix theory to the QCD partition function. We show that constraints imposed by chiral symmetry and its spontaneous breaking determine the structure of low-energy effective partition functions for the Dirac spectrum. We thus derive exact results for the low-lying eigenvalues of the QCD Dirac operator. We argue that the statistical pro

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37

Savvidy, George. "The gonihedric paradigm extension of the Ising model." Modern Physics Letters B 29, no. 32 (2015): 1550203. http://dx.doi.org/10.1142/s0217984915502036.

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In this paper we review a recently suggested generalization of the Feynman path integral to an integral over random surfaces. The proposed action is proportional to the linear size of the random surfaces and is called gonihedric. The convergence and the properties of the partition function are analyzed. The model can also be formulated as a spin system with identical partition functions. The spin system represents a generalization of the Ising model with ferromagnetic, antiferromagnetic and quartic interactions. Higher symmetry of the model allows to construct dual spin systems in three and fo

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38

CHATTERJEE, R. "EXACT PARTITION FUNCTION AND BOUNDARY STATE OF CRITICAL ISING MODEL WITH BOUNDARY MAGNETIC FIELD." Modern Physics Letters A 10, no. 12 (1995): 973–84. http://dx.doi.org/10.1142/s0217732395001071.

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We compute the exact partition function of the 2-D Ising Model at critical temperature but with nonzero magnetic field at the boundary. The model describes a renormalization group flow between the free and fixed conformal boundary conditions in the space of boundary interactions. For this flow the universal ground state degeneracy g and the full boundary state is computed exactly.

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39

Chang, Shu-Chiuan, and Robert Shrock. "Some exact results on the Potts model partition function in a magnetic field." Journal of Physics A: Mathematical and Theoretical 42, no. 38 (2009): 385004. http://dx.doi.org/10.1088/1751-8113/42/38/385004.

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40

Wood, D. W. "The exact location of partition function zeros, a new method for statistical mechanics." Journal of Physics A: Mathematical and General 18, no. 15 (1985): L917—L921. http://dx.doi.org/10.1088/0305-4470/18/15/003.

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41

Lee, Jae Hwan, Seung-Yeon Kim, and Julian Lee. "Parallel algorithm for calculation of the exact partition function of a lattice polymer." Computer Physics Communications 182, no. 4 (2011): 1027–33. http://dx.doi.org/10.1016/j.cpc.2011.01.004.

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42

Qian, Wei-Liang, Kai Lin, Rui-Hong Yue, Yogiro Hama, and Takeshi Kodama. "On the Partition Temperature of Massless Particles in High-Energy Collisions." Symmetry 15, no. 11 (2023): 2035. http://dx.doi.org/10.3390/sym15112035.

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Although partition temperature derived using the Darwin–Fowler method is exact for simple scenarios, the derivation for complex systems might reside in specific approximations whose viability is not ensured if the thermodynamic limit is not attained. This work elaborates on a related problem relevant to relativistic high-energy collisions. On the one hand, it is simple enough that closed-form expressions can be obtained precisely for the one-particle distribution function. On the other hand, the resulting expression is not an exponential form, and therefore, it is not straightforward that the

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43

Tamm, Mikhail V., Maxym Dudka, Nikita Pospelov, Gleb Oshanin, and Sergei Nechaev. "From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 3 (2022): 033201. http://dx.doi.org/10.1088/1742-5468/ac52a5.

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Abstract We expose a series of exact mappings between particular cases of four statistical physics models: (i) equilibrium 1D lattice gas with nearest-neighbor repulsion, (ii) (1 + 1)D combinatorial heap of pieces, (iii) directed random walks on a half-plane, and (iv) 1D totally asymmetric simple exclusion process (TASEP). In particular, we show that generating function of a 1D steady-state TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combinatio

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44

HANSEL, D., and J. M. MAILLARD. "FORMAL CONSTRAINTS ON SERIES ANALYSIS ON THE POTTS MODEL." Modern Physics Letters B 01, no. 04 (1987): 145–53. http://dx.doi.org/10.1142/s021798498700020x.

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It is shown that the low temperature expansion of the partition function, magnetization and nearest neighbour correlation functions of the q-state checkerboard Potts model in a magnetic field drastically simplify on a very simple algebraic variety. These four formal constraints on the expansions are also analysed in the framework of the resummed low temperature expansions and the large q expansions. These exact results are generalized straightforwardly to higher dimensional hypercubic lattices and also to some random problems.

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45

IGUCHI, KAZUMOTO, and KAZUHIKO AOMOTO. "INTEGRAL REPRESENTATION FOR THE GR AND PARTITION FUNCTION IN QUANTUM STATISTICAL MECHANICS OF EXCLUSION STATISTICS." International Journal of Modern Physics B 14, no. 05 (2000): 485–506. http://dx.doi.org/10.1142/s0217979200000455.

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We derive an exact integral representation for the gr and partition function for an ideal gas with exclusion statistics. Using this we show how the Wu's equation for the exclusion statistics appears in the problem. This can be an alternative proof for the Wu's equation. We also discuss that singularities are related to the existence of a phase transition of the system.

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46

Badasyan, Artem. "System Size Dependence in the Zimm–Bragg Model: Partition Function Limits, Transition Temperature and Interval." Polymers 13, no. 12 (2021): 1985. http://dx.doi.org/10.3390/polym13121985.

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Within the recently developed Hamiltonian formulation of the Zimm and Bragg model we re-evaluate several size dependent approximations of model partition function. Our size analysis is based on the comparison of chain length N with the maximal correlation (persistence) length ξ of helical conformation. For the first time we re-derive the partition function of zipper model by taking the limits of the Zimm–Bragg eigenvalues. The critical consideration of applicability boundaries for the single-sequence (zipper) and the long chain approximations has shown a gap in description for the range of exp

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47

Lee, Julian, and Koo-Chul Lee. "Exact zeros of the partition function for a continuum system with double Gaussian peaks." Physical Review E 62, no. 4 (2000): 4558–63. http://dx.doi.org/10.1103/physreve.62.4558.

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48

Lee, Jae Hwan, Seung-Yeon Kim, and Julian Lee. "Exact partition function zeros and the collapse transition of a two-dimensional lattice polymer." Journal of Chemical Physics 133, no. 11 (2010): 114106. http://dx.doi.org/10.1063/1.3486176.

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49

Kouzoudis, D. "Exact analytical partition function and spin gap for a 2×3 quantum spin ladder." Journal of Magnetism and Magnetic Materials 214, no. 1-2 (2000): 112–18. http://dx.doi.org/10.1016/s0304-8853(99)00809-4.

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50

Hedemann, Samuel R. "Integer Partition Function in Nonrecursive Form and Related Results." October 28, 2017. https://doi.org/10.5281/zenodo.1038438.

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