Mathematica Implementation Research Articles

Algorithmic boundedness-from-below conditions for generic scalar potentials

Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB conditions for any polynomial scalar potential. The method relies on elements of multivariate algebra, in particular, on resultants and on the spectral theory of tensors, which is being developed by the mathematical community. We give first a pedagogical introduction to this approach, illustrate it with elementary examples, and then present the working Mathematica implementation publicly available at GitHub. Due to the rapidly increasing complexity of the problem, we have not yet produced ready-to-use analytical BFB conditions for new multi-scalar cases. But we are confident that the present implementation can be dramatically improved and may eventually lead to such results.

Exploring the spectrum of planar AdS4/CFT3 at finite coupling

The Quantum Spectral Curve (QSC) equations for planar mathcal{N}=6 super-conformal Chern-Simons (SCS) are solved numerically at finite values of the coupling constant for states in the mathfrak{s}mathfrak{l}left(2Big|1right) sector.New weak coupling results for conformal dimensions of operators outside the mathfrak{s}mathfrak{l}(2) -like sector are obtained by adapting a recently proposed algorithm for the QSC perturbative solution. Besides being interesting in their own right, these perturbative results are necessary initial inputs for the numerical algorithm to converge on the correct solution.The non-perturbative numerical outcomes nicely interpolate between the weak coupling and the known semiclassical expansions, and novel strong coupling exact results are deduced from the numerics. Finally, the existence of contour crossing singularities in the TBA equations for the operator 20 is ruled out by our analysis.The results of this paper are an important test of the QSC formalism for this model, open the way to new quantitative studies and provide further evidence in favour of the conjectured weak/strong coupling duality between mathcal{N}=6 SCS and type IIA superstring theory on AdS4 × CP3. Attached to the arXiv submission, a Mathematica implementation of the numerical method and ancillary files containing the numerical results are provided.

Extending the functionality of a symbolic computational dynamic solver by using a novel term-tracking method

Symbolic computational dynamic solvers are currently under development in order to provide new and powerful tools for modelling nonlinear dynamical systems. Such solvers consist of two parts; the core solver, which comprises an approximate analytical method based on perturbation, averaging, or harmonic balance, and a specialised term-tracker. A term-tracking approach has been introduced to provide a powerful new feature into computational approximate analytical solutions by highlighting the many mathematical connections that exist, but which are invariably lost through processing, between the physical model of the system, the solution procedure itself, and the final result which is usually expressed in equation form. This is achieved by a highly robust process of term-tracking, recording, and identification of all the symbolic mathematical information within the problem. In this paper, the novel source and evolution encoding method is introduced for the first time and an implementation in Mathematica is described through the development of a specialised algorithm.

Systematics of aligned axions

We describe a novel technique that renders theories of N axions tractable, and more generally can be used to efficiently analyze a large class of periodic potentials of arbitrary dimension. Such potentials are complex energy landscapes with a number of local minima that scales as sqrt{N!} , and so for large N appear to be analytically and numerically intractable. Our method is based on uncovering a set of approximate symmetries that exist in addition to the N periods. These approximate symmetries, which are exponentially close to exact, allow us to locate the minima very efficiently and accurately and to analyze other characteristics of the potential. We apply our framework to evaluate the diameters of flat regions suitable for slow-roll inflation, which unifies, corrects and extends several forms of “axion alignment” previously observed in the literature. We find that in a broad class of random theories, the potential is smooth over diameters enhanced by N3/2 compared to the typical scale of the potential. A Mathematica implementation of our framework is available online.

MultivariateResidues : A Mathematica package for computing multivariate residues

Multivariate residues appear in many different contexts in theoretical physics and algebraic geometry. In theoretical physics, they for example give the proper definition of generalized-unitarity cuts, and they play a central role in the Grassmannian formulation of the S-matrix by Arkani-Hamed et al. In realistic cases their evaluation can be non-trivial. In this paper we provide a Mathematica package for efficient evaluation of multivariate residues based on methods from computational algebraic geometry. Program summaryProgram Title:MultivariateResiduesProgram Files doi:http://dx.doi.org/10.17632/7z87sf98xs.1Licensing provisions: GNU General Public License (GPLv3)Programming language: Wolfram Mathematica version 7.0 or higherNature of problem: Evaluation of multivariate complex residuesSolution method: Mathematica implementation

Azurite: An algebraic geometry based package for finding bases of loop integrals

For any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package Azurite (A ZUR ich-bred method for finding master I nTE grals), which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems Singular and Mathematica. It can moreover analytically calculate the part of the IBP identities that is supported on the cuts. In some cases, the basis obtained by Azurite may be slightly overcomplete. Program summaryProgram Title:AzuriteProgram Files doi:http://dx.doi.org/10.17632/g7p7z3w9dj.1Licensing provisions: GNU General Public License (GPL)Programming language: Wolfram Mathematica version 10.0 or higherSupplementary material: A manual in the form of a Mathematica notebookNature of problem: Determination of a basis of the space of loop integrals spanned by a given Feynman diagram and all of its subdiagramsSolution method:Mathematica implementation

Analytical method for calculation of deviations from intended dosages during multi-infusion

BackgroundIn this paper, a new method is presented that combines mechanical compliance effects with Poiseuille flow and push-out effects (“dead volume”) in one single mathematical framework for calculating dosing errors in multi-infusion set-ups. In contrast to existing numerical methods, our method produces explicit expressions that illustrate the mathematical dependencies of the dosing errors on hardware parameters and pump flow rate settings.MethodsOur new approach uses the Z-transform to model the contents of the catheter, and after implementation in Mathematica (Wolfram), explicit expressions are produced automatically. Consistency of the resulting analytical expressions has been examined for limiting cases, and three types of in-vitro measurements have been performed to obtain a first experimental test of the validity of the theoretical results.ResultsThe relative contribution of various factors affecting the dosing errors, such as the Poiseuille flow profile, resistance and internal volume of the catheter, mechanical compliance of the syringes and the various pump flow rate settings, can now be discerned clearly in the structure of the expressions generated by our method. The in-vitro experiments showed a standard deviation between theory and experiment of 14% for the delay time in the catheter, and of 13% for the time duration of the dosing error bolus.ConclusionsOur method provides insight and predictability in a large range of possible situations involving many variables and dependencies, which is potentially very useful for e.g. the development of a fast, bed-side tool (“calculator”) that provides the clinician with a precise prediction of dosing errors and delay times interactively for many scenario’s. The interactive nature of such a device has now been made feasible by the fact that, using our method, explicit expressions are available for these situations, as opposed to conventional time-consuming numerical simulations.

Correlation factors for impurity diffusion on the sublattice of dodecahedral sites in garnet

The method of Koiwa and Ishioka (Philos Mag A 47:927–938, 1983) is used, with slight modification, to evaluate the correlation factor for vacancy-mediated diffusion of impurity atoms on the sublattice of dodecahedral sites in garnet, as a function of the relevant vacancy-jump frequencies. The required values of the lattice Green’s function were obtained from multiple Monte Carlo simulations in lattices of progressively larger size, extrapolated to an infinite lattice using a model that linearizes the dependence of the functional value on lattice size. As Online Resources, codes are provided that permit evaluation of the correlation factor for any chosen set of vacancy-jump frequencies, for implementation in either Mathematica® or Matlab®.

Coding sequence density estimation via topological pressure

We give a new approach to coding sequence (CDS) density estimation in genomic analysis based on the topological pressure, which we develop from a well known concept in ergodic theory. Topological pressure measures the 'weighted information content' of a finite word, and incorporates 64 parameters which can be interpreted as a choice of weight for each nucleotide triplet. We train the parameters so that the topological pressure fits the observed coding sequence density on the human genome, and use this to give ab initio predictions of CDS density over windows of size around 66,000 bp on the genomes of Mus Musculus, Rhesus Macaque and Drososphilia Melanogaster. While the differences between these genomes are too great to expect that training on the human genome could predict, for example, the exact locations of genes, we demonstrate that our method gives reasonable estimates for the 'coarse scale' problem of predicting CDS density. Inspired again by ergodic theory, the weightings of the nucleotide triplets obtained from our training procedure are used to define a probability distribution on finite sequences, which can be used to distinguish between intron and exon sequences from the human genome of lengths between 750 and 5,000 bp. At the end of the paper, we explain the theoretical underpinning for our approach, which is the theory of Thermodynamic Formalism from the dynamical systems literature. Mathematica and MATLAB implementations of our method are available at http://sourceforge.net/projects/topologicalpres/ .

All one-loop NMHV gluon amplitudes in $ \mathcal{N} $ = 1 SYM

Abstract We compute the next-to-maximally-helicity-violating one-loop n-gluon amplitudes in $ \mathcal{N} $ = 1 super-Yang-Mills theory. These amplitudes contain three negative-helicity gluons and an arbitrary number of positive-helicity gluons, and constitute the first infinite series of amplitudes beyond the simplest, MHV, amplitudes. We assemble ingredients from the $ \mathcal{N} $ = 4 NMHV tree super-amplitude into previously unwritten double cuts and use the spinor integration technique to calculate all bubble coefficients. We also derive the missing box coefficients from quadruple cuts. Together with the known formula for three-mass triangles, this completes the set of NMHV one-loop master integral coefficients in $ \mathcal{N} $ = 1 SYM. To facilitate further use of our results, we provide their Mathematica implementation.

A Constructive Method for Computing Generalized Manley-Rowe Constants of Motion

AbstractThe Manley-Rowe constants of motion (MRC) are conservation laws written out for a dynamical system describing the time evolution of the amplitudes in resonant triad. In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants (gMRC) and give a constructive method how to compute them. We also give details of a Mathematica implementation of this method. While MRC provide integrability of the underlying dynamical system, gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.

The k-means clustering technique: General considerations and implementation in Mathematica

Data clustering techniques are valuable tools for researchers working with large databases of multivariate data. In this tutorial, we present a simple yet powerful one: the k-means clustering technique, through three different algorithms: the Forgy/Lloyd, algorithm, the MacQueen algorithm and the Hartigan & Wong algorithm. We then present an implementation in Mathematica and various examples of the different options available to illustrate the application of the technique. Data clustering techniques are descriptive data analysis techniques that can be applied to multivariate data sets to uncover the structure present in the data. They are particularly useful when classical second order statistics (the sample mean and covariance) cannot be used. Namely, in exploratory data analysis, one of the assumptions that is made is that no prior knowledge about the dataset, and therefore the dataset’s distribution, is available. In such a situation, data clustering can be a valuable tool. Data clustering is a form of unsupervised classification, as the clusters are formed by evaluating similarities and dissimilarities of intrinsic characteristics between different cases, and the grouping of cases is based on those emergent similarities and not on an external criterion. Also, these techniques can be useful for datasets of any dimensionality over three, as it is very difficult for humans to compare items of such complexity reliably without a support to aid the comparison.

Mathematica implementation of the high order finite element method applied to eigenproblems

In this paper an \(hp\)-FEM implementation on Mathematica is discussed. FEM-implementations on higher-level programming platforms are useful for prototyping new algorithms and ideas, but also serve as testing ground for interesting programming techniques. Here, an \(hp\)-adaptive algorithm for eigenproblems, and the use of precomputed data and generation of highly graded \(hp\)-meshes, are examples of the former and latter, respectively. The performance of the code is evaluated in relation to a suite of benchmark problems for the Laplacian and thin solids in elasticity.

Algorithm 923

Computing the Pfaffian of a skew-symmetric matrix is a problem that arises in various fields of physics. Both computing the Pfaffian and a related problem, computing the canonical form of a skew-symmetric matrix under unitary congruence, can be solved easily once the skew-symmetric matrix has been reduced to skew-symmetric tridiagonal form. We develop efficient numerical methods for computing this tridiagonal form based on Gaussian elimination, using a skew-symmetric, blocked form of the Parlett-Reid algorithm, or based on unitary transformations, using block Householder transformations and Givens rotations, that are applicable to dense and banded matrices, respectively. We also give a complete and fully optimized implementation of these algorithms in Fortran (including a C interface), and also provide Python, Matlab and Mathematica implementations for convenience. Finally, we apply these methods to compute the topological charge of a class D nanowire, and show numerically the equivalence of definitions based on the Hamiltonian and the scattering matrix.

A general framework for solving Riemann–Hilbert problems numerically

A new, numerical framework for the approximation of solutions to matrix-valued Riemann---Hilbert problems is developed, based on a recent method for the homogeneous Painleve II Riemann---Hilbert problem. We demonstrate its effectiveness by computing solutions to other Painleve transcendents. An implementation in Mathematica is made available online.

A recursive method to calculate UV-divergent parts at one-loop level in dimensional regularization

A method is introduced to calculate the UV-divergent parts at one-loop level in dimensional regularization. The method is based on the recursion, and the basic integrals are just the scaleless integrals after the recursive reduction, which involve no other momentum scales except the loop momentum itself. The method can be easily implemented in any symbolic computer language, and a implementation in Mathematica is ready to use. Program summaryProgram title: UVPartCatalogue identifier: AELY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AELY_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 26 361No. of bytes in distributed program, including test data, etc.: 412 084Distribution format: tar.gzProgramming language: MathematicaComputer: Any computer where the Mathematica is running.Operating system: Any capable of running Mathematica.Classification: 11.1External routines: FeynCalc (http://www.feyncalc.org/), FeynArts (http://www.feynarts.de/)Nature of problem: To get the UV-divergent part of any one-loop expression.Solution method: UVPart is a Mathematica package where the recursive method has been implemented.Running time: In general it is below one second.

Extension of HPL to complex arguments

In this paper we describe the extension of the Mathematica package HPL to treat harmonic polylogarithms of complex arguments. The harmonic polylogarithms have been introduced by Remiddi and Vermaseren [E. Remiddi, J.A.M. Vermaseren, Int. J. Modern Phys. A 15 (2000) 725, hep-ph/9905237] and have many applications in high energy particle physics. New version program summary Program title: HPL Catalogue identifier: ADWX_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWX_v2_0.html Program obtainable from: CPC Program Library, Queenʼs 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.: 13 610 No. of bytes in distributed program, including test data, etc.: 1 055 706 Distribution format: tar.gz Programming language: Mathematica 7/8. Computer: All computers running Mathematica. Operating system: Operating systems running Mathematica. Supplementary material: Additional “high weight” MinimalSet files available. Classification: 4.7. Catalogue identifier of previous version: ADWX_v1_0 Journal reference of previous version: Comput. Phys. Comm. 174 (2006) 222 Does the new version supersede the previous version?: Yes Nature of problem: Computer algebraic treatment of the harmonic polylogarithms which appear in the evaluation of Feynman diagrams. Solution method: Mathematica implementation. Reasons for new version: Added treatment of complex arguments. Details in arXiv:hep-ph/0703052. Summary of revisions: Added treatment of complex arguments. Details in arXiv:hep-ph/0703052. Running time: A few seconds for each function.

New recurrence algorithms for the nonclassic Adomian polynomials

In this article, we present new algorithms for the nonclassic Adomian polynomials, which are valuable for solving a wide range of nonlinear functional equations by the Adomian decomposition method, and introduce their symbolic implementation in MATHEMATICA. Beginning with Rach’s new definition of the Adomian polynomials, we derive the explicit expression for each class of the Adomian polynomials, e.g. A m = ∑ k = 1 m f ( k ) ( u 0 ) Z m , k for the Class II, III and IV Adomian polynomials, where the Z m , k are called the reduced polynomials. These expressions provide a basis for developing improved algorithmic approaches. By introducing the index vectors, the recurrence algorithms for the reduced polynomials are suitably deduced, which naturally lead to new recurrence algorithms for the Class II and Class III Adomian polynomials. MATHEMATICA programs generating these classes of Adomian polynomials are subsequently presented. Computation shows that for computer generation of the Class III Adomian polynomials, the new algorithm reduces the running times compared with the definitional formula. We also consider the number of summands of these classes of Adomian polynomials and obtain the corresponding formulas. Finally, we demonstrate the versatility of the four classes of Adomian polynomials with several examples, which include the nonlinearity of the form f ( t , u ) , explicitly depending on the argument t .

Numeric and symbolic evaluation of the pfaffian of general skew-symmetric matrices

Evaluation of pfaffians arises in a number of physics applications, and for some of them a direct method is preferable to using the determinantal formula. We discuss two methods for the numerical evaluation of pfaffians. The first is tridiagonalization based on Householder transformations. The main advantage of this method is its numerical stability that makes unnecessary the implementation of a pivoting strategy. The second method considered is based on Aitkenʼs block diagonalization formula. It yields to a kind of LU (similar to Choleskyʼs factorization) decomposition (under congruence) of arbitrary skew-symmetric matrices that is well suited both for the numeric and symbolic evaluations of the pfaffian. Fortran subroutines (FORTRAN 77 and 90) implementing both methods are given. We also provide simple implementations in Python and Mathematica for purpose of testing, or for exploratory studies of methods that make use of pfaffians. Program summaryProgram title:PfaffianCatalogue identifier: AEJD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJD_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2281No. of bytes in distributed program, including test data, etc.: 13 226Distribution format: tar.gzProgramming language: Fortran 77 and 90Computer: Any supporting a FORTRAN compilerOperating system: Any supporting a FORTRAN compilerRAM: a few MBClassification: 4.8Nature of problem: Evaluation of the pfaffian of a skew symmetric matrix. Evaluation of pfaffians arises in a number of physics applications involving fermionic mean field wave functions and their overlaps.Solution method: Householder tridiagonalization. Aitkenʼs block diagonalization formula.Additional comments: Python and Mathematica implementations are provided in the main body of the paper.Running time: Depends on the size of the matrices. For matrices with 100 rows and columns a few milliseconds are required.

Topological entropy of DNA sequences

Topological entropy has been one of the most difficult to implement of all the entropy-theoretic notions. This is primarily due to finite sample effects and high-dimensionality problems. In particular, topological entropy has been implemented in previous literature to conclude that entropy of exons is higher than of introns, thus implying that exons are more 'random' than introns. We define a new approximation to topological entropy free from the aforementioned difficulties. We compute its expected value and apply this definition to the intron and exon regions of the human genome to observe that as expected, the entropy of introns are significantly higher than that of exons. We also find that introns are less random than expected: their entropy is lower than the computed expected value. We also observe the perplexing phenomena that introns on chromosome Y have atypically low and bimodal entropy, possibly corresponding to random sequences (high entropy) and sequences that posses hidden structure or function (low entropy). A Mathematica implementation is available at http://www.math.psu.edu/koslicki/entropy.nb koslicki@math.psu.edu Supplementary data are available at Bioinformatics online.