MAST Institute for Scientific Computing (MAST-ISC)

Significant Issues in Scientific Computing

It is safe to say that almost every practicing scientist today uses a computer. Many use the computer to develop models, even among experimentalists. How many program such models efficiently? What does it mean to program efficiently, from a scientific point-of-view? How many are simply patching old code that has become somewhat worn out?
    In a recent survey [2][3] most respondents claimed to spend about 15 hours a week developing software and 20 hours a week using it; accounting for about 70% of their time. Almost half claim that they spend more time developing software now than they did five years ago, and 70% claim that they spend much more time using it. More than half reported developing their own software, while 35% reported working in groups of up to five people. The primary hardware are desktop machines and small clusters running small codes (usually less than 5000 lines of code) and most time is spent on coding and debugging. The biggest complaint is poor documentation. The biggest worry is that people are not sure how to verify their code, and second is that they do not know how to produce the most efficient code. The vast majority of scientists have little or no formal training in software design, nor do they feel that they have the time to acquire it; they take an informal, self-study approach to the problem.
    The bottom line of all of this is that most scientists use computers at a fundamentally primitive level. This impedes what work can be done through self-limitation in the scope of work believed possible, in the choice of methods to attack problems, and in the tools to implement those methods.
    What is needed is a way to produce top-quality research in the methods of scientific computation, ways to implement these methods that are as close to platform independent as possible, the application of such implementation to problems of scientific interest, the dissemination of these ideas to the scientific clients of such work, and the education of existing and future scientists in them.

Why a Not-For-Profit Institute of Scientific Computation?

The purpose of a scientific computation institute is to bring together experts in software design, mathematical and computational methods, and application fields into one place where they can discuss their respective issues, apply their respective talents, and solve problems of mutual interest. The emphasis is on solving scientific problems through computational means. Some of the participants should be from industry, some from government, some from academe, and some from the public at large. A non-profit organization is ideally suited for the task of merging these disparate populations.
    The goal of such an institute should be the development of research in both applications and methods, and the development of educational and operational resources for the individual scientist. This will result in a set of deliverables. Some deliverables will be research and review papers. Some will be tutorials in the effective use of software, hardware, specific computational methods, specific mathematical methods, and specific methods that are applied to specific problems. All of these will be available on the Internet through a dedicated web site. The institute will also provide several services:

  1. Courses, both in-person and over the web, to teach basic and advanced techniques in Mathematica, heuristics, algorithm development, software development, research methodology, computational techniques for specific types of problems, and mathematical/scientific principles relevant to specific types of problems.

  2. A Summer School to teach effective programming, heuristics, methodology, and research. To go to the Summer School web site, click here.

  3. Summer Internships to bring in students from among high school seniors,  undergraduate and graduate intuitions of higher learning (both foreign and domestic) to teach them to operate and develop software effectively, and to provide them with solid research experience. This will include frequent meetings for discussing research problems.

  4. Year-long internships for members of the general public (Citizen Scientists, as they are called), recent graduates and post-docs, and professional scientists, mathematicians, and engineers, to provide more substantial research opportunities.


Madison Area Science and Technology (MAST) is a not-for-profit scientific research and education organization that was established in 1999. During the last eleven years we have worked with government (the National Weather Service and local emergency management agencies), industry, and academic institutions. The senior personnel of MAST have extensive computational experience in a wide range of applications and methods. All senior personnel have years of programming experience in Mathematica.


MAST has chosen to use Mathematica, produced by Wolfram Research, as its primary software development system.

Some of the Scientific Problems to be Attacked

While the list of interesting problems that can be addressed by such a group as the MAST-ISC is almost endless, there are some problems that are of particular interest to our staff. The problems we can address are also chosen as those that can be explored effectively with desktop work stations, small clusters, and GPU-equipped systems; we are not in competition with supercomputer centers.

Compact Objects

Compact objects are white dwarf stars, neutron stars, and black holes. They are interesting because of the incredible states of compression of the matter that form them. White dwarfs experience electron degeneracy, neutron stars experience neutron degeneracy and the possibility of quark matter phase transitions at their cores, and black holes experience a compression that holds out the possibility of the matter within them being crushed out of physical existence through an artifact called a singularity. There are several good texts on compact objects that we use [7][8][9][10] [11][12][13][14]. Here are some specific problems of interest:

The equation of state for neutron star matter and neutron star structure [15], [16].

Neutron star propulsion.

The process of neutron star formation [17].

Black hole formation [18].

The fate of matter falling into a black hole [19].

Scattering of waves by black holes [20].

Shock Waves

Whether in diffuse matter in the interstellar medium, a blast wave in a fluid, or a compression wave in a dense solid, shock waves are a discontinuity that propagates through the medium within which they travel. There are several good texts on shock waves that we use [21][22][23][24]. Here are some specific problems of interest:

Shock wave dynamics in condensed and diffuse matter [25].

Detonation dynamics [26].

Shock waves in curved spacetimes [27].

Fluid Dynamics

Fluids account for as much as three quarters of the states of matter in the universe. If we include exotic states such a superfluid vortices, quantum gauge hydrodynamics, and treating the matter in the early universe as a hot-dense plasma, fluid models are finding wide application. There are several good texts on fluid dynamics that we use [28][29][30][31][32][33][34][35][36]. Here are some specific problems of interest:

Specialized hydrocodes for various purposes [4].

Thunderstorm modeling [37].

Tornado modeling [38].

Accretion disk modeling [39][40][41][42].

Asteroid impact dynamics [43].

Underwater oil leakage plume dynamics [43].

Pyroclastic flow dynamics [43].

Chaotic Dynamical Systems

Dynamical systems are systems that exhibit change with respect to a parameter, such as time. Chaotic dynamical systems exhibit chaotic change, that is they have a positive Lyapunov exponent, or they have a strange attractor. There are several good texts on dynamical systems and chaos theory that we use [44][45][46][47][48][49][50][51][52][53]. Here are some specific problems of interest:

Calculating Lyapunov exponents form first principles [54].

Differential equations and analysis on fractals [55][56].

Scattering Theory

The interaction between objects, on whatever scale, comes down to scattering cross sections for impact and absorption. The applications of scattering theory are very important and range from CT scan data analysis, to sonar and radar theory, to collider theory. There are several good texts on scattering theory that we use [57][58][59][60][61][62]. Here are some specific problems of interest:

The role of time delay in quantum scattering [63].

Scattering theory and unitary operators, orthogonal polynomials, and CMV matrices [64][65][66][67].

Casimir theory using scattering amplitudes [68].

Compton scattering in the astrophysical context [69].

Application of scattering theory to mesoscopic circuit dynamics [70].

Inverse scattering through the Lippmann-Schwinger equation [71][72][73].

Quantum Computation and Information Theory

Quantum computers hold a great deal of promise due to quantum effects, such as the superposition of states, should the problems of quantum algorithm design be overcome. There are several good texts on quantum computing that we use [74][75][76][77][78]. Here is a specific problem of interest:

Information theory.

Developing a quantum computer simulator [75][79][80].

Bioinformatics and Biophysics

Bioinformatics is the application of mathematical and computational tools to problems in molecular biology. One area of development is the incorporation of statistical mechanical ideas into bioinformatics, specifically the idea of entropy of information channels. Biophysics is the application of existing physics to biological systems and the invention of new physics based on biological systems. There are several good texts on bioinformatics and biophysics that we use [81][82][83][84][85]. Here are some specific problems of interest:

Computational modeling of HIV-therapy schemes [43].

Emergence of complexity [86].

Predicting the distribution of species [43].

Correlation of descent order genomic segments in viruses with their database accession date. [43].

Bayesian phylogenetic methods of cancer diagnosis given publicly available blood serum protein mass spectrometry data [43].

Bayesian neural network methods for diagnosing cancer based on micrographs [43].

Solution Methods for ODEs and PDEs

Most of the application of mathematics to the sciences is in the form of differential equations. We intend to develop symbolic and numerical methods of solving, approximating solutions, and estimating solutions of ODEs and PDEs. There are several good texts on ODEs and PDEs that we use [87][88][89][90][91][92][93][94][05][96][97][98][99]. Here are some specific problems of interest:

Apply techniques arising from partial Fourier transforms [100].

Devise tests to determine solvability or non-solvability of operators [101][102][103][104][105].

Automation of integral estimation [106][107].

ODE solution methods [108][109].

Complete integrability [110][111][112][113][114].

Linearizability [115][116][117][118].

Geometry of ODEs [119][120].

Solution methods for PDEs [121].

Data Analysis

Time series analysis of dinosaur extinction.

Educational Resources to be Developed

MAST expects to produce the following deliverables over the course of its first year:

  1. Calculus tutorial for High School seniors and Citizen Scientists.

  2. Mathematica tutorials, classes, and online videos for those not familiar with it.

  3. Mathematical tutorials covering ODEs, linear algebra, PDEs, and numerical methods.

  4. Proceedings of the Summer School.

These materials will be made available on the web and in hard copy through the Lulu print-on-demand publishing service. Development will not be limited to these, as all discussion sessions will be recorded and transcribed, and additional tutorials and writing will be developed as required.


[1] Greg Wilson, (2006), "Where's the Real Bottleneck in Scientific Computing?", American Scientist, online at

[2] Greg Wilson, (2009), "How Do Scientists Really Use Computers?", American Scientist, online at

[3] Jo Erskine Hannay, Carolyn MacLeod, Janice Singer, Hans Petter Langtangen, Dietmar Pfahl, Greg Wilson, "How do scientists develop and use scientific software?", Proceedings of the 2009 ICSE Workshop on Software Engineering for Computational Science and Engineering.

[4] George E. Hrabovsky, (2010), A Mathematica-Based Hydrodynamics Model, Proceedings of the 2010 International Conference on Scientific Computing, WorldComp 2010, Las Vegas, NV.

[5] Steven E. Koonan, Dawn C. Meredith, (1991), Computational Physics, Westview Press.

[6] Gerome H. Millgram, (2003), Numerical Methods in Incompressible Fluid Mechanics, Lecture Notes.

[7] Stuart L. Shapiro, Saul A. Teukolsky, (1983), Black Holes, White Dwarfs, and Neutron Stars, John Wiley and Sons, Inc.

[8] Norman K. Glendenning, (2000), Compact Stars: Nuclear Physics, Particle Physics, and General Relativity, Springer-Verlag, New York, Inc.

[9] Norman K. Glendenning, (2007), Special and General Relativity With Applications to White Dwarfs, Neutron Stars, and Black Holes, Springer-Verlag, New York, Inc.

[10] Andreas Schmitt, (2010), Dense matter in compact stars—A pedagogical introduction—, arXiv:1001.3294v1 [astro-ph.SR].

[11] Ya B. Zel'dovich, I. D. Novikov, (1971), Relativistic Astrophysics I, The University of Chicago Press, reprinted by Dover Publications, Inc. in 1996 as Stars and Relativity.

[12] S. D. Kawaler, I. Novikov, G. Srinivaan, (1997), Stellar Renmants, Springer-Verlag, New York, Inc.

[13] S. Chandrasekhar, (1983), The Mathematical Theory of Black Holes, Oxford University Press, reprinted in 1998 the Oxford Classics series.

[14] Valerii P. Frolov, Igor D. Novikov, (1998), Black Hole Physics, Kluwer Academic Publishers.

[15] T.Klahn, C.D.Roberts, D.B.Blaschke, F.Sandin, (2009), "Neutron Stars and the High Density Equation of State," Proc. 5th ANL/MSU/JINA/INT FRIB Workshop on Bulk Nuclear Properties, E. Lansing, Nov. 19-22, 2008

[16] S. Gandolfi, A. Yu Illarionov, S. Fantoni, J.C. Miller, F. Pederiva, K.E. Schmidt, (2010), "Microscopic calculation of the equation of state of nuclear matter and neutron star structure," Mon Not.Roy.Astron.Soc. 404:L35-L39.

[17] Hans-Thomas Janka, (2004), "Neutron Star Formation and Birth Properties," Young Neutron Stars and Their Environments, IAU Symposium, Vol. 218, 2004, F. Camilo and B. M. Gaensler, eds.

[18] Yuichiro Sekiguchi, Masaru Shibata, (2010), "Formation of black hole and accretion disk in collapsar," arXiv:1009.5303v1 [astro-ph.HE].

[19] Shuang Nan Zhang, Yuan Liu, (2008), "Observe matter falling into a black hole," AIPConf. Proc.968:384-391, arXiv:0710.2443v1 [astro-ph].

[20] K Glampedakis, N Andersson, (2001), "Scattering of scalar waves by rotating black holes," Class.Quant.Grav. 18 (2001) 1939-1966.

[21] Ya B. Ael'dovich, Yu P. Raizer, (1966),  Edited by Wallace D. Hayes, Ronald F. Probstein, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena I & II, Academic Press, reprinted as one volume in 2002 by Dover Publications, Inc.

[22] C. J. Chapman, (2000), High Speed Flow, Cambridge University Press.

[23] P. L. Sachdev, (2004), Shock Waves and Explosions, Chapman & Hall/CRC monographs and surveys in pure and applied mathematics ; 132.

[24] A. Lichnerowicz, (1994), Magnetohydrodynamics waves and shock waves 1n curved space-time, by Kluwer Academic Publishers, Mathematical physics studies . v. 14.

[25] Christophe Berthon, Frédéric Coquel, Philippe G. LeFloch, (2010), "Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems," arXiv:1006.1102v1 [math.AP].

[26] Hua-Shu Dou, Her Mann Tsai, Boo Cheong Khoo, Jianxian Qiu, (2010), "Simulations of Detonation Wave Propagation in Rectangular Ducts Using a Three-Dimensional WENO Scheme," Combustion & Flame, Vol.154, 2008, pp.644-659, arXiv:1006.1218v1[physics.flu-dyn].

[27] Konstadinos Sfetsos, (1995), "On Gravitational Shock Waves in Curved Spacetimes," Nucl. Phys. B436 (1995) 721-746.

[28] Tsutomu Kambe, (2007), Elementary Fluid Mechanics, World Scientific Publishing Pte. Ltd.

[29] F. Durst, (2008), Fluid Mechanics, Springer-Verlag Berlin Heidelberg.

[30] L. D. Landau, E. M. Lifschitz, (1987), Fluid Mechanics, Second Edition, Butterworth-Heinemann, reprinted with corrections in 2000.

[31] Cathie Clark, Bob Carswell, (2007), Principles of Astrophysical Fluid Dynamics, Cambridge University Press.

[32] Sergay P. Kirilov, Evgenii V. Vorozhtsov, Vasily M. Fomin, (1999), Foundations of Fluid Mechanics with Applications, Birkhãuser Boston.

[33] Antonia Romano, Renato Lancellotta, Addolorata Marasco, (2006), Continuum Mechanics using Mathematica, Birkhãuser Boston.

[34] Antonia Romano, Renato Lancellotta, Addolorata Marasco, (2010), Continuum Mechanics Advanced Topics and Research Tre3nds, Birkhãuser Boston.

[35] William F. Hughes, John A. Brighton, (1999), Fluid Dynamics, McGraw-Hill, Schaum's Outline Series.

[36] Davison E. Soper, (1976), Classical Field Theory, John Wiley and sons, Inc. New York. Reprinted by Dover Publications, Inc. in 2008.

[37] Paul M. Markowski, Jerry Y. Harrington, (2005), "A Simulation of a Supercell Thunderstorm with Emulated Radiative Cooling beneath the Anvil," J. Atmos. Sci 62, 2607-2617.

[38] D. C. Lewellen, W. S. Lewllen, (2007), "Near-Surface Intensification of Tornado Vortices", J. Atmos. Sci 64, 2176-2194.

[39] Julian Frank, Andrew King, Derek Raine, (2002), Accretion Power in Astrophysics, Third Edition, Cambridge University Press.

[40] J. C. B. Papaloizou, D. N. C. Lin, (1995), "Theory of Accretion Disks I: Angular Momentum Transport Processes," Annu. Rev. Astron. Astrophys, 1995. 33:505-4.

[41] J. C. B. Papaloizou, D. N. C. Lin, (1996), "Theory of Accretion Disks II: Application to Observed Systems," Annu. Rev. Astron. Astrophys. 1996. 34:703–47.

[42] H. C. Spruit, (2010), "Accretion Disks," arXiv:1005.5279v1 [astro-ph.HE].

[43] Jack K. Horner, (2010), Private conversation.

[44] Steven H. Strogatz, (1994), Nonlinear Dynamics and Chaos, Perseus Books.

[45] John Guckenheimer, Philip Holmes, (1983), Nonlinear Oscilaltions, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, Inc.

[46] Stephen Lynch, (2007), Dynamical Systems with Applications using Mathematica, Birkhãuser Boston.

[47] Predrag Cvitanovi´c, Roberto Artuso, Ronnie Mainieri, Gregor Tanner, G´abor Vattay, (2010), Chaos: Classical and Quantum,

[48] Jun Kigami, (2001), Analysis on Fractals, Cambridge University Press.

[49] Kenneth Falconer, (2003), Fractal Geometry, 2nd Ed., Jhn Wiley and Sons Ltd.

[50] Robert Herman, (1991), Geometric Structures in Nonlinear Physics, Math Sci Press.

[51] Tsutomu Kambe, (2004), Geometrical Theory of Dynamical Systems and Fluid Flows, World Scientific Publishing Pte. Ltd.

[52] Gerald Edgar, (2008), Measure, Topology, and Fractal Geometry, 2nd Ed., Springer Science + Business Media, LLC.

[53] Robert Gilmore, Christophe Letellier, (2007), The Symmetry of Chaos, Oxford University Press, Inc.

[54] Lando Caiani, Lapo Casetti, Cecilia Clementi, Marco Pettini, (1997), "Geometry of dynamics, Lyapunov exponents and phase transitions," Phys.Rev.Lett. 79 (1997) 4361-4364.

[55] Robert S. Strichartz, (2006), Differential Equations on Fractals A Tutorial, Princeton University Press.

[56] Santanu Raut, Dhurjati Prasad Datta, (2009), "Analysis on a Fractal Set," Fractals, vol. 17, No. 1, (2009), 45-52.

[57] Michael Reed, Barry Simon, (1979),Methods of Mathematical Physics III: Scattering Theory, Academic Press, Inc.

[58] Roy Pike, Pierre Sabatier, (2002), Scattering I, Academic Press, Inc.

[59] Roy Pike, Pierre Sabatier, (2002), Scattering II, Academic Press, Inc.

[60] Leung Tsang, Jin Au Kong, Kung-Hau Ding, (2000), Scattering of Electromagnetic Waves: Theories and Applications, John Wiley & Sons, Inc.

[61] Leung Tsang, Jin Au Kong, Kung-Hau Ding, Chi On Ao, (2001), Scattering of Electromagnetic Waves: Numerical Solutions, John Wiley & Sons, Inc.

[62] Leung Tsang, Jin Au Kong,  (2001), Scattering of Electromagnetic Waves: Advanced Topics, John Wiley & Sons, Inc.

[63] S. Richard, R. Tiedra de Aldecoa, (2010), "Time delay is a common feature of quantum scattering theory," arXiv:1008.3433v1 [math-ph].

[64] M. J. Cantero, L. Moral, L. Velazquez, (2004), "Minimal representations of unitary operators and orthogonal polynomials on the unit circle," arXiv:math/0405246v1 [math.CA].

[65] Irina Nenciu, (2006), "CMV matrices in random matrix theory and integrable systems: a survey,"J. Phys. A: Math. Gen. 39 (2006), 8811--8822.

[66] Barry Simon, (2006), "CMV matrices: Five years after,"arXiv:math/0603093v1 [math.SP].

[67] L. Golinskii, A. Kheifets, F. Peherstorfer, P. Yuditskii, (2010), "Scattering theory for CMV matrices: uniqueness, Helson--Szegö and Strong Szegö theorems," arXiv:1008.3284v1 [math.SP].

[68] Sahand Jamal Rahi, Thorsten Emig, Robert L. Jaffe, (2010), "Geometry and material effects in Casimir physics - Scattering theory," arXiv:1007.4355v1 [quant-ph]

[69] Juri Poutanen, Indrek Vurm, (2010), "Theory of Compton scattering by anisotropic electrons," Astrophys. J. Suppl. Ser.,189:286-308, 2010.

[70] Vered Ben-Moshe, Dhurba Rai, Spiros S. Skourtis, Abraham Nitzan, (2010), "Steady state current transfer and scattering theory." arXiv:1002.2775v1 [cond-mat.mes-hall].

[71] Christopher J. Winfield, (2005), "A study of resolvent and spectra for some unbounded quantum potentials," Rocky Mountain Journal of Mathematics vol. 35 (4).

[72] R. de la Madrid, (2006), "The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part I,"J. Phys. A: Math. Gen. 39 (2006) 3949-3979.

[73] R. de la Madrid, (2006), "The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part II: The analytic continuation of the Lippmann-Schwinger bras and kets,"J. Phys. A: Math. Gen. 39 (2006) 3981-4009.

[74] Michael A. Nielsen, Isaac L. Chuang, (2000), Quantum Computation and Quantum Information, Cambridge University Press.

[75] Colin P. Williams, Scott H. Clearwater, (1998), Explorations in Quantum Computing, Springer-Verlag, New York, Inc.

[76] Zdzislaw Meglicki, (2008), Quantum Computing without Magic, The MIT Press.

[77] S.M. Barnett, (2009), Quantum Information, Oxford university press.

[78] N. David Mermin, (2007), Quantum Computer Science, Cambridge University Press.

[79] N. Ratanje, S. Virmani, (2010), "Generalised state spaces and non-locality in the classical simulation of noisy quantum computers," arXiv:1007.3455v1 [quant-ph].

[80] Pablo Arrighi, Jonathan Grattage, (2010), "A Quantum Game of Life," arXiv:1010.3120v1 [quant-ph].

[81] Jonathan Pevsner, (2009), Bioninformatics and Functional Genomics, John Wiley & Sons, Inc.

[82] Ann Finney Batiza, (2006), Bioinformatics, Genomics, and Proteomics, Infobase Publishing.

[83] Russell Schwarz, (2008), Biological Modeling and Simulation, The MIT Press.

[84] Viktor V. Ivanov, Natalya V. Ivanova, (2006), Mathematical Models of the Cell and Cell Associated Objects, Elsevier B. V.

[85] Vasantha Pattabhi, N. Guatham, (2002), Biophysics, Kluwer Academic Publishers.

[86] Juan G. Restrepo, Edward Ott, Brian R. Hunt, (2006), "The emergence of coherence in complex networks of heterogeneous dynamical systems," PhysRevLett.96, 128101 (2006).

[87] C. F. Chan Man Fong, D. De Kee, P. N. Kaloni, (2003), Advanced Mathematics for Engineering and Science, World Scientific Publishing Co. Pte. Ltd.

[88] Michael D. Greenberg, (1998), Advanced Enegineering Mathematics, Prentice Hall, Inc.

[89] K. F. Riley, M. P. Hobson, S. J. Bence, (2006), Mathematical Methods for Physics and Engineering, 3rd Ed. Cambridge University Press.

[90] William E. Boyce, Richard C. DiPrima, (2001), Elementary Differential Equations and Boundary Value Problems, 7th Ed., John Wiley and Sons, inc.

[91] Addolorata Marasco, Antonio Romana, (2001), Scientific Computing with Mathematica Mathematical Problems for Ordinary Differential Equations, Birkhãuser Boston.

[92] Vladimir I. Arnol'd, (1992), Ordinary Differential Equations, Springer-Verlag Berlin Heidellberg New York.

[93] Walter G. Kelley, Allan C. Peterson, (2010), The Theory of Differential Equations, 2nd Ed., Springer Science +Business Media, LLC.

[94] Gerd Baumann, (2000), Symmetry Analysis of Differential Equations with Mathematica, Springer-Verlag, New York, Inc.

[95] Werner M. Seiler, (2010), Involution, The Formal Theory of Differential Equations and its Application in Computer Algebra, Springer-Verlag Berlin Heidellberg.

[96] Arthur David Snider, (1999) Partial Differential Equations Sources and Solutions, Prentice-Hall, inc. Republished in 2006 by Dover Publications, Inc.

[97] Michael E. Taylor, (1996), Partial Differential Equations I, Springer-Verlag, New York, Inc.

[98] Michael E. Taylor, (1996), Partial Differential Equations II, Springer-Verlag, New York, Inc.

[99] Michael E. Taylor, (1996), Partial Differential Equations III, Springer-Verlag, New York, Inc.

[100] Lexing Ying, Sergey Fomel, (2008), "Fast Computation of Partial Fourier Transforms." arXiv:0802.1554v1 [math.NA].

[101] Christopher J. Winfield, (2009), Personal conversation.

[102] Vitali Vougalter, Vitaly Volpert, (2009), "Solvability Conditions for Some non Fredholm Operators," arXiv:0902.3182v1 [math.AP].

[103] Claudia Garetto, (2009), "Sufficient conditions of local solvability for partial differential operators in the Colombeau context" arXiv:0902.0508v1 [math.AP].

[104] Nils Dencker, (2009), "The Solvability and Subellipticity of Systems of Pseudodifferential Operators," Advances in Phase Space Analysis of Partial Differential Equations, In Honor of Ferruccio Colombini's 60th Birthday (A. Bove, D. Del Santo, M.K.V. Murthy, eds.), Birkhauser, Boston, 2009, 73--94

[105] Nils Dencker, (2010), "On the solvability of systems of pseudodifferential operators," arXiv:0801.4043v4 [math.AP].

[106] Michael Ruzhansky, Jens Wirth, (2010), "Dispersive type estimates for Fourier integrals and applications to hyperbolic systems," arXiv:1006.5600v1 [math.AP].

[107] Joan Mateu, Joan Orobitg, Carlos Perez, Joan Verdera, (2010), "New estimates for the maximal singular integral," arXiv:0904.3379v2 [math.CA].

[108] C. Bervillier, (2008), "Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators," arXiv:0812.2262v1 [math-ph].

[109] Wrick Sengupta, (2009), "A novel analytical operator method to solve linear ordinary differential equations with variable coefficients," arXiv:0902.0910v1 [math-ph].

[110] V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, (2005), "On the complete integrability and linearization of certain second order nonlinear ordinary differential equations," arXiv:nlin/0408053v4 [nlin.SI].

[111] V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, (2005), "On the complete integrability and linearization of certain second order nonlinear ordinary differential equations- Part II: Third order equations," arXiv:nlin/0510036v1 [nlin.SI].

[112] V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, (2008), "On the complete integrability and linearization of certain second order nonlinear ordinary differential equations- Part III: Coupled first order equations," arXiv:0808.0958v2 [nlin.SI].

[113] V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, (2008), "On the complete integrability and linearization of certain second order nonlinear ordinary differential equations- Part IV: Coupled second order equations," arXiv:0808.0968v2 [nlin.SI].

[114] V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, (2008), "On the complete integrability and linearization of certain second order nonlinear ordinary differential equations- Part V: Linearization of coupled second order equations," arXiv:0808.0969v2 [nlin.SI].

[115] V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, (2005), "A unification in the theory of linearization of second order nonlinear ordinary differential equations," arXiv:nlin/0510045v1 [nlin.SI].

[116] Hector Giacomini, Jaume Gine, Maite Grau, (2007), "Linearizable ordinary differential equations," arXiv:0710.5143v1 [math.DS].

[117] Asghar Qadir, (2007), "Geometric Linearization of Ordinary Differential Equations," SIGMA 3 (2007), 103, 7 pages, arXiv:0711.0814v2 [math.CA].

[118] S. Ali, F. M. Mahomed, Asghar Qadir, (2007), "Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - I: Ordinary Differential Equations," arXiv:0711.4914v1 [math.CA].

[119] Raouf Dridi, Michel Petitot, (2008), "New classification techniques for ordinary differential equations," arXiv:0712.0002v2 [math.DG].

[120] Michal Godlinski, (2008), "Geometry of Third-Order Ordinary Differential Equations and Its Applications in General Relativity," arXiv:0810.2234v1 [math.DG].

[121] C. Le Bris, T. Lelievre, Y. Maday, (2008), "Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations," arXiv:0811.0474v1 [math.NA].

[122] Viktor Levandovskyy, Bernd Martin, (2010), "A Symbolic Approach to Generation and Analysis of Finite Difference Schemes of Partial Differential Equations," arXiv:1007.4443v1 [math-ph].

[123] Douglas Poole, Willy Hereman, (2010), "Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions," arXiv:1007.5119v1 [nlin.SI].

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