Robert S Maier
 Professor, Mathematics
 Professor, Physics
 Professor, StatisticsGIDP
Contact
 (520) 6212617
 Mathematics, Rm. 609B
 Tucson, AZ 85721
 rsm@math.arizona.edu
Degrees
 Ph.D. Mathematics
 Rutgers, The State University of New Jersey, New Brunswick, New Jersey, U.S.A.
 The Density of States of Random Schroedinger Operators
 M.S. Physics
 California Institute of Technology, Pasadena, California, U.S.A.
Work Experience
 University of Colorado, Boulder, Colorado (2013  Ongoing)
 University of Arizona, Tucson, Arizona (1986  Ongoing)
 University of Texas at Austin, Austin, Texas (1983  1986)
Interests
Teaching
applied analysis, statistics, discrete mathematics
Research
mathematical physics, special functions, applied probability, systems engineering, bioinformatics, computational statistics and data analysis
Courses
201819 Courses

Adv Applied Mathematics
MATH 422 (Fall 2018) 
Adv Applied Mathematics
MATH 522 (Fall 2018) 
Intro:Stat+Biostatistics
MATH 263 (Fall 2018)
201718 Courses

Topics In Applied Math
MATH 577 (Fall 2017)
201617 Courses

Adv Applied Mathematics
MATH 422 (Fall 2016) 
Adv Applied Mathematics
MATH 522 (Fall 2016) 
Intro:Stat+Biostatistics
MATH 263 (Fall 2016)
Scholarly Contributions
Books
 Maier, R. S. (2010). Algorithmic Probability and Combinatorics. Providence, RI: American Mathematical Society. doi:10.1090/conm/520More infoThis volume, which I coedited, appeared as number 520 in the series Contemporary Mathematics, published by the American Mathematical Society.
 Maier, R. S. (2008). Special Functions and Orthogonal Polynomials. Providence, RI: American Mathematical Society. doi:10.1090/conm/471More infoThis volume, which I coedited, appeared as number 471 in the series Contemporary Mathematics, published by the American Mathemaitcal Society.
Chapters
 Maier, R. S. (2018). Algebraic generating functions for Gegenbauer polynomials. In Frontiers In Orthogonal Polynomials and qseries(pp 425444). Hackensack, NJ: World Scientific. doi:10.1142/9789813228887_0022
 Maier, R. S. (2008). Psymbols, Heun identities, and F3(2) identities. In Special Functions and Orthogonal Polynomials(pp 139159). Providence, RI: American Mathematical Society.
Journals/Publications
 Maier, R. S. (2018). Associated Legendre functions and spherical harmonics of fractional degree and order. Constructive Approximation, 48(2), 235281. doi:10.1007/s0036501794035
 Maier, R. S. (2019). Extensions of the classical transformations of the hypergeometric function 3F2. Advances in Applied Mathematics, 105, 2547. doi:10.1016/j.aam.2019.01.002
 Melia, F., Wei, J., Maier, R. S., & Wu, X. (2018). Cosmological tests with the Joint Lightcurve Analysis. EPL (Europhysics Letters), 123(5), id 59002. doi:10.1209/02955075/123/59002
 Narayan, G., Zaidi, T., Soraisam, M. D., Wang, Z., Lochner, M., Matheson, T., Saha, A., Yang, S., Zhao, Z., Kececioglu, J. D., Scheidegger, C. E., Snodgrass, R. T., Axelrod, T., Jenness, T., Maier, R. S., Ridgway, S. T., Seaman, R. L., Evans, E. M., Singh, N., , Taylor, C., et al. (2018). Machinelearningbased brokers for realtime classification of the LSST alert stream. The Astrophysical Journal Supplement Series, 236(9), 26 pages. doi:doi:10.3847/15384365/aab781
 Suh, Y., Snodgrass, R. T., Kececioglu, J. D., Downey, P. J., Maier, R. S., & Yi, C. (2017). EMP: Execution time measurement protocol for compute‐bound programs. Software: Practice and Experience, 47(4), 559597. doi:10.1002/spe.2476
 Maier, R. S. (2016). Integrals of Lipschitz–Hankel type, Legendre functions and table errata. Integral Transforms and Special Functions, 27(5), 385391. doi:10.1080/10652469.2015.1132716
 Maier, R. S. (2016). Legendre functions of fractional degree: transformations and evaluations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472, 20160097 (26 pp.). doi:10.1098/rspa.2016.0097
 Wei, J., Wu, X., Melia, F., & Maier, R. S. (2015). A comparative analysis of the Supernova Legacy Survey Sample with Lambda CDM and the Rh = ct universe. Astronomical Journal, 149(3). doi:10.1088/00046256/149/3/102
 Maier, R. S. (2014). The uniformization of certain algebraic hypergeometric functions. Advances in Mathematics, 253, 86138. doi:10.1016/j.aim.2013.11.013More infoAbstract: The hypergeometric functions Fn1n are higher transcendental functions, but for certain parameter values they become algebraic, because the monodromy of the defining hypergeometric differential equation becomes finite. It is shown that many algebraic Fn1n's, for which the finite monodromy is irreducible but imprimitive, can be represented as combinations of certain explicitly algebraic functions of a single variable; namely, the roots of trinomials. This generalizes a result of Birkeland, and is derived as a corollary of a family of binomial coefficient identities that is of independent interest. Any tuple of roots of a trinomial traces out a projective algebraic curve, and it is also determined when this socalled Schwarz curve is of genus zero and can be rationally parametrized. Any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic Fn1n's. Many examples of such uniformizations are worked out explicitly. Even when the governing Schwarz curve is of positive genus, it is shown how it is sometimes possible to construct explicit singlevalued or multivalued parametrizations of individual algebraic Fn1n's, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials. © 2013 Elsevier Inc.
 Maier, R. S. (2013). The integration of threedimensional LotkaVolterra systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469(2158). doi:10.1098/rspa.2012.0693More infoAbstract: The general solutions of many threedimensional LotkaVolterra systems, previously known to be at least partially integrable, are constructed with the aid of special functions. Examples include certain ABC and MayLeonard systems. The special functions used are elliptic and incomplete beta functions. In some cases, the solution is parametric, with the independent and dependent variables expressed as functions of a 'new time' variable. This auxiliary variable satisfies a nonlinear thirdorder differential equation of a generalized Schwarzian type, and results of CartonLeBrun on the equations of this type that have the Painlevé property are exploited, so as to produce solutions in closed form. For several especially difficult LotkaVolterra systems, the solutions are expressed in terms of Painlevé transcendents. An appendix on incomplete beta functions and closedform expressions for their inverses is included. © 2013 The Author(s).
 Melia, F., & Maier, R. S. (2013). Cosmic chronometers in the Rh = ct universe. Monthly Notices of the Royal Astronomical Society, 432(4), 26692675. doi:10.1093/mnras/stt596More infoAbstract: The use of luminous red galaxies as cosmic chronometers provides us with an indispensable method of measuring the universal expansion rate H(z) in a modelindependent way. Unlike many probes of the cosmological history, this approach does not rely on integrated quantities, such as the luminosity distance, and therefore does not require the preassumption of any particular model, which may bias subsequent interpretations of the data. We employ three statistical tools . the Akaike, Kullback and Bayes Information Criteria (AIC, KIC and BIC) . to compare the δ cold dark matter (δCDM) model and the Rh = ct Universe with the currently available measurements of H(z), and show that the Rh = ct Universe is favoured by these model selection criteria. The parameters in each model are individually optimized by maximum likelihood estimation. The Rh =ct Universe fits the data with a reduced χ2 dof = 0.745 for a Hubble constant H0 = 63.2 ± 1.6 km s1 Mpc1, and H0 is the sole parameter in this model. By comparison, the optimal δCDM model, which has three free parameters (including H0 = 68.9 ± 3.3 km s.1 Mpc.1, σm = 0.32, and a darkenergy equation of state pde =ρde), fits the H(z) data with a reduced χ2 dof = 0.777. With these χ2 dof values, the AIC yields a likelihood of .82 per cent that the distance.redshift relation of the Rh = ct Universe is closer to the correct cosmology, than is the case for δCDM. If the alternative BIC criterion is used, the respective Bayesian posterior probabilities are 91.2 per cent (Rh = ct) versus 8.8 per cent (δCDM). Using the concordance δCDM parameter values, rather than those obtained by fitting δCDM to the cosmic chronometer data, would further disfavour δCDM. © 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.
 Maier, R. S. (2011). Nonlinear differential equations satisfied by certain classical modular forms. Manuscripta Mathematica, 134(1), 142. doi:10.1007/s0022901003789More infoAbstract: A unified treatment is given of lowweight modular forms on Γ0(N), N = 2,3,4, that have Eisenstein series representations. For each N, certain weight1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear thirdorder equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of qexpansions, and a transformation law under Γ0(4) for the second complete elliptic integral is derived. More generally, it is shown how PicardFuchs equations of triangle subgroups of PSL(2, R), which are hypergeometric equations, yield systems of nonlinear equations for weight1 forms, and generalized Chazy equations. Each triangle group commensurable with Γ(1) is treated. © 2010 SpringerVerlag.
 Maier, R. S. (2009). On rationally parametrized modular equations. Journal of the Ramanujan Mathematical Society, 24(1), 173.
 Maier, R. S. (2008). Lamé polynomials, hyperelliptic reductions and Lamé band structure. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366(1867), 11151153. doi:10.1098/rsta.2007.2063More infoPMID: 17588866;Abstract: The band structure of the Lamé equation, viewed as a onedimensional Schrödinger equation with a periodic potential, is studied. At integer values of the degree parameter l, the dispersion relation is reduced to the l=1 dispersion relation, and a previously published l=2 dispersion relation is shown to be partly incorrect. The HermiteKrichever Ansatz, which expresses Lamé equation solutions in terms of l=1 solutions, is the chief tool. It is based on a projection from a genusl hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lamé equation parameters take complex values, are investigated. If the Lamé equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lamé polynomials, i.e. bandedge solutions, are given. A table in the earlier literature is corrected. © 2007 The Royal Society.
 Maier, R. S. (2007). Algebraic hypergeometric transformations of modular origin. Transactions of the American Mathematical Society, 359(8), 38593885. doi:10.1090/S0002994707041281More infoAbstract: It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2F1 arises from a relation between modular curves, namely the covering of X0(3) by X0(9). In general, when 2 ≤ N ≤ 7, the Nfold cover of X0(N) by X0(N2) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmeticgeometric mean iterations, but the N = 5, 6, 7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X0(6),X0(7) are of genus 1. Since their quotients X+0 (6),X+0 (7) under the Fricke involution (an AtkinLehner involution) are of genus 0, the parametrization is by twovalued algebraic functions. The resulting hypergeometric transformations are closely related to the twovalued modular equations of Fricke and H. Cohn. © 2007 American Mathematical Society.
 Maier, R. S. (2007). Parametrized stochastic grammars for RNA secondary structure prediction. 2007 Information Theory and Applications Workshop, Conference Proceedings, ITA, 256260. doi:10.1109/ITA.2007.4357589More infoAbstract: We propose a twolevel stochastic contextfree grammar (SCFG) architecture for parametrized stochastic modeling of a family of RNA sequences, including their secondary structure. A stochastic model of this type can be used for maximum a posteriori estimation of the secondary structure of any new sequence in the family. The proposed SCFG architecture models RNA subsequences comprising paired bases as stochastically weighted Dycklanguage words, i.e., as weighted balancedparenthesis expressions. The length of each run of unpaired bases, forming a loop or a bulge, is taken to have a phasetype distribution: that of the hitting time in a finitestate Markov chain. Without loss of generality, each such Markov chain can be taken to have a bounded complexity. The scheme yields an overall family SCFG with a manageable number of parameters.
 Maier, R. S. (2007). The 192 solutions of the Heun equation. Mathematics of Computation, 76(258), 811843. doi:10.1090/S0025571806019399More infoAbstract: A machinegenerated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the Coxeter group Dn. Each of the 192 expressions is labeled by an element of D4. Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that the resulting order24 group of transformations of Hl is isomorphic to the symmetric group 54. The isomorphism encodes each transformation as a permutation of an abstract fourelement set, not identical to the set of singular points. © 2006 American Mathematical Society.
 Maier, R. S. (2006). A generalization of Euler's hypergeometric transformation. Transactions of the American Mathematical Society, 358(1), 3957.More infoAbstract: Euler's transformation formula for the Gauss hypergeometric function 2F 1 is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but not linearly. Its consequences for hypergeometric summation are explored. It has as a corollary a summation formula of Slater. From this formula new oneterm evaluations of 2F 1 (1) and 3F 2(1) are derived by applying transformations in the Thomae group. Their parameters are also constrained nonlinearly. Several new oneterm evaluations of 2F 1(1) with linearly constrained parameters are derived as well. © 2005 American Mathematical Society.
 Maier, R. S. (2005). On reducing the Heun equation to the hypergeometric equation. Journal of Differential Equations, 213(1), 171203. doi:10.1016/j.jde.2004.07.020More infoAbstract: The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heuntohypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its crossratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higherdegree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. © 2004 Elsevier Inc. All rights reserved.
 Maier, R. S. (2004). A theory of magnetization reversal in nanowires. Proceedings of SPIE  The International Society for Optical Engineering, 5471, 4857. doi:10.1117/12.553199More infoAbstract: Magnetization reversal in a ferromagnetic nanowire which is much narrower than the exchange length is believed to be accomplished through the thermally activated growth of a spatially localized nucleus, which initially occupies a small fraction of the total volume. To date, the most detailed theoretical treatments of reversal as a fieldinduced but noiseactivated process have focused on the case of a very long ferromagnetic nanowire, i.e., a highly elongated cylindrical particle, and have yielded a reversal rate per unit length, due to an underlying assumption that the nucleus may form anywhere along the wire. But in a boundedlength (though long) cylindrical particle with flat ends, it is energetically favored for nucleation to begin at either end. We indicate how to compute analytically the energy of the critical nucleus associated with either end, i.e., the activation barrier to magnetization reversal, which governs the reversal rate in the lowtemperature (Kramers) limit. Our treatment employs elliptic functions, and is partly analytic rather than numerical. We also comment on the Kramers prefactor, which for this reversal pathway does not scale linearly as the particle length increases, and tends to a constant in the lowtemperature limit.
 Maier, R. S. (2004). Algebraic solutions of the Lamé equation, revisited. Journal of Differential Equations, 198(1), 1634. doi:10.1016/j.jde.2003.06.006More infoAbstract: A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out (see Baldassarri, J. Differential Equations 41 (1) (1981) 44). It is shown that if the group is the octahedral group S4, then the degree parameter of the equation may differ by ± 1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation (see Churchill, J. Symbolic Comput. 28 (45) (1999) 521). The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group. © 2003 Elsevier Inc. All rights reserved.
 Maier, R. S. (2003). On Crossing Event Formulas in Critical TwoDimensional Percolation. Journal of Statistical Physics, 111(56), 10271048. doi:10.1023/A:1023006413433More infoAbstract: Several formulas for crossing functions arising in the continuum limit of critical twodimensional percolation models are studied. These include Watts's formula for the horizontalvertical crossing probability and Cardy's new formula for the expected number of crossing clusters. It is shown that for lattices where conformal invariance holds, they simplify when the spatial domain is taken to be the interior of an equilateral triangle. The two crossing functions can be expressed in terms of an equianharmonic elliptic function with a triangular rotational symmetry. This suggests that rigorous proofs of Watts's formula and Cardy's new formula will be easiest to construct if the underlying lattice is triangular. The simplification in a triangular domain of Schramm's "bulk Cardy's formula" is also studied.
 Maier, R. S., & Stein, D. L. (2003). The effects of weak spatiotemporal noise on a bistable onedimensional system. Proceedings of SPIE  The International Society for Optical Engineering, 5114, 6778.More infoAbstract: We treat analytically a model that captures several features of the phenomenon of spatially inhomogeneous reversal of an order parameter. The model is a classical GinzburgLandau field theory restricted to a bounded onedimensional spatial domain, perturbed by weak spatiotemporal noise having a flat power spectrum in time and space. Our analysis extends the Kramers theory of noiseinduced transitions to the case when the system acted on by the noise has nonzero spatial extent, and the noise itself is spatially dependent. By extending the LangerColeman theory of the noiseinduced decay of a metastable state, we determine the dependence of the activation barrier and the Kramers reversal rate prefactor on the size of the spatial domain. As this is increased from zero and passes through a certain critical value, a transition between activation regimes occurs, at which the rate prefactor diverges. Beyond the transition, reversal preferentially takes place in a spatially inhomogeneous rather than in a homogeneous way. Transitions of this sort were not discovered by Langer or Coleman, since they treated only the infinitevolume limit. Our analysis uses higher transcendental functions to handle the case of finite volume. Similar transitions between activation regimes should occur in other models of metastable systems with nonzero spatial extent, perturbed by weak noise, as the size of the spatial domain is varied.
 Maier, R. S., & Stein, D. L. (2001). Droplet nucleation and domain wall motion in a bounded interval. Physical Review Letters, 87(27 I), 27060112706014.More infoAbstract: A study on droplet nucleation and domain wall motion in a bounded interval was performed. A classical GinzburgLandau model as a spatially extended model of noiseinduced magnetization reversal was studied and restricted to a bounded interval and perturbed by a weak spatiotemporal noise. The dependence of the activation barrier and kramers rate prefactor on the interval length was determined by adapting the ColemanLanger approach to false vacuum decay. It was shown that a rich structure of activation regimes separated by phase transition was yielded.
 Maier, R. S., & Stein, D. L. (2001). Noiseactivated escape from a sloshing potential well. Physical Review Letters, 86(18), 39423945.More infoPMID: 11328066;Abstract: The noiseactivated escape from a onedimensional potential well of an overdamped particle was examined. Focus was on the case when the forcing does not die away in the weaknoise limit. The resulting data was analyzed in detail.
 Maier, R. S., & Stein, D. L. (2000). How an anomalous cusp bifurcates in a weaknoise system. Physical Review Letters, 85(7), 13581361.More infoAbstract: The unfolding of an anomalous cusp in a typical twodimensional noiseactivated system was analyzed. It was shown how the unfolding may bifurcate into conventional cusps. The scaling law for the bifurcation yielded a corresponding law for the divergence of the Kramers prefactor.
 Luchinsky, D. G., Maier, R. S., Mannella, R., McClintock, P. V., & Stein, D. L. (1999). Observation of SaddlePoint Avoidance in NoiseInduced Escape. Physical Review Letters, 82(9), 18061809.More infoAbstract: The first measurements of an exit location distribution are reported for an overdamped nonconservative system perturbed by weak white noise, modeled both numerically and by an analog electronic circuit. In the weaknoise limit the distribution is increasingly concentrated near a saddle point of the dynamics and is increasingly well approximated by a Weibull distribution, in agreement with theoretical predictions. A physical explanation for this behavior is given, which should facilitate the computation of corrections to the limiting form.
 Luchinsky, D. G., Maier, R. S., Mannella, R., McClintock, P. V., & Stein, D. L. (1997). Experiments on critical phenomena in a noisy exit problem. Physical Review Letters, 79(17), 31093112.More infoAbstract: We consider a noisedriven exit from a domain of attraction in a twodimensional bistable system lacking detailed balance. Through analog and digital stochastic simulations, we find a theoretically predicted bifurcation of the most probable exit path as the parameters of the system are changed, and a corresponding nonanalyticity of the generalized activation energy. We also investigate the extent to which the bifurcation is related to the local breaking of timereversal invariance.
 Maier, R. S., & Stein, D. L. (1997). Limiting exit location distributions in the stochastic exit problem. SIAM Journal on Applied Mathematics, 57(3), 752790.More infoAbstract: Consider a twodimensional continuoustime dynamical system, with an attracting fixed point S. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength ∈, the system state will eventually leave the domain of attraction Ωof S. We analyze the case when, as ∈ → 0, the exit location on the boundary ∂Ωis increasingly concentrated near a saddle point H of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on ∂Ω is generically nonGaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter μ equal to the ratio λs(H)/λu(H) of the stable and unstable eigenvalues of the linearized deterministic flow at H. If μ< 1, then the exit location distribution is generically asymptotic as ∈ → 0 to a Weibull distribution with shape parameter 2/μ, on the script O sign(∈μ/2) lengthscale near H. If μ > 1, it is generically asymptotic to a distribution on the script O sign(∈1/2) lengthscale, whose moments we compute. Our treatment employs both matched asymptotic expansions and stochastic analysis. As a byproduct of our treatment, we clarify the limitations of the traditional Eyring formula for the weaknoise exit time asymptotics.
 Smelyanskiy, V. N., Dykman, M. I., & Maier, R. S. (1997). Topological features of large fluctuations to the interior of a limit cycle. Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 55(3 SUPPL. A), 23692391.More infoAbstract: We investigate the pattern of optimal paths along which a dynamical system driven by weak noise moves, with overwhelming probability, when it fluctuates far away from a stable state. Our emphasis is on systems that perform selfsustained periodic vibrations, and have an unstable focus inside a stable limit cycle. We show that in the vicinity of the unstable focus, the flow field of optimal paths generically displays a pattern of singularities. In particular, it contains a switching line that separates areas to which the system arrives along optimal paths of topologically different types. The switching line spirals into the focus and has a selfsimilar structure. Depending on the behavior of the system near the focus, it may be smooth, or have finitelength branches. Our results are based on an analysis of the topology of the Lagrangian manifold for an auxiliary, purely dynamical, problem that determines the optimal paths. We illustrate our theory by studying, both theoretically and numerically, a van der Pol oscillator driven by weak white noise.
 Dykman, M. I., Smelyanskiy, V. N., Maier, R. S., & Silverstein, M. (1996). Singular features of large fluctuations in oscillating chemical systems. Journal of Physical Chemistry, 100(49), 1919719209.More infoAbstract: We investigate the way in which large fluctuations in an oscillating, spatially homogeneous chemical system take place. Starting from a master equation, we study both the stationary probability density of such a system far from its limit cycle and the optimal (most probable) fluctuational paths in its space of species concentrations. The flow field of optimal fluctuational paths may contain singularities, such as switching lines. A "switching line" separates regions in the space of species concentrations that are reached, with high probability, along topologically different sorts of fluctuational paths. If an unstable focus lies inside the limit cycle, the pattern of optimal fluctuational paths is singular and selfsimilar near the unstable focus. In fact, a switching line spirals down to the focus. The logarithm of the stationary probability density has a selfsimilar singular structure near the focus as well. For a homogeneous Selkov model, we provide a numerical analysis of the pattern of optimal fluctuational paths and compare it with analytic results. © 1996 American Chemical Society.
 Maier, R. S., & Stein, D. L. (1996). A scaling theory of bifurcations in the symmetric weaknoise escape problem. Journal of Statistical Physics, 83(34), 291357.More infoAbstract: We consider twodimensional overdamped doublewell systems perturbed by white noise. In the weaknoise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric doublewell system are varied, a unique MPEP may bifurcate into two equally likely MPEPs. At the bifurcation point in parameter space, the activation kinetics of the system become nonArrhenius. We quantify the nonArrhenius behavior of a system at the bifurcation point, by using the MaslovWKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our construction relies on a new scaling theory, which yields "critical exponents" describing weaknoise behavior at the bifurcation point, near the saddle.
 Maier, R. S., & Stein, D. L. (1996). Oscillatory behavior of the rate of escape through an unstable limit cycle. Physical Review Letters, 77(24), 48604863.More infoAbstract: Suppose a twodimensional dynamical system has a stable attractor that is surrounded by an unstable limit cycle. If the system is additively perturbed by white noise, the rate of escape through the limit cycle will fall off exponentially as the noise strength tends to zero. By analyzing the associated FokkerPlanck equation we show that in general the weaknoise escape rate is nonArrhenius: it includes a factor that is periodic in the logarithm of the noise strength. The presence of this slowly oscillating factor is due to the nonequilibrium potential of the system being nondifferentiable at the limit cycle. We point out the implications for the weaknoise limit of stochastic resonance models.
 MAIER, R. (1995). THE SHAPE OF STRETCHED PLANAR TREES. RANDOM STRUCTURES & ALGORITHMS, 6(23), 331340.More infoWe study the asymptotics of a ''stretched'' model of unlabeled rooted planar trees, in which trees are not taken equiprobable but are weighted exponentially, according to their height. By using standard methods for computing the probabilities of large deviations of random processes, we show that, as the number of vertices tends to infinity, the normalized shape of a random tree converges in distribution to a deterministic limit. We compute this limit explicitly. (C) 1995 John Wiley and Sons, Inc.
 Maier, R. S., & Stein, D. L. (1995). Optimal paths, caustics, and boundary layer approximations in stochastically perturbed dynamical systems. American Society of Mechanical Engineers, Design Engineering Division (Publication) DE, 84(3 Pt A/2), 903910.More infoAbstract: We study the asymptotic properties of overdamped dynamical systems with one or more point attractors, when they are perturbed by weak noise. In the weaknoise limit, fluctuations to the vicinity of any specified nonattractor point will increasingly tend to follow a welldefined optimal path. We compute precise asymptotics for the frequency of such fluctuations, by integrating a matrix Riccati equation along the optimal path. We also consider noiseinduced transitions between domains of attraction, in twodimensional double well systems. The optimal paths in such systems may focus, creating a caustic. We examine `critical' systems in which a caustic is beginning to form, and show that due to criticality, the mean escape time from one well to the other grows in the weaknoise limit in a nonexponential way. The analysis relies on a MaslovWKB approximation to the solution of the Smoluchowski equation.
 Maier, R. S. (1993). Phasetype distributions and the structure of finite Markov chains. Journal of Computational and Applied Mathematics, 46(3), 449453.More infoAbstract: Maier, R.S., Phasetype distributions and the structure of finite Markov chains, Journal of Computational and Applied Mathematics 46 (1993) 449453. We show that all discrete phasetype distributions arise as first passage times (i.e., absorption times) in finitestate Markov chains with a certain recursive internal structure. This arises from the special properties of an automatatheoretic algorithm which can be used to solve the inverse problem for phasetype distributions: the construction of a Markov chain with specified absorption time distribution. © 1993.
 Maier, R. S., & Stein, D. L. (1993). Effect of focusing and caustics on exit phenomena in systems lacking detailed balance. Physical Review Letters, 71(12), 17831786.More infoAbstract: We study the trajectories followed by a particle weakly perturbed by noise, when escaping from the domain of attraction of a stable fixed point. If the particle's stationary distribution lacks detailed balance, a focus may occur along the most probable exit path, leading to a breakdown of symmetry (if present). The exit trajectory bifurcates, and the exit location distribution may become ''skewed'' (nonGaussian). The weaknoise asymptotics of the mean escape time are also affected. Our methods extend to the study of skewed exit location distributions in stochastic models without symmetry. © 1993 The American Physical Society.
 Maier, R. S., & Stein, D. L. (1993). Escape problem for irreversible systems. Physical Review E, 48(2), 931938.More infoAbstract: The problem of noiseinduced escape from a metastable state arises in physics, chemistry, biology, systems engineering, and other areas. The problem is well understood when the underlying dynamics of the system obey detailed balance. When this assumption fails many of the results of classical transitionrate theory no longer apply, and no general method exists for computing the weaknoise asymptotic behavior of fundamental quantities such as the mean escape time. In this paper we present a general technique for analyzing the weaknoise limit of a wide range of stochastically perturbed continuoustime nonlinear dynamical systems. We simplify the original problem, which involves solving a partial differential equation, into one in which only ordinary differential equations need be solved. This allows us to resolve some old issues for the case when detailed balance holds. When it does not hold, we show how the formula for the asymptotic behavior of the mean escape time depends on the dynamics of the system along the most probable escape path. We also present results on shorttime behavior and discuss the possibility of focusing along the escape path. © 1993 The American Physical Society.
 MAIER, R., & OCINNEIDE, C. (1992). A CLOSURE CHARACTERIZATION OF PHASETYPE DISTRIBUTIONS. JOURNAL OF APPLIED PROBABILITY, 29(1), 92103.More infoWe characterise the classes of continuous and discrete phasetype distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary 'geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class.
 Maier, R. S., & Stein, D. L. (1992). Transitionrate theory for nongradient drift fields. Physical Review Letters, 69(26), 36913695.More infoAbstract: Classical transitionrate theory provides analytic techniques for computing the asymptotics of a weakly perturbed particle's mean residence time in the basin of attraction of a metastable state. If the dynamics of the particle are derivable from a potential, it typically escapes over a saddle point. In the nonpotential case exit may take place over an unstable point instead, leading to unexpected phenomena. These may include an anomalous preexponential factor, with a continuously varying exponent, in the residence time asymptotics. Moreover, the most probable escape trajectories may eventually deviate from the leastaction escape path.
 MAIER, R. (1991). COLLIDING STACKS  A LARGE DEVIATIONS ANALYSIS. RANDOM STRUCTURES & ALGORITHMS, 2(4), 379420.More infoWe analyze the performance of a prototypical scheme for shared storage allocation: two initially empty stacks evolving in a contiguous block of memory of size m. We treat the case in which the stacks are more likely to shrink than grow, but with the probabilities of insertion and deletion allowed to depend arbitrarily on stack height as a fraction of m. New results are obtained on the m > infinity asymptotics of the stack collision time, and of the final stack heights. The results of Wentzell and Freidlin on the large deviations of Markov chains are used, and the relation of their formalism to the hamiltonian formulation of classical mechanics is emphasized. Certain results on higherorder asymptotics follow from WKB expansions.
 Maier, R. S. (1991). A path integral approach to data structure evolution. Journal of Complexity, 7(3), 232260.More infoAbstract: A probabilistic analysis is presented of certain pointerbased implementations of dictionaries, linear lists, and priority queues; in particular, simple list and dheap implementations. Under the assumption of equiprobability of histories, i.e., of paths through the internal state space of the implementation, it is shown that the integrated space and time costs of a sequence of n supported operations converge as n → ∞ to Gaussian random variables. For list implementations the mean integrated spatial costs grow asymptotically as n2, and the standard deviations of the costs as n 3 2. For dheap implementations of priority queues the mean integrated space cost grows only as n2√log n, i.e., more slowly than the worstcase integrated cost. The standard deviation grows as n 3 2. These asymptotic growth rates reflect the convergence as n → ∞ of the normalized structure sizes to datatypedependent deterministic functions of time, as earlier discovered by Louchard. This phenomenon is clarified with the aid of path integrals. Path integral techniques, drawn from physics, greatly facilitate the computation of the cost asymptotics. This is their first application to the analysis of dynamic data structures. © 1991.
 FARIS, W., & MAIER, R. (1988). PROBABILISTIC ANALYSIS OF A LEARNING MATRIX. ADVANCES IN APPLIED PROBABILITY, 20(4), 695705.
 Maier, R. S. (1987). Bounds on the density of states of random Schrödinger operators. Journal of Statistical Physics, 48(34), 425447.More infoAbstract: Bounds are obtained on the unintegrated density of states ρ(E) of random Schrödinger operators H=Δ + V acting on L2(ℝd) or l2(ℤd). In both cases the random potential is {Mathematical expression}in which the {Mathematical expression} are IID random variables with density f. The χ denotes indicator function, and in the continuum case the {Mathematical expression} are cells of unit dimensions centered on y∈ℤd. In the finitedifference case Λ(y) denotes the site y∈ℤd itself. Under the assumption f ∈ L01+e{open}(ℝ) it is proven that in the finitedifference case p ∈ L∞(ℝ), and that in the d= 1 continuum case p ∈ Lloc∞(ℝ). © 1987 Plenum Publishing Corporation.
 MAIER, R., & WHALING, W. (1977). TRANSITIONPROBABILITIES IN ND(II) AND SOLAR NEODYMIUM ABUNDANCE. JOURNAL OF QUANTITATIVE SPECTROSCOPY & RADIATIVE TRANSFER, 18(5), 501507.
Presentations
 Maier, R. S. (2015, December). Legendre function identities and algebraic geometry. Math Research Blitz. Tucson, AZ: University of Arizona Mathematics Dept..
 Maier, R. S. (2015, May). The Ince equation and its solutions. International Conference on Orthogonal Polynomials and qSeries. Orlando, FL.More infoABSTRACT: The Ince equation is a generalization of the Lamé equation that has anextra free parameter. Like the Lamé equation, it can be expressed in manyforms (algebraic, trigonometric, elliptic), and the underlying Incedifferential operator is a tridiagonalizable secondorder one. The Inceequation may have polynomial solutions, but their relation to the theory oforthogonal polynomials is unclear. The equation can be viewed as aquasiexactlysolvable (QES) Schroedinger equation, the spectral theory ofwhich has unusual algebraic aspects. We begin by reviewing the spectraltheory of the (trigonometric) Ince equation, which originated with Magnusand Winkler, and which is a concrete application of Floquet theory. Incepolynomials, like Lamé polynomials, correspond to the edges of spectralbands. We go on to discuss recently found families of nonpolynomialsolutions that can be expressed in terms of the Gauss hypergeometricfunction. This includes solutions that generalize theBrioschiHalphenCrawford solutions of the Lamé equation, and algebraicones that can be reduced to hypergeometric functions of finite monodromy(dihedral, octahedral, etc.).
 Maier, R. S. (2015, September). The Ince equation, the elliptical vortex instability, and Painlevé VI. Analysis, Dynamics, and Applications Seminar. Tucson, AZ: University of Arizona Mathematics Dept..
 Maier, R. S. (2014, April). Threedimensional quadratic differential systems and the Painlevé property. American Mathematical Society, Spring Central Sectional Meeting. Lubbock, TX: American Mathematical Society.More infoAn invited talk at the Special Session on Applications of Special Functions in Combinatorics and Analysis.
 Maier, R. S. (2014, November). When special functions become algebraic. Math Research Blitz. Tucson, AZ: University of Arizona Mathematics Dept..