Robert S Maier

Interests

Research

mathematical physics, special functions and orthogonal polynomials, number theory and algebraic geometry, applied probability and stochastic modelling, statistical inference, information theory, quantum computing

Teaching

applied analysis, algorithmic statistics, discrete mathematics

Courses

Scholarly Contributions

Books

• Maier, R. S. (2010). Algorithmic Probability and Combinatorics. Providence, RI: American Mathematical Society. doi:10.1090/conm/520
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  This refereed proceedings volume, which I co-edited, 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/471
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  This refereed proceedings volume, which I co-edited, 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 q-Series(pp 425-444). Hackensack, NJ: World Scientific. doi:10.1142/9789813228887_0022
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  It is shown that several of Brafman's generating functions for the Gegenbauer polynomials are algebraic functions of their arguments, if the Gegenbauer parameter differs from an integer by one-fourth or one-sixth. Two examples are given, which come from recently derived expressions for associated Legendre functions with octahedral or tetrahedral monodromy. It is also shown that if the Gegenbauer parameter is restricted as stated, the Poisson kernel for the Gegenbauer polynomials can be expressed in terms of complete elliptic integrals. An example is given.
• Maier, R. S. (2008). P-symbols, Heun identities, and F-3(2) identities. In Special Functions and Orthogonal Polynomials(pp 139-159). Providence, RI: American Mathematical Society. doi:10.48550/arxiv.0712.4299
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  The usefulness of Riemann P-symbols in deriving identities involving the parametrized special function Hl is explored. Hl is the analytic local solution of the Heun equation, the canonical second-order differential equation on the Riemann sphere with four regular singular points. The identities discussed include ones coming from Möbius automorphisms and F-homotopies, and also quadratic and biquadratic transformations. The case when Hl is identical to a generalized hypergeometric function of 3F2 type is examined, and Pfaff and Euler transformations of 3F2(a1,a2,e+1;b1,e;x) are derived. They extend several 3F2 identities of Bailey and Slater.

Journals/Publications

• Maier, R. S. (2024). Boson operator ordering identities from generalized Stirling and Eulerian numbers. Advances in Applied Mathematics, 156, 102678 [34 pp.]. doi:10.1016/j.aam.2024.102678
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  Ordering identities in the Weyl-Heisenberg algebra generated by single-mode boson operators are investigated. A boson string composed of creation and annihilation operators can be expanded as a linear combination of other such strings, the simplest example being a normal ordering. The case when each string contains only one annihilation operator is already combinatorially nontrivial. Two kinds of expansion are derived: (i) that of a power of a string Ω in lower powers of another string Ω', and (ii) that of a power of Ω in twisted versions of the same power of Ω'. The expansion coefficients are shown to be, respectively, generalized Stirling numbers of Hsu and Shiue, and certain generalized Eulerian numbers. Many examples are given. These combinatorial numbers are binomial transforms of each other, and their theory is developed, emphasizing schemes for computing them: summation formulas, Graham-Knuth-Patashnik (GKP) triangular recurrences, terminating hypergeometric series, and closed-form expressions. The results on the first type of expansion subsume a number of previous results on the normal ordering of boson strings.
• Maier, R. S. (2023). Triangular recurrences, generalized Eulerian numbers, and related number triangles. Advances in Applied Mathematics, 146, 102485 [62 pp.]. doi:10.1016/j.aam.2023.102485
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  Many combinatorial and other number triangles are solutions of recurrences of the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted upon by a transformation group generated by two involutions: a left-right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu-Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.
• Maier, R. S. (2019). Extensions of the classical transformations of the hypergeometric function 3F2. Advances in Applied Mathematics, 105, 25-47. doi:10.1016/j.aam.2019.01.002
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  It is shown that the classical quadratic and cubic transformation identities satisfied by the hypergeometric function 3F2 can be extended to include additional parameter pairs, which differ by integers. In the extended identities, which involve hypergeometric functions of arbitrarily high order, the added parameters are nonlinearly constrained: in the quadratic case, they are the negated roots of certain orthogonal polynomials of a discrete argument (dual Hahn and Racah ones). Specializations and applications of the extended identities are given, including an extension of Whipple's identity relating very well poised 7F6(1) series and balanced 4F3(1) series, and extensions of other summation identities.
• Maier, R. S. (2018). Associated Legendre functions and spherical harmonics of fractional degree and order. Constructive Approximation, 48(2), 235-281. doi:10.1007/s00365-017-9403-5
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  Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it are worked out. It is a (generalized) Heun polynomial, not a hypergeometric one. For each of these families of algebraic associated Legendre functions, a representation of the rank-2 Lie algebra so(5,C) is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics. All such representations of so(5,C) are shown to have a common value for each of its two Casimir invariants. The Dirac singleton representations of so(3,2) are included.
• Melia, F., Wei, J., Maier, R. S., & Wu, X. (2018). Cosmological tests with the Joint Lightcurve Analysis. Europhysics Letters (EPL), 123(5), 59002 (7 pp.). doi:10.1209/0295-5075/123/59002
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  We examine whether a comparison between wCDM and R_h=ct using merged Type-Ia SN catalogs produces results consistent with those based on a single homogeneous sample. Using the Betoule et al. (Astron. Astrophys., 568 (2014) 22). Joint Lightcurve Analysis (JLA) of a combined sample of 613 events from SNLS and SDSS-II, we estimate the parameters of the two models and compare them. We find that the improved statistics can alter the model selection in some cases, but not others. In addition, based on the model fits, we find that there appears to be a lingering systematic offset of ∼0.04–0.08 mag between the SNLS and SDSS-II sources, in spite of the cross-calibration in the JLA. Treating wCDM, ΛCDM and R_h=ct as separate models, we find in an unbiased pairwise statistical comparison that the Bayes Information Criterion (BIC) favors the R_h=ct Universe with a likelihood of 82.8% vs. 17.2% for wCDM, but the ratio of likelihoods is reversed (16.2% vs. 83.8%) when w_de=-1 (i.e., ΛCDM) and strongly reversed (1.0% vs. 99.0%) if in addition k = 0 (i.e., flat ΛCDM). We point out, however, that the value of k is a measure of the net energy (kinetic plus gravitational) in the Universe and is not constrained theoretically, though some models of inflation would drive k->0 due to an expansion-enforced dilution. Since we here consider only the basic ΛCDM model, the value of k needs to be measured and, therefore, the pre-assumption of flatness introduces a significant bias into the BIC.
• 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). Machine-learning-based brokers for real-time classification of the LSST alert stream. The Astrophysical Journal Supplement Series, 236(9), 26 pages. doi:10.3847/1538-4365/aab781
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  The unprecedented volume and rate of transient events that will be discovered by the Large Synoptic Survey Telescope (LSST) demand that the astronomical community update its follow-up paradigm. Alert-brokers—automated software system to sift through, characterize, annotate, and prioritize events for follow-up—will be critical tools for managing alert streams in the LSST era. The Arizona-NOAO Temporal Analysis and Response to Events System (ANTARES) is one such broker. In this work, we develop a machine learning pipeline to characterize and classify variable and transient sources only using the available multiband optical photometry. We describe three illustrative stages of the pipeline, serving the three goals of early, intermediate, and retrospective classification of alerts. The first takes the form of variable versus transient categorization, the second a multiclass typing of the combined variable and transient data set, and the third a purity-driven subtyping of a transient class. Although several similar algorithms have proven themselves in simulations, we validate their performance on real observations for the first time. We quantitatively evaluate our pipeline on sparse, unevenly sampled, heteroskedastic data from various existing observational campaigns, and demonstrate very competitive classification performance. We describe our progress toward adapting the pipeline developed in this work into a real-time broker working on live alert streams from time-domain surveys.
• 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), 559-597. doi:10.1002/spe.2476
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  Measuring execution time is one of the most used performance evaluation techniques in computer science research. Inaccurate measurements cannot be used for a fair performance comparison between programs. Despite the prevalence of its use, the intrinsic variability in the time measurement makes it hard to obtain repeatable and accurate timing results of a program running on an operating system. We propose a novel execution time measurement protocol (termed EMP) for measuring the execution time of a compute-bound program on Linux, while minimizing that measurement's variability. During the development of execution time measurement protocol, we identified several factors that disturb execution time measurement. We introduce successive refinements to the protocol by addressing each of these factors, in concert, reducing variability by more than an order of magnitude. We also introduce a new visualization technique, what we term `dual-execution scatter plot' that highlights infrequent, long-running daemons, differentiating them from frequent and/or short-running daemons. Our empirical results show that the proposed protocol successfully achieves three major aspects—precision, accuracy, and scalability—in execution time measurement that can work for open-source and proprietary software.
• Maier, R. S. (2016). Integrals of Lipschitz–Hankel type, Legendre functions and table errata. Integral Transforms and Special Functions, 27(5), 385-391. doi:10.1080/10652469.2015.1132716
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  The complete Lipschitz-Hankel integrals (LHIs) include the Laplace transforms of the Bessel functions, multiplied by powers. Such Laplace transforms can be evaluated using associated Legendre functions. It is noted that there are errors in published versions of these evaluations, and a merged and emended list of seven transforms is given. Errata for standard reference works, such as the table of Gradshteyn and Ryzhik, are also given. Most of the errors are attributable to inconsistent normalization of the Legendre functions. These transforms can be viewed as limits of incomplete LHIs, which find application in communication theory.
• 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
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  Associated Legendre functions of fractional degree appear in the solution of boundary value problems on wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer, they can be expressed using complete elliptic integrals. In this study, many transformations are derived, which reduce the case when the degree differs from an integer by one-third, one-fourth, or one-sixth to the classical case. These transformations, or identities, facilitate the symbolic manipulation and evaluation of Legendre and Ferrers functions. They generalize both Ramanujan's transformations of elliptic integrals and Whipple's formula, which relates Legendre functions of the first and second kinds. The proofs employ algebraic coordinate transformations, specified by algebraic curves.
• Wei, J., Wu, X., Melia, F., & Maier, R. S. (2015). A comparative analysis of the Supernova Legacy Survey Sample with Lambda CDM and the R-h = ct universe. Astronomical Journal, 149(3), 102 (11 pp.). doi:10.1088/0004-6256/149/3/102
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  The use of Type Ia supernovae (SNe Ia) has thus far produced the most reliable measurement of the expansion history of the universe, suggesting that ΛCDM offers the best explanation for the redshift–luminosity distribution observed in these events. However, analysis of other kinds of sources, such as cosmic chronometers, gamma-ray bursts, and high-z quasars, conflicts with this conclusion, indicating instead that the constant expansion rate implied by the Rh = ct universe is a better fit to the data. The central difficulty with the use of SNe Ia as standard candles is that one must optimize three or four nuisance parameters characterizing supernova (SN) luminosities simultaneously with the parameters of an expansion model. Hence, in comparing competing models, one must reduce the data independently for each. We carry out such a comparison of ΛCDM and the Rh = ct universe using the SN Legacy Survey sample of 252 SN events, and show that each model fits its individually reduced data very well. However, since Rh = ct has only one free parameter (the Hubble constant), it follows from a standard model selection technique that it is to be preferred over ΛCDM, the minimalist version of which has three (the Hubble constant, the scaled matter density, and either the spatial curvature constant or the dark energy equation-of-state parameter). We estimate using the Bayes Information Criterion that in a pairwise comparison, the likelihood of Rh = ct is ∼90%, compared with only ∼10% for a minimalist form of ΛCDM, in which dark energy is simply a cosmological constant. Compared to Rh = ct, versions of the standard model with more elaborate parametrizations of dark energy are judged to be even less likely.
• Maier, R. S. (2014). The uniformization of certain algebraic hypergeometric functions. Advances in Mathematics, 253, 86-138. doi:10.1016/j.aim.2013.11.013
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  Abstract:The hypergeometric functions Fn-1n 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 Fn-1n'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 so-called 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 Fn-1n'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 single-valued or multivalued parametrizations of individual algebraic Fn-1n's, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials.
• Maier, R. S. (2013). The integration of three-dimensional Lotka-Volterra systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469(2158), 20120693 (26 pp.). doi:10.1098/rspa.2012.0693
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  The general solutions of many three-dimensional Lotka-Volterra systems, previously known to be at least partially integrable, are constructed with the aid of special functions. Examples include certain ABC and May-Leonard 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 third-order differential equation of a generalized Schwarzian type, and results of Carton-LeBrun 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 Lotka-Volterra systems, the solutions are expressed in terms of Painlevé transcendents. An appendix on incomplete beta functions and closed-form expressions for their inverses is included.
• Melia, F., & Maier, R. S. (2013). Cosmic chronometers in the Rh = ct universe. Monthly Notices of the Royal Astronomical Society, 432(4), 2669-2675. doi:10.1093/mnras/stt596
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  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 model-independent 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 pre-assumption 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 Λ cold dark matter (ΛCDM) 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 s-1 Mpc-1, and H0 is the sole parameter in this model. By comparison, the optimal ΛCDM model, which has three free parameters (including H0 = 68.9 ± 3.3 km s.1 Mpc.1, σm = 0.32, and a dark-energy 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 ΛCDM. If the alternative BIC criterion is used, the respective Bayesian posterior probabilities are 91.2 per cent (Rh = ct) versus 8.8 per cent (ΛCDM). Using the concordance ΛCDM parameter values, rather than those obtained by fitting ΛCDM to the cosmic chronometer data, would further disfavour ΛCDM.
• Maier, R. S. (2011). Nonlinear differential equations satisfied by certain classical modular forms. Manuscripta Mathematica, 134(1), 1-42. doi:10.1007/s00229-010-0378-9
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  A unified treatment is given of low-weight modular forms on Γ_0(N), N = 2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under Γ_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2, R), which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with Γ(1) is treated.
• Maier, R. S. (2009). On rationally parametrized modular equations. Journal of the Ramanujan Mathematical Society, 24(1), 1-73. doi:10.48550/arxiv.math/0611041
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  Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup Gamma_0(N), as an algebraic transformation of elliptic curve periods, parametrized by a Hauptmodul (function field generator). The periods satisfy a Picard-Fuchs equation, of hypergeometric, Heun, or more general type; so the new modular equations are algebraic transformations of special functions. When N=4,3,2, they are modular transformations of Ramanujan's elliptic integrals of signatures 2,3,4. This gives a modern interpretation to his theories of integrals to alternative bases: they are attached to certain families of elliptic curves. His anomalous theory of signature 6 turns out to fit into a general Gauss-Manin rather than a Picard-Fuchs framework.
• 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), 1115-1153. doi:10.1098/rsta.2007.2063
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  The band structure of the Lamé equation, viewed as a one-dimensional 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 Hermite-Krichever Ansatz, which expresses Lamé equation solutions in terms of l=1 solutions, is the chief tool. It is based on a projection from a genus-l 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. band-edge solutions, are given. A table in the earlier literature is corrected.
• Maier, R. S. (2007). Algebraic hypergeometric transformations of modular origin. Transactions of the American Mathematical Society, 359(8), 3859-3885. doi:10.1090/S0002-9947-07-04128-1
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  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 N-fold cover of X0(N) by X0(N2) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic-geometric 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 Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.
• Maier, R. S. (2007). Parametrized stochastic grammars for RNA secondary structure prediction. 2007 Information Theory and Applications Workshop, Conference Proceedings, ITA, 256-260. doi:10.1109/ITA.2007.4357589
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  We propose a two-level stochastic context-free 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 Dyck-language words, i.e., as weighted balanced-parenthesis expressions. The length of each run of unpaired bases, forming a loop or a bulge, is taken to have a phase-type distribution: that of the hitting time in a finite-state 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), 811-843. doi:10.1090/S0025-5718-06-01939-9
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  A machine-generated 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 D_n. Each of the 192 expressions is labeled by an element of D_4. Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that the resulting order-24 group of transformations of Hl is isomorphic to the symmetric group S_4. The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.
• Maier, R. S. (2006). A generalization of Euler's hypergeometric transformation. Transactions of the American Mathematical Society, 358(1), 39-57. doi:10.1090/s0002-9947-05-04045-6
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  Abstract: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 one-term 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 one-term 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), 171-203. doi:10.1016/j.jde.2004.07.020
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  Abstract:The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric 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 cross-ratio 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 higher-degree 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, 48-57. doi:10.1117/12.553199
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  Abstract: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 field-induced but noise-activated 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 bounded-length (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 low-temperature (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 low-temperature limit.
• Maier, R. S. (2004). Algebraic solutions of the Lamé equation, revisited. Journal of Differential Equations, 198(1), 16-34. doi:10.1016/j.jde.2003.06.006
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  Abstract: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 (4-5) (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 two-dimensional percolation. Journal of Statistical Physics, 111(5-6), 1027-1048. doi:10.1023/A:1023006413433
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  Abstract:Several formulas for crossing functions arising in the continuum limit of critical two-dimensional percolation models are studied. These include Watts's formula for the horizontal-vertical 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 one-dimensional system. Proceedings of SPIE - The International Society for Optical Engineering, 5114, 67-78. doi:10.1117/12.497641
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  Abstract: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 Ginzburg-Landau field theory restricted to a bounded one-dimensional spatial domain, perturbed by weak spatiotemporal noise having a flat power spectrum in time and space. Our analysis extends the Kramers theory of noise-induced 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 Langer-Coleman theory of the noise-induced 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 infinite-volume 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), 270601 (4 pp.). doi:10.1103/PhysRevLett.87.270601
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  Abstract:A study on droplet nucleation and domain wall motion in a bounded interval was performed. A classical Ginzburg-Landau model as a spatially extended model of noise-induced 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 Coleman-Langer 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). Noise-activated escape from a sloshing potential well. Physical Review Letters, 86(18), 3942-3945. doi:10.1103/PhysRevLett.86.3942
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  We treat the noise-activated escape from a one-dimensional potential well of an overdamped particle, to which a periodic force of fixed frequency is applied. Near the well top, the relevant length scales and the boundary layer structure are determined. We show how behavior near the well top generalizes the behavior determined by Kramers, in the case without forcing. Our analysis includes the case when the forcing does not die away in the weak noise limit. We discuss the relevance of scaling regimes, defined by the relative strengths of the forcing and the noise, to recent optical trap experiments.
• Maier, R. S., & Stein, D. L. (2000). How an anomalous cusp bifurcates in a weak-noise system. Physical Review Letters, 85(7), 1358-1361. doi:10.1103/PhysRevLett.85.1358
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  Abstract:The unfolding of an anomalous cusp in a typical two-dimensional noise-activated 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 saddle-point avoidance in noise-induced escape. Physical Review Letters, 82(9), 1806-1809. doi:10.1103/PhysRevLett.82.1806
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  Abstract: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 weak-noise 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), 3109-3112. doi:10.1103/PhysRevLett.79.3109
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  Abstract:We consider a noise-driven exit from a domain of attraction in a two-dimensional 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 time-reversal invariance.
• Maier, R. S., & Stein, D. L. (1997). Limiting exit location distributions in the stochastic exit problem. SIAM Journal on Applied Mathematics, 57(3), 752-790. doi:10.1137/S0036139994271753
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  Abstract:Consider a two-dimensional continuous-time 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 non-Gaussian 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 weak-noise exit time asymptotics.
• Smelyanskiy, V. N., Dykman, M. I., & Maier, R. S. (1997). Erratum - Topological features of large fluctuations to the interior of a limit cycle. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 56(2), 2332-2332. doi:10.1103/PhysRevE.56.2332
• 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), 2369-2391. doi:10.1103/PhysRevE.55.2369
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  Abstract: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 self-sustained 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 self-similar structure. Depending on the behavior of the system near the focus, it may be smooth, or have finite-length 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 nu-merically, 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), 19197-19209. doi:10.1021/jp962746i
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  Abstract: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 self-similar near the unstable focus. In fact, a switching line spirals down to the focus. The logarithm of the stationary probability density has a self-similar 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 weak-noise escape problem. Journal of Statistical Physics, 83(3-4), 291-357. doi:10.1007/bf02183736
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  Abstract:We consider two-dimensional overdamped double-well systems perturbed by white noise. In the weak-noise 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 double-well 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 non-Arrhenius. We quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB 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 weak-noise 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), 4860-4863. doi:10.1103/PhysRevLett.77.4860
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  Abstract:Suppose a two-dimensional 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 Fokker-Planck equation we show that in general the weak-noise escape rate is non-Arrhenius: 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 weak-noise limit of stochastic resonance models.
• Maier, R. S. (1995). The shape of stretched planar trees. Random Structures & Algorithms, 6(2-3), 331-340. doi:10.1002/rsa.3240060220
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  We 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), 903-910. doi:10.1115/DETC1995-0335
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  Abstract:We study the asymptotic properties of overdamped dynamical systems with one or more point attractors, when they are perturbed by weak noise. In the weak-noise limit, fluctuations to the vicinity of any specified non-attractor point will increasingly tend to follow a well-defined 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 noise-induced transitions between domains of attraction, in two-dimensional 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 weak-noise limit in a non-exponential way. The analysis relies on a Maslov-WKB approximation to the solution of the Smoluchowski equation.
• Maier, R. S. (1993). Phase-type distributions and the structure of finite Markov chains. Journal of Computational and Applied Mathematics, 46(3), 449-453. doi:10.1016/0377-0427(93)90040-i
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  We show that all discrete phase-type distributions arise as first passage times (i.e., absorption times) in finite-state Markov chains with a certain recursive internal structure. This arises from the special properties of an automata-theoretic algorithm which can be used to solve the inverse problem for phase-type distributions: the construction of a Markov chain with specified absorption time distribution.
• 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), 1783-1786. doi:10.1103/PhysRevLett.71.1783
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  Abstract: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'' (non-Gaussian). The weak-noise 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), 931-938. doi:10.1103/PhysRevE.48.931
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  Abstract:The problem of noise-induced 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 transition-rate theory no longer apply, and no general method exists for computing the weak-noise asymptotic behavior of fundamental quantities such as the mean escape time. In this paper we present a general technique for analyzing the weak-noise limit of a wide range of stochastically perturbed continuous-time 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 short-time behavior and discuss the possibility of focusing along the escape path. © 1993 The American Physical Society.
• Maier, R. S., & O'Cinneide, C. A. (1992). A closure characterization of phase-type distributions. Journal of Applied Probability, 29(1), 92-103. doi:10.2307/3214794
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  We characterise the classes of continuous and discrete phase-type 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). Transition-rate theory for nongradient drift fields. Physical Review Letters, 69(26), 3691-3695. doi:10.1103/PhysRevLett.69.3691
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  Abstract:Classical transition-rate 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 pre-exponential factor, with a continuously varying exponent, in the residence time asymptotics. Moreover, the most probable escape trajectories may eventually deviate from the least-action escape path.
• Faris, W. G., & Maier, R. S. (1987). Confirmation in experimental mathematics - A case study. Complex Systems, 5(2), 259-264.
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  On the basis of experiment, two groups arrived independently at similar conjectures on the nature of optimal strategies employed in large random games. The first conjecture was made by Kuhn and Quandt in 1963; the second was made by the present authors in 1987. In traditional mathematics such duplication is superfluous. In experimental mathematics it may provide mutual confirmation.
• Maier, R. S. (1991). A path integral approach to data structure evolution. Journal of Complexity, 7(3), 232-260. doi:10.1016/0885-064x(91)90035-v
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  Abstract:A probabilistic analysis is presented of certain pointer-based implementations of dictionaries, linear lists, and priority queues; in particular, simple list and d-heap 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 d-heap implementations of priority queues the mean integrated space cost grows only as n2√log n, i.e., more slowly than the worst-case 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 datatype-dependent 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.
• Maier, R. S. (1991). Colliding stacks - A large deviations analysis. Random Structures & Algorithms, 2(4), 379-420. doi:10.1002/rsa.3240020404
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  We 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 higher-order asymptotics follow from WKB expansions.
• Maier, R. S. (1991). The algebraic construction of phase-type distributions. Communications in Statistics. Stochastic Models, 7(4), 573-602. doi:10.1080/15326349108807207
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  A probability distribution of discrete phase-type must have a rational generating function. We give an algorithm that constructs, from a given rational function G(z), a Markov chain whose absorption-time distribution has G(z) as generating function. The algorithm, which is based on an automata-theoretic algorithm of Soittola, may be applied to any G(z) that satisfies the conditions on discrete phase-type generating functions discovered by O’Cinneide. So it provides an alternative, algebraic proof of O’Cinneide's characterisation of discrete phase-type distributions. We also clarify the relation between the classes of continuous and discrete phase-type distributions, and show that O’Cinneide's characterisation of continuous phase-type distributions is a corollary of his discrete characterisation. In conjunction with our discrete-time algorithm, this engenders an algorithm for constructing a Markov process representation for any distribution of continuous phase-type.
• Faris, W. G., & Maier, R. S. (1988). Probabilistic analysis of a learning matrix. Advances in Applied Probability, 20(4), 695-705. doi:10.2307/1427355
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  A learning matrix is defined by a set of input and output pattern vectors. The entries in these vectors are zeros and ones. The matrix is the maximum of the outer products of the input and output pattern vectors. The entries in the matrix are also zeros and ones. The product of this matrix with a selected input pattern vector defines an activity vector. It is shown that when the patterns are taken to be random, then there are central limit and large deviation theorems for the activity vector. They give conditions for when the activity vector may be used to reconstruct the output pattern vector corresponding to the selected input pattern vector.
• Faris, W. G., & Maier, R. S. (1987). The value of a random game - The advantage of rationality. Complex Systems, 1(2), 235-244.
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  Two players play against each other in a game with payoffs given by a random n-by-n matrix with mean zero. If one player adopts a uniform, purely random, strategy, then his loss is limited by the law of averages to a quantity proportional to sqrt(log n)/sqrt(n). On the other hand, if he plays an optimal strategy his losses will typically be considerably less. Numerical evidence is presented for the following conjecture: The standard deviation of the value of the game is asymptotically proportional to 1/n. This smaller loss exhibits the advantage of rationality over randomness. The rational player, moreover, tends as n->infinity to employ a strategy vector that has half its components zero.
• Maier, R. S. (1987). Bounds on the density of states of random Schrödinger operators. Journal of Statistical Physics, 48(3-4), 425-447. doi:10.1007/bf01019681
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  Abstract: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 finite-difference 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. S., & Whaling, W. (1977). Transition probabilities in Nd(II) and the solar neodymium abundance. Journal of Quantitative Spectroscopy & Radiative Transfer, 18(5), 501-507. doi:10.1016/0022-4073(77)90048-6
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  We have measured branching ratios for all the known transitions from nine levels in Nd^+. We use the known mean lifetimes of four of these levels to compute transition probabilities for 60 transitions. We have measured the solar equivalent widths of 12 of these lines on the Preliminary Edition of the Kitt Peak Solar Atlas and compute a photospheric Nd abundance log(NNd N_H)+12 = 1.26±0.14, using the solar model parameters of Righini and Rigutti based on the Mutschlecner model. The solar lines used in this computation are weak ( W λ

Proceedings Publications

• Maier, R. S., & Stein, D. L. (2000). The birth of a cusp - The unfolding of a ‘boundary catastrophe’. In AIP [American Institute of Physics] Conference Proceedings, 502, 26-33.
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  We study a phenomenon, analogous to a phase transition, that takes place in the theory of noise-activated transitions between attractors. Large fluctuations away from any attractor tend to be concentrated along certain optimal trajectories, and the flow field of optimal trajectories may contain focal points, or cusps. Optimal trajectories may be viewed as Hamiltonian trajectories, and a cusp may be viewed as a cusp catastrophe in the Lagrangian manifold that the optimal trajectories trace out. As the parameters of a noise-driven system are changed, a cusp may emerge from a saddle point of the underlying deterministic dynamics. This corresponds to a cusp catastrophe being formed at the boundary of the Lagrangian manifold, and moving inward. It is possible to find a nonpolynomial normal form that unfolds the corresponding ‘boundary catastrophe,’ in a space of higher dimensionality. Just as the quartic normal form for a cusp catastrophe resembles the scaling form for a classical (Ginzburg-Landau) phase trans...
• Maier, R. S. (1995, Oct). Effective bandwidth of Markov fluids with occupancy-based admission control. In Thirty-Third Annual Allerton Conference on Communication, Control, and Computing, 766-775.
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  We consider the performance of a single-buffer statistical multiplexer, when the aggregate traffic source is modelled as a Markov-modulated fluid. An occupancy-based admission control policy would take into account the current buffer occupancy when deciding whether or not to admit traffic of any specified priority class. We show that the now standard effective bandwidth concept can be extended to nontrivial occupancy-based admission control policies. The effective bandwidth, which is the minimum service rate needed to ensure a specified asymptotic exponential falloff of the buffer occupancy probability density, turns out to be additive over homogeneous sources, though not over heterogeneous ones. We also show how the pre-exponential factor in the probability density asymptotics can be computed. Our asymptotic analysis uses the method of multiple scales, applied to the system of ordinary differential equations that determines the stationary buffer occupancy distribution.
• Maier, R. S. (1992, Oct). Communications networks as stochastically perturbed nonlinear systems: A cautionary note. In Thirtieth Annual Allerton Conference on Communication, Control, and Computing, 674-681.
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  Lim, Meerkov and coworkers have studied large homogeneous communications networks from a stochastic point of view. If a Markovian access control discipline is used, the evolution of an M-node network, in which each node buffers up to N packets, is in the M->infinity limit modelled as a random walk (with drift) in an N-dimensional polyhedron. This random walk may be viewed as a stochastically perturbed nonlinear system. If this system has more than a single point of stable equilibrium, they argue that the study of the occasional transitions between them can be reduced to a one-dimensional first passage time problem. We show, by considering a related model, that this is an oversimplification which can result in arbitrarily large errors.
• Maier, R. S. (1990, Oct). Instability times of multiaccess broadcast channels. In Twenty-Eighth Annual Allerton Conference on Communication, Control, and Computing, 139-148.

Presentations

• Maier, R. S. (2022, June). Triangular recurrences and hypergeometric transformations. OPSFA16, The 16th International Symposium on Orthogonal Polynomials, Special Functions, and Applications. Montréal, Canada: Centre de Recherches Mathématiques.
• Maier, R. S. (2022, October). From triangular recurrences to operator ordering identities. American Mathematical Society, Fall Western Sectional Meeting. Salt Lake City, UT: American Mathematical Society.
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  An invited talk at the Special Session on Hypergeometric Functions and q-Series.
• Maier, R. S. (2021, August). Legendre functions: New formulas, identities, and transformations. NIST ACMD seminar series. Gaithersburg, MD: NIST [National Institute of Standards & Technology], ACMD [Applied & Computational Mathematics Division].
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  An invited talk, given to the NIST Applied & Computational Mathematics Division.
• Maier, R. S. (2019, August). Hypergeometric transformations based on Hahn and Racah polynomials. International Symposium on Orthogonal Polynomials, Special Functions, and Applications (OPSFA 2019). RISC-Linz, Hagenberg, Austria: Society for Industrial and Applied Mathematics.
• Maier, R. S. (2019, November). Transformation formulas and rational representations for hypergeometric functions. Analysis, Dynamics, and Applications Seminar (UA). Tucson, AZ: University of Arizona.
• 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 q-Series. Orlando, FL.
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  ABSTRACT: 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 second-order one. The Inceequation may have polynomial solutions, but their relation to the theory oforthogonal polynomials is unclear. The equation can be viewed as aquasi-exactly-solvable (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 non-polynomialsolutions that can be expressed in terms of the Gauss hypergeometricfunction. This includes solutions that generalize theBrioschi-Halphen-Crawford 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). Three-dimensional quadratic differential systems and the Painlevé property. American Mathematical Society, Spring Central Sectional Meeting. Lubbock, TX: American Mathematical Society.
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  An 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..