# Associate Professor Tim Langtry

**Associate Professor,**School of Mathematical and Physical Sciences

BAppSc (NSWIT), BA (Hons) (UNSW), MAppSc (NSWIT), PhD (UNSW)

Member, Australian Mathematical Society

**Phone**

+61 2 9514 2258

**Room**

CB07.05.38

### Research Interests

My main research interests are in numerical analysis and computational mathematics. Particular areas of interest include quasi-Monte Carlo methods for numerical multiple integration (eg rank 1 lattice rules) and, more recently, computational photonics.

My work in photonics has focussed on computational modelling of various aspects of photonic crystals - composite materials which exhibit a periodic variation in their refractive index, leading to novel and interesting optical properties. An example of some results of this work may be seen in the paper Effects of disorder in two-dimensional photonic crystal waveguides.

**Can supervise:**Yes

## Chapters

Botten, L.C., McPhedran, R.C., de Sterke, C.M., Nicorovici, N.A., Asatryan, A.A., Smith, G.H., Langtry, T., White, T., Fussell, D.P. & Kuhlmey, B. 2006, 'From Multipole Methods to Photonic Crystal Device modeling' in Yasumoto, K. (ed),

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*Electromagnetic Theory and Applications for Photonic Crystals*, Taylor & Francis, Florida, USA, pp. 47-122.View/Download from: UTS OPUS

NA

Botten, L.C., McPhedran, R.C., Nicorovici, N.A., Asatryan, A.A., de Sterke, C.M., Robinson, P.A., Busch, K., Smith, G.H. & Langtry, T.N. 2003, 'Rayleigh multipole methods for photonic crystal calculations', pp. 21-60.

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Multipole methods have evolved to be an important class of theoretical and computational techniques in the study of photonic crystals and related problems. In this chapter, we present a systematic and unified development of the theory, and apply it to a range of scattering problems including finite sets of cylinders, two-dimensional stacks of grating and the calculation of band diagrams from the scattering matrices of grating layers. We also demonstrate its utility in studies of finite systems that involve the computation of the local density of states.

Langtry, T.N. 1998, 'A generalisation of ratios of Fibonacci numbers with application to numerical quadrature', SPRINGER, pp. 239-253.

Langtry, T. 1997, 'Bounds on the figure of merit of some lattice rules' in Niederreiter, H., Hellekalek, P., Larcher, G. & Zinterhof, P. (eds),

*Monte Carlo and Quasi-Monte Carlo Methods 1996*, Springer Verlag, New York, pp. 308-320. Langtry, T. 1993, 'Applications of Fibonacci Numbers: Volume 5' in Bergum, G.E., Philippou, A.N. & Horadam, A.F. (eds),

*Applications of Fibonacci Numbers: Volume 5*, Kluwer Academic Publishers, Dordrecht, pp. 331-343.## Conferences

Zinder, Y., Nicorovici, N.A. & Langtry, T. 2011, 'Mathematica based platform for self-paced learning',

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*Proceedings of the 10th International Conference on Technology in Mathematics Teaching (ICTMT10)*, ICTMT10, University of Portsmouth, Portsmouth, United Kingdom, pp. 203-208.View/Download from: UTS OPUS

One of the major challenges in teaching applied mathematics is the large amount of calculation involved in many practical mathematical methods. On one hand, mastering these methods requires students to gain experience in performing all steps of a calculation. This experience is crucial for gaining an understanding of the methods, their capabilities and limitations, and cannot be replaced by black box type commercial software which simply displays the results of calculations and gives no insight into the nature of the operations performed. On the other hand, the calculations involved in many modern mathematical methods are tedious and time consuming with fatal results caused by even minor computational mistakes. The advent of computer algebra systems such as Mathematica opened new horizons in teaching mathematics. The ability to perform symbolic calculations in combination with powerful graphics and programming capabilities makes it possible to develop software that provides students with the opportunity for step by step exploration of mathematical procedures. We present the results of an ongoing research project aimed at the development of a Mathematica based platform which allows academics to develop software for self-paced learning.

Zinder, Y., Nicorovici, N. & Langtry, T. 2010, 'MATHEMATICA BASED PLATFORM FOR SELF-PACED LEARNING',

*EDULEARN10: INTERNATIONAL CONFERENCE ON EDUCATION AND NEW LEARNING TECHNOLOGIES*, 2nd International Conference on Education and New Learning Technologies (EDULEARN), IATED-INT ASSOC TECHNOLOGY EDUCATION A& DEVELOPMENT, Barcelona, SPAIN, pp. 323-330. Maher, A., Lake, M., Langtry, T. & Hill, C. 2007, 'A proteomics laboratory system and digital mentor',

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*APAC07*, Conference and Exhibition on Advanced Computing, Grid Application and eResearch, Australian Partnership for Advanced Computing, Perth, Australia, pp. 1-19.View/Download from: UTS OPUS

The typical small biological laboratory is cmposed of electronic scientific equipment and wet laboratory equipment where staff record reesults into paper workbooks. There is oftern a lack of expert staff to guide new users, and hard won experience is locked away with individual staff and their workbooks. The Proteomics Laboratory System and Digital Mentor aims to: facilitate the capture, analysis and sharing of results; facilitate the creation and documentation of standardised work flows and experimental procedures; and mentore inexperienced users in best practice when engaged in particular procedures in the laboratory. In this paper we discuss the design of the system and report on the initial production release. This is nased on a relational database and web-server framework. A novel aspect of the system is the exploitation of custom extensions to a wiki-style interface that add functionality for non-expert end-users.

Asatryan, A.A., Botten, L.C., McPhedran, R.C., de Sterke, C.M., Langtry, T. & Nicorovici, N.A. 2004, 'Conductance of photons and Anderson localization of light',

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*Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies*, Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, Optical Society of America, San Francisco, USA, pp. 860-861.View/Download from: UTS OPUS

Conductance properties of photons in disordered two-dimensional photonic crystals is calculated using exact multipole expansions technique. The Landauers two-terminal formula is used to calculate average of the conductance, its variance and the probability density distribution.

Langtry, T., Botten, L.C., Asatryan, A.A., Byrne, M.A. & Bourgeois, A. 2003, 'Localisation and disorder in the design of 2D photonic crystal devices',

*ANZIAM Journal - 11th Biennial Computational Techniques and Applications Conference: CTAC-2003*, Computational Techniques and Applications Conference, Cambridge University Press, Sydney, Australia, pp. 744-758. Photonic crystals are meta-materials that can inhibit the propagation of light in all directions for specific wavelength ranges. Material or structural defects can be introduced into the crystal to cause localised modes, providing the ability to mould the flow of light on the wavelength scale and allowing the development of miniaturised, integrated photonic devices. For this reason, photonic crystals will likely be key building blocks for future micro-optical and communication technology. In this paper, we examine the Bloch mode modelling of 2D photonic crystal structures with application to the analysis of photonic crystal waveguides and their susceptibility to disorder, which provides a framework for studying fabrication tolerances in realistic devices.

Botten, L.C., Asatryan, A.A., White, T.P., McPhedran, R.C., De Sterke, C.M. & Langtry, T.N. 2004, 'Modelling of extended photonic crystal devices using scattering matrix techniques',

*PIERS 2004 - Progress in Electromagnetics Research Symposium, Extended Papers Proceedings*, pp. 13-16. A rigorous semi-analytic approach to the modelling of coupling, guiding and propagation in complex microstructures embedded in photonic crystals is presented. The method, based on Bloch modes and generalized Fresnel coefficients, is outlined and a variety of applications of the design tool are presented.

Asatryan, A.A., Botten, L.C., White, T.P., de Sterke, C.M., McPhedran, R.C. & Langtry, T. 2003, 'Modelling of complex waveguides structures embedded in photonic crystals',

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*COIN/ACOFT 2003. Proceedings on the Optical Internet and Optical Fibre Technology*, Conference on the Optical Internet/Australian Conference on Optical Fibre Technology 2003, COIN/ACOFT 2003, Melbourne, Australia, pp. 145-148.View/Download from: UTS OPUS

de Sterke, C.M., Botten, L.C., White, T.P., McPhedran, R.C., Asatryan, A.A. & Langtry, T. 2003, 'Photonic bandgap effects as a basis for novel compact devices',

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*COIN/ACOFT 2003. Conference Proceedings on the Optical Internet and Optical Fibre Technology*, Conference on the Optical Internet / Australian Conference on Optical Fibre Technology 2003, COIN/ACOFT 2003, Melbourne, Australia, pp. 133-136.View/Download from: UTS OPUS

Langtry, T., Botten, L.C., Asatryan, A.A., Byrne, M.A., Bourgeois, A. & McPhedran, R.C. 2003, 'Computational modelling of photonic crystals',

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*Proceedings of the APAC Conference and Exhibition on Advanced Computing, Grid Application and eResearch*, Conference and Exhibition on Advanced Computing, Grid Application and eResearch, Australian Partnership for Advanced Computing (APAC, Gold Coast, Australia, pp. 1-19.View/Download from: UTS OPUS

Botten, L.C., Asatryan, A.A., Langtry, T.N., White, T.P., De Sterke, C.M. & McPhedran, R.C. 2003, 'An analytic treatment of propagation in straight and bent photonic crystal waveguides',

*OSA Trends in Optics and Photonics Series*, pp. 1021-1022. A semi-analytic model is developed for coupling and guiding in nano-structured waveguides embedded in two-dimensional photonic crystals. The method, based on Bloch modes and generalized Fresnel coefficients, is applied to waveguide junctions and bends. © 2003 Optical Society of America.

Botten, L.C., Asatryan, A.A., Langtry, T., de Sterke, C.M. & McPhedran, R.C. 2002, 'Propagation of Photonic Crystal Waveguides',

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*Proceedings of the Australian Institute of Physics 15th Biennial Congress 2002.*, Congress of the Australian Institute of Physics, Australian Institute of Physics, Sydney, Australia, pp. 289-291.View/Download from: UTS OPUS

Langtry, T., Botten, L.C., de Sterke, C.M., Asatryan, A.A. & McPhedran, R.C. 2002, 'Effects of Disorder in Photonic Crystal Waveguides',

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*Proceedings of the Australian Institute of Physics 15th Biennial Congress 2002.*, Congress of the Australian Institute of Physics, Australian Institute of Physics, Sydney, Australia, pp. 310-312.View/Download from: UTS OPUS

Langtry, T., Botten, L.C., Asatryan, A.A. & McPhedran, R.C. 2001, 'Local Density of States Calculations for Photonic Crystals',

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*Proceedings of HPC Asia 2001*, International Conference and Exhibition on High Performance Computing in the Asia-Pacific Region, Queensland Parallel Supercomputing Facilty and Australian Partnership for Advasnced Computing, Gold Coast, pp. 0-0.View/Download from: UTS OPUS

Langtry, T.N. 1998, 'A discrepancy-based analysis of figures of merit for lattice rules',

*MONTE CARLO AND QUASI-MONTE CARLO METHODS 1998*, 3rd International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQM 98), SPRINGER-VERLAG BERLIN, CLAREMONT GRAD UNIV, CLAREMONT, CA, pp. 296-310.## Journal articles

Groen, L., Coupland, M., Langtry, T., Memar, J., Moore, B.J. & Stanley, J. 2015, 'The Mathematics Problem and Mastery Learning for First-Year, Undergraduate STEM Students',

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*International Journal of Learning, Teaching and Educational Research*, vol. 11, no. 1, pp. 141-160.View/Download from: UTS OPUS

In the 2014 academic year Mastery Learning was implemented in four first-year mathematics subjects in an effort to address a lack of preparedness and poor outcomes of increasing numbers of undergraduate students in science, engineering and mathematics programs. This followed partial success in the use of diagnostic testing and pre-teaching, active learning, and a greater emphasis on problem solving in context - under-prepared students were still more likely to fail the pre-teaching subject and to struggle with subsequent mathematics subjects. Also, failure rates overall were higher than benchmarks required. This paper describes the learning design used, and the outcomes achieved, with implementing Mastery Learning – the positive: improved academic success, time management, and attitudes towards learning and Mathematics, an increased sense of independence, confidence and retention of content, and reduced stress and anxiety; and the negative: students having a sense of being taught how to pass a test rather than having a deeper understanding of the content. It will be seen that this negative is a consequence of a small but important difference in implementation.

Jay, K., Chaumet, P.C., Langtry, T. & Rahmani, A. 2010, 'Optical binding of electrically small magnetodielectric particles',

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*Journal of Nanophotonics*, vol. 4, pp. 1-9.View/Download from: UTS OPUS or Publisher's site

An ensemble of spherical particles with arbitrary dielectric permittivity and magnetic penneability was considered in the dipole approximation. Each particle was described by complex electric and magnetic polarizabilities. A computational approach based on the coupled dipole method, also called the discrete dipole approximation, was used to derive the optical force experienced by each particle due to an incident electromagnetiG..Ji.eld and the fields scattered by all other particles. This approach is general and can handle material dispersion and losses. In order to illustrate this approach, we studied the case of two spherical particles separated by a distance d, and illuminated by an incident plane wave whose wave vector is normal to the axis of the particles. We computed the optical force experienced by each particle in the direction of the beam (radiation pressure), and perpendicular to the beam (optical binding) for particles with positive and negative refractive indices. We also considered the effect of material losses.

Jay, K., Chaumet, P.C., Langtry, T.N. & Rahmani, A. 2010, 'Optical binding of electrically small magnetodielectric particles',

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*Journal of Nanophotonics*, vol. 4, no. 1.View/Download from: Publisher's site

An ensemble of spherical particles with arbitrary dielectric permittivity and magnetic permeability was considered in the dipole approximation. Each particle was described by complex electric and magnetic polarizabilities. A computational approach based on the coupled dipole method, also called the discrete dipole approximation, was used to derive the optical force experienced by each particle due to an incident electromagnetic field and the fields scattered by all other particles. This approach is general and can handle material dispersion and losses. In order to illustrate this approach, we studied the case of two spherical particles separated by a distance d, and illuminated by an incident plane wave whose wave vector is normal to the axis of the particles. We computed the optical force experienced by each particle in the direction of the beam (radiation pressure), and perpendicular to the beam (optical binding) for particles with positive and negative refractive indices. We also considered the effect of material losses. © 2010 Society of Photo-Optical Instrumentation Engineers.

Asatryan, A.A., Botten, L.C., Byrne, M.A., Langtry, T.N., Nicorovici, N.A., McPhedran, R.C., De Sterke, C.M. & Robinson, P.A. 2005, 'Conductance of photons in disordered photonic crystals',

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*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 71, no. 3.View/Download from: Publisher's site

The conductance of photons in two-dimensional disordered photonic crystals is calculated using an exact multipole-plane wave method that includes all multiple scattering processes. Conductance fluctuations, the universal nature of which has been established for electrons in the diffusive regime, are studied for photons, in both principal polarizations and for varying disorder. Our simulations show that universal conductance fluctuations can be observed in H? (TE) polarization for weak and intermediate disorder while, for E? (TM) polarization, we show that the conductance variance is essentially independent of sample size but strongly dependent on disorder. The probability distribution of the conductance is also calculated in the diffusive and localized regimes, and also at their transition, for which the distributions for both polarizations are seen to be very similar. © 2005 The American Physical Society.

Botten, L.C., White, T.P., De Sterke, C.M., McPhedran, R.C., Asatryan, A.A. & Langtry, T.N. 2004, 'Photonic crystal devices modelled as grating stacks: Matrix generalizations of thin film optics',

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*Optics Express*, vol. 12, no. 8, pp. 1592-1604.View/Download from: UTS OPUS or Publisher's site

A rigorous semi-analytic approach to the modelling of coupling, guiding and propagation in complex microstructures embedded in two-dimensional photonic crystals is presented. The method, which is based on Bloch mode expansions and generalized Fresnel coefficients, is shown to be able to treat photonic crystal devices in ways which are analogous to those used in thin film optics with uniform media. Asymptotic methods are developed and exemplified through the study of a serpentine waveguide, a potential slow wave device. © 2004 Optical Society of America.

White, T.P., Botten, L.C., de Sterke, C.M., McPhedran, R.C., Asatryan, A.A. & Langtry, T.N. 2004, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications',

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*PHYSICAL REVIEW E*, vol. 70, no. 5.View/Download from: UTS OPUS or Publisher's site

Langtry, T., Botten, L.C., Asatryan, A.A., Byrne, M.A. & Bourgeois, A. 2004, 'Localisation and disorder in the design of 2D photonic crystal devices',

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*ANZIAM Journal*, vol. 45, pp. 744-758.View/Download from: UTS OPUS

Photonic crystals are meta-materials that can inhibit the propagation of light in all directions for specific wavelength ranges. Material or structural defects can be introduced into the crystal to cause localised modes, providing the ability to mould the flow of light on the wavelength scale and allowing the development of miniaturised, integrated photonic devices. For this reason, photonic crystals will likely be key building blocks for future micro-optical and communication technology. In this paper, we examine the Bloch mode modelling of 2D photonic crystal structures with application to the analysis of photonic crystal waveguides and their susceptibility to disorder, which provides a framework for studying fabrication tolerances in realistic devices.

Botten, L.C., White, T.P., Asatryan, A.A., Langtry, T.N., de Sterke, C.M. & McPhedran, R.C. 2004, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory',

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*PHYSICAL REVIEW E*, vol. 70, no. 5.View/Download from: UTS OPUS or Publisher's site

White, T.P., Botten, L.C., de Sterke, C.M., McPhedran, R.C., Asatryan, A.A. & Langtry, T.N. 2004, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications.',

*Phys Rev E Stat Nonlin Soft Matter Phys*, vol. 70, no. 5 Pt 2, p. 056607. The Bloch mode scattering matrix method is applied to several photonic crystal waveguide structures and devices, including waveguide dislocations, a Fabry-Pérot resonator, a folded directional coupler, and a Y-junction design. The method is an efficient tool for calculating the properties of extended photonic crystal (PC) devices, in particular when the device consists of a small number of distinct photonic crystal structures, or for long propagation lengths through uniform PC waveguides. The physical insight provided by the method is used to derive simple, semianalytic models that allow fast and efficient calculations of complex photonic crystal structures. We discuss the situations in which such simplifications can be made and provide examples.

Botten, L.C., White, T.P., Asatryan, A.A., Langtry, T.N., de Sterke, C.M. & McPhedran, R.C. 2004, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory.',

*Phys Rev E Stat Nonlin Soft Matter Phys*, vol. 70, no. 5 Pt 2, p. 056606. We present a rigorous Bloch mode scattering matrix method for modeling two-dimensional photonic crystal structures and discuss the formal properties of the formulation. Reciprocity and energy conservation considerations lead to modal orthogonality relations and normalization, both of which are required for mode calculations in inhomogeneous media. Relations are derived for studying the propagation of Bloch modes through photonic crystal structures, and for the reflection and transmission of these modes at interfaces with other photonic crystal structures.

Asatryan, A.A., Robinson, P.A., McPhedran, R.C., Botten, L.C., de Sterke, C.M., Langtry, T.L. & Nicorovici, N.A. 2003, 'Diffusion and anomalous diffusion of light in two-dimensional photonic crystals',

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*PHYSICAL REVIEW E*, vol. 67, no. 3.View/Download from: UTS OPUS or Publisher's site

Langtry, T., Coupland, M. & Moore, B. 2003, 'Mathematicaâ„¢ in context',

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*International Journal of Mathematical Education in Science and Technology*, vol. 34, no. 5, pp. 699-718.View/Download from: UTS OPUS or Publisher's site

Langtry, T., Botten, L.C., Asatryan, A.A. & McPhedran, R.C. 2003, 'Monte Carlo modelling of imperfections in two-dimensional photonic crystals',

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*Mathematics And Computers In Simulation*, vol. 62, no. 3-6, pp. 385-393.View/Download from: UTS OPUS or Publisher's site

In this paper we describe a Monte Carlo simulation of imperfections in photonic crystals, a new class of materials with optical properties that offer promise in a range of potential applications in the areas of information and communications technology. We describe the relevant physical and structural properties of these materials and outline the derivation of a theoretical model. We then present a Monte Carlo investigation of the tolerance of these materials to fabrication defects.

Botten, L.C., Asatryan, A.A., Langtry, T.N., White, T.P., Martijn de Sterke, C. & McPhedran, R.C. 2003, 'Semianalytic treatment for propagation in finite photonic crystal waveguides',

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*Optics Letters*, vol. 28, no. 10, pp. 854-856.View/Download from: UTS OPUS

We present a semianalytic theory for the properties of two-dimensional photonic crystal waveguides of finite length. For single-mode guides, the transmission spectrum and field intensity can be accurately described by a simple two-parameter model. Analogies are drawn with Fabry-Perot interferometers, and generalized Fresnel coefficients for the interfaces are calculated. © 2003 Optical Society of America.

Langtry, T.N., Asatryan, A.A., Botten, L.C., de Sterke, C.M., McPhedran, R.C. & Robinson, P.A. 2003, 'Effects of disorder in two-dimensional photonic crystal waveguides',

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*PHYSICAL REVIEW E*, vol. 68, no. 2.View/Download from: Publisher's site

Thornton, B.S., Nguyen, H.T., Hung, A., Hirst, C., Thornton-Benko, E. & Langtry, T. 2001, 'Breast Screening Outcomes: Communications Problems, Chaos Relationship and Control Theory',

*Canadian Applied Mathematics Quarterly*, vol. 9, no. 4, pp. 377-401. Langtry, T.N. 2001, 'A construction of higher-rank lattice rules',

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*Mathematics and Computers in Simulation*, vol. 55, no. 1-3, pp. 103-111.View/Download from: UTS OPUS or Publisher's site

Lattice rules are quasi-Monte Carlo methods for numerical multiple integration that are based on the selection of an s-dimensional integration lattice. The abscissa set is the intersection of the integration lattice with the unit hypercube. It is well-known that the abscissa set of a lattice rule can be generated by a number of fixed rational vectors. In general, different sets of generators produce different integration lattices and rules, and a given rule has many different generator sets. The rank of the rule is the minimum number of generators required to span the abscissa set. A lattice rule is usually specified by a generator set, and the quality of the rule varies with the choice of generator set. This paper describes a new method for the construction of generator sets for higher-rank rules that is based on techniques arising from the theory of simultaneous Diophantine approximation. The method extends techniques currently applied in the rank 1 case. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.

Langtry, T.N. 1999, 'Lattice rules of minimal and maximal rank with good figures of merit',

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*JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS*, vol. 112, no. 1-2, pp. 147-164.View/Download from: UTS OPUS or Publisher's site

Langtry, T.N. 1996, 'An application of diophantine approximation to the construction of rank-1 lattice quadrature rules',

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*Mathematics of Computation*, vol. 65, no. 216, pp. 1635-1662.View/Download from: UTS OPUS or Publisher's site

Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in ?s and are generalizations of 'number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962) -themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variaole, these bounds are conveniently characterized by the figure of merit ?, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order N (that is, having N nodes) then becomes that of finding rules with large values of ?. This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.

Langtry, T.N. 1995, 'The determination of canonical forms for lattice quadrature rules',

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*Journal of Computational and Applied Mathematics*, vol. 59, no. 2, pp. 129-143.View/Download from: Publisher's site

Lattice rules are equal weight numerical quadrature rules for the integration of periodic functions over the s-dimensional unit hypercube Us = [0, 1)s. For a given lattice rule, say QL, a set of points L (the integration lattice), regularly spaced in all of Rs, is generated by a finite number of rational vectors. The abscissa set for QL is then P(QL)= L ? Us. It is known that P(QL) is a finite Abelian group under addition modulo the integer lattice Zs, and that QL(f) may be written in the form of a nonrepetitive multiple sum, QL(f)= 1 n1?nm ? j1=1 n1? ? jm=1 nmf j1 n1z1+?+ jm nmzm, known as a canonical form, in which + denotes addition modulo Zs. In this form, zi ? Zs, m is called the rank and n1, n2,..., nm are called the invariants of QL, and ni+1|ni for i = 1,2,..., m - 1. The rank and invariants are uniquely determined for a given lattice rule. In this paper we provide a construction of a canonical form for a lattice rule QL, given a generator set for the lattice L. We then show how the rank and invariants of QL may be determined directly from the generators of the dual lattice L?. © 1995.

Thornton, B.S., Thornton, B.S., Langtry, T.N. & Langtry, T.N. 1988, 'Prescheduling graphic displays for optimal cancer therapies to reveal possible tumor regression or stabilization',

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*Journal of Medical Systems*, vol. 12, no. 1, pp. 31-41.View/Download from: UTS OPUS or Publisher's site

The paper describes an adaptive control approach to the problem of the treatment of solid tumors. The evolution with time t of the state of a tumor is modelled by a two-compartment system, governed by two differential equations forming an autonomous system under therapy control u, {Mathematical expression} where y1 and y2 are the number of proliferating and nonproliferating cells, respectively. The output is analyzed in the phase plane y1y2. The control problem is that of restricting the tumor state to a predetermined region of the plane by selecting a suitable change in therapy control u, e.g., modality and dosage, when the state solution intersects the boundary of this region and the ratio y1/y2 of proliferating to nonproliferating cells is displayed together with an elapsed time scale. Then, consequent selection of a suitable therapeutic sequence may be assisted by the use of a data base as part of an expert system. The process is repeated at each intersection of the prescribed boundary. Such sequences may lead to stabilization of the system through the appearance on a computer display screen of a stable equilibrium point or a limit cycle. © 1988 Plenum Publishing Corporation.