The renormalization group: A perturbation method for the graduate curriculum

In this paper the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. Lett., 73 (1994), pp.1311-1315; Phys. Rev. E, 54 (1996), pp.376-394] is presented in a pedagogical way to increase its visibility in applied mathematics and to argue favorably for its incorporation into the corresponding graduate curriculum.The method is illustrated by some linear and nonlinear singular perturbation problems. Key word. © 2012 Society for Industrial and Applied Mathematics.

A survey in mathematics for industry: Two-timing and matched asymptotic expansions for singular perturbation problems

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124, 383-410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations. © 2011 Cambridge University Press.

Amplitude equations and asymptotic expansions for multi-scale problems

In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases. © 2010 - IOS Press and the authors. All rights reserved.

Reduction of amplitude equations by the renormalization group approach

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence { Zi } i=1 . Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation. © 2008 The American Physical Society.

Data collection techniques

The objective of this chapter is to provide an overview of traffic data collection that can and should be used for the calibration and validation of traffic simulation models. There are big differences in availability of data from different sources. Some types of data such as loop detector data are widely available and used. Some can be measured with additional effort, for example, travel time data from GPS probe vehicles. Some types such as trajectory data are available only in rare situations such as research projects.

Mathematical modeling of the cancer cell’s control circuitry : paving the way to individualized therapeutic strategies

Cancer is a disease of signal transduction in which the dysregulation of the network of intracellular and extracellular signaling cascades is sufficient to thwart the cells finely-tuned biochemical control mechanisms. A keen interest in the mathematical modeling of cell signaling networks and the regulation of signal transduction has emerged in recent years, and has produced a glimmer of insight into the sophisticated feedback control and network regulation operating within cells. In this review, we present an overview of published theoretical studies on the control aspects of signal transduction, emphasizing the role and importance of mechanisms such as ‘ultrasensitivity’ and feedback loops. We emphasize that these exquisite and often subtle control strategies represent the key to orchestrating ‘simple’ signaling behaviors within the complex intracellular network, while regulating the trade-off between sensitivity and robustness to internal and external perturbations. Through a consideration of these apparent paradoxes, we explore how the basic homeostasis of the intracellular signaling network, in the face of carcinogenesis, can lead to neoplastic progression rather than cell death. A simple mathematical model is presented, furnishing a vivid illustration of how ‘control-oriented’ models of the deranged signaling networks in cancer cells may enucleate improved treatment strategies, including patient-tailored combination therapies, with the potential for reduced toxicity and more robust and potent antitumor activity.

Using mathematical modelling to evaluate drivers and predict trajectories of HIV and STI epidemics in South East Asian and Australian populations

Objective To evaluate the potential impact of the current global economic crisis (GEC) on the spread of HIV. Design To evaluate the impact of the economic downturn we studied two distinct HIV epidemics in Southeast Asia: the generalized epidemic in Cambodia where incidence is declining and the epidemic in Papua New Guinea (PNG) which is in an expansion phase. Methods Major HIV-related risk factors that may change due to the GEC were identified and a dynamic mathematical transmission model was developed and used to forecast HIV prevalence, diagnoses, and incidence in Cambodia and PNG over the next 3 years. Results In Cambodia, the total numbers of HIV diagnoses are not expected to be largely affected. However, an estimated increase of up to 10% in incident cases of HIV, due to potential changes in behavior, may not be observed by the surveillance system. In PNG, HIV incidence and diagnoses could be more affected by the GEC, resulting in respective increases of up to 17% and 11% over the next 3 years. Decreases in VCT and education programs are the factors that may be of greatest concern in both settings. A reduction in the rollout of antiretroviral therapy could increase the number of AIDS-related deaths (by up to 7.5% after 3 years). Conclusions The GEC is likely to have a modest impact on HIV epidemics. However, there are plausible conditions under which the economic downturns can noticeably influence epidemic trends. This study highlights the high importance of maintaining funding for HIV programs.

Evaluation of resonant damping techniques for Z-source current-type inverter

For the renewable energy sources whose outputs vary continuously, a Z-source current-type inverter has been proposed as a possible buck-boost alternative for grid-interfacing. With a unique X-shaped LC network connected between its dc power source and inverter topology, Z-source current-type inverter is however expected to suffer from compounded resonant complications in addition to those associated with its second-order output filter. To improve its damping performance, this paper proposes the careful integration of Posicast or three-step compensators before the inverter pulse-width modulator for damping triggered resonant oscillations. In total, two compensators are needed for wave-shaping the inverter boost factor and modulation ratio, and they can conveniently be implemented using first-in first-out stacks and embedded timers of modern digital signal processors widely used in motion control applications. Both techniques are found to damp resonance of ac filter well, but for cases of transiting from current-buck to boost state, three-step technique is less effective due to the sudden intermediate discharging interval introduced by its non-monotonic stepping (unlike the monotonic stepping of Posicast damping). These findings have been confirmed both in simulations and experiments using an implemented laboratory prototype.

A three-phase to single-phase matrix converter based bi-directional IPT system for charging electric vehicles

This paper presents a novel three-phase to single-phase matrix converter (TSMC) based bi-directional inductive power transfer (IPT) system for vehicle-to-grid (V2G) applications. In contrast to existing techniques, the proposed technique which employs a TSMC to drive an 8th order high frequency resonant network, requires only a single-stage power conversion process to facilitate bi-directional power transfer between electric vehicles (EVs) and a three-phase utility power supply. A mathematical model is presented to demonstrate that both magnitude and direction of power flow can be controlled by regulating either relative phase angles or magnitudes of voltages generated by converters. The viability of the proposed mathematical model is verified using simulated results of a 10 kW bi-directional IPT system and the results suggest that the proposed system is efficient, reliable and is suitable for high power applications which require contactless power transfer.

Modeling of protein signaling networks in clinical proteomics

Molecular interactions that underlie pathophysiological states are being elucidated using techniques that profile proteomicend points in cellular systems. Within the field of cancer research, protein interaction networks play pivotal roles in the establishment and maintenance of the hallmarks of malignancy, including cell division, invasion, and migration. Multiple complementary tools enable a multifaceted view of how signal protein pathway alterations contribute to pathophysiological states.One pivotal technique is signal pathway profiling of patient tissue specimens. This microanalysis technology provides a proteomic snapshot at one point in time of cells directly procured from the native context of a tumor micro environment. To study the adaptive patterns of signal pathway events over time, before and after experimental therapy, it is necessary to obtain biopsies from patients before, during, and after therapy. A complementary approach is the profiling of cultured cell lines with and without treatment. Cultured cell models provide the opportunity to study short-term signal changes occurring over minutes to hours. Through this type of system, the effects of particular pharmacological agents may be used to test the effects of signal pathway inhibition or activation on multiple endpoints within a pathway. The complexity of the data generated has necessitated the development of mathematical models for optimal interpretation of interrelated signaling pathways. In combination,clinical proteomic biopsy profiling, tissue culture proteomic profiling, and mathematical modeling synergistically enable a deeper understanding of how protein associations lead to disease states and present new insights into the design of therapeutic regimens.

Short utterance PLDA speaker verification using SN-WLDA and variance modelling techniques

This paper proposes a combination of source-normalized weighted linear discriminant analysis (SN-WLDA) and short utterance variance (SUV) PLDA modelling to improve the short utterance PLDA speaker verification. As short-length utterance i-vectors vary with the speaker, session variations and phonetic content of the utterance (utterance variation), a combined approach of SN-WLDA projection and SUV PLDA modelling is used to compensate the session and utterance variations. Experimental studies have found that a combination of SN-WLDA and SUV PLDA modelling approach shows an improvement over baseline system (WCCN[LDA]-projected Gaussian PLDA (GPLDA)) as this approach effectively compensates the session and utterance variations.

Mathematical modelling and experimental validation of solar assisted heat pump system

Singapore is located at the equator, with abundant supply of solar radiation, relatively high ambient temperature and relative humidity throughout the year. The meteorological conditions of Singapore are favourable for efficient operation of solar energy based systems. Solar assisted heat pump systems are built on the roof-top of National University of Singapore’s Faculty of Engineering. The objectives of this study include the design and performance evaluation of a solar assisted heat-pump system for water desalination, water heating and drying of clothes. Using MATLAB programming language, a 2-dimensional simulation model has been developed to conduct parametric studies on the system. The system shows good prospect to be implemented in both industrial and residential applications and would give new opportunities in replacing conventional energy sources with green renewable energy.

Computational strategies for surface fitting using thin plate spline finite element methods

Thin plate spline finite element methods are used to fit a surface to an irregularly scattered dataset [S. Roberts, M. Hegland, and I. Altas. Approximation of a Thin Plate Spline Smoother using Continuous Piecewise Polynomial Functions. SIAM, 1:208--234, 2003]. The computational bottleneck for this algorithm is the solution of large, ill-conditioned systems of linear equations at each step of a generalised cross validation algorithm. Preconditioning techniques are investigated to accelerate the convergence of the solution of these systems using Krylov subspace methods. The preconditioners under consideration are block diagonal, block triangular and constraint preconditioners [M. Benzi, G. H. Golub, and J. Liesen. Numerical solution of saddle point problems. Acta Numer., 14:1--137, 2005]. The effectiveness of each of these preconditioners is examined on a sample dataset taken from a known surface. From our numerical investigation, constraint preconditioners appear to provide improved convergence for this surface fitting problem compared to block preconditioners.

Mathematical Models of Tumor-Immune System Dynamics

"This collection of papers offers a broad synopsis of state-of-the-art mathematical methods used in modeling the interaction between tumors and the immune system. These papers were presented at the four-day workshop on Mathematical Models of Tumor-Immune System Dynamics held in Sydney, Australia from January 7th to January 10th, 2013. The workshop brought together applied mathematicians, biologists, and clinicians actively working in the field of cancer immunology to share their current research and to increase awareness of the innovative mathematical tools that are applicable to the growing field of cancer immunology. Recent progress in cancer immunology and advances in immunotherapy suggest that the immune system plays a fundamental role in host defense against tumors and could be utilized to prevent or cure cancer. Although theoretical and experimental studies of tumor-immune system dynamics have a long history, there are still many unanswered questions about the mechanisms that govern the interaction between the immune system and a growing tumor. The multidimensional nature of these complex interactions requires a cross-disciplinary approach to capture more realistic dynamics of the essential biology. The papers presented in this volume explore these issues and the results will be of interest to graduate students and researchers in a variety of fields within mathematical and biological sciences."--Publisher website

Surface reconstruction of wheat leaf morphology from three-dimensional scanned data

Realistic virtual models of leaf surfaces are important for a number of applications in the plant sciences, such as modelling agrichemical spray droplet movement and spreading on the surface. In this context, the virtual surfaces are required to be sufficiently smooth to facilitate the use of the mathematical equations that govern the motion of the droplet. While an effective approach is to apply discrete smoothing D2-spline algorithms to reconstruct the leaf surfaces from three-dimensional scanned data, difficulties arise when dealing with wheat leaves that tend to twist and bend. To overcome this topological difficulty, we develop a parameterisation technique that rotates and translates the original data, allowing the surface to be fitted using the discrete smoothing D2-spline methods in the new parameter space. Our algorithm uses finite element methods to represent the surface as a linear combination of compactly supported shape functions. Numerical results confirm that the parameterisation, along with the use of discrete smoothing D2-spline techniques, produces realistic virtual representations of wheat leaves.