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On the number of composites less than a given number
by
Rob Akscyn
University of Waikato
We present a sieve of the composites based on their greatest proper factors, not the integers themselves (in contradistinction to the Sieve of Erasthothenes). By abandoning the traditional table of integers "problem space" (in which each integer is represented only once), we instead make each row a separate number line (going off the right margin towards infinity). This shift in problem space enables counting all the composites under n implicitly, by counting all the instances of their greatest proper factors instead (a one-to-one mapping under this multi-row representation).
This shift in viewpoint highlights additional structure in the integers: (a) the hyperbolic of n/p being an upper bound for greatest proper factors (strongly suggesting ln(n) is involved, as we already know from the Prime Number Theorem) and (b) that sieving of a finite n is a function of two factors (not just one!): a "thinning force" (sieving as we know it), and a "trimming force" (the hyperbolic serving as upper bound). In effect, each column integer is monotonically reused as a greatest proper factor over and over, all the way up to the hyperbolic curve, unless stopped before that limit because it has itself been sieved.
What this "Venetian Blind" model demonstrates is that "thinning" and "trimming" are not independent forces, and thus not 'compoundable', as use of Mertens product formula for estimating P(n) would imply. Instead, like sibling rivals, these two factors compete with one another; the trimming force eventually gaining the upper hand when further 'thinning' of the greatest proper factors completely ceases once the n/p hyperbolic trims inward to reach p squared.
Thereafter the remaining unsieved greatest proper factors (now all prime) are reduced solely by the trimming force of the hyperbolic (which continues until it reaches the square root of p).
In effect, separating these two forces for independent examination reveals that the thinning aspect of sieving is strongly-recursive, and consequently rapidly induce a pattern to the composites (and by duality, the primes) that is highly regular.
We believe this elementary approach represents a promising assault on key number theory problems such as Goldbach's Conjecture (as well as Mertens Paradox) by unveiling additional exploitable structure, and eventually could lead to stronger series formulas for upper and lower bounds on P(n) (on par with Riemann's formula for P(n)).
Date received: September 17, 2008
Mazur intersection property for Asplund spaces
by
Miroslav Bacak
University of Newcastle
The aim of this talk is to show that even purely geometric properties of Banach spaces can strongly depend on choice of set-theoretic axioms. Namely, we show that (under Martin's Maximum axiom) every Asplund space of density character w1 has a renorming with the Mazur intersection property. Combined with the previous result of Jim\' enez and Moreno, who (under CH) proved the negation of this statement, we obtain that the MIP renormability of Asplund spaces of density w1 is undecidable in ZFC. This result is contained in a recent joint paper with P. Hájek.
Date received: November 26, 2008
Monotone relaxation iterates and applications to singularly perturbed problems
by
Igor Boglaev
Massey University, Palmerston North
This talk deals with monotone relaxation iterative methods for solving nonlinear monotone difference schemes of elliptic type. The monotone w-Jacobi and SUR (Successive Under-Relaxation) methods are constructed. The monotone methods solve only linear discrete systems at each iterative step and converge monotonically to the exact solution of the nonlinear monotone difference schemes. Convergent rates of the monotone methods are estimated. The proposed methods are applied to solving singularly perturbed reaction-diffusion problems. Uniform convergence of the monotone methods is proved. Numerical experiments complement the theoretical results.
Date received: October 19, 2008
Groups in absolute algebraic geometry
by
James Borger
Australian National University
It is possible to build an enriched version of algebraic geometry based not on commutative rings but on lambda-rings. (Lambda-rings are commutative rings with certain extra operations. They originally arose in K-theory.) In a precise sense, this lambda-algebraic geometry is an algebraic geometry over a deeper base than the ring of integers, the usual base in arithmetic algebraic geometry.
I will make some remarks and raise some questions about the analogues of algebraic groups in lambda-algebraic geometry.
Date received: November 3, 2008
Testing irreducibility of sparse polynomials over GF(2)
by
Richard Brent
Australian National University
Coauthors: Paul Zimmermann
We consider three algorithms for testing irreducibility of sparse polynomials (for example, trinomials) over finite fields of small characteristic. In fact, we restrict our attention to characteristic two, though the algorithms generalise.
The first algorithm is straightforward and involves squaring of polynomials modulo the polynomial to be tested.
The second algorithm replaces squaring by modular composition, using an old algorithm of Brent and Kung which reduces the problem to matrix multiplication. In fact, there are several variations on this algorithm, depending on how the matrix multiplications are performed.
The third algorithm involves the recent "fast" modular composition of Kedlaya and Umans (2008).
We show that the theoretical (asymptotic) complexity analysis of the algorithms may be misleading in practice, and suggest a combination of the first two algorithms that may be faster than either.
Date received: September 21, 2008
The stability of Padé and generalized Padé approximations
by
John Butcher
University of Auckland
Numerical methods for the solution of ordinary differential equations possess an associated stability matrix whose characteristic polynomial determines stable behaviour of the method. If the principal eigenvalue is a maximally high-order approximation to the exponential function for given degrees of the coefficients in the characteristic polynomial, then it is a generalized Padé approximation. For the classical Padé approximations, the cases when A-stabilty is exhibited follows from a result of Hairer, Norsett and Wanner. This has recently been extended to generalized Padé approximations and aspects of this new result will be presented.
Date received: October 30, 2008
On the homology of finite dimensional Lie algebras
by
Grant Cairns
La Trobe University
This talk summarizes results in recent papers coauthored with Sebastian Jambor and Barry Jessup. This work is part of a program that seeks to better understand Lie algebra homology and its dependence on the characteristic of the underlying field.
Date received: October 30, 2008
Algebraic properties of chromatic roots
by
Peter J. Cameron
Queen Mary, University of London
The chromatic polynomial of a graph is the monic integer polynomial P(q) whose evaluation at a positive integer k is the number of proper k-colourings of the graph. A lot is known about the location of chromatic roots (roots of chromatic polynomials), but rather less about their algebraic properties.
A working group at the recent Newton Institute programme on Combinatorics and Statistical Mechanics looked at this. We made two conjectures:
Date received: August 6, 2008
Singularities of orders on surfaces
by
Kenneth Chan
University of New South Wales
Canonical orders are examples of noncommutative surfaces which have "mild" singularities. We show that such orders satisfy a numerical criterion which is analogous to the numerical criterion satisfied by commutative rational singularities.
Date received: November 18, 2008
Lean induced cycles
by
Yury Chebiryak
ETH, Zurich
Coauthors: Daniel Kroening, Igor Zinovik, Thomas Wahl
Covering, dominating, and induced paths in binary hypercubes are well-studied notions in combinatorics. For example, Blass et al. investigate lower bounds on the length of cube-dominating paths and cycles. In this talk, I will introduce a combinatorial problem of constructing lean induced cycles, which is defined to be longest chord-free cycles that span a minimum number of hypercube nodes. This problem is important in modeling gene networks in Systems Biology, as lean induced cycles correspond to stable network models. I will demonstrate how to use a SAT solver to compute lean induced cycles for hypercubes up to dimension 7 and classify the cycles with respect to the number of nodes they span. The classification is obtained using a custom-made All-SAT solver with blocking clauses. Efficient filtering of these clauses allows to reduce their number by two orders of magnitude for the 6-cube and thus to compute the classification in reasonable time.
Date received: October 31, 2008
Components in random planar graphs with n vertices and m edges
by
Chris Dowden
University of Canterbury
Let Pn, m denote a graph taken uniformly at random from the set of all labelled planar graphs with n vertices and m(n) edges. We shall use elementary counting arguments to investigate the probability that Pn, m has a component isomorphic to H, for various fixed H, as n →∞. We will provide a complete picture of exactly when the probability is bounded away from 0 and/or 1, showing that there is different behaviour depending on both the graph H and the ratio m/n.
Date received: November 2, 2008
Counting irreducible representations
by
Shannon Ezzat
University of Canterbury
Representation growth is quite a new area of mathematics. It concerns itself with counting the number of irreducible representations from a finitely generated group to matrices over a field, usually the complex numbers. This talk will give a brief introduction to the field, as well as compare it with its sister field, subgroup growth. Also, we will look at explicitly counting representations of the Heisenberg group over the rational integers and the Gaussian integers.
Date received: October 30, 2008
Combinatorial challenges in conservation biology
by
Beáta Faller
University of Canterbury
There is a diverse range of interesting mathematical questions that arise in evolutionary biology, including many of a combinatorial and probabilistic nature. This talk will present a few of the challenges that we have been facing when studying future biodiversity using extinction models and optimization methods. Its aim is to convince the audience how exciting applied mathematics can be.
Date received: November 1, 2008
Curious properties of Maximum Parsimony in estimating evolutionary trees and ancestral sequence states
by
Mareike Fischer
Allan Wilson Centre for Molecular Ecology and Evolution, and Biomathematics Research Centre, University of Canterbury
Coauthors: Bhalchandra D. Thatte
Maximum Parsimony (MP) and Maximum Likelihood (ML) are two of the most freqently used methods for inferring phylogenetic trees and for estimating ancestral root states. Both methods have been frequently discussed, and many scenarios are to-date well understood. For instance, it is well known that MP and ML can lead to different tree estimations (e.g. in the so-called Felsenstein zone) but that under a simple model of substitution, they always choose the same set of trees for sequences that developed ``under no common mechanism'' (as shown by Tuffley and Steel, 1997).
But some surprising properties of MP and ML have only recently been investigated: I will present examples for MP and ML favoring different sets of trees even under ``no common mechanism'' when the underlying model is changed slightly, for example, when substitution probabilities are subject to an upper bound or when a molecular clock condition is imposed. Additionally, I will show that the probability of MP choosing the correct ancestral state can, unlike ML, sometimes be increased by ignoring parts of the tree which may even be close to the root.
Thus, I will show that even 35 years after Fitch's parsimony algorithm was first introduced and 30 years after the discovery of the Felsenstein zone, there are still properties of MP and ML which are surprising, and that therefore both methods are still worth further investigation.
Date received: October 29, 2008
The finite volume method and Riemann problem for a mathematical model of a hydrothermal eruption
by
Luke Fullard
Massey University, Palmerston North
I will introduce the concept of the Finite Volume Method for solving a set of hyperbolic PDEs, (such as our current hydrothermal eruption model), and discuss its various advantages over finite difference methods in certain cases. Also, I will discuss the Riemann Problem for the situation of two fluids separated at an interface with a discontinuous pressure profile over the interface. A mathematical model for the initialization of a hydrothermal eruption will then be presented, making use of the previously mentioned methodologies.
Date received: November 2, 2008
Mathematical comparative analysis of syntax and semantic search engines for end user performance
by
Apakuki Gavoka
University of the South Pacific, Suva
This research investigates the effectiveness versus efficiency of search engines with respect to rankings of searches. It focuses on the mathematical formalization of where syntax searches and semantic search differ in effective and efficient discovery times of information on the web. The Page Rank algorithm drives the generic syntax search engine Google and semantic rank algorithm steers generic semantic search engine Hakia, these are the two search engines chosen for this research. Search cases were contrived and run through the two search engines and mathematically analysed. Interestingly, it is shown that a vector relationship exists in both the searches. The findings show that the popular belief of Google being the most efficient is highly questionable. However, the analyses establish that Google is more consistent in searches in contrast to Hakia.
Date received: October 23, 2008
The cyclic sliding operation in Garside groups
by
Volker Gebhardt
University of Western Sydney
Coauthors: Juan González-Meseses (University of Seville, Spain)
Garside groups are generalisations of the well-known Artin braid groups. Basically, the notion of Garside groups captures the fundamental algebraic properties of braid groups and separates them from properties arising from a specific geometric or topological context. The most fundamental characteristic is the existence of the greedy normal form.
I will recall some well-known invariants of conjugacy classes which were introduced to solve certain computational problems in Garside groups. We will see that the theoretical properties of these established invariants are in some sense unsatisfactory. This will lead us to the definition of what appears to be a more natural structure.
Date received: October 29, 2008
Method for calculating the spectra of self-adjoint extensions of simple symmetric operators
by
Yufang Hao
Department of Applied Mathematics, University of Waterloo, Canada
Coauthors: Achim Kempf
By the Cayley transform, a simple symmetric operator T with deficiency indices (1, 1) has a U(1)-family of self-adjoint extensions, which can be parameterized as T(a) with 0 ≤ a < 1. Under the assumption that one of these self-adjoint extensions, say at a = 0, has only a discrete set of eigenvalues with no accumulation point, all other self-adjoint extensions T(a) have only point spectra with no accumulation point as well, and their eigenvalues increase simultaneously in a continuous manner as a increases. Together these eigenvalues cover the real line exactly once. In addition with the knowledge of the derivatives of these eigenvalues with respect to a at a = 0, we provide an explicit formula for computing the eigenvalues of all other self-adjoint extensions T(a). This gives a computational realization of the abstract spectral theory of self-adjoint extensions of symmetric operators. As an application, we present a new generalized sampling theorem, in which samples are taken at a time-varying rate adjusted to the behaviour of the signal and the signal is stably reconstructed.
Date received: August 18, 2008
The Toeplitz-Hausdorff theorem in a constructive setting
by
Robin Havea
School of Computing, Information and Mathematical Sciences, University of the South Pacific, Suva
Coauthors: Douglas Bridges
The Toeplitz-Hausdorff theorem simply says that the numerical range of a Hilbert space operator is always convex. Most of the classical proof of this theorem is said to be `computational'. However, in Bishop's constructive mathematics (i.e. mathematics with intuitionistic logic this is not the case. We look at a proof given by Halmos where we identify and fix the nonconstructivity in it. By means of limiting examples, we also show that our result is the best we can hope for in a constructive setting.
Date received: October 19, 2008
Stability of variable stepsize BDF methods for initial value problems
by
Allison Heard
Dept of Mathematics, University of Auckland
Coauthors: John Butcher
The stability of a method is dependent on the formulation of a method as well as the method itself. Theoretical analysis of the underlying one-step method in Nordsieck form is considered and the ``scale and modify'' approach is applied to second and third order BDF methods.
Date received: October 28, 2008
Nesting polynomials in infinite radicals
by
Peter Humphries
University of Canterbury
A well-known problem of Ramanujan's asks for the evaluation
of the infinite nested radical
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Date received: November 25, 2008
Periodic initial value problems for (integrable) partial difference equations
by
Peter van der Kamp
La Trobe
For partial difference equations defined on a square, initial values can be given on staircases. By taking periodic initial conditions the equation reduces to a system of ordinary difference equations, or a mapping/correspondence. For integrable equations, integrals for these mappings/correspondences are obtained by taking the trace of the monodromy matrix. We show how to generalize the construction to more general (systems of) equations.
Date received: November 17, 2008
Unbounded functional calculus for bounded groups with applications
by
Mihály Kovács
Department of Mathematics and Statistics, University of Otago
Coauthors: Boris Baeumer and Markus Haase
We develop the unbounded extension of the Hille-Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Lévy-Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by non-odd fractional powers of group generators and a shifted abstract Grünwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines.
Date received: October 13, 2008
Scoring Bayes
by
Gerrard Liddell
University of Otago
If a utility value, or `truth score' is attached to correct statistical inferences, then it has been proved that every scientific experiment increases the truth score under the assumption of `reciprocal convexity'[1]. The relevant and invariant scoring functions that have been used by economists are not reciprocally convex, but reciprocal convexity is a necessary condition for proving that experimentation increases truth scores [2]. Other questions about the `credibility' of inferences also involve scoring functions. This talk will show how the analytic results needed can be reduced to decidable algebraic problems.
Date received: November 3, 2008
Glueing continuous functions constructively
by
Iris Loeb
University of Canterbury
Coauthors: Douglas Bridges
The glueing of (sequentially, pointwise, or uniformly) continuous functions that coincide on the intersection of their closed domains is examined in the light of Bishop-style constructive analysis. This requires us to pay attention to the way that the two domains intersect, not just to the existence of their intersection.
Date received: October 21, 2008
Külshammer's second problem
by
Daniel Lond
University of Canterbury
Let G be a linear algebraic group over an algebraically closed field k, G an arbitrary finite group and Gp ⊂ G a Sylow p-subgroup of G, where p=char(k). It is known that there may be infinitely many equivalence classes of representations of G into G. Külshammer asks the following:
Given an equivalence class of representations of Gp into G, are there only finitely many representations r: G→ G, up to equivalence, such that the restriction of r to Gp belongs to that given class?
The aim of this elementary talk will be to describe the problem to a general audience and to show how a cohomology argument may provide some answers.
Date received: November 12, 2008
Rank dominations in matroids
by
Arun Mani
Monash University
A well known property of a matroid is its rank submodularity. This states that for any two subsets A, B of the ground set, E, of a matroid with rank function r:2E → N, r(A) + r(B) ≥ r(A ∪B) + r(A ∩B). We recast the submodularity of matroid ranks as a bijective map property between appropriately defined sets whose members are pairs of subsets of E. The bijection maintains an additive rank inequality across these subset pairs. We call such bijections rank dominating bijections of matroids. In this talk, we will present some results on the existence of rank dominating maps for certain special cases, along with a conjecture on their existence for the general case. We will also discuss some consequences of the presence of such rank dominating maps in matroids.
Date received: November 2, 2008
Frequency parameters of non-symmetric box-type structures using the Rayleigh-Ritz method and penalty functions
by
Luis Monterrubio
University of Waikato
Coauthors: Sinniah Ilanko
In this work, frequency parameters of non-symmetric box-type structures with several combinations of classical boundary conditions are obtained using the Rayleigh-Ritz method and penalty parameters. In this work, two different types of penalty parameters were used to model constraints. These penalty functions represent either stiffness or inertia, and as demonstrated in recent publications in both cases the penalty parameters can be either positive or negative. In all cases, the same set of admissible functions is used in the Rayleigh-Ritz method, which can be used to model a completely free plate. All geometric boundary conditions, as well as joints between plates are modelled using penalty parameters. Results compare well with those in the existing literature when available or with results obtained from finite element analysis using commercial software.
Date received: October 29, 2008
Reich theorem and mappings with fixed points theorem on G-metric spaces
by
Zead Mustafa
Department of Mathematics, Hashemite University, Jordan
Coauthors: Hamed Obiedat
In 2006, Zead Mustafa and Brailey Sims introduced a more robust concept of a generalized metric spaces, called G-metric space, and they developed a topological structure in such spaces. Also they discussed the fixed point theory of contractive mappings, and mappings satisfying various related conditions in complete G-metric spaces.
In this talk we prove some fixed point results for mappings satisfying sufficient contractive conditions on a complete G-metric space, moreover we show that if the G-metric space (X, G) is symmetric, then the existence and uniqueness of these fixed point results follows from Reich theorems in the usual metric space (X, dG), where (X, dG) is the metric induced by the G-metric space (X, G).
Date received: September 28, 2008
A review of the internet congestion control via rate control algorithms with examples and simulations
by
Salsabil Nusair
School of Computing, Information & Mathematical Sciences, University of the South Pacific, Suva
Coauthors: Jito Vanualailai
One of the most recent and exciting areas of research in computing science and information systems deals with the need to control traffic in the ever-growing Internet in a more systematic, rigorous and efficient manner. The world started experiencing a series of ``Internet congestion collapses'' in the early days of the Internet. The work of Jacobson in 1988 (V. Jacobson, Congestion avoidance and control, ACM Communication Review, 18 (1988), pp.314-329) first recorded this phenomenon and proposed a congestion control and avoidance algorithm. This algorithm, which is implemented in the transport layer protocol called TCP (Transmission Control Protocol), has served the Internet well during a time of unprecedented growth. However, as Srikant noted (R. Srikant, The Mathematics of Internet Congestion Control, Birkhauser, Boston, 2003), it was designed during a time when the Internet was a relatively small network compared to its size today. Therefore, there has been much interest in re-examining the role of congestion control in the Internet with the goal of enhancing TCP to make it scalable to large networks.
In this presentation, for non-computer science audience, we provide a simple explanation of the Internet congestion control problem, viewed as a resource allocation problem, and then review a convex optimization algorithm for providing a solution. Furthermore, we review a technique by Kelly (F. Kelly, Mathematical modeling of the Internet, Mathematics Unlimited --- 2001 and Beyond, Springer-Verlag, 2001, pp.685-702), who uses the Direct Method of Lyapunov to design a model of the Internet governed by ODEs, the solutions of which are an approximation to the solutions obtained via convex optimization. Throughout the presentation, we will use simple examples of the Internet to illustrate concepts and applications of convex optimization and the Direct Method of Lyapunov.
Date received: October 27, 2008
Space of test functions for (w1, w2)-tempered ultradistributions via Fourier transform
by
Hamed M. Obiedat
Hashemite University, Jordan)
Coauthors: Wasfi Shatanawi and Mohd Yasin
We introduce the space Sw1, w2 of test functions for (w1, w2)-tempered ultradistributions where w1 and w2 are two weights satisfying the classical Beurling conditions. Moreover, we give a topological characterization of the space Sw1, w2 without conditions on the derivatives. For functional in the dual space Sw1, w2, we prove a structure theorem by using the classical F. Riesz representation theorem.
Date received: September 28, 2008
An algebraic approach to quantum and classical information theory
by
Manas K Patra
Department of Computer Science, University of York, UK
Coauthors: Samuel L Braunstein
The aim of this paper is to give a unified description of classical and quantum information in the language of C* algebras. We then interpret and analyze several important notions from information theory (both classical and quantum) in this context. Quantum information theory combines quantum and classical information. Classical information theory treats information as sets of random variables and studies their statistical properties under various transformations. Some of the most important quantities are expectations (e.g. Shannon entropy), conditional expectations, or correlations of functions of random variables. We treat the random variables as classical observables burrowing from the language of classical statistical mechanics: further the set of observables can be given the structure of a commutative C* algebra. Quantum observables on the other hand form a noncommutative C* algebra. Hence, we regard a quantum information system as a non-commutative C* algebra with the classical components lying in its centre. The notion of statistical correlations and independence are shown to be connected with entanglement and separability. Moreover, the notion of subsystems has a natural generalization which avoids the use of tensor product, although in many (but not all) situations our formulation is equivalent to a tensor structure. Interestingly, we can prove important theorems (no broadcasting, no cloning) in this general formulation. Further, our formalism covers situations (e.g. indistinguishable particles, infinite dimensions) where the standard formulation cannot applied without drastic modifications. Finally, we also explore connections with free probability theory.
Date received: October 29, 2008
The notion of "recursive" subset in Euclidean space and related questions
by
Petrus H Potgieter
Department of Decision Sciences, University of South Africa
Roger Penrose, in his The Emperor's New Mind (1989), challenged the mathematical community to find a definition of ``recursive'' subsets of Euclidean space that would allow one to determined whether the Mandelbrot set is ``recursive'' or not. The talk discusses the properties that a ``nice'' definition should have and the elementary observation that it is impossible to simultaneously realize all of these properties. We consider several notions of recursiveness that have been introduced for sets in Euclidean space and counter-examples separating these notions. Finally, the relation to computational geometry is briefly discussed.
Date received: November 10, 2008
New algorithm for motion planning and posture control of 3-trailer systems.
by
Krishna Sami Raghuwaiya
University of the South Pacific, Suva
Coauthors: Bibhya Nand Sharma and Prof Jito Vanualailai
This paper utilizes a new Lyapunov-based control scheme to extract an algorithm that improves upon, in general, the motion planning and posture control of 3-trailer systems. The control scheme inherently guarantees point and posture stabilities, convergence and collision avoidance properties of the articulated systems in a priori known environment. We employ the concepts of ghost walls and minimum distance technique (MDT) to attain point and posture stabilities, in the sense of Lyapunov, of our kinodynamical model. The effectiveness of the control scheme and its control laws are demonstrated via simulations of two traffic-like scenarios.
Date received: October 23, 2008
CARTopt: a random search method for non-smooth optimization
by
B. L. Robertson
University of Canterbury
Coauthors: C. J. Price and M. Reale
A random search optimization method for finding optima of unconstrained optimization problems is described. The method operates by using a batch of random points at each iteration. These points are used to partition the optimization region into sub-regions using Classification and Regression Trees (CART). Each sub-region is classified as either high or low with respect to function value. The next batch of points has an increased probability distribution in sub-regions which are classified as low. Although the method focuses on regions where objective function is relatively low, points in the high sub-regions are still sampled reducing the risk of missing the global optima. The method requires no gradient information and thus can be applied to non-smooth problems. Numerical results will be presented to show the performance of the algorithm on a selection of non-smooth test functions.
Date received: October 28, 2008
Formation control of a swarm of mobile manipulators
by
Bibhya Sharma
University of the South Pacific, Suva
Coauthors: Jito Vanualailai and Avinesh Prasad
This paper presents a new Lyapunov-based centralized formation control planner for a swarm of 2-link mobile manipulators in a priori known environment. To ensure a significant degree of formation stiffness along the flight-path, information on moving ghost targets, inter-robot bounds for aggregation, and heading for the mobile manipulators are captured in the control planner. The final desired orientation of the formation is by observing a minimum distance between every member of the swarm and ghost walls. The nonlinear control laws extracted from the Lyapunov-based control scheme is utilized to obtain collision-free trajectories of the swarm in a low-degree formation, whilst ensuring the stability of the kinodynamic system governing the swarm. The effectiveness of the control scheme and its controllers are demonstrated by simulating interesting traffic-like situations.
Date received: October 14, 2008
Orbital forcing over the Cenozoic Era
by
Philip Sharp
University of Auckland
Long-term changes in Earth's orbit about the Sun can cause long-term changes in Earth's climate through a process known as orbital forcing. Establishing a link between the orbit and climate requires the orbit be known accurately. A suitable orbit is found by specifying a detailed model of the gravitational forces acting on the Earth and then using an accurate numerical integration method on a computer to calculate the orbit from the model. An orbit of the required accuracy for the Neogene Period (0 - 23 million years ago) has been calculated and used to calibrate the timing of geological events over the Neogene Period.
I will describe the challenges in calculating an accurate orbit for the Cenozoic Era (0 - 65 million years ago).
Date received: October 28, 2008
Global stability of a mathematical model of the internet
by
Ronal Singh
School of Computing, Information & Mathematical Sciences, University of the South Pacific, Suva
Coauthors: Jito Vanualailai
Utilizing the Direct Method of Lyapunov (DML), we study the global stability of a mathematical model of the Internet. We follow the pioneering work of Kelly who, in 2001 (F. Kelly, Mathematical modeling of the Internet, Mathematics Unlimited---2001 and Beyond, Springer-Verlag, 2001, pp.685-702) began a new direction in enhancing TCP (Transmission Control Protocol) with a novel technique that uses a type of distributed control algorithm derived from the DML and convex optimization. Kelly showed that it was possible to optimize the benefits for a user (e.g. bandwidth, cost) while meeting constraints (e.g. capacity of a link) and maintaining the stability of a network regardless of its size. In other words, the network would not experience congestion problems.
In this presentation, firstly, we review one of Kelly's results on the primal algorithm and secondly, we will attempt to create a new Lyapunov-based congestion algorithm by generalizing Kelly’s result.
Date received: October 25, 2008
Conditionally invariant measures and sets with low escape rates.
by
Ognjen Stancevic
School of Mathematics and Statistics, University of New South Wales
Consider a dynamical system T: X → X, and let H ⊂ X be a "hole" such that almost every point eventually enters the hole. Such systems were first studied by Yorke and Pianigiani in 1979. An important quantity associated with these open systems is the "escape rate": how fast, asymptotically, do points enter the hole. In some cases, escape rates can be related to conditionally invariant measures as well as to eigenvalues of the Perron-Frobenius operator. In this talk I will give a brief review of escape rates and possible applications in detection of "almost invariant" sets.
Date received: October 29, 2008
3-unisolvent sets, 4-dimensional Laguerre planes and generalized quadrangles
by
Gunter Steinke
Dept. of Mathematics and Statistics, University of Canterbury
Geometrically 3-unisolvent sets of functions correspond to Laguerre type geometries. In this talk we look at 3-unisolvent sets of functions from the 2-sphere to 2-dimensional Euclidean space that solve the Hermite interpolation problem and their corresponding geometric objects, 4-dimensional Laguerre planes. We give a construction via associated generalized quadrangles of such Laguerre planes whose automorphism group is not transitive on the set of circles (the graphs of the function in the 3-unisolvent set).
Date received: October 15, 2008
The best approximation of a function by a sum of radial functions
by
Steve Taylor
University of Auckland
We consider the best (in the least squares sense) approximation of a function on R3 by a finite sum of radial functions, each with a different centre.
A calculus of variations approach leads to a pretty system of integral equations for the unknown functions. Traditional methods for solving these integral equations have problems. However the system has a surprising connection to an initial value problem which serves as a useful tool for solving the system.
This problem arose in quantum chemistry from a desire to approximate the electronic charge density of a molecule by a sum of atomic-like charge densities.
Date received: November 2, 2008
Multiple solutions for systems of differential equations with nonlinear boundary conditions
by
Bevan Thompson
University of Queensland
Coauthors: Jutarat Kongson and Yongwimon Lenbury
We discuss existence results for three solutions of systems of nonlinear differential equations of the form y"=f(x, y, y') satisfying fully nonlinear boundary conditions g((y(0), y(1));(y'(0), y'(1)))=0. The Dirichlet, Periodic and Sturm-Liouville boundary conditions are included as special cases. Here assumptions on f guarentee solutions in an admissible bounding region of (x, y) space are bounded in C1 and g is compatible with this region. Our results extend those of Frigon and Montoki, Schmitt and Thompson, and Agarwal, Thompson and Tisdell.
Date received: October 31, 2008
An interactive window for split/rejoin maneuvers of swarms
by
Rajneel Totaram
University of the South Pacific, Suva
Coauthors: Bibhya Sharma, Jito Vanualailai
In this paper we develop a Java-based interactive window that illustrates the split/rejoin maneuvers of non-holonomic car-like robot swarm fixed in a prescribed formation. The split/rejoin maneuvers are desired so that the swarm is able to avoid obstacles in its path. The application allows the user to choose the size of the swarm and the number of obstacles in a constrained workspace. The shape of the formation will, by default of the prototype, depend on the size of the swarm provided by the user. The underlying controls of the swarm will be governed by the Lyapunov-based control scheme. Upon receiving the necessary details, the control scheme will design a set of suitable continuous acceleration controllers that produces the desired split/rejoin maneuvers of the swarm.
Date received: October 15, 2008
Cell structures for finite subset spaces
by
Christopher Tuffley
Massey University, Palmerston North
The kth finite subset space of a topological space X consists of the nonempty subsets of X of size at most k. The most famous of these is the 3rd finite subset space of the circle, which Bott proved in 1952 is homeomorphic to the 3-sphere. We will look at methods for studying these spaces when the underlying space X is "nice"; in particular, we will look at methods for constructing cell structures for finite subset spaces from a cell structure for the underlying space.
Date received: November 2, 2008
Developement of a 3D numerical model for salinity intrusion in Brisbane
by
Gurudeo Anand Tularam
Griffith University, Brisbane
Coauthors: Roshan Singh
A three-dimensional (3D) density dependent seawater intrusion field scale model is developed in this paper for Pine Rivers Shire (Australia). The 3D model not only allows for complex flow behaviour and boundary conditions but also takes into account the influence of tidal behaviour that is modelled using Cartwright's (2001) approach. The new model was solved using the finite element FEMWATER package. Along with various longer term salinity profiles for the region, the results show that regions within a distance of 1300m from the seawater boundary will be affected by higher levels of salinity concentration after 100 years. For longer simulations, deviation of the concentration contours from the normal behaviour in the vadose zone is observed at the end of the diffusive zone i.e. at 1300m from seaward boundary for the Pine Rivers Shire aquifer. A diffusive zone rather than a sharp interface was observed and the intrusion process was dominated by diffusion, rather that advection as expected. Parts of Australia have been under drought conditions with significant water shortages and lack of rainfall generally. The simulations suggest that irrigation for farming, pumping for industrial and other uses of groundwater should carefully monitored to avoid salinization of coastal aquifers through excessive extraction in the longer term.
Date received: October 13, 2008
Industrial mathematics initiatives: an (inter)national need?
by
Graeme Wake
Massey University, Albany
There is a world-wide trend to introduce (mostly post-graduate) developments to produce graduates attuned to industry and society need. Further, Universities are enjoying an increasing opportunity for industrial partnerships with industry in sponsored research and consulting. The OECD Global Science Forum surveyed initiatives in this area and made some important recommendations to member countries: see their report <http://www.oecd.org/dataoecd/47/1/41019441.pdf>. The intention of this informal discussion session is to survey what cooperative opportunities exist and any response we as member countries of the OECD should or could make.
Date received: October 31, 2008
Some properties of conformal transformations of symmetrical spinors
by
Graham Weir
Industrial Research Ltd, New Zealand
A general linear transformation of symmetrical spinors corresponds to a conformal transformation. The corresponding matrices are n ×n, whose elements are homogeneous polynomials of degree n-1. The determinants and inverses of these matrices have remarkable properties. In addition, these n ×n matrices can be built up successively by recursion, using the 2 ×2 and (n-1) ×(n-1) matrices, but in doing this, we encounter a diagonal matrix whose entries are the binomial coefficients, C(n, i). We briefly describe the determinant of this diagonal matrix (the product over i). Finally, we show that the fundamental system of differential equations arising from this system of symmetrical spinors is Baba's equation.
Date received: September 2, 2008
Singularities arising from the stretching of threads with viscous heating
by
Jonathan Wylie
Dept of Mathematics, City University of Hong Kong
Coauthors: Huaxiong Huang
We investigate the role played by viscous heating in extensional flows of viscous threads with temperature-dependent viscosity. We develop a formal asymptotic theory based on a long-wavelength approximation of the Navier-Stokes equations to describe such flows. Using this framework we show that there exists an interesting interplay between the effects of viscous heating, which accelerates thinning, and inertia, which prevents pinch-off. We first consider steady drawing of a thread that is fed through a fixed aperture at given speed and pulled with a constant force at a fixed downstream location. For pulling forces above a critical value, we show that inertialess solutions cannot exist and inertia is crucial in controlling the dynamics. We also consider the unsteady stretching of a thread that is fixed at one end and pulled with a constant force at the other end. In contrast to the case in which inertia is neglected, the thread will always pinch at the end where the force is applied. Our results show that viscous heating can have a profound effect on the final shape and total extension at pinching.
Date received: October 28, 2008
Idempotents in Stone-Cech compactifications and homogeneous maximal spaces
by
Yevhen Zelenyuk
University of the Witwatersrand, South Africa
Let G be an infinite group and let bG be the Stone-Cech compactification of G as a discrete semigroup. We take the points of bG to be the ultrafilters on G. Being a compact right topological semigroup, bG has idempotents. Every idempotent p ∈ bG determines a left translation invariant Hausdorff topology Tp on G with a neighborhood base at the identity e ∈ G consisting of subsets A∪{e} where A ∈ p. An idempotent p ∈ bG is regular if p is uniform (= for every A ∈ p, |A|=|G|) and the topology Tp is regular. We show that for every infinite group G, there exists a regular idempotent in bG. As a consequence we obtain that for every infinite cardinal k, there exists a homogeneous regular maximal space of dispersion character k, which is the answer to an old difficult question. Another consequence tells us that there exists a translation invariant regular maximal topology on the real line of dispersion character c stronger than the natural topology.
Date received: August 8, 2008
Nonlinear theories for shock-induced fingering instabilities
by
Qiang Zhang
Department of Mathematics, City University of Hong Kong
It is well known that, when a shock propagates through a material interface that separates fluids of different densities, the interface becomes unstable and fingers develop. It is a long-standing problem to develop a theory to predict the size of these fingers. In this talk, we present an analytical nonlinear theory which provides predictions for the growth rate and the size of the fingers. Our theoretical approach is based on the methods of Padé approximation and asymptotic matching. The theoretical predictions are in remarkably good agreement with the results from full-scale numerical simulations.
Date received: October 28, 2008