A continuation of last week's talk.

We consider multi-objective convex optimal control problems. First we state a relationship between the (weakly or properly) efficient set of the multi-objective problem and the solution of the problem scalarized via a convex combination of objectives through a vector of parameters (or weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Next we consider an additional objective over the efficient set. Based on the main result, the new objective can instead be considered over the (parametric) solution set of the scalarized problem. For the purpose of constructing numerical methods, we point to existing solution differentiability results for parametric optimal control problems. We propose numerical methods and give an example application to illustrate our approach. This is joint work with Henri Bonnel (University of New Caledonia).

This talk will introduce fractal transformations and some of their remarkable properties. I will explain the mathematics that sustains them and how to construct them in simple cases. In particular I hope to demonstrate a very recent result, showing how they can be applied to generate convenient mutually-singular measures that enable the storage of multiple images within a single image. The talk will include some beautiful computer graphics.

I will describe four recent theorems, developed jointly with Andrew Vince and David C. Wilson (both of the University of Florida) that reveal a surprisingly rich theory associated with an attractor of a projective iterated function system (IFS). The first theorem characterizes when a projective IFS has an attractor that avoids a hyperplane. The second theorem establishes that a projective IFS has at most one attractor. In the third theorem the classical duality between points and hyperplanes in projective space leads to connections between attractors that avoid hyperplanes and repellers that avoid points and an associated index, which is a nontrivial projective invariant, is defined. I will link these results to the Conley decomposition theorem.

Hyperbolic geometry will be introduced, and visualised via the Cinderella software suite. Simple constructions will be performed and compared and contrasted to with Euclidean geometry. Constructions and examples will be quite elementary. Audience participation, specifically suggestions for constructions to attempt, during the demonstration is actively encouraged.The speaker apologises in advance for not being nearly as knowledgeable of the subject as he probably ought to be.

Scenario Trees are compact representations of processes by which information becomes available. They are most commonly used when solving stochastic optimization problems with recourse, but they have many other uses. In this talk we discuss two uses of scenario trees: computing the value of hydroelectricity in a regulated market; and updating parameters of epidemiological models based on observations of syndromic data. In the former, we investigate the impact that the size and shape of the tree has on the dual price of the first stage demand constraint. In the latter, we summarize a simulation of epidemics and syndromic behaviors on a tree, then identify the subtree most likely to match observed data.

Mineral freight volume increases are driving transport infrastructure investments on Australia's east and west coasts. New and upgraded railways, roads and ports are planned or are under construction -- to serve new mines, processing facilities and international markets. One of the fastest growing regions is Northern Queensland, central to which is the so-called Northern Economic Triangle that has Rockhampton, Mt Isa and Townsville at its vertices. CSIRO has been working with Queensland Government to construct a new GIS-based infrastructure planning optimisation system that is known as the Infrastructure Futures Analysis Platform (IFAP). IFAP can be used to build long-term (eg. 25 year) plans for infrastructure development in regions such as the Northern Economic Triangle. IFAP consists of a commercial Geographic Information System (MapInfo), a database and a network optimisation solver that has been constructed by CSIRO and will ultimately by open-sourced. The prototype IFAP is nearing completion and in this presentation I will discuss the development process and the underlying network optimisation problem.

Joint work with Kim Levy, Andreas Ernst, Gaurav Singh, Stuart Woodman, Andrew Higgins, Leorey Marquez, Olena Gavriliouk and Dhananjay Thiruvady

Nonconvex/nonsmooth phenomena appear naturally in many complex systems. In static systems and global optimization problems, the nonconvexity usually leads to multi-solutions in the related governing equations. Each of these solutions represents certain possible state of the system. How to identify the global and local stability and extremality of these critical solutions is a challenge task in nonconvex analysis and global optimization. The classical Lagrangian-type methods and the modern Fenchel-Moreau-Rockafellar duality theories usually produce the well-known duality gap. It turns out that many nonconvex problems in global optimization and computational science are considered to be NP-hard. In nonlinear dynamics, the so-called chaotic behavior is mainly due to nonconvexity of the objective functions. In nonlinear variational analysis and partial differential equations, the existence of nonsmooth solutions has been considered as an outstanding open problem.

In this talk, the speaker will present a potentially useful canonical duality theory for solving a class of optimization and control problems in complex systems. Starting from a very simple cubic nonlinear equation, the speaker will show that the optimal solutions for nonconvex systems are usually nonsmooth and cannot be captured by traditional local analysis and Newton-type methods. Based on the fundamental definitions of the objectivity and isotropy in continuum physics, the canonical duality theory is naturally developed, and can be used for solving a large class of nonconvex/nonsmooth/discrete problems in complex systems. The results illustrate the important fact that smooth analytic or numerical solutions of a nonlinear mixed boundary-value problem might not be minimizers of the associated variational problem. From a dual perspective, the convergence (or non-convergence) of the FDM is explained and numerical examples are provided. This talk should bring some new insights into nonconvex analysis, global optimization, and computational methods.

We shall continue from last week, performing more hyperbolic constructions, as well as some elliptic constructions for comparison. In particular, some alternating projection algorithms will be explored in hyperbolic space. With some luck we shall confound some more of our euclidean intuitions. Audience participation is actively encouraged.

Machine learning problems are a particularly rich source of applications for sparse optimization, giving rise to a number of formulations that require specialized solvers and structured, approximate solutions. As case studies, we discuss two such applications - sparse SVM classification and sparse logistic regression - and present algorithms that are assembled from different components, including stochastic gradient methods, random approximate matrix factorizations, block coordinate descent, and projected Newton methods. We also describe a third (distantly related) application to selection of captive breeding populations for endangered species using binary quadratic programming, a project started during a visit to Newcastle in June 2009.

CARMA reflects changes in the mathematical research being undertaken at Newcastle. Mathematics is "the language of high technology" which underpins all facets of modern life, while current Information and Communication Technology (ICT) has become ubiquitous. No other research centre exists which focuses on the implications of developments in ICT, present and future, for the practice of research mathematics. CARMA fills this gap through the exploitation and development of techniques and tools for computer-assisted discovery and disciplined data-mining including mathematical visualization. Advanced mathematical computation is equally essential to solution of real-world problems; sophisticated mathematics forms the core of software used by decision-makers, engineers, scientists, managers and those who design, plan and control the products and systems which are key to present-day life.

This talk is aimed at being accessible to all.

Sebastian will be presenting Lagrangian Relaxation and the Cost Splitting Dual on Monday July 5 at 4pm. He will discuss when the Cost Splitting Dual can provide a better bound than the Lagrangian Relaxation or regular linear relaxation and apply the theory learned to an example.

• Prof Frank Garvan, University of Florida
• Laureate Prof Jonathan Borwein, The University of Newcastle
• Prof Heng Huat Chan, National University of Singapore

In Euclidean space the medians of a triangle meet at a point that divides each median in the ratio 2 to 1. That point is called the centroid. Cinderella tells us that the medians of a triangle in hyperbolic space meet at a point, but the medians do not divide each other in any fixed ratio. What characterises that point? One answer is that it is the centre of mass of equal-mass particles placed at the vertices. I will outline how one can define the centre of mass of a set of particles (points) in a Riemannina manifold, and how one can understand this in terms of the exponential map. This centre of mass, or geometric mean, is sometimes called the Karcher mean (apparently first introduced by Cartan!). I will attempt to show what this tells us about the medians of a triangle.

In this talk we will present some results recently obtained in collaboration with B.F.Svaiter on maximal monotone operators in nonreflexive Banach spaces. The focus will be on the use of concept of convex representation of a maximal monotone operator for obtaining results on these operators of type: surjectivity of perturbations by duality mappings, uniqueness of the extension to the bidual, Brondsted-Rockafellar property, etc.

Several types of subdifferentials will be introduced on Riemannian manifolds. We'll show their properties and applications, including spectral functions, Borwein-Preiss variational principle, and distance functions.

We study the existence and approximation of fixed points of several types of Bregman nonexpansive operators in reflexive Banach spaces.

Please note the change in our usual day from Monday to Tuesday.

The Hu-Washizu formulation in elasticity is the mother of many different finite element methods in engineering computation. We present some modified Hu-Washizu formulations and their performance in removing locking effect in the nearly incompressible elasticity. The stabilisation of the standard Hu-Washizu formulation is used to obtain the stabilised nodal strain formulation or node-based uniform strain elements. However, we show that standard or stabilised nodal strain formulation should be modified to have a uniformly convergent finite element approximation in the nearly incompressible case.

The classical prolate spheroidal wavefunctions (prolates) arise when solving the Helmholtz equation by separation of variables in prolate spheroidal coordinates. They interpolate between Legendre polynomials and Hermite functions. In a beautiful series of papers published in the Bell Labs Technical Journal in the 1960's, they were rediscovered by Landau, Slepian and Pollak in connection with the spectral concentration problem. After years spent out of the limelight while wavelets drew the focus of mathematicians, physicists and electrical engineers, the popularity of the prolates has recently surged through their appearance in certain communication technologies. In this talk we discuss the remarkable properties of these functions, the ``lucky accident'' which enables their efficient computation, and give details of their role in the localised sampling of bandlimited signals.

In vehicle routing problems (VRPs), a fleet of vehicles must be routed to service the demands of a set of customers in a least-cost fashion. VRPs have been studied extensively by operations researchers for over 50 years. Due to their complexity, VRPs generally cannot be solved optimally, except for very small instances, so researchers have turned to heuristic algorithms that can generate high-quality solutions in reasonable run times. Along these lines, we develop novel integer programming-based heuristics for several different VRPs. We apply our heuristics to benchmark problems in the literature and report computational results to demonstrate their effectiveness.

We discuss a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. We give a general expression which permits numerical evaluation of members of the class of sums to arbitrary order. We use this to illustrate numerically the properties of trajectories along which the real and imaginary parts of the sums are zero, and we show results for the first two of a particular set of angular sums (denoted C(1, 4 m; s)) which indicate their density of zeros on the critical line of the complex exponent is the same as that for the product (denoted C(0, 1; s)) of the Riemann zeta function and the Catalan beta function. We then introduce a function which is the quotient of the angular lattice sums C(1, 4 m; s) with C(0, 1; s), and use its properties to prove that C(1, 4 m; s) obeys the Riemann hypothesis for any m if and only if C(0, 1; s) obeys the Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds, then C(1 ,4 m; s) and C(0, 1; s) have the same distribution of zeros on the critical line (in a sense made precise in the proof).

I will describe the history and some recent research on a subject with a remarkable pedigree.

We discuss the Hahn-Banach-Lagrange theorem, a generalized form of the Hahn--Banach theorem. As applications, we derive various results on the existence of linear functionals in functional analysis, on the existence of Lagrange multipliers for convex optimization problems, with an explicit sharp lower bound on the norm of the solutions (multipliers), on finite families of convex functions (leading rapidly to a minimax theorem), on the existence of subgradients of convex functions, and on the Fenchel conjugate of a convex function. We give a complete proof of Rockafellar's version of the Fenchel duality theorem, and an explicit sharp lower bound for the norm of the solutions of the Fenchel duality theorem in terms of elementary geometric concepts.

Computation of definite integrals to high precision (typically several hundred digit precision) has emerged as a particularly fruitful tool for experimental mathematics. In many cases, integrals with no known analytic evaluations have been experimentally evaluated (pending subsequent formal proof) by applying techniques such as the PSLQ integer relation algorithm to the output numerical values. In other cases, intriguing linear relations have been found in a class of related integrals, relations which have subsequently been proven as instances of more general results. In this lecture, Bailey will introduce the two principal algorithms used for high-precision integration, namely Gaussian quadrature and tanh-sinh quadrature, with some details on efficient computer implementations. He will also present numerous examples of new mathematical results obtained, in part, by using these methods.

In a subsequent lecture, Bailey will discuss the PSLQ algorithm and give the details of efficient multi-level and parallel implementations.

Historically, fossil fuels have been vital for our global energy needs. However climate change is prompting renewed interest in the role of fossil fuel production for our energy needs. In order to appropriately plan for our future energy needs, a new detailed model of fossil fuel supply is required. The modelling applied an algorithm-based approach to predict both supply and demand for coal, gas, oil and total fossil fuel resources. Total fossil fuel demand was calculated globally, based on world population and per capita demand; while production was calculated on a country-by-country basis and summed to obtain global production. Notably, production over the lifetime of a fuel source was not assumed to be symmetrical about a peak value like that depicted by a Hubbert curve. Separate production models were developed for mining (coal and unconventional oil) and field (gas and conventional oil) operations, which reflected the basic differences in extraction and processing techniques. Both of these models included a number of parameters that were fitted to historical production data, including: (1) coal for New South Wales, Australia; (2) gas from the North Sea, UK; and (3) oil from the North Sea, UK, and individual state data from the USA.

In this talk we will focus our attention on certain regularization techniques related to two operations involving monotone operators: point-wise sums of maximal monotone operators and pre-compositions of such operators with linear continuous mappings. These techniques, whose underlying idea is to obtain a bigger operator as a result, lead to two concepts of generalized operations--extended and variational sums of maximal monotone operators and, the corresponding to them, extended and variational compositions of monotone mappings with linear continuous operators. We will revue some of the basic results concerning these generalized concepts, as well as will present some recent important advances.

Given a pair of Banach spaces X and Y such that one is the dual of the other, we study the relationships between generic Fr´echet differentiability of convex continuous functions on Y (Asplund property), generic existence of linear perturbations for lower semicontinuous functions on X to have a strong minimum (Stegall variational principle), and dentability of bounded subsets of X (Radon-Nikod´ym Property).

The PSLQ algorithm is an algorithm for finding integer relations in a set of real numbers. In particular, if (x1, ..., xn) is a vector of real numbers, then PSLQ finds integers (a1, ..., an), not all zero, such that a1*x1 + a2*x2 + ... + an*xn = 0, if such integers exist. In practice, PSLQ finds a sequence of matrices B_n such that if x is the original vector, then the reduced vector y = x * B_n tends to have smaller and smaller entries, until one entry is zero (or a very small number commensurate with precision), at which point an integer relation has been detected. PSLQ also produces a sequence of bounds on the size of any possible integer, which bounds grow until either precision is exhausted or a relation has been detected.

The fundamental duality formula (see Zalinescu ”Convex Analysis in General Vector Spaces”, Theorem 2.7.1) is extended to functions mapping into the power set of a topological linear space with a convex cone which is not necessarily pointed. Pairs of linear functionals are used as dual variables instead of linear operators. The talk will consist of three parts. First, motivations and explanations are given for the infimum approach to set-valued optimization. It deviates from other approaches, and it seems to be the only way to obtain a theory which completely resembles the scalar case. In the second part, the main results are presented, namely the fundamental duality formula and several conclusions. The third part deals with duality formulas for set-valued risk measures, a cutting edge development in mathematical finance. It turns out that the proposed duality theory for set-valued functions provides a satisfying framework not only for set-valued risk measures, but also for no-arbitrage and superhedging theorems in conical market models.

An Asplund space is a Banach space which possesses desirable differentiability properties enjoyed by Euclidean spaces. Many characterisations of such spaces fall into two classes: (i) those where an equivalent norm possesses a particular general property, (ii) those where every equivalent norm possesses a particular property at some points of the space. For example: (i) X is an Asplund space if there exists an equivalent norm Frechet differentiable on the unit sphere of the space, (ii) X is an Asplund space if every equivalent norm is Frechet differentiable at some point of its unit sphere. In 1993 (F-P) showed that (i) X is an Asplund space if there exists an equivalent norm strongly subdifferentiable on the unit sphere of the space and in 1995 (G-M-Z) showed that (ii) X separable is an Asplund space if every equivalent norm is strongly subdifferentiable at a nonzero point of X. Problem: Is this last result true for non-separable spaces? In 1994 (C-P) showed (i) X is an Asplund space if there exists an equivalent norm with subdifferential mapping Hausdorff weak upper semicontinuous on its unit sphere. We show: (ii) X is an Asplund space if every continuous gauge on X has a point where its subdifferential mapping is Hausdorff weak upper semicontinuous with weakly compact image which is some way towards solving the problem.

It is an expository talk about (conjectural) hypergeometric evaluations of lattice sums

$F(a,b,c,d)=(a+b+c+d)^2\sum_{n_j=-\infty,\ j=1,2,3,4}^\infty \frac{(-1)^{n_1+n_2+n_3+n_4}}{(a(6n_1+1)^2+b(6n_2+1)^2+c(6n_3+1)^2+d(6n_4+1)^2)^2}$

which arise as the values of L-functions of certain elliptic curves.

Riemannian manifolds constitute a broad and fruitful framework for the development of different fields in mathematic, such as convex analysis, dynamical systems, optimization or mathematical programming, among other scientific areas, where some of its approaches and methods have successfully been extended from Euclidean spaces. The nonpositive sectional curvature is an important property enjoyed by a large class of differential manifolds, so Hadamard manifolds, which are complete simply connected Riemannian manifolds of nonpositive sectional curvature, have worked out a suitable setting for diverse disciplines.

On the other hand, the study of the class of nonexpansive mappings has become an active research area in nonlinear analysis. This is due to the connection with the geometry of Banach spaces along with the relevance of these mappings in the theory of monotone and accretive operators.

We study the problems that arise in the interface between the fixed point theory for nonexpansive type mappings and the theory of monotone operators in the setting of Hadamard manifolds. Different classes of monotone and accretive set-valued vector fields and the relationship between them will be presented, followed by the study of the existence and approximation of singularities for such vector fields. Then we analyze the problem of finding fixed points of nonexpansive type mappings and the connection with monotonicity. As a consequence, variational inequality and minimization problems in this setting will be discussed.

The term "closed form" is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms and also to give examples from modern research where the question of closure looms both important and elusive.

This talk accompanies a paper by Jonathan M. Borwein and Richard E. Crandall, to appear in the Notices of the AMS, which is available at http://www.carma.newcastle.edu.au/~jb616/closed-form.pdf

The term "closed form" is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms and also to give examples from modern research where the question of closure looms both important and elusive.

This talk accompanies a paper by Jonathan M. Borwein and Richard E. Crandall, to appear in the Notices of the AMS, which is available at http://www.carma.newcastle.edu.au/~jb616/closed-form.pdf.

The design of signals with specified frequencies has applications in numerous fields including acoustics, antenna beamforming, digital filters, optics, radar, and time series analysis. It is often desirable to concentrate signal intensity in certain locations and design methods for this have been intensively studied and are well understood. However, these methods assume that the specified frequencies consist of an interval of integers. What happens when this assumption fails is almost a complete mystery that this talk will attempt to address.

• Prof Joe Diestel (Kent State University)
• Prof Anthony Lau (University of Alberta)
• Prof George Willis (The University of Newcastle)

• Laureate Prof Jonathan Borwein (The University of Newcastle)
• Dr Jeff Hogan (The University of Newcastle)
• Dr Warren Moors (University of Auckland)

• Metric Invariance and Haar Measure
• Locally Compact Groups
• Marriage and Haar Measure
• Introduction to Topological Groups and the Birkhoff-Kakutani Theorem
• Asymptotic Geometric Analysis, Fall 2006

In the area of Metric Fixed Point Theory, one of the outstanding question was if the fixed point property implied reflexivity. This question was answered in the negative in 2008 by P.K.Lin, when he showed that certain renorm in the space of absolutely sumable sequences, had the fixed point property.

In this talk we will show a general way to renorm certain spaces in order to have the fixed point property. We also give general properties for a given Banach space to enjoy the f.p.p. And we will also show equivalences of geometrical properties to certain fixed point properties.

There are many algorithms with which linear programs (LPs) can be solved (Fourier-Motzkin, simplex, barrier, ellipsoid, subgradient, bundle, ...). I will provide a very brief review of these methods and their advantages and disadvantages. An LP solver is the main ingredient of every solution method.(branch&bound, cutting planes, column generation, ...) for (NP hard) mixed-integer linear programs (MIPs). What combinations of which techniques work well in practice? There is no general answer. I will show, by means of many practical examples from my research group (telecommunication, transport, traffic and logistics, energy, ...), how large scale LPs and MIPs are successfully attacked today.

The log-concavity of a sequence is a much studied concept in combinatorics with surprising links to many other mathematical fields. In this talk we discuss the stronger but much less studied notion of m-fold log-concavity which has recently recieved some attention after Boros and Moll conjectured that a "remarkable" sequence encountered in the integration of an inverse cubic is infinitely log-concave. In particular, we report on a recent result of Branden which implies infinite log-concavity of the binomial coefficients and other new developments. Examples and conjectures are promised. A PDF of the talk is available here.

Accurate computer recognition of handwritten mathematics offers to provide a natural interface for mathematical computing, document creation and collaboration. Mathematical handwriting, however, provides a number of challenges beyond what is required for the recognition of handwritten natural languages. For example, it is usual to use symbols from a range of different alphabets and there are many similar-looking symbols. Many writers are unfamiliar with the symbols they must use and therefore write them incorrectly. Mathematical notation is two-dimensional and size and placement information is important. Additionally, there is no fixed vocabulary of mathematical "words" that can be used to disambiguate symbol sequences. On the other hand there are some simplifications. For example, symbols do tend to be well-segmented. With these charactersitics, new methods of character recognition are important for accurate handwritten mathematics input.

An informal one-day workshop on Multi Zeta Values will be held on Wed 20th Oct, from 12.30 pm to 6:00 pm. There will be talks by Laureate Professor Jonathan Borwein (Newcastle), Professor Yasuo Ohno (Kinki University, Osaka), and A/Professor Wadim Zudilin (Newcastle), as well as by PhD students from the two universities, followed by a dinner. If you are interested in attending, please inform Juliane Turner Juliane.Turner@newcastle.edu.au so that we can plan for the event.

The Matching Polynomial is a topic in the area of mathematics, statistical physics (dimer-molomer problem) and chemistry (topological resonant energy). In this talk we will discuss the computation of matching polynomial and location of its roots. We show that the roots of matching generating polynomials of graphs are dense in (−∞, 0] and the roots of matching polynomials of graphs dense in (−∞,+∞) which answer a problem of Brown et. al. (see Journal of Algebraic Combinatorics, 19, 273–282, 2004). Some similar result in characteristic polynomial, independent polynomial and chromatic polynomial are also presented.
[Also speaking: Prof Weigen Yan]

We discuss the feasibility pump heuristic and we interpret it as a multi-start, global optimization algorithm that utilizes a fast local minimizer. The function that is minimized has many local minima, some of which correspond to feasible integral solutions. This interpretation suggests alternative ways of incorporating restarts one of which is the use of cutting planes to eliminate local optima that do not correspond to feasible integral solutions. Numerical experiments show encouraging results on standard test libraries.

We are looking at families of finite sets, more specifically subsets of [n]={1,2,...,n}. In particular, we are interested in antichains, that means no member of the family is contained in another one. In this talk we focus on antichains containing only sets of two different cardinalities, say k and l, and study the question what the smallest size of a maximal antichain is (maximal in the sense that it is impossible to add any k-set or l-set without violating the antichain property). This can be nicely reformulated as a problem in extremal (hyper)graph theory, looking similar to the Turán problem on the maximum number of edges in a graph without a complete subgraphs on l vertices. We sketch the solution for the case (k,l)=(2,3), conjecture an optimal construction for the case (k,l)=(2,4) and present some asymptotic bounds for this case.

The continued fraction: $${\cal R}_\eta(a,b) =\,\frac{{\bf \it a}}{\displaystyle \eta+\frac{\bf \it b^2}{\displaystyle \eta +\frac{4{\bf \it a}^2}{\displaystyle \eta+\frac{9 {\bf \it b}^2}{\displaystyle \eta+{}_{\ddots}}}}}$$
enjoys attractive algebraic properties such as a striking arithmetic-geometric mean relation and elegant links with elliptic-function theory. The fraction presents a computational challenge, which we could not resist.

The continued fraction:
$${\cal R}_\eta(a,b) =\,\frac{{\bf \it a}}{\displaystyle \eta+\frac{\bf \it b^2}{\displaystyle \eta +\frac{4{\bf \it a}^2}{\displaystyle \eta+\frac{9 {\bf \it b}^2}{\displaystyle \eta+{}_{\ddots}}}}}$$
enjoys attractive algebraic properties such as a striking arithmetic-geometric mean relation and elegant links with elliptic-function theory. The fraction presents a computational challenge, which we could not resist. In Part II will reprise what I need from Part I and focus on the dynamics. The talks are stored here.

We will give a brief overview and of the history of Ramanujan and give samplings of areas such as partitions, partition congruences, ranks, modular forms, and mock theta functions. For example: A partition of a positive number $n$ is a non-increasing sequence of positive integers whose sum is $n$. There are five partitions of the number four: 4, 3+1, 2+2, 2+1+1, 1,1,1,1. If we let $p(n)$ be the number of partitions of $n$, it turns out that $p(5n+4)\equiv \pmod{5}$. How does one explain this? Once the basics and context have been introduced, we will discuss new results with respect to mock theta functions and show how they relate to old and recent results.

What is a {\em random subgroup} of a group, and who cares? In a (non-abelian) group based cryptosystem, two parties (Alice and Bob) each choose a subgroup of some platform group "at random" -- each picks $k$ elements "at random" and takes the subgroup generated by their chosen elements.

But for some platform groups (like the braid groups, which were chosen first, being so complicated and difficult) a "random subgroup" is not so random after all. It turned out, pick $k$ elements of a braid group, and they will generate (almost always) a {\em free group} with your $k$ elements as the free basis. And if Alice and Bob are just playing with free groups, it makes their secrets easy to attack.

Richard Thompson's group $F$ is an infinite, torsion free group, with many weird and cool properties, but the one I liked for this project is that it has {\em no} free subgroups (of rank $>1$) at all, so a random subgroup of $F$ could not be free -- so what would it be?

This is joint work with Sean Cleary (CUNY), Andrew Rechnitzer (UBC) and Jeniffer Taback (Bowdoin).

In this talk we consider two-phase flow models in porous media as they occur in several applications like oil production, pollute transport or CO2-storage. After a general introduction, we focus on an enhanced model where the capillary pressure is rate-dependent. We discuss the consequences of this term for heterogeneous materials with and without entry pressure. In the case of entry pressures the problem can be reformulation as inequality constraint at the material interface. Suitable discretization schemes and solution algorithms are proposed and used in various numerical simulations.

The complete elliptic integrals of the first and second kinds (K(x) and E(x)) will be introduced and their key properties revised. Then, new and perhaps interesting results concerning moments and other integrals of K(x) and E(x) will be derived using elementary means. Diverse connections will be made, for instance with random walks and some experimental number theory.

See flyer [PDF]

Automorphism groups of locally finite trees form a significant class of examples of locally compact totally disconnected topological groups. In this talk I will discuss my honours research, which covered the various local properties of automorphism groups. I will provide methods of constructing such groups, in particular groups acting on regular trees, and discuss what conclusions we can make regarding the structure of these groups.

This talk will be an introduction to cycle decompositions of complete graphs in the context of Alspach's conjecture about the necessary and sufficient conditions for their existence. Several useful methods of construction based on algebra, graph products and modifying existing decompositions will be presented. The most up to date results on this problem will be mentioned and future directions of study may be suggested.

This talk will describe a number of dierent variational principles for self-adjoint eigenvalue problems that arose from considerations of convex and nonlinear analysis.
First some unconstrained variational principles that are smooth analogues of the classical Rayleigh principles for eigenvalues of symmetric matrices will be described. In particular the critical points are eigenvectors and their norms are related to the eigenvalues of the matrix. Moreover the functions have a nice Morse theory with the Morse indices describing the ordering of the eigenvector.
Next an unconstrained variational principle for eigenfunctions of elliptic operators will be illustrated for the classical Dirichlet Laplacian eigenproblem. The critical points of this problems have a Morse theory that plays a similar role to the classical Courant- Fischer-Weyl minimax theory.
Finally I will describe certain Steklov eigenproblems and indicate how they are used to develop a spectral characterization of trace spaces of Sobolev fundtions.

Rational numbers can be represented in many different ways: as a fraction, as a M"obius function, as a 2x2 matrix, as a string of L's and R's, as a continued fraction, as a disc in the plane, or as a point in the lattice Z^2. Converting between the representations involves interesting questions about computation and geometry. The geometries that arise are hyperbolic, inversive, or projective.

Various Vacation Scholars, HRD students and CARMA RAs will report on their work. This involves visualization and computation, practice and theory. Everyone is welcome to see what they have done and what they propose to do.

In this talk I will give some classical fixed point theorems and present some of their applications. The talk will be of a "Chalk and Talk" style and will include some elegant classical proofs. The down side of this is that the listener will be expected to have some familiarity with metric spaces, convexity and hopefully Zorn's Lemma.

Various Vacation Scholars, HRD students and CARMA RAs will report on their work. This involves visualization and computation, practice and theory. Everyone is welcome to see what they have done and what they propose to do.

We discuss a short-term revenue optimization problem that involves the optimal targeting of customers for a promotional sale in which a finite number of perishable items are offered on a last-minute offer. The goal is to select the subset of customers to whom the offer will be made available, maximizing the expected return. Each client replies with a certain probability and reports a specific value that depends on the customer type, so that the selected subset has to balance the risk of not selling all the items with the risk of assigning an item to a low value customer. Selecting all those clients with values above a certain optimal threshold may fail to achieve the maximal revenue. However, using a linear programming relaxation, we prove that such threshold strategies attain a constant factor of the optimal value. The achieved factor is ${1\over 2}$ when a single item is to be sold, and approaches 1 as the number of available items grows to infinity. Furthermore, for the single item case, we propose an upper bound based on an exponential size linear program that allows us to get a threshold strategy achieving at least ${2\over 3}$ of the optimal revenue. Computational experiments with random instances show a significantly better performance than the theoretical predictions.

Talk in [PDF]

We describe an integrated model for TCP/IP protocols with multipath routing. The model combines a Network Utility Maximization for rate control, with a Markovian Traffic Equilibrium for routing. This yields a cross-layer design in which sources minimize the expected cost of sending flows, while routers distribute the incoming flow among the outgoing links according to a discrete choice model. We prove the existence of a unique equilibrium state, which is characterized as the solution of an unconstrained strictly convex program of low dimension. A distributed algorithm for solving this optimization problem is proposed, with a brief discussion of how it can be implemented by adapting current Internet protocols.

Talk in [PDF]

Research, as an activity, is fundamentally collaborative in nature. Driven by the massive amounts of data that are produced by computational simulations and high resolution scientific sensors, data-driven collaboration is of particular importance in the computational sciences. In this talk, I will discuss our experiences in designing, deploying, and operating an Canada wide advanced collaboration infrastructure in the support of the computational sciences. In particular, I will focus on the importance of data in such collaborations and discuss how current collaboration tools are sorely lacking in their support of data-centric collaboration.

McCullough-Miller space X = X(W) is a topological model for the outer automorphism group of a free product of groups W. We will discuss the question of just how good a model it is. In particular, we consider circumstances under which Aut(X) is precisely Out(W).

The talk will explain joint work with Yehuda Shalom showing that the only homomorphisms from certain arithmetic groups to totally disconnected, locally compact groups are the obvious, or naturally occurring, ones. For these groups, this extends the supperrigidity theorem that G. Margulis proved for homomorphisms from high rank arithmetic groups to Lie groups. The theorems will be illustrated by referring to the groups $SL_3(\mathbb{Z})$, $SL_2(\mathbb{Z}[\sqrt{2}])$ and $SL_3(\mathbb{Q})$.

http://www.maths.usyd.edu.au/u/SemConf/InfGps/index.html

CARMA is currently engaged in several shared projects with the IRMACS Centre and the OCANA Group UBC-O, both in British Columbia, Canada. This workshop will be an opportunity to learn about irmacs, Centres and to experience the issues in collaborating for research and teaching across the Pacific.

• Introductions - Jon Borwein - 10 min
• Introduction to CARMA -David Allingham - 20 min
• Introduction to Irmacs -Pam Borghardt - 20 min
• Introduction to OCANA -Heinz Bauschke - 20 min

This will be followed by discussion and illustrations of collaboration, technology, teaching and funding etc.

Cross Pacific Collaboration pages at irmacs.

TBA

This is a discrete mathematics instructional seminar commencing 24 February--to meet on subsequent Thursdays from 3:00-4:00 p.m. The seminar will focus on "classical" papers and portions of books.

"In this talk I'll exhibit the interplay between Selberg integrals (interpreted broadly) and random matrix theory. Here an important role is played by the basic matrix operations of a random corank 1 projection (this reduces the number of nonzero eigenvalues by one) and bordering (this increases the number of eigenvalues by one)."

Concave utility functions and convex risk measures play crucial roles in economic and financial problems. The use of concave utility function can at least be traced back to Bernoulli when he posed and solved the St. Petersburg wager problem. They have been the prevailing way to characterize rational market participants for a long period of time until the 1970’s when Black and Scholes introduced the replicating portfolio pricing method and Cox and Ross developed the risk neutral measure pricing formula. For the past several decades the `new paradigm’ became the main stream. We will show that, in fact, the `new paradigm’ is a special case of the traditional utility maximization and its dual problem. Moreover, the convex analysis perspective also highlights that overlooking sensitivity analysis in the `new paradigm’ is one of the main reason that leads to the recent financial crisis. It is perhaps time again for bankers to learn convex analysis.

The talk will be divided into two parts. In the first part we layout a discrete model for financial markets. We explain the concept of arbitrage and the no arbitrage principle. This is followed by the important fundamental theorem of asset pricing in which the no arbitrage condition is characterized by the existence of martingale (risk neutral) measures. The proof of this gives us a first taste of the importance of convex analysis tools. We then discuss how to use utility functions and risk measures to characterize the preference of market agents. The second part of the talk focuses on the issue of pricing financial derivatives. We use simple models to illustrate the idea of the prevailing Black -Scholes replicating portfolio pricing method and related Cox-Ross risk-neutral pricing method for financial derivatives. Then, we show that the replicating portfolio pricing method is a special case of portfolio optimization and the risk neutral measure is a natural by-product of solving the dual problem. Taking the convex analysis perspective of these methods h

Concave utility functions and convex risk measures play crucial roles in economic and financial problems. The use of concave utility function can at least be traced back to Bernoulli when he posed and solved the St. Petersburg wager problem. They have been the prevailing way to characterize rational market participants for a long period of time until the 1970’s when Black and Scholes introduced the replicating portfolio pricing method and Cox and Ross developed the risk neutral measure pricing formula. For the past several decades the `new paradigm’ became the main stream. We will show that, in fact, the `new paradigm’ is a special case of the traditional utility maximization and its dual problem. Moreover, the convex analysis perspective also highlights that overlooking sensitivity analysis in the `new paradigm’ is one of the main reason that leads to the recent financial crisis. It is perhaps time again for bankers to learn convex analysis.

The talk will be divided into two parts. In the first part we layout a discrete model for financial markets. We explain the concept of arbitrage and the no arbitrage principle. This is followed by the important fundamental theorem of asset pricing in which the no arbitrage condition is characterized by the existence of martingale (risk neutral) measures. The proof of this gives us a first taste of the importance of convex analysis tools. We then discuss how to use utility functions and risk measures to characterize the preference of market agents. The second part of the talk focuses on the issue of pricing financial derivatives. We use simple models to illustrate the idea of the prevailing Black -Scholes replicating portfolio pricing method and related Cox-Ross risk-neutral pricing method for financial derivatives. Then, we show that the replicating portfolio pricing method is a special case of portfolio optimization and the risk neutral measure is a natural by-product of solving the dual problem. Taking the convex analysis perspective of these methods h

Professor Jonathan Borwein shares with us his passion for \(\pi\), taking us on a journey through its rich history. Professor Borwein begins with approximations of \(\pi\) by ancient cultures, and leads us through the work of Archimedes, Newton and others to the calculation of \(\pi\) in today's age of computers.

Professor Borwein is currently Laureate Professor in the School of Mathematical and Physical Sciences at the University of Newcastle. His research interests are broad, spanning pure, applied and computational mathematics and high-performance computing. He is also Chair of the Scientific Advisory Committee at the Australian Mathematical Sciences Institute (AMSI).

This talk will be broadcast from the Access Grid room V206 at the University of Newcastle, and will link to the West coast of Canada.

For more information visit AMSI's Pi Day website or read Jon Borwein's talk.

Co-author: Thomas Prellberg (Queen Mary, University of London)

Various kinds of paths on lattices are often used to model polymers. We describe some partially directed path models for which we find the exact generating functions, using instances of the `kernel method'. In particular, motivated by recent studies of DNA unzipping, we find and analyze the generating function for pairs of non-crossing partially directed paths with contact interactions. Although the expressions involving two-path problem are unweildy and tax the capacities of Maple and Mathematica, we are still able to gain an understanding of the singularities of the generating function which govern the behaviour of the model.

The Mahler measure of a polynomial of several variables has been a subject of much study over the past thirty years. Very few closed forms are proven but more are conjectured. We provide systematic evaluations of various higher and multiple Mahler measures using log-sine integrals. We also explore related generating functions for the log-sine integrals. This work makes frequent use of “The Handbook” and involves extensive symbolic computations.

In the 80's R.Grigorchuk found a finitely generated group such that the number of elements that can be written as a product of at most \(n\) generators grows faster than any polynomial in \(n\), but slower than any exponential in \(n\), so-called "intermediate" growth.

It can be described as an group of automorphisms of an infinite rooted binary tree, or in terms of abstract computing devices called "non-initial finite transducers".

In this talk I will describe what some of these short words/products of generators look like, and speculate on the asymptotic growth rate of all short words of length \(n\).

This is joint unpublished work with Mauricio Gutierrez (Tufts) and Zoran Sunic (Texas A&M).

The Mahler measure of a polynomial of several variables has been a subject of much study over the past thirty years. Very few closed forms are proven but more are conjectured. We provide systematic evaluations of various higher and multiple Mahler measures using log-sine integrals. We also explore related generating functions for the log-sine integrals. This work makes frequent use of “The Handbook” and involves extensive symbolic computations.

I will continue by showing relationships between Mahler measures and logsine integrals [PDF]. This should be comprehensible whether or not you heard Part 1.

This talk will present recent theoretical and experimental results contrasting quantum randomness with pseudo-randomness.

We shall conclude the discussion of some of the mathematics surrounding Birkhoff's Theorem about doubly stochastic matrices.

The stochastic Loewner evolution (SLE) is a one-parameter family of random growth processes in the complex plane introduced by the late Oded Schramm in 1999 which is predicted to describe the scaling limit of a variety of statistical physics models. Recently a number of rigorous results about such scaling limits have been established; in fact, Wendelin Werner was awarded the Fields Medal in 2006 for "his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory" and Stas Smirnov was awarded the Fields Medal in 2010 "for the proof of conformal invariance of percolation and the planar Ising model in statistical physics." In this talk, I will introduce some of these models including the Ising model, self-avoiding walk, loop-erased random walk, and percolation. I will then discuss SLE, describe some of its basic properties, and touch on the results of Werner and Smirnov as well as some of the major open problems in the area. This talk will be "colloquium style" and is intended for a general mathematics audience.

In my talk I will review some recent progress on evaluations of Mahler measures via hypergeometric series and Dirichlet L-series. I will provide more details for the case of the Mahler measure of $1+x+1/x+y+1/y$, whose evaluation was observed by C. Deninger and conjectured by D. Boyd (1997). The main ingredients are relations between modular forms and hypergeometric series in the spirit of Ramanujan. The talk is based on joint work with Mat Rogers.

The demiclosedness principle is one of the key tools in nonlinear analysis and fixed point theory. In this talk, this principle is extended and made more flexible by two mutually orthogonal affine subspaces. Versions for finitely many (firmly) nonexpansive operators are presented. As an application, a simple proof of the weak convergence of the Douglas-Rachford splitting algorithm is provided.

This week we shall start the classical paper by Jack Edmonds and DR Fulkerson on partitioning matroids.

On wednesday afternoon, we will be visited by Dr Stephen Hardy and Dr Kieran Larkin from Canon Information Systems Research Australia, Sydney. Drs Larkin and Hardy will be here to explore research opportunities with University of Newcastle researchers. To familiarise them with what we do and to help us understand what they do, there will be three short talks, giving information on the functions and activities of the CDSC, CARMA and Canon's group of 45 researchers. All are welcome to participate.

You are invited to celebrate the life and work of Paul Erdős!

NUMS and CARMA are holding a "Meet Paul Erdős Night" on Wednesday the 20th April starting at 4pm in V07 and we'd love you to come. You can view a poster with information about the night here.

Please RSVP by next Friday 15th April so that we can cater appropriately. To RSVP, reply to: nums@newcastle.edu.au

The Chaney-Schaefer $\ell$-tensor product $E\tilde{\otimes}_{\ell}Y$ of a Banach lattice $E$ and a Banach space $Y$ may be viewed as an extension of the Bochner space $L^p(\mu,Y) (1\leq p < \infty)$. We consider an extension of a classical martingale characterization of the Radon Nikodým property in $L^p(\mu,Y)$, for $1 < p < 1$, to $E\tilde{\otimes}_{\ell}Y$. We consider consequences of this extension, and time permitting, use it to represent set-valued measures of risk dened on Banach lattice-valued Orlicz hearts.

We introduce, assuming only a modest background in one variable complex analysis, the rudiments of infinite dimensional holomorphy. Approaches and some answers to elementary questions arising from considering monomial expansions in different settings and spaces are used to sample the subject.

We meet this Thursday at the usual time when I will show you a nice application of the Edmonds-Fulkerson matroid partition theorem, namely, I'll prove that Paley graphs have Hamilton decompositions (an unpublished result).

The elements of a free group are naturally considered to be reduced "words" in an certain alphabet. In this context, a palindrome is a group element which reads the same from left-to-right and right-to-left. Certain primitive elements, elements that can be part of a basis for the free group, are palindromes. We discuss these elements, and related automorphisms.

We resolve some recent and fascinating conjectural formulae for $1/\pi$ involving the Legendre polynomials. Our mains tools are hypergeometric series and modular forms, though no prior knowledge of modular forms is required for this talk. Using these we are able to prove some general results regarding generating functions of Legendre polynomials and draw some unexpected number theoretic connections. This is joint work with Heng Huat Chan and Wadim Zudilin. The authors dedicate this paper to Jon Borwein's 60th birthday.

In the late seventies, Bill Thurston defined a semi-norm on the homology of a 3-dimensional manifold which lends itself to the study of manifolds which fibre over the circle. This led him to formulate the Virtual Fibration Conjecture, which is fairly inscrutable and implies almost all major results and conjectures in the field. Nevertheless, Thurston gave the conjecture "a definite chance for a positive answer" and much research is currently devoted to it. I will describe the Thurston norm, its main properties and applications, as well as its relationship to McMullen’s Alexander norm and the geometric invariant for groups due to Bieri, Neumann and Strebel.

The aim of this talk is to demonstrate how cyclic division algebras and their orders can be used to enhance wireless communications. This is done by embedding the information bits to be transmitted into smart algebraic structures, such as matrix representations of order lattices. We will recall the essential algebraic definitions and structures, and further familiarize the audience with the transmission model of fading channels. An example application of major current interest is digital video broadcasting. Examples suitable to this application will be provided.

The Landau-Lifshitz-Gilbert equation (LLGE) comes from a model for the dynamics of the magnetization of a ferromagnetic material. In this talk we will first describe existing finite element methods for numerical solution of the deterministic and stochastic LLGEs. We will then present another finite element solution to the stochastic LLGE. This is a work in progress jointly with B. Goldys and K-N Le.

Let $T$ be a topological space (a compact subspace of ${\mathbb R^m}$, say) and let $C(T)$ be the space of real continuous functions on $T$, equipped with the uniform norm: $||f|| = \text{max}_{t\in T}|f(t)|$ for all $f \in C(T)$. Let $G$ be a finite-dimensional linear subspace of $C(T)$. If $f \in C(T)$ then $$d(f,G) = \text{inf}\{||f−g|| : g \in G\}$$ is the distance of $f$ from $G$, and $$P_G(f) = \{g \in G : ||f−g|| = d(f,G)\}$$ is the set of best approximations to $f$ from $G$. Then $$P_G : C(T) \rightarrow P(G)$$ is the set-valued metric projection of $C(T)$ onto $G$. In the 1850s P. L. Chebyshev considered $T = [a, b]$ and $G$ the space of polynomials of degree $\leq n − 1$. Our concern is with possible properties of $P_G$. The historical development, beginning with Chebyshev, Haar (1918) and Mairhuber (1956), and the present state of knowledge will be outlined. New results will demonstrate that the story is still incomplete.

High-dimensional integrals come up in a number of applications like statistics, physics and financial mathematics. If explicit solutions are not known, one has to resort to approximative methods. In this talk we will discuss equal-weight quadrature rules called quasi-Monte Carlo. These rules are defined over the unit cube $[0,1]^s$ with carefully chosen quadrature points. The quadrature points can be obtained using number-theoretic and algebraic methods and are designed to have low discrepancy, where discrepancy is a measure of how uniformly the quadrature points are distributed in $[0,1]^s$. In the one-dimensional case, the discrepancy coincides with the Kolmogorov-Smirnov distance between the uniform distribution and the empirical distribution of the quadrature points and has also been investigated in a paper by Weyl published in 1916.

The talk will focus on recent results or work in progress, with some open problems which span both Combinatorial Design and Sperner Theory. The work focuses upon the duality between antichains and completely separating systems. An antichain is a collection $\cal A$ of subsets of $[n]=\{1,...,n\}$ such that for any distinct $A,B\in\cal A$, $A$ is not a subset of $B$. A $k$-regular antichain on $[m]$ is an antichain in which each element of $[m]$ occurs exactly $k$ times. A CSS is the dual of an antichain. An $(n,k)CSS \cal C$ is a collection of blocks of size $k$ on $[n]$, such that for each distinct $a,b\in [n]$ there are sets $A,B \in \cal C$ with $a \in A-B$ and $b \in B-A$. The notions of $k$-regular antichains of size $n$ on $[m]$ and $(n,k)CSS$s in $m$ blocks are dual concepts. Natural questions to be considered include: Does a $k$-regular antichain of size $n$ exist on $[m]$? For $k

The concept of orthogonal double covers (ODC) of graphs originates in questions concerning database constraints and problems in statistical combinatorics and in design theory. An ODC of the complete graph $K_n$ by a graph $G$ is a collection of $n$ subgraphs of $K_n$, all isomorphic to $G$, such that any two of them share exactly one edge, and every edge of $K_n$ occurs in exactly two of the subgraphs. We survey some of the main results and conjectures in the area as well as constructions, generalizations and modifications of ODC.

This paper studies combinations of the Riemann zeta function, based on one defined by P.R. Taylor, and shown by him to have all its zeros on the critical line. With a rescaled complex argument, this is denoted here by ${\cal T}_-(s)$, and is considered together with a counterpart function ${\cal T}_+(s)$, symmetric rather than antisymmetric about the critical line. We prove by a graphical argument that ${\cal T}_+(s)$ has all its zeros on the critical line, and that the zeros of both functions are all of first order. We also establish a link between the zeros of ${\cal T}_-(s)$ and of ${\cal T}_+s)$ with zeros of the Riemann zeta function $\zeta(2 s-1)$, and between the distribution functions of the zeros of the three functions.

This talk concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness and Lipschitzian assumptions, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and $B$-differentiable versions of Newton's method for nonsmooth Lipschitzian equations.

One of the most effective avenues in recent experimental mathematics research is the computational of definite integrals to high precision, followed by the identification of resulting numerical values as compact analytic formulas involving well-known constants and functions. In this talk we summarize several applications of this methodology in the realm of applied mathematics and mathematical physics, in particular Ising theory, "box integrals", and the study of random walks.

We will investigate the existence of common fixed points for point-wise Lipschitzian semigroups of nonlinear mappings $Tt : C - C$ where $C$ is a bounded, closed, convex subset of a uniformly convex Banach space $X$, i.e. a family such that $T0(x) = x$, $Ts+t = Ts(Tt(x))$, where each $Tt$ is pointwise Lipschitzian, i.e. there exists a family of functions $at : C - [0;x)$ such that $||Tt(x)-Tt(y)|| < at(x)||x-y||$ for $x$, $y \in C$. We will also demonstrate how the asymptotic aspect of the pointwise Lipschitzian semigroups can be expressed in terms of the respective Frechet derivatives. We will discuss some questions related to the weak and strong convergence of certain iterative algorithms for the construction of the stationary and periodic points for such semigroups.

These talks are aimed at extending the undergraduates' field of vision, or increase their level of exposure to interesting ideas in mathematics. We try to present topics that are important but not covered (to our knowledge) in undergraduate coursework. Because of the brevity and intended audience of the talks, the speaker generally only scratches the surface, concentrating on the most interesting aspects of the topic.

In my talk I will try to overview ideas behind (still recent) achievements on arithmetic properties of numbers $\zeta(s)=\sum_{n=1}^\infty n^{-s}$ for integral $s\ge2$, with more emphasis on odd $s$. The basic ingredients of proofs are generalized hypergeometric functions and linear independence criteria. I will also address some "most recent" results and observations in the subject, as well as connections with other problems in number theory.

This paper considers designing permission sets to influence the project selection decision made by a better-informed agent. The project characteristics are two-dimensional. The principal can verify the characteristics of the project selected by the agent. However, the principal cannot observe the number and characteristics of those projects that the agent could, but does not, propose. The payoffs to the agent and the principal are different. Using calculus of variations, we solve the optimal permission set, which can be characterized by a threshold function. We obtain comparative statics on the preference alignment and expected number of projects available. When outcome-based incentives are feasible, we discuss the use of financial inducement to maximize the social welfare. We also extend our analysis to two cases: 1) when one of the project characteristics is unobservable; and 2) when there are multiple agents with private preferences and the principal must establish a universal permission set.
Key words: calculus of variations, optimal permission set, project management.

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this talk, we prove the maximal monotonicity of the sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function satisfying Rockafellar's constraint qualification. Moreover, we show that this sum operator is of type (FPV).

Infinite index subgroups of integer matrix groups like $SL(n,Z)$ which are Zariski dense in $SL(n)$ arise in geometric diophantine problems (e.g., integral Apollonian packings) as well as monodromy groups associated with families of varieties. One of the key features needed when applying such groups to number theoretic problems is that the congruence graphs associated with these groups are "expanders". We will introduce and explain these ideas and review some recent developments especially those connected with the affine sieve.

It is shown that, for maximally monotone linear relations defined on a general Banach space, the monotonicities of dense type, of negative-infimum type, and of Fitzpatrick-Phelps type are the same and equivalent to monotonicity of the adjoint. This result also provides affirmative answers to two problems: one posed by Phelps and Simons, and the other by Simons.

We continue looking at the 1960 Hoffman-Singleton paper about Moore graphs and related topics.

Given a mixed-up sequence of distinct numbers, say 4 2 1 5 7 3 6, can you pass them through an infinite stack (first-in-last-out) from right to left, and put them in order?

                                2 1 5 7 3 6
                     |   |
                     | 4 |
                     |___|

                                1 5 7 3 6
                     |   |
                     | 2 |
                     | 4 |
                     |___|
 
    1                           5 7 3 6
                     |   |
                     | 2 |
                     | 4 |
                     |___|
 
    1 2                         5 7 3 6
                     |   |
                     | 4 |
                     |___|

20-minute presentation followed by 10 minutes of questions and discussion.

We introduce the concept and several examples of q-analogs. A particular focus is on the q-binomial coefficients, which are characterized in a variety of ways. We recall classical binomial congruences and review their known q-analogs. Finally, we establish a full q-analog of Ljunggren's congruence which states that (a*p choose b*p) is congruent to (a choose b) mod p^3.

This week we shall conclude the proof of the uniqueness of the Hoffman-Singleton graph.

This Thursday is your chance to start anew! I shall be starting a presentation of the best work that has been done on Lovasz's famous 1979 problem (now a conjecture) stating that every connected vertex-transitive graph has a Hamilton path. This is good stuff and requires minimal background.

Basically, a function is Lipschitz continuous if it has a bounded slope. This notion can be extended to set-valued maps in different ways. We will mainly focus on one of them: the so-called Aubin (or Lipschitz-like) property. We will employ this property to analyze the iterates generated by an iterative method known as the proximal point algorithm. Specifically, we consider a generalized version of this algorithm for solving a perturbed inclusion $$y \in T(x),$$ where $y$ is a perturbation element near 0 and $T$ is a set-valued mapping. We will analyze the behavior of the convergent iterates generated by the algorithm and we will show that they inherit the regularity properties of $T$, and vice versa. We analyze the cases when the mapping $T$ is metrically regular (the inverse map has the Aubin property) and strongly regular (the inverse is locally a Lipschitz function). We will not assume any type of monotonicity.

We resolve and further study a sinc integral evaluation, first posed in The American Mathematical Monthly in [1967, p. 1015], which was solved in [1968, p. 914] and withdrawn in [1970, p. 657]. After a short introduction to the problem and its history, we give a general evaluation which we make entirely explicit in the case of the product of three sinc functions. Finally, we exhibit some general structure of the integrals in question.

The topic is Lovasz's Conjecture that all connected vertex-transitive graphs have Hamilton paths.

We are interested in local geometrical properties of a Banach space which are preserved under natural embeddings in all even dual spaces. An example of this behaviour which we generalise is:

if the norm of the space $X$ is Fréchet differentiable at $x \in S(X)$ then the norm of the second dual $X^{**}$ is Fréchet differentiable at $\hat{x}\in S(X)$ and of $X^{****}$ at $\hat{\hat{x}} \in S(X^{****})$ and so on....

The results come from a study of Hausdorff upper semicontinuity properties of the duality mapping characterising general differentiability conditions satisfied by the norm.

One of the most intriguing problems in metric fixed point theory is whether we can find closed convex and unbounded subsets of Banach spaces with the fixed point property. A celebrated theorem due to W.O. Ray in 1980 states that this cannot happen if the space is Hilbert. This problem was so poorly understood that two antagonistic questions were raised: the first one was if this property characterizes Hilbert spaces within the class of Banach spaces, while the second one asked if this property characterizes any space at all, that is, if Ray's theorem states in any Banach space. The second problem is still open but the first one has recently been answered in the negative by T. Domínguez Benavides after showing that Ray's theorem also holds true in the classical space of real sequences $c_0$.

The situation seems, however, to be completely different for CAT(0) spaces. Although Hilbert spaces are a particular class of CAT(0) spaces, there are different examples of CAT(0) spaces, including $\mathbb{R}$-trees, in the literature for which we can find closed convex and unbounded subsets with the fixed point property. In this talk we will look closely at this problem. First, we will introduce a geometrical condition inspired in the Banach-Steinhaus theorem for CAT(0) spaces under which we can still assure that Ray's theorem holds true. We will provide different examples of CAT(0) spaces with this condition but we will notice that all these examples are of a very strong Hilbertian nature. Then we will look at $\delta$-hyperbolic geodesic spaces. If looked from very far these spaces, if unbounded, resemble $\mathbb{R}$-trees, therefore it is natural to try to find convex closed and unbounded subsets with the fixed point property in these spaces. We will present some partial results in this direction.

This talk is based a joint work with Bożena Piątek.

This week the discrete mathematics instructional seminar will continue with a consideration of the Lovasz problem. This is the last meeting of the seminar until 13 October.

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). If the quadrilateral is given with its four vertices $A$, $B$, $C$, and $D$ in order, then the theorem states that: $$|AC| \cdot |BD| = |AB| \cdot |CD| + |AD| \cdot |BC|.$$ Furthermore, it is well known that in every Euclidean (or Hilbert) space $H$ we have that $$||x - y|| \cdot ||z - w|| \leq ||x - z|| \cdot ||y - w|| + ||z - y|| \cdot ||x - w||$$ for any four points $w, x, y, z \in H$. This is the classical Ptolemy inequality and it is well-known that it characterizes the inner product spaces among all normed spaces. A Ptolemy metric space is any metric space for which the same inequality holds, replacing norms by distances, for any four points. CAT(0) spaces are geodesic spaces of global nonpositive curvature in the sense of Gromov. Hilbert spaces are CAT(0) spaces and, even more, CAT(0) spaces have many common properties with Hilbert spaces. In particular, although a Ptolemy geodesic metric space need not be CAT(0), any CAT(0) space is a Ptolemy metric space. In this expository talk we will show some recent progress about the connection between Ptolemy metric spaces and CAT(0) spaces.

In this seminar talk we will recall results on the Drop property in Banach spaces and study it in geodesic spaces. In particular, we will show a variant of Danes Drop Theorem in Busemann convex spaces and derive well-posedness results about minimization of convex functions. This talk is based a joint work with Adriana Nicolae.

The choice of a plan for radiotherapy treatment for an individual cancer patient requires the careful trade-off between the goals of delivering a sufficiently high radiation dose to the tumour and avoiding irradiation of critical organs and normal tissue. This problem can be formulated as a multi-objective linear programme (MOLP). In this talk we present a method to compute a finite set of non-dominated points that can be proven to uniformly cover the complete non-dominated set of an MOLP (a finite representation). This method generalises and improves upon two existing methods from the literature. We apply this method to the radiotherapy treatment planning problem, showing some results for clinical cases. We illustrate how the method can be used to support clinician’s decision making when selecting a treatment plan. The treatment planner only needs to specify a threshold for recognising two treatment plans as different and is able to interactively navigate through the representative set without the trial-and-error process often used in practice today.

The talk will begin with a reminder of what a triangulated category is, the context in which they arose, and why we care about them. Then we will discuss the theory of compactly generated triangulated categories, again illustrating the applications. This theory is old and well understood. Finally we will come to well generated categories, where several open problems remain very mysterious.

Noncommutative geometry is based on fairly sophisticated methods: noncommutative C*-algebras are called noncommutative topological spaces, noncommutative von Neumann algebras are noncommutative measure spaces, and Hopf algebras and homological invariants describe the geometry. Standard topology, on the other hand, is based on naive intuitions about discontinuity: a continuous function is one whose graph does not have any gaps, and cutting and gluing are used to analyse and reconstruct geometrical objects. This intuition does not carry over to the noncommutative theory, and the dictum from quantum mechanics that it does not make sense any more to think about point particles perhaps explains a lack of expectation that it should. The talk will describe an attempt to make this transfer by computing the polar decompositions of certain operators in the group C*-algebras of free groups. The computation involves some identities and evaluations of integrals that might interest the audience, and the polar decomposition may be interpreted as a noncommutative version of the double angle formula familiar from high school geometry.

Mean periodic functions of a single real variable were an innovation of the mid-20th century. Although not as well known as almost periodic functions, they have some nice properties, with applications to certain mean value theorems.

We shall start looking at Dave Witte's (now Dave Morris) proof that connected Cayley digraphs of prime power order have Hamilton directed cycles.

This week we shall conclude our look at the paper by Dave Witte on Hamilton directed cycles in Cayley digraphs of prime power order.

I shall describe highlights of my two decades of experience with Advanced Collaborative Environments (ACEs) in Canada and Australia, running shared seminars, conferences, resources and courses over the internet. I shall also describe the AMSI Virtual Lab proposal which has just been submitted to NeCTAR. The slides for much of this talk are at http://www.carma.newcastle.edu.au/jon/aces11.pdf.

We interpret the Hamiltonian Cycle problem (HCP) as a an optimisation problem with the determinant objective function, naturally arising from the embedding of HCP into a Markov decision process. We also exhibit a characteristic structure of the class of all cubic graphs that stems from the spectral properties of their adjacency matrices and provide an analytic explanation of this structure.

We look at (parts of) the survey paper Dependent Random Choice by Jacob Fox and Benny Sudakov: http://arxiv.org/abs/0909.3271
The abstract of the paper says "We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss some of them."
My plan for the seminar is to start with a quick recap of the classics of extremal (hyper)graph theory (i.e. Turan, Ramsey, Ramsey-Turan), then look at some simple examples for the probabilistic method in action, and finally come to the striking applications mentioned in the quoted abstract.
Only elementary probability is required.

The arrival of online casinos in 1996 brought games that you would find at land-based casinos to the computer screens of gamblers all over the world. A major benefit in online casinos is in the automation of systems across several computers for favourable games; as this has the potential to make a significant amount of profit. This article applies this concept to online progressive video poker games. By establishing a set of characteristics to compare different games, analyses are carried out to identify which game should be the starting point for building an automated system. Bankroll management and playing strategies are also analyzed in this article, and are shown to be important components if profiting from online gambling is going to be a long term business.

Within the topic of model-based forecasting with exponential smoothing, this paper seeks to contribute to the understanding of the property of certain stochastic processes to converge almost surely to a constant. It provides a critical discussion of the related views and ideas found in the recent forecasting literature and aims at elucidating the present confusion by review and study of the classical and less known theorems of probability theory and random processes. The paper then argues that a useful role of exponential smoothing for modelling and forecasting sequential count data is limited and methods that are either not based on exponential smoothing or use exponential smoothing in a more flexible way are worthy of exploration. An approach to forecasting such data based on applying exponential smoothing to the probabilities of each count outcome is thus introduced and its merits are discussed in the context of pertinent statistical literature.

In this talk we consider a problem of scheduling several jobs on multiple machines satisfying precedence and resource constraints. Each job has a due date and the objective is to minimize the cumulative weighted tardiness across all jobs. We investigate how to efficiently obtain heuristic solutions on multi-core computers using an Ant Colony Systems framework for the optimisation. The talk will discuss some of the challenges that arise in designing a multi-threaded heuristic and provided computational results for some alternative algorithm variants. The results showing that theACS heuristic is more effective particularly for large problem instances than other methods developed to date.

The relation between mechanics and optimization goes back at least to Euler and was further strengthened by the Lagrangian and Hamiltonian formulations of Newtonian mechanics. Since then, numerous variational formulations of mechanical phenomena have been proposed and although the link to optimization often has been somewhat obscured in the subsequent development of numerical methods, it is in fact as strong as ever. In this talk, I will summarize some of the recent developments in the application of modern mathematical programming methods to problems involving the simulation mechanical phenomena. While the methodology is quite general, emphasis will be on the static and dynamic deformation processes in civil engineering,geomechanics and the earth sciences.

The Feasibility Pump (FP) has proved to be an effective method for finding feasible solutions to Mixed-Integer Programming problems. We investigate the benefits of replacing the rounding procedure with a more sophisticated integer line search that efficiently explores a larger set of integer points with the aim of obtaining an integer feasible solution close to an FP iterate. An extensive computational study on 1000+ benchmark instances demonstrates the effectiveness of the proposed approach.

A common issue when integrating airline planning processes is the long planning horizon of the crew pairing problem. We propose a new approach to the crew pairing problem through which we retain a significant amount of flexibility. This allows us to solve an integrated aircraft routing, crew pairing, and tail number assignment problem only few days before the day of operations and with a rolling planning horizon. The model simultaneously schedules appropriate rest periods for all crews and maintenance checks for all tail numbers.
A Branch-and-Price method is proposed in which each tail number and each 'crew block' is formulated as a subproblem.

A water and sewage system, a power grid, a telecommunication network, are all examples of network infrastructures. Network infrastructures are a common phenomenon in many industries. A network infrastructure is characterized by physical links and connection points. Examples of physical links are pipes (water and sewage system), fiber optic cables (telecommunication network), power lines (power grid), and tracks (rail network). Such network infrastructures have to be maintained and, often, have to be upgraded or expanded. Network upgrades and expansions typically occur over a period of time due to budget constraints and other considerations. Therefore, it becomes important to determine both when and where network upgrades and expansions should take place so as to minimize the infrastructure investment as well as current and future operational costs.
We introduce a class of multi-period network infrastructure expansion problems that allow us to study the key issues related to the choice and timing of infrastructure expansions and their impact on the costs of the activities performed on that infrastructure. We focus on the simplest variant, an incremental shortest path problem (ISPP). We show that even ISPP is NP-hard, we introduce a special case that is polynomially solvable, we derive structural solution properties, we present an integer programming formulation and classes of valid inequalities, and discuss the results of a computational study.

The classical single period problem (SPP) has wide applicability especially in service industries which dominates the economy. In this paper a single period production problem, is considered, as a specific type of SPP. The SPPmodel is extended by considering the probability of scrap and rework in production at the beginning and during the period. The optimal solution which maximizes the expected value of total profit obtained. In the case of producing the scrap items and defective items which should rework, the optimal profit of system in comparison to ideal production system reduces. Also, the reduction of profit is more sensitive by increasing the probability of producing scrap items in comparison with the probability of producing defective items. These results would help the managers in order to make the right decision about changing or revising machines or technologies.

In this presentation I briefly discuss practical and philosophical issues related to the role of the peer-review process in maintaining the quality of scientific publications. The discussion is based on, among other things, my experience over the past eight years in containing the spread of voodoo decision theories in Australia. To motivate the discussion, I ask: how do you justify the use a model of local robustness (operating in the neighborhood of a wild guess) to manage Black Swans and Unknown Unknowns?

Design of a complex system needs both micro and macro level competencies to capture the underlying structure of complex problem satisfying convergence to good solution point. Systems such as complex organizations, complex New Product Development (NPD) and complex network of firms (Supply Chains or SC) require competencies at both macro (coordination and integration) and micro (capable designers, teams for NPD and capable firms in SC) entities. Given high complexity in such problems at both macro and micro levels, a couple of different errors can happen at each: 1) Either acceptance of a wrong solution or rejection of a right solution at micro level. 2) Either coordination of a couple of entities that do not need any coordination [e.g. teams or designers working in NPD might put too much time in meetings and firms in SC might lose their flexibility due to limitations from powerful and leader firms in SC] or lack of deployed resources for entities that need coordination [e.g. inconsistencies in decisions made in decentralized systems such as NPD and SC]. In this paper a simple and parsimonious Agent Based Model (ABM) of NK type is build and simulated to study these complex interactive systems. The results of simulations provide some insights on imperfect management of above mentioned complex systems. For instance, we found that asymmetry in any of the above mentioned errors favours a particular policy in management of these systems.

A problem that frequently arises in environmental surveillance is where to place a set of sensors in order to maximise collected information. In this article we compare four methods for solving this problem: a discrete approach based on the classical k-median location model, a continuous approach based on the minimisation of the prediction error variance, an entropy-based algorithm, and simulated annealing. The methods are tested on artificial data and data collected from a network of sensors installed in the Springbrook National Park in Queensland, Australia, for the purpose of tracking the restoration of biodiversity. We present an overview of these methods and a comparison of results.

This talk presents an innovative model for describing the effects of QM on organizational productivity, traditionally researched by statistical models. Learning inside organizations combined with the information processing metaphor of organizations is applied to build a computational model for this research. A reinforcement learning (RL) algorithm is implemented in the computational model to characterize the effects of quality leadership on productivity. The results show that effective quality leadership, being a balanced combination of exploration of new actions and exploitation of previous good actions, outperform pure exploration or exploitation strategies in the long run. However, pure exploitation outperforms the exploration and RL algorithms in the short term. Furthermore, the effects of complexity of customer requirements on productivity are investigated. From the results it can be argued that more complexity usually leads to less productivity. Also, the gap between random action algorithm and RL is reduced when the complexity of customer requirements increases. As regards agent types, it can be inferred that well- balanced business processes comprised of similar agents (in terms of agents’ processing time and accuracy) perform better than other scenarios.

Modular forms have had an important role in number theory for over one hundred years. Modular forms are also of interest in areas such as topology, cryptography and communications network theory. More recently, Peter Sarnak’s August talk, "Chaos, Quantum Mechanics and Number Theory" strongly suggested a link between modular forms and quantum mechanics. In this talk we explain modular forms, in the context of seeking a formula for the number of representations of an integer as the sum of four squares.

We look at (parts of) the survey paper Dependent Random Choice by Jacob Fox and Benny Sudakov: http://arxiv.org/abs/0909.3271. The abstract of the paper says "We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss some of them." My plan for the seminar is to start with a quick recap of the classics of extremal (hyper)graph theory (i.e. Turan, Ramsey, Ramsey-Turan), then look at some simple examples for the probabilistic method in action, and finally come to the striking applications mentioned in the quoted abstract. Only elementary probability is required.

In this talk I attempt to explain a general approach in proving irrationality and linear independence results for q-hypergeometric series. An explicit Pade construction is introduced with some (quantitative) arithmetic implications for well-known q-mathematical constants.

Probability densities are a major tool in exploratory statistics and stochastic modelling. I will talk about a numerical technique for the estimation of a probability distribution from scattered data using exponential families and a maximum a-posteriori approach with Gaussian process priors. Using Cameron-Martin theory, it can be seen that density estimation leads to a nonlinear variational problem with a functional defined on a reproducing kernel Hilbert space. This functional is strictly convex. A dual problem based on Fenchel duality will also be given. The (original) problem is solved using a Newton-Galerkin method with damping for global convergence. In this talk I will discuss some theoretical results relating to the numerical solution of the variational problem and the results of some computational experiments. A major challenge is of course the curse of dimensionality which appears when high-dimensional probability distributions are estimated.

Thomas will be finishing his talks this Thursday where he will finish looking at (parts of) the survey paper Dependent Random Choice by Jacob Fox and Benny Sudakov: http://arxiv.org/abs/0909.3271

We discuss the asymmetric sandwich theorem, a generalization of the Hahn–Banach theorem. As applications, we derive various results on the existence of linear functionals in functional analysis that include bivariate, trivariate and quadrivariate generalizations of the Fenchel duality theorem. We consider both results that use a simple boundedness hypothesis (as in Rockafellar’s version of the Fenchel duality theorem) and also results that use Baire’s theorem (as in the Robinson–Attouch–Brezis version of the Fenchel duality theorem).

Lattice paths effectively model phenomena in chemistry, physics and probability theory. Techniques of analytic combinatorics are very useful in determining asymptotic estimates for enumeration, although asymptotic growth of the number of Self Avoiding Walks on a given lattice is known empirically but not proved. We survey several families of lattice paths and their corresponding enumerative results, both explicit and asymptotic. We conclude with recent work on combinatorial proofs of asymptotic expressions for walks confined by two boundaries.

A Hamilton surface decomposition of a graph is a decomposition of the collection of shortest cycles in such a way that each member of the decomposition determines a surface (with maximum Euler characteristic). Some sufficient conditions for Hamilton surface decomposition of cartesian products of graphs are obtained. Necessary and sufficient conditions are found for the case when factors are even cycles.

The minimal degree of a finite group $G$ is the smallest non-negative integer $n$ such that $G$ embeds in $\Sym(n)$. This defines an invariant of the group $\mu(G)$. In this talk, I will present some interesting examples of calculating $\mu(G)$ and examine how this invariant behaves under taking direct products and homomorphic images.

In particular, I will focus on the problem of determining the smallest degree for which we obtain a strict inequality $\mu(G \times H) < \mu(G) + \mu(H)$, for two groups $G$ and $H$. The answer to this questions also leads us to consider the problem of exceptional permutation groups. These are groups $G$ that possess a normal subgroup $N$ such that $\mu(G/N) > \mu(G)$. They are somewhat mysterious in the sense that a particular homomorphic image becomes 'harder' to faithfully represent than the group itself. I will present some recent examples of exceptional groups and detail recent developments in the 'abelian quotients conjecture' which states that $\mu(G/N) < \mu(G)$, whenever $G/N$ is abelian.

We prove the it is NP-hard for a coalition of two manipulators to compute how to manipulate the Borda voting rule. This resolves one of the last open problems in the computational complexity of manipulating common voting rules. Because of this NP-hardness, we treat computing a manipulation as an approximation problem where we try to minimize the number of manipulators. Based on ideas from bin packing and multiprocessor scheduling, we propose two new approximation methods to compute manipulations of the Borda rule. Experiments show that these methods significantly outperform the previous best known approximation method. We are able to find optimal manipulations in almost all the randomly generated elections tested. Our results suggest that, whilst computing a manipulation of the Borda rule by a coalition is NP-hard, computational complexity may provide only a weak barrier against manipulation in practice.

We also consider Nanson’s and Baldwin’s voting rules that select a winner by successively eliminating candidates with low Borda scores. We theoretically and experimentally demonstrate that these rules are significantly more difficult to manipulate compared to Borda rule. In particular, with unweighted votes, it is NP-hard to manipulate either rule with one manipulator, whilst with weighted votes, it is NP-hard to manipulate either rule with a small number of candidates and a coalition of manipulators.

We consider the problem of packing ellipsoids of different size and shape in an ellipsoidal container so as to minimize a measure of total overlap. The motivating application is chromosome organization in the human cell nucleus. A bilevel optimization formulation is described, together with an algorithm for the general case and a simpler algorithm for the special case in which all ellipsoids are in fact spheres. We prove convergence to stationary points of this nonconvex problem, and describe computational experience. The talk describes joint work with Caroline Uhler (IST, Vienna).

Having been constructed as trading strategies, option spreads are also used in margin calculations for offsetting positions in options. All option spreads that appear in trading and margining practice have two, three or four legs. As shown in Rudd and Schroeder (Management Sci, 1982), the problem of margining option portfolios where option spreads with two legs are used for offsetting can be solved in polynomial time by network flow algorithms. However, spreads with only two legs do not provide sufficient accuracy in measuring risk. Therefore, margining practice also employs spreads with three and four legs. A polynomial-time solution to the extension of the problem where option spreads with three and four legs are also used for offsetting is not known. We propose a heuristic network-flow algorithm for this extension and present a computational study that demonstrates high efficiency of the proposed algorithm in margining practice.

We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem. We introduce a gradient-based method, called the minimal norm gradient method, for solving this class of problems, and establish the convergence of the sequence generated by the algorithm as well as a rate of convergence of the sequence of function values. A portfolio optimization example is given in order to illustrate our results.

Graph closures became recently an important tool in Hamiltonian Graph Theory since the use of closure techniques often substantially simplifies the structure of a graph under consideration while preserving some of its prescribed properties (usually of Hamiltonian type). In the talk we show basic ideas of construction of some graph closures for claw-free graphs and techniques that allow to reduce the problem to cubic graphs. The approach will be illustrated on a recently introduced closure concept for Hamilton-connectedness in claw-free graphs and, as an application, an asymptotically sharp Ore-type degree condition for Hamilton-connectedness in claw-free graphs will be obtained.

Two sets of functions are studied to ascertain whether they are Stieltjes functions and whether they are completely monotonic. The first group of functions are all built from the Lambert $W$ function. The $W$ function will be reviewed briefly. It will be shown that $W$ is Bernstein and various functions containing $W$ are Stieltjes. Explicit expressions for the Stieltjes transforms are obtained. We also give some new results regarding general Stieltjes functions.

The second set of functions were posed as a challenge by Christian Berg in 2002. The functions are $(1+a/x)^{(x+b)}$ for various $a$ and $b$. We show that the functions is Stieltjes for some ranges of $a,b$ and investigate experimentally complete monotonicity for a larger range. We claim an accurate experimental value for the range.

My co-authors are Rob Corless, Peter Borwein, German Kalugin and Songxin Liang.

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p > 0$. Let $L/K$ be a finite Galois extension with Galois group $G = \text{Gal}(L/K)$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. Let $W_n(.)$ denote the ring of $p$-typical Witt vectors of length $n$. Hesselholt [Galois cohomology of Witt vectors of algebraic integers, Math. Proc. Cambridge Philos. Soc. 137(3) (2004), 551557] conjectured that the pro-abelian group ${H^1(G,W_n(O_L))}_{n>0}$ is isomorphic to zero. Hogadi and Pisolkar [On the cohomology of Witt vectors of $p$-adic integers and a conjecture of Hesselholt, J. Number Theory 131(10) (2011), 17971807] have recently provided a proof of this conjecture. In this talk, we present a simplified version of the original proof which avoids many of the calculations present in that version.

Integrability theory is the area of mathematics in which methods are developed for the exact solution of partial differential equations, as well as for the study of their properties. We concentrate on PDEs appearing in Physics and other applications. Darboux transformations constitute one of the important methods used in integrability theory and, as well as being a method for the exact solution of linear PDEs, they are an essential part of the method of Lax pairs, used for the solution of non-linear PDEs. A large series of Darboux transformations may be constructed using Wronskians built from some number of individual solutions of the original PDE. In this talk we prove a long-standing conjecture that this construction captures all possible Darboux transformations for transformations of order two, while for transformations of order one the construction captures everything but two Laplace transformations. An introduction into the theory will be provided.

Power line communication has been proposed as a possible solution to the "last mile" problem in telecommunications i.e. providing economical high speed telecommunications to millions of end users. As well as the usual background interference (noise), two other types of noise must also be considered for any successful practical implementation of power line communication. Coding schemes have traditionally been designed to deal only with background noise, and in such schemes it is often assumed that background noise affects symbols in codewords independently at random. Recently, however, new schemes have been proposed to deal with the extra considerations in power line communication. We introduce neighbour transitive codes as a group theoretic analogue to the assumption that background noise affects symbols independently at random. We also classify a family of neighbour transitive codes, and show that such codes have the necessary properties to be useful in power line communication.

We present a technique for enhancing a progressive hedging-based metaheuristic for a network design problem that models demand uncertainty with scenarios. The technique uses machine learning methods to cluster scenarios and, subsequently, the metaheuristic repeatedly solves multi-scenario subproblems (as opposed to single-scenario subproblems as is done in existing work). With a computational study we see that solving multi-scenario subproblems leads to a significant increase in solution quality and that how you construct these multi-scenario subproblems directly impacts solution quality. We also discuss how scenario grouping can be leveraged in a Benders' approach and show preliminary results of its effectiveness. This is joint work with Theo Crainic and Walter Rei at University of Quebec at Montreal.

We start this talk by introducing some basic definitions and properties relative to geodesic in the setting of metric spaces. After showing some important examples of geodesic metric spaces (which will be used through this talk), we shall define the concept of firmly nonexpansive mappings and we shall prove the existence, under mild conditions, of periodic points and fixed points for this class of mappings. Some of these results unify and generalize previous ones. We shall give a result relative to the $\Delta$-convergence to a fixed point of Picard iterates for firmly nonexpansive mappings, which is obtained from the asymptotic regularity of this class of iterates. Moreover, we shall get an effective rate of asymptotic regularity for firmly nonexpansive mappings (this result is new, as far as we know, even in linear spaces). Finally, we shall apply our results to a minimization problem. More precisely, we shall prove the $\Delta$-convergence to a minimizer of a proximal point-like algorithm when applied to a convex proper lower semi-continuous function defined on a CAT(0) space.

The Discrete Mathematics Instructional Seminar will be getting underway again this Thursday.

Parabolic obstacle problems find applications in the financial markets for pricing American put options. We present a mixed and an equivalent variational inequality hp-interior penalty DG (IPDG) method combined with an hp-time DG (TDG) method to solve parabolic obstacle problems approximatively. The contact conditions are resolved by a biorthogonal Lagrange multiplier and are component-wise decoupled. These decoupled contact conditions are equivlent to finding the root of a non-linear complementary function. This non-linear problem can in turn be solved efficiently by a semi-smooth Newton method. For the hp-adaptivity a p-hierarchical error estimator in conjunction with a local analyticity estimate is employed. For the considered stationary problem, this leads to exponential convergence, and for the instationary problem to greatly improved convergence rates. Numerical experiments are given demonstrating the strengths and limitations of the approaches.

Network infrastructures are a common phenomenon. Network upgrades and expansions typically occur over time due to budget constraints. We introduce a class of incremental network design problems that allow investigation of many of the key issues related to the choice and timing of infrastructure expansions and their impact on the costs of the activities performed on that infrastructure. We focus on the simplest variant: incremental network design with shortest paths, and show that even its simplest variant is NP-hard. We investigate structural properties of optimal solutions, we analyze the worst-case performance of natural greedy heuristics, we derive a 4-approximation algorithm, and we present an integer program formulation and conduct a small computational study.

Selection theorems assert that one can pick a well behaved function from a corresponding multifunction. They play a very important role in modern optimization theory. I will survey their structure and some applications before sketching some important open research problems.

The celebrated Littlewood conjecture in Diophantine approximation concerns the simultaneous approximation of two real numbers by rationals with the same denominator. A cousin of this conjecture is the mixed Littlewood conjecture of de Mathan and Teulié, which is concerned with the approximation of a single real number, but where some denominators are preferred to others.

In the talk, we will derive a metrical result extending work of Pollington and Velani on the Littlewood conjecture. Our result implies the existence of an abundance of numbers satisfying both conjectures.

• 10.00-10.30 am: Michael Rose
• 11.00-11.30 am: Daniel Sutherland
• 12.00 noon-12.30 pm - Chris Banks

Selection theorems assert that one can pick a well behaved function from a corresponding multifunction. They play a very important role in modern optimization theory. In Part I, I will survey their structure and some applications before sketching some important applications and open research problems in Part II.

In this talk, we present a numerical method for a class of generalized inequality constrained integer linear programming (GILP) problems that includes the usual mixed-integer linear programming (MILP) problems as special cases. Instead of restricting certain variables to integer values as in MILP, we require in these GILP problems that some of the constraint functions take integer values. We present a tighten-and-branch method that has a number of advantages over the usual branch-and-cut algorithms. This includes the ability of keeping the number of constraints unchanged for all subproblems throughout the solution process and the capability of eliminating equality constraints. In addition, the method provides an algorithm framework that allows the existing cutting-plane techniques to be incorporated into the tightening process. As a demonstration, we will solve a well-known "hard ILP problem".

Symbolic and numeric computation have been distinguished by definition: numeric computation puts numerical values in its variables as soon as possible, symbolic computation as late as possible. Chebfun blurs this distinction, aiming for the speed of numerics with the generality and flexibility of symbolics. What happens when someone who has used both Maple and Matlab for decades, and has thereby absorbed the different fundamental assumptions into a "computational stance", tries to use Chebfun to solve a variety of computational problems? This talk reports on some of the outcomes.

The Mathematics and Statistics Learning Centre was established at the University of Melbourne over a decade ago, to respond to the needs of, initially, first year students of mathematics and statistics. The role of the centre and its Director has grown. The current Director, Dr Deborah King, will expound upon her role in the Centre.

The modernization of infrastructure networks requires coordinated planning and control. Considering traffic networks and electricity grids raises similar issues on how to achieve substantial new capabilities of effectiveness and efficiency. For instance, power grids need to integrate renewable energy sources and electric vehicles. It is clear that all this can only be achieved by greater reliance on systematic planning in the presence of uncertainty and sensing, communications, computing and control on an unprecedented scale, these days captured in the term "smart grids". This talk will outline current research on planning future grids and control of smart grids. In particular, the possible roles of network science will be emphasized and the challenges arising.

The problem posed by Hilbert in 1900 was resolved in the 1930s independently by A. Gelfond and Th. Schneider. The statement is that $a^b$ is transcendental for algebraic $a \ne 0,1$ and irrational algebraic $b$. The aim of the two 2-hour lectures is to give a proof of this result using the so-called method of interpolation determinants.

In this paper, we construct maximally monotone operators that are not of Gossez's dense-type (D) in many nonreflexive spaces. Many of these operators also fail to possess the Brønsted-Rockafellar (BR) property. Using these operators, we show that the partial inf-convolution of two BC-functions will not always be a BC-function. This provides a negative answer to a challenging question posed by Stephen Simons. Among other consequences, we deduce that every Banach space which contains an isomorphic copy of the James space J or its dual $J^*$, or of $c_0$ or its dual $l^1$ admits a non type (D) operator.

In this talk, we consider the automorphism groups of the Cayley graph with respect to the Coxeter generators and the Davis complex of an arbitrary Coxeter group. We determine for which Coxeter groups these automorphism groups are discrete. In the case where they are discrete, we express them as semidirect products of two obvious families of automorphisms. This extends a result of Haglund and Paulin.

We investigate various properties of the sublevel set $\{x : g(x) \leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$ are positively homogeneous functions. For instance, the latter integral reduces to integrating $h\exp(- g)$ on the whole space $\mathbb{R}^n$ (a non-Gaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of its coefficients.

In fact, whenever $h$ is non-negative, the functional $\int \phi(g)h dx$ is a convex function of $g$ for a large class of functions $\phi:\mathbb{R}_{+} \to \mathbb{R}$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently, to approximate the associated non-Gaussian integral). We also show that finding the sublevel set $\{x : g(x) \leq 1\}$ of minimum volume that contains some given subset $K$ is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non-Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.

Simultaneous Localisation and Mapping (SLAM) has become prominent in the field of robotics over the last decade, particularly in application to autonomous systems. SLAM enables any system equipped with exteroceptive (and often inertial) sensors to simultaneously update its own positional estimate and map of the environment by utilising information collected from the surroundings. The solution to the probabilistic SLAM problem can be derived using Bayes Theorem to yield estimates of the system state and covariance. In recursive form, the basic prediction-correction algorithm employs an Extended Kalman Filter (EKF) with Cholesky decomposition for numerical stability during inversion. This talk will present the mathematical formulation and solution of the SLAM problem, along with some algorithms used in implementation. We will then look at some applications of SLAM in the real world and discuss some of the challenges for future development.

In my opinion, the most significant unsolved problem in graph decompositions is the cycle double conjecture. This begins a series of talks on this conjecture in terms of background, relations to other problems and partial results.

This will be an introductory talk which begins by describing the four colour theorem and finite projective planes in the setting of graph decompositions. A problem posed by Ringel at a graph theory meeting in Oberwolfach in 1967 will then be discussed. This problem is now widely known as the Oberwolfach Problem, and is a generalisation of a question asked by Kirkman in 1850. It concerns decompositions of complete graphs into isomorphic copies of spanning regular graphs of degree two.

In this talk, we consider the structure of maximally monotone operators in Banach space whose domains have nonempty interior and we present new and explicit structure formulas for such operators. Along the way, we provide new proofs of the norm-to-weakstar closedness and property (Q) of these operators (recently established by Voisei). Various applications and limiting examples are given. This is the joint work with Jon Borwein.

Brian Alspach will continue with "The Anatomy of a Famous Conjecture" this Thursday. One can easily pick up the thread this week without having attended last week, but if you miss this week it will not be easy to join in next week.

I have embarked on a project of looking for Hamilton paths in Cayley graphs on finite Coxeter groups. This talk is a report on the progress thus far.

Exceptional Lie group $G_2$ is a beautiful 14-dimensional continuous group, having relations with such diverse notions as triality, 7-dimensional cross product and exceptional holonomy. It was found abstractly by Killing in 1887 (complex case) and then realized as a symmetry group by Engel and Cartan in 1894 (real split case). Later in 1910 Cartan returned to the topic and realized split $G_2$ as the maximal finite-dimensional symmetry algebra of a rank 2 distribution in $\mathbb{R}^5$. In other words, Cartan classified all symmetry groups of Monge equations of the form $y'=f(x,y,z,z',z'')$. I will discuss the higher-dimensional generalization of this fact, based on the joint work with Ian Anderson. Compact real form of $G_2$ was realized by Cartan as the automorphism group of octonions in 1914. In the talk I will also explain how to realize this $G_2$ as the maximal symmetry group of a geometric object.

Brian Alspach will continue with "The Anatomy Of A Famous Conjecture" this Thursday. One can easily pick up the thread this week without having attended last week, but if you miss this week it will not be easy to join in next week.

In this talk, we consider a general convex feasibility problem in Hilbert space, and analyze a primal-dual pair of problems generated via a duality theory introduced by Svaiter. We present some algorithms and their convergence properties. The focus is a general primal-dual principle for strong convergence of some classes of algorithms. In particular, we give a different viewpoint for the weak-to-strong principle of Bauschke and Combettes. We also discuss how subgradient and proximal type methods fit in this primal-dual setting.

Joint work with Maicon Marques Alves (Universidade Federal de Santa Catarina-Brazil)

The talk will outline some topics associated with constructions for Hadamard matrices, in particular, a relatively simple construction, given by a sum of Kronecker products of ingredient matrices obeying certain conditions. Consideration of the structure of the ingredient matrices leads, on the one hand, to consideration of division algebras and Clifford algebras, and on the other hand, to searching for multisets of {-1,1} ingredient matrices. Structures within the sets of ingredient matrices can make searching more efficent.

We consider some fundamental generalized Mordell-Tornheim-Witten (MTW) zeta-function values along with their derivatives, and explore connections with multiple-zeta values (MZVs). To achieve these results, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory.

Our original motivation was to represent previously unresolved constructs such as Eulerian log-gamma integrals. Indeed, we are able to show that all such integrals belong to a vector space over an MTW basis, and we also present, for a substantial subset of this class, explicit closed-form expressions. In the process, we significantly extend methods for high-precision numerical computation of polylogarithms and their derivatives with respect to order. That said, the focus of our paper is the relation between MTW sums and classical polylogarithms. It is the adumbration of these relationships that makes the study significant.

The associated paper (with DH Bailey and RE Crandall) is at http://carmasite.newcastle.edu.au/jon/MTW1.pdf.

Approximation theory is a classical part of the analysis of functions defined on an Euclidean space or its subset and the foundation of its applications, while the problems related to high or infinite dimensions create known challenges even in the setting of Hilbert spaces. The stability (uniform continuity) of a mapping is one of the traditional properties investigated in various branches of pure and applied mathematics and further applications in engineering. Examples include analysis of linear and non-linear PDEs, (short-term) prediction problems and decision-making and data evolution.

We describe the uniform approximation properties of the uniformly continuous mappings between the pairs of Banach and, occasionally, metric spaces from various wide parameterised and non-parameterised classes of spaces with or without the local unconditional structure in a quantitative manner. The striking difference with the finite-dimensional setting is represented by the presence of Tsar'kov's phenomenon. Many tools in use are developed under the scope of our quasi-Euclidean approach. Its idea seems to be relatively natural in light of the compressed sensing and distortion phenomena.

We consider some fundamental generalized Mordell-Tornheim-Witten (MTW) zeta-function values along with their derivatives, and explore connections with multiple-zeta values (MZVs). To achieve these results, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory.

Our original motivation was to represent previously unresolved constructs such as Eulerian log-gamma integrals. Indeed, we are able to show that all such integrals belong to a vector space over an MTW basis, and we also present, for a substantial subset of this class, explicit closed-form expressions. In the process, we significantly extend methods for high-precision numerical computation of polylogarithms and their derivatives with respect to order. That said, the focus of our paper is the relation between MTW sums and classical polylogarithms. It is the adumbration of these relationships that makes the study significant.

The associated paper (with DH Bailey and RE Crandall) is at http://carmasite.newcastle.edu.au/jon/MTW1.pdf.

Brian Alspach will continue his discussion "The Anatomy Of A Famous Conjecture."

This talk will discuss opportunities and challenges related to the development and application of operations research techniques to transportation and logistics problems in non-profit settings. Much research has been conducted on transportation and logistics problems in commercial settings where the goal is either to maximize profit or to minimize cost. Significantly less work has been conducted for non-profit applications. In such settings, the objectives are often more difficult to quantify since issues such as equity and sustainability must be considered, yet efficient operations are still crucial. This talk will present several research projects that introduce new approaches tailored to the objectives and constraints unique to non-profit agencies, which are often concerned with obtaining equitable solutions given limited, and often uncertain, budgets, rather than with maximizing profits.

This talk will assess the potential of operations research to address the problems faced by non-profit agencies and attempt to understand why these problems have been understudied within the operations research community. To do so, we will ask the following questions: Are non-profit operations problems rich enough for academic study? and Are solutions to non-profit operations problems applicable to real communities?

Brian Alspach will continue his discussion "The Anatomy Of A Famous Conjecture."

This talk will survey some of the classical and recent results concerning operators composed of a projection onto a compact set in time, followed by a projection onto a compact set in frequency. Such "time- and band-limiting" operators were studied by Landau, Slepian, and Pollak in a series of papers published in the Bell Systems Tech. Journal in the early 1960s identifying the eigenfunctions, providing eigenvalue estimates, and describing spaces of "essentially time- and band-limited signals."

Further progress on time- and band-limiting has been intermittent, but genuine recent progress has been made in terms of numerical analysis, sampling theory, and extensions to multiband signals, all driven to some extent by potential applications in communications. After providing an outline of the historical developments in the mathematical theory of time- and bandlimiting, some details of the sampling theory and multiband setting will be given. Part of the latter represents joint work with Jeff Hogan and Scott Izu.

Brian Alspach will continue his discussion "The Anatomy Of A Famous Conjecture."

This involves (in pre-nonstandard analysis times) the development of a simple system of infinites and infinitesmals that help to clarify Cantor's Ternary Set, nonmeasurable sets and Lebesgue integration. The talk will include other memories as a maths student at Newcastle University College, Tighes Hill, from 1959 to 1961.

This week Brian Alspach concludes his series of talks entitled "The Anatomy Of A Famous Conjecture." We shall be in V27 - note room change.

A graph on v vertices is called pancyclic if it contains cycles of every length from 3 to v. Obviously such graphs exist — the complete graph on v vertices is an example. We shall look at the question, what is the minimum number of edges in a pancyclic graph? Interestingly, this question was "solved", incorrectly, in 1978. A complete solution is not yet known.

This week the speaker in the Discrete Mathematics Instructional Seminar is Judy-anne Osborn who will be discussing Hadamard matrices.

There is a high prevalence of tuberculosis (TB) in Papua New Guinea (PNG), which is exacerbated by the presence of drug-resistant TB strains and HIV infection. This is an important public health issue not only locally within PNG, but also in Australia due to the high cross-border traffic in the Torres Strait Island–Western Province (PNG) treaty region. A metapopulation model is used to evaluate the effect of varying control strategies in the region, and some initial cost-benefit analysis figures are presented.

The double zeta values are one natural way to generalise the Riemann zeta function at the positive integers; they are defined by $\zeta(a,b) = \sum_{n=1}^\infty \sum_{m=1}^{n-1} 1/n^a/m^b$. We give a unified and completely elementary method to prove several sum formulae for the double zeta values. We also discuss an experimental method for discovering such formulae.

Moreover, we use a reflection formula and recursions involving the Riemann zeta function to obtain new relations of closely related functions, such as the Witten zeta function, alternating double zeta values, and more generally, character sums.

TBA

This week the speaker in the Discrete Mathematics Instructional Seminar is Judy-anne Osborn who will be discussing Hadamard matrices.

I will give a brief introduction to the theory of self-similar groups, focusing on a couple of pertinent examples: Grigorchuk's group of intermediate growth, and the basilica group.

Based on generalized backward shift operators we introduce adaptive Fourier decomposition. Then we discuss its relations and applications to (i) system identification; (2) computation of Hilbert transform; (3) algorithm for the best order-n rational approximation to functions in the Hardy space H2; (4) forward and backward shift invariant spaces; (5) band preserving in filter designing; (6) phase retrieving; and (7) the Bedrosian identity. The talk also concerns possible generalizations of the theory and applications to higher dimensional spaces.

The Douglas-Rachford algorithm is an iterative method for finding a point in the intersection of two (or more) closed sets. It is well-known that the iteration (weakly) converges when it is applied to convex subsets of a Hilbert space. Despite the absence of a theoretical justification, the algorithm has also been successfully applied to various non-convex practical problems, including finding solutions for the eight queens problem, or sudoku puzzles. In particular, we will show how these two problems can be easily modelled.

With the aim providing some theoretical explanation of the convergence in the non-convex case, we have established a region of convergence for the prototypical non-convex Douglas-Rachford iteration which finds a point on the intersection of a line and a circle. Previous work was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given.

PS: Bring your hardest sudoku puzzle :)

• 10 am -- Rachel Bunder
• 11 am -- Simranjit Kaur
• 12 noon -- Reena Kapoor

A body moves in a rarefied medium of resting particles and at the same time very slowly rotates (somersaults). Each particle of the medium is reflected elastically when hitting the body boundary (multiple reflections are possible). The resulting resistance force acting on the body depends on the time; we are interested in minimizing the time-averaged value of resistance (which is called $R$). The value $R(B)$ is well defined in terms of billiard in the complement of $B$, for any bounded body $B \subset \mathbb{R}^d$, $d\geq 2$ with piecewise smooth boundary.

Let $C\subset\mathbb{R}^d$ be a bounded convex body and $C_1\subset C$ be another convex body with $\partial C_1 \cap C=\varnothing$. It would be interesting to get an estimate for $$R(C1_,C)= \inf_{C_1\subset B \subset C} R(B) .................. (1)$$ If $\partial C_1$ is close to $\partial C$, problem (1) can be referred to as minimizing the resistance of the convex body $C$ by "roughening" its surface. We cannot solve problem (1); however we can find the limit $$\lim_{\text{dist}(\partial C_1,\partial C)\rightarrow 0} \frac{R(C_1,C)}{R(C)}. .................. (2) $$

It will be explained that problem (2) can be solved by reduction to a special problem of optimal mass transportation, where the initial and final measurable spaces are complementary hemispheres, $X=\{x=(x_1,...,x_d)\in S^{d-1}: x_1\geq 0\}$ and $Y=\{x\in S^{d-1}:x_1\leq 0\}$. The transportation cost is the squared distance, $c(x,y)=\frac{1}{2}|x-y|^2$, and the measures in $X$ and $Y$ are obtained from the $(d-1)$-dimensional Lebesgue measure on the equatorial circle $\{x=(x_1,...,x_d):|x|\leq 1,x_1=0\}$ by parallel translation along the vector $e_1=(1,0,...,0)$. Let $C(\nu)$ be the total cost corresponding to the transport plan $\nu$ and let $\nu_0$ be the transport plan generated by parallel translation along $e_1$; then the value $\frac{\inf C(\nu)}{C(\nu_0)}$ coincides with the limit in (2).

Surprisingly, this limit does not depend on the body $C$ and depends only on the dimension $d$.

In particular, if $d=3$ ($d=2$), it equals (approximately) 0.96945 (0.98782). In other words, the resistance of a 3-dimensional (2-dimensional) convex body can be decreased by 3.05% (correspondingly, 1.22%) at most by roughening its surface.

Motivated by questions of algorithm analysis, we provide several distinct approaches to determining convergence and limit values for a class of linear iterations.

This is joint work with D. Borwein and B. Sims.

We consider the bipartite version of the degree/diameter problem; namely, find the maximum number Nb(d,D) of vertices in a bipartite graph of maximum degree d>2 and diameter D>2. The actual value of Nb(d,D) is still unknown for most (d,D) pairs.

The well-known Moore bound Mb(d,D) gives a general upper bound for Nb(d,D); graphs attaining this bound are called Moore (bipartite) graphs. Moore bipartite graphs are very scarce; they may only exist for D=3,4 or 6, but no other diameters. Interest has then shifted to investigate the existence or otherwise of graphs missing the Moore bound by a few vertices. A graph with order Mb(d,D)-e is called a graph of defect e.

It has been proved that bipartite graphs of defect 2 do not exist when D>3. In our paper we 'almost' prove that bipartite graphs of defect 4 cannot exist when D>4, thereby establishing a new upper bound on Nb(d,D) for more than 2/3 of all (d,D) combinations.

We present a nonconvex bundle technique where function and subgradient values are available only up to an error tolerance which remains unknown to the user. The challenge is to develop an algorithm which converges to an approximate solution which, despite the lack of information, is as good as one can hope for. For instance, if data are known up to the error $O(\epsilon)$, the solution should also be accurate up to $O(\epsilon)$. We show that the oracle of downshifted tangents is an excellent tool to deal with this difficult situation.

Dr Koerber will speak about the experience of using MapleTA extensively in undergraduate teaching at the University of Adelaide, and demonstrate how they have been using the system there. Bio: Adrian Koerber is Director of First Year Studies in Mathematics at the University of Adelaide. His mathematical research is in the area of modelling gene networks.

We consider the bipartite version of the degree/diameter problem; namely, find the maximum number Nb(d,D) of vertices in a bipartite graph of maximum degree d>2 and diameter D>2. The actual value of Nb(d,D) is still unknown for most (d,D) pairs.

The well-known Moore bound Mb(d,D) gives a general upper bound for Nb(d,D); graphs attaining this bound are called Moore (bipartite) graphs. Moore bipartite graphs are very scarce; they may only exist for D=3,4 or 6, but no other diameters. Interest has then shifted to investigate the existence or otherwise of graphs missing the Moore bound by a few vertices. A graph with order Mb(d,D)-e is called a graph of defect e.

It has been proved that bipartite graphs of defect 2 do not exist when D>3. In our paper we 'almost' prove that bipartite graphs of defect 4 cannot exist when D>4, thereby establishing a new upper bound on Nb(d,D) for more than 2/3 of all (d,D) combinations.

Snarks are 3-regular graphs that are not 3-edge-colourable and are cyclically 4-edge-connected. They exist but are hard to find. On the other hand, it is believed that Cayley graphs can never be snarks. The latter is the subject of the next series of talks.

Hajek proved that a WUR Banach space is an Asplund space. This result suggests that the WUR property might have interesting consequences as a dual property. We show that

(i) every Banach Space with separable second dual can be equivalently renormed to have WUR dual,

(ii) under certain embedding conditions a Banach space with WUR dual is reflexive.

Snarks are 3-regular graphs that are not 3-edge-colourable and are cyclically 4-edge-connected. They exist but are hard to find. On the other hand, it is believed that Cayley graphs can never be snarks. The latter is the subject of the next series of talks.

Snarks are 3-regular graphs that are not 3-edge-colourable and are cyclically 4-edge-connected. They exist but are hard to find. On the other hand, it is believed that Cayley graphs can never be snarks. The latter is the subject of the next series of talks.

Let $F(z)$ be a power series, say with integer coefficients. In the late 1920s and early 1930s, Kurt Mahler discovered that for $F(z)$ satisfying a certain type of functional equation (now called Mahler functions), the transcendence of the function $F(z)$ could be used to prove the transcendence of certain special values of $F(z)$. Mahler's main application at the time was to prove the transcendence of the Thue-Morse number $\sum_{n\geq 0}t(n)/2^n$ where $t(n)$ is either 0 or 1 depending on the parity of the number of 1s in the base 2 expansion of $n$. In this talk, I will talk about some of the connections between Mahler functions and finite automata and highlight some recent approaches to large problems in the area. If time permits, I will outline a new proof of a version of Carlson's theorem for Mahler functions; that is, a Mahler function is either rational or it has the unit circle as a natural boundary.

(Joint speakers, Jon Borwein and Michael Rose)

Burnside's Theorem characterising transitive permutation groups of prime degree has some wonderful applications for graphs. This week we start an exploration of this topic.

We are holding an afternoon mini-conference, in conjunction with the School of Mathematical and Physical Sciences.

If you are engaged in any of the many Outreach Activities in the Mathematical Sciences that people from CARMA, our School and beyond contribute to, for example visiting primary or secondary schools, presenting to schools who visit us, public lectures, media interviews, helping run maths competitions, etc etc, and would like to share what you're doing, please let us know. Also if you're not currently engaged in an outreach activity but have an idea that you would like to try, and want to use a talk about your idea as a "sounding board", please feel free to do so.

There will be some very short talks: 5 minutes, and some longer talks: 20 minutes, with time for discussion in between. We'll be serving afternoon tea throughout the afternoon; and will have an open discussion forum near the end of the day. If you're interested in giving a talk please contact Judy-anne.Osborn@newcastle.edu.au, indicating whether you'd prefer a 5-minute or a 20-minute slot. If you're simply interested in attending, please let us know as well for catering purposes. The event will be held in one of the function rooms in the Shortland building.

12:05 — Begin, with welcome and lunch
15:45 — Last talk finishes
15:45-16:15 — Open discussion

George is going to start giving some talks on the embedding of Higman-Thompson groups into the groups of almost automorphisms of trees, and over several weeks will look at the structure theorems and scale calculations for these examples.

We shall continue exploring implications of Burnside's Theorem for vertex-transitive graphs.

Groundwater makes up nearly 30% of the entire world’s freshwater but the mathematical models for the better understanding of the system are difficult to validate due to the disordered nature of the porous media and the complex geometry of the channels of flow. In this seminar, after establishing the statistical macroscopic equivalent of the Navier-Stokes equations for the groundwater hydrodynamic and its consequences in term of Laplace and diffusion equations, some cases will be solved in term of special functions by using the modern Computer Algebra System.

Variational methods have been used to derive symmetric solutions for many problems related to real world applications. To name a few we mention periodic solutions to ODEs related to N-body problems and electrical circuits, symmetric solutions to PDEs, and symmetry in derivatives of spectral functions. In this talk we examine the commonalities of using variational methods in the presence of symmetry.

This is an ongoing collaborative research project with Jon Borwein. So far our questions still outnumber our answers.

George is going to continue his series of talks on the embedding of Higman-Thompson groups into the groups of almost automorphisms of trees, and over several weeks look at the structure theorems and scale calculations for these examples.

We shall continue exploring implications of Burnside's Theorem for vertex-transitive graphs.

Given a positive integer b, we say that a mathematical constant alpha is "b-normal" or "normal base b" if every m-long string of digits appears in the base-b expansion of alpha with precisely the limiting frequency 1/b^m. Although it is well known from measure theory that almost all real numbers are b-normal for all integers b > 1, nonetheless proving normality (or nonnormality) for specific constants, such as pi, e and log(2), has been very difficult.

In the 21st century, a number of different approaches have been attempted on this problem. For example, a recent study employed a Poisson model of normality to conclude that based on the first four trillion hexadecimal digits of pi, it is exceedingly unlikely that pi is not normal. In a similar vein, graphical techniques, in most cases based on the digit-generated "random" walks, have been successfully employed to detect certain nonnormality in some cases.

On the analytical front, it was shown in 2001 that the normality of certain reals, including log(2) and pi (or any other constant given by a BBP formula), could be reduced to a question about the behavior of certain specific pseudorandom number generators. Subsequently normality was established for an uncountable class of reals (the "Stoneham numbers"), the simplest of which is: alpha_{2,3} = Sum_{n >= 0} 1/(3^n 2^(3^n)), which is provably normal base 2. Just as intriguing is a recent result that alpha_{2,3}, for instance, is provably NOT normal base 6. These results have now been generalized to some extent, although many open cases remain.

In this talk I will present an introduction to the theory of normal numbers, including brief mention of new graphical- and statistical-based techniques. I will then sketch a proof of the normality base 2 (and nonnormality base 6) of Stoneham numbers, then suggest some additional lines of research. Various parts of this research were conducted in collaboration with Richard Crandall, Jonathan and Peter Borwein, Francisco Aragon, Cristian Calude, Michael Dinneen, Monica Dumitrescu and Alex Yee.

A frequent theme of 21st century experimental math is the computer discovery of identities, typically done by means of computing some mathematical entity (a sum, limit, integral, etc) to very high numeric precision, then using the PSLQ algorithm to identify the entity in terms of well known constants.

Perhaps the most successful application of this methodology has been to identify integrals arising in mathematical physics. This talk will present numerous examples of this type, including integrals from quantum field theory, Ising theory, random walks, 3D lattice problems, and even mouse brains. In some cases, it is necessary to compute these integrals to 3000-digit precision, and developing techniques to do such computations is a daunting technical challenge.

George continues his series of talks on the embedding of Higman-Thompson groups into the groups of almost automorphisms of trees, structure theorems and scale calculations for these examples.

This week Brian Alspach will complete the discussion on Burnside's Theorem and vertex-transitive graphs of prime order.

Recently the Alternating Projection Algorithm was extended into CAT(0) spaces. We will look at this and also current work on extending the Douglas Rachford Algorithm into CAT(0) spaces. By using CAT(0) spaces the underlying linear structure of the space is dispensable and this allows certain algorithms to be extended to spaces such as classical hyperbolic spaces, simply connected Riemannian manifolds of non-positive curvature, R-trees and Euclidean buildings.

In this talk, we study the properties of integral functionals induced on $L_\text{E}^1(S,\mu)$ by closed convex functions on a Euclidean space E. We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We show that in this generality functions such as the Boltzmann-Shannon entropy and the Fermi-Dirac entropy are strongly rotund. We also study convergence in measure and give various limiting counter-example.

This is joint work with Jon Borwein.

George continues his series of talks on the embedding of Higman-Thompson groups into the groups of almost automorphisms of trees, structure theorems and scale calculations for these examples.

I will discuss four much abused words Interdisciplinarity, Innovation, Collaboration and Creativity. I will describe what they mean for different stakeholder groups and will speak about my own experiences as a research scientist, as a scientific administrator, as an educator and even as a small high-tech businessman. I will also offer advice that can of course be ignored.

Linear Water Wave theory is one of the most important branches on fluid mechanics. Practically it underpins most of the engineering design of ships, offshore structures, etc. It also has a very rich history in the development of applied mathematics. In this talk I will focus on the connection between solutions in the frequency and time-domains and show how we can use various formulations to make numerical calculations and to construct approximate solutions. I will illustrate these methods with application to some simple wave scattering problems.

We consider the problem of characterising embeddings of an abstract group into totally disconnected locally compact (tdlc) groups. Specifically, for each pair of nonzero integers $m,n$ we construct a tdlc group containing the Baumslag-Solitar group $BS(m,n)$ as a dense subgroup, and compute the scales of elements and flat rank of the tdlc group.

This is joint work with George Willis.

In this talk, we study the properties of integral functionals induced on the Banach space of integrable functions by closed convex functions on a Euclidean space.

We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We show that in this generality functions such as the Boltzmann-Shannon entropy and the Fermi-Dirac entropy are strongly rotund. We also study convergence in measure and give various limiting counter-example.

In this talk projection algorithms for solving (nonconvex) feasibility problems in Euclidian spaces are considered. Of special interest are the Method of Alternating Projections (MAP) and the Averaged Alternating Reflection Algorithm (AAR) which cover some of the state of the art algorithms for our intended application, the phase retrieval problem. In the case of convex feasibility, firm nonexpansiveness of projection mappings is a global property that yields global convergence of MAP, and, for consistent problems, AAR. Based on epsilon-delta-regularity of sets (Bauschke, Luke, Phan, Wang 2012) a relaxed local version of firm nonexpansiveness with respect to the intersection is introduced for consistent feasibility problems. This combined with a type of coercivity condition, which relates to the regularity of the intersection, yields local linear convergence of MAP for a wide class of nonconvex problems, and even local linear convergence of AAR in more limited nonconvex settings.

If some arithmetical sums are small then the complex zeroes of the zeta-function are linearly dependent. Since we don't believe the conclusion we ought not to believe the premise. I will show that the zeroes are 'almost linearly independent' which implies, in particular, that the Mertens conjecture fails more drastically than was previously known.

In this talk, we will show that a D-finite Mahler function is necessarily rational. This gives a new proof of the rational-transcendental dichotomy of Mahler functions due to Nishioka. Using our method of proof, we also provide a new proof of a Pólya-Carlson type result for Mahler functions due to Randé; that is, a Mahler function which is meromorphic in the unit disk is either rational or has the unit circle as a natural boundary. This is joint work with Jason Bell and Eric Rowland.

In 1966 Gallai conjectured that a connected graph of order n can be decomposed into n/2 or fewer paths when n is even, or (n+1)/2 or fewer paths when n is odd. We shall discuss old and new work on this as yet unsolved conjecture.

Motivated by the desire to visualise large mathematical data sets, especially in number theory, we offer various tools for representing floating point numbers as planar walks and for quantitatively measuring their “randomness”.

What to expect: some interesting ideas, many beautiful pictures (including a 108-gigapixel picture of π), and some easy-to-understand maths.
What you won’t get: too many equations, difficult proofs, or any “real walking”.

This is a joint work with David Bailey, Jon Borwein and Peter Borwein.

Many cognitive models derive their predictions through simulation. This means that it is difficult or impossible to write down a probability distribution or likelihood that characterizes the random behavior of the data as a function of the model's parameters. In turn, the lack of a likelihood means that standard Bayesian analyses of such models are impossible. In this presentation we demonstrate a procedure called approximate Bayesian computation (ABC), a method for Bayesian analysis that circumvents the evaluation of the likelihood. Although they have shown great promise for likelihood-free inference, current ABC methods suffer from two problems that have largely revented their mainstream adoption: long computation time and an inability to scale beyond models with few parameters. We introduce a new ABC algorithm, called ABCDE, that includes differential evolution as a computationally efficient genetic algorithm for proposal generation. ABCDE is able to obtain accurate posterior estimates an order of magnitude faster than a popular rejection-based method and scale to high-dimensional parameter spaces that have proven difficult for the current rejection-based ABC methods. To illustrate its utility we apply ABCDE to several well-established simulation-based models of memory and decision-making that have never been fit in a Bayesian framework.

AUTHORS: Brandon M. Turner (Stanford University) Per B. Sederberg (The Ohio State University)

In 1966 Gallai conjectured that a connected graph of order n can be decomposed into n/2 or fewer paths when n is even, or (n+1)/2 or fewer paths when n is odd. We shall discuss old and new work on this as yet unsolved conjecture.

We discuss how the title is related to π.

I will give an extended version of my talk at the AustMS meeting about some ongoing work with Pierre-Emmanuel Caprace and George Willis.

Given a locally compact topological group G, the connected component of the identity is a closed normal subgroup G_0 and the quotient group is totally disconnected. Connected locally compact groups can be approximated by Lie groups, and as such are relatively well-understood. By contrast, totally disconnected locally compact (t.d.l.c.) groups are a more difficult class of objects to understand. Unlike in the connected case, it is probably hopeless to classify the simple t.d.l.c. groups, because this would include for instance all simple groups (equipped with the discrete topology). Even classifying the finitely generated simple groups is widely regarded as impossible. However, we can prove some general results about broad classes of (topologically) simple t.d.l.c. groups that have a compact generating set.

Given a non-discrete t.d.l.c. group, there is always an open compact subgroup. Compact totally disconnected groups are residually finite, so have many normal subgroups. Our approach is to analyse a t.d.l.c. group G (which may itself be simple) via normal subgroups of open compact subgroups. From these we obtain lattices and Cantor sets on which G acts, and we can use properties of these actions to demonstrate properties of G. For instance, we have made some progress on the question of whether a compactly generated topologically simple t.d.l.c. group is abstractly simple, and found some necessary conditions for G to be amenable.

We study the problem of finding an interpolating curve passing through prescribed points in the Euclidean space. The interpolating curve minimizes the pointwise maximum length, i.e., L∞-norm, of its acceleration. We re-formulate the problem as an optimal control problem and employ simple but effective tools of optimal control theory. We characterize solutions associated with singular (of infinite order) and nonsingular controls. We reduce the infinite dimensional interpolation problem to an ensuing finite dimensional one and derive closed form expressions for interpolating curves. Consequently we devise numerical techniques for finding interpolating curves and illustrate these techniques on examples.

Infecting aedes aegypti with Wolbachia has been proposed as an alternative in reducing dengue transmission. If Wolbachia-infected mosquitoes can invade and dominate the population of aedes aegypti mosquitoes, they can reduce dengue transmission. Cytoplasmic Incompatibility (CI) provides the reproductive advantage for Wolbachia-infected mosquitoes with which they can reproduce more and dominate the population. A mosquito population model is developed in order to determine the survival of Wolbachia-infected mosquiotes when they are released into the wild. The model has two physically stable realistic steady states. The model reveals that once the Wolbachia-infected mosquitoes survive, they ultimately dominate the population.

Giuga's conjecture will be introduced, and we will discuss what's changed in the computation of a counterexample in the last 17 years.

Automata groups are a class of groups generated by recursively defined automorphisms of a regular rooted tree. Associated to each automata group is an object known as the self-similarity graph. Nekrashevych showed that in the case where the group satisfies a natural condition known as contracting, the self-similarity graph is Gromov-hyperbolic and has boundary homeomorphic to the limit space of the group action. I will talk about self-similarity graphs of automata groups that do not satisfy the contracting condition.

In this talk, we present our ongoing efforts in solving a number of continuous facility location problems that involve sets using recently developed tools of variational analysis and generalized differentiation. Subgradients of a class of nonsmooth functions called minimal time functions are developed and employed to study these problems. Our approach advances the applications of variational analysis and optimization to a well-developed field of facility location, while shedding new light on well-known classical geometry problems such as the Fermat-Torricelli problem, the Sylvester smallest enclosing circle problem, and the problem of Apollonius.

I will discuss a new algorithm for counting points on hyperelliptic curves over finite fields.

This talk is an introduction to symbolic convex analysis.

Multi-linear functions appear in many global optimization problems, including reformulated quadratic and polynomial optimization problems. There is a extended formulation for the convex hull of the graph of a multi-linear function that requires the use of an exponential number of variables. Relying on this result, we study an approach that generates relaxations for multiple terms simultaneously, as opposed to methods that relax the nonconvexity of each term individually. In some special cases, we are able to establish analytic bounds on the ratio of the strength of the term-by-term and convex hull relaxations. To our knowledge, these are the first approximation-ratio results for the strength of relaxations of global optimization problems. The results lend insight into the design of practical (non-exponentially sized) relaxations. Computations demonstrate that the bounds obtained in this manner are competitive with the well-known semi-definite programming based bounds for these problems.

Joint work with Jim Luedtke, University of Wisconsin-Madison, and Mahdi Namazifar, now with Opera Solutions.

Nonexpansive operators in Banach spaces are of utmost importance in Nonlinear Analysis and Optimization Theory. We are concerned in this talk with classes of operators which are, in some sense, nonexpansive not with respect to the norm, but with respect to Bregman distances. Since these distances are not symmetric in general, it seems natural to distinguish between left and right Bregman nonexpansive operators. Some left classes have already been studied quite intensively, so this talk is mainly devoted to right Bregman nonexpansive operators and the relationship between both classes.

This talk is based on joint works with Prof. Simeon Reich and Shoham Sabach from Technion-Israel Institute of Technology, Haifa.