Some generalization of Desargues and Veronese configurations

2000 Mathematics Subject Classification: 51E14, 51E30.

Development of rapid and highly resolving combinatorial genotyping schemes for Campylobacter jejuni and Campylobacter coli

Campylobacter jejuni followed by Campylobacter coli contribute substantially to the economic and public health burden attributed to food-borne infections in Australia. Genotypic characterisation of isolates has provided new insights into the epidemiology and pathogenesis of C. jejuni and C. coli. However, currently available methods are not conducive to large scale epidemiological investigations that are necessary to elucidate the global epidemiology of these common food-borne pathogens. This research aims to develop high resolution C. jejuni and C. coli genotyping schemes that are convenient for high throughput applications. Real-time PCR and High Resolution Melt (HRM) analysis are fundamental to the genotyping schemes developed in this study and enable rapid, cost effective, interrogation of a range of different polymorphic sites within the Campylobacter genome. While the sources and routes of transmission of campylobacters are unclear, handling and consumption of poultry meat is frequently associated with human campylobacteriosis in Australia. Therefore, chicken derived C. jejuni and C. coli isolates were used to develop and verify the methods described in this study. The first aim of this study describes the application of MLST-SNP (Multi Locus Sequence Typing Single Nucleotide Polymorphisms) + binary typing to 87 chicken C. jejuni isolates using real-time PCR analysis. These typing schemes were developed previously by our research group using isolates from campylobacteriosis patients. This present study showed that SNP + binary typing alone or in combination are effective at detecting epidemiological linkage between chicken derived Campylobacter isolates and enable data comparisons with other MLST based investigations. SNP + binary types obtained from chicken isolates in this study were compared with a previously SNP + binary and MLST typed set of human isolates. Common genotypes between the two collections of isolates were identified and ST-524 represented a clone that could be worth monitoring in the chicken meat industry. In contrast, ST-48, mainly associated with bovine hosts, was abundant in the human isolates. This genotype was, however, absent in the chicken isolates, indicating the role of non-poultry sources in causing human Campylobacter infections. This demonstrates the potential application of SNP + binary typing for epidemiological investigations and source tracing. While MLST SNPs and binary genes comprise the more stable backbone of the Campylobacter genome and are indicative of long term epidemiological linkage of the isolates, the development of a High Resolution Melt (HRM) based curve analysis method to interrogate the hypervariable Campylobacter flagellin encoding gene (flaA) is described in Aim 2 of this study. The flaA gene product appears to be an important pathogenicity determinant of campylobacters and is therefore a popular target for genotyping, especially for short term epidemiological studies such as outbreak investigations. HRM curve analysis based flaA interrogation is a single-step closed-tube method that provides portable data that can be easily shared and accessed. Critical to the development of flaA HRM was the use of flaA specific primers that did not amplify the flaB gene. HRM curve analysis flaA interrogation was successful at discriminating the 47 sequence variants identified within the 87 C. jejuni and 15 C. coli isolates and correlated to the epidemiological background of the isolates. In the combinatorial format, the resolving power of flaA was additive to that of SNP + binary typing and CRISPR (Clustered regularly spaced short Palindromic repeats) HRM and fits the PHRANA (Progressive hierarchical resolving assays using nucleic acids) approach for genotyping. The use of statistical methods to analyse the HRM data enhanced sophistication of the method. Therefore, flaA HRM is a rapid and cost effective alternative to gel- or sequence-based flaA typing schemes. Aim 3 of this study describes the development of a novel bioinformatics driven method to interrogate Campylobacter MLST gene fragments using HRM, and is called ‘SNP Nucleated Minim MLST’ or ‘Minim typing’. The method involves HRM interrogation of MLST fragments that encompass highly informative “Nucleating SNPS” to ensure high resolution. Selection of fragments potentially suited to HRM analysis was conducted in silico using i) “Minimum SNPs” and ii) the new ’HRMtype’ software packages. Species specific sets of six “Nucleating SNPs” and six HRM fragments were identified for both C. jejuni and C. coli to ensure high typeability and resolution relevant to the MLST database. ‘Minim typing’ was tested empirically by typing 15 C. jejuni and five C. coli isolates. The association of clonal complexes (CC) to each isolate by ‘Minim typing’ and SNP + binary typing were used to compare the two MLST interrogation schemes. The CCs linked with each C. jejuni isolate were consistent for both methods. Thus, ‘Minim typing’ is an efficient and cost effective method to interrogate MLST genes. However, it is not expected to be independent, or meet the resolution of, sequence based MLST gene interrogation. ‘Minim typing’ in combination with flaA HRM is envisaged to comprise a highly resolving combinatorial typing scheme developed around the HRM platform and is amenable to automation and multiplexing. The genotyping techniques described in this thesis involve the combinatorial interrogation of differentially evolving genetic markers on the unified real-time PCR and HRM platform. They provide high resolution and are simple, cost effective and ideally suited to rapid and high throughput genotyping for these common food-borne pathogens.

M-ary trees for combinatorial asset management decision problems

A novel m-ary tree based approach is presented to solve asset management decisions which are combinatorial in nature. The approach introduces a new dynamic constraint based control mechanism which is capable of excluding infeasible solutions from the solution space. The approach also provides a solution to the challenges with ordering of assets decisions.

Non-combinatorial library screening reveals subsite cooperativity and identifies new high-efficiency substrates for kallikrein-related peptidase 14

An array of substrates link the tryptic serine protease, kallikrein-related peptidase 14 (KLK14), to physiological functions including desquamation and activation of signaling molecules associated with inflammation and cancer. Recognition of protease cleavage sequences is driven by complementarity between exposed substrate motifs and the physicochemical signature of an enzyme's active site cleft. However, conventional substrate screening methods have generated conflicting subsite profiles for KLK14. This study utilizes a recently developed screening technique, the sparse matrix library, to identify five novel high-efficiency sequences for KLK14. The optimal sequence, YASR, was cleaved with higher efficiency (k(cat)/K(m)=3.81 ± 0.4 × 10(6) M(-1) s(-1)) than favored substrates from positional scanning and phage display by 2- and 10-fold, respectively. Binding site cooperativity was prominent among preferred sequences, which enabled optimal interaction at all subsites as indicated by predictive modeling of KLK14/substrate complexes. These simulations constitute the first molecular dynamics analysis of KLK14 and offer a structural rationale for the divergent subsite preferences evident between KLK14 and closely related KLKs, KLK4 and KLK5. Collectively, these findings highlight the importance of binding site cooperativity in protease substrate recognition, which has implications for discovery of optimal substrates and engineering highly effective protease inhibitors.

Combinatorial design of key distribution mechanisms for wireless sensor networks

Secure communications in wireless sensor networks operating under adversarial conditions require providing pairwise (symmetric) keys to sensor nodes. In large scale deployment scenarios, there is no prior knowledge of post deployment network configuration since nodes may be randomly scattered over a hostile territory. Thus, shared keys must be distributed before deployment to provide each node a key-chain. For large sensor networks it is infeasible to store a unique key for all other nodes in the key-chain of a sensor node. Consequently, for secure communication either two nodes have a key in common in their key-chains and they have a wireless link between them, or there is a path, called key-path, among these two nodes where each pair of neighboring nodes on this path have a key in common. Length of the key-path is the key factor for efficiency of the design. This paper presents novel deterministic and hybrid approaches based on Combinatorial Design for deciding how many and which keys to assign to each key-chain before the sensor network deployment. In particular, Balanced Incomplete Block Designs (BIBD) and Generalized Quadrangles (GQ) are mapped to obtain efficient key distribution schemes. Performance and security properties of the proposed schemes are studied both analytically and computationally. Comparison to related work shows that the combinatorial approach produces better connectivity with smaller key-chain sizes.

Combinatorial design of key distribution mechanisms for wireless sensor networks

Key distribution is one of the most challenging security issues in wireless sensor networks where sensor nodes are randomly scattered over a hostile territory. In such a sensor deployment scenario, there will be no prior knowledge of post deployment configuration. For security solutions requiring pairwise keys, it is impossible to decide how to distribute key pairs to sensor nodes before the deployment. Existing approaches to this problem are to assign more than one key, namely a key-chain, to each node. Key-chains are randomly drawn from a key-pool. Either two neighboring nodes have a key in common in their key-chains, or there is a path, called key-path, among these two nodes where each pair of neighboring nodes on this path has a key in common. Problem in such a solution is to decide on the key-chain size and key-pool size so that every pair of nodes can establish a session key directly or through a path with high probability. The size of the key-path is the key factor for the efficiency of the design. This paper presents novel, deterministic and hybrid approaches based on Combinatorial Design for key distribution. In particular, several block design techniques are considered for generating the key-chains and the key-pools.

Combinatorial design of key distribution mechanisms for wireless sensor networks

Key distribution is one of the most challenging security issues in wireless sensor networks where sensor nodes are randomly scattered over a hostile territory. In such a sensor deployment scenario, there will be no prior knowledge of post deployment configuration. For security solutions requiring pair wise keys, it is impossible to decide how to distribute key pairs to sensor nodes before the deployment. Existing approaches to this problem are to assign more than one key, namely a key-chain, to each node. Key-chains are randomly drawn from a key-pool. Either two neighbouring nodes have a key in common in their key-chains, or there is a path, called key-path, among these two nodes where each pair of neighbouring nodes on this path has a key in common. Problem in such a solution is to decide on the key-chain size and key-pool size so that every pair of nodes can establish a session key directly or through a path with high probability. The size of the key-path is the key factor for the efficiency of the design. This paper presents novel, deterministic and hybrid approaches based on Combinatorial Design for key distribution. In particular, several block design techniques are considered for generating the key-chains and the key-pools. Comparison to probabilistic schemes shows that our combinatorial approach produces better connectivity with smaller key-chain sizes.

A combinatorial approach to anonymous membership broadcast

A set system (X, F ) with X= {x 1,...,x m}) and F = {B1...,B n }, where B i ⊆ X, is called an (n, m) cover-free set system (or CF set system) if for any 1 ≤ i, j, k ≤ n and j ≠ k, |B i >2 |B j ∩ B k | +1. In this paper, we show that CF set systems can be used to construct anonymous membership broadcast schemes (or AMB schemes), allowing a center to broadcast a secret identity among a set of users in a such way that the users can verify whether or not the broadcast message contains their valid identity. Our goal is to construct (n, m) CF set systems in which for given m the value n is as large as possible. We give two constructions for CF set systems, the first one from error-correcting codes and the other from combinatorial designs. We link CF set systems to the concept of cover-free family studied by Erdös et al in early 80’s to derive bounds on parameters of CF set systems. We also discuss some possible extensions of the current work, motivated by different application.

Plasma–surface interactions at nanoscales : a combinatorial theoretical, process diagnostics and surface microanalysis approach

This paper introduces an integral approach to the study of plasma-surface interactions during the catalytic growth of selected nanostructures (NSs). This approach involves basic understanding of the plasma-specific effects in NS nucleation and growth, theoretical modelling, numerical simulations, plasma diagnostics, and surface microanalysis. Using an example of plasma-assisted growth of surface-supported single-walled carbon nanotubes, we discuss how the combination of these techniques may help improve the outcomes of the growth process. A specific focus here is on the effects of nanoscale plasma-surface interactions on the NS growth and how the available techniques may be used, both in situ and ex situ to optimize the growth process and structural parameters of NSs.

Effect of elemental composition and size on electron confinement in self-assembled SiC quantum dots : a combinatorial approach

A high level of control over quantum dot (QD) properties such as size and composition during fabrication is required to precisely tune the eventual electronic properties of the QD. Nanoscale synthesis efforts and theoretical studies of electronic properties are traditionally treated quite separately. In this paper, a combinatorial approach has been taken to relate the process synthesis parameters and the electron confinement properties of the QDs. First, hybrid numerical calculations with different influx parameters for Si1-x Cx QDs were carried out to simulate the changes in carbon content x and size. Second, the ionization energy theory was applied to understand the electronic properties of Si1-x Cx QDs. Third, stoichiometric (x=0.5) silicon carbide QDs were grown by means of inductively coupled plasma-assisted rf magnetron sputtering. Finally, the effect of QD size and elemental composition were then incorporated in the ionization energy theory to explain the evolution of the Si1-x Cx photoluminescence spectra. These results are important for the development of deterministic synthesis approaches of self-assembled nanoscale quantum confinement structures.

Second-order quantile methods for experts and combinatorial games

We aim to design strategies for sequential decision making that adjust to the difficulty of the learning problem. We study this question both in the setting of prediction with expert advice, and for more general combinatorial decision tasks. We are not satisfied with just guaranteeing minimax regret rates, but we want our algorithms to perform significantly better on easy data. Two popular ways to formalize such adaptivity are second-order regret bounds and quantile bounds. The underlying notions of 'easy data', which may be paraphrased as "the learning problem has small variance" and "multiple decisions are useful", are synergetic. But even though there are sophisticated algorithms that exploit one of the two, no existing algorithm is able to adapt to both. In this paper we outline a new method for obtaining such adaptive algorithms, based on a potential function that aggregates a range of learning rates (which are essential tuning parameters). By choosing the right prior we construct efficient algorithms and show that they reap both benefits by proving the first bounds that are both second-order and incorporate quantiles.

On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Combinatorial Design of Secure Lightweight Data Dispersal Schemes (Work in Progress)

Dispersing a data object into a set of data shares is an elemental stage in distributed communication and storage systems. In comparison to data replication, data dispersal with redundancy saves space and bandwidth. Moreover, dispersing a data object to distinct communication links or storage sites limits adversarial access to whole data and tolerates loss of a part of data shares. Existing data dispersal schemes have been proposed mostly based on various mathematical transformations on the data which induce high computation overhead. This paper presents a novel data dispersal scheme where each part of a data object is replicated, without encoding, into a subset of data shares according to combinatorial design theory. Particularly, data parts are mapped to points and data shares are mapped to lines of a projective plane. Data parts are then distributed to data shares using the point and line incidence relations in the plane so that certain subsets of data shares collectively possess all data parts. The presented scheme incorporates combinatorial design theory with inseparability transformation to achieve secure data dispersal at reduced computation, communication and storage costs. Rigorous formal analysis and experimental study demonstrate significant cost-benefits of the presented scheme in comparison to existing methods.

A Combinatorial Construct for Keyless Message Dispersal (Work in Progress)

We propose a keyless and lightweight message transformation scheme based on the combinatorial design theory for the confidentiality of a message transmitted in multiple parts through a network with multiple independent paths, or for data stored in multiple parts by a set of independent storage services such as the cloud providers. Our combinatorial scheme disperses a message into v output parts so that (k-1) or less parts do not reveal any information about any message part, and the message can only be recovered by the party who possesses all v output parts. Combinatorial scheme generates an xor transformation structure to disperse the message into v output parts. Inversion is done by applying the same xor transformation structure on output parts. The structure is generated using generalized quadrangles from design theory which represents symmetric point and line incidence relations in a projective plane. We randomize our solution by adding a random salt value and dispersing it together with the message. We show that a passive adversary with capability of accessing (k-1) communication links or storage services has no advantage so that the scheme is indistinguishable under adaptive chosen ciphertext attack (IND-CCA2).

A Combinatorial Optimization Problem for High Order PODs with Few Sensors

Experimental characterization of high dimensional dynamic systems sometimes uses the proper orthogonal decomposition (POD). If there are many measurement locations and relatively fewer sensors, then steady-state behavior can still be studied by sequentially taking several sets of simultaneous measurements. The number required of such sets of measurements can be minimized if we solve a combinatorial optimization problem. We aim to bring this problem to the attention of engineering audiences, summarize some known mathematical results about this problem, and present a heuristic (suboptimal) calculation that gives reasonable, if not stellar, results.