Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods

Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym- metry consisting of only pentagons and hexagons faces. It has been the object of interest for chemists and mathematicians due to its widespread application in various fields, namely including electronic and optic engineering, medical sci- ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751 isomers. The Fries and Clar numbers are stability predictors of a Fullerene molecule. These number can be computed by solving a (possibly N P -hard) combinatorial optimization problem. We propose several ILP formulation of such a problem each yielding a solution algorithm that provides the exact value of the Fries and Clar numbers. We compare the performances of the algorithm derived from the proposed ILP formulations. One of this algorithm is used to find the Clar isomers, i.e., those for which the Clar number is maximum among all isomers having a given size. We repeated this computational experiment for all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774 isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path following (PDIPF) interior point (IP) algorithm for separable convex quadratic minimum cost flow network problem. In each iteration of PDIPF algorithm, the main computational effort is solving the underlying Newton search direction system. We concentrated on finding the solution of the corresponding linear system iteratively and inexactly. We assumed that all the involved inequalities can be solved inexactly and to this purpose, we focused on different approaches for distributing the error generated by iterative linear solvers such that the convergences of the PDIPF algorithm are guaranteed. As a result, we achieved theoretical bases that open the path to further interesting practical investiga- tion

Brownian Motion of Decaying Particles: Transition Probability, Computer Simulation, and First-Passage Times

Recent developments in the measurement of radioactive gases in passive diffusion motivate the analysis of Brownian motion of decaying particles, a subject that has received little previous attention. This paper reports the derivation and solution of equations comparable to the Fokker-Planck and Langevin equations for one-dimensional diffusion and decay of unstable particles. In marked contrast to the case of stable particles, the two equations are not equivalent, but provide different information regarding the same stochastic process. The differences arise because Brownian motion with particle decay is not a continuous process. The discontinuity is readily apparent in the computer-simulated trajectories of the Langevin equation that incorporate both a Wiener process for displacement fluctuations and a Bernoulli process for random decay. This paper also reports the derivation of the mean time of first passage of the decaying particle to absorbing boundaries. Here, too, particle decay can lead to an outcome markedly different from that for stable particles. In particular, the first-passage time of the decaying particle is always finite, whereas the time for a stable particle to reach a single absorbing boundary is theoretically infinite due to the heavy tail of the inverse Gaussian density. The methodology developed in this paper should prove useful in the investigation of radioactive gases, aerosols of radioactive atoms, dust particles to which adhere radioactive ions, as well as diffusing gases and liquids of unstable molecules

Computing the forcing spectrum of outerplanar graphs in polynomial time

The forcing number of a graph with a perfect matching $M$ is the minimum number of edges in $M$ whose endpoints need to be deleted, such that the remaining graph only has a single perfect matching. This number is of great interest in theoretical chemistry, since it conveys information about the structural properties of several interesting molecules. On the other hand, in bipartite graphs the forcing number corresponds to the famous feedback vertex set problem in digraphs. Determining the complexity of finding the smallest forcing number of a given planar graph is still a widely open and important question in this area, originally proposed by Afshani, Hatami, and Mahmoodian in 2004. We take a first step towards the resolution of this question by providing an algorithm that determines the set of all possible forcing numbers of an outerplanar graph in polynomial time. This is the first polynomial-time algorithm concerning this problem for a class of graphs of comparable or greater generality.Comment: 22 pages, 3 figure

Stochastic methods and models for neuronal activity and motor proteins

The main topic of this thesis is to specialize mathematical methods and construct stochastic models for describing and predict biological dynamics such as neuronal firing and acto-myosin sliding. The apparently twofold theme on which we focused the research is part of a strictly unitary context as the methods and tools of investigation adapt naturally to both issues. Models describing the stochastic evolution in the case of neuronal activity or that of the acto-myosin dynamics are governed by stochastic differential equations whose solutions are diffusion processes and Gauss-Markov processes. Of fundamental importance in the study of these phenomena is the determination of the density of the first passage times through one or two time-depending boundaries. For this purpose the development or use of numerical solution algorithms and the comparison of the results with those obtained by simulation algorithms is essential. A particular type of Gauss-Markov process (the time-inhomogeneous Ornstein-Uhlenbeck process) and its first passage time through suitable boundaries are fundamental for modeling phenomena subject to additional (external) time-dependent forces

Mathematical modelling of corneal epithelium maintenance and recovery

The cornea is the clear shield that covers the eye, estimated to contribute approximately two thirds of the eye’s optical power. Therefore, cornea protection is highly important and is achieved by maintaining its outermost layer, the corneal epithelium. Corneal epithelium maintenance depends on a peripheral population of stem cells, the limbal stem cells, that continuously replenish the basal epithelium layer through generating transient amplifying cells (TAC). TACs move from the periphery to the centre, undergoing several rounds of cell division, before terminal differentiation (TD). TD cells lose contact with the basal layer and move up through the epithelium until shed from the surface. A better understanding of corneal epithelium maintenance and recovery processes are crucial and may contribute to better treatments. We develop and analyse mathematical models to investigate these underlying complex biological processes with the aim of a better understanding and possibly prediction of what cannot be currently obtained from laboratories and clinical trials. The proliferation events are modeled by a stochastic mathematical model, based on an analogy to chemical reactions. Using this model we are aiming to: (i) clarify the main factors involved in the maintaining process; (ii) determine the constraints placed on the proliferation process for healthy maintenance; (iii) investigate robustness via noise analysis. We then consider the spatial migration of TACs from the corneal epithelium periphery to the centre. In the absence of no precise biological data on the exact mechanism governing centripetal migration of TAC cells, we consider two simple prototype models: (i) TACs simply migrate randomly, i.e. a diffusion type process; (ii) random migration with a centrally-directed bias, i.e. a diffusion-advection type process. We first use a first passage time approach to understand the constraints on the parameters for TAC cells to reach the corneal epithelial centre and then investigate prototypical models to investigate the mechanisms that account for TAC cell redistribution from the periphery to the centre. Behaviour of the models is investigated under wound healing scenarios, exploring whether different healing is observed according to model and position of the wound.Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016508/01Scottish Funding Counci

Developing reliable anomaly detection system for critical hosts: a proactive defense paradigm

Current host-based anomaly detection systems have limited accuracy and incur high processing costs. This is due to the need for processing massive audit data of the critical host(s) while detecting complex zero-day attacks which can leave minor, stealthy and dispersed artefacts. In this research study, this observation is validated using existing datasets and state-of-the-art algorithms related to the construction of the features of a host's audit data, such as the popular semantic-based extraction and decision engines, including Support Vector Machines, Extreme Learning Machines and Hidden Markov Models. There is a challenging trade-off between achieving accuracy with a minimum processing cost and processing massive amounts of audit data that can include complex attacks. Also, there is a lack of a realistic experimental dataset that reflects the normal and abnormal activities of current real-world computers. This thesis investigates the development of new methodologies for host-based anomaly detection systems with the specific aims of improving accuracy at a minimum processing cost while considering challenges such as complex attacks which, in some cases, can only be visible via a quantified computing resource, for example, the execution times of programs, the processing of massive amounts of audit data, the unavailability of a realistic experimental dataset and the automatic minimization of the false positive rate while dealing with the dynamics of normal activities. This study provides three original and significant contributions to this field of research which represent a marked advance in its body of knowledge. The first major contribution is the generation and release of a realistic intrusion detection systems dataset as well as the development of a metric based on fuzzy qualitative modeling for embedding the possible quality of realism in a dataset's design process and assessing this quality in existing or future datasets. The second key contribution is constructing and evaluating the hidden host features to identify the trivial differences between the normal and abnormal artefacts of hosts' activities at a minimum processing cost. Linux-centric features include the frequencies and ranges, frequency-domain representations and Gaussian interpretations of system call identifiers with execution times while, for Windows, a count of the distinct core Dynamic Linked Library calls is identified as a hidden host feature. The final key contribution is the development of two new anomaly-based statistical decision engines for capitalizing on the potential of some of the suggested hidden features and reliably detecting anomalies. The first engine, which has a forensic module, is based on stochastic theories including Hierarchical hidden Markov models and the second is modeled using Gaussian Mixture Modeling and Correntropy. The results demonstrate that the proposed host features and engines are competent for meeting the identified challenges

A hierarchical adaptive model for robust short-term visual tracking

Visual tracking is a topic in computer vision with applications in many emerging as well as established technological areas, such as robotics, video surveillance, human-computer interaction, autonomous vehicles, and sport analytics. The main question of visual tracking is how to design an algorithm (visual tracker) that determines the state of one or more objects in a stream of images by accounting for their sequential nature. In this doctoral thesis we address two important topics in single-target short-term visual tracking. The first topic is related to construction of an object appearance model for visual tracking. The modeling and updating of the appearance model is crucial for successful tracking. We introduce a hierarchical appearance model which structures object appearance in multiple layers. The bottom layer contains the most specific information and each higher layer models the appearance information in a more general way. The hierarchical relations are also reflected in the update process where the higher layers guide the lower layers in their update while the lower layers provide a source for adaptation to higher layers if their information is reliable. The benefits of hierarchical appearance models are demonstrated with two implementations, primarily designed to tackle tracking of non-rigid and articulated objects that present a challenge for many existing trackers. The first example of appearance model combines local and global visual information in a coupled-layer appearance model. The bottom layer contains a part-based appearance description that is able to adapt to the geometrical deformations of non-rigid targets and the top layer is a multi-modal global object appearance model that guides the model during object appearance changes. The experimental evaluation shows that the proposed coupled-layer appearance model excels in robustness despite the fact that is uses relatively simple appearance descriptors. Our evaluation also exposed several weaknesses that were reflected in a decreased accuracy. Our second presented appearance model extends the hierarchy by introducing the third layer and a concept of template anchors. The first two layers are conceptually similar to the original two-layer appearance model, while the third layer is a memory system that is composed of static templates that provide a strong spatial cue when one of the templates is matched to the image reliably, thus assisting in quick recovery of the entire appearance model. In the experimental evaluation we show that this addition indeed improves the accuracy, as well as the overall performance of a tracker. The second question that we are addressing is the performance evaluation of single-target short-term visual tracking algorithms. In contrast to the dominant trend in the past decades, we claim that visual tracking is a complex process and that the performance of visual trackers cannot be reduced to a single performance measure, nor should it be described by an arbitrary set of measures where the relationship between measures is not well understood. In our research we investigate performance measures that are traditionally used in performance evaluation of single-target short-term visual trackers, through theoretical and empirical analysis, and show that some of them are measuring the same aspect of tracking performance. Based on our analysis we propose a pair of two weakly correlated measures to measure the accuracy and robustness of a tracker, propose a visualization of the results as well as the analysis of the entire methodology using the theoretical trackers that exhibit extreme tracking behaviors. This is followed by an extension of the methodology on ranking of multiple trackers where we also take into account the potentially stochastic nature of visual trackers and test the statistical significance of performance differences. To support the proposed evaluation methodology we have developed an open-source software tool that implements the methodology and a simple communication protocol that enables a straightforward integration of trackers. The proposed evaluation methodology and the evaluation system have been adopted by several Visual Object Tracking (VOT) challenges

Journal of Telecommunications and Information Technology, 2017, nr 1

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