Branching random walks and Minkowski sum of random walks

We show that the range of a critical branching random walk conditioned to survive forever and the Minkowski sum of two independent simple random walk ranges are intersection-equivalent in any dimension $d\ge 5$, in the sense that they hit any finite set with comparable probability, as their common starting point is sufficiently far away from the set to be hit. Furthermore, we extend a discrete version of Kesten, Spitzer and Whitman's result on the law of large numbers for the volume of a Wiener sausage. Here, the sausage is made of the Minkowski sum of $N$ independent simple random walk ranges in $\mathbb{Z}^d$, with $d>2N$, and of a finite set $A\subset \mathbb{Z}^d$. When properly normalised the volume of the sausage converges to a quantity equivalent to the capacity of $A$ with respect to the kernel $K(x,y)=(1+\|x-y\|)^{2N-d}$. As a consequence, we establish a new relation between capacity and {\it branching capacity}.Comment: 25 page

On the minimum size of subset and subsequence sums in integers

Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$. For a nonnegative integer $\alpha \le rk$, let $\Sigma _{\alpha } (\mathcal{A})$ be the set of all subsequence sums of $\mathcal{A}$ that correspond to the subsequences of length $\alpha$ or more. When $r=1$, we call the subsequence sums as subset sums and we write $\Sigma _{\alpha } (A)$ for $\Sigma _{\alpha } (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of $\Sigma _{\alpha } (A)$ and $\Sigma _{\alpha } (\mathcal{A})$. As special cases, we also obtain some already known results in this study

On the minimum size of subset and subsequence sums in integers

Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$. For a nonnegative integer $\alpha \le rk$, let $\Sigma _{\alpha } (\mathcal{A})$ be the set of all subsequence sums of $\mathcal{A}$ that correspond to the subsequences of length $\alpha$ or more. When $r=1$, we call the subsequence sums as subset sums and we write $\Sigma _{\alpha } (A)$ for $\Sigma _{\alpha } (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of $\Sigma _{\alpha } (A)$ and $\Sigma _{\alpha } (\mathcal{A})$. As special cases, we also obtain some already known results in this study

Developing an ontological sandbox : investigating multi-level modelling’s possible Metaphysical Structures

One of the central concerns of the multi-level modelling (MLM) community is the hierarchy of classifications that appear in conceptual models; what these are, how they are linked and how they should be organised into levels and modelled. Though there has been significant work done in this area, we believe that it could be enhanced by introducing a systematic way to investigate the ontological nature and requirements that underlie the frameworks and tools proposed by the community to support MLM (such as Orthogonal Classification Architecture and Melanee). In this paper, we introduce a key component for the investigation and understanding of the ontological requirements, an ontological sandbox. This is a conceptual framework for investigating and comparing multiple variations of possible ontologies – without having to commit to any of them – isolated from a full commitment to any foundational ontology. We discuss the sandbox framework as well as walking through an example of how it can be used to investigate a simple ontology. The example, despite its simplicity, illustrates how the constructional approach can help to expose and explain the metaphysical structures used in ontologies, and so reveal the underlying nature of MLM levelling

Phase Transitions and Symmetry Breaking in Genetic Algorithms with Crossover

In this paper, we consider the role of the crossover operator in genetic algorithms. Specifically, we study optimisation problems that exhibit many local optima and consider how crossover affects the rate at which the population breaks the symmetry of the problem. As an example of such a problem, we consider the subset sum problem. In so doing, we demonstrate a previously unobserved phenomenon, whereby the genetic algorithm with crossover exhibits a critical mutation rate, at which its performance sharply diverges from that of the genetic algorithm without crossover. At this critical mutation rate, the genetic algorithm with crossover exhibits a rapid increase in population diversity. We calculate the details of this phenomenon on a simple instance of the subset sum problem and show that it is a classic phase transition between ordered and disordered populations. Finally, we show that this critical mutation rate corresponds to the transition between the genetic algorithm accelerating or preventing symmetry breaking and that the critical mutation rate represents an optimum in terms of the balance of exploration and exploitation within the algorithm

Search Heuristics, Case-Based Reasoning and Software Project Effort Prediction

This paper reports on the use of search techniques to help optimise a case-based reasoning (CBR) system for predicting software project effort. A major problem, common to ML techniques in general, has been dealing with large numbers of case features, some of which can hinder the prediction process. Unfortunately searching for the optimal feature subset is a combinatorial problem and therefore NP-hard. This paper examines the use of random searching, hill climbing and forward sequential selection (FSS) to tackle this problem. Results from examining a set of real software project data show that even random searching was better than using all available for features (average error 35.6% rather than 50.8%). Hill climbing and FSS both produced results substantially better than the random search (15.3 and 13.1% respectively), but FSS was more computationally efficient. Providing a description of the fitness landscape of a problem along with search results is a step towards the classification of search problems and their assignment to optimum search techniques. This paper attempts to describe the fitness landscape of this problem by combining the results from random searches and hill climbing, as well as using multi-dimensional scaling to aid visualisation. Amongst other findings, the visualisation results suggest that some form of heuristic-based initialisation might prove useful for this problem