In this paper, we study a finite connected graph which admits a quasi-monomorphism to hyperbolic spaces and give a geometric bound for the Cheeger constants in terms of the volume, an upper bound of the degree, and the quasi-monomorphism.

The form of information presented can influence its utility for the conveying of knowledge by affecting an interpreter’s ability to reason with the information. There are distinct types of representational systems (for example, symbolic versus diagrammatic logics), various sub-systems (for example, propositional versus predicate logics), and even within a single representational system there may be different means of expressing the same piece of information content. Thus, to display information, choices must be made between its different representations, depending upon many factors such as: the context, the reasoning tasks to be considered, user preferences or desires (for example, for short symbolic sentences or minimal clutter within diagrammatic systems). The identification of all equivalent representations with the same information content is a sensible precursor to attempts to minimise a metric over this class. We posit that defining notions of semantic redundancy and identifying the syntactic properties that encapsulate redundancy can help in achieving the goal of completely identifying equivalences within a single notational system or across multiple systems, but that care must be taken when extending systems, since refinements of redundancy conditions may be necessary even for conservative system extensions. We demonstrate this theory within two diagrammatic systems, which are Euler-diagram-based notations. Such notations can be used to represent logical information and have applications including visualisation of database queries, social network visualisation, statistical data visualisation, and as the basis of more expressive diagrammatic logics such as constraint languages used in software specification and reasoning. The development of the new associated machinery and concepts required is important in its own right since it increases the growing body of knowledge on diagrammatic logics. In particular, we consider Euler diagrams with shading, and then we conservatively extend the system to include projections, which allow for a much greater degree of flexibility of representation. We give syntactic properties that encapsulate semantic equivalence in both systems, whilst observing that the same semantic concept of redundancy is significantly more difficult to realise as syntactic properties in the extended system withprojections.

We introduce a class of discrete-time stochastic processes, called disjunctive processes, which are important for reliable simulations in random iteration algorithms. Their definition requires that all possible patterns of states appear with probability 1. Sufficient conditions for nonhomogeneous chains to be disjunctive are provided. Suitable examples show that strongly mixing Markov chains and pairwise independent sequences, often employed in applications, may not be disjunctive. As a particular step towards a general theory we shall examine the problem arising when disjunctiveness is inherited under passing to a subsequence. An application to the verification problem for switched control systems is also included.

We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node, where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight ϑ>0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For ϑ=1, the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length, and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n→∞. Mainly expectations, variances, and asymptotic distributions of these parameters are derived.

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok in [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. For a given polytope 𝔭 with facets parallel to rational hyperplanes and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope 𝔭 parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step-polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.

The triple product property (TPP) for subsets of a finite group was introduced by Henry Cohn and Christopher Umans in 2003 as a tool for the study of the complexity of matrix multiplication. This note records some consequences of the simple observation that if (S1,S2,S3) is a TPP triple in a finite group G, then so is (dS1a,dS2b,dS3c) for any a,b,c,d∈G.

Let si:=∣Si∣ for 1≤i≤3. First we prove the inequality s1(s2+s3−1)≤∣G∣ and show some of its uses. Then we show (something a little more general than) that if G has an abelian subgroup of index v, then s1s2s3 ≤v2 ∣G∣.

In this paper we study the number of random records in an arbitrary split tree (or, equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for the convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1-stable. This work is a generalization of our earlier results for the random binary search tree in Holmgren (2010), which is one specific case of split trees. Other important examples of split trees include m-ary search trees, quad trees, medians of (2k + 1)-trees, simplex trees, tries, and digital search trees.

Bipartivity is an important network concept that can be applied to nodes, edges and communities. Here we focus on directed networks and look for subnetworks made up of two distinct groups of nodes, connected by ‘one-way’ links. We show that a spectral approach can be used to find hidden substructures of this form. Theoretical support is given for the idealized case where there is limited overlap between subnetworks. Numerical experiments show that the approach is robust to spurious and missing edges. A key application of this work is in the analysis of high-throughput gene expression data, and we give an example where a biologically meaningful directed bipartite subnetwork is found from a cancer microarray dataset.

Let P=A×A⊂𝔽p×𝔽p, p a prime. Assume that P=A×A has n elements, n<p. See P as a set of points in the plane over 𝔽p. We show that the pairs of points in P determine lines, where c is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of n points and a set of n lines in the projective plane over 𝔽p (n<p)is bounded by , where C is an absolute constant.

We analyse an ALOHA-type random multiple-access protocol where users have local interactions. We show that the fluid model of the system workload satisfies a certain differential equation. We obtain a sufficient condition for the stability of this differential equation and deduce from that a sufficient condition for the stability of the protocol. We discuss the necessary condition. Furthermore, for the underlying Markov chain, we estimate the rate of convergence to the stationary distribution. Then we establish an interesting and unexpected result showing that the main diagonal is locally unstable if the input rate is sufficiently small. Finally, we consider two generalisations of the model.

In this work we give precise asymptotic expressions for the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.

In this paper we give an analytic solution for graphs with n nodes and E = cn log n edges for which the probability of obtaining a given graph G is µn (G) = exp (- β ∑i=1 nd i 2), where d i is the degree of node i. We describe how this model appears in the context of load balancing in communication networks, namely peer-to-peer overlays. We then analyse the degree distribution of such graphs and show that the degrees are concentrated around their mean value. Finally, we derive asymptotic results for the number of edges crossing a graph cut and use these results (i) to compute the graph expansion and conductance, and (ii) to analyse the graph resilience to random failures.

Let Bn(x) denote the number of 1’s occurring in the binary expansion of an irrational number x>0. A difficult problem is to provide nontrivial lower bounds for Bn(x) for interesting numbers such as , e or π: their conjectural simple normality in base 2 is equivalent to Bn(x)∼n/2. In this article, amongst other things, we prove inequalities relating Bn(x+y), Bn(xy) and Bn(1/x) to Bn(x) and Bn(y) for any irrational numbers x,y>0, which we prove to be sharp up to a multiplicative constant. As a by-product, we provide an answer to a question raised by Bailey et al. (D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, ‘On the binary expansions of algebraic numbers’, J. Théor. Nombres Bordeaux16(3) (2004), 487–518) concerning the binary digits of the square of a series related to the Fibonacci sequence. We also obtain a slight refinement of the main theorem of the same article, which provides a nontrivial lower bound for Bn(α) for any real irrational algebraic number. We conclude the article with effective or conjectural lower bounds for Bn(x) when x is a transcendental number.

We studythe k-epistasis of a fitness function over a search space. This concept is a natural generalization of that of epistasis, previously considered by Davidor, Suys and Verschoren and Van Hove and Verschoren [Y. Davidor, in: Foundations of genetic algorithms, Vol. 1, (1991), pp. 23–25; D. Suys and A. Verschoren, ‘Proc Int. Conf. on Intelligent Technologies in Human-Related Sciences (ITHURS’96), Vol. II (1996), pp. 251–258; H. Van Hove and A. Verschoren, Comput. Artificial Intell.14 (1994), 271–277], for example. We completely characterize fitness functions whose k-epistasis is minimal: these are exactly the functions of order k. We also obtain an upper bound for the k-epistasis of nonnegative fitness functions.

Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let V n = {X 1, X 2, …, X n }, where X 1, X 2, … are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in V n that cover all of V n .

The iterative division of a triangle by chords which join a randomly-selected vertex of a triangle to the opposite side is investigated. Results on the limiting random graph which eventuates are given. Aspects studied are: the order of vertices; the fragmentation of chords; age distributions for elements of the graph; various topological characterisations of the triangles. Different sampling protocols are explored. Extensive use is made of the theory of branching processes.

Given a finite collection of strings of letters from a fixed alphabet, it is of interest, in the contexts of data compression and DNA sequencing, to find the length of the shortest string which contains each of the given strings as a consecutive substring. In order to analyze the average behavior of the optimal superstring length, substrings of specified lengths are considered with the letters selected independently at random. An asymptotic expression is obtained for the savings from compression, i.e. the difference between the uncompressed (concatenated) length and the optimal superstring length.

Three differently defined classes of two-symbol sequences, which we call the two-distance sequences, the linear sequences and the characteristic sequences, have been discussed by a number of authors and some equivalences between them are known. We present a self-contained proof that the three classes are the same (when ambiguous cases of linear sequences are suitably in terpreted). Associated with each sequence is a real invariant (having a different appropriate definition for each of the three classes). We give results on the relation between sequences with the same invariant and on the symmetry of the sequences. The sequences are closely related to Beatty sequences and occur as digitized straight lines and quasicrystals. They also provide examples of minimal word proliferation in formal languages.

The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small excluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed.