A Fano configuration is the hypergraph of 7 vertices and 7 triplets defined by the points and lines of the finite projective plane of order 2. Proving a conjecture of T. Sós, the largest triple system on $n$ vertices containing no Fano configuration is determined (for $n> n_1$). It is 2-chromatic with $\binom{n}{3}-\binom{\lfloor n/2 \rfloor}{3} -\binom{\lceil n/2 \rceil}{3}$ triples. This is one of the very few nontrivial exact results for hypergraph extremal problems.

Suppose that $q$ is a prime power exceeding five. For every integer $N$ there exists a 3-connected GF($q$)-representable matroid that has at least $N$ inequivalent GF($q$)-representations. In contrast to this, Geelen, Oxley, Vertigan and Whittle have conjectured that, for any integer $r > 2$, there exists an integer $n(q,\, r)$ such that if $M$ is a 3-connected GF($q$)-representable matroid and $M$ has no rank-$r$ free-swirl or rank-$r$ free-spike minor, then $M$ has at most $n(q,\, r)$ inequivalent GF($q$)-representations. The main result of this paper is a proof of this conjecture for Zaslavsky's class of bias matroids.

We give new formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay [Europ. J. Combin. 4 149–160] for regular graphs. The general answer involves a quantity for infinite graphs that we call ‘tree entropy’, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasi-transitive amenable graphs, extending a result of Burton and Pemantle [Ann. Probab. 21 1329–1371].

We study the Lovász number $\vartheta$ along with two related SDP relaxations $\vartheta_{1/2}$, $\vartheta_2$ of the independence number and the corresponding relaxations $\bar\vartheta$, $\bar\vartheta_{1/2}$, $\bar\vartheta_2$ of the chromatic number on random graphs $G_{n,p}$. We prove that $\vartheta,\vartheta_{1/2},\vartheta_2(G_{n,p})$ are concentrated about their means, and that $\bar\vartheta,\bar\vartheta_{1/2},\bar\vartheta_2(G_{n,p})$ in the case $p<n^{-1/2-\varepsilon}$ are concentrated in intervals of constant length. Moreover, extending a result of Juhász [28], we estimate the probable value of $\vartheta,\vartheta_{1/2},\vartheta_2(G_{n,p})$ for edge probabilities $c_0/n\leq p\leq 1-c_0/n$, where $c_0>0$ is a constant. As an application, we give improved algorithms for approximating the independence number of $G_{n,p}$ and for deciding $k$-colourability in polynomial expected time.

We show that symmetry, represented by a graph's automorphism group, can be used to greatly reduce the computational work for the substitution method. This allows application of the substitution method over larger regions of the problem lattices, resulting in tighter bounds on the percolation threshold $p_c$. We demonstrate the symmetry reduction technique using bond percolation on the $(3,12^2)$ lattice, where we improve the bounds on $p_c$ from (0.738598,0.744900) to (0.739399,0.741757), a reduction of more than 62% in width, from 0.006302 to 0.002358.

Let $G$ be a finite group of order $n$ and let $k$ be a natural number. Let $\{x_i : i\in I\}$ be a family of elements of $G$ such that $|I|= n+k-1$. Let $v$ be the most repeated value of the family. Let $ \{ \sigma_i : 1\leq i \leq k \} $ be a family of permutations of $G$ such that $\sigma_i(1)=1$ for all $i$. We obtain the following result.

There are pairwise distinct elements $i_1, i_2, \dots ,i_k\in I$ such that \[ \prod_{1\leq j\leq k } \sigma_j \big(v^{-1}x_ {i_j }\big) =1.\]

We study self-avoiding walks (SAWs) on non-Euclidean lattices that correspond to regular tilings of the hyperbolic plane (‘hyperbolic graphs’). We prove that on all but at most eight such graphs, (i) there are exponentially fewer $N$-step self-avoiding polygons than there are $N$-step SAWs, (ii) the number of $N$-step SAWs grows as $\mu_w^N$ within a constant factor, and (iii) the average end-to-end distance of an $N$-step SAW is approximately proportional to $N$. In terms of critical exponents from statistical physics, (ii) says that $\gamma=1$ and (iii) says that $\nu=1$. We also prove that $\gamma$ is finite on all hyperbolic graphs, and we prove a general identity about non-reversing walks that had previously been discovered for certain special cases.

The mathematical formulation of many physical problems results in the task of inverting a compact operator. The only known sensible solution technique is regularization which poses a severe problem in itself. Classically one dealt with deterministic noise models and required the knowledge of smoothness of the solution or the overall error behaviour. We will show that we can guarantee an asymptotically almost optimal regularization for a physically motivated noise model under no assumptions for the smoothness and rather weak assumptions on the noise behaviour. An application to the determination of the gravitational field out of satellite data will be shown.

Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations.

Square grid circle patterns with prescribed intersection angles, mimicking holomorphic maps $z^{\gamma }$ and ${\rm log}(z)$ are studied. It is shown that the corresponding circle patterns are embedded and described by special separatrix solutions of discrete Painlevé and Riccati equations. The general solution of this Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solutions, as well as of the discrete $z^{\gamma }$ and ${\rm log}(z)$, are established.

We consider the discrete Boussinesq integrable system and the compatible set of differential difference, and partial differential equations. The latter not only encode the complete hierarchy of the Boussinesq equation, but also incorporate the hyperbolic Ernst equations for an Einstein-Maxwell-Weyl field in general relativity. We demonstrate a specific symmetry reduction of the partial differential equations, to a six-parameter, second order coupled system of ordinary differential equations, which is conjectured to be of Garnier type.

A closer look at Laguerre and Meixner polynomials shows that they interact, with a more active role played by the latter. We intend here to expound on this development of the story up to some level of abstraction.

We consider a special class of solutions of the BKP hierarchy which we call $\tau$-functions of hypergeometric type. These are series in Schur $Q$-functions over partitions, with coefficients parameterised by a function of one variable $\xi$, where the quantities $\xi(k)$, $k\in\mathbb{Z^+}$, are integrals of motion of the BKP hierarchy. We show that this solution is, at the same time, a infinite soliton solution of a dual BKP hierarchy, where the variables $\xi(k)$ are now related to BKP higher times. In particular, rational solutions of the BKP hierarchy are related to (finite) multi-soliton solution of the dual BKP hierarchy. The momenta of the solitons are given by the parts of partitions in the Schur $Q$-function expansion of the $\tau$-function of hypergeometric type. We also show that the KdV and the NLS soliton $\tau$-functions coinside the BKP $\tau$-functions of hypergeometric type, evaluated at special point of BKP higher time; the variables $\xi$ (which are BKP integrals of motions) being related to KdV and NLS higher times.

The main goal of the paper is to find the discrete analogue of the Bianchi system in spaces of arbitrary dimension together with its geometric interpretation. We show that the proper geometric framework for such generalization is the language of dual quadrilateral lattices and of dual congruences. After introducing the notion of the dual Koenigs lattice in a projective space of arbitrary dimension, we define the discrete dual congruences and we present, as an important example, the normal dual discrete congruences. Finally, we introduce the dual Bianchi lattice as a dual Koenigs lattice allowing for a conjugate normal dual congruence, and we find its characterization in terms of a system of integrable difference equations.

The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.

A $q$-discrete analog of the Toda molecule equation and its $N$-soliton solution are constructed by using the bilinear method. The solution is expressed in the Casorati determinant form whose elements are given in terms of the $q$-orthogonal polynomials.

We consider a class of deformations of Laplace-Beltrami operators on flat and constant curvature spaces, which possess a family of commuting operators. These are built as deformations of the symmetries of the underlying geometric space. In flat spaces it is also possible to extend some symmetries into ladder operators. In all cases it is possible to choose sub-classes which are super-integrable.

A construction of differential constraints compatible with the Gibbons-Tsarev equation is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations that are used in searching for Lie operators. We introduce the notion of an invariant solution under an involutive distribution and give sufficient conditions for existence of such solutions.