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A Heron triangle is one that has all integer side lengths and integer area, which takes its name from Heron of Alexandria's area formula. From a more relaxed point of view, if rescaling is allowed, then one can define a Heron triangle to be one whose side lengths and area are all rational numbers. A perfect triangle is a Heron triangle with all three medians being rational. According to a longstanding conjecture, no such triangle exists, so perfect triangles are as rare as unicorns. However, if perfect is the enemy of good, then perhaps it is best to insist on only two of the medians being rational. Buchholz and Rathbun found an infinite family of Heron triangles with two rational medians, which they were able to associate with the set of rational points on an elliptic curve E(Q). Here we describe a recently discovered explicit formula for the sides, area and medians of these (almost perfect) triangles, expressed in terms of a pair of integer sequences: these are Somos sequences, which first became popular thanks to David Gale's column in Mathematical Intelligencer.

In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus g. As well as proving that each such discrete system is an integrable map in the Liouville sense, we also showed it to be an algebraic completely integrable system. In the discrete setting, the latter means that the generic level set of the invariants is an affine part of an abelian variety, in this case the Jacobian of the hyperelliptic curve, and each iteration of the map corresponds to a translation by a fixed vector on the Jacobian. In addition, we demonstrated that, by combining the discrete integrable dynamics with the flow of one of the commuting Hamiltonian vector fields, these maps provide genus g algebro-geometric solutions of the infinite Volterra lattice, which justified naming them Volterra maps, denoted V_g. The original motivation behind our work was the fact that, in the particular case g=2, we could recover an example of an integrable symplectic map in four dimensions found by Gubbiotti, Joshi, Tran and Viallet, who classified birational maps in 4D admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Hence, in this particular case, the map V_2 yields genus two solutions of the Volterra lattice. The purpose of this note is to point out how two of the other 4D integrable maps obtained in the classification of Gubbiotti et al. correspond to genus two solutions of two different forms of the modified Volterra lattice, being related via a Miura-type transformation to the g=2 Volterra map V_2. We dedicate this work to a dear friend and colleague, Decio Levi.

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4 we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions of the discrete sine-Gordon equation.

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E(Q) with Mordell-Weil group Z × Z/2Z, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E(Q) with Mordell-Weil group Z × Z/2Z, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4 we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions of the discrete sine-Gordon equation.

Quispel-Roberts-Thompson (QRT) maps are a family of birational maps of the plane which provide the simplest discrete analogue of an integrable Hamiltonian system, and are associated with elliptic fibrations in terms of biquadratic curves. Each generic orbit of a QRT map corresponds to a sequence of points on an elliptic curve. In this preliminary study, we explore versions of the elliptic curve method (ECM) for integer factorization based on performing scalar multiplication of a point on an elliptic curve by iterating three different QRT maps with particular initial data. Pseudorandom number generation and other possible applications are briefly discussed.

A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. Recurrences with this property appear in diverse areas of mathematics and physics, ranging from Lie theory and supersymmetric gauge theories to Teichmuller theory and dimer models. In many cases where such recurrences appear, there is a common structural thread running between these different areas, in the form of Fomin and Zelevinsky's theory of cluster algebras. Laurent phenomenon algebras, as defined by Lam and Pylyavskyy, are an extension of cluster algebras, and share with them the feature that all the generators of the algebra are Laurent polynomials in any initial set of generators (seed). Here we consider a family of nonlinear recurrences with the Laurent property, referred to as "Little Pi", which was derived by Alman et al. via a construction of periodic seeds in Laurent phenomenon algebras, and generalizes the Heideman-Hogan family of recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. We derive the latter from linear relations with periodic coefficients, which were found recently by Kamiya et al. from travelling wave reductions of a linearizable lattice equation on a 6-point stencil. By making use of the periodic coefficients, we further show that the birational maps corresponding to the Little Pi family are maximally superintegrable. We also introduce another linearizable lattice equation on the same 6-point stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the 6-point lattice equations considered, we use the formalism of van der Kamp to construct a broad class of initial value problems with the Laurent property.

Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the context of cluster mutation. In particular, we give examples of birational maps that are integrable in the Liouville sense and arise from cluster algebras with periodicity, as well as examples of discrete Painleve equations that are derived from Y-systems.

A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered.

We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case.

We construct the explicit solution of the initial value problem for sequences generated by the general Somos-6 recurrence relation, in terms of the Kleinian sigma-function of genus two. For each sequence there is an associated genus two curve \(X\), such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of \(X\). The construction is based on a Lax pair with a spectral curve \(S\) of genus four admitting an involution \(\sigma\) with two fixed points, and the Jacobian of \(X\) arises as the Prym variety Prym \((S,\sigma)\).

We consider certain types of discretization schemes for differential equations with quadratic nonlinearities, which were introduced by Kahan, and considered in a broader setting by Mickens. These methods have the property that they preserve important structural features of the original systems, such as the behaviour of solutions near to fixed points, and also, where appropriate (e.g. for certain mechanical systems), the property of being volume-preserving, or preserving a symplectic/Poisson structure. Here we focus on the application of Kahan's method to models of biological systems, in particular to reaction kinetics governed by the Law of Mass Action, and present a general approach to birational discretization, which is applied to population dynamics of Lotka-Volterra type.

We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations.

We consider series of the form $p/q + \sum_{j=2}^\infty 1/x_j$, where $x_1=q$ and the integer sequence $x_n$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for n?1. It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.

The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multi-soliton solutions - "peakons" - with a peaked shape corresponding to a discontinuous first derivative. There is a one-parameter family of generalized Camassa-Holm equations, most of which are not integrable, but which all admit peakon solutions. Numerical studies reported by Holm and Staley indicate changes in the stability of these and other solutions as the parameter varies through the family. In this article, we describe analytical results on one of these bifurcation phenomena, showing that in a suitable parameter range there are stationary solutions - "leftons" - which are orbitally stable.

We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure. Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville–Arnold sense.

We consider a family of non-linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearizable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrödinger operators is also explained.

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented.

We study the integrability of mappings obtained as reductions of the discrete Korteweg–de Vries (KdV) equation and of two copies of the discrete potential KdV (pKdV) equation. We show that the mappings corresponding to the discrete KdV equation, which can be derived from the latter, are completely integrable in the Liouville–Arnold sense. The mappings associated with two copies of the pKdV equation are also shown to be integrable.

We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map.

Recently Vladimir Novikov found a new integrable analogue of the Camassa-Holm equation, admitting peaked soliton (peakon) solutions, which has nonlinear terms that are cubic, rather than quadratic. In this paper, the explicit formulas for multipeakon solutions of Novikov's cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the Degasperis-Procesi equation

We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a two-field coupling between the Camassa-Holm and Degasperis-Procesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation with Painlev\'e analysis provides strong evidence that the Popowicz system is non-integrable. Nevertheless, we are able to construct exact travelling wave solutions in terms of an elliptic integral, together with a degenerate travelling wave corresponding to a single peakon. We also describe the dynamics of N-peakon solutions, which is given in terms of an Hamiltonian system on a phase space of dimension 3N.

Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable bi-Hamiltonian system in three dimensions. The Hirota-Kimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the two-dimensional Lotka-Volterra system. The Euler top is naturally written in terms of the so(3) Lie-Poisson algebra. Here we consider algebraically integrable systems that are associated with pairs of Lie-Poisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets that are one-parameter deformations of the original Lie-Poisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three bi-Hamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd's Diophantine integrability criterion.

The Yablonskii-Vorob'ev polynomials yn(t), which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions of the second Painlev\'{e} equation (PII). Here we define two-variable polynomials Yn(t,h) on a lattice with spacing h, by considering rational solutions of the discrete time Toda lattice as introduced by Suris. These polynomials are shown to have many properties that are analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce when h=0. They also provide rational solutions for a particular discretisation of PII, namely the so called {\it alternate discrete} PII, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation (PIII). It is shown that B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is a symplectic map, and the shift in time is also symplectic. Finally we present a Lax pair for the alternate discrete PII, which recovers Jimbo and Miwa's Lax pair for PII in the continuum limit h?0.

We consider some bilinear recurrences that have applications in number theory. The explicit solution of a general three-term bilinear recurrence relation of fourth order is given in terms of the Weierstrass sigma function for an associated elliptic curve. The recurrences can generate integer sequences, including the Somos 4 sequence and elliptic divisibility sequences. An interpretation via the theory of integrable systems suggests the relation between certain higher order recurrences and hyperelliptic curves of higher genus. Analogous sequences associated with a q-discrete Painlev\'e I equation are briefly considered.

Artificial immune systems (AIS) constitute a relatively new area of bio-inspired computing. Biological models of the natural immune system, in particular the theories of clonal selection, immune networks and negative selection, have provided the inspiration Cor AIS algorithms. Moreover, such algorithms have been successfully employed in a wide variety of different application areas. However, despite these practical successes, until recently there has been a dearth of theory to justify their Use. In this paper, the existing theoretical work oil AIS is reviewed. After the presentation of a simple example of each of the three main types of AIS algorithm (that is, clonal selection, immune network and negative selection algorithms respectively), details of the theoretical analysis for each of these types are given. Some of the future challenges in this area are also highlighted.

This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property. Subsequently a family of fourth-order recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations.

The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we prove that the two subsequences of odd/even index terms each satisfy a Somos 4 (fourth order) recurrence. This leads directly to the explicit solution of the initial value problem for the Somos 5 sequences in terms of the Weierstrass sigma function for an associated elliptic curve.

We consider a one-parameter family of third order nonlinear recurrence relations. Each member of this family satisfies the singularity confinement test, has a conserved quantity, and moreover has the Laurent property: all of the iterates are Laurent polynomials in the initial data. However, we show that these recurrences are not Diophantine integrable according to the definition proposed by Halburd. Explicit bounds on the asymptotic growth of the heights of iterates are obtained for a special choice of initial data. As a by-product of our analysis, infinitely many solutions are found for a certain family of Diophantine equations, studied by Mordell, that includes Markoff's equation.

We consider a family of integro-differential equations depending upon a parameter b as well as a symmetric integral kernel g(x). When b=2 and g is the peakon kernel (i.e. g(x)=exp(?|x|) up to rescaling) the dispersionless Camassa-Holm equation results, while the Degasperis-Procesi equation is obtained from the peakon kernel with b=3. Although these two cases are integrable, generically the corresponding integro-PDE is non-integrable. However,for b=2 the family restricts to the pulson family of Fringer & Holm, which is Hamiltonian and numerically displays elastic scattering of pulses. On the other hand, for arbitrary b it is still possible to construct a nonlocal Hamiltonian structure provided that g is the peakon kernel or one of its degenerations: we present a proof of this fact using an associated functional equation for the skew-symmetric antiderivative of g. The nonlocal bracket reduces to a non-canonical Poisson bracket for the peakon dynamical system, for any value of b?1.

The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous family of sequences associated with an hyperelliptic curve of genus two defined by the affine model y2=4x5+c4x4+...+c1x+c0. We show that the recurrence sequences associated with such curves satisfy bilinear recurrences of order 8. The proof requires an addition formula which involves the genus two Kleinian sigma function with its argument shifted by the Abelian image of the reduced divisor of a single point on the curve. The genus two recurrences are related to a B\"{a}cklund transformation (BT) for an integrable Hamiltonian system, namely the discrete case (ii) H\'{e}non-Heiles system.

Both simple and hybrid genetic algorithms encounter difficulties when presented with a function which has multiple values. Similarly, changing environments or functions which change rapidly present other problems. This paper presents an algorithm that is capable of coping with both of these scenarios: it can accommodate multiple solutions simultaneously and can track changes in optima efficiently. The proposed B-cell algorithm is inspired by the natural immune system, which itself displays similar capabilities of tracking multiple, moving targets in the form of infectious agents. This paper employs two nonlinear mappings which display chaotic behaviour to demonstrate the effectiveness of the B-cell algorithm in tracking multiple, moving targets. A number of experiments are conducted and results reported from the B-cell algorithm and standard hybrid genetic algorithm approaches. These results show the benefit of the B-cell algorithm approach when compared against these heuristic approaches.

We consider a pair of coupled Painlevé equations arising as a scaling similarity reduction of the Hirota-Satsuma system of partial differential equations. Bäcklund transformations constructed in a previous work are presented explicitly as discrete shifts in a two-dimensional parameter space. ?-functions derived from a Hamiltonian description are also presented, which satisfy multilinear lattice equations built from Hirota bilinear operators, and these are used to calculate polynomial ?-functions for rational solutions.

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under M\"obius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDEs for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the {\it full} Painlev\'e VI equation, i.e. with four arbitary parameters.

A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or B\"acklund transformations. We describe such a chain for the sixth Painlev\'e equation (\pvi), containing, apart from \pvi itself, a Schwarzian version as well as a second-order second-degree ordinary differential equation (ODE). As a byproduct we derive an auto-B\"acklund transformation, relating two copies of \pvi with different parameters. We also establish the analogous ordinary difference equations in the discrete counterpart of the chain. Such difference equations govern iterations of solutions of \pvi under B\"acklund transformations. Both discrete and continuous equations constitute a larger system which include partial difference equations, differential-difference equations and partial differential equations, all associated with the lattice Korteweg-de Vries equation subject to similarity constraints.

We present Backlund transformations (BTs) with parameter for certain classical integrable n-body systems, namely the many-body generalised Henon-Heiles, Garnier and Neumann systems. Our construction makes use of the fact that all these systems may be obtained as particular reductions (stationary or restricted flows) of the KdV hierarchy; alternatively they may be considered as examples of the reduced sl(2) Gaudin magnet. The BTs provide exact time-discretizations of the original (continuous) systems, preserving the Lax matrix and hence all integrals of motion, and satisfy the spectrality property with respect to the Backlund parameter.