Andrew N. W. Hone
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View article: Special solutions of a discrete Painlevé equation for quantum minimal surfaces
Special solutions of a discrete Painlevé equation for quantum minimal surfaces Open
We consider solutions of a discrete Painlevé equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe and Kontsevich, and in earlier work of Cornalba and Taylor on static membranes. While the discrete equation adm…
View article: Confluent Darboux Transformations and Wronskians for Algebraic Solutions of the Painlevé III (D7) Equation
Confluent Darboux Transformations and Wronskians for Algebraic Solutions of the Painlevé III (D7) Equation Open
Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of …
View article: Growth of Mahler Measure and Algebraic Entropy of Dynamics with the Laurent Property
Growth of Mahler Measure and Algebraic Entropy of Dynamics with the Laurent Property Open
We consider the growth rate of the Mahler measure in discrete dynamical systems with the Laurent property, and in cluster algebras, and compare this with other measures of growth. In particular, we formulate the conjecture that the growth …
View article: Casting more light in the shadows: dual Somos-5 sequences
Casting more light in the shadows: dual Somos-5 sequences Open
Motivated by the search for an appropriate notion of a cluster superalgebra, incorporating Grassmann variables, Ovsienko and Tabachnikov considered the extension of various recurrence relations with the Laurent phenomenon to the ring of du…
View article: New cluster algebras from old: integrability beyond Zamolodchikov periodicity
New cluster algebras from old: integrability beyond Zamolodchikov periodicity Open
We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodch…
View article: Casting more light in the shadows: dual Somos-5 sequences
Casting more light in the shadows: dual Somos-5 sequences Open
Motivated by the search for an appropriate notion of a cluster superalgebra, incorporating Grassmann variables, Ovsienko and Tabachnikov considered the extension of various recurrence relations with the Laurent phenomenon to the ring of du…
View article: A family of integrable maps associated with the Volterra lattice
A family of integrable maps associated with the Volterra lattice Open
Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Th…
View article: Performance of Higher-Order Networks in Reconstructing Sequential Paths: from Micro to Macro Scale
Performance of Higher-Order Networks in Reconstructing Sequential Paths: from Micro to Macro Scale Open
Activities such as the movement of passengers and goods, the transfer of physical or digital assets, web navigation and even successive passes in football, result in timestamped paths through a physical or virtual network. The need to anal…
View article: Heron Triangles and the Hunt for Unicorns
Heron Triangles and the Hunt for Unicorns Open
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 b…
View article: New cluster algebras from old: integrability beyond Zamolodchikov periodicity
New cluster algebras from old: integrability beyond Zamolodchikov periodicity Open
We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodch…
View article: Deformed cluster maps of type $A_{2N}$
Deformed cluster maps of type $A_{2N}$ Open
We extend recent work of the third author and Kouloukas by constructing deformations of integrable cluster maps corresponding to the Dynkin types $A_{2N}$, lifting these to higher-dimensional maps possessing the Laurent property and demons…
View article: Integrable maps in 4D and modified Volterra lattices
Integrable maps in 4D and modified Volterra lattices Open
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 dis…
View article: Heron triangles and the hunt for unicorns
Heron triangles and the hunt for unicorns Open
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 b…
View article: Integrable maps in 4D and modified Volterra lattices
Integrable maps in 4D and modified Volterra lattices Open
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 dis…
View article: A family of integrable maps associated with the Volterra lattice
A family of integrable maps associated with the Volterra lattice Open
Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Th…
View article: Introduction to special feature dedicated to Prof. Allan Fordy on the occasion of his 70th birthday
Introduction to special feature dedicated to Prof. Allan Fordy on the occasion of his 70th birthday Open
View article: Review of: "Relations between e, π and golden ratios"
Review of: "Relations between e, π and golden ratios" Open
Potential competing interests: No potential competing interests to declare.I'm afraid there is nothing new or worth publishing here: some of the identities, such as ( 13),( 14),( 15) are just obvious tautologies that follow from Euler's id…
View article: Deformations of cluster mutations and invariant presymplectic forms
Deformations of cluster mutations and invariant presymplectic forms Open
View article: Heron triangles with two rational medians and Somos-5 sequences
Heron triangles with two rational medians and Somos-5 sequences Open
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 …
View article: Similarity reductions of peakon equations: integrable cubic equations
Similarity reductions of peakon equations: integrable cubic equations Open
We consider the scaling similarity solutions of two integrable cubically nonlinear partial differential equations (PDEs) that admit peaked soliton (peakon) solutions, namely the modified Camassa–Holm (mCH) equation and Novikov’s equation. …
View article: Similarity reductions of peakon equations: the $$b$$-family
Similarity reductions of peakon equations: the $$b$$-family Open
The $b$-family is a one-parameter family of Hamiltonian partial differential equations of non-evolutionary type, which arises in shallow water wave theory. It admits a variety of solutions, including the celebrated peakons, which are weak …
View article: Основанные на подобии редукции пиконного уравнения: $b$-семейство
Основанные на подобии редукции пиконного уравнения: $b$-семейство Open
$b$-Семейство - это однопараметрическое семейство гамильтоновых дифференциальных уравнений неэволюционного типа в частных производных, возникающее в теории волн на мелкой воде. Оно имеет разнообразные решения, в том числе знаменитые пиконы…
View article: Casting light on shadow Somos sequences
Casting light on shadow Somos sequences Open
Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurr…
View article: Casting light on shadow Somos sequences
Casting light on shadow Somos sequences Open
Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurr…
View article: Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property
Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property Open
We consider the growth rate of the Mahler measure in discrete dynamical systems with the Laurent property, and in cluster algebras, and compare this with other measures of growth. In particular, we formulate the conjecture that the growth …
View article: Deformations of cluster mutations and invariant presymplectic forms
Deformations of cluster mutations and invariant presymplectic forms Open
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-cy…
View article: Heron triangles with two rational medians and Somos-5 sequences
Heron triangles with two rational medians and Somos-5 sequences Open
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 …
View article: Continued fractions for strong Engel seriesand Lüroth series with signs
Continued fractions for strong Engel seriesand Lüroth series with signs Open
An Engel series is a sum of reciprocals $\sum_{j\geq 1} 1/x_j$ of a non-decreasing sequence of positive integers $x_n$ with the property that $x_n$ divides $x_{n+1}$ for all $n\geq 1$. In previous work, we have shown that for any Engel ser…
View article: Linear relations for Laurent polynomials and lattice equations
Linear relations for Laurent polynomials and lattice equations Open
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, r…
View article: Efficient ECM factorization in parallel with the lyness map
Efficient ECM factorization in parallel with the lyness map Open
The Lyness map is a birational map in the plane which provides one of the simplest discrete analogues of a Hamiltonian system with one degree of freedom, having a conserved quantity and an invariant symplectic form. As an example of a symm…