We consider a fourth order nonlinear partial differential equation in n-dimensional space introdu... more We consider a fourth order nonlinear partial differential equation in n-dimensional space introduced by Abreu in the context of Kähler metrics on toric varieties. Rotation invariant similarity solutions, depending only on the radial coordinate in R n , are determined from the solutions of a second order ordinary differential equation (ODE), with a non-autonomous Lagrangian formulation. A local asymptotic analysis of solutions of the ODE in the neighbourhood of singular points is carried out, and the existence of a class of solutions on an interval of the positive real semi-axis is proved using a nonlinear integral equation. The integrability (or otherwise) of Abreu's equation is discussed.
A parameter-dependent class of Hamiltonian (generalized) Lotka-Volterra systems is considered. We... more A parameter-dependent class of Hamiltonian (generalized) Lotka-Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.
Journal of Physics A: Mathematical and Theoretical
We introduce a two-parameter family of birational maps, which reduces to a family previously foun... more 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 classify integrable scalar polynomial partial differential equations of second order generaliz... more We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.
Recently the associated Camassa-Holm (ACH) equation, related to the Fuchssteiner-Fokas-Camassa-Ho... more Recently the associated Camassa-Holm (ACH) equation, related to the Fuchssteiner-Fokas-Camassa-Holm equation by a hodograph transformation, was introduced by Schiff, who derived B\"{a}cklund transformations by a loop group technique and used these to obtain some simple soliton and rational solutions. We show how the ACH equation is related to Schr\"{o}dinger operators and the KdV hierarchy, and use this connection to obtain
Photocopying permitted by license only a member of the Old City Publishing Group Conceptual Frameworks for Artificial Immune Systems
In a decision diagram, the average path length (APL) is the average number of nodes on a path fro... more In a decision diagram, the average path length (APL) is the average number of nodes on a path from the root node to a terminal node over all assignments of values to variables. Smaller APL values result in faster evaluation of the ...
Using a set of genetic logic gates (AND, OR and XOR), we constructed a binary full-adder. The opt... more Using a set of genetic logic gates (AND, OR and XOR), we constructed a binary full-adder. The optimality analysis of the full-adder showed that, based on the position of the regulation threshold, the system displays different optimal configurations for speed and accuracy under fixed metabolic cost. In addition, the analysis identified an optimal trade-off curve bounded by these two optimal configurations. Any configuration outside this optimal trade-off curve is sub-optimal in both speed and accuracy. This type of analysis represents a useful tool for synthetic biologists to engineer faster, more accurate and cheaper genes.
Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV ... more Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations, which are time-dependent generalizations of the well-known integrable Hénon-Heiles systems. The (time-dependent) Hamiltonians are given by logarithmic derivatives of the tau-functions (inherited from the original PDEs). The ODEs for the similarity solutions also have inherited Bäcklund transformations, which may be used to generate sequences of rational solutions as well as other special solutions related to the first Painlevé transcendent.
Artificial Immune Systems, 9th International Conference, ICARIS 2010, Edinburgh , UK, July 26-29, 2010. Proceedings
Idc, 2010
... Mario Pavone Ed Clark Fiona Polack Martin Drozda Peter Ross Andries Engelbrecht M. Zubair Sha... more ... Mario Pavone Ed Clark Fiona Polack Martin Drozda Peter Ross Andries Engelbrecht M. Zubair Shafiq Stephanie Forrest Susan Stepney Maoguo Gong Thomas Stibor Fabio Gonzalez Alexander Tarakanov Emma Hart Jon Timmis Andy Hone Andy Tyrrell Christian Jacob Neil ...
Discrete and Continuous Dynamical Systems, Aug 1, 2008
We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz... more 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 Painlevé 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.
We consider a fourth order nonlinear partial differential equation in n-dimensional space introdu... more We consider a fourth order nonlinear partial differential equation in n-dimensional space introduced by Abreu in the context of Kähler metrics on toric varieties. Rotation invariant similarity solutions, depending only on the radial coordinate in R n , are determined from the solutions of a second order ordinary differential equation (ODE), with a non-autonomous Lagrangian formulation. A local asymptotic analysis of solutions of the ODE in the neighbourhood of singular points is carried out, and the existence of a class of solutions on an interval of the positive real semi-axis is proved using a nonlinear integral equation. The integrability (or otherwise) of Abreu's equation is discussed.
A parameter-dependent class of Hamiltonian (generalized) Lotka-Volterra systems is considered. We... more A parameter-dependent class of Hamiltonian (generalized) Lotka-Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.
Journal of Physics A: Mathematical and Theoretical
We introduce a two-parameter family of birational maps, which reduces to a family previously foun... more 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 classify integrable scalar polynomial partial differential equations of second order generaliz... more We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.
Recently the associated Camassa-Holm (ACH) equation, related to the Fuchssteiner-Fokas-Camassa-Ho... more Recently the associated Camassa-Holm (ACH) equation, related to the Fuchssteiner-Fokas-Camassa-Holm equation by a hodograph transformation, was introduced by Schiff, who derived B\"{a}cklund transformations by a loop group technique and used these to obtain some simple soliton and rational solutions. We show how the ACH equation is related to Schr\"{o}dinger operators and the KdV hierarchy, and use this connection to obtain
Photocopying permitted by license only a member of the Old City Publishing Group Conceptual Frameworks for Artificial Immune Systems
In a decision diagram, the average path length (APL) is the average number of nodes on a path fro... more In a decision diagram, the average path length (APL) is the average number of nodes on a path from the root node to a terminal node over all assignments of values to variables. Smaller APL values result in faster evaluation of the ...
Using a set of genetic logic gates (AND, OR and XOR), we constructed a binary full-adder. The opt... more Using a set of genetic logic gates (AND, OR and XOR), we constructed a binary full-adder. The optimality analysis of the full-adder showed that, based on the position of the regulation threshold, the system displays different optimal configurations for speed and accuracy under fixed metabolic cost. In addition, the analysis identified an optimal trade-off curve bounded by these two optimal configurations. Any configuration outside this optimal trade-off curve is sub-optimal in both speed and accuracy. This type of analysis represents a useful tool for synthetic biologists to engineer faster, more accurate and cheaper genes.
Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV ... more Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations, which are time-dependent generalizations of the well-known integrable Hénon-Heiles systems. The (time-dependent) Hamiltonians are given by logarithmic derivatives of the tau-functions (inherited from the original PDEs). The ODEs for the similarity solutions also have inherited Bäcklund transformations, which may be used to generate sequences of rational solutions as well as other special solutions related to the first Painlevé transcendent.
Artificial Immune Systems, 9th International Conference, ICARIS 2010, Edinburgh , UK, July 26-29, 2010. Proceedings
Idc, 2010
... Mario Pavone Ed Clark Fiona Polack Martin Drozda Peter Ross Andries Engelbrecht M. Zubair Sha... more ... Mario Pavone Ed Clark Fiona Polack Martin Drozda Peter Ross Andries Engelbrecht M. Zubair Shafiq Stephanie Forrest Susan Stepney Maoguo Gong Thomas Stibor Fabio Gonzalez Alexander Tarakanov Emma Hart Jon Timmis Andy Hone Andy Tyrrell Christian Jacob Neil ...
Discrete and Continuous Dynamical Systems, Aug 1, 2008
We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz... more 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 Painlevé 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.
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