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In this thesis we focus on questions of stability, existence and uniqueness for PDE con- strained problems in both dynamics and statics under appropriate convexity conditions. In the first part, we establish a Gårding-type inequality for quantities associated to (A, 0)- quasiconvex functions, where A is a constant-coefficient, linear differential operator with constant rank. In dynamics, we initially apply a simplified version of our derived inequality to prove weak-strong uniqueness results for conservation laws possessing involutions i.e. a differential constraint A propagated by the initial data, provided that the system is endowed with an A-quasiconvex entropy. In addition to this, combining our Gårding-type inequality with the relative entropy method, we establish a weak-strong uniqueness result for the hyperbolic system of adiabatic thermoelasticity under quasiconvexity conditions. In particular, we show that classical solutions of that system are unique within a suitable class of dissipative measure-valued solutions, provided that the internal energy is stronly (curl, 0)-quasiconvex. In statics, we investigate the so-called Weierstrass problem of finding necessary and sufficient conditions for local minimisers. More precisely, we prove an A-quasiconvexity based sufficiency theorem for local minimisers for general problems constrained by an operator A. An additional contribution of our result is that we infer uniqueness of these local minimisers and quantify the difference in energy between them and arbitrary comperitors. In the second part, in the context of image processing, we study a class of PDE constrained variational problems whose regularising terms depend on the differential operator. We prove the lower semicontinuity of the functionals in question and existence of minimisers for the corresponding variational problems. Then, we embed the latter into a bilevel scheme in order to automatically compute the space-dependent regularisation parameters, and we establish existence of optima for the scheme. We finally substantiate its feasibility by numerical examples in image denoising.

3-RRR planar parallel robots are utilized for solving precise material-handling problems in industrial automation applications. Thus, robust and stable control is required to deliver high accuracy in comparison to the state of the art. The operation of the mechanism is achieved based on three revolute (3-RRR) joints which are geometrically designed using an open-loop spatial robotic platform. The inverse kinematic model of the system is derived and analyzed by using the geometric structure with three revolute joints. The main variables in our design are the platform base positions, the geometry of the joint angles, and links of the 3-RRR planar parallel robot. These variables are calculated based on Cayley-Menger determinants and bilateration to determine the final position of the platform when moving and placing objects. Additionally, a proposed fractional order proportional integral derivative (FOPID) is optimized using the bat optimization algorithm to control the path tracking of the center of the 3-RRR planar parallel robot. The design is compared with the state of the art and simulated using the Matlab environment to validate the effectiveness of the proposed controller. Furthermore, real-time implementation has been tested to prove that the design performance is practical.

Kahan introduced an explicit method of discretization for systems of first order differential equations with nonlinearities of degree at most two (quadratic vector fields). Kahan’s method has attracted much interest due to the fact that it preserves many of the geometrical properties of the original continuous system. In particular, a large number of Hamiltonian systems of quadratic vector fields are known for which their Kahan discretization is a discrete integrable system. In this note, we introduce a special class of explicit order-preserving discretization schemes that are appropriate for certain systems of ordinary differential equations of higher order and higher degree.

We consider two-parameter bifurcation of equilibrium states of an elastic rod on a deformable foundation. Our main theorem shows that bifurcation occurs if and only if the linearization of our problem has nontrivial solutions. In fact our proof, based on the concept of the Brouwer degree, gives more, namely that from each bifurcation point there branches off a continuum of solutions., 16 pages, 2 figures

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 revisit an example of a semi-Riemannian geodesic that was discussed by Musso, Pejsachowicz and Portaluri in 2007 to show that not every conjugate point is a bifurcation point. We point out a mistake in their argument, showing that on this geodesic actually every conjugate point is a bifurcation point. Finally, we provide an improved example which yields that the claim in our title is nevertheless true.

The chapter will illustrate how concepts in differential geometry arise\ud naturally in different areas of mathematical physics. We will describe\ud manifolds, fibre bundles, (co)tangent bundles, metrics and symplectic\ud structures, and their applications to Lagrangian mechanics, field theory\ud and Hamiltonian systems, including various examples related to inte-\ud grable systems and topological solitons.

Lyapunov functions are functions with negative orbital derivative, whose existence guarantee the stability of an equilibrium point of an ODE. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. In this thesis, we improve an established numerical method to construct Lyapunov functions using the radial basis functions (RBF) collocation method. The RBF collocation method approximates the solution of linear PDE's using scattered collocation points, and one of its applications is the construction of Lyapunov functions. More precisely, we approximate Lyapunov functions, that satisfy equations for their orbital derivative, using the RBF collocation method. Then, it turns out that the RBF approximant itself is a Lyapunov function. \ud Our main contributions to improve this method are firstly to combine this construction method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of collocation points needed to construct Lyapunov functions. Moreover, we design two modified refinement algorithms to deal with the issue of the early termination of the original refinement algorithm without constructing a Lyapunov function. These algorithms uses cluster centres to place points where the Voronoi vertices failed to do so. \ud Secondly, we derive two verification estimates, in terms of the first and second derivatives of the orbital derivative, to verify if the constructed function, with either a regular grid of collocation points or with one of the refinement algorithms, is a Lyapunov function, i.e., has negative orbital derivative over a given compact set. Finally, the methods are applied to several numerical examples up to 3 dimensions.

Networks are often characterised in terms of their degree distribution and global clustering coefficient. It is assumed that these provide a sufficient parametrisation of networks. However, since the global clustering coefficient is only sensitive to the total number of triangles found in the network, it is evident that two networks could have the same number of triangles but significantly different higher-order structure, i.e., the topologies that result from the placement of closed subgraphs around nodes. The two main objectives of my work are: (1) developing network generating algorithms and network based epidemic models with controllable higher-order structure and (2) investigating the impact of higher-order structure on dynamics on networks. This thesis is based on three papers, corresponding to Chapters. 3, 4 and 5. Chapter 3 presents a novel higher-order structure based network generating algorithm and subgraph counting algorithm. Chapter. 4, generalises a previously proposed ODE model that accurately captures the time evolution of the susceptible-infected-recovered (SIR) dynamics on networks constructed using arbitrary subgraphs. Chapter. 5, improves, extends and generalises the network generating algorithms proposed in the previous two papers. All three chapters demonstrate that for a fixed degree distribution and global clustering, diverse higher-order structure is still possible and that this structure will impact significantly on dynamics unfolding on networks. Hence, we suggest that higher-order structure should receive more attention when analysing network-based systems and dynamics.

This work is focused on the approximation of sets of attractive solutions of planar dynamical systems. Existing work has shown that for many dynamical systems a Riemannian contraction metric can be used to determine sets of solutions with certain attraction properties. For autonomous dynamical systems in R² it has been shown that the Riemannian contraction metric can be reduced to a scalar weight function W. In this work we show that a similar result holds true for finite-time dynamical systems with one spatial dimension. We show how meshless collocation can be used to construct an approximation of W. The approximated weight function can then be used to determine subsets of the area of exponential attraction. This is the first time a method has been introduced to approximate finite-time areas of exponential attraction. We also give a convergence proof for the method. For autonomous dynamical systems in R² there already exists a method that uses W to determine a subset of the basin of attraction of an exponentially stable periodic orbit, Ω. However that method relies on properties of Ω being known. We show that the existing equation for W can be manipulated so that no knowledge of the periodic orbit is required to approximate W. We present a method that utilises meshless collocation to approximate W and show that the method is convergent. The approximant of W is then used to determine subsets of the basin of attraction of Ω.

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., Comment: 17 pages

Se reportan los resultados obtenidos al implementar actividades didácticas disociadas con el propósito de que los estudiantes den sentido a conceptos de la física, como: velocidad, aceleración, entre otros. Uno de los conceptos, aparentemente, sencillo, y que tiene fuerte impacto, es el referente a los Sistemas de referencia, mediante al uso de estos sistemas, se ha propuesto a los estudiantes que analizaran el movimiento de objetos físicos dándole la gráfica bosquejo que en un sistema cartesiano; en el que se daba un mismo origen del movimiento de los objeto. 665

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., Comment: 8 pages, AIMS class file. Proceedings of AIMS conference on Dynamical Systems, Differential Equations and Applications, Arlington, Texas, 2008

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., Comment: 41 pages, LaTeX + AMS packages + pstricks

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 $b\neq 1$., Contribution to volume of Journal of Nonlinear Mathematical Physics in honour of Francesco Calogero

We consider the fifth order partial differential equation (PDE) u4x,t?5uxxt+4ut+uu5x+2uxu4x?5uu3x?10uxuxx+12uux=0, which is a generalization of the integrable Camassa-Holm equation. The fifth order PDE has exact solutions in terms of an arbitrary number of superposed pulsons, with geodesic Hamiltonian dynamics that is known to be integrable in the two-body case N=2. Numerical simulations show that the pulsons are stable, dominate the initial value problem and scatter elastically. These characteristics are reminiscent of solitons in integrable systems. However, after demonstrating the non-existence of a suitable Lagrangian or bi-Hamiltonian structure, and obtaining negative results from Painlev\'{e} analysis and the Wahlquist-Estabrook method, we assert that the fifth order PDE is not integrable.

Elementary, one- and two-point, Backlund transformations are constructed for the generic case of the sl(2) Gaudin magnet. The spectrality property is used to construct these explicitly given, Poisson integrable maps which are time-discretizations of the continuous flows with any Hamiltonian from the spectral curve of the 2x2 Lax matrix.

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., LaTeX2e, 8 pages

We present a novel approach for probabilistic clustering of white matter fibre pathways using curve-based regression mixture modelling techniques in 3D curve space. The clustering algorithm is based on a principled method for probabilistic modelling of a set of fibre trajectories as individual sequences of points generated from a finite mixture model consisting of multivariate polynomial regression model components. Unsupervised learning is carried out using maximum likelihood principles. Specifically, conditional mixture is used together with expectation-maximisation (EM) algorithm to estimate cluster membership. The result of clustering is the probabilistic assignment of fibre trajectories to each cluster and an estimate of the cluster parameters. A statistical model is calculated for each clustered fibre bundles using fitted parameters of the probabilistic clustering. We illustrate the potential of our clustering approach on synthetic data and real data.

The enhancement of skin image in optical coherence tomography (OCT) imaging can help\ud dermatologists to investigate tissue layers more accurately, hence the more efficient diagnosis. In this paper, we\ud propose an image enhancement technique including speckle reduction, attenuation compensation and cleaning to\ud improve the quality of OCT skin images. A weighted median filter is designed to reduce the level of speckle\ud noise while preserving the contrast. A novel border detection technique is designed to outline the main skin layers,\ud stratum corneum, epidermis and dermis. A model of the light attenuation is then used to estimate the absorption\ud coefficient of epidermis and dermis layers and compensate the brightness of the structures at deeper levels. The\ud undesired part of the image is removed using a simple cleaning algorithm. The performance of the algorithm has\ud been evaluated visually and numerically using the commonly used no-reference quality metrics. The results shows\ud an improvement in the quality of the images.\ud Keywords: Optical coherence tomography (OCT), Skin, Image enhancement, Speckle reduction, Attenuation\ud compensation.