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In the procedure of finding nonlocally related systems, subsystems are obtained by excluding dependent variables of a given PDE system, or after ``hodograph'' type transformations. More generally, one can further extend this procedure to more general invertible point transformations. As an example, I have investigated the nonlinear wave equation and have shown that a more general point transformation of the potential system of the nonlinear wave equation does yield a subsystem that can yield previously unknown nonlocal symmetries of the nonlinear wave equation. Moreover, some open problems will be posed.

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Cells and tissues rearrange under the action of chemical signals. Numerous examples are found in eggshell development, wing disc remodeling, dorsal closure, wound healing, etc. In many cases, this can be attributed to changing local mechanical properties of cytoskeleton due to motor attachment/detachment and rearrangement of the actin network triggered by signaling. I consider in more detail the action of myosin motors on nonlinear viscoelastic properties of cytoskeleton. It turns out that motors activity may either stiffen the network due to stronger prestress or soften it due to motor agitation, in accordance with experimental data. Prestress anisotropy, which may be induced by redistribution of motors triggered by either external force or a chemical signal, causes anisotropy of elastic moduli. Based on this assumption, we developed a cellular mechano-diffusive model cell that describes reshaping of the Drosophila wing disc. Similar models may be applicable to other processes where mechanics is influenced by chemical signals through the action of myosin motors.

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In classical physics, especially electrodynamics, often one must consider the behaviour of waves as they move between two media in which the speed of the waves differs. In the past this has often been approached through studying a system in which this change occurs instantaneously. Using symmetry methods it is possible to extend this result to obtain an exact solution for the wave behaviour in a system in which there is a smooth transition between the two media. The details of this method will be discussed, as well as the use of the solution in obtaining relevant physical quantities, such as the reflection coefficient, in these new systems.

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A striking feature of symmetries of DEs or other geometric objects is that the group action may project to subsets of the variables.  For example, the 3-d Navier-Stokes equations, with independent variables (x,y,z,t) and dependent variables (u,v,w,p) (velocity components and pressure), has a symmetry group that projects naturally to actions on t, or on (x,y,z,t) or on (t,u,v,w) or various other subsets of the variables.  These projections can be characterised automatically from the determining equations by constructing and condensing a certain directed graph. The digraph suggests a block elimination ranking to assist in solving the determining system. The methods work in both finite- and infinite-dimensional cases.

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I will discuss Elie Cartan's paper "Les sous-groupes des groupes continus de transformations. Annales scientifiques de l'Ecole Normale Suprieure, Sr. 3, 25 (1908), p. 57-194" and relate it to the equivalence problem of partial differential equations completely characterized by their symmetries.