Related papers: Evolutionary forms: The generation of differential…
Extending our previous work on 2D growth for the Laplace equation we study here {\it multidimensional} growth for {\it arbitrary elliptic} equations, describing inhomogeneous and anisotropic pattern formations processes. We find that these…
Equations for dislocation evolution bridge the gap between dislocation properties and continuum descriptions of plastic behavior of crystalline materials. Computer simulations can help us verify these evolution equations and find their…
Conservation laws are time-invariant properties that constrain many physical systems. For systems of chemical reactions, the law of mass conservation constrains how atoms flow between chemical species. Chemical reaction networks can display…
The one-dimensional shallow water equations in Eulerian coordinates are considered. Relations between symmetries and conservation laws for the potential form of the equations, and symmetries and conservation laws in Eulerian coordinates are…
We present a general formalism which allows us to derive the evolution equations describing one-dimensional (1D) and isotropic 2D interfacelike systems, that is based on symmetries, conservation laws, multiple scale arguments, and exploits…
An intriguing unanswered question about the evolution of bilateral animals with internal skeletons is how an internal skeleton evolved in the first place. Computational modeling of the development of bilateral symmetric organisms suggests…
A construction of conservation laws for $\sigma$-models in two dimensions is generalized in the framework of noncommutative geometry of commutative algebras. This is done by replacing the ordinary calculus of differential forms with other…
A method for constructing evolution equations admitting a master symmetry is proposed. Several examples illustrating the method are presented. It is also noted that for certain evolution equations master symmetries can be useful for…
Modeling the spontaneous evolution of morphology in natural systems and its preservation by proportionate growth remains a major scientific challenge. Yet, it is conceivable that if the basic mechanisms of growth and the coupled kinetic…
In the works by the author it has been shown that the conservation laws for material media (the conservation laws for energy, linear momentum, angular momentum, and mass, that establish a balance between the variation of a physical quantity…
Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined…
In the present paper geometric aspects of relationship between non-Noether symmetries and conservation laws in Hamiltonian systems is discussed. It is shown that integrals of motion associated with continuous non-Noether symmetry are in…
Conformal defects -- extended objects in conformal field theories -- carry localised excitations inherited from symmetry currents, known as the displacements and tilts. They capture the linear response of the defect to deformations of its…
We analyse the evolution of cosmological perturbations which leads to the formation of large voids in the distribution of galaxies. We assume that perturbations are spherical and all components of the Universe - radiation, matter and dark…
In this survey on local additive invariants of real and complex definable singular germs we systematically present classical or more recent invariants of different nature as emerging from a tame degeneracy principle. For this goal, we…
A class of dissipative dynamical systems evolving on smooth constraint hypersurfaces endowed with degenerate induced bilinear forms is studied. The intrinsic evolution is generated by constraint--preserving vector fields on manifolds whose…
We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences…
Here we postulate three laws which form a mathematical framework to capture the essence of Darwinian evolutionary dynamics. The second law is most quantitative and is explicitly expressed by a unique form of stochastic differential…
Plasticity is governed by the evolution of, in general anisotropic, systems of dislocations. We seek to faithfully represent this evolution in terms of density-like variables which average over the discrete dislocation microstructure.…
Observations are inconsistent with a homogeneous and isotropic universe with ordinary matter and gravity. The universe is far from exact homogeneity and isotropy at late times, and the effect of the non-linear structures has to be…