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Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney’s approach to such a case are discussed. Hierarchy of singularities is analyzed by the double-scaling expansion method for the simplest $2$-component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by $(k+1,k+2)$ curves which are of cusp type for $k=2n+1$, $n=1,2,3,\dots .$ Regularization of these singularities by deformation of plane into plane mappings into surface $S^{2+k}(\subset {ℝ}^{2+k})$ to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted. We finally compare the results obtained for the parabolic case with non-generic gradient catastrophes for hyperbolic systems.

The question "What lies beyond the Quantized String or Superstring Theory?" and the question "What lies beyond Quantum Mechanics itself?" might have one common answer: a discretized, classical version of string theory, which lives on a lattice in Minkowski space. The size a of the meshes on this lattice in Minkowski space is determined by the string slope parameter, α′.

Asian options incorporate the average stock price in the terminal payoff. Examination of the Asian option partial differential equation (PDE) has resulted in many equations of reduced order that in general can be mapped into each other, although this is not always shown. In the literature these reductions and mappings are typically acquired via inspection or ad hoc methods. In this paper, we evaluate the classical Lie point symmetries of the Asian option PDE. We subsequently use these symmetries with Lie's systematic and algorithmic methods to show that one can obtain the same aforementioned results. In fact we find a familiar analytical solution in terms of a Laplace transform. Thus, when coupled with their methodic virtues, the Lie techniques reduce the amount of intuition usually required when working with differential equations in finance.

We introduce two types of mappings that preserve nonnull helices in Minkowski spaces. The first type constructs helices in the $n$-dimensional Minkowski space from helices in the same Minkowski space. The second type constructs helices in the $(n+1)$-dimensional Minkowski space from helices in the $n$-dimensional Minkowski space. Furthermore, we study invariants of these mappings and present examples.