Einstein's theory of general relativity has not only proven to be physically accurate [1], it also sets a standard for mathematical beauty and elegance—geometrically. When viewed as a dynamical system of equations for evolving initial data, however, these equations have a serious flaw: they cannot be proven to be well-posed (except in special coordinates [2–4]). That is, they do not produce uni...

Let X be the Dynkin diagram of a symmetrizable Kac-Moody algebra, and X0 a subgraph with all vertices of degree 1 or 2. Using the crystal structure on the components of quiver varieties for X , we show that if we expand X by extending X0, the branching multiplicities and tensor product multiplicities stabilize, provided the weights involved satisfy a condition which we call “depth” and are supp...

In this paper, we develop a finite element model for a cable suspended in water. Global existence and uniqueness of solutions of the truncated system is shown for a slightly simplified equation describing the motion of a cable having negligible added mass and supported by fixed end-points. Based on this, along with well known results on local existence and uniqueness of solutions for symmetriza...

The Littelmann path model gives a realisation of the crystals of integrable representations of symmetrizable Kac-Moody Lie algebras. Recent work of Gaussent-Littelmann [GL] and others [BG] [GR] has demonstrated a connection between this model and the geometry of the loop Grassmanian. The alcove walk model is a version of the path model which is intimately connected to the combinatorics of the a...

The tetrad-based equations for vacuum gravity published by Estabrook, Robinson, and Wahlquist are simplified and adapted for numerical relativity. We show that the evolution equations as partial differential equations for the Ricci rotation coefficients constitute a rather simple first-order symmetrizable hyperbolic system, not only for the Nester gauge condition on the acceleration and angular...

We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity ε, ut +F (u )x = ε(B(u)ux)x. When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that uε converges to a solution of the inviscid system before the formation of shocks if th...

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and MalekMadani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic–parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributio...

The analog of the principal SO(3) subalgebra of a finite dimensional simple Lie algebra can be defined for any hyperbolic Kac Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact group SO(1, 2). We exhibit the decomposition of g(A) into representations of SO(1, 2); with the exception of the adjoint SO(1, 2) algebra itself, all of these represent...

A cluster algebra is a commutative whose structure decided by skew-symmetrizable matrix or quiver. When invariant under an action of finite group and this admissible, the folded obtained from original one. Any non-simply-laced affine type can be folding simply-laced with specific $G$-action. In paper, we study combinatorial properties quivers in type. We prove that for any quiver type, $G$-inva...