A Turaev–Viro invariant is a state sum, i.e., a polynomial that can be read off from a special spine or a triangulation of a compact 3-manifold. If the polynomial is evaluated at the solution of a certain system of polynomial equations (Biedenharn–Elliott equations) then the result is a homeomorphism invariant of the manifold (" numerical Turaev-Viro invariant "). The equation system defines an...

We study the effect of rotation on life-span solutions to 3D hydrostatic Euler equations with and inviscid Primitive (PEs) torus. The space analytic functions appears be natural initial value problem for PEs general data, as they have been recently shown exhibit Kelvin–Helmholtz type instability. First, a short interval time that is independent rate $$|\Omega |$$ , we establish local well-posed...

Smale’s α-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale presented a bound for the higher order derivatives of a system of polynomial equations based in part on the degrees of the equations. For a given system of polynomial-...

The linearized Primitive Equations with vanishing viscosity are considered. Some new boundary conditions (of transparent type) are introduced in the context of a modal expansion of the solution which consist of an infinite sequence of integral equations. Applying the linear semi-group theory, existence and uniqueness of solutions is established. The case with nonhomogeneous boundary values, enc...

It is well known that a system of power polynomial equations can be reduced to a single-variable polynomial equation by exploiting the so-called Newton’s identities. In this work, by further exploring Newton’s identities, we discover a binomial decomposition rule for composite elementary symmetric polynomials. Utilizing this decomposition rule, we solve three types of systems of composite power...