Adjoining to the language of rings function symbols for splitting coefficients, relative $p$-coordinate functions, and division predicate a valuation, some theories pseudo-algebraically closed non-trivially valued fields admit quantifier elimination. It is also shown that in same theory given exponent imperfection does not elimination, due Galois theoretic obstructions.

In (Arch. Math. 57 (1991), pp. 446–455), R. Farré proved a positivstellensatz for real-series closed fields. Here we consider p-valued fields 〈K, vp〉 with a non-trivial valuation v which satisfies a compatibility condition between vp and v. We use this notion to establish the p-adic analogue of real-series closed fields; these fields are called henselian residually p-adically closed fields. Fir...

This paper studies nonlinear deformations of the linear gauge theory of any number of spin-2 and spin-3/2 fields with general formal multiplication rules in place of standard Grassmann rules for manipulating the fields, in four spacetime dimensions. General possibilities for multiplication rules and coupling constants are simultaneously accommodated by regarding the set of fields equivalently a...

This paper studies the stability of complex-valued nonlinear differential system. The stability criteria of complex-valued nonlinear autonomous system are established. For the general complex-valued nonlinear non-autonomous system, the comparison principle in the context of complex fields is given. Those derived stability criteria not only provide a new method to analyze complex-valued differen...

This book addresses a gap in the model-theoretic understanding of valued fields that has, until now, limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a st...

We develop a level set method for the computation of multi-valued velocity and electric fields of one-dimensional Euler-Poisson equations. The system of these equations arises in the semiclassical approximation of SchrödingerPoisson equations and semiconductor modeling. This method uses an implicit Eulerian formulation in an extended space — called field space, which incorporates both velocity ...

In this thesis, we propose one approach to compute the multi-valued density via the existing level set method in computing multi-valued velocity of WKB system in high frequency wave dynamics and a superposition property is proved and numerically validated, too. Several novel level set methods are developed and analyzed for computing multi-valued solutions to the Euler-Poisson equations and the ...