We consider a finite-dimensional vector space $W\subset K^E$ over a field K and a set E. We show that the set $\mathcal {C}(W)\subset 2^E$ of minimal supports of W are the circuits of a matroid on E. When the cardinality of K is large (compared to that of E), then the family of supports of W is a matroid. Afterwards we apply these results to tropical differential algebraic geometry (tdag), studying the set of supports of spaces of formal power series solutions $\text {Sol}(\Sigma )$ of systems of linear differential equations (ldes) $\Sigma$ in variables $x_1,\ldots ,x_n$ having coefficients in . If $\Sigma $ is of differential type zero, then the set $\mathcal {C}(Sol(\Sigma ))\subset (2^{\mathbb {N}^{m}})^n$ of minimal supports defines a matroid on $E=[n]\times \mathbb {N}^{m}$, and if the cardinality of K is large enough, then the set of supports is also a matroid on E. By applying the fundamental theorem of tdag (fttdag), we give a necessary condition under which the set of solutions $Sol(U)$ of a system U of tropical ldes is a matroid. We give a counterexample to the fttdag for systems $\Sigma $ of ldes over countable fields for which is not a matroid.

Let $k$ be field of characteristic zero. Let $f\in k[X,Y]$ be a nonconstant polynomial. We prove that the space of differential (formal) deformations of any formal general solution of the associated ordinary differential equation $f(y^{\prime },y)=0$ is isomorphic to the formal disc $\text{Spf}(k[[Z]])$ .

The design optimization and analysis of charged particle beam systems employing intense beams requires a robust and accurate Poisson solver. This paper presents a new type of Poisson solver which allows the effects of space charge to be elegantly included into the system dynamics. This is done by casting the charge distribution function into a series of basis functions, which are then integrated with an appropriate Green's function to find a Taylor series of the potential at a given point within the desired distribution region. In order to avoid singularities, a Duffy transformation is applied, which allows singularity-free integration and maximized convergence region when performed with the help of Differential Algebraic methods. The method is shown to perform well on the examples studied. Practical implementation choices and some of their limitations are also explored.

In this paper we deal with the model theory of differentially closed fields of characteristic zero with finitely many commuting derivations. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types which are orthogonal to fields. Then we classify the subgroups of the additive group of Lascar rank omega with differential-type 1 which are nonorthogonal to fields. The last parts consist of an analysis of the quotients of the heat variety. We show that the generic type of such a quotient is locally modular. Finally, we answer a question of Phylliss Cassidy about the existence of certain Jordan-Hölder type series in the negative.