This note covers the following
topics: Conventions, Differential graded algebras, Differential graded
modules, The homotopy category, Cones, Admissible short exact sequences,
Distinguished triangles, Cones and distinguished triangles, The homotopy
category is triangulated, Projective modules over algebras, Injective modules
over algebras, P-resolutions, I-resolutions, The derived category, The canonical
delta-functor, Linear categories, Graded categories, Differential graded
categories, Obtaining triangulated categories, Tensor product, 24. Derived
tensor product, Variant of derived tensor product, Characterizing compact
objects and Equivalences of derived categories.

This PDF book covers the following topics related to Differential
Algebra : Foundations of Differential Algebra, the Ring of
Differential Polynomials and Its Ideals, Differential Algebraic Extensions,
Differential Galois Theory, Linear Algebraic Groups, Liouvillian
Extensions.

Author(s): Alexey Ovchinnikov, Maxwell Shapiro, Peter
Thompson

The goal of this note is to contribute to the qualitative theory of
differential-algebraic systems by providing new asymptotic stability criteria
for a class of nonlinear, fully implicit DAEs with tractability index two.
Topics covered includes: State space analysis of differential-algebraic
equations, Properly formulated DAEs with tractability index 2, The state space
form, Index reduction via differentiation, Stability criteria for
differential-algebraic systems, Asymptotic stability of periodic solutions,
Lyapunov’s direct method regarding DAEs.

This note covers the following
topics: Conventions, Differential graded algebras, Differential graded
modules, The homotopy category, Cones, Admissible short exact sequences,
Distinguished triangles, Cones and distinguished triangles, The homotopy
category is triangulated, Projective modules over algebras, Injective modules
over algebras, P-resolutions, I-resolutions, The derived category, The canonical
delta-functor, Linear categories, Graded categories, Differential graded
categories, Obtaining triangulated categories, Tensor product, 24. Derived
tensor product, Variant of derived tensor product, Characterizing compact
objects and Equivalences of derived categories.

This note introduces
both, state some of their basic properties, and explain connections to o-minimal
structures. Also describe a common algebraic framework for these examples: the
category of H-fields. This unified setting leads to a better understanding of
Hardy fields and transseries from an algebraic and model-theoretic perspective.