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Notes on Differential Algebra by Reid Dale
This PDF book covers the following topics related to Differential Algebra : Basic Differential Algebra, Derivations and Dual Numbers, Differential Ideals and Ritt Noetherianity, Characteristic Sets and the Partial Ritt-Raudenbush, Basic Differential Algebraic Geometry: Properties of the Kolchin Topology, Differentially Closed Fields, Differential Dimension Polynomials, Differential Galois Theory, Binding Groups and Internality, Pillay’s X-strongly-normal theory, Galois Theory of Linear Differential Equations, Algebraic D-Groups and Logarithmic Derivatives, Constrained Cohomology, The Galois Groupoid, Differential Algebraic Groups, Preliminaries from Model Theory.
Author(s): Reid Dale
47 Pages 
Differential Geometry by Balazs Csikos
The aim of this textbook is to give an introduction to differential geometry. Topics covered includes: Categories and Functors, Linear Algebra, Geometry, Topology, Multivariable Calculus, Ordinary Differential Equations, The Notion of a Curve, The Length of a Curve, Plane Curves, Osculating Spheres, Hypersurfaces in R n, Manifolds, Differentiation of Vector Fields and Integration of Differential Forms.
Author(s): Balazs Csikos
354 Pages
Stability criteria for nonlinear fully implicit differential algebraic systems
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.
Author(s): Michael Menrath
191 Pages
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.
Author(s): Columbia University
57 Pages
Asymptotic Differential Algebra
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.
Author(s): Matthias Aschenbrenner
38 PagesAdvertisement
