We introduce normal cores, as well the more general action in context of a semi-abelian category, and further generalise those to split extension cores homological category. prove that, if category is moreover well-powered with (small) joins, then existence equivalent condition that change-of-base functors fibration points are geometric. call finitely complete satisfies this an algebraic logos....

Abstract In this chapter we briefly introduce important concepts from homology theory and highlight how some results Chap. 3 can be understood through the lens of algebraic topology. This accumulates in showing that Euler characteristic by means CW complexes, or intersection theory.

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Differential-algebraic equations (DAEs) arise in a variety of applications. Their analysis and numerical treatment, therefore, plays an important role in modern mathematics. The paper gives an introduction to the topics of DAEs. Examples of DAEs are considered showing their importance for practical problems. Some essential concepts that are really essential for understanding the DAE systems are...

This paper studies the contraction properties of nonlinear differential-algebraic equation (DAE) systems. Specifically we develop scalable techniques for constructing the attraction regions associated with a particular stable equilibrium, by establishing the relation between the contraction rates of the original systems and the corresponding virtual extended systems. We show that for a contract...

It is shown that the structural index of a linear differential–algebraic equation (DAE) with constant coefficients and with index 1 may be arbitrarily high, contrary to a previous result from the literature. This demonstrates that Pantelides’ algorithm applied to DAEs of index 1 may perform an arbitrarily high number of iterations and differentiations.