An Abstraction of Whitney's Broken Circuit Theorem

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $\sum_{A\subseteq S} f(A)$ where $S$ is a finite set and $f$ is a mapping from the power set of $S$ into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the M\"obius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical M\"obius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).Comment: 18 page

An Inductive Proof of Whitney's Broken Circuit Theorem

We present a new proof of Whitney's broken circuit theorem based on induction on the number of edges and the deletion-contraction formula

On the Number of Regular Elements in Zn\mathbb{Z}_n

We give a combinatorial proof to a formula by Alkam, Osba and T\'oth on the number of regular elements in $\mathbb{Z}_n$ based on the inclusion-exclusion principle

Higher Education Landscape 2030

This open access Springer Brief provides a systematic analysis of current trends and requirements in the areas of knowledge and competence in the context of the project “(A) Higher Education Digital (AHEAD)—International Horizon Scanning / Trend Analysis on Digital Higher Education.” It examines the latest developments in learning theory, didactics, and digital-education technology in connection with an increasingly digitized higher education landscape. In turn, this analysis forms the basis for envisioning higher education in 2030. Here, four learning pathways are developed to provide a glimpse of higher education in 2030: Tamagotchi, a closed ecosystem that is built around individual students who enter the university soon after secondary education; Jenga, in which universities offer a solid foundation of knowledge to build on in later phases; Lego, where the course of study is not a monolithic unit, but consists of individually combined modules of different sizes; and Transformer, where students have already acquired their own professional identities and life experiences, which they integrate into their studies. In addition, innovative practice cases are presented to illustrate each learning path