We prove a Lefschetz formula L(T ) = ∑ x∈F iT (x) for graph endomorphisms T : G→ G, where G is a general finite simple graph and F is the set of simplices fixed by T . The degree iT (x) of T at the simplex x is defined as (−1)dim(x)sign(T |x), a graded sign of the permutation of T restricted to the simplex. The Lefschetz number L(T ) is defined similarly as in the continuum as L(T ) = ∑ k(−1)tr...

An amino acid sequence of a protein may be decomposed into consecutive overlapping strings of length K. How unique is the converse, i.e., reconstruction of amino acid sequences using the set of K-strings obtained in the decomposition? This problem may be transformed into the problem of counting the number of Eulerian loops in an Euler graph, though the well-known formula must be modified. By ex...

We compute (algebraically) the Euler characteristic of a complex of sheaves with constructible cohomology. A stratified Poincaré-Hopf formula is then a consequence of the smooth Poincaré-Hopf theorem and of additivity of the Euler-Poincaré characteristic with compact supports, once we have a suitable definition of index. AMS classification: 55N33 57R25

We construct sequences of finite sums [Formula: see text] and [Formula: see text] converging increasingly and decreasingly, respectively, to the Euler-Mascheroni constant γ at the geometric rate 1/2. Such sequences are easy to compute and satisfy complete monotonicity-type properties. As a consequence, we obtain an infinite product representation for [Formula: see text] converging in a monotone...