It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P , RP is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreove...

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Parameterized telescoping Given f1(k), ....

A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, mixed characteristic minimal model program, Cohen–Macaulayness absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem whether property spreads from p...

Let f be a subhyperbolic entire transcendental function of finite order and let z0 be a repelling periodic point of f . We show that there exists at least one dynamic ray (injective curve to ∞ consisting of escaping points) that lands at z0. In fact, our result holds more generally, namely for any subhyperbolic entire function f for which each periodic address is realised by some dynamic ray in...

A graph G has a representation modulo r if there exists an injective map f : V (G)→ {0, 1, ..., r − 1} such that vertices u and v are adjacent if and only if f(u) − f(v) is relatively prime to r. The representation number rep(G) is the smallest r such that G has a representation modulo r. Following earlier work on stars, we study representation numbers of complete bipartite graphs and more gene...

A graph G has a representation modulo n if there exists an injective map f : V (G) → {0, 1, . . . , n} such that vertices u and v are adjacent if and only if |f(u) − f(v)| is relatively prime to n. The representation number rep(G) is the smallest n such that G has a representation modulo n. We present new results involving representation numbers for stars, split graphs, complements of split gra...

Let F be a family of graphs and let d be large enough. For every d-regular graph G, we study the existence of a spanning F -free subgraph of G with large minimum degree. This problem is wellunderstood if F does not contain bipartite graphs. Here we provide asymptotically tight results for many families of bipartite graphs such as cycles or complete bipartite graphs. To prove these results, we s...