We find that a compatible graded left-symmetric algebra structure on the Witt algebra induces an indecomposable module of the Witt algebra with 1-dimensional weight spaces by its left multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebra structures on the Witt algebra are classified. All of them are simple and they in...

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph which is reducible by some finite sequence of these moves, to a graph with no edges is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycl...

We use weights on objects in an abelian category to define what we call a path metric. introduce three special classes of weight: those compatible with short exact sequences; induced by their metric; and which bound prove that these conditions are fact equivalent, such exact. As case metric, obtain distance for generalized persistence modules whose indexing is measure space. this Wasserstein di...

We present the programming language Keris, an extension of Java with explicit support for software evolution. Keris introduces extensible modules as the basic building blocks for software. Modules are composed hierarchically revealing explicitly the architecture of systems. A distinct feature of the module design is that modules do not get linked manually. Instead, the wiring of modules gets in...

Standard ML is a statically typed programming language that is suited for the construction of both small and large programs. \Programming in the small" is captured by Standard ML's Core language. \Programming in the large" is captured by Standard ML's Modules language that provides constructs for organising related Core language deenitions into self-contained modules with descriptive interfaces...

In this article, we study certain local cohomology modules over F-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula computing these invariants when ring is standard graded and, by its means, provide new examples tables. associated primes Ext-modules, showing that they are all compatible ideals. Finally, focus on properties numbers detect glo...