Algorithm for Testing the Leibniz Algebra Structure

Algorithmic method to obtain combinatorial structures associated with Leibniz algebras

“…Hence, studying the computational data for this family is relevant enough; especially if we take into consideration that the study of complexity for Steps 1 to 3 has been done by considering any possible configuration (not necessarily a (pseudo)digraph), since our algorithm runs fine even if the Leibniz algebra is associated with a configuration different from a (pseudo)digraph. As we have commented previously, when the edge weight of the (pseudo)digraph (or equivalently the structure constants of the Leibniz algebra) consists of fixed (non-parametric) values, the computing time (and consequently, the complexity) of the algorithm here presented can be reduced by using the algorithm given in [4]. However, this is not the case considered in this paper in which the weights (i.e.…”

“…By using the algorithm given in [4], we propound a very efficient algorithm to deal with the converse problem to that proposed in this section: testing directly if a given (pseudo)digraph is associated with a Lie or Leibniz algebra, instead of computing the combinatorial structure associated with a Leibniz algebra. More concretely, we consider a (pseudo)digraph as input determined starting from its vertex-set and edge weights (which are not parameters, but fixed values).…”

“…This involves to use an algorithm less efficient than others existing in the literature. For example, the algorithm introduced by Casas et al [4] to test if a given law corresponds to a Leibniz or Lie algebra can be used as replacement of the three first steps of our algorithm. This alternative is more efficient because it uses Gröbner bases to decide if the law corresponds to a Lie or Leibniz algebra.…”

“…More concretely, we consider a (pseudo)digraph as input determined starting from its vertex-set and edge weights (which are not parameters, but fixed values). Starting from the weights, the law of the possible Leibniz (or Lie) algebra is constructed by reversion in the association explained in Section 3 and hence we can apply the algorithm in [4] to this law in order to check if it corresponds to a Leibniz (or Lie) algebra or, on the contrary, it is not a law of such a type of algebra.…”

Separation of Powers: Indiana Constitutional Law to the Forefront -- 2021-2022

“…ARTICLE 4, SECTION 24 -RIGHT TO SUE THE STATE In Ladra v. State, 241 the Indiana Supreme Court modified its rule in Catt v. Board of Commissioners, 242 concluding "when the government knows of an existing defect in a public thoroughfare, and when it has ample opportunity to respond, immunity does not apply simply because the defect manifests during recurring inclement weather." 243 In making this determination, the court analyzed the history of actions against the state, including Article 4, Section 24 of the Indiana Constitution, which authorizes "general statutes permitting a party to…”