We extend the notion of absolute subsets Betti moduli spaces smooth algebraic varieties to case normal varieties. As a consequence we prove that twisted cohomology jump loci in rank one over variety are finite union translated subtori. show same holds for by unitary local system where underlying X is projective with H1(X,Q) pure weight one. Lastly, study interaction these Hodge theoretic data n...

We describe an algorithm for classifying the closed subsets of a root system, up to conjugation by associated Weyl group. Our is implemented in language computer algebra system GAP 4. discuss implementation and give runtimes on some sample inputs. The classification irreducible closely related regular subalgebras, inner automorphism, corresponding simple Lie algebra. show how obtain subalgebras...

In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts ...

We hypothesize that optimal deep neural networks (DNN) class-conditional posterior probabilities live in a union of lowdimensional subspaces. In real test conditions, DNN posteriors encode uncertainties which can be regarded as a superposition of unstructured sparse noise over the optimal posteriors. We aim to investigate different ways to structure the DNN outputs by exploiting low-rank repres...

The rank ordered distribution of the codon usage frequencies for 123 bacteriae is best fitted by a three parameters function that is the sum of a constant, an exponential and a linear term in the rank n. The parameters depend (two parabolically) from the total GC content. The rank ordered distribution of the amino acids is fitted by a straight line. The Shannon entropy computed over all the cod...

We hypothesize that optimal deep neural networks (DNN) class-conditional posterior probabilities live in a union of lowdimensional subspaces. In real test conditions, DNN posteriors encode uncertainties which can be regarded as a superposition of unstructured sparse noise over the optimal posteriors. We aim to investigate different ways to structure the DNN outputs by exploiting low-rank repres...

In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-fre...

In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-fre...

The query for retrieving the rank of all neighbors of a moving object at any given time, a continuous rank query, is an important case of continuous nearest neighbor (CNN) queries. An application for ranking queries is given by an ambulance driver who needs to keep track of the closest hospitals at all times. We present a set of incremental algorithms that facilitate efficient rank updates for ...