On Balance for Relative Homology

The Hodge spectrum is an important analytic invariant of singularities encoding the Hodge filtration and the monodromy of the Milnor fiber. However, explicit formulas exist in only a few cases. In this article, the main result is a combinatorial formula for the Hodge spectrum of any homogeneous polynomials in three variables whose zero locus is a projective curve arrangement having only ordinary multiple points.     

Strongly Flat Modules over Matlis Domains

Strongly flat modules were introduced by Bazzoni–Salce [3 Bazzoni , S. , Salce , L. ( 2003 ). Almost perfect domains . Colloq. Math. 95 : 285 – 301 .[Crossref] , [Google Scholar]] and used to characterize almost perfect domains. Here we wish to study strongly flat modules, more generally, over Matlis domains; these are integral domains R such that the field of quotients Q has projective dimension 1. In Section 2, criteria are proved for strong flatness. We also prove that over arbitrary domains, strongly flat submodules of projective modules are projective (Theorem 3.2), in particular, strongly flat ideals are projective (Corollary 3.4) and use these results to show that the strongly flat dimension (which makes sense over Matlis domains) coincides with the projective dimension whenever it is > 1.     

Dimension Formulas for the Free Nonassociative Algebra

The free nonassociative algebra has two subspaces which are closed under both the commutator and the associator: the Akivis elements and the primitive elements. Every Akivis element is primitive, but there are primitive elements which are not Akivis. Using a theorem of Shestakov, we give a recursive formula for the dimension of the Akivis elements. Using a theorem of Shestakov and Umirbaev, we prove a closed formula for the dimension of the primitive elements. These results generalize the Witt dimension formula for the Lie elements in the free associative algebra.     

Unified min-max and interlacing theorems for linear operators

There exist striking analogies in the behaviour of eigenvalues of Hermitian compact operators, singular values of compact operators and invariant factors of homomorphisms of modules over principal ideal domains, namely diagonalization theorems, interlacing inequalities and Courant–Fischer type formulae. Carlson and Sá [D. Carlson and E.M. Sá, Generalized minimax and interlacing inequalities, Linear Multilinear Algebra 15 (1984) pp. 77–103.] introduced an abstract structure, the s-space, where they proved unified versions of these theorems in the finite-dimensional case. We show that this unification can be done using modular lattices with Goldie dimension, which have a natural structure of s-space in the finite-dimensional case, and extend the unification to the countable-dimensional case.     

Circular‐Arc Bigraphs and Its Subclasses

In this article, we obtain two new characterizations of circular‐arc bigraphs. One of them is the representation of a circular‐arc bigraph in terms of two two‐clique circular‐arc graphs while another one represents the same as a union of an interval bigraph and a Ferrers bigraph. Finally, we introduce the notions of proper and unit circular‐arc bigraphs, characterize them and show that, as in the case of circular‐arc graphs, unit circular‐arc bigraphs form a proper subclass of the class of proper circular‐arc bigraphs.     

Using thresholding difference-based estimators for variable selection in partial linear models

A commonly used semiparametric model is considered. We adopt two difference based estimators of the linear component of the model and propose corresponding thresholding estimators that can be used for variable selection. For each thresholding estimator, variable selection in the linear component is developed and consistency of the variable selection procedure is shown. We evaluate our method in a simulation study and implement it on a real data set.