we investigate buchbaum and eisenbud's construction of the second symmetric power $s_r(x)$ of a chain complex $x$ of modules over a commutative ring $r$. we state and prove a number of results from the folklore of the subject for which we know of no good direct references. we also provide several explicit computations and examples. we use this construction to prove the following ...

To every subspace arrangement X we will associate symmetric functions P[X] and H[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant H[X] specializes to the Tutte polynomial T [X]. Billera, Jia and Reiner recently introduced ...

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces ...

0. Introduction This paper following a geometric approach proves new, and reproves old, vanishing and nonvanishing results on the space of twisted symmetric differentials, H 0 (X, S m Ω 1 X ⊗ O X (k)) with k ≤ m, on subvarieties X ⊂ P N. The case of k = m is special and the nonvanishing results are related to the space of quadrics containing X and lead to interesting geometrical objects associa...

Addressing covariate imbalance in causal analysis will be reformulated as an elimination of the nuisance variables problem. We show, within a counterfactual balanced setting, how averaging, conditioning, and marginalization techniques can be used to reduce bias due to a possibly large number of imbalanced baseline confounders. The notions of X-sufficient and X-ancillary quantities are discussed...

This paper is concerned with the approximation of the effective conductivity σ(A) associated to an elliptic operator ∇xA(x, η)∇x where for x ∈ R d, d ≥ 1, A(x, η) is a bounded elliptic random symmetric d × d matrix and η takes value in an ergodic probability space. Writing AN (x, η) the periodization of A(x, η) on the torus T d N of dimension d and side N we prove that η-a.s. lim N→+∞ σ(A (x, η...

Matrix transpose in parallel systems typically involves costly all-to-all communications. In this paper, we provide a comparative characterization of various efficient algorithms for transposing small and large matrices using the popular symmetric multiprocessors (SMP) architecture, which carries a relatively low communication cost due to its large aggregate bandwidth and lowlatency inter-proce...