The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the matrix and its adjoint are available or the entries of the matrix can be sampled individually. For an N ×N matrix, the resulting factorization is a product of O...

In a paper of 1974, Brian Day employed a notion of factorization system in the context of enriched category theory, replacing the usual diagonal lifting property with a corresponding criterion phrased in terms of hom-objects. We set forth the basic theory of such enriched factorization systems. In particular, we establish stability properties for enriched prefactorization systems, we examine th...

A simple definition of torsion theory is presented, as a factorization system with both classes satisfying the 3–for–2 property. Comparisons with the traditional notion are given, as well as connections with the notions of fibration and of weak factorization system, as used in abstract homotopy theory.

classes of‎ ‎abaffy-broyden-spedicato (abs) methods have been introduced for‎ ‎solving linear systems of equations‎. ‎the algorithms are powerful methods for developing matrix‎ ‎factorizations and many fundamental numerical linear algebra processes‎. ‎here‎, ‎we show how to apply the abs algorithms to devise algorithms to compute the wz and zw‎ ‎factorizations of a nonsingular matrix as well as...

classes of‎ ‎abaffy-broyden-spedicato (abs) methods have been introduced for‎ ‎solving linear systems of equations‎. ‎the algorithms are powerful methods for developing matrix‎ ‎factorizations and many fundamental numerical linear algebra processes‎. ‎here‎, ‎we show how to apply the abs algorithms to devise algorithms to compute the wz and zw‎ ‎factorizations of a nonsingular matrix as well as...

Graph bundles generalize the notion of covering graphs and graph products. In Imrich et al. (Discrete Math. 167=168 (1998) 393) authors constructed an algorithm that 5nds a presentation as a nontrivial cartesian graph bundle for all graphs that are cartesian graph bundles over triangle-free simple base using the relation ∗ having the square property. An equivalence relation R on the edge set of...

We extend recent results on the quantum Hoeffding bound from the i.i.d. setting to a setting when both the null-hypothesis and the counter-hypothesis can be correlated states of a spin chain. The key property in the extension is a certain factorization property of the states. Typical examples with this property are the global Gibbs states of translationinvariant finite-range interactions.

We argue that the evolution of coloured partons into colour-singlet hadrons has approximate factorization into an extended parton-shower phase and a colour-singlet resonance–pole phase. The amplitude for the conversion of colour connected partons into hadrons necessarily resembles Regge-pole amplitudes since qq̄ resonance amplitudes and Regge-pole amplitudes are related by duality. A ‘Regge-casc...

We study the interpolation properties of weakly compact bilinear operators by real method and also complex method. factorization property through reflexive Banach spaces.