Recognizing sparse perfect elimination bipartite graphs

When applying Gaussian elimination to a sparse matrix, it is desirable to avoid turning zeros into non-zeros to preserve the sparsity. The class of perfect elimination bipartite graphs is closely related to square matrices that Gaussian elimination can be applied to without turning any zero into a non-zero. Existing literature on the recognition of this class and finding suitable pivots mainly focusses on time complexity. For $n \times n$ matrices with m non-zero elements, the currently best known algorithm has a time complexity of $O(n^3/\log n)$. However, when viewed from a practical perspective, the space complexity also deserves attention: it may not be worthwhile to look for a suitable set of pivots for a sparse matrix if this requires $\Omega(n^2)$ space. We present two new algorithms for the recognition of sparse instances: one with a $O(n m)$ time complexity in $\Theta(n^2)$ space and one with a $O(m^2)$ time complexity in $\Theta(m)$ space. Furthermore, if we allow only pivots on the diagonal, our second algorithm can easily be adapted to run in time $O(n m)$

Monetary policy and inflation

Inflation

Bisimplicial edges in bipartite graphs

Bisimplicial edges in bipartite graphs are closely related to pivots in Gaussian elimination that avoid turning zeroes into non-zeroes. We present a new deterministic algorithm to nd such edges in bipartite graphs. The expected time complexity of our new algorithm is $O(n^2 \log n)$ on random bipartite graphs in which each edge is present with a fixed probability p, a polynomial improvement over the fastest algorithm found in the existing literature

A note on perfect partial elimination

In Gaussian elimination it is often desirable to preserve existing zeros (sparsity). This is closely related to perfect elimination schemes on graphs. Such schemes can be found in polynomial time. Gaussian elimination uses a pivot for each column, so opportunities for preserving sparsity can be missed. In this paper we consider a more flexible process that selects a pivot for each nonzero to be eliminated and show that recognizing matrices that allow such perfect partial elimination schemes is NP-hard

Immanence and irreconcilability: on the character of public law as political jurisprudence

In Foundations of Public Law, Martin Loughlin constructs an intricate conceptual triangle made up out of religion, law, and politics, in order to offer an account of the character of public law as a secular, political, jurisprudence. In this essay, I argue that this account takes neither religion nor law seriously - and this in revealingly similar ways. Loughlin’s book presents public law as an irreducibly paradoxical discourse, devoted to sustaining ‘the irreconcilable’ within society. Relative to this discourse, religion - understood by Loughlin as absolutist dogma - appears in the book only as a threat, whereas law - seen as a mere tool - becomes a necessary and innocent means for its support. I offer a critique of both these lines of argument, as not sufficiently attentive to religion and law as fields with their own histories, internal dynamics, and forms of efficacy. Religion, when seen in terms of practice, discipline, and ritual, in fact has much in common with - and considerable resources to offer to - Loughlin’s own vision of public law. And attention to the legalism so deeply embedded in juristic discourse reveals law as posing precisely the absolutist threat that Loughlin fears (but only associates with religion and ‘the social’). Bringing these two lines of argument together, I argue that Loughlin’s own ambition for public law as a prudential discourse of contained irreconcilability, is better served, not by striving for radical - and impossible - immanence, but by acknowledging that discourse’s dual character as a phenomenon both immanent and transcendent. I also suggest that ritual practice may have an important role to play in maintaining this dual, and paradoxical, character

Consistency of a system of equations: What does that mean?

The concept of (structural) consistency also called structural solvability is an important basic tool for analyzing the structure of systems of equations. Our aim is to provide a sound and practically relevant meaning to this concept. The implications of consistency are expressed in terms of explicit density and stability results. We also illustrate, by typical examples, the limitations of the concept

Proportionality in comparative law

orthcoming as an entry on ‘Proportionality’ in the Elgar Encyclopedia of Comparative Law (2nd edition), Jan Smits, Catherine Valcke, Jaakko Husa & Madalena Narciso, eds. Investigations of proportionality’s role in contemporary public law are complicated by the way the topic straddles so many binaries familiar within the discipline of comparative law. These include those of substance and form, discourse and practice, ‘function’ and ‘culture’, and – perhaps most importantly – similarity and difference. Comparative legal scholarship, this entry argues, will have to grapple with the contradictory tasks of simultaneously investigating and questioning proportionality’s (real or purported) hegemony. To this end, the paper presents brief overviews of work concerned with the (1) identification, (2) explanation, (3) interpretation, and (4) critique, of proportionality’s global diffusion and ‘success’