Counting Proper Mergings of Chains and Antichains

A proper merging of two disjoint quasi-ordered sets $P$ and $Q$ is a quasi-order on the union of $P$ and $Q$ such that the restriction to $P$ and $Q$ yields the original quasi-order again and such that no elements of $P$ and $Q$ are identified. In this article, we consider the cases where $P$ and $Q$ are chains, where $P$ and $Q$ are antichains, and where $P$ is an antichain and $Q$ is a chain. We give formulas that determine the number of proper mergings in all three cases, and introduce two new bijections from proper mergings of two chains to plane partitions and from proper mergings of an antichain and a chain to monotone colorings of complete bipartite digraphs. Additionally, we use these bijections to count the Galois connections between two chains, and between a chain and a Boolean lattice respectively.Comment: 36 pages, 15 figures, 5 table

Asymptotics of the Euler number of bipartite graphs

We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing the Euler number of certain subgraphs of the Cartesian product of a graph $G$ with the path $P_m$ in terms of self adjoint operators. The asymptotic expansion of the Euler number is given in terms of the eigenvalues of the associated operator. For two classes of graphs, the comb graphs and the Cartesian product $P_2 \Box P_m$, we numerically solve the eigenvalue problem.Comment: 13 pages, 6 figure, submitted to JCT

Community detection for correlation matrices

A challenging problem in the study of complex systems is that of resolving, without prior information, the emergent, mesoscopic organization determined by groups of units whose dynamical activity is more strongly correlated internally than with the rest of the system. The existing techniques to filter correlations are not explicitly oriented towards identifying such modules and can suffer from an unavoidable information loss. A promising alternative is that of employing community detection techniques developed in network theory. Unfortunately, this approach has focused predominantly on replacing network data with correlation matrices, a procedure that tends to be intrinsically biased due to its inconsistency with the null hypotheses underlying the existing algorithms. Here we introduce, via a consistent redefinition of null models based on random matrix theory, the appropriate correlation-based counterparts of the most popular community detection techniques. Our methods can filter out both unit-specific noise and system-wide dependencies, and the resulting communities are internally correlated and mutually anti-correlated. We also implement multiresolution and multifrequency approaches revealing hierarchically nested sub-communities with `hard' cores and `soft' peripheries. We apply our techniques to several financial time series and identify mesoscopic groups of stocks which are irreducible to a standard, sectorial taxonomy, detect `soft stocks' that alternate between communities, and discuss implications for portfolio optimization and risk management.Comment: Final version, accepted for publication on PR

Pitfalls in QCA's consistency measure

Background. As a consequence of difficulties in movement initiation and execution, people with Parkinson's disease (PD) are typically encouraged to consciously monitor and control the mechanics of their actions. This is described as 'reinvestment' and has been shown to help mediate effective motor output. Paradoxically, in situations where people with PD are particularly motivated to move effectively, reinvestment may exacerbate existing movement deficits. Objective. To examine the propensity for reinvestment in a sample of people with PD. Methods. A volunteer sample of 55 people with PD was asked to complete a previously validated measure, the Reinvestment Scale. A sub-sample (and age matched controls) was asked to complete a recently developed, movement specific, version of the Scale. Data was collected on Mini Mental State Examination and the Hoehn & Yahr Scale. Participant demographics, including age of onset and duration of disease, were also collated. Results. Multiple regression analyses showed that duration of disease was associated with reinvestment score on both the Reinvestment Scale and the Movement Specific Reinvestment Scale. Conclusions. Participants appeared to become more aware of the mechanics of their actions over time. Possible explanations for this finding are discussed with reference to rehabilitation. Copyright © 2007 The American Society of Neurorehabilitation.link_to_subscribed_fulltex

Good faith and fair dealing as an underenforced legal norm

American contract law includes a duty of good faith and fair dealing in the performance of every contract. The duty appears, on first reading, to authorize judges to attach sanctions whenever one party to a contract acts unreasonably towards another. But judicial practice very often falls short of such an expansive standard. This article proposes a novel interpretation of the doctrine that accommodates both the rhetoric of good faith and fair dealing and the reality of judicial enforcement. Good faith and fair dealing, the article contends, is an underenforced legal norm. The duty is valid as a legal norm to the fullest extent, even though courts engage only in partial enforcement of that norm. This article is the first to bring the idea of underenforced legal norms into private law, drawing on the extensive literature on underenforced legal norms in constitutional law, and on analogous ideas in corporate law. The article explores the reasons why legislatures and courts might want to announce a duty whose scope extends beyond what the courts enforce. In private law, as elsewhere, the underenforcement idea allows courts to lend their expressive support to the broader norm while avoiding the negative side effects that attempted full enforcement would entail

Mesoscopic Community Structure of Financial Markets Revealed by Price and Sign Fluctuations

The mesoscopic organization of complex systems, from financial markets to the brain, is an intermediate between the microscopic dynamics of individual units (stocks or neurons, in the mentioned cases), and the macroscopic dynamics of the system as a whole. The organization is determined by "communities" of units whose dynamics, represented by time series of activity, is more strongly correlated internally than with the rest of the system. Recent studies have shown that the binary projections of various financial and neural time series exhibit nontrivial dynamical features that resemble those of the original data. This implies that a significant piece of information is encoded into the binary projection (i.e. the sign) of such increments. Here, we explore whether the binary signatures of multiple time series can replicate the same complex community organization of the financial market, as the original weighted time series. We adopt a method that has been specifically designed to detect communities from cross-correlation matrices of time series data. Our analysis shows that the simpler binary representation leads to a community structure that is almost identical with that obtained using the full weighted representation. These results confirm that binary projections of financial time series contain significant structural information.Comment: 15 pages, 7 figure

“Come, what, a siege?” : Metarepresentation in Lady Jane Cavendish and Lady Elizabeth Brackley’s The Concealed Fancies

The Concealed Fancies is a play written by Lady Jane Cavendish and Lady Elizabeth Brackley, the two eldest daughters of William Cavendish, marquis (and later duke) of Newcastle, during the English Civil War. We can feel reasonably certain that the sisters wrote it in the hope that it could be performed; however, that would have required the presence of their father, whose return is the climax of the story, from the Continental exile to which he had fled after his comprehensive defeat at the Battle of Marston Moor. Closet drama was in any case a genre with which the play had several features in common, and it is at least possible that the sisters envisaged any performance of the play at one of the two family homes of Bolsover or Welbeck as taking place in a promenade style which would have involved moving to an actual closet for part of this scene. This link between the literal, the symbolic, and the material conditions of possible performance is typical of the radical metatheatricality which characterises and configures the play. In this essay, which like the play itself is co-authored, we combine historicist and formalist approaches to the play with a cognitive pragmatic account of its verbal and non-verbal interactions to fully reveal the extent to which the play’s internal logic writes back against the external circumstances of its authors, and to highlight what a skilled and sophisticated piece of writing it is. First, we introduce theoretical frameworks from cognitive psychology and pragmatics which are used later to explore the intricacies of cognition and communication in the play. After this we offer an account of the play in its wider context before examining the detail and nature of its metarepresentations. The article ends by drawing connections between the play’s apparent preoccupation with metarepresentation and the historical context and conditions in which it was written

Spectrum of injuries associated with paediatric ACL tears: an MRI pictorial review

OBJECTIVE: Magnetic resonance imaging (MRI) findings in anterior cruciate ligament (ACL) injury are well known, but most published reviews show obvious examples of associated injuries and give little focus to paediatric patients. Here, we demonstrate the spectrum of MRI appearances at common sites of associated injury in adolescents with ACL tears, emphasising age-specific issues. METHODS: Pictorial review using images from children with surgically confirmed ACL tears after athletic injury. RESULTS: ACL injury usually occurs with axial rotation in the valgus near full extension. The MRI findings can be obvious and important to management (ACL rupture), subtle but clinically important (lateral meniscus posterior attachment avulsion), obvious and unimportant to management (femoral condyle impaction injury), or subtle and possibly important (medial meniscocapsular junction tear). Paediatric-specific issues of note include tibial spine avulsion, normal difficulty visualising a thin ACL and posterolateral corner structures, and differentiation between incompletely closed physis and impaction fracture. CONCLUSION: ACL tear is only the most obvious sign of a complex injury involving multiple structures. Awareness of the spectrum of secondary findings illustrated here and the features distinguishing them from normal variation can aid in accurate assessment of ACL tears and related injuries, enabling effective treatment planning and assessment of prognosis. TEACHING POINTS: • The ACL in children normally appears thin or attenuated, while thickening and oedema suggest tear. • Displaced medial meniscal tears are significantly more common later post-injury than immediately. • The meniscofemoral ligaments merge with the posterior lateral meniscus, complicating tear assessment. • Tibial plateau impaction fractures can be difficult to distinguish from a partially closed physis. • Axial MR sequences are more sensitive/specific than coronal for diagnosis of medial collateral ligament (MCL) injury