Minimal Hilbert series for quadratic algebras and the Anick conjecture

We study the question on whether the famous Golod–Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper ‘Generic algebras and CW-complexes’, Princeton Univ. Press, where he proved that the estimate is attained for the number of quadratic relations $d\leq n^2/4$
and $d\geq n^2/2$, and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to $n(n-1)/2$ was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional. We announce here the result that over any infinite field, the Anick conjecture holds for $d \geq 4(n2+n)/9$ and an arbitrary number of generators. We also discuss the result that confirms the Vershik conjecture over any field of characteristic 0, and a series of related
asymptotic results.

The proof of Kontsevich's periodicity conjecture on noncommutative birational transformations

For an arbitrary associative unital ring RR, let J1J1 and J2J2 be the following noncommutative, birational, partly defined involutions on the set M3(R)M3(R) of 3×33×3 matrices over RR: J1(M)=M−1J1(M)=M−1 (the usual matrix inverse) and J2(M)jk=(Mkj)−1J2(M)jk=(Mkj)−1 (the transpose of the Hadamard inverse).

We prove the surprising conjecture by Kontsevich that (J2∘J1)3(J2∘J1)3 is the identity map modulo the DiagL×DiagRDiagL×DiagR action (D1,D2)(M)=D−11MD2(D1,D2)(M)=D1−1MD2 of pairs of invertible diagonal matrices. That is, we show that, for each MM in the domain where (J2∘J1)3(J2∘J1)3 is defined, there are invertible diagonal 3×33×3 matrices D1=D1(M)D1=D1(M) and D2=D2(M)D2=D2(M) such that (J2∘J1)3(M)=D−11MD2(J2∘J1)3(M)=D1−1MD2.

Graded Relative Evidence

Relative Evidential Supports (RES) was developed and justified several years ago as a non-numeric apparatus that allows us to compare evidential supports for alternative conclusions when making a decision. An extension called Graded Relative Evidence (GRE) of the RES concept of pairwise balancing and trading-off of evidence is reported here which keeps its basic features of simplicity and perspicacity but enriches its modelling fidelity by permitting very modest and intuitive variations in degrees of outweighing (which the essentially binary RES does not). The formal justification is very simply based on linkages to RES and to the Dempster - Shafer theory of evidence. The use of the simple extension is illustrated and to a small degree further justified empirically by application to a topical scientific debate about what is called the Congo Crossover Conjecture here. This decision-making instance is chosen because of the wealth of evidence that has been accumulated on both sides of the debate and the range of evidence strengths manifested in it. The conjecture is that the advent of Aids was in the late 1950s in the Congo when a vaccine for polio was allegedly cultivated in the kidneys of chimpanzees which allowed the Aids infection to cross over to humans from primates. © 2005 Springer.

Positronium formation in positron-noble gas collisions

Distorted-wave Born approximation calculations for Ps formation in positron impact on He, Ne, Ar, Kr and Xe are reported for the energy range up to 200 eV. Capture into the n = 1, 2 and 3 states of Ps is calculated explicitly and 1/n(3) scaling is used to estimate capture into states with n > 3. The calculations for the heavier noble gases allow for capture not only from the outer np(6) shell of the atom but also from the first inner ns(2) shell. However, the inner shell capture is found to be very small. Although by no means unambiguous, the calculations provide some support to the conjecture of Larrichia et al. [J. Phys. B 35 (2002) 2525] that the double peak and shoulder structures observed experimentally for Ps formation in Ar, Kr and Xe arise from formation in excited states. (C) 2004 Elsevier B.V. All rights reserved.

Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement

A beam splitter is a simple, readily available device which can act to entangle output optical fields. We show that a necessary condition for the fields at the output of the beam splitter to be entangled is that the pure input states exhibit nonclassical behavior. We generalize this proof for arbitrary (pure or impure) Gaussian input states. Specifically, nonclassicality of the input Gaussian fields is a necessary condition for entanglement of the field modes with the help of a beam splitter. We conjecture that this is a general property of beam splitters: Nonclassicality of the inputs is a necessary condition for entangling fields in a beam splitter.

Graphs of relations and Hilbert series

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by $n$ generators and $\frac {n(n-1)}{2}$ relations for $n \leq 7$. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.

Placing political economy: organising opposition to free trade before the abolition of the Corn Laws

The unfurling of global capitalism – and its attendant effects – has long been fertile intellectual terrain for geographers. But whilst studies of the processes and mechanisms of globalisation undoubtedly assume a talismanic importance in the discipline, geographers, with few exceptions, have left examinations of early economic liberalism to historians. One such critically important episode in the evolution of the liberal economic project was the repeal of the so-called 'Corn Laws' in 1846. Whilst the precise impact of the Manchester-based Anti-Corn Law League (ACLL) continues to be a matter of conjecture, Eric Sheppard has asserted that their particular take on political economy managed to assume a 'truth-like status' and worldwide universality. But the ACLL's campaign represents only one, albeit decisive, stage in the long intellectual and practical struggle between 'protectionists' and the disciples of free trade. Studies of the non-'Manchester' components have tended to focus squarely upon national politics. This paper examines a pivotal attempt in 1838 by Lord Melbourne's Government to experiment with the effective elimination of import duties on fresh fruit. Unlike most agricultural commodities, table fruit was produced in a tightly defined area, thus allowing the Government's experiment to play out, in theory, without national political fallout. Whilst the Government's clandestine actions left little time for a concerted opposition to develop, Kentish fruit growers soon organised. A formidable lobby was forged that drew wide local support yet also evolved beyond the original 'epistemic community'. Whilst the coalition failed in their efforts to reintroduce protective duties, their actions allow us to see how protectionist ideologies and policies were vivified through practices at many different spatial scales and to better understand the complex spatiality of protectionist takes on political economy. Their campaign also changed – at least in the short term – the course of British mercantile policy.

Drivers of sales performance: a contemporary meta-analysis. Have salespeople become knowledge brokers?

It has been 25 years since the publication of a comprehensive review of the full spectrum of salesperformance drivers. This study takes stock of the contemporary field and synthesizes empirical evidence from the period 1982–2008. The authors revise the classification scheme for sales performance determinants devised by Walker et al. (1977) and estimate both the predictive validity of its sub-categories and the impact of a range of moderators on determinant-sales performance relationships. Based on multivariate causal model analysis, the results make two major observations: (1) Five sub-categories demonstrate significant relationships with sales performance: selling-related knowledge (ß=.28), degree of adaptiveness (ß=.27), role ambiguity (ß=-.25), cognitive aptitude (ß=.23) and work engagement (ß=.23). (2) These sub-categories are moderated by measurement method, research context, and salestype variables. The authors identify managerial implications of the results and offer suggestions for further research, including the conjecture that as the world is moving toward a knowledge-intensive economy, salespeople could be functioning as knowledge-brokers. The results seem to back this supposition and indicate how it might inspire future research in the field of personal selling.

Quotients, exactness, and nuclearity in the operator system category

We continue our study of tensor products in the operator system category. We define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products. One of our main goals is to relate these refinements of nuclearity to the Kirchberg conjecture. In particular, we prove that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient. We prove that an operator system is exact for the min tensor product if and only if it is (min,el)-nuclear. We give many characterizations of operator systems that are (min,er)-nuclear, (el,c)-nuclear, (min,el)-nuclear and (el,max)-nuclear. These characterizations involve operator system analogues of various properties from the theory of C*-algebras and operator spaces, including the WEP and LLP.

Neural pathways mediating cross education of motor function

Cross education is the process whereby training of one limb gives rise to enhancements in the performance of the opposite, untrained limb. Despite interest in this phenomenon having been sustained for more than a century, a comprehensive explanation of the mediating neural mechanisms remains elusive. With new evidence emerging that cross education may have therapeutic utility, the need to provide a principled evidential basis upon which to design interventions becomes ever more pressing. Generally, mechanistic accounts of cross education align with one of two explanatory frameworks. Models of the 'cross activation' variety encapsulate the observation that unilateral execution of a movement task gives rise to bilateral increases in corticospinal excitability. The related conjecture is that such distributed activity, when present during unilateral practice, leads to simultaneous adaptations in neural circuits that project to the muscles of the untrained limb, thus facilitating subsequent performance of the task. Alternatively, 'bilateral access' models entail that motor engrams formed during unilateral practise, may subsequently be utilised bilaterally - that is, by the neural circuitry that constitutes the control centres for movements of both limbs. At present there is a paucity of direct evidence that allows the corresponding neural processes to be delineated, or their relative contributions in different task contexts to be ascertained. In the current review we seek to synthesise and assimilate the fragmentary information that is available, including consideration of knowledge that has emerged as a result of technological advances in structural and functional brain imaging. An emphasis upon task dependency is maintained throughout, the conviction being that the neural mechanisms that mediate cross education may only be understood in this context. © 2013 Ruddy and Carson.

Decentralised formation control and stability analysis for cooperative multi-vehicle manoeuvre

Multi-vehicle cooperative formation control problem is an important and typical topic of research on multi-agent system. This paper presents a formation stability conjecture to conceive a new methodology for solving the decentralised multi-vehicle formation control problem. It employs the “extension-decomposition-aggregation” scheme to transform the complex multi-agent control problem into a group of sub-problems which is able to be solved conveniently. Based on this methodology, it is proved that if all the individual augmented subsystems can be stabilised by using any approach, the overall formation system is not only asymptotically but also exponentially stable in the sense of Lyapunov within a neighbourhood of the desired formation. Simulation study on 6-DOF aerial vehicles (Aerosonde UAVs) has been performed to verify the achieved formation stability result. The proposed multi-vehicle formation control strategy can be conveniently extended to other cooperative control problems of multi-agent systems.

Next generation femtocells: An enabler for high efficiency multimedia transmission

The first generation of femtocells is evolving to the next generation with many more capabilities in terms of better utilisation of radio resources and support of high data rates. It is thus logical to conjecture that with these abilities and their inherent suitability for home environment, they stand out as an ideal enabler for delivery of high efficiency multimedia services. This paper presents a comprehensive vision towards this objective and extends the concept of femtocells from indoor to outdoor environments, and strongly couples femtocells to emergency and safety services. It also presents and identifies relevant issues and challenges that have to be overcome in realization of this vision.

Rent: Prostitution and the Irish apartment block

In 1974, pursuing his interest in the infra-ordinary – ‘the banal, the quotidian, the obvious, the common, the ordinary, the back-ground noise, the habitual’ – Georges Perec wrote about an idea for a novel:

‘I imagine a Parisian apartment building whose façade has been removed … so that all the rooms in the front, from the ground floor up to the attics, are instantly and simultaneously visible’.

In Life A User’s Manual (1978) the consummation of this precis, patterns of existence are measured within architectural space with an archaeological sensibility that sifts through narrative and décor, structure and history, services and emotion, the personal and the system, ascribing commensurate value to each. Borrowing methods from Perec, to move somewhere between conjecture, analysis and other documentation and tracing relationships between form, structure, materiality, technology, organisation, tenure and narrative use, this paper interrogates the late twentieth-century speculative apartment block in Britain and Ireland arguing that its speculative and commodified purpose often allows a series of lives that are less than ordinary to inhabit its spaces.