The norm of polynomials in large random and deterministic matrices

Let X_N= (X_1^(N), ..., X_p^(N)) be a family of N-by-N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices Y_N =(Y_1^(N), ..., Y_q^(N)), possibly random but independent of X_N, for which the operator norm of P(X_N, Y_N, Y_N^*) converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y_N and of the polynomials P we get for a large class of matrices the "no eigenvalues outside a neighborhood of the limiting spectrum" phenomena. We give examples of diagonal matrices Y_N for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.Comment: 41 pages, with an appendix by D. Shlyakhtenk

Business Cycle Persistence in Developing Countries: How Successful is a DSGE Model with a Vertical Production Chain and Sticky Prices?

It is well documented that business cycles of developed countries are characterised by persistent output fluctuations, and this has been the subject of much theoretical interest. However, the case for developing countries has been somewhat neglected in the literature. This paper addresses this imbalance, revealing that whilst both developed and developing countries exhibit persistent output fluctuations, there is a significant positive relationship between output persistence and level of economic development. This relationship was successfully modelled using a vertical production chain DSGE model (Huang and Liu, 2001). This model lends itself to such an analysis, as by altering the number of production stages (N) it is possible to represent economies at different levels of development. However, calibration of low input-output (γ) parameter values for the US and UK effectively inhibited the model from generating enough persistence to match that observed in these countries. Nonetheless, after abstracting from the US and UK results, there was found to be a strong significant positive relationship between the magnitude of output persistence generated by the model and economic development. A final very significant finding of this analysis is that the model overestimates output persistence in high inflation countries and underestimates output persistence in low inflation countries. This has important implications not only for this model, but also for any economist attempting to construct a business cycle model capable of replicating the observed patterns of output persistence.Output persistence, Vertical production chain, Staggered price contracts, Economic development, Inflation

Developing Country Business Cycles: Characterising the Cycle

Classical business cycles, following Burns and Mitchell (1946), can be defined as the sequential pattern of expansions and contractions in aggregate economic activity. Recently, Harding and Pagan (2002, 2006) have provided an econometric toolkit for the analysis of these cycles, and this has resulted in a recent surge in researchers using these methods to analyse developing country business cycles. However, the existing literature consists of diminutive samples and the majority fail to consider the statistical significance of the concordance statistics. To address this shortfall, this paper examines the business cycle characteristics and synchronicity for thirty-two developing countries. Furthermore, the US, the UK and Japan are included; this provides benchmarks upon which to compare the characteristics of the developing country cycles and also to examine the degree of synchronisation between developed and developing countries. Significantly, this research reveals that business cycles of developing countries are not, as previously believed, significantly shorter than those of the developed countries. However, the amplitude of both expansion and contraction phases tends to be greater in the developing countries. Furthermore a clear relationship between the timing of business cycle fluctuations and periods of significant regional crises, such as the Asian Financial Crisis, is exhibited. However, the more specific timing of the onset of these fluctuations appears to be determined by country-specific factors. Moreover, there are no clear patterns of concordance either within regions or between developed and developing country business cycles.Classical business cycle, Turning points, Synchronisation, Concordance, Contagion, Developing economies

Uniform regular weighted graphs with large degree: Wigner's law, asymptotic freeness and graphons limit

For each $N\geq 1$, let $G_N$ be a simple random graph on the set of vertices $[N]=\{1,2, ..., N\}$, which is invariant by relabeling of the vertices. The asymptotic behavior as $N$ goes to infinity of correlation functions: $$\mathfrak C_N(T)= \mathbb E\bigg[ \prod_{(i,j) \in T} \Big(\mathbf 1_{\big(\{i,j\} \in G_N \big)} - \mathbb P(\{i,j\} \in G_N) \Big)\bigg], \ T \subset [N]^2 \textrm{finite}$$ furnishes informations on the asymptotic spectral properties of the adjacency matrix $A_N$ of $G_N$. Denote by $d_N = N\times \mathbb P(\{i,j\} \in G_N)$ and assume $d_N, N-d_N\underset{N \rightarrow \infty}{\longrightarrow} \infty$. If $\mathfrak C_N(T) =\big(\frac{d_N}N\big)^{|T|} \times O\big(d_N^{-\frac {|T|}2}\big)$ for any $T$, the standardized empirical eigenvalue distribution of $A_N$ converges in expectation to the semicircular law and the matrix satisfies asymptotic freeness properties in the sense of free probability theory. We provide such estimates for uniform $d_N$-regular graphs $G_{N,d_N}$, under the additional assumption that $|\frac N 2 - d_N- \eta \sqrt{d_N}| \underset{N \rightarrow \infty}{\longrightarrow} \infty$ for some $\eta>0$. Our method applies also for simple graphs whose edges are labelled by i.i.d. random variables.Comment: 21 pages, 7 figure

Self-assembly of Escherichia coli phage shock protein A

The Phage shock protein (Psp) response is an extracytoplasmic stress response. The central component of this system is PspA, a protein that mediates the physiological response to membrane stress. PspA is also involved in regulating its own transcription and that of the psp operon, forming a positive feedback loop. PspA has been previously shown to oligomerise into higher-order species, including a 36-meric species with ring-like structure. In this study, we demonstrate that the ring-like PspA structures further self-assemble into rod-shaped complexes. These rod-like structures may play a scaffolding role in the maintenance of membrane integrity during phage shock protein response

The strong asymptotic freeness of Haar and deterministic matrices

In this paper, we are interested in sequences of q-tuple of N-by-N random matrices having a strong limiting distribution (i.e. given any non-commutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having this property, and we show that this property pertains if the q-tuple is enlarged with independent unitary Haar distributed random matrices. Besides, the limit of norms and traces in non-commutative polynomials in the enlarged family can be computed with reduced free product construction. This extends results of one author (C. M.) and of Haagerup and Thorbjornsen. We also show that a p-tuple of independent orthogonal and symplectic Haar matrices have a strong limiting distribution, extending a recent result of Schultz.Comment: 12 pages. Accepted for publication to Annales Scientifique de l'EN

Magnetic resonance and susceptibility studies of some intermetallic compounds and rare-earth alloys

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The Professional Induction of Headteachers in England: Findings from the IBPS project

This paper reports on the data gathered through a self-completion questionnaire administered to a sample of headteachers in England who took up post in September, 2000. The survey is part of a larger, international study (The International Beginning Principals Study – IBPS) which aims to investigate and report on the experiences of beginning headteachers and principals during their first two years in post. Research teams are undertaking parallel studies in other countries, including Belgium, Canada, Netherlands and USA. Four research questions underpin the design of questionnaires used in all countries, thus allowing for data to be compared across the countries whilst allowing each questionnaire to reflect cultural, structural and linguistic differences between school systems. The four basic research questions are: 1. How does the process of professional development evolve, what are the main determinants and can we distinguish different patterns? 2. To what extent is it possible for a headteacher – given some external constraints – to develop autonomously a local policy? 3. How (and to what degree) can a beginning headteacher influence the existing school (daily organisation; structures; culture) and the development of individual teachers? 4. What encourages and discourages people from aspiring to headship

The Impact of National Culture on School Leadership in England

This paper reports on the survey findings of an international project to examine the impact of national culture on school leadership. The survey instrument was based on the work of Geert Hofstede, with the data collected for this paper emerging from semi-structured interviews with headteachers in England