Spacings and pair correlations for finite Bernoulli convolutions

We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$ supported on a discrete point set, generically of size $2^N$. These sequences are uniformly distributed with respect to the infinite Bernoulli convolution measure $\nu_r$, as $N$ tends to infinity. Numerical evidence suggests that for a generic $r$, the distribution of spacings between appropriately rescaled points is Poissonian. We obtain some partial results in this direction; for instance, we show that, on average, the pair correlations do not exhibit attraction or repulsion in the limit. On the other hand, for certain algebraic $r$ the behavior is totally different.Comment: 17 pages, 6 figure

Scaling and interleaving of sub-system Lyapunov exponents for spatio-temporal systems

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a sub-system in a very cost effective way. In this work we present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. It is inspired by the stability analysis of the homogeneous evolution in a one-dimensional coupled map lattice but appears to be equally valid in a much wider range of cases. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative sub-system volume) for one and two-dimensional lattices in spatio-temporal chaotic regimes. In doing so we notice that the Lyapunov spectra for consecutive sub-system sizes are interleaved and we discuss the possible ways in which this may arise. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension and Kolmogorov-Sinai entropy) finding better convergence as the sub-system size is increased than with conventional rescaling.Comment: 18 pages, double column, REVTeX, 27 embedded postscript figures with psfig. Submitted to Chao

Combinatorics of linear iterated function systems with overlaps

Let $\bm p_0,...,\bm p_{m-1}$ be points in ${\mathbb R}^d$, and let $\{f_j\}_{j=0}^{m-1}$ be a one-parameter family of similitudes of ${\mathbb R}^d$: $$f_j(\bm x) = \lambda\bm x + (1-\lambda)\bm p_j, j=0,...,m-1,$$ where $\lambda\in(0,1)$ is our parameter. Then, as is well known, there exists a unique self-similar attractor $S_\lambda$ satisfying $S_\lambda=\bigcup_{j=0}^{m-1} f_j(S_\lambda)$. Each $\bm x\in S_\lambda$ has at least one address $(i_1,i_2,...)\in\prod_1^\infty\{0,1,...,m-1\}$, i.e., $\lim_n f_{i_1}f_{i_2}... f_{i_n}({\bf 0})=\bm x$. We show that for $\lambda$ sufficiently close to 1, each $\bm x\in S_\lambda\setminus\{\bm p_0,...,\bm p_{m-1}\}$ has $2^{\aleph_0}$ different addresses. If $\lambda$ is not too close to 1, then we can still have an overlap, but there exist $\bm x$'s which have a unique address. However, we prove that almost every $\bm x\in S_\lambda$ has $2^{\aleph_0}$ addresses, provided $S_\lambda$ contains no holes and at least one proper overlap. We apply these results to the case of expansions with deleted digits. Furthermore, we give sharp sufficient conditions for the Open Set Condition to fail and for the attractor to have no holes. These results are generalisations of the corresponding one-dimensional results, however most proofs are different.Comment: Accepted for publication in Nonlinearit

Similar dissection of sets

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner's questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let $D\subset \mathbb{R}^d$ be a given set and let $f_1,...,f_k$ be injective continuous mappings. Does there exist a set $X$ such that $D = X \cup f_1(X) \cup ... \cup f_k(X)$ is satisfied with a non-overlapping union? We prove that such a set $X$ exists for certain choices of $D$ and $\{f_1,...,f_k\}$. The solutions $X$ often turn out to be attractors of iterated function systems with condensation in the sense of Barnsley. Coming back to Gardner's setting, we use our theory to prove that an equilateral triangle can be dissected in three similar copies whose areas have ratio $1:1:a$ for $a \ge (3+\sqrt{5})/2$

Don't bleach chaotic data

A common first step in time series signal analysis involves digitally filtering the data to remove linear correlations. The residual data is spectrally white (it is ``bleached''), but in principle retains the nonlinear structure of the original time series. It is well known that simple linear autocorrelation can give rise to spurious results in algorithms for estimating nonlinear invariants, such as fractal dimension and Lyapunov exponents. In theory, bleached data avoids these pitfalls. But in practice, bleaching obscures the underlying deterministic structure of a low-dimensional chaotic process. This appears to be a property of the chaos itself, since nonchaotic data are not similarly affected. The adverse effects of bleaching are demonstrated in a series of numerical experiments on known chaotic data. Some theoretical aspects are also discussed.Comment: 12 dense pages (82K) of ordinary LaTeX; uses macro psfig.tex for inclusion of figures in text; figures are uufile'd into a single file of size 306K; the final dvips'd postscript file is about 1.3mb Replaced 9/30/93 to incorporate final changes in the proofs and to make the LaTeX more portable; the paper will appear in CHAOS 4 (Dec, 1993

Self-Consistent Asset Pricing Models

We discuss the foundations of factor or regression models in the light of the self-consistency condition that the market portfolio (and more generally the risk factors) is (are) constituted of the assets whose returns it is (they are) supposed to explain. As already reported in several articles, self-consistency implies correlations between the return disturbances. As a consequence, the alpha's and beta's of the factor model are unobservable. Self-consistency leads to renormalized beta's with zero effective alpha's, which are observable with standard OLS regressions. Analytical derivations and numerical simulations show that, for arbitrary choices of the proxy which are different from the true market portfolio, a modified linear regression holds with a non-zero value $\alpha_i$ at the origin between an asset $i$'s return and the proxy's return. Self-consistency also introduces ``orthogonality'' and ``normality'' conditions linking the beta's, alpha's (as well as the residuals) and the weights of the proxy portfolio. Two diagnostics based on these orthogonality and normality conditions are implemented on a basket of 323 assets which have been components of the S&P500 in the period from Jan. 1990 to Feb. 2005. These two diagnostics show interesting departures from dynamical self-consistency starting about 2 years before the end of the Internet bubble. Finally, the factor decomposition with the self-consistency condition derives a risk-factor decomposition in the multi-factor case which is identical to the principal components analysis (PCA), thus providing a direct link between model-driven and data-driven constructions of risk factors.Comment: 36 pages with 8 figures. large version with 6 appendices for the Proceedings of the 5th International Conference APFS (Applications of Physics in Financial Analysis), June 29-July 1, 2006, Torin

The DDX6-4E-T interaction mediates translational repression and P-body assembly

This is the final version of the article. Available from the publisher via the DOI in this record.4E-Transporter binds eIF4E via its consensus sequence YXXXXLΦ, shared with eIF4G, and is a nucleocytoplasmic shuttling protein found enriched in P-(rocessing) bodies. 4E-T inhibits general protein synthesis by reducing available eIF4E levels. Recently, we showed that 4E-T bound to mRNA however represses its translation in an eIF4E-independent manner, and contributes to silencing of mRNAs targeted by miRNAs. Here, we address further the mechanism of translational repression by 4E-T by first identifying and delineating the interacting sites of its major partners by mass spectrometry and western blotting, including DDX6, UNR, unrip, PAT1B, LSM14A and CNOT4. Furthermore, we document novel binding between 4E-T partners including UNR-CNOT4 and unrip-LSM14A, altogether suggesting 4E-T nucleates a complex network of RNA-binding protein interactions. In functional assays, we demonstrate that joint deletion of two short conserved motifs that bind UNR and DDX6 relieves repression of 4E-T-bound mRNA, in part reliant on the 4E-T-DDX6-CNOT1 axis. We also show that the DDX6-4E-T interaction mediates miRNA-dependent translational repression and de novo P-body assembly, implying that translational repression and formation of new P-bodies are coupled processes. Altogether these findings considerably extend our understanding of the role of 4E-T in gene regulation, important in development and neurogenesis.BBSRC [BB/J00779X/1 to N.S.]; CNRS PICS (to D.W.); Agence Nationale pour la Recherche [ANR-14-CE09-0013-01ANR to D.W.]; Gates Cambridge Foundation (to A.K.); Fondation Wiener – Anspach of the Université Libre de Bruxelles and the Cambridge Newton Trust (C.V.). Funding for open access charge: BBSRC

Phases of M2-brane Theories

We investigate different toric phases of 2+1 dimensional quiver gauge theories arising from M2-branes probing toric Calabi-Yau 4 folds. A brane tiling for each toric phase is presented. We apply the 'forward algorithm' to obtain the toric data of the mesonic moduli space of vacua and exhibit the equivalence between the vacua of different toric phases of a given singularity. The structures of the Master space, the mesonic moduli space, and the baryonic moduli space are examined in detail. We compute the Hilbert series and use them to verify the toric dualities between different phases. The Hilbert series, R-charges, and generators of the mesonic moduli space are matched between toric phases.Comment: 60 pages, 28 figures, 6 tables. v2: minor correction