Block Sensitivity of Minterm-Transitive Functions

Boolean functions with symmetry properties are interesting from a complexity theory perspective; extensive research has shown that these functions, if nonconstant, must have high `complexity' according to various measures. In recent work of this type, Sun gave bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f) = Omega(N^{1/3}), and that there exists such a function for which bs(f) = O(N^{3/7}ln N). His example function belongs to a subclass of transitively invariant functions called the minterm-transitive functions (defined in earlier work by Chakraborty). We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy bs(f) = Omega(N^{3/7}). Thus Sun's example function has nearly minimal block sensitivity for this subclass. Second, we give an improved example: a minterm-transitive function for which bs(f) = O(N^{3/7}ln^{1/7}N).Comment: 10 page

Sensitivity of the limit shape of sample clouds from meta densities

The paper focuses on a class of light-tailed multivariate probability distributions. These are obtained via a transformation of the margins from a heavy-tailed original distribution. This class was introduced in Balkema et al. (J. Multivariate Anal. 101 (2010) 1738-1754). As shown there, for the light-tailed meta distribution the sample clouds, properly scaled, converge onto a deterministic set. The shape of the limit set gives a good description of the relation between extreme observations in different directions. This paper investigates how sensitive the limit shape is to changes in the underlying heavy-tailed distribution. Copulas fit in well with multivariate extremes. By Galambos's theorem, existence of directional derivatives in the upper endpoint of the copula is necessary and sufficient for convergence of the multivariate extremes provided the marginal maxima converge. The copula of the max-stable limit distribution does not depend on the margins. So margins seem to play a subsidiary role in multivariate extremes. The theory and examples presented in this paper cast a different light on the significance of margins. For light-tailed meta distributions, the asymptotic behaviour is very sensitive to perturbations of the underlying heavy-tailed original distribution, it may change drastically even when the asymptotic behaviour of the heavy-tailed density is not affected.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ370 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Limits on the Network Sensitivity Function for Multi-Agent Systems on a Graph

This report explores the tradeoffs and limits of performance in feedback control of interconnected multi-agent systems, focused on the network sensitivity functions. We consider the interaction topology described by a directed graph and we prove that the sensitivity transfer functions between every pair of agents, arbitrarily connected, can be derived using a version of the Mason's Direct Rule. Explicit forms for special types of graphs are presented. An analysis of the role of cycles points out that these structures influence and limit considerably the performance of the system. The more the cycles are equally distributed among the formation, the better performance the system can achieve, but they are always worse than the single agent case. We also prove the networked version of Bode's integral formula, showing that it still holds for multi-agent systems

Magnetic-field symmetries of mesoscopic nonlinear conductance

We examine contributions to the dc-current of mesoscopic samples which are non-linear in applied voltage. In the presence of a magnetic field, the current can be decomposed into components which are odd (antisymmetric) and even (symmetric) under flux reversal. For a two-terminal chaotic cavity, these components turn out to be very sensitive to the strength of the Coulomb interaction and the asymmetry of the contact conductances. For both two- and multi-terminal quantum dots we discuss correlations of current non-linearity in voltage measured at different magnetic fields and temperatures.Comment: 9 pages, 4 figure

Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?

We study the contour dependence of the out-of-time-ordered correlation function (OTOC) both in weakly coupled field theory and in the Sachdev-Ye-Kitaev (SYK) model. We show that its value, including its Lyapunov spectrum, depends sensitively on the shape of the complex time contour in generic weakly coupled field theories. For gapless theories with no thermal mass, such as SYK, the Lyapunov spectrum turns out to be an exception; their Lyapunov spectra do not exhibit contour dependence, though the full OTOCs do. Our result puts into question which of the Lyapunov exponents computed from the exponential growth of the OTOC reflects the actual physical dynamics of the system. We argue that, in a weakly coupled $\Phi^4$ theory, a kinetic theory argument indicates that the symmetric configuration of the time contour, namely the one for which the bound on chaos has been proven, has a proper interpretation in terms of dynamical chaos. Finally, we point out that a relation between these OTOCs and a quantity which may be measured experimentally --- the Loschmidt echo --- also suggests a symmetric contour configuration, with the subtlety that the inverse periodicity in Euclidean time is half the physical temperature. In this interpretation the chaos bound reads $\lambda \leq \frac{2\pi}{\beta}= \pi T_{\text{physical}}$.Comment: Comment on regularization dependence in 2d-CFTs added. Published versio

Slow relaxation and sensitivity to disorder in trapped lattice fermions after a quench

We consider a system of non-interacting fermions in one dimension subject to a single-particle potential consisting of (a) a strong optical lattice, (b) a harmonic trap, and (c) uncorrelated on-site disorder. After a quench, in which the center of the harmonic trap is displaced, we study the occupation function of the fermions and the time-evolution of experimental observables. Specifically, we present numerical and analytical results for the post-quench occupation function of the fermions, and analyse the time-evolution of the real-space density profile. Unsurprisingly for a non-interacting (and therefore integrable) system, the infinite-time limit of the density profile is non-thermal. However, due to Bragg-localization of the higher-energy single-particle states, the approach to even this non-thermal state is extremely slow. We quantify this statement, and show that it implies a sensitivity to disorder parametrically stronger than that expected from Anderson localization.Comment: 15 pages, 11 figure

New Dirac points and multiple Landau level crossings in biased trilayer graphene

Recently a new high-mobility Dirac material, trilayer graphene, was realized experimentally. The band structure of ABA-stacked trilayer graphene consists of a monolayer-like and a bilayer-like pairs of bands. Here we study electronic properties of ABA-stacked trilayer graphene biased by a perpendicular electric field. We find that the combination of the bias and trigonal warping gives rise to a set of new Dirac points: in each valley, seven species of Dirac fermions with small masses of order of a few meV emerge. The positions and masses of the emergent Dirac fermions are tunable by bias, and one group of Dirac fermions becomes massless at a certain bias value. Therefore, in contrast to bilayer graphene, the conductivity at the neutrality point is expected to show non-monotonic behavior, becoming of the order of a few e^2/h when some Dirac masses vanish. Further, we analyze the evolution of Landau level spectrum as a function of bias. Emergence of new Dirac points in the band structure translates into new three-fold-degenerate groups of Landau levels. This leads to an anomalous quantum Hall effect, in which some quantum Hall steps have a height of 3e^2/h. At an intermediate bias, the degeneracies of all Landau levels get lifted, and in this regime all quantum Hall plateaus are spaced by e^2/h. Finally, we show that the pattern of Landau level crossings is very sensitive to certain band structure parameters, and can therefore provide a useful tool for determining their precise values.Comment: 11 pages, 6 figures; v2: expanded introduction, new references added, a few typos correcte

An Introduction to Rule-based Modeling of Immune Receptor Signaling

Cells process external and internal signals through chemical interactions. Cells that constitute the immune system (e.g., antigen presenting cell, T-cell, B-cell, mast cell) can have different functions (e.g., adaptive memory, inflammatory response) depending on the type and number of receptor molecules on the cell surface and the specific intracellular signaling pathways activated by those receptors. Explicitly modeling and simulating kinetic interactions between molecules allows us to pose questions about the dynamics of a signaling network under various conditions. However, the application of chemical kinetics to biochemical signaling systems has been limited by the complexity of the systems under consideration. Rule-based modeling (BioNetGen, Kappa, Simmune, PySB) is an approach to address this complexity. In this chapter, by application to the Fc$\varepsilon$RI receptor system, we will explore the origins of complexity in macromolecular interactions, show how rule-based modeling can be used to address complexity, and demonstrate how to build a model in the BioNetGen framework. Open source BioNetGen software and documentation are available at http://bionetgen.org.Comment: 5 figure