Shifted Power Method for Computing Tensor Eigenpairs

Recent work on eigenvalues and eigenvectors for tensors of order m >= 3 has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form Ax^{m-1} = \lambda x subject to ||x||=1, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex eigenpairs

Noncooperatively Optimized Tolerance: Decentralized Strategic Optimization in Complex Systems

We introduce noncooperatively optimized tolerance (NOT), a generalization of highly optimized tolerance (HOT) that involves strategic (game theoretic) interactions between parties in a complex system. We illustrate our model in the forest fire (percolation) framework. As the number of players increases, our model retains features of HOT, such as robustness, high yield combined with high density, and self-dissimilar landscapes, but also develops features of self-organized criticality (SOC) when the number of players is large enough. For example, the forest landscape becomes increasingly homogeneous and protection from adverse events (lightning strikes) becomes less closely correlated with the spatial distribution of these events. While HOT is a special case of our model, the resemblance to SOC is only partial; for example, the distribution of cascades, while becoming increasingly heavy-tailed as the number of players increases, also deviates more significantly from a power law in this regime. Surprisingly, the system retains considerable robustness even as it becomes fractured, due in part to emergent cooperation between neighboring players. At the same time, increasing homogeneity promotes resilience against changes in the lightning distribution, giving rise to intermediate regimes where the system is robust to a particular distribution of adverse events, yet not very fragile to changes

Fronts in randomly advected and heterogeneous media and nonuniversality of Burgers turbulence: Theory and numerics

A recently established mathematical equivalence--between weakly perturbed Huygens fronts (e.g., flames in weak turbulence or geometrical-optics wave fronts in slightly nonuniform media) and the inviscid limit of white-noise-driven Burgers turbulence--motivates theoretical and numerical estimates of Burgers-turbulence properties for specific types of white-in-time forcing. Existing mathematical relations between Burgers turbulence and the statistical mechanics of directed polymers, allowing use of the replica method, are exploited to obtain systematic upper bounds on the Burgers energy density, corresponding to the ground-state binding energy of the directed polymer and the speedup of the Huygens front. The results are complementary to previous studies of both Burgers turbulence and directed polymers, which have focused on universal scaling properties instead of forcing-dependent parameters. The upper-bound formula can be heuristically understood in terms of renormalization of a different kind from that previously used in combustion models, and also shows that the burning velocity of an idealized turbulent flame does not diverge with increasing Reynolds number at fixed turbulence intensity, a conclusion that applies even to strong turbulence. Numerical simulations of the one-dimensional inviscid Burgers equation using a Lagrangian finite-element method confirm that the theoretical upper bounds are sharp within about 15% for various forcing spectra (corresponding to various two-dimensional random media). These computations provide a new quantitative test of the replica method. The inferred nonuniversality (spectrum dependence) of the front speedup is of direct importance for combustion modeling.Comment: 20 pages, 2 figures, REVTeX 4. Moved some details to appendices, added figure on numerical metho

Theorem-Proving Analysis of Digital Control Logic Interacting with Continuous Dynamics

AbstractThis work outlines an equation-based formulation of a digital control program and transducer interacting with a continuous physical process, and an approach using the Coq theorem prover for verifying the performance of the combined hybrid system. Considering thermal dynamics with linear dissipation for simplicity, we focus on a generalizable, physically consistent description of the interaction of the real-valued temperature and the digital program acting as a thermostat. Of interest in this work is the discovery and formal proof of bounds on the temperature, the degree of variation, and other performance characteristics. Our approach explicitly addresses the need to mathematically represent the decision problem inherent in an analog-to-digital converter, which for rare values can take an arbitrarily long time to produce a digital answer (the so-called Buridan's Principle); this constraint ineluctably manifests itself in the verification of thermostat performance. Furthermore, the temporal causality constraints in the thermal physics must be made explicit to obtain a consistent model for analysis. We discuss the significance of these findings toward the verification of digital control for more complex physical variables and fields

Field theory of the inverse cascade in two-dimensional turbulence

A two-dimensional fluid, stirred at high wavenumbers and damped by both viscosity and linear friction, is modeled by a statistical field theory. The fluid's long-distance behavior is studied using renormalization-group (RG) methods, as begun by Forster, Nelson, and Stephen [Phys. Rev. A 16, 732 (1977)]. With friction, which dissipates energy at low wavenumbers, one expects a stationary inverse energy cascade for strong enough stirring. While such developed turbulence is beyond the quantitative reach of perturbation theory, a combination of exact and perturbative results suggests a coherent picture of the inverse cascade. The zero-friction fluctuation-dissipation theorem (FDT) is derived from a generalized time-reversal symmetry and implies zero anomalous dimension for the velocity even when friction is present. Thus the Kolmogorov scaling of the inverse cascade cannot be explained by any RG fixed point. The beta function for the dimensionless coupling ghat is computed through two loops; the ghat^3 term is positive, as already known, but the ghat^5 term is negative. An ideal cascade requires a linear beta function for large ghat, consistent with a Pad\'e approximant to the Borel transform. The conjecture that the Kolmogorov spectrum arises from an RG flow through large ghat is compatible with other results, but the accurate k^{-5/3} scaling is not explained and the Kolmogorov constant is not estimated. The lack of scale invariance should produce intermittency in high-order structure functions, as observed in some but not all numerical simulations of the inverse cascade. When analogous RG methods are applied to the one-dimensional Burgers equation using an FDT-preserving dimensional continuation, equipartition is obtained instead of a cascade--in agreement with simulations.Comment: 16 pages, 3 figures, REVTeX 4. Material added on energy flux, intermittency, and comparison with Burgers equatio

Scaling of Huygens-front speedup in weakly random media

Front propagation described by Huygens' principle is a fundamental mechanism of spatial spreading of a property or an effect, occurring in optics, acoustics, ecology and combustion. If the local front speed varies randomly due to inhomogeneity or motion of the medium (as in turbulent premixed combustion), then the front wrinkles and its overall passage rate (turbulent burning velocity) increases. The calculation of this speedup is subtle because it involves the minimum-time propagation trajectory. Here we show mathematically that for a medium with weak isotropic random fluctuations, under mild conditions on its spatial structure, the speedup scales with the 4/3 power of the fluctuation amplitude. This result, which verifies a previous conjecture while clarifying its scope, is obtained by reducing the propagation problem to the inviscid Burgers equation with white-in-time forcing. Consequently, field-theoretic analyses of the Burgers equation have significant implications for fronts in random media, even beyond the weak-fluctuation limit.Comment: 7 pages, 3 figures, elsart5p. v2: additional discussion of Hamiltonian formalism; v3: clarification of transient behavio

“We Don’t Get Into All That”: an analysis of how teachers uphold heteronormative sex and relationship education

Legislation that applies to UK SRE currently advocates inclusive provision. Given the nonstatutory status of SRE, however, it is unclear how teachers incorporate sexual inclusivity, especially as research has shown that teachers’ discursive practices can promote a heteronormative SRE climate (Renn, 2010). Using a discursive psychological approach to analyze interview data, this study examined how teachers account for their provision as inclusive. It was revealed that even when promoting their inclusivity, teachers’ SRE provision upholds heteronormativity. In doing this, they positioned LGB and same-sex practices outside of the classroom, potentially leaving these young people without a sufficient sex education

Autoimmune encephalomyelitis in NOD mice is not initially a progressive multiple sclerosis model.

OBJECTIVE: Despite progress in treating relapsing multiple sclerosis (MS), effective inhibition of nonrelapsing progressive MS is an urgent, unmet, clinical need. Animal models of MS, such as experimental autoimmune encephalomyelitis (EAE), provide valuable tools to examine the mechanisms contributing to disease and may be important for developing rational therapeutic approaches for treatment of progressive MS. It has been suggested that myelin oligodendrocyte glycoprotein (MOG) peptide residues 35-55 (MOG35-55 )-induced EAE in nonobese diabetic (NOD) mice resembles secondary progressive MS. The objective was to determine whether the published data merits such claims. METHODS: Induction and monitoring of EAE in NOD mice and literature review. RESULTS: It is evident that the NOD mouse model lacks validity as a progressive MS model as the individual course seems to be an asynchronous, relapsing-remitting neurodegenerative disease, characterized by increasingly poor recovery from relapse. The seemingly progressive course seen in group means of clinical score is an artifact of data handling and interpretation. INTERPRETATION: Although MOG35-55 -induced EAE in NOD mice may provide some clues about approaches to block neurodegeneration associated with the inflammatory penumbra as lesions form, it should not be used to justify trials in people with nonactive, progressive MS. This adds further support to the view that drug studies in animals should universally adopt transparent raw data deposition as part of the publication process, such that claims can adequately be interrogated. This transparency is important if animal-based science is to remain a credible part of translational research in MS.Stichting MS ResearchWellcome TrustMedical Research CouncilNational Multiple Sclerosis Society. Grant Number: RG4132A5/

Cholesterol Perturbation in Mice Results in p53 Degradation and Axonal Pathology through p38 MAPK and Mdm2 Activation

Perturbation of lipid metabolism, especially of cholesterol homeostasis, can be catastrophic to mammalian brain, as it has the highest level of cholesterol in the body. This notion is best illustrated by the severe progressive neurodegeneration in Niemann-Pick Type C (NPC) disease, one of the lysosomal storage diseases, caused by mutations in the NPC1 or NPC2 gene. In this study, we found that growth cone collapse induced by genetic or pharmacological disruption of cholesterol egress from late endosomes/lysosomes was directly related to a decrease in axonal and growth cone levels of the phosphorylated form of the tumor suppressor factor p53. Cholesterol perturbation-induced growth cone collapse and decrease in phosphorylated p53 were reduced by inhibition of p38 mitogen-activated protein kinase (MAPK) and murine double minute (Mdm2) E3 ligase. Growth cone collapse induced by genetic (npc1−/−) or pharmacological modification of cholesterol metabolism was Rho kinase (ROCK)-dependent and associated with increased RhoA protein synthesis; both processes were significantly reduced by P38 MAPK or Mdm2 inhibition. Finally, in vivo ROCK inhibition significantly increased phosphorylated p53 levels and neurofilaments in axons, and axonal bundle size in npc1−/− mice. These results indicate that NPC-related and cholesterol perturbation-induced axonal pathology is associated with an abnormal signaling pathway consisting in p38 MAPK activation leading to Mdm2-mediated p53 degradation, followed by ROCK activation. These results also suggest new targets for pharmacological treatment of NPC disease and other diseases associated with disruption of cholesterol metabolism