Anomalous heat-kernel decay for random walk among bounded random conductances

We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded random conductances $\omega_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition that the probability of $\omega_{xy}>0$ exceeds the threshold for bond percolation on $\Z^d$. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the $2n$-step return probability $P_\omega^{2n}(0,0)$. We prove that $P_\omega^{2n}(0,0)$ is bounded by a random constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)$ in $d=4$. By producing examples with anomalous heat-kernel decay approaching $1/n^2$ we prove that the $o(n^{-2})$ bound in $d\ge5$ is the best possible. We also construct natural $n$-dependent environments that exhibit the extra $\log n$ factor in $d=4$. See also math.PR/0701248.Comment: 22 pages. Includes a self-contained proof of isoperimetric inequality for supercritical percolation clusters. Version to appear in AIHP + additional correction

Generalized Koszul properties for augmented algebras

Under certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra E(A) := Ext_A(K, K). We show that there exists a bigraded algebra monomorphism from gr E(A) to E_Gr(gr A), where E_Gr(gr A) is the graded Yoneda algebra of gr A. This monomorphism can be applied in the case where A is connected graded to determine that A has the K_2 property recently introduced by Cassidy and Shelton.Comment: 14 page

Mental Imagery Induces Cross-Modal Sensory Plasticity and Changes Future Auditory Perception

Can what we imagine in our minds change how we perceive the world in the future? A continuous process of multisensory integration and recalibration is responsible for maintaining a correspondence between the senses (e.g., vision, touch, audition) and, ultimately, a stable and coherent perception of our environment. This process depends on the plasticity of our sensory systems. The so-called ventriloquism aftereffect—a shift in the perceived localization of sounds presented alone after repeated exposure to spatially mismatched auditory and visual stimuli—is a clear example of this type of plasticity in the audiovisual domain. In a series of six studies with 24 participants each, we investigated an imagery-induced ventriloquism aftereffect in which imagining a visual stimulus elicits the same frequency-specific auditory aftereffect as actually seeing one. These results demonstrate that mental imagery can recalibrate the senses and induce the same cross-modal sensory plasticity as real sensory stimuli

Measuring Underemployment at the County Level

As labor markets tightened in the last half of the nineties, economic development and community leaders sought to identify more locally available workers than were indicated by published statistics. Using results from commissioned surveys, they pointed to large numbers of part-time workers who desired full-time work, and to full-time workers who were qualified for better jobs. These statistics were often used to negate low official unemployment rates that deterred firms, concerned by the ostensible shortage of workers, from locating in their counties. We have conducted a larger, statewide, survey of underemployment and linked it to the detailed demographic and labor force data from the 2000 Census. We used the results to identify variations in the number and type of underemployed persons around the state, with emphasis on the differences between urbanized and rural areas. Over a quarter of full-time workers reported underemployment, including a third of workers in exurban counties. However, forty to fifty percent of underemployment is reportedly by choice, with the highest rates in the small urban and exurban regions. Of those that are not underemployed by choice, over ninety percent of respondents in some regions cited lack of job opportunities. We find that between fourteen and forty percent of part-time workers prefer full-time work, with the highest rates in rural Appalachian counties. We provide some of the reasons underemployed people cite as constraints to better employment. Also, we used the survey results and the recent Census information to predict the number and type of underemployed persons in each county. The model can be used to update predictions as new local demographic and labor force estimates are released annually from the Census Bureau’s forthcoming American Community Surveys

Bayesian Calibrated Significance Levels Applied to the Spectral Tilt and Hemispherical Asymmetry

Bayesian model selection provides a formal method of determining the level of support for new parameters in a model. However, if there is not a specific enough underlying physical motivation for the new parameters it can be hard to assign them meaningful priors, an essential ingredient of Bayesian model selection. Here we look at methods maximizing the prior so as to work out what is the maximum support the data could give for the new parameters. If the maximum support is not high enough then one can confidently conclude that the new parameters are unnecessary without needing to worry that some other prior may make them significant. We discuss a computationally efficient means of doing this which involves mapping p-values onto upper bounds of the Bayes factor (or odds) for the new parameters. A p-value of 0.05 ($1.96\sigma$) corresponds to odds less than or equal to 5:2 which is below the `weak' support at best threshold. A p-value of 0.0003 ($3.6\sigma$) corresponds to odds of less than or equal to 150:1 which is the `strong' support at best threshold. Applying this method we find that the odds on the scalar spectral index being different from one are 49:1 at best. We also find that the odds that there is primordial hemispherical asymmetry in the cosmic microwave background are 9:1 at best.Comment: 5 pages. V2: clarifying comments added in response to referee report. Matches version to appear in MNRA

On the effective strain tensor in heterogeneous materials

ArticleThis is the author's accepted manuscript. The final published version of record is available at https://dx.doi.org/10.1177/1081286514521092This paper considers effective strain tensors within the context of linear elastic equilibrium theory. The elastic properties of structured materials are often averaged over subvolumes of various scales inside the material. For subvolumes smaller than a representative volume element, simple volume-averaging of the stress and strain may not preserve the elastic energy. We introduce an averaging process which preserves the energy for all boundary conditions. This averaging process emphasizes the parts of the material which carry the most stress. Here the effective strain is weighted by the local stress, and can be interpreted as an average strain over all paths taken by loads and forces through the volume. This alternative effective strain may be especially appropriate for materials with voids, such as foams and granular matter, as the averaging only involves the material itself. For uniform boundary conditions the weighted strain matches the volume-averaged strain. This paper investigates the properties of this weighted strain tensor. First, for each path taken by loads and forces through the volume we can measure a net length as well as a net extension due to the linear deformation. The weighted effective strain equals the ratio of average length to average extension, where the averaging is over all possible force paths. Thus this method provides a connection to load path analysis. Secondly, even when the average rotation within the subvolume is zero, there may be local fluctuations in the rotation field. These rotations can act like a mechanism, transferring elastic energy between boundaries or degrees of freedom. The effective strain defined here highlights this mechanism effect