Moment Estimates of the Cloud of a Planar Measure

With a proper function theoretic definition of the cloud of a positive measure with compact support in the real plane, a computational scheme of transforming the moments of the original measure into the moments of the uniformly distributed mass on the cloud is described. The main limiting operation involves exclusively truncated Christoffel-Darboux kernels, while error bounds depend on the spectral asymptotics of a Hankel kernel belonging to the Hilbert-Schmidt class.

Validation of a Wireless Sensor System for the Estimation of Cumulative Lumbar Loads in Occupational Settings

This study aimed to evaluate the accuracy of 3D L5/S1 moment estimates from a wearable inertial motioncapture system during manual lifting tasks. Reference L5/S1 moments were calculated using inversedynamics bottom-up and top-down laboratory models, based on the data from a measurement systemcomprising optical motion capture and force plates. Nine groups of four subjects performed tasks consistingof lifting and lowering 10 lbs. load with three different heights and asymmetry angles. As a measure ofsystem performance, the root means square errors and absolute peak errors between models werecompared. Also, repeated measures analyses of variance were calculated comparing the means and theabsolute peaks of the estimated moments. The results suggest that most of the estimates obtained from thewireless sensor system are in close correspondence when comparing the means, and more variability isobserved when comparing peak values to other models calculating estimates of L5/S1 moments.

Moment estimates for invariant measures of stochastic Burgers equations

In this paper, we study moment estimates for the invariant measure of the stochastic Burgers equation with multiplicative noise. Based upon an a priori estimate for the stochastic convolution, we derive regularity properties on invariant measure. As an application, we prove smoothing properties for the transition semigroup by introducing an auxiliary semigroup. Finally, the m-dissipativity of the associated Kolmogorov operator is given.

Edge-statistics on large graphs

The inducibility of a graph H measures the maximum number of induced copies of H a large graph G can have. Generalizing this notion, we study how many induced subgraphs of fixed order k and size ℓ a large graph G on n vertices can have. Clearly, this number is $\left( {\matrix{n \cr k}}\right)$ for every n, k and $\ell \in \left\{ {0,\left( {\matrix{k \cr 2}} \right)}\right\}$. We conjecture that for every n, k and $0 \lt \ell \lt \left( {\matrix{k \cr 2}}\right)$ this number is at most $ (1/e + {o_k}(1)) {\left( {\matrix{n \cr k}} \right)}$. If true, this would be tight for ℓ ∈ {1, k − 1}.In support of our ‘Edge-statistics Conjecture’, we prove that the corresponding density is bounded away from 1 by an absolute constant. Furthermore, for various ranges of the values of ℓ we establish stronger bounds. In particular, we prove that for ‘almost all’ pairs (k, ℓ) only a polynomially small fraction of the k-subsets of V(G) have exactly ℓ edges, and prove an upper bound of $ (1/2 + {o_k}(1)){\left( {\matrix{n \cr k}}\right)}$ for ℓ = 1.Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun’s sieve, as well as graph-theoretic and combinatorial arguments such as Zykov’s symmetrization, Sperner’s theorem and various counting techniques.