Square root meadows

Let Q_0 denote the rational numbers expanded to a meadow by totalizing inversion such that 0^{-1}=0. Q_0 can be expanded by a total sign function s that extracts the sign of a rational number. In this paper we discuss an extension Q_0(s ,\sqrt) of the signed rationals in which every number has a unique square root.Comment: 9 page

Square Root Singularity in Boundary Reflection Matrix

Two-particle scattering amplitudes for integrable relativistic quantum field theory in 1+1 dimensions can normally have at most singularities of poles and zeros along the imaginary axis in the complex rapidity plane. It has been supposed that single particle amplitudes of the exact boundary reflection matrix exhibit the same structure. In this paper, single particle amplitudes of the exact boundary reflection matrix corresponding to the Neumann boundary condition for affine Toda field theory associated with twisted affine algebras $a_{2n}^{(2)}$ are conjectured, based on one-loop result, as having a new kind of square root singularity.Comment: 10 pages, latex fil

Approximate square-root-time relaxation in glass-forming liquids

We present data for the dielectric relaxation of 43 glass-forming organic liquids, showing that the primary (alpha) relaxation is often close to square-root-time relaxation. The better an inverse power-law description of the high-frequency loss applies, the more accurately is square-root-time relaxation obeyed. These findings suggest that square-root-time relaxation is generic to the alpha process, once a common view, but since long believed to be incorrect. Only liquids with very large dielectric losses deviate from this picture by having consistently narrower loss peaks. As a further challenge to the prevailing opinion, we find that liquids with accurate square-root-time relaxation cover a wide range of fragilities

Square root kalman filter with contaminated observations

The algorithm of square root Kalman filtering for the case of contaminated observations is described in the paper. This algorithm is suitable for the parallel computer implementation allowing to treat dynamic linear systems with large number of state variables in a robust recursive way

Sharp Oracle Inequalities for Square Root Regularization

We study a set of regularization methods for high-dimensional linear regression models. These penalized estimators have the square root of the residual sum of squared errors as loss function, and any weakly decomposable norm as penalty function. This fit measure is chosen because of its property that the estimator does not depend on the unknown standard deviation of the noise. On the other hand, a generalized weakly decomposable norm penalty is very useful in being able to deal with different underlying sparsity structures. We can choose a different sparsity inducing norm depending on how we want to interpret the unknown parameter vector $\beta$. Structured sparsity norms, as defined in Micchelli et al. [18], are special cases of weakly decomposable norms, therefore we also include the square root LASSO (Belloni et al. [3]), the group square root LASSO (Bunea et al. [10]) and a new method called the square root SLOPE (in a similar fashion to the SLOPE from Bogdan et al. [6]). For this collection of estimators our results provide sharp oracle inequalities with the Karush-Kuhn-Tucker conditions. We discuss some examples of estimators. Based on a simulation we illustrate some advantages of the square root SLOPE