In this paper, we prove some general results about Lagrangian symmetric ordinary differential expressions [·] of order n when the coefficients of [·] are sufficiently smooth. In particular, for a natural number j , we show that under increased smoothness conditions on the coefficients, the j th composite power j [·] of [·] is also Lagrangian symmetric. More generally, we show that if w is a sym...

It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on matrices that are invariants by the action of conjugation by general linear group. We generalize this result showing that the abelianization of the algebra of the symmetric tensors of fixed order over a free associative algebra is isomo...

all of them of degree −1. The symmetric powers should be interpreted in the graded sense: for example the bracket [ , ] restricts to a map ΛL1 → L1. We shall identify the 1-ary bracket [ ] with a differential ∂ on L. The various axioms satisfied by these brackets can be summarized as follows: Let L be the graded vector space with Hom(Li,R) in degree i, and let C(L) := Sym(L) be its symmetric al...

A Pareto distribution has the property that any tail of the distribution has the same shape as the original distribution. The exponential distribution and the uniform distribution have the tail property too. The tail property characterizes the univariate generalized Pareto distributions. There are three classes of univariate GPDs: Pareto distributions, power laws, and the exponential distributi...

Rheological characteristics of Al2O3, CuO and TiO2 nano particles were investigated in oil asthe base fluid at 1 and 2 wt.%. Constitutive relations for non-Newtonian fluid were discussedbased on the power-law model. Measured viscosities of each nanofluid were used to evaluatethe power-law and consistency index. Results indicated that the nanofluid viscosity decreasedby increasing the concentrat...

State estimation has been introduced to power systems and implemented in the 60s, using a single frequency, balanced and symmetric power system model under steady state conditions. This implementation is still prevalent today. The single frequency, balanced and symmetric system assumptions have simplified the implementation but have generated practical problems. This paper examines these simpli...

Power-law spectra have been used to interpret scintillation data for decades. Twocomponent power-law spectra were invoked to explain observed equatorial scintillation from VHF to S-Band. Carrano and Rino developed new strong-scatter theoretical results described in a companion IES 2015 paper. The new theoretical results have been used to estimate two-component power-law parameters from multi-fr...