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...

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...

Let k be a commutative ring, A a commutative k-algebra and D the filtered ring of k-linear differential operators of A. We prove that: (1) The graded ring grD admits a canonical embedding θ into the graded dual of the symmetric algebra of the module ΩA/k of differentials of A over k, which has a canonical divided power structure. (2) There is a canonical morphism θ from the divided power algebr...

Image hashing allows compression, enhancement or other signal processing operations on digital images which are usually acceptable manipulations. Whereas, cryptographic hash functions are very sensitive to even single bit changes in image. Image hashing is a sum of important quality features in quantized form. In this paper, we proposed a novel image hashing algorithm for authentication which i...

Let p be a prime and n a positive integer. In [J. Austral. Math. Soc. 81 (2006), 153–164], Feng and Kwak showed that if p > 5 then every connected cubic symmetric graph of order 2p is a Cayley graph. Clearly, this is not true for p = 5 because the Petersen graph is non-Cayley. But they conjectured that this is true for p = 3. This conjecture is confirmed in this paper. Also, for the case when p...