Let L be a free Lie algebra of finite rank over a field K and let Ln denote the degree n homogeneous component of L. Formulae for the dimension of the subspaces [Lm, Ln] for all m and n were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form [Lm, Ln, L k ] = [[Lm, Ln], L k ]. Surprisingly, in contrast to the case of a product of two homogeneous...

This report gives an overview of the main ideas in Chapter 9 of the studied book, about the dimension of a variety. After recalling some definitions and basic properties about projective varieties, homogeneous ideals and Gröbner bases, we will begin our study of the dimension of a variety with the observation of a geometric way to define the dimension of a variety defined by a monomial ideal. T...

We prove that a unital simple approximately homogeneous C*-algebra with no dimension growth absorbs the Jiang–Su algebra tensorially without appealing to the classification theory of these algebras. Our main result continues to hold under the slightly weaker hypothesis of exponentially slow dimension growth.

Canonical dimension is an integral-valued invariant of algebraic structures. We are mostly interested in understanding the canonical dimension of projective homogeneous varieties under semisimple affine algebraic groups over arbitrary fields. Known methods, results, applications, and open problems are reviewed, some new ones are provided. Mathematics Subject Classification (2010). Primary 14L17...

We show that it is possible to construct arbitrary order stable schemes for the homogeneous and heterogeneous wave equation in any dimension. The construction is elementary and uses formal power series techniques. We shall also calculate exact stability limits in various cases, and apparently this limit depends only on the dimension of the space.

We consider a commutative ring R graded by an arbitrary abelian group G, and define the grade of a G-homogeneous ideal I on R in terms of vanishing of C̆ech cohomology. By defining the dimension in terms of chains of homogeneous prime ideals and supposing R satisfies a.c.c. on G-homogeneous ideals and has a unique G-homogeneous maximal ideal, we can define graded versions of the depth of R and C...

The propagation of singularities for the system of homogeneous thermoelasticity in one space dimension is studied. Linear and a class of semilinear Cauchy problems are considered.

We prove that a unital simple approximately homogeneous (AH) C-algebra with no dimension growth absorbs the Jiang-Su algebra tensorially without appealing to the classification theory of these algebras. Our main result continues to hold under the slightly weaker hypothesis of exponentially slow dimension growth.

The dimension of the third homogeneous component of a matrix quantum bialgebra, determined by pair of quantum spaces, is calculated. The Poincaré series of some deformations of GL(n) is calculated. A new deformation ofGL(3) with the correct dimension is given.

A Bäcklund transformation(BT) and a recurrence formula are derived by the homogeneous balance(HB) method. A initial problem of Burgers equations is reduced to a initial problem of heat equation by the BT, the initial problem of heat equation is resolved by the Fourier transformation method, substituting various solutions of the initial problem of the heat equation will yield solutions of the in...