Given any fixed integer $m \ge 3$, the author presents simple formulas for $m - 2$ algebraically independent polynomials over $\mathbb{Q}$ having the remarkable property, with respect to transformations of homogeneous linear differential equations of order $m$, that each polynomial is both a semi-invariant of the first kind (with respect to changes of the dependent variable) and a semi-invariant of the second kind (with respect to changes of the independent variable). These relative invariants are suitable for global studies in several different contexts and do not require Laguerre-Forsyth reductions for their evaluation. In contrast, all of the general formulas for basic relative invariants that have been proposed by other researchers during the last 113 years are merely local ones that are either much too complicated or require a Laguerre-Forsyth reduction for each evaluation.

The problem of deducing the basic relative invariants possessed by monic homogeneous linear differential equations of order $m$ was initiated in 1879 with Edmund Laguerre's success for the special case $m = 3$. It was solved in number 744 of the Memoirs of the AMS (March 2002), by a procedure that explicitly constructs, for any $m \geq3$, each of the $m - 2$ basic relative invariants. During that 123-year time span, only a few results were published about the basic relative invariants for other classes of ordinary differential equations. With respect to any fixed integer $\, m \geq 1$, the author begins by explicitly specifying the basic relative invariants for the class $\, \mathcal{C {m,2 $ that contains equations like $Q {m = 0$ in which $Q {m $ is a quadratic form in $y(z), \, \dots, \, y{(m) (z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y{(m) (z) \bigr){2 $ is $1$.Then, in terms of any fixed positive integers $m$ and $n$, the author explicitly specifies the basic relative invariants for the class $\, \mathcal{C {m, n $ that contains equations like $H {m, n = 0$ in which $H {m, n $ is an $n$th-degree form in $y(z), \, \dots, \, y{(m) (z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y{(m) (z) \bigr){n $ is $1$.These results enable the author to obtain the basic relative invariants for additional classes of ordinary differential equa

Includes a paper that provides necessary and sufficient conditions on a couple of Banach lattices of measurable functions $(X_{0}, X_{1})$ which ensure that, for all weight functions $w_{0}$ and $w_{1}$, the couple of weighted lattices $(X_{0, w_{0}}, X_{1, w_{1}})$ is a Calderon-Mityagin cou

This memoir develops, discusses and compares a range of commutative and non-commutative invariants defined for projection method tilings and point patterns. The projection method refers to patterns, particularly the quasiperiodic patterns, constructed by the projection of a strip of a high dimensional integer lattice to a smaller dimensional Euclidean space. In the first half of the memoir the acceptance domain is very general - any compact set which is the closure of its interior - while in the second half the authors concentrate on the so-called canonical patterns. The topological invariants used are various forms of $K$-theory and cohomology applied to a variety of both $C DEGREES*$-algebras and dynamical systems derived from such a p

Intersection homology theory provides a way to obtain generalized Poincare duality, as well as a signature and characteristic classes, for singular spaces. For this to work, one has had to assume however that the space satisfies the so-called Witt condition. We extend this approach to constructing invariants to spaces more general than Witt spaces.

An $n$-dimensional $\mu$-component boundary link is a codimension $2$ embedding of spheres $L=\sqcup_{\mu}S DEGREESn \subset S DEGREES{n+2}$ such that there exist $\mu$ disjoint oriented embedded $(n+1)$-manifolds which span the components of $L$. This title proceeds to compute the isomorphism class of $C_{

The authors analyse two topological invariants of an embedding of an arrangement of rational plane curves in the projective complex plane, namely, the cohomology ring of the complement and the characteristic varieties. Their main result states that the cohomology ring of the complement to a rational arrangement is generated by logarithmic 1 and 2-forms and its structure depends on a finite number of invariants of the curve (its combinatorial type).

Presents the problem of the splitting of invariant manifolds in multidimensional Hamiltonian systems, stressing the canonical features of the problem. This book offers introduction of a canonically invariant scheme for the computation of the splitting matrix.

Concerned with the nonnegative solutions of $\Delta u = u^2$ in a bounded and smooth domain in $\mathbb{R}^d$, this title intends to prove that they are uniquely determined by their fine trace on the boundary as defined in [DK98a], answering a major open question of [Dy02].