Reduction theorems for general linear connections

Compactness Theorems for Invariant Connections

The Palais-Smale Condition C holds for the Yang-Mills functional on principal bundles over compact manifolds of dimension ≤ 3. This was established by S. Sedlacek [17] and C. Taubes [18] Proposition 4.5 using the compactness theorem of K. Uhlenbeck [20]; see also [23]. It is well known that Condition C fails for Yang-Mills over compact manifolds of dimension ≥ 4. The example of SO(3)-invariant ...

Adiabatic theorems for general linear operators and well-posedness of linear evolution equations

Reduction theorems for Noether’s problem

Let K be any field, G be a finite group. Let G act on the rational function field K(x(g) : g ∈ G) by K-automorphisms and h · x(g) = x(hg). Denote by K(G) = K(x(g) : g ∈ G) the fixed field. Noether’s problem asks whether K(G) is rational (= purely transcendental) over K. We will give several reduction theorems for solving Noether’s problem. For example, let G̃ = G × H be a direct product of finit...

Connections for general group actions

Principal bundles serve as a powerful and elegant geometric framework for analyzing group actions and symmetry. Beyond their geometric origins, principal bundles play significant roles in the analysis of mechanical systems with symmetry, as well as the design of appropriate computational algorithms. A connection on a principal bundle is defined as an equivariant decomposition of vectors into in...

Decomposition theorems for linear programs

It is well known that any feasible arc-flow solution to a network problem defined on a graph G = (N,A), where N is the set of nodes whereas A is the set of arcs, can be expressed using at most |A| + |N | paths and cycles having nonzero flow, out of these, at most |A| cycles. This existence theorem is used in a number of proofs establishing the complexity of strongly polynomial algorithms for ne...