For $n=2m\ge 4$, let $\Omega \in {ℝ}^n$ be a bounded smooth domain and $𝒩\subset {ℝ}^L$ a compact smooth Riemannian manifold without boundary. Suppose that $\left\{u_k\right\}\in W^{m,2}(\Omega ,𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}{E}_m\left(\Pi (u+t\xi )\right)=0$$ with ${\Phi }_k\to 0$ in ${\left(W^{m,2}(\Omega ,𝒩)\right)}^{*}$ and $u_k\rightharpoonup u$ weakly in $W^{m,2}(\Omega ,𝒩)$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-$W^{m,2}$ topology.

Introduction: This article will present just one example of a general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution. It was the motivating example from two lectures on the BGG resolution given at the 19th Czech Winter School on Geometry and Physics held in Srní in January 1999. This article may be seen as a technical example to go with a more elementary introduction which will appear elsewhere [M. Eastwood, Notices Am. Math. Soc. 46, No. 11, 1368-1376 (1999)]. In fact, there...

Let $G$ be a bundle functor of order $(r,s,q)$, $s\ge r\le q$, on the category $ℱ{ℳ}_{m,n}$ of $(m,n)$-dimensional fibered manifolds and local fibered diffeomorphisms. Given a general connection $\Gamma$ on an $ℱ{ℳ}_{m,n}$-object $Y\to M$ we construct a general connection $𝒢(\Gamma ,\lambda ,\Lambda )$ on $GY\to Y$ be means of an auxiliary $q$-th order linear connection $\lambda$ on $M$ and an $s$-th order linear connection $\Lambda$ on $Y$. Then we construct a general connection $𝒢(\Gamma ,{\nabla }_1,{\nabla }_2)$ on $GY\to Y$ by means of auxiliary classical linear connections ${\nabla }_1$ on $M$ and ${\nabla }_2$ on $Y$. In the case $G=J^1$ we determine all general connections $𝒟(\Gamma ,\nabla )$ on $J^{1}Y\to Y$ from...

Summary: The Ado theorem is a fundamental fact, which has a reputation of being a `strange theorem'. We give its natural proof.

We construct series of examples of non-flat non-homogeneous parabolic geometries that carry a symmetry of the parabolic geometry at each point.

We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned...

We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned...

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t+{\nabla }_g·f(x,u)=0$ on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}}$ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$