On the Jordan–Hölder property for geometric derived categories

We prove that the semiorthogonal decompositions of the derived category of the classical Godeaux surface X   do not satisfy the Jordan–Hölder property. More precisely, there are two maximal exceptional sequences in this category, one of length 11, the other of length 9. Assuming the Noetherian property for semiorthogonal decompositions, one can define, following Kuznetsov, the Clemens–Griffiths component CG(D)$\mathrm{CG}(\mathfrak{D})$ for each fixed maximal decomposition D$\mathfrak{D}$. We then show that D^b(X)${\mathrm{D}}^b(X)$ has two different maximal decompositions for which the Clemens–Griffiths components differ. Moreover, we produce examples of rational fourfolds whose derived categories also violate the Jordan–Hölder property.     

Inclusions of Waterman–Shiba spaces into generalized Wiener classes

The characterization of the inclusion of Waterman–Shiba spaces ΛBV^(p)$\Lambda {\mathit{BV}}^{(p)}$ into generalized Wiener classes of functions BV(q;δ)$\mathit{BV}(q;\delta )$ is given. It uses a new and shorter proof and extends an earlier result of U. Goginava.     

A reduction method for semilinear elliptic equations and solutions concentrating on spheres

We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A⊆R^2m$A\subseteq {\mathbb{R}}^{2m}$, m?2$m?2$, invariant by the action of a certain symmetry group can be reduced to a nonhomogeneous similar problem in an annulus D⊂R^m+1$D\subset {\mathbb{R}}^{m+1}$, invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A   which concentrate on one or two (m−1)$(m-1)$ dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration tends to infinity.     

The adjoint representation inside the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra g$\mathfrak{g}$ we study the space of invariants A=(?g^?⊗g^?)^g$A={(?{\mathfrak{g}}^{?}\otimes {\mathfrak{g}}^{?})}^{\mathfrak{g}}$, which describes the isotypic component of type g$\mathfrak{g}$ in ?g^?$?{\mathfrak{g}}^{?}$, as a module over the algebra of invariants (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$. As main result we prove that A   is a free module, of rank twice the rank of g$\mathfrak{g}$, over the exterior algebra generated by all primitive invariants in (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$, with the exception of the one of highest degree.