International Journal of Mathematics

Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain types of generic distributions and partially integrable almost CR-structures of hypersurface type. The aim of this article is to develop for a large class of (semi-) linear overdetermined systems of partial differential equations on regular infinitesimal flag manifolds M a conceptual method to rewrite these systems as systems of the form , where is a linear connection on some vector bundle V over M and C : V → T* M ⊗ V is a (vector) bundle map. In particular, if the overdetermined system is linear, will be a linear connection on V and hence the dimension of its solution space is bounded by the rank of V. We will see that the rank of V can be easily computed using representation theory.

We derive Harnack-type inequalities for non-negative solutions of the porous medium equation on a complete Riemannian manifold with non-negative Ricci curvature. Along with gradient estimates, reparametrization of a geodesic and time rescaling of a solution are key tools to get the results.

In this paper we consider spectral representation of infinite dimensional stationary random fields over an abelian locally compact (or LCA) group, and then extend the results of earlier authors who consider wide sense Markov random fields over a Euclidean group ℝⁿ to the general LCA group context and obtain minimality properties. We also indicate possibilities of some extensions of these results to certain nonstationary classes.

In this work we define a semialgebraic set S ⊂ ℝⁿ to be irreducible if the noetherian ring of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring : Substitution Theorem, Positivstellensätze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

In this paper we give some versions of the Łojasiewicz inequality on non-compact domains for polynomial functions. We also point out some relations between the existence Łojasiewicz inequality and the phenomenon of singularities at infinity.

In this paper, we consider a periodic Hunter–Saxton equation with weak dissipation. We obtain the existence of global weak solutions to the equation for any initial data in H¹(𝕊).

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙ^r-1)^rg//PGL(r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle E_G over M admits a Hermitian–Einstein structure if and only if E_G is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.

We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor s from a category G to Top^op and show that it defines what we call a skew category algebra A ⋊^σ G. We study the connection between topological freeness of s and, on the one hand, ideal properties of A ⋊^σ G and, on the other hand, maximal commutativity of A in A ⋊^σ G. In particular, we show that if G is a groupoid and for each e ∈ ob(G) the group of all morphisms e → e is countable and the topological space s(e) is Tychonoff and Baire. Then the following assertions are equivalent: (i) s is topologically free; (ii) A has the ideal intersection property, i.e. if I is a nonzero ideal of A ⋊^σ G, then I ∩ A ≠ {0}; (iii) the ring A is a maximal abelian complex subalgebra of A ⋊^σ G. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.

We will prove a uniqueness theorem for the Riemann zeta-function in terms of the preimages of three complex values.

For a smooth, irreducible projective surface S over ℂ, the number of r-nodal curves in an ample linear system (where is a line bundle on S) can be expressed using the rth Bell polynomial P_r in universal functions a_i, 1 ≤ i ≤ r, of (S, ), which are ℤ-linear polynomials in the four Chern numbers of S and . We use this result to establish a proof of the classical shape conjectures of Di Francesco–Itzykson and Göttsche governing node polynomials in the case of ℙ². We also give a recursive procedure which provides the -term of the polynomials a_i.