On the Alexander-Hirschowitz theorem

The Alexander-Hirschowitz theorem says that a general collection of k double points in imposes independent conditions on homogeneous polynomials of degree d with a well-known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on the previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d=3, where our proof is shorter. We end with an account of the history of the work on this problem.     

A simple algebraic proof of the equational interpolation theorem

An algebraic proof is given of a natural version of Craig's interpolation lemma for equational logic.Presented by Bjarni Jónsson.     

A new proof of the Caffarelli-Kohn-Nirenberg theorem

Here we give a self-contained new proof of the partial regularity theorems for solutions of incompressible Navier-Stokes equations in three spatial dimensions. These results were originally due to Scheffer and Caffarelli, Kohn, and Nirenberg. Our proof is much more direct and simpler. © 1998 John Wiley & Sons, Inc.     

A new proof of the Eilenberg-Zilber-Cartier theorem

Let T and D denote respectively the functors which assign to every semi-simplicial double object in an abelian category with infinite direct sums, the total and the diagonal complex. The idea of the proof of the theorem in the title of this note is to show that H_nT and H_nD are left-satellites and that HT=HD. A proof of this theorem was first given in[2].The author is a recipient of an Alexander von Humboldt fellowship at the Department of Mathematics of the University of Heidelberg.     

A new proof of the Abhyankar-Moh-Suzuki theorem

We give a new proof of the famous result that any two embeddings of the affine lineA ¹ inA ² are equivalent by an automorphism ofA ². We also prove that iff(X,Y) is a polynomial overC with one place at infinity, then for every C,f– also has one place at infinity. The proof uses basic results about algebraic surfaces.     

A new proof of the Gitik-Shelah theorem

A purely combinatorial proof of a recent result of M. Gitik and S. Shelah is presented.     

A new proof of the Solomon-Tits theorem

We give a new proof of the Solomon-Tits Theorem which asserts that the Tits building of a finite group of Lie type has the homotopy type of a bouquet of spheres.

     

A new proof of the Maurey-Pisier theorem

In this paper we give a new proof of the Maurey-Pisier theorem about the finite representation ofl _pE andl _qE in any infinite dimensional Banach spaceE.     

A new proof of Poltoratskii's theorem

We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on and any a.e. w.r.t. μ_sing, where μ_sing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, and F_μ(z)=∫(x−z)⁻¹dμ(x).     

A new proof of Mayer''s theorem

This paper gives a new proof of Mayer's theorem concerning the convergence of powers of an interval matrix.     

A new proof of the stable manifold theorem

We give a new proof of the stable manifold theorem for hyperbolic fixed points of smooth maps. This proof shows that the local stable and unstable manifolds are projections of a relation obtained as a limit of the graphs of the iterates of the map. The same proof generalizes to the setting of stable and unstable manifolds for smooth relations.