A new correctness criterion for multiplicative non-commutative proof nets

 We introduce a new correctness criterion for multiplicative non commutative proof nets which can be considered as the non-commutative counterpart to the Danos-Regnier criterion for proof nets of linear logic. The main intuition relies on the fact that any switching for a proof net (obtained by mutilating one premise of each disjunction link) can be naturally viewed as a series-parallel order variety (a cyclic relation) on the conclusions of the proof net. Received: 8 November 2000 / Revised version: 21 June 2001 / Published online: 2 September 2002 Research supported by the EU TMR Research Programme ``Linear Logic and Theoretical Computer Science'. Mathematics Subject Classification (2000): 03F03, 03F07, 03F52, 03B70 Key words or phrases: Linear and non-commutative logic – Proof nets – Series-parallel orders and order varieties     

A new proof of the Garsia-Wachs algorithm

A new proof of correctness for the Garsia-Wachs algorithm is presented. Like the well-known Hu-Tucker algorithm, the Garsia-Wachs algorithm constructs minimum cost binary trees in O(n log n) time. The new proof has a simpler structure and is free of the difficult inductions of the original.     

A new proof for the first-fit decreasing bin-packing algorithm

This paper provides a new proof of a classical result of bin-packing, namely the $\text{11}\text{9}$ performance bound for the first-fit decreasing algorithm. In bin-packing, a list of real numbers in (0,1] is to be packed into a minimal number of bins, each of which holds a total of at most 1. The first-fit decreasing (FFD) algorithm packs each number in order of nonincreasing size into the first bin in which it fits. In his doctoral dissertation, D. S. Johnson (“Near-Optimal Bin Packing Algorithms,” Doctoral thesis, MIT, Cambridge, Mass., 1973) proved that for every list L, $\text{FFD}\text{(L) ?}\text{11}\text{9}\text{OPT}\text{(L) + 4}$, where FFD(L) and OPT(L) denote the number of bins used by FFD and an optimal packing, respectively. Unfortunately, his proof required more than 100 pages! This paper contains a much shorter and simpler proof that $\text{FFD}\text{(L) ?}\text{11}\text{9}\text{OPT}\text{(L) + 3}$.     

A new area-maximization proof for the circle

Conclusion  Finally, we compare the trianglesT _i with the isosceles triangles which would be obtained by slicing the circle by the same procedure. Since they all have the same short base length and the same angle opposite the base, the isosceles triangles composing the circle have more area than the trianglesT _i. Because the latter triangles cover all ofR ₁ (perhaps even with overlap and/or extension beyondR ₁), we see that the area of the quarter-circle is greater than that ofR ₁, and thus the circle's entire area is greater than that of our original regionR.     

A new proof of A. F. Timan’s approximation theorem

This paper gives a new proof of A. F. Timan’s approximation theorem. It seems to be of considerable advantage that for a fixedn our polynomialG _n(f) is of degree ≦n−1 and depends onn values off(x) only.     

A new proof of the formula for the Bernoulli numbers

This note presents a new proof of Gould's famous formula for the Bernoulli numbers.     

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 Howe Conjecture

The Howe Conjecture, which has formulations for both a reductive -adic group and its Lie algebra, is a statement about the finite dimensionality of certain spaces of -invariant distributions. Howe proved the algebra version of the conjecture for via a method of descent. Harish-Chandra extended Howe's method, when the characteristic is zero, to arbitrary reductive Lie algebras. Harish-Chandra then used the conjecture, in both its Lie algebra and group formulations, as a fundamental underpinning of his approach to harmonic analysis on the group and Lie algebra. Many properties of -invariant distributions, which for real Lie groups follow from differential equations, in the -adic case are consequences of the Howe Conjecture and other facts, e.g. properties of orbital integrals. Clozel proved the group Howe Conjecture in characteristic zero via a method very different than Howe's and Harish-Chandra's descent methods. We give a new proof of the group Howe Conjecture via the Bruhat-Tits building. A key tool in our proof is the geodesic convexity of the displacement function. Highlights of the proof are that it is valid in all characteristics, it has similarities to Howe's and Harish-Chandra's methods, and it has similarities to the existence proof of an unrefined minimal K-type.

     

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 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.