Geometric proof of a conjecture of Fulton

We give a geometric proof of a conjecture of Fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for GL(r).     

A proof of the Hoggatt-Bergum conjecture

It is proved that if and are positive integers such that the product of any two distinct elements of the set

increased by is a perfect square, then has to be . This is a generalization of the theorem of Baker and Davenport for .

     

Towards the Willmore conjecture

We develop a variety of approaches, mainly using integral geometry, to proving that the integral of the square of the mean curvature of a torus immersed in must always take a value no less than . Our partial results, phrased mainly within the -formulation of the problem, are typically strongest when the Gauss curvature can be controlled in terms of extrinsic curvatures or when the torus enjoys further properties related to its distribution within the ambient space (see Sect. 3). Corollaries include a recent result of Ros [20] confirming the Willmore conjecture for surfaces invariant under the antipodal map, and a strengthening of the expected results for flat tori. The value arises in this work in a number of different ways – as the volume (or renormalised volume) of or , and in terms of the length of shortest nontrivial loops in subgroups of SO(4). Received April 26, 1999 / Accepted January 14, 2000 / Published online June 28, 2000     

A proof of a consequence of Dirac's conjecture

Geometriae Dedicata -     

A proof of MIKOTA's conjecture

Coming from MIKOTA's conjecture [1] a proof is given by examinating the properties of the occuring matrices M , K and M ^–1 K and deducing the natural frequencies of the chain structured mass-spring vibration system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A proof of Nogura's conjecture

Answering a question of T. Nogura (1985), we show using the Open Coloring Axiom that the weak diagonal sequence property is preserved by taking products whenever the products themselves are Fréchet. As an application we show, using the same assumption, that the product of two Fréchet groups is Fréchet provided it is sequential. Recall that the product of two Fréchet groups may not be sequential.

     

The proof of hermann''s conjecture

This note proves Thomas Hermann's conjecture on the comparison between twoboundaries of the derivatives of rational cubic Bezier curves. The result is valuable for computer aided geometric design.     

A proof of Coleman's conjecture

Almost thirty years ago Coleman made a conjecture that for any convex lattice polygon with v vertices, g (g?1) interior lattice points and b boundary lattice points we have b?2g-v+10. In this note we give a proof of the conjecture. We also aim to describe all convex lattice polygons for which the bound b=2g-v+10 is attained.     

Short proof of a conjecture concerning split-by-nilpotent extensions

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E. We provide a short proof to a conjecture by Assem and Zacharia concerning properties of \(\mathop {\text {mod}}B\) inherited by \(\mathop {\text {mod}}C\). We show if B is a tilted algebra, then C is a tilted algebra.     

On the Nochka-Chen-Ru-Wong proof of Cartan's conjecture

In 1982-1983, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in Pn relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his 1987 thesis, and subsequently simplified by M. Ru and P.-M. Wong in 1991. The proof involved assigning weights to the hyperplanes. This paper provides further simplification of the proof of the construction of the weights, by bringing back the use of the convex hull in working with the “Nochka diagram.”     

A proof of the bounded graph conjecture

Summary An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph. We prove an old conjecture of Halin, which characterizes the bounded graphs in terms of four forbidden topological subgraphs.Oblatum 17-IV-1991 & 25-X-1991     

Squaring a tournament: A proof of Dean's conjecture

Let the square of a tournament be the digraph on the same nodes with arcs where the directed distance in the tournament is at most two. This paper verifies Dean's conjecture: any tournament has a node whose outdegree is at least doubled in its square. © 1996 John Wiley & Sons, Inc.