Remark on Wielandt's visibility theorem

A short proof is given of Wielandt's visibility theorem, using a special case of a theorem of Rédei, which was proved in an elementary way by Lóvasz and Schriver.     

Solution of an extremal problem for sets using resultants of polynomials

A new, short proof is given of the following theorem of Bollobás: LetA ₁,..., A_h andB ₁,..., B_h be collections of sets with _i ¦A _i¦=r,¦B_i¦=s and ¦A _iB_j¦=Ø if and only ifi=j, thenh( _s ^r+s ). The proof immediately extends to the generalizations of this theorem obtained by Frankl, Alon and others.     

Proof of a conjecture by Đoković on the Poincaré series of the invariants of a binary form

?okovi? (2006)  [3] gave an algorithm for the computation of the Poincaré series of the algebra of invariants of a binary form, where the correctness proof for the algorithm depended on an unproven conjecture. Here we prove this conjecture.     

A Fast Method to Compute the MSD and MWD of Polymer Populations Formed by Step‐Growth Polymerization of Polyfunctional Monomers With Identical Reactive Groups

A fast method is presented for the calculation of the MSD and the MWD of polymers obtained via step‐growth polymerization of polyfunctional monomers bearing identical reactive groups (i.e., systems of type “Afi”). Using this method, the complete distribution can be calculated rapidly, not just the statistical averages of the polymer population such as or . The computed MSD and MWD give more insight than these averages and can be compared to similar data measured on actual polymer systems. The low‐ and intermediate molecular size/weight part of the distribution curves are calculated using a recurrence scheme, while the high‐molecular tail (large and very large polymers) of the distributions is derived from an asymptotic approximation of the associated generating functions.

     

A Fast Method to Compute the MSD and MWD of Polymer Populations Formed by Step‐Growth Polymerization of Polyfunctional Monomers Bearing A and B Coreactive Groups

General step‐growth polymerization systems of order 2 are considered, i.e., systems of type “AfiBgi”, and a fast algorithmic method is presented to compute, at a given degree of conversion, the MSD and the MWD. The complete distribution is calculated; not just statistical averages of the polymer population such as or . For the computation of the low‐ and intermediate size/weight parts of the distribution curves, a set of recurrence relations is used. The high‐molecular size/weight parts of the curves (right tails) are computed using an accurate approximation derived from generating functions. In a previous paper, we applied our method to general order‐1 systems, i.e., systems of type “Afi”.

     

The graph with spectrum 141 240 (?4)10?(?6)9

We show that there is a unique graph with spectrum as in the title. It is a subgraph of the McLaughlin graph. The proof uses a strong form of the eigenvalue interlacing theorem to reduce the problem to one about root lattices.     

On the chromatic number of q-Kneser graphs

We show that the q-Kneser graph qK _2k:k (the graph on the k-subspaces of a 2k-space over GF(q), where two k-spaces are adjacent when they intersect trivially), has chromatic number q ^k ?+?q ^k?1 for k?=?3 and for k < q log q ? q. We obtain detailed results on maximal cocliques for k = 3.     

On a generalization of a theorem of B. Segre

An extension is given of Segre's generalization of Menelaus' theorem to an arbitrary collection of lines.