Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let k be an algebraically closed field and A the polynomial algebra in r variables with coefficients in k. In case the characteristic of k is 2, Carlsson [9] conjectured that for any DG-A-module M of dimension N as a free A-module, if the homology of M is nontrivial and finite dimensional as a k-vector space, then $2^r\le N$. Here we state a stronger conjecture about varieties of square-zero upper triangular $N\times N$ matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N<8$ or $r<3$ without any restriction on the characteristic of k. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N>4$ and $r=2$.     

Existential monadic second order logic of undirected graphs: The Le Bars conjecture is false

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ? such that $\mathsf{P}(G_n??)$ does not converge as $n\to \infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1,\dots ,n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.     

Integral points, divisibility between values of polynomials and entire curves on surfaces

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first examples, to our knowledge, of a simply connected smooth variety whose sets of integral points are never Zariski-dense. Some of our results are connected with divisibility problems, i.e. the problem of describing the integral points in the plane where the values of some given polynomials in two variables divide the values of other given polynomials.     

The Darmon-Dasgupta units over genus fields and the Shimura correspondence

We study a special case of the Gross-Stark conjecture (Gross, 1981 [Gr]), namely over genus fields. Based on the same idea we provide evidence of the rationality conjecture of the elliptic units for real quadratic fields over genus fields, which is a refinement of the Gross-Stark conjecture given by Darmon and Dasgupta (2006) [DD]. Then a relationship between these units and the Fourier coefficients of p-adic Eisenstein series of half-integral weight is explained.     

An improvement to multiple criteria ABC inventory classification

In this paper we presented an extended version of the Ng-modelg [W.L. Ng, A simple classifier for multiple criteria ABC analysis, European Journal of Operational Research 177 (2007) 344–353] for multi-criteria inventory classification. The proposed model is a nonlinear programming model which determines a common set of weights for all the items. Our model not only incorporates multiple criteria for ABC classification, but also maintains the effects of weights in the final solution, an improvement over the model proposed by Ng. An illustrative example is presented to compare our model and the Ng-model.     

Elliptic curves of rank 1 satisfying the 3-part of the Birch and Swinnerton–Dyer conjecture

Let E be an elliptic curve over Q of conductor N and K be an imaginary quadratic field, where all prime divisors of N split. If the analytic rank of E over K is equal to 1, then the Gross and Zagier formula for the value of the derivative of the L-function of E over K, when combined with the Birch and Swinnerton–Dyer conjecture, gives a conjectural formula for the order of the Shafarevich–Tate group of E over K. In this paper, we show that there are infinitely many elliptic curves E such that for a positive proportion of imaginary quadratic fields K, the 3-part of the conjectural formula is true.     

Families of pairs of graphs with a large number of common cards

The vertex‐deleted subgraph G?v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertex‐deleted subgraphs. The number of common cards of G and H (or between G and H) is the cardinality of the multiset intersection of the decks of G and H. In this article, we present infinite families of pairs of graphs of order n ≥ 4 that have at least \begin{eqnarray*}2\lfloor\frac{1}{3}(n-1)\rfloor\end{eqnarray*} common cards; we conjecture that these, along with a small number of other families constructed from them, are the only pairs of graphs having this many common cards, for sufficiently large n. This leads us to propose a new stronger version of the Reconstruction Conjecture. In addition, we present an infinite family of pairs of graphs with the same degree sequence that have \begin{eqnarray*}\frac{2}{3}(n+5-2\sqrt{3n+6})\end{eqnarray*} common cards, for appropriate values of n, from which we can construct pairs having slightly fewer common cards for all other values of n≥10. We also present infinite families of pairs of forests and pairs of trees with \begin{eqnarray*}2\lfloor\frac{1}{3}(n-4)\rfloor\end{eqnarray*} and \begin{eqnarray*}2\lfloor\frac{1}{3}(n-5)\rfloor\end{eqnarray*} common cards, respectively. We then present new families that have the maximum number of common cards when one graph is connected and the other disconnected. Finally, we present a family with a large number of common cards, where one graph is a tree and the other unicyclic, and discuss how many cards are required to determine whether a graph is a tree. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 146–163, 2010     

An approximate version of Hadwiger's conjecture for claw‐free graphs

Hadwiger's conjecture states that every graph with chromatic number χ has a clique minor of size χ. In this paper we prove a weakened version of this conjecture for the class of claw‐free graphs (graphs that do not have a vertex with three pairwise nonadjacent neighbors). Our main result is that a claw‐free graph with chromatic number χ has a clique minor of size $\lceil\frac{2}{3}\chi\rceil$. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 259–278, 2010     

Hadamard matrices of small order and Yang conjecture

We show that 138 odd values of n<10000 for which a Hadamard matrix of order 4n exists have been overlooked in the recent handbook of combinatorial designs. There are four additional odd n=191, 5767, 7081, 8249 in that range for which Hadamard matrices of order 4n exist. There is a unique equivalence class of near‐normal sequences NN(36), and the same is true for NN(38) and NN(40). This means that the Yang conjecture on the existence of near‐normal sequences NN(n) has been verified for all even n⩽40, but it still remains open. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 254–259, 2010