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A (0, 1) matrix A is strongly unimodular if A is totally unimodular and every matrix obtained from A by setting a nonzero entry to 0 is also totally unimodular. Here we consider the linear discrepancy of strongly unimodular matrices. It was proved by Lováz, et.al. [5] that for any matrix A, lindisc(A) ≤ herdisc(A). (1) When A is the incidence matrix of a… (More)

In his paper (1942), Ore found necessary and sufficient conditions under which the modular and distributive laws hold in the lattice of equivalence relations on a set S. In the present paper, we consider commuting equivalence relations. It has been proved by J6nsson (1953) that the modular law holds in the lattice of commuting equivalence relations. We give… (More)

- Joseph P S Kung, Xinyu Sun, Catherine Yan
- 2006

In this paper we extend the work of [1] to study combinatorial problems via the theory of biorthogonal polynomials. In particular, we give a unified algebraic approach to several combinatorial objects, including order statistics of a real sequence, parking functions, lattice paths, and area-enumerators of lattice paths, by describing the properties of the… (More)

- Joseph P S Kung, Xinyu Sun, Catherine Yan
- 2007

We describe an involution on a set of sequences associated with lattice paths with north or east steps constrained to lie between two arbitrary boundaries. This involution yields recursions (from which determinantal formulas can be derived) for the number and area enumerator of such paths. An analogous involution can be defined for parking functions with… (More)

- B P Bernays, Bg, V Blomer, A Granville, Bf, J Bourgain +8 others
- 2014

A proof of the positive density conjecture for integer Apollonian circle packings,

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