An intercalate matrix M of type [r, s, n] is an r × s matrix with entries in {1, 2, . . . , n} such that all entries in each row are distinct, all entries in each column are distinct, and all 2 × 2 submatrices of M have either 2 or 4 distinct entries. Yuzvinsky’s Conjecture on intercalate matrices claims that the smallest n for which there is an intercalate matrix of type [r, s, n] is the Hopf-...

in this paper, we mainly investigate the uniqueness of the entire function sharing a small entire function with its high difference operators. we obtain one results, which can give a negative answer to an uniqueness question relate to the bruck conjecture dealt by liu and yang. meanwhile, we also establish a difference analogue of the bruck conjecture for entire functions of order less than 2, ...

We propose a conjecture refining the Stark conjecture St(K/k,S) in the function field case. Of course St(K/k,S) in this case is a theorem due to Deligne and independently to Hayes. The novel feature of our conjecture is that it involves tworather than onevariable algebraic functions over finite fields. Because the conjecture is formulated in the language and framework of Tate’s thesis we have p...

We present previously unpublished elementary proofs by Dekker and Ottens (1991) and Boyce (private communication) of a special case of the Dinitz conjecture. We prove a special case of a related basis conjecture by Rota, and give a reformulation of Rota's conjecture using the Nullstellensatz. Finally we give an asymptotic result on a related Latin square conjecture.

A Boolean constraint satisfaction problem (CSP) is called approximation resistant if independently setting variables to 1 with some probability α achieves the best possible approximation ratio for the fraction of constraints satisfied. We study approximation resistance of a natural subclass of CSPs that we call Symmetric Constraint Satisfaction Problems (SCSPs), where satisfaction of each const...

a(p) = 2p~ @ ) cos 0(p). Since we know the truth of the Ramanujan-Petersson conjecture, it follows that the 0(p)'s are real. Inspired by the Sato-Tate conjecture for elliptic curves, Serre [14] conjectured that the 0(p)'s are uniformly distributed in the interval [0, rc] with respect to the 1 measure -sin2OdO. Following Serre, we shall refer to this as the Sato-Tate r~ conjecture, there being n...

Let S be a nonsingular projective K3 surface. Motivated by the study of the Gromov-Witten theory of the Hilbert scheme of points of S, we conjecture a formula for the GromovWitten theory (in all curve classes) of the Calabi-Yau 3-fold S×E where E is an elliptic curve. In the primitive case, our conjecture is expressed in terms of the Igusa cusp form χ10 and matches a prediction via heterotic du...

We show that if the L-function of an irreducible 2-dimensional complex Galois representation over Q is not automorphic then it has infinitely many poles. In particular, the Artin conjecture for a single representation implies the corresponding strong Artin conjecture. Introduction Let ρ : Gal(Q/Q) → GLn(C) be an irreducible continuous representation of the absolute Galois group of Q. Brauer [2]...

A hypergraph H is linear if no two distinct edges of H intersect in more than one vertex and loopless if no edge has size one. A q-edge-colouring of H is a colouring of the edges of H with q colours such that intersecting edges receive different colours. We use ∆H to denote the maximum degree of H. A well known conjecture of Erdös, Farber and Lovász is equivalent to the statement that every loo...

For an abelian extension L/K of number fields, the Equivariant Tamagawa Number Conjecture at s = 0, which is equivalent to the Lifted Root Number Conjecture, implies Rubin’s Conjecture by work of Burns. We show that, for relative biquadratic extensions L/K satisfying a certain condition on the splitting of places, Rubin’s Conjecture in turn implies the ETNC/LRNC. We conclude with some examples....