Fi(r, y1, . . . , yn) = 0 ∀i has a solution (y1, . . . , yn) ∈ R iff r ∈ D. Equivalently, if there is a (possibly reducible) algebraic variety XR over R and a morphism π : XR → A 1 R such that D = π(XR(R)). In this situation we call dioph(XR, π) := π(XR(R)) ⊂ R the diophantine set corresponding to XR and π. A characterization of diophantine subsets of Z was completed in connection with Hilbert’...

The quadrance between two points A1 = (x1, y1) and A2 = (x2, y2) is the number Q(A1, A2) = (x1 − x2) + (y1 − y2). Let q be an odd prime power and Fq be the finite field with q elements. The unit-quadrance graph Dq has the vertex set F 2 q , and X,Y ∈ F 2 q are adjacent if and only if Q(A1, A2) = 1. In this paper, we study some colouring problems for the unit-quadrance graph Dq and discuss some ...

1 Department of Pathology, University of Cambridge, Cambridge, United Kingdom, 2 Division of Infection and Immunity, School of Medicine, Cardiff University, Cardiff, United Kingdom, 3 Cambridge Institute for Medical Research, University of Cambridge School of Clinical Medicine, Cambridge, United Kingdom, 4 Institut für Virologie und Immunbiologie, JuliusMaximilians-Universität Würzburg, Würzbur...

The correct formulation of a characteristic problem and a Darboux type problem in the special weighted functional spaces for an ultrahyperbolic equation is investigated. In the space of variables x1, x2, y1 and y2 we consider the ultrahyperbolic equation uy1y1 + uy2y2 − ux1x1 − ux2x2 = F. (1) Denote by D : −y1 < x1 < y1, 0 < y1 < +∞, a dihedral angle bounded by the characteristic surfaces S1 : ...

Let Y be an algebraic manifold of dimension 3 with H(Y,ΩjY ) = 0 for all j ≥ 0, i > 0 and h0(Y,OY ) > 1. Let X be a smooth completion of Y such that the boundary X − Y is the support of an effective divisor D on X with simple normal crossings. We prove that the D-dimension of X cannot be 2, i.e., either any two nonconstant regular functions are algebraically dependent or there are three algebra...

We obtain a local uniqueness result for bubbling solutions of the prescribed scalar curvature problem in R . Such result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y). In particular, we prove in this paper that if K(y) is periodic in y1 with period 1 and has a local maximum point at 0, then a bubbling solution whose blow-up set is {(jL, 0, · · · , 0) ...

Let f (z1, . . . ,zk)∈ C(Ik,I) be a given function, where I is (bounded or unbounded) subinterval of R, and k ∈ N. Assume that f (y1, y2, . . . , yk) ≥ f (y2, . . . , yk, y1) if y1 ≥ max{y2, . . . , yk}, f (y1, y2, . . . , yk) ≤ f (y2, . . . , yk, y1) if y1 ≤ min{y2, . . . , yk}, and f is nondecreasing in the last variable zk. We then prove that every bounded solution of an autonomous differenc...

OBJECTIVES To determine whether HLA-DP genes are involved in determining susceptibility to systemic lupus erythematosus (SLE). METHODS HLA-DPA1 and DPB1 genes were amplified by PCR of DNA samples from a panel of patients with SLE and normal controls. Amplified DNA was blotted on to nylon filters and probed with sequence-specific oligonucleotide (SSO) probes. RESULTS No DPA1 or DPB1 allele w...