Interactive mathematics over the Internet is particularly appealing in that it can take advantage of the wealth of resources available online. Among the prerequisites to cooperation among systems for computer mathematics is the meaningfulness of the objects that are exchanged. Strong OpenMath is a sublanguage of OpenMath for which meaningfullness of the objects is exactly de ned. The paper addr...

Let E/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N . Assume that the number of primes dividing N and inert in K is odd, and let Hc be the ring class field of K of conductor c prime to ND with Galois group Gc over K. Fix a complex character χ of Gc. Our main result is that if LK(E,χ, 1) 6= 0 then Selp(...

Much attention has been paid in the literature to total-variation-diminishing (TVD) numerical processes in the solution of nonlinear hyperbolic differential equations. For special Runge– Kutta methods, conditions on the stepsize were derived that are sufficient for the TVD property; see, e.g., Shu and Osher [J. Comput. Phys., 77 (1988), pp. 439–471] and Gottlieb and Shu [Math. Comp., 67 (1998),...

We study classical solutions (ic–instantons) in N = 4 SYM in 4D which, in the strong coupling limit, correspond to complex two–dimensional manifolds. Asymptotically in time the latter have boundaries represented by compact real three–manifolds. Therefore they lend themselves to an interpretation in terms of 3–brane scattering. We suggest that these solutions may represent scattering of D3–brane...

In this letter a generic counterexample to the strong cosmic censor conjecture is exhibited. More precisely—taking into account that the conjecture lacks any precise formulation yet—first we make sense of what one would mean by a “generic counterexample” by introducing the mathematically unambigous and logically stronger concept of a “robust counterexample”. Then making use of Penrose’ nonlinea...

In this paper, we present a new model and mechanism for auctions in two-sided markets of buyers and sellers, with budget constraints imposed on buyers. Our mechanism is viewed as a two-sided extension of the polyhedral clinching auction by Goel et al., and enjoys various nice properties, such as incentive compatibility of buyers, individual rationality, pareto optimality, strong budget balance....

We introduce the concept of a geometric categorical sl2 action and relate it to that of a strong categorical sl2 action. The latter is a special kind of 2-representation in the sense of Rouquier. The main result is that a geometric categorical sl2 action induces a strong categorical sl2 action. This allows one to apply the theory of strong sl2 actions to various geometric situations. Our main e...

On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p...

Suppose that λ = μ+ and μ is singular. We consider two aspects of the square property on subsets of λ. First, we have results which show e.g. that for א0 < κ = cf(κ) < μ, the equality cf([μ]≤κ,⊆) = μ is a sufficient condition for the set of elements of λ whose cofinality is ≤ κ to be split into the union of μ sets with squares. Secondly, we introduce a certain weak version of the square propert...

Given an elliptic curve E over a finite field Fq of q elements, we say that an odd prime l ∤ q is an Elkies prime for E if tE − 4q is a quadratic residue modulo l, where tE = q+1−#E(Fq) and #E(Fq) is the number of Fq-rational points on E. These primes are used in the presently most efficient algorithm to compute #E(Fq). In particular, the bound Lq(E) such that the product of all Elkies primes f...