We use multivariate splines to investigate linear diophantine equations and related problems in graph theory. In particular, we solve a conjecture of Stanley about symmetric magic squares. 0 Elsevier Science Inc., 1997

We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary real-valued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian mechanics and dynamical systems theory. Our examples deal with the algorithmic impossibility of d...

We present an algorithm to decide the intruder deduction problem (IDP) for a class of locally stable theories enriched with normal forms. Our result relies on a new and efficient algorithm to solve a restricted case of higher-order associative-commutative matching, obtained by combining the Distinct Occurrences of AC-matching algorithm and a standard algorithm to solve systems of linear Diophan...

In this report, we present an algorithm for solving directly linear Diophantine systems of both equations and inequations. Here directly means without adding slack variables for encoding inequalities as equalities. This algorithm is an extension of the algorithm due to Contejean and Devie 10] for solving linear Diophantine systems of equations, which is itself a generalization of the algorithm ...

We describe mean value estimates for exponential sums of degree exceeding 2 that approach those conjectured to be best possible. The vehicle for this recent progress is the efficient congruencing method, which iteratively exploits the translation invariance of associated systems of Diophantine equations to derive powerful congruence constraints on the underlying variables. There are application...

Brualdi et al. introduced the concept of poset codes. In this paper, we consider the problem of classifying all perfect linear codes when the set of coordinate positions is endowed with crown poset structure. We derive a Ramanujan–Nagell type diophantine equation which is satis ed by parameters of perfect linear P-code. Solving this equation, we characterize parameters of one and two error corr...

The diophantine equation x3 _ 3xy2 _y3 = ?3no 17 n l9n2 is completely solved as follows. First, a large upper bound for the variables is obtained from the theory of linear forms in p-adic and real logarithms of algebraic numbers. Then this bound is reduced to a manageable size by p-adic and real computational diophantine approximation, based on the L3-algorithm. Finally the complete list of sol...