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Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. Let x = (x 1 ,. .. , x n) be a vector in K n. The vector x has an integer relation if there exists a vector m = (m 1 ,. .. , m n) ∈ O(K) n , m = 0, such that m 1 x 1 + m 2 x 2 +. .. + m n x n = 0. In this paper we define the parameterized integer… (More)

Let x = (x 1 , x 2 , · · · , x n) be a vector of real numbers. x is said to possess an integer relation if there exist integers a i not all zero such that a 1 x 1 + a 2 x 2 + · · · + a n x n = 0. Beginning in 1977 several algorithms (with proofs) have been discovered to recover the a i given x. The most efficient of these existing integer relation… (More)

Let x = (x 1 ; x 2 ; ; x n) be a vector of real numbers. x is said to possess an integer relation if there exist integers a i not all zero such that a 1 x 1 + a 2 x 2 + + a n x n = 0. Beginning ten years ago, algorithms were discovered by one of us which, for any n, are guaranteed to either nd a relation if one exists or else establish bounds within which… (More)

Lumbosacral conditions are now recognized as compromising a significant percentage of problems for dogs and cats with rear limb abnormalities. Diagnosis is based on logically ruling out the more common maladies and then performing specific diagnostic procedures that definitively show the lesion(s).

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