In the paper we describe the subcategory of the category of Zgraded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the TKK construction. For instance, we prove that L = L −1 ⊕ L 0 ⊕ L 1 can be constructed from a Jordan pair if and only if L 0 = [L −1 , L 1 ] and the second graded homology group H gr 2 (L) is trivial. Similar descriptions are obtained for Jordan triple systems and Jordan algebras. New functorial versions of the TKK construction are given for pairs and algebras.

It is well known that array deformations can distort beam patterns and introduce bearing errors if the beamformer assumes linearity. It is also known that deformed arrays can resolve left-right ambiguities, provided the shape is known. In this work, these two effects are studied for undamped and damped sinusoidally deformed arrays with small deformation amplitudes in the horizontal (x,y) plane only. By use of fixed arc-length separations along the array, the hydrophone (x,y) coordinates are determined numerically and the error in assuming equal x spacing is summarized for a sample array. Array-response patterns are analyzed for two conditions: ͑1͒ when the deformed array shape is assumed linear and ͑2͒ when the deformed array shape is known exactly. Degradations resulting from assuming linearity and the ability to resolve left-right ambiguities are discussed in terms of reduced gain, degraded angular resolution, and bearing errors. Shape-unknown signal-gain degradation ranges to 7 dB at broadside, but is less than 1 dB near endfire. For the shape-known case, signal gain for the true peak is greater than signal gain for the ambiguous peak by up to 9 dB for sources at broadside and to just over 2.5 dB for arrivals near endfire.

Abstract.In 1937, Franklin and Schneider generalized the Gelfond-Schneider result on the transcendence of a.P . They proved the following theorem: If /? is an algebraic, irrational number and a is "suitably well-approximated by algebraic numbers of bounded degree", then aP is transcendental. In 1964, Feldman established the algebraic independence of a and a.P under similar conditions. We use results concerning linear forms in logarithms to give quantitative versions of the Franklin-Schneider and Feldman results.

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