Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt [“Bounds in the theory of polynomial rings over fields. A nonstandard approach.”, Inventionnes Mathematicae, 76:77–91, 1984] are here extended to differential polynomial rings over differential fields. Amon...

W. L. Ayres, C. R. Ballantine, J. P. Ballantine, A. A. Bennett, Eisenhart, Fiske, Fite, Fort, Frink, Gafafer, Garretson, Gill, R. L. Green, Gronwall, Hansell, Hedlund, Hille, Huntington, Joffe, Kasner, Kormes, Lamond, Lamson, Langford, Langman, Littauer, Maria, Meyer, H. H. Mitchell, L. T. Moore, T. W. Moore, Murnaghan, Norwood, B. C. Patterson, Pfeiffer, Pierpont, R. G. Putnam, Rainich, Rauden...

State-of-the-art methods in convex and non-convex optimization employ higher-order derivative information, either implicitly or explicitly. We explore the limitations of higher-order optimization and prove that even for convex optimization, a polynomial dependence on the approximation guarantee and higher-order smoothness parameters is necessary. As a special case, we show Nesterov’s accelerate...