Powering product discovery in ChatGPT

A discovery system for detecting correspondences in data is described, based on the familiar induction methods of J. S. Mill. Given a set of observations, the system induces the ``causally'' related facts in these observations. Its application to empirical linguistic discovery is described.

We show that for a C*-algebra A and a discrete group G with an action of G on A, the reduced crossed product C*-algebra possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product. We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.

Twentyfive titanium isotopes have so far been observed; the discovery of these isotopes is discussed. For each isotope a brief summary of the first refereed publication, including the production and identification method, is presented.