ABSTRACT. We consider the dyadic model, which is a toy model to test issues of well–posedness and blow–up for the Navier–Stokes and Euler equations. We prove well–posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier–Stokes. Likewise we

structure collectively revealed by known labeled and unlabeled points. We present a simple algorithm to obtain such a smooth solution. Our method yields encouraging experimental results on a number of classification problems and demonstrates effective use of unlabeled data. 1

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This paper shows that coprime right factorizations exist for the input to state mapping of a continuous time nonlinear system provided that the smooth feedback stabilization problem be solvable for this system. In particular, it follows that feedback linearizable systems admit such factorizations

We prove that the space of smooth initial data and the set of smooth solutions of the Liouville equation are homeomorphic. 1 1

How do people know as much as they do with as little information as they get? The problem takes many forms; learning vocabulary from text is an especially dramatic and convenient case for research. A new general theory of acquired similarity and knowledge representation, latent semantic analysis (LSA), is presented and used to successfully simulate such learning and several other psycholinguistic phenomena. By inducing global knowledge indirectly from local co-occurrence data in a large body of representative text, LSA acquired knowledge about the full vocabulary of English at a comparable rate to schoolchildren. LSA uses no prior linguistic or perceptual similarity knowledge; it is based solely on a general mathematical learning method that achieves powerful inductive effects by extracting the right number of dimensions (e.g., 300) to represent objects and contexts. Relations to other theories, phenomena, and problems are sketched.

Abstract In this paper, the global existence and uniqueness of a smooth solution to the periodic initial-value problem of the spin equations of antiferromagnets in 1 dimension are proved.

A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this "variance equation " completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary

We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting