A generalized Winternitz Theorem

A Generalized Winternitz Theorem

We prove that, for every simple polygon P having k ≥ 1 reflex vertices, there exists a point q ∈ P such that every half-polygon that contains q contains nearly 1/2(k + 1) times the area of P . We also give a family of examples showing that this result is the best possible. Mathematics Subject Classification (2010). 51E99.

A Fixed Point Theorem in Generalized D-metric Spaces

A Generalized Cantor Theorem

A well known theorem of Cantor asserts that the cardinal of the power-set of a given set always exceeds the cardinal of the original set. An analogous result for sets having additional structure is the well known theorem that the set of initial segments of a well ordered set always has order type greater than the original set. These two theorems suggest that there should be a similar result for...

A Generalized Representer Theorem

Wahba’s classical representer theorem states that the solutions of certain risk minimization problems involving an empirical risk term and a quadratic regularizer can be written as expansions in terms of the training examples. We generalize the theorem to a larger class of regularizers and empirical risk terms, and give a self-contained proof utilizing the feature space associated with a kernel...

A Generalized Resolution Theorem

W.C. Rounds and G.-Q. Zhang have recently proposed to study a form of resolution on algebraic domains [1]. This framework allows reasoning with knowledge which is hierarchically structured and forms a (suitable) domain, more precisely, a coherent algebraic cpo as studied in domain theory. In this paper, we give conditions under which a resolution theorem — in a form underlying resolution-based ...