We provide a new proof of a theorem of Saks which is an extension of Greene’s Theorem to acyclic digraphs, by reducing it to a similar, known extension of Greene and Kleitman’s Theorem. This suggests that the Greene-Kleitman Theorem is stronger than Greene’s Theorem on posets. We leave it as an open question whether the same holds for all digraphs, that is, does Berge’s conjecture concerning pa...

We present a model of truthful elicitation which generalizes and extends mechanisms, scoring rules, number related settings that do not qualify as one or the other. Our main result is characterization theorem, yielding characterizations for all these settings. This includes new rules non-convex sets distributions. combine theorem with duality to give simple construction convert between randomiz...

School of Law. The author would like to thank Michael J. Saks and Mark P. Denbeaux for helpful comments on earlier drafts of this Article. 1. See Richard H. Underwood, "X-Spurt" Witnesses, 19 AM. J. TRIAL ADVOC. 343, 34548 (1995) (discussing increasing role of expert testimony). 2. See FED. R. EVID. 704(a) (stating that "testimony . . . is not objectionable because it embraces an ultimate issue...

We introduce the sequence space ℓpλ(B) of none absolute type which is a p-normed space and BK space in the cases 0<p<1 and 1≤p≤∞, respectively, and prove that ℓpλ(B) and ℓ p are linearly isomorphic for 0<p≤∞. Furthermore, we give some inclusion relations concerning the space ℓpλ(B) and we construct the basis for the space ℓpλ(B), where 1≤p<∞. Furthermore, we determine the alpha-, beta- and gamm...

A recent approach to the Beck–Fiala conjecture, a fundamental problem in combinatorics, has been understand when random integer matrices have constant discrepancy. We give complete answer this question for two natural models: with Bernoulli or Poisson entries. For matrices, we further characterize discrepancy any rectangular aspect ratio. These results sharp answers questions of Hoberg and Roth...

We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways. 1. We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (cl...

We give an efficient structural decomposition theorem for formulas that depends on their negation complexity and demonstrate its power with the following applications. We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that ...