The following is a proof which is independent of this characterisation. First assume that ‖ ‖ is non-archimedean. Let x, y ∈ K. Using that ‖ ‖ extends | | we then obtain |x + y| = ‖x + y‖ ≤ max{‖x‖, ‖y‖} = max{|x|, |y|} which shows that | | is non-archimedean. Now assume that | | is non-archimedean. Let x, y ∈ K̂. Let ε > 0. Since K is dense in K̂ there exist u, v ∈ K such that ‖x − u‖ < ε and ‖y...

Let P be a set of n points in R. In the projective clustering problem, given k, q and norm ρ ∈ [1,∞], we have to compute a set F of k q-dimensional flats such that ( ∑ p∈P d(p,F)) is minimized; here d(p,F) represents the (Euclidean) distance of p to the closest flat in F . We let f k (P, ρ) denote the minimal value and interpret f k (P,∞) to be maxr∈P d(r,F). When ρ = 1, 2 and ∞ and q = 0, the ...

We prove that for every convex body K with the center of mass at the origin and every ε ∈ ( 0, 12 ) , there exists a convex polytope P with at most eO(d)ε− d−1 2 vertices such that (1− ε)K ⊂ P ⊂ K.

Mohammad Zaki, MN State Univ, Mankato We define an algebraic number α as a root of an algebraic equation, a0∗x+a1∗x+· · ·+an−1∗x+an = 0 where ao, a1, · · · , an are rational integers, not all zero. We say α is an algebraic integer if a0 = 0. If an algebraic number α satisfies an algebraic equation of degree n with rational coefficients, and none of lower degree, then we say α is of degree n. If...

The drops are used to control the descents,  stabilize the bed level, and control the upstream water level in sloping channels with less slope than the ground slope. The current study presents a numerical analysis of hydraulic characteristics in the vertical drop using computational fluid dynamics. At first, the laboratory models were used for verification and choosing the best model of turbule...

In this paper we extend the study of bipartite graphs with the induced ε-density property introduced by Frankl, Rödl, and the author. For a given kpartite k-uniform hypergraph G we say that a k-partite k-uniform hypergraph R = (W1, . . . ,Wk,F) has the induced ε-density property if every subhypergraph ofR with at least ε|F| edges contains a copy of G which is an induced subhypergraph of R. We s...

We explore a number of problems related to learning and covering structured distributions. Hypothesis Selection: We provide an improved and generalized algorithm for selecting a good candidate distribution from among competing hypotheses. Namely, given a collection of N hypotheses containing at least one candidate that is ε-close to an unknown distribution, our algorithm outputs a candidate whi...

We prove that for any positive integer k and for ε = 1/((k + 2)(3k2 + 1)), the edges of any graph whose fractional arboricity is at most k + ε can be decomposed into k forests and a matching.