Families Of Finite Sequences Research Articles

Some greedyt-intersecting families of finite sequences

Letn, s1,s2, ... ands n be positive integers. Assume\(\mathcal{M}(s_1 ,s_2 , \cdots ,s_n ) = \{ (x_1 ,x_2 , \cdots x_n )|0 \leqslant x_i \leqslant s_i ,x_i\) is an integer for eachi}. For\(a = (a_1 ,a_2 , \cdots a_n ) \in \mathcal{M}(s_1 ,s_2 , \cdots ,s_n )\),\(\mathcal{F} \subseteq \mathcal{M}(s_1 ,s_2 , \cdots ,s_n )\), and\(A \subseteq \{ 1,2, \cdots ,n\}\), denotes p (a)={j|1≤j≤n,a j ≥p},\(S_p (\mathcal{F}) = \{ s_p (a)|a \in \mathcal{F}\}\), and\(W_p (A) = p^{n - |A|} \prod\limits_{i \in A} {(s_i - p)}\).\(\mathcal{F}\) is called anI t p -intersecting family if, for any a,b∈\(\mathcal{F}\),a i Λb i =min(a i ,b i )≥p for at leastt i's.\(\mathcal{F}\) is called a greedyI t P -intersecting family if\(\mathcal{F}\) is anI t p -intersecting family andW p (A)≥W p (B+A c ) for anyAeS p (\(\mathcal{F}\)) and any\(B \subseteq A\) with |B|=t−1.

Asymptotically Neutral Distributions of Electrons and Polynomial Approximation

Let D be a bounded simply connected region (connected open set) in the complex plane. We wish to approximate holomorphic functions in D by polynomials; the error must become uniformly small on every fixed closed subset of D. By Runge's theorem such approximation is always possible. However, we impose an additional condition: the zeros of the approximating polynomials must belong to a prescribed set P. Polynomials whose zeros lie in P will be referred to as F-polynomials. It will be assumed for simplicity that F and D are disjoint. P will be called a polynomial approximation set relative to D if and only if every zero free holomorphic function in D can be approximated by F-polynomials. It may be observed that, if F is a polynomial approximation set and J is a simple arc in D, then every continuous function on J can be uniformly approximated by F-polynomials. G. R. MacLane [4] has shown that every rectifiable Jordan curve is a polynomial approximation set relative to its interior. Recently M. D. Thompson [5] gave a new proof of this result, extending it to a different special class of Jordan curves. M.D. Thompson [5] and the author [3] have also considered certain unbounded sets F. In the present paper we determine all bounded sets F that can serve as polynomial approximation sets relative to D. The discussion will be based on the notion of asymptotically neutral families. A family of finite sequences