Let P denote the set of all primes. P1, P2, P3 are three subsets of P . Let δ(Pi) (i = 1, 2, 3) denote the lower density of Pi in P , respectively. It is proved that if δ(P1) > 5/8, δ(P2) ≥ 5/8, and δ(P3) ≥ 5/8, then for every sufficiently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3. The condition is the best possible.

This paper studies compositional semantics of nonmonotonic logic programs. We suppose the answer set semantics of extended disjunctive programs and consider the following problem. Given two programs P1 and P2, which have the sets of answer sets AS(P1) and AS(P2), respectively; find a program Q which has answer sets as minimal sets S∪T for S from AS(P1) and T from AS(P2). The program Q combines ...

This paper proposes an MPC - based (model predictive control) scheme to control active and reactive powers of DERs (distributed energy resources) in a grid - connected mode (either through a bus with its associated loads as a PCC (point of common coupling) or an MG (micro - grid)). DER may be a DG (distributed generation) or an ESS (energy storage system). In the proposed scheme, the set - poin...

A property of graphs is any class of graphs closed under isomorphism. Let P1,P2, . . . ,Pn be properties of graphs. A graph G is (P1,P2, . . . ,Pn)-partitionable if the vertex set V (G) can be partitioned into n sets, {V1, V2, . . . , Vn}, such that for each i = 1, 2, . . . , n, the graph G[Vi] ∈ Pi. We write P1◦P2◦ · · · ◦Pn for the property of all graphs which have a (P1,P2, . . . ,Pn)-partit...

In this paper, we suppose an agent which has a knowledge base represented by a logic program under the answer set semantics. We then consider the following two problems: given two programs P1 and P2, which have the sets of answer sets AS(P1) and AS(P2), respectively; (i) find a program Q which has the answer sets as the minimal elements of {S ∩ T | S ∈ AS(P1) and T ∈ AS(P2) }; (ii) find a progr...

Let k be an algebraically closed field of characteristic zero. Denote the projective plane over k by P2 = Proj k[x, y, z]. Let K be the field of rational functions on P2. We view K as the set of all rational functions of the form f/g ∈ k(x, y, z) where f and g are homogeneous forms in k[x, y, z] of the same degree. Since k is algebraically closed, the Brauer group of P2, denoted B(P2), is trivi...

abstract let ‎g=(v,e) ‎be a‎ ‎simple ‎connected ‎graph ‎with ‎vertex ‎set ‎v‎‎‎ ‎and ‎edge ‎set ‎‎‎e. ‎the szeged index ‎of ‎‎g is defined by ‎ where ‎ respectively ‎ ‎ is the number of vertices of ‎g ‎closer to ‎u‎ (‎‎respectively v)‎ ‎‎than ‎‎‎v (‎‎respectively u‎).‎ ‎‎if ‎‎‎‎s ‎is a‎ ‎set ‎of ‎size‎ ‎ ‎ ‎let ‎‎v ‎be ‎the ‎set ‎of ‎all ‎subsets ‎of ‎‎s ‎of ‎size ‎3. ‎then ‎we ‎define ‎three ‎...