For every mV 3, let n ˆ R…L3…m†† be the least integer such that for every 2-coloring of the set S ˆ f1; 2; . . . ; ng, there exists in S a monochromatic solution to the following system. L3…m†: x1 ‡ x2 ‡ ‡ xmÿ1 < xm x1 < x2 < < xm: The main result of this paper is that R…L3…m†† ˆ 9 16 m ÿm2 ‡m‡ 1 if m1 0 …mod 4† 9 16 m ÿm2 ‡ 13 16 m‡ 13 8 if m1 1 …mod 4† 9 16 m ÿm2 ‡m‡ 1 2 if m1 2 …mod 4† 9 16 ...

For integers k, r > 0, a (k, r)-coloring of a graph G is a proper k-coloring c such that for any vertex v with degree d(v), v is adjacent to at least min{d(v), r} different colors. Such coloring is also called as an r-hued coloring. The r-hued chromatic number of G, χr (G), is the least integer k such that a (k, r)-coloring of G exists. In this paper, we proved that if G is a planar graph with ...

u n d o u b t e d l y, i n t h e l a s t d e c a d e s, t h e a s s e t p r i c i n g m o d e l i so n e o f m o s t i m p o r t a n t a n d, a t t h e s a m e t i m e, a t t r a c t i v e, f i n a n c e a r e a s. c a p m h a s b e e n t h e d o m i n a n t m o d e l i n t h i s s e c t i o n f o r m o r e t h a n 40 y e a r s. o n e r e a s o n i s i t s s t r o n g b a s e, w h i c h h a s b...

Let V and W be vector spaces over a common field, and let S and T be sets of subspaces of V and W, respectively. A linear function φ : V → W is called a (Grassmann) homomorphism from S to T if φ(S) ∈ T holds for every S ∈ S. Coloring of graphs and hypergraphs and flows in graphs can be reformulated in terms of these homomorphisms. The main results are constructive characterizations of the sets ...

For positive integers k and r , a (k, r)-coloring of a graphG is a proper coloring of the vertices with k colors such that every vertex of degree i will be adjacent to vertices with at least min{i, r} different colors. The r-dynamic chromatic number of G, denoted by χr (G), is the smallest integer k for which G has a (k, r)-coloring. For a k-list assignment L to vertices of G, an (L, r)-colorin...

shortage of irrigation water, being a crucial problem in arid and semiarid regions, makes it necessary to use treated wastewater in agriculture. a field experiment was conducted to investigate the effect of treated municipal wastewater (tmw) on the yield and fiber quality of cotton (gossypium hirsutum l.) (mehr variety). treated municipal wastewater was supplied from shahrake ghods sewage treat...

abstract introduction: the mineral content of dentin increases with age and its primary structure is altered. these changes may have unknown effects on efficiency of adhesive systems. the aim of this study was to find the effect of aging on dentin bonding agents. methods and materials: fourty eight extracted human premolars were divided into 2 groups under 15 years old & 2 groups over 45 years ...