Let f be an increasing and convex (concave) function on [0, 1) and φ a positive increasing concave function on [0,∞) such that φ(0) = 0 and the sequence { φ(i+1) ( φ(i+1) φ(i) − 1 )} i∈N decreases ( the sequence { φ(i) ( φ(i) φ(i+1) − 1 )} i∈N increases ) . Then the sequence { 1 φ(n) ∑n−1 i=0 f ( φ(i) φ(n) )} n∈N is increasing. Mathematics subject classification (2000): 26D15, 26A51.

Proof. I will check the four axioms of the root system. (R1) Since |Φ∨| = |Φ|, then Φ∨ is finite. Since Φ spans E and α∨ is a non-zero scalar multiple of α for all α ∈ Φ, then Φ∨ spans E and 0 / ∈ Φ∨. (R2) Since (−α)∨ = 2(−α) ((−α),(−α)) = − 2α (α,α) = −α∨, then the only multiples of α∨ are ±α∨. (R3) I need to show that if α∨ ∈ Φ∨ then the reflection σα∨ leaves Φ∨ invariant. Let α in Φ, then fo...

1 2 ∆ φ t = φ t − φ 1 , ∆ ψ t = ψ t − ψ 1 , ∆φ t = φ t − φ 1 , ∆ψ t = ψ t − ψ 1 Further, we define cos x (u, v) = u Sxv uxvx , the cosine of the angle between two vectors induced by the inner product u, v = u S x v. Similarly, we define cos y (u, v) = u Syv uyvy. To prove the theorem, we will repeatedly use the following two lemmas. Proof of Lemma 1. The proof is in the main paper. Lemma 2. ∆φ ...

We define a total k-labeling φ of graph G as combination an edge labeling φe:E(G)→{1,2,…,ke} and vertex φv:V(G)→{0,2,…,2kv}, such that φ(x)=φv(x) if x∈V(G) φ(x)=φe(x) x∈E(G), where k= max {ke,2kv}. The is called irregular reflexive every two different edges has distinct weights, the weight defined summation label itself its labels. Thus, smallest value k for which strength G. In this paper, we ...

Let A be a unital complex semisimple Banach algebra, and MA denote its maximal ideal space. For matrix M∈An×n, Mˆ denotes the obtained by taking entry-wise Gelfand transforms. M∈Cn×n, σ(M)⊂C set of eigenvalues M. It is shown that if A∈An×n B∈Am×m are such for all φ∈MA, σ(Aˆ(φ))∩σ(Bˆ(φ))=∅, then C∈An×m, Sylvester equation AX−XB=C has unique solution X∈An×m. As an application, Roth's removal rule...

Let A, B be algebras and a∈A, b∈B a fixed pair of elements. We say that map φ:A→B preserves products equal to b if for all a1,a2∈A the equality a1a2=a implies φ(a1)φ(a2)=b. In this paper we study bijective linear maps φ:I(X,F)→I(X,F) preserving primitive idempotents I(X,F), where I(X,F) is incidence algebra finite connected poset X over field F. fully characterize situation, when such φ exists,...

In order to extend the Schützenberger’s factorization to general perturbations, the combinatorial aspects of the Hopf algebra of the φ-deformed stuffle product is developed systematically in a parallel way with those of the shuffle product. and in emphasizing the Lie elements as studied by Ree. In particular, we will give an effective construction of pair of bases in duality.

Let Φ be an irreducible root system (sometimes we denote Φ by Φ(X) to indicate its type X). Choose a simple root system Π in Φ. Let Φ + (resp. Φ~) be the corresponding positive (resp. negative) root system of Φ. By a subsystem Φ' of Φ (resp. of Φ + ) , we mean that Φ' is a subset of Φ (resp. of Φ) which itself forms a root system (resp. a positive root system). We refer the readers to Bourbaki'...