A coloring of a digraph is partition its vertex set such that each class induces with no directed cycles. k k -chromatic if ...

Let σZ(k) be the smallest n such that there exists an identity (x1 + x 2 2 + · · ·+ xk) · (y 1 + y 2 + · · ·+ y k) = f 1 + f 2 + · · ·+ f n , with f1, . . . fn being polynomials with integer coefficients in the variables x1, . . . , xk and y1, . . . , yk. We prove that σZ(k) ≥ Ω(k).

We consider the fourth order boundary value problem (ry′′)′′+(py′)′+ qy = λwy, y(a) = y′(a) = y(b) = y′(b) = 0, which is used in a variety of physical models. For such models, the extremal values of the smallest eigenvalue help answer certain optimization problems, such as maximizing the fundamental frequency of a vibrating elastic system or finding the tallest column that will not buckle under...

Let σZ(k) be the smallest n such that there exists an identity (x 1 + x 2 2 + · · · + x 2 k) · (y 2 1 + y 2 2 + · · · + y 2 k) = f 2 1 + f 2 2 + · · · + f 2 n , with f1, . . . , fn being polynomials with integer coefficients in the variables x1, . . . , xk and y1, . . . , yk. We prove that σZ(k) ≥ Ω(k 6/5).

Given non-negative integers j and k, an L( j, k)-labeling of a graph G is a function f from the vertex set V (G) to the set of all non-negative integers such that | f (x) − f (y)| ≥ j if d(x, y) = 1 and | f (x) − f (y)| ≥ k if d(x, y) = 2. The L( j, k)-labeling number λ j,k is the smallest number m such that there is an L( j, k)-labeling with the largest value m and the smallest label 0. This p...

The algebraic connectivity ofG is the second smallest eigenvalue of its Laplacianmatrix. Let Un be the set of all unicyclic graphs of order n. In this paper, wewill provide the ordering of unicyclic graphs inUn up to the last seven graphs according to their algebraic connectivities when n ≥ 13. This extends the results of Liu and Liu [Y. Liu, Y. Liu, The ordering of unicyclic graphswith the sma...

We have isolated the Hansenula polymorpha CPY gene encoding carboxypeptidase Y (Hp-CPY). The deduced amino acid sequence revealed that Hp-CPY consists of 541 amino acids and has a calculated Mr of 60,793. The protein is highly similar to Saccharomyces cerevisiae CPY (61.8% identity). At the N-terminus of Hp-CPY signals for the entry into the secretory pathway and subsequent sorting to the vacuo...

We generalize the Davenport transform to prove that, for A = (A,+) a cancellative unital semigroup and X,Y subsets of A such that the smallest subsemigroup generated by Y is commutative, one has that |X + Y | ≥ Ω(X,Y ) := min ( |X|+ |Y | − 1, sup y0∈Y× min y∈Y \{y0} ord(y − y0) ) if 2 ≤ |X|, |Y | < ∞. While extending the Cauchy-Davenport theorem to the broader and abstract setting of (possibly ...

THEOREM I. Let X be a small category. Then the following assertions are equivalent: (1) The inverse limit proj limx: AB-^AB is exact (2) For every abelian category SÏ with exact direct products y the inverse limit proj lim* : %—»3I is exact. (3) Every connected component Y of X contains an object y together with an endomorphism eÇz Y (y, y) such that the following properties are satisfied: (i) ...