An equitable k-coloring of a graph G is proper such that the sizes any two color classes differ by at most one. (q,r)-tree-coloring an q-coloring subgraph induced each class forest maximum degree r. Let strong vertex r-arboricity G, denoted var≡(G), be minimum p has for every q≥p. The values va1≡(Kn,n) were investigated Tao and Lin Wu, Zhang, Li where exact found in some special cases. In this ...

Let bα(R 1+n + ) be the space of solutions to the parabolic equation ∂tu+ (−△)u = 0 (α ∈ (0, 1]) having finite L(R 1+n + ) norm. We characterize nonnegative Radon measures μ on R + having the property ‖u‖Lq(R1+n + ,μ) . ‖u‖ Ẇ1,p(R + ) , 1 ≤ p ≤ q < ∞, whenever u(t, x) ∈ bα(R 1+n + ) ∩ Ẇ 1.p(R + ). Meanwhile, denoting by v(t, x) the solution of the above equation with Cauchy data v0(x), we chara...

A monotonous polyomino is formed by all lattice unit squares met the graph of some fixed continuous function $$f:[a,b] \to \mathbb{R}$$ with $$f(k) \notin \mathbb{Z}$$ whenever $$k \in . Our main result says that least cardinality a covering $$(m \times n)$$ -rectangle polyominoes $$\big\lceil \frac{2}{3}(m+n-\sqrt{m^2+n^2-mn})\big\rceil.$$ The paper motivated problem on arrangements straight l...

Let E be an optimal elliptic curve over Q of conductor N having analytic rank zero, i.e., such that the L-function LE(s) of E does not vanish at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to E modulo a power r of a prime p. The theory of vis...

Elaydi and Yakubu showed that a globally asymptotically stable(GAS) periodic orbit in an autonomous difference equation must in fact be a fixed point whenever the phase space is connected. In this paper we extend this result to periodic nonautonomous difference equations via the concept of skew-product dynamical systems. We show that for a k-periodic difference equation, if a periodic orbit of ...

with coefficients in +. It is an interesting problem in number theory to determine when such a system possesses a nontrivial +-rational solution. In particular, we define Γ*(k,R,+) to be the smallest natural number such that any system of R equations of degree k in N variables with coefficients in + has a nontrivial +-rational solution provided only that N&Γ*(k,R,+). For example, when k ̄ 1, ord...

We study pointwise convergence of the solutions to Schrödinger equations with initial datum f ∈ H(R). The conjecture is that the solution ef converges to f almost everywhere for all f ∈ H(R) if and only if s ≥ 1/4. The conjecture is known true for one spatial dimension and the convergence when s > 1/2 was verified for n ≥ 2. Recently, concrete progresses have been made in R for some s < 1/2. Ho...

Abstract We study rational generating functions of sequences $\{a_n\}_{n\geq 0}$ that agree with a polynomial and investigate symmetric decompositions the numerator for subsequences $\{a_{rn}\}_{n\geq 0}$. prove if is degree $s$ its coefficients satisfy set natural linear inequalities, then decomposition real-rooted whenever $r\geq \max \{s,d+1-s\}$. Moreover, symmetric, we show interlacing. ap...