Call a sequence of fc Boolean variables or their negations a fc-tuple.For a set V of n Boolean variables,let T_k(V) denote the set of all 2~kn^k possible fc-tuples on V.Randomly generate a set C of fc-tuples by including every fc-tuple in T_k(V) independently with probability p,and let Q be a given set of q "bad" tuple assignments.An instance I=(C,Q) is called satisfiable if there exists an assignment that does not set any of the fc-tuples in C to a bad tuple assignment in Q.Suppose that θ,q > 0 are fixed and ε= ε(n) > 0 be such that ε In n/In In n→∞.Let k≥(1+θ) log_2 n and let p_0=ln2/qn^(k-1).We prove that lim(n→∞) P[I is satisfiable]={1,p≤(1-ε)p0,0,p≥(1+ε)p0.
In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients. For the above equation, the order of growth, the exponents of convergence of zeros and poles of its transcendental meromorphic solution f(z), and the exponents of convergence of poles of difference △f(z) and divided difference △f(z)/f(z)are estimated. Furthermore, we study the forms of rational solutions of the above equation.
In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equationsf(z)n+ P_(n-1)(f) = 0,where n ≥ 2 and P_(n-1)(f) is a difference polynomial of degree at most n- 1 in f with small functions as coefficients. Moreover, we give two examples to show that one conjecture proposed by Yang and Laine [2] does not hold in general if the hyper-order of f(z) is no less than 1.
In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k + 12. If fn+ af(k)and gn+ ag(k)share b CM and the b-points of fn+ af(k)are not the zeros of f and g, then f and g are either equal or closely related.
