Let P ⊂ R and Q ⊂ R be integral convex polytopes of dimension d and e which contain the origin of R and R, respectively. In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of P and Q to possess the integer decomposition property will be presented. Introduction A convex polytope is called integral if any of its vertices has integer coordinates. ...

Let f be a function transcendental and meromorphic in the plane with lim inf r→∞ T (r, f) (log r)2 = 0. Let q ∈ C with |q| > 1. It is shown that at least one of the functions F (z) = f(qz)− f(z), G(z) = F (z) f(z) has infinitely many zeros. This result is sharp. MSC 2000: 30D35.

with T (z) = D 1 (z)D 2 (z 01)0 D 2 (z)D 1 (z 01): The sign of Q(z) is the same as that of Re fG(z)=D1(z)g on the unit circle; an identical consideration can be made for R(z) and D 2 (z): Therefore, G(z) can be computed by finding Q(z) = Q(z 01) and R(z) = R(z 01) positive on the unit circle and such that R(z)D 1 (z) 0 Q(z)D 2 (z) = 0 8z such that T (z) = 0; jzj 1: The roots of T (z) outside th...

In this paper, we study the following first-order nonperiodic Hamiltonian system ż = JHz(t, z), where H ∈ C1(R× R ,R) is the form H(t, z) = 1 2 L(t)z · z + R(t, z). Under weak superquadratic condition on the nonlinearitiy. By applying the generalized Nehari manifold method developed recently by Szulkin and Weth, we prove the existence of homoclinic orbits, which are ground state solutions for a...

Abstract For $R(z, w)\in \mathbb {C}(z, w)$ of degree at least 2 in w , we show that the number rational functions $f(z)\in {C}(z)$ solving difference equation $f(z+1)=R(z, f(z))$ is finite and bounded just terms degrees R two variables. This complements a result Yanagihara, who showed any finite-order meromorphic solution to this sort must be function. We prove similar for differential $f'(z)=...

This paper establishes two new connections between the familiar function ring functor ${mathfrak R}$ on the category ${bf CRFrm}$ of completely regular frames and the category {bf CR}${mathbf sigma}${bf Frm} of completely regular $sigma$-frames as well as their counterparts for the analogous functor ${mathfrak Z}$ on the category {bf ODFrm} of 0-dimensional frames, given by the integer-valued f...