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Hopf bifurcations in a three-species food chain system with multiple delays

Xiaoliang XieWen Zhang — 2017

Open Mathematics

This paper is concerned with a three-species Lotka-Volterra food chain system with multiple delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the stability of the positive equilibrium and existence of Hopf bifurcations are investigated. Furthermore, the direction of bifurcations and the stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations....

On super-weakly compact sets and uniformly convexifiable sets

Lixin ChengQingjin ChengBo WangWen Zhang — 2010

Studia Mathematica

This paper mainly concerns the topological nature of uniformly convexifiable sets in general Banach spaces: A sufficient and necessary condition for a bounded closed convex set C of a Banach space X to be uniformly convexifiable (i.e. there exists an equivalent norm on X which is uniformly convex on C) is that the set C is super-weakly compact, which is defined using a generalization of finite representability. The proofs use appropriate versions of classical theorems, such as James' finite tree...

Chebyshev bounds for Beurling numbers

Harold G. DiamondWen-Bin Zhang — 2013

Acta Arithmetica

The first author conjectured that Chebyshev-type prime bounds hold for Beurling generalized numbers provided that the counting function N(x) of the generalized integers satisfies the L¹ condition 1 | N ( x ) - A x | d x / x 2 < for some positive constant A. This conjecture was shown false by an example of Kahane. Here we establish the Chebyshev bounds using the L¹ hypothesis and a second integral condition.

Optimality of Chebyshev bounds for Beurling generalized numbers

Harold G. DiamondWen-Bin Zhang — 2013

Acta Arithmetica

If the counting function N(x) of integers of a Beurling generalized number system satisfies both 1 x - 2 | N ( x ) - A x | d x < and x - 1 ( l o g x ) ( N ( x ) - A x ) = O ( 1 ) , then the counting function π(x) of the primes of this system is known to satisfy the Chebyshev bound π(x) ≪ x/logx. Let f(x) increase to infinity arbitrarily slowly. We give a construction showing that 1 | N ( x ) - A x | x - 2 d x < and x - 1 ( l o g x ) ( N ( x ) - A x ) = O ( f ( x ) ) do not imply the Chebyshev bound.

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