Trans Amer Math Soc Research Articles

Homotopy Gerstenhaber Structures and Vertex Algebras

We provide a simple construction of a G ∞ -algebra structure on an important class of vertex algebras V, which lifts the Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications to algebraic topology: the construction of a sheaf of G ∞ algebras on a Calabi–Yau manifold M, extending the operations of multiplication and bracket of functions and vector fields on M, and of a Lie ∞ structure related to the bracket of Courant (Trans Amer Math Soc 319:631–661, 1990).

Viability for Differential Inclusions on Graphs

Let X be a Banach space, I a nonempty, bounded interval and let \(K:I\leadsto X\) be a given multi-valued function. We rephrase the concept of tangent set introduced by Cârjă et al. (Trans Amer Math Soc, 2008; Viability, Invariance and Applications. North-Holland Mathematics Studies, vol. 207. Elsevier, 2007) by saying that a bounded set \(E\subseteq X\) is right tangent to Open image in new window at Open image in new window if \(\liminf_{h\downarrow0}{1}/{h}\,\text{\rm dist}(\xi+hE;K(\tau+h))=0\). Next, by using a tangency condition expressed in the terms of this concept, we establish several necessary and sufficient conditions for viability referring to differential inclusions on graphs.

Global existence of periodic solutions in a special neural network model with two delays

A simple neural network model with two delays is considered. By analyzing the associated characteristic transcendental equation, it is found that Hopf bifurcation occurs when the sum of two delays passes through a sequence of critical values. Using a global Hopf bifurcation theorem for FDE due to Wu [Wu J. Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 1998;350:4799–838], a group of sufficient conditions for this model to have multiple periodic solutions are obtained when the sum of delays is sufficiently large. Numerical simulations are presented to support the obtained theoretical results.

On the Calculus of Limiting Subhessians

We discuss various qualification assumptions that allow calculus rules for limiting subhessians to be derived. Such qualification assumptions are based on a singular limiting subjet derived from a sequence of efficient subsets of symmetric matrices. We introduce a new efficiency notion that results in a weaker qualification assumption than that introduced in Ioffe and Penot (Trans Amer Math Soc 249: 789–807, 1997) and prove some calculus rules that are valid under this weaker qualification assumption.

Simplicial volume of locally symmetric spaces covered by $${\rm SL}_{3}{\mathbb{R}}/{\rm SO}(3)$$

We give the first complete proof of the strict positivity of the simplicial volume of compact locally symmetric spaces covered by \({\rm SL}_{3}{\mathbb{R}}/{\rm SO}(3)\) and show why the proof in [Savage RP, Trans Amer Math Soc 274(1):239–263, 1982] is incorrect.

Strange distributionally chaotic triangular maps III

In the class T of triangular maps of the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [Trans Amer Math Soc 1994;344:737–854] for continuous maps of the interval. We show that a map F ∈ T is DC1 if F has a periodic orbit with period ≠ 2 n , for any n ⩾ 0. Consequently, a map in T is DC1 if it has a homoclinic trajectory. This result is important since in general systems like T , positive topological entropy itself does not imply DC1. It contributes to the solution of a long-standing open problem of A. N. Sharkovsky concerning classification of triangular maps of the square.

Strong associativity of a group algebra

The concept of strongly associative algebra was introduced in Dokuchaev and Exel [Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans Amer Math Soc, 2005] roughly 2 years ago. In Section 3 of Dokuchaev and Exel (to appear), the authors gave an example of a non-strongly associative algebra. Following that example, the strong associativity of the group algebra of the cyclic group of order four was claimed. In this paper, we will prove that this group algebra is not strongly associative.

Analysis of stability and Hopf bifurcation for a delayed logistic equation

Abstract The dynamics of a logistic equation with discrete delay are investigated, together with the local and global stability of the equilibria. In particular, the conditions under which a sequence of Hopf bifurcations occur at the positive equilibrium are obtained. Explicit algorithm for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.]. Global existence of periodic solutions is also established by using a global Hopf bifurcation result of Wu [Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 350:1998;4799–38.]