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Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of A-infinity algebras built of differential forms on the symplectic manifold. We show that these symplectic A-infinity algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these A-infinity algebras are equivalent to the standard de Rham differential graded algebra on certain odd dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yau's symplectic A-infinity algebras satisfy the Calabi-Yau property, and importantly, that they can be used to define an intersection theory for coisotropic/isotropic chains. We further demonstrate that these symplectic A-infinity algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality and invariance in the smooth category.

In this paper, we analyze the minimization of seminorms ∥L · ∥ on R^n under the constraint of a bounded I-divergence D(b,H·) for rather general linear operators H and L. The I-divergence is also known as Kullback–Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data but also in the case of multiplicative Gamma noise. Often H represents, e.g., a linear blur operator and L is some discrete derivative or frame analysis operator. A central part of this paper consists in proving relations between the parameters of I-divergence constrained and penalized problems. To solve the I-divergence constrained problem, we consider various first-order primal–dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. One of these proximation problems is an I-divergence constrained least-squares problem which can be solved based on Morozov’s discrepancy principle by a Newton method. We prove that these algorithms produce not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which converges to a regularization parameter so that the corresponding penalized problem has the same solution. Furthermore, we derive a rule for automatically setting the constraint parameter for data corrupted by multiplicative Gamma noise. The performance of the various algorithms is finally demonstrated for different image restoration tasks both for images corrupted by Poisson noise and multiplicative Gamma noise.

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This paper studies the dynamics of a network-based SIS epidemic model with nonmonotone incidence rate. This type of nonlinear incidence can be used to describe the psychological effect of certain diseases spread in a contact network at high infective levels. We first find a threshold value for the transmission rate. This value completely determines the dynamics of the model and interestingly, the threshold is not dependent on the functional form of the nonlinear incidence rate. Furthermore, if the transmission rate is less than or equal to the threshold value, the disease will die out. Otherwise, it will be permanent. Numerical experiments are given to illustrate the theoretical results. We also consider the effect of the nonlinear incidence on the epidemic dynamics.