 Award ID(s):
 1714237
 Publication Date:
 NSFPAR ID:
 10297093
 Journal Name:
 Mathematical Models and Methods in Applied Sciences
 Volume:
 28
 Issue:
 14
 Page Range or eLocationID:
 2763 to 2801
 ISSN:
 02182025
 Sponsoring Org:
 National Science Foundation
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This paper studies two classes of variational problems introduced in Bressan and Sun (On the optimal shape of tree roots and branches. arXiv:1803.01042), related to the optimal shapes of tree roots and branches. Given a measure μ describing the distribution of leaves, a sunlight functional S(μ) computes the total amount of light captured by the leaves. For a measure μ describing the distribution of root hair cells, a harvest functional H(μ) computes the total amount of water and nutrients gathered by the roots. In both cases, we seek a measure μ that maximizes these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk or from the trunk to the leaves. Compared with Bressan and Sun, here we do not impose any a priori bound on the total mass of the optimal measure μ, and more careful a priori estimates are thus required. In the unconstrained optimization problem for branches, we prove that an optimal measure exists, with bounded support and bounded total mass. In the unconstrained problem for tree roots, we prove that an optimal measure exists, with bounded support but possibly unbounded total mass. The last section of the paper analyzes howmore »

This work concerns the asymptotic behavior of solutions to a (strictly) subcritical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidthsharing policy. Here we consider fair bandwidthsharing policies that are a slight generalization of the [Formula: see text]fair policies introduced by Mo and Walrand [Mo J, Walrand J (2000) Fair endtoend windowbased congestion control. IEEE/ACM Trans. Networks 8(5):556–567.]. Since the year 2000, it has been a standing problem to prove stability of the data communications network model of Massoulié and Roberts [Massoulié L, Roberts J (2000) Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15(1):185–201.], with general file sizes and operating under fair bandwidth sharing policies, when the offered load is less than capacity (subcritical conditions). A crucial step in an approach to this problem is to prove stability of subcritical fluid model solutions. In 2012, Paganini et al. [Paganini F, Tang A, Ferragut A, Andrew LLH (2012) Network stability under alpha fair bandwidth allocation with general file size distribution. IEEE Trans. Automatic Control 57(3):579–591.] introduced a Lyapunov function for this purpose and gave an argument, assuming that fluid model solutions are sufficiently smooth in timemore »

We consider the periodic review dynamic pricing and inventory control problem with fixed ordering cost. Demand is random and price dependent, and unsatisfied demand is backlogged. With complete demand information, the celebrated [Formula: see text] policy is proved to be optimal, where s and S are the reorder point and orderupto level for ordering strategy, and [Formula: see text], a function of onhand inventory level, characterizes the pricing strategy. In this paper, we consider incomplete demand information and develop online learning algorithms whose average profit approaches that of the optimal [Formula: see text] with a tight [Formula: see text] regret rate. A number of salient features differentiate our work from the existing online learning researches in the operations management (OM) literature. First, computing the optimal [Formula: see text] policy requires solving a dynamic programming (DP) over multiple periods involving unknown quantities, which is different from the majority of learning problems in OM that only require solving singleperiod optimization questions. It is hence challenging to establish stability results through DP recursions, which we accomplish by proving uniform convergence of the profittogo function. The necessity of analyzing actiondependent state transition over multiple periods resembles the reinforcement learning question, considerably more difficult thanmore »

The goal of this paper is to study limiting behavior of a selforganized continuous flock evolving according to the 1D hydrodynamic Euler Alignment model. We provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution. The key quantity that controls density distortion is the entropy [Formula: see text], and the measure of deviation from uniformity is given by a wellknown conserved quantity [Formula: see text], where [Formula: see text] is velocity and [Formula: see text] is the communication operator with kernel [Formula: see text]. The cases of Lipschitz, singular geometric, and topological kernels are covered in the study.

Within the pycnocline, where diapycnal mixing is suppressed, both the vertical movement (uplift) of isopycnal surfaces and upward motion along sloping isopycnals supply nutrients to the euphotic layer, but the relative importance of each of these mechanisms is unknown. We present a method for decomposing vertical velocity w into two components in a Lagrangian frame: vertical velocity along sloping isopycnal surfaces [Formula: see text] and the adiabatic vertical velocity of isopycnal surfaces [Formula: see text]. We show that [Formula: see text], where [Formula: see text] is the isopycnal slope and [Formula: see text] is the geometric aspect ratio of the flow, and that [Formula: see text] accounts for 10%–25% of the total vertical velocity w for isopycnal slopes representative of the midlatitude pycnocline. We perform the decomposition of w in a process study model of a midlatitude eddying flow field generated with a range of isopycnal slopes. A spectral decomposition of the velocity components shows that while [Formula: see text] is the largest contributor to vertical velocity, [Formula: see text] is of comparable magnitude at horizontal scales less than about 10 km, that is, at submesoscales. Increasing the horizontal grid resolution of models is known to increase vertical velocity; thismore »