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1. Variational problems for tree roots and branches

   https://doi.org/10.1007/s00526-019-1666-1  

   Bressan, Alberto; Palladino, Michele; Sun, Qing (November 2019, Calculus of variations and partial differential equations)

   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 how the size of the optimal tree depends on the parameters defining the various functionals. 

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2. On the structure of limiting flocks in hydrodynamic Euler Alignment models

   https://doi.org/10.1142/S0218202519500507  

   Leslie, Trevor M.; Shvydkoy, Roman (December 2019, Mathematical Models and Methods in Applied Sciences)

   The goal of this paper is to study limiting behavior of a self-organized 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 well-known 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. 

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3. A Family of Banach Spaces Over ℝ∞

   https://doi.org/10.1142/S2591722621400020  

   Gill, Tepper L.; Kalita, Hemanta; Hazarika, Bipan (March 2021, Proceedings of the Singapore National Academy of Science)

   null (Ed.)

   In T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016), the topology of [Formula: see text] was replaced with a new topology and denoted by [Formula: see text]. This space was then used to construct Lebesgue measure on [Formula: see text] in a manner that is no more difficult than the same construction on [Formula: see text]. More important for us, a new class of separable Banach spaces [Formula: see text], [Formula: see text], for the HK-integrable functions, was introduced. These spaces also contain the [Formula: see text] spaces and the Schwartz space as continuous dense embeddings. This paper extends the work in T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016) from [Formula: see text] to [Formula: see text]. 

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4. Dynamic Pricing and Inventory Control with Fixed Ordering Cost and Incomplete Demand Information

   https://doi.org/10.1287/mnsc.2021.4171  

   Chen, Boxiao; Simchi-Levi, David; Wang, Yining; Zhou, Yuan (December 2021, Management Science)

   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 order-up-to level for ordering strategy, and [Formula: see text], a function of on-hand 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 single-period optimization questions. It is hence challenging to establish stability results through DP recursions, which we accomplish by proving uniform convergence of the profit-to-go function. The necessity of analyzing action-dependent state transition over multiple periods resembles the reinforcement learning question, considerably more difficult than existing bandit learning algorithms. Second, the pricing function [Formula: see text] is of infinite dimension, and approaching it is much more challenging than approaching a finite number of parameters as seen in existing researches. The demand-price relationship is estimated based on upper confidence bound, but the confidence interval cannot be explicitly calculated due to the complexity of the DP recursion. Finally, because of the multiperiod nature of [Formula: see text] policies the actual distribution of the randomness in demand plays an important role in determining the optimal pricing strategy [Formula: see text], which is unknown to the learner a priori. In this paper, the demand randomness is approximated by an empirical distribution constructed using dependent samples, and a novel Wasserstein metric-based argument is employed to prove convergence of the empirical distribution. This paper was accepted by J. George Shanthikumar, big data analytics. 

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5. Finally: The X-ray crystal structure of the illusive unsubstituted iron(III) phthalocyanine μ-oxo(1) dimer. DFT-predicted Mössbauer quadrupole splitting and antiferromagnetic coupling constants for X-ray geometry

   https://doi.org/10.1142/s1088424624500263  

   Nemykin, Victor N; Gerasimchuk, Nikolay N; Muldowney, Breanna E (May 2024, Journal of Porphyrins and Phthalocyanines)

   The molecular structure of the unsubstituted iron(III) phthalocyanine [Formula: see text]-oxo(1) dimer ((PcFe)₂O) was determined by single crystal X-ray diffraction. In agreement with the earlier speculations, the dimer has a bent (Fe-O-Fe angle is 152.4[Formula: see text]) structure. The interplay between the [Formula: see text]-[Formula: see text] interactions and steric hindrances caused by the isoindole units led to the observed staggering angle of [Formula: see text]24[Formula: see text] between two phthalocyanine ligands. The high-spin iron(III) centers are located significantly above the phthalocyanine N4 planes (0.57–0.58 Å). Several DFT exchange-correlation functionals were used to calculate the absolute value and sign of the Mössbauer quadrupole splitting and antiferromagnetic coupling constant for X-ray determined geometry of (PcFe)₂O. It was demonstrated that the hybrid functionals provide the correct sign of the electric field gradient and the magnitude of the antiferromagnetic coupling constant compared to the pure functionals. 

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