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1. Equilibria of time‐inconsistent stopping for one‐dimensional diffusion processes

   https://doi.org/10.1111/mafi.12385  

   Bayraktar, Erhan ; Wang, Zhenhua ; Zhou, Zhou ( July 2023 , Mathematical Finance)

   Abstract We consider three equilibrium concepts proposed in the literature for time‐inconsistent stopping problems, including mild equilibria (introduced in Huang and Nguyen‐Huu (2018)), weak equilibria (introduced in Christensen and Lindensjö (2018)), and strong equilibria (introduced in Bayraktar et al. (2021)). The discount function is assumed to be log subadditive and the underlying process is one‐dimensional diffusion. We first provide necessary and sufficient conditions for the characterization of weak equilibria. The smooth‐fit condition is obtained as a by‐product. Next, based on the characterization of weak equilibria, we show that an optimal mild equilibrium is also weak. Then we provide conditions under which a weak equilibrium is strong. We further show that an optimal mild equilibrium is also strong under a certain condition. Finally, we provide several examples including one showing a weak equilibrium may not be strong, and another one showing a strong equilibrium may not be optimal mild. 

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2. Critical value asymptotics for the contact process on random graphs

   https://doi.org/10.1090/tran/8399  

   Nam, Danny ; Nguyen, Oanh ; Sly, Allan ( January 2022 , Transactions of the American Mathematical Society)

   Recent progress in the study of the contact process (see Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly [Ann. Probab. 49 (2021), pp. 244–286]) has verified that the extinction-survival threshold λ 1 \lambda _1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution ξ \xi has an exponential tail. In this paper, we derive the first-order asymptotics of λ 1 \lambda _1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if ξ \xi is appropriately concentrated around its mean, we demonstrate that λ 1 ( ξ ) ∼ 1 / E ξ \lambda _1(\xi ) \sim 1/\mathbb {E} \xi as E ξ → ∞ \mathbb {E}\xi \rightarrow \infty , which matches with the known asymptotics on d d -regular trees. The same results for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well. 

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3. Data for Printability Prediction in Projection Two-Photon Lithography via Machine Learning Based Surrogate Modeling of Photopolymerization

   https://doi.org/10.17632/8z29gzf6rd.1  

   Saha, Sourabh ( January 2023 , Mendeley)

   Data description This dataset presents the raw and augmented data that were used to train the machine learning (ML) models for classification of printing outcome in projection two-photon lithography (P-TPL). P-TPL is an additive manufacturing technique for the fabrication of cm-scale complex 3D structures with features smaller than 200 nm. The P-TPL process is further described in this article: “Saha, S. K., Wang, D., Nguyen, V. H., Chang, Y., Oakdale, J. S., and Chen, S.-C., 2019, "Scalable submicrometer additive manufacturing," Science, 366(6461), pp. 105-109.” This specific dataset refers to the case wherein a set of five line features were projected and the printing outcome was classified into three classes: ‘no printing’, ‘printing’, ‘overprinting’. Each datapoint comprises a set of ten inputs (i.e., attributes) and one output (i.e., target) corresponding to these inputs. The inputs are: optical power (P), polymerization rate constant at the beginning of polymer conversion (kp-0), radical quenching rate constant (kq), termination rate constant at the beginning of polymer conversion (kt-0), number of optical pulses, (N), kp exponential function shape parameter (A), kt exponential function shape parameter (B), quantum yield of photoinitiator (QY), initial photoinitiator concentration (PIo), and the threshold degree of conversion (DOCth). The output variable is ‘Class’ which can take these three values: -1 for the class ‘no printing’, 0 for the class ‘printing’, and 1 for the class ‘overprinting’. The raw data (i.e., the non-augmented data) refers to the data generated from finite element simulations of P-TPL. The augmented data was obtained from the raw data by (1) changing the DOCth and re-processing a solved finite element model or (2) by applying physics-based prior process knowledge. For example, it is known that if a given set of parameters failed to print, then decreasing the parameters that are positively correlated with printing (e.g. kp-0, power), while keeping the other parameters constant would also lead to no printing. Here, positive correlation means that individually increasing the input parameter will lead to an increase in the amount of printing. Similarly, increasing the parameters that are negatively correlated with printing (e.g. kq, kt-0), while keeping the other parameters constant would also lead to no printing. The converse is true for those datapoints that resulted in overprinting. The 'Raw.csv' file contains the datapoints generated from finite element simulations, the 'Augmented.csv' file contains the datapoints generated via augmentation, and the 'Combined.csv' file contains the datapoints from both files. The ML models were trained on the combined dataset that included both raw and augmented data. 

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4. Imaging of 3D objects with experimental data using orthogonality sampling methods

   https://doi.org/10.1088/1361-6420/ac3d85  

   Le, Thu ; Nguyen, Dinh-Liem ; Schmidt, Hayden ; Truong, Trung ( December 2021 , Inverse Problems)

   Abstract This paper is concerned with imaging of 3D scattering objects with experimental data from the Fresnel database. The first goal of the paper is to investigate a modified version of the orthogonality sampling method (OSM) by Harris and Nguyen [2020 SIAM J. Sci. Comput. 42 B72–737] for the imaging problem. The advantage of the modified OSM over its original version lies in its applicability to more types of polarization vectors associated with the electromagnetic scattering data. We analyze the modified OSM using the factorization analysis for the far field operator and the Funk–Hecke formula. The second goal is to verify the performance of the modified OSM, the OSM, and the classical factorization method for the 3D Fresnel database. The modified OSM we propose is able to invert the sparse and limited-aperture real data in a fast, simple, and efficient way. It is also shown in the real data verification that the modified OSM performs better than its original version and the factorization method. 

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5. The local sum conjecture in two dimensions

   https://doi.org/10.1142/S1793042120500888  

   Fraser, Robert ; Wright, James ( September 2020 , International Journal of Number Theory)

   The local sum conjecture is a variant of some of Igusa’s questions on exponential sums put forward by Denef and Sperber. In a remarkable paper by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture in two dimensions which follows Varčenko’s treatment of Euclidean oscillatory integrals based on Newton polyhedra for good coordinate choices. Another elementary proof is given by Veys from an algebraic geometric perspective. 

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