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This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case $U(1,1)$ not considered there. The main idea to embed a $U(1, 1)$ Shimura curve to many $U(n-1, 1)$ Shimura varieties for big $n$, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modulartiy result on $U(1, 1)$ Shimura curves.

In this article, we prove that a version of Tate conjectures for certain Deligne-Lusztig varieties implies the Kudla-Rapoport conjecture for unitary Shimura varieties with maximal parahoric level at unramified primes. Furthermore, we prove that the Kudla-Rapoport conjecture holds unconditionally for several new cases in any dimension., Comment: 93 pages

We construct local Arthur packets associated with a dihedral long root $A$-parameter of a split reductive group of type $G_2$ over a nonarchimedian local field of characteristic zero. The construction relies on an exceptional correspondence for the pair $(\mathrm{PU}_3\rtimes \mathbb{Z}/2\mathbb{Z}, G_2)$., Comment: 41 pages. Comments welcome

Repurposing existing natural gas pipelines is a promising solution for large-scale transportation of mixed hydrogen-methane gas. However, it remains debatable whether gravitational stratification can notably affect hydrogen partial pressure in the gas mixture. To address this issue, we combined molecular dynamics simulation with thermodynamic and diffusion theories. Our study systematically examined the equilibrium distribution of hydrogen-methane mixtures in gravity fields. We demonstrated that partial pressures of both gases decrease with altitude, with hydrogen showing slower decrease due to its smaller molar mass. As a result, the volume fraction of hydrogen is maximized at the top end of pipes. The stratification is more favorable at low temperature and large altitude drops, with notable gas stratification only occurring at extremely large drops in altitude, being generally negligible even at a drop of 1500 m. Furthermore, we showed that the diffusion time required to achieve the equilibrium distribution is proportional to gas pressure and the square of pipeline height. This requires approximately 300 years for a 1500 m pipeline at 1 bar. Therefore, temporary interruptions in pipeline gas transportation will not cause visible stratification. Our work clarifies the effect of gravity on hydrogen-methane gas mixtures and provides quantitative insights into assessing the stratification of gas mixtures in pipelines., Comment: 14 pages, 8 figures

In this paper, we prove an intersection-theoretic result pertaining to curves in certain Hilbert modular surfaces in positive characteristic. Specifically, we show that given two appropriate curves C,D parameterizing abelian surfaces with real multiplication, the set of points (x,y) in the product CxD with surfaces parameterized by x and y isogenous to each other is Zariski dense in C x D, thereby proving a case of a just-likely intersection conjecture. We also compute the change in Faltings height under appropriate p-power isogenies of abelian surfaces with real multiplication over characteristic p global fields., Comment: Accepted version

In this paper, we propose a modified Kudla-Rapoport conjecture for the Kr\"amer model of unitary Rapoport-Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas of Hermitian local density polynomials, we prove the modified Kudla-Rapoport conjecture when $n=3$. Our conjecture, combining with known results at inert and infinite primes, implies arithmetic Siegel-Weil formula for all non-singular coefficients when the level structure of the corresponding unitary Shimura variety is defined by a self-dual lattice., Comment: Various typos fixed. To appear in Compositio Mathematica

We prove the Kudla--Rapoport conjecture for Kr\"amer models of unitary Rapoport--Zink spaces at ramified places. It is a precise identity between arithmetic intersection numbers of special cycles on Kr\"amer models and modified derived local densities of hermitian forms. As an application, we relax the local assumptions at ramified places in the arithmetic Siegel--Weil formula for unitary Shimura varieties, which is in particular applicable to unitary Shimura vartieties associated to unimodular hermitian lattices over imaginary quadratic fields., Comment: 82pp, final version, to appear in Invent. Math

Uncertainties from experiments and models render multi-modal difficulties in model calibrations. Bayesian inference and \textsc{mcmc} algorithm have been applied to obtain posterior distributions of model parameters upon uncertainty. However, multi-modality leads to difficulty in convergence criterion of parallel \textsc{mcmc} sampling chains. The commonly applied $\widehat{R}$ diagnostic does not behave well when multiple sampling chains are evolving to different modes. Both partitional and hierarchical clustering methods has been combined to the traditional $\widehat{R}$ diagnostic to deal with sampling of target distributions that are rough and multi-modal. It is observed that the distributions of binding parameters and pore diffusion of particle parameters are multi-modal. Therefore, the steric mass-action model used to describe ion-exchange effects of the model protein, lysozyme, on the \textsc{sp} Sepharose \textsc{ff} stationary phase might not be fully capable in certain experimental conditions, as model uncertainty from steric mass-action would result in multi-modality., Comment: 25 pages

In this paper, we proved a local arithmetic Siegel-Weil formula for a $U(1, 1)$-Shimura variety at a ramified prime, a.k.a. a Kudla-Rapoport conjecture at a ramified case. The formula needs to be modified from the original Kudla-Rapoport conjecture. In the process, we also gives an explicit decomposition of the special divisors of the Rapoport-Zink space of unitary type $(1, 1)$ (Kr\"amer model). A key ingredient is to relate the Rapoport-Zink space to the Drinfeld upper plane., Comment: 34pp, final version, published in Transactions of the AMS

The real challenge in drone-logistics is to develop an economically-feasible Unmanned Aerial Mobility Network (UAMN). In this paper, we propose an integrated airport location (strategic decision) and routes planning (operational decision) optimization framework to minimize the total cost of the network, while guaranteeing flow constraints, capacity constraints, and electricity constraints. To facility expensive long-term infrastructure planning facing demand uncertainty, we develop a data-driven risk-averse two-stage stochastic optimization model based on the Wasserstein distance. We develop a reformulation technique which simplifies the worst-case expectation term in the original model, and obtain a fractable Min-Max solution procedure correspondingly. Using Lagrange multipliers, we successfully decompose decision variables and reduce the complexity of computation. To provide managerial insights, we design specific numerical examples. For example, we find that the optimal network configuration is affected by the "pooling effects" in channel capacities. A nice feature of our DRO framework is that the optimal network design is relatively robust under demand uncertainty. Interestingly, a candidate node without historical demand records can be chosen to locate an airport. We demonstrate the application of our model for a real medical resources transportation problem with our industry partner, collecting donated blood to a blood bank in Hangzhou, China., Comment: 31pages

In this article, we will derive the differential equations considering the concentration polarization (\textsc{cp}) in membrane separations and purifications, which involve convective-diffusive equations of the concentration polarization layer and the Spiegler-Kedem-Katchalsky (\textsc{skk}) model of the membrane layer. Analytical solutions of these differential equations are straightforward and given. The film theory coupled \textsc{skk} model can be used to evaluate the degree of concentration polarization, and predict flux and rejection rates., Comment: There is an error in Eq. (8) on the page 8

Prominent features of simulated moving bed (SMB) chromatography processes in the downstream processing is based on the determination of operating conditions. However, effects of different types of uncertainties have to be studied and analysed whenever the triangle theory or numerical optimization approaches are applied. In this study, a Bayesian inference based method is introduced to consider the uncertainty of operating conditions on the performance assessment, of a glucose-fructose SMB unit under linear condition. A multiple chain Markov Chain Monte Carlo (MCMC) algorithm (i.e., Metropolis algorithm with delayed rejection and adjusted Metropolis) is applied to generate samples. The proposed method renders versatile information by constructing from the MCMC samples, e.g., posterior distributions, uncertainties, credible intervals of the operating conditions, and posterior predictive check and Pareto fronts between each pair of the performance indicators. Additionally, the MCMC samples can be mapped onto the $(m_\text{II}, m_\text{III})$ and $(m_\text{IV}, m_\text{I})$ plains to show the actually complete separation region under uncertainties. The proposed method is a convenient tool to find both optimal values and uncertainties of the operating conditions. Moreover, it is not limited to SMB processes under the linear isotherm; and it should be more powerful in the nonlinear scenarios., Comment: 32 pages, 8 figures, 4 tables

Model-based process design of ion-exchange simulated moving bed (IEX-SMB) chromatography for center-cut separation of proteins is studied. Use of nonlinear binding models that describe more accurate adsorption behaviours of macro-molecules could make it impossible to utilize triangle theory to obtain operating parameters. Moreover, triangle theory provides no rules to design salt profiles in IEX-SMB. In the modelling study here, proteins (i.e., ribonuclease, cytochrome and lysozyme) on the chromatographic columns packed with strong cation-exchanger SP Sepharose FF is used as an example system. The general rate model with steric mass-action kinetics was used; two closed-loop IEX-SMB network schemes were investigated (i.e., cascade and eight-zone schemes). Performance of the IEX-SMB schemes was examined with respect to multi-objective indicators (i.e., purity and yield) and productivity, and compared to a single column batch system with the same amount of resin utilized. A multi-objective sampling algorithm, Markov Chain Monte Carlo (MCMC), was used to generate samples for constructing the Pareto optimal fronts. MCMC serves on the sampling purpose, which is interested in sampling the Pareto optimal points as well as those near Pareto optimal. Pareto fronts of the three schemes provide the full information of trade-off between the conflicting indicators of purity and yield. The results indicate the cascade IEX-SMB scheme and the integrated eight-zone IEX-SMB scheme have the similar performance that both outperforms the single column batch system., Comment: 26 pages, 14 figures, 7 tables

Many statistical estimators for high-dimensional linear regression are M-estimators, formed through minimizing a data-dependent square loss function plus a regularizer. This work considers a new class of estimators implicitly defined through a discretized gradient dynamic system under overparameterization. We show that under suitable restricted isometry conditions, overparameterization leads to implicit regularization: if we directly apply gradient descent to the residual sum of squares with sufficiently small initial values, then under some proper early stopping rule, the iterates converge to a nearly sparse rate-optimal solution that improves over explicitly regularized approaches. In particular, the resulting estimator does not suffer from extra bias due to explicit penalties, and can achieve the parametric root-n rate when the signal-to-noise ratio is sufficiently high. We also perform simulations to compare our methods with high dimensional linear regression with explicit regularization. Our results illustrate the advantages of using implicit regularization via gradient descent after overparameterization in sparse vector estimation., Comment: Accepted by Biometrika

In this paper, we investigate the energy system design problems with the multi-generation technologies, i.e., simultaneous generation of multiple types of energy. We propose a long-term planning model which integrates macro-level strategic decisions such as facility location and multi-generation technology investment, and micro-level operational decisions such as production planning and energy transportation. Our results illustrate the economic value of multi-generation technologies in lieu of the spatio-temporal demand uncertainty for energy. In particular, we show that the multi-generation technologies can reduce demand uncertainty by risk pooling both within and across different facilities.