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1. Cophonologies by Ph(r)ase

   https://doi.org/10.1007/s11049-020-09467-x  

   Sande, Hannah; Jenks, Peter; Inkelas, Sharon (November 2020, Natural Language & Linguistic Theory)

   null (Ed.)

   Abstract Phonological alternations are often specific to morphosyntactic context. For example, stress shift in English occurs in the presence of some suffixes, -al , but not others, -ing : "Equation missing" , "Equation missing" , "Equation missing" . In some cases a phonological process applies only in words of certain lexical categories. Previous theories have stipulated that such morphosyntactically conditioned phonology is word-bounded. In this paper we present a number of long-distance morphologically conditioned phonological effects, cases where phonological processes within one word are conditioned by another word or the presence of a morpheme in another word. We provide a model, Cophonologies by Phase, which extends Cophonology Theory, intended to capture word-internal and lexically specified phonological alternations, to cyclically generated syntactic constituents. We show that Cophonologies by Phase makes better predictions about the long-distance morphologically conditioned phonological effects we find across languages than previous frameworks. Furthermore, Cophonologies by Phase derives such effects without requiring the phonological component to directly reference syntactic features or structure. 

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2. Stochastic Bridges of Linear Systems

   https://doi.org/10.1109/TAC.2015.2440567  

   Chen, Yongxin; Georgiou, Tryphon (September 2019, IEEE Transactions on Automatic Control)

   We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two end-points of a given interval. Their motion in phase space can be modeled as an Ornstein–Uhlenbeck process conditioned at the two end-points—a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a time-varying drift is sufficient to modify the prior SDE and meet the end-point conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions. 

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3. Representer Theorems in Banach Spaces: Minimum Norm Interpolation, Regularized Learning and Semi-Discrete Inverse Problems

   Wang, Rui; Xu, Yuesheng (September 2021, Journal of machine learning research)

   Shawe-Taylor, John (Ed.)

   Learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation (MNI) problem, a regularized learning problem or, in general, a semi-discrete inverse problem (SDIP), in either Hilbert spaces or Banach spaces. The goal of this paper is to systematically study solutions of these problems in Banach spaces. We aim at obtaining explicit representer theorems for their solutions, on which convenient solution methods can then be developed. For the MNI problem, the explicit representer theorems enable us to express the infimum in terms of the norm of the linear combination of the interpolation functionals. For the purpose of developing efficient computational algorithms, we establish the fixed-point equation formulation of solutions of these problems. We reveal that unlike in a Hilbert space, in general, solutions of these problems in a Banach space may not be able to be reduced to truly finite dimensional problems (with certain infinite dimensional components hidden). We demonstrate how this obstacle can be removed, reducing the original problem to a truly finite dimensional one, in the special case when the Banach space is ℓ1(N). 

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4. A quantum-classical Liouville formalism in a preconditioned basis and its connection with phase-space surface hopping

   https://doi.org/10.1063/5.0124835  

   Wu, Yanze; Subotnik, Joseph E. (January 2023, The Journal of Chemical Physics)

   We revisit a recent proposal to model nonadiabatic problems with a complex-valued Hamiltonian through a phase-space surface hopping (PSSH) algorithm employing a pseudo-diabatic basis. Here, we show that such a pseudo-diabatic PSSH (PD-PSSH) ansatz is consistent with a quantum-classical Liouville equation (QCLE) that can be derived following a preconditioning process, and we demonstrate that a proper PD-PSSH algorithm is able to capture some geometric magnetic effects (whereas the standard fewest switches surface hopping approach cannot capture such effects). We also find that a preconditioned QCLE can outperform the standard QCLE in certain cases, highlighting the fact that there is no unique QCLE. Finally, we also point out that one can construct a mean-field Ehrenfest algorithm using a phase-space representation similar to what is done for PSSH. These findings would appear extremely helpful as far as understanding and simulating nonadiabatic dynamics with complex-valued Hamiltonians and/or spin degeneracy. 

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5. Brownian bridges for stochastic chemical processes—An approximation method based on the asymptotic behavior of the backward Fokker–Planck equation

   https://doi.org/10.1063/5.0080540  

   Wang, Shiyan; Venkatesh, Anirudh; Ramkrishna, Doraiswami; Narsimhan, Vivek (May 2022, The Journal of Chemical Physics)

   A Brownian bridge is a continuous random walk conditioned to end in a given region by adding an effective drift to guide paths toward the desired region of phase space. This idea has many applications in chemical science where one wants to control the endpoint of a stochastic process—e.g., polymer physics, chemical reaction pathways, heat/mass transfer, and Brownian dynamics simulations. Despite its broad applicability, the biggest limitation of the Brownian bridge technique is that it is often difficult to determine the effective drift as it comes from a solution of a Backward Fokker–Planck (BFP) equation that is infeasible to compute for complex or high-dimensional systems. This paper introduces a fast approximation method to generate a Brownian bridge process without solving the BFP equation explicitly. Specifically, this paper uses the asymptotic properties of the BFP equation to generate an approximate drift and determine ways to correct (i.e., re-weight) any errors incurred from this approximation. Because such a procedure avoids the solution of the BFP equation, we show that it drastically accelerates the generation of conditioned random walks. We also show that this approach offers reasonable improvement compared to other sampling approaches using simple bias potentials. 

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