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Results for “biorthogonality”
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Isogeometric Bézier dual mortaring: The Kirchhoff–Love shell problem
Here in this paper we develop an isogeometric Bézier dual mortar method for coupling multi-patch Kirchhoff–Love shell structures. The proposed approach weakly enforces the continuity of the solution at patch interfaces through a dual mortar method and can be applied to both conforming and non-conforming discretizations. As the employed dual basis functions have local supports and satisfy the biorthogonality property, the resulting stiffness matrix is sparse. In addition, the coupling accuracy is optimal because the dual basis possesses the polynomial reproduction property. We also formulate the continuity constraints through the Rodrigues’ rotation operator which gives a unified framework for coupling patches that are intersected with G 1 continuity as well as patches that meet at a kink. Several linear and nonlinear examples demonstrated the performance and robustness of the proposed coupling techniques.
Scripting a new dialogue between diazotrophs and crops
Diazotrophs are bacteria and archaea that can reduce atmospheric dinitrogen (N 2 ) into ammonium. Plant–diazotroph interactions have been explored for over a century as a nitrogen (N) source for crops to improve agricultural productivity and sustainability. This scientific quest has generated much information about the molecular mechanisms underlying the function, assembly, and regulation of nitrogenase, ammonium assimilation, and plant–diazotroph interactions. This review presents various approaches to manipulating N fixation activity, ammonium release by diazotrophs, and plant–diazotroph interactions. Here, we discuss the research avenues explored in this area, propose potential future routes, emphasizing engineering at the metabolic level via biorthogonal signaling, and conclude by highlighting the importance of biocontrol measures and public acceptance.
Methods to Calculate Electronic Excited-State Dynamics for Molecules on Large Metal Clusters with Many States: Ensuring Fast Overlap Calculations and a Robust Choice of Phase
Here, we present an efficient set of methods for propagating excited-state dynamics involving a large number of configuration interaction singles (CIS) or Tamm-Dancoff approximation (TDA) single-reference excited states. Specifically, (i) following Head-Gordon et al., we implement an exact evaluation of the overlap of singly-excited CIS/TDA electronic states at different nuclear geometries using a biorthogonal basis and (ii) we employ a unified protocol for choosing the correct phase for each adiabat at each geometry. For many-electron systems, the combination of these techniques significantly reduces the computational cost of integrating the electronic Schrodinger equation and imposes minimal overhead on top of the underlying electronic structure calculation. As a demonstration, we calculate the electronic excited-state dynamics for a hydrogen molecule scattering off a silver metal cluster, focusing on high-lying excited states, where many electrons can be excited collectively and crossings are plentiful. Interestingly, we find that the high-lying, plasmon-like collective excitation spectrum changes with nuclear dynamics, highlighting the need to simulate non-adiabatic nuclear dynamics and plasmonic excitations simultaneously. In the future, the combination of methods presented here should help theorists build a mechanistic understanding of plasmon-assisted charge transfer and excitation energy relaxation processes near a nanoparticle or metal surface.
Nicotinamide Cofactor Biomimetics: Design and Structure Activity Relationships
Biocatalysis is an attractive methodology due to its sustainable metrics, combined with its vaunted regio-, chemo-, and stereoselectivity. Among biocatalysts, oxidoreductases catalyze redox transformations on a wide variety of substrates and are thus particularly attractive for synthetic applications. Many oxidoreductases depend on nicotinamide-based cofactors; however, the high cost and poor atom economy of such cofactors can negate the advantage of biosynthetic processes. The native cofactors (i.e., NADH or NADPH) can be replaced by nicotinamide cofactor biomimetics (NCBs) to address these issues. Additionally, NCBs can enhance enzyme activity due to structures with expanded redox properties. These capabilities open possibilities for reactivity and biorthogonality. Thus, the field has responded with new NCB structures, including variation in their electronic and molecular recognition properties. This perspective focuses on structure-activity relationships of simple NCBs, including their function, stability, and other properties, along with methods for their regeneration in biosynthesis.
Stability of PT and anti- PT -symmetric Hamiltonians with multiple harmonics
Hermitian Hamiltonians with time-periodic coefficients can be analyzed via Floquet theory, and have been extensively used for engineering Floquet Hamiltonians in standard quantum simulators. Generalized to non-Hermitian Hamiltonians, time periodicity offers avenues to engineer the landscape of Floquet quasienergies across the complex plane. We investigate two-level non-Hermitian PT and anti- PT -symmetric Hamiltonians with coefficients that have multiple harmonics using Floquet theory. By analytical and numerical calculations, we obtain their regions of stability, defined by real Floquet quasienergies, and contours of exceptional point (EP) degeneracies. We extend our analysis to study the phases that accompany these cyclic changes with the biorthogonality approach. Our results demonstrate that these time-periodic Hamiltonians generate a rich landscape of stable (real) and unstable (complex) regions. Published by the American Physical Society 2024
On the Definition of the Prompt Neutron Lifetime
There are myriad definitions for the mean neutron lifetime, neutron generation time, and other related quantities to characterize a neutron and its progeny’s propagation through an assembly. In this document, we consider lifetime definitions that are associated with eigenvalue forms of the neutron transport equation. Specifically, we focus on the two most widely used eigenvalues: the $\kappa$ eigenvalue and the $\alpha$ eigenvalue. These eigenvalues are used more often than other eigenvalues due to the physical phenomenon they capture. The $\kappa$ eigenvalue allows for the ability to determine if the system can sustain a chain reaction, while the $\alpha$ eigenvalue captures the asymptotic time dependent behavior of the system. An advantage of using these eigenvalues resides in the biorthogonality of the solutions with their adjoint counterpart. This feature allows us to simplify the expressions and to obtain appropriately weighted definitions that highlight important physics and regions of an assembly.
Abstracting models of strong normalization for classical calculi
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Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks
One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs), especially with arbitrary initial data. We address this problem by taking advantage of recent advances in scientific machine learning and the spectral dynamically orthogonal (DO) and borthogonal (BO) methods for representing stochastic processes. The recently introduced DO/BO methods reduce the SPDE to solving a system of deterministic PDEs and a system of stochastic ordinary differential equations. Specifically, we propose two new physics-informed neural networks (PINNs) for solving time-dependent SPDEs, namely the neural network (NN)-DO/BO methods. The proposed methods incorporate the DO/BO constraints into the loss function (along with the modal decomposition of the SPDE) with an implicit form instead of generating explicit expressions for the temporal derivatives of the DO/BO modes. Hence, the NN-DO/BO methods can overcome some of the drawbacks of the original DO/BO methods. For example, we do not need the assumption that the covariance matrix of the random coefficients is invertible as in the original DO method, and we can remove the assumption of no eigenvalue crossing as in the original BO method. Moreover, the NN-DO/BO methods can be used to solve time-dependent stochastic inverse problems with the same formulation and same computational complexity as for forward problems. Furthermore, we demonstrate the capability of the proposed methods via several numerical examples, namely: (1) A linear stochastic advection equation with deterministic initial condition: we obtain good results with the proposed methods, while the original DO/BO methods cannot be applied directly in this case. (2) Long-time integration of the stochastic Burgers' equation: we show the good performance of NN-DO/BO methods, especially the effectiveness of the NN-BO approach for such problems with many eigenvalue crossings during the whole time evolution, while the original BO method fails. (3) Nonlinear reaction diffusion equation: we consider both the forward problem and the inverse problems, including very noisy initial point values, to investigate the flexibility of the NN-DO/BO methods in handling inverse and mixed type problems. Taken together, these simulation results demonstrate that the NN-DO/BO methods can be employed to effectively quantify uncertainty propagation in a wide range of physical problems, but future work should address the efficiency issue of PINNs for forward problems.