Spin-Orbit Matrix Elements for a Combined Spin-Flip and IP/EA Approach

**mean**

**field**spin-orbit approximation was used to account for the two-electron contributions. Expand abstract.

**mean**

**field**spin-orbit approximation was used to account for the two-electron contributions. Basis set dependence as well as the effect of including additional excitations is presented. The effect of correlating the core and semi-core orbitals is also examined. Surprisingly accurate results are obtained for spin-orbit coupling constants, despite the fact that the efficient wavefunction approximations we explore neglect the bulk of dynamical correlation.

4/10 relevant

chemRxiv

Ab Initio Non-Covalent Crystal **Field** Theory for Lanthanide Complexes: A Multiconfigurational Non-Orthogonal Group Functions Approach

**field**energy levels and magnetic properties in lanthanide complexes, implementing a systematic description of non-covalent contributions to metal-ligand bonding. The approach has two steps. In the first step, appropriate ab initio wavefunctions for the various ionic fragments (lanthanide ion and coordinating ligands) are separately optimized, accounting for the electrostatic influence of the surrounding environment, within various approximations. In the second and final step, the scalar relativistic (DKH2) electrostatic Hamiltonian of the whole molecule is represented on the basis of the optimized metal-ligand multiconfigurational non-orthogonal group functions (MC-NOGF), and reduced to an effective (2J+1)-dimensional non-orthogonal Configuration Interaction (CI) problem via L{\"o}wdin-partitioning. Within the proposed formalism, the projected Hamiltonian can be implemented to any desired order of perturbation theory in the fragment-localised excitations out of the degenerate space, and its eigenvalues and eigenfunctions are systematic approximations to the crystal

**field**energies and wavefunctions. We present a preliminary implementation of the proposed MC-NOGF method to first-order degenerate perturbation theory within our own ab initio code CERES, and compare its performance both with the simpler non-covalent orthogonal ab initio approach Fragment Ab Initio Model Potential (FAIMP) approximation, and with the full CAHF/CASCI-SO method, accounting for metal-ligand covalency in a

**mean**-

**field**manner. We find that energies and magnetic properties for 44 complexes obtained via an iteratively optimized version of our MC-NOGF first-order non-covalent method, compare remarkably well to the full CAHF/CASCI-SO method including metal-ligand covalency, and are superior to the best purely electrostatic results achieved via an iteratively optimized version of the FAIMP approach.

4/10 relevant

chemRxiv

A **mean**-**field** analysis of two-player zero-sum games

**mean**-

**field**dynamics. Expand abstract.

**mean**-

**field**dynamics. Our method identifies mixed equilibria in high dimensions and is demonstrably effective for training mixtures of GANs.

10/10 relevant

arXiv

Traveling waves in a **mean** **field** learning model

**mean**-

**field**type interacting system of individual agents. Expand abstract.

**mean**-

**field**type interacting system of individual agents. In this paper, we prove existence of traveling wave solutions to this system. They correspond to what is known in economics as balanced growth path solutions. We also study the dependence of the solutions and their propagation speed on various economic parameters of the system.

10/10 relevant

arXiv

Experimental Design in Two-Sided Platforms: An Analysis of Bias

**mean**

**field**limit to capture dynamics in such systems, and use our model to investigate how performance of these estimators is affected by interference effects between listings and between customers. Expand abstract.

**mean**

**field**limit to capture dynamics in such systems, and use our model to investigate how performance of these estimators is affected by interference effects between listings and between customers. Good experimental design depends on market balance: we show that in highly demand-constrained markets, CR is unbiased, while LR is biased; conversely, in highly supply-constrained markets, LR is unbiased, while CR is biased. We also study a design based on two-sided randomization (TSR) where both customers and listings are randomized to treatment and control, and show that appropriate choices of such designs can be unbiased in both extremes of market balance, and also yield low bias in intermediate regimes of market balance.

4/10 relevant

arXiv

Mean-**Field** Analysis of Two-Layer Neural Networks: Non-Asymptotic Rates
and Generalization Bounds

**mean**

**field**analysis and the neural tangent kernel based analysis. Expand abstract.

**mean**-

**field**analysis to demonstrate the global convergence of noisy (stochastic) gradient descent for training over-parameterized two-layer neural networks. However, existing results in the

**mean**-

**field**setting do not provide the convergence rate of neural network training, and the generalization error bound is largely missing. In this paper, we provide a

**mean**-

**field**analysis in a generalized neural tangent kernel regime, and show that noisy gradient descent with weight decay can still exhibit a "kernel-like" behavior. This implies that the training loss converges linearly up to a certain accuracy in such regime. We also establish a generalization error bound for two-layer neural networks trained by noisy gradient descent with weight decay. Our results shed light on the connection between

**mean**

**field**analysis and the neural tangent kernel based analysis.

10/10 relevant

arXiv

Analytical solutions of a CQNLSE with a source

**mean**-

**field**description, we investigate the existence of cnoidal waves in a driven cubic-quartic nonlinear Schr\"odinger equation (CQNLSE), as the most simplistic modeling of beyond

**mean**-

**field**formalism of Bose-Einstein condensate (BEC) can be traced in the framework of CQNLSE. To the best of our knowledge, a comprehensive analytical description of CQNLSE is nonexistent. We check for the existence of the analytical solutions of forced CQNLSE. The stabilization of the system necessitates the inclusion of the source term. We derive analytical cnoidal wave solutions which are both localized as well as nonlocalized type. Further we reveal the existence of bright soliton like structure, a flat valley like structure and $w$-type solitons. We are also able to see the non existence of sinusoidal modes.

4/10 relevant

arXiv

Strategic advantages in **mean** **field** games with a major player

**mean**

**field**games involving a major player which has a strategic advantage, while only allowing closed loop markovian strategies for all the players. Expand abstract.

**mean**

**field**games theory. We show how to model

**mean**

**field**games involving a major player which has a strategic advantage, while only allowing closed loop markovian strategies for all the players. We illustrate this property through three examples.

10/10 relevant

arXiv

Central limit theorem over non-linear functionals of empirical measures
with applications to the **mean**-**field** fluctuation of interacting particle
systems

**mean**-

**field**limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear. Expand abstract.

**mean**-

**field**limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear.

10/10 relevant

arXiv

**Mean** **Field** Theory for the Quantum Rabi Model, Inconsistency to the
Rotating Wave Approximation

**mean**

**field**theory (MFT) was applied to replace the operators by equivalent expectation values. Expand abstract.

**mean**

**field**theory (MFT) was applied to replace the operators by equivalent expectation values. The Rabi model was reduced to a fourth orders NDE describing atoms position. Solution by the harmonic balance method (HBM) showed good accuracy and consistency to the numerical results, which introduces it as a useful tool in the quantum dynamics studies.

10/10 relevant

arXiv