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Factorization method is developed for a family of discretely spiked harmonic oscillators. Two sets of intertwining and ladder operators are presented to algebraically generate eigenstates with energies isomorphic to those of the ordinary…
In-Context Learning (ICL) has emerged as a new paradigm in large language models (LLMs), enabling them to perform novel tasks by conditioning on a few examples embedded in the prompt. Yet, the highly nonlinear behavior of ICL for NLP tasks…
The plethora of spectra of OB-type stars in observatory archives and the much larger numbers to come from the WEAVE and 4MOST spectroscopic facilities require efficient, but also accurate and precise methods for (semi)automatic quantitative…
We present new observations of the center-to-limb variation of spectral lines in the quiet Sun. Our long-slit spectra are corrected for scattered light, which amounts to 4-8 % of the continuum intensity, by comparison with a Fourier…
We theoretically investigate the convergence rate and support consistency (i.e., correctly identifying the subset of non-zero coefficients in the large sample limit) of multiple kernel learning (MKL). We focus on MKL with block-l1…
This paper studies a low-communication algorithm for solving elliptic partial differential equations (PDE's) on high-performance machines, the nested iteration with range decomposition algorithm (NIRD). Previous work has shown that NIRD…
Neural networks are typically trained with a single learning rate across all layers. While recent empirical evidence suggests that assigning layer-specific learning rates can accelerate training, a principled understanding of the conditions…
Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations.…
Modern deep networks are heavily overparameterized yet often generalize well, suggesting a form of low intrinsic complexity not reflected by parameter counts. We study this complexity at initialization through the effective rank of the…
We explore a variational Ansatz for lattice quantum systems -- named long-range entangled-plaquette state -- in which pairs of clusters of adjacent lattice sites are correlated at any distance. The explicit scale-free structure of…
The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as…
Accurately known stellar lithium abundances may be used to shed light on a variety of astrophysical phenomena such as Big Bang nucleosynthesis, radial migration, ages of stars and stellar clusters, and planet engulfment events. We present a…
We systematically investigate the finite size scaling behavior of the R\'enyi entanglement entropy (EE) of several representative 2d quantum many-body systems between a subregion and its complement, with smooth boundaries as well as…
The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential $V(x) =\beta x^{2}+x^{2m}$ ($m>0$). The…
We propose a novel variational ansatz for the ground-state preparation of the $\mathbb{Z}_2$ lattice gauge theory (LGT) in quantum simulators. It combines dissipative and unitary operations in a completely deterministic scheme with a…
We propose a technique for reformulation of state and parameter estimation problems as that of matching explicitly computable definite integrals with known kernels to data. The technique applies for a class of systems of nonlinear ordinary…
The objective of the present paper is to introduce the concept of a spatially inhomogeneous linear inverse problem where the degree of ill-posedness of operator $Q$ depends not only on the scale but also on location. In this case, the rates…
Kernelized Gram matrix $W$ constructed from data points $\{x_i\}_{i=1}^N$ as $W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {\sigma^2} )$ is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how…
This paper investigates the stochastic linear-quadratic control problems with affine constraints, in which both equality and inequality constraints are involved. With the help of the Pontryagin maximum principle and Lagrangian duality…
Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate…