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Rotary positional embedding has become the state-of-the-art approach to encode position information in transformer-based models. While it is often succinctly expressed in complex linear algebra, we note that the actual implementation of…

A series of recent papers have modified the classical variational phase-field fracture models to successfully predict both the nucleation and propagation of cracks in brittle fracture under general loading conditions. This is done through…

The variance--covariance matrix plays a central role in the inferential theories of high-dimensional factor models in finance and economics. Popular regularization methods of directly exploiting sparsity are not directly applicable to many…

Computational physics simulation can be a powerful tool to accelerate industry deployment of new scientific technologies. However, it must address the challenge of computationally tractable, moderately accurate prediction at large industry…

Conditional Random Fields (CRFs) are undirected graphical models, a special case of which correspond to conditionally-trained finite state machines. A key advantage of these models is their great flexibility to include a wide array of…

Recent studies suggest that context-aware low-rank approximation is a useful tool for compression and fine-tuning of modern large-scale neural networks. In this type of approximation, a norm is weighted by a matrix of input activations,…

Fine-tuning has become a popular approach to adapting large foundational models to specific tasks. As the size of models and datasets grows, parameter-efficient fine-tuning techniques are increasingly important. One of the most widely used…

We develop a fractional return-mapping framework for power-law visco-elasto-plasticity. In our approach, the fractional viscoelasticity is accounted through canonical combinations of Scott-Blair elements to construct a series of well-known…

The resolvent analysis of McKeon & Sharma (2010) recasts the Navier-Stokes equations into an input/output form in which the nonlinear term is treated as a forcing that acts upon the linear dynamics to yield a velocity response. The…

The principle of Maximal Coding Rate Reduction (MCR$^2$) has recently been proposed as a training objective for learning discriminative low-dimensional structures intrinsic to high-dimensional data to allow for more robust training than…

Kernel approximation via nonlinear random feature maps is widely used in speeding up kernel machines. There are two main challenges for the conventional kernel approximation methods. First, before performing kernel approximation, a good…

This paper derives new results for the analysis of nonlinear systems by extending contraction theory in the framework of vector distances. A new tool, vector contraction analysis utilizing a notion of the vector-valued norm which evidently…

This paper proposes a novel neural network architecture inspired by the nonstandard form proposed by Beylkin, Coifman, and Rokhlin in [Communications on Pure and Applied Mathematics, 44(2), 141-183]. The nonstandard form is a highly…

We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged…

We derive in a simple manner and from first principles the Inglis semi-classical phenomenological cranking model for nuclear collective rotation. The derivation transforms the nuclear Schrodinger equation (instead of the Hamiltonian) to a…

This short note gives a new framework for dealing with nonlinear sampled-data systems. We introduce a new idea of lifting, which is well known for linear systems, but not successfully generalized to nonlinear systems. This paper introduces…

Alternating projection method has been used in a wide range of engineering applications since it is a gradient-free method (without requiring tuning the step size) and usually has fast speed of convergence. In this paper, we formalize two…

In many areas of engineering, nonlinear numerical analysis is playing an increasingly important role in supporting the design and monitoring of structures. Whilst increasing computer resources have made such formerly prohibitive analyses…

A generalized covariant method of analysis applicable to frames for which time is not orthogonal to space, such as spacetime around a star possessing angular momentum or on a rotating disk, is presented. Important aspects of such an…

Modeling of non-rigid object launching and manipulation is complex considering the wide range of dynamics affecting trajectory, many of which may be unknown. Using physics models can be inaccurate because they cannot account for unknown…