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The increasing volume of space objects in Earth's orbit presents a significant challenge for Space Situational Awareness (SSA). And in particular, accurate orbit prediction is crucial to anticipate the position and velocity of space…
We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs. The invariant objects considered in this paper are equilibrium points, traveling waves, periodic orbits and invariant…
Transformer architectures rely on position encodings to model the spatial structure of input data. Rotary Position Encoding (RoPE) is a widely used method in language models that encodes relative positions through fixed, block-diagonal,…
For a commutative cocommutative Hopf algebra we study the relationship between a certain linear map defined via a bicharacter, an exponential of a quadratic differential operator and a $\bullet$ product obtained via twisting by a…
We have derived orbital basis sets from scattering theory. They are expressed as polynomial approximations to the energy dependence of a set of partial waves, in quantized form. The corresponding matrices, as well as the Hamiltonian and…
Parameter estimation for ordinary differential equations (ODEs) plays a fundamental role in the analysis of dynamical systems. Generally lacking closed-form solutions, ODEs are traditionally approximated using deterministic solvers.…
Spatially localized one-electron orbitals, orthogonal and nonorthogonal, are widely used in electronic structure theory to describe chemical bonding and speed up calculations. In order to avoid linear dependencies of localized orbitals, the…
We define the e\~ne product for the multiplicative group of polynomials and formal power series with coefficients on a commutative ring and unitary constant coefficient. This defines a commutative ring structure where multiplication is the…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
For two matrices $A$ and $B$, and large $n$, we show that most products of $n$ factors of $e^{A/n}$ and $n$ factors of $e^{B/n}$ are close to $e^{A + B}$. This extends the Lie-Trotter formula. The elementary proof is based on the relation…
The presence of second-order smoothness for objective functions of optimization problems can provide valuable information about their stability properties and help us design efficient numerical algorithms for solving these problems. Such…
The attention mechanism in a Transformer architecture matches key to query based on both content -- the what -- and position in a sequence -- the where. We present an analysis indicating that what and where are entangled in the popular RoPE…
The Neumann-Poincar\'{e} operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the…
The present work develops a framework to derive piecewise polynomial measures arising from invariant measures on adjoint orbits in the context of compact and semisimple Lie groups. These measures are computed from orbital integrals via…
Neural NLI models overfit dataset artifacts instead of truly reasoning. A hypothesis-only model gets 57.7% in SNLI, showing strong spurious correlations, and 38.6% of the baseline errors are the result of these artifacts. We propose…
The Lie algebra of the Poincar\'e-Maxwell group is derived in a manner that provides the interpretation of the equations of motion. It is clarified that the dynamics obtained from the orbit method is exactly equivalent to the classical…
We investigate properties of exponential operators preserving the particle number using combinatorial methods developed in order to solve the boson normal ordering problem. In particular, we apply generalized Dobinski relations and methods…
The asymptotic performance of a polar code under successive cancellation decoding is determined by the exponent of its polarizing matrix. We first prove that the partial distances of a polarizing matrix constructed from the Kronecker…
Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor-coefficients of the…
Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible…