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Solving linear systems of polynomial equations is a ubiquitous problem in both mathematics and physics. The standard approach, Gaussian elimination, scales cubically with system size and often constitutes a computational bottleneck. The…
Gauge invariance is essential for making physically meaningful predictions. In superconductors, mean-field Hamiltonians that explicitly break $U(1)$ symmetry often yield gauge-dependent results. While this issue has been resolved for linear…
Invertible transformation of large graphs into fixed dimensional vectors (embeddings) remains a challenge. Its overcoming would reduce any operation on graphs to an operation in a vector space. However, most existing methods are limited to…
In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with…
Gaussian building blocks are essential for photonic quantum information processing, and universality can be practically achieved by equipping Gaussian circuits with adaptive measurement and feedforward. The number of adaptive steps then…
Classical vector analysis is the predominant formalism used by engineers of computational electromagnetism, despite the fact that manifold as a theoretical concept has existed for a century. This paper discusses the benefits of manifolds…
Graph signal processing extends spectral analysis to data supported on irregular domains. Existing fractional transforms for two-dimensional graph signals, including the two-dimensional graph fractional Fourier transform (GFRFT), typically…
Sinusoidal excitation is particularly popular for testing structures in the nonlinear regime. Due to the nonlinear behavior and the inevitable feedback of the structure on the exciter, higher harmonics in the applied excitation are…
A dispersive analysis of $\eta\to 3\pi$ decays has been performed in the past by many authors. The numerical analysis of the pertinent integral equations is hampered by two technical difficulties: i) The angular averages of the amplitudes…
What makes a class of quantum circuits efficiently classically simulable on average? I present a framework that applies harmonic analysis of groups to circuits with a structure encoded by group parameters. Expanding the circuits in a…
The problem of synchronization over a group $\mathcal{G}$ aims to estimate a collection of group elements $G^*_1, \dots, G^*_n \in \mathcal{G}$ based on noisy observations of a subset of all pairwise ratios of the form $G^*_i {G^*_j}^{-1}$.…
We numerically model key building blocks of a phononic integrated circuit that enable phonon routing in high-acoustic-index waveguides. Our particular focus is on Gallium Nitride-on-sapphire phononic platform which has recently demonstrated…
In this paper we discuss the numerical solution on a simple 2D domain of the Helmoltz equation with mixed boundary conditions. The so called radiation problem depends on the wavenumber constant parameter k and it is inspired here by medical…
Data analysis often requires methods that are invariant with respect to specific transformations, such as rotations in case of images or shifts in case of images and time series. While principal component analysis (PCA) is a widely-used…
Geometric integration of non-autonomous classical engineering problems, such as rotor dynamics, is investigated. It is shown, both numerically and by backward error analysis, that geometric (structure preserving) integration algorithms are…
Modern understanding of symmetry in quantum field theory includes both invertible and non-invertible operations. Motivated by this, we extend Nielsen's geometric approach to quantum circuit complexity to incorporate non-invertible gates.…
In these lecture notes, partly based on a course taught at the Karpacz Winter School in March 2014, we explore the close connections between non-adiabatic response of a system with respect to macroscopic parameters and the geometry of…
This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…
Quadratic surfaces gain more and more attention among the Geometric Algebra community and some frameworks were proposed in order to represent, transform, and intersect these quadratic surfaces. As far as the authors know, none of these…
The development of emerging technologies in quantum optics demands accurate models that faithfully capture genuine quantum effects. Mature semiclassical approaches reach their limits when confronted with quantized electromagnetic fields,…