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Symmetry properties of the evolution equation and the state to be controlled are shown to determine the basic features of the linear control of unstable orbits. In particular, the selection of control parameters and their minimal number are…
We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The…
Symmetry plays a fundamental role in understanding complex quantum matter, particularly in classifying topological quantum phases, which have attracted great interests in the recent decade. An outstanding example is the time-reversal…
We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we…
Models of disorder with a direction (constant imaginary vector-potential) are considered. These non-Hermitian models can appear as a result of computation for models of statistical physics using transfer matrix technique or describe…
The article reviews recent developments in the theory of fluctuations and correlations of energy levels and eigenfunction amplitudes in diffusive mesoscopic samples. Various spatial geometries are considered, with emphasis on…
When representing convective instability mechanisms with the streamwise BiGlobal stability approach, results suffer from a sensitivity to the streamwise domain truncation length and boundary conditions. The methodology proposed in this…
Initially introduced as a machine translation model, the Transformer architecture has now become the foundation for modern deep learning architecture, with applications in a wide range of fields, from computer vision to natural language…
In this article we find explicit formulae for spherically symmetric solutions of the multidimensional zero-pressure gas dynamics system and its adhesion approximation. The asymptotic behaviour of the explicit solutions of the adhesion…
This work presents a theoretical investigation of an active photonic grating of the parity-time (PT) symmetric architecture. The analytical study of the free-space mode propagation from the grating structure indicates the unique bifurcation…
We explore the scattering of waves in designed asymmetric one-dimensional waveguide networks. We show that the reflection between two ports of an asymmetric network can be identical over a broad frequency range, as if the network was…
These notes offer a unified introduction to spectral methods for the study of complex systems. They are intended as an operative manual rather than a theorem-proof textbook: the emphasis is on tools, identities, and perspectives that can be…
Exact self-consistent particle-like solutions with spherical and/or cylindrical symmetry to the equations governing the interacting system of scalar, electromagnetic and gravitational fields have been obtained. As a particular case it is…
We investigate the field equations in the Einstein-aether theory for static spherically symmetric spacetimes and a perfect fluid source and subsequently with the addition of a scalar field (with an exponential self-interacting potential).…
Non-Hermitian dynamics in open systems can give rise to a variety of fascinating non-equilibrium phenomena, ranging from symmetry-breaking transitions to directional energy flow. Parity-time (PT) symmetry breaking determines the occurrence…
We consider an aggregation-diffusion equation modelling particle interaction with non-linear diffusion and non-local attractive interaction using a homogeneous kernel (singular and non-singular) leading to variants of the Keller-Segel model…
Transformer architecture has shown impressive performance in multiple research domains and has become the backbone of many neural network models. However, there is limited understanding on how it works. In particular, with a simple…
We consider the self-attention model - an interacting particle system on the unit sphere, which serves as a toy model for Transformers, the deep neural network architecture behind the recent successes of large language models. We prove the…
We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points. This model arises naturally in various physical contexts, such…
Transport-induced self-sustained oscillations in electromechanical systems convert a static electrochemical bias into robust, autonomous oscillatory motion in the absence of any external periodic drive. However, an exact description of such…