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We describe a framework for estimating Hilbert-Samuel multiplicities $e_XY$ for pairs of projective varieties $X \subset Y$ from finite point samples rather than defining equations. The first step involves proving that this multiplicity…
We show explicitly how a strongly coupled fixed point can be constructed in scalar $g\varphi^4$ theory from the solutions to a non-linear eigenvalue problem. The fixed point exists only for $d< 4$, is unstable and characterized by $\nu=2/d$…
We study the spectral properties of certain non-self-adjoint matrices associated with large directed graphs. Asymptotically the eigenvalues converge to certain curves, apart from a finite number that have limits not on these curves.
A model Hamiltonian describing a two-level system with a crossing plus a pairing force is investigated using technique of large-amplitude collective motion. The collective path, which is determined by the decoupling conditions, is found to…
This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with…
For the orthogonal-unitary and symplectic-unitary transitions in random matrix theory, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion…
A holomorphic endomorphism of $\mathbb{CP}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study…
Exceptional points associated with non-hermitian operators, i.e. operators being non-hermitian for real parameter values, are investigated. The specific characteristics of the eigenfunctions at the exceptional point are worked out. Within…
This paper investigates the asymptotics of eigenstructure of sample covariance matrix under the spiked covariance matrix model in ultra-high-dimensional settings, where the dimensionality can grow much faster than the sample size with $ p…
Entanglement of high-dimensional quantum systems has become increasingly important for quantum communication and experimental tests of nonlocality. However, many effects of high-dimensional entanglement can be simulated by using multiple…
We present a new method to diagnose strong coupling in multi-mode open systems. Our method presents a non-trivial extension of exceptional point (EP) analysis employed for such systems; specifically, we show how eigenvectors can not only…
In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum…
Within the effective mass approximation an adiabatic description of spheroidal and dumbbell quantum dot models in the regime of strong dimensional quantization is presented using the expansion of the wave function in appropriate sets of…
The eigenmodes of resonating structures, e.g., electromagnetic cavities, are sensitive to deformations of their shape. In order to compute the sensitivities of the eigenpair with respect to a scalar parameter, we state the Laplacian and…
We study a class of holomorphic matrix models. The integrals are taken over middle dimensional cycles in the space of complex square matrices. As the size of the matrices tends to infinity, the distribution of eigenvalues is given by a…
Very few works exist to date on development of a consistent energy-based coupling of atomistic and continuum models of materials in more than one dimension. The difficulty in constructing such a coupling consists in defining a coupled…
In the present paper, first the mathematical basic properties of the exceptional points are discussed. Then, their role in the description of real physical quantum systems is considered. Most interesting value is the phase rigidity of the…
For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here ``eigenvalue''…
We study the entanglement properties of deconfined quantum critical points. We show not only that these critical points may be distinguished by their entanglement structure but also that they are in general more highly entangled that…
In this note, we consider arbitrary finite-dimensional real algebras containing a copy of complex numbers. It is proved that matrices with entries from an arbitrary finite-dimensional real algebra containing a square root of negative one in…