Quadratic Operator Research Articles

A general expression for vibrational Hamiltonians expressed in oblique coordinates

We examine the properties of oblique coordinates. The coordinates, introduced by Zúñiga et al. [J. Phys. B: At., Mol. Opt. Phys. 52, 055101, (2019)], reduce vibrational mode-mixing and enhance the quality of vibrational assignments in quantum mechanical investigations of two-dimensional model Hamiltonians. Oblique coordinates are obtained by making non-orthogonal rotations of the original coordinates that convert the matrix representation of the quadratic Hamiltonian operator into a block-diagonal matrix where the blocks are distinguished by the total quanta of vibrational excitation. Using techniques for the polar decomposition of matrices, we present a scheme for finding these coordinates for systems of arbitrary dimensions. Several molecular examples are presented that highlight the advantages of these coordinates.

A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients

Traditional linear subspace-based reduced order models (LS-ROMs) can be used to significantly accelerate simulations in which the solution space of the discretized system has a small dimension (with a fast decaying Kolmogorov n-width). However, LS-ROMs struggle to achieve speed-ups in problems whose solution space has a large dimension, such as highly nonlinear problems whose solutions have large gradients. Such an issue can be alleviated by combining nonlinear model reduction with operator learning. Over the past decade, many nonlinear manifold-based reduced order models (NM-ROM) have been proposed. In particular, NM-ROMs based on deep neural networks (DNN) have received increasing interest. This work takes inspiration from adaptive basis methods and specifically focuses on developing an NM-ROM based on Convolutional Neural Network-based autoencoders (CNNAE) with iteration-dependent trainable kernels. Additionally, we investigate DNN-based and quadratic operator inference strategies between latent spaces. A strategy to perform vectorized implicit time integration is also proposed. We demonstrate that the proposed CNN-based NM-ROM, combined with DNN-based operator inference, generally performs better than commonly employed strategies (in terms of prediction accuracy) on a benchmark advection-dominated problem. The method also presents substantial gain in terms of training speed per epoch, with a training time about one order of magnitude smaller than the one associated with a state-of-the-art technique performing with the same level of accuracy.

Detector operators for celestial symmetries

This paper presents a systematic cataloging of the generators of celestial symmetries on phase space. Starting from the celestial OPEs, we first show how to extract a representation of the general-spin analog of the wedge subalgebra of w1+∞ on the phase space of massless matter fields of arbitrary helicity. These generators can be expressed as light-sheet operators that are quadratic in the matter fields at future or past null infinity. We next show how to extend these symmetries beyond the wedge. Doing so requires us to augment the quadratic operators with: 1) linear terms corresponding to primary descendants of the negative helicity gauge fields the matter modes couple to, and 2) a tower of higher-particle composite operator contributions. These modes can be realized as light-ray operators supported on generators of null infinity, but local on the celestial sphere. Finally, we construct a representation of the celestial symmetries that captures how the positive helicity gauge fields transform. We close by discussing how these celestial symmetries inform our choice of detector operators.

On the dynamics of non-Volterra quadratic operators corresponding to permutations

The present paper deals to a family of non-Volterra quadratic stochastic operators depending on a parameter and a permutation and their trajectory behaviours. We find all fixed and periodic points for such non-Volterra quadratic stochastic operator on a finite-dimensional simplex. We proved that the ω- limit set of any trajectory of such non-Volterra quadratic stochastic operator consists of either single point or finite points.

A Note on the Jacobian Problem of Coifman, Lions, Meyer and Semmes

Coifman, Lions, Meyer and Semmes asked in 1993 whether the Jacobian operator and other compensated compactness quantities map their natural domain of definition onto the real-variable Hardy space H1(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^1({\\mathbb {R}}^n)$$\\end{document}. We present an axiomatic, Banach space geometric approach to the problem in the case of quadratic operators. We also make progress on the main open case, the Jacobian equation in the plane.

On the resilience of the gravitational variational principle under renormalization

A well-defined variational principle for gravitational actions typically requires to cancel boundary terms produced by the variation of the bulk action with a suitable set of boundary counterterms. This can be achieved by carefully balancing the coefficients multiplying the bulk operators with those multiplying the boundary ones. A typical example of this construction is the Gibbons-Hawking-York boundary action that needs to be added to the Einstein-Hilbert one in order to have a well-defined metric variation for General Relativity with Dirichlet boundary conditions. Quantum fluctuations of matter fields lead to a renormalization of these coefficients which may or may not preserve this balance. Indeed, already at the level of General Relativity, the resilience of the matching between bulk and boundary constants is far from obvious and it is anyway incomplete given that matter generically induces quadratic curvature operators. We investigate here the resilience of the matching of higher-order couplings upon renormalization by a non-minimally coupled scalar field and show that a problem is present. Even though we do not completely solve the latter, we show that it can be greatly ameliorated by a wise splitting between dynamical and topological contributions. Doing so, we find that the bulk-boundary matching is preserved up to a universal term (present for any Weyl invariant matter field content), whose nature and possible cancellation we shall discuss in the end.

Spectral and steady-state properties of fermionic random quadratic Liouvillians

We study spectral and steady-state properties of generic Markovian dissipative systems described by quadratic fermionic Liouvillian operators of the Lindblad form. The Hamiltonian dynamics is modeled by a generic random quadratic operator, i.e., as a featureless superconductor of class D, whereas the Markovian dissipation is described by MM random linear jump operators. By varying the dissipation strength and the ratio of dissipative channels per fermion, m=M/(2N_F)m=M/(2NF), we find two distinct phases where the support of the single-particle spectrum has one or two connected components. In the strongly dissipative regime, this transition occurs for m=1/2m=1/2 and is concomitant with a qualitative change in both the steady-state and the spectral gap that rules the large-time dynamics. Above this threshold, the spectral gap and the steady-state purity qualitatively agree with the fully generic (i.e., non-quadratic) case studied recently. Below m=1/2m=1/2, the spectral gap closes in the thermodynamic limit and the steady-state decouples into an ergodic and a nonergodic sector yielding a non-monotonic steady-state purity as a function of the dissipation strength. Our results show that some of the universal features previously observed for fully random Liouvillians are generic for a sufficiently large number of jump operators. On the other hand, if the number of dissipation channels is decreased the system can exhibit nonergodic features, rendering it possible to suppress dissipation in protected subspaces even in the presence of strong system-environment coupling.

Hermite spectral method for the inelastic Boltzmann equation

We propose a Hermite spectral method for the inelastic Boltzmann equation, which makes two-dimensional periodic problem computation affordable by the hardware nowadays. This new approach involves utilizing a Hermite expansion, whereby the expansion coefficients for the variable hard sphere model are simplified into a series of summations that can be precisely derived. Additionally, a new collision model is built with a combination of the quadratic collision operator and a simplified collision operator, which helps us to balance the computational cost and the accuracy. Various numerical experiments, including spatially two-dimensional simulations, demonstrate the accuracy and efficiency of this numerical scheme.

Periodic and Regular Trajectories of Non-Volterra Quadratic Stochastic Operators

Periodic and Regular Trajectories of Non-Volterra Quadratic Stochastic Operators

Nuclear Shape-Phase Transition From Spherical U(5) to Deformed gamma-Unstable O(6) Dynamical Symmetries of Interacting Boson Model Applied to Ru, Pd, and Xe Isotopic Chains

The shape transition from spherical vibrator U(5) to gamma-unstable deformed rotor O(6) in even-even Ru, Pd, and Xe isotopic chains are studied in framework of sd interacting boson model (IBM1) using the coherent state formalism to obtain the potential energy surfaces (PES’s). The location of critical points in the transition are identified by analysis the PES’s in terms of the deformation parameter beta and by using the catastrophe theory in terms of the two essential parameters (r_1, r_2). By using the most general IBM1 Hamiltonian in Casimir form with neglecting the U(5) and SU(3) quadratic Casimir operators and introducing only one control parameter the PES leads to the same energy surface as the Q-consistent IBM Hamiltonian at gamma =0. For the studied isotopic chains, the chi ^2-test is used to perform the fitting between the experimental and the corresponding calculated IBM for some selected energy levels and electric quadrupole transition probabilities B(E2) values using a simulated search program. A good agreement is produced for both energies and B(E2) transition rates. The present model calculations suggest that ^{100}Ru, ^{102}Pd and ^{130}Xe nuclei are good candidates for the E(5) critical point symmetry.

Truncated quadratic norm minimization for bilinear factorization based matrix completion

Low-rank matrix completion is an important research topic with a wide range of applications. One prevailing way for matrix recovery is based on rank minimization. Directly solving this problem is NP hard. Therefore, various rank surrogates are developed, like nuclear norm. However, nuclear norm regularization minimizes the sum of all the singular values, and hence the rank is not well approximated. We propose a new rank substitution named truncated quadratic norm that performs the corresponding truncated quadratic operation on the singular values. This function takes the square of the minor singular values and maps large singular values to one. In order to reduce computational complexity, the original target matrix is factorized into two small matrices on which the truncated quadratic norm is imposed. The resultant problem is then solved by alternating minimization. We also prove that the solution sequence is able to converge to a critical point. Experimental results on synthetic data and real-world images demonstrate the excellent performance of our method in terms of recovery accuracy.

Gains of integrability and local smoothing effects for quadratic evolution equations

We characterize geometrically the semigroups generated by non-selfadjoint quadratic differential operators (e−tqw)t≥0 enjoying local smoothing effects and providing gains of integrability. More precisely, we prove that the evolution operators e−tqw map Lp on Lq∩C∞, for all 1≤p≤q≤∞, if and only if the singular space of the quadratic operator qw is included in the graph of a linear map. We also provide quantitative estimates for the associated operator norms in the short-time asymptotics 0<t≪1.

Decoding quantum field theory with machine learning

We demonstrate how one can use machine learning techniques to bypass the technical difficulties of designing an experiment and translating its outcomes into concrete claims about fundamental features of quantum fields. In practice, all measurements of quantum fields are carried out through local probes. Despite measuring only a small portion of the field, such local measurements have the capacity to reveal many of the field’s global features. This is because, when in equilibrium with their environments, quantum fields store global information locally, albeit in a scrambled way. We show that neural networks can be trained to unscramble this information from data generated from a very simple one-size-fits-all local measurement protocol. To illustrate this general claim we will consider three non-trivial features of the field as case studies: a) how, as long as the field is in a stationary state, a particle detector can learn about the field’s boundary conditions even before signals have time to propagate from the boundary to the detector, b) how detectors can determine the temperature of the quantum field even without thermalizing with it, and c) how detectors can distinguish between Fock states and coherent states even when the first and second moments of all their quadrature operators match. Each of these examples uses the exact same simple fixed local measurement protocol and machine-learning ansatz successfully. This supports the claim that the framework proposed here can be applied to nearly any kind of local measurement on a quantum field to reveal nearly any of the field’s global properties in a one-size-fits-all manner.

Algebraic analysis of non-Hermitian quadratic Hamiltonians

We study a general one-mode non-Hermitian quadratic Hamiltonian that does not exhibit PT-symmetry. By means of an algebraic method we determine the conditions for the existence of real eigenvalues as well as the location of the exceptional points. We also put forward an algebraic alternative to the generalized Bogoliubov transformation that enables one to convert the quadratic operator into a simpler form in terms of the original creation and annihilation operators. We carry out a similar analysis of a two-mode oscillator that consists of two identical one-mode oscillators coupled by a quadratic term.

A Convex Combination of Two Quadratic Stochastic Operators Acting in the 2D Simplex

A Convex Combination of Two Quadratic Stochastic Operators Acting in the 2D Simplex

Discrete time model of sexual systems

Small factors are the biggest contributors to the biggest changes in an ecosystem which may lead to its demise. Simple things such as tweaking the humidity or tampering with the temperature can be the downfall of the habitants and the ecosystem itself. These changes can be observed right now with the consistent temperature increase on Earth along with the slew of issues that come with global warming. A small change in a few degrees in temperature can perish an entire food supply and the habitat of many already endangered beings forcing them to either adapt or die. Within such an ecosystem the main reason it withstands the test of time boils down to one simple thing; reproduction. This creates the natural question of asking oneself: How does an ecosystem replenish itself and is able to resist collapsing? To even begin considering this thought, one must realise that the size of the population is the key to gaining a deeper understanding as two constituents that are important from evolutionary theory are survival and reproduction. The case of asexual reproducers, it is not difficult to see any alterations that can be introduced to increase reproductivity. Meanwhile, for sexual reproducers, the evolution to increase reproductivity can be observed by looking into the generational data of the species. A certain group of time-based dynamic systems that are connected to a sexual system are the point of contention. The suggested model is a dynamic representation of a hermaphrodite population which is described through quadratic stochastic operators. The key findings offer fresh insights into the future of hermaphrodite populations, that is perhaps a probable solution to prevent the decline of endangered or at-risk species. This demonstrates a fresh perspective on reproduction, which is explored through a purely mathematical approach.

Behavior of Trajectories of a Quadratic Operator

Behavior of Trajectories of a Quadratic Operator

Genetic Algebras Associated with ξ(a)-Quadratic Stochastic Operators.

The present paper deals with a class of ξ(a)-quadratic stochastic operators, referred to as QSOs, on a two-dimensional simplex. It investigates the algebraic properties of the genetic algebras associated with ξ(a)-QSOs. Namely, the associativity, characters and derivations of genetic algebras are studied. Moreover, the dynamics of these operators are also explored. Specifically, we focus on a particular partition that results in nine classes, which are further reduced to three nonconjugate classes. Each class gives rise to a genetic algebra denoted as Ai, and it is shown that these algebras are isomorphic. The investigation then delves into analyzing various algebraic properties within these genetic algebras, such as associativity, characters, and derivations. The conditions for associativity and character behavior are provided. Furthermore, a comprehensive analysis of the dynamic behavior of these operators is conducted.

Lyapunov functions and dynamics of infinite dimensional Volterra operators

The majority of research on quadratic stochastic operators (QSOs) was done on a finite-dimensional set of all probability distributions (also known as a simplex), thus it is intriguing to extend them for an infinite case. In particular, the infinite dimensional Volterra operators are discussed in the current paper. Due to the fact non-compactness of infinite dimensional simplex (unlike a finite case) makes the general study challenging, therefore, a sub-class of infinite dimensional Volterra operators is introduced. Furthermore, we construct Lyapunov functions which allow us to explore the dynamics of the introduced operators. Several examples are given with a full description of the limiting set.

$L^p$-bounds for semigroups generated by non-elliptic quadratic differential operators

In this note, we establish L^p -bounds for the semigroup e^{-tq^w(x,D)} , t \ge 0 , generated by a quadratic differential operator q^w(x,D) on \mathbb{R}^n that is the Weyl quantization of a complex-valued quadratic form q defined on the phase space \mathbb{R}^{2n} with non-negative real part \operatorname{Re} q \ge 0 and trivial singular space. Specifically, we show that e^{-tq^w(x,D)} is bounded from L^p(\mathbb{R}^n) to L^q(\mathbb{R}^n) for all t &gt; 0 whenever 1 \le p \le q \le \infty , and we prove bounds on \parallel { e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel in both the large t \gg 1 and small 0 &lt; t \ll 1 time regimes.Regardin g L^p \rightarrow L^q bounds for the evolution semigroup at large times, we show that \parallel{e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel is exponentially decaying as t \rightarrow \infty , and we determine the precise rate of exponential decay, which is independent of (p,q) . At small times 0 &lt; t \ll 1 , we establish bounds on \parallel {e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel for (p,q) with 1 \le p \le q \le \infty that are polynomial in t^{-1} .