Linear Realization Research Articles

Arrangements, Matroids and Logarithmic Vector Fields

The focus of this workshop was on the ongoing interaction between geometric aspects of matroid theory with various directions in the study of hyperplane arrangements. A hyperplane arrangement is exactly a linear realization of a (loop-free, simple) matroid. While a matroid is a purely combinatorial object, though, an arrangement is associated with a range of algebraic and geometric constructions that connect closely with the combinatorics of matroids.The meeting brought together researchers involved with complementary angles on the subject, many of whom had not met before, so an important underlying objective was to make introductions between groups with overlapping interests in order to facilitate new collaborations and advances in the subject.

Electric dipole moments in 5+3 flavor weak effective theory

A fully generic treatment of electric dipole moments (EDMs) is presented in the CP-violating and flavor-conserving weak effective field theory (WET) with five flavors of quarks and three flavors of leptons. We systematically analyze leading contributions to EDMs originating from QCD and QED renormalization group running between the electroweak scale and low energy scales of about 2 GeV. We include the full one-loop anomalous dimension and a subset of two-loop corrections, as well as threshold corrections at the bottom, charm and τ masses. This allows us to derive master formulae in the space of generic WET for the neutron and proton EDMs, for EDMs of diamagnetic atoms, and for the precession frequencies constrained in molecular EDM experiments, from which bounds on the electron EDM are extracted. In particular, our master formulae capture the contributions of WET CP-violating operators with heavy quark and lepton flavors. As an application, we study EDM constraints on the Yukawa couplings of the Higgs boson, in both the linear and non-linear realizations of electroweak symmetry breaking.

Parity doublet model for baryon octets: Ground states saturated by good diquarks and the role of bad diquarks for excited states

Parity doublet model is an effective chiral model that includes the chiral-variant and -invariant masses of baryons. The chiral-invariant mass has large impacts on the density dependence of models which can be constrained by neutron star observations. In the previous work, models of two flavors have been considered up to a few times nuclear saturation density, but in such dense regions it is also necessary to consider hyperons. With the chiral-invariant masses baryons can stay massive in extreme environments (e.g., neutron stars) where the chiral symmetry restoration takes place. In this work, we generalize the previous SU(2)L×SU(2)R parity models of nucleons to SU(3)L×SU(3)R models of the baryon octet within the linear realization of the chiral symmetry. The major problem in constructing such models has been too many candidates for the chiral representations of baryons. Motivated by the concepts of diquarks and the mended symmetry, we choose the (3L,3¯R)+(3¯L,3R), (3L,6R)+(6L,3R), and (1L,8R)+(8L,1R) representations and use quark diagrams to constrain the possible types of Yukawa interactions. The masses of the baryon octets for positive and negative baryons up to the first excitations are successfully reproduced. As expected from the diquark considerations, the ground-state baryons are well dominated by (3L,3¯R)+(3¯L,3R) and (1L,8R)+(8L,1R) representations, while the excited states require (3L,6R)+(6L,3R) representations. Important applications of our model are the chiral restoration for strange quarks at large density and the continuity of diquarks from hadronic to quark matter. We also address the problem of large Yukawa couplings which are enhanced in three-flavor construction. Published by the American Physical Society 2024

Optimal IOFL-based economic model predictive control technique for boiler-turbine system

The optimal control design of the boiler-turbine system is vital to ensure feasibility and high responsiveness over desired load variations. Using the traditional linear control techniques realization of this task is difficult, as the boiler-turbine mechanism has strong nonlinearities. Besides, environmental and economic concerns have replaced existing tracking control ones as the primary concerns of advanced power plants. Thus, this study proposes an optimal economic model predictive controller (EMPC) scheme for this unit on the basis of the input/output feedback linearization (IOFL) method. By employing the IOFL method, this unit is decoupled into a new linearized model that is utilized for developing the suggested optimal IOFL EMPC technique. The proposed control scheme is formulated in an economic quadratic programming form that considers the input-rate and input limits of the unit for optimal economic performance. In addition, an adaptive iterative algorithm is utilized for constraints mapping with guaranteeing a feasible solution in a finite number of steps without violation of original constraints over the entire predictive horizon. The outcomes of the simulation show that the suggested optimal IOFL EMPC scheme offers an improved dynamic and economic output performance over fuzzy hierarchical MPC, fuzzy EMPC, and nonlinear EMPC techniques during various load variations.

On unitary/strong linear realizations: Buratti–Horak–Rosa conjecture

On unitary/strong linear realizations: Buratti–Horak–Rosa conjecture

Q-Markov Covariance equivalent realizations for unstable and marginally stable systems

An observer-based q-Markov Covariance equivalent realization (QMC) formulation is presented in the paper. It is shown that by inserting an observer in the input–output relationship associated with the dynamical system, a QMC approach can be developed that is applicable to systems with marginally stable or unstable equilibrium points. A system of equations governing all linear state-space realizations, along with the corresponding observers, is derived from matching a set of Markov and Covariance parameters. The solution to this equation system is shown to parameterize all state-space realizations that match the pre-specified correlation functions. Four numerical examples are used to show the utility of the proposed approach. The numerical results show that the observer-based QMC presented in the paper can identify linear SISO and MIMO systems with stable, marginally stable, and unstable characteristics.

Chiral Restoration of Nucleons in Neutron Star Matter: Studies Based on a Parity Doublet Model

We review the chiral variant and invariant components of nucleon masses and the consequence of their existence on the chiral restoration in extreme conditions, particularly in neutron star matter. We consider a model of linear realization of chiral symmetry with the nucleon parity doublet structure that permits the chiral invariant mass, m0, for positive and negative parity nucleons. The nuclear matter is constructed with the parity doublet nucleon model coupled to scalar fields σ, vector fields (ω,ρ), and mesons with strangeness through the U(1)A anomaly. In models with a large m0, the nucleon mass is insensitive to the medium, and the nuclear saturation properties can be reproduced without demanding strong couplings of the nucleons to the scalar fields σ and vector fields ω. We confront the resulting nuclear equations of state with nuclear constraints and neutron star observations and delineate the chiral invariant mass and effective interactions. To further examine the nuclear equations of state beyond the saturation density, we supplement quark models to set the boundary conditions from the high-density side. The quark models are constrained by the two-solar-mass conditions, and such constraints are transferred to nuclear models through the causality and thermodynamic stability conditions. We also calculate various condensates and the matter composition from nuclear to quark matter in a unified matter by constructing a generating functional that interpolates the nuclear and quark matter with external fields. Two types of chiral restoration are discussed: one due to the positive scalar charges of nucleons and the other triggered by the evolution of the Dirac sea. We found that the U(1)A anomaly softens equations of state from low to high density.

Linear time-continuous state-space realization of flame transfer functions by means of a propagation equation.

Low-order network models, commonly used to assess the thermo-acoustic stability of combustors, can be cast in a linear, time-continuous state-space representation. A standard linear eigenvalue problem for the system modes results, which can be solved in a robust and efficient manner. To represent the linear dynamics of any time-invariant flame in the state-space framework, this study presents an approximation of the distributed time-delayed flame response to acoustic velocity perturbations based on a spatially discretized propagation equation (PE). We derive the rational flame transfer function of a first-order-upwind-PE state-space model and discuss its relation to the Tustin approximation of flame transfer functions. For an exemplary discrete finite impulse response of a flame, a third-order-upwind-PE state-space model is shown to match the discrete flame frequency response with comparable accuracy as a rational approximation found by non-linear optimization. The numerical dissipation introduced by discretization of the PE ensures zero gain above the Nyquist frequency of the underlying discrete flame impulse response. Finally, we apply the PE state-space flame model to a generic Rijke tube and show that the predicted thermo-acoustic modes agree well with results obtained from a classical non-linearly optimized rational approximation of the frequency response function of the flame.

Different Methods of Linear Phase IIR Filer Realization

Signal processing has become one of the most popular research topics. Researchers have designed a variety of digital filters to exclude unwanted random noises during the transmission or extract part of the signal in the desired range. There is a prevalent trend for digital filters to replace analog ones since they do not require hardwires. All the performances are operated on a single processor and are free from the effect of external factors. Finite impulse response (FIR) and infinite impulse response (IIR) filters in digital filters respectively provide infinite and finite impulse responses. In the application, it is preferred to have a linear phase digital filter, and FIR filters are naturally linear. However, FIR filters have higher orders and group delay than IIR filters, so researchers found various ways to implement linear phase IIR filters for improvement. This paper introduces and compares Powell and Chau Linear Phase IIR Filter, Kwan Linear Phase IIR Filter, and Xiao, Oliver, and Agathoklis Linear Phase IIR Filter.

Deformed Quantum Phase Spaces, Realizations, Star Products and Twists

We review deformed quantum phase spaces and their realizations in terms of undeformed phase space. In particular, methods of calculation for the star product, coproduct of momenta and twist from realizations are presented, as well as their properties and the relations between them. Lie deformed quantum phase spaces and Snyder type spaces are considered. Examples of linear realizations of the κ-Minkowski spacetime are elaborated. Finally, some new results on quadratic deformations of quantum phase spaces and a generalization of Yang and triply special relativity models are presented.

Electroweak Higgs effective field theory after LHC run 2

We analyze the electroweak interactions in the framework of the Higgs effective field theory using the available Higgs and electroweak diboson production results from LHC run 2 as well as the electroweak precision data. Assuming universality of the weak current, our study considers 25 possible anomalous couplings. To unveil the nature of the Higgs boson, i.e., isosinglet versus part of $SU(2{)}_{L}$ doublet, we explore the correlation effects between observables that are predicted to exist in the linear realization of the electroweak gauge symmetry but not in its nonlinear counterpart. This improves previous studies aimed at investigating the Higgs nature and the origin of the electroweak symmetry breaking.

Generalized Heisenberg Algebra, Realizations of the gI(N) Algebra And Applications

We introduce the generalized Heisenberg algebra appropriate for realizations of the $\mathfrak{gl}(n)$ algebra. Linear realizations of the $\mathfrak{gl}(n)$ algebra are presented and the corresponding star product, coproduct of momenta and twist are constructed. The dual realization and dual $\mathfrak{gl}(n)$ algebra are considered. Finally, we present a general realization of the $\mathfrak{gl}(n)$ algebra, the corresponding coproduct of momenta and two classes of twists. These results can be applied to physical theories on noncommutative spaces of the $\mathfrak{gl}(n)$ type.

A non-intrusive method to inferring linear port-Hamiltonian realizations using time-domain data

Port-Hamiltonian systems have gained a lot of attention in recent years due to their inherent valuable properties in modeling and control. In this paper, we are interested in constructing linear port-Hamiltonian systems from time-domain input-output data. We discuss a non-intrusive methodology that is comprised of two main ingredients–(a) inferring frequency response data from time-domain data and (b) constructing an underlying port-Hamiltonian realization using the inferred frequency response data. We illustrate the proposed methodology by means of two numerical examples and also compare it with two other system identification methods to infer the frequency response from the input-output data.

Autonomous Driving using Linear Model Predictive Control with a Koopman Operator based Bilinear Vehicle Model

This paper presents a real-time Model Predictive Control (MPC) formulation for autonomous driving based on a lifted bilinear vehicle model developed using the Koopman operator. Koopman operator based models can closely mimic the original nonlinear behaviors with a higher dimensional linear structure, which is attractive for computationally efficient linear MPC formulations for controlling nonlinear systems. However, current linear models based on linear Koopman realizations cannot capture the control-affine dynamics in nonlinear systems. This may result in large discrepancies between the original nonlinear system and the data-driven linear model, hindering its use in MPC. To address this gap, first, a novel Koopman bilinear vehicle model that takes control-affine dynamics into consideration is constructed and tested in open-loop simulations. This bilinear Koopman model is then linearized to serve as a prediction model in MPC, and is shown to have higher accuracy compared to the state-of-the-art linear models. The model is then used to develop a linear MPC formulation for simultaneous planning and control of an autonomous vehicle. The formulation is tested on lane change scenarios with obstacles against the nonlinear MPC and standard linear MPC benchmarks. The results show that the new formulation can achieve a lane change performance closer to the nonlinear MPC with a computational performance similar to the standard linear MPC. The new formulation is observed to be successful in handling high speeds where the standard linear MPC fails.

Electroweak skyrmions in the HEFT

We study the existence of skyrmions in the presence of all the electroweak degrees of freedom, including a dynamical Higgs boson, with the electroweak symmetry being non-linearly realized in the scalar sector. For this, we use the formulation of the Higgs Effective Field Theory (HEFT). In contrast with the linear realization, a well-defined winding number exists in HEFT for all scalar field configurations. We classify the effective operators that can potentially stabilize the skyrmions and numerically find the region in parameter spaces that support them. We do so by minimizing the static energy functional using neural networks. This method allows us to obtain the minimal-energy path connecting the vacuum to the skyrmion configuration and calculate its mass and radius. Since skyrmions are not expected to be produced at colliders, we explore the experimental and theoretical bounds on the operators that generate them. Finally, we briefly consider the possibility of skyrmions being dark matter candidates.

New methods to attack the Buratti-Horak-Rosa conjecture

The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list L of v−1 positive integers not exceeding ⌊v2⌋ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set {0,1,…,v−1} if and only if, for every divisor d of v, the number of multiples of d appearing in L is at most v−d. In this paper we present new methods that are based on linear realizations and can be applied to prove the validity of this conjecture for a vast choice of lists. As example of their flexibility, we consider lists whose underlying set is one of the following: {x,y,x+y}, {1,2,3,4}, {1,2,4,…,2x}, {1,2,4,…,2x,2x+1}. We also consider lists with many consecutive elements.

Estimating extreme characteristics of stochastic non-linear systems

The nature of the ocean is random and, as a result, marine structures face extreme, non-linear load effects. Estimating extreme responses of said structures often reduces to purely probabilistic approaches involve a considerable amount of conjecture. A method is developed in this paper to conserve original system information without the need for high fidelity Monte Carlo Simulations. Here, an iterative linearization of a non-linear system is developed to estimate extremes for the non-linear system. The Matched Upcrossing Equivalent Linear System (MUELS) method finds linear systems with zero-upcrossing periods equivalent to that of the non-linear system under a specified forcing spectrum. These linear systems are used as input into the Design Loads Generator, where ensembles of extreme linear realizations at the exposure period of interest, and the input time series that lead to those extremes, are generated. These input time series are valid inputs into the non-linear system and create a set of non-linear realizations that approximate the set of non-linear extremes at the exposure period of interest. As an example, this paper introduces and applies this method to the Duffing Oscillator at various levels of increasing cubic stiffness, forced by an ITTC wave spectrum, for exposure periods as long as 108 hours representing O(1010) zero-crossing maxima. Quantitative comparisons are made with Monte Carlo simulations, GEVD estimates, and expected values of time series near extreme local maxima.

Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems With Unknown Dynamics

Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This letter describes how the Koopman operator can be used to generate approximate linear, bilinear, and nonlinear model realizations from data, and argues in favor of bilinear realizations for characterizing systems with unknown dynamics. Necessary and sufficient conditions for a dynamical system to have a valid linear or bilinear realization over a given set of observable functions are presented and used to show that every control-affine system admits an infinite-dimensional bilinear realization, but does not necessarily admit a linear one. Therefore, approximate bilinear realizations constructed from generic sets of basis functions tend to improve as the number of basis functions increases, whereas approximate linear realizations may not. To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm is constructed from data. In a trajectory following task, the bilinear realization exceeds the prediction accuracy of the linear realization and the computational efficiency of the nonlinear realization when incorporated into a model predictive control framework.

Matroid connectivity and singularities of configuration hypersurfaces

Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.

The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization

The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian) Lanczos algorithm. Analogously, the Gauss quadrature for linear functionals is connected with formal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm with look-ahead strategy; moreover, it is related to the minimal partial realization problem. We will review these connections pointing out the relationships between several results established independently in related contexts. Original proofs of the Mismatch Theorem and of the Matching Moment Property are given by using the properties of formal orthogonal polynomials and the Gauss quadrature for linear functionals.