Representation Of Operators Research Articles

Blockchain-Empowered Digital Twins Collaboration: Smart Transportation Use Case

Digital twins (DTs) is a promising technology in the revolution of the industry and essential for Industry 4.0. DTs play a vital role in improving distributed manufacturing, providing up-to-date operational data representation of physical assets, supporting decision-making, and avoiding the potential risks in distributed manufacturing systems. Furthermore, DTs need to collaborate within distributed manufacturing systems to predict the risks and reach consensus-based decision-making. However, DTs collaboration suffers from single failure due to attack and connection in a centralized manner, data interoperability, authentication, and scalability. To overcome the above challenges, we have discussed the major high-level requirements for the DTs collaboration. Then, we have proposed a conceptual framework to fulfill the DTs collaboration requirements by using the combination of blockchain, predictive analysis techniques, and DTs technologies. The proposed framework aims to empower more intelligence DTs based on blockchain technology. In particular, we propose a concrete ledger-based collaborative DTs framework that focuses on real-time operational data analytics and distributed consensus algorithms. Furthermore, we describe how the conceptual framework can be applied using smart transportation system use cases, i.e., smart logistics and railway predictive maintenance. Finally, we highlighted the future direction to guide interested researchers in this interesting area.

Gottesman Types for Quantum Programs

The Heisenberg representation of quantum operators provides a powerful technique for reasoning about quantum circuits, albeit those restricted to the common (non-universal) Clifford set H, S and CNOT. The Gottesman-Knill theorem showed that we can use this representation to efficiently simulate Clifford circuits. We show that Gottesman's semantics for quantum programs can be treated as a type system, allowing us to efficiently characterize a common subset of quantum programs. We also show that it can be extended beyond the Clifford set to partially characterize a broad range of programs. We apply these types to reason about separable states and the superdense coding algorithm.

Bulk reconstruction and Bogoliubov transformations in AdS2

In the bulk reconstruction program, one constructs boundary representations of bulk fields. We investigate the relation between the global/Poincare and AdS-Rindler representations for AdS2. We obtain the AdS-Rindler smearing function for massive and massless fields and show that the global and AdS-Rindler boundary representations are related by conformal transformations. We also use the boundary representations of creation and annihilation operators to compute the Bogoliubov transformation relating global modes to AdS-Rindler modes for both massive and massless particles.

An open-source framework for the implementation of large-scale integral operators with flexible, modern high-performance computing solutions: Enabling 3D Marchenko imaging by least-squares inversion

Numerical integral operators of the convolution type form the basis of most wave-equation-based methods for the processing and imaging of seismic data. Because several of these methods require the solution of an inverse problem, multiple forward and adjoint passes of the modeling operator are generally required to converge to a satisfactory solution. We have highlighted the memory requirements and computational challenges that arise when implementing such operators on 3D seismic data sets and their use for solving large systems of integral equations. A Python framework is presented that leverages libraries for distributed storage and computing, and it provides a high-level symbolic representation of linear operators. A driving goal for our work is not only to offer a widely deployable, ready-to-use high-performance computing framework, but also to demonstrate that it enables addressing research questions that are otherwise difficult to tackle. To this end, the first example of 3D full-wavefield target-oriented imaging, which comprises two subsequent steps of seismic redatuming, is presented. The redatumed fields are estimated by means of gradient-based inversion using the full data set as well as spatially decimated versions of the data set as a way to investigate the robustness of both inverse problems to spatial aliasing in the input data set. Our numerical example shows that when one spatial direction is finely sampled, satisfactory redatuming and imaging can also be accomplished when the sampling in the other direction is coarser than a quarter of the dominant wavelength. Although aliasing introduces noise in the redatumed fields, they are less sensitive to well-known spurious artifacts compared to cheaper, adjoint-based redatuming techniques. These observations are shown to hold for a relatively simple geologic structure, and although further testing is needed for more complex scenarios, we expect them to be generally valid while possibly breaking down for extreme cases (e.g., highly heterogeneous/scattering media).

A Digitized Representation of the Modified Prandtl-Ishlinskii Hysteresis Model for Modeling and Compensating Piezoelectric Actuator Hysteresis.

Piezoelectric actuators are widely used in micromanipulation and miniature robots due to their rapid response and high repeatability. The piezoelectric actuators often have undesired hysteresis. The Prandtl–Ishlinskii (PI) hysteresis model is one of the most popular models for modeling and compensating the hysteresis behaviour. This paper presents an alternative digitized representation of the modified Prandtl–Ishlinskii with the dead-zone operators (MPI) hysteresis model to describe the asymmetric hysteresis behavior of piezoelectric actuators. Using a binary number with n digits to represent the classical Prandtl–Ishlinskii hysteresis model with n elementary operators, the inverse model can be easily constructed. A similar representation of the dead-zone operators is also described. With the proposed digitized representation, the model is more intuitive and the inversion calculation is avoided. An experiment with a piezoelectric stacked linear actuator is conducted to validate the proposed digitized MPI hysteresis model and it is shown that it has almost the same performance as compared to the classical representation.

N-point correlators of twist-2 operators in SU(N) Yang-Mills theory to the lowest perturbative order

We compute, to the lowest perturbative order in SU(N) Yang-Mills theory, n-point correlators in the coordinate and momentum representation of the gauge-invariant twist-2 operators with maximal spin along the p+ direction, both in Minkowskian and — by analytic continuation — Euclidean space-time. We also construct the corresponding generating functionals. Remarkably, they have the structure of the logarithm of a functional determinant of the identity plus a term involving the effective propagators that act on the appropriate source fields.

Life cycle assessment of electricity generation: A systematic review of spatiotemporal methods

• Life cycle assessments of electricity increasingly include spatiotemporal methods. • Studies tend to focus on climate impacts while other impact categories may benefit. • Advancements in temporal methodology lag spatial improvements. • Results present a roadmap, identifying areas that warrant future research. • GIS and optimization increasingly used for advances in LCA. Life cycle assessments (LCAs) of electricity generation are increasingly incorporating more granular spatial and temporal information, enhancing the accuracy of both inventories and results. This systematic review determined contributions to LCA that improved spatial, temporal, or spatiotemporal resolution from 2009-2018. We analyzed 251 articles screened from an initial review of 6,519 to identify such contributions and determine areas in need of research. The geographic focus of the studies leans towards Europe, Asia, and North America, suggesting many regions remain understudied. As the impact categories were heavily weighted towards greenhouse gas emissions, the impacts that may benefit most from more granular analyses reflecting local environmental conditions were less studied (e.g., land use and eutrophication). While studies tend to focus more on spatial rather than temporal information, those that examine the most granular spatial and temporal scales (for this review, site and hourly) can result in more effective interventions that improve both environmental and economic outcomes. The two most common analysis tools used in the screened articles were optimization and Geographic Information Systems. The increasing use of these tools supported diverse improvements in LCA, such as more detailed investigations of grid interactions, enhanced characterization of impacts, and improved evaluation of resource availability. Analyses conducted at more refined spatiotemporal resolutions can provide more realistic representations of electricity generation, grid operations, and environmental impacts, supporting more effective interventions.

Effect of impurity on modulational instability of localized modes in a discrete quantum ferromagnetic spin chain

The modulational instability (MI) of localized modes in a discrete quantum ferromagnetic spin chain with the effect of impurity is studied in this paper. The model of Heisenberg ferromagnetic spin chain is considered and dynamical equation is derived by the use of Glauber’s coherent state method along with Holstein-Primakoff (H-P) bosonic representation of spin operators. The dynamical equation is in the form of discrete nonlinear Schro¨dinger (DNLS) equation which describes the complete dynamics of ferromagnetic spin chain. The MI technique is applied to the DNLS equation and obtains the dispersion relation which relates the frequency and wavenumber of the modulating perturbations. Using dispersion relation, we establish the stability criteria of the spin waves in ferromagnetic spin chain and explore how the presence of impurity supports the formation of localized modes as shown in the stability/instability profile. With the use of graphical illustrations of MI, the strength of impurity on the stability/instability of localized modes in ferromagnetic spin chain is discussed. Through numerical simulation, we explore how the role of impurity supports the excitation of localized modes in ferromagnetic chain and also analyze the short time and long time evolution of localized wave in ferromagnetic spin chain.

Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

AbstractWe introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two‐ (2D) and three‐dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D spacial grids. Using the diagonalization of the arising matrix‐valued functions in the eigenbasis of the one‐dimensional Sturm–Liouville operators, we construct the rank‐structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right‐hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low‐rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an ‐threshold. This method reduces the numerical cost for solving the control problem to (plus the quadratic term with a small weight), which outperforms traditional approaches with complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by . This essentially outperforms the traditional methods operating with fully populated matrices and vectors in . Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n.

On Lie group representations and operator ranges

In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R \mathcal {R} be an operator range of a Hilbert space H \mathcal {H} . Given the set Λ \Lambda of R \mathcal {R} -invariant operators, and given a Lie group representation ρ : G → GL ( H ) \rho :G\rightarrow \text {GL}(\mathcal {H}) , we discuss the induced semigroup homomorphism ρ ~ : ρ − 1 ( Λ ) → B ( R ) \widetilde {\rho }: \rho ^{-1}(\Lambda ) \rightarrow \mathcal {B(R)} for the operator range topology on R \mathcal {R} . In one direction, we work under the assumption ρ − 1 ( Λ ) = G \rho ^{-1} (\Lambda ) = G , so ρ ~ : G → B ( R ) \widetilde {\rho }:G\rightarrow \mathcal {B}(\mathcal {R}) is in fact a group representation. In this setting, we prove that ρ ~ \widetilde {\rho } is continuous (and smooth) if and only if the tangent map d ρ d\rho is R \mathcal {R} -invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H \mathcal {H} , the set ρ − 1 ( Λ ) \rho ^{-1}(\Lambda ) is neither a group for a large set of nonclosed operator ranges R \mathcal {R} nor closed for all nonclosed operator ranges R \mathcal {R} . Both results are proved by means of explicit counterexamples.

Study of fourth-order boundary value problem based on Volterra–Fredholm equation: numerical treatment

This paper presents a study of the performance of the Tau method using Chebyshev basis functions for solving fourth-order differential equation with boundary conditions. Existence and uniqueness of the solution of this equation are investigated transforming it into the Volterra–Fredholm integral equation. We use the operational Tau matrix representation with Chebyshev basis functions for constructing the algebraic equivalent representation of the problem.This representation is an special semi lower triangular system whose solution gives the components of the vector solution. Applying Gronwall’s and the generalized Hardy’s inequality, convergence analysis and error estimation of the Tau method are discussed. The error analysis indicates that the numerical errors decay exponentially when the source function are sufficiently smooth. Illustrative examples are given to represent the efficiency and the accuracy of the proposed method. Also, some comparisons are made with existing results such that the results obtained by Tau method are more accurate than the proposed methods in this case.

Modelling Oil Price with Lie Algebras and Long Short-Term Memory Networks

In this paper, we propose hybrid models for modelling the daily oil price during the period from 2 January 1986 to 5 April 2021. The models on S2 manifolds that we consider, including the reference ones, employ matrix representations rather than differential operator representations of Lie algebras. Firstly, the performance of LieNLS model is examined in comparison to the Lie-OLS model. Then, both of these reference models are improved by integrating them with a recurrent neural network model used in deep learning. Thirdly, the forecasting performance of these two proposed hybrid models on the S2 manifold, namely Lie-LSTMOLS and Lie-LSTMNLS, are compared with those of the reference LieOLS and LieNLS models. The in-sample and out-of-sample results show that our proposed methods can achieve improved performance over LieOLS and LieNLS models in terms of RMSE and MAE metrics and hence can be more reliably used to assess volatility of time-series data.

On the group generated by C, P and T: I 2 = T 2 = P 2 = ITP = −1, with applications to pseudo-scalar mesons

We study faithful representations of the discrete Lorentz symmetry operations of parity P and time reversal T , which involve complex phases when acting on fermions. If the phase of P is a rational multiple of π then P 2n = 1 for some positive integer n and it is shown that, when this is the case, P and T generate a discrete group, a dicyclic group (also known as a generalised quaternion group) which are generalisations of the dihedral groups familiar from crystallography. Charge conjugation C introduces another complex phase and, again assuming rational multiples of π for complex phases, TC generates a cyclic group of order 2m for some positive integer m. There is thus a doubly infinite series of possible finite groups labelled by n and m. Demanding that C commutes with P and T forces n = m = 2 and the group generated by P and T is uniquely determined to be the quaternion group. Neutral pseudo-scalar mesons can be simultaneous C and P eigenstates, T commutes with P and C when acting on fermion bi-linears so neutral pseudo-scalar mesons can also be T eigenstates. The T -parity should therefore be experimentally observable and the CPT theorem dictates that T = CP.

Exact k -body representation of the Jaynes-Cummings interaction in the dressed basis: Insight into many-body phenomena with light

Analog quantum simulation - the technique of using one experimentally well-controlled physical system to mimic the behavior of another - has quickly emerged as one of the most promising near term strategies for studying strongly correlated quantum many-body systems. In particular, systems of interacting photons, realizable in solid-state cavity and circuit QED frameworks, for example, hold tremendous promise for the study of nonequilibrium many-body phenomena due to the capability to locally create and destroy photons. These systems are typically modeled using a Jaynes-Cummings-Hubbard (JCH) Hamiltonian, named due to similarities with the Bose-Hubbard (BH) model. Here, we present a non-perturbative procedure for transforming the JC Hamiltonian into a dressed operator representation that, in its most general form, admits an infinite sum of bosonic $k$-body terms where $k$ is bound only by the number of excitations in the system. We closely examine this result in both the dispersive and resonant coupling regimes, finding rapid convergence in the former and contributions from $k\gg1$ in the latter. Through extension to a two-site JCH system, we demonstrate that this approach facilitates close inspection of the analogy between the JCH and BH models and its breakdown for resonant light-matter coupling. Finally, we use this framework to survey the many-body character of a two-site JCH for general system parameters, identifying four unique quantum phases and the parameter regimes in which they are realized, thus highlighting phenomena realizable with finite JCH-based quantum simulators beyond the BH model. More broadly, this work is intended to serve as a clear mathematical exposition of bosonic many-body interactions underlying JC-type systems, often postulated through analogy to Kerr-like nonlinear susceptibilities or by matching coefficients to obtain the appropriate eigenvalue spectrum.

Compact fermion to qubit mappings

Mappings between fermions and qubits are valuable constructions in physics. To date only a handful exist. In addition to revealing dualities between fermionic and spin systems, such mappings are indispensable in any quantum simulation of fermionic physics on quantum computers. The number of qubits required per fermionic mode, and the locality of mapped fermionic operators strongly impact the cost of such simulations. We present a fermion to qubit mapping that outperforms all previous local mappings in both the qubit to mode ratio and the locality of mapped operators. In addition to these practically useful features, the mapping bears an elegant relationship to the toric code, which we discuss. Finally, we consider the error mitigating properties of the mapping---which encodes fermionic states into the code space of a stabilizer code. Although there is an implicit tradeoff between low weight representations of local fermionic operators, and high distance code spaces, we argue that fermionic encodings with low-weight representations of local fermionic operators can still exhibit error mitigating properties which can serve a similar role to that played by high code distances. In particular, when undetectable errors correspond to ``natural'' fermionic noise. We illustrate this point explicitly both for this encoding and the Verstraete-Cirac encoding.

Matrix representation of operations of soft partial orderings on a generalized soft poset

A soft set is a parametrized family of subsets of an initial universal set. Soft set theory is a generalization of fuzzy set theory and is a mathematical tool for dealing with uncertainty and vagueness. Posets are used in many applications of Mathematics and Computer Science. Matrix representations are more applicable for handling data in computer programs. In this paper, we introduce the ordinary matrix representation of a soft relation, soft matrix representation of soft partial ordering and operations of soft partial orderings on a generalized soft poset.

A more accurate half-discrete Hilbert-type inequality in the whole plane and the reverses

A more accurate half-discrete Hilbert-type inequality in the whole plane with multi-parameters is established by the use of Hermite–Hadamard’s inequality and weight functions. Furthermore, some equivalent forms and some special types of inequalities and operator representations as well as reverses are considered.

Continental Hydrologic Intercomparison Project, Phase 1: A Large‐Scale Hydrologic Model Comparison Over the Continental United States

AbstractHigh‐resolution, coupled, process‐based hydrology models, in which subsurface, land‐surface, and energy budget processes are represented, have been applied at the basin‐scale to ask a wide range of water science questions. Recently, these models have been developed at continental scales with applications in operational flood forecasting, hydrologic prediction, and process representation. As use of large‐scale model configurations increases, it is exceedingly important to have a common method for performance evaluation and validation, particularly given challenges associated with accurately representing large domains. Here, we present phase 1 of a comparison project for continental‐scale, high‐resolution, processed‐based hydrologic models entitled the Continental Hydrologic Intercomparison Project (CHIP). The first phase of CHIP is based on past Earth System Model intercomparisons and comprised of a two‐model proof of concept comparing the ParFlow‐CONUS hydrologic model, version 1.0 and a NOAA US National Water Model configuration of WRF‐Hydro, version 1.2. The objectives of CHIP phase 1 are: (a) describe model physics and components, (b) design an experiment to ensure a fair comparison, and (b) assess simulated streamflow with observations to better understand model bias. To our knowledge, this is the first comparison of continental‐scale, high‐resolution, physics‐based models which incorporate lateral subsurface flow. This model intercomparison is an initial step toward a continued effort to unravel process, parameter, and formulation differences in current large‐scale hydrologic models and to engage the hydrology community in improving hydrology model configuration and process representation.

Hyperquaternion Conformal Groups

The paper gives a new representation of conformal groups in n dimensions in terms of hyperquaternions defined as tensor products of quaternion algebras (or a subalgebra thereof). Being Clifford algebras, hyperquaternions provide a good representation of pseudo-orthogonal groups such as $$O(p+1,q+1)$$ isomorphic to the nD conformal group with $$n=p+q.$$ The representation yields simple expressions of the generators, independently of matrices or operators. The canonical decomposition and the invariants are discussed. As application, the 4D relativistic conformal group is detailed together with a worked example. Finally, the formalism is compared to the operator representation. Potential uses include in particular, conformal geometry, computer graphics and conformal field theory.

Ripser: efficient computation of Vietoris\u2013Rips persistence barcodes

We present an algorithm for the computation of Vietoris–Rips persistence barcodes and describe its implementation in the software Ripser. The method relies on implicit representations of the coboundary operator and the filtration order of the simplices, avoiding the explicit construction and storage of the filtration coboundary matrix. Moreover, it makes use of apparent pairs, a simple but powerful method for constructing a discrete gradient field from a total order on the simplices of a simplicial complex, which is also of independent interest. Our implementation shows substantial improvements over previous software both in time and memory usage.