Representation Of Operators Research Articles

On 2-variable $q$-Legendre polynomials: the view point of the $q$-operational technique

In this work, we exploit the methods of an operational formality and extension of quasi-monomials to describe and realize 2-variable $q$-Legendre polynomials. We introduce the generating function of 2-variable $q$-Legendre polynomials with a context of $0^{\text{th}}$ order $q$-Bessel Tricomi functions and obtain their properties such as series definition and $q$-differential equations. Also, we establish the $q$-multiplicative and $q$-derivative operators of these polynomials. The operational representations of 2-variable $q$-Legendre polynomials are obtained.

Thermalization in Krylov basis

We study thermalization in closed non-integrable quantum systems using the Krylov basis. We demonstrate that for thermalization to occur, the matrix representation of typical local operators in the Krylov basis should exhibit a specific tridiagonal form with all other elements in the matrix being exponentially small, reminiscent of the eigenstate thermalization hypothesis. Within this framework, we propose that the nature of thermalization, whether weak or strong, can be examined by the infinite time average of the Krylov complexity. Moreover, we analyze the variance of Lanczos coefficients as another probe for the nature of thermalization. One observes that although the variance of Lanczos coefficients may capture certain features of thermalization, it is not as effective as the infinite time average of complexity.

Heavy-quark mass relation from a standard-model boson operator representation in terms of fermions

Heavy-quark mass relation from a standard-model boson operator representation in terms of fermions

Optimization of Bulk Cargo Terminal Unloading and Outbound Operations Based on a Deep Reinforcement Learning Framework

This study addresses the integrated scheduling problem of dry bulk cargo terminal yards, which includes three components: transportation planning, yard selection optimization, and equipment scheduling. Additionally, the research integrates safety considerations and addresses the complexities of dynamic transportation planning. This work presents two innovations. Firstly, this study develops a sophisticated modeling framework that integrates graph structures for precise yard mapping with mixed-integer programming to enforce operational constraints. This integrated approach facilitates a more accurate and comprehensive representation of yard operations, capturing diverse operational aspects while maintaining model clarity and computational efficiency. Secondly, this study proposes an advanced solution methodology that employs a reinforcement learning technique integrating a Dueling Deep Q-Network and Double Deep Q-Network. This hybrid algorithm significantly enhances optimization performance and accelerates the learning process, thereby improving the efficiency of the solutions. The experimental results demonstrate that the proposed model effectively manages the integrated scheduling of bulk material ingress, storage, and egress within the yard. The operational plans generated by the approach outperform traditional first-come, first-served strategies, showcasing substantial improvements in port operational efficiency and reliability. This comprehensive solution underscores the potential for significant advancements in the overall management and performance of dry bulk cargo ports.

Representation and normality of hyponormal operators in the closure of $$\mathcal{AN}$$-operators

Representation and normality of hyponormal operators in the closure of $$\mathcal{AN}$$-operators

Spectral operator representations

Machine learning in atomistic materials science has grown to become a powerful tool, with most approaches focusing on atomic geometry, typically decomposed into local atomic environments. This approach, while well-suited for machine-learned interatomic potentials, is conceptually at odds with learning complex intrinsic properties of materials, often driven by spectral properties commonly represented in reciprocal space (e.g., band gaps or mobilities) which cannot be readily partitioned in real space. For such applications, methods that represent the electronic rather than the atomic structure could be more promising. In this work, we present a general framework focused on electronic-structure descriptors that take advantage of the natural symmetries and inherent interpretability of physical models. We apply this framework first to material similarity and then to accelerated screening, where a model trained on 217 materials correctly labels 75% of entries in the Materials Cloud 3D database, which meet common screening criteria for promising transparent-conducting materials.

Minimal length scale in quantum mechanics from a deformation quantization perspective

Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.

Treelike process tensor contraction for automated compression of environments

The algorithm “automated compression of environments” (ACE) [M. Cygorek , ] provides a versatile way of simulating an extremely broad class of open quantum systems. This is achieved by encapsulating the influence of the environment, which is determined by the interaction Hamiltonian(s) and initial states, into compact process tensor matrix product operator (PT-MPO) representations. The generality of the ACE method comes at high numerical cost. Here, we demonstrate that orders-of-magnitude improvement of ACE is possible by changing the order of PT-MPO contraction from a sequential to a treelike scheme. The problem of combining two partial PT-MPOs with large inner bonds is solved by a preselection approach. The drawbacks of the preselection approach are that the MPO compression is suboptimal and that it is more prone to error accumulation than sequential combination and compression. We therefore also identify strategies to mitigate these disadvantages by fine-tuning compression parameters. This results in a scheme that is similar in compression efficiency and accuracy to the original ACE algorithm, yet is significantly faster. Our numerical experiments reach similar conclusions for bosonic and fermionic test cases, suggesting that our findings are characteristic of the combination of PT-MPOs more generally. Published by the American Physical Society 2024

Quasi-radial operators on the Fock space and C*-algebras of entire functions

In this article, we consider quasi-radial operators on the Fock space of holomorphic functions on C n that are square integrable with respect to the Gaussian measure on C n . We provide an integral representation for quasi-radial operators. As a consequence, we study several operator theoretic properties of these operators and the Toeplitz operators with quasi-radial defining symbols. In addition to this, we construct various examples of commutative C ∗ -algebras of analytic functions on C n each associated with a particular class of quasi-radial operators.

Separable Integral Neural Networks

Integral neural networks adopt continuous integral operators instead of conventional discrete convolutional operations to perform deep learning tasks. As this integral operator is the continuous representation of the regular convolutional operation, it is not suitable for representing the separable convolutional operations widely deployed on mobile devices. To address this issue, a separable integral layer composed of a depth-wise integral operator and a point-wise integral operator is proposed in this paper to represent discrete depth-wise and point-wise convolutional operations in continuous manner. According to the fabric units of five classical convolutional neural networks(NIN, VGG11, GoogleNet, ResNet18, ResNet50), we design five kinds of separable integral blocks(SIBs) to encapsulate separable integral layers in different manner. Using the proposed SIBs as basic blocks, a family of lightweight separable integral neural networks(SINNs) are constructed and deployed on resource-constrained mobile devices. SINNs have the characteristics of integral neural networks, i.e., performing structural pruning without fine-tuning, and also inherit the advantages of separable convolutional operations, i.e., reducing the computational cost while keeping a competitive performance. The experimental results show that SINNs achieve the similar performance with the state-of-the-art integral neural networks(INNs), while reducing the computational cost to up to 1/1.79 times that of INN(1.74× fewer parameters than INN using ResNet101 backbone framework) on ImageNet dataset. The code will be released at https://github.com/ljh3832-ccut/SINN.

Quantum Magnetic Skyrmion Operator.

We propose a variational wave function to represent quantum skyrmions as bosonic operators. The operator faithfully reproduces two fundamental features of quantum skyrmions: their classical magnetic order and a "quantum cloud" of local spin-flip excitations. Using exact numerical simulations of the ground states of a 2D chiral magnetic model, we find two regions in the single-skyrmion state diagram distinguished by their leading quantum corrections. We use matrix product state simulations of the adiabatic braiding of two skyrmions to verify that the operator representation of skyrmions is valid at large interskyrmion distances. Our work demonstrates that skyrmions can be approximately coarse-grained and represented by bosonic quasiparticles, which paves the way toward a field theory of many-skyrmion quantum phases and, unlike other approaches, incorporates the microscopic quantum fluctuations of individual skyrmions.

Discrete-type restricted representations of exponential groups and differential operators

Let [Formula: see text] be an exponential solvable Lie group with Lie algebra [Formula: see text], [Formula: see text] an analytic subgroup of [Formula: see text] with Lie algebra [Formula: see text] and [Formula: see text] an irreducible unitary representation of [Formula: see text]. The focus here is on the study of the restriction [Formula: see text] of discrete type, especially the algebra [Formula: see text] of [Formula: see text]-invariant differential operators in the space of [Formula: see text]. Provided that the coadjoint orbit of [Formula: see text] corresponding to [Formula: see text] is of maximal dimension, we diagonalize these operators by means of Penney’s distributions so that the algebra [Formula: see text] turns out to be commutative when [Formula: see text] is of discrete type, and show that the converse may fail to hold as a developed example reveals. Furthermore, we show that [Formula: see text] is isomorphic to the algebra [Formula: see text] of the [Formula: see text]-invariant polynomial functions on [Formula: see text]. We also produce a process to provide some generators of the kernel [Formula: see text] in the enveloping algebra of [Formula: see text].

On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel

Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the α-th power of a differential operator corresponds precisely to the α-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures.

Spent Time of Orbiting Eletron Generating the Quantized Energy En in the Hydrogen Like Atom

Despite the large literature on hydrogen-like atoms, one cannot find any mention as to how long an electron spends in the orbit to generate the quanized radiation energy En. Here, the Heisenberg representation of operators is exteded so as to be able to deal also with imverse oprators. This is necessary when dealing with Coulomb potentials. This then allows to write down the general hydrogen-like atom in a compact form. With the help of operator algebra that includes also the inverse operators, one arrives at the differential equation of the electron trajectory with respect to time.The time solution is achieved through established coupled integral equations. While already established quantized radiation electron energy En is proportional to 1/n2, the newely derived quantized electron circular time tn(e) is proportional to n3 with n=1,2,3,..,and tn(e) being of the order of 10−16 sec in magnitude.

Study on electric towing tractor quantity based on cellular automata

Abstract Aircraft ground taxiing contributes significantly to carbon emissions and engine wear. The electric towing tractor (ETT) addresses these issues by towing the aircraft to the runway end, thereby minimising ground taxiing. As the complexity of ETT towing operations increases, both the towing distance and time increase significantly, and the original method for estimating the number of ETTs is no longer applicable. Due to the substantial acquisition cost of ETT and the need to reduce waste while ensuring operational efficiency, this paper introduces for the first time an ETT quantity estimation model that combines simulation and vehicle scheduling models. The simulation model simulates the impact of ETT on apron operations, taxiing on taxiways and takeoffs and landings on runways. Key timing points for ETT usage by each aircraft are identified through simulation, forming the basis for determining the minimum number of vehicles required for airport operations using a hard-time window vehicle scheduling model. To ensure the validity of the model, simulation model verification is conducted. Furthermore, the study explores the influence of vehicle speed and airport scale on the required number of ETTs. The results demonstrate the effective representation of real-airport operations by the simulation model. ETT speed, airport runway and taxiway configurations, takeoff and landing frequencies and imbalances during peak periods all impact the required quantity of ETTs. A comprehensive approach considering these factors is necessary to determine the optimal number of ETTs.

Wigner state and process tomography on near-term quantum devices

We present an experimental scanning-based tomography approach for near-term quantum devices. The underlying method has previously been introduced in an ensemble-based NMR setting. Here we provide a tutorial-style explanation along with suitable software tools to guide experimentalists in its adaptation to near-term pure-state quantum devices. The approach is based on a Wigner-type representation of quantum states and operators. These representations provide a rich visualization of quantum operators using shapes assembled from a linear combination of spherical harmonics. These shapes (called droplets in the following) can be experimentally tomographed by measuring the expectation values of rotated axial tensor operators. We present an experimental framework for implementing the scanning-based tomography technique for circuit-based quantum computers and showcase results from IBM quantum experience. We also present a method for estimating the density and process matrices from experimentally tomographed Wigner functions (droplets). This tomography approach can be directly implemented using the Python-based software package DROPStomo.

An interior penalty method for a parabolic complementarity problem involving a fractional Black-Scholes operator

In this paper, an interior penalty method is proposed to solve a parabolic complementarity problem involving fractional Black–Scholes operator arising in pricing American options under a geometric Lévy process. The complementarity problem is first reformulated as a fractional partial differential variational inequality problem using the representations of fractional order operators and appropriate mathematical techniques. A penalty equation is then proposed to approximate the variational inequality problem by introducing a novel interior-point based penalty term. The existence and uniqueness of the solution to the penalized problem are proved, and an upper bound on the distance between the solutions to the penalty equation and the variational inequality problem is established. To test our method, we discretize the penalty equation by a finite difference method in space and the Crank–Nicolson method in time. We then present numerical experimental results to demonstrate the usefulness and effectiveness for the interior penalty method.

Quantum computing and tensor networks for laminate design: A novel approach to stacking sequence retrieval

Quantum computing and tensor networks for laminate design: A novel approach to stacking sequence retrieval

An Elliptic Generalization of A1 Spherical DAHA at K = 2

Abstract We construct an algebra that is an elliptic generalization of $A_1$ spherical DAHA acting on its finite-dimensional module at $t=-q^{-K/2}$ with $K=2$. We prove that $PSL(2,\mathbb{Z})$ acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by $3 \times 3$ matrices with matrix elements given by products of elliptic functions. A relation of this construction to the $K$-theory character of affine Laumon space is conjectured. We point out two potential applications, respectively to $SL(3,\mathbb{Z})$ symmetry of Felder–Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds.

Moyal product and generalized Hom-Lie-Virasoro symmetries in Bloch electron systems

Moyal product and generalized Hom-Lie-Virasoro symmetries in Bloch electron systems