Invariant operator theory for the single-photon energy in time-varying media

After the birth of quantum mechanics, the notion in physics that the frequency of light is the only factor that determines the energy of a single photon has played a fundamental role. However, under the assumption that the theory of Lewis-Riesenfeld invariants is applicable in quantum optics, it is shown in the present work that this widely accepted notion is valid only for light described by a time-independent Hamiltonian, i.e., for light in media satisfying the conditions, ε（t） = ε（0）, u（t） = u（0）, and σ（t） = 0 simultaneously. The use of the Lewis Riesenfeld invariant operator method in quantum optics leads to a marvelous result： the energy of a single photon propagating through time-varying linear media exhibits nontrivial time dependence without a change of frequency.

The Construction and Analysis of Linear Ring Spaces

This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi-linearity is discussed, in addition to the examination of properties and classifications of linear ring spaces. Particularly, the ring of holomorphic functions on a region of the complex plane is examined, and the manner in which it generates an iterated linear ring space under the complex derivative operator. This notion is then generalized to all rings with nth order linear and surjective operators. Basic operator theory regarding the classifications of linear ring maps is also covered.

Operator-Based Robust Nonlinear Free Vibration Control of a Flexible Plate With Unknown Input Nonlinearity

In this paper,a robust nonlinear free vibration control design using an operator based robust right coprime factorization approach is considered for a flexible plate with unknown input nonlinearity.With considering the effect of unknown input nonlinearity from the piezoelectric actuator,operator based controllers are designed to guarantee the robust stability of the nonlinear free vibration control system.Simultaneously,for ensuring the desired tracking performance and reducing the effect of unknown input nonlinearity,operator based tracking compensator and estimation structure are given,respectively.Finally,both simulation and experimental results are shown to verify the effectiveness of the proposed control scheme.

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU（1,1） algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld＇s invariant operator theory of treating timedependent harmonic oscillators.

Introducing the general condition for an operator in curved space to be unitary

We investigate the general condition for an operator to be unitary.This condition is introduced according to the definition of the position operator in curved space.In a particular case,we discuss the concept of translation operator in curved space followed by its relation with an anti-Hermitian generator.Also we introduce a universal formula for adjoint of an arbitrary linear operator.Our procedure in this paper is totally different from others,as we explore a general approach based only on the algebra of the operators.Our approach is only discussed for the translation operators in one-dimensional space and not for general operators.

Data-driven Static Equivalence with Physics-informed Koopman Operators

With deployment of measurement units,fitting static equivalent models of distribution networks(DNs)by linear regression has been recognized as an effective method in power flow analysis of a transmission network.Increasing volatility of measurements caused by variable distributed renewable energy sources makes it more difficult to accurately fit such equivalent models.To tackle this challenge,this letter proposes a novel data-driven method to improve equivalency accuracy of DNs with distributed energy resources.This letter provides a new perspective that an equivalent model can be regarded as a mapping from internal conditions and border voltages to border power injections.Such mapping can be established through 1)Koopman operator theory,and 2)physical features of power flow equations at the root node of a DN.Performance of the proposed method is demonstrated on the IEEE 33-bus and IEEE 136-bus test systems connected to a 661-bus utility system.

Quantum entangled fractional Fourier transform based on the IWOP technique

In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

ON THE SPH-DISTRIBUTION CLASS

Following up Neuts' idea, the SPH-distribution class associated with bounded Q matrices for infinite Markov chains is denned. The main result in this paper is to characterize the SPH class through the derivatives of the distribution functions. Based on the characterization theorem, closure properties, the expansion, uniform approximation, and the matrix representations of the SPH class are also discussed by the derivatives of the distribution functions at origin.

DCS devices based non-linear process control system design for plants with distributed time-delay using particle filter

Remote control process system with distributed time-delay has attracted much attention in different fields.In this paper,non-linear remote control of a single tank process system with wireless network is considered.To deal with the distributed time-delay in a large-scale plant,the time-delay compensation controller based on DCS devices is designed by using operator theory and particle filter.Distributed control system(DCS)device is developed to monitor and control from the central monitoring room to each process.The particle filter is a probabilistic method to estimate unobservable information from observable information.First,remote control system and experimental equipment are introduced.Second,control system based on an operator theory is designed.Then,process system with distributed time-delay using particle filter is carried out.Finally,the actual experiment is conducted by using the proposed time-delay compensation controller.When estimating with the proposed method,the result is close to the case in which the distributed time-delay does not exist.The effectiveness of the proposed control system is confirmed by experiment results.

A rigorous criterion to identify the validity of the Born approximation

This paper intends to identify the validity of the orn approximation by a new universal criterion, which is ultimately reduced to the calculation of an operator norm. With the purpose of enabling the criterion to be applicable to general scattering problems, a method is proposed to estimate the norm of the operator concerned. Compared with the conventional criterion, this method excels in its ability to acquire a quantificational upper bound of the relative error by Born approximation as well as to extend its valid frequency to a wider range. Two canonical scattering examples are given as evidence for the validity of the criterion.

On Kadison’s Similarity Problem for Homomorphisms of the Algebra of Complex Polynomials

We prove that every homomorphism of the algebra Pn into the algebra of operators on a Hilbert space is completely bounded. We show that the contractive homomorphism introduced by Parrott, which is not completely contractive, is completely bounded (similar to a completely contractive homomorphism). We also show that homomorphisms of the algebra Pn generate completely positive maps over the algebras C(Tn)and M2(C(Tn)).

Spectral properties and geometric interpretation of R-filters

By applying the Fourier analysis, we study the spectral properties of R- filters. Further, we prove that R-filters are a generalization of least squares polynomial adjustment, and we give the geometric interpretation of R-filters.

On Von Neumann’s Inequality for Matrices of Complex Polynomials

We prove that every matrix F∈Mk (Pn) is associated with the smallest positive integer d (F)≠1 such that d (F)‖F‖∞ is always bigger than the sum of the operator norms of the Fourier coefficients of F. We establish some inequalities for matrices of complex polynomials. In application, we show that von Neumann’s inequality holds up to the constant 2n for matrices of the algebra Mk (Pn).

Model of tunnelling through periodic array of quantum dots in a magnetic field

A two-dimensional periodic array of quantum dots with two laterally coupled leads in a magnetic field is considered. The model of electron transport through the system based on the theory of self-adjoint extensions of symmetric operators is suggested. We obtain the formula for the transmission coefficient and investigate its dependence on the magnetic field.

Knotted Picture of Single Qubit Quantum Logic Gate

This paper briefly introduces the five types of the surgical operations in knot theory and obtains the expression of single qubit quantum logic gate in terms of these surgical operations.

Knotted Pictures of Hadamard Gate and CNOT Gate

This paper obtains the knotted pictures of Hadamard gate and CNOT gate in terms of surgical operations described in knot theory.

On a Boundary Value Problem for a Polynomial Pencil of the Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions

The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.

Asset allocation： Evidence From China

As a key part of a corporate＇s operation, Asset allocation is critical to its survival and development This paper uses Markowitz financial security portfolio theory on corporate＇s asset allocation, to derive the optimal asset allocation for an corporate in China through case study.

An Operator-based Nonlinear Vibration Control System Using a Flexible Arm with Shape Memory Alloy

In the past,arms used in the fields of industry and robotics have been designed not to vibrate by increasing their mass and stiffness.The weight of arms has tended to be reduced to improve speed of operation,and decrease the cost of their production.Since the weight saving makes the arms lose their stiffness and therefore vibrate more easily,the vibration suppression control is needed for realizing the above purpose.Incidentally,the use of various smart materials in actuators has grown.In particular,a shape memory alloy(SMA)is applied widely and has several advantages:light weight,large displacement by temperature change,and large force to mass ratio.However,the SMA actuators possess hysteresis nonlinearity between their own temperature and displacement obtained by the temperature.The hysteretic behavior of the SMA actuators affects their control performance.In previous research,an operator-based control system including a hysteresis compensator has been proposed.The vibration of a flexible arm is dealt with as the controlled object;one end of the arm is clamped and the other end is free.The effectiveness of the hysteresis compensator has been confirmed by simulations and experiments.Nevertheless,the feedback signal of the previous designed system has increased exponentially.It is difficult to use the system in the long-term because of the phenomenon.Additionally,the SMA actuator generates and radiates heat because electric current passing through the SMA actuator provides heat,and strain on the SMA actuator is generated.With long-time use of the SMA actuator,the environmental temperature around the SMA actuator varies through radiation of the heat.There exists a risk that the ambient temperature change dealt with as disturbance affects the temperature and strain of the SMA actuator.In this research,a design method of the operator-based control system is proposed considering the long-term use of the system.In the method,the hysteresis characteristics of the SMA actuator and the temperature change around the actuator are considered.The effectiveness of the proposed method is verified by simulations and experiments.

Time Operator in Relativistic Quantum Mechanics

It is first shown that the Dirac＇s equation in a relativistic frame could be modified to allow discrete time, in agreement to a recently published upper bound. Next, an exact self-adjoint 4 x 4 relativistic time operator for spin-l/2 particles is found and the time eigenstates for the non-relativistic case are obtained and discussed. Results confirm the quantum mechanical speculation that particles can indeed occupy negative energy levels with vanishingly sma/l but non- zero probablity, contrary to the general expectation from classical physics. Hence, Wolfgang Pauli＇s objection regarding the existence of a self-adjoint time operator is fully resolved. It is shown that using the time operator, a bosonic field referred here to as energons may be created, whose number state representations in non-relativistic momentum space can be explicitly found.