CO-ORDINATE TRANSFORMATIONS FOR SECOND ORDER SYSTEMS. PART II: ELEMENTARY STRUCTURE-PRESERVING TRANSFORMATIONS

It has been shown in a previous paper that there is a real-valued transformation from the generalN -degree-of-freedom second order system to a second order system characterized by diagonal matrices. An immediate extension of this fact is that for any second order system, there is a set of real-valued transformations (thestructure-preserving transformations) which transform this system to a different second order system having identical characteristic behaviour. There are several possible reasons why it may be very useful to achieve a particular structure in the transformed system. It is obvious that a diagonal structure is extremely useful and a method has been devised for determining the diagonalizing transformation from the solution of the usual (complex) eigenvalue-eigenvector problem.This paper begins by outlining the usefulness of some other structures. Then it defines a class of elementary structure-preserving co-ordinate transformations that transform from one N -degree-of-freedom second order system to another. The termelementary is applied because any one of these transformations is the minimum-rank modification of the identity transformation. The changes occurring in the system matrices as a result of the application of one such elementary transformation transpire to be very simple in form, they are low rank, and they can be computed very efficiently.This paper provides the fundamental tools to enable the design of structure-preserving co-ordinate transformations which transform a second order system originally characterized by three general matrices in stages into a mathematically similar second order system characterized by three diagonal matrices. The procedure by which the individual elementary transformations are obtained is still under development and it is not discussed in this paper. However, an illustration is given of a five-degree-of-freedom self-adjoint system being transformed into tridiagonal form.     

The decoupling of damped linear systems in configuration and state spaces

It has recently been reported that any viscously damped linear system can be decoupled in the configuration space by a real, nonlinear, time-dependent transformation. The purpose of this rapid communication is to provide a few clarifying remarks about the decoupling operation. It is shown that, for homogeneous systems, the time-dependent configuration-space decoupling transformation is real, linear and time-invariant when cast in state space. In addition, the configuration-space transformation generates a diagonalizing structure-preserving transformation. In non-homogeneous systems, both the configuration and associated state transformations are nonlinear and depend continuously on the excitation. An example is given of a linear system that can be decoupled in configuration but not in state space.     

高斯轨迹参量的线性变换对于电子光学象差的影响

本文从比较普遍的观点出发,采用矢量形式描写高斯轨迹,运用矩阵形式表示象差,从而讨论了普遍的高斯轨迹参量线性变换下的电子光学系统的三级象差和象差系数及其偏微商。由此得到了电子光学广义的Fraunhofer条件(即消除各向同性和各向异性彗差的条件),证明了在此条件下,象散和场曲取极值或稳定值。利用上述的普遍公式和结果,讨论了固定束和扫描束电子光学系统中光阑位置变化的影响,还讨论了阴极发射系统中磁浸没的影响。 关键词：     

A SYSTEMATIC METHOD OF FINDING LINEARIZING TRANSFORMATIONS FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS I: SCALAR CASE

In this paper we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and efficient and helps to unearth several new types of linearizing transformations besides the known ones in the literature. To make our studies systematic we divide our analysis into two parts. In the first part we confine our investigations to the scalar ODEs and in the second part we focus our attention on a system of two coupled second-order ODEs. In the case of scalar ODEs, we consider second and third-order nonlinear ODEs in detail and discuss the method of deriving maximal number of linearizing transformations irrespective of whether it is local or nonlocal type and illustrate the underlying theory with suitable examples. As a by-product of this investigation we unearth a new type of linearizing transformation in third-order nonlinear ODEs. Finally the study is extended to the case of general scalar ODEs. We then move on to the study of two coupled second-order nonlinear ODEs in the next part and show that the algorithm brings out a wide variety of linearization transformations. The extraction of maximal number of linearizing transformations in every case is illustrated with suitable examples.     

FOLDING TRANSFORMATIONS AND HKY MAPPINGS

We present a new method for the derivation of mappings of HKY type. These are second-order mappings which do not have a biquadratic invariant like the QRT mappings, but rather an invariant of degree higher than two in at least one of the variables. Our method is based on folding transformations which exist for some discrete Painlevé equations. They are transformations which relate the variable of a discrete Painlevé equation to the square of the variable of some other one. By considering the autonomous limit of these relations we derive folding-like transformations which relate QRT mappings to HKY ones. We construct the invariants of the latter mappings and show how they can be extended beyond the ones given by the strict application of the folding transformation.     

On the nature of “strong” Bogoliubov transformations for fermions

Properties of the unitary operator implementing a general Bogoliubov transformation on a system of fermions are examined. The nature of this operator (and that of the transformed vacuum state) is exhibited for the case in which the operators appearing in the “diagonal” part of the transformation do not have an inverse. This case turns out to be very different from either the case where the inverses exist or the case with Bogoliubov transformations on systems of bosons. In particular, such transformations can be unitarily implementable in the initial Fock-Hilbert space even though the new vacuum is orthogonal to the initial vacuum. Particles are then created with certainty (i.e. probability 1) in the initial vacuum state and even the new charge operator could be different from the initial one.     

A SYSTEMATIC METHOD OF FINDING LINEARIZING TRANSFORMATIONS FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS II: EXTENSION TO COUPLED ODEs

In this second paper on the method of deriving linearizing transformations for nonlinear ODEs, we extend the method to a set of two coupled second-order nonlinear ODEs. We show that beside the conventional point, Sundman and generalized linearizing transformations one can also find a large class of mixed or hybrid type linearizing transformations like point-Sundman, point-generalized linearizing transformation and Sundman-generalized linearizing transformation in coupled second-order ODEs using the integrals of motion. We propose suitable algorithms to identify all these transformations (with maximal in number) in a straightforward manner. We illustrate the method of deriving each one of the linearizing transformations with a suitable example.     

约束系统的对称变换

对于用非独立态函数描述的受约束系统,在对称变换下将导致推广的Noether定理,一般说,此时不产生Noether定理的守恒量,在有限约束下内部对称变换的推广Noether定理,就化为通常Noether定理的结果。 关键词：     

Computation in finitary stochastic and quantum processes

We introduce stochastic and quantum finite-state transducers as computation-theoretic models of classical stochastic and quantum finitary processes. Formal process languages, representing the distribution over a process’ behaviors, are recognized and generated by suitable specializations. We characterize and compare deterministic and nondeterministic versions, summarizing their relative computational power in a hierarchy of finitary process languages. Quantum finite-state transducers and generators are a first step toward a computation-theoretic analysis of individual, repeatedly measured quantum dynamical systems. They are explored via several physical systems, including an iterated-beam-splitter, an atom in a magnetic field, and atoms in an ion trap—a special case of which implements the Deutsch quantum algorithm. We show that these systems’ behaviors, and so their information processing capacity, depends sensitively on the measurement protocol.     

Critical behavior of RMn2O5 oxides near the magnetic phase transition to a structure incommensurate in two spatial directions

A phase transition from the paramagnetic state to the long-period magnetic structure in RMn₂O₅ oxides with the star of the wave vector determining the incommensurability of long-range magnetic order in two spatial directions has been investigated. An effective Hamiltonian of the system that allows one to describe this transition in the framework of the renormalization group approach has been constructed. It has been shown that there is a stable critical point of transformations of this group at which there occurs a second-order phase transition. The critical indices have been found. The obtained results have been compared with the results for phase transitions occurring in these oxides in accordance with the star of the wave vector, which provides incommensurability in one of the spatial directions. It has been found that fluctuations of the four-component order parameter due to the low spatial symmetry of these compounds do not change the order of the phase transition, which was found in terms of the Landau theory.     

THE EQUIVALENCE OF TWO TRANSFORMATIONS OF THE INVARIANCE OF LAGRANGIAN FUNCTION IN NON-ABELIAN GAUGE FIELD

In this paper,the general transformation of non-abelian gauge field is derived from the in variance reguirement of the effective lagrangian function.It is shown that when the group parameter θ(x) is the product of the anti-commuting ghost field and an infinitesimal number anticommuting ξ which is independent of the component indices space,the B.R.S.transformation is obtained.When θ(x) is equal to the ghost field C(x) times an infinitesimal tensor ξ_i which is commuting with respect to the upper index b,we get another transformation.Both transformations are proved to be equivalent.     

Manifold-informed state vector subset for reduced-order modeling

Reduced-order models (ROMs) for turbulent combustion rely on identifying a small number of parameters that can effectively describe the complexity of reacting flows. With the advent of data-driven approaches, ROMs can be trained on datasets representing the thermo-chemical state-space in simple reacting systems. For low-Mach flows, the full state vector that serves as a training dataset is typically composed of temperature and chemical composition. The dataset is projected onto a lower-dimensional basis and the evolution of the complex system is tracked on a lower-dimensional manifold. This approach allows for substantial reduction of the number of transport equations to solve in combustion simulations, but the quality of the manifold topology is a decisive aspect in successful modeling. To mitigate manifold challenges, several authors advocate reducing the state vector to only a subset of major variables when training ROMs. However, this reduction is often done ad hoc and without giving detailed insights into the effect of removing certain variables on the resulting low-dimensional data projection. In this work, we present a quantitative manifold-informed method for selecting the subset of state variables that minimizes unwanted behaviors in manifold topologies. While many authors in the past have focused on selecting major species, we show that a mixture of major and minor species can be beneficial to improving the quality of low-dimensional data representations. The desired effects include reducing non-uniqueness and spatial gradients in the dependent variable space. Finally, we demonstrate improvements in regressibility of manifolds built from the optimal state vector subset as opposed to the full state vector.     

Role of surface integrals in the Hamiltonian formulation of general relativity

It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincaré transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions.A Hamiltonian formalism which is manifestly covariant under Poincaré transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g_ij, π^ij. In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincaré transformations. In that case all ten generators of the Poincaré group are obtained by inserting the solution of the constraints into corresponding surface integrals.     

Covariant formulation of non-equilibrium statistical thermodynamics

The Fokker Planck equation is considered as the master equation of macroscopic fluctuation theories. The transformation properties of this equation and quantities related to it under general coordinate transformations in phase space are studied. It is argued that all relations expressing physical properties should be manifestly covariant, i.e. independent of the special system of coordinates used. The covariance of the Langevin-equations and the Fokker Planck equation is demonstrated. The diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space. Covariant drift vectors associated to the Langevin- and the Fokker Planck equation are found. It is shown that special coordinates exist in which the covariant drift vector of the Fokker Planck equation and the usual non-covariant drift vector are equal.The physical property of detailed balance and the equivalent potential conditions are given in covariant form. Finally, the covariant formulation is used to study how macroscopic forces couple to a system in a non-equilibrium steady state. A general fluctuation-dissipation theorem for the linear response to such forces is obtained.     

Entanglement transformation between ensembles of three-qubit GHZ-type states by LOCC

We present a novel method to verify whether an ensemble of three-qubit GHZ-type state could be transformed to another one by local operations and classical communication (LOCC). This result highlights intriguing similarity compared with the case in the transformation between two ensembles of two-qubit states. Our approach may provide a splendid insight into the transformations between two general multipartite states.     

Investigation of elastic modes propagating in multi-wire helical waveguides

Elastic guided waves have some potential for non-destructive inspection of civil engineering multi-wire steel cables. However, wave propagation inside such structures is not yet fully understood. This paper investigates multi-wire helical waveguides with special attention to the common seven-wire strand configuration (one straight core surrounded by one layer of six helical wires). A helical coordinate system is first proposed. Though non-orthogonal, this system preserves translational invariance along the helix centreline to explicitly perform a spatial Fourier transform. Then, it is shown that for the analysis of multi-wire helical strands a twisting system—which is a special case of helical systems—is translationally invariant. A semi-analytical finite element method is developed reducing the problem on the cross-section only. A straightforward computation of energy velocity is proposed. Dispersion curves for a single straight wire and a helical wire are first computed to verify the adequacy of the twisting system. Finally the seven-wire strand is analysed using simplified contact conditions. Theoretical dispersion curves are compared to low-frequency magnetostrictive measurements. Good agreement is found for the first compressional-like mode and its associated veering central frequency (‘notch frequency’).     

Grassmann phase space theory and the Jaynes–Cummings model

The Jaynes-Cummings model of a two-level atom in a single mode cavity is of fundamental importance both in quantum optics and in quantum physics generally, involving the interaction of two simple quantum systems—one fermionic system (the TLA), the other bosonic (the cavity mode). Depending on the initial conditions a variety of interesting effects occur, ranging from ongoing oscillations of the atomic population difference at the Rabi frequency when the atom is excited and the cavity is in an n$n$-photon Fock state, to collapses and revivals of these oscillations starting with the atom unexcited and the cavity mode in a coherent state. The observation of revivals for Rydberg atoms in a high-Q microwave cavity is key experimental evidence for quantisation of the EM field. Theoretical treatments of the Jaynes-Cummings model based on expanding the state vector in terms of products of atomic and n$n$-photon states and deriving coupled equations for the amplitudes are a well-known and simple method for determining the effects.     

Phase diagram for the <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) model with defects of “random local field” type and verity of the Imry–Ma theorem

It is shown that the Imry–Ma theorem stating that in space dimensions d < 4 the introduction of an arbitrarily small concentration of defects of the “random local field” type in a system with continuous symmetry of the n-component vector order parameter (O(n) model) leads to long-range order collapse and to the occurrence of a disordered state is not true if the anisotropic distribution of the defect-induced random local field directions in the space of the order parameter gives rise to the effective anisotropy of the “easy axis” type. In the case of a weakly anisotropic field distribution, in space dimensions 2 ≤ d < 4 there exists some critical defect concentration, above which the inhomogeneous Imry–Ma state can exist as an equilibrium one. At a lower defect concentration, long-range order takes place in the system. In the case of a strongly anisotropic field distribution, the Imry–Ma state is suppressed completely and long-range order state takes place at any defect concentration.     

Giant paramagnetism in Li-doped Mo-S nanostructures

Magnetic measurements of Li-doped MoS_2−y nanostructures show a peculiarly large T-independent paramagnetic signal, which cannot be understood in terms of either Curie- or Pauli paramagnetism, or—for that matter—in terms of any simple spin-correlated state, such as antiferromagnetism or superparamagnetic spin clustering. This behaviour appears to be a result of a near-ideal one-dimensional (1D) strongly correlated state, due to the extraordinarily weak inherent coupling between 1D subunits (nanotubes or nanowires), which is an order of magnitude weaker even than in carbon nanotube ropes. In spite of clear evidence for strong electronic correlations from a giant paramagnetic susceptibility, no transition to an ordered state is observed down to very low temperatures and the system appears to be stabilised in a paramagnetic state by fluctuations characteristic of 1D systems. Down to 2 K there is no evidence of a low-temperature quantum critical point.