Global versus local superintegrability of nonlinear oscillators

Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. Therefore, it seems natural to investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.     

联合分数变换相关器的分数相关特性分析

从分数傅里叶变换（FRFT）的定义出发，理论分析了联合分数变换相关器（JFRTC）的分数相关特性.从所得JFRTC的数学表达式中可以看出，将FRFT应用到联合变换相关器（JTC）中得到的JFRTC具有与传统JTC不同的性质.对于传统JTC，一旦输入平面上参考图像与目标图像之间的距离给定，相关输出峰的位置即确定，而JFRTC的相关输出峰的位置则可以由分数级次p₁和p₂来自由调节，这个特性在实际模式识别中非常有用.另一方面，JFRTC的相关输出峰值在大多数情况下低于传统JTC的相关峰值，却是JFRTC的一大缺点.最后，从FRFT的比例性质出发，给出了FRFT谱畸变不变的实现条件，并由此预言了JFRTC畸变不变模式识别的功能.     

One-Dimensional Traps,Two-Body Interactions,Few-Body Symmetries: I. One,Two, and Three Particles

This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the N particle case.     

One-Dimensional Traps,Two-Body Interactions,Few-Body Symmetries. II. N Particles

This is the second in a pair of articles that classify the configuration space and kinematic symmetry groups for N identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly–unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article analyzes the symmetries of N particles in asymmetric, symmetric, and harmonic traps; the prequel article treats the one, two and three particle cases.     

Fast Car Physics,by Chuck Edmondson

An introduction to the theory of modular symmetries in two-dimensional materials, and its application to ‘relativistic’ group IV materials like graphene, silicene, germanene and stanene, is given. Universal properties of the magneto-electric Hall effect are extracted by projecting experimental transport data directly onto the phase diagram. When families of data depending on the dominant scale parameter (usually temperature) are available, we can extract flow lines that chart the geometry of the phase diagram, including the location of quantum critical points and phase boundaries connecting these. The universal data are used to identify emergent modular symmetries, which are infinite discrete groups of fractional linear (Möbius) transformations. Such symmetries are extremely rigid, and therefore spawn a host of sharp predictions that are easy to falsify, but so far they have failed to fail. The unique topology of the Fermi surface in the graphene family gives a robust gapless mode with linear dispersion (relativistic Dirac cones) that shifts the spectrum of Landau levels that appear when the material is placed in a strong magnetic field. The modular analysis can be extended to this case, and where reliable data are available, there appears to be agreement. A convincing case for the ‘relativistic’ quantum Hall group is hampered by the paucity of fractional quantum Hall data, the absence of scaling data and the crossover between different scaling regimes. This is likely to change in the near future, as scaling data for graphene are just now becoming available.     

n维固体带间光学性质与振子模型 n/2次积分关系

本文证明,n维固体临界点附近的带间介电函数及其微分谱可用n/2次积分分别从振子的介电函数及其微分谱求出。n维固体带间光谱中的振子物理特征通过n/2次积分算符的作用表现出来。文中讨论了带间跃迁与振子模型,光学性质与维度性之间的物理联系。在固体带间光谱的分数次积分表达式中,维度n可以推广到分数维数。 关键词：     

Irreversible quantum graphs

Irreversibility is introduced to quantum graphs by coupling the graphs to a bath of harmonic oscillators. The interaction which is linear in the harmonic oscillator amplitudes is localized at the vertices. It is shown that for sufficiently strong coupling, the spectrum of the system admits a new continuum mode which exists even if the graph is compact, and a single harmonic oscillator is coupled to it. This mechanism is shown to imply that the quantum dynamics is irreversible. Moreover, it demonstrates the surprising result that irreversibility can be introduced by a ‘bath’ which consists of a single harmonic oscillator.     

Riemann–Liouville and Weyl fractional oscillator processes

Two types of oscillator processes can be obtained as solutions to fractional Langevin equation based on Riemann–Liouville and Weyl fractional integro-differential operators. The relation between these fractional oscillator processes and the corresponding fractional Brownian motion is considered. Generalization of the Weyl fractional oscillator process to positive temperature can be carried out and its partition function can be calculated using the zeta function regularization method.     

Nonlinear Dynamical Symmetries of Some Two-Dimensional Quantum Systems

In this paper nonlinear dynamical symmetries of three quantum systems are studied in detail, such as the Kepler-Coulomb system and the isotropic harmonic oscillator in a two-dimensional curved space, and the generalized pseudo-oscillators in the two-dimensional fiat space. Their nonlinear spectrum generating algebras are shown to be relevant to polynomial angular momentum algebras.     

Superconformal deformation,zero-norm states and space-time symmetries of the superstring theories

We present a formalism to study type II and Heterotic superstrings with massless and massive background fields in the bosonic sector. This formalism is appropriate to study high energy symmetries of the superstring. As an example, we explicitly relate all massless symmetries to the massless zero-norm states in the spectrum. This includes theE ₈ ?E ₈ andSO (32) gauge symmetries in the ten-dimensional Heterotic string. The first (evenG-parity) massive level is briefly described. We then argue the existence of new symmetries for the massive Yang-Mills-like gauge bosons and tensor fields at each fixed mass level. These enlarged stringy symmetries correspond to the decoupling of massive zero-norm states in the spectrum.     

Flat directions,string compactification and three generation models

We show how identification of absolutely flat directions allows the construction of a new class of compactified string theories with reduced gauge symmetry that may or may not be continuously connected to the original theory. We use this technique to construct a class of three generation models with just the Standard Model gauge group after compactification. We discuss the low-energy symmetries necessary for a phenomenologically viable low-energy model and construct an example in which these symmetries are identified with string symmetries which remain unbroken down to the supersymmetry breaking scale. Remarkably the same symmetry responsible for stabilising the nucleon is also responsible for ensuring one and only one pair of Higgs doublets is kept light. We show how the string symmetries also lead to textures in the quark and lepton mass matrices which can explain the hierarchy of fermion masses and mixing angles.     

Irreversible quantum graphs

Irreversibility is introduced to quantum graphs by coupling the graphs to a bath of harmonic oscillators. The interaction which is linear in the harmonic oscillator amplitudes is localized at the vertices. It is shown that for sufficiently strong coupling, the spectrum of the system admits a new continuum mode which exists even if the graph is compact, and a single harmonic oscillator is coupled to it. This mechanism is shown to imply that the quantum dynamics is irreversible. Moreover, it demonstrates the surprising result that irreversibility can be introduced by a 'bath' which consists of a single harmonic oscillator.     

Supersymmetric large extra dimensions and the cosmological constant: an update

This article critically reviews the proposal for addressing the cosmological constant problem within the framework of supersymmetric large extra dimensions (SLED), as recently proposed in hep-th/0304256. After a brief restatement of the cosmological constant problem, a short summary of the proposed mechanism is given. The emphasis is on the perspective of the low-energy effective theory in order to see how it addresses the problem of why low-energy particles like the electron do not contribute too large a vacuum energy. This is followed by a discussion of the main objections, which are grouped into the following five topics: (1) Weinberg’s No-Go Theorem. (2) Are hidden tunings of the theory required, and are these stable under renormalization? (3) Why should the mechanism apply only now and not rule out possible earlier epochs of inflationary dynamics? (4) How big are quantum effects, and which are the most dangerous? and (5) Even if successful, can the mechanism be consistent with cosmological or current observational constraints? It is argued that there are plausible reasons why the mechanism can thread the potential objections, but that a definitive proof that it does depends on addressing well-defined technical points. These points include identifying what fixes the size of the extra dimensions, checking how topological obstructions renormalize and performing specific calculations of quantum corrections. More detailed studies of these issues, which are well within reach of our present understanding of extra-dimensional theories, are currently underway. As such, the jury remains out concerning the proposal, although the prospects for acquittal still seem good. (An abridged version of this article appears in the proceedings of SUSY 2003.)     

The role of interplay between the potential and the ambient energies on the vibration energy harvesting

In this paper we have demonstrated how the conversion of the ambient energy into the electrical energy depends on the properties of the ambient energy and the mechanical oscillator. We have observed that the conversion of the vibration energy into the electrical energy may be good if the voltage can follow closely the evolution of of the amplitude of the oscillator. Thus if the voltage and the position fluctuate in a correlated manner then the conversion of the ambient energy into the electrical energy is good. Our another observation is that for a given capacitance, the power transferred (PT) from the oscillator to the transducer may be maximum in the variation of PT with increase in resistance, R. In other words, the power transferred changes with a maximum as the capacitance, C grows for a fixed value of the resistance. Along with these we have investigated how the other relevant quantities such as the efficiency of the energy transferred process depends on the characteristics of the oscillating systems, the environment and the piezoelectric dynamics.     

Measuring the momentum of a nanomechanical oscillator through the use of two tunnel junctions

We propose a way to measure the momentum p of a nanomechanical oscillator. The p detector is based on two tunnel junctions in an Aharonov-Bohm-type setup. One of the tunneling amplitudes depends on the motion of the oscillator, the other one not. Although the coupling between the detector and the oscillator is assumed to be linear in the position x of the oscillator, it turns out that the finite-frequency noise output of the detector will in general contain a term proportional to the momentum spectrum of the oscillator. This is a true quantum phenomenon, which can be realized in practice if the phase of the tunneling amplitude of the detector is tuned by the Aharonov-Bohm flux Phi to a p-sensitive value.     

Higher-Dimensional Integrable Models with a Common Recursion Operator

Starting from any one of hereditary symmetries, we can construct a type of integrable models with arbitrary dimensions. The models with different dimensions obtained from a same hereditary symmetry possess a common recursion operator. The symmetry structures of the models are studied in their potential forms. Using the formal series symmetry approach, we can get various sets of formal series symmetries with some arbitrary functions. Generally, these sets of series symmetries are not truncated for arbitrary functions. The series symmetries wiU all be truncated if the arbitrary functions are fixed as polynomials. Some sets of nontruncated symmetries constitute generalized Virasoro algebras. The more details about the symmetries and algebras are discussed for a concrete (3+1)-dimensional KdV equation.     

第一类Poschl-Teller势的非线性代数结构

在形变李代数理论的基础上,利用哈密顿算符和自然算符,构造出第一类Poschl-Teller势的非线性谱生成代数。该非线性代数能够完全确定势场的能量本征态集合和本征值谱,在适当的非线性算符变换下可以化为谐振子代数,显示了该系统具有新的对称性。     

Lattice Wess-Zumino-Witten model and quantum groups

Quantum groups play the role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as Poisson-Lie symmetries of the corresponding phase spaces. We discuss specifically the Wess-Zumino-Witten conformally invariant quantum field model combining two chiral parts which describe the left- and right-moving degrees of freedom. On one hand, the quantum group plays the role of the symmetry of the chiral components of the theory. On the other hand, the model admits a lattice regularization (in Minkowski space) in which the current algebra symmetry of the theory also becomes quantum, providing the simplest example of a quantum group symmetry coupling space-time and internal degrees of freedom. We develop a free field approach to the representation theory of the lattice sl (2)-based current algebra and show how to use it to rigorously construct an exact solution of the quantum SL (2) WZW model on lattice.     

Spectral Properties of the Renormalization Group at Infinite Temperature

The renormalization group (RG) approach is largely responsible for the considerable success that has been achieved in developing a quantitative theory of phase transitions. Physical properties emerge from spectral properties of the linearization of the RG map at a fixed point. This article considers RG for classical Ising-type lattice systems. The linearization acts on an infinite-dimensional Banach space of interactions. At a trivial fixed point (zero interaction), the spectral properties of the RG linearization can be worked out explicitly, without any approximation. The results are for the RG maps corresponding to decimation and majority rule. They indicate spectrum of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, only residual spectrum. This may serve as a lesson in what one might expect in more general situations.     

Analysis of retroreflector-introduced spectrum distortion in birefringent crystal-based interleaver filters

Spectrum distortion in a birefringent crystal-based interleaver filter introduced by the right-angle total internal reflection retroreflector is studied. The phase shift between two perpendicularly polarized components of the polarized light incident onto the retroreflector is calculated. The sensitivity of the phase shift to the incident angle for different materials from which the retroreflector is made is also considered. This study indicates that the phase shift is less sensitive to the incident angle for retroreflectors having a higher index of refraction, suggesting a fixed-phase compensator that can be used to cancel out the phase shift introduced by the retroreflector to recover the interleaver filter's spectrum. The fold-back 50 GHz channel spacing birefringent crystal interleaver filter is fabricated to demonstrate that the spectrum distortion due to the phase shift introduced by retroreflector can be eliminated by a phase compensator.