The modal interpretation of algebraic quantum field theory

Recently, the robot with many links and degrees of freedoms is actively produced. Generally, the robot that has many degrees of freedoms owns many redundant joints. So we make a suggestion for using redundant joints effectively. The paper proposes using redundant joint as a counter weight. We make a trajectory in momentum space, and we use Fourier Basis Algorism as the method for an optimization. This trajectory generating method has an effect on reducing a driving energy for throwing motion.

In this thesis, we present a dynamical systems approach to adaptive controllers for locomotion control. The approach is based on a rigorous mathematical framework using nonlinear dynamical systems and is inspired by theories of self-organization. Nonlinear dynamical systems such as coupled oscillators are an interesting approach for the on-line generation of trajectories for robots with many degrees of freedom (e.g. legged locomotion). However, designing a nonlinear dynamical system to satisfy a given specification and goal is not an easy task, and, hitherto no methodology exists to approach this problem in a unified way. Nature presents us with satisfactory solutions for the coordination of many degrees of freedom. One central feature observed in biological subjects is the ability of the neural systems to exploit natural dynamics of the body to achieve efficient locomotion. In order to be able to exploit the body properties, adaptive mechanisms must be at work. Recent work has pointed out the importance of the mechanical system for efficient locomotion. Even more interestingly, such well suited mechanical systems do not need complicated control. Yet, in robotics, in most approaches, adaptive mechanisms are either missing or they are not based on a rigorous framework, i.e. they are based on heuristics and ad-hoc approaches. Over the last three decades there has been enormous progress in describing movement coordination with the help of Synergetic approaches. This has led to the formulation of a theoretical framework: the theory of dynamic patterns. This framework is mathematically rigorous and at the same time fully operational. However, it does not provide any guidelines for synthetic approaches as needed for the engineering of robots with many degrees of freedom, nor does it directly help to explain adaptive systems. We will show how we can extend the theoretical framework to build adaptive systems. For this purpose, we propose the use of multi-scale dynamical systems. The basic idea behind multi-scale dynamical systems is that a given dynamical system gets extended by additional slow dynamics of its parameters, i.e. some of the parameters become state variables. The advantages of the framework of multi-scale dynamical systems for adaptive controllers are 1) fully dynamic description, 2) no separation of learning algorithm and learning substrate, 3) no separation of learning trials or time windows, 4) mathematically rigorous, 5) low dimensional systems. However, in order to fully exploit the framework important questions have to be solved. Most importantly, methodologies for designing the feedback loops have to be found and important theoretical questions about stability and convergence properties of the devised systems have to be answered. In order to tackle this challenge, we first introduce an engineering view on designing nonlinear dynamical systems and especially oscillators. We will highlight the important differences and freedom that this engineering view introduces as opposed to a modeling one. We then apply this approach by first proposing a very simple adaptive toy-system, consisting of a dynamical system coupled to a spring-mass system. Due to its spring-mass dynamics, this system contains clear natural dynamics in the form of resonant frequencies. We propose a prototype adaptive multi-scale system, the adaptive frequency oscillator, which is able to adapt its intrinsic frequency to the resonant frequency of the body dynamics. After a small sidetrack to show that we can use adaptive frequency oscillators also for other applications than for adaptive controllers, namely for frequency analysis, we then come back to further investigation of the adaptive controller. We apply the same controller concept to a simple spring-mass hopper system. The spring-mass system consists of a body with two legs attached by rotational joints. The legs contain spring-damper elements. Finally, we present results of the implementation of the controller on a real robot, the experimental robot PUPPY II. This robot is a under-actuated robot with spring dynamics in the knee joints. It will be shown, that due to the appropriate simplification and concentration on relevant features in the toy-system the controller concepts works without a fundamental change on all systems from the toy system up to the real robot.

Fluctuations in the Dynamics of Single Quantum Systems

We consider the problem of potential interaction between a finite-dimensional linear Lagrangian system and an infinite-dimensional one (a system of linear oscillators and a thermostat). We study the final dynamics of the system. Under natural assumptions, this dynamics turns out to be very simple and admits an explicit description because the thermostat produces an effective dissipation despite the energy conservation and the Lagrangian nature of the system. We use the methods of [1], where the final dynamics of the finite-dimensional subsystem is studied in the case when it has one degree of freedom and a linear potential or (under additional assumptions) polynomial potential. We consider the case of finite-dimensional subsystems with arbitrarily many degrees of freedom and a linear potential and study the final dynamics of the system of oscillators and the thermostat. The necessary assertions from [1] are given with proofs adapted to the present situation.

We address the impossibility of achieving exact time reversal in a system with many degrees of freedom. This is a particular example of the difficult task of “aiming” an initial classical state so as to become a specific final state. We also comment on the classical-to-quantum transition in any non-separable closed system of \(n \ge 2\) degrees of freedom. Even if the system is initially in a well-defined WKB, semi-classical state, quantum evolution and, in particular, multiple reflections at classical turning points make it completely quantum mechanical with each particle smeared almost uniformly over all the configuration space. The argument, which is presented in the context of \(n\) hard discs, is quite general. Finally, we briefly address more complex quantum systems with many degrees of freedom and ask when can they provide an appropriate environment to the above simpler systems so that quantum spreading is avoided by continuously leaving “imprints” in the environment. We also discuss the possible connections with the pointer systems that are needed in the quantum-to-classical “collapse” transitions.

A methodology to capture and post-process bat flight 3-D Stereo Triangulation data to formulate an approximated rigid body kinematic model was investigated. Bat flight is unique in nature due to the bats inherent agility and many degrees of freedom when compared to other flying animals. This complexity makes capturing accurate aerodynamic data very difficult. Unlike insects, which utilize few degrees of freedom and a high flap frequency for sustained flight and maneuverability, the agility of bats comes in part from the many degrees of freedom present in the bat wing. In order to better understand the aerodynamics present in bat flight, bats Hipposideridae (Old World leaf-nosed bats) were examined. The trajectories of critical points along the bat wings were recorded using 3D stereo triangulation techniques to capture the complexities of the bat flight. Markers were placed at all the joint locations along the bat wing. The resulting trajectories were then translated into a periodic kinematic model for future computational use.

Evaluation of the absolute configurational entropy of tri-O-methyl-β-cyclodextrin, a molecule with many degrees of freedom

The standard C*-formalism of representations of commutations relations for systems of infinitely many degrees of freedom over a real numbers field is extended to the case of a p-adic numbers field. It is proved that any positive function on the Weyl algebra over a p-adic symplectic space is regular and restriction of the generating function of a state of the Weyl algebra on any nongenerate finite-dimensional subspace of p-adic symplectic space is continuous. Some nontrivial class of nonequivalent representations of commutations relations is described and analogies of Fock representations and representations with number operators are constructed. The criteria for the existence of free-field unitary quantization of symplectic transformations is given.

A fast algorithm is presented that can handle the motion planning problem for articulated figures with branches and many degrees of freedom. The algorithm breaks down the degrees of freedom of the figure into Cspace groups and compute the free motion for each of these groups in a sequential fashion. It traverses the tree in a depth first order to compute the motion for all the branches. A special playback routine is then used to traverse the tree again in a reverse order to playback the final motion. The planner runs in linear time with respect to the total number of Cspace groups without backtracking. We believe that the planner would find a path in most cases and is fast enough for practical use in a wide range of applications.

Many real-world systems are often regarded as weakly coupled limit-cycle oscillators, in which each oscillator corresponds to a dynamical system with many degrees of freedom that have collective oscillations. One of the most practical methods for investigating the synchronization properties of such a rhythmic system is to statistically extract phase coupling functions between limit-cycle oscillators directly from observed time-series data. In Particular, using a method that combines phase reduction theory and Bayesian inference, the phase coupling functions can be extracted from the time-series data of even just one variable in each oscillatory dynamical system with many degrees of freedom. However, it remains unclear how the choice of the observed variables affects the statistical inference for the phase coupling functions. In this study, we examine the influence of observed variable types on the extraction of phase coupling functions using some typical dynamical elements under various conditions. We demonstrate that our method can consistently extract the macroscopic phase coupling functions between two phases representing collective oscillations in a fully locked state, regardless of the observed variable types; for example, even using one variable of any element in one system and the mean-field value over all the elements in another system. We also study the case of globally coupled phase oscillators in a partially locked state. Our results reveal directional asymmetry in the robustness of extracting the macroscopic phase coupling function between two networks. For instance, when an asynchronous oscillator in network $A$ and the macroscopic collective oscillation of network $B$ is observed, the macroscopic phase coupling function from network $A$ to network $B$ can be extracted more robustly than in the opposite direction.

The method of multiple scales is used to analyze parametrically excited, linear systems having many degrees of freedom and distinct natural frequencies. Explicit second-order expressions are obtained for the characteristic exponents which yield the transition curves. Various combination resonances are treated. The results are applied to the buckling of free-clamped columns under the influence of sinusoidally varying axial loads.

From metastable to stable states in a magnetic system with many degrees of freedom

In this paper, we address the autonomous control of a 3-dimensional snake-like robot by using reinforcement learning, and we apply it in the case of rubble. In general, snake-like robots have high mobility that is realized by many degrees of freedom, and they can move on rubble. However, the many degrees of freedom cause the state explosion problem, and the complexity of the rubble results in incomplete learning. Therefore, it is impossible to apply reinforcement learning to conventional snake-like robots that move on rubble. In this paper, to solve these problems, we focus on properties of the real environment and the dynamics of a mechanical body. We design the body of the robot for abstracting the necessary small state-action space by considering real-world properties, and we make it possible to apply reinforcement learning. To demonstrate the effectiveness of the proposed snake-like robot, we conducted experiments where learning was completed within reasonable time and the robot effectively adapted itself to an unknown 3-dimensional environment.

Thermal states of quantum systems with many degrees of freedom are subject to a bound on the rate of onset of chaos, including a bound on the Lyapunov exponent, $\lambda_L\leq 2\pi /\beta$. We harness this bound to constrain the space of putative holographic CFTs and their would-be dual theories of AdS gravity. First, by studying out-of-time-order four-point functions, we discuss how $\lambda_L=2\pi/\beta$ in ordinary two-dimensional holographic CFTs is related to properties of the OPE at strong coupling. We then rule out the existence of unitary, sparse two-dimensional CFTs with large central charge and a set of higher spin currents of bounded spin; this implies the inconsistency of weakly coupled AdS$_3$ higher spin gravities without infinite towers of gauge fields, such as the $SL(N)$ theories. This fits naturally with the structure of higher-dimensional gravity, where finite towers of higher spin fields lead to acausality. On the other hand, unitary CFTs with classical $W_{\infty}[\lambda]$ symmetry, dual to 3D Vasiliev or hs[$\lambda$] higher spin gravities, do not violate the chaos bound, instead exhibiting no chaos: $\lambda_L=0$. Independently, we show that such theories violate unitarity for $|\lambda|>2$. These results encourage a tensionless string theory interpretation of the 3D Vasiliev theory. We also perform some CFT calculations of chaos in Rindler space in various dimensions.

Interactively creating and editing 3D content requires the manipulation of many degrees of freedom (DoF). For instance, docking a virtual object involves 6 DoF (position and orientation). Multi-touch surfaces are good candidates as input devices for those interactions: they provide a direct manipulation where each finger contact on the table controls 2 DoF. This leads to a theoretical upper bound of 10 DoF for a single-handed interaction. With a new hand parameterization, we investigate the number of DoF that one hand can effectively control on a multi-touch surface. A first experiment shows that the dominant hand is able to perform movements that can be parameterized by 4 to 6 DoF, and no more (i.e., at most 3 fingers can be controlled independently). Through another experiment, we analyze how gestures and tasks are associated, which enable us to discover some principles for designing 3D interactions on tabletop.