Arbitrary High Precision Research Articles

Exact solution for wave scattering from black holes: Formulation

AbstractWe establish an exact formulation for wave scattering of a massless field with spin and charge by a Kerr–Newman–de Sitter black hole. Our formulation is based on the exact solution of the Teukolsky equation in terms of the local Heun function, and does not require any approximation. It serves as simple exact formulae with arbitrary high precision, which realize fast calculation without restrictions on model parameters. We highlight several applications including quasinormal modes, cross section, reflection/absorption rate, and Green function.

Robust cooperative learning control for directed networks with nonlinear dynamics

This paper studies a class of robust cooperative learning control problems for directed networks of agents (a) with nonidentical nonlinear dynamics that do not satisfy a global Lipschitz condition and (b) in the presence of switching topologies, initial state shifts and external disturbances. All uncertainties are not only time-varying but also iteration-varying. It is shown that the relative formation of nonlinear agents achieved via cooperative learning can be guaranteed to converge to the desired formation exponentially fast as the number of iterations increases. A necessary and sufficient condition for exponential convergence of the cooperative learning process is that at each time step, the network topology graph of nonlinear agents can be rendered quasi-strongly connected through switching along the iteration axis. Simulation tests illustrate the effectiveness of our proposed cooperative learning results in refining arbitrary high precision relative formation of nonlinear agents.

High‐precision formation control of nonlinear multi‐agent systems with switching topologies: A learning approach

SummaryArbitrary high precision is considered one of the most desirable control objectives in the relative formation for many networked industrial applications, such as flying spacecrafts and mobile robots. The main purpose of this paper is to present design guidelines of applying the iterative schemes to develop distributed formation algorithms in order to achieve this control objective. If certain conditions are met, then the control input signals can be learned by the developed algorithms to accomplish the desired formations with arbitrary high precision. The systems under consideration are a class of multi‐agent systems under directed networks with switching topologies. The agents have discrete‐time affine nonlinear dynamics, but their state functions do not need to be identical. It is shown that the learning processes resulting from the relative output formation of multi‐agent systems can converge exponentially fast with the increase of the iteration number. In particular, this work induces a distributed algorithm that can simultaneously achieve the desired relative output formation between agents and regulate the movement of multi‐agent formations as desired along the time axis. The illustrative numerical simulations are finally performed to demonstrate the effectiveness and performance of the proposed distributed formation algorithms. Copyright © 2014 John Wiley & Sons, Ltd.

Adaptive and Phase Selective Spike Timing Dependent Plasticity in Synaptically Coupled Neuronal Oscillators

We consider and analyze the influence of spike-timing dependent plasticity (STDP) on homeostatic states in synaptically coupled neuronal oscillators. In contrast to conventional models of STDP in which spike-timing affects weights of synaptic connections, we consider a model of STDP in which the time lags between pre- and/or post-synaptic spikes change internal state of pre- and/or post-synaptic neurons respectively. The analysis reveals that STDP processes of this type, modeled by a single ordinary differential equation, may ensure efficient, yet coarse, phase-locking of spikes in the system to a given reference phase. Precision of the phase locking, i.e. the amplitude of relative phase deviations from the reference, depends on the values of natural frequencies of oscillators and, additionally, on parameters of the STDP law. These deviations can be optimized by appropriate tuning of gains (i.e. sensitivity to spike-timing mismatches) of the STDP mechanism. However, as we demonstrate, such deviations can not be made arbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights. Thus if accurate phase-locking in the system is required then an additional tuning mechanism is generally needed. We found that adding a very simple adaptation dynamics in the form of slow fluctuations of the base line in the STDP mechanism enables accurate phase tuning in the system with arbitrary high precision. Adaptation operating at a slow time scale may be associated with extracellular matter such as matrix and glia. Thus the findings may suggest a possible role of the latter in regulating synaptic transmission in neuronal circuits.

Precise Self-tuning of Spiking Patterns in Coupled Neuronal Oscillators

In this work we discuss and analyze spiking patterns in a generic mathematical model of two coupled non-identical nonlinear oscillators supplied with a spike-timing dependent plasticity (STDP) mechanism. Spiking patterns in the system are shown to converge to a phase-locked state in a broad range of parameters. Precision of the phase locking, i.e. the amplitude of relative phase deviations from a given reference, depends on the natural frequencies of oscillators and, additionally, on parameters of the STDP law. These deviations can be optimized by appropriate tuning of gains (i.e. sensitivity to spike-timing mismatches) of the STDP mechanisms. The deviations, however, can not be made arbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights. Thus if accurate phase-locking in the system is required then an additional tuning mechanism is generally needed. We found that adding a very simple adaptation dynamics in the form of slow fluctuations of the base line in the STDP mechanism enables accurate phase tuning in the system with arbitrary high precision. The scheme applies to systems in which individual oscillators operate in the oscillatory mode. If the dynamics of oscillators becomes bistable then relative phase may fail to converge to a given value giving rise to the emergence of complex spiking sequences.

Selection of the learning gain matrix of an iterative learning control algorithm in presence of measurement noise

Arbitrary high precision output tracking is one of the most desirable control objectives found in industrial applications regardless of measurement errors. The main purpose of this paper is to supply to the iterative learning control (ILC) designer guidelines to select the corresponding learning gain in order to achieve this control objective. For example, if certain conditions are met, then it is necessary for the learning gain to converge to zero in the learning iterative domain. In particular, this paper presents necessary and sufficient conditions for boundedness of trajectories and uniform tracking in presence of measurement noise and a class of random reinitialization errors for a simple ILC algorithm. The system under consideration is a class of discrete-time affine nonlinear systems with arbitrary relative degree and arbitrary number of system inputs and outputs. The state function does not need to satisfy a Lipschitz condition. This work also provides a recursive algorithm that generates the appropriate learning gain functions that meet the arbitrary high precision output tracking objective. The resulting tracking output error is shown to converge to zero at a rate inversely proportional to square root of the number of learning iterations in presence of measurement noise and a class of reinitialization errors. Two illustrative numerical examples are presented.

Development of an automatic test equipment for nano gauging displacement transducers

In order to satisfy the increasing demands on the precision in manufacturing technology, nanaometrology gradually becomes more important in manufacturing process. To ensure the precision of manufacture, precise measuring instruments and sensors play a decesive role for the accurate characterization and inspection of products. For linear length inspection, high precision gauging displacement transducers, i.e. nano gauging displacement transducers (NGDT), have been often utilized, which have been often utilized, which have the resolution in the nanometer range and can achieve an accuracy of less than 100 nm. Such measurement instruments include transducers based on electronic as well as optical measurement principles, e.g. inductive, incremental-optical or interference optical. To guarantee the accuracy and the traceability to the definition of the meter, calibration and test of NGDT are essential. Currently, there are some methods and machines for test of NGDT, but they suffer from various disadvantages. Some of them permit only manual test procedures which are time-consuming, e.g. with high accurate gauge blocks as material measures. Other tests can reach higher accuracy only in the micrometer range or result in uncertainties of more than 100 nm in the large measuring ranges. To realize the test of NGDT with a high resolution as well as a large measuring range, an automatic test equipment was constructed, that has a resolution of 1.24 nm, a measuring range of up to 20 nm (60 mm) and a measuring uncertainty of approximate ±10 nm can fulfil the requirements of high resolution within the nanometer range while simultaneously covering a large measuring range in the order of millimeters. The test system includes a stable frame, a polarization interferometer, an angle sensor, an angular control, a drive system and piezo translators. During the test procedure, the angular control and piezo translators minimize the Abbe error. For the automation of the test procedure a measuring program adhering to the measurement principle outlined in VDI/VDE 2617 guidelines was designed. With this program NGDT can be tested in less than thirty minutes with eleven measuring points and five repetitions. By mean of theoretical and experimental investigations it can be proved that the automatic test system achieves a test uncertainty of approx. ±10 nm at the measuring range of 18 mm, that corresponds to a relative uncertainty of approximately ±5 × 10−7. With small uncertainty, the minimization of the Abbe error and short test time, this system can be regarded as a universal and efficient precision test equipment, which is available for the accurate test of arbitrary high precision gauging displacement transducers.

How to compute one-loop Feynman diagrams in lattice QCD with Wilson fermions

We describe an algebraic algorithm which allows to express every one-loop lattice integral with gluon or Wilson-fermion propagators in terms of a small number of basic constants which can be computed with arbitrary high precision. Although the presentation is restricted to four dimensions the technique can be generalized to every space dimension. We also give a method to express the lattice free propagator for Wilson fermions in coordinate space as a linear function of its values in eight points near the origin. This is an essential step in order to apply the recent methods of Lüscher and Weisz to higher-loop integrals with fermions.

Learning Temporally Encoded Patterns in Networks of SpikingNeurons

Networks of spiking neurons are very powerful and versatile models for biological and artificial information processing systems. Especially for modelling pattern analysis tasks in a biologically plausible way that require short response times with high precision they seem to be more appropriate than networks of threshold gates or models that encode analog values in average firing rates. We investigate the question how neurons can learn on the basis of time differences between firing times. In particular, we provide learning rules of the Hebbian type in terms of single spiking events of the pre- and postsynaptic neuron and show that the weights approach some value given by the difference between pre- and postsynaptic firing times with arbitrary high precision.

Algebraic algorithm for the computation of one-loop Feynman diagrams in lattice QCD with Wilson fermions

We describe an algebraic algorithm which allows us to express every one-loop lattice integral with gluon or Wilson-fermion propagators in terms of a small number of basic constants which can be computed with arbitrary high precision. Although the presentation is restricted to four dimensions the technique can be generalized to every space dimension. Various examples are given, including the one-loop self-energies of the quarks and gluons and the renormalization constants for some dimension-three and dimension-four lattice operators. We also give a method to express the lattice free propagator for Wilson fermions in coordinate space as a linear function of its values in eight points near the origin. This is an essential step in order to apply the recent methods of Lüscher and Weisz to higher-loop integrals with fermions.

Effective Noether Irreducibility Forms and Applications

Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polynomial of a certain degree is absolutely irreducible, if and only if all the corresponding irreducibility forms vanish when evaluated at the coefficients of the specific polynomial. Our forms have much smaller degrees and coefficients than the forms derived originally by Emmy Noether. We can also apply our estimates to derive more effective versions of irreducibility theorems by Ostrowski and Deuring and of the Hilbert irreducibility theorem. We also give an effective estimate on the diameter of the neighborhood of an absolutely irreducible polynomial with respect to the coefficient space in which absolute irreducibility is preserved. Furthermore, we can apply the effective estimates to derive several factorization results in parallel computational complexity theory: we show how to compute arbitrary high precision approximations of the complex factors of a multivariate integral polynomial and how to count the number of absolutely irreducible factors of a multivariate polynomial with coefficients in a rational function field, both in the complexity class NC. The factorization results also extend to the case where the coefficient field is a function field.