Iteration Of Map Research Articles

Reduction of cluster iteration maps

We study iteration maps of difference equations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and presymplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix representing the quiver is singular. The reduced iteration map is explicitly computed for several periodic quivers using either the presymplectic reduction or a Poisson reduction via log-canonical Poisson structures.

Time irreversibility analysis of converter based on symbolic dynamics

A new method based on symbobic dynamics and relative entropy theory is proposed to examine the nonlinear behaviours of converters. Firstly, the discrete numerical sequence is obtained from iteration map, which is then transferred to a symbol-time series according to the topological conjugation, and the relative entropy is calculated by means of forward and backward probabilities. This paper takes a first one-order voltage feedback DCM Boost converter as an example, and the result shows that the relative entropy, which can measure quantitatively the distance apart from equilibrium when converter lies in a chaotic state, is a new and quantified nonlinear dynamic behaviours which has not been used in converters yet.

A Class of Fejér Convergent Algorithms, Approximate Resolvents and the Hybrid Proximal-Extragradient Method

A new framework for analyzing Fejer convergent algorithms is presented. Using this framework, we define a very general class of Fejer convergent algorithms and establish its convergence properties. We also introduce a new definition of approximations of resolvents, which preserves some useful features of the exact resolvent and use this concept to present an unifying view of the Forward-Backward splitting method, Tseng’s Modified Forward-Backward splitting method, and Korpelevich’s method. We show that methods, based on families of approximate resolvents, fall within the aforementioned class of Fejer convergent methods. We prove that such approximate resolvents are the iteration maps of the Hybrid Proximal-Extragradient method, which is a generalization of the classical Proximal Point Algorithm.

Global attractor alphabet of neural firing modes

The elementary set, or alphabet, of neural firing modes is derived from the widely accepted conductance-based rectified firing-rate model. The firing dynamics of interacting neurons are shown to be governed by a multidimensional bilinear threshold discrete iteration map. The parameter-dependent global attractors of the map morph into 12 attractor types. Consistent with the dynamic modes observed in biological neuronal firing, the global attractor alphabet is highly visual and intuitive in the scalar, single-neuron case. As synapse permeability varies from high depression to high potentiation, the global attractor type varies from chaotic to multiplexed, oscillatory, fixed, and saturated. As membrane permeability decreases, the global attractor transforms from active to passive state. Under the same activation, learning and retrieval end at the same global attractor. The bilinear threshold structure of the multidimensional map associated with interacting neurons generalizes the global attractor alphabet of neuronal firing modes to multineuron systems. Selective positive or negative activation and neural interaction yield combinatorial revelation and concealment of stored neuronal global attractors.

Qutrit magic state distillation

Magic state distillation (MSD) is a purification protocol that plays an important role in fault-tolerant quantum computation. Repeated iteration of the steps of an MSD protocol generates pure single non-stabilizer states, or magic states, from multiple copies of a mixed resource state using stabilizer operations only. Thus mixed resource states promote the stabilizer operations to full universality. MSD was introduced for qubit-based quantum computation, but little has been known concerning MSD in higher-dimensional qudit-based computation. Here, we describe a general approach for studying MSD in higher dimensions. We use it to investigate the features of a qutrit MSD protocol based on the five-qutrit stabilizer code. We show that this protocol distils non-stabilizer magic states, and identify two types of states that are attractors of this iteration map. Finally, we show how these states may be converted, via stabilizer circuits alone, into a state suitable for state-injected implementation of a non-Clifford phase gate, enabling non-Clifford unitary computation.

Complex Earthquake Cycle Simulations Using a Two-Degree-of-Freedom Spring-Block Model with a Rate- and State-Friction Law

Numerical simulations of complex earthquake cycles are conducted using a two-degree-of-freedom spring-block model with a rate- and state-friction law, which has been supported by laboratory experiments. The model consisted of two blocks coupled to each other and connected by elastic springs to a constant-velocity, moving driver. By widely and systematically varying the model parameters, various slip patterns were obtained, including the periodic recurrence of seismic and aseismic slip events, and several types of chaotic behaviour. The transition in the slip pattern from periodic to chaotic is examined using bifurcation diagrams. The model system exhibits typical period-doubling sequences for some parameter ranges, and attains chaotic motion. Simple relationships are found in iteration maps of the recurrence intervals of simulated earthquakes, suggesting that the simulated slip behaviour is deterministic chaos. Time evolutions of the cumulative slip distance in chaotic slip patterns are well approximated by a time-predictable model. In some cases, both seismic and aseismic slip events occur at a block, and aseismic slip events complicate the earthquake recurrence patterns.

Noninvertibility, Chaotic Coding, and Chaotic Multiplexity of Synaptically Modulated Neural Firing

Widely accepted neural firing and synaptic potentiation rules specify a cross-dependence of the two processes, which, evolving on different timescales, have been separated for analytic purposes, concealing essential dynamics. Here, the morphology of the firing rates process, modulated by synaptic potentiation, is shown to be described by a discrete iteration map in the form of a thresholded polynomial. Given initial synaptic weights, a firing activity is triggered by conductance. Elementary dynamic modes are defined by fixed points, cycles, and saddles of the map, building blocks of the underlying firing code. Showing parameter-dependent multiplicity of real polynomial roots, the map is proved to be noninvertible. The incidence of chaos is then implied by the parameter-dependent existence of snap-back repellers. The highly patterned geometric and statistical structures of the associated chaotic attractors suggest that these attractors are an integral part of the neural code. It further suggests the chaotic attractor as a natural mechanism for statistical encoding and temporal multiplexing of neural information. The analytic findings are supported by simulation.

Specific heat study on a spin-one-half frustrated diamond chain

The specific heat of S = 1 / 2 frustrated diamond-spin chain has been examined by means of the Green's function theory combined with the Jordan–Wigner transformation, and the particular emphasis is laid on the investigation of the effect of geometric frustration, which is generated by the competition between the exchange interactions along the spin chain. It is found that the interplay between the geometric frustration and quantum effects gives rise to non-trivial behaviors in the curves of the specific heat: in the absence of the external magnetic field, the temperature dependence of the specific heat shows a double-peak structure for different strength of the frustration, and in the presence of the external field, the curves of the specific heat display as a three-peak structure due to the field-induced splitting of the energy levels related to the frustrated spins, which are in good agreement with the exact analytical approach based on the generalized decoration–iteration map method.

Geometric frustration in the class of exactly solvable Ising–Heisenberg diamondchains

Ground-state and finite-temperature properties of the mixed spin- and spin-S Ising–Heisenberg diamond chains are examined within an exact analytical approachbased on the generalized decoration–iteration map. A particular emphasis is laidon the investigation of the effect of geometric frustration, which is generatedby the competition between Heisenberg- and Ising-type exchange interactions.It is found that an interplay between the geometric frustration and quantumeffects gives rise to several quantum ground states with entangled spin states inaddition to some semi-classically ordered ones. Among the most interesting results toemerge from our study one could mention rigorous evidence for quantized plateauxin magnetization curves, an appearance of the round minimum in the thermaldependence of susceptibility times temperature data, double-peak zero-field specific heatcurves, or an enhanced magnetocaloric effect when the frustration comes intoplay. The triple-peak specific heat curve is also detected when applying smallexternal field to the system driven by the frustration into the disordered state.

Iterates of the core model

AbstractLetNbe a transitive model of ZFC such that “N⊂NandP(ℝ) ⊂N. Assume that bothVandNsatisfy “the core modelKexists.” ThenKNis an iterate ofK, i.e., there exists an iteration treeFonKsuch thatFhas successor length and. Moreover, if there exists an elementary embedding π:V→Nthen the iteration map associated to the main branch ofFequals π յK. (This answers a question of W. H. Woodin, M. Gitik, and others.) The hypothesis thatP(ℝ) ⊂Nis not needed if there does not exist a transitive model of ZFC with infinitely many Woodin cardinals.

Wave propagation in RTD-based cellular neural networks

This work investigates the existence of monotonic traveling wave and standing wave solutions of RTD-based cellular neural networks in the one-dimensional integer lattice Z 1 . For nonzero wave speed c, applying the monotone iteration method with the aid of real roots of the corresponding characteristic function of the profile equation, we can partition the parameter space ( γ, δ)-plane into four regions such that all the admissible monotonic traveling wave solutions connecting two neighboring equilibria can be classified completely. For the case of c=0, a discrete version of the monotone iteration scheme is established for proving the existence of monotonic standing wave solutions. Furthermore, if γ or δ is zero then the profile equation for the standing waves can be viewed as an one-dimensional iteration map and we then prove the multiplicity results of monotonic standing waves by using the techniques of dynamical systems for maps. Some numerical results of the monotone iteration scheme for traveling wave solutions are also presented.

Distillation of continuous-variable entanglement with optical means

We present an event-ready procedure that is capable of distilling Gaussian two-mode entangled states from a supply of weakly entangled states that have become mixed in a decoherence process. This procedure relies on passive optical elements and photon detectors distinguishing the presence and the absence of photons, but does not make use of photon counters. We identify fixed points of the iteration map, and discuss in detail its convergence properties. Necessary and sufficient criteria for the convergence to two-mode Gaussian states are presented. On the basis of various examples we discuss the performance of the procedure as far as the increase of the degree of entanglement and two-mode squeezing is concerned. Finally, we consider imperfect operations and outline the robustness of the scheme under non-unit detection efficiencies of the detectors. This analysis implies that the proposed protocol can be implemented with currently available technology and detector efficiencies.

CHAOTIC STATIONARY SOLUTIONS OF CELLULAR NEURAL NETWORKS

This study describes the chaotic stationary solutions of one-dimensional Cellular Neural Networks (CNN) without inputs with a specific term by applying the iteration map method. Under perfectly determined specific parameters, the map which corresponds to the stationary solution of CNN is two-dimensional and has a hyperbolic invariant Cantor set on which it is topologically conjugate to a two-sided shift of symbols space. The used main tool is the Conley–Moser conditions.

Acceleration Schemes of the Discrete Velocity Method: Gaseous Flows in Rectangular Microchannels

The convergence rate of the discrete velocity method (DVM), which has been applied extensively in the area of rarefied gas dynamics, is studied via a Fourier stability analysis. The spectral radius of the continuum form of the iteration map is found to be equal to one, which justifies the slow convergence rate of the method. Next the efficiency of the DVM is improved by introducing various acceleration schemes. The new synthetic-type schemes speed up significantly the iterative convergence rate. The spectral radius of the acceleration schemes is also studied and the so-called H1 acceleration method is found to be the optimum one. Finally, the two-dimensional flow problem of a gas through a rectangular microchannel is solved using the new fast iterative DVM. The number of required iterations and the overall computational time are significantly reduced, providing experimental evidence of the analytic formulation. The whole approach is demonstrated using the BGK and S kinetic models.

SPATIAL DISORDER OF CELLULAR NEURAL NETWORKS — WITH BIASED TERM

This study describes the spatial disorder of one-dimensional Cellular Neural Networks (CNN) with a biased term by applying the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a staircase function.

Spatial disorder of Cellular Neural Networks

This paper shows the spatial disorder of one-dimensional Cellular Neural Networks (CNN) using the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a devil-staircase like function.

SPATIAL DISORDER OF CNN — WITH ASYMMETRIC OUTPUT FUNCTION

This investigation will describe the spatial disorder of one-dimensional Cellular Neural Networks (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function.

Convergence of iterations of Euler family under weak condition

The iteration maps of Euler family for finding zeros of an operator f in Banach spaces is defined as the partial sum of Taylor expansion of the local inverse f 1 z of f at z . The unified convergence theorem is established for the iterations of Euler family under the assumption that α ≤3-2, while the strong condition that f is analytic in Smale's criterion αis replaced by the weak condition that f is of finite order derivative.

SMALE HORSESHOE OF CELLULAR NEURAL NETWORKS

The paper shows the spatial disorder of one-dimensional Cellular Neural Networks (CNN) using the iteration map method. Under certain parameters, the map is two-dimensional and the Smale horseshoe is constructed. Moreover, we also illustrate the variant of CNN, closely related to Henón-type and Belykh maps, and discrete Allen–Cahn equations.

A weak Dodd-Jensen lemma

AbstractWe show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice.