Prediction of amorphous alloy formation by ion beam mixing

In order to predict the glass forming ability of binary alloys produced by ion beam mixing, a two dimensional map is constructed. The two coordinates of this map are the ratio of atomic radii r_A/r_B, and the heat of formation ΔH of an equiatomic compound, calculated using Miedema's model. In this map, binary systems which form in the amorphous phase are clearly separated from the others. The same map is valid both for metal-metal and metal-semiconductor binaries.     

Vehicular motion through a sequence of traffic lights controlled by logistic map

We study the dynamical behavior of a single vehicle through the sequence of traffic lights controlled by the logistic map. The phase shift of traffic lights is determined by the logistic map and varies from signal to signal. The nonlinear dynamic model of the vehicular motion is presented by the nonlinear map including the logistic map. The vehicle exhibits the very complex behavior with varying both cycle time and logistic-map parameter a. For a>3, the dependence of arrival time on the cycle time becomes smoother and smoother with increasing a. The dependence of vehicular motion on parameter a is clarified.     

Stability of minimal periodic orbits

Symplectic twist maps are obtained from a Lagrangian variational principle. It is well known that nondegenerate minima of the action correspond to hyperbolic orbits of the map when the twist is negative definite and the map is two-dimensional. We show that for more than two dimensions, periodic orbits with minimal action in symplectic twist maps with negative definite twist are not necessarily hyperbolic. In the proof we show that in the neighborhood of a minimal periodic orbit of period n, the nth iterate of the map is again a twist map. This is true even though in general the composition of twist maps is not a twist map.     

Nonlinear Channels of Werner States

We study the nonlinear positive map of the density matrix of two-qubit Werner states, called the nonlinear channel. The map ρ → Φ(ρ) is realized by the rational function Φ. We discuss the influence of the map on the entanglement properties of the transformed density matrix. We investigate the violation of the Bell inequality (CHSH inequality) for the two-qubit state Φ(ρ). The nonlinear channels under discussion create the entangled state from a separable Werner state. We study the quantum spin tomograms of the states.     

Supertracks,supertrack functions and chaos in the quadratic map

Chaotic behavior for the quadratic map is conjectured to be a bounding phenomenon based on state-space trajectories starting from the maximum in the map. These trajectories, called supertracks, have no sensitivity to initial conditions. The chaotic regime is found to be characterized recursively by these trajectories as functions of the map parameter.     

High-resolution scanning tunneling microscopy and dI/dV map studies of peptidenucleic acid and fluorescein isothiocyanate

We used scanning tunneling microscopy (STM) to spatially map the local density of states of individual PNA molecules labeled with fluorescein isothiocyanate (FITC) on a Cu(1 1 1) surface. From the observed bias voltage dependences of the topographic height and the dI/dV map of FITC and individual PNA molecules, we confirmed that FITC and PNA have different electrical properties. We clearly differentiated the FITC and PNA molecules by mapping the density of states feature. This study shows that STM with the dI/dV map method is useful in FITC mapping.     

The role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables

In this Letter we analyse the behaviour of the probability density function of the sum of N deterministic variables generated from the triangle map of Casati-Prosen. For the case in which the map is both ergodic and mixing the resulting probability density function quickly concurs with the Normal distribution. However, when the map is weakly chaotic, and fuzzily not mixing, the resulting probability density functions are described by power-laws. Moreover, contrarily to what it would be expected, as the number of added variables N increases the distance to Gaussian distribution increases. This behaviour goes against standard central limit theorem. By extrapolation of our finite size results we preview that in the limit of N going to infinity the distribution has the same asymptotic decay as a Lorentzian (or a q=2-Gaussian).     

Distribution of periodic orbits for the Casati-Prosen map on rational lattices

We study numerically the periodic orbits of the Casati-Prosen map, a two-parameter reversible map of the torus, with zero entropy. For rational parameter values, this map preserves rational lattices, and each lattice decomposes into periodic orbits. We consider the distribution function of the periods over prime lattices, and its dependence on the parameters of the map. Based on extensive numerical evidence, we conjecture that, asymptotically, almost all orbits are symmetric, and that for a set of rational parameters having full density, the distribution function approaches the gamma-distribution R(x)=1−e^−x(1+x). These properties, which have been proved to hold for random reversible maps, were previously thought to require a stronger form of deterministic randomness, such as that displayed by rational automorphisms over finite fields. Furthermore, we show that the gamma-distribution is the limit of a sequence of singular distributions which are observed on certain lines in parameter space. Our experiments reveal that the convergence rate to R is highly non-uniform in parameter space, being slowest in sharply-defined regions reminiscent of resonant zones in Hamiltonian perturbation theory.     

Accelerating quantitative MR imaging with the incorporation of B1 compensation using deep learning

Quantitative magnetic resonance imaging (MRI) attracts attention due to its support to quantitative image analysis and data driven medicine. However, the application of quantitative MRI is severely limited by the long data acquisition time required by repetitive image acquisition and measurement of field map. Inspired by recent development of artificial intelligence, we propose a deep learning strategy to accelerate the acquisition of quantitative MRI, where every quantitative T₁ map is derived from two highly undersampled variable-contrast images with radiofrequency field inhomogeneity automatically compensated. In a multi-step framework, variable-contrast images are first jointly reconstructed from incoherently undersampled images using convolutional neural networks; then T₁ map and B₁ map are predicted from reconstructed images employing deep learning. Thus, the acceleration includes undersampling in every input image, a reduction in the number of variable contrast images, as well as a waiver of B₁ map measurement. The strategy is validated in T₁ mapping of cartilage. Acquired with a consistent imaging protocol, 1224 image sets from 51 subjects are used for the training of the prediction models, and 288 image sets from 12 subjects are used for testing. High degree of acceleration is achieved with image fidelity well maintained. The proposed method can be broadly applied to quantify other tissue properties (e.g. T₂, T_1ρ) as well.     

Linear integral equations and nonlinear difference-difference equations

This paper studies the dynamics of members of the two-parameter family of maps x → μx(1 ? x^v), emphasizing the evolution from snapback repeller to crisis bifurcations. The example of the square root map $\text{v =}\text{1}\text{2}$ is taken to represent the subfamily where v is fixed and taken from the range $\text{1}\text{2}\text{≤ v ≤ 1}$. A map from such a subfamily is shown to be conjugate with a map with negative Schwarzian derivative. This allows a characterization of crisis as the demise of a snapback repeller on a proper subinterval.     

Universal scaling behaviour for iterated maps in the complex plane

According to the theory of Schröder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ _n ^th iterates of the map, whereQ _n is then ^th Fibonacci number.     

Info-quantifiers’ map-characterization revisited

We highlight the potentiality of a special Information Theory (IT) approach in order to unravel the intricacies of nonlinear dynamics, the methodology being illustrated with reference to the logistic map. A rather surprising dynamic feature→plane-topography map becomes available.     

Evolution of Wigner's phase-space density under a nonintegrable quantum map

A nonintegrable area-preserving map for a system with one freedom is quantized, and the evolution of Wigner's function W(q,p) illustrated by contour plots of W in the paase plane. In the classical limit, propagation is governed by Liouville's equation and the contours of W rapidly develop an intricate structure of whorls and tendrils. When Planck's constant ? is not zero, the quantum map smooths out classical detail in phase-space areas smaller than ?. The quantum-mechanical distributions spread more slowly than their classical counterparts.     

p-Harmonic morphisms,biharmonic morphisms,and nonharmonic biharmonic maps

We study the transformation of a p-harmonic morphism into a q-harmonic morphism via biconformal change of the domain metric and/or conformal change of the codomain metric. As an application of p-harmonic morphisms, we characterize a twisted product among doubly twisted products and a warped product among twisted products using p-harmonicity of their projection maps. We describe those p-harmonic morphisms which are also biharmonic morphisms and give a complete classification of polynomial biharmonic morphisms between Euclidean spaces. Finally, we show that a horizontally homothetic harmonic morphism with harmonic energy density pulls back a nonharmonic biharmonic map to a nonharmonic biharmonic map and that totally geodesic immersing the target manifold of a nonharmonic biharmonic map into an ambient manifold produces a new nonharmonic biharmonic map. These methods are used to construct many examples of nontrivial biharmonic maps.     

On ergodic and mixing properties of the triangle map

In this paper, we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters defining the map. The dynamics is studied numerically by means of two different symbolic encoding schemes, both relying on the fact that it maps polygons to polygons: in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons, in the second we consider the standard binary partition of the torus induced by the discontinuity set. These encoding schemes are studied in detail and shown to be compatible, although not equivalent. The ergodic properties of the triangle map are then investigated in terms of the Markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the Koopman operator in L²(T²). Finally, a stochastic version of the triangle map is introduced and studied. A simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map.     

Entropy increase for a class of dynamical maps

The dynamical evolution of a quantum system is described by a one parameter family of linear transformations of the space of self-adjoint trace class operators (on the Hilbert space of the system) into itself, which map statistical operators to statistical operators. We call such transformations dynamical maps. We give a sufficient condition for a dynamical map A not to decrease the entropy of a statistical operator. In the special case of an N-level system, this condition is also necessary and it is equivalent to the property that A preserves the central state.     

Reductions related with Hopf maps

We consider the reductions of 2p-dimensional particle system (p = 2, 4, 8) associated with the Hopf map. For the third Hopf map we explicitly construct the functions associated to the symmetry related to the rotations in the fiber.     

Elliptic behaviour in the sawtooth standard map

This paper examines the standard map with sawtooth nonlinearity when the eigenvalues of the Jacobian lie on the unit circle. This is an area-preserving map of the torus to itself that is linear except on a line on which it is discontinuous. We discuss the closure of the set of images of the discontinuity and present numerical evidence that its Lebesgue measure is positive. Moreover, we present evidence that the measure of the closure of images of the discontinuity changes continuously with the parameter k. This means that the sawtooth standard map may exhibit coexistence of two positive measure subsets on which the dynamics is respectively regular and irregular in a certain sense. In the appendix we show that this map is equivalent to a map studied by electronics engineers as a model for the quiescent behaviour of a linear lossless digital filter with “two's complement” overflow.     

Solutions of the Multivariate Inverse Frobenius–Perron Problem

We address the inverse Frobenius–Perron problem: given a prescribed target distribution ρ, find a deterministic map M such that iterations of M tend to ρ in distribution. We show that all solutions may be written in terms of a factorization that combines the forward and inverse Rosenblatt transformations with a uniform map; that is, a map under which the uniform distribution on the d-dimensional hypercube is invariant. Indeed, every solution is equivalent to the choice of a uniform map. We motivate this factorization via one-dimensional examples, and then use the factorization to present solutions in one and two dimensions induced by a range of uniform maps.     

Forbidden ordinal patterns in higher dimensional dynamics

Forbidden ordinal patterns are ordinal patterns (or rank blocks) that cannot appear in the orbits generated by a map taking values on a linearly ordered space, in which case we say that the map has forbidden patterns. Once a map has a forbidden pattern of a given length L₀, it has forbidden patterns of any length L≥L₀ and their number grows superexponentially with L. Using recent results on topological permutation entropy, in this paper we study the existence and some basic properties of forbidden ordinal patterns for self-maps on n-dimensional intervals. Our most applicable conclusion is that expansive interval maps with finite topological entropy have necessarily forbidden patterns, although we conjecture that this is also the case under more general conditions. The theoretical results are nicely illustrated for n=2 both using the naive counting estimator for forbidden patterns and Chao’s estimator for the number of classes in a population. The robustness of forbidden ordinal patterns against observational white noise is also illustrated.