Pure Point Research Articles

Reducibility for Schrödinger Operator with Finite Smooth and Time-Quasi-periodic Potential

In this paper, the author establishes a reduction theorem for linear Schrodinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM (Kolmogorov-Arnold-Moser) technique. Moreover, it is proved that the corresponding Schrodinger operator possesses the property of pure point spectra and zero Lyapunov exponent.

Pure point measures with sparse support and sparse Fourier–Bohr support

Fourier-transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier--Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.

Three variations on a theme by Fibonacci

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalization structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.

(2+1)-dimensional static cyclic symmetric traversable wormhole: quasinormal modes and causality

In this paper we study a static cyclic symmetric traversable wormhole (WH) in (2+1)-dimensional gravity coupled to nonlinear electrodynamics in anti-de Sitter spacetime. The solution is characterized by three parameters: mass M, cosmological constant and one electromagnetic parameter, . The causality of this spacetime is studied, determining its maximal extension and constructing then the Penrose diagram. The quasinormal modes (QNMs) that result from considering a massive scalar test field in the WH background are determined by solving in exact form the Klein–Gordon equation; the effective potential resembles the one of a harmonic oscillator shifted from its equilibrium position and, consequently, the QNMs have a pure point spectrum.

Singular continuous Cantor spectrum for magnetic quantum walks

In this note, we consider a physical system given by a two-dimensional quantum walk in an external magnetic field. In this setup, we show that both the topological structure as well as its type depend sensitively on the value of the magnetic flux $\Phi$: while for $\Phi/(2{\pi})$ rational the spectrum is known to consist of bands, we show that for $\Phi/(2{\pi})$ irrational the spectrum is a zero-measure Cantor set and the spectral measures have no pure point part.

Diffraction of a model set with complex windows

The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals.

Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings

Primitive inflation tilings of the real line with finitely many tiles of natural length and a Pisot-Vijayaraghavan unit as inflation factor are considered. We present an approach to the pure point part of their diffraction spectrum on the basis of a Fourier matrix cocycle in internal space. This cocycle leads to a transfer matrix equation and thus to a closed expression of matrix Riesz product type for the Fourier transforms of the windows for the covering model sets. In general, these windows are complicated Rauzy fractals and thus difficult to handle. Equivalently, this approach permits a construction of the (always continuously representable) eigenfunctions for the translation dynamical system induced by the inflation rule. We review and further develop the underlying theory, and illustrate it with the family of Pisa substitutions, with special emphasis on the classic Tribonacci case.

Sparse Bayesian learning for network structure reconstruction based on evolutionary game data

Network structure reconstruction is a fundamental problem for understanding, predicting and controlling the behaviors of complex networked systems and has received growing attention due to the potentials in a wide range of fields. Recent years have witnessed dramatic advances in the field of network structure reconstruction, especially the famous compressed sensing-based methods. However, some neglected disadvantages still exist in the existing works, such as the high measurement correlation existing in the solution matrix, reconstruction behaviors subject to model-based constraints and pure point estimate of the reconstruction results without credibility, which inevitably drag down the reconstruction performance. To address these problems, we propose a new framework of sparse Bayesian learning for network structure reconstruction based on evolutionary game data from the perspective of Bayesian and statistics. Specifically, we formulate the problem of network structure reconstruction as a Bayesian compressed sensing problem. Then, a hierarchical prior model is invoked for conjugated Bayesian inference to obtain the posterior distribution of the reconstructed result, including the reconstructed mean and covariance. Finally, the parameters in the reconstructed results are updated by an iterative estimation procedure. Results from numerical experiments have demonstrated applicability and efficiency of the proposed method and presented superiority over other reconstruction methods.

On the Fourier analysis of measures with Meyer set support

In this paper we show the existence of the generalized Eberlein decomposition for Fourier transformable measures with Meyer set support. We prove that each of the three components is also Fourier transformable and has Meyer set support. We obtain that each of the pure point, absolutely continuous and singular continuous components of the Fourier transform is a strong almost periodic measure, and hence is either trivial or has relatively dense support. We next prove that the Fourier transform of a measure with Meyer set support is norm almost periodic, and hence so is each of the pure point, absolutely continuous and singular continuous components. We show that a measure with Meyer set support is Fourier transformable if and only if it is a linear combination of positive definite measures, which can be chosen with Meyer set support, solving a particular case of an open problem. We complete the paper by discussing some applications to the diffraction of weighted Dirac combs with Meyer set support.

Heisenberg pseudo-exchange and emergent anisotropies in field-driven pinwheel artificial spin ice

Rotating all islands in square artificial spin ice (ASI) uniformly about their centres gives rise to the recently reported pinwheel ASI. At angles around 45$^\mathrm{o}$, the antiferromagnetic ordering changes to ferromagnetic and the magnetic configurations of the system exhibit near-degeneracy, making it particularly sensitive to small perturbations. We investigate through micromagnetic modelling the influence of dipolar fields produced by physically extended islands in field-driven magnetisation processes in pinwheel arrays, and compare the results to hysteresis experiments performed in-situ using Lorentz transmission electron microscopy. We find that magnetisation end-states induce a Heisenberg pseudo-exchange interaction that governs both the inter-island coupling and the resultant array reversal process. Symmetry reduction gives rise to anisotropies and array-corner mediated avalanche reversals through a cascade of nearest-neighbour (NN) islands. The symmetries of the anisotropy axes are related to those of the geometrical array but are misaligned to the array axes as a result of the correlated interactions between neighbouring islands. The NN dipolar coupling is reduced by decreasing the island size and, using this property, we track the transition from the strongly coupled regime towards the pure point dipole one and observe modification of the ferromagnetic array reversal process. Our results shed light on important aspects of the interactions in pinwheel ASI, and demonstrate a mechanism by which their properties may be tuned for use in a range of fundamental research and spintronic applications.

Topological complexity of photons' paths in biological tissues.

In the present contribution, three means of measuring the geometrical and topological complexity of photons' paths in random media are proposed. This is realized by investigating the behavior of the average crossing number, the mean writhe, and the minimal crossing number of photons' paths generated by Monte Carlo (MC) simulations, for different sets of optical parameters. It is observed that the complexity of the photons' paths increases for increasing light source/detector spacing, and that highly "knotted" paths are formed. Due to the particular rules utilized to generate the MC photons' paths, the present results may have an interest not only for the biomedical optics community, but also from a pure mathematical point of view.

Towards a coupled structural and economical design procedure of unstiffened isotropic shell structures

Abstract Thin-walled structures such as those used for space launch vehicles are prone to buckling when axial load is applied. These structures can be designed as imperfection-sensitive structures in the form of unstiffened shell structures or else as imperfection-tolerant structures by applying frames and stringers to the shell structure. Weight-Strength-Curves can be applied to compare structural concepts from a pure technological point of view, where neither costs of manufacturing nor the manufacturing signature are taken into account. Within this paper, a cost model for unstiffened isotropic shell structures based on the assembly out of multiple panels is introduced. Furthermore, a relation between the manufacturing process and the load carrying capacity of an isotropic shell structure is derived. On this basis a coupled structural and economical design procedure is evolved. In a final step, an example is introduced in order to illustrate the application of the coupled design procedure and to derive a structure which is optimized with regard to its structural mass and manufacturing costs. Based on this example it is shown that the chosen panel size affects not only the manufacturing costs, but also the load carrying capacity.

Spectral mapping theorem of an abstract quantum walk

Given two Hilbert spaces, $\mathcal{H}$ and $\mathcal{K}$, we introduce an abstract unitary operator $U$ on $\mathcal{H}$ and its discriminant $T$ on $\mathcal{K}$ induced by a coisometry from $\mathcal{H}$ to $\mathcal{K}$ and a unitary involution on $\mathcal{H}$. In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpi\'nski lattice, which is pure point and has a Cantor-like structure.

Agnotologie, die Grenzen der Gewissheit und die Theorie der Demokratie

Abstract Inspired to some extent by the work of Thomas Kuhn, Ignorance Studies seeks to grapple with the implications that follow from the idea that on a theoretical level we can hardly ever know that we have approached truth. In order to test theories, we would need to have access to empirically raw, theoretically unadorned facts and to confront them with theoretically diverse or contradictory accounts of how to make sense of them. It’s the nature of the case that we are not able to penetrate to such a pure point of origin for things. The words that we use to describe them are already suffused with biases and distortions that limit what we can say about them. Our relationship to phenomena inside us and to things outside us always subsists in a symbolically mediated, indirect medium. Our efforts to be either before or after this state of symbolic mediation generally land us in a state of deepened awareness of the extent of the prolongation of the middle. The conceptualized entity is for all intents and purposes the entity we relate to. The paper argues that given the limits to certainty highlighted by Ignorance Studies, our political institutions need to be responsive to the priorities enshrined in democratic theory to ensure that the policies pursued by any given government are reflective of the needs and the wishes of the majority of the people.

On Bernstein processes generated by hierarchies of linear parabolic systems in [formula omitted

In this article we investigate the properties of Bernstein processes generated by infinite hierarchies of forward–backward systems of decoupled linear deterministic parabolic partial differential equations defined in Rd, where d is arbitrary. An important feature of those systems is that the elliptic part of the parabolic operators may be realized as an unbounded Schrödinger operator with compact resolvent in standard L2-space. The Bernstein processes we are interested in are in general non-Markovian, may be stationary or non-stationary and are generated by weighted averages of measures naturally associated with the pure point spectrum of the operator. We also introduce time-dependent trace-class operators which possess several attributes of density operators in Quantum Statistical Mechanics, and prove that the statistical averages of certain bounded self-adjoint observables usually evaluated by means of such operators coincide with the expectation values of suitable functions of the underlying processes. In the particular case where the given parabolic equations involve the Hamiltonian of an isotropic system of quantum harmonic oscillators, we show that one of the associated processes is identical in law with the periodic Ornstein–Uhlenbeck process.

Pure point spectrum for the Maryland model: a constructive proof

We develop a constructive method to prove and study pure point spectrum for the Maryland model with Diophantine frequencies.

Shape Verification Neural Network to Detection Personal Signature

This paper was concerned to study the standard features of the personal signature image. Independent features such as the area, height ,width, maximum vertical, maximum horizontal , and fractal dimension were taken to isomorphic of the original image . for more efficiently of this work the degree of complexity of image was studied and introduced as extra property of signature. Proposed algorithm was introduced for calculating the features of signature image. Many test image were taken for test this algorithm The executive appears that this algorithm is very sensitive and accurate when we have different and similar signature images. In this paper a method of verifying signatures using a technique of neural network is presented. static of various and dynamic of signature features are added in this model. Our proposal algorithm enhanced via neural network and Elman network. In our modified our method, many parameters of detection of digital images were added. These image parameters named as Width, Area, Maximum Vertical Projection, and Fractal Dimension. Fractal dimension is considered as the degree of complexity of the shape. Fractal dimension calculated the condensation of complexity of pure points of the shape without its boundaries. The dimension of the signature is varying between 1.0025 to 1.0584. Originally the signature is belonging to R2-Space (with dimension 2). The signature is considered as one of to the fractal shape. The fractal dimension is calculated by the Theorem of Boxes Squares Counting. The fractal signature dimension added an accuracy to our method for detection many signature shapes by the number of similarities in certain properties.

Reducibility, Lyapunov Exponent, Pure Point Spectra Property for Quasi-periodic Wave Operator

In the present paper, it is shown that the linear wave equation subject to Dirichlet boundary condition \[ u_{tt} - u_{xx} + \varepsilon V(\omega t, x) u = 0, \quad u(t,-\pi) = u(t,\pi) = 0 \] can be changed by a symplectic transformation into \[ v_{tt} - v_{xx} + \varepsilon M_{\xi} v = 0, \quad v(t,-\pi) = v(t,\pi) = 0, \] where $V$ is finitely smooth and time-quasi-periodic potential with frequency $\omega \in \mathbb{R}^n$ in some Cantor set of positive Lebeague measure and where $M_{\xi}$ is a Fourier multiplier. Moreover, it is proved that the corresponding wave operator $\partial_t^2 - \partial_x^2 + \varepsilon V(\omega t, x)$ possesses the property of pure point spectra and zero Lyapunov exponent.

Steady point vortex pair in a field of Stuart-type vorticity

A new family of exact solutions to the two-dimensional steady incompressible Euler equation is presented. The solutions provide a class of hybrid equilibria comprising two point vortices of unit circulation – a point vortex pair – embedded in a smooth sea of non-zero vorticity of ‘Stuart-type’ so that the vorticity $\unicode[STIX]{x1D714}$ and the stream function $\unicode[STIX]{x1D713}$ are related by $\unicode[STIX]{x1D714}=a\text{e}^{b\unicode[STIX]{x1D713}}-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})-\unicode[STIX]{x1D6FF}(\boldsymbol{x}+\boldsymbol{x}_{0})$, where $a$ and $b$ are constants. We also examine limits of these new Stuart-embedded point vortex equilibria where the Stuart-type vorticity becomes localized into additional point vortices. One such limit results in a two-real-parameter family of smoothly deformable point vortex equilibria in an otherwise irrotational flow. The new class of hybrid equilibria can be viewed as continuously interpolating between the limiting pure point vortex equilibria. At the same time the new solutions continuously extrapolate a similar class of hybrid equilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).

Modeling of Hydrate Dissociation Curves with a Modified Cubic-Plus-Association Equation of State

A modified cubic-plus-association equation of state (CPA EoS) has been presented in recent works that enforces the representation of the pure component critical point, is based on an extended Mathias–Copeman α function, and uses a volume shift to improve liquid densities. In this work, this same version of the CPA model is applied together with the hydrate van der Waals and Platteeuw model for the description of hydrate dissociation curves. The model is applied to the description of both single and mixed guest hydrates, the results being compared with the available experimental data, and the predictions obtained using the commercial hydrates model from the Multiflash software. This analysis shows that, for most cases, this approach is accurate, with results similar to those of Multiflash. The accuracy of the model for describing the three phase compositions in hydrate + liquid + gas systems is also discussed.