Pure Point Research Articles

Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation

In this paper we study the multi-frequency quasi-periodic operator with a Gevrey type perturbation. We first establish the large deviation theorem (LDT) for the multi-dimensional operator with a sub-exponential (or Gevrey) long-range hopping, and then prove the pure point spectrum property. Based on the LDT and the Aubry duality, we show the absence of a point spectrum for the 1D exponential long-range operator with a multi-frequency and a Gevrey potential. We also prove the spectrum has positive Lebesgue measure.

On the spectrum of the Kronig–Penney model in a constant electric field

We are interested in the nature of the spectrum of the one-dimensional Schr\"odinger operator $$ - \frac{d^2}{dx^2}-Fx + \sum_{n \in \mathbb{Z}}g_n \delta(x-n) \qquad\text{in } L^2(\mathbb{R}) $$ with $F>0$ and two different choices of the coupling constants $\{g_n\}_{n\in \mathbb{Z}}$. In the first model $g_n \equiv \lambda$ and we prove that if $F\in \pi^2 \mathbb{Q}$ then the spectrum is $\mathbb{R}$ and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model the $g_n$ are independent random variables with mean zero and variance $\lambda^2$. Under certain assumptions on the distribution of these random variables we prove that almost surely the spectrum is $\mathbb{R}$ and it is dense pure point if $F < \lambda^2/2$ and purely singular continuous if $F> \lambda^2/2$.

Biodegradability and Cytocompatibility of 3D-Printed Mg-Ti Interpenetrating Phase Composites.

Orthopedic hybrid implants combining both titanium (Ti) and magnesium (Mg) have gained wide attraction nowadays. However, it still remains a huge challenge in the fabrication of Mg-Ti composites because of the different temperatures of Ti melting point and pure Mg volatilization point. In this study, we successfully fabricated a new Mg-Ti composite with bi-continuous interpenetrating phase architecture by infiltrating Mg melt into Ti scaffolds, which were prepared by 3D printing and subsequent acid treatment. We attempted to understand the 7-day degradation process of the Mg-Ti composite and examine the different Mg2+ concentration composite impacts on the MC3T3-E1 cells, including toxicity, morphology, apoptosis, and osteogenic activity. CCK-8 results indicated cytotoxicity and absence of the Mg-Ti composite during 7-day degradation. Moreover, the composite significantly improved the morphology, reduced the apoptosis rate, and enhanced the osteogenic activity of MC3T3-E1 cells. The favorable impacts might be attributed to the appropriate Mg2+ concentration of the extracts. The results on varying Mg2+ concentration tests indicated that Mg2+ showed no cell adverse effect under 10-mM concentration. The 8-mM group exhibited the best cell morphology, minimum apoptosis rate, and maximum osteogenic activity. This work may open a new perspective on the development and biomedical applications for Mg-Ti composites.

Understanding the dual formulation for the hedging of path-dependent options with price impact

We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (SIAM J. Control Optim. 57 (2019) 4125–4149), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong regularity conditions. Using only probabilistic arguments, we prove, in a path-dependent setting and under weak regularity conditions, that any solution to this dual problem actually allows one to construct explicitly a perfect hedging portfolio. From a pure probabilistic point of view, our approach also allows one to exhibit solutions to a specific class of second order forward backward stochastic differential equations, in the sense of Cheridito et al. (Comm. Pure Appl. Math. 60 (2007) 1081–1110). Existence of a solution to the dual optimal control problem is also addressed in particular settings. As a by-product of our arguments, we prove a version of Itô’s lemma for path-dependent functionals that are only C0,1 in the sense of Dupire.

Absolutely continuous and pure point spectra of discrete operators with sparse potentials

We consider the discrete Schrödinger operator with a sparse potential and find conditions guaranteeing either existence of wave operators for the pair and or presence of dense purely point spectrum of the operator on some interval with .

The continuous maximum capacity path interdiction problem

This paper studies the continuous maximum capacity path interdiction problem, where two players, user and interdictor, compete in a capacitated network. The user wants to send the maximum possible amount of flow through a path, whose capacity is given by the minimum capacity among its arcs. The budget-constrained interdictor decreases arc capacities by any continuous amount to reduce the quality of the user’s chosen path. We present an efficient algorithm based on a discrete version of the Newton’s method, which helps us solve the problem in polynomial time. We also prove that the problem can be transformed into a zero-sum game, which has always a pure Nash equilibrium point. We demonstrate the performance of our algorithm over a set of randomly generated networks.

Expressions of Friction Force of a Four-Point Contact Linear Ball Bearing With Two Rows Under Low-Speed Operation

Abstract This article deals with the expressions of friction force of a four-point contact linear ball bearing with two rows under low-speed operation. First, the slip velocity on a contact area and the equilibrium of moments acting on a ball were theoretically examined. Then, an expression for the location of the pure rolling point (where the slip velocity is zero) was derived. Using the location of the pure rolling point, expressions of the friction force were developed. Next, to confirm the validity of the developed expressions, the friction force of four-point contact linear ball bearings with two rows was measured. The results showed that the developed expressions for the friction force in this study are effective. The developed expression would be applicable for estimating the friction force as a basic performance in the design procedure and for checking the product quality in bearing manufacturing.

Quartic Kerr cavity combs: bright and dark solitons.

We theoretically investigate the dynamics, bifurcation structure, and stability of localized states in Kerr cavities driven at the pure fourth-order dispersion point. Both the normal and anomalous group velocity dispersion regimes are analyzed, highlighting the main differences from the standard second-order dispersion case. In the anomalous regime, single and multi-peak localized states exist and are stable over a much wider region of the parameter space. In the normal dispersion regime, stable narrow bright solitons exist. Some of our findings can be understood using a new, to the best of our knowledge, scenario reported here for the spatial eigenvalues, which imposes oscillatory tails to all localized states.

Localisation and Delocalisation for a Simple Quantum Wave Guide with Randomness

In this paper, we consider Schrödinger operators on Mtimes {mathbb {Z}}^{d_{2}}, with M={M_{1},ldots ,M_{2}}^{d_{1}} (‘quantum wave guides’) with a ‘Gamma -trimmed’ random potential, namely a potential which vanishes outside a subset Gamma which is periodic with respect to a sub-lattice. We prove that (under appropriate assumptions) for strong disorder these operators have pure point spectrum outside the set Sigma _{0}=sigma (H_{0,Gamma ^{c}}) where H_{0,Gamma ^{c}} is the free (discrete) Laplacian on the complement Gamma ^{c} of Gamma . We also prove that the operators have some absolutely continuous spectrum in an energy region {mathcal {E}}subset Sigma _{0}. Consequently, there is a mobility edge for such models. We also consider the case -M_{1}=M_{2}=infty , i.e. Gamma -trimmed operators on {mathbb {Z}}^{d}={mathbb {Z}}^{d_{1}}times {mathbb {Z}}^{d_{2}}. Again, we prove localisation outside Sigma _{0} by showing exponential decay of the Green function G_{E+ieta }(x,y) uniformly in eta >0 . For all energies Ein {mathcal {E}} we prove that the Green’s function G_{E+ieta } is not (uniformly) in ell ^{1} as eta approaches 0. This implies that neither the fractional moment method nor multi-scale analysis can be applied here.

On Numerical Approximations of the Koopman Operator

We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. Since Krylov sequence-based approximations can mitigate the curse of dimensionality, this result indicates that they may also have low spectral error without an exponential-in-dimension increase in the number of functions needed.

Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one

In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let $P\\colon L^2(\\mathbb{R})\\to L^2(\\mathbb{R})$ have the form $$ P:=-\\frac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $\\mathbf{1}{(-\\infty,\\lambda^2]}(P)$ has a full asymptotic expansion in powers of $\\lambda$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, the class of potentials includes certain potentials with \_dense pure point spectrum. The proof combines the gauge transform methods of Parnovski–Shterenberg and Sobolev with Melrose's scattering calculus.

Spectral dimensions of Kreĭn–Feller operators and Lq-spectra

We study the spectral dimensions and spectral asymptotics of Kreĭn–Feller operators for arbitrary finite Borel measures on (0,1). Connections between the spectral dimension, the Lq-spectrum, the partition entropy and the optimized coarse multifractal dimension are established. In particular, we show that the upper spectral dimension always corresponds to the fixed point of the Lq-spectrum of the corresponding measure. Natural bounds reveal intrinsic connections to the Minkowski dimension of the support of the associated Borel measure. Further, we give a sufficient condition on the Lq-spectrum to guarantee the existence of the spectral dimension. As an application, we confirm the existence of the spectral dimension of self-conformal measures with or without overlap as well as of certain measures of pure point type. We construct a simple example for which the spectral dimension does not exist and determine explicitly its upper and lower spectral dimension.

Trojan Horse Investigation for AGB Stellar Nucleosynthesis

Asymptotic Giant Branch (AGB) stars are among the most important astrophysical sites influencing the nucleosynthesis and the chemical abundances in the Universe. From a pure nuclear point of view, several processes take part during this peculiar stage of stellar evolution thus requiring detailed experimental cross section measurements. Here, we report on the most recent results achieved via the application of the Trojan Horse Method (THM) and Asymptotic Normalization Coefficient (ANC) indirect techniques, discussing the details of the experimental procedure and the deduced reaction rates. In addition, we report also on the on going studies of interest for AGB nucleosynthesis.

Singular rank one perturbations

In this paper, A = B + V represents a self-adjoint operator acting on a Hilbert space H. We set a general theoretical framework and obtain several results for singular perturbations of A of the type Aβ = A + βτ*τ for τ being a functional defined in a subspace of H. In particular, we apply these results to Hβ = −Δ + V + β|δ⟩⟨δ|, where δ is the singular perturbation given by δ(φ) = ∫Sφ dσ, where S is a suitable hypersurface in Rn. Using the fact that the singular perturbation τ*τ is a sort of rank one perturbation of the operator A, it is possible to prove the invariance of the essential spectrum of A under these singular perturbations. The main idea is to apply an adequate Krein’s formula in this singular framework. As an additional result, we found the corresponding relationship between the Green’s functions associated with the operators H0 = Δ + V and Hβ, and we give a result about the existence of a pure point spectrum (eigenvalues) of Hβ. We also study the case β goes to infinity.

Bayesian surface regression versus spatial spectral nonparametric curve regression

COVID-19 incidence is analyzed at the provinces of the Spanish Communities in the Iberian Peninsula during the period February–October, 2020. Two infinite-dimensional regression approaches, surface regression and spatial curve regression, are proposed. In the first one, Bayesian maximum a posteriori (MAP) estimation is adopted in the approximation of the pure point spectrum of the temporal regression residual autocorrelation operator. Thus, an alternative to the moment-based estimation methodology developed in Ruiz-Medina, Miranda and Espejo (2019) is derived. Additionally, spatial curve regression is considered. A nonparametric estimator of the spectral density operator, based on the spatial periodogram operator, is computed to approximate the spatial correlation between curves. Dimension reduction is achieved by projection onto the empirical eigenvectors of the long-run spatial covariance operator. Cross-validation procedures are implemented to test the performance of the two functional regression approaches.

Constraints on pure point diffraction on aperiodic point patterns of finite local complexity* *Dedicated to the memory of Uwe Grimm.

We consider the pure point part of the diffraction on families of aperiodic point sets obeying common local rules. It is shown that imposing such rules results in linear constraints on the partial diffraction amplitudes. These relations can be explicitly derived from the geometry of the prototile space representing the local rules.

Beyond immediacy and transparency. A semiotic approach to discursive and rhetorical strategies in media visualization and data visualization

Media visualization based on big cultural data, as “visualization without reduction” (Manovich 2010) is supposed to make data immediately and completely available, in contrast to classic data visualization, which visually translates information by means of “graphical primitives.” On the other hand, from a pure functionalist point of view, also the visual form of diagrams, charts, and graphs, being fully proportional to the data values it conveys, is transparent with respect to its object (cf. Tufte 1990, 1997, 2001, and Card, Mackinlay, Shneiderman 1999). In this paper, we will try to consider both media visualization and data visualization (across several examples, including some of Manovich’s and Accurat’s projects and New York Times graphics) as complex visual communication artifacts, not only from a purely informational point of view but from a semiotic point of view, by introducing a semiotic reflection on what we have proposed to call “discourse of data” (Manchia 2020a). From our perspective, situated in the methodological framework of visual semiotics and of the semiotics of scientific discourse, it might be interesting to pay attention to the whole process of constructing knowledge (and visual information) from data, understood as a chain of “devices of visualization” (Bastide 1985a, 1990 [1985b], 2001), investigating data as a channeled result, and also visualization strategies of specific–and oriented–discourses across data.

Effect of MgO/Al&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; on Fluidity of SFCA-Based Binder Phase

The fluidity of sinter binder phase is one of the main properties that determine the quality of sinter, and it is also an important index to characterize the quality of sinter. So far, iron and steel enterprises mainly produce high alkalinity sinter, and its binder phase is mainly composite calcium ferrite based binder phase. Therefore, the research object of this paper is the composite calcium ferrite based binder phase of high alkalinity sinter. The effect of MgO/Al₂O₃ on the fluidity of high calcium composite calcium ferrite based binder phase was investigated from three aspects: fluidity index, liquid phase formation temperature and melting time by using chemical pure reagent and melting point velocimeter. The results show that in the range of MgO/Al₂O₃ from 0.46 to 0.68, with the increase of the ratio of MgO to Al₂O₃, the fluidity index first increases and then decreases, and the maximum value is obtained at MgO/Al₂O₃=0.65; The melting time decreases first and then increases. At the same time, the minimum value is obtained at MgO/Al₂O₃=0.65, and the influence on the melting time becomes more and more obvious with the increase of MgO/Al₂O₃ ratio; the characteristic temperature of liquid phase formation increases briefly and then decreases.

Highly Efficient and Accurate Gas-Alkane Binary Mixture Interfacial Tension Equations for a Broad Range of Temperatures, Pressures, and Compositions

Summary Gas-alkane interfacial tension (IFT) is an important parameter in the enhanced oil recovery (EOR) process. Thus, it is imperative to obtain an accurate gas-alkane mixture IFT for both chemical and petroleum engineering applications. Various empirical correlations have been developed in the past several decades. Although these models are often easy to implement, their accuracy is inconsistent over a wide range of temperatures, pressures, and compositions. Although statistical mechanics-based models and molecular simulations can accurately predict gas-alkane IFT, they usually come with an extensive computational cost. The Shardt-Elliott (SE) model is a highly accurate IFT model that for subcritical fluids is analytic in terms of temperature T and composition x. In applications, it is desirable to obtain IFT in terms of temperature T and pressure P, which requires time-consuming flash calculations, and for mixtures that contain a gas component greater than its pure species critical point, additional critical composition calculations are required. In this work, the SE model is combined with a machine learning (ML) approach to obtain highly efficient and highly accurate gas-alkane binary mixture IFT equations directly in terms of temperature, pressure, and alkane molar weights. The SE model is used to build an IFT database (more than 36,000 points) for ML training to obtain IFT equations. The ML-based IFT equations are evaluated in comparison with the available experimental data (888 points) and with the SE model, as well as with the less accurate parachor model. Overall, the ML-based IFT equations show excellent agreement with experimental data for gas-alkane binary mixtures over a wide range of T and P, and they outperform the widely used parachor model. The developed highly efficient and highly accurate IFT functions can serve as a basis for modeling gas-alkane binary mixtures for a broad range of T, P, and x.

Contact interactions II; Gross-Pitaewskii equation, Bose-Einstein condensate, Fermi sea

In Dell’Antonio (Eur Phys J Plus 13:1–20, 2021), we explored the possibility to analyse contact interaction in Quantum Mechanics using a variational tool, Gamma Convergence. Here, we extend the analysis in Dell’Antonio (Eur Phys J Plus 13:1–20, 2021) of joint weak contact of three particles to the non-relativistic case in which the free one particle Hamiltonian is H_0 = - frac{Delta }{2M} . We derive the Gross–Pitaevskii equation for a system of three particles in joint weak contact. We then define and study strong contact and show that the Gross–Pitaevskii equation is also the variational equation for the energy of the Bose–Einstein condensate (strong contact in a four-particle system). We add some comments on Bogoliubov’s theory. In the second part, we use the non-relativistic Pauli equation and weak contact to derive the spectrum of the conduction electrons in an infinite crystal. We prove that the spectrum is pure point with multiplicity two and eigenvalues that scale as frac{1}{log {n}}.