Discrete Levels Research Articles

I have three more than you, you have three less than me? Levels of flexibility in dealing with additive situations

Assessment and intervention in the early years should ideally be based on evidence-based models describing the structure and development of students’ skills. Mathematical word problems have been identified as a challenge for mathematics learners for a long time and in many countries. We investigate flexibility in dealing with additive situations as a construct that develops during grades 1 through 3 and contributes to the development of students’ word problem solving skills. We introduce the construct based on prior research on the difficulty of different situation structures entailed in word problems. We use data from three prior empirical studies with N = 383 German grade 2 and 3 students to develop a model of discrete levels of students’ flexibility in dealing with additive situations. We use this model to investigate how the learners in our sample distribute across the different levels. Moreover, we apply it to describe students’ development over several weeks in one study comprising three measurements. We derive conclusions about the construct in terms of determinants of task complexity, and about students’ development and then provide an outlook on potential uses of the model in research and practice.

Tractability of linear ill-posed problems in Hilbert space

We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.

Transport properties of the resonant tunneling structures based on the pseudospin-1 Dirac–Weyl fermions

Transmission, conductivity, and shot noise are investigated in multi-barrier resonant tunneling structures based on the pseudospin-1 Dirac–Weyl model. Super Klein tunneling, perfect transmission plateau, and resonant tunneling collectively contribute to the transport properties. While resonant tunneling occurs when the energy of the passing quasiparticle coincides with the discrete quasibound energy level of the multi-barrier structure and is widely observed in parabolic and Dirac-cone-shape dispersive materials, super Klein tunneling and perfect transmission plateau are unique in pseudospin-1 systems and barrier-structure independent. As a result, the conductivity and shot noise in pseudospin-1 resonant-tunneling multi-barrier structures demonstrate remarkably different properties from their counterparts of both parabolic-dispersion semiconductors and pseudospin-1/2 Dirac–Weyl materials, typically graphene. It is found that the zero conductivity minimum remains for multi-barrier structures and the relative shot noise indicated by the Fano factor in the three-barrier system is larger than that in the double-barrier system when the Klein tunneling effect weakens.

Second-order, positive, and unconditional energy dissipative scheme for modified Poisson–Nernst–Planck equations

First-order energy dissipative schemes in time are available in literature for the Poisson–Nernst–Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.

Racial disparities in debt collection

This paper documents that Black and Hispanic borrowers are 52% more likely to experience a debt collection judgment compared to their counterparts, controlling for socioeconomic variables including income, credit scores, delinquent debt balances, and other relevant credit characteristics. Statistical discrimination mechanisms such as spatially targeting collection efforts based on the likelihood that an attorney represents the defendant or that the defendant will contest the debt in court do not explain the racial gap in judgments. We find support that the judgment gap is partially driven by taste-based discrimination, as evidenced by minorities having less 90-day past due debt balances than non-minority borrowers when controlling for model covariates. Furthermore, the disparity primarily affects non-delinquent borrowers, indicating higher levels of creditor discretion in initiating judgment proceedings. A back-of-the-envelope calculation suggests that the racial wealth gap explains at most 48% of the racial disparity in judgments.

Pressure‐Driven Intrinsic Quantum Confinement and Semiconducting‐to‐Metallic Transition in the Topological Flat Bands Kagome Nb3Cl8 Compounds

AbstractNb3Cl8 is a unique subset of 2D crystalline materials renowned for their Kagome structure and distinctive flat energy bands. These bands contribute to the distinct electronic behavior, rendering Nb3Cl8 a compelling subject for study. In this study, by employing ab initio calculations, the impact of pressure on the electronic properties of Nb3Cl8is scrutinized, yielding valuable insights. Under high pressures, Nb3Cl8 undergoes a transition from a semiconductor state (with a 1.23 eV bandgap) to a metallic one, accompanied by electronic band restructuration. Notably, the flat energy bands are suppressed with increasing pressure. Furthermore, this investigation underscores the distinctive nature of flat energy bands in Nb3Cl8, delving into quantum effects, particularly their association with the quantum confinement of electronic states. The confinement of charge carriers in Nb3Cl8 results in discrete energy levels, corresponding to specific orbitals in the density of states, indicating robust electron confinement. Pressure‐induced changes in the energy gap between these quantized levels suggest a decrease in confinement strength. Consequently, the study lays a robust foundation for future exploration, contributing to the development of innovative electronic devices grounded in quantum confinement effects.

Enhancement of Upconversion Luminescence through Three-Dimensional Design of Plasma Ti3O5-Coupled Structural Domain-Limiting Effects.

Upconversion luminescence plays a crucial role in various technological applications, and among the various valence states of lanthanide elements, Ln3+ has the highest stability. The 4f orbitals of these elements are in a fully empty, semifull, or full state. This special 4f electron configuration allows them to exhibit rich discrete energy levels. However, the 4f-4f transition of Ln3+ rare earth ions itself is prohibited, resulting in a lower luminescence efficiency. This limitation greatly hinders the practical application of upconversion luminescence. In this study, we report nanostructured luminescence-enhanced substrate platforms with both semiconductive local surface plasmons and spatially confined domain effects on a single defect semiconductor substrate. By coupling NaYF4:Yb-Er nanoparticle emitters to the surface of Ti3O5 NC-arrays plasmonic nanostructures, an ultrabright luminescence with a 32-fold increase in green emission and a 40-fold increase in red emission was achieved. Furthermore, the fluorescence resonance energy transfer characteristics observed in the R6G/NaYF4/Ti3O5 NC-array composite film enable accurate detection of fluorescent molecules. The results provide an innovative and intelligent approach to enhance the upconversion luminescence intensity of rare-doped nanoparticles and develop highly sensitive molecular detection systems based on the above luminescence enhancement.

Modelling and measuring of temperature distribution in explosive cladded materials under induction heating

The work focuses on the modelling and experimental verification of induction heating systems with multilayered loads made of paramagnetic materials obtained through explosive cladding. The research is driven by the industry’s increasing interest in multilayered materials in electrical engineering and electronics. In this case, two- and three-layered configurations, as well as a cylindrical-shaped work coil, were considered. The main parameter subjected to verification was the surface temperature of the batch and its distribution. Due to relatively high temperatures, three measurement methods were used for temperature verification and effectiveness comparison: thermal imaging camera, pyrometer, and thermocouple. Additionally, the developed numerical models will provide a better understanding of the phenomena occurring in multilayered materials under the influence of rapidly changing electromagnetic fields and their impact on the heating process of the entire load. The article also addresses the issue of constructing models with a high discretisation level.

Coulomb blockade and Coulomb staircases in CoBi nanoislands on SrTiO3 (001)

We successfully fabricated two-dimensional metallic CoBi nanoislands on SrTiO3 (001) substrate by molecular beam epitaxy, and systematically investigated their electronic structures by scanning tunneling microscopy and spectroscopy in situ at 4.2 K. Coulomb blockade and Coulomb staircases with discrete and well-separated levels are observed for the individual nanoisland, which is attributed to single-electron tunneling via two tunnel junction barriers. They are in excellent agreement with the simulations based on orthodox theory. Furthermore, we demonstrated that the Coulomb blockade becomes weaker with increasing temperature and almost disappears at ∼22 K in our variable temperature experiment, and its full-width at half-maximum of dI/dV peaks with temperature is ∼6 mV. Our results provide a new platform for designing single-electron transistors that have potential applications in future microelectronics.

Evolution of Excited-State Behaviors of Gold Complexes, Nanoclusters and Nanoparticles.

Metal nanomaterials have been extensively investigated owing to their unique properties in contrast to bulk counterparts. Gold nanoparticles (e. g., 3-100 nm) show quasi-continuous energy bands, while gold nanoclusters (<3 nm) and complexes exhibit discrete energy levels and display entirely different photophysical properties than regular nanoparticles. This review summarizes the electronic dynamics of these three types of gold materials studied by ultrafast spectroscopy. Briefly, for gold nanoparticles, their electronic relaxation is dominated by heat dissipation between the electrons and the lattice. In contrast, gold nanoclusters exhibit single-electron transitions and relatively long excited-state lifetimes being analogous to molecules. In gold complexes, the excited-state dynamics is dominated by intersystem crossing and phosphorescence. A detailed understanding of the photophysical properties of gold nanocluster materials is still missing and thus calls for future efforts. The fundamental insights into the discrete electronic structure and the size-induced evolution in quantum-sized nanoclusters will promote the exploration of their applications in various fields.

Space-Time Least-Squares Finite Element Methods for Parabolic Distributed Optimal Control Problems

Abstract We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrange multiplier. Our final formulation fulfills the Babuška–Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a posteriori error estimator which is efficient and reliable up to an additional discretization error of the adjoint problem. Numerical experiments confirm our theoretical findings.

Rates of severe neutropenia and infection risk in patients treated with deferiprone: 28 years of data

AbstractPatients treated with deferiprone for transfusional iron overload may experience idiosyncratic drug–induced neutropenia (IDIN) that may increase risk of infection. This analysis examined the rates of severe IDIN and risk of serious infections at different absolute neutrophil count (ANC) levels during deferiprone treatment. Events of severe IDIN (ANC <0.5 × 109/L) and associated serious infections from clinical trials and postmarketing setting were analyzed by discrete ANC levels: group 1, 0.2 × 109/L to 0.5 × 109/L; group 2, 0.1 × 109/L to 0.199 × 109/L; group 3, <0.1 × 109/L. In clinical trials, 22 events of severe IDIN occurred (group 1, n = 9; group 2, n = 3; group 3, n = 10), and rates of severe IDIN per 100 patient-years were 0.45 in group 1; 0.15 in group 2; and 0.50 in group 3 (1990.26 patient-years deferiprone exposure). All serious infections were in group 3 (3/10 [30.0%]). In the postmarketing setting, 176 events of severe IDIN were reported (group 1, n = 65; group 2, n = 20; group 3, n = 91) and rates of severe IDIN per 100 patient-years were 0.06 in group 1; 0.02 in group 2; and 0.08 in group 3 (111 570.24 patient-years deferiprone exposure). Rates of serious infection were 7.7% (5/65) in group 1; 10% (2/20) in group 2; and 13.2% (12/91) in group 3. Our findings suggest a high risk of serious infections with ANC <0.2 × 109/L during deferiprone treatment, a level consistent with the recent neutropenia guidelines.

Convergent scheme for a non-local transport system modeling dislocations dynamics

In this paper, we consider a diagonal hyperbolic, not necessarily strictly hyperbolic, nonlinear and non-local system of transport equations that arises in the theory of dislocations densities dynamics. Under certain monotony assumptions on the initial data, global existence and uniqueness results of Hloc1 solutions for this system was proved in Hajj (2007). We propose a natural implicit scheme verifying a gradient L2 estimate at the discrete level compatible with that obtained in the continuous case in Hajj (2007). Which leads us to prove the convergence of the numerical solution to the continuous one. To our knowledge, this is the first convergent scheme proposed for this model.

An entropy conserving/stable discontinuous Galerkin solver in entropy variables based on the direct enforcement of entropy balance

Numerical schemes that guarantee the preservation of flow properties as prescribed by physical laws, ensure high-fidelity results and improved computational robustness. This work presents a highly accurate entropy conserving/stable solver for the solution of the compressible Euler equations. The method uses a modal Discontinuous Galerkin approximation and takes advantage of both entropy variables and the Direct Enforcement of Entropy Balance (DEEB), proposed by Abgrall [1] to fulfill the second law of thermodynamics at the discrete level. Thanks to the use of an orthonormal modal basis, the cost of adding the DEEB correction to the spatially discretized equations is negligible. Entropy variables are used directly, as unknowns, or to assemble the discrete operators while solving for the conservative variables. In the latter case, the conservative variables are “converted” into the entropy ones through an L2 “entropy projection”. If this strategy is coupled with DEEB, a very efficient alternative to the direct use of the entropy variables is obtained, with a dramatic improvement in robustness over the direct use of the conservative variables. For the time integration both an explicit Strong Stability Preserving Runge-Kutta scheme and the implicit Generalized Crank-Nicolson method are considered. The latter allows for the construction of a discrete entropy conservative/stable scheme both in space and time to be used as a reference. Convergence and conservation properties are demonstrated together with robustness and computational efficiency on a suite of two-dimensional test cases, showing that by coupling DEEB with entropy projection the solver robustness is improved while not affecting the stability limit.

Tunable quantized spin Hall effect of light in graphene

The spin Hall effect of light is an appealing toolset for metrology that holds incredible potential for precision measurements due to its high sensitivity to the refractive index and nanostructure physical parameters. Here, we theoretically examine the quantized spin Hall effect of light on the surface of a monolayer graphene–substrate system subjected to an external perpendicular magnetic field. We discuss the impact of Landau levels induced by the external magnetic field B and the photon energy on magneto-optical conductivities and Fresnel reflection coefficients. We find that the spin-dependent splittings in graphene exhibit quantized characteristics due to the Landau levels quantization of the magneto-optical conductivities and Fresnel’s reflection coefficients near the Brewster angle. Moreover, the spin-dependent shifts are quantized and show oscillatory behavior with changes in the magnetic field and the photon energy. Our results manifest that the quantized photonic spin-dependent separations are sensitive to variations in the incidence angle, magnetic field, and incident photonic energies. The quantized spin-dependent splitting opens a promising route to find the quantized Hall conductivity and discrete Landau levels in the monolayer graphene by a direct optical weak measurement.

Thermodynamically consistent numerical modeling of immiscible two‐phase flow in poro‐viscoelastic media

AbstractNumerical modeling of immiscible two‐phase flow in deformable porous media has become increasingly significant due to its applications in oil reservoir engineering, geotechnical engineering and many others. The coupling between two‐phase flow and geomechanics gives rise to a major challenge to the development of physically consistent mathematical models and effective numerical methods. In this article, based on the concept of free energies and guided by the second law of thermodynamics, we derive a thermodynamically consistent mathematical model for immiscible two‐phase flow in poro‐viscoelastic media. The model uses the fluid and solid free energies to characterize the fluid capillarity and solid skeleton elasticity, so that it rigorously follows an energy dissipation law. The thermodynamically consistent formulation of the pore fluid pressure is naturally derived for the solid mechanical equilibrium equation. Additionally, the model ensures the mass conservation law for both fluids and solids. For numerical approximation of the model, we propose an energy stable and mass conservative numerical method. The method herein inherits the energy dissipation law through appropriate energy approaches and subtle treatments for the coupling between two phase saturations, the effective pore pressure and porosity. Using the locally conservative cell‐centered finite difference methods on staggered grids with the upwind strategies for saturations and porosity, we construct the fully discrete scheme, which has the ability to conserve the masses of both fluids and solids as well as preserve the energy dissipation law at the fully discrete level. In particular, the proposed method is an unbiased algorithm, that is, treating the wetting phase, the non‐wetting phase and the solid phase in the same way. Numerical results are also given to validate and verify the features of the proposed model and numerical method.

On the homogeneous torsion problem for heterogeneous and orthotropic cross-sections: Theoretical and numerical aspects

For many years, torsion of arbitrary cross-sections has been a subject of numerous investigations from theoretical and numerical points of view. As it is well known, the resulting boundary value problem (BVP) governing such phenomenon happens to be a pure Neumann BVP and, therefore, its solutions are determined up to a constant. Among a large plethora of finite element method (FEM) techniques that can be used in this context, most of FEM practitioners resolve this uniqueness issue by fixing the candidate solution to a node of the domain. Although such popular and pinpointing technique is widely spread and works well for practical purposes, it does not have a continuous counterpart and therefore its justification remains a matter of debate. Hence, this self-contained work aims to address the modeling of arbitrary heterogeneous and orthotropic cross-sections as well as the theoretical and numerical aspects of their solutions. In particular, we discuss the existence of weak solutions, well-posedness, regularity of solutions, and convergence of Galerkin’s method for different variational settings (with special focus on a regularized variational approach). Moreover, we establish a connection, at a discrete level, between the convergence of solutions of well-posed variational settings and those solutions coming from the usual practice of fixing a datum at a node. Finally, we discuss some numerical aspects of all the FEM discrete formulations proposed here by performing convergence analysis in L2 and H1 norms. The section of numerical results is closed by presenting a series of study cases ranging from a square cross-section composed of two different materials to an isotropic bridge cross-section for which no analytical solution exists.

Formation of one-dimensionally self-aligned Si-QDs and their local electron discharging properties

Self-aligned Si-quantum-dots (Si-QDs) with an areal density as high as ∼1011 cm−2 have been fabricated on ultrathin SiO2 by using a ∼4.5 nm thick poly-Si on insulator (SOI) substrate, and controlling low-pressure CVD using monosilane (SiH4), and followed by thermal oxidation. By controlling the thermal oxidation processes of Si-QDs and the poly-Si layer, we have successfully demonstrated the vertical alignment of Si-QDs, where the Si-QDs are also used as a shadow mask of the underlying poly-Si layer. We also demonstrated in-plane alignment of the one-dimensionally self-aligned Si-QDs on line-patterned SiO2. In addition, from surface potential measurements by using atomic force microscopy/Kelvin probe force microscopy, we confirmed that the initial surface potential change caused by valence electron extraction from the dots to the tip was stably maintained until ∼120 min, implying the quantum confinement effect at discrete energy levels of the upper and lower-QDs.

Unipolar optical transitions in nanoclusters of ellipsoidal geometry

The quantum states of an ellipsoidal nanocluster of a heterophase system InAs/GaAs are studied using an exact analytical approach, in contrast to the generally accepted theoretical model based on the adiabatic approximation. It is shown that the spectrum of a nanoobject is formed from local groups, consisting of discrete levels, separated by terahertz frequency intervals. A double random degeneration of certain spectrum states is revealed. Allowed mid-infrared (IR) optical transitions between different spectral states are analyzed. The role of dimensional parameters and features of the shape of a nanoobject in the characteristics of unipolar transitions is assessed. The absorption spectrum for the transition in the lower part of the substructure spectrum is calculated.

Red Persistent Luminescence and Trap Properties of Mg2SiO4: Mn2+, M3+ (M3+ = B3+; Al3+; Ga3+; In3+) Material

Abstract Persistent luminescence (PersL), also called long-lasting phosphorescence or simply afterglow, is a luminescence characterised by the emission of radiation from a few seconds to several days after the excitation source has been switched off. Over the past two decades, research on PersL materials, both in fundamental and applied physics, has developed rapidly; however, the explanation for the physical processes that cause afterglow still needs to be clarified. Today, PersL materials are used mainly for luminescent paints, safety signs and decorations. At the same time, research into using such materials in medicine, information storage, anti-counterfeiting technology, etc., is underway. Currently, information on the long persistent luminescence materials with emission in the blue and green spectral range is widely available. In contrast, the number of publications on the afterglow in the red and near-infrared spectral range is considerably lower. Within the framework of this research, Mg2SiO4: Mn2+; M3+ (M3+ = B3+; Al3+; Ga3+; In3+) materials were synthesised using solid state reaction synthesis. When excited with X-rays, the materials exhibited a broad Mn2+ PersL band with two maxima at approximately 625 nm and 730 nm. After cessation of irradiation, an afterglow of at least 6 hours could be observed. The research focuses on the trap properties of the materials. It was concluded that at least three discrete trap levels with activation energies approximately between 0.4–1.6 eV were present in the samples. Additionally, co-doping with Al3+; Ga3+; In3+ ions improved PersL longevity of the Mg2SiO4: Mn2+ material.