Delta Function Research Articles

Wigner function dynamics with boundaries expressed as convolution

In the present paper a method of finding the dynamics of the Wigner function of a particle in an infinite quantum well is developed. Starting with the problem of a reflection from an impenetrable wall, the obtained solution is then generalized to the case of a particle confined in an infinite well in arbitrary dimensions. It is known, that boundary value problems in the phase space formulation of the quantum mechanics are surprisingly tricky. The complications arise from nonlocality of the expression involved in calculation of the Wigner function. Several ways of treating such problems were proposed. They are rather complicated and even exotic, involving, for example, corrections to the kinetic energy proportional to the derivatives of the Dirac delta–function. The presented in the manuscript approach is simpler both from analytical point of view and regarding numerical calculation. The solution is brought to a form of convolution of the free particle solution with some function, defined by the shape of the well. This procedure requires calculation of an integral, which can be done by developed analytical and numerical methods.

Generation of incident wave in two-phase flow simulation based on field decomposition

Field decomposition is an effective strategy for reducing numerical dissipation and dispersion. This strategy was employed by Li et al. (2021) to generate incident waves in two-phase flow simulations. This study attempts to improve previous methods in two ways. First, the density gradient in the additional source term, i.e. a delta function at the interface, is explicitly discretised. Although the explicit calculation simplifies the implementation, an additional pressure translation correction method is proposed to ensure numerical stability and accuracy. Second, the coupled level-set and volume-of-fluid method is used for interface capture. The calculation of the additional source term is more precise using the level-set function. The two proposed improvements result in a second-order spatial accuracy for the wave amplitude. A test on wave propagation over a flat bottom shows that the proposed method provides more accurate predictions of the wave amplitude compared with the previous method. In other test cases, including wave propagation over two-dimensional breakwater and three-dimensional shoal, the simulation results show good agreement with the experimental data.

Development of a New Zeta Formula and Its Role in Riemann Hypothesis and Quantum Physics

In this study, we investigated a new zeta formula in which the zeta function can be expressed as the sum of an infinite series of delta and cosine functions. Our findings demonstrate that this formula possesses duality characteristics and we established a direct connection between the Riemann hypothesis and this new formula. Additionally, we explored the behavior of energy or particles in quantum physics within the proposed mathematical model framework based on the new formula. Our model provides a valuable understanding of several important physics inquiries, including the collapse of the wave function during measurement and quantum entanglement, as well as the double slits experiment.

An improved continuum surface tension model in SPH for simulating free-surface flows and heat transfer problems

The continuum surface force (CSF) model in conjunction with the smoothed particle hydrodynamics (SPH) method is commonly used in the simulation of multi-phase flows with surface tension effect. However, it cannot be directly applied for modeling free-surface flows since the normalization requirement, which states that the integral of the surface delta function through the interface should be equal to one, is violated due to the particle deficiency issue. In this work, we propose an improved CSF model based on the SPH method with the aim of recovering the normalization requirement. The particle deficiency issue at the free surface is tackled by introducing the jump in color function into the calculation of the surface delta function. As a result, the accuracy of the surface tension measurement is significantly improved without using the virtual particles. Besides, this improved CSF model also allows an accurate and efficient implementation of the Robin boundary condition which is frequently encountered in heat transfer problems. Several benchmark problems are investigated to validate the performance of the proposed method. The numerical results verify that the improved CSF model in SPH method is well capable of simulating free-surface flows and heat transfer problems.

Free vibration analysis of FG‐GPL and FG‐CNT hybrid laminated nano composite truncated conical shells using systematic differential quadrature method

AbstractIn the present study, the free vibration of functionally graded graphene platelet‐reinforced (FG‐GPLs) and functionally graded carbon nanotube‐reinforced (FG‐CNTs) hybrid laminated nanocomposite truncated conical shells and panels are analyzed. Multi‐layers truncated conical shell and panel of pure FG‐CNTs, pure FG‐GPLs and hybrid CNTs‐GPLs reinforcement were evaluated. In light of its high accuracy in the calculation of thin and thick shells, a third‐order shear deformation theory is adopted. The governing equations and boundary conditions is derived using Hamilton's principle and is solved numerically using the systematic differential quadrature method (DQM) which uses Kronecker delta function. The effective mechanical properties of the CNT‐reinforced nanocomposite layers are estimated using the rule of mixtures, whereas those of the GPL‐reinforced nanocomposite layers is calculated using the Halpin‐Tsai micromechanical model. Convergence and accuracy evaluation of the presented study are confirmed and a number of parameters, including the CNTS volume fraction, GPLs mass fraction, distribution patterns (i.e., Uniform distribution (UD), Functionally graded O‐distribution (FG‐O), Functionally graded X‐distribution (FG‐X), Functionally graded V‐distribution (FG‐V) and FG‐A), different boundary conditions and the vertex angle of the cone, are investigated. The results obtained in this article for the combination of two different materials, GPLs and CNTs, showed that in controlling the natural frequency of the system, without changing the percentage of fiber and only by changing the arrangement, very wonderful results can be achieved.

Sufficient Solvability Conditions for the Problem of Pursuit under an Impulse Action

The article considers a linear differential pursuit game under the condition that an integral constraint is imposed on the evader’s control and the pursuer uses an impulse control. These impulse actions on the object are made at predetermined time points, and the corresponding control is represented using the Dirac delta function. The article studies linear conflicts described by a system of ordinary differential equations whose trajectories have jumps at certain times. The terminal set is represented by a cylinder in an n-dimensional Euclidean space. The problem is solved using the resolving function method. The fact that the lower bound is reached is proved using the theory of support functions. As a result, the quasi-strategy is replaced by an almost stroboscopic strategy and a method for building this strategy is presented. An example of a nonlinear right-hand side is given.

Exact Treatment of the Infinite Square Well in One Dimension with λδ^' (x) Potential

Abstract: This work considered the infinite square well in one dimension with a contact potential. The Dirac delta derivative function potential λδ^' where λ is a coupling constant was used to represent the contact potential. Using Green’s function technique, exact implicit expressions of the energy eigenvalues and eigenfunctions were obtained. The energy eigenvalues were expressed using a transcendental equation. The energy eigenfunctions satisfy the Schrödinger equation and the infinite square well boundary conditions. Also, the eigenfunctions and their first derivative were shown to be discontinuous. The values of these discontinuity jumps agreed with the required conditions for a self-adjoint extension Hamiltonian. In the weak coupling region, the energy eigenvalues are close to that of the even parity solution before adding the contact potential. The energy eigenvalues in the strong coupling regime reveal the energy eigenvalues of the odd parity solution. Keywords: Point interactions, Infinite square well, Green’s function technique. PACS numbers: 03.65.-w, 03.65.Db, 03.65.Ge

Calculation of Dynamical Response Functions Using a Bound-State Method

We investigate a method to extract response functions (dynamical polarisabilities) directly from a bound-state approach applied to calculations of perturbation-induced reactions. The use of a square-integrable basis leads to a response in the form of a sum of delta functions. We integrate this over energy and fit a smooth function to the resulting stepwise-continuous one. Its derivative gives the final approximation to the physical response function. We show that the method reproduces analytical results where known, and analyse the details for a variety of models. We apply it to some simple models, using the stochastic variational method as the numerical method. Albeit we find that this approach, and other numerical techniques, have some difficulties with the threshold behavior in coupled-channel problems with multiple thresholds, its stochastic nature allows us to extract robust results even for such cases.

Level Sets Guided by SoDEF-Fitting Energy for River Channel Detection in SAR Images

To achieve river channel detection in SAR (synthetic aperture radar) images, we developed a level-set-based model (LSBM) guided by a designed data-fitting energy which is called the SoDEF (sum of dual exponential functions)-fitting energy. Firstly, we designed a function by computing the sum of dual exponential functions to substitute for the quadratic function, and used it to construct the data-fitting energy. Secondly, the adaptive area-fitting centers (AFCs) were computed based on two kinds of grayscale characteristics, which are more accurate and more stable. Thirdly, the Dirac function in gradient descent flow was displaced by an edge indicator function to help the evolving level sets stop at the target edges. Moreover, some regularized terms were incorporated into the objective function to guarantee the model’s stability. The river channel detection experiments conducted with real SAR images indicated that the developed model is superior to the related state-of-the-art methods in its detection accuracy and efficiency.

Considering the Dirac Delta function DDF as an analogy of the behavior of the COLCAP index 2008- 2019.

Considering the Dirac Delta function DDF as an analogy of the behavior of the COLCAP index 2008- 2019.

Analysis of the anomalous diffusion in comb structure with absorbing boundary conditions

The diffusion in comb structure is an important kind of anomalous diffusion with widespread applications. The special structure corresponds to a novel characteristic of anomalous diffusion, which is characterised by the Dirac delta function in the governing equation. By considering the memory characteristic, the fractional derivative is introduced into the constitutive relation, and a new fractional governing equation in the infinite regions is constructed. Instead of simply truncating for the infinite regions, the exact absorbing boundary conditions are deduced by using the (inverse) Laplace transform technique and the stability is analysed. To deal with the governing equation containing the Dirac function, the finite difference method is proposed and the term with the Dirac function is handled using an integration method. The stability and convergence of the numerical scheme are discussed in detail. A fast algorithm is presented that the normal L1-scheme is approximated via a sum-of-exponentials approximation. Three examples are conducted, in which the particle distributions and the mean square displacement for the anomalous diffusion in comb structure are discussed. The computational time between the normal numerical scheme and the fast numerical scheme is compared and the rationality and validity of absorbing boundary conditions are analysed. An important finding is that the distribution of the mean square displacement with the absorbing boundary conditions can match the exact one accurately, which demonstrates the effectiveness of the method.

Multiplication Operators by White Noise Delta Functions and Associated Differential Equations

Multiplication Operators by White Noise Delta Functions and Associated Differential Equations

Mathematical analysis on the propagation of Griffith crack in an initially stressed strip subjected to punch pressure

The purpose of this study is to analyze the features of a moving Griffith crack in an initially stressed infinitely long and finitely thick isotropic strip with moving parallel punches of constant load acting on its boundaries owing to plane wave propagation under point loading. Coupled singular integral equations and singularities of the Cauchy-type are used to formulate the present model, Dirac delta function is employed to analyze point load located at the moving crack edge, and Hilbert transformation properties are used for obtaining stress intensity factor (SIF) with constant point loading. Numerical simulations and graphical illustrations are performed to analyze the influences of the prevalent parameters, viz. initial stresses, punch pressure, distinct positions of point load, length and speed of the crack on the SIF for the considered isotropic material strip.

Linear analysis of the gravitational beam–plasma instability

We investigate the well-known phenomenon of the beam–plasma instability in the gravitational sector when a fast population of particles interacts with the massive scalar mode of a Horndeski theory of gravity, resulting in linear growth of the latter amplitude. Following the approach used in the standard electromagnetic case, we start from the dielectric representation of the gravitational plasma, as introduced in a previous analysis of the Landau damping for the scalar Horndeski mode. We then set up the modified Vlasov–Einstein equation, using a Dirac delta function to describe the fast beam distribution. We thus provide an analytical expression for the dispersion relation, and we demonstrate the existence of a nonzero growth rate for the linear evolution of the Horndeski scalar mode. A numerical investigation is then performed with a trapezoidal beam distribution function, which confirms the analytical results and allows us to demonstrate how the growth rate decreases as the beam spread increases.

A novel insight into the interface fracture between magnesium phosphate cement mortar and cement concrete

ABSTRACT This paper focused on fracture problem about interfacial transition zone (ITZ) formed after repairing between magnesium phosphate cement (MPC) mortar and old Portland cement concrete (PCC) pavements. Firstly, the mode Ⅱ cracks on the ITZ were hypothesized and the interface fracture model with dislocation theory was proposed. The stress field at the crack tip was represented by Airy stress potential function and Fourier Transform, and the structural characteristics of the kernel solution of Cauchy were retained. Then, combined with the oscillation term, the Dirac function was replaced by the normal distribution function to obtain stable fracture parameters. Besides, the ϵ about 8 ‰ of the crack length in model was analyzed. Secondly, the elastic moduli of ITZs were obtained by nanoindentation experiments. Results showed the elastic modulus of ITZ between MPC mortar and ITZ-1 (the interface in concrete) was lowest. Such weakest position in repair interface was further verified by finite element analysis (FEA). Finally, a twodimensional model was developed to validate the fracture model. Simulation results were found consistent with the theoretical solution of interface fracture model. The developed interface fracture model can provide insights for evaluating the shear behavior of the interface between MPC mortar and concrete.

Mehler–Fock transforms and retarded radiative Green functions in hyperbolic and spherical spaces

We develop the theory of causal radiation Green functions on hyperbolic and hyperspherical spaces using a constructive approach based on generalized Mehler–Fock transforms. This approach focuses for HRd on the kernel of the transformation expressed in terms of hyperbolic angles θ with 0 ≤ θ < ∞. The kernel provides an explicit representation for the generalized delta distribution, which acts as the source term for the radiation, and allows for an easy implementation of the causality or retardation condition and determination of the Green function. We obtain the corresponding kernel distribution on SRd by analytic continuation of the kernel distribution of an appropriate Helmholtz equation on HRd and then show that this construction leads to the proper retarded Green function for the wave equation. That result is then used to establish the validity of a new generalized Mehler–Fock transformation for 0 ≤ θ < π. The present results clarify and extend those obtained recently by Cohl et al. [SIGMA 14, 136 (2018)].

Two-dimensional probability distribution of the solution to the random Burgers–Riemann problem

We investigate Burgers’ partial differential equation with random Riemann initial states and random-field solution. Previous works obtained the generalized probability density function in the one-dimensional case, by using different methods. We explore a technique that has not been employed in this context before, which fully develops the potential of the Dirac delta function. We start with the one-dimensional scenario and afterward extend to two dimensions. The generalized probability density function is derived in those cases, both for non-independent and independent initial conditions, and general statistics are obtained as a consequence.

About the quantum Talbot effect on the sphere

The Schrödinger equation on a circle with an initially localized profile of the wave function is known to give rise to revivals or replications, where the probability density of the particle is partially reproduced at rational times. As a consequence of the convolutional form of the general solution it is deduced that a piecewise constant initial wave function remains piecewise constant at rational times as well. For a sphere instead, it is known that this piecewise revival does not necessarily occur, indeed the wave function becomes singular at some specific locations at rational times. It may be desirable to study the same problem, but with an initial condition being a localized Dirac delta instead of a piecewise constant function, and this is the purpose of the present work. By use of certain summation formulas for the Legendre polynomials together with properties of Gaussian sums, it is found that revivals on the sphere occur at rational times for some specific locations, and the structure of singularities of the resulting wave function is characterized in detail. In addition, a partial study of the regions where the density vanishes, named before valley of shadows in the context of the circle, is initiated here. It is suggested that, differently from the circle case, these regions are not lines but instead some specific set of points along the sphere. A conjecture about the precise form of this set is stated and the intuition behind it is clarified.

Coronavirus Classification based on Enhanced X-ray Images and Deep Learning

In light of the fact that the global pandemic of Coronavirus Disease 2019 (COVID-19) is still having a significant impact on the health of people all over the world, there is a growing need for testing diagnosis and treatment that can be completed quickly. The primary imaging modalities used in the respiratory disease diagnostic process are the Chest X-ray (CXR) and the computed tomography scan. In this context, this paper aims to design a new Convolutional Neural Network (CNN) to diagnose COVID-19 in patients based on CXR images and determine whether they are COVID or healthy. We have tested the performance of our CNN on the COVID-19 Radiography Database with three classes (COVID, Pneumonia, and Normal). Also, we proposed a new enhancement technique to enhance the CXR image using the Laplacian kernel with Delta Function and Contrast-Limited Adaptive Histogram Equalization. The proposed CNN has been trained and tested on 15153 enhanced and original images, COVID (3616), Pneumonia (1345), and Normal (10192). Our enhancement technique increased the performance metrics scores of the proposed CNN. Hence, the proposed method obtained better results than the state-of-the-art methods in accuracy, sensitivity, precision, specificity, and F measure.

Infrared effects and the Unruh state

Detailed behaviors of the modes of quantized scalar fields in the Unruh state for various eternal black holes in two dimensions are investigated. It is shown that the late-time behaviors of some of the modes of the quantum fields and of the symmetric two-point function are determined by infrared effects. The nature of these effects depends upon whether there is an effective potential in the mode equation and what form this potential takes. Here, three cases are considered, one with no potential and two with potentials that are nonnegative everywhere and are zero on the event horizon of the black hole and zero at either infinity or the cosmological horizon. Specifically, the potentials are a delta function potential and the potential that occurs for a massive scalar field in Schwarzschild–de Sitter spacetime. In both cases, scattering effects remove infrared divergences in the mode functions that would otherwise arise from the normalization process. When such infrared divergences are removed, it is found that the modes that are positive frequency with respect to the Kruskal time on the past black hole horizon approach zero in the limit that the radial coordinate is fixed and the time coordinate goes to infinity. In contrast, when there is no potential and thus infrared divergences occur, the same modes approach nonzero constant values in the late-time limit when the radial coordinate is held fixed. The behavior of the symmetric two-point function when the field is in the Unruh state is investigated for the case of a delta function potential in certain asymptotically flat black hole spacetimes in two dimensions. The removal of the infrared divergences in the mode functions results in the elimination of terms that grow linearly in time.